ANALYSIS AND DESIGN OPTIMIZATION OF INPARALLEL HAPTIC DEVICES
by
LENG-FENG LEE
DECEMBER 2010
A dissertation submitted to the Faculty of the Graduate School of the State
University of New York at Buffalo in partial fulfillment of the requirements for
the degree of
DOCTOR OF PHILOSOPHY
Department of Mechanical and Aerospace Engineering
State University of New York at Buffalo
Buffalo, New York 14260
To my family and friends
Without whom, I am nothing
ii
Abstract
The overall kinesthetic immersive experience in a haptic interactive virtual
environment is the synthesis of the human user, the haptic user interface (HUI), and
virtual environment (VE) – all of which playing critical roles. Our focus will be on the
development of advanced HUIs for providing the users with sophisticated tactile or force
feedback during interaction with virtual environments with innovative features including
active and passive manipulation assists. Desirable features for high-performance HUIs
include: human matched force capabilities, sizeable workspace, low inertia, high stiffness,
low friction, back-drivability, near-zero backlash, and gravitational counterbalancing.
Parallel-architecture haptic devices offer significant advantages over serialarchitecture counterparts in applications requiring high stiffness and high accuracy. To
this end, many haptic devices have been created and deployed by modularly piecing
together multiple serial-chain arms to form an in-parallel system. Furthermore, recent
haptic device designs such as the Sensable‟s Phantom Premium line of haptic devices and
Quanser‟s High Definition Haptic Device (HD)2 transfer the distal actuation to the base
via a parallelogram/fourbar linkage in order to reduce the moving inertia in the system.
However, such design choices can affect the overall system performance which
now depends both on the nature of the individual arms as well as the interactions. The
multiple closed-kinematic chains constrains effective degree-of-freedom and require
careful selection of type, number, location and actuation of the individual articulations
iii
(within the chain) completes the determination of the workspace, mobility, controllability,
and overall performance of the system.
In this work, we build on the rich theoretical background of constrained
articulated mechanical systems to provide a systematic framework for formulation of
system-level performance from individual module characteristics. Specifically, we
discuss: (i) development of pertinent symbolic equations; (ii) generalization to arbitrary
architectures; and (iii) perform combined symbolic/numeric analyses, focusing on salient
zeroth order (workspace), first order (manipulability), and second order (stiffness)
kinematics performance measures. We demonstrated our studies using Sensable Phantom
Premium line of haptic devices and Quanser‟s High Definition Haptic Device (HD)2. In
particular, we highlight the effect of the added parallelogram sub-system to the overall
system-level manipulability and stiffness measure.
Finally, we note that traditional performance analysis on haptic devices focuses
solely on the device. However, haptic devices are typically employed in close coupling
with a human user creating a need to include their characteristics in the design and
analysis. We therefore examined the use of a musculoskeletal analysis framework to
study the performance of haptic devices, extracted biomechanically relevant performance
measures from the human user, and use this to tailor the ergonomics and regimen within a
rehabilitation program.
iv
Acknowledgment
三人行, 必有我师
When there are three people walking, my teacher is bound to be among them.
-- Confucius
The first paragraph in the Acknowledgement section in all dissertations you will
ever read is reserved for their advisor. This is no exception. This journey begins 9 years
ago when I knock on his office door and tell him I wanted to „do research‟. He opened
the door and I never left. I would like to thank my advisor Professor Venkat Krovi. His
guidance and encouragement have been invaluable throughout my undergraduate and
graduate study. He helped me passed through some of the tough time as a friend, and
enlightened me in the methods of scientific research as a researcher. He also sees my
teaching „potential‟ and always recommends me whenever there a teaching opportunity
in the department.
I would also like to express my sincere thanks to Dr. Mayne and Dr. Soom for
serving on my thesis committee. I „enjoyed‟ Dr. Mayne graphics programming class, and
valued his encouragement, guidance, and especially the freedom in my teaching of two
courses (MAE 377 Product Design in CAD Environment, and MAE477/577 CAD
Applications) in the department. I was Dr. Soom class‟ TA once and he is the chair of the
department when I first teach the MAE377 class. He has been very helpful and
supportive throughout. Special thanks to Dr. Frank Mendel, whose sense of humor often
v
relief some of the pressure during those project meetings, for serving as the outside
reader.
I also like to thank Dr. John Rasmussen from Aalborg University, who invited me
to visit his lab and company in Denmark for two weeks, working with the „AnyBuddies‟.
It was a very unique experience. I also wish to thank Dr. Dargursh, who is the current
department chair for attending my dissertation defense and offered numerous suggestions.
I also appreciate the trust he had in me, in which he assigned me to teach a
senior/graduate level course during my final semester with the department.
Financially, this work is supported through NSF‟s Computer & Information
Science & Engineering Computing Research Infrastructure (CRI) grant. Beside this, the
teaching assistantship from the department of Mechanical & Aerospace Engineering is
the major source of funding for most part of my graduate studies. I also like to thank Dr.
Rajan Batta, from School of Engineering and Applied Science, who allocated some
funding that allowed me to attend the „2010 North American Summer School in Surgical
Robotics and Simulation‟ in Seattle. The „Future Faculty Program‟ initiated by Dr. Batta,
is also a very valuable experience for me. I felt privillages and always appreciate the fact
that I am funded throughout my graduate studies.
Yes, as a graduate student, your friends are basically your labmate. For this reason,
I would like to thank the current and previous (long-list of) members of ARM Lab: Rajan,
Chin-Pei (CP), Seung Kook (SK), Pravin, Harpreet, Ajay, Daniel, Chris, Chetan, Annu,
Talib, Glenn, Mike, Tao, Kiran, Chi-Han, Anand, Kun, Patrick, Hao, Qiushi, Yao, Madu,
Srikanth, Xiaobo, John, Shajan, Sophie, and Vanina. Recent addition including Dan,
Priyanshu, and Suren are also great to talk to. They are great colleagues, who made the
vi
lab so interactive and fun. I am sure we shared some great moments together, but forgive
me for my short-term memory (or because I have been in the lab for too long), I cannot
recall those moments now. Visiting scholars Ilya, from Russia and Luca, from Italy are
fun to work with.
I valued the friendship and encouragements from CP, friend I have known since
day one of my undergraduate study long before we both came to the States. Xiaobo, who
shared the same hobby as me were great friend inside and outside the lab. The long drive
to Montreal attending conference is especially memorable. SK and his wife, especially
their adorable 4 years old daughter Seo Young, and those never-seen-before Korean food,
are great companion during the very stressful months preparing my dissertation.
Being a graduate student and 10,000 miles away from home sometime also means
your „friends‟ now existed only on the internet. So thanks to those „friends‟ who send
couple of words over from time to time checking in on me by asking „have you
graduated?‟. My long absent from home is not possible without the support from my
parents, brother, and sister. Thanks for their trust in me. For the first time in many years,
my answer to their most frequently asked question – „when are you going to graduate?‟ is
finally not „hm… probably next year‟.
Everyone who knows me well will probably agree that this long journey will not
be possible without the company of my best friend/girlfriend Lye Theng. This has been
an amazing journey so far for both of us, and its time to begin the next chapter of our lifelong journey.
vii
Contents
Abstract .............................................................................................................................. iii
Acknowledgment ................................................................................................................ v
Contents ........................................................................................................................... viii
List of Figures ................................................................................................................... xii
List of Tables ............................................................................................................... xxviii
1
Introduction .............................................................................................. 1
1.1
Fundamental Issues at Core of Haptic Realism .................................................3
1.2
Problem Statement .............................................................................................5
1.3
Serial vs. Parallel Architecture Haptic Devices .................................................6
1.4
Contributions....................................................................................................11
1.5
Thesis Organization .........................................................................................12
2
Background ............................................................................................ 14
2.1
Parallel Architecture Manipulator....................................................................14
2.2
Parallel Architecture Manipulator Control ......................................................16
2.3
Virtual and Physical Prototyping Implementation ...........................................18
3
Technical Background ........................................................................... 20
3.1
Serial Manipulator Jacobian Formulation ........................................................21
3.1.1
Conventional Method..............................................................................21
3.1.2
Spatial Vector Formulation .....................................................................23
3.1.3
Twist based Jacobian Formulation..........................................................27
viii
3.1.4
3.2
3.3
Reciprocal Screws ...................................................................................36
Conventional Jacobian Formulation – Parallel Manipulator ...........................38
3.2.1
Method I ..................................................................................................40
3.2.2
Method II ................................................................................................42
3.2.3
Method III ...............................................................................................43
Jacobian Formulation for Parallel Systems......................................................44
3.3.1
4
System Jacobian Matrix of Parallelogram sub-system ...........................46
Performance Measures ........................................................................... 49
4.1
Background ......................................................................................................50
4.2
Zeroth Order Kinematic Performance Measure: Workspace...........................51
4.3
4.2.1
Workspace Computation .........................................................................54
4.2.2
Examples .................................................................................................57
First Order Kinematics Performance Measure: Manipulability.......................59
4.3.1
4.4
Second Order Kinematics Performance Measures: Stiffness ..........................71
4.4.1
4.5
Manipulability Formulation ....................................................................62
Stiffness Formulation ..............................................................................73
Chapter Summary ............................................................................................84
5
Serial-Architecture Haptic Devices ....................................................... 85
5.1
Parallelogram Sub-Structure ............................................................................85
5.1.1
5.2
Formulation of Planar Parallelogram Sub-System .................................88
Phantom 1.5 Haptic Device .............................................................................92
5.2.1
Workspace Analysis................................................................................94
5.2.2
Manipulability Measures ........................................................................96
ix
5.3
5.2.3
Isotropy Index .........................................................................................97
5.2.4
Stiffness Analysis..................................................................................100
Manipulability Optimization / Regulation .....................................................102
6
Parallel-Architecture Haptic Devices .................................................. 107
6.1
High Definition Haptic Device (HD2) ...........................................................107
6.2
Forward Kinematics .......................................................................................108
6.3
Numerical Inverse Kinematics .......................................................................111
6.4
Workspace Analysis.......................................................................................113
6.4.1
Constant Orientation Workspace ..........................................................113
6.4.2
Constant Orientation Spatial Workspace ..............................................116
6.5
Manipulability Measure .................................................................................119
6.6
Isotropy Index ................................................................................................122
6.7
Stiffness Measures .........................................................................................124
6.8
Trajectory Tracking .......................................................................................127
6.9
SimMechanics / MapleSim Model.................................................................129
6.10
Chapter Summary ..........................................................................................130
7
Musculoskeletal Analysis-based Performance Measures .................... 132
7.1
Haptic-based Virtual Rehabilitation Systems ................................................135
7.2
Critical Challenges .........................................................................................137
7.2.1
7.3
Computational Musculoskeletal Analysis Tools ..................................138
Performance Measures ...................................................................................143
7.3.1
Muscle Forces .......................................................................................144
7.3.2
Muscle Activity.....................................................................................144
x
7.4
Case Study: Haptic Motor Rehabilitation Programs ......................................146
7.4.1
7.5
8
Central Interface....................................................................................147
Musculoskeletal Models ................................................................................148
7.5.1
Parametric Studies ................................................................................150
7.5.2
Optimization Studies.............................................................................152
7.5.3
Discussion .............................................................................................154
Discussion and Future Work ................................................................ 156
8.1
List of Contributions ......................................................................................158
8.2
Future Work ...................................................................................................159
Bibliography ................................................................................................................... 161
Appendix A ..................................................................................................................... 179
A.1
Jacobian Formulation Example: Stanford Manipulator .................................179
A.2
Numerical Workspace Computation ..............................................................184
A.3
Workspace Computation Examples ...............................................................196
A.4
Manipulability of Parallel Systems ................................................................199
A.5
Stiffness Computation....................................................................................205
A.6
Workspace of HD2 .........................................................................................208
A.7
Inverse Kinematics of HD2 ............................................................................209
xi
List of Figures
Figure 1: Immersive and interactive rendering of “touch” in virtual environments depends
in equal part on: (i) the human user‟s sensory abilities; (ii) the selected haptic user
interface; and (iii) fidelity of computed virtual environment responses. .................... 1
Figure 2: The fundamental issues at the core of haptic realism are tied closely to
complexity of Haptic User Interfaces and Haptic Modeling Fidelity. In this thesis,
we focus on studying various Haptic User Interfaces. ................................................ 4
Figure 3: Typical tasks involved in a knee replacement surgery (a) drilling through bones;
(b) hammering pins; (c) sawing through bones; and (d) suturing. .............................. 5
Figure 4: Examples of commercial haptic devices based on serial architecture................. 7
Figure 5: Examples of commercial haptic devices design based on parallel architecture. . 8
Figure 6: In-parallel haptic devices by modularly combining two or more serial
architecture haptic devices. ....................................................................................... 10
Figure 7: Modularity creates design choices. The performance of the overall system
depends on the nature of the individual module as well as their interactions........... 12
Figure 8: Using a 2-DOF RR manipulator as building block, many interesting haptic
devices can be developed by modularly piecing the individual small system in
different ways to create multi DOF haptic devices. .................................................. 14
Figure 9: Kinetostatic design optimization to improve both workspace volume and
quality (stiffness, accuracy, velocity and force performance). ................................. 15
Figure 10: Rapid virtual and physical prototyping application developed for 5-DOF
parallel haptic device. ............................................................................................... 18
Figure 11: Reference frames assignment for a RRR manipulator. ................................... 22
xii
Figure 12: Homogeneous Transformation between two frames. ...................................... 28
Figure 13: Surface plot of isotropy index over the entire workspace of a two link RR
manipulator computed using the Jacobian formulated using (top left) analytical
method, (top right) spatial-twist expressed in {W } frame, (bottow left) spatial twist
expressed in {F } frame, and (bottom right) body-fixed twist expressed in {E } frame.
................................................................................................................................... 35
Figure 14: Manipulability ellipsoid computed from Jacobian formulated using (a) spatial
twist, expressed in {F} frame; (b) body-fixed twist, express in {E} ; and (c) spatial
twist, expressed in {W} frame. ................................................................................. 36
Figure 15: Locations and directions of twists and wrenches of a three link RRR planar
manipulator, red color arrows represent twists and blue color arrows represent
wrenches. (a) Isometric view; and (b) top view of the manipulator. ........................ 38
Figure 16: Nomenclature of 3RRR planar manipulator used in the analysis. .................. 39
Figure 17: Closed-loop constraints selection for a parallel manipulator, reproduce from
[62] ............................................................................................................................ 45
Figure 18: A planar RR manipulator system that consists of a parallelogram structure can
be modeled as a constrained system consisting a RRR and a RR manipulator. ....... 47
Figure 19: Example of the computing the area of a circular planar workspace using
contour and polyarea algorithm. ............................................................................... 55
Figure 20: Example of computing the volume workspace of a sphere using mesh
generated from data points. ....................................................................................... 56
xiii
Figure 21: Constant orientation workspace of a 2-RRR planar manipulator changes as the
distance of their bases varied from (s) L
1.5 to (c) L
4.5 with end-effector
orientation held at zero.............................................................................................. 57
Figure 22: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
0.5 to L
zero. Largest workspace exist when L
1.7321 . ....................................................... 58
4.5 with end-effector angle held at
Figure 23: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
0.5 to L
4.5 with end-effector angle held at
45 . Largest workspace no longer exist at L
1.7321 . ....................................... 58
Figure 24: Constant orientation workspace of a 4-RRR planar manipulator changes as the
distance of their bases varied from L
2 to L
3.25 with end-effector angle held at
zero............................................................................................................................ 58
Figure 25: Velocity and force manipulability ellipsoid of a planar two link serial RR
manipulator at various locations. .............................................................................. 59
Figure 26: An example of a manipulability ellipsoid. ...................................................... 63
Figure 27: Manipulability ellipsoid of a five-bar manipulator with two different actuated
joints compared with (a) result obtained from [119], (b) five-bar with both 1st joint
actuated; and (c) five-bar with both 2nd joint actuated. ............................................. 66
Figure 28: Manipulability ellipsoid of a six-bar manipulator with two different actuation
schemes compared with (a) results reproduced from [119], (b) six-bar with 1st and
2nd joint of the left chain actuated; and (c) six-bar with both 1st and 2nd joint of the
right chain actuated. .................................................................................................. 66
xiv
Figure 29: Five-bar systems manipulability ellipsoid at two singular positions (top and
bottom) with two actuation schemes: (a) both actuated joints located at the 1st joint;
and (b) both actuated joints located at the 2nd joint. ................................................. 67
Figure 30: Six-bar system manipulability ellipsoid at singular position. ......................... 67
Figure 31: For the same end-effector position and angle, manipulability ellipsoid can be
different due to inverse solution of each RRR chain. ............................................... 68
Figure 32: Isotropy index/ Inverse of condition number over the entire workspace of a 3RRR parallel manipulator changes for two selected inverse kinematics
configurations. .......................................................................................................... 69
Figure 33: Manipulability ellipsoid changes as well as workspace as configuration
changes. ..................................................................................................................... 69
Figure 34: Comparison of Isotropy index / Inverse of condition number for a (a) 2-RRR,
(b) 3-RRR, and (c) 4-RRR system at selected configuration.................................... 70
Figure 35: Stiffness Ellipsoid of a two link RR manipulator as the two joint stiffness
constants k1, k2 varied from k1
0.1, k2
0.9 to k1
0.9, k2
0.1 from (a) to (i). ...... 75
Figure 36: Stiffness maps of the 3-RPR manipulator in the (a)
direction; and (c) y –direction with
–direction; (b) x –
0 . Comparing result with comparing results
with [143] (top row). ................................................................................................. 76
Figure 37: Stiffness maps of 3RRR manipulator in the (a) x-direction, (b) y- direction,
and (c)
– direction, with end effector angle
0 . Comparing result with [153]
(bottom row). ............................................................................................................ 77
xv
Figure 38: Comparison of the stiffness map of a 3-RRR manipulator in x-direction (2nd
row), y-direction (3rd row), and
-direction (4th row) with different joints actuated:
(a) 3-RRR; (b) 3-RRR; and (c) 3-RRR manipulator. End-effector angle
0 . .... 78
Figure 39: With the modular formulation that we have shown, we can look at how adding
an additional chain to the 3-RRR system (shown in (s)), now forms a 4-RRR system
(as shown in (b)) changes the overall stiffness of the system. .................................. 79
Figure 40: The stiffness map in the (a) x-direction, (b) y-direction, and (c)
-direction of
a 3-RRR system described in Figure 39(a) (top row), and the stiffness map after
adding another chain (bottom row), the 4-RRR system shown in Figure 39(b). ...... 80
Figure 41: (a) A 4-RRR planar parallel manipulator with only three chains actuated (the
top left chain is passive); compare with (b) a 3-RRR planar parallel manipulator
with same link lengths and locations as the 4-RRR manipulator, but with all chains
actuated. Shown here with their corresponding constant orientation workspace. .... 81
Figure 42: The stiffness map of a 4-RRR planar parallel manipulator with only three
chains actuated (top row), compare with a 3-RRR planar parallel manipulator
(bottom row): (a) x-direction, (b) y-direction, and (c)
-direction stiffness. ........... 82
Figure 43: Comparison of stiffness map of a (a) 2-RRR, (b) 3-RRR, and (4) 4-RRR
manipulator in x-direction (2nd row), y-direction (3rd row), and
with end-effector orientation
-direction (4th row)
0 . ......................................................................... 83
Figure 44: Examples of commercial haptic devices with parallelogram sub-system within
their kinematic chains. .............................................................................................. 86
xvi
Figure 45: Using PHANToM 1.5 system as a design block, various types of in-parallel
haptic devices can be design by combining them in different ways. Note that device
with „?‟ on the side is not a currently available haptic device. ................................. 87
Figure 46: (a) A mechanism with parallelogram structure can be modeled as a
combination of a RRR manipulator with another RR manipulator. (b) The
manipulability ellipsoid of the combined system is shown with the manipulability of
individual RR and RRR manipulator (assuming all joints actuated). ....................... 89
Figure 47: Comparison of parallelogram structure manipulability ellipsoid formulation
with a five-bar mechanism: (a) and (c) is the parallelogram structure where the last
link is zero, and (b) and (d) is the five-bar mechanism. ........................................... 90
Figure 48: (a) Surface plot of isotropy index of RR linkage with parallelogram structure,
RR manipulator, and RRR manipulator for comparison. The contour plot of each of
their isotropy index is given in (b) for RR linkage with parallelogram structure; (c)
for RR Manipulator and (d) for RRR manipulator. .................................................. 91
Figure 49: The parallelogram structure in the Phantom 1.5 system modeled as
combination of one (a) spatial RR manipulator with another (b) spatial RRR
manipulator. D-H parameters of both systems are given in Table 2 and Table 3 .... 92
Figure 50: (a) The reachable workspace of PHANToM 1.5 haptic device is about 0.08m3,
(b), (c), (d) are side view, top view, and front view of the workspace along with the
haptic device. ............................................................................................................ 95
Figure 51: 3D view of the manipulability ellipsoid of Phantom 1.5 system, compared
with the manipulability ellipsoid of a spatial RR system (without the parallelogram
sub-system). .............................................................................................................. 96
xvii
Figure 52: With the addition of the parallelogram linkages, the manipulability ellipsoid of
the Phantom 1.5 system at x = (a) 1.5; (b) 2.5, and (c) 3.5, compared with the
manipulability ellipsoid of a spatial RR system. In all these cases, the sizes of the
manipulability ellipsoids were scaled to half of its actual size. ................................ 97
Figure 53: Isotropy Index of (a) a PHANToM 1.5 system compared to a (b) spatial RR
manipulator over the entire workspace on the x-z plane (y = 0). ............................. 98
Figure 54: (a) Isotropy index of Phantom 1.5 plotted against a spatial RR manipulator.
