A Geometric Study of
Single Gimbal Control Moment Gyros
— Singularity Problems and Steering Law —
Haruhisa Kurokawa
Mechanical Engineering Laboratory
Report of Mechanical Engineering Laboratory, No. 175, p.108, 1998.
A Geometric Study of
Single Gimbal Control Moment Gyros
— Singularity Problems and Steering Law —
by
Haruhisa Kurokawa
Abstract
In this research, a geometric study of singularity
characteristics and steering motion of single gimbal
Control Moment Gyros (CMGs) was carried out in order
to clarify singularity problems, to construct an effective
steering law, and to evaluate this law’s performance.
Passability, as defined by differential geometry
clarified whether continuous steering motion is possible
in the neighborhood of a singular system state.
Topological study of general single gimbal CMGs
clarified conditions for continuous steering motion over
a wider range of angular momentum space. It was shown
that there are angular momentum vector trajectories such
that corresponding gimbal angles cannot be continuous.
If the command torque, as a function of time, results in
such a trajectory in the angular momentum space, any
steering law neither can follow the command exactly
nor can be effective.
A more detailed study of the symmetric pyramid type
of single gimbal CMGs clarified a more serious problem
of continuous steering, that is, no steering law can follow
all command sequences inside a certain region of the
angular momentum space if the command is given in
real time. Based on this result, a candidate steering law
effective for rather small space was proposed and verified
not only analytically, but also using ground experiments
which simulated attitude control in space.
Similar evaluation of other steering laws and
comparison of various system configurations in terms
of the allowed angular momentum region and the
system’s weight indicated that the pyramid type single
gimbal CMG system with the proposed steering law is
one of the most effective candidate torquer for attitude
control, having such advantages as a simple mechanism,
a simpler steering law, and a larger angular momentum
space.
Keywords
Attitude control, Singularity, Momentum exchange device, Inverse kinematics, Steering law
––– i –––
Acknowledgments
This research work is a result of projects conducted
at the Mechanical Engineering Laboratory, Agency of
Industrial Technology and Science, Ministry of
International Trade and Industry, Japan. Related projects
are, “Development of Attitude Control Equipment
(FY1982–1987)“, “Attitude Control System for Large
Space Structures (FY1988–1993)”, and “High Precision
Position and Attitude Control in Space (FY1993–1997)”.
I wish to acknowledge my debt to many people. Prof.
Nobuyuki Yajima of the Institute of Space and
Astronautical Science (ISAS) are earnestly thanked for
inspiring me with this theme, as well as for collaborations
during his tenure as a division head of our laboratory. I
would extend thanks to the late Prof. Toru Tanabe,
formerly of the University of Tokyo for his guidance in
the culmination of this work into a dissertation. In
finishing this work, the following professors guided me,
Assoc. Prof. Shinichi Nakasuka of the University Tokyo,
Prof. Hiroki Matsuo of ISAS, Prof. Shinji Suzuki, Prof.
Yoshihiko Nakamura,Assoc. Prof. Ken Sasaki of the
University of Tokyo.
Many discussions with Dr. Shigeru Kokaji of our
laboratory proved invaluable. He patiently listened to
my abstract explanation of geometry and provided
valuable suggestions. Furthermore, he assisted me by
soldering and checking circuits, and reviewed this paper
from cover-to-cover, providing constructive criticism.
I would also like to thank my colleague Akio Suzuki
who constructed most of the experimental apparatus, and
designed and installed controllers for the attitude control.
Prof. Tsuneo Yoshikawa of Kyoto University helped
me when we started the project of attitude control by
CMGs. Discussions held with Dr. Nazareth Bedrossian
and Dr. Joseph Paradiso of the Charles Stark Draper
Laboratory (CSDL) were invaluable. They gave me
valuable suggestions with various research papers in this
field.
Dr. Mark Lee Ford as a visiting researcher of our
laboratory spent his precious hours for me to correct
expressions in English.
I would like to thank all the above people, other
colleagues sharing other research projects, and the
Mechanical Engineering Laboratory (MEL) and the
directors especially the Director General Dr. Kenichi
Matuno and the former Department Head Dr. Kiyofumi
Matsuda for allowing me to continue this research.
Finally, I thank my wife and daughters for their patience
particularly during some hectic months.
––– ii –––
Haruhisa Kurokawa
June 7, 1997
Contents
Abstract ............................................................................................................................................................ i
Acknowledgments ........................................................................................................................................... ii
Terms ........................................................................................................................................................... viii
Nomenclature ................................................................................................................................................. ix
List of Figures ................................................................................................................................................. x
List of Tables ............................................................................................................................................... xiii
Chapter 1 Introduction .............................................................................................. 1
1.1 Research Background ..................................................................................................................................... 1
1.2 Scope of Discussion ........................................................................................................................................ 3
1.3 Outline of this Thesis ...................................................................................................................................... 4
Chapter 2 Characteristics of Control Moment Gyro Systems ............................... 5
2.1 CMG Unit Type ............................................................................................................................................. 5
2.2 System Configuration .................................................................................................................................... 5
2.2.1 Single Gimbal CMGs ............................................................................................................................ 6
2.2.2 Two Dimensional System and Twin Type System ................................................................................ 7
2.2.3 Configuration of Double Gimbal CMGs ............................................................................................... 7
2.3 Three Axis Attitude Control ........................................................................................................................... 7
2.3.1 Block Diagram ...................................................................................................................................... 8
2.3.2 CMG Steering Law ............................................................................................................................... 8
2.3.3 Momentum Management ...................................................................................................................... 8
2.3.4 Maneuver Command ............................................................................................................................. 8
2.3.5 Disturbance ........................................................................................................................................... 8
2.3.6 Angular Momentum Trajectory ............................................................................................................. 8
2.4 Comparison and Selection ............................................................................................................................. 9
2.4.1 Performance Index ................................................................................................................................ 9
2.4.2 Component Level Comparison ............................................................................................................. 9
2.4.3 System Level Comparison .................................................................................................................... 9
2.4.4 Work Space Size and Weight ................................................................................................................ 9
Chapter 3 General Formulation .............................................................................. 11
3.1 Angular Momentum and Torque ................................................................................................................... 11
3.2 Steering Law ................................................................................................................................................. 12
3.3 Singular Value Decomposition and I/O Ratio ............................................................................................... 12
3.4 Singularity ..................................................................................................................................................... 13
––– iii –––
3.5 Singularity Avoidance ................................................................................................................................... 13
3.5.1 Gradient Method ................................................................................................................................. 14
3.5.2 Steering in Proximity to a Singular State ............................................................................................. 14
Chapter 4 Singular Surface and Passability .......................................................... 15
4.1 Singular Surface ........................................................................................................................................... 15
4.1.1 Continuous Mapping ........................................................................................................................... 15
4.1.2 Envelope .............................................................................................................................................. 16
4.1.3 Visualization Method of the Surface ................................................................................................... 16
4.2 Differential Geometry .................................................................................................................................. 17
4.2.1 Tangent Space and Subspace ............................................................................................................... 17
4.2.2 Gaussian Curvature ............................................................................................................................. 17
4.3 Passability .................................................................................................................................................... 18
4.3.1 Quadratic Form ................................................................................................................................... 18
4.3.2 Signature of Quadratic Form ............................................................................................................... 19
4.3.3 Passability and Singularity Avoidance ................................................................................................ 19
4.3.4 Discrimination ..................................................................................................................................... 20
4.4 Internal Impassable Surface ......................................................................................................................... 21
4.4.1 Impassable Surface of an Independent Type System .......................................................................... 21
4.4.2 Impassable Surface of a Multiple Type System .................................................................................. 21
4.4.3 Minimum System ................................................................................................................................ 22
Chapter 5 Inverse Kinematics ................................................................................. 23
5.1 Manifold ....................................................................................................................................................... 23
5.2 Manifold Path .............................................................................................................................................. 24
5.3 Domain and Equivalence Class ................................................................................................................... 24
5.4 Terminal Class and Domain Type ................................................................................................................ 25
5.5 Class Connection ......................................................................................................................................... 25
5.5.1 Type 2 Domain .................................................................................................................................... 25
5.5.2 Type 1 Domain .................................................................................................................................... 26
5.5.3 Class Connection Rules ....................................................................................................................... 27
5.5.4 Continuous Steering over Domains .................................................................................................... 28
5.5.5 Manifold Selection .............................................................................................................................. 28
5.5.6 Discussion of the Critical Point ........................................................................................................... 29
5.6 Topological Problem ..................................................................................................................................... 29
Chapter 6 Pyramid Type CMG System ................................................................... 31
6.1 System Definition ........................................................................................................................................ 31
6.2 Symmetry ..................................................................................................................................................... 31
6.3 Singular Manifold for the H Origin ............................................................................................................. 33
––– iv –––
6.4 Singular Surface Geometry .......................................................................................................................... 35
Chapter 7
Global Problem, Steering Law Exactness and Proposal ................... 41
7.1 Global Problem ............................................................................................................................................ 41
7.1.1 Control Along the z Axis ..................................................................................................................... 41
7.1.2 Global problem .................................................................................................................................... 45
7.1.3 Details of the Problem ......................................................................................................................... 45
7.1.4 Possible Solutions ............................................................................................................................... 47
7.2 Steering Law with Error .............................................................................................................................. 47
7.2.1 Geometrical Meaning .......................................................................................................................... 47
7.2.2 Exactness of Control ........................................................................................................................... 48
7.3 Path Planning ............................................................................................................................................... 49
7.4 Preferred Gimbal Angle ............................................................................................................................... 49
7.5 Exact Steering Law ...................................................................................................................................... 51
7.5.1 Workspace Restriction ......................................................................................................................... 51
7.5.2 Repeatability and Unique Inversion .................................................................................................... 51
7.5.3 Constrained Control ............................................................................................................................ 52
7.5.4 Reduced Workspace ............................................................................................................................ 52
7.5.5 Characteristics of Constrained Control ............................................................................................... 54
Chapter 8 Ground Experiments .............................................................................. 57
8.1 Attitude Control ........................................................................................................................................... 57
8.1.1 Dynamics ............................................................................................................................................. 57
8.1.2 Exact Linearization ............................................................................................................................. 57
8.1.3 Control Method ................................................................................................................................... 58
8.2 Experimental Facility and Procedure ........................................................................................................... 58
8.2.1 Facility ................................................................................................................................................. 58
8.2.2 Design of Control Command Sequence .............................................................................................. 59
8.2.3 Experimental Procedure ...................................................................................................................... 59
8.3 Experimental Results ................................................................................................................................... 60
8.3.1 Attitude Keeping under Constant Disturbance .................................................................................... 60
8.3.2 Rotation About the z Axis ................................................................................................................... 64
8.3.3 Maneuver after Momentum Accumulation ......................................................................................... 67
8.3.4 Mode Selection and Switching ............................................................................................................. 69
8.4 Summary of Experiments ............................................................................................................................ 69
Chapter 9 Evaluation ............................................................................................... 71
9.1 Conditions for Comparison .......................................................................................................................... 71
9.2 Spherical Workspace .................................................................................................................................... 71
9.3 Evaluation by Weight ................................................................................................................................... 72
––– v –––
9.4 Ellipsoidal Workspace ................................................................................................................................. 73
9.5 Summary of Evaluation ................................................................................................................................ 75
Chapter 10 Conclusions .......................................................................................... 77
Appendix A
Double Gimbal CMG System .............................................................. 79
A.1 General Formulation ................................................................................................................................... 79
A.2 Singularity ................................................................................................................................................... 79
A.3 Steering Law and Null Motion ................................................................................................................... 80
A.4 Passability ................................................................................................................................................... 80
A.4.1 Two Unit System ................................................................................................................................ 80
A.4.2 Three Unit System .............................................................................................................................. 81
A.5 Workspace ................................................................................................................................................... 81
Appendix B Proofs of Theories ............................................................................... 83
B.1 Basis of Tangent Spaces .............................................................................................................................. 83
B.2 Gaussian Curvature ..................................................................................................................................... 83
B.3 Inverse Mapping Theory ............................................................................................................................. 84
B.4 Impassable condition for two negative signs .............................................................................................. 85
Appendix C Internal Impassability of Multiple Type Systems .............................. 87
C.1 Roof Type System M(2, 2) .......................................................................................................................... 87
C.1.1 Evaluation of Singular Surface (2) ..................................................................................................... 87
C.1.2 Evaluation of Singular Surface (3) ..................................................................................................... 87
C.1.3 Evaluation of Singular Surface (5) ..................................................................................................... 88
C.1.4 Evaluation of Singular Surface (7) ..................................................................................................... 88
C.1.5 Conclusion .......................................................................................................................................... 88
C.2 M(3, 2): M(2, 2)+1 ...................................................................................................................................... 88
C.2.1 Condition (3) of M(2,2) ...................................................................................................................... 88
C.2.2 Condition (5) of M(2,2) ...................................................................................................................... 89
C.3 M(3, 3): M(2, 2)+2 ...................................................................................................................................... 89
C.4 M(2, 2, 1): M(2, 2)+1 .................................................................................................................................. 89
C.5 M(2, 2, 2): M(2, 2)+2 .................................................................................................................................. 89
C.6 Minimum System ........................................................................................................................................ 89
Appendix D Six and Five Unit Systems .................................................................. 91
D.1 Symmetric Six Unit System S(6) ................................................................................................................. 91
D.1.1 System Definition ................................................................................................................................ 91
––– vi –––
D.1.2 Fault Management ............................................................................................................................... 91
D.1.3 Four out of Six Control ....................................................................................................................... 92
D.2 Five Unit Skew System ................................................................................................................................ 92
Appendix E Specification of Experimental Apparatus and Experimental Procedure
. ................................................................................................... 95
E.1 Experimental Apparatus .............................................................................................................................. 95
E.2 Specifications .............................................................................................................................................. 97
E.3 Attitude Control System .............................................................................................................................. 97
E.4 Steering Law Implementation ..................................................................................................................... 99
E.5 Code Size and Calculation Time ................................................................................................................. 99
E.6 Parameter Estimation .................................................................................................................................. 99
Appendix F General kinematics ............................................................................ 101
F.1 Analogy with a Spatial Link Mechanism ................................................................................................... 101
F.2 Spatial Link Mechanism Kinematics ......................................................................................................... 101
F.3 Singularity .................................................................................................................................................. 102
F.4 Passability .................................................................................................................................................. 102
References ............................................................................................................... 105
––– vii –––
Terms
Class : A set of manifolds which correspond to a certain
domain and are equivalent to each other.
Null motion : Gimbal angle motion which keeps the
angular momentum vector constant.
Domain : A region in the angular momentum space which
is surrounded by singular surfaces and does not
contain any singular surface.
Single gimbal CMG : Fig. 2–1
Double gimbal CMG : Fig. 2–1
Singular surface : A surface formed by the total angular
momentum vector point, H, which corresponds
to singular point.
Singular vector : A unit vector to the plane spanned by
all torque vectors when the system is singular.
Gimbal vector : A unit vector of gimbal direction.
Independent type : A single gimbal CMG system without
parallel gimbal direction pair.
Manifold : A connected subspace of gimbal angle space
whose element corresponds to the same total
angular momentum.
Manifold equivalence : Two manifolds corresponding
to a certain domain are equivalent if there is an
angular momentum path which corresponds to a
continuous manifold path between these two
manifolds.
Multiple type : A single gimbal CMG system composed
of groups each of which elements possess
identical gimbal direction.
Skew type : A single gimbal CMG system with gimbal
directions axially symmetric about one direction.
Symmetric type : A single gimbal CMG system with
gimbal directions arranged normal to surfaces of
a regular polyhedron.
Torque vector : A unit vector of a component CMG to
which direction the CMG can generate an output
torque.
Workspace: Allowed region of the angular momentum
vector of a CMG system.
––– viii –––
Nomenclature
Symbol
Definition
Section number
–––––––––––––––––––––––––––––––––––––––––––
α:
Skew angle of the symmetric pyramid type
system
6.1
β:
Euler parameter of satellite orientation
8.1.1
β* :
Vector part of β
8.1.1
B:
Strip like surface of impassable surface called
branch
6.4
c* :
= cosα
6.1
ci :
= gi × hi. Torque vector
3.1
C:
Jacobian matrix of the kinematic function,
H = f (θ)
Mi : Manifold
5.1
MSj : Singular manifold
5.1
n :
Number of CMG units in the system
3.1
pi :
= 1 / (u ⋅ hi)
4.1.3
P:
Diagonal matrix of pi .
4.1.3
θi :
Gimbal angle of ith CMG unit
θ:
=(θi.). A state variable of the system. Point of n
dimensional torus T (n)
3.1
dθS
∈ΘS.
4.2.1
dθN
∈ΘN
4.2.1
dθT
∈ΘT
4.2.1
D:
Domain in the H space surrounded by singular
Θ S:
3.1
3.1
Singularly constrained tangent space of the θ
space (two dimensional).
4.2.1
ΘN:
Null space of C (n−2 dimensional).
4.2.1
ΘT:
Complementary subspace of ΘN (two
dimensional).
4.2.1
rg :
Symmetric transformation in the θ space.
= {εi }. Sign parameter of the singular surface.
Rg :
Symmetric transformation in the H space. 6.2
4.1.1
s* :
= sinα
surfaces
ε:
Symbol
Definition
Section number
–––––––––––––––––––––––––––––––––––––––––––
M(2, 2): Roof type system
2.2.1
5.3
6.2
6.1
gi :
Gimbal vector
3.1
G:
Equivalence class in a domain.
5.3
hi :
Normalized angular momentum vector
3.1
Sε :
A region of the singular surface of sign ε. 4.1.1
H:
= Σ hi.= f (θ). Total angular momentum vector.
T:
Total output torque of the system
3.1
u:
Singular vector. Unit vector normal to all
κ:
Gaussian curvature of the singular surface.4.2.2
LA :
Segment included by a manifold of H=(0,0,0)t
6.3
S(n) : Symmetric type single gimbal CMG system.
2.2.1
torque vectors.
3.1
3.4
ω:
Gimbal rate vector. Time derivative of θ. 3.1
ωN :
Null motion,
––– ix –––
3.2
List of Figures
Chapter 2
2–1
2–2
2–3
2–4
5–1
Two types of CMG units
Configurations of single gimbal CMGs
Twin type system
Block diagram of three axis attitude
control
5–2
5–3
5–4
5–5
Chapter 3
3–1
3–2
3–3
3–4
3–5
3–6
Orthonormal vectors of a CMG unit
Gimbal angle and vectors
Input ⁄Output ratio
Singularity condition and singular vector
Typical vector arrangement for a 2D
system
Steering at a singular condition
5–6
5–7
5–8
5–9
5–10
Manifolds in the neighborhood of a
singular point.
Continuous change of manifolds.
An example of a continuous manifold
path.
Relations between H space, manifold
space and θ space.
Domains and manifolds of the pyramid
type system
Class connection graph around domains
An illustration of class connection rule
(1).
An illustration of class connection rule
(2).
An illustration of motion by the gradient
method.
Manifold relations around critical point
Chapter 4
4–1
4–2
4–3
4–4
4–5
4–6
4–7
4–8
4–9
4–10
4–11
4–12
Vectors at a singularity condition
Examples of the singular surfaces for the
pyramid type system.
Envelope of a roof type system M(2, 2).
Cross sections of a singular surface of the
pyramid type system.
Infinitesimal motion from a singular point
of 2D system.
Second order infinitesimal motion from
singular surface.
Possible motions in both direction of u at
a singular point.
Local shape of an impassable singular
surface.
Impassable surface of S(6)
Impassable surface of Skew(5) with skew
angle α = 0.6 rad.
Impassable surface of another Skew(5),
with skew angle α = 1.2 rad.
Impassable surface of S(4).
Chapter 6
6–1
6–2
6–3
6–4
6–5
6–6
6–7
6–8
6–9
6–10
6–11
6–12
6–13
Schematic of a pyramid type system
Transformation in H space and in θ space
Line segments for singular manifold
Definition of the cross sectional plane and
the distance d
Saddle like part of the envelope
Cross sections of singular surface
Internal impassable singular surface
Analytical line on an impassable surface
Equilateral parallel hexahedron of
impassable branches
Overall structure of impassable branches
Internal impassable surface with envelope
cutaway
Cross section through the xz plane
Cross section through the xy plane
Chapter 7
7–1
7–2
Chapter 5
––– x –––
Candidate of workspace
Cross section nearly crossing P
7–3
7–4
7–5
7–6
7–7
7–8
7–9
7–10
7–11
7–12
7–13
7–14
7–15
7–16
7–17
7–18
7–19
7–20
Manifold bifurcation and termination
from DA
Simplified class connection diagram
around domain DA
Manifolds of eight domains around the z
axis
Singular manifold of a point U on the z−
axis
Manifold of H near the origin
Continuos change of manifold for H
nearly along the z axis
Manifold connection over several
domains
Cross sections of domains
Possible motion following an example of
singular surface
Illustration of H trajectory of the CMG
system for the example maneuver
Avoidance of an impassable surface
Problems of movement on an impassable
surface
Change in manifolds for H moving along
the x axis
Estimation of reduced workspace for
exact steering
Discontinuity in the maximum of
det(CCt)
Cross section of possible workspace by
constrained steering law
Reduced workspace of the constrained
system
Reduced workspace of three modes
8–12
8–13
Chapter 9
9–1
9–2
9–3
9–4
9–5
9–6
9–7
9–8
9–9
A–1
A–2
A–3
8–2
8–3
8–4
8–5
8–6
8–7
8–8
8–9
8–10
8–11
Vectors and variables relevant to a double
gimbal CMG
Vectors at singularity conditions
Infinitesimal motion at a singular point
of condition (b)
Appendix D
D–1
D–2
D–3
D–4
Experimental test rig showing the center−
mount suspending mechanism
Target trajectory
Block diagram of the control system
Results of Experiment A
Results of Experiment B
Results of Experiment C
Results of Experiment D
Results of Experiment E
Results by Experiment F
Results of Experiment G
Results of Experiment H
System configurations for comparison
Spherical workspace size for various
system configurations
Trade-off between workspace size and
system weight
Definition of ellipsoidal workspace
Average radius vs. skew angle
Workspace radius as a function of aspect
ratio
Combined plot of radii as a function of
aspect ratio
Converted weight as a function of aspect
ratio
Radius as a function of aspect ratio for a
degraded system with one faulty unit
Appendix A
Chapter 8
8–1
Command sequence of Experiment J
Results of Experiment J
Envelopes of S(6) and degraded systems
Four unit subsystem of MIR type system
Restricted workspace of a constrained
MIR-type system
Concept of singularity avoidance by an
additional torquer
Appendix E
––– xi –––
E–1
E–2
E–3
E–4
E–5
E–6
E–7
E–8
Experimental apparatus
Block diagram of experimental apparatus
Three axis gimbal mechanism
Single gimbal CMG
Balance adjuster
Onboard computer
Block diagram of the model matching
controller.
Block diagram of the tracking controller.
E–9
E–10
Block diagram of the gradient method.
Block diagram of the constrained method.
Appendix F
F−1
Analogy to a parallel link mechanism
––– xii –––
List of Tables
Chapter 2
2–1
2–2
8–2
Component Level Comparison
System Level Comparison
Appendix E
E–1
E–2
Chapter 6
6–1
6–2
Symmetric Transformations
Segment Transformation Rule
Specification of experimental apparatus
Code size and calculation time of process
Appendix F
F–1
Chapter 8
8–1
Condition and Results of Experiments (2)
Condition and Results of Experiments (1)
––– xiii –––
Similarity between CMGs and link
mechanism
––– xiv –––
–– 2. Characteristics of Control Moment Gyro Systems ––
Chapter 1
Introduction
A Control Moment Gyro (CMG) is a torque generator
for attitude control of an artificial satellite in space. It
rivals a reaction wheel in its high output torque and rapid
response. It is therefore used for large manned satellites,
such as a space station, and is also a candidate torquer
for a space robot.
There are two types of CMGs, single gimbal and
double gimbal. Though single gimbal CMGs are better
in terms of mechanical simplicity and higher output
torque than double gimbal CMGs, the control of single
gimbal CMGs has inherent and serious singularity
problem. At a singularity condition, a CMG system
cannot produce a three axis torque. Despite various
efforts to overcome this problem, the problem still
remains, especially in the case of the pyramid type CMG
system.
This research aims to elucidate this singularity
problem. Detailed study of the pyramid type system
leads to a global problem of singularity. The final
objective of this work involves evaluation of various
steering laws and the proposal of an effective steering
law. As all the geometric studies are either theoretical
or analytical and based on computer calculations, ground
experiments were carried out to support those results.
1.1 Research Background
Research of CMG systems started in the mid 1960s.
This was intended for later application to the large
satellite of the USA, “Skylab”, and its high precision
component, ”Apollo Telescope Mount (ATM)” 1, 2, 3).
The studies included hardware studies of a gyro bearing
and gyro motor, and software studies for attitude control
and CMG steering control. Evaluation of various types
and configurations was made in terms of weight and
power consumption4). At that time, an onboard computer
lacked the ability to perform real time matrix inversion
calculation. One of the candidates was a twin type
system made of two single gimbal CMGs driven in
opposite directions. Control of this system requires only
simple calculation5). If another system was chosen, a
simple computation scheme was required using an analog
circuit. For example, a method using an approximation
with some feedback was proposed6, 7, 8). For the three
double gimbal CMG system9) applied to the Skylab, an
approximated inverse using the transposed Jacobian was
used10). This CMG system successfully completed its
mission, though one of the CMGs became nonfunctional
during the flight11). After that, studies of double gimbal
CMGs have continued for eventual application to the
space shuttle and the space station “Freedom” which is
now called ISS12, 13).
Another CMG type, i.e., a single gimbal one, was
studied for use in satellites such as the “High Energy
Astronomical Observatory (HEAO)” and the “Large
Space Telescope (LST)”. One of the configurations
intensively studied was a pyramid type, which consists
of four single gimbal CMGs in a skew configuration.
Comparing six different independently developed
steering laws indicated that an exact inverse calculation
was necessary14). It was also observed from various
simulations that the singularity problem could not be
ignored. It was concluded that some sort of singularity
avoidance control using system redundancy was required
for this type system.
A roof type system, which is another four unit system
of single gimbal CMGs, was also a candidate for the
HEAO. As its mathematical formulation is simpler than
that of the pyramid type, singularity avoidance was
originally included in a steering law15). An improvement
of this law involved a new approach in which the nature
of numerical calculation and discrete time control were
utilized16).
Singularity avoidance has been studied for all CMG
types. This was a simple matter for double gimbal CMG
systems17)–20). Typically used was a gradient method,
which maximized a certain objective function by using
redundancy21). While this method was effective in the
evaluation of double gimbal CMG systems, it was not
successful for single gimbal CMG systems. For
––– 1 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
example, optimization of a redundant variable resulted
in discontinuity 16) or an optimized value became
singular15) in the case of a roof type system. In the case
of a pyramid type system, various problems were found
in computer simulations even when a gradient method
was used.
Margulies was the first to formulate a theory of
singularity and control22). His paper included geometric
theory of a singular surface, a generalized solution of
the output equation and null motion, and the possibility
of singularity avoidance for a general single gimbal CMG
system. Also, some problems of the gradient method
were pointed out using an example of a two dimensional
system.
Works by the Russian researcher, Tokar, were
published in the same year, and included a description
of the singular surface shape 23) , the size of the
workspace24) and some considerations of the gimbal
limits25). In his next paper26), passability of a singular
surface was introduced. It was made clear that a system
such as a pyramid type has an ‘impassable’ surface inside
its workspace. Moreover, the problems of steering near
such an impassable surface were described. In spite of
those important results, his work was not widely
received, because the original papers were published in
Russian. Even though an English translation appeared,
several terms were used for a CMG, such as “gyroforce”,
“gyro stabilizer” and “gyrodyne”. His conclusion was
that a system with no less that six units would provide
an adequately sized workspace including no impassable
surfaces. After this work, a six unit symmetric system
was designed for the Russian space station “MIR”27).
Some years later after Tokar’s studies, Kurokawa
formulated passability again28) in terms of the geometric
theory given by Margulies. Most of these results
coincided with Tokar’s work. In addition, the existence
of impassability in the roof type system was clarified29)
and a discrimination method using the surface curvature
was presented30,31,32,33). In the last paper, the theory
was expanded to a general system including a double
gimbal CMG system. It was made clear that multiple
systems of no less than six units do not have any internal
impassable surface, while any system of less that six
units must have such a surface. Various configurations,
even containing faulty units, were compared with regard
to their workspace size as an extension of Tokar’s work.
Along with these theoretical and general research
works, intensive efforts continued to find an effective
steering law regarding the passability problem as a local
problem. Most of these dealt with the pyramid type
system. The reason this type was selected was because
a six unit system was considered too large and too
complicated. Many proposals suggested a type of
gradient method34, 35, 36). The method utilized for the
four unit subsystem of the “MIR” was also of this
kind27). Another method used global optimization28),
and nearly all methods showed some problems in
computer simulations.
Passability is defined locally and its problem reported
first was a kind of local problem28). Later, Bauer showed
difficulty in steering as a global problem37). He found
two different command sequences, both of which could
not be realized by the same steering method. After this,
Vadali proposed a method to overcome this problem
using a preferred state38). Finally, the problem by Bauer
was formulated exactly, stating that no steering law can
follow an arbitrary command sequence inside certain
wide region of the workspace39). Under this limitation,
an effective method was proposed.
The research described above dealt with exact
control, but other research has also been carried out. One
research effort permitted an error in the output if required.
Generalized inverse Jacobian22) minimizes the error.
Extension of this method, called the SR inverse method,
was first proposed for control of a manipulator and later
applied to CMG control40,41). Another research type
dealt with path planning. If the command sequence in
the near future is given, steering can be planned
beforehand which realizes not only singularity avoidance
but also some degree of optimization42, 43, 44). In one
of the research papers43), some paths were chosen by
Kurokawa in consideration of impassable surfaces. Since
all these tended to take a heuristic approach, evaluation
was made by computer simulation considering attitude
control of a given satellite.
More realistic studies have also been made which
dealt with attitude control using a CMG system,
considering disturbance and other torquers. The largest
problem may be a precision control using a CMG system.
Since a CMG system can generate a large output torque
and its output resolution is critical for precision control,
various analyses and simulations have shown that
pointing control by a CMG system can result in a limit
cycle because of friction in gimbal motion45, 46, 47). In
spite of efforts such as improvement of motor control48)
and torque cancellation by additional reaction wheels49),
the problem of precision control has not been overcome.
For application to the space station, another studies were
carried out such as an effective combination of a CMG
and RCS 50) and integration of CMGs and power
––– 2 –––
–– 2. Characteristics
Moment
–– of
1. Control
Introduction
–– Gyro Systems ––
storage51). In order to evaluate its attitude control
performance, not only numerical simulations, but also
some experiments using real mechanisms have been
made, such as a platform supported by a spherical air
bearing44, 52). The author also developed ground test
equipment using normal ball bearings53) and attempted
robust attitude control using a CMG system54,55).
The motion of a CMG system with regard to the
motion of the angular momentum vector is similar to
the motion of a link mechanism22). Analysis of the
motion and control of such a mechanism has been widely
studied. Those results were, therefore, used for CMG
control40, 41). On the other hand, some researchers first
studied CMG control and then applied their results to a
robot control56, 57, 58). In spite of various researches in
robot kinematics 59, 60, 61) , generalized theory for
singularity and inverse kinematics has not been
formulated yet.
1.2 Scope of Discussion
This research effort deals with the following subjects:
(1) General formulation of an arbitrarily configured
CMG system, especially of single gimbal CMGs.
(2) Geometric study of the singularity problem of a
general single gimbal CMG system.
(3) Problem of exact and real-time steering of the
pyramid type CMG system.
(4) Proposal and evaluation of steering laws for the
pyramid type CMG system.
(5) Evaluation of various CMG systems.
The main purposes of this work are to clarify the
singularity problems, to construct an exact and strictly
real-time steering law, and to specify and evaluate its
performance. Among all, singularity problems are the
most important relating to the others. A singularity can
degrade a CMG system, even causing the system to loose
control, and this situation might be fatal for an artificial
satellite. Therefore, a CMG system must have
redundancy and it must be controlled to avoid
singularities by using an appropriate steering law.
