A Nonlinearly Compliant Transmission Element for Force
Sensing and Control
by
Andrew W. Curtis
B.S., Mechanical Engineering
Rice University, 1992
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 2000
@2000 Massachusetts Institute of Technology
All rights reserved
NSTITUTE
MASSACHUSETTS INSTITUTE
OF TECHNOL OGY
SE P 2 0
LIBRARI ES
Signature of Author......
Depa rtment of Mechanical Engineering
January 31, 2000
Certified by
........
Pt.
Kenneth Salisbury, Jr.
P iciple Research Scientist
Thesis Supervisor
Accepted by ................
Professor Ain A. Sonin
Chairman Committee on Graduate Students
A Nonlinearly Compliant Transmission Element for Force
Sensing and Control
by
Andrew W. Curtis
Submitted to the Department of Mechanical Engineering
on January 31, 2000 in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
Abstract
Recently, other researchers have demonstrated that force sensing and control capabilities
can be implemented on a robot manipulator without the use of an explicit force/torque
sensor. Instead, a measured flexible, or compliant, mechanism is added to an otherwise
rigid joint transmission assembly and the joint forces are determined by measuring the
This thesis
compliant mechanism displacement [Williamson, 1995; Shah, 1997].
design
mechanism
transmission
joint
compliant
nonlinearly
an
alternative
investigates
manipulators.
that may be used to improve the dexterity of future planetary exploration
The proposed nonlinear compliant element is based on the concept of a wrapping spring
that undergoes a continuous decrease in spring free length and a resulting increase in
stiffness for greater displacements. Multiple generations of prototypes were constructed
to explore and evaluate mechanism design issues regarding the stiffening profile, the
robustness, and the manufacturing, assembly, and integration methods. A four degree-offreedom robot arm was designed and constructed in conjunction with [Katz, 1999] to
demonstrate integration of the proposed elements into a manipulator and to conduct
evaluations of system level trajectory and force control techniques.
The results indicate that the desired stiffening behavior can be achieved with this design
and that it can be relatively easily integrated into a manipulator design, especially one
with a cable transmission. However, the research also indicates that the mechanism
performance can be significantly altered by small changes in part or assembly
dimensions, thus indicating a potential lack of robustness as currently implemented.
Thesis supervisor: Dr. J. Kenneth Salisbury, Jr.
Title: Principal Research Scientist
5
Acknowledgements
I would like to thank and acknowledge the following people for their time,
encouragement, and assistance that contributed to my ability to complete this work:
Dr. Ken Salisbury for inviting me to join his research group, where I have been able to
explore several robotics topics while pursuing my specific interest in space robotics;
The members of my research group, Arrin, Brian, Jesse, Ela, and Mark, for providing
valuable technical advice and moral support on the numerous occasions when I
encountered obstacles during this research;
My friends and colleagues at Lockheed Martin Space Operations for providing me an
opportunity to stay involved in ongoing Shuttle RMS activities as I work to improve my
space robotics skills and knowledge base;
My friends at Stewart Automotive Research for continuing to inspire me with their
ingenuity, persistence, and intelligence;
My parents and grandparents for teaching me at a young age that I can do anything if I
put my mind to it; and
My brothers, Adam, Byron, and Matthew.
7
Contents
A B STRA C T ........................................................................................................................
3
A CKN O WLED G EM EN TS ..........................................................................................
5
C ON TEN T S........................................................................................................................7
FIG U R E S............................................................................................................................9
TA BLE S..............................................................................................................................9
IN TRO D U C TION ...............................................................................................
1
1.1
1.2
1.3
11
O VERVIEW AND SCOPE .......................................................................................
11
BRIEF SURVEY OF PRIOR TRANSMISSION COMPLIANCE WORK ..........................
O UTLINE OF THESIS ................................................................................................ 13
THEO RY ..................................................................................................................
2
2.1
11
15
THE U TILITY OF EXPONENTIAL JOINT COM PLIANCE ............................................... 15
2.1.1
Exponential vs. Linear Compliance............................................................
16
17
18
2.2 INTERPRETING JOINT TORQUES...............................................................................
2.3 CONTROL ................................................................................................................ 19
19
2.3.1 Position (Trajectory) Control .....................................................................
20
2.3.2 Force Control.............................................................................................
20
2.3.3 Hybrid Position/ForceControl...................................................................
23
2.3.4 Controller Options......................................................................................
2.1.2
3
Physicalvs. Virtual Compliance................................................................
H A RD W A R E D E SIG N ........................................................................................
25
REQUIREM ENTS AND G OALS...............................................................................
25
3.1.1
NonlinearStiffness......................................................................................
26
3.1.2
3.1.3
3.1.4
3.1.5
Compact.....................................................................................................
Modular..........................................................................................................26
Robust (Durable) .......................................................................................
Scaleable.....................................................................................................
26
3.2.1
3.2.2
Description.................................................................................................
Satisfaction of Requirements ....................................................................
27
28
3.1
26
27
3.2 W RAPPING SPRING CONCEPT ................................................................................27
3.3
M ODIFICATIONS & IMPLEMENTATION ISSUES.........................................................30
3.3.1
3.3.2
3.3.3
3.3.4
3.4
Addition of a Cam to Improve the Stiffening Characteristic.....................
Spring Term inations .................................................................................
Manufacturingtolerances.........................................................................
Measuringthe D isplacement ....................................................................
D ESIGN VARIABLES ............................................................................................
3.4.1
3.4.2
Maxim um Torque........................................................................................
Torque Dynamic Range .............................................................................
30
30
30
31
33
33
33
8
3.5
CONFIGURATIONS ...............................................................................................
3.5.1
3.5.2
3.6
4
Capstan Configuration.............................................................................
Coupling Configuration..............................................................................
W RAPPING SPRING A NALYTICAL M ODEL ...............................................................
DESIGN EVOLUTION AND RESULTS..............................................................37
4.1
PROOF OF CONCEPT .............................................................................................
4.1.1 Sim ulation of the Wrapping Spring Concept.............................................
4.1.2 Initial Elem ent Prototype...........................................................................
4.2 ONE D EGREE-OF-FREEDOM TEST STAND ...........................................................
4.2.1 Description.................................................................................................
4.2.2 Experim ental Procedures...........................................................................
4.3 V ERSION 1 COM PLIANT ELEMENT PROTOTYPE...................................................
4.3.1 Objective ........................................................................................................
4.3.2 Description.................................................................................................
4.3 .3 R e s u lts ............................................................................................................
4.4 V ERSION 2 COM PLIANT ELEMENT PROTOTYPE...................................................
4.4.1 Objective ........................................................................................................
4.4.2 Description.................................................................................................
4 .4 .3 R e su lts ............................................................................................................
4.5 COMPLIANT ELEMENT FINAL DESIGN .................................................................
4.5.1
4.5.2
5
Objective ........................................................................................................
Description.................................................................................................
37
37
38
39
39
39
40
40
41
42
44
44
44
45
49
49
49
COMPLIANT ARM DESIGN FOR DIGGING (CADD).................51
5.1
5.2
6
33
34
34
34
O BJECTIVE ...........................................................................................................
D ESCRIPTION .......................................................................................................
C O N C LU SIO N S ..................................................................................................
6.1
6.2
REVIEW OF THESIS W ORK ......................................................................................
FUTURE W ORK........................................................................................................53
BIBLIO G R A PH Y ............................................................................................................
51
51
53
53
55
MATLAB CODE OF ANALYTICAL WRAPPING SPRING
APPENDIX A.
57
M O D EL .............................................................................................................................
APPENDIX B. COMPLIANT ELEMENT PART DRAWINGS..............61
9
Figures
Figure 1. Series Elastic Actuator Spring Designed by Williamson.............................12
13
Figure 2. Compliant Capstan Designed by Shah .........................................................
13
Figure 3. NSCA Element Designed by Katz ...............................................................
Figure 4. Phase Portrait and Equations for a Mass-Linear Spring System...................17
Figure 5. Phase Portrait and Equations for a Mass-Exponential Spring System...........17
Figure 6. Hybrid Position/Force Controller for a Compliant Joint Manipulator..........21
28
Figure 7. W rapping Spring Concept.............................................................................
Figure 8. Inner Shaft with Original (Concentric) Shaft Adapter and Cam...................30
Figure 9. Original Potentiometer Signal Conditioning Circuit....................................32
32
Figure 10. Improved Potentiometer Signal Conditioning Circuit ...............................
34
Figure 11. Model Predictions and Sampled Data .........................................................
37
Figure 12. Working Model 3D Simulation Images ......................................................
38
Figure 13. Working Model 3D Simulation Results ....................................................
39
Figure 14. 1-D O F Test Stand........................................................................................
41
Figure 15. Version 1 Compliant Element Prototype ....................................................
42
Figure 16. Shaft Pin D etail............................................................................................
42
Figure 17. "Top Hat" Retaining Ring Clasp Detail ....................................................
Cylinder
without
Secure
Figure 18. Version 1 Static Test Results without Cam and
43
T erm in atio n ................................................................................................................
Figure 19. Version 1 Static Test Results with Cam.....................................................43
44
Figure 20. Version 2 Compliant Element Prototype ....................................................
45
Figure 21. Tw o V iew s of the T-block...........................................................................
Figure 22. Version 2 Static Displacement vs. Torque Test Results with Cam.............46
Figure 23. Typical Torque Sensitivity Test Data.........................................................47
Figure 24. Version 2 Torque Sensitivity vs. Applied Torque......................................47
48
Figure 25. Version 2 Motor Driven Step Response Test Data .....................................
Simulations............49
Test
Data
and
Response
Offset
Step
Figure 26. Version 2 Initial
50
Figure 27. Final Element Design - Exploded View .....................................................
Figure 28. Compliant Arm Design for Digging CAD Model......................................52
Figure 29. Compliant Arm Design for Digging...........................................................52
Figure 30. Compliant Element Part Overview..............................................................61
Tables
Table 1. Linear and Exponential Spring Displacements for the Same Force...............17
Table 2. Summary of Encoder and Potentiometer Attributes......................................31
Table 3. Index of Compliant Element Part Drawings...................................................61
11
1 Introduction
1.1
Overview and Scope
The commonly accepted robot design methodology for serial chain manipulators is to
make the mechanism as stiff as possible so that the endpoint position can be calculated
with reasonable accuracy given measurements of the joint angles. To perform force
controlled tasks with these manipulators, some type of compliance is usually incorporated
in to the system. This compliance may be implemented using hardware or software
methods and can be active (i.e. adapting to sensor data) or passive (e.g. compliant
coverings). Besides enabling force control, hardware based compliance techniques can
reduce peak shock loads on the motor/transmission assembly and have the potential to
store energy, thus making the manipulator safer and more efficient. The consequence of
adding compliance to the manipulator is that the trajectory control bandwidth and
possibly the positioning accuracy will be reduced. For certain natural tasks, such as the
process of extraterrestrial excavation that drives this research, this trade can be justified.
This thesis presents the design of a nonlinearly compliant transmission element
developed to improve robot force sensing and control capabilities. It was designed to
meet requirements regarding nonlinear compliance, compactness, modularity, robustness,
and scalability. The choice of a nonlinear, stiffening compliance characteristic was made
to increase the force dynamic range of the mechanism without introducing unduly large
displacements. The other requirements are intended to drive the development of a
mechanism that may be integrated into both new and existing robot manipulators with
minimal redesign effort.