The (b) maximum and average isotropy index of Phantom 1.5 is 0.7907 and 0.5353;
and (c) the maximum and average isotropy index of a spatial RR manipulator is
0.8541 and 0.3939, over the workspace. ................................................................... 99
Figure 55: Stiffness maps of spatial RR system in the (a) x direction; (b) y direction;
and (c) z
x
direction, compare with stiffness map of PHANToM 1.5 in the (d)
direction; (e) y direction; and (f) z
direction, on the x z plane ( y
0 ). .. 101
Figure 56: Comparison of stiffness map of PHANToM 1.5 system with a spatial RR
system in the (a) x-direction; (b) y-direction; and (c) z-direction. The addition of
parallelogram structure reduces the overall stiffness in the system but at the same
time allow a more uniform stiffness distribution. ................................................... 102
Figure 57: Manipulability ellipsoid (in blue color) of Phantom 1.5, (a) – (c) side view,
and (d)- (f) isometric view. The size of the parallelogram does not change the
magnitude or the direction of the manipulability ellipsoid. .................................... 103
Figure 58: Study the effect of the parallelogram structure by replacing it with a fourbar
linkage. .................................................................................................................... 103
xviii
Figure 59: The manipulability of the PHANToM 1.5 system changes with varying length
of L3a from (a) L3a
0.8 ; (b) L3a
1.2 ; to (c) L3a
1.45 . ........................................ 104
Figure 60: (a) Result of optimization to improve the isotropy index of the selected region
shown in (b). The contour plots of the original and optimized isotropy index are
given in (c) and (d). ................................................................................................. 106
Figure 61: (a) Actual High Definition Haptic Device (HD)2 system; and (b) CAD model
created in SolidWorks. ............................................................................................ 107
Figure 62: The HD2 Haptic Device and DH Frames assignment on the upper arm. ...... 108
Figure 63: Passive attachment (via universal joint and revolute joint) of the serial-chain‟s
end-effector to the common handle. ....................................................................... 109
Figure 64: DH frame assignment of the „inner chain‟ of HD2, to include the parallelogram
structure in the formulation. .................................................................................... 110
Figure 65: (a) The end effector position trajectory to be solve numerically; (b) the root
mean square error of the actual position and solutions obtained using inverse
solution; (c) the angles in the upper chain and (d) angles in the lower chain obtained
from the numerical inverse computation. ............................................................... 111
Figure 66: The end effector position points to be solve numerically; (b) the root mean
square error of the actual position and solutions obtained using inverse solution; (c)
the angles in the upper chain and (d) angles in the lower chain obtained from the
numerical inverse computation. .............................................................................. 113
Figure 67: The circular arcs that describe the constant orientation workspace boundaries
of the HD2 device at
1
1'
0 can be identified with circular arc equations
(with radius and range of angle) for this position. .................................................. 114
xix
Figure 68: Isometric view of the constant orientation workspace of the HD2 haptic device
(handle is held vertically) at various heights. ......................................................... 115
Figure 69: Constant orientation workspace of HD2 shown with (a) side view; (b) front
view; (c) top view, and (d) isometric view. At this orientation, the minimum and
maximum reachable distance is from 0.08m to 0.56m in the x-direction, -0.66m to
0.42m in the y-direction, and 0.08m to 0.61m in the z-direction. The workspace
volume is computed to be 0.956m3. ........................................................................ 116
Figure 70: (a) a biopsy needle were attached to the handle of the HD2 haptic device
performing virtual biopsy training; and (b) the corresponding constant orientation of
the HD2 handle that approximate the needle angle that is being used for workspace
study. ....................................................................................................................... 117
Figure 71: The constant orientation workspace of the HD2 haptic reduced to about 0.76m3
when the handle is held at an angle approximating the biopsy needle insertion angle
(-30 degree in the x-axis). Shown here (a) and (b) are two different view angles of
the resulting workspace........................................................................................... 118
Figure 72: The constant orientation workspace of the HD2 haptic further reduced to about
0.51m3 when the handle is held at an rotated angle of -60 degree along the x-axis.
Shown here are (a) isometric view and (b) front view of the resulting workspace.
More views are presented in the Appendix............................................................. 118
Figure 73: Scaled manipulability ellipsoid of (HD)2 shown in (a) isometric view; (b) front
view; (c) top view; and (d) side view...................................................................... 119
Figure 74: With the inclusion of parallelogram structure, the manipulability ellipsoid
changes considerably for the HD2 haptic device. ................................................... 120
xx
Figure 75: Comparison of manipulability ellipsoids of HD2 haptic device with endeffector positioned at (2.5, 0, 3.44), modeled with (in blue color) and without (in
yellow color) the parallelogram structure at various view (a) front view; (b) side
view; (c) top view; and (d) Isometric view. ............................................................ 121
Figure 76: Comparison of manipulability ellipsoids of HD2 haptic device with endeffector positioned at (2.8, 2, 3.44) and tilted 60 deg along the x-axis, modeled with
(in blue color) and without (in yellow color) the parallelogram structure at various
view (a) front view; (b) side view; (c) top view; and (d) Isometric view. .............. 122
Figure 77: Inverse condition number/ Isotropy index of HD2 haptic (a) with and (b)
without including the parallelogram structure in the Jacobian formulation, evaluated
at plane z=0.344m and the handle is held vertically. (c) Although the condition
number without the parallelogram structure is better (higher number), the isotropy
index with the addition of parallelogram structure is more uniform throughout the
workspace. .............................................................................................................. 123
Figure 78: Stiffness of (HD)2 in each directions, with formulation without considering the
parallelogram structure in the upper and lower linkages. ....................................... 125
Figure 79: Stiffness of (HD)2 in each directions, with formulation including the
parallelogram structure in the upper and lower linkages. ....................................... 126
Figure 80: (a) The trajectory tracking result shows the desired path and the traced path; (b)
show the HD2 tracking the trajectory; (c) angle profile of all the joints in the system;
and (d) the length of the handle throughout the trajectory tracking show the position
and angular constraints were satisfied. ................................................................... 128
xxi
Figure 81: Hardware-in-the-loop (HIL) design and testing framework of the HD2 haptic
device. ..................................................................................................................... 129
Figure 83: SimMechanics model of HD2 haptic device with its corresponding VRML
model for visualization. .......................................................................................... 130
Figure 84: (a) MapleSim model of the HD2 haptic device with its corresponding 3D
visualization model. ................................................................................................ 130
Figure 85: When a human user holding on a haptic device, a geometric loop-constraint
between them and the bi-directional transmission of forces and moments create
challenges to evaluate the performance of the combined human-device interactions.
................................................................................................................................. 132
Figure 86: (a) A simple planar human kinematic model created to include human user in
optimizing the haptic device workspace, given in [174-175]; and (b) Collision
avoidance and motion planning by modeling both the human and haptic device as
linkages, given in [173]........................................................................................... 134
Figure 87: Example of haptic rehabilitation devices:(a) the MIT-MANUS for upper-body
function rehabilitation after stroke, developed in MIT [207]; (b) the “Rutgers Ankle”
Rehabilitation Interface for ankle rehabilitation developed in Rutgers University
[208]; and (c) The haptic rehabilitation device for upper-limb motor function
rehabilitation developed in Shibaura Institute of Technology [209]. ..................... 136
Figure 88: Examples of adjustable, reconfigurable rehabilitation exercise machines from
BIODEX: (a) Upper Body Cycle; (b) System 4, and (c) BioStep Clinical Pro. ..... 136
Figure 89: (a) AnyBody musculoskeletal modeling environment [202]; and (b) OpenSim
open source software system interface [215]. ......................................................... 138
xxii
Figure 90: Systematic and parametric musculoskeletal analysis based study can be used
to better understand bipedal walking frequency [200]. .......................................... 141
Figure 91: (a) Parametric bicep curl study on a simplified upper-arm/shoulder
musculoskeletal model and muscle force profiles for both (b) biceps short and (c)
biceps long muscles. ............................................................................................... 142
Figure 92: Right muscle activity profile (solid) and average muscle activity (dashed) for
one pendulum swing cycle used in [200]. ............................................................... 145
Figure 93: (a) the actual setup that is being modeled and studied[212]; (b) Front view of
the user and driving wheel (rehabilitation device) arrangement in the case study; and
(c) the corresponding top view of the same arrangement. ...................................... 146
Figure 94: Paradigm of our framework for VP of rehabilitation device: A MATLAB GUI
that serves as the Center Interface that allows the therapist to examine the effects of
different regimen and determines the „best‟ regimen based on user‟s geometric
information. The AnyBody engine is responsible for the computation of the muscle
forces, while the optimization routine is handle using MATLAB‟s optimization
toolbox. ................................................................................................................... 148
Figure 95: The human musculoskeletal model, the rehabilitation device, and their
interaction used in the case study were modeled using AnyBody Modeling System.
Shown here is the (a) Side view and (b) Front view of the model.......................... 149
Figure 96: (a)-(d) The hand placement on the steering wheel at different time instant in
the parametric study, view from the top of the virtual model. (a) The initial position;
(b)-(c) the intermediate positions; and (d) the final position. ................................. 150
xxiii
Figure 97: (a) Surface and (b) 2D plot of the maximum combine muscles force for a
patient to turn a wheel at a constant angular velocity of 30deg/sec (with 330 deg
total movement), for a 0 to 60 degree tilted wheel angle, with wheel center located at
(0.25, -0.1, 0.25), measured in meters. .................................................................. 151
Figure 98: (a) Surface and (b) 2D plot of the maximum combine doing the same motion,
this time with 0 deg tilted angle (no tilt) , for wheel x position varied from 0.1m to
0.45m (far or close in front of the user). ................................................................ 152
Figure 99: (a) Function space of maximum combined muscle forces fluctuation as the
objective function and steering wheel‟s x location and tilt angle as the design
variables; and (b) Function space of average muscle force as the objective function
and steering wheel‟s x location and wheel‟s tilt angle as the design variables. ..... 153
Figure 100: Stanford Manipulator, a R-R-P-R-R-R serial manipulator and its D-H frame
assignment............................................................................................................... 179
Figure 101: (a) raw data points; (b) interior points are filtered; (c) the boundary points are
not arranged sequentially around the boundary; (d) re-ordered points sequentially
envelop the boundary. ............................................................................................. 186
Figure 102: (a) Surface plot that represent the workspace; (b) Contour at specific level
can be generated; (c) the workspace boundary point extracted from the contour plot.
................................................................................................................................. 186
Figure 103: Process of computing area using Algorithm 1. (a) shows the original data
points that are inside the workspace; (b) the filtered and re-organized boundary data
points; and (c) the area computed using these data point using polyarea algorithm,
xxiv
for grid size 0.1. (d)-(f) shows the same process for data points generated with grid
size 0.01. ................................................................................................................. 189
Figure 104: Process of computing area using Algorithm 2. (a) shows the surface plot of
points that are inside the workspace; (b) the level at which each contour can be
generated, and subsequently points can be extracted; and (c) the area computed
using these data point using polyarea algorithm, for grid size 0.1. (d)-(f) shows the
same process for data points generated with grid size 0.01. ................................... 190
Figure 105: Percent difference of estimating workspace area from actual workspace at
different contour levels (from 0 to 1.0, with increment of 0.1), for grid size of 0.1,
0.05, and 0.01. The horizontal line represents perfect estimation. ......................... 191
Figure 106: (a)-(e) Raw data points representing the constant orientation workspace of
HD2 haptic device computed using Algorithm 1 at grid size 0.2, 0.1, 0.05 and 0.02;
and (f)-(i)their corresponding workspace evaluated using the filtered and reorganized boundary points. ..................................................................................... 193
Figure 107: (a)-(d)Constant orientation workspace of HD2 haptic device computed using
Algorithm 2 at grid size 0.2, 0.1, 0.05 and 0.02; and (e)-(g) their corresponding
workspace area computed with contour evaluated at level 0.5. .............................. 194
Figure 108: Workspace area approximation with Algorithm 2 evaluated at various
contour levels, with grid size of 0.02, 0.05 and 0.1. Workspace area approximated
with Algorithm 1 also plotted for the same grid sizes for comparison. .................. 195
Figure 109: Constant orientation workspace of a 2-RRR planar manipulator changes as
the distance of their bases varied from (s) L
1.5 to (f) L
4.5 with end-effector
angle held at zero. ................................................................................................... 196
xxv
Figure 110: Constant orientation workspace of a 3-RRR planar manipulator changes as
the distance of their bases varied from L
held at zero. Largest workspace exist when L
0.5 to L
4.5 with end-effector angle
1.7321 . ......................................... 197
Figure 111: Constant orientation workspace of a 3-RRR planar manipulator changes as
the distance of their bases varied from L
held at
0.5 to L
45 . Largest workspace no longer exist at L
4.5 with end-effector angle
1.7321 . ......................... 198
Figure 112: Constant orientation workspace of a 4-RRR planar manipulator changes as
the distance of their bases varied from L
2 to L
3.25 with end-effector angle held
at zero. ..................................................................................................................... 199
Figure 113: Manipulability ellipsoid of a Five-bar manipulator with two different
actuated joints compared with the result obtained from [119], at three different end
effector locations (a), (b) and (c). ........................................................................... 200
Figure 114: Manipulability ellipsoid of a Six-bar manipulator with two different actuation
schemes compared with the result obtained from [119], at three different end
effector locations (a), (b) and (c). ........................................................................... 201
Figure 115: For the same end-effector position and angle, manipulability ellipsoid can be
different due to inverse solution of each RRR chain. ............................................. 202
Figure 116: Inverse of Condition Number map across the workspace of a 3RRR parallel
manipulator changes for different configurations. .................................................. 203
Figure 117: Manipulability elliposoid changes as well as workspace as configuration
changes. ................................................................................................................... 204
Figure 118: Stiffness maps of the 3-RPR manipulator in the (a)
direction; and (c) y –direction with
–direction; (b) x –
0 , (d), (e), and (f) are stiffness map with
xxvi
end-effector angle
90 . Comparing result with comparing results with [143] (top
row). ........................................................................................................................ 205
Figure 119: Stiffness maps of 3RRR manipulator in the (a) x-direction, (b) y- direction,
and (c)
– direction, with end effector angle
map with end-effector angle
0 . (d), (e), and (f) are stiffness
90 . Comparing result with [153] (bottom row). 206
Figure 120: Comparison of the stiffness map of a 3-RRR manipulator in x-direction (2nd
row), y-direction (3rd row), and
-direction (4th row) with different joints actuated:
(a) 3-RRR; (b) 3-RRR; and (c) 3-RRR manipulator. End-effector angle
45 . . 207
Figure 121: Constant orientation workspace of (HD)2 at various heights: (a) isometric
view; (b) side view; and (c) front view. .................................................................. 208
Figure 122: The constant orientation workspace of the HD2 haptice further reduced to
about 0.51m3 when the handle is held at an rotated angle of 60 degree along the xaxi. Shown here are two different view angles of the resulting workspace. ........... 209
xxvii
List of Tables
Table 1: Table summarizing the Jacobian matrix formulated using various methods. .... 34
Table 2: Geometric Parameters for Phantom 1.5 .............................................................. 92
Table 3: Geometric Parameters for the Parallelogram sub-system in Phantom 1.5 ......... 92
Table 4: Measured joint limits of the PHANToM 1.5 haptic system. .............................. 93
Table 5: DH Frame assignment of the upper chain of HD2. ........................................... 109
Table 6: DH frame assignment of the upper chain of HD2, included the parallelogram
sub-system............................................................................................................... 109
Table 7: Measured joint limits for the HD2 haptic device. ............................................. 114
Table 8: DH Frame assignment for Stanford manipulator.............................................. 180
Table 9: Comparison of computation time and number of boundary points generated in
each of the two algorithms, for various grid size. ................................................... 187
Table 10: Comparison of computed are and % difference with actual area at various grid
size, in each of the two algorithms.......................................................................... 188
Table 11: Measured joint limits for the HD2 haptic device. ........................................... 192
Table 12: Comparison of computation time and number of boundary points generated in
each of the two algorithms, for various grid size used in estimating the constant
orientation workspace of HD2. ................................................................................ 195
xxviii
1 Introduction
Haptic Immersive Virtual Environments (HIVE) hold considerable promise for
ability to expand, assist, train and monitor human sensorimotor capabilities, for
improving physical strength, redirecting undesirable motions and forces, augmenting the
manual precision, and improving overall dexterity.
Figure 1: Immersive and interactive rendering of ‚touch‛ in virtual environments depends in
equal part on: (i) the human user’s sensory abilities; (ii) the selected haptic user interface; and (iii)
fidelity of computed virtual environment responses.
The overall kinesthetic immersive experience in a haptic interactive virtual
environment is the synthesis of the human user, the haptic user interface (HUI), and
virtual environment (VE) – all of which playing critical roles. The virtual environment
refers to the computer-based model, capable of interactively generating cues for
immersion of multiple senses of the user. While early VEs were restricted to generation
1
of visual and auditory cues, contemporary VEs have been extended (using “haptic
models”) to encompass the generation of kinesthetic cues. The HUI is to denote the
computer-controlled electro-mechanical system (“haptic device”), the feedback control
laws (“haptic control laws”), as well as all the intermediate elements (A/D, D/A,
conditioning electronics) that help interface the motions and forces between the human
operator and the virtual environment. The effectiveness of the interface – in
communicating human-user intent to the virtual environment and rendering of results
back to the users - can be judged using performance benchmarks such as the fidelity,
transparency, stability, accuracy, and real-time interactivity. The differences from the
traditional paradigm of human-computer interaction arise due to: (i) the far higher
bandwidth of kinesthetic and tactile perception (over 1000Hz vs. the 30Hz for visual
perception); and (ii) the bi-directional exchange of information and energy occurring at
the kinesthetic user interface. These requirements come at significant computational cost
and often conflict with the requirement for maintaining real-time interactivity and overall
system stability. Critical tradeoffs, in terms of bandwidth, fidelity and accuracy, are
required for development of “suitable haptic models” for generating responses to
dynamic user interactions with the virtual world as well as “suitable haptic interfaces” to
render the resulting computation without attenuation or interference.
It is noteworthy that despite the obvious benefits and availability of several
commercial simulators, this approach has yet to become mainstream. The limited
capability for sensorimotor “realism” offers a serious obstacle for successful
implementation of simulation-based learning. Significant technical challenges, in terms
2
of limits on quality, immersivity, intuitiveness, and realism of visual and haptic
simulation need to be overcome before today„s Haptic UIs pass the haptic “Turing Test”.
1.1 Fundamental Issues at Core of Haptic Realism
In order to realize the full set of benefits, further basic research is required in the
areas of optimal designs and advanced control algorithms for (i) advanced haptic user
interfaces; (ii) algorithm and architectures for physics based high performance distributed
computation of the graphic and haptic rendering; and (iii) assessment of simulator
validity, suitable user interface design and automated user-evaluation methods as shown
in Figure 4. Among these our focus will be on the development of Advanced Haptic User
Interfaces.
Haptic User Interface (HUI) devices offer potential for providing the users with
sophisticated tactile or force feedback during interaction with virtual environments with
innovative features including active and passive manipulation assist. However, realizing
these benefits requires the HUI be able to generate and render a wide range of high
fidelity dynamic behavior. Hence as electro – mechanically actuated articulated
mechanical systems, high-performance HUI should possess: human matched force
capabilities, sizeable workspace, low inertia, high stiffness, low friction, back-drivability,
near-zero backlash, and gravitational counterbalancing,. Particular stress is place on the
transparency of the device so that the HUI does not distort the reflected forces/torques by
its electro-mechanical characteristics. To this end, we are proposing to design, develop,
implement and validate HUI, based on parallel mechanism architecture.
3
Figure 2: The fundamental issues at the core of haptic realism are tied closely to complexity of
Haptic User Interfaces and Haptic Modeling Fidelity. In this thesis, we focus on studying various
Haptic User Interfaces.
The physical design of the haptic device, in terms of selection of the type, number,
location and actuation of the individual articulations plays an important role in
determining the capabilities and overall performance of the system. The system must be
designed and controlled carefully in order to minimize singular configurations, perform a
wide range of constrained motion-force manipulation tasks with dexterity and strength,
while remaining robust to local controller lapses and environmental disturbances.
4
1.2 Problem Statement
Consider the illustrative case-scenerio of creating a haptic device to be used for
simulating virtual knee replacement surgery. In any such typical orthopedic surgery, the
surgeon needs to perform gross motor tasks (such as cutting through tissues, drilling and
sawing through bones, hammering in pins to attach a jig), as well as finer sensorimotor
tasks (such as paring, shaving and suturing). Figure 3 is a representative illustration of
some of these typical tasks. In seeking to create a haptically-enabled virtual simulator to
train these sensorimotor skills, a designer needs to first translate these various tasks
specifications into design specifications. For example, the size of the typical working
region, the ranges of motion needed for the various tasks, the ranges of forces (from
suturing to drilling and sawing) that need to be rendered are important task-specifications.
(a)
(b)
(c)
(d)
Figure 3: Typical tasks involved in a knee replacement surgery (a) drilling through bones; (b)
hammering pins; (c) sawing through bones; and (d) suturing.
The operation region required to perform the surgery defines the „workspace‟ of a
haptic device (although proper scaling can be used to reduce the required space). The
requirement to move in various directions and orientations defines the degree-of-freedom
of the haptic devices. The ability to render high force and high stiffness for drilling and
5
sawing simulation probably favors a parallel-architecture haptic device design over a
serial-architecture one.