Problems include whether such singularity avoidance is
globally possible and which steering law can realize such
control. Even if a steering law cannot avoid all the
singularities, the system’s working range of the angular
momentum must be specified in which singularity
avoidance is strictly guaranteed because such
specification is necessary for designing the total attitude
control system. Thus, this work deals with CMG systems
alone, but it is made in consideration with the attitude
control of artificial satellites. Exactness and strict real
time feature of steering laws are essential for the realtime attitude control.
For this aim, a geometric approach was taken. As
described above, there have been various research works
dealing with singularity and steering laws. Most used
computer simulations to evaluate their steering laws, for
lack of other methods. As simulations alone cannot
guarantee the performance of a system as nonlinear as a
CMG system, it is necessary to clarify the problem of
singularity by other means. A geometrical approach is a
more effective way of simplification and qualitative
comprehension. The theoretical portion of this work
aims for general formulation of singularity problems.
Under consideration of these general results,
extensive study was made for a specific type of system,
that is, the pyramid type. The reasons why this system
was chosen are:
1) A three-unit system does not need further study
because it has no redundancy. Systems with no less
than six units also do not need detailed study for
singularity avoidance, a fact described in more detail
in this work. Thus, four and five unit systems remain
for further study.
2) Most previous research works dealt with this
pyramid type system. Four units are the minimum
having one degree of redundancy. The number of
units is important in the real situation. By a
simplified evaluation, a system with fewer units is
lighter for a given total storage of angular
momentum. Also, steering law calculation is less
complicated for a system with fewer units.
3) The pyramid type system has symmetry, which
enables easier analysis. Numerical data and
analytical expression of some geometric
characteristics can be reduced by using this
symmetry. This fact is useful for actual
implementation.
As geometric study is more qualitative rather than
quantitative, ground experiments were performed to
demonstrate the performance of the steering laws. Also
for evaluation, various system types are compared in
terms of the size of the possible angular momentum
vector operational space and the systems’ weight.
As mentioned above, specific studies of an attitude
control are beyond the scope of this work. Such studies
involve optimal maneuvering and angular momentum
management, which are possible only after the
––– 3 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
specification of CMG systems are given by using the
results of this work. In addition, this work does not treat
in detail steering laws of double gimbal CMG systems,
combinations of single and double gimbal CMGs,
combination of CMGs and other torquers, passive type
CMGs62), and systems with different size or controllable
size CMGs63). Also, the effect of gimbal limit is not
considered in general except in the proposed method for
the pyramid type system.
1.3 Outline of this Thesis
Chapter 2 will represent a general description of a
CMG unit and CMG systems. The difference between
three types of torquers, reaction wheels, single gimbal
CMGs and double gimbal CMGs, will be described.
Also an important parameter termed ‘workspace’ in this
paper will be defined in terms of an attitude control
system.
Chapter 3 will represent a general formulation of an
arbitrary system of single gimbal CMGs, which includes
the kinematic equation and the torque equation. The
general steering law, singularity and singularity
avoidance will be outlined. This chapter is analytical
while the following chapters, from Chapter 4 to 7, are
mainly geometrical.
Chapter 4 will detail singularity. A singular surface
which includes the angular momentum envelope will
be examined. For this surface, ‘passability’ which is
one of the most important characteristics of a singular
surface will be defined. Passability and surface geometry
will be related by the curvature of the surface.
Chapter 5 will introduce a way of understanding the
steering motion as to whether continuous control is
possible, or how the impassable situation can be avoided,
if possible.
From Chapter 6 to 8, the pyramid type system will
be detailed. Chapter 6 will offer analytical and geometric
system results without considering a steering law. The
impassable singular surface of this system will be fully
defined.
Chapter 7 will prove the ‘global’ problem. After
various proposals are evaluated based on this result, a
new proposal will be offered.
Chapter 8 will demonstrate the performance of the
proposed method by using a ground test apparatus.
Chapter 9 will offer evaluation not only of the
proposed method for the pyramid type system but also
of various system configurations.
Chapter 10 will conclude this work.
Because double gimbal CMG systems and various
systems other than the pyramid type system will not be
detailed in the main text, Appendices A and D will
provide these details. Appendices B and C present
detailed proofs of some theories given in Chapter 4.
Appendix E will give the specifications and
implementation of the ground test apparatus. Appendix
F will detail the kinematics of a general spatial link
mechanism which is analogous to the CMG kinematics.
––– 4 –––
–– 2. Characteristics of Control Moment Gyro Systems ––
Chapter 2
Characteristics of Control Moment Gyro
Systems
A control moment gyro (CMG) system is a torquer
for three axis attitude control of an artificial satellite.
There are two types of CMG units and various
configurations of three axis torquer systems. Designing
a CMG system therefore includes a process of selecting
a unit type and a system type defined by configuration.
Among two unit types and various system types, a
single gimbal CMG system of pyramid configuration is
mainly described in this work. For the simple
comparison, this chapter gives an outline of CMG system
characteristics with consideration paid to its use in an
attitude control system. The angular momentum
workspace, torque output, steering law and singularity
problems are the important factors for evaluation of a
CMG system.
axes in the case of single gimbal CMGs and the outer
gimbal axes in the case of double gimbal CMGs. In the
following figures, these principal axes are indicated by
arrows denoted by gi .
The system of each configuration is named as a system
type such as twin type system or the pyramid type system.
Gyro Effect Torque
T
Gimbal Motor
AA
AA
Gyro Motor
ω
Flywheel
Gimbal
Mechanism
2.1 CMG Unit Type
Angular Momentum Vector
A CMG consists of a flywheel rotating at a constant
speed, one or two supporting gimbals, and motors which
drive the gimbals. A rotating flywheel possesses angular
momentum with a constant vector length. Gimbal
motion changes the direction of this vector and thus
generates a gyro−effect torque.
There are two types of CMG units, as shown in Fig.
2–1, single gimbal and double gimbal. A single gimbal
CMG generates a one axis torque and a double gimbal
CMG generates a two axis torque. In both cases, the
direction of the output torque changes in accordance with
gimbal motion. For this reason, a system composed of
several units is usually required to obtain the desired
torque.
(a) Single gimbal CMG
Outer Gimbal
Motor
Flywheel
A
AA
AA
A
Gyro
Motor
Outer
Gimbal
Inner
Gimbal
Motor
Inner Gimbal
2.2 System Configuration
(b) Double gimbal CMG
Typical system configurations will now be discussed.
The configuration is defined by a set of principal axes
of all the component CMG units, which are the gimbal
––– 5 –––
Fig. 2–1
Two types of CMG units
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
2.2.1 Single Gimbal CMGs
Typical single gimbal CMG systems have certain kinds
of symmetries, which can be classified into two types,
‘independent’ and ‘multiple’. They are somewhat
different in their mathematical description.
(1) Independent Type
Independent type CMGs have no parallel axis pairs.
Two categories of independent type CMGs, ‘symmetric
types’ and ‘skew types’, have been mainly studied.
Symmetric Type G i m b a l a x e s a r e a r r a n g e d
symmetrically according to a regular polyhedron. There
are five regular polyhedrons with 4, 6, 8, 12 and 20
surfaces. Possible configurations of this type are three,
four, six and ten unit systems, because only surfaces not
g4
θ4
θ3
h3
parallel to each other are considered and because a
tetrahedron and hexahedron are complementary or
“dual” to each other. The three, four, six, and ten unit
systems are denoted as S(3), S(4), S(6) and S(10). The
four unit or S(4) system, shown in Fig. 2–2(a), is called
the symmetric ‘pyramid type’. Most of this work deals
with this type of system. An example of the six unit or
S(6) system, shown in Fig. 2–2(b), is now in use on the
Russian space station “MIR”.
Skew Type
All individual units are arranged
in axial symmetry about a certain axis as depicted in
Fig. 2–2(c). Skew three and four unit systems of certain
skew angles are the same as the S(3) and the S(4).
(2) Multiple Type
In this type some number of individual units possess
g3
g4
h4
g5
Z
h2
α
X
g1
g6
g2
h1
α
g3
h3
h5
θ2
Y
h4
2π⁄n
θ1
g2
h2
g1
h6
h1
(c) Skew type
(a) Pyramid type S(4)
g1
g2
θ 12
h6
θ 23
g1
h1
h3
g2
g1
g6
h2
θ 13
g1
g2
θ11
g5
θ 22
h5
g3
h4
g2
θ 21
g4
(b) Symmetric type S(6)
(d) Multiple type M(3, 3)
Fig. 2–2 Configurations of single gimbal CMGs
––– 6 –––
–– 2. Characteristics of Control Moment Gyro Systems ––
identical gimbal directions. These are denoted as M(m1,
m2, ...) hereafter, where mi is the number of the units
with the same gimbal direction. As an example, the
system in Fig. 2–2(d) is denoted by M(3,3). A similar
system called ‘roof type’15, 16) would be denoted as
M(2,2) with this notation.
Gimbal Motor
2.2.2 Two Dimensional System and Twin Type
System
A single gimbal CMG system of an arbitrary number
of units all having a common gimbal direction will be
called a two dimensional system in this work. In such a
system, the angular momentum vector and output torque
vector are always on a certain plane normal to the gimbal
direction. Though this type of system is not ordinarily
used by itself for attitude control, it can easily be
visualized and understood. It is, therefore, used for some
examples in this work.
If a pair of single gimbal CMG units with a common
gimbal direction are driven in opposite directions by the
same angle, the direction of the output torque is always
kept constant, as shown in Fig. 2–3. This type of system
is called a ‘twin type’ or a ‘V−pair’ system5).
Though a three axis system is easily designed by
combining several twin type CMGs, such a system is
not so much advantageous. A three or more V−pair
system is identical to a multiple system, M(2, 2, ..., 2),
whose state variables are constrained, but its workspace
is smaller than that of the original multiple system.
Though a V−pair system is the easiest to control, a
multiple system can be also simply controlled as will be
described later.
Fig. 2−3 Twin type system
2.2.3 Configuration of Double Gimbal CMGs
Two typical configurations of double gimbal CMGs
are an orthogonal type and a parallel type. The
orthogonal type consists of three orthogonally positioned
units. This type of system was used for the ‘Skylab’
space vehicle. The parallel type consists of an arbitrary
number of units all having a common axis20).
2.3 Three Axis Attitude Control
The design requirement of a CMG system is
determined by the specification of a spacecraft attitude
control. There are various kinds of attitude control
techniques such as spin stabilization, bias momentum
stabilization and zero momentum active control. The
last is also called three axis attitude control. Reaction
wheels and CMGs are commonly used torquers for this
Disturbance
A
Maneuver
Command
Generator
B
+
T com
Vehicle
Control
Law
-
D
C
CMG
Steering
Law
Momentum
Management
Control Logic
TCMG
CMG
System
Unloading
Torquers
Attitude & Rate
Sensors
Fig. 2–4 Block diagram of three axis attitude control
––– 7 –––
Spacecraft
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
attitude control.
2.3.1 Block Diagram
A functional block diagram of a three axis attitude
control is shown in Fig. 2–4. Most of the blocks are the
same when either reaction wheels or CMGs are used.
The attitude and rotational velocity commands are
generated by a maneuver command generator denoted
by A in Fig. 2–4. The command and sensor information
are the inputs to the vehicle control law block, B. This
block calculates the torque necessary for control. The
next block, C, shows the CMG steering law which
calculates the CMG motion for the torque calculated by
block B. In this manner the actual CMG system is driven
and an output torque to the satellite is generated. The
blocks relating to CMG control are the CMG steering
law, C, and the momentum management block, D. Those
two blocks are described first in the following sections.
Then, relating subjects, i.e., maneuver commands,
disturbances and the motion of angular momentum
vector will be explained.
management control block, D, because such torquers
have their own limitations, i.e., a gas jet does not have
enough resolution and it have a limit of storage, and a
magnetic torquer’s output depends on orbit position.
For effective management of angular momentum, the
space of allowed angular momentum of a CMG system
must be defined beforehand. This space is termed
‘workspace’ in this paper. The workspace must be
included by the possible angular momentum space of
the CMG itself. Moreover, a simple shaped space such
as a sphere tends to result in more simplified
management.
2.3.4 Maneuver Command
The command issued by a maneuver command
generator depends on the mode of operation. Typical
operational modes are pointing, maneuvering, scanning
and tracking. In the pointing mode, precision is of
primary importance and is affected by disturbances,
torque response and resolution. The speed of
maneuvering as well as momentum accumulation while
pointing is a matter of workspace size of the torquer.
2.3.2 CMG Steering Law
2.3.5 Disturbance
The steering law block computes a set of gimbal angle
rates which produce the required torque. The steering
law is usually realized in two parts, one being simply a
solution to a linear equation and the other for singularity
avoidance by using system redundancy.
This block is usually designed independent of the
particulars of the total attitude control system. This
implies that the vehicle control law (B in Fig. 2–4) is
designed under the assumption that the output of the
CMG system corresponds exactly to the command. The
CMG steering law must satisfy this requirement. The
meaning of this exactness is described in a later chapter.
The time dependence of disturbances vary according
to orbit parameters and a mission type, such as earth
pointing or inertial pointing. In any case, a disturbance
may have cyclic terms and offset terms. The following
function is an example of disturbance used for the
simulation of HEAO with a pyramid type CMG
system14);
Tg = (Txsin ωt, Ty(cos ωt − 1), Tzsin ωt)t,
where ω denotes orbital angular rate. Because there is
an offset in the y direction, angular momentum will be
accumulated in this direction while pointing.
2.3.3 Momentum Management
2.3.6 Angular Momentum Trajectory
A CMG and a reaction wheel are called momentum
exchange devices because they don’t actually “produce”
angular momentum but rather exchange it with the
satellite. Such torquers have limits to their accumulation
of angular momentum, because the rotational speed of a
flywheel is limited. Therefore, another type torquer is
needed when it becomes necessary to offload excess
accumulated momentum. This unloading is usually done
by gas jets or magnetic torquers. The unloading process
must be carefully managed by the momentum
The size and shape of the workspace determines the
maximum accumulation of disturbances or the maximum
speed of maneuvering. A disturbance or a maneuvering
command can be expressed as a function of time by a
trajectory of the angular momentum vector of the
satellite. Since the total angular momentum of the system
is equal to the time integral of the disturbance, the angular
momentum trajectory of a CMG system can be expressed
using the spacecraft’s momentum and disturbance. The
––– 8 –––
–– 2. Characteristics of Control Moment Gyro Systems ––
workspace of a CMG system must include any possible
angular momentum trajectory when the unloading
torquers are not operating.
2.4 Comparison and Selection
CMG systems and a reaction wheel system are all
examples of the same type of torquers. In order to design
an attitude control system, some sort of selection criteria
is needed. By using the following performance indices,
a brief comparison will be made, first at the component
level then at the system level.
2.4.1 Performance Index
The performance of a CMG systems depends not only
on elements of hardware design, such as the CMG unit
type and the system configuration, but also on the design
of the steering law. These factors all affect the maximum
workspace and the magnitude of the output torque, two
nonscalar performance indices. Another performance
index is the steering law complexity, which affects the
attitude control cycle time and the capacity of an onboard
computer.
2.4.2 Component Level Comparison
Table 2−1 clarifies the main differences among these
three torquers64). A reaction wheel has only one motor
which is used not only for accumulation of angular
momentum but also for generation of torque. On the
other hand, the CMGs use either two or three motors,
one for accumulation of angular momentum and the
others for torque generation. Since the torque of a motor
depends on its speed and the same maximum torque
cannot be generated over the motor’s working speed
range, both angular momentum and output torque of a
reaction wheel are much smaller than for CMGs.
Size and weight of a CMG depends on the size of the
flywheel and complexity of the mechanism. A double
gimbal CMG is the most complicated at the unit level,
but less so at the system level because this unit generates
Table 2–1 Component Level Comparison
––––––––––––––––––––––––––––––––––––––––––––
Angular Momentum Torque
Reaction Wheel
1 to 1000
1
Double Gimbal CMG 1000 to 3000
100
Single Gimbal CMG 10 to 2000
1000
––––––––––––––––––––––––––––––––––––––––––––
a two axis torque.
Maximum output torque is much different. A single
gimbal CMG can produce more output torque than a
double gimbal CMG. The reason is as follows. The
output torque of a single gimbal CMG appears on the
flywheel and is then transferred directly to the satellite
across the gimbal bearings. The output torque can be
much larger than the gimbal motor torque required to
drive the gimbal. This is called ‘torque amplification’.
By contrast, some part of the output torque of a double
gimbal CMG must be balanced by the gimbal motors.
Thus, in this case, the output toque is limited by the motor
torque limit.
2.4.3 System Level Comparison
Table 2−2 shows a system level comparison for the
three types of torquers being compared. Difference in
the first two indices, torque and weight, are derived from
component level differences. The other two indices
relate to each other. The steering law of any reaction
wheel system is linear and no singularity problems arise.
Steering law complexity and singularity problems of
CMG systems, especially single gimbal CMGs, can be
serious and thus form the main subject of the present
work.
2.4.4 Work Space Size and Weight
The size and shape of the maximum workspace are
not compared in the above table because they depend
on the number of units and system configuration.
Workspace size as a scalar value, and the weigh of the
CMG system can be roughly evaluated in terms of the
number of units. Let’s consider similarly shaped
Table 2–2 System Level Comparison
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Torque Weight
Steering Law
Singularity
Reaction Wheel
1
1
simple
none
Double Gimbal CMG 100
2
not simple
slight
Single Gimbal CMG 1000
2
most complex
serious
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
––– 9 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
flywheels of diameter d and thickness t. Similarity
implies t ∝ d. Then, the weight and the size of maximum
workspace of an n unit system, denoted as W and H,
follow the following relation if the rotational rate of the
gyro is the same:
W ∝ n d2 t ∝ n d3,
(2–1)
H ∝ n ∫ (t d d2) dr ∝ n d5 .
(2–2)
If H is set constant, W is given by;
W ∝ n d3 ∝ n 2/5 .
(2–3)
This implies that the system with fewer units is lighter
but can still realize the same workspace size. Despite
the fact that other factors are ignored in estimating the
weight, it can generally be concluded that the systems
of less units have advantages in weight.
In this evaluation, it is assumed that the size of the
work space is proportional to the number of units by the
same multiplier for any system. From the comparison
in Chapter 9, this is almost true for systems of no less
than 6 units in the case of single gimbal CMGs. This,
however, is not true in the case of less that 6 units.
Therefore it is better to evaluate some configuration
composed of 4 to 6 units.
––– 10 –––
–– 3. General Formulation ––
Chapter 3
General Formulation
This chapter first defines vectors, variables and
parameters of a single gimbal CMG system in an
arbitrary configuration, after which a basic
mathematical description of several system
characteristics are made. These characteristics are the
kinematic equation, the steering law, the torque output
performance index, and singularity avoidance. The
shape of the maximum workspace and singularity
problem are described in the next chapter. Similar
descriptions for double gimbal systems are given in
Appendix A.
dependent upon the gimbal angle θi. Once the initial
vectors are defined as in Fig. 3–2, the other vectors are
obtained as follows;
hi = hi0cosθi + ci0sinθi ,
ci = − hi0sinθi + ci0cosθi .
(3–2)
The total angular momentum is the sum of all hi
multiplied by the unit’s angular momentum value which
is denoted by h. In this work, H denotes the total angular
momentum without the multiplier h:
H = Σ hi .
3.1 Angular Momentum and Torque
A generalized system is considered consisting of n
identically sized single gimbal CMG units. The number
n is not less than 3 to enable three axis control. The
system configuration is defined by the relative
arrangement of the gimbal directions. The system state
is defined by the set of all gimbal angles, each of which
are denoted by θi. Three mutually orthogonal unit
vectors are shown in Fig. 3–1 and defined as follows:
gi : gimbal vector,
hi : normalized angular momentum vector,
ci : torque vector,
where
ci = ∂hi / ∂θi = gi × hi .
This relation is simply written as a nonlinear mapping
from the set of θi to H;
H = f (θ) .
(3–4)
The variable, θ=(θ1, θ2, ..., θn), is a point on an n
dimensional torus denoted by T(n) which is the domain
of this mapping. The mapping range is a subspace of
the physical Euclidean space and is denoted by H. This
space is the maximum workspace.
By the analogy of this relation with a spatial link
mechanism, this relation will be called “kinematics” or
“kinematic equation” in this work (see Appendix F).
The output torque without the multiplier h is obtained
by taking the time derivative as follows.
T = − dH / dt = − Σ ∂hi/∂θi dθi/dt .
(3–1)
The gimbal vectors are constant while the others are
(3–3)
(3–5)
Any additional gyro effect torques generated by the
satellite motion are omitted because they are usually
gi
θ
hi0
θi
g
hi
h
ci0
c
Fig. 3–1 Orthonormal vectors of a CMG unit
ci
Fig. 3–2 Gimbal angle and vectors
––– 11 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
treated in the overall satellite system dynamics, which
includes the CMG system (see Chapter 8).
Because the total output torque is a sum of output of
each unit, it is also given as,
where [a b c] denotes the vector triple product, a⋅(b×c).
I/O Ratio
(3–6)
where
ωi = dθi/dt, and ω = (ω1, ω2, ..... , ωn)t .
(3–7)
The variable ωi is the rotational rate of each gimbal.
The vector ω is a component vector of a tangent space
of T(n). The matrix C is a Jacobian of Eq. 3–4 and is
given by,
C = (c1 c2 .... cn) .
(3–11)
3.3 Singular Value Decomposition and
T = − Σ ci ωi
= −Cω ,
[c3 c1 c2], −[c1 c2 c3]) ,
The magnitude of the total output torque is not a
simple sum of the output of each unit. An elements of
each output, ωici, normal to T cancels each other. The
ratio of input and output norms, |ω|/|T|, can be evaluated
by a singular value of the matrix C.
The matrix C can be decomposed into a diagonal
matrix by two orthonormal matrices, Q (3×3) and R
(n×n) as follows;
(3–8)
0
σ 1 0
QCR =  0 σ 2 0
 0
0 σ3
As the unit’s angular momentum value is omitted in
Eqs. 3–5 and 3–6, the real output is obtained by
multiplying h.
3.2 Steering Law
The ‘steering law’ functions to compute the gimbal
rates, ω, necessary to produce the desired torque, Tcom,
and is generally given as a solution of the linear equation
given in Eq. 3–6:
0 . . 0
0 . . 0  , (3–12)
0 . . 0 
where σi is called a singular value of C. As shown in
Fig. 3–3, the maximum ratio of the input and output
norms is given by the radius of the ellipsoid whose
principal diameters are the singular values. Thus, the
ω3 ... ωn
ω = −Ct(CCt)−1Tcom + (I − Ct(CCt)−1C) k .
|ω|=1
(3–9)
where I is the n × n identity matrix and k is an arbitrary
vector of n elements.
The first term has the minimum norm among all
solutions to the equation. The matrix Ct(CCt)−1 is called
a pseudo-inverse matrix. The second term, denoted by
ωN, is a solution of the homogeneous equation;
C ωN = 0 .
ω2
ω1
n - sphere
(a) Gimbal rate
(3–10)
This implies that the motion by this ωN does not generate
a torque (T) and keeps the angular momentum (H)
constant. In this sense, this term is called a ‘null motion’.
The null motion has n−3 degrees of freedom because it
is an element of the kernel of the linear transformation
represented by C.
An effective method of calculating a null motion is
given in Ref. 22. For example, a null motion of a four
unit system is generally given as,
ωN = ([c2 c3 c4], −[c3 c4 c1],
σ3
H
σ1
σ2
(b) Angular momentum ellipsoid
Fig. 3–3 Input ⁄ Output ratio
––– 12 –––
–– 3. General Formulation ––
size of this ellipsoid represents the performance index
of the output torque. The following relations are derived
from the fact that all row vectors of C are unit vectors.
σ12 + σ22 + σ32 = Trace(CCt) = n ,
(3–13)
det(CCt) = (σ1 ⋅ σ2 ⋅ σ3)2 .
(3–14)
h3
h2
H
h1
Singular
Line
O
S1
S0
A
B
3.4 Singularity
Angular Momentum
Envelope
The steering law function in Eq. 3–9 is invalid at lower
ranks of C where the following condition is satisfied:
det(CCt) =0 .
(3–15)
(a) Angular momentum
Referring to Fig. 3–4, degeneration of rank implies that
all the possible output, T of Eq. 3–6, does not span three
dimensional space. Since all the row vectors, ci, of
matrix C become coplanar, the output T does not have a
component normal to this plane. Let u denote the unit
normal vector of this plane and be called a ‘singular
vector’. It is defined by
u ⋅ ci = 0, where i = 1, 2, ...., n ,
O
A
S0
(3–16)
and may also be written in the matrix form as;
B
ut C = 0 .
(3–17)
The rank of C does not generally reduce to 1. The
rank is unity only when all ci are aligned in the same
direction. This can only happen if all the gi are on the
same plane, as the case for a roof type system.
When the system is singular, one of the singular
values reduces to 0. In this sense, the minimum singular
value can be used as a singularity measure. But the
determinant, det(CCt), is also useful as such a measure
and is more easily calculated.
Figure 3–5 shows two types of singularity of a two
dimensional, three unit system. The border of the
maximum workspace, termed the ‘angular momentum
Singular Vector u
c1
c2
cn
Fig. 3–4 Singularity condition and singular vector
S1
(b) Vector arrangement
Fig. 3–5 Typical vector arrangement
for a 2D system
envelope’ is clearly singular. The singular H other than
this envelope are called ‘internal’.
3.5 Singularity Avoidance
Any steering law is based on the solution of Eq. 3–
9. Among all solutions, the pseudo-inverse solution with
no null motion was regarded effective. However, the
fact that the pseudo-inverse solution has a minimum
norm implies that once the torque vector is nearly normal
to the required output then this unit hardly moves. If
the required torque maintains its direction, such a unit
keeps its state so the system sometimes approaches a
singular state. In order to avoid such a situation,
singularity avoidance is usually included in the steering
law.
––– 13 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
3.5.1 Gradient Method
A system containing more than three units possesses
null motion redundancy. Freedom in determining null
motion can realize singularity avoidance while keeping
the output torque exactly equal to the command. The
gradient method is a general method in which some
objective function is maximized. The following
formulation of a gradient method is taken from Ref. 21.
The objective function, W(θ), is chosen as a
continuous function of θ. It is zero in the singular state
and otherwise positive. The dependence of W on CMG
motion is:
∆W = Σ ξiωi ,
(3–18)
ξi = ∂W/∂θi .
(3–19)
where
In order to obtain the objective function extremum,
the motion ω should be determined so that ∆W is
positive. This ∆W has two parts, one given by the
pseudo-inverse solution and the other by a null motion.
The first depends on the command torque Tcom, while
the latter depends on the selection of a null motion.
Though the first part cannot be changed, the latter can
be freely determined. The latter part is evaluated as
follows;
∆WN = ξt (I − Ct(CCt)−1C) k .
3.5.2 Steering in Proximity to a Singular State
There is no solution to Eq. 3–6 in a singular state
except when Tcom is orthogonal to the singular vector
u. Even when Tcom is normal to u, the solution is not
given by Eq. 3–9 because the linear equation is
mathematically singular. A generalized solution can be
obtained which is the exact solution when Tcom is
normal to u otherwise minimizes the output error. The
minimum error is realized when the output is equal to
the projection of the torque command onto the plane
normal to the singular direction (Fig. 3–6). Such motion
is given as22):
ω = − Ct(CCt + k uut)−1 Tcom .
(3–21)
then ∆W N becomes a semi-positive quadratic form.
Thus, the null motion by this k results in non-negative
∆W N, so it is expected that singularity is avoided.
Various objective functions have been proposed, such
as:
(3–22)
Derivation of this is explained by supposing that there
is a virtual CMG unit whose torque vector c equals u.
Another method called the SR (Singularity Robust)
inverse steering law is proposed as a smooth extension
of this41). This method minimizes the weighted sum of
the input norm, |ω|, and the norm of the error. The SR
solution is given as:
ω = − Ct(CCt + W)−1 Tcom,
(3–20)
It is easily observed that the matrix (I − Ct(CCt)−
1 C) is semi-positive symmetric. If the vector k is
selected as:
k = k ξ, where k >0 ,
pyramid type single gimbal CMG systems, various
simulations showed that a gradient method is not
effective. Details of this problem is described in
Chapters 4, 5 and 7.
where W is a n×n matrix .
(3–23)
In both methods, the solution is zero if the command,
Tcom, is either zero or parallel to the u direction. This
method, therefore, cannot always guarantee avoidance
of a singular state nor can it escape from one. Moreover,
this kind of control is effective only if the attitude control
is not totally degraded by the error in torque. Details
are described in Section 7.2.
Tcom
Possible Output
(1) (det(CCt))−1/2, 21)
u
(2) min(σi), 36)
(3) min(1/|di|),
c1
where di is a row vector of the matrix
Ct(CCt)−1, 35)
c2
cn
(4) Σi,j |ci × cj |2, 27).
This gradient method has been successful for double
gimbal CMG systems 21). However, in the case of
––– 14 –––
Fig. 3–6 Steering at a singular condition
–– 4. Singular Surface and Passability ––
Chapter 4
Singular Surface and Passability
Angular momentum vectors in a singular condition
form a smooth surface which includes the angular
momentum envelope. This chapter first summarizes the
geometric theory of the singular surface of a general
single gimbal CMG system by following the research
work in Ref. 22. It includes a definition of a singular
surface, a mapping from a sphere to the surface, and
techniques for drawing the surface by computer
calculation. By using these techniques, the workspace
is visualized for various system configurations. Also,
geometric characteristics such as Gaussian curvature of
a singular surface is defined.
The passability of a singular surface is then defined.
The existence of an impassable surface explains why
most steering laws fail to generate output starting from
certain initial states. A gradient method works well for
avoiding passable singular points but not for avoiding
impassable ones.
The passability can be determined by the curvature
of the singular surface. It is demonstrated that any
independent type system has an internal impassable
surface while multiple type systems of no less than six
units have no internal impassable surfaces.
4.1 Singular Surface
4.1.1 Continuous Mapping
εi = sign( u ⋅ hi) .
(4–1)
Thus there are 2n combinations of singular points
for the given direction u. This combination is denoted
by ε or by a set of signs, such as {+ + − + ... +}.
For the given singular direction u and the given set
of signs, each torque vector in the singular condition is
determined by:
cSi = εi gi × u / |gi × u | .
(4–2)
From this point, variables subscripted by S denote
singular point values. The total angular momentum HS
is obtained as follows:
HS = Σ εi (gi × u) × gi / |gi × u | .
(4–3)
This defines a continuous mapping from u to HS
while the εi are fixed as parameters. The domain of u is
a unit sphere except ±g i direction, because the
denominator of Eq. 4–3 is zero when u = ±gi . Thus HS
with fixed εi form a two dimensional surface with u
covering this sphere. This surface is denoted as Sε. If
all the εi are reversed and the vector u is changed to −u,
HS remains the same. This implies that the surface of
{εi} and the surface of all the εi reversed are identical.
For example S{− + +} is the same as S{+ − −}. One may
thus suppose that no less than half of the εi are positive.
Thus, the number of different surfaces is 2n−1.
In case that u = ±gi, any state of this ith unit satisfies
Let’s examine all the singular points and their H
vectors. First, an independent type system is assumed
in the following discussion.
The torque vectors, ci, satisfy the condition given by
Eq.(3–16) when the system is singular. On each singular
point, a singular vector u is defined. As a reverse relation
of this, singular points are obtained from a given u vector.