The proposed mechanism is based on the concept of a wrapping spring that undergoes a
continuous decrease in spring free length and a resulting increase in stiffness as the
displacement is increased. The mechanism was evaluated using a one degree-of-freedom
test stand and a four degree-of-freedom robot arm was designed to demonstrate
integration of the element into a manipulator and to conduct evaluations of manipulator
control techniques.
1.2
Brief Survey of Prior Transmission Compliance Work
Most research in the area of transmission compliance and joint flexibility has focused on
developing control methods to compensate for this compliance with the objective of
improving the trajectory tracking performance [Book, 1991; Hung, 1991; Readman,
1994; Spong, 1987]. The general assumptions associated with these research efforts are
that the joint flexibility can be modeled by a linear spring, that the magnitude of the
displacement caused by the flexibility is small, and that the natural frequency of the
compliance is high compared to the bandwidth of the total arm motion. While these
research efforts can provide useful guidance for modeling, analysis, and control
techniques, the results are not directly applicable to the current research, which is
interested in force control and which violates all of these assumptions to some degree.
12
Many other researchers have investigated active control of joint compliance using
additional motors or other devices. [Sugano, et. al., 1992] designed a three actuator
mechanism to simultaneously control the position, compliance, and damping
characteristics of a robotic finger joint. [Morrell, 1996] investigated the use of separate
but coupled position and force control motors for each joint.
Recently, there have been some efforts to use passive joint compliance devices to provide
improved force control capabilities. By measuring the deflection of a properly designed
passive element, much of the benefit of the actively controlled methods can be achieved
without a significant increase in the complexity of the original control system.
In an effort to improve the capabilities of the humanoid robot Cog at the MIT Artificial
Intelligence Lab, Williamson developed a passive compliant transmission mechanism
called a series elastic actuator [Williamson, 1995] and integrated it in to Cog's arms to
facilitate "natural" movements. These series elastic actuators are essentially linear
torsion springs with a cross shaped cross section (see Figure 1) that are used as a torque
coupling between the drive trains and the respective arm links. The deflection of the
spring is measured using strain gauges. In practice, the series elastic elements in Cog's
arms have demonstrated a successful implementation of a passive joint compliance
technique.
Figure 1. Series Elastic Actuator Spring Designed by Williamson
The series elastic element concept was carried forward by the MIT Leg Lab during their
development of several walking robots. [Matteo, 1997] provides a comprehensive
overview of their efforts that resulted in a self contained linear actuator that uses
compression springs to produce compliance.
During research to develop a more capable and modular robotic finger, [Shah, 1997]
designed a compliant capstan that provides a nearly exponential torque vs. angular
displacement characteristic. This performance was achieved through the use of 6 Buna-N
rubber balls in pie shaped slots formed by extrusions from opposite sides of the element
(see Figure 2). The balls are compressed in the slots when the two sides of the element
are rotated relative to each other. This relative rotation is measured with a potentiometer.
While this is an elegant and compact mechanism that exhibits the desired stiffening
behavior, there are a few drawbacks. First, in the high torque region, the two sides of the
element are separated by the axial forces exerted by the compressed balls, resulting in
inconsistent behavior. Second, the use of rubber may be appropriate for terrestrial
13
applications, but the potential pressure and temperature extremes of extraterrestrial
applications would greatly degrade the performance of the rubber. Third, under repeated
high load conditions, the rubber is likely to undergo some permanent deformation that
would adversely effect the performance.
Extrusions
Figure 2. Compliant Capstan Designed by Shah
In concurrent work to this research, [Katz, 1999] designed a nonlinear series compliance
This
actuator (NSCA) with a stiffening force vs. displacement characteristic.
transmission
drive
cable
a
in
pairs
in
used
be
to
is
intended
concept
spring
compression
to achieve joint compliance. The element uses a conical spring to achieve the desired
nonlinear stiffening behavior and the displacement is measured using a linear
potentiometer.
II
Figure 3. NSCA Element Designed by Katz
Outline of Thesis
1.3
Section 2 discusses the theoretical issues associated with the use of exponentially
stiffening compliant elements in a robot. Following a description of the value of
exponentially stiffening springs relative to linear springs in Section 2.1, Section 2.2
provides an overview of the issues associated with the interpretation of joint forces to
14
calculate applied loads. Section 2.3 provides an overview of control techniques that are
applicable to flexible joint manipulators.
Section 3 provides a description of the hardware design process of the nonlinearly
compliant transmission element. Section 3.1 defines the requirements used to develop
the hardware and Section 3.2 describes the wrapping spring concept that is the basis for
The following sections elaborate on some of the
the new mechanism design.
implementation and integration issues associated with incorporating the element in to
manipulator hardware. Section 3.6 presents an analytical model of the wrapping spring
mechanism that provides a reasonable estimate of actual performance.
Section 4 describes the evolution of the design from the initial simulations and the
evaluation of prototypes through to the final design of the element that will be integrated
into the Compliant Arm Designed for Digging described in Section 5.
Finally, Section 6 presents some conclusions of this investigation along
recommendations for follow up activities.
with
15
2 Theory
2.1
The Utility of Exponential Joint Compliance
In general, the benefits of adding compliance to a serial manipulator include the potential
to improve force control, to improve safety, to reduced wear, and to increase energy
efficiency. The potential improvements in force control capabilities are derived from the
dynamic stability characteristics of the combined manipulator and environment system.
When both the robot and the contacted environment are rigid, small motions can generate
large contact forces. Joint compliance alleviates these loads. However, the original rigid
manipulator controller must be appropriately modified to prevent introducing new
unstable behaviors through excitation of the added flexibility. Safety and reduced wear
of the gearhead and motor are a direct result of the lower impact loads experienced by a
serial manipulator that incorporates compliance. Furthermore, since the compliant
components can store energy, the energy efficiency of the manipulator may be improved
if an appropriate control strategy is employed.
The two primary methods of adding compliance to robot manipulators, other than
through joint compliance, are to use flexible links or to add a compliant covering to the
parts of the manipulator that contact the environment. The major benefit of joint
compliance compared to these other methods is that a relatively simple measurement of
the joint compliance displacement is all that is required to calculate the endpoint position
with the same accuracy as a rigid robot. With flexible links, a sophisticated combination
of strain gauges and dynamic models are usually required to calculate accurate endpoint
positions. The use of compliant coverings tends to insulate the manipulator from the
environment and can introduce uncertainty about the exact point of contact even if
contact sensing capabilities are incorporated into the covering.
The key attribute of an exponentially stiffening compliant spring is that the incremental
displacement, and thus the measurable incremental force, is a constant percentage of the
applied load [Salisbury, in Mason and Salisbury, 1985]. To put it another way, an
exponentially compliant element can provide a constant force resolution over the entire
force range of the joint. This relationship can be derived by examining the following
torque versus displacement equation for an exponential spring:
e
Equation 1.
O
where 0 is measured from the neutral, or zero displacement position. An incremental
change in the torque is represented by:
Equation 2.
6t = AeA06 ,
and thus the torque resolution is given by the ratio:
Sz,
=
AeA196
eAO
Se
A
= A80.
Equation 3.
16
Equation 3 expresses the torque resolution as a linear function of displacement and
illustrates how the compliant element displacement angle measurements will provide
torque data with a constant resolution over the entire torque range.
2.1.1
Exponential vs. Linear Compliance
The force dynamic range of a mechanism is defined as the ratio of the maximum
controllable force divided by the minimum controllable force. It serves as an indication
of the sensitivity and the range of the force sensing capability, with larger values being
better. For reference, few multi-degree-of-freedom manipulators have a dynamic range
better than 100 while the dynamic range of a human finger is on the order of 10000.
When attempting to achieve a large dynamic range using a compliant force detection
mechanism, linear springs are less practical than stiffening springs since they require a
significantly larger displacement to measure high forces, assuming the sensitivities of the
two are the same at low forces.
The following equations demonstrate this property for the ideal case of an exponentially
stiffening spring. Without loss of generality, the linear and exponential springs can be
assumed to start at a neutral position where zero displacement corresponds to zero force.
Equation 4 and Equation 5 represent the force equations for a linear spring and an
exponential spring, respectively. Equation 6 and Equation 7 show the results of solving
the first two equations for the displacement as a function of force.
Equation 4.
f= kx
g
x
k(e
=
- 1)
Equation 6.
f/k
y=IlnC1
A
Assuming that k
Equation 5.
+1)
Equation 7.
k
=
1 and solving for a force of f= g = 1, the resulting displacements are:
x = 1/1 =1
Equation 8.
y = ln(1/1 + 1)/A = ln(2)/A = .6931/A
Equation 9.
In Equation 9, the parameter A can be chosen to be .6931 so that the applied force of I
causes a displacement of 1 in the exponential case to match the displacement in the linear
case. Using these values for k and A, Equation 6 and Equation 7 can be used to
demonstrate the drastically greater displacements necessary to measure or apply larger
forces. Some results are shown in Table 1. By the time the dynamic range reaches 1000,
the displacement of the linear spring is approximately 100 times greater than the
displacement of the exponential spring.
17
Table 1. Linear and Exponential Spring Displacements for the Same Force
Force
X (Linear
Y (Exponential
1
10
100
1000
Spring)
1
10
100
1000
Spring)
1
3.4597
6.6587
9.9679
Some insight in to the relative stability of an exponentially compliant mechanism can be
obtained by examining phase portraits. As with a mass-linear spring system, the phase
portrait of a mass-exponential spring system is of the form of a center point, indicating
marginal stability (stable in the sense of Lyapunov) and the potential for limit cycle
behavior (see Figure 4 and Figure 5). However, the difference in the case of the
exponential spring is that the curves are not circular (or elliptical in the general case), but
instead follow a more rectangular path. Of course, near the origin, the phase portrait of
the exponential spring approaches a circular appearance, as would be expected since a
linear approximation is applicable over this region.
X
m3 + kx = 0
x2 +x2 =C
Figure 4. Phase Portrait and Equations
for a Mass-Linear Spring System
m+k(eAx -1)=0
+ 2(A-LeAx
- x)=
C
Figure 5. Phase Portrait and Equations
for a Mass-Exponential Spring System
The addition of damping, primarily due to friction, to either of these cases will tend to
change the phase portraits into stable foci (asymptotically stable).
2.1.2 Physical vs. Virtual Compliance
While the use of virtual compliance in the form of a control algorithm based on
force/torque sensor data can be used to artificially create a wide range of compliance
behaviors in a manipulator, the addition of physical compliance has a number of
advantages in the context of a planetary exploration robot. Most of these benefits are
18
derived from the fact that the use of physical compliant mechanisms changes the
fundamental open loop dynamics of the manipulator. Virtual compliance relies on closed
loop routines and control algorithms that may be limited by sampling and control
frequencies, noise, and actuator saturation.
First, virtual compliance algorithms require the use of force/torque data to gather the data
needed to issue force control commands. The use of an explicit force torque sensor
becomes redundant when the proposed compliant elements are used since each joint
torque can be measured directly.
Second, physical compliance enables safer, more robust interaction with the environment.
While software implemented compliance is dependent on the continued correct operation
of all sensor and motor control hardware and circuitry, a physically compliant robot will
maintain its compliant behavior when these components fail and even when power is
removed from the control system entirely. The trade is an introduction of some physical
component failure modes, which are likely to be more manageable on long duration space
flights than failure modes for electronic components.