Ideally, these design specifications can be captured and characterized by
appropriate quantitative performance measures (workspace, manipulability, and stiffness),
wherein it is desirable to retain a parametric relationship to critical device parameters
(link lengths, types of joints, etc.). In this setting, the process of design of a haptic device
to meet desired specifications can be reduced to an optimization problem with
appropriate performance measures to serve as objective function over the design space of
the unknown device parameters (which we adopt).
1.3 Serial vs. Parallel Architecture Haptic Devices
Parallel architecture manipulators consist of a moving platform that is attached to
a fixed base by several articulated kinematic chains, which create one or more closed
kinematic loops within the system [1-2]. While serial architectures possess a large work
volume and high dexterity, numerous disadvantages such as low precision, poor force
exertion capability, low payload-to-weight ratio, and high inertias tend to limit its use in
such applications. On the other hand, parallel mechanisms based devices are well-known
for their high stiffness, low inertia, high rigidity and accuracy, and high payload-toweight ratios, which enable large bandwidth transmission of forces. Other desired
properties can be achieved through optimizing the underlying mechanical design and
selection of suitable control scheme. The Moore‟s law driven improvements in
computation have now almost overcome the limitations on real-time placed by the
complicated input-output relationships. Hence, application settings such as the multi-axis
grinding machining centers [3-6]. Other applications to benefit from the high motion
6
agility and payload capacity of parallel architectures include flight simulators [7-8], agileeyes [9], human motion simulators [10], and force reflecting hand controllers [11-13].
Figure 4: Examples of commercial haptic devices based on serial architecture.
Parallel-architecture haptic devices offer significant advantages over serialarchitecture counterparts in applications requiring high stiffness and high accuracy. To
this end, many haptic devices have been created and deployed by modularly piecing
together several serial-chain arms to form an in-parallel system. However, the overall
system performance depends both on the nature of the individual arms as well as their
interactions. We build on the rich theoretical background of constrained articulated
mechanical systems to provide a systematic framework for formulation of system-level
kinematic performance from individual-arm characteristics. Specifically, we develop the
system-level kinematic model in a symbolic (yet algorithmic) fashion that facilitates: (i)
7
computational development of pertinent symbolic equations; (ii) generalization to
arbitrary architectures; and (iii) combined symbolic/numeric analyses of performance
(workspace, singularities, and design sensitivities). These various aspects are illustrated
using the example of the Quanser High Definition Haptic Device (HD)2 – an in-parallel
haptic device formed by coupling two 3-link Phantom 1.5 type serial chain manipulators
with appropriate passive joints.
Figure 5: Examples of commercial haptic devices design based on parallel architecture.
Recently, there has been considerable interest in creating parallel-architecture
haptic devices as in-parallel systems by a modular composition approach, wherein
multiple articulated serial-chain arms cooperate to control a common end-effector. Such a
composite-system can now potentially allow for increased redundancy, robustness, and
reliability and even active reconfigurability for different tasks. Further, in promoting
8
reuse of components, such a modular “building-block” approach is also a favorable
engineering practice.
However, such modularity creates increased design-choices, in terms of methods
to realize given tasks, and requires a design-selection process to determine the best
designs. However, the system performance in a modularly composed system depends
both the nature of the individual modules as well as their interactions, which creates
challenges. The physical layout of the parallel haptic device, in terms of selection of the
type and number of the in-parallel articulated chains, and their attachment to the mobile
platform determines the topology of the system. The subsequent selection of type,
number, location and actuation of the individual articulations (within the chain)
completes the determination of the workspace, mobility, controllability, and overall
performance of the system.
This is an aspect that we examine in the context of the Quanser High Definition
Haptic Device (HD)2 – an in-parallel haptic device formed by coupling two 3-link
Phantom 1.5 type serial chain manipulators to a common end-effector handle. However,
the two three-link manipulators cannot be arbitrarily coupled to the handle due to the
potential kinematic incompatibility (of the velocities at the end effector). Hence, further
passive articulations are necessary in order to accommodate the rigid body constraints. In
(HD)2 system, this role is played by the passive revolute and universal joints. Hence, a
systematic (and preferably computational) framework for evaluation of the designchoices on individual module- and system-level characteristics is desirable.
9
Figure 6: In-parallel haptic devices by modularly combining two or more serial architecture haptic
devices.
Performance measures play a critical role in design, optimization and control of
robotic systems. See [14-17] for surveys of various local performance measures derived
from the generalized Jacobian matrix. The singular value decomposition of the
generalized kinematic Jacobian and the geometric relationship (volume, eccentricity, etc.)
of the manipulability ellipsoid has been investigated. Other researchers have examined
the development of dynamics based measures by extending the notion of manipulability
and isotropy to the dynamics domain [18-19]. Equivalent force performance measures for
the local velocity based measures may be computed by virtue of the principle of virtual
work and used for the synthesis and design of multi-DOF mechanisms.
The resulting performance measures have been employed as a means to evaluate
the workspace quality [20], measure of kinematic accuracy [21], selection of operating
regions [22], design of modular kinematic structures [23], and improvement of numerical
properties for resolved velocity and acceleration control of manipulators [24]. While
10
many such measures have tended to be local, several authors have obtained global
measures by spatial integration of local measures over a region [25] or by integration
over time. However, as Park [16] and van den Doel and Pai [17] note, the sheer diversity
of measures to quantify qualitative measures is partially due to the adoption of a local
coordinate representation and hence advocate the formulation of coordinate independent
measures using the unifying framework of differential geometry.
Extensions to manipulability analysis have been studied both in the context of
multiple cooperating robot arms as well as parallel manipulators [26-28]. Park and
Bobrow [29] examine the optimal base positioning of two cooperating robot manipulators
in a geometric framework. Park and Kim [30-31] present an elegant differentialgeometric framework for analysis of manipulability of both redundantly-actuated and
exactly-actuated parallel manipulator systems in a unified fashion, which we compared
our results with.
1.4 Contributions
To this end, we leverage the rich theoretical analysis background for constrained
articulated mechanical systems. In particular, we exploit a twist-based analysis of inparallel systems [32] to create the underlying performance-characterization framework.
The novel contributions of this work come from: (i) algorithmic modeling of the
individual arms, especially exploiting symbolic computational tools; (ii) systematic
system-level motion analysis, by composing contributions from the individual modules;
and (iii) validation within a combined simulation and hardware-in-the-loop environment.
11
Figure 7: Modularity creates design choices. The performance of the overall system depends on
the nature of the individual module as well as their interactions.
1.5 Thesis Organization
The remainder of this thesis is organized as follows:
In Chapter 2, we discussed some of the design issues for parallel-architecture
haptic devices. In Chapter 3, we outlined several existing methods to formulate the
Jacobian of both serial and parallel manipulator. Using an RRR and a 3-RRR system as
example, we motivate the use of twist-based Jacobian formulation for serial and parallel
manipulators, which allows systematic Jacobian formulation that we use in this work.
In Chapter 4, performance measures we use in this thesis were presented and their
formulations were given. Several planar in-parallel manipulators were use as examples to
illustrate the presented ideas.
In Chapter 5, we study the PHANToM 1.5 system, which is a serial type
manipulator but with a parallelogram sub-system (some authors called it „hybrid‟
manipulator). Specifically, we study the effect of the added parallelogram sub-system to
12
the overall performance of the system. In Chapter 6, we study the Quanser‟s High
Definition Haptic Device – HD2, a 5-DOF in-parallel haptic device created from
modularly piecing together two PHANToM 1.5 haptic devices. We show the workspace,
manipulability and stiffness of this device. Similar to the PHANToM 1.5 system, we also
study the effect of the two added parallelogram sub-systems to the overall performance of
the device.
In Chapter 7, we look at the design of haptic device from another perspective.
Since a haptic device is always coupled with a human user while in use, we proposed a
musculoskeletal analysis framework in which co-simulate of a human user with a device
is possible. Using this framework, we can parametrically study various design variables
and perform optimization to study design parameters that affect human-device
interactions. Chapter 8 summarizes the contributions in this work, and concludes the
work by providing directions for future research.
13
2 Background
Parallel architecture haptic devices feature multiple closed-kinematic loops and
kinematic/
actuation
redundancy which create
opportunities for
performance
enhancement as well as challenges in modeling and control of such systems
2.1 Parallel Architecture Manipulator
Figure 8: Using a 2-DOF RR manipulator as building block, many interesting haptic devices can
be developed by modularly piecing the individual small system in different ways to create multi
DOF haptic devices.
14
Immense possibilities exist for realization the design of parallel-architecture
manipulators based on number of supporting chains, the type and number of joints in
each chain and the location of their attachment to the fixed and moving platforms - see
[33] and [34-35] for detailed listing of parallel-architecture robots and related
bibliography.
From a design perspective, three overlapping task decomposition stages will be
considered. First, the designer has to select between competing component designs such
as selection of type, location and number of articulations within each parallel chain and
then number of chains – a process which we will loosely term “type synthesis”. Second,
considerable freedom still exists from the viewpoint of selection of the characteristic
dimensions of the sub-chains, locations of attachment for the manipulators – a process
which we will term “dimensional synthesis”. Finally, since the system is redundant, it can
still assume several infinities of postures/configurations permitting the selection of an
“optimal” configuration – a process we will term “configuration synthesis”.
Figure 9: Kinetostatic design optimization to improve both workspace volume and quality
(stiffness, accuracy, velocity and force performance).
15
Major design variants are possible based on nature of the intermediate chains
connecting the fixed and moving platform. Pantograph legs, five-bar linkages, serial
chain linkages, spherical linkages, or even cable tendon transmission are all viable
candidates and choice of leg design and its parameters significantly influence systemworkspace and performance (See Figure 8). For a parallel mechanism, the workspace of a
PKM has not a simple geometric shape, and its functional volume is reduced, compared
to the space occupied by the machine [36]. The velocity and force transmission ratios and
end-effector stiffness may vary significantly in the workspace (at various locations and
along different directions) because the displacement of the tool is not linearly related to
the displacement of the actuators. Further, unlike serial chain manipulators, singularities
appear inside the workspace due to individual-chain or system-level effects [37]. Hence,
we will pay special attention to improving both the size and quality of the workspace by
consideration of kinetostatic criteria at every stage in the design process. The
development and use of quantitative kinetostatic measures, based on the conditioning of
the combined-system Jacobian matrix, allow a transformation of the design problem into
a design optimization problem. The overall optimal design can thus by realized by a
combined design-synthesis and simulation-based design-refinement approach [38-39]. It
is noteworthy that the framework of Park and Kim [30-31] permits treatment of these
issues, including the effects of redundancy and indeterminacy, in a coordinate-free
differential-geometric setting.
2.2 Parallel Architecture Manipulator Control
The system must be subsequently designed and controlled carefully due to the
presence of closed kinematic chains with configurations chosen in order to: (i) minimize
16
singular configurations of the system [37]; (ii) enhance mutual cooperation (motions and
forces) to take advantage of redundancy during task performance; and (iii) improved
robustness to local controller lapses and environmental disturbances.
A major difficulty that prevents application of the vast control literature
developed for the serial counterparts to redundantly actuated parallel manipulators is the
lack of an efficient dynamical model for real-time control. Unlike unconstrained or nonholonomically constrained mechanical systems, Yun and Sarkar [40] survey several
methods for modeling systems with holonomic constraints, and present an elegant unified
state-space formulation for such systems we will adopt. The configuration kinematic
model will be determined and extended to the configuration dynamic model by a
Lagrangian reduction onto the tangent velocity distribution [40-41] permitting us to
connect this parallel manipulator to the standard control literature.
Redundant actuation provides an effective means for eliminating singularities of a
parallel manipulator, thereby improving its performance such as Cartesian stiffness and
homogeneous output forces. The use of redundant actuation also gives rise to more
homogeneous output forces and minimized internal loading torques of the actuators of the
manipulator [42-44]. There have been papers on dynamics of parallel manipulators with
normal or non-redundant actuation [45-47] but, research on redundantly actuated parallel
manipulators is scarce [48].
The redundant actuation, in the articulations, can be used to actively control both
the relative configuration and the interaction forces between the modules simultaneously.
Most of the developed schemes (in other contexts), seek to achieve this by decoupling
position control from the force control, either in the task/operational space or joint space.
17
While the decoupling of the motions and force control in operational space [49] yields
algorithms that lack coordinate and frame invariances [50] configuration space
decoupling approaches [51-52] do not suffer from similar problems. We will develop a
hybrid velocity/force control method of Z. Li [48] in the configuration space. The
velocity controller will be designed in the constrained configuration motion space while
projected interaction forces, in the orthogonal joint force space, may be controlled by
controlling the integral of the force error by implicit, explicit force control or impedancebased approaches.
2.3 Virtual and Physical Prototyping Implementation
Time domain simulation of ordinary- and partial-differential equations will play a
major role both in the initial design, test and integration of haptic simulation systems and
subsequently in the scenario-based training of normal and abnormal procedures.
Figure 10: Rapid virtual and physical prototyping application developed for 5-DOF parallel
haptic device.
18
However, it is highly desirable to be able to completely simulate a wide variety of
scenarios digitally initially and to subsequently be able to validate models against actual
field results. Such an approach is again both simulation intensive and incorporates
elements of HIL data acquisition. Our approach emphasizes: (a) Development of the
models and algorithms in a graphical, high-level block diagrammatic language, Simulink
[53] that preserves design intent but permits hierarchical abstraction and encapsulation;
(b) Simulation, testing and refinement of the models and algorithms by rapid distributed
virtual prototyping; and (c) Rapid conversion of the refined algorithms into real-time
executable, using RT-LAB for operation on the distributed computing environment for
hardware-in-the-loop testing with the sensors and actuators. We anticipate that these will
play a significant role in speeding up the process of development and testing of our
systems of interest.
19
3 Technical Background
Parallel-architecture haptic devices offer significant advantages over serialarchitecture counterparts in applications requiring high stiffness and high accuracy. To
this end, many haptic devices have been created and deployed by modularly piecing
together several serial-chain arms to form an in-parallel system. The concept of
modularly piecing serial chains or simple parallel chain (i.e. the Five-bar) to form parallel
haptic systems with multiple degree-of-freedom (DOF) such as the di-tetrahedral
mechanism, Hayward mechanism, and Compass mechanism, discussed by Hui, et al [54]
as early as 1995. The 6-DOF Pen-based Force Display [55] and 6-DOF Haptic Master
[56] developed by Hiroo Iwata of University of Tsukuba in Japan were created by
modularly piecing together two and three PHANToM 1.5 like manipulator, are examples
of such systems.
However, the overall system performance depends both on the nature of the
individual arms as well as their interactions. In order to study the performance of the
haptic devices, we need to formulate the Jacobian of the system. We build on the rich
theoretical background of constrained articulated mechanical systems to provide a
systematic framework for formulation of system-level kinematic performance from
individual-arm characteristics. Specifically, we develop the system-level kinematic model
in a symbolic (yet algorithmic) fashion that facilitates: (i) computational development of
20
pertinent symbolic equations; (ii) generalization to arbitrary architectures; and (iii)
combined symbolic/numeric analyses of performance (workspace, manipulability, and
stiffness).
In this Chapter, we show various exiting formulations for obtaining the Jacobian
of serial manipulator, including Conventional method (by differentiating position vector),
the Spatial Vector Formulation shown in [34], and finally the Twist-based Jacobian
Formulation (including Body-Fixed Twist and Spatial Twist). Followed by using a 3RRR manipulator as an example, we show the process of obtaining system-level Jacobian
for parallel manipulator using various methods and with Twist-based Jacobian
Formulation method. In particular, we show the process of formulating system-level
Jacobian for in-parallel system from Jacobian of individual chain, which we use in our
work to study and analyze various in-parallel haptic devices.
3.1 Serial Manipulator Jacobian Formulation
3.1.1 Conventional Method
The Jacobian of a serial manipulator shown for a three link RRR manipulator can
be obtained by first expressing the position of a point on the end-effector of the
manipulator with respect to the global frame of reference, and subsequently
differentiating the equation to obtain the velocity-level relationship between end-effector
velocity and joint rate [24].
21
Figure 11: Reference frames assignment for a RRR manipulator.
The position vector to the end-effector expressed in the fixed frame {0} can be
written as:
e
1
xe
ye
L1 cos( 1 )
L1 sin( 1 )
L2 cos(
L2 sin(
1
1
2
3
)
2
)
2
L3 cos( 1
L3 sin( 1
)
)
3
2
(1)
3
2
The Jacobian can be obtained by simply differentiating the position vectors with respect
to time:
e
xe
ye
where sin
-L1 sin
L1 cos 1
123
sin(
1
1
1
- L2 sin
L2 cos
12
2
3
12
- L3 sin 123
L3 cos 123
) , and cos
123
22
1
-L2 sin 12 - L3 sin
L2 cos 12 L3 cos
cos(
1
2
123
123
3
).
1
-L3 sin
L3 cos
1
123
2
123
3
(2)
3.1.2 Spatial Vector Formulation
While the above formulation works well for planar system, for spatial system, the
analytic Jacobian does not yield the system Jacobian. Instead we adopt a systematic
approach, which utilizes the DH frame assignments to determine various columns of the
Jacobian matrix (the twist contribution from each joint). Using the origin of the endeffector reference frame as reference point to describe the velocity state of the endeffector, the end-effector velocity can be expressed as [34]:
n
vn
i
i
zi
pn*
zidi
(3)
i 1
n
z
n
i i
(4)
i 1
where
and d are the rate of rotation about i th joint and rate of translation along i th
joint, z i is a unit vector along i th joint axis, and i pn* is a vector defined from the
reference frame of i th link to the origin of the end-effector frame. Note that all vectors in
Eqn. (3) and (4) are expressed in the fixed reference frame. Note that the expression of
subscript i in both z i and i pn* in Eqn. (3) and (4) is related to the DH convention used
(Here we show a variant using the modified DH convention outlined in [57], if the
conventional DH is used, refer to [34], their resulting Jacobian matrix will be the same).
Writing Eqn. (3) and (4) in matrix form, we obtain
x
vn
n
where
23
Jq
(5)
J
Ji
Ji
J 1, J 2 ,
i
zi
, Jn ,
pn*
for revolute joint,
zi
zi
0
(6)
for prismatic joint.
For spatial system, the left-hand side of Eqn. (5) is a 6 1 ( 3 1 for planar
system) vector with velocity vn and angular
n
component. The right-hand side of Eqn.
(5) is the product of the Jacobian matrix with the vector of all the actuated joint rates. The
joint rates are given as qi
, qn , and the i th column of the Jacobian matrix,
q1, q2,
J i , represent the contribution from i th joint rate to the velocity of the end-effector.
Equation (5) allows the systematic symbolic computation of the Jacobian for a
given system. To compute the Jacobian, Eqn. (5) shows that we need to compute the
location and direction of each joint axis, and this can be accomplished using the
following relations:
0
Ri 0 ,
1
0
zi
i
Ri i ri
pn*
(7)
i 1
0
1
pn*
where
ai
i
ri
1
di sin
di cos
i
(8)
i
Which is the position vector defined in the DH convention used to create the
homogeneous transformation matrix. Equation (7) and (8) shows that, once we assigned
coordinate frames according to DH convention to a robotic system, the homogeneous
24
transformation matrices ( iAi 1 ) that generated from these frame assignment contain all
the necessary information we need to compute Eqn. (7) and (8), thereby allow the
automatic generation of the Jacobian matrix symbolically, since
i
i
Ai
1
i
Ri
1
0
ri
1
(9)
1
We demonstrate this process with a planar 3 link manipulator. The DH frame
assignment and the corresponding DH parameters are given in Figure 11. The
corresponding homogenous transformation matrices are given as:
0
cos
sin
0
0
3
1
0
0
0
A1
A4
1
1
0
1
0
0
sin 1
cos 1
0
0
0
0
1
0
0
0 1
A
0 2
0
cos
sin
0
0
sin 2
cos 2
0
0
2
2
0 L1
0 0 2
A3
1 0
0 0
cos
sin
0
0
3
3
sin 3
cos 3
0
0
0 L2
0 0
1 0
0 0
0 L3
0 0
1 0
0 0
The direction of each joint axis (as expressed in the base frame) is given by
0
z1
0
R1 0
1
0
0 , z2
1
0
0
R2 0
1
The vector i pn* , is given by:
25
0
0 , z3
1
0
0
R3 0
0
1
1
0
(10)
3
2
1
*
4
0
3
*
4
0
2
*
4
0
p
p
p
L3 cos( 1 + 2 + 3 )
L3 sin( 1 + 2 + 3 )
0
0
R3 0
L3
0
R3 r4
3
R2 r3
1
2
R1 r2
L2
R2 0
0
*
4
p
p
L3 cos( 1 + 2 + 3 )
L3 sin( 1 + 2 + 3 )
L3
0
L1
R1 0
0
*
4
0
2
*
4
p
L1 cos( 1 )
L1 sin( 1 )
L2 cos(
L2 sin(
L2 cos(
L2 sin(
2
1
L3 cos( 1
L3 sin( 1
)
)
2
0
1
2
1
L3 cos( 1
L3 sin( 1
0
)
)
2
1
)
) (11)
3
2
3
2
)
)
3
2
3
2
Since all three joints are revolute joints, the columns of the Jacobian can then be
calculated from Eqn. (6):
z1
J1
1
p4*
z1
J2
p4*
z2
2
3
2
)
)
3
-L2 sin( 1
L2 cos( 1
) - L3 sin( 1
) L3 cos( 1
2
2
2
2
)
)
3
3
(12)
1
3
z3
J3
) - L3 sin( 1
) L3 cos( 1
2
2
1
2
z2
-L1 sin( 1 ) - L2 sin( 1
L1 cos( 1 ) L2 cos( 1
p4*
z3
-L3 sin( 1
L3 cos( 1
)
)
3
2
3
2
1
Note that in Eqn. (12), the resulting J i are 6 1 vectors but reduced to 3 1 since it is a
planar system. Finally, the Jacobian is given as:
J
J 1, J 2 ,
, Jn
-L1 sin 1 - L2 sin
L1 cos 1 L2 cos
1
12
12
- L3 sin 123
-L2 sin 12 - L3 sin
L3 cos 123 L2 cos 12 L3 cos
1
123
123
Which is identical to the one obtain in Eqn. (2), when we express x
of x
vn
T
n
.