Given any singular vector u, there are two
possibilities of singularity condition for each unit as hS
and –hS in Fig. 4–1. The two cases are distinguished by
the following sign variable;
u
hS
g
ε=1
cS
ε=–1
–cS
– hS
Fig. 4−1 Vectors at a singularity condition
––– 15 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
z
g1
g2
x
y
–g3
–g4
(a) Lattice points of the unit sphere of vector u
Unit Circle C1
Envelope
g1
z
x
y
the singular condition. As the vector hi rotates about gi,
these singular H form a unit circle which appears as a
hole or a window of the surface Sε as shown in Fig. 4–2
(b). As there is a hole for each gi, or −gi, direction, the
surface has 2n holes in total. Surfaces of different {εi}
are connected by these unit circles (for example, C1 in
Fig. 4–2 (b) and (c)). Thus all the surfaces form a closed
surface. This closed surface is called a ‘singular surface’.
It may be noted that the same kind of continuous mapping
is defined from u to θS with all the θS forming a two
dimensional surface in the n dimensional torus of θ. Such
a surface, however, is not termed a singular surface in
this paper.
An independent type system is assumed in the above
discussion. In the case of a multiple type system, the
number of different singular surface is 2m−1 where m is
the number of groups. Each surface has 2m holes of
diameter of several values which is determined by the
number of units in a group and sign ε. In case that u =
±gi, any state of units of this group satisfies the singular
condition. Thus, all singular H of this u form a circular
plate which fills the hole. Another singular surface of
different sign connects to this plate by a circle of different
diameter.
4.1.2 Envelope
(b) Singular surface of all sign positive denoted by
S {++++}
Unit Circle C 1
Envelope Portion
Internal Portion
Unit Circle C 2
(c) Singular surface of one minus sign denoted by
S{−+++}
Fig. 4–2 Examples of the singular surfaces for the
pyramid type system.
Each dot of Figs. (b) & (c) corresponds to the lattice
point of Fig. (a). The unit circle indicated by C1 connects
two singular surfaces S{+ + + +} & S{– + + +}. Other
circles of the surface S{– + + +}, C2 for example, are connections to other singular surfaces such as S{– – + +}
The angular momentum envelope, which is the
border of the maximum workspace, is most definitely
singular. The surface corresponding to all εi positive is
clearly a part of the envelope. Surfaces with one negative
sign which is connected to this surface by the holes share
the envelope surface in the case of an independent type
system.
The envelope of a multiple type system consists of a
singular surface of all positive signs and circular plate
which fills 2m holes22). The one negative sign surfaces
do not share the envelope surface and is fully internal.
The singular surface of a M(2, 2) roof type system shown
in Fig. 4–3 is part of an envelope of all positive signs.
There are four circular holes of diameter 2. The circular
plates filling these four circles share the envelope. The
singular surface of one negative sign is connected at the
center of these plates.
4.1.3 Visualization Method of the Surface
The singular surface and envelope are visualized by
taking θ at each lattice points of the unit sphere and
calculating the angular momentum using Eq. 4–3.
––– 16 –––
–– 4. Singular Surface and Passability ––
z
y
x
Fig. 4–3 Envelope of a roof type system M(2, 2).
Dots are obtained from Eq. 4–3 for lattice points
Unit Circle C 1
Fig. 4–4 Cross sections of a singular surface of
the pyramid type system.
The outermost unit circle is the same as C1 in
Fig. 4–2. The other lines are cross sections of
planes orthogonal to the gimbal axis g1.
of a u sphere with all positive signs. Circles are
filled by plates.
Figures 4–2 and 4–3 are such examples. A singular
surface and an envelope may also be visualized using
various cross sections. The following inverse mapping
theory22) is available to obtain a cross section of the
singular surface.
4–4 is such an example.
The proof of this theory is given in Appendix B.3.
4.2 Differential Geometry
Inverse Mapping Theory
Suppose that θ is constrained singular and V is an
arbitrary vector normal to u. If the differential dH
along the singular surface satisfies,
dH = V × u ,
Geometric theory presented in Ref. 22 formulated
fundamental forms of the singular surface and clarified
geometric characteristics. Other than Gaussian
curvature, details are given in the original paper.
(4–4)
4.2.1 Tangent Space and Subspace
then the differential of u is given by
du = κ ( CPCtV) × u ,
(4–5)
where κ is the Gaussian curvature of the singular
surface, which is described in Section 4.2.2. The
matrix P is a diagonal matrix whose nonzero element
Pii is given by:
Pii = pi = 1 / (u ⋅ hi) .
(4–6)
Using this theory, a cross section of the singular
surface is calculated by the following procedure. First,
obtain a singular point on the cross sectional plane and
its u vector by some means. Second, obtain dH on the
intersection of the surface tangential plane and the cross
sectional plane. Third, obtain V by Eq. 4–4 and du by
Eq. 4–5 after which dθ is obtained by the relation dθ =
pi ci⋅du (Appendix B). Finally, H on the cross sectional
Suppose that θ is on a singular point. The differential
dθ is a tangent vector of the θ space. The following
three subspaces are defined in the tangent space of the θ
space32):
ΘS: Singularly constrained tangent space of the θ
space (two dimensional).
ΘN: Space of null motion, i.e., the null space of C (n−
2 dimensional).
Θ T : Complementary subspace of Θ N (two
dimensional). The solution given by Eq. 3–22
for all Tcom belongs to this space.
The elements of these three subspaces are denoted by
dθS, dθN and dθT. These are illustrated in Fig. 4–5 for
a two dimensional three-unit system, for example. The
general bases of subspaces are given in Appendix B.1.
4.2.2 Gaussian Curvature
plane is obtained by numerical integration of dθ. Figure
––– 17 –––
The Gaussian curvature, κ, of a singular surface is
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
By using the relation as,
Singular Direction
u
∂H / ∂θi = ci ,
B
dθ N1 =(∆ , 0, ∆ ) t
∂2H / ∂θi∂θj = – hi, if i = j otherwise 0 ,(4–9)
the difference ∆H as
h1
B
h2
B
dθ N2 =(0, ∆ , ∆ ) t
h3
(a) Singular state
∆H = H(θS+dθ) − H(θS) ,
(4–10)
is expressed as,
∆H = Σ ci dθi − 1 / 2 Σi hi(dθi)2 ,
(b) Null motion Θ N
(4–11)
where the third order terms are omitted.
The first order difference is a linear combination of
ci and has no component in the u direction. On the other
hand, the second order term may have a component to
this direction. More specifically,
B
B
dθ S =(∆ , ∆ , ∆ ) t
dθ T =(∆ , ∆ , – ∆ ) t
(c) Θ S
∆H⋅u = 1 / 2 u ⋅(–Σihi (dθi)2) ,
= − 1 / 2 Σi(dθi)2 / pi .
(d) Θ T
Fig. 4–5 Infinitesimal motion from a singular
point of 2D system.
This may also be expressed in matrix form as:
∆H⋅u = −1 / 2 dθt P−1 dθ .
Four independent motions, dθN1, dθN2, dθS and
dθT, are members of three subspaces, ΘN, ΘS
and ΘT.
dθ = dθS + dθN ,
(4–14)
the quadratic form (4–12) is also similarly decomposed:
(4–7)
The proof of this is detailed in Appendix B.2.
The sign of Gaussian curvature has an important role
in determining the following passability of the surface.
4.3 Passability
∆H⋅u = −1 / 2 dθSt P−1 dθS
− 1 / 2 dθNt P−1 dθN .
(4–15)
This is derived by the fact that P−1dθS is an element
of ΘT hence dθNt P−1 dθS = 0 (See Appendix B.1).
Let QS and QN denote the two quadratic forms on
the right of Eq. 4–15:
QS = −1 / 2 dθSt P−1 dθS ,
4.3.1 Quadratic Form
Suppose that the system state is singular, that is, θS
is a singular point and HS is on the singular surface. It
is instructive to examine an infinitesimal change in θ
from this singular point and the resulting infinitesimal
change in H. A second order Taylor’s series expansion
of H(θ) in the neighborhood of the θS is given by:
H(θS+dθ)
QN = −1 / 2 dθNt P−1 dθN .
(4–16)
The vectors, dθS and dθN, are elements of the tangent
subspace ΘS and ΘN, and they can be represented by
using bases of each subspaces:
dθS = φ1eS1 + φ2eS2 ,
dθN = ψ1eN1 + ... + ψn-2eNn-2 ,
(4–17)
where eSi and e Ni are bases of Θ S and ΘN. These
expressions are expressed simply as,
= H(θS) + Σi∂H / ∂θi dθi
+ 1 / 2 ΣiΣj(∂2H / ∂θi∂θj)dθidθj
+ O(dθi3) .
(4–13)
This is a quadratic form of dθi.
If any dθ are decomposed as follows,
given by:
1 / κ = 1/2 ΣiΣj pi pj [ci cj u ]2 .
(4–12)
(4–8)
––– 18 –––
dθS = ES φ ,
where ES : n×2, φ : 1×2 ,
–– 4. Singular Surface and Passability ––
dθN= ENψ ,
where ES : n×n–2, ψ : 1×n–2 .
(4–18)
Substituting these into two quadratic forms in Eq. 4–16
results in the following:
QS = −1 / 2 φt ESt P−1 ES φ ,
QN = −1 / 2 ψt ENt P−1 EN ψ .
(4–19)
The first quadratic form is of order 2 and expresses
the curvature of the singular surface, because dθ S
represents a motion on the surface. The second quadratic
form is of order n−2. By the definition, this QN is
∆H(dθN)⋅u. Therefore, if this quadratic form is not zero,
this null motion moves the vector H away from the
surface as shown in Fig. 4–6.
Note that the decomposition in Eq. 4–15 is not always
possible, for example if CdθS = 0. This case, however,
can be treated by similarity with another neighborhood.
u
QN
∆H(θN)
∆
H
QS
∆H(
θS)
Singular surface
Fig. 4−6 Second order infinitesimal motion
from singular surface
4.3.2 Signature of Quadratic Form
Any quadratic form, Σaij xixj, can be transformed to
Σbi yi2 by using a regular transformation from {xi} to
{yi}. The set of two numbers of positive and negative bi
are called “signature” of the quadratic form. Any
quadratic form has a unique signature, that is, the
signature does not depend on the transformation, as is
Sylvester’s law of inertia.
By the signature, a quadratic form is categorized as
definite, semi-definite or indefinite. Definite form have
only the same signs, while an indefinite form has both
positive and negative signs. A semi-definite form has
only the same sign but their number is less than the order
of the form.
If the quadratic form, QN is definite or semi-definite,
it implies that any motion away from the surface is
limited to a certain side of the surface. Thus, no motion
from this side of the surface to the other side is possible.
With indefinite forms, some motions result on one side
of the surface while others appear on the other side.
The signature is the characteristics of the form itself,
independent of the variables which is dθN in this case.
Thus, any singular point is categorized by this
characteristics such as definite or indefinite.
4.3.3 Passability and Singularity Avoidance
Since the quadratic form and its derivatives are
continuous with respect to θ, its eigenvalues which
determine the signature are also continuous along the
surface. This implies that if a point has a definite form
then its neighborhood likewise does. The points of
definite form make up a certain area of the surface, near
which it is not possible to pass from one side to the other
if θ is in the neighborhood of this singular point. In this
sense, such an area is called ‘impassable’, while that of
an indefinite form is termed ‘passable’. This notation
follows that of Tokar26) who pioneered this work. Other
notation used in other references are elliptic/
hyperbolic28, 65) and definite /indefinite32).
Another aspect of this form category is as follows.
If the singular point is passable, i.e., having an indefinite
form, a certain value of dθN results in a zero value of
the quadratic form. The motion by this dθN keeps H on
the singular surface but θ does not stay singular. This
implies that escape from the passable singular point is
possible while keeping H the same. On the contrary, no
motion can keep H at an impassable singular point.
The internal singular point of a two dimensional
system is passable. Figure 4–7 shows two motions in
opposite directions at the singular point. The null motion
Singular direction u
B
B
dθ =dθ N1– dθ N2
=( ∆ , – ∆ , 0)t
dθ =dθ N1+ dθ N2
=( ∆ , ∆ , 2∆ ) t
(a) ∆ H⋅u < 0
(b) ∆ H⋅u > 0
Fig. 4–7 Possible motions in both direction of u at a
singular point. This is the case of an internal singular
state of a 2D system. Infinitesimal null motions dθN1
and dθN2 are defined in Fig. 4–5.
––– 19 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
in Fig. 4–5 (b) allows the system to escape from the
singular point while keeping H the same.
From the above discussion, it is clear that no steering
law can avoid an impassable singularity if the command
is to approach the surface and the initial θ is in the
neighborhood of the impassable singular point. On the
other hand, a steering law such as the gradient method
is effective in avoiding a passable singularity because
escape from such a singular point is always possible even
when H is on the singular surface.
4.3.4 Discrimination
From these results, passability is discriminated for
any singular point as follows. First obtain the sign {εi}.
Reverse all the sign if required so that the number of
negative signs is less than that of positive signs. If they
are all positive, this surface is impassable. If more than
two signs are negative, this surface is passable. The
remaining cases may correspond to the above two cases
(2) or (3). The next procedure is to calculate the Gaussian
curvature κ by Eq. 4–7. If there is only one negative
sign, it is impassable when κ is negative otherwise
passable. If there is two negative signs and κ is positive,
we need additional calculation to determine passability.
Passability of a surface is defined by the signature.
The following discussion gives a discrimination method
of this by the sign {εi} and the curvature of the surface.
Equations 4–18 represents a basis change for each
subspace. As mentioned above, the signature is
conserved by any basis change. Thus the signature of
the total quadratic form is conserved and is simply
obtained by the signs of pi, which is εi, because of Eq.
4–12. Thus, passability which is defined by the signature
of Q N, is determined by the total signature and the
signature of Q S . The signature of Q S indicates
characteristics such as concavity/convexity of the surface
because this quadratic form expresses curvature of the
singular surface. Thus the following three conditions
for an impassable surface are obtained in terms of the
sign and the curvature of the surface32).
Condition (1) ε={+ + ... +}.
Both QN and QS have only positive signs. This
singular point is on the surface S{+ + ... +} which is
a part of the envelope and is trivially impassable.
Gaussian curvature is positive and the surface is
convex to u. (Fig. 4–8(a))
Condition (2) All the εi but one are positive and the
signature of QS is {− +}.
This surface is partially on the envelope and
impassable. Some part of this impassable surface
is possibly inside the envelope. The Gaussian
curvature is negative and the surface is a hyperbolic
saddle point. (Fig. 4–8(b))
Condition (3) All the εi but two are positive and the
signature of QS is {− −}.
The surface is fully inside the envelope. The
surface is concave to u and the Gaussian curvature
is positive. Note that positive κ is not a necessary
condition for this because there is a possibility that
the signature of QS is {+ +}. (Fig. 4–8(c))
u
Singular Surface
(a) ε = {+ + + ... +}
u
(b) ε = {– + + ... +}
u
(c) ε = {– – + ... +}
Fig. 4−8 Local shape of an impassable singular
surface.
A side of the surface in –u direction is an allowed
H region while another side to u direction is
unreachable through the surface. (a) is a concave
part of the envelope, (b) is a saddle point of the
envelope and internal surface, and (c) is convex
and fully internal.
––– 20 –––
–– 4. Singular Surface and Passability ––
By removing one unit whose sign is negative and by
checking the above condition (2) for this subsystem,
passability of the original system is determined. This is
proven in Appendix B. 4.
4.4 Internal Impassable Surface
4.4.1 Impassable Surface of an Independent Type
System
In the case of an independent type system, the
singular surface with all εi positive except one joins
smoothly into the envelope, as depicted in Fig. 4–4.
Because the surface and the curvature are continuous,
any impassable portion also goes into the envelope.
Thus, any independent type system has an impassable
surface distinct from the envelope.
Figures 4–9, 4–10, 4–11 and 4–12 show examples
of internal impassable surfaces, along with the envelopes.
In Fig. 4–9 it is clear that although the symmetric six
unit system S(6) has internal impassable surfaces, they
are very near the envelope. On the contrary, Figs. 4–
10, 4–11 and 4–12 show that the skew five unit system
and the S(4) system have internal impassable surface
considerably further inside their envelope. If the
workspace of these systems is defined such that it does
not include these impassable surfaces in order to assure
singularity avoidance, it becomes much smaller than the
workspace given by the envelope.
4.4.2 Impassable Surface of a Multiple Type
System
The analysis of a multiple type system is quite
different from the above discussion. For multiple
systems, every surface corresponding to all εi positive
except one are totally inside the envelope. It is instructive
to examine passability conditions (2) and (3) of section
4.3.4. Here, each variable is subscripted by the group
number, because all the variables of the same group can
be represented by only one member. The number of
units in the ith group is denoted by mi.
z
z
x
y
x
(a) Envelope. Singular surface S {+ + + + +}
(a) Envelope. Singular surface S{+ + + + + +}
Envelope
Envelope
Internal Part
z
Internal Part
z
x
x
Unit of H
Unit of H
(b) Impassable surface S{− + + + +}
(b) Impassable surface S{− + + + + +}
Fig. 4–10 Impassable surface of Skew(5)
with skew angle α = 0.6 rad.
Fig. 4–9 Impassable surface of S(6)
––– 21 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
the right is the Gaussian curvature of the subsystem
envelope excluding the ith group, and thus is positive.
The first term is zero if mi=2 and otherwise positive.
Thus the overall curvature is positive and condition (2)
is not satisfied.
z
Condition (3) Suppose that two signs are negative.
If the two units corresponding to these two negative signs
belong to different groups, condition (3) simply results
in condition (2) of the subsystem by removing one of
the two units, so the condition is not satisfied. Suppose
then that the two units corresponding to the two negative
signs belong to the same group, this being the ith one.
If the unit number, mi, is larger than two, the above
reasoning is applied and the condition is not satisfied.
If mi=2,
x
(a) Envelope.
Singular surface S{+ + + + +}
Envelope
1 / κ = 1/2 (−pi − pi )Σj pj [ci cj u ]2
Internal Part
+ Σj≠iΣk≠i pj pk [cj ck u ]2 . (4–21)
z
If the overall system is definite, the subsystem
without one unit of the ith group is also definite and
condition (2) is satisfied for this subsystem, so 1 / κ is
negative. In this case, Eq. 4–21 in its entirety is also
negative, so the condition is not satisfied.
x
(b) Impassable surface S{− + + + +}
Fig. 4–11 Impassable surface of another Skew(5),
with skew angle α = 1.2 rad.
Condition (2) Suppose that only one of the signs is
negative and it is in the ith group. The Gaussian
curvature of (4–7) is written as:
1/κ = 1/2(−pi +pi +...+pi)Σj pj [ci cj u ]2
+Σj≠iΣk≠i pj pk [cj ck u ]2 ,
(4–20)
The discussion presented here does not hold for
systems of fewer units, such as M(2,2). Further details
are examined in Appendix C and the results lead to the
following conclusion
4.4.3 Minimum System
The conclusion is as follows.
Any multiple type system with no less than six units
has no impassable singular surface other than the
envelope, while any independent type system has internal
impassable surfaces.
where pi and all the pj are positive. The second term on
––– 22 –––
–– 5. Inverse Kinematics ––
Chapter 5
Inverse Kinematics
The impassable surfaces defined in the previous
chapter cause steering law problems. It is possible to
leave this problem unsolved and define a workspace
which excludes the impassable surfaces, but the resulting
space would be much smaller in the case of a four or
five unit system. Impassability, however, is defined
locally only in the neighborhood of an impassable
singular point. There is a possibility to avoid an
impassable situation by using some kind of global
control. For this, a geometric approach was taken in
order to understand the CMG control qualitatively by
ignoring the factor of time. The sequence of torque
commands is represented as a trajectory of the angular
momentum vector, while the possible gimbal angles are
represented by a manifold. By using equivalence
relations of manifolds and their connections, conditions
necessary for continuous control are formulated.
5.1 Manifold
A steering law is a method to obtain gimbal rates
which corresponds to a given torque command. If we
ignore the factor of time, the steering law is regarded as
a method to obtain gimbal angles by a given change of
the angular momentum. This is the reverse relation of
the kinematic equation 3–4. The (forward) kinematics
is a one-to-one mapping but the reverse relation, which
is called an ‘inverse kinematics’, is generally a one-tomulti mapping. Therefore, possible θ having the same
H is given by an inverse image of this mapping.
The inverse image from H to θ is a set of sub-spaces
disjoint to each other. Supposing that a sub-space has
no singular state, an n−3 dimensional tangent space is
defined at each point of this space as a linear space of
null motion. Thus, this sub-space is a n−3 dimensional
manifold. Supposing that a sub-space has singular
points, no tangent is defined there, but even in this case,
tangent spaces are defined at all other points of this space.
Thus, this sub-space is nearly the same as a manifold
and in this work will be termed a ‘singular manifold’.
The inverse image is a sum of manifolds and singular
manifolds, which are denoted by M i and M Sj
respectively. Note that this manifold should be termed
rather ‘null motion manifold’ or ‘self-motion
manifold’60). In this work however, no other manifold
is used and it is simply called ‘manifold’.
The shape of manifolds in the neighborhood of a
singular point is characterized by the quadratic
relationship given by Eq. 4–12. Suppose that H is in the
neighborhood of a singular surface where H = HS + eu.
By the same discussion as Eqs. from 4–10 to 4–15,
possible θ in the neighborhood of the singular point, θS,
satisfies the following quadratic relation;
− 1⁄2 Σ (dθNi)2 ⁄ pi ≈ e,
where θ = θS + dθN .
(5–1)
In this equation, the motions, dθN, is a tangent vector
at the singular point θS. In the case of an impassable
singular state, this quadratic form is definite, so this
manifold resembles a super-ellipsoid. The quadratic
form of an impassable singular point is indefinite, so
the shape of the manifold resembles super-hyperbolic
surfaces in the neighborhood of this singular point. This
is illustrated in Fig. 5–1 for a four unit system, for which
the manifolds are loops in the four dimensional torus.
MS(HS) = {θS}
M0(HS+e u)
M0(HS+2 e u)
(a) Ellipsoidal
manifolds around
an impassable
singular point.
M01(HS–e u)
θS
M0(HS+e u)
MS(HS)
M1(HS+e u)
(b) Manifolds crossing
near a passable
singular point.
Fig. 5−1 Manifolds in the neighborhood of a
singular point.
These manifolds are one-dimensional loops if
the number of units is four.
––– 23 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
H
5.2 Manifold Path
HS–e u
HS
As H changes continuously, each manifold changes
its shape continuously as shown in Fig. 5–2. A manifold
may deform its shape into a point when H crosses an
impassable surface and may bifurcate when H crosses a
passable surface. These continuous change of a manifold
can be simplified as a continuous path in the manifold
space, where each manifold is regarded as a point (Fig.
5–3). If we need an exact definition of manifold
continuity, it can be made based on the distance d
between manifolds which is defined as follows;
HS+e u
θ
M1
M0
(c) Continuous change of manifolds in the
neighborhood of impassable H.
HS
= max( min(| θ − φ ); ∀θ ∈ MA ) ;∀φ ∈ MB ) ,
HS+e u
(5–2)
5.3 Domain and Equivalence Class
The angular momentum space is divided into several
domains by one or more continuous singular surfaces.
These will hereafter be simply termed ‘domains’. Let
each domain have no singular surface inside and its
border be a set of singular surfaces. Each domain will
be denoted by Di. Any continuous path of H inside a
domain corresponds to a finite number of continuous
manifold paths with neither bifurcation nor termination.
Thus, the number of manifolds for each point in the
domain is constant.
M2
HS–e u
d(MA, MB)
θ
(d) Continuous change of manifolds in the
neighborhood of passable H.
Fig. 5–2 Continuous change of manifolds
θS1
H0
M1
M3
Domain
m=2
M5
m=3
M4
m=2
M2
θS2
D3
H1
M6
D2
H2
D1
corresponding to H path across a singular point.
H path
where MA and MB are two manifolds and the norm | θ −
φ | is defined appropriately in the gimbal angle space.
By this definition, a manifold becomes discontinuous at
a bifurcation point.
The meaning of a continuous manifold path can be
thought of in the following terms. If the manifold (M1
in Fig. 5–3 for example), including an initial θ, is on a
continuous manifold path for a given H path (the path
H0H1 in Fig. 5–3), then any θ of the manifold on the
other side of the path (M2 in Fig. 5–3) can be reached by
some continuous steering method using an appropriate
null motion, while any θ of another manifold (M4 in
Fig. 5–3 for example) cannot be reached. If the manifold
path bifurcates, path selection (from M3 either to M4 or
to M5) depends on the null motion hence on the steering
method. If the manifold path including the initial θ
terminates somewhere for a given H path (θS2 for the
path H1H2 in Fig. 5–3 for example), no steering method
can realize this motion.
M1
M0
H
Manifold Paths
Fig. 5−3
An example of a continuous
manifold path.
A passable singular point θS1 is a bifurcating
point and an impassable point θS2 is a
terminal of the path.
A manifold equivalence relation is defined as follows:
definition: Two manifolds of a domain are considered
‘equivalent’ when there is a path from one manifold
to the other which corresponds to an H path inside
the domain.
All the equivalent manifolds form a domain in the
––– 24 –––
–– 5. Inverse Kinematics ––
Equivalence Class
Singular Surface
H path
Manifold Path
G1
θA
MA
HA
θB
MB
D2
MB
θ’B
HB
M ’B
Domain D1
G2
Manifold space
H space
Fig. 5−4
MA
θ space
Relations between H space, manifold space and θ space.
manifold space which is isomorphic to the original
domain of H. As the representation of this set of
equivalent manifolds, an ‘equivalence class’ is defined.
The number of the classes is called the ‘order’ of the
domain. Let Gi and m denote the class and order,
respectively. The order of each domain is obtained in a
step by step fashion. The outermost domain next to the
envelope is of order 1. Two domains facing each other
have an order difference of 1, because only one manifold
path either bifurcates or terminates.
The definition of equivalence class may be extended
to different domains connected by a certain H path.
Classes of different domains are termed equivalent when
the H path connecting the domains corresponds to a
continuous manifold path which includes these classes.
In Fig. 5–3 for example, the path from M1 to M6 via M2
is continuous through domains D1, D2 and D3, so these
manifolds and classes on this path are equivalent. On
the other hand, M3 in domain D1 is not equivalent to
any manifold of the domain D 2. The relationship
between manifold and class is illustrated in Fig. 5–4.
The equivalence among two domains implies that
the manifold path is continuous. If there is bifurcation
on the manifold path, classes are not equivalent. In this
case, classes can be termed ‘connected’, because a
continuous θ path can be chosen.
5.4 Terminal Class and Domain Type
For continuous steering, it is important to know
whether or not each class has equivalent or connected
classes for any H path exiting of the domain. The class
is called a ‘terminal class’ if there is an H path exiting to
a neighbor domain which results in termination from
this class. By this definition, the class of the domain
just inside the envelope is not a terminal class even
though it terminates on the envelope, because an H path
exiting the envelope has no meaning.
Each domain is classified into one of the following
three types by considering the order and number of
terminal classes, k:
Type 1: m = k > 0,
Type 2: m > k > 0,
Type 3: k = 0.
The outermost domain nearest the envelope is Type 3
by the above definition. Type 3 domains have no
terminal class, and as such, no difficulty arises as far as
steering inside of itself and its neighboring domains.
5.5 Class Connection
Class connection around Type 1 and 2 domains is
described by examples in the following sections. A Type
2 domain is examined first. By introducing a graph of
class connection, a Type 1 domain is next examined.
5.5.1 Type 2 Domain
The following examples are obtained by computer
calculations for the S(4) system. Figure 5–5(a) shows
a part of a cross section of a singular surface near the
envelope. The curved triangle is where the surface was
cut, and the bold line indicates an impassable edge. This
triangle divides the H space into two parts; the domain
outside is denoted by D0 and the domain inside by D1.
Domain D0 is a Type 3 domain just inside the envelope
––– 25 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Impassable Surface S0
S
C
Domain D1
(Type 2)
B
Q
R
P
A
Domain D0
Passable Surface S1
(a) Cross section of an internal
singular surface and H path
Bifurcation at A
R
M1
M 2∈ G 2
Q
H
θ2
Bifurcation at B
P
θ1
(b) Manifolds for path PQR
Impassable
Singular Point
S
C
M2
H
M1
θ2
Q
P
and D1 is a Type 2 domain of m = 2. Figures 5–5(b) and
(c) show manifolds of some points in these two domains.
The manifolds are drawn using a two-dimensional
projection of the skew coordinates. While the actual θ
space is a four-dimensional torus, the illustration is made
in a quadrilateral whose edges parallel to each other are
regarded as the same points.
Consider now the angular momentum path PAQBR,
which traverses the domain D1 in Fig. 5–5(a). There is
no equivalent class for this path which traverses three
domains, from D0 back to D0 via D1. The manifold path
bifurcates at points A and B. Though the classes are not
equivalent, they are connected and continuous steering
is very much possible by some gradient method because
such bifurcation points (passable singular points) are
easily avoidable.
Consider next the path PAQCS penetrating the
impassable surface. Figure 5–5(c) shows that the
manifold M2 is of the terminal class in this domain. Once
this class is selected when going into D1 from P through
A, there is no continuous way to reach another manifold,
such as M1, so steering fails. On the contrary, if manifold
M1 is selected, continuous motion egressing this domain
along QCS is strictly guaranteed without any special
steering methods. In this case, the main question is how
to select an appropriate class.
The above discussion is more easily understood by
utilizing a class connection graph, as shown in Fig. 5–
6(a). The jagged lines represent the cross section of a
singular surface obtained from numerical computation.
Various circles drawn inside each domain represent
equivalence classes of domains and the color of the
circles (white and gray) indicates whether they are a
terminal class or not. Circles drawn on the edge of the
domain represent a class of singular manifolds. Curved
lines connecting the circles represent class connections.
This connection graph makes it easily understood that
all classes would be connected even after the omission
of the terminal class G2. Therefore, necessary class
selection is unique for any H path crossing this domain.
θ1
5.5.2 Type 1 Domain
(c) Manifolds for path QCS
Fig. 5-5 Domains and manifolds of the pyramid
type system
A cross section of a Type 2 domain and manifolds
for several points are obtained by computer
calculation. Manifolds are drawn as a two
dimensional projection on (θ1, θ2) coordinates in
a quadrilateral whose right and left edges (space
2π apart) are regarded as the same points.
Figure 5–6(b) shows another cross section of a
singular surface and a class connection graph. The
triangular domain D3 is Type 1. In this case, no class
remains if we omit the terminal classes of this domain.
Thus, there is no contiguous connection of classes for
an H path such as FG. This implies that no steering law
can realize this angular momentum path. On the other
––– 26 –––
–– 5. Inverse Kinematics ––
Domain D3
(Type 1)
G2
G1
Domain D1
(Type 2)
Domain D2
(a) Type 2 domain
Domain D4
(Type 1)
(c) Another type 1 domain
Domain D3 (Type 1)
D'
D
Impassable surface (Bold)
Passable surface
E
E'
Class of manifold
Class connection
G
F
Bifurcation
Termination
H path
(b) Type 1 domain
Envelope part
Fig. 5−6 Class connection graph around domains
hand, continuous motion along the path DE is possible
by selecting the appropriate class prior to bifurcation.
This selection, however, is not always effective. If
another path, D’E’ for example, is taken, the class to be
selected is different. This implies that there is no unique
selection rule for entering a Type 1 domain. Figure 5–
6(c) shows another Type 2 domain example including
various class connections. The above discussion also
holds in this case.
5.5.3 Class Connection Rules
The class connections in Fig. 5–6 can be derived
without calculation of manifolds but by considering
continuity. In the cross section shown in Fig. 5–7, the
sharp point R represents the borderline between an
impassable and a passable sides of the singular surface.
Since the singular point θ along the surface is continuous,
passable points θP and θQ smoothly change to impassable
points θS and θT. This implies that after bifurcation by
the passable surface, one of the two manifolds must
terminate at the impassable surface. Thus, class
connections such as those drawn in Fig. 5–6(a) are
general for this type of domain even if its order is not 2.
Suppose that two impassable surfaces cross as shown
in Fig. 5–8. Singular points are continuous along each
surface. Manifolds MP and MQ which are the terminal
manifolds of each surface are equivalent and manifolds
MR and MS also. Therefore, there can be no connection
between two manifolds of the different group, MP and
MR for example. Thus, class connections such as those
drawn in Fig. 5–6(b) are derived. With increased
complexity in surface crossings however, finding class
connections is more difficult.