Third, the physical compliant elements have the capacity to store energy, which can not
be done using virtual compliance methods. This stored energy may be useful during
digging or shoving tasks and may even augment striking tasks if a back swing motion of
the appropriate amplitude and frequency is used.
2.2
InterpretingJoint Torques
Given the kinematic configuration of a particular robot, the measured joint torques
represent a set of wrenches located at the joint positions and aligned with the joint axes.
The resultant force represented by the summation of these wrenches can be transformed
in to the end effector coordinates, the base coordinates, or any other applicable coordinate
system using kinematic equations, joint angle measurements, and transformation matrices
[Bicchi, et. al., 1990, Murray, et. al., 1994]. The calculation of an applied load using
joint angle torque measurements is dependent on the kinematic configuration, knowledge
of the unloaded dynamic characteristics of the manipulator, and the number of applied
loads.
First, by their nature, joint torque measurements are in the same directions as the joint
motion. Therefore, to acquire a complete six degree-of-freedom characterization of an
endpoint force/torque couple requires that the manipulator have at least six joints and that
it is not in a singular configuration. Using the manipulator Jacobian (Equation 10) and
the duality principle, the joint torques are seen to be related to the endpoint forces by the
transpose of the Jacobian matrix (Equation 11). For loads applied at locations other than
the endpoint, the same process applies with an appropriately modified Jacobian matrix.
=
Equation 10.
A
,r = j
F
Equation 11.
19
For many tasks, such as the basic trenching tasks expected for future planetary missions,
the three components of the endpoint force are of primary interest and the endpoint
torques are usually not particularly useful. In these cases, a minimum of three torque
sensing compliant joints could provide complete force data.
Second, the measured joint loads are the sum of the internal loads due to gravity, inertia,
and motion effects (Coriolis and centrifugal forces) and of the externally applied loads.
One way to separate these components to identify the contribution of the external load is
to use an analytical dynamic model of the manipulator, such as Equation 12, to predict
the internal loads. Alternatively, empirical data may be gathered by maneuvering an
unloaded manipulator throughout its workspace at different speeds and in different
directions to characterize the system before using it with applied loads.
H(O6+
C(6, 0)0 + G
Equation 12.
Third, to accurately determine the magnitude of external loads on the manipulator from
joint torque measurements, the number and location of the external loads must be
determined or assumed. For multiple external loads on a single link of the manipulator,
the joint torques can only be used to calculate the magnitude of an equivalent resultant
load applied at some point on that link. For the cases in which there are at least one joint
between two applied loads, the in-between joint data will reflect the effects of one load
while the remaining joints will measure the resultant of the two loads. If the kinematic
alignment is favorable and if there are enough joints between the two loads, both may be
calculated. The control system could be designed to suspect multiple forces if the
measurements from the last one or more joints indicate a sufficiently different force than
the other joint measurements.
2.3
Control
This section provides a brief description of the fundamental control methods used with
flexible joint manipulators. The method chosen for a particular manipulator will depend
on the task description.
2.3.1
Position (Trajectory) Control
Most techniques for controlling joint flexibility during trajectory commanding boil down
to a separation of the controller in to a fast inner force control loop and a slower outer
position control loop [Book, 1991; Hung, 1991; Readman, 1994; Spong, 1987]. The goal
of the inner loop is to counteract the joint flexibility and thus virtually stiffen the joint.
This control technique is based on the assumption that the joint dynamics are relatively
fast and of small amplitude when compared to the dynamics of the entire arm. This is not
a good assumption for the designed compliant element, which introduces enough
compliance to dominate most other sources of manipulator flexibility. Furthermore,
since virtually stiffening the joints is counter productive to the objective of providing
compliance to facilitate safe interactions with the environment, a different strategy must
be employed.
20
Another difference between the proposed compliant element and the standard treatment
of flexible joints is that the compliance is measured. This provides a direct measurement
of the joint angle for use in the control algorithm rather than relying on a model based
estimation. Thus, the dual nature of the measured compliant element potentially enables
better force control as well as better positioning accuracy than previous flexible joint
concepts.
2.3.2 Force Control
When the manipulator is constrained by its environment in one or more directions, it
becomes desirable to control the contact forces in those directions using a force
controller. The successful use of another compliant element design to perform force
control was demonstrated by [Shah, 1997]. As with the current compliant element, the
measured feedback consisted of the motor angle and the compliant element displacement
angle for each joint. The controller consisted of an outer, slower, PID torque control loop
and an inner, faster, PD motor position control loop.
2.3.3 Hybrid Position/Force Control
The fundamental concept of hybrid control schemes is to combine the utility of both
position and force controllers in to a single controller that will automatically issue
trajectory commands in the unconstrained directions and force commands in the
constrained directions.
Figure 6 illustrates one such control scheme. The inputs to this system are the vectors
defining the commanded position of the manipulator endpoint in the fixed (base)
reference frame and the desired endpoint force vector, also expressed in the reference
frame. The measured outputs are the motor angles (from the motor encoder) and the
compliant element displacement angles (from the potentiometer). The controlled state
variables are the joint angles and the compliant element displacement angles. Use of
these state variables allows decoupled control of the endpoint position and force through
the kinematic and compliant element stiffness relationships respectively.
21
0
Xd
Inverse
Kinematics
AO
Od
A'AO
Trajectory
Control
(3p
+
T
cmd
=
cto Ir
1m 'Tm
N
tT
gemTra
Motor
Equation
oo
V.
Robot
+
faT
J
Td
-- > -Ce
-1
Ce
Limiter
Tf
Oced+,~
e'~'A,
A0~ KlA
Gf
-
Force
Fre
Control
-
Environment
t
0~0
Ke
ce
Forward
Dynamics
+
'Txt
Q4
x
Tn
'y0O
2
rY
'=HO+GS
0
A
Figure 6. Hybrid Position/Force Controller for a Compliant Joint Manipulator
The following is a description of the components of this hybrid controller concept.
Inverse Kinematics - The inverse kinematic relationship from manipulator endpoint
position to joint angles.
Manipulator Jacobian, J 1 - This is the standard Jacobian that relates the joint rates to
the endpoint rates.
= J1 6
Equation 13.
Transmission Jacobian, Jt - This matrix is the transfer function from the output side of
the compliant elements to the joint angles. When the compliant elements are collocated
with the joints, the transmission Jacobian reduces to the identity matrix.
0 = J,0t
Equation 14.
Compliant Element Stiffness Function, Kce - The compliant element stiffness has been
deliberately designed to be nonlinear. In the ideal case, it can be represented by an
exponential function of the displacement angle,
T = sign(O,,) * A(e BIO,,l _ 1)
Equation 15.
22
where A and B are constants. The specific function used will be determined by the
characteristics of the actual stiffening compliant element used.
Inverse Compliant Element Stiffness Function, Kee-' - This function is necessary to
transform the commanded forces into pseudo joint angle commands for combination with
the position control inputs to generate the composite motor control commands. In the
ideal case of an exponentially stiffening compliant element, the inverse stiffness
relationship is a logarithmic function of the torque applied to the compliant element.
Oce
= sign(r) * [1-%ln
B (A)
I
Equation 16.
Gear Ratio, Ng - This matrix represents the total gear ratio between the motor output and
the input to the compliant element, including any intermediate transmission coupling
effects.
Motor Equation - The motor equation is derived from the motor specifications to
transform the commanded torques into appropriate voltage signals to drive the motors.
For the simplified case when the motor dynamics can be neglected, the motor voltage is a
linear function of the commanded torque derived from the motor constant, the rotor
inertia, and the terminal resistance. For the full dynamics case, the motor equation will
also include rotor velocity and acceleration dependent terms.
Forward Dynamics - A forward dynamics calculation of the joint torques from the
measured joint angles and their derivatives provide a torque vector that is associated with
the actual velocity of the manipulator endpoint. As shown in Figure 6, the traditional
serial chain dynamics equation has been simplified to eliminate the Coriolis and
centrifugal terms based on the assumption that the joint velocities will be relatively slow.
Selection Matrices, a'f and a'p This hybrid control scheme employs a slightly different implementation of the selection
matrices than a typical hybrid controller. Rather than being in Cartesian space, these
selection matrices are in joint space and represent the relative weights on the torque and
position controller commands passed to each joint motor.
The torque calculated by the forward dynamics equation is subtracted from the measured
joint torques to determine the component of the measured torque that is generated by
external forces on the robot. This component is assumed to be in the direction that must
be force controlled. Thus, the 'f selection matrix is calculated as the normalized
diagonalization of the external torques calculated for each joint. The u's selection matrix
is then calculated by subtracting O'f from the Identity matrix.
[
Te'=
I
ext
0
0
0
*.
0
0
Equation 17.
23
2.3.4 Controller Options
In the above discussions of position, force, and hybrid control, the specific control laws
were not identified. While PID control laws are common and can generate adequate
control commands under many circumstances, some model based control laws may be
able to improve performance. It is not the intent of this section to review all possible
methods for controlling manipulators with compliant joints, but to point out a few of the
options. The four degree-of-freedom arm described in Section 5 has been designed and
constructed to further explore control techniques for digging, trenching, scooping, and
shoving tasks that might be expected of a planetary exploration manipulator.
PID Controllers
The use of PID controllers with the proposed nonlinear compliant element requires the
application of some method to perform gain scheduling based on the current
displacement angle (the current torque on the joint). One approach, demonstrated by
[Shah, 1997], is to select gains for the current cycle that will produce a (constant) desired
control bandwidth. This gain selection process relies on a locally linear approximation of
the compliant element performance in its current configuration. Alternatively, one could
adapt this process to choose optimized gains that exhibit increased bandwidth when the
actuator is subjected to higher loads and the compliant element is relatively stiff.
Model Based Controllers
Control systems that incorporate a dynamic model of the manipulator, such as computed
torque control, sliding control, and adaptive control, can often provide better trajectory
tracking performance than PID controllers, but at the cost of increased computational
workload. However, for a control system like the hybrid controller described above that
already incorporates a dynamic model of the manipulator (to differentiate internal from
external forces), the additional workload is minimal.
25
3 Hardware Design
3.1
Requirements and Goals
This research effort was undertaken to develop a nonlinearly compliant transmission
element that is compact, modular, robust, and scaleable. The target application for this
mechanism is to improve the capabilities of interplanetary exploration robots. The
fundamental goals are threefold:
*
Improve force control over a wide dynamic range,
*
Simplify actuator design and instrumentation, and
*
Increase overall manipulator durability.
Improvements in force control are necessary to facilitate dexterous interaction of
manipulators with the environment. To date, operational space manipulators such as the
Shuttle Remote Manipulator System and the experiment arm on the Mars Sojourner rover
have been position controlled devices without any force feedback capabilities. In
recognition of the utility of force control to enable more dexterous operations, the
partners in the International Space Station program plan to incorporate force/torque
sensors in to the smaller, precision task robots such as the Special Purpose Dexterous
Manipulator (SPDM) and the Japanese Small Fine Arm (SFA) [Brimley, et. al., 1994].
Additionally, improved manipulator dexterity through design as well as through software
algorithms is one goal of NASA's Planetary Dexterous Manipulator program at the Jet
Propulsion Laboratory (JPL) to enable more productive remote geology operations on
Mars and possibly on other celestial bodies [Das, 1999].