26
-L3 sin
L3 cos
1
n
vn
123
(13)
123
T
instead
3.1.3 Twist based Jacobian Formulation
Screw theory was introduced by Sir Robert S. Ball in 1876 to generalize two
important theorems in the study of rigid body mechanics. Chasles Theorem states that a
rigid body moves from one position to another by means of a rotation about an axis (line),
followed by translation that is parallel to that axis (line). Analogously, Poinsot Theorem
states that any rigid body acted under a force system can be replaced by a single force
along an axis (line) combined with a torque that is about that axis (line). Hence, a rigid
body motion can be described by the notion of screw motion, and the infinitesimal
version of such motion is called a twist. Due to the duality relationship, the force system
can be referred as a wrench that is dual to twist. Hence, the analysis of motion and force
system can be analyzed in a unified manner using screw theory. An interested reader may
refer to Hunt [58] for the more traditional line-based screw-theoretic modeling.
The adopted twist-based modeling approach emphasizes the linkage to matrix-Lie
group based modeling and analysis of rigid-body mechanics[59]. Specifically, the 3-step
process discussed below emphasizes: (i) systematically constructing the twist matrix for
each joint in the local proximal frame, from a homogeneous transformation-matrix
representation; (ii) transforming the joint twist-matrices into a common reference frame,
using adjoint transformations; and (iii) extracting twist vectors and composing into a
Twist-Assembled Jacobian Matrix (TAJM).
27
Figure 12: Homogeneous Transformation between two frames.
The relative configuration of a moving frame {E } relative to a fixed frame {F }
is defined by the homogeneous transformation expressed in a 4
F
F
where F RE
F
3
d
4
SO(3)
AE
T
0
3 3
{R
RE
F
d
F
1
: RRT
AE
4 matrix form as:
SE 3
I , det R
(14)
1} is a rotation matrix, and
is a displacement vector. A twist matrix T
se(3) can be represented by a
4 matrix of the form:
T
v
0T
0
0
,
z
0
z
y
where
T
T
vT
T
6
x
3
is a skew-symmetric matrix,
the form of t
y
, and v
so 3
x
(15)
0
3
. The twist vector then takes
is the angular velocity and v is a linear
where
28
velocity vector. An unskew operation may also be defined on the set of twist matrices
that allows the extraction of twist vectors as:
Unskew(T )
t
T
T
vT
T
(16)
3.1.3.1 Spatial Twist
In general, a spatial twist matrix corresponding to the motion of the moving frame
{E } with respect to its immediately preceding frame {F } (as expressed in the frame
{F } ) can be formed as:
F
F
F
TE
F
AE
AE
1
(17)
Such a twist matrix can then be transformed to any arbitrary frame {N } by a
similarity transformation as:
N
F
TE
N
AF
F
F
N
TE
AF
1
(18)
The forward kinematics of an articulated system with N joints can be formulated
in terms of homogeneous transforms as:
0
AN
0
A1 1A2
N 1
AN
(19)
The total twist can be considered as a linear combination of various twist
contributions of individual articulated degrees of freedom expressed in a common frame:
29
0
0
0
TN
0
1
0
T1
A1
1
2
T2
A1
1
2
0
A2
2
T2
T1
0
0
AN
0
T3
A2
1
T3
N 1
1
TN
1
0
AN
(20)
1
TN
Using Eq. (18), this can be transformed to inertial frame {W } as:
W
0
TN
W
W
T1
0
A0
0
T7
W
A0 T1
T2
A0
W
A0
1
1
W
A0 T2
W
A0
(21)
1
W
7
A0 TN
W
A0
TN
Finally, these twist matrices can be rewritten as linear combinations of twist
vectors
rates
parameterized
[
1
N
2
by
the
corresponding
manipulation
variable
]T . A twist-assembled Jacobian matrix ( JT ) may now be
constructed in the form of:
1
w
t
t1 t2
tN
JT
where ti
rates of
2
(22)
N
Unskew(Ti ) are the twist vectors corresponding to the manipulation variable
1
,
2
, …,
N
, respectively. The Jacobian matrix ( JT ) can be interpreted as the
linear operator that maps the contribution of the manipulation variable rate at each
actuated joint in the system to the inertial twist at frame {W } .
30
Again, using a RRR manipulator as our example, with the DH frame assignment
shown in Figure 1, the overall transformation matrix for this 3 link system can be written
as:
0
0
A1 1A2 2A3 3A4
A4
(23)
Taking the derivative of Eqn (23), we obtain:
0
0
A1 1A2 2A3 3A4
A4
0
A1 1A2 2A3 3A4
+ 0A1 1A2 2A3 3A4
(24)
0
A1 1A2 2A3 3A4
Post-multiplying Eqn (24) by 0A4 1 , we obtain:
0
A4 0A4 1
0
A1 1A2 2A3 3A4 0A4 1
0
A1 1A2 2A3 3A4 0A4 1
+ 0A1 1A2 2A3 3A4 0A4 1
Where 0A4 1
0
A1 1A2 2A3 3A4
1
3
A4 1
2
A3 1
(25)
0
A1 1A2 2A3 3A4 0A4 1
1
A2 1
0
A1 1 , recognizing AA
1
I ,
we obtained the following expression:
0
A4 0A4 1
0
A1 0A1 1
0
A1 1A2 1A2 1 0A1 1
+ 0A2 2A3 2A3 1 0A2 1
0
A3 3A4 3A4 1 0A3 1
(26)
The Jacobian obtained by assemble the twists contributed from all joint screws
(the last term in Eqn (26) is zero since 3A4
0 ), expressed in fixed frame {F } [60], is
given as:
e
1
1
1
xe
0
L1 sin( 1 )
L2 sin(
ye
0 -L1 cos( 1 ) -L1 cos( 1 ) - L2 cos(
L1 sin( 1 )
31
1
1
)
2
2
1
2
)
3
(27)
On the other hand, the Jacobian obtained by assembles the twists vectors
contributed from all joint screws, expressed in frame {W } ( {W } is located at the endeffector while having the same orientation as the fixed frame {F } ), is given as:
1
e
xe
-L1 sin
ye
L1 cos
1
1
- L2 sin
12
L2 cos
12
1
- L3 sin
-L2 sin
123
L3 cos
123
L2 cos
12
1
1
- L3 sin
123
-L3 sin
123
2
L3 cos
123
L3 cos
123
3
12
(28)
Comparing Eqn (2) with Eqn (28), we see that the Jacobian matrix formulated
using the conventional method and the Jacobian matrix formulated using spatial twist
expressed in end-effector frame {W } are identical.
3.1.3.2 Body Fixed Twist
A body-fixed twist corresponding to the motion of the moving frame {E } , with
respect to its immediate preceding frame {F } , expressed in the moving frame {E } is
given by:
E
E
where
0
x
y
F
TE
AE
x
F
E
E
F
1
F
AE
y
0
E
and
z
z
E
F
E
0 0 0
F
vE
vx
F
vE
0
vy
vz
(29)
T
. Using the unskew
0
operation, the twist vector can be extracted as:
E
F
tE
T
T
32
vT
T
(30)
where one can interpret a body-fixed twist as follow: v
( x,
y
,
z
(vx , vy , vz ) and
) are instantaneous linear and angular velocities of frame {E } with
respect to frame {F } , as expressed in moving frame {E } .
The Jacobian of a serial RRR formulated using body-fixed twist (expressed in the
end-effector frame {E } ) can be obtained by pre-multiplying 0A4 1 to the derivative of the
position transformation matrix (Eqn (24)):
0
A4 1 0A4
0
A4 1 0A1 1A2 2A3 3A4
0
A4 1 0A1 1A2 2A3 3A4
+ 0A4 1 0A1 1A2 2A3 3A4
where 0A4 1
and mAn 1
1
0
A1 1A2 2A3 3A4
n
3
A4 1
2
A3 1
(31)
0
A4 1 0A1 1A2 2A3 3A4
1
A2 1
0
A1 1 , recognizing AA
1
I
Am , we obtained the following expression:
0
A4 1 0A4
1
A4 1 0A1 1 0A1 1A4
2
A4 1 1A2 1 1A2 2A4
+ 3A4 1 2A3 1 2A3 3A4
4
A1 0A1 1 0A1 4A1 1
3
A4 1 3A4
(32)
4
A2 1A2 1 1A2 4A2 1
+ 4A3 2A3 1 2A3 4A3 1
3
A4 1 3A4
The resulting Jacobian matrix, formulated using body-fixed twist, is given as:
1
e
xe
ye
L1 sin
L1 cos
23
23
1
L2 sin
L2 cos
L2 sin
3
L3
3
L2 cos
3
3
L3
1
1
0
2
L3
3
(33)
Summarizing the Jacobian obtained using all methods discussed above in the
following table:
33
Summary
Differentiating Position
Vector
Spatial Vector
Formulation
-L1 sin
L1 cos 1
-L1 sin
L1 cos 1
1
1
1
- L2 sin
L2 cos
1
- L2 sin
L2 cos
12
12
12
12
- L3 sin 123
L3 cos 123
1
-L2 sin 12 - L3 sin
L2 cos 12 L3 cos
- L3 sin 123
L3 cos 123
1
-L2 sin 12 - L3 sin
L2 cos 12 L3 cos
1
1
1
TAJM with Spatial Twist 0 L1 sin( 1 ) L1 sin( 1 ) L2 sin(
0 -L1 cos( 1 ) -L1 cos( 1 ) - L2 cos(
TAJM with Spatial Twist
in {W } Frame
-L1 sin
L1 cos 1
1
1
- L2 sin
L2 cos
12
12
- L3 sin 123
L3 cos 123
1
TAJM with Body-Fixed
Twist
L1 sin
L1 cos 23
23
L2 sin
L2 cos 3
3
L3
123
123
123
1
-L3 sin
L3 cos
123
123
123
123
)
)
2
1
2
1
1
-L2 sin 12 - L3 sin
L2 cos 12 L3 cos
1
L2 sin
L2 cos 3
123
1
-L3 sin
L3 cos
3
L3
123
123
1
-L3 sin
L3 cos
123
123
1
0
L3
Table 1: Table summarizing the Jacobian matrix formulated using various methods.
The body and spatial velocities are physically interpreted as the instantaneous
translational and rotational velocity written relative to the body or spatial frame,
respectively. Body-fixed twist and Spatial twist of a manipulator can be related by
Adjoint Transformation [61]. While the various twist formulations all represent the endeffector twist in terms of joint rates, care has to be taken when use the resulting Jacobian
matrix to compute performance measures such as manipulability ellipsoid, stiffness
ellipsoid of a manipulator. The linear velocity portion of the Jacobian formulated using
spatial twist represent the velocity of a point on the end-effector that is located at frame
{F } (imagine the end-effector as a rigid body such that part of this body include a point
that is located on the origin of frame {F } ). The body-fixed twist, on the other hand,
represent the linear and angular velocity of the end-effector but expressed in the end34
effector frame, let say {E } . It can be used to compute performance measures of the
system such as manipulability indices and stiffness indices (where orientation is not
considered, see Figure 13). However, to compute the manipulability ellipsoid and
stiffness ellipsoid, the ellipsoid needs to compensate for the angular difference between
frame {F } and frame {E } to express them properly (see Figure 14). This tends to be a
tedious process for spatial manipulator. In particular, Jacobian matrix resulting from
using spatial twist and expressed in {W } frame is the formulation we used in this thesis
to obtain the Jacobian of in-parallel manipulator as it represents the velocity of the endefffector expressed in a frame that is common to all chains.
Figure 13: Surface plot of isotropy index over the entire workspace of a two link RR manipulator
computed using the Jacobian formulated using (top left) analytical method, (top right) spatialtwist expressed in {W } frame, (bottow left) spatial twist expressed in {F } frame, and (bottom
right) body-fixed twist expressed in {E } frame.
35
(a)
(b)
(c)
Figure 14: Manipulability ellipsoid computed from Jacobian formulated using (a) spatial twist,
expressed in {F} frame; (b) body-fixed twist, express in {E} ; and (c) spatial twist, expressed in
{W} frame.
3.1.4 Reciprocal Screws
The wrenches combine naturally with twist to define instantaneous work done by
the mechanical system. Given a twist tT
[wT
are
velocities,
wT
vectors
[
T
of
fT ]
angular
6
and
linear
3
, where f
and m
vT ]
3
6
3
, where w
respectively
and
and v
a
3
wrench
are linear force and angular moment,
respectively. We can ten define the infinitesimal work as:
W
wT t
f v
(34)
A wrench w is said to be reciprocal twist t if the value of W vanishes. This
means the action of wrench w on the twist t produces no work. Given a twist system T
spanned the set of twists {t1, t2,
, tn } , we can form the Jacobian matrix J . The
reciprocal wrench system of T , can be determine using linear algebra approach, where
Wr
{wr 1, wr 2,
, wr ,6 n } , W
[wr 1, wr 2,
36
, wr ,6 n ] , and W is the left nullspace of J .
As outlined in [61], the twists and the wrenches can be visualized using a RRR
manipulator as an example. In this example, twist are expressed as : wt
wrenches are expressed as: wW
[M z
Fx
[wz
vy ]T ,
vx
Fy ]T . The twist vectors, expressed in frame
{W } , are:
w
t1
-L3 sin 123
L3 cos 123
1
- L2 sin
L2 cos
12
12
w
- L1 sin 1
L1 cos 1
1
-L3 sin 123 - L2 sin
L3 cos 123 L2 cos
t2
w
12
1
-L3 sin
L3 cos
t3
12
123
123
The wrenches are:
w
W1
L3 sin( 3 )
cos( 1
) wW2
2
sin( 1
)
2
L3 [L2 sin( 3 )
L2 cos( 1
L2 sin( 1
L1 sin( 2
)]
3
) L1 cos( 1 ) wW3
2
) L1 sin( 1 )
2
L3 sin(
2
) L2 sin( 2 )
cos( 1 )
sin( 1 )
3
where:
w
W1
null ([ wt2, wt3 ]T )
w
W2
null ([ wt1, wt3 ]T )
w
W3
null ([ wt1, wt2 ]T )
One can verify the resulting wrenches by making sure
w
W1
w
t3
0 , etc.
37
w
W1
w
t2
0,
RRR Manipulator and location of its Twists and Wrenches
RRR Manipulator and location of its Twists and Wrenches
4
3
Wr
1
Wr
2
2
2
Wr
Wr 1
Y
2
T
3
Wr
3
1
3
T
2
T
0
X
Wr
Y
2
2
1
1
3
T
3
T
T
y-position
4
1
1
X
0
-1
0
-1
0
1
-1
2
y-position
3
4
5
-2
-1
-2
x-position
0
1
2
3
4
5
x-position
(a)
(b)
Figure 15: Locations and directions of twists and wrenches of a three link RRR planar
manipulator, red color arrows represent twists and blue color arrows represent wrenches. (a)
Isometric view; and (b) top view of the manipulator.
3.2 Conventional Jacobian Formulation – Parallel Manipulator
In this section, using an 3-RRR parallel manipulator as example, we show that
various Jacobian that could resulted by using choosing different loop in the system. One
could end up with different sets of equations using this method. Method I gives the
Jacobian in terms of three actuated joints only; Method II give use the Jacobian in terms
of all the joint rates (including three active and all six passive joints); while Method III,
resulted in a Jacobian with three active joints and three passive joints. This serves as our
motivation, as twist-based formulation allows for consistent way of Jacobian formulation
for parallel system.
38
Figure 16: Nomenclature of 3RRR planar manipulator used in the analysis.
Refer to Figure 16, let xe , ye , e , be the end-effector position and angle with
respect to the inertia frame {F } .
RRR manipulator,
2i
, i
1i
, i
1,2, 3 and
1,2, 3 , are the active joints on leg i of the 33i
, i
1,2, 3 are the passive joints on leg i of
the 3-RRR manipulator, a1i , a2i , a 3i represent the length of the 3 links on leg i of the 3RRR manipulator.
39
3.2.1 Method I
Here we want to find the relation between the end-effector position xe , ye , e , and
active joints
1i
, i
1,2, 3 . Looking at the first chain A1B1C 1 in Figure 16:
The vector P21 can be found by:
P21
P11
a11 cos(
11
)
a11 sin(
11
)
(35)
Similarly, the vector P31 can be found by:
P31
xe
a 31 cos(
e
31
)
ye
a 31 sin(
e
31
)
(36)
P21 , along with P31 , form the following relationship:
P31
P21
T
P31
P21
a21
2
(37)
Equation (37) can be written explicitly as:
P31,x
P21,x
2
P31,y
Which gave the relation between xe , ye , e , and
P21,y
1i
2
, i
a21
2
(38)
1,2, 3 .
The same relationship holds for chain A2B2C 2 and A3B3C 3 , where we can now
generalized Eq. (35) as:
P2i
P1i
a1i cos( 1i )
a1i sin( 1i )
Similarly, we can generalize Eq. (36) as:
40
,
i
1,2, 3
(39)
P3i
xe
a 3i cos(
ye
a 3i sin(
)
e
3i
e
)
3i
i
,
1,2, 3
(40)
i
(41)
And finally, Eq. (38) can be generalized as:
P3i,x
Where P2i
f
P2i,x
2
and P3i
1i
P3i,y
f xe , ye ,
P2i,y
e
2
2
a2i ,
1,2, 3
.
Note that P1i and a1i are known variables in Eq. (39), and a 3i and
i
are known
variables in Eq. (40). Hence, from Eq. (41), we now have three equations and six
unknown in xe , ye ,
e
and
1i
, i
1,2, 3 . Since these are nonlinear equations, numerical
method is needed to solve for either the forward position problem (given
1i
solve for xe , ye ,
e
1i
, i
e
) or the inverse position problem (given xe , ye ,
, i
1,2, 3 ,
, solve for
1,2, 3 ).
To find the Jacobian for the 3RRR system, differentiate Eq. (41), and one can
obtain the following linear relation:
A
Where A
f xe , ye , e ,
,
11
12
,
11
,
13
12
T
,
13
and B
B xe , ye ,
T
0
e
f xe , ye , e ,
,
11
12
(42)
,
13
.
Hence, the Jacobian of the system is:
xe , ye ,
T
B 1A
e
xe , ye ,
T
e
J
,
11
11
,
,
12
12
,
T
13
T
13
Using MATLAB Symbolic Toolbox, one can obtain the matrix A and B, given as:
41
(43)
A
2a11 sin( 11 )xe - 2a11a 31 sin(- 11
2a12 sin( 12 )xe - 2a12a 32 sin(- 12
2a13 sin( 13 )xe - 2a13a 33 sin(- 13
B
) - 2a11 sin(
)
- 2a12 sin(
2
) - 2a13 sin(
3
e
1
e
e
)P11x - 2a11 cos( 11 )ye
)P12x - 2a12 cos( 12 )ye
12
)P13x - 2a13 cos( 13 )ye
13
2a11 cos( 11 )P11y
2a12 cos( 12 )P12y (44)
2a13 cos( 13 )P13y
11
B(:,1) B(:,2) B(:, 3)
(45)
where:
B(:,1)
B(:,2)
B(:, 3)
2xe
2xe
2xe
2a 31 cos(
2a 32 cos(
2a 33 cos(
2ye
2ye
2ye
2a 31 sin(
2a 32 sin(
2a 33 cos(
2a 31 sin( e
2a 32 sin( e
2a 33 sin( e
e
e
e
e
e
e
) - 2P11x - 2a11 cos( 11 )
) - 2P12x - 2a12 cos( 12 )
2
) - 2P13x - 2a13 cos( 13 )
3
1
) - 2P11y - 2a11 sin( 11 )
) - 2P12y - 2a12 sin( 12 )
2
) - 2P13y - 2a13 sin( 13 )
3
1
)(P11x
)(P12x
2
)(P13x
3
1
xe )
xe )
xe )
2a11a 31 sin(2a12a 32 sin(2a13a 33 sin(-
11
e
12
e
13
e
2a 31 cos(
2a 32 cos(
2a 32 cos(
)
)
2
)
3
1
e
e
e
)(ye
)(ye
2
)(ye
3
1
P11y )
P12y )
P13y )
3.2.2 Method II
Here we want to find the relation between the end-effector position xe , ye , e , and
active joints
1i
, i
1,2, 3 . Looking at the first chain A1B1C 1 in Figure 16:
The vector P21 can be found by:
xe
ye
P11
a11 cos(
11
)
a21 cos(
21
)
a 31 cos(
31
)
a11 sin(
11
)
a21 sin(
21
)
a31 sin(
31
)
which directly gave the relation between xe , ye , e , and
1i
, i
(46)
1,2, 3 .
The same relationship holds for chain A2B2C 2 and A3B3C 3 , where we can now
generalized Eq. (35) as:
42
xe
ye
P1i
a1i cos( 1i )
a2i cos( 2i )
a 3i cos( 3i )
a1i sin( 1i )
a2i sin( 2i )
a 3i sin( 3i )
i
,
1,2, 3
(47)
Differentiate this position equation and one can obtain the relationship between
passive joints and active joints. Then form the Jacobian. Note that P1i and a1i are known
variables in Eq. (39), and a 3i and
i
are known variables in Eq. (40). Hence, from Eq.