––– 27 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Impassable Surface
5.5.4 Continuous Steering over Domains
S
T
R
Q
A
Passable Surface
P
(a) H space
MR
MS
θR
θQ
θS
MQ
θT MA2
MT
θP
MA1
(b) θ space
MP
Fig. 5−7 An illustration of class connection rule (1).
Impassable Surface
R
P
T
S
Q
(a) H space
Lines of Singular Point
MR
MP
θT1
MQ
θT2
MS
(b) θ space
Fig. 5−8 An illustration of class connection rule (2).
If two impassable surfaces cross each other, both
terminal classes are different.
From the above two examples, the following general
facts are observed.
(1) Continuous steering around a Type 2 domain
depends upon manifold selection prior to
bifurcation. If any class other than a terminal
class is selected, continuous control is guaranteed.
(2) Selecting a manifold other than the terminal classes
is impossible while entering a Type 1 domain.
(3) Some paths of H which cross a Type 1 domain do
not have a connected manifold path.
The item (3) implies that there is no continuous θ path
for a certain H path. The item (2) implies that even if a
continuous θ path exists for any given H path, real-time
steering is not guaranteed when H path is not given
beforehand. Those two items, therefore, implies that no
steering law can maintain continuous steering over the
entire work space if the system contains Type 1 domains.
On the other hand, an impassable surface of a Type 2
domain does not cause any problem if an appropriate
manifold is selected before bifurcation as the item (1).
5.5.5 Manifold Selection
There has been no simple method to select an
appropriate manifold prior to bifurcation. Although a
gradient method avoids a passable singular point (a
bifurcation point), judicious manifold selection depends
on the control values of θ before bifurcation. The
gradient method is unsuitable because of the following
reason.
The objective function of a gradient method is
defined zero at a singular point and otherwise positive.
Thus the singular point is the minimum of the objective
function along the manifold. There must therefore be
local maxima on both side of the singular point, such as
at A and B in Fig. 5–9. Knowing that the objective
function is continuous, the manifolds before and after
bifurcation have local maxima A’, B’, A”, and B” in the
neighborhood of A and B. The gradient method only
maintains the local maximum and its motion may be
either A’AA” or B’BB”. Even if one of the manifolds
after bifurcation belongs to a terminal class, this method
can not move θ from one maximum point to the other.
If sufficient time computing power were available, a
method like a path planning42) could be utilized (See
Chap. 8). If a number of possible H paths of a certain
length are assumed, calculations along those paths may
then be carried out in order to determine whether there
––– 28 –––
–– 5. Inverse Kinematics ––
Passable Singular Point
Terminal Class
Impassable
A
Passable
B"
V
B
A"
A
A'
B'
C
B
Local Minimum
(a) Cross section orthogonal to gi direction.
Fig. 5−9 An illustration of motion by the gradient
method.
A, A', A", B, B' and B" are the local maxima of an
objective function. Motion may be either A'AA" or
B'BB" in the neighborhood of the passable
singular surface.
θ2, θ3, θ4
Manifolds
0
are any bifurcations or intersections with impassable
surfaces. It would then be possible to determine an
appropriate motion. For this strategy, a question still
remains whether such manifold selection can be
consistent for all possible H path. This will be discussed
in Chapter 7.
A
θ1
2π
Singular
Manifold
Ｖ
C
B
MSV
Terminal
Manifold
5.5.6 Discussion of the Critical Point
(b) Manifold of points A, B and C
The above discussion pertains to manifolds and
classes within domains. An arbitrary angular momentum
point not on a singular surface is discussed. It was
assumed that anything on the singular surface and on
the singular manifold is qualitatively the same as that in
the H neighborhood and in the neighbor manifold. There
is however some exceptions.
If the H path starts at an intersection of singular
surfaces, there is a problem of manifold selection. Three
triangular domains in Fig. 5–10 are Type 2 of order 2.
Having the same kind of class relations as the Type 2
domain in Fig. 5–6(a) means any trajectory across one
of them can be continuously realized by an appropriate
control. However, if one were to commence at the
crossing point V, the possibility of selecting a terminal
class cannot be omitted once the initial θ is selected on
the manifold. This situation is depicted in Fig. 5–10(b).
5.6 Topological Problem
A steering law can be represented by a mapping from
the H space to the θ space. It would be nice if there is a
continuous mapping which uniquely determines θ from
H. However, it is clearly observed from the examples
Fig. 5–10 Manifold relations around critical
point.
The distance between the origin and the cross
sectional plane of (a) is 0.875 of the maximum
distance.
of Type 1 domains that there is no such mapping. This
fact is explained directly by the topology of kinematic
mapping.
Consider the circle on the envelope which
corresponds to the case u=g i . Suppose there is a
candidate mapping from H to θ. The image of the circle
by this mapping is a loop on the torus where θi changes
from 0 to 2π. Consider a deformation of the circle in
the H space and the image in the θ space. The circle can
be deformed to a point in the H space but their image in
the θ space cannot, because continuity requires θi to vary
from 0 to 2π. (Note: A similar statement for robot
kinematics was generally proven by topological theories
in Ref. 59.)
The above is true as long as the mapping covers the
H space in its entirety. If a small enough region of H
space, a domain for example, is considered, any
––– 29 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
continuous steering law can obviously realize such a
continuous mapping. Even if such a region includes a
Type 2 domain, we can realize such a mapping by using
an appropriate manifold selection. Thus, the question is
how large a region of H space can be covered by such a
continuous mapping, and what steering law actually can
realize such a mapping.
It is to be noted that the examples of Type 1 domains
and critical points in the previous section are only found
near the envelope by various computer calculations of
the S(4) system. Therefore, the above problems may
not be serious. On the other hand, the discussion in this
chapter only suggested that there is a possibility of
continuous steering in the presence of a Type 2 domain.
The discussion in the above pertains to the local area
around one domain. In actuality, the candidate
workspace may involve various domains, so a study of
global class connections is necessary. In the following
chapters, a relatively specific problem will be studied
for the symmetric pyramid type system using geometric
tools given in this chapter.
––– 30 –––
–– 6. Pyramid Type CMG System ––
Chapter 6
Pyramid Type CMG System
This chapter and the next two deal with a symmetric
pyramid type system of single gimbal CMGs. This
chapter describes system characteristics obtained by
analysis and computer calculations. These are kinematic
equations, system symmetry, expression of the gimbal
angles when the angular momentum is at its origin,
internal singular surface details, and impassable surface
geometry. All of them will be utilized for the analysis
of steering motion in the next chapter.
s* = sinα =
2 ⁄ 3,
c* = cosα = 1 ⁄ 3 .
(6–1)
Angular momentum vectors and torque vectors of all
the units are given as:
− c * sin θ1 
h1 =  cos θ1 ,
 s * sin θ1 
 c * sin θ 3 


h 3 =  − cos θ 3 ,
 s * sin θ 3 
6.1 System Definition
The pyramid type CMG system consists of four
single gimbal CMGs in a skew configuration, as depicted
in Fig. 6–1. An example of this type is the S(4)
symmetric system, where each gimbal axis lies in the
direction normal to each surface of a regular octahedron.
The pyramid shown in Fig. 6–1 is the upper half of an
octahedron. The skew angle of this type denoted by α
 − cos θ 2 
h 2 = − c * sin θ 2 ,
 s * sin θ 2 
 cos θ 4 
h 4 =  c * sin θ 4 ,
 s * sin θ 4 
(6–2)
− c * cos θ1 
c1 =  − sin θ1 ,
 s * cos θ1 
 c * cos θ 3 


c 3 =  sin θ 3 ,
 s * cos θ 3 
is given as cosα = 1 ⁄ 3 , and is about 53.7 degrees. It
is expedient to define additional parameters:
 − sin θ 2 
c 2 = − c * cos θ 2 ,
 s * cos θ 2 
 − sin θ 4 
c 4 =  c * cos θ 4 ,
 s * sin θ 4 
(6–3)
g4
c4
h3
g3
θ3
where the origin and the direction of each gimbal angle
are defined by Fig. 6–1.
c3
θ4
h4
6.2 Symmetry
Z
α
c1
X
g1
Y
h1
θ1
h2
θ2
g2
c2
Fig. 6−1 Schematic of a pyramid type system.
The origin of each θ i is defined when h i is on the
square in the xy plane. The symmetric type S(4) is
the case where the skew angle α is set as
cosα = 1/ 3 .
The pyramid type CMG system has symmetry in its
kinematics. This symmetry is useful for understanding
the geometry of the singular surface and will be used
for deriving a global problem in the next chapter. This
symmetry is derived by the rotational transformations
of regular octahedron, which is a well-known example
of finite group theory. There are 48 symmetric
transformations of an octahedron including the mirror
transformations.
The symmetry of the pyramid type system is
represented by two groups of transformations and an
––– 31 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Z
Z
θ3
X
θ4
θ4
θ1
θ3
θ2
Y
X
θ1
θ2
Y
(a) Rotation about z axis
Z
Y'
θ2
θ3
θ4
X'
θ2
θ3
Z
θ1
θ1
New
Coordinate
System
θ4
Y'
X'
(b) Rotation about g 1 axis
Fig. 6−2 Transformation in H space and in θ space.
A new coordinate system (X', Y', Z) is defined in (b) for a simple expression.
equivalence relationship of those. Transformations in
the H space represent rotation of a vector, while the
others in the θ space represent permutation with
translation, i.e., a kind of an affine transformation. The
meaning of the equivalence is as follows. As all four
CMG units are arranged on the surfaces of the
hexahedron, a certain rotation in the H space preserves
the hexahedron and thus results in the exchange of four
units. Therefore such a rotation in the H space is
equivalent to the transformation in the θ space.
An example of the H transformation is a 1 ⁄4 reverse
rotation about the z axis shown in Fig. 6–2 (a), after
which CMG unit i is replaced by unit i +1. This
transformation of θ is expressed by:
rz(θ=(θ1, θ2, θ3, θ4)) = (θ2, θ3, θ4, θ1) ,(6–4)
and the transformation of the angular momentum vector
by:
Rz((x, y, z)t) = (y, −x, z)t .
(6–5)
By those two transformations, the following equivalence
relationship is satisfied.
Rz(H(θ)) = H(rz(θ)) .
(6–6)
Another example is a 1 ⁄3 rotation about a gimbal
axis shown in Fig. 6–2 (b). Two transformations are as
follows:
rg (θ) = (θ1+2 π ⁄ 3, −θ3−2 π ⁄ 3,
−θ4 +2π ⁄ 3, θ2−2 π ⁄ 3) ,
Rg((x’, y’, z)t) = (z, x’, y’)t .
(6–7)
(6–8)
The latter transformation is expressed in a different
coordinate system from the original; one which is rotated
45° about the z axis as in Fig. 6–2 (b). This is because
expressions based on these new coordinates are simpler
than those based on the original coordinates. The
equivalence Eq. 6–6 is also maintained by the
transformations rg and Rg. Applicable notation of all
will now be defined.
Rotations about g1 are first defined. The identical
transformation and the g1-z plane reflection are denoted
by Re1 and RE1, respectively. A 1 ⁄ 3 rotation about the
g1 axis after Re1 (or RE1) is denoted by Rr1 (or RR1). A
reverse 1 ⁄ 3 rotation after Re1 (or RE1) is denoted by
Rq1 (or RQ1). Thus six transformations are defined.
Successive 1 ⁄ 4 rotations about the z axis are simply
denoted by increasing the indices. For example, Re2 is
a 1 ⁄ 4 rotation about the z axis and Rr3 is a 1 ⁄ 2 rotation
––– 32 –––
–– 6. Pyramid Type CMG System ––
Table 6−1 Symmetric Transformations
Notation* H
θ
H Transformation
(A)
Transformation Transformation **
of Mirror (MA )
_________________________________________________________________________
e1
( x, y, z)
( θ1 , θ2 , θ 3, θ4)
( −x, −y, −z)
E1
( y, x, z)
( π−θ 1 , π−θ 4 , π−θ 3 , π−θ 2 )
( −y, −x, −z)
r1
( z, x, y)
(2 σ+θ1 , −2σ−θ 3 , 2σ−θ 4 , −2σ +θ2 ) ( −z, −x, −y)
R1
( z, y, x)
( −σ−θ 1 , σ+θ 3, −σ+θ 2 , σ−θ 4)
( −z, −y, − x)
q1
( y, z, x)
( −2σ+θ 1 , 2σ+θ 4 , −2σ−θ 2 , 2σ−θ 3) ( −y, −z, −x)
Q1
( x, z, y)
( σ−θ 1, −σ−θ 2, σ+θ 4, −σ+θ 3)
( −x, −z, −y)
e2
E2
r2
R2
q2
Q2
(
(
(
(
(
(
−y, x, z)
e3
E3
r3
R3
q3
Q3
(
(
(
(
(
(
−x, −y, z)
−x, y, z)
−x, z, y)
−y, z, x)
−z, y, x)
−z, x, y)
−y, −x, z)
−z, −x, y)
−z, −y, x)
−y, −z, x)
−x, −z, y)
( θ4, θ1 , θ2 , θ3)
( y, −x, −z)
( π−θ 2 , π−θ 1 , π−θ 4 , π−θ 3 )
( x, −y, −z)
( σ−θ 4, −σ−θ 1, σ+θ 3, −σ+θ 2)
(2 σ−θ 3 , −2σ+θ 1, 2 σ+θ 4 , −2σ−θ 2)
( z, −y, −x)
( −2σ+θ 2 , 2σ+θ 1, − 2σ−θ 3, 2σ−θ 4) ( x, −z, −y)
( −σ+θ 3 , σ−θ 1, −σ−θ 2 , σ+θ 4)
( θ3 , θ4 , θ 1, θ2)
( y, −z, −x)
( z, −x, −y)
( x, y, −z)
( π−θ 3 , π−θ 2 , π−θ 1 , π−θ 4 )
( y, x, −z)
( −2σ−θ 2, 2σ−θ 3, −2σ+θ 1, 2σ+θ 4)
( σ+θ 4, −σ+θ 3, σ−θ 1, −σ−θ 2)
( y, z, −x)
(2 σ−θ 4 , −2σ+θ 2, 2 σ+θ 1 , −2σ−θ 3)
( −σ+θ 2 , σ−θ 4, −σ−θ 1 , σ+θ 3)
( z, x, −y)
( z, y, −x)
( x, z, −y)
e4
( y, −x, z)
( θ2 , θ3 , θ 4, θ1)
( −y, x, −z)
E4
( x, −y, z)
( π−θ 4 , π−θ 3 , π−θ 2 , π−θ 1 )
( −x, y, −z)
r4
( x, −z, y)
( −2σ−θ 3, 2σ−θ 4, −2σ+θ 2, 2σ+θ 1) ( −x, z, −y)
R4
( y, −z, x)
( σ+θ 3, −σ+θ 2, σ−θ 4, −σ−θ 1)
( −y, z, −x)
q4
( z, −y, x)
(2 σ+θ 4 , −2σ−θ 2, 2 σ−θ 3 , −2σ+θ 1) ( −z, y, −x)
Q4
( z, −x, y)
( −σ−θ 2 , σ+θ 4, −σ+θ 3 , σ−θ 1)
( −z, x, −y)
_________________________________________________________________________
Note;
*: Each transformation is represented only by its suffix.
**: σ = π ⁄ 3
about z after a 1 ⁄ 3 rotation about g1. So far, these total
24 transformations. Subsequent point symmetric
transformations by the origin are denoted by adding M
to the left of the original notation, MRe1 for example.
After including these, all 48 transformations are defined.
Before continuing, it should be noted that Rr2 is not a
simple rotation about the g2 axis.
Table 6−1 presents a list of all 48 symmetric
transformations. The first row shows the notation, the
second row gives the H transformation, the third shows
the θ transformation and the last row gives the H
transformation of the point symmetric image. Both the
H transformation and the θ transformation are expressed
as the right hand side only. So, for example, the
expressions 6–4 and 6–5 are given by Re2 and 6–7 and
6–8 by R r1 in Table 6−1. Note again that all H
transformations are expressed in the new coordinate
system rotated 45° for simplicity. Τransformation of all
the point symmetric images in θ space is omitted in Table
6−1 but is simply accomplished by adding π to each θi.
6.3 Singular Manifold for the H Origin
The origin of H, (0, 0, 0)t, is used as a nominal state
of control. This H corresponds to one singular manifold
with 6 singular points. The 6 singular points divides the
singular manifold into 12 line segments. The 12
segments are classified into two groups. These segments
and groups are used to explain the global problem and a
steering law in the next chapter.
The singular manifold for this H origin has analytical
expressions. It consists of four lines, which are straight
lines but closed in the torus space. Two of them are
given by,
––– 33 –––
(1) θ = ( φ, −φ, φ, −φ),
where −π < φ ≤ π,
(6–9)
(2) θ = ( φ+π⁄2, φ−π⁄2, φ+π⁄2, φ−π⁄2)t,
where −π < φ ≤ π .
(6–10)
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
π
π
Singular
e
l
k
(h)
(k)
(g)
a
0
θj
θ2 , θ 4
f
0
g
b
(l)
c
h
d
−π
−π
−π
π
0
θ1, θ 3
(a) (θi, θj) =(φ, −φ)
−π
0
θi
(b) (θ i , θj )=(φ+ψ, φ−ψ)
0≤φ≤π, ψ= π⁄2, –5π⁄6
π
Fig. 6−3 Line segments for singular manifold.
Arrows a to l are parametric line segments of a pair of coordinates.
The remaining two lines can be obtained as symmetric
images of the latter. Let’s define notation of total 12
line segments and derive their symmetric relations.
The first line by Eq. 6–9 includes 6 singular points
hence 6 line segments. Let line segment of θ be
expressed by the combination of two coordinate sets.
Referring to Fig. 6–3 (a), each segment will have a
parameter along it and the direction in which the
parameter increases will be expressed by an arrow. Six
line segments from LA to LF in a four-dimensional torus
Table 6−2 Segment Transformation Rule
_____________________________________________________________
Transformation A
Transformation MA
Α
G H K L M N
G H K L M N
-------------------------------------------------------------------------------------------e1 G H K L M N
−H −G −L −K −N −M
E1 H G M N K L
−G −H −N −M −L −K
r1
L K N M G H
−K −L −M −N −H −G
R1 K L G H N M
−L −K −H −G −M −N
q1 M N H G L K
−N −M −G −H −K −L
Q1 N M L K H G
−M −N −K −L −G −H
e2 −H −G −M
E2 −G −H −L
r2 −N −M −Κ
R2 −M −N −H
q2 −L −K −G
Q2 −K −L −N
e3
E3
r3
R3
q3
Q3
G
H
K
L
N
M
H
G
L
K
M
N
L
N
M
G
H
K
K
M
N
H
G
L
−N
−K
−L
−G
−H
−M
N
L
G
M
K
H
−L
−M
−H
−K
−N
−G
M
K
H
N
L
G
−K
−N
−G
−L
−M
−H
G
H
M
N
K
L
H
G
N
M
L
K
−H
−G
−L
−K
−M
−N
N
K
L
G
H
M
−G
−H
−K
−L
−N
−M
M
L
K
H
G
N
−K
−M
−N
−H
−G
−L
K
N
G
L
M
H
−L
−N
−M
−G
−H
−K
L
M
H
H
N
G
−M
−K
−H
−N
−L
−G
−N
−L
−G
−M
−K
−H
e4 −H −G −N −M −K −L
G H M N L K
E4 −G −H −K −L −N −M
H G L K M N
r4 −M −N −L −K −H −G
N M K L G H
R4 −N −M −H −G −L −K
M N G H K L
q4 −K −L −G −H −M −N
L K H G N M
Q4 −L −K −M −N −G −H
K L N M H G
_____________________________________________________________
Note:Each transformation and each segment are represented by
their suffices.
––– 34 –––
–– 6. Pyramid Type CMG System ––
are then given by a pair of those segments (from a to f)
as follows:
{LA, LB, LC, LD, LE, LF}
= { {a,a}, {b,b}, {c,c}, {d,d}, {e,e}, {f,f};
{(θ1, θ2),(θ3, θ4)} } .
(6–11)
Segment LA in this expression, for example, is given as
follows:
LA={θ:=(ϕ, −ϕ, ϕ, −ϕ), −π ⁄ 6≤ ϕ < π ⁄ 6} .
(6–12)
The line by Eq. 6–10 orthogonally crosses the first
line at two singular points. The two singular points divide
the line into two segments. Referring to Fig. 6–3 (b),
the two segments are defined as follows:
LG = {(θ1, θ2)=g, (θ3, θ4)=g} ,
LH = {(θ1, θ2)=h, (θ3, θ4)=h} .
transformation of LA, while any segment from LG to
LN by some transformation of LG. This implies that
any characteristics of the segments must accordingly be
derived from characteristics of either LA or LG.
6.4 Singular Surface Geometry
Singular surface has been described by its curvature
or by an example of a cross section in the previous
chapters. Now the total geometry of the singular surface
especially of the impassable surface will be examined
by using a series of cross sections66).
Here, cross sectional planes orthogonal to the g1 axis
are mainly used as shown in Fig. 6–4. Each plane has a
parameter d which is a distance from the H origin to the
plane. All distances will henceforth be normalized by
its maximum value, which is the distance to the unit circle
(6–13)
Other two lines, i.e., four line segments are defined
similarly:
Sectional Plane
Orthogonal to g 1
d : Distance from
the H Origin
A
LK = {(θ1, θ4)=k, (θ3, θ2)=l} ,
LL = {(θ1, θ4)=l, (θ3, θ2)=k} ,
LM = {(θ2, θ1)=k, (θ4, θ3)=l} ,
LN = {(θ2, θ1)=l, (θ4, θ3)=k} .
(6–14)
The set of segments from LA to LF are transformed
to the same set of the segments by any symmetric
transformation. The followings are examples of
transformed results, where a minus sign before the
segment implies that the direction is reversed.
RE1(LA, LB, LC, LD, LE, LF)
Envelope
Portion of Internal
Impassable Surface
=(−LD, −LC, −LB, −LA, −LF, −LE) ,
(a) Envelope and sectional plane
Rr1(LA, LB, LC, LD, LE, LF)
= (LC, LD, LE, LF, LA, LB) ,
=(−LA, −LF, −LE, −LD, −LC, −LB) ,
g1
MRe1(LA, LB, LC, LD, LE, LF)
= (LD, LE, LF, LA, LB, LC) .
g2
g4
Re2(LA, LB, LC, LD, LE, LF)
(6–15)
Similarly, the other segments from LG to LN are also
transformed to the same segments. The results of all
such transformations are listed in Table 6−2 in which
each segment is represented by its suffix.
From the above analysis, it is observed that any
segment from L A to L F can be obtained by some
––– 35 –––
g3
(b) Cross section on plane A
Fig. 6–4 Definition of the cross sectional plane
and the distance d.
The distance d is divided by 2 2 for
normalization such that d=1 for the unit circle
on the envelope.
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
A
Internal
impassable part
Domain D1
D1
A
B
A
Internal
passable
part
Envelope part
(a)d=0.9
d = 1.0
Unit Circle
d = 0.99
(b)d=0.8919
D1
d = 0.92
A
B
B
A
Fig. 6–5 Saddle like part of the envelope.
Curves are various cross sections of S{− + + +}. For
d<0.94, some portions of surfaces are inside the
envelope.
D2
(d)d=0.88
(c)d=0.885
on the envelope connecting two singular surface of
different signs.
Figure 6–5 illustrates a singular surface of one
negative sign, a portion of which is shared with the
envelope. The outermost unit circle is the case of u=gi
and d=1. Other curves are cross sections with d<1. It
is of interest to observe what happens to this surface
when d is made smaller and smaller.
As d decreases in size, the curve deforms like a
triangle. After sharp edges appear, folding and then
small triangles appear on the curve as shown by A in
Fig. 6–5. As a result of this folding, each curve is
divided into six smoothly curved segments. As the
curvature changes sign at the folding point, the
Gaussian curvature also changes its sign. This folding
is therefore a connecting point between a passable and
an impassable surface. From continuity to the
envelope, passability of each curve is determined as
shown in Fig. 6–6. Observing the cross sections of
various d in this figure, it can be seen that both passable
and impassable surfaces penetrate the envelope like
“strips” or “belts”.
Folding also appears in the cross sectional line at d
≈ 0.65 and as d becomes smaller, the strip bifurcates.
By repetitive calculation for various d values,
impassable portions of the singular surface are obtained
as in Fig. 6–7. This figure does not show all of the
internal impassable surfaces. All surfaces are obtained
by successive 1⁄4 rotations about the z axis.
Let’s make a simplification of these impassable
surfaces. The following analytical expressions are
found29) which corresponds to a smooth line shown
B
A
D3
B
A
D3
B
D2
(e)d=0.87
(f)d=0.857
B
A
A
D3
B
D4
(h)d=0.64
(g)d=0.85
A
B
(i)d=0.6
Fig. 6–6 Cross sections of singular surface.
Bold curves are impassable and thin curves are
passable. All are drawn at the same scale except the
last two, (h)d=0.64 and (i)d=0.6. The impassable
curve segments AB have envelope parts as shown in
(a), are totally internal as shown in (b) and are
divided into two as shown in (h).
––– 36 –––
–– 6. Pyramid Type CMG System ––
A
Z
Z
g1
Q
B
P (0,0,2s*)
C
F
D
α
E
D’
Y
X
O
X
Y
A
4c*
P’
C’
B’
F’
Fig. 6–7 Internal impassable singular surface.
Cross sections orthogonal to g 1 are drawn at a d
Q’
Fig. 6–8 Analytical line on an impassable surface.
The surface near this line is called a branch.
step size of 0.05. The detail of the region indicated
by A is in Fig. 6−6.
(4) Straight line DE
H = (− c*, c*, s*)t + c*sinφ(1, −1,
in Fig. 6–8. This line is on the impassable surface shown
in Fig. 6–7 hence can represent the surface. This line
has the following four parts:
2)
t ,
(6–19)
where:
θ = (φ +π ⁄ 3, π ⁄ 6, 5π ⁄ 6, φ −π ⁄ 3 ) ,
(1) Elliptic arc AB
H = (2(c*− cosφ ), 0, 2s*sinφ )t ,
(D) 5π ⁄ 6 ≤ φ ≤ π (E) ,
(6–16)
u = (− ( c*cosφ + sinφ ),
where:
( c*cosφ − s*cosφ ), c*cosφ )t .
θ = (− π ⁄ 2, φ , π ⁄ 2, π − φ ) ,
π ≥ φ ≥ π ⁄ 2 (B in Fig. 6–8) ,
u = ( − s*cosφ , 0, sinφ )t .
(2) Straight line BC
H = (2c*sinφ , 0, 2s*)t ,
(6–17)
where:
θ = (φ , π ⁄ 2, − φ , π ⁄ 2 ) ,
(B) − π ⁄ 2 ≥ φ ≥ − 5π ⁄ 6 (C) ,
u = ( 0, − s*cosφ , − sinφ )t .
(3) Circular arc CD
H =(c*−cosφ, c*(1−sinφ ), c*(1+sinφ ))t ,
(6–18)
where:
θ = (− 5π ⁄ 6, φ , 5π ⁄ 6, π ⁄ 2 ) ,
(C) π ⁄ 2 ≥ φ ≥ π ⁄ 6 (D) ,
u = g2 = ( 0, s*, c*)t .
Let point F on the arc AB be the location at which
the line AB is divided to an envelope side (AF) and an
internal side (FB). The line FBCDE is connected and
continuous both in the H space and in the θ space. By
the transformation MRR1 in the notation of Sec. 6.2, a
line F’B’C’D’E can be obtained which is continuously
connected to the original line. Thus, referring to Fig. 6–
8, the line FBCDED’C’B’F’ connects two points on
opposite sides of the envelope.
The surface of Fig. 6–7 is composed of several strips
of impassable surface and a portion of it is represented
by this line. This particular strip will be called an
impassable branch and be denoted by B e1 . In the
following figures, the analytical line FBCDED’C’B’F’
is simplified by using a broken line QPP’Q’, as shown
in Fig. 6–8.
This branch, along with its symmetric images by
transformations Rr1, Rq1, RE1, RR1 and RQ1, form a
frame of parallel hexahedron shown in Fig. 6–9, which
fits all the surface shown in Fig. 6–7. Each branch is
denoted by B and the subscript denotes the
––– 37 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Z
transformation, Br1 for example. In this figure, some
lines are shared by two branches. This is partially the
result of simplification of the curved line. The other
reason is that two branches share the same surface near
the envelope and this surface bifurcates as depicted in
Fig. 6–6.
By the successive rotation about the z axis, the stellar
hexahedron in Fig. 6–10 is obtained. In this figure, only
the suffix is shown for each branch. The same stellar
hexahedron is drawn in Fig. 6–11 with a cut envelope to
reveal the size and the shape of the internal impassable
surface. In summary, all impassable surfaces of the
pyramid type CMG system are described by the envelope
P
B e1, B E1
,B
1
Be
B E1
Bq
1
B r1
Q
,B
1
BE
1
B r1, B R1
Q
1
B q1
Be1, BQ1
r1
B Q0
BR
X
Y
BR1, Bq1
Fig. 6–9 Equilateral parallel hexahedron of
impassable branches.
E4
2
e1, Q1
e4, E3
e3
2,
E
,R
Y
q2
R4, q3
3
1
r4, E4
3
q3, R
r1, q1
e
R4 4,
, q4 Q
4
X
O
R
1
q2
g1
r4
1,
R2, q
r2,
Z
E
1
r3, Q
1
e3, Q3
g3
q2, Q
e1
e2
,R
E
E2
3
Q
4,
E
r2
q4
2
E2, r
3
r
3,
,
q3
Q2,
2
,R
Q4
2,
Q
Q3, r1
E1, e2
1
,Q
q1
1
r1, R
X
e3, E3
e2,
e1, E1
r2,
Z
r3, R3
4
,R
r4
q4, Q4
e4,
Envelope
Fig. 6–10 Overall structure of impassable branches
Fig. 6–12 Cross section through the xz plane.
Cutaway of Envelope
Envelope
Z
P
Passable
P’
Y
X
Y
Impassable
Simplified Branches
Fig. 6–11 Internal impassable surface with
envelope cutaway.
X
Fig. 6–13 Cross section through the xy plane.
––– 38 –––
–– 6. Pyramid Type CMG System ––
and the frame like structure of the branches.
Figures 6–12 and 6–13 are examples of other cross
sections which are not orthogonal to the gimbal axes.
In these figures, all singular surface, passable and
impassable, are drawn. Two figures show that there are
relatively large region with no singular surface and
impassable surfaces are narrow strips compared with the
maximum workspace. Nevertheless, impassable
surfaces cannot be ignored because they surround the
origin which is the nominal point for the control.
Moreover, some of the impassable surfaces crosses z
axis and others lie on the x-y plane. As the CMG
system’s axes coincide with those of an attitude control
system, angular momentum of the CMG system tend to
travel near such axes.
––– 39 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 40 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
Chapter 7
Global Problem, Steering Law Exactness
and Proposal
This chapter deals with the question whether it is
possible to steer the pyramid type CMG system to avoid
any terminal class of Type 2 domains. In Chapter 5, it is
clarified that any steering law will fail if it aims to cover
the workspace in its entirety. Moreover in this chapter,
it will be shown by examples that any steering law fails
continuous and real-time control for the wide variation
of command inputs even if an appropriate manifold
selection is tried.
Based on this global problem, various steering laws
are evaluated. In so doing, the CMG motion by each
steering law is analyzed geometrically. Three groups of
steering laws are examined and their performance and
problems are clarified. The first of those permits errors
in the output. The second is realized as a path planning.
The third one is effective for a certain fixed direction.
By those evaluation, importance of steering law
exactness is clarified.
Finally, a new type steering law will be proposed
which assures exact and real time control inside a reduced
workspace. This steering law uses a simple constraint
and determines uniquely the system state from the
angular momentum. The reduced workspace is larger
than the spherical one which excludes all impassable
surfaces, but has the same length in one direction as the
original maximal workspace.