With this recognition of the value of force control comes the engineering challenge of
designing improvements. The power, mass, and volume limitations as well as the harsh
operating environment of space and on foreign planets, etc., lead to the adoption of the
goal for a simple design. The compliance should be implemented in a minimal package
(mass and volume), that uses a minimal amount of power (for sensors and computation),
and that has built in redundancy or graceful degradation.
For space manipulators to become a more useful tool for exploration and investigation,
they need to become more robust to external disturbances and capable of sustaining
extensive intentional and unintentional contact with their surroundings. The current
generation of arms is often operated to maximize clearance between the manipulator and
the surrounding structure except for occasions when an end effector interaction is
required. Operations would be facilitated by not being concerned about incidental
contacts because the manipulator has enough compliance to absorb impacts with minimal
physical damage to itself or to the environment.
The specific requirements for this research effort are documented in the following
sections.
26
3.1.1
Nonlinear Stiffness
The case for exponential stiffness is made in Section 2.1 above. While this characteristic
is a reasonable goal, it is not necessary to match an exponential curve exactly to realize
the benefits of nonlinear compliance. Thus, the requirements are:
" The compliant element stiffness shall increase nonlinearly such that at high loads, an
incremental displacement measurement corresponds to a greater incremental force
than the same displacement measurement at a lower applied load. The ideal
characteristic for the nonlinear characteristic is exponential.
*
The compliant element stiffness characteristics shall be known (able to be modeled),
symmetric, and constant for the lifetime of the element.
3.1.2 Compact
Since the target application of this project is interplanetary robotics, it is important to
develop a component package that is sufficiently compact and power efficient to be
integrated into small, lightweight manipulators. Specifically:
" The compliant element shall have a size and mass comparable to a conventional
coupling or capstan used in current (space) manipulator designs.
" The compliant element shall not add significant power, data, or computation
requirements to the manipulator.
3.1.3 Modular
As part of the effort to meet the simplification goal, it should be possible to retrofit
compliant elements into existing hardware. Thus:
*
The compliant element shall be designed to be similar enough to existing capstans or
couplings to allow for retrofitting into existing robotic devices.
Additionally, satisfaction of this requirement should lead to a design that can be readily
replaced without significant disturbance to the rest of the manipulator, thus facilitating
repairs and upgrades or change outs to update the specific nonlinear compliance
characteristics for each joint.
3.1.4 Robust (Durable)
In order to survive the harsh environment of space for long periods of time, either in
operation or in transit to its destination, the compliant elements must be designed using
appropriate materials and techniques for minimizing performance degradation.
"
The compliant element shall use appropriate materials for space based applications.
*
The compliant element shall be designed to have minimal performance variation due
to ambient thermal and pressure conditions.
27
*
The compliant element shall be designed to withstand anticipated operational and
non-operational (i.e. launch and landing) dynamic and shock loads.
3.1.5 Scaleable
The two fundamental characteristics of the compliant element that must be scaleable are
the overall package size and the force range. The range of displacement for a rotary
element will be determined by the dynamic range and the maximum force requirements.
*
The compliant element shall be designed to be scaleable such that it can be
incorporated into robots from rover scale to space station scale.
3.2
Wrapping Spring Concept
Upon examination, none of the previously developed compliant element mechanisms
fully satisfied the design requirements just presented. The [Williamson, 1995] element
has a linear torque vs. displacement characteristic and is not particularly compact. The
[Shah, 1997] element has an appropriate nonlinear characteristic and is very compact, but
it uses inappropriate materials for space applications. Thus, a new mechanism concept
was sought that could satisfy all of the requirements, leading ultimately to the wrapping
spring concept that is employed by the compliant elements evaluated in this research.
3.2.1
Description
Figure 7 illustrates the fundamental components of the wrapping spring concept,
including the central shaft (blue), a concentric cylinder (red), and two opposing springs
(yellow and green). The springs are semicircular wire forms with 90' hook terminations
on both ends. The bottom terminator must be securely fastened to the center shaft while
the top terminator is fastened to the inside of the cylinder. As the cylinder rotates
concentrically with respect to the shaft, one spring will begin to wrap around the shaft as
the other is pressed against the inside wall of the cylinder. Both of these results have the
effect of shortening the free length of the springs. Thus, while the springs themselves
have a linear stiffness characteristic, the progressively shorter free length causes a
nonlinear increase in torque per unit displacement in the mechanism. The maximum
relative rotation of the cylinder relative to the shaft is limited by the spring being
compressed against the inner wall of the cylinder.
28
Figure 7. Wrapping Spring Concept
3.2.2 Satisfaction of Requirements
Nonlinear Stiffness
While the modeling and initial simulation efforts predicted a desirable spring behavior,
the actual performance of the Version 1 prototype did not exhibit as much of a stiffening
profile as desired (see Section 4.3). This motivated the examination of the addition of a
non-concentric cam element to the inner shaft (see Section 3.3.1). This design
modification succeeded in creating a more desirable stiffening characteristic.
The stiffness behavior of the mechanism does not fully satisfy the requirement for
symmetry or constant behavior. As discussed further below, the test results indicate a
definite asymmetry in the torque vs. displacement performance of the element that is
likely due to a combination of part dimension uncertainty and/or assembly misalignment.
Also, the mechanism has been observed to exhibit hysteresis -- the displacement
measurements can be affected by the previous state of the apparatus and whether it is
being loaded or unloaded.
Compact
The final design of the nonlinearly compliant element is 1.5" in diameter by 1.25" long,
including clearance for the potentiometer wiper element. At this size, it is no larger than
a standard capstan element that would have otherwise been used to implement the cable
transmission that was chosen for the four degree-of-freedom manipulator described in
Section 5.
The power requirements for the designed element are minimal. Only a 5 Volt power
supply using less than 10 gA is required to drive the potentiometer signal. The analog
output voltage signal from the potentiometer does require some signal conditioning
(described in Section 3.3.4) before it can be read by a typical A/D converter.
29
Modular
The most significant aspect of the final compliant element design that reduces its
modularity is the need to modify the inner shaft with holes for the springs. For
manipulator designs with readily removable shafts, such as the shoulder yaw joint of the
four degree-of-freedom manipulator described in Section 5, the designed element is
entirely modular.
When used in the capstan configuration, the element is of an appropriate size to be
For use in a shaft coupling
directly interchangeable with standard capstans.
configuration, the design requires a modification to the end cap to incorporate a shaft
clamp, pin, or set screw feature. At 1.5" in diameter and approximately 1.75" long (to
accommodate the end cap modification), this compliant element is about 1.5 to 3 times
larger than a standard shaft coupling for a 0.25" diameter shaft.
While different sets of springs (different spring wire diameters) will require
modifications to the termination hardware (the T-block, shaft, and shaft adapter parts), all
other parts are reusable.
Robust (Durable)
This design uses metal springs to meet the materials requirement for robots designed for
operation in space. While [Shah, 1997] demonstrated promising results with compliant
elements using rubber balls as the compliance mechanism, rubber may not be a wise
choice for use in space where the temperature and pressure extremes can make the rubber
brittle.
While metal springs are less affected by thermal and pressure conditions, the prototypes
have revealed that the performance of the mechanism is very sensitive to the exact
relative geometries of the parts. Therefore, further thermal analysis would be prudent to
evaluate whether the use of metals with different thermal coefficients (steel and
aluminum) is acceptable, or if a single material (steel) must be used.
Scalable
The physical package size of the compliant element is inherently scalable to fit virtually
any shaft size. The force range of the element is scaleable by choosing different spring
wire diameters, by using multiple spring pairs, or by using flat springs of various widths.
Several design details are worth noting. First, the specific means for terminating the ends
of the springs must be carefully planned for each spring wire cross section and size.
Second, an appropriate rotary encoder that accommodates the selected shaft size must be
identified. Third, the minimum outer diameter of the compliant element will be
determined either by the chosen spring geometry (force dynamic range and maximum
displacement) or by the rotary potentiometer diameter. Fourth, the minimum length of
the element will be determined by the spring wire diameter (or width), the number of
spring pairs, the shaft bearing width, and the potentiometer width.
30
3.3
Modifications & Implementation Issues
3.3.1 Addition of a Cam to Improve the Stiffening Characteristic
The initial tests of the compliant element prototypes (see Section 4.3.3) indicated that the
torque versus displacement characteristic of the element was not exhibiting as much
stiffening as desired. To increase the stiffening behavior, a cam part was added to the
assembly. This cam is a cylindrical part with a 5/8" diameter (the original shaft adapter is
1/2" diameter) that is mounted on the inner shaft adapter in a non-concentric fashion as
shown in Figure 8. Thus, as the inner shaft rotates with respect to the cylinder, the radius
of the cam part in contact with the spring increases and causes a more rapid increase in
the stiffening behavior than the original design. Test results demonstrate the success of
this modification (Sections 4.3.3 and 4.4.3).
Figure 8. Inner Shaft with Original (Concentric) Shaft Adapter and Cam
3.3.2 Spring Terminations
The springs in this mechanism must have rigid terminations that will accommodate the
maximum anticipated torque loads. Secure terminations are vital to maintaining the
required component geometry that is responsible for generating the nonlinear
performance. Depending on the spring wire diameter and the relative dimensions of the
shaft, the cam, and the wire curvature and cylinder diameters; it may be necessary to use
different means to adequately terminate the springs. Several methods were explored
using the prototypes, including a pin restraint, a retaining ring restraint, and a setscrew
restraint. Descriptions of these methods and of how well they worked are included in
Sections 4.3.3 and 4.4.3.
3.3.3 Manufacturing tolerances
Since the stiffening characteristic of the element is a function of the geometry of the
springs and of the parts they contact, the manufacturing tolerances of all these elements
must be considered. While the spring wire diameter is extremely uniform (+/- 0.001" or
less for wire diameters of 0.1" or less), the wire forming process to create the spring
curvature and the termination segments can not be controlled with such precision. It was
necessary to select closely matched pairs of springs following the bending process,
31
especially for the Version 2 prototype, which used a larger spring wire diameter and thus
had smaller clearances between the springs and the other parts.
3.3.4 Measuring the Displacement
Choice of Sensor
The usefulness of the compliant element as a force sensor is dependent on the ability to
accurately measure the displacement angle. Both a through-shaft encoder and a rotary
potentiometer were considered for this purposes. Both were implemented on the one
degree-of-freedom test stand to evaluate their relative performance characteristics. A
summary of the attributes of both pieces of hardware is provided in Table 2.
Table 2. Summary of Encoder and Potentiometer Attributes
Attribute
Package Size *
Package Mass *
Encoder (HEDS 6505)
2.6" X 2.2" X .81"
30 g
4.63 in 3
Power Requirement
5 Volts, 5 mA max -> 25 mW
Resolution
1024 CPR
Absolute Zero
Output Signal
Additional Equipment
No
Digital
Mounting Hardware
Potentiometer (Novotechnik
Model P45a502)
1.5" Diam. X .32" = 2.26 in 3
3.3 g
5 Volts, 10 tA -> 50 jiW
Limited only by A/D
conversion process
Yes
Analog (Voltage)
Mounting Hardware
Signal Conditioning Circuitry
*Without mounting hardware.
PotentiometerIntegration
With instrument resolution and size as the most important attributes, the potentiometer
was chosen for use in the final design. The technical challenge associated with the
integration of a potentiometer in to the compliant element mechanism is to amplify the
output signal so that the compliant element range of motion (+/- Degrees) corresponds to
the A/D converter input range (+/- Volts) without introducing excessive noise.