(41), we now have three equations and six unknown in xe , ye ,
e
and
1i
, i
1,2, 3 .
Since these are nonlinear equations, numerical method is needed to solve for either the
forward position problem (given
position problem (given xe , ye ,
e
1i
, i
, solve for
1,2, 3 , solve for xe , ye ,
1i
, i
e
) or the inverse
1,2, 3 ).
3.2.3 Method III
Similar to Method II, instead of joining the loop at the end-effector position, we
join the loop at the passive joints. Here we want to find the relation between the endeffector position xe , ye , e , and active joints
1i
, i
1,2, 3 . Looking at the first chain
A1B1C 1 in Figure 16, the vector P31 can be found by:
P31
P11
a11 cos(
11
)
a21 cos(
21
)
a11 sin(
11
)
a21 sin(
21
)
(48)
Similarly, the vector P31 can be found by:
P31
xe
a 31 cos(
e
1
)
ye
a 31 sin(
e
1
)
Equation (48) and (49) form the following relationship:
43
(49)
P11
a11 cos(
11
)
a21 cos(
21
)
xe
a 31 cos(
a11 sin(
)
11
a21 sin(
)
21
ye
a 31 sin(
)
e
1
e
)
1
(50)
The same relationship holds for chain A2B2C 2 and A3B3C 3 , where we can now
generalized Eq. (50) as:
P1i
a1i cos( 1i )
a2i cos( 2i )
xe
a 3i cos(
e
i
)
a1i sin( 1i )
a2i sin( 2i )
ye
a 3i sin(
e
i
)
which gave the relation between xe , ye , e , and
1i
, i
i
,
1,2, 3
(51)
1,2, 3 .
Note that P1i and a1i are known variables in Eq. (39), and a 3i and
i
are known
variables in Eq. (40). Hence, from Eq.(51), we now have six equations and nine unknown
in xe , ye ,
and
e
1i
,
2i
i
1,2, 3 . Since these are nonlinear equations, numerical method
is needed to solve for either the forward position problem (given
for xe , ye ,
1i
,
2i
i
e
and
2i
i
1i
, i
1,2, 3 , solve
1,2, 3 ) or the inverse position problem (given xe , ye ,
e
, solve for
1,2, 3 ). Differentiate Eqn. (51), we obtain:
a1i sin( 1i )
a1i cos( 1i )
a2i sin( 2i )
a2i cos( 2i )
1i
1 0
a 3i sin(
2i
0 1
a 3i cos(
e
e
i
i
)
)
xe
ye ,
i
1,2, 3 (52)
e
3.3 Jacobian Formulation for Parallel Systems
The differential-kinematic model of the closed-loop constrained system can then
be written as:
w
t
JT
subject to the general velocity-level constraint equations
44
(53)
JC
0
(54)
The number of independent loops that forms the Jacobian matrix can be
determined by the Euler equation by viewing the system as a network (with the links as
nodes and the joints as edges), following [62].
Figure 17: Closed-loop constraints selection for a parallel manipulator, reproduce from [62]
Note that within a parallel chain, not all the joints in the system need to be active.
The mixture of active and passive joint components can help partition the rate vector as
T
[
T
a
T
p
].
T
a
,
T
p
are the subvectors of the active and passive manipulation rates
variables within the entire constrained mechanical system. JT and JC can then be
partitioned accordingly, rewriting Eqs (53) and (54) as:
45
JT
a
JC
a
w
JT
a
p
p
JC
a
A general solution of Eq. (56) for
(55)
0
(56)
can be found as:
p
JC# JC
p
p
p
t
p
a
JC
a
(57)
p
where the superscript “#” denotes the Moore-Penrose inverse of the matrix, JC is the
p
right annihilator of JC , i.e. JC JC
p
p
0 and
p
is any arbitrary vector parametrizing the
nullspace of JC . Eq. (55) can now be written as:
p
w
t
JT JC# JC
JT
a
JT
p
a
p
JT J
p
a
a
JT JC#
p
(58)
p
#
Cp
JT is the system Jacobian matrix that now relates the actuated joint rates of the
system to the end-effector twists. Depending on the nature of the actuation of individual
module that forms the system, the overall system can be under-actuated, redundantlyactuated, or fully-actuated. Resulting in different sizes of JC and JC . See [63] for a
a
p
detailed discussion on these various cases.
3.3.1 System Jacobian Matrix of Parallelogram sub-system
The system Jacobian JT for in-parallel systems with closed-subloops can now be
formulated very systematically and algorithmically. We illustrate this using a general
planar RR 2-DOF manipulator with one parallelogram subloop as an example. We model
46
P   Lx
Ly 
T
L2
2
P
2
P
L2
1
y
2
L1
0
1'
y
L1
1
this system as a constrained system that combines a planar RRR manipulator with
0
x
1
x
another RR manipulator, both actuated only at the base joint, as shown in Figure 18.
w
y
w
w
3
2
2 '
x
P   Lx
L2
Ly 
3
L2
T
L2
2
P
P
2
y
0
1
1
L1
2
P
L2
y
2
L1
1
0
1
0
x
x
1'
y
L1
1
x
(a)
(b)
w
Figure 18: A planar RR manipulator system that consists of a parallelogram structure can be
modeled as a constrained system consisting a RRR and a RR manipulator.
2
For a system that consist of N
linkage, denoting each linkage as
L2
k
and
I , II , III ,
, N (with N
P
1 vector loop-closure constraints),
and there are
2
-active
-passive DOF in each linkage, the configurationy of the overall system
can be
1
L1
completely described by N (
) generalized coordinates.
 The generalized velocity
0
1
x
can then be partition into:
a
I
a
p
I
p
T
T
N
a
N
p
T
(59)
T
(60)
where the subscript a and p indicate active and passive joint respectively. The endeffector task equation can be obtained by selecting one linkage as the primary linkage of
47
the system and setting contribution from other linkage to zero. JT and JT can be
a
p
determine as:
JT
a
JaI
06
m
06
m
(61)
J pI
06
n
06
n
(62)
JT
p
While there are many ways to formulate the constraints Jacobian matrix JC and
a
JC (see [62], reproduce here in Figure 17), one such possibility is to use the velocityp
level loop closure equations existed within each two linkages:
J aI
JC
J aII
J
m
06
m
J pII
06
06
m
J
06
m
06
06
06
m
06
m
m
06
m
a
J pI
JC
06
m
III
a
II
a
II
p
06
m
J
06
m
06
J aN
1
J aN
06
m
06
m
J
m
III
p
06
m
06
m
06
m
J pN
p
m
1
(63)
(64)
J pN
The procedures shown here allow one to formulate the Jacobian of the in-parallel
system formed by multiple serial chain systematically. This process also allows symbolic
computation of the system Jacobian.
48
4 Performance Measures
…when you can measure what you are speaking about, and express it in numbers,
you know something about it; but when you cannot measure it, when you cannot
express it in numbers, your knowledge is of a meagre and unsatisfactory kind…
-- William Thomson
From a design standpoint, the development of quantitative performance measures
is an important precursor to applying computational tools to engineering design problem
[64]. The performance measures that we adopt here will allow us to systematically study
evaluate and subsequently optimize in-parallel haptic devices for specific task
requirements. Over the years various performance measures have been developed by the
robotics community, initially for serial chain manipulator and subsequently extended to
parallel-architecture systems. We leverage and build upon the rich literature of
performance measures for articulated mechanical systems to study in-parallel haptic
devices. Among the many criteria, a variety of zeroth-order, first-order, and second-order
kinematic measures have been studied. In our work, we will focus our efforts on a subset
of such criteria, i.e. workspace, manipulability (velocity and force), and stiffness.
49
4.1 Background
A variety of performance measures have been used in the recent past for design of
parallel haptic devices. For example, Lee et al. [65] use workspace and isotropy index to
design and optimize their 6-DOF parallel haptic device. Similarly, Kim et al. use the
same design optimization criteria to study their 7-DOF parallel haptic device, PATHOSII. Gosselin et al. [66]design a 6-DOF parallel device with a maximal singularity-free
workspace. Kim et al. [67] design a 3-DOF parallel haptic device with redundant
actuation by optimizing the stiffness and workspace of the device. Yoon and Ryu [68]
study the workspace, isotropy index, and payload index of their 6-DOF parallel haptic
device and subsequently optimize their device.
Numerous other examples of design of haptic device designs based on similar
performance measures can be found in the literature [69-77]. The unique feature that
separates a haptic device from a typical robotic manipulator is that it involves
bidirectional forces and motions flow between human and device. For this reason, Hale
and Stanney [78] suggest that the design of haptic device needs to incorporate aspect of
human physiology, psychophysics, and neurological performance which were
summarized into: (i) Psychophysical Tactile Interaction Design Guidelines; (ii)
Psychophysical Kinesthetic Interaction Design Guidelines; and (iii) Multimodal
Interaction Design Guidelines. Hayward and Astley [79] on the other hand suggest
performance measures based mostly on device mechanical properties. They provided
guidelines in three broad areas: (i) Gross Features (include Degree-Of-Freedom, DeviceBody Interface, Motion Range, Peak Force, Peak Acceleration, Inertia and Damping), (ii)
Detail Features (include Resolution, Precision, Bandwidth, Structural Response, Dynamic
50
Precision, and Closed Loop Performance); and (iii) Environment Factors (Including
Weight, Noice, Volume, and Safefy). Kirkpatrick and Douglas [80], suggest haptic
devices to be evaluated using a standardized virtual reality shape recognition test. The
device is evaluated based on how quick a user can use it to recognize shape haptically.
Khan et al. summarize these performance measures (partially quantitative and qualitative
measures), including some statistical based performance measures in [81]. While these
performance measures are useful for the design of haptic devices, they are either
qualitative measures or can only be evaluated after the device is manufactured, and are
hence not considered further.
Beside the workspace, manipulability, and stiffness measures, there are other
performance measures used in the robotic community. For example, manipulability
polytope [82-83], force polytope [84], dynamic manipulability [85], force transmissibility
[86], among other. Pashkevich et al. [87] provide a summary of these other performance
measures.
4.2 Zeroth Order Kinematic Performance Measure: Workspace
The determination of the reachable workspace of a reference point chosen on the
end-effector of an in-parallel robotic manipulators is a important issue in the context of
kinematic design of haptic devices. As a position-level performance measure, the
workspace determination is a zeroth order kinematic measure of system performance. It
is even more important for parallel manipulators which already have a small workspace
compared to serial manipulator. Hence is it crucial to tailor/ customize the workspace of a
given parallel-manipulators, based on specific application. Workspace computation for
parallel manipulator is generally more complicated than its serial counterpart due to
51
multiple closed-loop constraints existing in the system. As outlined by Pernkopf and
Husty [88], various types of workspace for in-parallel systems have been studied in the
literature. Reachable workspace – all position that can be reached by a point on the endeffector in some orientation, Orientation workspace - all possible orientation of the endeffector about all axes at a point of interest, etc. Out of these, the two that are most
commonly studied are (i) constant orientation workspace – the set of points which a point
on an end-effector can reach while keeping its orientation fixed, as studied in [89-90];
and (ii) constant position workspace (or called the dexterous workspace, as studied in
[91-93]) – the set of orientations possible for a point on the end-effector of a manipulator
while keeping the position of the center fixed.
The challenge of workspace determination of parallel manipulators arises
principally from the lack of analytical solution of the forward kinematics. The inverse
position kinematics based approach for determining workspace tends to be inefficient and
time consuming. The most common technique for determining the reach of a point on the
end-effector of an in-parallel system is by direct numerical simulation, for example [9496]. The workspace is discretized and then each point is being evaluated to determine if it
is reachable by all the chains in an in-parallel system. This method is iterative and usually
only required minimal theoretical understanding of the system characteristic. Postprocessing is usually required to compute the overall workspace of the system for a given
set of systems parameters.
Another approach, used by Agrawal [91], Jo and Haug [97], Landsberger and
Shanmugasundram [98], using the insight that on the border of the workspace, the
velocity of the manipulator along the normal direction to the border must be zero.
52
However, this method uses the Jacobian matrix of the robot for which no closed-form is
known. In addition, this method is not convenient to introduce the constraints of link
interference and mechanical limits on the passive joints [89]. Improved upon Jo and Haug
work [97], Snyman, Plessis, and Duffy [99] formulated the problem as a constraint
optimization problem in which the process to determine workspace boundaries can be
automated and solve numerically.
The determination of analytical expression for the workspace of an in-parallel
system in terms of its design parameters (including link lengths, ground locations), are
attractive from the viewpoint of performing design optimization for such system. For
certain planar parallel manipulator, analytical workspace computation is possible [100],
by first determining the parametric expression for the boundaries of the workspace, and
then applying the Gauss Divergence Theorem. The procedures of finding the parametric
curves that define the workspace boundary can also be automated, resulting in a very fast
computation of the workspace. However, this process is not easily extendable to spatial
system with three-dimensional workspace. Efforts by Gosselin [101], and Masouleh [102]
showed that for three-dimensional workspace, one can obtain the expression of the
boundary of the workspace for a specific height (a slice of the workspace) and the overall
workspace can be obtained by integrating the workspace area at each slice over its height.
But the process become very involved and is also problem dependent.
The above three methods were broadly classified as Discretization method,
Jacobian matrix method, and Geometrical method by Merlet [103]. In recent time,
Computer-Aided Design (CAD) packages are also found to be very useful tools to
compute the workspace volume of a parallel manipulator [104-106]. The workspace
53
volume of individual chain can be plotted, and their common reachable workspace can be
computed using Boolean operations. It is also possible to perform optimization within the
CAD package itself. The disadvantage of these CAD-based approaches is that they are
limited to optimize the workspace of the parallel manipulator alone, while other
performance measures are not easy to incorporate into the optimization process.
Nonetheless, this provides a very quick way to compute the workspace of a parallel
manipulator.
However, it is important to note that a parallel manipulator designed for only for
maximizing the workspace for a given set of parameters may not be a good design in
practice [107-109]. It is possible that the design with maximum workspace has
undesirable kinematic characteristics such as poor dexterity, stiffness, or manipulability.
It is for this reason, optimization of workspace is usually accompany with other
performance
measures
such
as
singularity-free
workspace,
uniform
isotropy
(manipulability) workspace, or workspace with highest stiffness measure.
4.2.1 Workspace Computation
As shown in previous section, analytical workspace computation (spatial or planar)
is not straight forward especially for parallel manipulator. In this thesis, we look at
constant orientation workspace of parallel haptic device. Similar to the work by Wang et
al. [106], Tanaka et al. [95], Zhen Gao and Dan Zhang [110], the workspace is computed
numerically. For planar parallel manipulator, given a workspace grid, we evaluate its
reachable workspace for a given end-effector position and orientation by evaluating the
inverse kinematics of each chain, followed by checking if the joint angles satisfy the joint
limits. The workspace area can be computed by multiplying the total reachable points in
54
Spatial workspace volume computation using contour and polyarea algorithm
the workspace by the grid size. Alternately, in this work, the workspace is computed by
the boundary point generated by contour of the workspace and computed using the
polyarea algorithm, as shown in Figure 19.
1. Original data points.
2. Filtered surface points.
3. Create mesh & compute volume.
Planar workspace area computation using contour and polyarea algorithm
1. Original data points.
2. Create contours.
3. Extract contour points &
compute area w/ polyarea.
Figure 19: Example of the computing the area of a circular planar workspace using contour and
polyarea algorithm.
This alternate method allows one to compute the workspace using the selected
value at which the contour is being evaluated, and allow one to extract points represent
the workspace boundary efficiently. But accuracy is still depends on the grid size that is
being used to evaluate the workspace. We also use this method to compute the area of a
slice of a workspace volume at a particular level. The other advantage of having the
points that describe the workspace boundary is that these points can be used to export to
CAD package to generate the corresponding 3D workspace for efficient computation.
This process is illustrated by Cao et al. in [111].
55
Spatial workspace volume computation using triangulated mesh generated from data points
1. Original data points.
2. Filtered surface points.
3. Create mesh & compute volume.
Figure 20: Example of computing the volume workspace of a sphere using mesh generated from
data points.
Planar workspace area computation using contour and polyarea algorithm
For spatial workspace, the process is almost the same, except now one need to
evaluate a 3D grid. The total volume of the workspace can then be compute by
multiplying the total reachable points by the grid volume. Again, the accuracy of this
method is determined by the grid size. An alternate method that was attempted here were
1. Original data points.
2. Create contours.
3. Extract contour points &
compute area w/ polyarea.
to convert the cloud of data points that are inside the workspace to triangulated mesh. The
algorithm first filter the data points such that only those points on the surface are kept.
Using the Delauyney algorithm, one can create triangular mesh and the normal vector to
each triangulated face, and the volume of the workspace can be computed using the
triangular mesh. However, this alternate method did not significantly improve accuracy
or efficiency of the workspace volume computation. Nonetheless, the mesh generated
from the workspace volume gives a better visualization of the spatial workspace volume
to the user and hence is used here.
56
4.2.2 Examples
In this section, workspaces of several planar parallel manipulators were computed
using the method described in the previous section. Specifically, the constant orientation
workspaces of a 2-RRR (shown in Figure 21), 3-RRR (Figure 22 and Figure 23), and a 4RRR (shown in Figure 24) manipulator were plotted to show how modularly adding
additional chain to a parallel manipulator further restricted the workspace of the parallel
system. It is also important to notice that the pivoting location of the chain changes the
workspace significantly, if other geometric parameters of the system stay the same. This
can be seen visually from Figure 21 - Figure 24.
From Figure 22 and Figure 23, we show the constant orientation workspace of a
3-RRR manipulator at two different end-effector orientation:
end-effector orientation
0 , and
45 . For
0 , the largest workspace occur when the spacing between
each chain is around 1.732, but with
45 , the largest workspace occur when the
spacing between each chain is around 1.5. Also notice that even though the workspace is
largest at 1.732, that particular placement might not be the best since the singularity occur
right at the center of the workspace (see Figure 22(d)).
(a)
(b)
(c)
Figure 21: Constant orientation workspace of a 2-RRR planar manipulator changes as the
distance of their bases varied from (s) L
1.5 to (c) L
zero.
57
4.5 with end-effector orientation held at
(a)
(b)
(c)
Figure 22: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
Largest workspace exist when L
0.5 to L
4.5 with end-effector angle held at zero.
1.7321 .
(a)
(b)
(c)
Figure 23: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
0.5 to L
Largest workspace no longer exist at L
1.7321 .
(a)
(b)
4.5 with end-effector angle held at
45 .
(c)
Figure 24: Constant orientation workspace of a 4-RRR planar manipulator changes as the
distance of their bases varied from L
2 to L
3.25 with end-effector angle held at zero.
58
4.3 First Order Kinematics Performance Measure: Manipulability
The notion of manipulability – the ability to move and apply forces in arbitrary
directions - has shown to be very useful in the analysis and design of both serial and
parallel manipulators. Yoshikawa [27, 44] provided one of the first comprehensive
mathematical treatment of manipulability for general open chains system (shown in
Figure 25), while Salisbury and Craig [29] are credited with the first use of the
manipulability measures in the design context in robotic mechanical systems. Since then,
alternate formulations with different aspects of open chain manipulability have been
investigated and proposed by a number of other authors in various applications.
(a)
(b)
Figure 25: Velocity and force manipulability ellipsoid of a planar two link serial RR manipulator
at various locations.
Lee and Lee extended the concept of manipulability ellipsoid to a dual-arm
system [112-113]. Since then, researchers including Chiaccio, Chiaverini, Sciavicco, and
Siciliano [27, 114], Bicchi, Melchiorri, and Balluchi [26, 115-116], Wen and Wilfinger
59
[117-118], Park and Kim [31, 119] attempt to extend the idea of manipulability to general
multi-limbs parallel systems.
Lee [112] and Chiaccio [27] examined manipulability in systems when all
cooperating arms have full mobility in their task space. Bicchi at al. [115] then extended
the manipulability study to general cooperating arms, with arbitrary number of joints per
arm. Melchiorri [120] then extended the framework to study force manipulability in fully
actuated systems. Three papers presented at a same conference session look at
manipulability for cooperative systems. Wen and Wilfinger [117] extended the concept of
kinematic manipulability to general constrained rigid multibody systems that includes
closed kinematic chains. Park and Kim [31] studied manipulability of closed chains that
includes passive joints, under an elegant differential geometric framework. Bicchi and
Prattichizzo [121] looked at the problem that including effects of redundancy and
indeterminancy of kinematics, and introduced the notion of active and passive force
manipulability. Later, they developed a systematic way of building the kinematic model
of a system of cooperating arms with passive joints, and study the ill-posed kinematic
redundancy and indeterminacy resulting from these system as an optimization problem
[26].
For parallel robots, however, the optimization of a single performance index is
often not sufficient. Usually, a combination of performance indices is used. The
combined optimization of workspace and condition number of the manipulator Jacobian
has been performed by several researchers, for example [122-124]. Application of these
performance measures into design of parallel manipulator is not without any problem.
Khan and Angeles [125], note the challenges arising from the lack of dimensional
60
homogeneity in the Jacobian matrix (from which performance indices such as
manipulability, condition number, and isotropy index are derived). One way to solve this
lack of dimensional homogeneity in the entries of the Jacobian matrix is to introduce the
characteristic length [125]. Several researchers propose ways of determining this
characteristic length, including Stocco, Salcudean, and Sassani [126], Mansouri and
Quali [127]. Interesting enough, effort by Kim and Ryu to formulate a dimensionally
homogenous Jacobian matrix of a Steward Platform shown that the optimal configuration
of the Steward Platform has no difference than those using Jacobian that lack of
dimensionally homogeneity [128].