First, an H path and its deviations along the z axis
from O to P are considered. For continuous control on
these paths, the necessary condition of θ at the origin O
is obtained. Then, by consideration of an H path from
O to Q, the impossibility of continuous control is derived.
7.1.1 Control Along the z Axis
Let’s find necessary conditions for continuous realtime steering along the z axis as shown by OP in Fig. 7–
1. Consider now only H in the neighborhood of P. Fig.
7–2 (a) shows a cross section of the singular surface by
a plane orthogonal to z axis and which crosses near the
point P. In the close-up view of Fig. 7–2(b), it can be
seen that there are eight domains around the center and
four pairs of impassable branches cross each other.
One of the eight domains, the domain DA in Fig. 7–
2(b), has 2 equivalence classes whose elements (which
are manifolds of those classes) are MA0 and MA1 in Fig.
7–3. The class GA0 including MA0 bifurcates into two
classes G00 and G01 when entering the neighbor domain
D1 in Fig. 7–2(b). Two classes are represented by two
manifolds M00 and M01 in Fig. 7–3. The classes G00
Candidate
workspace
Z
7.1 Global Problem
P
Following the discussion of Section 5.7, the
workspace size must be slightly reduced from the
maximum to enable continuous control. Here, it is shown
that continuous control is impossible if the workspace
includes certain domains. An example of the workspace
is a sphere around the H origin O in Fig. 7–1 including
the hexahedron made of impassable branches which is a
part of the stellar hexahedron in Fig. 6–11. Domains
around two vertices, P and Q, are relevant for the
following discussion.
––– 41 –––
T
Q
X
O
S
R
Y
Fig. 7–1 Candidate of workspace
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
around these domains, as in Fig. 7–4. The situation is
similar to that around a Type 2 domain described in
Chapter 5 and hence, only one class GA0 must be selected
in this domain DA.
As illustrated in Fig. 7–5, domains and manifolds
symmetric about the z axis are obtained by successive
1/4 rotations. Though there are two equivalence classes
to each of the four domains, i.e., DA to DD, one of them
must be selected with the consideration above.
Manifolds to be selected are MA1, MB1, MC1, and MD1
in Fig. 7–5. All the four domains are connected with
each other by a line segment lying on the z axis which is
shown as a cross point U in Fig. 7–5 (b). Figure 7–6
shows a singular manifold for this point U. By
comparing the manifolds MA1, MB1, MC1, and MD1 in
Fig. 7–5 and the singular manifold MU in Fig. 7–6, it is
observed that all the manifolds are continuously
connected by two curved line segments, φ1φ2 and φ3φ4,
shown bold in Fig. 7–6. Thus, the necessary condition
for continuous steering from the point U to either DA,
D B , D C or D D is controlling θ on one of the two
segments, either φ1φ2 or φ3φ4, when H is at U.
If a cross sectional plane orthogonal to the z axis is
moved towards the H origin, the topology of domain
connections, the intersection of the singular surface and
class connections over domains become different from
those in Figs. 7–2 and Fig. 7–4. However, two segments
of the singular manifold in Fig. 7–6 are analytically
defined for any H= (0,0,Hz)t on the z axis as follows:
Y
Envelope
Passable
Impassable
X
|H| = 1
(a) Cross section normal to the z axis.
Y
Domain D1
Branch BE4
X
H path 1
Branch Be3
HHpath
path2 2
Domain D A
(b) Magnified view inside the dotted square
shown in (a).
θ = (φ+ψ, φ−ψ, φ+ψ, φ−ψ),
where Hz = 4 s* cosψ sinφ .
Fig. 7–2 Cross section nearly crossing P .
The distance from O to the plane is 1.4 (not
normalized). This plane crosses the z axis
nearer the origin than P, as OP = 2 s* ≈ 1.63.
and G01 are equivalent to the terminal classes of the
impassable branches BE4 and Be3 respectively. This
implies that M00 continuously changes to a singular point
(θE4 in Fig. 7–3(c)) when H follows the path 1 in Fig.
7–2, and M00 changes to another singular point (θe3) by
the H path 2.
Thus the class containing MA1 in Fig. 7–3 should be
selected in the domain DA for continuous steering in
consideration of these H paths. This is more easily
understood by making a simplified class connection map
(7–1)
One of the segments includes the point (φ, φ, φ, φ)
where φ = sin-1(Hz ⁄(4 s*)) and both of its edges are
singular. The other segment can be obtained as a mirror
image of this.
Because the segment (and its edges) given by Eq. 7–
1 are continuous with respect to Hz, θ must be located
on this segment, or on its mirror image, for any Hz. At
the H origin, this segment and its mirror image take the
following simplified form:
––– 42 –––
θ(H=(0,0,0)) = (ψ, −ψ, ψ, −ψ) ,
where −π/6 ≤ ψ ≤ π/6 and 5π/6 ≤ ψ ≤ 7π/6 .
(7–2)
–– 7. Global Problem, Steering Law Exactness and Proposal ––
–3π⁄2
–3π⁄2
θ2
θ2
MA0 ∈G A0
0
M A1
0
–π⁄2
–π⁄2
–π⁄2
0
θ1
–3π⁄2
–π⁄2
0
–3π⁄2
θ1
(a) Two manifold of H = (−0.02, 0.02, 1.4) t in domain DA .
–3π⁄2
–3π⁄2
θ2
θ e3
θ2
M 01
∈G01
M 00
∈G00
θ E4
0
0
–π⁄2
–π⁄2
–π⁄2
0
θ1
–π⁄2
–3π⁄2
0
θ1
–3π⁄2
(c) Impassable singular points of branches BE4
and B e3.
θ E4 is connected with M 00 in (b) through Path 1
in Fig. 7−2 and θ e3 is connected with M01 in (b)
through Path 2.
(b) Two manifolds of H = (−0.05, 0.05, 1.4) t in
domain D1. Both are connected with M A0 in (a).
Another manifold connected with MA1 is not
drawn.
Fig. 7–3 Manifold bifurcation and termination from DA .
Manifolds are drawn using (θ 1, θ 2) coordinates. The θ origin is not on the center
to avoid a manifold drawn separately.
G 01
BE4
G A0
Domain DA
D1
G A1
G12
G00
Be3
Fig. 7–4 Simplified class connection diagram around domain D A.
For clarity, this figure of domains has been simplified by omitting some singular surfaces.
––– 43 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Y
Ε3
e1
Ε2
e2
U
X
Ε4
DA
DC
DA
DB
U(0, 0, 1.4)
e4
e3
DD
(b) Four domains around the point U
Ε1
(a) Branches and domains
MD1
M C0∈ GC0
MD0
∈ GD0
M C1
DD
DA
MA1
DC
DB
MB1
MA0 ∈ GA0
M B0
∈ GB0
(c) Manifolds of eight domains.
Fig. 7−5 Manifolds of eight domains around the z axis.
In Fig. (a), each branch is indicated by its suffix such as E1 for BE1 . In Fig. (c), each
domain, from DA to DD, has two manifolds. They are drawn using (θ1 , θ 2) coordinates.
Though four manifolds, from MA1 to M D1 , are congruent, they look different in two
dimensional projections.
––– 44 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
These crossings are not singular points
but only due to the 2D projection .
φ4
φ3
Singular
Point
7.1.3 Details of the Problem
φ1
θ2
This is a more general conclusion than the difficulty
reported in the former research work37), which dealt only
with the specific examples of motion along the z axis.
By the geometric analysis made above, it is understood
that not only the H path on the z axis but also a variety
of other paths cannot be realized by any steering law.
φ2
θ1
MU
Segment
Fig. 7–6 Singular manifold of a point U on the
z-axis.
Two curved line segments drawn bold, φ1 φ2 and
φ3 φ 4, are the segments continuously connected
to manifolds, M A1 to M D1 in Fig. 7–5.
7.1.2 Global problem
The same discussion can be made for the H path from
O to Q. All θ and H are simply transformed by a
rotational transformation such as the 1 ⁄3 rotation about
the g1 axis. This transformation is denoted by Rr1 in the
notation of the previous chapter. By the corresponding
θ transformation, the above conditions, Eq. 7–2, is now
transformed to the following segment:
θ(H=(0,0,0)) = (ψ, −ψ, ψ, −ψ) ,
Suppose that θ is on the latter segment of Eq. 7–3,
when H = 0. This segment is denoted by LF after the
notation of Section 6.3. Referring to Fig. 7–7, an
infinitesimal motion of H towards (1, 1, 1)t moves H
away from a singular surface and θ moves onto a
manifold which is originally a rectangle outlined by
segments LF, LM, LC and LH when H = 0. As H moves
closely along the z axis in the same domain, the manifold
changes equivalently as shown in Fig. 7–8, with neither
bifurcation nor termination. Finally, near the point U in
Fig. 7–5 (b) the manifold connects with either MA0, MB0
or MD0 (Fig. 7–9). Because the impending H path is
not given, there is a possibility of these manifolds being
selected once θ is determined on manifold MV in Fig.
7–9. The three manifolds inevitably bifurcate into
terminal classes if the H path crosses certain branches,
for example branches BE4 and Be3 if manifold MA0 is
selected, branches BE1 and Be4 for MB0 and branches
BE3 and Be2 for MD0.
The manifolds in Fig. 7–8 are all inside one domain,
denoted by DV. Though Fig. 7–10 in the next page
indicates that this domain is not large by itself, some
where π/2 ≤ ψ ≤ 5π/6 and −π/2 ≤ ψ ≤ −π/6 .
M V (H=(0.02, 0.02, 0.02)t)
(7–3)
Since the two sets of segments given by Eqs. 7–2
and 7–3 have no common θ, continuous control from O
to P and from O to Q cannot be satisfied simultaneously.
It is clear that once the condition imposed by Eq. 7–3 is
satisfied, the system will meet an impassable singularity
on the H path nearly along the z axis while crossing
some of the impassable branches BEi and Bei in Fig. 7–
5. Thus it is concluded that continuous steering over
this entire workspace, including O, P and Q is not
possible. These two sets of segments defined by the
above two equations are (LA, LD) and (LC, LF) in the
notation of Section 6.3. The remaining segments LB
and LE are the condition for the continuous control in
the OR direction.
––– 45 –––
θ2
U
LH
T
V
LF
LC
LM
S
θ1
Fig. 7–7 Manifold of H near the origin.
MV is one of the manifolds for H = (0.02, 0.02,
0.02)t which continuously deformed from the rectangle STUV. Four edges of the rectangle are LF,
LM, LC and LH by the notation of Section 6.4.
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
a = 0.02
Y
DV
a = 0.25
a = 0.5
a = 0.75
X
a = 1.0
a = 1.3
|H|=1
Fig. 7–8 Continuous change of manifold for H
nearly along the z axis.
These six manifolds are for H=(0.02, 0.02, a)t
where a = 0.02, 0.25, 0.5, 0.75, 1.0 and 1.3. The
last manifold for a = 1.3 is MV of the next figure.
Filled circles and blank circles are maximum and
minimum of det(CCt) along the manifolds.
Envelope
(a) HZ= 0.1
DV
MV
θ2
Envelope
θ1
(b) HZ= 0.5
(a) Manifold M V
DV
M D0
MB0
θ2
θ1
MA0
Envelope
(c) HZ= 1.0
(b) Manifolds in the neighborhood
Fig. 7−10 Cross sections of domains.
Fig. 7−9 Manifold connection over several
domains.
The manifold M V connects partially with M A0,
MB0 and M D0.
––– 46 –––
Domain DV corresponding to
manifolds in Fig. 7−8.
These domains have
manifolds equivalent to M V.
–– 7. Global Problem, Steering Law Exactness and Proposal ––
neighboring domains have equivalent classes to this
manifold. Once the segments at H = 0 given by Eq. 7–
3 are selected, H then moves inside these domains, there
is no way to escape from the manifold equivalent to this
MV. The branches mentioned above pass along edges
of the “top” half of an octahedron, namely PQ, PR, PS
and PT in Fig. 7–1. Since it is safe to assume continuity
of the surfaces and manifolds, it can be expressed that
some parts of this manifold connect to manifolds
belonging to terminal classes of these branches.
Therefore there is a possibility of termination for any H
path crossing such branches. Moreover, these domains
are so large that this problem cannot be neglected.
Manifold MV, however, is not connected to MC so it
does not present a problem with respect to branches BE2
and Be1 when this manifold is selected.
7.1.4 Possible Solutions
The above discussion is made without consideration
for any specific steering law. The problem applies to
any steering law which aims an exact and strictly real
time control. Exactness implies that an output is always
equal to the command input. Strict real time feature
implies that information of future command is not used.
Possible methods to overcome the problem could
involve either of the following.
1) relaxing an exactness condition.
2) relaxing a real time condition.
3) restricting the workspace.
In the following sections, from Section 7.2 to 7.4, various
proposals of the former two kinds will be evaluated. By
these evaluations, importance of steering law exactness
and real time feature will be clarified. Then, a new
steering law using workspace restriction will be proposed
in Section 7.5.
surface. Because a passable surface is generally avoided
by steering laws using a gradient method, such a steering
motion will take place on an impassable surface. Its
solution is obtained so that the output torque lies on a
plane tangential to the singular surface. Therefore, this
steering motion can be imagined as a ‘sliding’ motion
of H on the singular surface.
As depicted in Fig. 7–11, there are four possibilities
of motion along the singular surface when the command,
Tcom, is fixed. The case (d) can be ignored straight off
because it is not stable. The case (c) is possible when
the surface is convex to u and this is the case of an
envelope (see Section 4.3.4 and Fig. 4–8). For an internal
surface, only the cases (a) and (b) are possible. Thus, a
motion is always possible in response to the command
as long as the command is fixed.
This discussion ignores how large the torque error
is. If the area of the impassable surface is excessively
large, this steering law is not effective in practical use.
As shown previously in Figs. 4–10, 11 and 12, the
impassable surface of a 4 or 5 unit CMG system is shaped
like a narrow strip and the curvature of the surface is
negative to its narrow direction. Because of this, such
(3)
Singular Surface
(3)
(2)
(2)
(1)
(a) Smooth Break Away
(1)
(b) Folding
7.2 Steering Law with Error
(2)
The steering laws described in Section 3.5.2 enable
calculation of the inverse Jacobian even on a singular
point. This is made possible by permitting a minimum
error in output torque. This kind of method has so far
been evaluated only by a limited number of simulations.
7.2.1 Geometrical Meaning
The CMG motion by a steering method accompanied
by error is understood from the shape of an impassable
Stop
Stop
(1)
(c) Stop at Convex
(d) Unstable Stop
Fig. 7−11 Possible motion following an example
of singular surface.
Motions may be (1) reach the singular surface,
(2) go along the surface, then (3) break away from
the surface, if possible.
––– 47 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
escape motion as in the cases (a) and (b) will take place
approximately to the narrow direction. Possible error,
therefore, can be roughly estimated by the width of the
singular suface stip.
If a faster ‘sliding’ motion is required, judicious
knowledge of the direction of narrow width is very
useful. This direction can be approximately obtained as
an eigenvector of the negative curvature of the surface.
Supposing that this direction is obtained as v, the θ
motion, dθS, that will realize this ‘sliding’ motion is
obtained by movement along the singular surface as
follows:
dθS = PCt (CCt + k u ut )−1 v .
(7–4)
H motion due to
Vehicle's Rotation
H Trajectory
Acceleration
Deceleration
H0
H1
Fig. 7−12
Illustration of H trajectory of the
CMG system for the example maneuver.
This is derived by Eq. 3–22 and Eq. B–9 in Appendix B.
Impassable Surface
7.2.2 Exactness of Control
Desired Path
As mentioned previously, the steering of a CMG
system is similar to the kinematic control of a multijoint manipulator (also see Appendix F). If a CMG is
used by itself and the objective is to realize a certain H
trajectory, the steering law problem is analogous to the
kinematic control of a manipulator. In this situation, the
above method gives a possible solution whose H deviates
slightly from the desired trajectory. The difference
between CMG control and control of a manipulator is
that a CMG is used for the attitude control. The objective
is not to control the actual CMG but rather to control the
vehicle’s attitude. If there is an output torque error, not
only is there deviation in the path of H but also the
attitude of the satellite changes from that intended. This
attitude error changes the command issued by the
feedback control and then the desired H path also
changes. The above method should therefore be
evaluated in consideration of the attitude control.
Suppose that the angular momentum of the satellite
is zero and the angular momentum of the CMG system
on board is not zero. Suppose further that the control
command is to maneuver the satellite and finally stop
the rotation. This implies that the final angular
momentum of the satellite is zero and the angular
momentum of the CMG system is not zero. By the
conservation law, the initial and final angular momentum
must be the same in the inertial coordinates. Since the
coordinate frame of the CMG system rotates, the initial
and final angular momentum will be different in the
rotating coordinate frame.
Let H0 and H1 denote the initial and final H in
Detour
H0
H1
Fig. 7−13 Avoidance of an impassable surface
coordinates fixed on the CMG system. The H trajectory
of the CMG system will be some path from H0 to H1, as
depicted in Fig. 7–12. Though the exact path varies for
different control methods, H0 and H1 will not vary if
the control is successful and if there are no disturbances.
Suppose that there is an impassable surface somewhere
along this path. If this surface is located sufficiently far
from the goal and the surface are small enough, it might
be possible to make a ‘detour’ as shown in Fig. 7–13,
and reach the goal by the above steering law.
If the surface is near enough the goal, H will stay on
the singular surface (at the point A in Fig. 7–14(a))
despite the negative surface curvature, because the point
A is the nearest to H1 . Since the residual angular
momentum H−H1 of the spacecraft is not zero, there
remains some rotation of the spacecraft when H is on
this surface. Though this rotation depends on the inertia
matrix of the body and the direction of the residual
angular momentum, H of the CMG may stay inside some
area of the impassable surface, as shown in Fig. 7–14(b).
––– 48 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
43, 44).
No sliding motion possible
This problem is similar to the path planning and
its realization of a robot manipulator.
In Section 7.1, it was concluded that some of the
various possible command sequences cannot be realized
simultaneously by the same steering law. This is true as
long as the future H path is not specified. On the
contrary, manifold selection and continuous θ path is
possible for a given H path, even for one of the two
paths in Fig. 7–2 for example. Of course, continuous
control is not possible if the H path starts from the
singular surface as described in Section 5.5.6 or if the H
path crosses a Type 2 domain as described in Section
5.5. Thus, conditions of successful path planning can
be clarified by this geometric study. If we permit a
minimum error in the solution42,43), path planning is
always possible because of the nature of such motions.
Geometric study also reveals some problem of this
manner of path planning. Since different manifolds may
be selected for different H paths, the θ path may be
completely different even though the H path is very
similar, which may degrade the robustness of the control
system. Moreover, optimization is limited only to the
given H path but no future situations are considered.
Moreover, this method is too complicated for actual
implementation.
H1
A
Impassable
Surface
H Trajectory
(a) Impassability
This residual angular momentum
causes rotation of spacecraft
H1
Possible H Trajectory
7.4 Preferred Gimbal Angle
(b) Motion on the impassable surface
Fig. 7−14 Problems of movement on an
impassable surface.
In this case, it is impossible to reach the goal and the
spacecraft will continue its rotation.
If an attitude keeping problem under some
disturbance is considered, such a situations as in Fig. 7–
14 is not avoided by this steering law. Because of these,
it is better not to use the above steering law and better to
keep steering law exactness.
7.3 Path Planning
Another steering law approach takes advantage of
rapid maneuvering to enable off-line planning. If the
maneuver occurs fast enough, the period of maneuver
will be short enough that any disturbance can be
neglected and hence the maneuver trajectory and H path
can be designed beforehand. For this given path, the
CMG motion can be planned by off-line calculations42,
Another method38) is similar to path planning but
supposes that the direction of a near future maneuver
can be known, and this direction is one of
certain predefined possibilities. This method introduces
‘preferred gimbal angles’ from the maneuver direction
and adjusts the system to the preferred angles before the
maneuver motion. An examples of preferred gimbal
angles and their corresponding maneuver directions are
given as follows38):
––––––––––––––––––––––––––––––––––––––––––
Direction of Maneuver
Preferred Gimbal Angles
z-axis, (1, 1, 1)t direction (0, 0, 0, 0)
x-axis, (4, 2, 0)t direction (−π⁄3, π⁄3, 2π⁄3, −2π⁄3 )
y-axis, (2, 4, 0)t direction (−2π⁄3, −π⁄3, π⁄3, 2π⁄3 )
––––––––––––––––––––––––––––––––––––––––––––
It is guaranteed from the discussion of Section 7.1.1
that an initial gimbal angle of (0, 0, 0, 0) is suitable for
z-axis maneuvers. However, evaluation of the others
are not simple.
––– 49 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
A(θ 0), B
θ4
θ3
θ1
θ2
H X =1.0
H X=1.0
B
H X=0
Hx=0
A(θ 0)
(a) 0.0 ≤ HX ≤ 1.0
HX =1.0
B, C
HX =1.2
HX =1.4
HX =1.0
H X=1.2
C
H X=1.4
B
Connection
(b) 1.0 ≤ HX ≤ 1.4
H X=1.4
D
H X=2.6
HX=2.6
H X=1.4
C
(c) 1.4 ≤ HX ≤ 2.6
Fig. 7−15 Change in manifolds for H moving along the x axis.
(H = (HX , 0, 0)t, HX step size = 0.2). Manifolds are drawn in (θ1 , θ 3) and (θ 2, θ4 ) coordinates
as shown in (a). Manifold bifurcations are observed in (b). Motion of θ from the preferred
angles θ 0=(−π ⁄ 3, π ⁄ 3, 2π ⁄ 3, −2π ⁄ 3) follows the line ABCD, which is a trace of the maxima of
det(CCt ) and is indicated by dots.
In the (1, 1, 1)t direction, there are two impassable
branches, i.e., BE2 and Be1. The discussion in Section
7.1.3 suggests that the θ on the segments LF, LM, LC or
LH may be better than preferred angle (0, 0, 0, 0). The
second and third θ are on segments LL and LM. The
second preferred θ in the above list will next be evaluated
by observing the change in manifolds as H moves along
the x axis.
Figure 7–15 shows manifolds corresponding several
H on the x axis. A gradient method is used and θ is
maintained at the local maximum of det(CCt). Starting
with θ set to the values in line 2 of the above list, θ
subsequently follows the path ABCD. There are only
two bifurcations in this manifold path, as shown in Fig.
7–15(b), and these correspond to passable surfaces in
the neighborhood of two impassable branches. The bold
curve in the left plot of Fig. 7–15(b) indicates a
connection, and this shows that the manifolds are
––– 50 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
connected by this part before and after the bifurcations.
The robustness of this motion is evaluated by the length
of this part, which is more than π ⁄2 in this case.
The preceding discussion verifies the performance
of this steering method but also clarifies its limitations.
This method is valid only when H is initially on the
origin. No method was specified to obtain θ when H is
not zero. If a gradient method is applied, this method is
valid as long as the maneuver is carried out exactly along
the defined direction. But if the maneuvering path
deviates, various gradient method problems may occur,
which will be described in the next section.
Conceptually, this method may be effective for exact
and real-time steering as long as the system does not
meet impassable singularity before H returns to the
origin. The main question is whether this can be assured.
By extension of the first set of preferred angles and by
making the motion exact, a more effective method will
be proposed in the following section.
7.5 Exact Steering Law
The evaluation above clarified that steering law
exactness and real time feature are important for the real
usage of the CMG system in the attitude control of a
satellite. In this section, a new steering law is proposed
which assures its exactness and real time feature39).
Though, the idea of manifold selection prior to
bifurcation is important for analyzing continuous control,
suitable algorithms for actually doing this have not been
developed. Segments defined by Eqs. 7–1 and 7–2 only
defines θ where H is on the z axis. While geometric
concepts such as class connection around domains are
useful for evaluating the steering law, the actual steering
law algorithm must determine the θ value at any H point
so that the desired manifold selection is made.
7.5.1 Workspace Restriction
Because of the problem in Section 7.1, the workspace
must be restricted in order to keep exact steering. One
way of workspace restriction is to exclude all impassable
surfaces from the workspace. This however is a too strict
way of restriction. The condition imposed by Eq. 7–1 is
effective with regard to motion nearly along the z axis,
while it is not applicable for control in the neighborhood
of Q. Thus, a new workspace of a two-lobe shape may
be obtained by excluding some of the impassable
branches crossing near Q, R, S and T as shown in Fig.
4 s*
Possibly Passable
Impassable
Z
P
T
Q
S
X
R
Y
2 c*
Estimated
Workspace
Fig. 7–16 Estimation of reduced workspace for
exact steering. Branches drawn by bold lines are
impassable but those drawn by thin lines may be
made passable.
7–16. The shape of this workspace is not defined by the
discussion in Section 7.1, however it does include the z
axis and its neighborhood, and an area on the xy plane
slightly smaller than the square QRST as shown in Fig.
7–16.
7.5.2 Repeatability and Unique Inversion
It is required that θ remains on the segment given by
Eq. 7–1 whenever H is on the z axis. This is a matter of
repeatability of inverse kinematics. Generally,
repeatability is realized only when the system has an
inverse mapping67). The following example illustrates
that an ordinary gradient method does not possess
repeatability over the workspace in Fig. 7–16.
Figure 7–17 shows manifolds of several H points
on the line from O to Q. Each jagged-edge rounded
rectangle is a computer output of the manifold drawn in
(θ1, θ2) coordinates. Dots on the manifolds indicate local
maxima of det(CCt) along each manifold. This implies
that θ may be controlled on these points by a gradient
method. If the initial θ meets the condition of Eq. 7–2,
θ will follow line AB for commands on the H path
approaching Q from O. The line linking the local
––– 51 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
θ1 − θ2 + θ3 − θ4 = 0 ,
Local Maxima of det(CCt )
θ2
Manifold of
H=(0,0,0)t
A
θ1
D
C
B
E
(7–5)
because of the following reasons. When H is on the z
axis, this condition gives the center of the segment described by Eq. 7–1 which has the maximum det(CCt) on
the segment. This condition preserves some of the
system’s z axis symmetry. Moreover, it is simple and
the constrained kinematics are also simple. Finally, it
will be shown in the next section that the workspace by
this constraint is an appropriate realization of the expected workspace in Section 7.5.1.
The constrained kinematics has a following analytical
form;
 − c * cos φ sin ψ + sin φ sin γ 
H = 2 − sin φ sin ψ − c * cos φ sin γ 


 s * sin φ (cos ψ + cos γ ) 
Manifold of H=(0.5,0.5,0)t
Fig. 7–17
Discontinuity in the maximum of
det(CCt).
Rounded rectangles are parts of the manifold
for H on the z axis. Dots indicate the local
maxima of det(CCt ).
,
where θ = (φ+ψ, φ+γ, φ−ψ, φ−γ) . (7–6)
7.5.4 Reduced Workspace
maxima is discontinuous at point B where H ≈ (0.3, 0.3,
0) t . After passing this H, θ approaches another
maximum, either C or E. Suppose the case of C here. If
after this motion the command path of H is reversed
back to O, θ never goes back to B but follows CD, the
other line of maxima. Finally, θ does not satisfy Eq. 7–
2 when H returns to O. Thus, such a method is not a
possible candidate.
This problem is derived from the fact that an
equilibrium point by a gradient method, i.e., nominal θ
for a given H is neither unique nor continuous. The
condition of Eq. 7–1 requires that θ must be uniquely
determined by an inversion from H to θ. Thus, this
unique inversion is required for the exact steering law.
The allowed workspace of this system is defined by
keeping unique inversion feature within the domain of
three variables, φ, ψ, γ, to [−π ⁄2, π ⁄2]. Figure 7–18
shows possible regions of H in several cross sectional
plane orthogonal to the z axis. As an envelope of each
region is not simple enough to be handled by the
momentum management procedure of a controller, an
approximation is required. An example approximation
is made where |Hz|≤ 2s*, which is shown by rounded
squares in the same figure. These rounded squares are
defined by the following equation;
 −2(c * cp − sq)
H =  −2( sp − c * cq )


Hz


7.5.3 Constrained Control
In kinematics, characteristics of unique inversion are
realized by utilizing direct constraints of variables68).
By using some algebraic relation of variables as
constraints, inversion of the constrained kinematics
becomes a one-to-one and continuous inside of some
range, which specifies the workspace. For a four
dimensional system, it is adequate to constrain one
degree of freedom.
Though various constraining conditions were
possible, the following was applied39):
,
(7–7)
where
|Hz| ≤ 2s*, s = Hz ⁄ (2s*), c = (1 − s2)−1⁄2,
p = sin−1⁄2 s and q = cos−1⁄2 s .
By using this approximation, a workspace of the
constrained steering law is defined as illustrated in 3dimension in Fig. 7–19.
The reduced workspace has the same maximum
length as the maximum workspace in the z axis, but the
––– 52 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
y
Angular Momentum Envelope
y
Internal Singular
Surface
2c*
x
x
2(1+c*)
Cross Section of
Allowed
Workspace
(a) Hz=0.0
(b) Hz=0.4
y
y
Approximated
Workspace by Eq. 7–7
x
x
(d) Hz=1.0
(c) Hz =0.75
y
y
y
(e) Hz=1.4
x
x
x
(f) Hz =2.0
(g) Hz =2.6
Fig. 7−18 Cross section of possible workspace by constrained steering law.
Possible region of angular momentum given by Eq. 7–5 is drawn by two parameter net of (ψ, γ) under
condition that Hz is constant and the determinant of the Jacobian is positive. Approximated workspace
is defined by Eq. 7–7.
––– 53 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
z
type of mechanism but flexible wires can be used for
power supply.
4s*
2c*
x
(3) Simplicity of Calculation
Even though the proposed steering law involves
calculation of an inverse Jacobian, which in turn involves
inversion of a 3 × 3 matrix, this is simpler than calculation
of the pseudo-inversion of a 3 × 4 matrix. Also, no
complex calculations for the gradient method are needed.
The actual implementation is also simple, though it does
use some feedback. Implementation details are described
in Appendix E. Because of the simplicity, this method
actually installed on the experimental computer needed
about 2⁄ 3 memory storage and about 1 ⁄ 2 calculation
time compared with the gradient method with an
objective function det(CCt) (see Appendix E.5).
2c*
y
Fig. 7−19 Reduce workspace of the constrained
system.
minor diameter on the xy plane is only about 1 ⁄3 that of
the maximum workspace.
7.5.5 Characteristics of Constrained Control
The followings are characteristics of this method.
Most of all are useful for the real usage in the attitude
control system.
(1) Exactness and Repeatability
The inversion of Eq. 7–6 is not exactly one-to-one.
In order to maintain a one-to-one feature, H must be
kept inside the previously defined workspace. By
adhering to this limitation, continuous control over this
space is strictly guaranteed. Moreover, unique inversion
characteristic of the steering law assures repeatability.
(2) Gimbal Limits
Because of the uniqueness, each gimbal angle is
exactly within a certain domain. The domains of φ, ψ,
and γ are included in the domain [−π ⁄2, π ⁄2]. Each
gimbal angle, θi, is therefore inside the domain [−π, π].
This is very advantageous compared with other steering
laws. With a gradient method for example, the domain
of θ is not defined. Gimbal angles greater than one
revolution are observed in results of some computer
simulations. As a result, mechanisms such as a slip ring
is needed to permit free rotation of the gimbal. In
contrast, the method described here does not require this
(4) Modes and Mode Changing
The constraint of Eq. 7–5, the kinematics of Eq. 7–6
and the workspace of Eq. 7–7 defines one constrained
system. As the original unconstrained system has
symmetry, this constrained system can be symmetrically
transformed. There are six possible transformations
whose representations in the H space are the identical
transformation, a mirror transformation about x-z plane
and ±2/3π rotation about g1 with or without the mirror
transformation. By those transformations, six
constrained systems are defined which have their own
constraint condition and own workspace, and have the
similar properties such as exactness.
The six constrained systems makes three pairs. These
pairs are called “modes” and termed M1, M2 and M3.