Figure 9 provides an illustration of the original potentiometer signal conditioning circuit
used with the Version 1 prototype. The potentiometer acts as a voltage divider with R 1,
which was chosen to be large enough to keep the current below the 10 gA rating of the
potentiometer. The first operational amplifier circuit amplifies the potentiometer output
by -R1/R 2 and the second one inverts the signal so that the output is positive. Since R 2 is
the same resistance as the maximum potentiometer resistance, the output will be in the
range of 0 to 5 volts.
In practice, this circuit exhibited a significant amount of noise during the static tests of
the prototype. Figure 19 shows the wide variation of 40 consecutive potentiometer data
32
samples taken at each load condition.
without much success.
R,,
Various filtering techniques were attempted
R = 505 kQ
R2= 5 kQ
24 kQ
Vout = 0 to 5 V
R3
rR2
+R3
R2
:Pot,
Gain = RI/R2
Figure 9. Original Potentiometer Signal Conditioning Circuit
A second potentiometer circuit, modeled after the circuit used by [Shah, 1997] and shown
in Figure 10, was used with the Version 2 prototype. The first operational amplifier
divides and inverts the supply voltage to provide a voltage difference of -5 to 0 volts
across the potentiometer. The second operational amplifier provides a unity gain and
prevents any current from flowing through the output lead of the potentiometer. The last
operational amplifier provides a gain of -R4/R 3. The variable resistor R 2 is used to set the
zero bias of the output voltage as shown in Equation 18.
Equation 18.
R
R
R3
'
2
The performance of this improved circuit proved to be quite good during the static tests
of the Version 2 prototype. Figure 22 shows the dramatic reduction in the variation of 40
consecutive potentiometer samples at each load condition compared to those in Figure
19. However, a significant amount of noise was observed in the potentiometer signal
Since this amplifier and circuit board
when the motor amplifier was activated.
configuration are not the same as the one that will be used in the four degree-of-freedom
manipulator described in Section 5, further trouble shooting will be conducted with the
real hardware.
R= 138 kQ
R 2 = Oto 100
R1/3
R2
Ri
R4
kg
R3= 4 kQ
R4 = 37 kQ
Vout = -I to I V
+r
-
-R-1
Gain = R4/R3
15V'
-ot
Zero bias set by
adjusting R 2
Figure 10. Improved Potentiometer Signal Conditioning Circuit
33
Calibration
Calibration of the potentiometer output (using the circuit in Figure 10) consists of setting
the zero bias and establishing the conversion between output volts and the displacement
angle. The compliant element can be set to the zero displacement position by either
removing all applied loads or by applying a symmetric torque to the element. At this
position, the variable resistor R 2 is adjusted to establish the zero bias. It should be
emphasized that this zero output voltage is dependent on the supply voltage and on the
potentiometer wiper position. If either of these change, the zero bias must be reestablished.
Resistors R 3 and R4 will determine the conversion factor between the amplified output
voltage and the displacement angle. The gain (R4/R 3) should be selected so that the
maximum range of element displacement fills the A/D signal detection range. This will
maximize the resolution of the displacement measurements. A test set that includes an
encoder, such as the one degree-of-freedom test stand described in Section 4.2, can be
used to determine the calibration ratio of the selected resistors.
3.4
Design Variables
The two primary variables used to define a compliant element design are the maximum
torque and the torque dynamic range, as described in the following sections.
3.4.1
Maximum Torque
Since the proposed compliant element is intended to be installed at the joints, it must be
properly designed to accommodate the maximum dynamic loading conditions expected at
each joint. This is a significant factor in selecting the spring wire diameter or, in the case
of multiple or non-circular springs, the total spring cross sectional area. The maximum
load, in combination with the spring cross section, will define a minimum inner shaft
diameter for the spring to wind on.
3.4.2 Torque Dynamic Range
Selection of the desired torque dynamic range following the identification of the
maximum torque will drive many of the remaining design parameters. Most notably, the
dynamic range will drive the relative dimensions of the inner shaft diameter, the cylinder
diameter, and the spring radius of curvature. Consequently, these parameters will define
the outer dimensions of the compliant element as well as the details of the T-Block and
inner shaft termination designs.
3.5
Configurations
The compliant element proposed in this thesis can be used either in a capstan or in a shaft
coupling configuration as described in the following sections.
34
3.5.1
Capstan Configuration
The capstan configuration is intended for use in a cable transmission robotic system such
as the four degree-of-freedom manipulator described in Section 5. In this configuration,
the motor or gearhead output is rigidly coupled to the inner shaft and the cable
transmission uses the outer cylinder as a capstan.
3.5.2 Coupling Configuration
The coupling configuration is intended to be used as a replacement for a rigid shaft
coupling. Instead of extending the inner shaft through both end caps as in the capstan
configuration, the inner shaft terminates at the bearing on one side and the end cap is
modified to provide a rigid coupling between the cylinder and a second shaft.
Wrapping Spring Analytical Model
3.6
An analytical model was developed using geometric considerations and beam bending
theory to predict the performance of a mechanism based on the wrapping spring concept.
The model contains calculations for the contributions of both the winding and the
unwinding spring segments. For both cases, the model assumes ideal spring terminations
and that the entire induced torque is due to the bending moment of the remaining free
length of the spring at the current displacement angle. The model predictions and
sampled data for the .045" diameter springs of the Version 1 prototype and for the .092"
diameter springs of the Version 2 prototype are shown in Figure 11.
Displacement Torque: Version 1 CE with Cam, Model Prediction and Sampled Data
30
Displacement vs. Torque Version 2 CE with Cam, Model Prediction and Sampled Data
20
15
20
10
10
1D
-
0 -
c"-5In
* -10
-10
-20
-15
-500
-d00
-300
-200
0
100
-100
Static Torque (mNm)
200
300
400
500
-2000
-1500
-1000
0
500
-500
Static Torque (mNm)
1000
1500
2000
Figure 11. Model Predictions and Sampled Data
While the correlation between the model predictions and the sampled data is reasonably
good, there are two discrepancies that merit discussion. First, the measured data indicates
a nearly constant linear behavior of the mechanism near zero displacement that is not
reflected in the model prediction. This behavior is likely caused by some bending of the
spring termination segments and by preload stresses in the springs caused by part and
assembly variations. Second, there is a noticeable asymmetry of the measured data about
the zero displacement configuration. This deviation is also attributed to part variation,
35
most likely of the cam (not ideally aligned or circular), since the same asymmetry was
observed with two different sets of springs. Other part variations that could contribute to
this type of discrepancy include the spring radius of curvature and the precise alignment
of the T-block to terminate the springs symmetrically on the cylinder.
For reference, a Matlab script file of the wrapping spring analytical model is included as
Appendix A.
37
4 Design Evolution and Results
4.1
Proof of Concept
4.1.1 Simulation of the Wrapping Spring Concept
The initial proof of concept was accomplished using the Working Model 3D software
package from MSC.Software Corporation to conduct a quasi-static torque vs.
displacement simulation of the wrapping spring concept. The model was designed to
evaluate the winding half of the concept about the center shaft. The spring was modeled
as eight cylindrical solid bodies joined by linear torsion springs with neutral positions
chosen to create the semicircular spring shape (see Figure 12A). The top of the spring
was fixed while the bottom of the spring was constrained to move with the center shaft as
it was rotated. The surfaces of the center shaft and of the spring elements were defined as
solid contact surfaces.
A. Theta
0
B. Theta
18
Figure 12. Working Model 3D Simulation Images
The results of slowly rotating the center shaft and measuring the resulting axial torque
generated by the springs on the fixed support (corresponding to the outer cylinder) are
shown in Figure 13. Co-plotted with the measured data is the closest fit exponential
curve. While the results are not exactly exponential, they do trend in the desired
direction. One apparent artifact of the non-continuous spring model used in this
simulation is the appearance of noticeable increases in the stiffness as subsequent spring
elements make initial contact with the inner cylinder (at approximately 6, 13, and 18
degrees.) This behavior validates the wrapping spring concept as a means to generate a
nonlinearly stiffening mechanism. The other artifacts that appear as spikes between 8
and 10 seconds were caused by the contact model dynamics of the Working Model 3D
simulation that were occasionally excited as the shaft rotated.
38
8 Element Winding Spring Model -- Linear Torsion Spring/Damper Joints
--
Diameters: 6,12,1.5,9
Wnrking Madel 3D
Y k n.04865*exp(0.2601*theta)
0.5
0.4
0.3
0.2
0.1
-
0
2
4
6
12
8
10
Relative Twist Theta (Degrees)
14
16
18
20
Figure 13. Working Model 3D Simulation Results
4.1.2 Initial Element Prototype
An initial, uninstrumented prototype element was constructed to help identify the
assembly and manufacturing issues for the compliant element. It was constructed using a
0.5" diameter aluminum shaft, an acrylic, semi-transparent cylinder with an inner
diameter of 1.0", two .045" diameter wire springs with a radius of curvature of 0.375",
and two bearings. The ends of the springs were only passively secured in the shaft and in
the cylinder by the geometry of the spring termination segments.
The initial element prototype provided some useful insights:
"
Secure spring termination techniques are required for proper performance;
*
Performance is dependent on the spring mounting angle -- springs that are not
mounted with the plane of curvature perpendicular to the inner shaft have less contact
with the inner wall of the cylinder and thus will have a greater range of motion;
"
The method for assembling the springs with the inner shaft relies on the flexibility of
the springs to expand around the shaft during assembly - modifications to the inner
shaft will be required to accommodate less flexible springs;
"
The approximate range of motion (± 300) for the geometric configuration used
(0.1:1:1.5:2 spring wire, shaft, spring curvature, cylinder diameter ratio) was verified.
39
4.2
One Degree-of-Freedom Test Stand
4.2.1 Description
Figure 14 shows the one degree-of-freedom test stand. Starting from the left in the figure
are the vertical support, the compliant element, an encoder, the motor support, a 72.38:1
gearhead, the motor, and a motor encoder. The potentiometer can bee seen on the left
side of the compliant element. The encoder attached to the compliant element is a
Hewlett Packard HEDS 6505 model through-shaft encoder with 1024 counts per
revolution and was used to calibrate the compliant element potentiometer circuit to
provide the best possible resolution for the displacement range of the compliant element.
Figure 14. 1-DOF Test Stand
4.2.2 Experimental Procedures
Static Displacementvs. Torque Test
The one degree-of-freedom test stand was used to perform static tests to determine the
displacement vs. torque characteristics of the compliant element prototypes. This was
accomplished by immobilizing the inner shaft (with the clamps visible in Figure 14) and
then applying a torque load to the element using hanging weights (not visible in Figure
14). With the static load applied to the element, one encoder measurement and forty
The mean, maximum,
consecutive potentiometer measurements were sampled.
data were
potentiometer
of
samples
forty
the
of
minimum, and standard deviation
of this
results
The
signal.
calculated to evaluate the amount of noise in the potentiometer
an
and
11)
test were compared to the analytical model predictions (Section 3.6, Figure
exponential curve fit of the data was used to perform some simulations (Section 4.4.3).
40
Torque Dynamic Range Test
With an exponential curve fit of the data generated by the static displacement vs. torque
test and with an estimate of the noise level in the potentiometer signal, predictions can be
made about the expected torque resolution of the compliant element. This prediction can
be verified by observing the potentiometer signal when the incremental torque is applied.