Recently, Merlet [129] provide an unique perspective to study the performance of
parallel robot. He showed that the performance indices that were used for serial
manipulator such as manipulability, condition number, may not be suitable for parallel
robot. For parallel robot, the determination of maximal positioning error (including their
average value and their variance) were suggested as more appropriate global performance
indices. Cadou, Bouchard, and Gosselin [130] proposed to use maximum rotation
sensitivity and the maximum point-displacement sensitivity to study parallel manipulator.
These two indices provide upper bound to the end-effector rotation and pointdisplacement sensitivity under a unit-magnitude array of actuated-joint displacement.
However, this performance index can only apply to uniformly actuated manipulator
(including redundantly actuated system), which limits its usage.
From the above discussion, it is clear that performance indices derived from
Jacobian of the robotic systems remains an active research topic, especially for parallel
manipulators. From a design standpoint, however, manipulator design based on these
61
performance indices had shown success for general parallel manipulator design, as well
as parallel haptic device design.
4.3.1 Manipulability Formulation
In Chapter 3, we outlined the steps to formulate the Jacobian of an in-parallel
manipulator using Twist-based Jacobian formulation method. The formulation to
compute the manipulability can be found in any robotics textbooks and some of the
references in the papers we referred to in previous section, hence we will only briefly
discuss it here. The Singular Value Decomposition (SVD) of the Jacobian matrix and its
geometric relationship offer further insights in characterizing the “(velocity)
manipulability” of a manipulator. A m n Jacobian matrix JT can be transformed into
the form of:
JT
with U TU
UU T
singular values
i
Im
m
and V TV
1
2
VT
VV T
(65)
I n n , and
is an m n matrix with
on the diagonal:
m n
and
U
k
diag
, 2,
1
,
k
, 0k 1,
, where k is the rank of JT , and k
, 0n
k
(66)
min(m, n) .
The columns of U are the orthonormal eigenvectors of JT JTT , while the columns
of V are the orthonormal eigenvectors of JTT JT . The decomposed matrices allow us to
graphically interpret the geometric aspects of the manipulability characteristic of a
specific manipulator at the task space. The Jacobian matrix JT is assumed to map a unit
62
sphere in the joint space to the corresponding ellipsoid in the task space, which we call
the manipulability ellipsoid, as shown in Figure 26.
Figure 26: An example of a manipulability ellipsoid.
The columns of the U matrix, u1, u2,
, uk can be interpret as the direction of the
principal axes of the ellipsoid and the singular values,
, 2,
1
,
k
is the corresponding
magnitudes of the principal axes. The vector of the principal axes of the manipulability
ellipsoid are
u , 2u2,
1 1
, k uk . The u1 is the direction in which the manipulator can
move most easily, and lease easily in uk direction. It is easy to realize that when this
ellipsoid becomes a sphere, the end-effector can move with uniform ease in all directions,
such configuration is known as the isotropic configuration. Hence, such SVD and the
subsequent geometric interpretation as the manipulability ellipsoid become a very useful
performance measures in manipulator design and optimization. There are several
performance measures that derive from manipulability: Yoshikawa manipulability
measures, condition number, and isotropy index.
Yoshikawa manipulability measure [131] is the product of all the singular values.
Hence it is proportional to the volume of the manipulability ellipsoid. The condition
63
number [132] is the ratio of the minimum and maximum singular values of JT ,
1
/
k
.
It is the ratio of the length of the major axis to the length of the minor axis on the
manipulability ellipsoid. Hence it has a smallest value of 1 and tends to go infinity near
singular configurations. In other words, the manipulability measure related to the
magnitude of the ellipsoid and the condition number concerns about the shape of the
ellipsoid. The isotropy index [133] (or Inverse of the condition number), on the other
hand, is the reciprocal of the condition number,
k
/
1
. It is the ratio of the length of the
minor axis to the length of the major axis of the manipulability ellipsoid. Hence it has an
upper bound of 1 and a lower bound of 0, which is preferable. In our work, we mainly
look at manipulability ellipsoid, and isotropy index.
The manipulability measure, condition number, and isotropy index can be
determined by eigenvalues of JT JTT . If
is the vector of joint torques and F is the
vector of external force applied at the tip, then from virtual work considerations,
J T F . Assuming the manipulator is not at singular configuration, one can then write
F
J
T
. If we look at the mechanism as a device whose input torque produce output
forces, the force manipulability ellipsoid can be determined by the egenvectors and
eigenvalues of (JTJTT ) 1 . The eigenvectors of this force manipulability will be the same
with the manipulability ellipsoid we defined previously, but the eigenvalues will now be
1/
,1 /
1
2
,
,1 /
k
. It is clear, then the velocity and force manipulability is dual to
each other, the direction of the largest eigenvalues of the velocity manipulability is the
direction of the smallest eigenvalues of the force manipulability. The velocity/ standard
manipulability reflects the uniformity in the mechanism‟s velocity gain; the force
64
manipulability ellipsoid reflects the uniformity in the force-torque gain [119]. This
relationship can be seen from Figure 25. Because of this relation, we did not explicitly
study the force manipulability, but its effect should be clear from the reciprocity
relationship with velocity manipulability.
We compared our manipulability ellipsoids with those given in Park and Kim [31,
119, 133], who study the manipulability of closed kinematic chain using a differential
geometry framework and found that our result matched very well. Following work by
Kim et al. [134], we separate the Jacobian matrix into orientation and position subJacobian and study them separately, rather than weighting them in some adhoc fashion
(i.e. using the characteristic length). In particular, we study the position manipulability of
five-bar system at two different actuated joint locations (shown in Figure 27) and a sixbar system with one fully actuated case and one over actuated case (shown in Figure 30),
with comparison to results given in [119]. The size of the manipulability difference
between them is due to scaling that used in the plotting of our results. The two cases were
chosen to show that one can study the manipulability of a parallel on the joint, as well as
any point on a rigid body of a link. Note that the five-bar system, is the best known
parallel mechanism in robotics, having been popularized by Asada et al. [135]. Sometime
also called the Pantograph is a popular 2-DOF planar haptic device, for example [136138]. The Pantograph is also the modular pieces that form the Quanser‟s 3-DOF planar
Pantograph and 5-DOF Haptic Wand.
Two different actuation schemes of a six-bar parallel manipulator (or can be
viewed as a 2-RRR manipulator) are shown in Figure 28, one is fully actuated (Figure
28(b)) and the other one is redundantly actuated (Figure 28(c)). This shows that the
65
results match very well for fully actuated system as well as over actuated systems. More
comparison of results can be found in the Appendix.
st
Result from [119]
Both 1 joints actuated
(a)
(b)
Both 2
nd
joints actuated
(c)
Figure 27: Manipulability ellipsoid of a five-bar manipulator with two different actuated joints
compared with (a) result obtained from [119], (b) five-bar with both 1st joint actuated; and (c)
five-bar with both 2nd joint actuated.
Result from [119]
Fully Actuated
Redundantly Actuated
(a)
(b)
(c)
Figure 28: Manipulability ellipsoid of a six-bar manipulator with two different actuation schemes
compared with (a) results reproduced from [119], (b) six-bar with 1st and 2nd joint of the left chain
actuated; and (c) six-bar with both 1st and 2nd joint of the right chain actuated.
We also plot the manipulability ellipsoid of the five-bar system (with actuated
joints located at two different locations) at two singular positions, shown in Figure 29.
The manipulability ellipsoid of the six-bar system (with two different actuation schemes)
at singular position is shown in Figure 30. The singular position of a five-bar system is
66
studied in [139]. Note that at these singular positions, the Jacobian is ill-conditioned and
usually do not consider these solutions in analyzing the device performance.
Both 2
Both base joints actuated
nd
(a)
joints actuated
(b)
Figure 29: Five-bar systems manipulability ellipsoid at two singular positions (top and bottom)
with two actuation schemes: (a) both actuated joints located at the 1st joint; and (b) both
actuated joints located at the 2nd joint.
Fully actuated
Redundantly actuated
(a)
(b)
Figure 30: Six-bar system manipulability ellipsoid at singular position.
67
Next, we systematically show the isotropy index of several common planar inparallel manipulators, namely the 2-RRR, 3-RRR, and 4-RRR manipulator. Before we do
that, it is important to realize that the isotropy index changes as the choice of inverse
solutions. For a RRR chain, given an end-effector position and orientation, there exist
two solutions for each chain (conveniently called the „elbow up‟ and „elbow down‟
solution). The combinations of these two solutions can led to different configurations to
the system, and the direction and magnitude of the manipulability ellipsoid depends on
these configurations.
Same end-effector position and orientation w/ different inverse solution
(a)
(b)
Figure 31: For the same end-effector position and angle, manipulability ellipsoid can be different
due to inverse solution of each RRR chain.
As shown in Figure 31, two configurations are possible for a given end-effector
position and orientation for a 3-RRR manipulator. Figure 31 (a) shows the „elbow‟ of
each chain is pointing towards the „clockwise‟ direction (if we look at the center of the
end-effector as the center of a clock), whereas in Figure 31 (b) the chain at the lower left
is pointing toward the opposite direction. These two configurations led to different
direction and magnitude to the manipulability ellipsoid computation, as shown in the
68
dotted blue line. The corresponding plot of isotropy index over the entire workspace is
given in Figure 32. Note that the isotropy index is symmetry in Figure 32(a) and nonsymmetry in Figure 32 (b) due to the configurations selected in the inverse solution. In
addition, the manipulability ellipsoid also changes with the ground location of the chain,
as shown in Figure 33. So in this thesis, we try to also show the configurations whenever
isotropy indices and manipulability ellipsoid are plotted to avoid such confusion and to
have consistent comparison.
Isotropy Index of 3-RRR manipulator
(a)
(b)
Figure 32: Isotropy index/ Inverse of condition number over the entire workspace of a 3-RRR
parallel manipulator changes for two selected inverse kinematics configurations.
Isotropy Index of 3-RRR manipulator
(a)
(b)
(c)
Figure 33: Manipulability ellipsoid changes as well as workspace as configuration changes.
69
Isotropy Index of a 2-RRR, 3-RRR, 4-RRR manipulator
(a)
(b)
(c)
Figure 34: Comparison of Isotropy index / Inverse of condition number for a (a) 2-RRR, (b) 3RRR, and (c) 4-RRR system at selected configuration.
70
The isotropy index of a 2-RRR, 3-RRR, and 4-RRR manipulator (of selected
configuration) is shown in Figure 34. To ensure consistent measures, all link lengths were
set to 1 unit and spaces between each chain is 3.5 unit. We can see that at the center of
their workspace, the 4-RRR has the highest isotropy index, followed by 3-RRR, and then
2-RRR. Hence, as we increase the number of chain in the planar RRR system, the
manipulability increases. However, this gain in manipulability comes with the cost of
reduction in workspace. Wu et al. [140] performed similar study using the condition
number as their performance measures and obtained similar conclusion.
4.4 Second Order Kinematics Performance Measures: Stiffness
When there is an external force and/or moment applied to the end-effector of a
manipulator, it causes the end-effector to be deflected away from its desired location. The
amount of deflection is a function of the applied force and stiffness of the manipulator
[34]. The overall stiffness of a manipulator depends on several factors including size and
the material used for the link, the mechanical transmissions, the actuators, and the
controller. In our study, we assume that the major links are perfectly rigid and consider
the mechanical transmission mechanisms and the servo systems as the main source of
compliance. Generally, the stiffness analysis evaluates the effect of the applied external
torque and forces on the compliant displacement of the end-effector. This property is
defined through the stiffness matrix, which gives the relationship between the
translational/ rotational displacement and the static force/ torque causing this transition.
Generally, the approaches exist for the computation of the stiffness matrix is categorized
into three groups: (i) Finite Element Analysis (FEA); (ii) Matrix Structural Analysis
71
(MSA); and (iii) Virtual Joint Method (VJM) [141-142]. The Virtual Joint Method, also
referred to as “lumped modeling”, is based on the expansion of the traditional rigid body
model by adding virtual joints that describe the elastic deformations of the manipulator
components (links, joints, and actuators). This method is originated by the work of
Gosselin [143], who evaluate parallel manipulator stiffness taking into account only the
actuator compliance. Since we only look at haptic devices with rigid links, we will
formulate the stiffness matrix using the Virtual Joint Method. More recent work by Chen,
Ngo, and Kao, through series of investigations [144-146], proposed the Conservative
Congruence Transform (CCT) as the appropriate mapping between Cartesian stiffness
and Joint stiffness of a manipulator. The original joint and Cartesian stiffness mapping is
only valid when there is no external load applied on the system. When there is external
loading, there is an additional component in the CCT that account for the change in
geometry via the differential Jacobian (Hessian matrix) of the robot manipulator. This
component is capture in the effective stiffness matrix of the CCT.
Stiffness and workspace are important design factors for a haptic device. Haptic
device usually required software routine to control their stiffness as well as maintain a
large workspace [71, 147]. These two factors are related to each other, and stiffness is
expected to increase through kinematic parameters optimization [148-149]. However, the
relationship between the stiffness and workspace is contrary to each other from previous
findings [150]. Hence, optimization is usually needed to determine the optimum solution
[148].
72
4.4.1 Stiffness Formulation
Let m be the dimension of the joint space, n be the dimension of the end-effector
space,
i
be the toque or force transmitted through the i th joint, and
corresponding deflection at the joint. For small deflection, we can relate
i
qi be the
and
qi by
linear approximation:
i
ki qi
(67)
where ki is the joint stiffness constant. In matrix form, we can write Eqn.(67) as:
k q
[ 1, 2,
where
n
,
n
]T ,
q
[ q1, q2,
, qn ]T and k
(68)
diag[k1, k2,
, kn ] is an
n diagonal matrix. For serial manipulator (and parallel manipulator formulated using
twist assembled Jacobian matrix), the joint displacement
effector displacement
q is related to the end-
x by the Jacobian matrix J :
x
J q
and the end-effector output force F is related to the joint torque
(69)
by the transpose of the
conventional Jacobian matrix:
JT F
and q from Eqn. (68) and Eqn. (69) into Eqn. (70), one can get:
Substituting
F
where K
J
T
kJ
(70)
1
K x
(71)
is called the stiffness matrix. To compute the stiffness ellipsoid, one
would evaluate the eigenvectors and eigenvalues of the matrix K . One can easily see that
73
if k
I , i.e. if the all the joint in the manipulator have the same joint stiffness of
1 N / m , the stiffness ellipsoid essentially overlapping the force manipulability ellipsoid.
The equation relating the Joint space and Cartesian space stiffness is given by the
conservative congruence transformation [144]:
JT Kp J
K
Kg
(72)
where K is given by Eq. (71) and Kg is define as:
JT
n
Kg
i 1
T
J
f
i
JT
f
1
f
(73)
JT
f
n
2
Kg is the product of the changes in geometry via the differential Jacobian matrix,
JT /
i
and the externally applied load f . The Cartesian stiffness, K p , can be
compute from Eq. (72):
Kp
JT
1
K
Kg J
1
(74)
The stiffness matrix is a mapping between the end-effector forces and its
displacement. The ellipsoid associated with the stiffness matrix K p can be interpreted in
a similar fashion to the manipulability ellipsoid. The major and minor axes are given by
the eigenvalues and eigenvectors of K p , and indicate directions along which the
mechanism as a structure is the most and least stiff. As with manipulability, it is
inconsistent to combine quantities with different physical units, and hence they should be
analyzed separately, as suggested in [134]. It is apparent there are two parts in the
stiffness matrix of a mechanism. The stiffness that corresponding to K , is called the
74
„structural stiffness‟ by Li and Gosselin [151] (or „passive stiffness‟ by Kim et al. in
[152] ), it is configuration dependent and proportional to the joint stiffness. The other part
that corresponding to Kg , is call „force-affected‟ stiffness [151] (or „active stiffness‟ in
the case of redundantly actuated parallel manipulator, where one can modulate the
stiffness by changing the internal torque with additional motor, as in [152] ), it is as if the
mechanism can be „tightened‟ or „relaxed‟ when there is external forces exerted on the
mechanism such that the mechanism gain or lost some stiffness.
Stiffness Ellipsoid of Two link RR Manipulator
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 35: Stiffness Ellipsoid of a two link RR manipulator as the two joint stiffness constants
k1, k2 varied from k1
0.1, k2
0.9 to k1
0.9, k2
75
0.1 from (a) to (i).
Obviously, stiffness ellipsoid of a manipulator is also configuration dependent
and proportional to the actuating forces. In Figure 35, we show the stiffness ellipsoid of a
two link planar RR manipulator at a given point, and shows that one can modulate the
magnitude and direction of the stiffness ellipsoid by varying the stiffness value in the
joint.
Using our formulation, the „structural‟ or the „passive‟ stiffness map of a 3-RPR
and a 3-RRR planar parallel manipulator were computed and the results were compared
with those given by Gosselin [143], and Li and Gosselin [153]. Figure 36 shows the
–, x –, and y –direction with end-
result matches very well for the stiffness map in the
effector orientation
0 for the 3-RPR system. Comparison of stiffness map with end-
effector orientation
45 can be found in Appendix A.5.
Stiffness Mapping of 3RPR Manipulator
Stiffness in  -direction
Stiffness in x-direction
0.5
0.45
0.5
0.4
1.8309
1.2226
1.7394
98
80
02
00
0.35
0.35
1.3166
1.6478
1.4107
y-position
7
0.25
0.3
y-position
0.
5
0.45
1.1285
0.4
19
0.000 56
0.3
7
0.
00
05
61
9
0.0
01
40
49
0.35
0.
00
08
42
9
88
52
9
02
0.0 1 966
0
0.0
49
40
01
0.0
0.0
0.0
02
01
52
96
88
69
0.45
0.4
y-position
Stiffness in y-direction
0.5
1.5047
1.5988
0.25
1.5563
0.3
1.4647
0.25
1.3732
1.6928
97
0.000 561
0.2
0.2
0.15
1.2816
0.2
1.7868
0.0011239
0.0014049
1.1901
1.8809
0.15
0.15
1.0986
0.0019669
0.0025288
0.1
0.05
0.25
0.1
0.3
0.35
0.4
0.45
0.5
x-position
0.55
(a)
0.6
0.65
0.7
0.75
0.05
0.25
0.1
0.3
0.35
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.7
0.75
(b)
Figure 36: Stiffness maps of the 3-RPR manipulator in the (a)
(c) y –direction with
0.05
0.25
0.3
0.35
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.75
(c)
–direction; (b) x –direction; and
0 . Comparing result with comparing results with [143] (top row).
76
0.7
Stiffness Mapping of 3RRR Manipulator
Stiffness in x-direction
Stiffness in y-direction
0.5
0.5
22
35 .1
24
.87
8
14 .63
14
.63
41
0.25
41
17
.56
1
15 .80
65
18.0645
22 .58 06
68
20.32 26
29.354
8
0.2
33.871
0.15
0.3
15 .8
0 65
20
.3
22
6
27 .09
29.2683
7
70
.1
13
11.70 73
0.2
0.3
61
17 .5
0.25
13
.54
84
65
.80
15
14
.63
41
17
.56
1
21
.95
12
16 .0
976
20
.3
22
6
0.35
78
24 .8
24
.87
8
1
.56
17
12
.95
2 44
21
19 .0
y-position
51
.19
32
0.3
18 .0
645
15.80 65
0.4
19 .0
2 44
18
.0
64
5
8 78
20 .4
0.35
87
.83
24
24
.87
8
y-position
0.4
24 .83
22 .58 87
06
0.45
27 .8
0 49
49
.80
27
21.9512
26 .3
4 15
0.45
29 .35
27.0968
48
0.15
0.35
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.7
(a)
0.3
0.35
0.4
0.45
0.5
x-position
0.55
0.6
(b)
0.65
0.7
(c)
Figure 37: Stiffness maps of 3RRR manipulator in the (a) x-direction, (b) y- direction, and (c)
direction, with end effector angle
–
0 . Comparing result with [153] (bottom row).
Figure 37 shows the stiffness map comparison of a 3-RRR system in the
and y –direction with end-effector orientation
–, x –,
0 , and we can see that both results are
similar. See Appendix A.5 for more comparison results for the 3-RRR system. Once we
formulated the stiffness indices for these systems, we can study how the locations of the
actuated joints affect the stiffness in the system. Using a 3-RRR planar parallel
manipulator as an example, we can study the stiffness map of 3-RRR, 3RRR, and 3RRR
system. Figure 38 (a)-(c) compare the stiffness map in these 3 systems, with end-effector
angle
0 (and the kinematic configurations shown. With this configuration, the
stiffness in the
-direction is largest when the actuator is place at the 3rd joints, and is
smallest when the actuator is placed at the 2nd joint.
77
Stiffness Map of 3RRR, 3RRR, 3RRR Manipulator (End-effector angle = 0 deg)
(a)
(b)
(c)
Figure 38: Comparison of the stiffness map of a 3-RRR manipulator in x-direction (2nd row), ydirection (3rd row), and
-direction (4th row) with different joints actuated: (a) 3-RRR; (b) 3-
RRR; and (c) 3-RRR manipulator. End-effector angle
78
0 .
The stiffness in both the x- and y-direction, on the other hand, is largest when 2nd
joints are actuated, but is roughly the same for actuating the 1st or 3rd joints. However, if
the end-effector angle is held at
45 , the stiffness map changes (the resulting
stiffness maps can be found in Appendix A.5). Such analysis could help a designer decide
where to place the actuator (among other design considerations) to improve the stiffness
of the parallel system mechanism. We can also look at how modularly adding additional
chain to the system affects the overall stiffness of the entire system.