The workspace of each pair has a shape similar that
shown in Fig. 7–19, and the dominant direction lies along
the z-axis for the M1 mode, along (1, 1, 0)t for the M2
mode, and along (1, –1, 0)t for the M3 mode as shown
in Fig. 7–20. The nominal gimbal angles, which
correspond to H=(0, 0, 0)t are of the form of (ψ, −ψ, ψ,
−ψ), where ψ=0 or π for the M1 mode, 1/3π or −2/3π
for the M2 mode, and −1/3π or 2/3π for the M3 mode.
Since the dominant directions of all the workspaces are
orthogonal to each other, attitude control performance
will be improved by introducing mode switching.
Different modes share a region in H space inside of
which we can select and change modes. When it is
required to change the steering law mode, gimbal angles
must be changed to satisfy another constraint while
keeping the same H. There is, however, no continuous
path from θ of one mode to θ of another mode without a
––– 54 –––
–– 7. Global Problem, Steering Law Exactness and Proposal ––
z
z
y
x
z
(1, -1, 0)t
y
y
x
x
(1, 1, 0)t
(a) M1
(b) M2
(c) M3
Fig. 7−20 Reduce workspaces of three modes.
change in H, except for H=0. (When H=0, there exists
a mode connection path given by θ ∝ (1, −1, 1, −1).)
Therefore, operations like feedback attitude control
should be deferred until the switching process is
completed.
In the experiments, the following simple method was
applied. Here, one specifies a condition such that H is
on the dominant direction of the newer mode (e.g. the zaxis of the M1 mode). The gimbal angle for H along the
z-axis is acquired by a direct inverse calculation of Eq.
7–6, as follows:
φ = sin−1(Hz ⁄ 2s*) , ψ = γ = 0 .
(7–8)
The simplest way of changing θ from the current to the
above is a motion along a line.
This gimbal motion causes undesired torque but its
effect can be made small. Since H is the same for the
initial and the final θ in the CMG coordinate frame, the
initial and the final angular momentum of the spacecraft
alone may be similar when this motion is made fast
enough. Thus, this motion will result in small deviation
of the spacecraft’s orientation and this deviation can be
easily corrected by the feedback attitude control once
the mode is changed.
Though mode changing while H is not on any
principal axes cannot be specified by an analytical
solution, iterative numerical solution can be applied to
find a goal gimbal angles by using the solution 7–8.
(5) General Skew Case and the Maximum Spherical
Workspace
The proposed method does not depend on a specific
configuration symmetry. Equation 7–6 is satisfied with
the constraint of Eq. 7–5 for any value of the skew angle
α as long as the four units are set symmetrically about
the z axis. For any α, the workspace size to the z direction
is 4 sinα, while that of x or y direction is a little less than
2 cosα as shown in Fig. 7–19. If a smaller skew angle α
is used, the workspace becomes shorter in the z direction
and wider in the x and y directions. In this manner an
arbitrary design of the workspace shape can be obtained.
Of course, the original symmetry of the regular
octahedron is lost in an arbitrary skew angle α and only
one mode in the item (4) is available.
If the skew angle α = tan−1(1 ⁄2), the size of the
workspace along the x, y and z axes is almost identical.
This configuration therefore gives the maximum
unidirectional workspace size. If a spherical workspace
is desired for convenience of the attitude control, this is
the best configuration of four unit systems.
Application of the constrained control is not limited
to skew type systems. Any four unit system can be
controlled using one constraint. If the system does not
have symmetry, however, a simple constraint as Eq. 7–
5 may not be effective. In Appendix D, the same
constraint as Eq. 7–5 will be applied to the four unit
subsystem of the MIR-type, i.e., S(6) system69).
(6) Performance
Performance of the proposed steering law was
demonstrated by using ground-based test equipment.
The results are detailed in the following chapter. Also,
the pyramid type system controlled by this steering law
was evaluated by comparing with other type CMG
systems in terms of the workspace size. The results are
detailed in Chapter 9.
––– 55 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 56 –––
–– 8. Ground Experiments ––
Chapter 8
Ground Experiments
The previous chapters dealt only with CMG systems
while aiming primarily at a geometric understanding of
the CMG motion and leaving quantitative matters
neglected. In this chapter, a CMG-based total attitude
control system is briefly formulated and in order to
quantitatively verify the steering law performance,
ground experiments were carried out. Test results
showed clearly the problems found with typical steering
laws and the performance of the steering law proposed
in the previous chapter.
8.1 Attitude Control
The attitude, namely the orientation of a body can
be represented by various ways70) such as the direction
cosines, the Euler angles, Roll-Pitch-Yaw angles,
Rodrigues parameters and Euler parameters. In this
work, representation is made using Euler parameters.
Any attitude is defined as is caused by a single
rotation. The Euler parameters β = (β0, β1, β2, β3)
represent an attitude caused by a single rotation of angle
φ about the axis e = (e1, e2, e3)t:
(8–1)
βi = ei sin(φ / 2) ,
where i = 1, 2, 3 and |e|=1. As the rotation has three
degree of freedom, there is a constraining condition that
Σ βi2 = 1.
Any attitude, which is the result of a rotation a =
(a0, a1, a2, a3) after a rotation b = (b0, b1, b2, b3), is
expressed as a multiplication by a⋅b in the sense of a
Hamiltonian quarternion as follows:
a⋅b = (a0b0 – a1b1 − a2b2 – a3b3,
a0b1 + a1b0 + a2b3 − a3b2,
a0b2 − a1b3 + a2b0 + a3b1,
a0b3 + a1b2 – a2b1 + a3b0) .
dβ* ⁄ dt = 1 ⁄ 2 (β0 + β* × ωV) ,
(8–3)
β* = (β1, β2, β3)t .
(8–4)
where
This β* is called a vector part of β and regarded as a
usual vector in three-dimensional physical space.
The attitude dynamics of a rigid body is represented
in the body’s coordinates by the following Euler
equation:
IV dω ⁄ dt = τ − ωV × p ,
8.1.1 Dynamics
β0 = cos(φ / 2),
The time derivative of β, i.e., dβ ⁄dt is expressed by
angular velocity denoted by ωV:
(8–2)
(8–5)
where IV denotes satellite’s moment of inertia and τ
denotes the torque applied to the satellite. This torque
comes from both the outside as a disturbance torque and
from the inside by the CMG system. The vector p is the
total angular momentum of both the satellite and the
CMG system and is given by:
p = IV ωV + HCMG .
(8–6)
This total angular momentum is conserved in the
inertial coordinates if there is no disturbance torque. By
substituting Eq. 8–6 into Eq. 8–5, the ωV×HCMG term
appears. This term is omitted in Eq. 3–5, i.e., the output
equation of the CMG but is evaluated here. Both the
kinematic equation and the dynamic equation, Eqs. 8–3
and 8–5, are the describing functions of the system.
8.1.2 Exact Linearization
The system has six independent variables, three
components of β* and three components of ωV. The
dynamics is nonlinear as seen by the term ωV×(IVωV)
when Eq. 8–6 is substituted into Eq. 8–5. Nevertheless,
it is well known that this nonlinear system can be exactly
linearized with a suitable feedback71, 72).
Let (β*, dβ* ⁄ dt) be state variables and let v be a
new input variable. If a real input τ is given as:
––– 57 –––
–– 8. Ground Experiments ––
Chapter 8
Ground Experiments
The previous chapters dealt only with CMG systems
while aiming primarily at a geometric understanding of
the CMG motion and leaving quantitative matters
neglected. In this chapter, a CMG-based total attitude
control system is briefly formulated and in order to
quantitatively verify the steering law performance,
ground experiments were carried out. Test results
showed clearly the problems found with typical steering
laws and the performance of the steering law proposed
in the previous chapter.
8.1 Attitude Control
The attitude, namely the orientation of a body can
be represented by various ways70) such as the direction
cosines, the Euler angles, Roll-Pitch-Yaw angles,
Rodrigues parameters and Euler parameters. In this
work, representation is made using Euler parameters.
Any attitude is defined as is caused by a single
rotation. The Euler parameters β = ( β0, β1, β2, β3)
represent an attitude caused by a single rotation of angle
φ about the axis e = (e1, e2, e3)t:
(8–1)
βi = ei sin(φ/ 2) ,
where i = 1, 2, 3 and |e|=1. As the rotation has three
degree of freedom, there is a constraining condition that
Σ βi2 = 1.
Any attitude, which is the result of a rotation a =
(a0, a1, a2, a3) after a rotation b = (b0, b1, b2, b3), is
expressed as a multiplication by a⋅b in the sense of a
Hamiltonian quarternion as follows:
a⋅b = (a0b0 – a1b1 − a2b2 – a3b3,
a0b1 + a1b0 + a2b3 − a3b2,
a0b2 − a1b3 + a2b0 + a3b1,
a0b3 + a1b2 – a2b1 + a3b0) .
dβ* ⁄ dt = 1⁄2β0ωV + β*×ωV,
(8–2)
(8–3)
corrected July, 2011
where
β* = (β1,β2,β3)t .
(8–4)
This β* is called a vector part ofβ and regarded as a
usual vector in three-dimensional physical space.
The attitude dynamics of a rigid body is represented
in the body’s coordinates by the following Euler
equation:
IV dω⁄dt = τ − ωV × p ,
8.1.1 Dynamics
β0 = cos(φ/ 2),
The time derivative of β, i.e., dβ⁄dt is expressed by
angular velocity denoted by ωV:
(8–5)
where IV denotes satellite’s moment of inertia and τ
denotes the torque applied to the satellite. This torque
comes from both the outside as a disturbance torque and
from the inside by the CMG system. The vector p is the
total angular momentum of both the satellite and the
CMG system and is given by:
p = IV ωV + HCMG .
(8–6)
This total angular momentum is conserved in the
inertial coordinates if there is no disturbance torque. By
substituting Eq. 8–6 into Eq. 8–5, the ωV×HCMG term
appears. This term is omitted in Eq. 3–5, i.e., the output
equation of the CMG but is evaluated here. Both the
kinematic equation and the dynamic equation, Eqs. 8–3
and 8–5, are the describing functions of the system.
8.1.2 Exact Linearization
The system has six independent variables, three
components of β* and three components of ωV. The
dynamics is nonlinear as seen by the term ωV×(IVωV)
when Eq. 8–6 is substituted into Eq. 8–5. Nevertheless,
it is well known that this nonlinear system can be exactly
linearized with a suitable feedback71, 72).
Let (β*, dβ* ⁄d t ) be state variables and let v be a
new input variable. If a real inputτis given as:
––– 57 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
τ = ωV×IVωV
Support Rod
+ (2 ⁄ β0) IV(β02µ + β*β*tµ − β0β*×µ) ,
AA
A
A
AA
Rate Gyroscope
Three Axis
Gimbal
(8–7)
where
g3
µ = v + 1 ⁄ 4 ωVt ωV β* ,
CMG3
(8–8)
the system becomes a set of second order linear systems
described by:
d2β* ⁄ dt 2 = v .
750mm
g1
g4
Several controllers given by linear control theory can
be applied to the above linearized system. In the ground
tests done in the experimental portion of this work, a
model matching controller and a tracking controller were
used. Design of the model matching controller was made
by applying a model transfer function in order to satisfy
specified steady and transient properties54).
The tracking controller allowed an appropriate
motion of the angular momentum vector to be designed.
Tracking PD control of a given trajectory is realized by
the following input:
v = f1 ·(β* – r) + f2 ·(dβ* ⁄ dt – r1) + r2 ,(8–10)
where r(t) is the trajectory to be followed and r1 and r2
are its time derivatives, defined by:
(8–11)
Details of both are described in Appendix E.
8.2
CMG2
(8–9)
8.1.3 Control Method
r1(t)=dr ⁄ dt , r2(t)=d2r ⁄ dt2 .
g2
Experimental Facility and
Procedure
In order to quantitatively demonstrate the problems
and performance of steering laws, a ground test facility
was constructed54, 55) and a set of ground tests was
carried out.
8.2.1 Facility
The ground test facility shown in Fig. 8–1 simulates
the attitude dynamics of a spacecraft. The main structure
is a cubic frame made of steel pipes and joints. Triangular
plates on several surfaces holds such devices as the CMG
units, the CMG driver circuits, balance adjusters and a
computer. The system in its entirety was suspended from
Onboard
Computer
Balance
Adjusters
h4
z
CMG1
Rotary
Encoder
y
x
CMG4
Fig. 8–1
Experimental test rig showing the
center-mount suspending mechanism.
This
figure
shows
that
the
pyramid
configuration can be realized so that all four
units fit the
parallelepiped.
surfaces
of
a
rectangular
the ceiling by a three axis gimbal mechanism in its center.
If the center of the gimbal coincides with the body’s
center of gravity, no torque appears at any orientation
due to the gravitational force. In this way motion in
space can be simulated. This situation was realized by
using three balance adjusters, which could control their
weight along three orthogonal axes, which allowed the
center of gravity to be controlled. These mechanisms
were also used for initial set up without CMG control,
and generation of disturbance torque and unloading.
The orientation and angular rate of the body were
measured by rotary encoders at the three axis gimbal
and rate gyroscopes. In actual satellites these quantities
are measured by various sensors such as star/sun/earth
sensors and rate gyroscopes.
The main torquer was a pyramid type single gimbal
CMG system. All attitude control and the steering law
processes were installed in an onboard computer. A
wireless link was used for command transfer from the
stationary computer. All power was supplied from the
laboratory by a pair of thin wires which caused little
disturbance force. Additional details are presented in
Appendix E.
––– 58 –––
–– 8. Ground Experiments ––
Rotational angle φ
8.2.2 Design of Control Command Sequence
Figure 8–2 shows a typical tracking control
trajectory. This function is continuous with regard to
the first time derivative. It consists of a constant
acceleration, a constant speed rotation, a constant
deceleration, a constant attitude and then the same
sequence in reverse.
The maximum rotation of this trajectory was set in
consideration with the limit of rotation of the supporting
gimbal. The magnitude of the acceleration and the
deceleration which are almost proportional to the CMG
output torque was set as large as possible so that the
friction torque of the supporting gimbal can be neglected.
(3)
(4)
(5)
(6)
(2)
(7) (8)
(1)
t1
t2
t1
t3
t1
t2
t1 t3
time
Fig. 8–2 Target trajectory.
This trajectory has eight parts as,
(1) constant acceleration by d2 φ⁄dt2 = a,
(2) constant rate rotation, dφ⁄dt=at1 ,
(3) constant deceleration by d 2φ⁄dt2 = −a,
(4) pointing control at φ =at1 2+at1t2 ,
(5) to (8) are the reverse of (1) to (4).
8.2.3 Experimental Procedure
Tests were conducted using the software whose block
diagram is shown in Fig. 8–3. Additional details of each
block are described in Appendix E. The control
command sequence for each experiment was a sequence
of a number of maneuver motions given either as
reference attitude or as a trajectory in Fig. 8–2. After
each maneuver, the attitude returned to the original
position and rotation of the body ceased.
To allow comparison, three types of steering laws
were tested: a gradient method (abbreviated to GM
hereafter), a SR inverse method (abbreviated to SR) and
the constrained method (abbreviated to CM) proposed
in Chapter 7. The GM uses procedure described in
Section 3.4.1. Its objective function is det(CCt) and the
free parameter is the gain k defined in Eq. 3–21. The
Reference Attitude
or
Target Trajectory
Attitude
Command
Generator
Torque
Command
T COM
Attitude
Controller
SR is represented by Eq. 3–23. The CM method was
introduced in Section 7.5 and allows one free parameter,
a feedback gain denoted by k. Further details of the GM
and the CM implementations are given in Appendix E.
The test procedure can be outlined as follows: First,
the body was controlled at a nominal attitude of β*=(0,
0, 0)t by using only the balance adjusters along with the
PID controller. This control mode was done in order to
wait until the pendulum motion of the body and the
supporting rod stopped. Then the CMG control started
with a sequence of attitude commands consisting of
either an attitude reference in the model matching
controller case, or a trajectory in the tracking controller
case. The balance adjusters were controlled so that they
generated an expected disturbance torque during the
experiment, but which was zero otherwise.
Momentum and
Disturbance
Management
Torque
Output
T
Desired
Motion
ω
CMG
Steering
Law
Balance Adjusters
Proportional
Limiter
Pyramid
Type
CMG
System
Body
Attitude and Rotational Rate
Fig. 8–3 Block diagram of the control system.
Proportional limiter is used to limit the gimbal rate, ωcom, to the maximum gimbal
rate so that the real rate vector is proportional to the desired vector. By using this,
the real output becomes proportional to the torque command, i.e., Tcom ⁄⁄ T.
––– 59 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
8.3 Experimental Results
The experiments were carried out to demonstrate the
following problems and performance characteristics of
the CMG steering laws69).
(1) Performance and problems of a singularity
avoidance steering law such as SR-inverse
law.
(2) Problem of z axis rotation and advantage of
preferred gimbal angles
(3) The gradient method’s inability to keep a
nominal condition
(4) Performance in various modes of constrained
control
(5) Advantages of mode switching.
The results of the following experiments are drawn
in five graphical forms with time on the horizontal axis.
The first graph shows the ideal and measured attitude
variation on the vertical axis. Other variables on the
vertical include the measured gimbal angles, the output
torque level and the angular momentum of the CMG
system. Also shown is the determinant det(CCt), which
provides information regarding the proximity of the
system to a singular point. Note that the output torque
T here is an actual value with the multiplier h but the
angular momentum H is still without the multiplier (see
Section 3.1).
The third experiment was made from another initial
gimbal angles, which are one of the preferred angles for
this direction. The results in Fig. 8–6 shows that this
initial angles are appropriate for this situation.
Table 8-1 Condition and Results of Experiments (1)
Experiment
Initial θ
Steering Results
law
——————————————————————
Experiment A ( 0, 0, 0, 0)
GM
Fig. 8–4
Experiment B ( 0, 0, 0, 0)
SR
Fig. 8–5
Experiment C (−π/3, π/3, −π/3, π/3) GM
Fig. 8–6
——————————————————————
8.3.1 Attitude Keeping under Constant Disturbance
By the following three experiments, the item (1) was
demonstrated. The H path along (−1, 1, 0)t direction
was planned and pointing control under the constant
disturbance about this direction was carried out by the
model matching controller. The conditions of the three
experiments are listed in Table 8–1.
The reference attitude and the disturbance were kept
constant and the angular momentum of the CMG system,
i.e., H continuously increased along (−1, 1, 0)t direction
when the pointing control was successful.
The first two experiments clarifies the impassable
singularity problem of the gradient method and
performance of singularity avoidance by the SR-inverse
steering law. By the GM, the system became singular
(AB in Fig. 8–4 (f)), after that no torque was generated
and the body rotated by the disturbance torque (AA in
Fig. 8–4 (a)). On the contrary, the SR worked well with
slight degradation of pointing accuracy (BA in Fig. 8–
5).
––– 60 –––
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β
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
β2
β3
0
(a) attitude
2
θ
1
0
-1
10
T (N m)
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0
-0.5
-1
H
1
0
-1
-2
det
1
θ1
θ2
10
Tcom
time(s)
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30
time(s)
20
30
20
30
20
30
Tout
Hy
Hz
Hx
0
10
(d) CMG momentum (normalized)
1.5
30
θ3
0
10
(c) torque command & output
2
20
time(s)
θ4
0
(b) gimbal angle
1
β1
AA
time(s)
AB
0.5
0
0
(e) determinant
10
time(s)
Fig. 8−4 Results of Experiment A.
The attitude keeping by the gradient method under constant disturbance
torque about (−1,1,0)t direction from initial θ of (0, 0, 0, 0).
––– 61 –––
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–– Technical Report of Mechanical Engineering Laboratory No.175 ––
0.01
β
0.005
0
β2
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0
(a) attitude
2
θ
1
0
-1
T (N m)
-2
0
(b) gimbal angle
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0
-1
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-3
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β3
β1
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time(s)
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1
0
-1
-2
θ1
θ2
θ3
10
time(s)
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1
0.5
0
0
(e) determinant
20
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Tout
Tcom
time(s)
20
30
Hy
Hz
Hx
0
10
(d) CMG momentum (normalized)
1.5
30
θ4
0
10
(c) torque command & output
2
20
10
time(s)
time(s)
20
30
20
30
Fig. 8−5 Results by Experiment B.
The attitude keeping by the SR method under constant disturbance torque
about (−1,1,0)t direction from initial θ of (0, 0, 0, 0).
––– 62 –––
–– 8. Ground Experiments ––
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0.00125
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0
β2
-0.00125
β3
β1
-0.0025
0
(a) attitude
2
1
θ
0
-1
-2
-3
0
(b) gimbal angle
T (N m)
1.5
1
0.5
0
10
20
time(s)
30
θ2
θ1
10
time(s)
Tcom
θ4
θ3
20
30
20
30
Tout
-0.5
-1
0
(c) torque command & output
H
3
2
1
0
-1
-2
10
Hy
Hz
Hx
0
10
(d) CMG momentum (normalized)
2
det
1.5
1
0.5
0
0
(e) determinant
time(s)
10
time(s)
time(s)
20
30
20
30
Fig. 8−6 Results of Experiment C.
The attitude keeping by the gradient method under constant disturbance torque
about (−1,1,0)t direction from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3).
––– 63 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
8.3.2 Rotation About the z Axis
By the next three experiments, the above item (2)
was demonstrated. The H path along the z axis was
planned which is symmetric to the H path of the previous
three experiments. This time, maneuvering motion was
performed. For attitude control about the z axis, the H
trajectory is also on the z axis. This H path intersects a
singular surface. There is no singularity problem from
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0.1
0
β1, β2
DA
β
-0.1
β3
-0.2
Reference
-0.3
0
(a) attitude
θ
1
0
the nominal θ = (0, 0, 0, 0), while there is a problem of
impassability from another initial θ. The conditions of
the three experiments are listed in Table 8–2.
In the experiments, a command sequence consisting
of two maneuver motions was used when the model
matching controller was operating. The reference
attitude to the controller was changed twice, at first to
an orientation rotated 30 degrees about the z axis for t ≤
10 seconds then to the initial orientation for 10 seconds
10
time(s)
20
10
time(s)
20
-1
T (N m)
0
(b) gimbal angle
20
0
Tcom
Tout
-20
0
(c) torque command & output
2
10
time(s)
20
10
time(s)
20
10
time(s)
20
Hz
H
DB
0
Hx Hy
-2
0
(d) CMG momentum (normalized)
2
det
1.5
1
0.5
0
0
(e) determinant
Fig. 8−7 Results of Experiment D.
The z-axis maneuver from initial θ of (0, 0, 0, 0) by the gradient method.
––– 64 –––
–– 8. Ground Experiments ––
≤ t.
Table 8-2 Condition and Results of Experiments (2)
As the initial gimbal angles of Experiment D is
preferred gimbal angles for this direction, the system
did not approach any singular point as shown in Fig. 8–
7 and smooth maneuvering was performed as is the
analytical result of Section 7.1. On the contrary, in
Experiment E, the CMG system became singular at t ≈
2.3 second (EC in Fig. 8–8 (e)). For this reason, GM
Initial θ
Steering Results
law
——————————————————————
Experiment D ( 0, 0, 0, 0)
GM Fig. 8–7
Experiment E (−π/3, π/3, −π/3, π/3) GM* Fig. 8–8
Experiment F (−π/3, π/3, −π/3, π/3) SR Fig. 8–9
——————————————————————
(GM* is a modified GM)
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Experiment
β3
EA
β1, β2
-0.2
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-0.4
Reference
0
(a) attitude
4
θ4
θ
20
10
time(s)
20
θ2
2
θ3
0
θ1
-2
0
(b) gimbal angle
T (N m)
10
time(s)
Tcom
20
Tout
0
-20
0
(c) torque command & output
20
10
time(s)
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10
time(s)
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Hz
2
H
10
time(s)
EB
0
Hx Hy
-2
0
(d) CMG momentum (normalized)
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0.5
0
0
(e) determinant
Fig. 8−8 Results of Experiment E.
The z-axis maneuver from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3) by the modified gradient
method.
––– 65 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
was modified so that the transpose Jacobian was used
on behalf of the pseudo-inverse Jacobian when the
system was nearly singular. By this, the above singular
state was avoided but the element of H to the z axis, i.e.,
H z once saturated at smaller value than that of the
maximum in Experiment D (EB in Fig. 8–8 (d) and DB
in Fig. 8–7 (d)). As the result, the transient motion of
the control was degraded compared with the Experiment
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0.1
0
β
-0.1
-0.2
-0.3
β1, β2
FA
β3
Reference
-0.4
0
(a) attitude
4
θ
0
10
time(s)
20
10
time(s)
20
θ2
2
θ4
θ3
θ1
-2
0
(b) gimbal angle
T (N m)
D (EA in Fig. 8–8 (a) and DA in Fig. 8–7 (a)).
This time, the SR was not effective as shown in Fig.
8–9. The CMG system approached a singular point
similar to the above experiment (FC in Fig. 8–9 (e)).
Moreover, no singularity avoidance motion was realized
because the command changed faster than that in
Experiment B.
20
Tout
Tcom
0
-20
0
(c) torque command & output
10
time(s)
20
10
time(s)
20
10
time(s)
20
2
H
1
0
-1
FB
Hz
Hx Hy
-2
0
(d) CMG momentum (normalized)
2
det
1.5
1
FC
0.5
0
0
(e) determinant
Fig. 8−9 Results of Experiment F.
The z-axis maneuver from initial θ of (−π ⁄ 3, π ⁄ 3, −π ⁄ 3, π ⁄ 3) by the SR
method.
––– 66 –––
–– 8. Ground Experiments ––
Two experiments were carried out, i.e., Experiment
G by CM and Experiment H by GM for the same control
sequence. The results of these are shown in Fig. 8–10
and Fig. 8–11. The control sequence consists of five
parts. The reference orientation for the regulator is
shown in Fig. 8–10(a). The first part of the sequence is
a maneuver about the z axis, shown as the part A in Fig.
8–10(a). This command is similar to that of Section
8.3.2. In the second part B, a disturbance torque was
8.3.3 Maneuver after Momentum Accumulation
In Section 7.1, it was concluded that a gradient
method cannot always keep the nominal θ given by Eq.
7–2 after H travels around and back to the origin. In
order to demonstrate this problems, i.e., the above item
(3), a control sequence including maneuvering motions
as well as momentum accumulation was used while the
system was controlled by the model matching controller.
0.1
β
0
-0.1
-0.2
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(a) attitude
1
θ
0.5
0
-0.5
-1
-1.5
0
(b) gimbal angle
T (N m)
20
10
0
-10
-20
B
C
D
β3
reference(β
reference( 3)
20
GA
40 time(s) 60
H
1
0
-1
-2
det
1
0.5
0
0
(e) determinant
100
θ4
θ2
20
GB
θ1
40
time(s)
60
Tcom
Hx
Hy
20
80
100
80
100
80
100
80
100
Tout
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GC
0
20
40
60
time(s)
(d) CMG momentum (normalized)
1.5
80
θ3
0
20
40 time(s) 60
(c) torque command & output
2
E
40 time(s) 60
Fig. 8−10 Results of Experiment G. Control by the proposed constrained method.
––– 67 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
applied to move H to (0.6, 0.4, 0.)t while the body’s
orientation was fixed (see GC in Fig. 8–10(d)). This
motion is similar to that of Section 8.3.1. The third part,
indicated by C, is the same as the first part. Then in the
part D a reversed disturbance was applied to move H
back to its origin similarly to the part B. Both processes
B and D resulted in a similar H path as described in
Section 7.5.2. Finally, the same maneuver as the part A
was tried (E in Fig. 8–10(a)).
Those results in Fig. 8–10 and Fig. 8–11 show that
the motions of the first three parts of the two experiments
are almost the same. This implies that both the steering
laws have similar control performances. In the fourth
part, however, motions of θ are different. The gimbal
angle θ returned to the original in the case of CM (GB
in Fig. 8–10) but to the different point in the case of GM
(HB in Fig. 8–11). This is because GM controlled the
CMGs to another local maximum of det(CC t ) as
described in 7.5.2. As the result of this, the last part was
successful in the case of CM (GA in Fig. 8–10), while it
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A
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C
β
β3
-0.1
-0.2
reference
0
(a) attitude
4
θ
2
0
20
θ3
HA
40
0
(b) gimbal angle
20
10
0
time(s)
60
80
θ4
100
HB
θ2
-2
T (N m)
E
D
θ1
20
40
time(s)
60
80
100
80
100
Tout
Tcom
-10
-20
0
20
(c) torque command & output
2
H
1
0
40
Hx
-1
time(s)
60
Hz
HC
Hy
-2
0
20
40
(d) CMG momentum (normalized)
time(s)
60
80
100
1.6
det
1.2
0.8
HD
0.4
0
HE
0
(e) determinant
20
40
time(s)
60
80
100
Fig. 8−11 Results of Experiment H. Control by the gradient method.
––– 68 –––
–– 8. Ground Experiments ––
0.8, 0.)t and finally in the part G the maneuver about the
(1,1,0)t direction was repeated.
The results are shown in Fig. 8–13. The first three
parts A, B and C, show almost the same motion as in the
previous experiment. In the part C there was a slight
degradation of control compared with the response in
the part A.
In the part D, the mode of the CM was changed
directly, without considering the attitude control and by
using the fastest direct path to the other mode as
described in Section 7.5.5 (4). This motion inevitably
generated an undesired torque and there was some
attitude deviation of the body, as indicated by JA in Fig.
8–13. However, because the CMG was moved as fast
as possible, this deviation was somewhat minimized.
The attitude control immediately following this motion
easily corrected such a deviation.
As the theory in item (4) of Section 7.5.5 predicts,
the two maneuver motions in the M2 mode, shown in
the parts E and G of Fig. 8–13, were successful and did
not meet a singularity.
This experiment demonstrated the advantages of the
constrained method proposed in the previous chapter.
In addition, trajectory tracking control was used this time.
The results in Fig. 8–13(c) show that almost constant
torque was realized during the period of constant
acceleration or deceleration. The proposed method can
also cope with a change of maneuver direction by
switching between modes. Even by using a direct change
of the mode, deviation in attitude can be made small
enough to be corrected by the attitude control.
was not in the case of GM (HA in Fig. 8–11). In the
case of GM, the determinant went to zero and H saturated
(HC in Fig. 8–11) by hitting or approaching some
singular surface (HD and HE in Fig. 8–11).
These two results clarifies that the unique inversion
characteristic of CM is important even for such a simple
maneuver.
8.3.4 Mode Selection and Switching
The next test was carried out using only the CM in
order to demonstrate the above items (4) and (5). The
different modes of the constrained method have different
workspaces of H, as described in Section 7.5.4(4). In
Experiment J, a maneuver motion resulting in two kinds
of the H paths were planned for which different control
modes were required and mode switching was
performed.
In this experiment, trajectory tracking control was
used for the attitude control. The target trajectory for
this tracking control is shown in Fig. 8–12. This consisted
of the following motions: A maneuver about the z axis
shown by the part A in Fig. 8–12 was planned by a
trajectory defined in Section 8.2.2. After this maneuver,
a disturbance was applied in the part B so that the angular
momentum was accumulated to H=(0.5, 0.5, 0.)t. Then
in the part C, the z axis maneuver was repeated. As the
H path for these three parts are in the workspace of the
M1 mode, this mode is selected. The maneuver
command of the latter parts was designed so that the
resulting H path went out of the M1 mode workspace
but was inside the M2 mode one. For this reason, mode
switching in the part D was conducted before the next
maneuver. Then, the new principal axis was in the
(1,1,0) t direction. In the part E, a maneuver was
performed about this new direction, then in the part F
another disturbance was applied to accumulate H to (1.2,
8.4 Summary of Experiments
From experiments A to F, it is observed that an
appropriate combination of the initial gimbal angles and
the maneuver direction (or momentum accumulation
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E
F
β1, β2
G
β3
β1, β2
-0.1
0
20
40
60
time(s)
80
100
120
Fig. 8−12 Command sequence of Experiment J.
Maneuver motions in A and C are the same rotations about the z axis. Maneuver motions in E and G
are the same rotation about the (1,1,0)t axis. In periods B and F, a disturbance torque was applied so
that the final H becomes (0.5,0.5,0.)t in B and (1.2, 0.8,0.)t in F. The CM mode is changed at D.