The procedure for this test is to apply the desired initial load, begin data recording, and
add the incremental load. If the load is not detectable, then a slightly greater load is
applied until the signal difference is clearly discernable from the noise.
Motor Driven Step Response Test
This test is intended to assist in the evaluation of the dynamic characteristics of the
compliant element prototypes. In an attempt to generate a step input with the motor, a
maximum motor drive command is issued to the test set for a user-specified length of
time when the test is initiated. When the time limit is reached, a PD controller that uses
the motor encoder is activated to hold the current motor position for the remainder of a
four-second window. Then, the same process is executed in the opposite direction. The
results of this test provide a dynamic characterization of the combined motor, gearhead,
and compliant element system.
Initial Offset Step Response Test
This is another test to evaluate the dynamic characteristics of the compliant element
prototype. In this test, the center shaft is immobilized and the cylinder is displaced by
applying an external torque. After data recording is started, the external load is released
and the dynamic response of the prototype returning its neutral position (zero
displacement) is recorded. Unlike the motor driven step response test, the compliant
element prototype is isolated from the motor and gearhead dynamics, permitting dynamic
characterization of just the prototype.
4.3
Version 1 Compliant Element Prototype
4.3.1
Objective
Version 1 of the 1-DOF element prototype was constructed to accomplish the following
objectives:
*
Evaluate the performance of the selected potentiometer hardware;
*
Evaluate methods for securing the spring ends to the center shaft and to the cylinder;
*
Characterize the performance of one or more spring pairs with different spring wire
diameters for comparison with predicted behavior;
*
Evaluate the performance of the mechanism with the addition of a non-concentric
cam added to the inner shaft to increase the nonlinear behavior.
41
4.3.2 Description
The Version 1 compliant element prototype is shown in Figure 15. The inner shaft
consists of a .25" aluminum shaft and a .50" aluminum shaft that are joined using a rigid
shaft coupling. The outer cylinder was machined out of semi-transparent acrylic to
permit some visibility of the internal components. A Hewlett Packard HEDS 6505, 1024
CPR encoder is installed on one side of the element to provide redundant displacement
data for calibration of the potentiometer data. The rotary potentiometer is installed on the
opposite side of the element (on the left side in Figure 15).
The Version 1 prototype was assembled using .045" diameter springs mounted at a 60'
angle to the perpendicular plane. This configuration minimizes contact between the
spring and the inside of the cylinder wall, resulting in a nearly pure wrapping behavior.
Following the disappointing initial results (see below), this prototype was modified to
include a non-concentric cam on the inner shaft to amplify the nonlinear stiffening
behavior.
Figure 15. Version 1 Compliant Element Prototype
Two spring attachment techniques were investigated with this prototype. The spring was
attached to the inner cylinder using a keyway technique. Notches were cut in to the
terminator section of the spring and a pin was used to lock them in place (see Figure 16).
The spring was attached to the cylinder using a ring and groove technique. The ends of
the spring were inserted through the "brim" of a top hat shaped retention clasp and were
held in place by a retaining ring that fit into groves cut in to the spring terminator section
(see Figure 17).
42
Figure 16. Shaft Pin Detail
Figure 17. "Top Hat" Retaining Ring
Clasp Detail
4.3.3 Results
First, the displacement vs. torque performance of this prototype indicates that the .045"
diameter springs do not provide a stiff enough mechanism for use in the four degree-offreedom robotic digging arm under construction as a follow-on activity (see Section 5).
Second, the retaining ring technique for securing the ends of the spring was only
marginally successful and may not be adequately strong for larger diameter springs and
correspondingly larger loads. Third, the keyway method for attaching the spring to the
inner shaft was dependent on the geometry of this specific case (.045" diameter springs
attached at a 600 angle) and may be more difficult to implement with other geometries.
Fourth, some difficulties were experienced during assembly that led to the cylinder
design modifications made in the Version 2 prototype. Finally, the potentiometer signal
conditioning circuitry was observed to be unacceptably noisy.
Figure 18 shows the results of the Version 1 compliant element static displacement vs.
torque test without the cam part -- the winding surface was a concentric .50" diameter
concentric shaft. A slight asymmetry can be observed, but no stiffening behavior. The
lack of stiffening behavior is speculated to be due to part and assembly variations. First,
the shape of the spring wire forms -- both the radius of curvature and the correct location
of the termination segments along the circumference are vital to the formation of the
correct geometry to induce the stiffening behavior. The wire forming process leaves
some variation among the springs and there may have been additional deformation
caused during assembly, when the springs were integrated with the shaft. Second, any
gaps between the springs and the shaft or the cylinder at the termination locations would
tend to delay the appearance of the stiffening behavior. Third, any slippage of the spring
terminations would prevent the spring from properly wrapping around the shaft to create
the nonlinear profile. Slippage of this nature was occasionally observed with the cylinder
terminations during these tests.
43
Displacement vs. Torque: Version 1 CE without Cam and without Secure Cylinder Termination
40
-
30 2010CX
x
0
E
a -10 -
x
CX
X
-20 -30 -40
-300
-200
-100
0
Static Torque (mNm)
100
300
200
Figure 18. Version 1 Static Test Results without Cam and without Secure Cylinder
Termination
Figure 19 shows the results of the Version 1 compliant element static displacement vs.
torque test with the cam part. A similar asymmetry of the data still exists, but definite
stiffening behavior can be observed. This plot also shows the excessively noisy range of
potentiometer values measured using the original signal conditioning circuitry described
in Section 3.3.4.
Displacement vs. Torque: Version 1 CE with Cam, 40 Sample Potentiometer Data (Low, High, and Mean)
30
20
10
(Di
~-10
-IPI
o)I
-20-
-30 -40
-500
-400
-300
-200
100
0
Static Torque (mNm)
-100
200
300
400
Figure 19. Version 1 Static Test Results with Cam
500
44
4.4
Version 2 Compliant Element Prototype
4.4.1 Objective
The Version 2 compliant element prototype was designed to the same external and
interface dimensions as the Version 1 element to facilitate direct substitution in to the one
degree-of-freedom test stand. It was constructed to improve on the performance of the
Version 1 element in the following ways:
" Increase the element stiffness by an order of magnitude and reduce the maximum
displacement angle;
" Improve the method for terminating the springs in the cylinder;
"
Investigate alternative methods for terminating the springs in the shaft;
"
Make modifications to facilitate easier assembly;
"
Reduce the level of noise in the potentiometer signals.
4.4.2 Description
The Version 2 compliant element prototype shown in Figure 20 was designed to use
.092" diameter springs to increase the stiffness and to decrease the maximum
displacement angle. A quick calculation using the beam bending equation (Equation 19)
and the cross section inertia equation for a cylinder (Equation 20) indicates that this
spring should be about 17 times stiffer than the .045" springs.
Equation 19.
where: M -- bending moment
EI
E -- elastic modulus
M =R
I-- cross section moment of inertia
R -- bending radius of curvature
I =7
Equation 20.
4
4
The cylinder of the Version 2 prototype was machined out of two pieces of aluminum
that are fastened together with countersunk screws. This design change was implemented
to facilitate easier assembly of the final element within a joint assembly and to permit the
use of a pin retention technique to secure the spring to the cylinder.
Figure 20. Version 2 Compliant Element Prototype
45
A T-block retaining part (see Figure 21) was designed to retain the springs in the
cylinder. This part is similar in concept to the top hat retaining device in that the spring
ends penetrate the T-block, but in this case, they are retained by a pin, rather than by a
retaining ring. The use of a pin is facilitated by the redesign of the cylinder to be a two
piece assembly that is assembled around the shaft after the springs have been assembled
with both the shaft and with the T-block.
Figure 21. Two Views of the T-block
The Version 2 prototype was also used to investigate the use of set screws to secure the
springs to the inner shaft. Circular grooves on the spring termination segment were
found to work better than flats and retained the springs as well as the pin technique.
Finally, the potentiometer signal processing circuitry was modified (see Section 3.3.4) to
reduce the level of noise in the output.
4.4.3 Results
Static Displacement vs. Torque Test Results
Figure 22 shows the static displacement vs. torque test results for the Version 2 compliant
element prototype with the cam part. The data indicates that this mechanism is about 10
times stiffer than the Version 1 prototype. The maximum range of motion has also been
reduced to about 2/3 of its previous value. The T-block and set screw retention
techniques were found to work well and the new potentiometer circuit successfully
reduced the noise levels, as can be seen in the much narrower spread of displacement
values of Figure 22 compared to Figure 19.
However, there still exists an asymmetry in the displacement vs. torque data. Since the
asymmetry is similar to that observed for the Version 1 prototype, the most likely cause is
the alignment or circularity of the cam part, which was used in both prototypes. It is
desirable to correct this asymmetry so that the performance of the mechanism is the same
in both directions and so that a constant calibration can be used to convert displacement
measurements to torque estimates.
Assuming an exponentially stiffening profile as discussed in Section 2.1, a curve fit of
this displacement vs. torque data results in Equation 21.
46
,= 270(e0. 130
-
where: t is measured in mNm
e is measured in degrees
I)
Equation 21.
Equation 21 is the basis for a model of the compliant element that is used to help analyze
the additional test results presented in the following sections.
Displacement vs. Torque: Version 2 CE with Cam, 40 Sample Potentiometer Data (Low, High, and Mean)
20
15 -
10 -
S5
0
E
'a
-5
-10
-15
-201
-2500
-2000
-1500
-1000
500
0
-500
Static Torque (mNm)
1000
1500
2000
2500
Figure 22. Version 2 Static Displacement vs. Torque Test Results with Cam
Torque Dynamic Range Test Results
Following the methodology described in Section 2.1, the derivative of Equation 21 is
examined to estimate the torque sensitivity of the Version 2 compliant element (Equation
22). To evaluate Equation 22, the minimum detectable change in displacement angle
(60) is estimated to be 0.14", corresponding to 5 potentiometer counts. This choice is
based on the noise levels observed (-3 counts) and on the calibration setting (36.7 counts
per degree) of the potentiometer measurements.
5,
=
(270X0.13Xe 0)130
Equation 22.
Figure 23 shows a typical sample of the data gathered to determine the torque sensitivity
of the prototype under different initial load conditions. In this case, an additional
incremental load of 3.7 mNm was added (in the negative measured direction) to the
unloaded prototype at approximately 0.75 seconds, causing a momentary spike in the
load response. The measured load is observed to stabilize at a new value (-2 mNm) that
is distinct from the original value, even though it is not quite the same as the actual load
added.
47
Multiple tests of the type shown in Figure 23 were used to find the minimum detectable
incremental torque under different initial load conditions and are plotted in Figure 24.
Also plotted in Figure 24, as the solid line, is the expected relationship between these
values based on the model expressed by Equation 21 and Equation 22. This line has a
constant slope of .049, indicating that at any given applied load, the torque resolution is
4.9%. Figure 24 demonstrates that the actual torque resolution was observed to be
slightly better than predicted for most cases.
Typica I Torque
Sens itivity Test D ata
Torque Sensitivty vs. Applied Torque
-
Predicted and Measured
650
z
40
z
UX
-2
30
-4
0
20-
-6
E
10
-8
-10
0
0.5
1
1.5
Time (ueconds)
Figure 23. Typical Torque Sensitivity
Test Data
0
500
1500
1000
Applied Torque (mNm)
2000
2500
Figure 24. Version 2 Torque Sensitivity
vs. Applied Torque
The force dynamic range of the element, taken as the ratio of the maximum controllable
force to the minimum force, can be taken from Figure 24 to be 2500:5, or about 500.