(a)
(b)
Figure 39: With the modular formulation that we have shown, we can look at how adding an
additional chain to the 3-RRR system (shown in (s)), now forms a 4-RRR system (as shown in
(b)) changes the overall stiffness of the system.
A 3-RRR manipulator and its workspace is shown in Figure 39 (a), by adding
another chain to the system, essentially making it a 4-RRR system, as shown in Figure 39
(b), reduces its workspace. Figure 40 (a)-(c) show the stiffness map of the 3-RRR system
and compare it with the 4-RRR system, when the end-effector angle is held at
0 . As
one would expect, adding additional chain to the system increases the overall stiffness in
79
the x-, y- and
-directions. The location of the added chain corresponding to the original
chains, the location of actuated joints, and geometric parameters (link lengths, inverse
kinematics) of the added chain affect the overall workspace, manipulability, and stiffness
of the new system. These are the design parameters that one needs to decide based on its
intended use. Note that for this comparison, we only evaluate the common workspace
covered by both the 3-RRR and the 4-RRR system.
Stiffness Index of a 3-RRR vs. 4-RRR manipulator
(a)
(b)
Figure 40: The stiffness map in the (a) x-direction, (b) y-direction, and (c)
(c)
-direction of a 3-
RRR system described in Figure 39(a) (top row), and the stiffness map after adding another
chain (bottom row), the 4-RRR system shown in Figure 39(b).
80
The „passive‟ or „structural‟ stiffness measure that we compute here is determined
by the end-effector position in static state. This stiffness is also proportional to the
actuating joints only (Since the formulated Jacobian is in terms of active joints only in the
Virtual Joint Method). Hence, the stiffness of a 4-RRR parallel manipulator with only
three actuated chains (the remaining chain is passive) and a fully actuated 3-RRR parallel
manipulator shown in Figure 41 has identical stiffness measure. The stiffness
computation result is given in Figure 42. For this reason, it is not able to study the
contribution the passive chains (or passive joints) to the overall stiffness of the system.
(a)
(b)
Figure 41: (a) A 4-RRR planar parallel manipulator with only three chains actuated (the top left
chain is passive); compare with (b) a 3-RRR planar parallel manipulator with same link lengths
and locations as the 4-RRR manipulator, but with all chains actuated. Shown here with their
corresponding constant orientation workspace.
81
Stiffness Index of a 4-RRR (w/ one passive chain) vs. a 3-RRR manipulator
(a)
(b)
(c)
Figure 42: The stiffness map of a 4-RRR planar parallel manipulator with only three chains
actuated (top row), compare with a 3-RRR planar parallel manipulator (bottom row): (a) xdirection, (b) y-direction, and (c)
-direction stiffness.
Finally, we compare the stiffness map of 2-RRR, 3-RRR, and 4-RRR system in
Figure 43. It is however, not easy to compare the performance across among them as the
workspace as well as the configurations of the chains affect the overall stiffness
performance. Nonetheless, we can just look at the center of their workspace, 4-RRR
generally has the highest stiffness, followed by 3-RRR, and 2-RRR manipulator.
82
Stiffness Map of 2-RRR, 3-RRR, 4-RRR Manipulator (End-effector angle = 0 deg)
(a)
(b)
(c)
Figure 43: Comparison of stiffness map of a (a) 2-RRR, (b) 3-RRR, and (4) 4-RRR manipulator
in x-direction (2nd row), y-direction (3rd row), and
orientation
0 .
83
-direction (4th row) with end-effector
4.5 Chapter Summary
In this chapter, we reviewed some of the performance measures that are widely
used in the design of in-parallel haptic devices. In particular, using several planar inparallel manipulators as examples, we looked at workspace, manipulability (including
velocity and force), isotropy index, and stiffness indices of these systems. We also
showed design parameters (ground locations, link lengths, inverse solutions) in these
systems that determine the performance measures that we studied here. In addition, we
also showed how modularly adding addition chain to a system changes the overall
workspace, manipulability, and stiffness of the system, by comparison of a 2-RRR, 3RRR, and 4-RRR system. In the following two chapters, we use these performance
measures to study two haptic devices, the PHANToM 1.5 and the HD2. The examples we
shown in this chapter include five-bar, six-bar, 3-RPR, 2-RRR, 3-RRR, and 4-RRR
systems. We did not explicitly show the twist-based Jacobian formulation, as well as their
inverse and forward kinematics since they are relatively straight forward.
84
5 Serial-Architecture Haptic Devices
Study of commercial haptic devices using performance measures from the robotic
community were performed by several researchers. Cavusuglu, Feygin, and Tendick [154]
formulated the kinematics and dynamic equations for Sensable PHANToM 1.5 system,
Silva et al. [155] study the kinematics, workspace, and manipulability of PHANToM
OMNI system, Martin and Hillier study the kinematic and workspace of Novint Falcon
[156]. We performed similar performance analysis with the PHANToM 1.5 system. In
particular, we study the workspace, manipulability, and stiffness of this system. In
particular, we look at the parallelogram sub-system that exist in the PHANToM 1.5
system and study the effects of this structure on the overall performance. Although some
authors call this type of system a „hybrid‟ system, we refer it as a serial-type manipulator
with added parallelogram sub-system.
5.1 Parallelogram Sub-Structure
Recently, there has been considerable interest in constructing devices with
additional closed kinematic-loop (e.g. planar parallelogram subsystem) to improve the
overall stiffness, remove singularity and permitting relocation of the actuation [157-159].
Many commercial serial- and parallel-type haptic devices, shown in Figure 44, now
incorporate such parallelogram linkages. Examples range from the Sensable‟s Phantom
85
Premium Series serial-architecture haptic devices and the parallel-architecture systems
like Quanser‟s High Definition Haptic Device (HD)2 and MPB Technologies‟s The Cubic.
A more comprehensive list of haptic devices (more than 20) with parallelogram structure
included in their design, is tabulated in paper by Arata et al. [74].
Figure 44: Examples of commercial haptic devices with parallelogram sub-system within their
kinematic chains.
The inclusion of parallelogram structure in the linkages provides higher stiffness
and improved rotational capability [157, 160], and has been used widely in the design of
haptic devices [161-168].
Using the PHANToM 1.5 as an example, with this
parallelogram structure, one can place the actuators at the base of the structure instead of
placing it at the base of each links, thereby shifting most of the weight of the device to
the base. As we show in next section, the inclusion of this parallelogram structure also
changes the manipulability of the device.
86
To study the effect of parallelogram structure on manipulability, we compare the
manipulability ellipsoid of a parallelogram linkage with a traditional two-link RR
manipulator that it replaced. Using the modular composite approach that we shown in
Figure 6 (In Chapter 2), there are other candidate haptic devices can be designed by
modularly piecing together a PHANToM 1.5 haptic device, as shown in Figure 45. These
haptic systems, can be systematically study using the method we shown here.
Figure 45: Using PHANToM 1.5 system as a design block, various types of in-parallel haptic
devices can be design by combining them in different ways. Note that device with ‘?’ on the side
is not a currently available haptic device.
87
5.1.1 Formulation of Planar Parallelogram Sub-System
Similar to [169-170], we formulate the planar parallelogram linkage shown in
Figure 46(a) as constraint system of a RRR manipulator with an RR manipulator. The
twist-based Jacobian of the RRR manipulator is given as:
W
J RRR
W
t1
W
t2
t3
(75)
Where
1
W
t1
L1s
L1c
1
L2s
1
L2c
1
L3s
12
L3c
12
W
123
t2
L2s
L2c
123
W
L3s
12
L3c
12
123
t3
1
L3s
L3c
123
123
123
The twist-based Jacobian of the RR manipulator is given as:
W
J RR
W
t1
t2
(76)
Where
1
W
t1
'
1
Ls
'
1
Lc
'
1
'
1
1
'
2
Ls
'
2
Lc
W
'
12
L'2s
t2
'
2
Lc
'
12
'
12
'
12
To obtain the Jacobian of the combine system for computation of manipulability:
JTa
JTp
L1s
L1c
1
L2s
12
L2c
12
1
L2s
L2c
12
L3s
L3c
12
123
123
L3s
L3c
123
0
123
L3s
L3c
0
123
123
0
0
For constraints, we include the angular portion of the Jacobian [118]:
88
(77)
(78)
1
JTa
L1s
L1c
L2s
1
L2c
1
1
L3s
12
L3c
12
L2s
L2c
1
L3s
12
L3c
12
(L'1c
123
1
JTp
( L'1s
123
L3c
123
'
1
L'2s ' )
(79)
12
L'2c ' )
12
1
L3s
123
'
1
( L'2s ' )
123
(80)
12
(L'2c ' )
123
12
The combine Jacobian of the system, is given by:
J Sys
where
a
1
JT JC# JC
JT
a
p
p
a
(81)
a
.
1'
Parallelogram sub-system formulation
w
y
w
3
2 '
x
P   Lx
Ly 
3
L3
T
L2'
2
P
2
P
L2
1
y
2
y
L1'
0
1
L1
0
1
x
1'
x
(a)
(b)
w
Figure 46: (a) A mechanism with parallelogram structure can be modeled as a combination of a
RRR manipulator with
2 another RR manipulator. (b) The manipulability ellipsoid of the
combined system is shown with the manipulability of individual RR and RRR manipulator
(assuming all joints actuated).
L2
P
2
1
When the last linkL of the RRR manipulator is zero, with L2
y
1
0
L3
L1' L2 '
L1 , and
1
x
0 , the parallelogram structure reduced to a five-bar mechanism, which is what we
89
formulated previously using two RR manipulator. As such, the manipulability of this
special case should be the same as the manipulability of a five-bar mechanism that have
the same link lengths with the parallelogram structure and the base joint locate at the
same position. Figure 47 verify that they are in fact the same.
Left-RRR + Right-RR
Left-RR + Right-RR
(a)
(b)
(c)
(d)
Figure 47: Comparison of parallelogram structure manipulability ellipsoid formulation with a fivebar mechanism: (a) and (c) is the parallelogram structure where the last link is zero, and (b) and
(d) is the five-bar mechanism.
90
The isotropy index of the planar parallelogram linkage is computed over the entire
workspace, with L1
0.5 , L2
2 , L3
1.5 , L1'
1 , and L2 '
2 , the isotropy index
of the parallelogram linkage is given in Figure 48. Figure 48(a) show the surface plot of
the isotropy index of parallelogram linkage and compare it with the isotropy index of an
RR and RRR manipulator, computed using the same link length that made up the
parallelogram linkage (assuming all the joints are actuated). It is apparent that the added
parallelogram structure improved the isotropy index, compare to a RR manipulator.
Isotropy Index
Parallelogram Structure Isotropy Index
3
0.15
0.3
0.4
5
0.3
0.4
0.4
5
0.3 0.2
5
0.2
0.25
0.05
0.35
0.4
0.4
-1
0
x-position
(a)
0.
1
0.
25
0.1
50
.15
0.
3
0.
2
0.1
0.2
5
0.2
0.05
0.1
0.3
0.15 0.2
0.2
5
0.2
5
y-position
0.10.
05
0.1
5
0.2
0.1 0.05
0.15
(c)
-3
-2
-1
0
x-position
1
0.3
0
3
0.05
0.2
2
0.05
0.1
0.15
0.25
0.2
1
0.
3
0.2
0.25
0.5
0
x-position
25
0.
5
0.2
1
0.3
0.3
0.2
0.2
5
0.05
y-position
1.5
0.
05
0.15
0.
25
0.25
-1
0.25
0.1
5
0.2
0.2
0.1
0.05
-2
0.2
2
0.2
0.5
-3
0.
51
0.0
0.1
0.1
5
0.1
5
0.0
0.25
0.15
05
0.
2.5
5
0.0.1
0
0.15
1
0.1
5
5
0.2
1.5
0
0.2
0
0. .05
1
15
0.
2
0.
3
3
5
0.1
2
2
(b)
0.05
0.1
2.5
1
RRR Manipulator Isotropy Index
RR Manipulator Isotropy Index
3
0.4
0.1
-2
0.3
0.5
-3
0.
35
0.25 0.25
0.1
0.25
0.2
3
0.
1
0
0.4
5
0.3
5
0.4
y-position
45
0.
5
0.4
1.5
5
0.2
45
0.
2
25
0.
0.
25
0.
4
0.
1
2.5
0.3
5
0.4
0.45
25
0.
2
3
(d)
Figure 48: (a) Surface plot of isotropy index of RR linkage with parallelogram structure, RR
manipulator, and RRR manipulator for comparison. The contour plot of each of their isotropy
index is given in (b) for RR linkage with parallelogram structure; (c) for RR Manipulator and (d)
for RRR manipulator.
91
5.2 Phantom 1.5 Haptic Device
The DH parameters that describe the Phantom 1.5 system are given in Table 2 and
Figure 49(a). To include the parallelogram structure in the formulation of the Jacobian of
the Phantom Premium 1.5, another linkage that has one additional joint than the Phantom
1.5 is given in Table 3 and shown in Figure 49(b).
(a)
(b)
Figure 49: The parallelogram structure in the Phantom 1.5 system modeled as combination of one
(a) spatial RR manipulator with another (b) spatial RRR manipulator. D-H parameters of both
systems are given in Table 2 and Table 3 .
i
i
ai
di
i
i ,ini
1
 /2
0
L1
1
0
2
0
L2
0
2
0
3
0
L3
0
3
0
Note:
1 ,2  Active joints; 3  Passive joints.
Table 2: Geometric Parameters for Phantom 1.5
i
i
ai
di
i
i ,ini
1
 /2
0
L1
1'
0
0

'
2
 / 2
0

'
2a
 /2
0

2
3
4
Note:
L2a
0
L2
0
L3a
0
'
3
0
 ,  Active joints;  ,  Passive joints.
'
1
'
2
'
2a
'
3
Table 3: Geometric Parameters for the Parallelogram sub-system in Phantom 1.5
92
Physical (Measured) Joint Limits on PHAMToM 1.5
1
2
Maximum Angle
Minimum Angle
Range
90 deg
-90 deg
180 deg
115 deg
-65 deg
180 deg
50 deg
3
50 ,
50
50 deg
30
100 / 95 ( 65
2
),
65
100 deg
115
2
2
30
Table 4: Measured joint limits of the PHANToM 1.5 haptic system.
The Jacobian of the Phantom system form by a spatial RR manipulator is given by:
w
[ wt1
J RR
w
w
t2
t3 ]
(82)
where
w
s
0
0
1
t1
-s (L2c
1
c (L2c
1
2
c
w
L3c )
t2
c (L2s
23
1
s (L2s
23
1
0
L2c
1
w
2
L3s )
2
L3s )
1
0
c (L 3 s )
t3
23
1
23
s (L 3 s )
23
L3c
2
1
c
0
L3c )
2
s
1
1
L3 c
23
23
23
combined with a spatial RRR manipulator of the form:
w
J RRR
[ wt1
w
w
t2
w
t3
t4 ]
(83)
where
w
t1
-s (L2ac
1'
c (L2ac
1'
2a
2a
0
0
1
L2c
L2c
s
1'
c
2a 2 '
2a 2 '
L3ac
L3ac
2a 2 ' 3 '
2a 2 ' 3 '
)
w
t2
c (L2as
2a
s (L2as
2a
L2s
1'
)
1
0
L2ac
93
1'
0
L2s
2a
L2c
2a 2 '
L3as
2a 2 ' 3 '
2a 2 '
L3as
2a 2 ' 3 '
2a 2 '
L3ac
2a 2 ' 3 '
)
)
s
c
w
t3
s
1'
1'
0
c (L2s
s (L2s
1
L2c
w
2a 2 '
L3as
2a 2 ' 3 '
2a 2 '
L3as
2a 2 ' 3 '
1'
2a 2 '
1'
c
L3ac
)
t4
1'
0
c (L3as
2a 2 ' 3 '
s (L3as
2a 2 ' 3 '
1'
)
1
L3ac
2a 2 ' 3 '
)
)
2a 2 ' 3 '
The Jacobian of the PHANToM 1.5 can be formulated following the steps
outlined in Section 3.3.1. In the following sections, we will study the workspace,
manipulability, and stiffness map of this haptic device.
5.2.1 Workspace Analysis
The workspace of the PHANToM 1.5 system is computed numerically using the
link lengths given in [154], and the measured joints limits given in Table 4. The inverse
kinematic solution is straight forward in this case but due to the parallelogram sub-system,
the PHANToM 1.5 system can only have the „elbow-up‟ solution. Using the method
outlined in Section 4.2.1 with a grid size of 0.01, the workspace of the device is
computed and visualized in Figure 50. The total reachable workspace of the device is
about 0.08m3.
94
(a)
(b)
(c)
(d)
Figure 50: (a) The reachable workspace of PHANToM 1.5 haptic device is about 0.08m 3, (b), (c),
(d) are side view, top view, and front view of the workspace along with the haptic device.
95
5.2.2 Manipulability Measures
From [171], we obtained L2  0.215m , L2a  0.0325m and L3  0.170m . L1 can
be arbitrary value (since it is just an offset from the base). Using these values, the
manipulability ellipsoid of the Phantom 1.5 system at one location is shown in Figure 52.
From this figure, one can see that the resulting manipulability ellipsoid that include the
parallelogram structure in the formulation differ from the spatial RR manipulator‟s
manipulability in the x-z plane, as the ability to move in x-z plane is reduced, but have
the same manipulability ellipse on the y-z plane (as they should).
Figure 51: 3D view of the manipulability ellipsoid of Phantom 1.5 system, compared with the
manipulability ellipsoid of a spatial RR system (without the parallelogram sub-system).
96
(a)
(b)
(c)
Figure 52: With the addition of the parallelogram linkages, the manipulability ellipsoid of the
Phantom 1.5 system at x = (a) 1.5; (b) 2.5, and (c) 3.5, compared with the manipulability
ellipsoid of a spatial RR system. In all these cases, the sizes of the manipulability ellipsoids were
scaled to half of its actual size.
5.2.3 Isotropy Index
To study the manipulability of the PHANToM 1.5 over its entire workspace on
the x-z plane (y = 0), the isotropy index of the system is computed and is shown in Figure
53(a). The isotropy index of a spatial RR manipulator with same geometry parameters is
also plotted for comparison. We note that our results are consistent with the study
performed by Cavusuglu et al [154]. Compare this result with the parallelogram subsystem formulation in planar case, it seem that the added subsystem did not improved the
manipulability much. This is because now the manipulability ellipsoid is spatial, and the
isotropy index does not take into account directional consideration, i.e. one cannot
determine which direction the manipulator can move most easily by isotropy index alone.
Nonetheless, we see that the addition of parallelogram sub-system did not drop the
manipulability much compare to a spatial RR system. From the contour plot of the
isotropy index shown in Figure 53 and the 3D surface plots shown in Figure 54(a)-(c),
one can realize that with the addition of the parallelogram structure, the maximum
97
isotropy index moved to the center of the workspace instead of at the inner area of the
workspace. This is a preferable characteristic as most of the haptic manipulations tend to
remain at the center of the workspace to avoid hitting workspace singularity. Also notice
that although the maximum isotropy index drop from 0.8541 to 0.791 with added
parallelogram structure, the overall manipulability of the haptic device over the entire
workspace is improved – the average isotropy index is 0.5353 for PHANToM compare to
0.3939 for a spatial RR manipulator.
Phantom 1.5 Isotropy Index
Spatial RR Isotropy Index
4
4
0.1
0.2
0.3 0.2 0.1
0.4
0.5 0.
40.
3
0.6
0.2
0.7
0.
3
0.7
0.2
0.3
z-position
0.1
0.6
0.4
2
0.
4
0.
0.
5
0.4
0
0.5
1
0.5
0.1
0.1
0.7 6
0.
0.2
0.04.3
1
0.
0.3
0.2
0.3
0.2
00..13
0.2
0.6
0.1
z-position
-3
1
0. .2
0
4
0.
0.6
3
0.
-2
4
0.
3
0.
-1
-2
-4
-0.5
0
.240.5
00.
7
0.
5
0.103.
-1
0.
8
0.
6
0..8
1.7
00
0.2
0
0.6
00.2
.04.5
1
0.2
0.6
0.
6
0.3
0.4
00..13
0.5
0.5
0.7
0.5
1
2
0.20.1
0.3
0.4
0.1
0.1
0.2
2
3
0.5
0.1
3
0.1
-3
0.2
0.1
1.5
2
x-position
2.5
3
3.5
-4
-0.5
4
0
0.5
(a)
1
1.5
2
x-position
2.5
3
3.5
4
(b)
Figure 53: Isotropy Index of (a) a PHANToM 1.5 system compared to a (b) spatial RR
manipulator over the entire workspace on the x-z plane (y = 0).
98
(a)
(b)
(c)
Figure 54: (a) Isotropy index of Phantom 1.5 plotted against a spatial RR manipulator. The (b)
maximum and average isotropy index of Phantom 1.5 is 0.7907 and 0.5353; and (c) the maximum
and average isotropy index of a spatial RR manipulator is 0.8541 and 0.3939, over the workspace.
99
5.2.4 Stiffness Analysis
Using the stiffness formulation outlined in Section 4.4.1, the stiffness maps of the
PHANToM 1.5 system in the x-, y-, z-directions are plotted in Figure 55 (d)-(f). To study
the effect of added parallelogram sub-system in the PHANToM 1.5, the stiffness map of
a spatial RR manipulator is plotted in Figure 55 (a)-(c). In the y direction, their stiffness
map are identical simply because in those direction the stiffness only come from the first
joint in the system, which is the same for both systems.