––– 69 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
proposed constraint method. This method not only
preserves the merit of the above preferred gimbal angle
method but also realizes continuous steering and
repeatability. Though the workspace of each mode is
restricted, it can maintain nearly continuous steering for
various directional maneuver and/or momentum
accumulation events by using the proposed mode
switching operation. This was successfully demonstrated
by Experiment J.
direction) is the most important. This implies that the
method of ‘preferred gimbal angles’ described in Section
7.4 is effective. Though the experiments showed the
capability of singularity avoidance of the SR inverse
method, its performance was worse than that when the
initial gimbal angles were set appropriately. Moreover,
its performance of singularity avoidance depends on the
speed of momentum accumulation and it sometimes fails.
Experiments G and H clarified the advantage of the
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E
G
F
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JA
0.1
0
-0.1
0
(a) attitude
20
40
60
time(s)
80
100
120
2
θ
1
0
θ1
θ3
θ4
-1
θ2
-2
-3
0
20
(b) gimbal angle
40
60 time(s) 80
100
120
60 time(s) 80
100
120
60 time(s) 80
100
120
60 time(s) 80
100
120
T (N m)
20
10
Tcom
0
H
Tout
-10
0
20
40
(c) torque command & output
3
2
1
0
-1
-2
-3
Hx
Hz
Hy
0
20
40
(d) CMG momentum (normalized)
1.6
det
1.2
0.8
0.4
0
0
20
(e) determinant
40
Fig. 8−13 Results of Experiment J. Tracking control for the command given in Fig. 8−7,
illustrating use of the proposed constrained method.
––– 70 –––
–– 9. Evaluation ––
Chapter 9
Evaluation
The previous chapters dealt mainly with a specific
pyramid type CMG system. In order to evaluate its
performance, comparison with regard to the workspace
and weight was made for various system configurations.
9.1 Conditions for Comparison
The previous chapters revealed that it is generally
difficult to avoid impassable singularities. Most steering
laws have various problems. The only exception is the
constrained steering law proposed in Section 7.5, whose
performance is verified within a certain workspace. But
this method is only effective for the pyramid type CMG
system, and if another configuration is used, a gradient
method is the only candidate.
Thus, evaluation of various systems was made under
the assumption that a gradient method is used and the
work space is determined so that it does not include any
impassable surfaces.
S(4), S(6) and S(10). (b) Skew Type of 5 and 6 units,
denoted by Skew(n). (c) Multiple Type M(m,m) and
M(m,m,m) with orthogonal gimbal axes.
In addition to those, the following systems were
selected for comparison. A system denoted by
2×Skew(n) is a doubled skew configuration. The system
denoted by 1+Skew(n) and shown in Fig. 9–1 (a)
indicates that one unit is added to the symmetric axis of
a skew type system. The system denoted by S(3,4) is a
combination of two symmetric configurations, S(3) and
S(4) as shown in Fig. 9–1 (b). The units are arranged in
the surface directions and the vertex directions of a
regular octahedron.
g4
g5
9.2 Spherical Workspace
g6
g3
α
gn+1
2π⁄n
g2
Several CMG systems were examined including
double gimbal CMGs32, 73) with the following criteria
r and χ :
g1
(a) Example of 1+Skew(n).
r:
χ:
Maximum radius of a sphere in the angular
momentum space, centered on the H origin, and
including no impassable surfaces.
= r ⁄n.
Obviously, r = n for all double gimbal CMG systems
because their work space is a unit sphere of radius n
(Appendix A). The radius of any multiple type system
is obtained simply, because its envelope has circular
plates which the maximum sphere touches. Thus, the
radius r of M(m, m) with orthogonal axes is m and that
of M(m, m, m) with orthogonal axes is 2m.
Various CMG configurations of up to ten units were
examined. These included: (a) Symmetric Type , S(3),
g7
g3
g4
g6
g2
g1
(b) S(3, 4).
g5
Fig. 9−1 System configurations for comparison.
––– 71 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
r : Radius of Spherical Workspace
10.
2×SKEW(5)
5
0.7
χ=
2×S(4)
S(6)
1+SKEW(6)
SKEW(6)
7
0.6
χ=
S(3,4)
M(2,2,2)
S(10)
Double
Gimbal
CMG
5.
M(3,3,3)
M(5,5)
1+SKEW(7)
Envelope
of S(4)
0.
0
M(4,4)
M(3,3)
1+SKEW(4)
SKEW(5)
S(4)
5
10
n : Number of Units
Fig. 9−2 Spherical workspace size for various system configurations.
Filled circles indicates the workspace size of the original system, while attached open circles
indicate the workspace size of degraded systems. As a reference, a square indicates the
envelope size of the S(4) system.
The radius r was commonly obtained by computer
calculation. It was searched by calculating |H| for
impassable surfaces of all lattice points of u on the unit
sphere at a given increment. In the case of skew type
system, calculations were made using various values of
the skew angle and the maximum radius r is sought. The
various possibilities of unit’s breakdown are also
considered and the worst cases are taken, except in the
case of multiple systems for which all possible cases are
considered.
The results are shown in Fig. 9–2 as a graph of the
number of units n versus workspace radius r. The ratio
χ is the slope of the line from the origin. Filled circles
indicate values of original systems, while blank circles
connected to them by straight lines indicate the
performance of degraded systems. Conclusion from
these results are as follows:
a) As the number of units increases, the shape of the
angular momentum envelope approaches a sphere and
χ also approaches a limiting value given by:
∫ (h S ⋅ u)dS
χ∞ =
u ∈S 2
∫ dS
u ∈S 2
= ∫ π0⁄ 2 sin 2 φdφ
=π/4
≈ 0.765 ,
(9–1)
which is for an infinite number of units arranged equally
in all directions.
b) Most systems of no less than six units have
respectable χ values ranging from 0.67 to 0.75. This is
because although such systems have internal impassable
surfaces, they are only near the envelope. On the
contrary, four and five unit systems have smaller χ values
because they have internal impassable surfaces much
further inside.
c) Although any multiple type system composed of
no less than six units has no internal impassable surfaces,
its radius r is considerably smaller than that of other
independent systems.
d) Degradation of the system due to unit’s break
down becomes smaller as the number of units increases.
9.3 Evaluation by Weight
The workspace size and system weight will be
evaluated in light of the preceding results. Suppose that
the work space size H and system weight W satisfy the
similar relation as Eqs. 2–1 and 2–2, which are W ∝ n
d3 and H ∝ r d5 where d is the diameter of the flywheel.
Then, the following relationship is obtained by setting
––– 72 –––
–– 9. Evaluation ––
Z
W as a parameter while H is set constant:
r 3 W 5 ⁄ n 5 = constant .
(9–2)
Figure 9–3 shows results of this comparison. Dotted
curves indicate the relationship of n and r which satisfy
the condition 9–2. While the W = a curve passes the
S(6) point, the other points are under this curve. This
implies that the S(6) is the lightest for the same spherical
workspace size among all systems evaluated above. As
the W = 1.5a curve passes the S(4) point, the S(4) system
is 50% heavier than the S(6) system.
The data point labeled Skew(4, αopt) shows the
workspace of a skew type four unit system with the skew
angle α = tan-1(1 ⁄2) described in Section 7.5.4(5). As
the W = 1.15a curve passes this data point, this particular
system with the proposed steering law can realize
Skew(4) system only 15% heavier than S(6).
9.4 Ellipsoidal Workspace
An ellipsoidally shaped workspace may be required
when the attitude control has a certain principal axis. In
this section, skew type CMG systems of 4, 5 and 6 units
are evaluated in terms of their workspace size. Systems
of more than 6 units are omitted because they have
disadvantages in weight as the results above.
An evaluation similar to that done in Section 9.2 was
made under the same condition that the workspace does
not include any impassable surface. The shape of the
workspace is defined axially symmetric with a fixed
aspect ratio µ as shown in Fig. 9–4. Evaluation criteria
r : Radius of Spherical Workspace
10
r 3 W 5 ⁄ n 5 = const
W =a
W = 1.15a
W = 1.5a
S(6)
5
SKEW(5)
S(4)
SKEW(4, αopt)
0
0
2
4
6
8
10
n : Number of Units
Fig. 9−3 Trade-off between workspace size and
system weight.
Dotted curves indicate equal workspaces with
equal weight (W).
r2 = µr1
r1
r1
X
Y
Fig. 9−4 Definition of ellipsoidal workspace.
Aspect ratio µ is defined as r2 ⁄r1 .
r1 and r2 (=µr1) are the minimum and maximum radii
of the ellipsoid. Skew type systems of 4, 5 and 6 units
were examined at various skew angle values. As we
have two parameters, i.e., the skew angle α and the aspect
ratio µ, comparison will be made by keeping one
parameter constant.
By keeping the aspect ratio constant, the radii r1 and
r2 were calculated with respect to the skew angle α as
shown in Fig. 9–5. In these figures, the radii are
represented by an average radius defined as follows;
rA =
3 r 2r
1 2
.
(9–3)
This value represents the radius of a sphere which has
the same volume as the ellipsoid.
For four unit systems, each resulting curve has a
maximum at a different skew angle, as shown in Fig. 9–
5 (a). Noteworthy is the system corresponding to the
maximum of the µ=1.0 case, the S(4) symmetric type
system. For the other aspect ratios, the optimum skew
angle increases as the ratio increases. The results of
five unit skew systems are different. Though there are
local maxima, the global maxima for any aspect ratio
are given when the skew angle is π ⁄2. At this skew
angle, all gimbal axes are on the same plane.
In the case of six unit systems, we can find two groups
of candidates result in the largest workspace, one has a
skew angle of π ⁄2 and the other 0.26≤α≤0.33.
Next, the radii r1 and r2 were calculated with respect
to the aspect ratio by keeping the skew angle constant
(Fig. 9–6). In each figure, the maximum and minimum
radii are plotted with respect to the aspect ratio, for
various skew angles. Fig. 9–6 (a), for the Skew(4)
arrangements, shows that there is an optimal skew angle
––– 73 –––
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µ=2.0
µ=2.5
µ=1.0
1
0
0.1
0.2
0.3
0.4
Skew Angle α (radian)
3
radius r
2
µ=1.0
1
µ=1.5
0
0.1
(b) Skew(5, α)
0.2
0.3
0.4
Skew Angle α (radian)
4
µ=1.0
2
µ=2.0
0
0.1
0.2
µ=1.5
0.3
0.4
Skew Angle α (radian)
1.5
2
2.5
Aspect Ratio µ
3
(a) Skew(4, α)
6
0.45
r2
α=0.5
0.4
4
2
r1
01
1.5
2
2.5
Aspect Ratio µ
3
(b) Skew(5, α)
0.5
6
0.5
0.4
0.45
r2
α=0.32
0.26
4
2
r1
(c) Skew(6, α)
01
1.5
2
Aspect Ratio µ
2.5
3
(c) Skew(6,α)
giving the largest workspace for each value of the aspect
ratio. For example, the optimal skew angle is 0.35 π at
an aspect ratio of 1.75. On the contrary, a π ⁄2 skew
angle is optimum for any aspect ratio in the case of
Skew(5). For Skew(6), a 0.32 π skew angle is optimum
for aspect ratio less than 1.4, while π ⁄2 is optimum for
larger aspect ratios. These optimum values are selected
and plotted in Fig. 9–7.
Radius values in this figure can be converted to
indicate the system weight as discussed in Section 9.3.
This ‘converted weight’ W is equivalent to the weight of
a system whose workspace size is a certain fixed value.
By a relationship similar to Eq. 9–2, this converted
weight is defined as follows;
Fig. 9−6 Workspace radius as a function of
aspect ratio.
radius r1 and r2
Fig. 9−5 Average radius vs. skew angle.
W = n ⁄(r12r 2) −1 / 5 .
0.475
r1
01
0.5
µ=2.5
0.45
0.5
0.5
µ=2.0, 2.5
2
r2 0.4
0.375
0.35
1
radius r
Average Radius rA
α=0.3
0.325
1.5
µ=1.5
(a) Skew(4, α)
Average Radius rA
2
radius r
Average Radius rA
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
α=0.5
6
α=0.32
Skew(6, α)
Skew(5, π ⁄ 2)
r2
4
r1
2
0
α=0.3
1
M(2, 2)=Roof(4)
Skew(4, α)
0.35
0.325
1.5
0.375
2
2.5
0.4
3
Aspect Ratio µ
Fig. 9−7 Combined plot of radii as a function of
aspect ratio.
Envelope size of multiple systems, M(2, 2), is
drawn in addition to the results in Fig. 9−6.
(9–4)
The results are plotted in Fig. 9–8. The converted
weight of the Skew(4, α) system, controlled by the
proposed constrained method, is also included. In this
evaluation, the workspace of this system is approximated
by an ellipsoid whose radii are given by 2cosα and 4sinα
(See Fig. 7–19). The results in Fig. 9–8 show that the
weight is much larger in the case of the original Skew(4)
system than the other systems. All the other systems,
including the 4-unit skew system with the constrained
steering law, have similar weight values.
––– 74 –––
–– 9. Evaluation ––
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Skew(6, π ⁄ 2)
Skew(6)
Skew(5)
Skew(4)
Skew(4)’
2
01
1.5
2
Aspect Ratio µ
2.5
3
radius r
Weight
4
The next figure, Fig. 9–9, shows a relation similar to
Fig. 9–7, when one unit becomes nonfunctional. This
figure shows that degradation of the Skew(6) system is
much less than the others. On the contrary, degradation
of the Skew(4, α) with the constrained method is serious.
9.5 Summary of Evaluation
(1) For the spherical workspace, the S(6) system is
superior in terms of the system weight. The Skew(4)
system driven by the constrained method is only 15 %
heavier in the simplified comparison.
(2) For the ellipsoidal workspace, skew type systems
Skew(5, π ⁄ 2)
Skew(4, α)
1
01
3
Fig. 9−8 Converted weight as a function of
aspect ratio.
The original Skew(4) system is shown by the
solid line and the Skew(4) system controlled by
the proposed constrained method is shown by
the heavy dashed line of Skew(4)'.
Skew(4, 0.4 π )
2
1.5
2
Aspect Ratio µ
2.5
3
Fig. 9−9 Radius as a function of aspect ratio
for a degraded system with one faulty unit.
of 4, 5 and 6 units with optimal skew angle, and with
constrained control in the case of the 4 unit system, have
similar workspace size with the same weight.
(3) If fault tolerance is required, the Skew(6) system
is much better than Skew(4) and Skew(5) in terms of
degradation of the workspace due to loss of one unit.
In the evaluation of this chapter, the three modes of
a symmetric pyramid type system were not considered.
Since the workspace of each mode is a similarly shaped
ellipsoid, this pyramid type system becomes more
promising by considering the three modes. In addition
to this, other factors are also important, such as
mechanical complexity and steering law complexity. By
considering these, the Skew(4) system with the proposed
constrained method is advantageous for actual use,
especially the S(4) system.
––– 75 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 76 –––
–– 9. Evaluation ––
Chapter 10
Conclusions
In this paper, the control of a single gimbal CMG
system has been investigated, with emphasis on the
symmetric pyramid type. Specifically, the singularity
problems have been examined using geometric theories
and computer calculations. The “global” problem of the
pyramid type system has been clarified, and a new
steering law approach has been proposed and verified
using ground experiments.
In Chapter 2, single gimbal CMGs were described in
comparison with double gimbal CMGs and reaction
wheels. Then in Chapter 3, an analytical formulation of
general single gimbal CMGs was presented.
In Chapter 4, the singularity problem was described.
Methods for obtaining singular surfaces, especially the
envelope, were presented. The passability of a singular
surface was defined. Then, examples of some impassable
surfaces of various CMG systems were given. The
results showed that impassable singularity is a serious
problem for the steering law of 4 and 5 unit CMG
systems. In Chapter 5, continuous steering under the
existence of an impassable singular surface has been
generally examined by using a topological study. A
method to overcome some types of impassable
singularities were described in a geometric manner.
Some example conditions were presented in which no
steering law can realize continuous motion.
In Chapter 6, the symmetric pyramid type CMG
system was defined. Analytical results including
symmetry and singular surface structure were presented.
Chapter 7 clarified the global steering problem that
continuous real-time steering cannot be realized over
most of the workspace. By the consequences of this
and by geometric theories, typical steering laws were
evaluated. This evaluation showed that steering law
exactness is the most important. An alternative steering
law was proposed which maintains exactness but which
is valid in a restricted workspace. Then in Chapter 8,
this proposed method was evaluated using ground
experiments. First, the problems described in Chapter 7
were demonstrated. Then, the proposed method and
other steering laws were tested using some attitude
control sequences. The performance of the proposed
method was verified, especially for a realistic sequence
including maneuvering and pointing under a specified
directional disturbance.
In Chapter 9, a pyramid type system with the proposed
steering law was compared with other types of CMG
systems. Evaluation according to workspace size
showed that the symmetric six unit system was superior
in terms of weight. However, the proposed method, with
spherically shaped workspace, showed significant
improvement. Moreover, it was shown that the
workspace size was almost equal to that of the five or
six unit skew system when an ellipsoidal workspace is
considered. Because of this result and the fact that a
symmetric pyramid type system has three modes, as well
as mechanical and steering law simplicity, it was
concluded that the pyramid type CMG system with the
proposed steering law would be an ideal candidate for
three axis attitude control.
This paper does not include studies of more realistic
attitude control problems, which should be investigated
in consideration with the results of this paper. Evaluation
may require more detailed characteristics with regard to
mission requirement, disturbance profiles and unloading
torquer specifications.
––– 77 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 78 –––
Appendix A
Double Gimbal CMG System
cOi = ∂hi/∂θOi = gOi × hi,
Formulation of an arbitrarily configured double
gimbal CMG system is made in accordance with the
formulation presented in Chapters 3 and 4.
A.1 General Formulation
Consider a system of n equally sized double gimbal
CMGs in an arbitrary configuration. For each unit, one
fixed vector, two variable gimbal angles, and four other
vectors are defined. These are diagrammed in Fig. A–1
and defined by:
cIi = ∂hi/∂θIi = gIi × hi,
Note that the vector cIi is a unit vector while the vector
cOi is not.
The system configuration is then defined by the set
of {gOi}. The dependent system variables, namely the
total angular momentum H and the output torque T are
given by:
H = Σi hi = H(θ)
T = Σi (cOi ωOi + cIi ωIi )
= C ω,
gOi: Fixed unit vector along the outer gimbal axis.
θOi: Outer gimbal angle with origin located as
shown in Fig. A–1
θIi: Inner gimbal angle with origin located as shown
in Fig. A–1
gIi: Unit vector along inner gimbal axis (a function
of θOi)
Angular momentum vector, normalized to
|hi|=1
cOi: Outer gimbal torque vector
where θ is a point on a 2n dimensional torus whose
coordinates are given by:
θ = ( θO1 θI1 θO2 θI2 . . . θOn θIn).
The 2n dimensional vector ω is defined by:
ω = (dθi/dt).
hi:
cIi: Inner gimbal torque vector
where both torque vectors are defined as follows:
The difference between these expressions and those
for the single gimbal system are that the vector cOi is not
a unit vector and that some vector variables are not
always independent to another θ variable as follows:
∂cIi / ∂θIi = −hi ,
∂cIi / ∂θOi = (gOi × gIi) × hi + gIi × (gOi × hi),
∂cOi /∂θIi = gOi × (gIi × hi),
∂cOi /∂θOi = gOi × (gOi × hi).
gI
θO
cI
gO
cO
θI
h
The last expression implies that the vector ∂cOi /∂θOi is
parallel to gIi .
A.2 Singularity
When the system is singular, the following relation
is satisfied.
det(CCt) = 0
Fig. A−1 Vectors and variables relevant
to a double gimbal CMG
––– 79 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
previous section, suppose further that some unit is in
condition (b), then any dθOi results in the same H. This
implies that outer gimbal motion of this unit alone is a
null motion. When there are two units both in condition
(a) but different signs of ε, the angular momentum
vectors of these two units are in opposite directions and
they move on the unit sphere. Motion of the two units
can be chosen exactly canceling each other. Thus, this
motion is a null motion.
h=u
gO
gI
(a) Condition (a).
gI
A.4 Passability
Passability of a singular surface can be defined by a
quadratic form similar to Eq. 4–15. However, another
approach is also possible.
u
A.4.1 Two Unit System
h = gO
(b) Condition (b).
Fig. A −2 Vectors at singularity conditions.
Geometric comprehension of this fact is easier than the
case of a single gimbal system. Since both torque vectors
are orthogonal to each other and to h, a singular vector
u is determined as u parallels h. An exception arises
when the vector hi is parallel to the vector gOi, where cIi
is a zero vector. Thus, there are four possible conditions
of singularity for each unit when u is specified:
(a) hi = εi u, where εi=1 or −1
(b) hi = εi gOi, and (gIi × gOi).u = 0,
First, a two unit system will be considered. For an
arbitrary configuration of gO1 and gO2, there are five
spherical singular surfaces. One surface is of diameter
2 and is an angular momentum envelope. The remaining
four are unit spheres with their centers at gO1, −gO1, gO2
and −gO2. It is adequate to check the following two cases:
The first case, CASE I, is that one unit satisfies condition
(b). Singular surface for this case is a unit sphere. The
second case, CASE II, is that both units satisfy condition
(a). In this case H is at its origin.
CASE I Infinitesimal motion of the unit satisfying (a)
is exactly on the singular surface. On the other hand,
infinitesimal motion of another unit, say unit 1 for
example, includes a motion out of the surface with regard
to the second order differential. Second order
differentials such as dθI1 dθO1 are orthogonal to gO1 but
where εi=1 or −1.
These are drawn in Fig. A–2. Condition (b) is called
‘gimbal lock’ because such a unit looses one degree of
freedom autonomously.
Because the domain of u is a unit sphere, the total
angular momentum H forms a number of spheres in
accordance with the set of εi.
gI
u
h = gO
dθ Idθ O
A.3 Steering Law and Null Motion
dθ I
Any steering law has the same expression as in Eq.
3–9. When the system is singular, some null motions
are easily obtained as follows. Referring back to the
––– 80 –––
Fig. A−3 Infinitesimal motion at
a singular point of condition (b).
–– A. Double Gimbal CMG System ––
not always to u in Fig. A–3. This motion therefore can
realize a second order motion in both directions away
from the singular surface. Thus, internal part of this
surface is passable. The exception is the case that u =
±gO1 but this is either the case that H is on the envelope
or it is regarded as the following CASE II.
CASE II This condition is simply expressed as:
h1 = −h2 = u or −u.
Any null motion satisfies the following:
dh1 = −dh2,
so the differential is exactly zero for any null motion.
This implies that the quadratic form is exactly zero here.
This is similar to the H origin of the roof type system
M(2,2) (see Section C.1).
whose center is on the origin and additional spheres of
diameter 2. The unit sphere corresponds to the case that
all the units satisfy condition (a). In this case, the
following motion of h vectors is realized by null motions.
(1) dh1 = dh2 = − dh3/2,
(2) dh1 = − dh2, and dh3 = 0,
where it is supposed that the third unit’s ε is negative.
Clearly, the second order motion by (1) is in the direction
of −u and that by (2) is in the direction of u. Therefore,
this singular point is passable.
The sphere of diameter 2 represents the case that one
of the units satisfies condition (b). In this case, the two
unit subsystem is equivalent to CASE I in the above and
this singular point is passable. Thus, a three (or more)
unit system has no internal impassable surface.
A.5 Workspace
A.4.2 Three Unit System
The internal singular surface of an arbitrarily
configured three unit system consists of a unit sphere
There is no internal impassable surface for a double
gimbal CMG system of no less than three units. The
available work space is a sphere of diameter n.
––– 81 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 82 –––
Appendix B
Proofs of Theories
A full description and proof of the formulation given
in Chapter 4 is made here.
B.1 Basis of Tangent Spaces
For an independent type system in a singular state,
there are two independent torque vectors. Suppose that
they are described by c1 and c2. The three sets of bases,
ei, of the subspaces ΘS, ΘN and ΘT, are obtained as
follows.
Basis of ΘS
eSi = ( p1 qi, 1, p2 qi, 2,. . . ,pn qi, n)t ,
The two candidates of eSi given by Basis B–1 satisfy
Eq. B–5 with the above dui for i value in the same order.
Thus, the basis candidates B–1 constitutes the bases of
Θ S.
The following is a general relationship for four
arbitrary vectors in three dimensional space:
a[ b c d ] – b[ c d a ] + c[ d a b ] – c[ a b c ]=0 ,
(B–7)
where [ ] is a box product of three vectors. By
substituting c1, c2, ci and u into these four vectors, the
following relationship is obtained:
(B–1)
q1, 2 ci + q2, i c1 + qi, 1c2 = 0 .
where i = 1 or 2.
Hence, the candidates for eNi in Eq. B–2 satisfy the
definition of null motion since CeNi = 0. Furthermore,
they are independent, because q 1, 2 is not zero by
definition.
The scalar product of a null motion and n dimensional
vector in Eq. B–5 is described by:
Basis of ΘN
eN1 = ( q2, 3, q3, 1, q1, 2, 0, . . . ,0)t ,
eN2 = ( q2, 4, q4, 1, 0, q1, 2, 0, . . . ,0)t ,
...
eNn−2 = ( q2, n, qn, 1, 0, . . . ,0, q1, 2)t ,
Σi dθNi (dθSi ⁄ pi)
(B–2)
= Σi dθNi (cSi ⋅ du)
Basis of ΘT
eT2 = ( qi, 1, qi, 2, . . . ,qi, n)t ,
= (dθN)t Ct du
(B–4)
These are derived as follows.
The definition of dθS is given by differentiation of
the singularity relation (Eq. 3–16) as,
cSi ⋅du = dcSi ⋅ u = dθSi ⁄ pi .
(B–5)
Since the vector du lies on a plane orthogonal to u, there
are two independent vectors du1 and du2. These can be
defined as:
dui = ci × u, where i = 1 and 2 .
= (C dθN)t du
(B–3)
where qi, j is defined as the following vector triple
product;
qi, j = [ ci cj u ] .
(B–8)
(B–6)
=0 .
(B–9)
Thus, the n dimensional vector dθSi ⁄ pi is orthogonal to
the null motion and belongs to ΘT. Basis B–3 is obtained
by simple substitution.
B.2 Gaussian Curvature
Gaussian curvature of a surface in three dimensional
space is defined by the second fundamental form of a
surface, for which there are various definitions. Among
them is the area ratio of the surface and a Gaussian
sphere. By this ratio, the Gaussian curvature κ is defined
as:
––– 83 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
1 ⁄ κ = [u dH(du1) dH(du1)] ⁄ [u du1 du2] ,
(B–10)
where du 1 and du 2 are selected such that they are
independent, as Eq. B–6, for example. The two
differentials dθS1 and dθS2 corresponding to these du
are defined by Eq. B–5 as:
dθSj = {pi (cSi ⋅ duj)} ,
(B–11)
= P Ct duj ,
paper22) requires knowledge of the theory of dyadics.
Here this is proven in vector form.
The differential of H by du given by Eq. 4–5 is
obtained by substituting this du into Eq. B–12, giving:
dH(du) = CPCt du
= κ CPCt((CPCtV)×u) .
(B–17)
A part of the term on the right-hand side can be rewritten
as follows:
Ct((CPCtV)×u)
where j = 1 or 2 .
= (ci )t ((CPCtV)× u)
Therefore:
= (|ci (CPCtV) u|)t
dH(dui) = C dθSi
(B–12)
= ( ci × u )t(CPCtV)
= CPCt dui
= ( ci × u )tC (PCtV) ,
= Σj pj (cSj ⋅ dui)cj ,
where expressions (xi) and (vi) denote a row vector and
a matrix,
where i = 1 or 2 .
(xi) = (x1 x2 .... xn) ,
Substituting this into the first triple product on the
right-hand side of Eq. B–10 results in the following:
[u dH(du1) dH(du2)]
= ΣiΣjpipj(ci⋅du1)(cj⋅du2)[ci cj u] , (B–13)
From the four vector relationship, the following is
obtained;
This notation as well as a matrix notation such as (xij)
are used from this point in this section.
The first term of Eq. B–18, ( c i × u ) t C, is the
following matrix:
( ci × u )tC =(( ci × u )t(cj))
= ([ci cj u]) ,
(B–14)
As vectors such as ci and dui are orthogonal to the unit
vector u, the expression on the right with both two terms
multiplied by u is unchanged.
(B–20)
Multiplying both sides by P, a new matrix R is defined.
R = P([ci cj u])P = (pi pj [ci cj u]) .
(B–21)
Thus, the right hand side of Eq. B–17 can be rewritten
as:
(ci×cj)⋅(du1×du2)
= (ci×cj)⋅u (du1×du2)⋅u
= [ci cj u] [u du1 du2] .
(B–19)
(vi) = (v1 v2 ..... vn) .
(ci⋅du1)(cj⋅du2) − (ci⋅du2)(cj⋅du1)
= (ci×cj)⋅(du1×du2) .
(B–18)
κ CPCt((CPCtV)×u)
(B–15)
= κ CRCtV,
= κ C(rij )CtV.
By substituting this into Eq. B–13 and using [ci cj u]
= −[cj ci u], the following result is obtained:
= κ ( ci )(rij )(cj ⋅V))
[u dH(du1) dH(du2)]
= κ Σij rij (cj ⋅V) ci ,
= 1⁄2 ΣiΣjpipj[u du1 du2][ci cj u]2 .(B–16)
Thus, the expression of Gaussian curvature in Eq. 4–7
is obtained.
(B–22)
where rij is the element of R given by the preceding
equation. Taking advantage of the fact that matrix B–
21 is skew symmetric (rji = −rij):
κ Σij rij (cj ⋅V) ci
B.3 Inverse Mapping Theory
= 1⁄2 Σij rij ( (cj ⋅V) ci − (ci ⋅V) cj ) .(B–23)
Proof of the inverse mapping theory in the original
From the vector product rule and the fact that u is normal
––– 84 –––
–– B. Proofs of Theories ––
to (ci ×cj),
and
(cj ⋅V) ci − (ci ⋅V) cj
 a11 a12
 a21 a22
Ak = 
:
 :
a
 k1 a k 2
= (cj × ci ) × V
= [cj ci u] (u × V ) .
(B–24)
From Eqs. B–21 and 4–7,
κ CPCt((CPCtV)×u)
= κ 1⁄2 Σij pi pj [ci cj u]2 ( V × u )
(B–25)
From Eq. B–17,
dH = V×u .
(B–26)
Thus, Eq. 4–4 is derived and the theory is proven.
B.4 Impassable condition for two
negative signs
Passability is defined by the signature of the quadratic
form, QN, in Eq. 4–19. Let AN denote the matrix of this
quadratic form as;
AN = ENt P−1 EN .
(B–27)
Let’s find a condition of impassable state in case that
two of the signs, εi, are negative and remaining signs
are positive.
We have the following linear algebra theory for
definite matrix.
Theory Consider a symmetric m×m matrix denoted by
A and its sub-matrix Ak.
 a11
 a21
A=
 :

 am1
a12
a22
:
am 2
... a1m 
... a2 m 

:
: ,

... amm 
.
A is positive definite if and only if the ‘minor’, i.e., detAk
is positive for all k = 1, 2, ..., m.
= κ 1⁄2 Σij rij [ci cj u] ( V × u )
= V×u .
... a1k 
... a2 k 

:
: 
... akk 
Though all the minors must be examined in general,
in case of checking passability, only two of them should
be examined because we can suppose that εn and εn−1
are negative without loosing generality, .
Suppose that the condition (3) in Section 4.3.4 is
satisfied, that is, the Gaussian curvature κ is positive. In
this case, there are two possibilities where the quadratic
form QN is positive definite or its signature has two
negative terms. In both cases, determinant of the matrix
AN is positive because this determinant is a product of
all eigenvalues and its sign is determined by the
signature.
Consider the subsystem without the nth unit which
has the negative sign. For this subsystem, ANn−3 is the
matrix of the quadratic form, because the original matrix
A N is n−2×n−2. As there is one negative sign, the
condition (2) in Section 4.3.4 that the Gaussian curvature
is negative is equal to that the matrix ANn−3 is positive
definite. Thus, only this condition is enough to assure
that minors for all k = 1, 2, ..., n−3 are positive.