While this range is quite large compared to most other robot manipulators, it is clear that
it could be further increased if the noise in the potentiometer signal can be reduced
further.
Motor Driven Step Response Test Results
Figure 25 shows the results of a typical motor driven step response test. The gearhead
output angle response is the trajectory of the inner shaft of the prototype compliant
element while the joint angle output is the trajectory of the cylinder and reflects the
contribution of the compliant element. This test revealed that the combination of motor
and gearhead inertias with the limited current capability of the test set motor power
amplifier dominates the dynamic response characteristics. While it is difficult to quantify
the dynamic behavior of the compliant element given that the driving function has been
significantly altered from the intended step, the results do provide some insight in to its
general dynamic behavior.
48
Version 2 CE Motor Driven Step Response Test Data
160
-
-
-
- -
140 --
ointAngle Command
Gearhead Output Angle
Joint 0Outp ut A ng le
/
120
100
-
80-
S
60
0
40
20 -
0
0
/
0.1
0.2
0.3
Time (Seconds)
0.4
0.5
Figure 25. Version 2 Motor Driven Step Response Test Data
Initial Offset Step Response Test Results
Figure 26A shows a typical result from an initial offset step response test. Based on the
settling time, a value for the damping coefficient was calculated and was used in a
simulation with the exponential spring model (Equation 21) and the value of the applied
load to generate plot B of Figure 26. The measured response in plot A exhibits a higher
frequency oscillation than can be accounted for by this original model. Based on an
observation that the hanging masses of the test set did not appear to move during the final
oscillation, a modified model was developed that employs variable damping and variable
inertia schemes.
The results of the modified model are shown in plot C. Plot D shows the variable
damping values used in the model as a function of displacement angle. Damping was set
at the nominal value for displacements greater than 5' and was scaled down by a factor of
50 for displacements less than 50, except for a notch of +/- 0.05' around zero that was
only scaled down by a factor of 20. The greater energy loss of this center notch can be
reasonably interpreted as being caused by a "dead zone" around the zero displacement
position when the springs change from wrapping to unwrapping.
Plot E shows how the load inertia was assumed to vary as a function of displacement
angle in the modified model. For small displacement angles, the hanging masses of the
test set did not appear to move and the compliant element was observed to oscillate at a
frequency consistent with an unloaded condition. Physically, this may be accounted for
by some flexibility in the cable used to hang the weights and/or by the tendency for the
weights to swing a little bit, rather than move straight up and down. As shown in plot E,
the modeled inertia is decreased linearly from the nominal value above 12' displacement
angles to the unloaded value (very small compared to the load, but not zero) at zero
displacement.
49
B. Simulation -- Constant Damping, Constant Inertia
A. Version 2 CE Initial Offset Step Test Data
20
15
15
2 10
10
5
c
5
-a
CO0
0
-5
-5
-10
0
0.2
0.6
0.4
Time (Seconds)
0.8
016
0 '4
0.2
0
.
.
.
0
1
0.8
Time (Seconds)
D Variable Dam png Coefficient
0 07
0 06
C. Simulation -- Variable Damping, Variable Inertia
20.
E0
-5
15
2D
10
-20
2D
.15
-10
5
.5
5
E. Variable
X 10-
20
2
-5
0
15
35
0
.101
10
Inertia
0.2
0.4
0.6
Time (Seconds)
0.8
-
1
0
-20
-15
-10
-5
0
10
15
20
Figure 26. Version 2 Initial Offset Step Response Test Data and Simulations
4.5
Compliant Element Final Design
4.5.1 Objective
The final design of the compliant element is as a component in the four degree-offreedom manipulator described in Section 5 that will be used to investigate robotic
excavation techniques. This element will be integrated in to the shoulder yaw and pitch
joints of the robot while a second compliant element concept developed concurrently by
[Katz, 1999] will be used to add compliance to the elbow pitch and roll joints.
4.5.2 Description
Figure 27 provides an exploded view of the final compliant element design. With only
eight different machined parts (10 parts total), it can be regarded as a relatively simple
mechanism. However, as indicated by the test results above, the performance of the
mechanism requires a high degree of precision in the manufacturing and assembly of
these parts.
50
Potentiometer End Cap with Bearing -
T-Block
Cam
Shaft
- Cylinder (Half)
Spring
Shaft Adapter
End Cap with Bearing
Figure 27. Final Element Design - Exploded View
The final design is modeled after the Version 2 prototype with the following
modifications:
* The final design uses a single .25" diameter shaft and a shaft adapter rather than using
a rigid coupling to join a .25" shaft to a .50" shaft;
* The length of the compliant element was minimized to facilitate integration in to the
CADD manipulator;
* The hardware for mounting an encoder was eliminated (on the prototypes, the
encoder provided a redundant, lower resolution measurement of the compliant
element displacement angle to validate and calibrate the potentiometer).
51
5 Compliant Arm Design for Digging (CADD)
Objective
5.1
The Compliant Arm Design for Digging (CADD) is a four degree-of-freedom robot arm
designed and constructed to meet the following objectives:
* Evaluate the performance of two compliant transmission strategies in a manipulator
as a follow up to the one degree-of-freedom evaluations;
" Investigate excavation control strategies for a manipulator that employs a nonlinearly
compliant joint transmission;
" Design a manipulator that is approximately interplanetary rover scale that uses readily
available components when possible.
Description
5.2
The kinematic configuration of the manipulator was chosen to be Yaw-Pitch-Pitch-Roll.
The first three joints match the configuration of a typical industrial backhoe. The elbow
roll joint was implemented to facilitate the use of oscillatory motions that have been
observed to aid in excavation tasks by reducing stiction between the tool and the soil
[Hong, 1999].
Figure 28 shows a CAD representation of the four degree-of-freedom manipulator and
Figure 29 shows the manipulator itself. The yaw guide track on the base segment is used
with the yaw compliant element to produce an additional 3:1 output speed reduction with
an output range of motion of 1800. At the top of the base segment is a tapered roller
bearing pack to support the rest of the robot. All four motor and gearhead assemblies are
mounted on the first link that moves with the yaw joint. While it was recognized that this
motor placement adds a significant amount of additional inertia that must be moved by
the shoulder yaw joint, the choice is justified by the resulting simplification of the
transmission and the planned use of the robot to conduct relatively slow maneuvers. A
short cable transmission is used to connect the shoulder pitch gearhead output to the
shoulder pitch compliant element, which is located along the axis of the shoulder pitch
joint. Longer cable transmissions are used to drive the elbow pitch and roll joints through
a differential assembly located at the elbow joint. The long runs of cable along the first
link are necessary to accommodate the in-line compliant elements contributed by [Katz,
1999]. Not shown in Figure 28, but visible in Figure 29 are a one degree-of-freedom
(pitch) end effector mechanism that will be used with a scoop to interact with the soil and
a six degree-of-freedom force/torque sensor that will measure loads at the interface
between the end effector and the arm link. Additionally, some preliminary designs have
been examined to add counterbalance masses to the elbow and shoulder pitch joints so
the compliant elements will operate around their zero displacement states when the
manipulator is unloaded.
52
Figure 28. Compliant Arm Design for Digging CAD Model
Figure 29. Compliant Arm Design for Digging
53
6 Conclusions
6.1
Review of Thesis Work
The focus of this research was to develop a nonlinear, compact, modular, robust, and
scalable compliant transmission element that can be used to add compliance to both new
and existing robot manipulators with minimal redesign effort. A nonlinear, stiffening
compliance characteristic was sought to increase the force dynamic range of the
mechanism without unduly large displacements. To preserve positioning accuracy and to
measure the applied forces, the element must have a known compliance characteristic and
must provide a means to measure the displacement.
The final design of the compliant element presented in this thesis exhibits the desired
nonlinear stiffening behavior and a force dynamic range of about 500. However, the
design appears to be overly sensitive to part and assembly tolerances. An analytical
model is provided that may be used to predict the torque vs. displacement profile of
While the use of a
alternative spring, shaft (or cam), and cylinder diameters.
potentiometer was successfully demonstrated during static tests, some concerns about
noise were raised during the dynamic tests. The compliant element design has been
integrated in to the design of a four degree-of-freedom manipulator that will be used to
further investigate excavation control strategies using a robot with a compliant
transmission.
6.2
Future Work
The next step in the evaluation of the compliant element presented in this thesis is to
evaluate its performance as a component of the Compliant Arm Design for Digging.
Through this apparatus, various position, force, and hybrid controllers can be evaluated.
Furthermore, the accuracy of the joint torque calculations made from the compliant
element displacement measurements can be compared to the results indicated by an
independent force/torque sensor.
While this research effort has largely demonstrated the potential usefulness of the
proposed wrapping spring concept, there are several issues that merit further
investigation. First, efforts could be made to better characterize the effects of part and
assembly tolerances on element performance. Second, the design may benefit from a
rigorous evaluation of the mechanism parts to identify ways to simplify the design and
the manufacturing processes. Third, while the current analytical model provides a
reasonable prediction of the expected behavior of a wrapping spring mechanism, it could
benefit from the addition of a component that can account for the linear behavior of the
mechanism near zero displacement. An extension of this model could also be used as a
design tool to recommend geometric properties (relative shaft (or cam), spring, and
cylinder diameters) to meet desired maximum torque and dynamic range requirements.
Fourth, the potentiometer signal noise issue must be addressed for any future application.