From the above theory, we have the following
conclusion.
Conclusion
A singular surface of all the εi but two are positive is
impassable if and only if κ > 0 and κ’ < 0, where κ’ is
the Gaussian curvature of the subsystem without one
unit of negative sign.
––– 85 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 86 –––
Appendix C
Internal Impassability of Multiple Type
Systems
In this appendix, the minimum systems with no
internal impassable surfaces are found which are M(3,
3) and M(2, 2, 2). First, discrimination of singularity
for the four unit roof type system is made. Then, the
minimum systems are searched by adding units to this
system.
Though the subspace Θ N of the null motion is
generally two-dimensional, it is one-dimensional when
u lies on the plane spanned by the two gimbal vectors
g1 and g2. In this case, there are three independent null
vectors. From this point, each variable pi is defined for
each group so that the condition pi ≥ 0 is satisfied.
C.1 Roof Type System M(2, 2)
C.1.1 Evaluation of Singular Surface (2)
The four unit roof type system is shown in Fig. 2–2.
All singular surfaces are given as follows.
Independent null motions are;
φ1 = (1, −1, 0, 0),
u≠g1, u≠g2
(C–1)
φ2 = (0, 0, 1, −1) .
(1) θ11 = θ12, θ21 = θ22, ε={+ + + +}:
The quadratic form dθNtP−1dθN by dθN = a1φ1+a2φ2
Envelope
is;
(2) θ11 = θ12, θ21 = θ22,
dθNtP−1dθN
ε={+ + − −} or {− − + +}:
= −1⁄2(a12⁄p1+a12⁄p1−a22⁄p2 − a22⁄p2)
Internal Surface
= −a12⁄p1 + a22⁄p2 ,
(3) θ11 = θ12+π, θ21 = θ22, ε={+ − + +}:
Circle orthogonal to g1 centered on H origin.
(C–2)
Thus, this is indefinite and passable.
(4) θ11 = θ12, θ21 = θ22+π, ε={+ + + −}:
Circle orthogonal to g2 centered on H origin.
C.1.2 Evaluation of Singular Surface (3)
(5) θ11 = θ12+π, θ21 = θ22+π, ε={+ − + −}:
Independent null motions are;
H origin
φ1 = (1, 1, 0, 0) ,
u = g1
(C–3)
φ2 = (0, 0, 1, −1) .
(6) θ21 = θ22:
Envelope
(7) θ21 = θ22+π:
Inside Circle (3)
The quadratic form dθNtP−1dθN by dθN = a1φ1+a2φ2
is;
dθNtP−1dθN
u = g2
(8) θ11 = θ12:
Envelope
= −1⁄2(a12⁄p1−a12⁄p1+a22⁄p2 + a22⁄p2)
(9) θ11 = θ12+π:
Inside Circle (4)
= −a22⁄p2 ≤ 0 ,
It is only necessary to evaluate internal surfaces (2),
(3), (5) and (7), because pairs such as (3) & (4) or (7) &
(9) are identical when the group numbers are changed.
(C–4)
Thus, this is semi-definite and impassable.
If u happens to be on the plane spanned by g1 and g2,
another null vector φ3 (dependent on φ1 and φ2) is given
––– 87 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
by;
Thus, this is indefinite and passable.
φ3 = (1, 0, −1, 0)
(C–5)
C.1.4 Evaluation of Singular Surface (7)
The quadratic form by dθN = a1φ1+a2φ2+a3φ3 is;
Independent null motions are;
dθNtP−1dθN
φ1 = (0, 0, 1, 1) ,
= −1⁄2{(a1+a3)2⁄p1−a12⁄p1 +(a2−a3)2⁄p2
(C–11)
φ2 = (a, b, 1, −1) .
+ a22⁄p2}
= −1⁄2{ a32⁄p1 + 2a1a3⁄p1 + 2a22⁄p2
The quadratic form by dθN = a1φ1+a2φ2 is;
− 2a2a3⁄p2 + a32⁄p2 }
dθNtP−1dθN = −1⁄2{(a1+a2)2⁄p2− (a1−a2)2⁄p2} ,
= −1⁄2(ra32 + 2rsa1a3 + 2(a2−a3⁄2)2⁄p2 )
(C–12)
= −1⁄2{r(a32 + sa1)2 − rs2a12
Thus, this is indefinite and passable.
+ 2(a2−a3⁄2)2⁄p2 },
where r = 1⁄p1+3⁄(4p2) >0, 2rs = 2⁄p1 .
(C–6)
C.1.5 Conclusion
The roof type system M(2, 2) has internal impassable
surfaces given by (3) and (5) with u not on the plane
spanned by g1 and g2. Both are not fully definite but
the quadratic form of (3) is semi-definite and that of (5)
is zero.
Thus, this is indefinite and passable.
C.1.3 Evaluation of Singular Surface (5)
Independent null motions are;
φ1 = (1, 1, 0, 0) ,
(C–7)
φ2 = (0, 0, 1, 1) .
The system M(3, 2), which results from adding an
additional unit to M(2, 2), will now be analyzed. The
passability of sub-system M(2, 2) was evaluated above,
and so only those impassable conditions should be tested.
The quadratic form by dθN = a1φ1+a2φ2 is;
dθNtP−1dθN
= −1⁄2(a12⁄p1−a12⁄p1+a22⁄p2 −a22⁄p2)
=0 .
(C–8)
Thus, this is zero for any null motion and impassable.
If u is on the plane spanned by g1 and g2, another
null vector φ3 (dependent on φ1 and φ2) is given by;
φ3 = (1, 0, −1, 0) .
C.2 M(3, 2): M(2, 2)+1
C.2.1 Condition (3) of M(2,2)
System M(3, 2) is asymmetric and so both conditions
(3) and (4) should be tested. The singular points are
given by;
(3') θ13 = θ11 = θ12+π, θ21 = θ22,
(C–9)
ε={+ − + + +} ,
The quadratic form by dθN = a1φ1+a2φ2+a3φ3 is;
(4') θ13 = θ11 = θ12, θ21 = θ22+π,
dθNtP−1dθN
ε={+ + + + −} .
=−1⁄2{(a1+a3)2⁄p1−a12⁄p1+(a2−a3)2⁄p2
− a22⁄p2}
Independent null motions are then;
= −1⁄2 {(1⁄p1+1⁄p2)a32 + 2(a1⁄p1−a2⁄p2)a3}
φ1 = (1, ±1, 0, 0, 0) ,
=−1⁄2 [(1⁄p1+1⁄p2)
φ2 = (1, ±1, 1, 0, 0) ,
⋅ {a3+(a1⁄p1−a2⁄p2)⁄(1⁄p1+1⁄p2) }2
− (a1⁄p1−a2⁄p2)2⁄(1⁄p1+1⁄p2) ] .
(C–10)
(C–13)
φ2 = (0, 0, 0, 1, −(±1)) .
where upper and lower sign of the multiple sign ±
correspond to (3') and (4') respectively. The quadratic
––– 88 –––
–– C, Internal Impassability of Multiple Type Systems ––
dθNtP−1dθN
form by dθN = a1φ1+a2φ2+a3φ3 is;
(3')
= −1⁄2 {a12⁄p1 − (a1+a2)2⁄p1 + a22⁄p1 + a32⁄p2
dθNtP−1dθN
− a32⁄p2}
= −1⁄2{a12⁄p1 − (a1+a2)2⁄p1 + a22⁄p1
= −1⁄4(−(a1+a2)2 + (a1−a2)2)⁄p1 ,
+ a32⁄p2 + a32⁄p2}
= −1⁄2(−2a1a2⁄p1 + 2a22⁄p2)
= −1⁄2{−1⁄2(a1+a2)2⁄p1 + 1⁄2(a1−a2)2⁄p1
+ 2a22⁄p2} ,
(C–17)
Thus, this is passable.
It is concluded that M(3, 2) has an impassable surface
corresponding to condition (4').
(C–14)
(4')
dθNtP−1dθN
= −1⁄2{a12⁄p1 + (a1+a2)2⁄p1 + a22⁄p1 + a32⁄p2
C.3 M(3, 3): M(2, 2)+2
Even if a certain sub-system M(3,2) of M(3, 3)
satisfies condition (4'), the same condition will be (3') in
the other sub-system. This implies that there is no
internal impassable surface.
− a32⁄p2}
= −1⁄2(a12⁄p1 + (a1+a2)2⁄p1 + a22⁄p1) .
C.4 M(2, 2, 1): M(2, 2)+1
(C–15)
Thus, (3') is passable and (4') is impassable. The singular
H of (4') forms a circle centered on the origin and having
a diameter of 3.
C.2.2 Condition (5) of M(2,2)
This configuration is not a multiple system, so there
is an internal impassable surface.
C.5 M(2, 2, 2): M(2, 2)+2
If u is not parallel to gi, all internal surfaces are
passable from the discussion of Section 3.2.5. If u is
parallel to gi, condition (7) is satisfied by sub-system
M(2, 2) which includes the ith group. Thus all internal
surfaces are passable.
The singular points are given by;
(5') θ13 =θ11 = θ12+π, θ21 = θ22+π,
ε={+ − + + −} .
Independent null motions are then;
C.6 Minimum System
φ1 = (1, 1, 0, 0, 0) ,
φ2 = (0, 1, 1, 0, 0) ,
(C–16)
φ3 = (0, 0, 1, 1) .
The multiple systems M(3, 3) and M(2, 2, 2) are the
minimum systems with no internal impassable singular
surfaces. Both systems consist of six units.
The quadratic form by dθN = a1φ1+a2φ2+a3φ3 is;
––– 89 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 90 –––
Appendix D
Six and Five Unit Systems
D.1 Symmetric Six Unit System S(6)
z
D.1.1 System Definition
The S(6) system is a symmetric type with six units
arranged in the surface directions of a regular
dodecahedron. Its work space possesses symmetry and
can be approximated by a sphere. Being an independent
type, the system has internal impassable surfaces, which
are very near the envelope. Control over most of the
entire workspace shown in Fig. D–1 (a) can be
accomplished using a gradient method. The diameter
of the controllable spherical workspace is about 4.27
times larger than the angular momentum of the unit (see
Fig. 9–2 in Chapter 9).
y
x
(a) Original S(6) system
z
D.1.2 Fault Management
(1) Loss of One Unit
The S(6) system without any one unit is a congruent
five unit skew type system with a different major axis
direction. Thus, we need only one steering law for this
type of failure. The original and degraded system
envelopes are shown in Fig. D–1. The original envelope
is similar to a sphere but that of the degraded system is
more similar to an ellipsoid. Figure D–1 (b) corresponds
to the failure of the unit arranged in the z direction. If
another unit fails, the envelope has the same shape but
its major axis is different.
As the skew angle of this system is not optimized,
there is a more serious internal impassable surface
problem than the optimized system described in Section
9.4 has. Moreover, even though we can use this
workspace, its major axis is unknown before the accident,
and there are six possibilities. Thus, it is safe to consider
all possible situations and to evaluate a spherical
workspace which is included by all six possible
envelopes (see Fig. 9–2).
y
x
(b) After loss of one unit
z
x
y
(c) After loss of two units
Fig. D–1 Envelopes of S(6) and degraded systems.
All are drawn in the same scale.
––– 91 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
(2) Loss of Two Units
Any failure of two units also results in a congruent
configuration of four units. However, the system is not
at all symmetric and the envelope is like a skew ellipsoid,
as shown in Fig. D–1 (c). Because of reasons similar to
those given above, the workspace size must be evaluated
by a spherical workspace.
D.1.3 Four out of Six Control
A gradient method applied to the MIR system uses
the following objective function27):
W = Σij |ci × cj |2 .
(D–1)
As described in Section 7.5.5, the concept of the
constrained control can be applied to this subsystem69).
The four unit subsystem can be regarded as a deformed
pyramid configuration in which two units have a skew
angle α1 (= sin −1 1 / 2(1 + cos(π / 5)) ) and the other
The CMG system installed on the space station
“MIR” is a S(6) system but only four units out of six are
operating simultaneously. The subsystem of four units
is the same as that in the above section. As mentioned
there, any four unit subsystem is geometrically
congruent. Therefore, any fault up to two units can be
simply covered by exchange of faulty unit with a backup
unit without change of the steering law.
two, a skew angle α2 (= π/2 − α1) (Fig. D–2). As the
kinematic equation of this system is similar to that of
the pyramid type system, the same constraining condition
as Eq. 7–5 can be applied and nonredundant kinematics
similar to Eq. 7–6 can be obtained. This constrained
system has a restricted workspace (Fig. D–3), inside
which exact steering is assured. Of course, this
configuration is not rotationally symmetric about any
gimbal axis, so there is no additional mode.
g4
Z
D.2 Five Unit Skew System
g3
g1
α1
α2
X Y
g2
g6
g5
Fig. D–2
Four unit subsystem of MIR type system.
Original Workspace
As described in Chapter 9, various ellipsoidal
workspaces can be designed by selecting the skew angle.
The workspace size is given in the figures of Chapter 9.
The fault management is similar to that of the S(6) system
described in Appendix D.1.1, except that the skew angle
is different.
As this type of systems have not studied well, no
effective steering laws have been proposed except the
gradient method. Application of the constraint method
is possible with two independent constraining equations,
Constrained workspace
Impassable
Surface
Actual
Motion of H
H'
Desired
Motion
of H
Additional Angular
Momentum Imposed
by Another Torquer
H
H path
Fig. D−3 Restricted workspace of a constrained
MIR-type system.
Fig. D−4 Concept of singularity avoidance by
an additional torquer
––– 92 –––
–– D. Six and Five Unit Systems ––
but the symmetry of this system cannot be preserved by
using any two linear equations. Therefore, finding
appropriate constraints may need exhaustive calculations
and evaluations with some criteria of work space size
and shape.
Here, a potential steering law with an additional
torquer will be briefly outlined, which may be effective
for four or five unit system.
Impassable surfaces of these systems are shaped like
surface strips as shown in Figs. 4–10 to 4–12 and 6–7 to
6–10. In Section 6.4, these strips were called ‘impassable
branches’ for the S(4) system. They can be approximated
by the analytical expression given by Eqs. 6–16 to 6–
19. Though there is no such expression for 5 unit
systems, they can be expressed by some numerical lookup table. This look-up table can be reduced in its size
with the aid of system symmetry. By this knowledge,
impassable surface strip, the system can avoid the
surface, as shown in Fig. D–4. This mechanism is very
simple and has the following characteristics.
we can distinguish whether H is approaching an
impassable surface.
If appropriate angular momentum is added using
another torquer when H is nearly crossing some
can not use this knowledge of manifold connections,
and hence is very simple but requires additional
torquing more often than necessary.
(1) The avoidance movement can be planned so that it is
towards the narrower direction of the impassable
strip. Thus, the required angular momentum for
avoidance can be minimized. Moreover, this motion
only aims to avoid the surface, therefore an ON/OFF
type torquer such as a gas jet would be adequate.
(2) This motion must take place only once to avoid a
singular surface. Thus it may be accomplished by a
gas jet system.
(3) Sometimes, no singularity avoidance is necessary
when H is approaching an impassable surface, as is
described in Sections 7.1 and 7.4 on manifold
connections. The method described here, however,
––– 93 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 94 –––
Appendix E
Specification of Experimental Apparatus and
Experimental Procedure
E.1 Experimental Apparatus
The experimental apparatus, as shown in Figs. 8–1
and E–1, is composed of a body structure, a three axis
gimbal, attitude sensors and related circuitry, a CMG
system, balance adjusters and an onboard computer. The
block diagram is shown in Fig. E–2.
The body is a truss structure made of steel pipes. It
is designed sufficiently stiff so that deformation caused
by its weight can be neglected when the system changes
orientation. The three axis gimbal, which uses normal
ball bearings, permits free rotation of the body (Fig. E–
3). A precision rotary encoder is installed on each gimbal
axis. The encoder’s output pulses are converted to an
angle value by a decoder circuit, then supplied to the
onboard computer. The rotational speed of the body is
measured by rate gyroscopes, whose outputs are analog
signals that are converted to digital values by an Analog
to Digital (A/D) conversion circuit.
TG DCM
TG DCM
RE
P/D
Rate
Gyroscope
A/D
Balance RE
Adjusters
P/D
Rate Servo
Circuit
D/A
RE
P/D
Rate Servo
Circuit
D/A
CMGs
RE: Rotary Encoder
P/D: Pulse Decoder
A/D: Analog to Digital Converter
Wireless
Modem
Onboard Computer
Three Axis
Gimbal
Fig. E−1 Experimental apparatus
Wireless
Modem
Stationary
Computer
D/A: Digital to Analog Converter
DCM: DC Servo Motor
TG: Tachogenerator
Fig. E–2 Block diagram of experimental apparatus
––– 95 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
Fig. E−3 Three axis gimbal mechanism
Fig. E−4 Single gimbal CMG
Fig. E−5 Balance adjuster
Fig. E−6 Onboard computer
The CMG system is a S(4) type system composed of
four single gimbal CMGs and a unit CMG is shown in
Fig. E–4. A wheel motor and a driver circuit installed
inside the casing drives the flywheel at constant speed.
Slip rings installed through the gimbal axis enable free
rotation of the gimbal. The gimbal angle is measured
by a rotary encoder. The rotational speed of each gimbal
motor is controlled by a servo driver circuit.
The balance adjusters are composed of a moving
weight driven by a linear ball screw mechanism as shown
in Fig. E–5. The speed of the moving weight is
controlled by a DC motor and a rate servo circuit, and
––– 96 –––
–– E. Specification of Experimental Apparatus and Experimental Procedure ––
its position is measured by a rotary encoder and a decoder
circuit.
The onboard computer is composed of a 32 bit
microprocessor, an interface circuit for the decoder
circuits, an A/D converter board, a D/A converter board,
and a wireless modem driver (Fig. E–6). The wireless
modem enables serial communication with the stationary
computer.
E.2 Specifications
The specifications of the experimental apparatus are
listed in Table E–1.
E.3 Attitude Control System
Two types of controllers were installed in the onboard
computer. One was a model matching controller and
the other was a PD tracking controller. The block
diagram of the model matching controller is shown in
Fig. E–7. Each parameter was set by a kind of a pole
assignment method called the ‘model matching
method54, 55). These were obtained so that the overall
transfer function matched the given function. In the
experiments, the pole of the function was set to 1.0. The
block diagram of the tracking controller is shown in Fig.
E–8. Each parameter was set by a similar method above.
Table E–1 Specification of experimental apparatus
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Test Rig
Size
750mm cube
Moment of inertia
{ 38, 38, 42 } Kg m2 about x, y and z axes
Weight
Approx. 250 Kg
CMG
made at the Mechanical Engineering Laboratory
Flywheel Diameter
130 mm
Rotational Rate
5,000 rpm
Angular Momentum
3.8 Nms
Gimbal Motor
ESCAP – 34HL11-219E/204-2
Gimbal Motor Reduction Gear
P42 (17.7:1) + 62:13
Gimbal Rotation Sensor
Heiden Hein–ROD456.015B3600 + EXE601/5F
Resolution
3600×5 pulse⁄rev (0.02 deg ⁄1 pulse)
Attitude Sensor
Resolution
Rate gyroscope
Resolution
Optical Rotary Encoder Canon – R10
81,000 pulse ⁄rev. × 4
(1pulse ⁄10 arc sec)
JAE – DARS
±0.5 deg⁄sec
Onboard computer
CPU
i80386SX16MHz with i80387
Memory
640KB
Operating System
MS-DOS Ver.3.3 in ROM
Peripherals
A/D, D/A, Pulse Decoder, Wireless Modem
Cycle time
10 ms – 12 ms
Software Development
MS-DOS Ver.3.0, Optimizing C Compiler, Turbo C
Environment
Compiler
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
––– 97 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
k1
r
+
1
s
−
k
+
v
−
−
+
Body dynamics
.
β
β
1
1
s
s
f2
model transfer function
1
(s+p)
f1
Fig. E–7 Block diagram of the model matching controller.
r⋅⋅
1
s
r⋅
1
s
r +
−
f1
+
+
Body dynamics
.
β
1
1 β
s
s
v
+
+
−
f2
Fig. E–8 Block diagram of the tracking controller.
Tcom
ω
–C t(CCt)-1
+
C
θ
(3-11)
ωN
ξ
1
det(CCt )
×
k
(3-19)
Fig. E–9 Block diagram of the gradient method.
E.4 Steering Law Implementation
Three types of steering laws were installed on the
onboard computer, the gradient method, the SR inverse
method, and the constraint method proposed in Chapter
7. The gradient method was exactly the same as
described in Chapter 2. Its block diagram is shown in
 c * sin φ sin ψ + cos φ sin γ
1
T=−
− cos φ sin ψ + c * sin φ sin γ
h
 s * cos φ (cos ψ + cos γ )
Fig. E–9. The SR inverse method is defined in Section
3.5.2.
The constraint method used the kinematic equation
7–6. In actual implementation, numerical inversion of
this equation is inappropriate because of nonlinearity.
Therefore, the steering law was realized as a solution of
linear equations which are obtained by differentiation:
− c * cos φ cos ψ
− sin φ cos ψ
− s * sin φ sin ψ
––– 98 –––
sin φ cos γ 
− c * cos φ cos γ  ω *

− s * sin φ sin γ 
(E–1)
–– E. Specification of Experimental Apparatus and Experimental Procedure ––
where h is angular momentum of each CMG unit and
ω∗ is speed of three variable vector that is (dφ/dt, dψ/dt,
dγ/dt)t.
In real situations, the constraint condition is not
guaranteed because the values of the gimbal rates derived
from the above equation are used for a finite sampling
time and are not renewed continuously. If the constraint
condition is not satisfied, neither the following variable
transformation nor the constrained kinematics is valid.
θ = (φ+ψ, φ+γ, φ−ψ, φ−γ) ,
(E–2)
To cope with this, an approximated solution with
feedback was adopted in which null motion of the
original system was added to make residual (i.e., the
left of Eq. 7–5) vanish. An approximation of (φ, ψ, γ) is
defined by the following equation.
φ = (θ1 + θ2 + θ3 + θ4)/4 ,
ψ = (θ1 − θ3)/2 ,
(E–3)
γ = (θ2 − θ4)/2 .
By using this, an approximated motion is obtained as a
solution of Eq. E–1. The Jacobian matrix in Eq. E–1 is
a 3×3 matrix and its inverse can easily be obtained. With
this inverse matrix and the command torque Tcom, the
transformed gimbal rate ω∗ is obtained. By the
coordinates transformation given by Eq. E–2, the real
gimbal rates are obtained.
After that, feedback terms are added. Null motion
has one degree of freedom and is generally obtained as
kωN where |ωN| = 1 (normalized after Eq. 3–11). For
the stable feedback, the multiplier k is determined with
an appropriate feedback gain a as follows:
k = −a (θ1 − θ2 + θ3 − θ4)
⋅ ( ωN1− ωN2+ωN3−ωN4) .
(E–4)
The block diagram of this steering law is shown in
Tcom
J-1
d(φ, ψ, γ)
dt
Fig. E–10.
E.5 Code Size and Calculation Time
Control laws and steering laws were implemented
in the onboard computer. All the programs are coded by
C language and compiled by the Turbo-C compiler
version 2.0. Their code size and calculation time are
listed in Table E–2. The constrained method needed
about 2 ⁄ 3 memory storage and about 1 ⁄ 2 calculation
time of the gradient method.
Table E–2 Code size and calculation time
of process
process
code size
calculation time
(bytes)*
(ms)
––––––––––––––––––––––––––––––––––––––––––
MM-Controller
6,700
0.85
Tracking Controller
8,800
1.1
Gradient Method
3,800
3.8
Constrained Method
2,800
1.8
––––––––––––––––––––––––––––––––––––––––––
* Code size is an approximate value
E.6 Parameter Estimation
The system has various parameters. In order to
design an attitude controller, the inertia matrix of the
body and the size of the angular momentum of each
CMG unit must be given. Since precise evaluation of
such parameters by calculation was not enough, they
were estimated by experiments.
First, the weight of moving mass of each balance
adjuster was estimated by measuring the torque with a
Transform
(E-1)
ω1
J
Jacobian of (E-3)
Transform
(E-2)
θ
(3-11)
(E-4)
ωN
×
k
Fig. E-10 Block diagram of the constrained method.
––– 99 –––
ω
+
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
scale when the position of the mass was moved in a stepby-step manner. Then a certain fixed torque was applied
by this mechanism while the body was stabilized by the
CMG system. The total angular momentum of the CMG
system linearly increased by the constant disturbance
torque. From the kinematic relation and measured
gimbal angles, the size of angular momentum of each
unit was thereby estimated.
Then, the body was rotated by the CMG system,
generating a constant torque on the principal axis of the
body. It was presumed that the principal axes (eigenaxes
of the inertia matrix) were the same as the structure’s
frame directions. Trials about three axes were then made.
By comparing the measured angular velocity of the body
with the CMG angular momentum, the moment of inertia
about each axis was estimated. The estimated values
are included in the specification of Table E–1.
––– 100 –––
Appendix F
General kinematics
F.1
Analogy with a Spatial Link
Mechanism
The total angular momentum of a CMG system, H,
is a three dimensional vector and is given as the sum of
all hi by Eq. 3–3. Each hi has unit length and rotates
about gi. This is then very similar to a spatial link
mechanism such as a multi-joint manipulator22, 33, 40).
The total angular momentum, H, corresponds to the point
of the “hand”, i.e., the tip of the manipulator and the
gimbal angles, θi, correspond to the joint angles.
A parallel link mechanism shown in Fig. F−1
corresponds exactly to a single gimbal CMG system. In
the case of a link mechanism, the study of the relationship
between the input joint angle and the output hand point
is called kinematics, since it is an instantaneous
relationship and thus does not explicitly include time.
In this sense, the system equation giving H from θ (in
Eq. 3–4) is called a kinematic equation of a CMG system.
Table F−1 shows the similarity of a CMG system
and a manipulator.
F.2 Spatial Link Mechanism Kinematics
State variable of a link mechanism is a set of n joint
displacements denoted by q = {qi}. Output variable is a
set of a position vector p(∈R3) and orientation γ(∈a
subset of SO(3)).
θ3
θ2
g3
h3
h2
g2
Z
X
H
h1
θ1
g1
Y
(a) Parallel link
mechanism
(b) CMG
vectors
Fig. F–1 Analogy to a parallel link mechanism
Orientation has various representation, such as
orthogonal matrix, quarternion and Euler angles. Any
not redundant representation is enough for describing
local geometry here, because discussion is limited in the
neighborhood of a singular point. Thus, output variable
can be represented as follows;
 p
x = ( p1 p2 p3 γ1 γ2 γ3)t =  γ  ,
(F–1)
Table F-1 Similarity between CMGs and link mechanism
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
CMG
Link mechanism
State variable
θ : gimbal angles
q : joint angles
Output variable
H : angular momentum
x : end point location (and orientation)
Kinematics
H = H(θ)
x = x(q)
Kinematics
nonlinear without cross
nonlinear with cross coupling of qi
complexity
coupling of each θi
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
––– 101 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
x(q+dq) − x(q)
Kinematic equation is a nonlinear equation from joint
displacements q to the output x. Exact expression of
this equation is found in most literature.
x = f(q) .
(F–2)
Differentiation of the equation leads to,
dx = dq = J dq .
(F–3)
= Σi ∂f/∂qi dqi + 1/2 Σij ∂f2/∂qi∂qj dqi dqj
+ O(dq3) .
Then, component of the singular direction is extracted
and the 3rd and higher order terms are omitted.
As taking scalar product with ζ, first term vanishes,
∆x = ζt {x(q+dq) − x(q) }
Usual definition of Jacobian with axial velocity of
the end-effector ω is given as follows,
dp / dt
 ω  = J*dq / dt .
= 1/2 dqt Q dq .
(F–10)
where matrix Q is,
(F–5)
Therefore,
 I3
∂f / ∂q = J = Γ∗ -1 J∗ , where Γ∗ =  0

= 1/2 Σij (ζ ⋅ ∂f2/∂qi∂qj ) dqi dqj
(F–4)
From assumption above, there is a non-singular
transformation between ω and dγ/dt as,
ω = Γ dγ/dt .
(F–9)
0
G  .
(F–6)
Q = ( ζt ∂f2/∂qi∂qj ) .
(F–11)
This matrix is not so simple as a diagonal matrix P
in the case of CMGs. Moreover, signature is not
explicitly obtained from this matrix Q.
The next step is to decompose this quadratic form
into two sub-quadratic forms. One rises from the curved
hyper surface of singular state and the other from the
displacement from this hyper surface. This is obtained
by decomposing dq into two parts, dqS which keeps
singularity and homogeneous motion dqN:
Keeping generality, we can take the origin where the
transformation Γ∗ is identity hence J = J∗.
dq = dqN + dqS .
(F–12)
The definition of dqN and dqS is,
F.3 Singularity
det ( J( q+dqS) )=0 ,
Singular state is a case where the matrix J does not
have full rank. This means that,
det ( J ) = 0 ,
(F–7)
and there exists at least one direction, denoted as ζ, in
the tangent space of the x space, which satisfies,
ζt J = 0 , where | ζ | = 1 .
(F–8)
This direction can be called a singular direction. The
difference between CMGs and a manipulator is that there
is no explicit expression which gives singular state
variable q from this singular direction ζ as in Eqs. 4–2
and 4–3.
F.4 Passability
J dqN = 0 .
If a singular state is characterized by one direction
ζ, the dimension of the singular hyper surface is 5 and
so is the vector space of dqS. In this case the rank of J is
5. The kernel of J, that is a vector space of dqN, is n-5
dimensional.
Substituting (F–12) into (F–10) leads to,
∆x = 1/2d(qS)tQdqS + 1/2 (dqN)tQdqN
+ (dqS)tQdqN .
(F–14)
The third term in the right will be shown zero. In order
to prevent complication, let partial derivatives of the
function f = (fi) be denoted by index with comma, such
as,
fi,j = ∂fi / ∂qj .
In order to classify singularity, small displacement
from the singular point is expressed as a Taylor series,
(F–13)
Singular direction is defined again as,
––– 102 –––
(F–15)
–– F. General Kinematics ––
Σi ζi fi,j = 0 ,
(F–16)
and its differential is
Σi dζi fi,j + Σi ζi Σj fi,jk dqSk = 0 .
(F–17)
Then,
(dqN)t Q dqS
= 1/2 Σi ζi ΣjΣk fi,jk dqNj dqSk
= 1/2 Σj{Σi ζi Σk fi,jk dqSk }dqNj
= − 1/2 ΣjΣi dζi fi,j dqNj
= − 1/2 Σi dζi {Σj fi,j dqNj}
= − 1/2 dζt (J dqN)
=0 .
(F–18)
Thus the quadratic form ∆x is divided into two as,
∆x = 1/2(dqS)tQdqS + 1/2 (dqN)tQdqN
= 1/2 Σijk fi,jk ζi dqSj dqSk
+ 1/2 Σijk fi,jk ζi dqNj dqNk .
(F–19)
hyper surface. The second quadratic form gives
passability. There is a restriction for the above
discussion, i.e., the expression (F–12) is not always
possible. It is neither possible nor the last equation be
assured, when the product space of two linear spaces
spanned by {dqS} and {dqN} do not cover whole tangent
space.
A similar expression of passability is thus obtained
generally as CMGs but the following differences and
problems are remaining for further study;
(1) No expression of Gaussian curvature is obtained
(2) Neither simple description of whole quadratic form
nor whole signature is obtained. This is because there is
no general solution of singular state from the direction ζ
and sign {εi}.
(3) Dimension of the singular hyper surface (= 5) is
greater than that of remaining space ( =n − 5), in usual
case, because the number of the joints is not greater than
10. Therefore it seems easier to deal with the quadratic
form directly.
The first term is a kind of curvature of the singular
––– 103 –––
–– Technical Report of Mechanical Engineering Laboratory No.175 ––
––– 104 –––
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