55
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57
Appendix A. Matlab Code of Analytical Wrapping Spring Model
58
%
%
%
Wrapping Spring Model
Andrew W. Curtis
January, 2000
% Constants
E = 200e9;
rsp = (.092/2)*2.54/100;
I
(pi/4)*rspA4;
RO = (.375)*2.54/100;
MO
E*I/RO;
=
rc = (1.0/2)*2.54/100;
rsh = (.5/2)*2.54/100;
rcam = ((5/8)/2)*2.54/1 00; %
cO = rcam - rsh;
%
0
1s0 = pi*RO;
%
0
oung's
Modulus for steel
pring wire radius
ross section moment of inertia
nitial spring radius of curvature
nitial state of the wire form
(Pa)
(m)
(m^4)
(m)
(Nm)
ylinder radius
haft radius
am radius
enter offset between cam and shaft
nitial spring free length
(m)
(in)
% Set up shaft rotation angle
theta = (0:.001:.4)';
td = theta.*180/pi;
% t heta converted to degrees
(in)
(in)
(in)
(deg)
% Calculate the Moment generated by the wrapping side of the mechanism
% Assumption: point of contact moves in opposite direction of
the shaft rotation by same angle, theta
%0
a = rc - rsp;
b = rsh + rsp;
c = rcam+rsp;
% Calculate cam effect
beta = abs(asin((c0/c)*sin(pi-2.*theta)));
eta = 2.*theta - beta;
b2 = sqrt(c0^2 + cA2 - 2*c0*c.*cos(eta));
% Calculate the chord length
%lcf = sqrt(a^2 + bA2 - 2*a*b.*cos(pi-theta));
lcf = sqrt(aA2 + b2.A2 - 2*a.*b2.*cos(pi-theta));
without cam
with cam
% Calculate length of spring wrapped around the cylinder
% without cam
%lsw = b*2.*theta;
% with cam
lsw = c.*eta;
% Calculate new free length of spring
lsf
=
1s0
-
lsw;
% Calculate new radius of curvature for the free length of spring
for i=l:length(lcf),
I =
= solve('''num2str(lcf(i))
[R(i),psi(i)]
eval(['
2*R*(sin(psi/2))'','''num2str(lsf(i)) ' = R*psi'');']);
end
for i=l:length(lcf),
R2(i,1) =
psi2(i,l)
end
str2num(char(R(i)));
= str2num(char(psi(i)));
59
% Calculate the moment due to the free length of spring
Mwf = MO - (E*I./R2);
Calculate the component of Mwf that acts as a torque at the fixed
point on the cylinder
%without cam
%phil = abs(asin((b./lcf).*sin(theta)));
%with cam
phil = abs(asin((b2./lcf).*sin(theta)));
alpha = ((pi-psi2)./2)-phil;
Mwc = (a.*Mwf./R2).*cos(alpha);
% Calculate the Moment generated by the unwrapping side of
the mechanism
% Approximation: Free segment of spring maintains nearly constant curvature
% Calculate change in spring endpoint position due to shaft rotation:
dlsp = b.*theta;
% Calculate angle required to accomodate this change at the radius of
curvature when in contact with the cylinder
%
gamma = dlsp./(a-RO);
% Calculate the new free length of the spring
lsfu = lso - a.*gamma;
% Calculate the chord length of the free length of spring
lcfu = sqrt(a^2 + b^2 -2*a*b.*cos(pi-theta-gamma));
% Calculate the actual radius of curvature for the free length of spring
for i=l:length(lcfu),
=
eval(['[Ru(i),psiu(i)1 = solve('''num2str(lcfu(i))
2*Ru*(sin(psiu/2))'', ''num2str(lsfu(i)) ' = Ru*psiu'');']);
end
for i=l:length(lcfu),
Ru2(i,l) = str2num(char(Ru(i)));
psiu2(i,l) = str2num(char(psiu(i)));
end
% Compensate for possible sign inversion of psiu2
psiu3=abs(psiu2);
Ru3 = lsfu./psiu3;
% Calculate the moment due to the free length of spring
Muf = (E*I./Ru3) - MO;
% Calculate the component of Muf that acts as a moment at the fixed
point on the shaft
% Assumption: The radius vector Ru is nearly the same as the radius
vector b
% (to the fixed point on the shaft)
Mus = b.*Muf./Ru3;
% Sum the two torques
Mt = Mwc + Mus;
% End of Wrapping Spring Model
61
Appendix B. Compliant Element Part Drawings
Table 3. Index of Compliant Element Part Drawings
Part
Cam
Quantity
1
CSpring 092
Cylinder 92 (Half)
End Cap
End Cap Pot
Inner Shaft
Shaft Adapter
T-Block
2
2
1
1
1
1
1
Potentiometer End Cap with Bearing
T-Block
Cam
Cylinder
(Half)
GShaft
Shaft Adapter
End Cap with Bearing
Figure 30. Compliant Element Part Overview
H
DWG. NO-
REV.
REVISIONS
ZONE
REV
DESCRIPTION
DATE
APPROVED
.0625
.625
500
1875
.062
5
.375
Co noad
r
1/32'
Typical
.150
4*
. 31 2
1875
045
.280
0TY REO
FSCM
NO
NOMENCLATURE
PART OR
IDENTIFYING NO
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN INCHES
ARE:
FRACTIONS
DECIMALS
ANGLES
MATERIAL
SPECIFICATION
OR DESCRIPTION
PARTS LIST
CONTRACT NO.
IOLERANCES
YES-
DO NOT
TREATMENT
SCAL E
APPROVALS
-
DRAWI NG
DATE
ITLE
DRAWN
Cor
CHECKED
F INISH
SIZE
A
ISSUED
SIMILAR TO
AC W1
OWG
NO
CALC WT
SCALE
I
FSCM NO.
4
SHEET
ITEM
NO
SAC.
REV.
SH
NI
REVISIONS
REV
ZONE
.092
(Wire
DATE
DESCRIPTION
APPROVED
Diameter)
.12
0165
.0625
_2
093
.375
4-
30
'T
0625
-.
.204
.093
rCM
D?
PESO
PART OR
IDENTIFYIN
No
ITEM
MATERIAL
SPECI FICAT ION
NOMENCLAIURE
OF DE SCR IPT ION
PARTS LISI
G NO
NO
CONTRACT NO.
UNLESS OTHERWISE SPECIFIES
DIMENSIONS ARE IN INCHES
TOLERANCES ARE:
FRACTIONS
DECIMALS
ANCLES
T-.
DATE
APPROVALS
DO NOT SCALE DRAWING
TI TI.E
OSpring
DRAWN
TRPEAT MENT
CHECK<ED
F INISH
SIZE
FSCM
092
DWO
NO,
NO.
ISSUEDA
SIMILAR TO
I
ACT. PT
CALC
PT
SHEET
SCALE
I __
_
_
__
_
_
I
__
I
_
_
_ I___
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
1
2
ING. NO.
4
IREV
[S
REVISIONS
ZONE
.250
D
REV
DESCRIPTION
DATE
APPRDVED
.225
D
.047
0-80 T op .117 -
.0625 Radius
Typical
--
-
.300
16D
550
.102
C
.100
.1
1 .500
C
100
NJ
115
4---
25
.150
1 .125
.0 25
.062 5
B
B
1 .000
.116 Thru
.1875 Countersink
4-40 Clear
4
.0635 Thru
.125 Countersink
0-80 Clear
7
-
.047
0-80 Tap
2 Places
2 PlIa Ce S
Places
N
oty lwaiE0
&Ni~N
tENlRACT No.
IN IICHE
NSN3M
RANCES
ARE,
0IM
TOLE
TRACI 10"
D[CIMALS
NO
"MCL EI
2 :% -
A
DO N0l
PAT SCLIUSE
CP DE SN? IP1 IN
PARTS L IST
SCALE DRAWIrNG
APPRFOVALS
DATEr
Cylinder 92
11
2
2
rALE
NO.
TO
Ill, "I 1 A"W,
S
3
r I ATlION
L
O~A
F INISH
SIMILAR
PEVI
SCA LE
A
(Half)
oEe
ET
1
N
HEE T
4
4
-
3H
DWG. NO.
REV.
REVISIONS
ZONE
REV
DESCRIPTION
DATE
APPROVED
1.000
625
1 .125
.313
.063
.196
4-. 100
.250
.086 Thru
4-40 Top
4 P iaces
FSCM
OTY
REDO
NOMENCLAIURE
MATERIAL
ITEM
OR DESCRIPTION
SPECIF ICATION
NO
PART OR
IDENT IFYING
NC
NO
C
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN INCHES
TOLERANCES ARE:
FRACTIONS DECIMALS ANGLES
555DO NOT SCALE DRAWING
TREATMENT
C
CONTRACT
PARTS L IST
NO.
APPROVALS
DATE
TITLE
End Cap
ORAWN
CHECKED
SIZE
F IN ISWT
ISSUEDA
SIMILAR TO
I
I
ACl
1
FSCM
NO.
DWG
NO.
CADET
SCALE
4
SHEET
SH
GWG.
NO.
079
1 88
.157
REV.
REVISIONS
ZONE
REV
DESCRIPTION
DATE
APPROVED
438
.196
.100
188
.750
1.000
1 .000
.625
1 .500
.313
1.12 5
4.567
.125
1 .339
.086 Thru
4-40 Top
4 Places
Outside
Inside
0-1Y
REDO
|
PART OR
IIENTIFYINC NO
FSCM
NO
MATERIAL
SPECIFICATION
CONTRACT NO.
UNLESS O-HERWISE SPECIFIED
DIMENSIONS ARE IN INCHES
TOLERAkNCES ARE;
FRACTIONS DECIMALS
ANGLES
.
NOMENCLATURE
OR OESCRIPTION
PARTS LIST
I
APPROVALS
DO NOT SCALE DRAWING
DATE
End Cap
DRAWN
TREATMFENT
Pot
CHECKED
F INIS
SIZE
ISSUEDA
SIMILAR TO
AC'
W
CALC
FSCM
No.
OWG
NO.
WT
SCALE
4
SHEET
ITEM
NO
DWG0
~H PLY.
I
NO.
NO.
DWG
ISH
REVISIONS
ZONE
REV
DATE
DESCRIPTION
IFLv.
I
APPROVED
.093
.092
4
1
.093
.092
.125-
.250
4I
DrY
REO
FSCM
NO
PART
OR
IENTIFYINC
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN INCHES
1OLERANCES ARE:
FRACTIONS DECIMALS
ANGLES
.15NOT SCALE DRAWING
TREATMENT
DO
NOMENCLATURE
OR DESCRIPTION
PARTS LIST
NO
I
MATERIAL
ITEM
SPECIFICAlION
NO
CONTRACT NO.
APPROVALS
DATE
IITLE
Inner
DRAWN
Shaft
CHECKED
F INISW
CHSUED
5
FSCM NO.
owG
NO.
ISSUED97:
SIMILAR
TO
ACT
W1
CALC WT
SCALE
SHEET
ON
DWG.
SH
NO.
IS5H
REV.
IHV
00
REVISIONS
092
T hru
12u
2 Places
NE
REV
DESCRIPTION
DATE
APPROVED
.0 93
-
Groove
-
to
allow
ing assembly
______spr
.093
375
.106
6-32 lap
2 Places
xtX~f2K
4-
.0625
45 Deg
Chamfer
2 Places
-P----
.125
.5
.125
.25
FSCM
ETY
REO
NOMENCLAIURE
OR DESCRIPTION
PART DR
IDENTAIFYINC NO
NO
MATERIAL
SPECIFICAT ION
PARTS LIST
UNLESS OTHERWISE SPECIFIED
DIMENSIONS ARE IN INDIES
TOLERANCES ARE:
FRACTIONS
DECIMALS
ANGLES
DO NOT
TREATMENT
CONTRAC-T
.TTSCALE DRAWING
ND.
DATE
APPROVALS
Shaft
DRAWN
Adapter
CHECKED
SIZE
E INISPl
FSCM NO.
DWG
NO.
ISSUEDA
SIMILAR TO
ACT
T
CALM 1T
5CAL E
I
SHEETi
ITEM
ND
SH
wOG. NO.
.0635 Thru
.1094 Counter si nk I
REV.
RE VIS IONS
ZONE
REVI
DESCR
IPT
DATE
ION
APPROVED
0-80 C Ie ar
2 Places
0625
.092
.275
.0625
0930
-
180
-.
1 .50
.360
.080
115
4-
1 .00
313
.125
.0725
.125
0
15
REDO
FSCM
NO
NOMENCLATURE
OR DESCRIPTION
PART OR
IDENTIFYINC NO
MATERIAL
SPECIFICA~ ION
PARTS LIST
UNLESS ITHERWI SE SPECIF lET
DIMENSIONS ARE IN INCHES
CONTRACT NO.
ITOLERA MCIES ARE:
FRACTIONS
DO
TREATMENT
NOT
DECIMALS
ANGLES
APPROVALS
SCALE DRAWING
DATE
CHECKED
V INISW
ISSUED
ISSUEDA
TITLE
T - BIo ck
DRAWN
SZE AI
FSCM NO.
SCAL E
T
ISHEE
WG
NO.
T
ITEM
NO