However, their stiffness maps are different in x, z directions. In general, the
addition of the parallelogram structure reduces the overall stiffness on the end-effector, as
shown in the surface plot of the stiffness map in Figure 56 (a) and (c), where the stiffness
in x- and z- direction is reduced as parallelogram structure is added. This is because the
joints that contribute to the overall stiffness computations (which are the active joints)
now moved to the same location located at the base. However, it should also note that the
stiffness map across the entire workspace is now more uniform with the addition of
parallelogram structure in the system.
100
Stiffness Map
of Spatial RR Manipulator
Spatial RR System Stiffness in Y-direction
Spatial RR System Stiffness in X-direction
4
0.734 69
0.62973
3
0.2381
0
0.5
1
1.5
2
x-position
2.5
3
3.5
0.5
1
0.11905
0.15476
z-position
0.083333
-2
0.9446
2.5
3
3.5
-4
-0.5
4
-4
-0.5
z-position
(d)
1
0
0.
24
43
5
0.10714
0.071429
0.095238
z-position
0.32598
0.264 86
0.1
62 9
9
3 98
0.22
98
0.223
-2
0.16667
0.
18
33
7
0.2
03 7
4
0.20238
0.17857
0.5
0.32
5 79
-1
71
0.264
0.214
29
0
1.5
2
x-position
43
0.305
0.32579
-3
2.5
3
3.5
4
-4
-0.5
0
0.5
1
(e)
Figure 55: Stiffness maps of spatial RR system in the (a) x
1.5
2
x-position
direction; (b) y
y
direction; and (f) z
101
3
3.5
4
direction; and (c)
direction, compare with stiffness map of PHANToM 1.5 in the (d) x
z plane ( y
2.5
(f)
z
direction, on the x
4
6
83 2
0.1
3 62
0.20
4
3.5
2 54
0.14
9
62
0.1
3.5
0.20
3 62
0.2
23 9
8
07
85
0.2 6
61
34
0.
0.40724
3
0.083333
2.5
26
83
0.1
2
0.095238
1.5
2
x-position
0.11905
0.13095
0.19048
-3
0.14262
1
3
0.22398
1
0.15476
-2
0.18337
0.16299
2.5
Parallelogram Structure Vs. RR System Stiffness in Z-direction
4
0.13095
-1
-2
0.5
0.11905
0
0.14286
1
98
25
0.35 24
8
0.2
62
42
0.1
0
1.5
2
x-position
0.203 62
0.15476
49
44
6
0.2 264 8
0. 0.28524
2
0.
24
44
9
-1
-4
-0.5
1
0.28507
0.28507
0.2381
0.
22
41
1
-3
0.5
3
0.21429
0.19048
0.16667
3
0.
22
41
1
1
z-position
0.178 57
0.22619
4
3
0
0
(c)
Stiffness
Map of PHAMToM 1.5
Parallelogram Structure Vs. RR System Stiffness in Y-direction
2
4
0.8396
-3
(b)
Parallelogram Structure Vs. RR System Stiffness in X-direction
4
0.203 74
0.524 78
0.62973.787 16
0
1.5
2
x-position
(a)
0.095238
0.21429
0.19048
0.17857
0
0.157 43
0.3
67
0.26239
34 0
.314
87
0.36734
0.41982
0.472 3
0.19
0 48
z-position
z-position
-4
-0.5
4
0.071429
-3
0.10
4 96
0.157
43
0.20
9 91
0
-1
0.11905
-4
-0.5
0.22619
96
04
0.1
-3
1
0.13095
-2
0.209 91
0.15743
6
0.1049
0.10714
-1
9 91
0.20
-2
0.15476
87
14
0.3
0.
15
74
3
25
0.577
23
0.47
-1
0.41982
0.367 34 487
0.31
9
0.2623
0
0.17857
0.3
67
34
0.2
62 3
9
0.4723
2
0.14286
0
1
64
39
0.689
34 73
0.7 629
0.
0.41
9 82
8
47
52
0.
1
0.22619
82
19
0.4
2
4
96
83
0.
5
72
57
0.
4
67 3
0.3
3
0.472
2
0.095238
3
0.16667
0.15476
87
14
0.3
91
09
0.2
3
Spatial RR System Stiffness in Z-direction
4
0.22619
4
0 ).
direction; (e)
(a)
(b)
(c)
Figure 56: Comparison of stiffness map of PHANToM 1.5 system with a spatial RR system in the
(a) x-direction; (b) y-direction; and (c) z-direction. The addition of parallelogram structure
reduces the overall stiffness in the system but at the same time allow a more uniform stiffness
distribution.
5.3 Manipulability Optimization / Regulation
In previous sections, we fixed joint
1
0 and study the manipulability and
stiffness map of PHANToM 1.5 system on over its entire workspace in the x-z plane. If
one would want to optimize the system, the size of the parallelogram seems to be the
natural choice for optimization – how the size of the parallelogram affects the
manipulability or stiffness of the system.
To study the effect of the size of the parallelogram sub-system to the
manipulability of the system, a parametric study were performed by changing the size of
the parallelogram from L2a
0.2 , L2a
0.6 , to L2a
Figure 57 (a)-(f).
102
1.0 . The results are shown in
(a)
(b)
(c)
(d)
(e)
(f)
Figure 57: Manipulability ellipsoid (in blue color) of Phantom 1.5, (a) – (c) side view, and (d)- (f)
isometric view. The size of the parallelogram does not change the magnitude or the direction of
the manipulability ellipsoid.
2
P   Lx
Ly 
L3
3
T
L2
L3a
y
1
2
x
L2a
L2'2
z
L1
3 w
P
y
1
0
x
Figure 58: Study the effect of the parallelogram structure by replacing it with a fourbar linkage.
3
L2
2
1
2
3'
L3
103
3
y
 2a
1
L2a
L2
x
z
2
 2'
3
L3a
y
From the parametric study shown in Figure 57, we notice that the Jacobian of the
system does not depends on size of the parallelogram, namely L2a and L3a . The other
way to modulate the manipulability is to change the parallelogram structure to a fourbar
linkage, as shown in Figure 58.
(a)
(b)
(c)
Figure 59: The manipulability of the PHANToM 1.5 system changes with varying length of
L3a from (a) L3a
0.8 ; (b) L3a
1.2 ; to (c) L3a
1.45 .
To keep the study to a single variable design problem, we select L2
L2 ' , and
vary the length of L3a . However, as we do this, the workspace of the PHANToM is
constrained by the fourbar linkage. So we further constrain our problem to a region that is
reachable within the design range of the L3a . A complete design problem can be set up to
104
include all the link lengths as design variable, and subsequently optimize the workspace,
as well as manipulability, similar to the design of in-parallel manipulator [20, 29, 124,
134]. The optimization problem can be formulated as:
Min
f L3a
L3a
s.t :
g1 : 0.95
L3a
(84)
1.55
(85)
where f L3a is the average isotropy index over the workspace shown in Figure 60(b):
1.75
x
3.0 ,
0.5
z
2.5 , and y
0.
The optimization problem is solved using standard optimization routine given in
MATLAB (fmincon()). The result of the optimization shows that the optimized
L3a
1.422 improved the average isotropy index in this selected region. The
comparison of the isotropy index surface plot before and after the optimization is given in
Figure 60(a). The average isotropy index improve to 0.6486, compare to 0.6242 with the
parallelogram sub-system, with L3a
1.5 . From the contour plot shown in Figure 60(c)
and (d), we can also see there is an overall improvement of the isotropy index in the
selected workspace. From this simple example, we can see that it is possible to improve
the overall isotropy index of the workspace by carefully design the link length parameters.
More design variables can be included in to perform such optimization.
105
(a)
PHAMToM 1.5 Isotropy Index
0
0.5
1
1.5
2
x-position
2.5
3
3.5
4
-0.5
1.8
2
(b)
3
-0.5
1.8
2
0.73747
33
36
0.8
0.
85
6
1
0.67816
0.65
8 39
0.71
77
2.2
2.4
2.6
x-position
0.6
38
62
0.816
56
0.
55
95
3
0.5
32
53
0.6
14
49
2.8
0.777 02
0.
73
74
7
1
0.618
84
0.56531
0.61449
0.647
28
0.663 68
0.59
8
77
71
0.
16
78
0.6
7
17
0.7
0
2.2
2.4
2.6
x-position
(c)
93
57
0.
0.
65
83
9
0.
74
56
4
0.6
30
89
0.53
2 53
z-position
y-position
-3
-0.5
0.678 16
0.7177
64
45
0.7
25
29
0.7
7 93
0.69
89
30
0.6
0.
6
0.7 96 4
6
12
86
84
18
9
0.6 58 3
0.6
-2
0.757
25
0.5
0.777 02
0
07
80
0.6
89
30
0.6
1
0.73747
1
17
58
0.
-1
7
00
68
0.
68
63
0.6
0.5
25
0.757
7 47
0.73
16
0.678
0
9 07
0.59
9 53
0.55
1
0.63089
28
0.647
0.61449
31
0.565
1
1.5
0.5981
2
2
86
63
0.
7
80 0
0.6
7 02
0.77
92
81
59
0.
1.5
0.
67
81
6
8
0.54
3
79
96
0.7
2
49
0.614
46
96
0.6
7
17
0.7
2
6
97
53
0.
4
88
61
0.
4
97
49
0.
9
30 8
0.6
25
29
0.7
Workspace Area for Optimization
0.87587
1
56
3
0.8
63
3
8
0.
6
65
1
8
0.
0.7
77
02
0.7
17
7
31
65
0.5
0.6
80
07
0.6
47
28
4
PHAMToM 1.5 Isotropy Index
2.5
0.7
62
04
0.5
39 7
6
2.5
z-position
PHAMToM 1.5 Isotropy Index
5
2.8
3
(d)
Figure 60: (a) Result of optimization to improve the isotropy index of the selected region shown
in (b). The contour plots of the original and optimized isotropy index are given in (c) and (d).
106
6 Parallel-Architecture Haptic Devices
6.1 High Definition Haptic Device (HD2)
The Quanser High Definition Haptic Device (HD2) can be viewed as an inparallel manipulator created by coupling the end-effectors of two Sensable PHANToM
Premium 1.5 devices placed side-by-side to a common handle through a set of universal
and revolute joints. The HD2 can now provide controlled motion- and force-feedback
along 5-DOF compared to 3-DOF force-feedback provided by the original Sensable
PHANToM Premium 1.5 device. The uncontrolled degree-of-freedom is the rotational
direction of the handle about the axis along its length (i.e. z 6 direction in Figure 2).
(a)
(b)
Figure 61: (a) Actual High Definition Haptic Device (HD)2 system; and (b) CAD model created in
SolidWorks.
107
6.2 Forward Kinematics
We assign rigid-body frames for each link following the convention for modified
DH parameters. As shown in Figure 62, these frames were assigned as {0},{1},
,{6} ,
from the base of the upper chain to the mid-point of the handle. For clarity, we only show
the frame assignment for the links of the upper chain and describe the formulation
process. A similar process was followed for frame assignment on the lower chain as
{0'},{1'},
,{6'} . We then develop the table of DH parameters (as shown in Table 1)
which facilitates creation of the relative homogeneous transforms between the various
frames in a systematic manner.
x  axis
L1
z  axis
0
p  p xiˆ  p y ˆj  p z kˆ
{0}
{1}
L2
z0
z1
L6
y0
z5
L3
x0
x1
z3
{2},{3}
L0
x2 , x3
p
zw
z2
{4}
{5}
x5
z4
z6 , y w
x4
{w},{6}
Z0
zF
yF
{F }
xF
Figure 62: The HD2 Haptic Device and DH Frames assignment on the upper arm.
Formulated using Conventional DH-Parameter frame assignment.
Last Update: 10/18/2010
108
L7
x6 , xw
L6
z5 Revolute Joint
L3
x5
z3
Revolute Joint
{4}
{5}
z4
{2},{3}
x2 , x3
x4
Universal Joint
L7
z6 , y w
z2
zw
{6},{w}
x6 , xw
Figure 63: Passive attachment (via universal joint and revolute joint) of the serial-chain’s endeffector to the common handle.
i
i
ai
di
i
i ,ini
1
 /2
0
L1
1
0
2
0
L2
0
2
0
3
 /2
0
0
3
0
4
 /2
0
L3
4
 /2
5
 L6

 /2
0
Formulated using Conventional DH-Parameter frame assignment.5
 L7
6
0
0
Last Update: 04/18/2010 6
 / 2
Note:
0
1 ,2 ,3  Active joints; 4 ,5 ,6  Passive joints.
Table 5: DH Frame assignment of the upper chain of HD2.
i
i
ai
di
i
i ,ini
1
 /2
0
L1
1
0
2
0
L2a
0
2
 / 2
3
0
L2
0
 2a
 /2
4
 /2
0
0
3
0
5
 /2
0
L3
4
 /2
6
 /2
 L6
0
5
 / 2
7
0
0
 L7
6
0
Note:
1 ,2  Active joints; 2a ,3 ,4 ,5 ,6  Passive joints.
Table 6: DH frame assignment of the upper chain of HD2, included the parallelogram sub-system.
109
L2a
x  axis
L1
z  axis
0
p  p xiˆ  p y ˆj  p z kˆ
{0}
{1}
L2
z0
L6
y0
z1
z5
L3a
x0
x1
z
z2
{2},{3}
3
x2 , xp3
{4}
{5}
x5
z4
z6 , y w
x4
L7
L0
zw
z2
{w},{6}
x6 , xw
Z0
zF
yF
{F }
xF
Figure 64: DH frame assignment of the ‘inner chain’ of HD2, to include the parallelogram
Formulated using Conventional DH-Parameter frame assignment.
structure
in the 10/18/2010
formulation.
Last Update:
The DH parameters of the lower PHANToM 1.5 are same as those given in Table
5 and Table 6 except at the last row, of which it is  L7 instead of  L7 . The Jacobian of
the system can be obtained using the twist-based Jacobian formulation method described
in Section 3.3 systematically and symbolically. To include the parallelogram sub-system
in the formulation, we treat the HD2 system as if it is made up of four chains. The DH
parameters of the „inner chain‟, shown in Figure 64, are given in Table 6.
110
6.3 Numerical Inverse Kinematics
Due to the small offset linkage ( L6 shown in Figure 63) at the end of the
PHANToM that connects the universal joint and the handle, the analytical inverse
solution cannot be obtained, this is further discuss in Appendix. This situation is similar
to the case of a double universal joints with an offset wrist mentioned in [172]. For this
reason, we solved the inverse kinematic numerically.
-3
7
Cartesian Error at each location
x 10
Upper chain error
Lower chain error
6
Cartesian Error
5
4
3
2
1
0
0
10
20
30
(a)
40
50
60
70
80
90
(b)
Angle profile of upper chain
Angle profile of lower chain
2
2
1.5
1.5
1
1
Angle Profile (rad)
Angle Profile (rad)
1
0.5
0
1
2
-0.5
3
-1
3
4
5
0.5
6
0
-0.5
-1
4
5
-1.5
2
-1.5
6
-2
0
1
2
3
4
5
6
7
(c)
-2
0
1
2
3
4
5
6
7
(d)
Figure 65: (a) The end effector position trajectory to be solve numerically; (b) the root mean
square error of the actual position and solutions obtained using inverse solution; (c) the angles in
the upper chain and (d) angles in the lower chain obtained from the numerical inverse
computation.
111
This solution process is not ideal as it cannot be implemented in real time, but is
adequate for performing analysis required for computing performance measures that we
outlined in previous sections, where we only need to know the joint variables for a given
end-effector position and orientation.
The constraint equations consist of two parts. The first part required that the
x, y, z position of the end-effector from both chains must be equal. The second part
required that the z -direction of the rotation matrix from both chain must be equal, since
they are along the longtitunal direction of the handle of the HD2. These six constraint
equations can be obtained from the homogeneous transformation matrices. Fortunately,
the way the two chains where combined together in the HD2 haptic allow only one
solution in the numerical solution, which reduce the solution search considerably.
As an example, the inverse solution of a circle with varying height is use to verify
the effectiveness of the numerical inverse solution. The circle with the HD2 device is
shown in Figure 65 (a). The results of the all the joint angles are given in Figure 65 (c)
and (d), and the root-mean-square (RMS) error is given in Figure 65 (b). Another solution
of a straight line in 3D is given in Figure 66. From these two examples, we can see that
the numerical inverse kinematic provide very good results in most of the interior
workspace region, while at places close to the workspace boundary the error increases.
112
RMS Error
0.035
Upper Chain
Lower Chain
0.03
RMS Error
0.025
0.02
0.015
0.01
0.005
0
0
10
20
(a)
30
40
50
Angle profile of lower chain
Angle profile of upper chain
2
2
1
1
2
1.5
2
1.5
3
3
4
4
5
1
5
1
6
0.5
Angle Profile (rad)
Angle Profile (rad)
6
0
-0.5
0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
0
10
20
30
60
(b)
40
50
60
-2
0
(c)
10
20
30
40
50
60
(d)
Figure 66: The end effector position points to be solve numerically; (b) the root mean square
error of the actual position and solutions obtained using inverse solution; (c) the angles in the
upper chain and (d) angles in the lower chain obtained from the numerical inverse computation.
6.4 Workspace Analysis
6.4.1 Constant Orientation Workspace
We look at the constant orientation workspace of the HD2 by numerically
compute the reachable workspace of a point located at the middle of the handle of the
HD2 while it is being held vertically. The physical joint limits of the system, shown given
113
in Table 7, were included in the computation. Similar to PHANToM 1.5 system, the
limits on the
3
is dependent on
(at
2
60
2
30
), due to the parallelogram structure.
Physical (Measured) Joint Limits on (HD)
1
2
Maximum Angle
Minimum Angle
Range
30 deg
-30 deg
60 deg
105 deg
-60 deg
165 deg
55 deg
3
50 ,
55
55 deg
4
5
6
2
105 / 90 ( 60
30
2
),
105
2
60
2
105 deg
30
No limit
No limit
NA
0 deg
-180 deg
180 deg
No limit
No limit
NA
Table 7: Measured joint limits for the HD2 haptic device.
1
Arc generated when
2
3
30
105
4
4
3,min
1
Arc generated when
2
2,min
3
3,max
2,max
Arc generated when
3
3
2
3
2
60
( )
3,min 2
30
55
105 / 90 ( 60
2
)
Arc generated when
2
2
2,max
3
3,min
3,max
Figure 67: The circular arcs that describe the constant orientation workspace boundaries of the
HD2 device at
1
1'
0 can be identified with circular arc equations (with radius and range
of angle) for this position.
As an attempt to compute the workspace using Gauss Divergence Theorem as in
[100], the parametric circular arcs that describe the boundaries of the workspace of the
114
HD2 device can be uniquely defined at this position (
1
vertically), shown in Figure 67. However, at other heights (
1'
1
0 , and handle is held
1'
0 ), this relation
cannot be determined easily and hence we resort to compute the workspace numerically.
The planar constant orientation workspace of the HD2 device at various heights are
computed numerically and plotted along with the device in Figure 68. As one would
expect, the largest workspace is at the a height of 0.344 m, where
1
1'
0 , at
around 0.2975m2. Moving away from this plane the planar workspace reduces. Figures
showing the workspace from the side view and front view of the HD2 device can be
found in the Appendix.
Figure 68: Isometric view of the constant orientation workspace of the HD2 haptic device (handle
is held vertically) at various heights.
115
6.4.2 Constant Orientation Spatial Workspace
The constant orientation workspace of the HD2 haptic device is solved
numerically and iteratively over the entire workspace. First we study the constant
orientation workspace of HD2 where the handle is held vertically. The result is shown in
Figure 69 for various views of the workspace. The workspace, computed numerically is
about 0.956 m3. Note that this is also the largest constant orientation workspace attainable
by the HD2 haptic device.
(a)
(b)
(c)
(d)
Figure 69: Constant orientation workspace of HD2 shown with (a) side view; (b) front view; (c)
top view, and (d) isometric view. At this orientation, the minimum and maximum reachable
distance is from 0.08m to 0.56m in the x-direction, -0.66m to 0.42m in the y-direction, and 0.08m
to 0.61m in the z-direction. The workspace volume is computed to be 0.956m3.
116
One of the intended applications of the HD2 haptic device is to perform biopsy
simulation and training. We attached a biopsy needle at the handle and surgeon is
expected to perform the biopsy operation, shown in Figure 70(a). In this case, it is
worthwhile to know the workspace of the device when the handle is held in this position
so that we could use this information pre-plan a training session or experiment. Hence,
the workspace of the HD2 device where the handle is tilted 30 (shown in Figure 70(b)
along the x-axis is computed. The results are shown in Figure 71(a) and (b). In this
orientation, the total workspace is reduced to about 0.76m3 compared to 0.96m3 when the
handle is held vertically.
(a)
(b)
Figure 70: (a) a biopsy needle were attached to the handle of the HD2 haptic device performing
virtual biopsy training; and (b) the corresponding constant orientation of the HD2 handle that
approximate the needle angle that is being used for workspace study.
If the handle is further rotated by 60 along the x-axis, the workspace is further
reduced to about 0.51m3, as shown in Figure 72 (a)-(b). Other views of this workspace
can be found in the Appendix. The workspace study is useful for pre-planning for
preparation of haptic applications, and is an important design criteria based on it specific
application [173-175].
117
(a)
(b)
Figure 71: The constant orientation workspace of the HD2 haptic reduced to about 0.76m3 when
the handle is held at an angle approximating the biopsy needle insertion angle (-30 degree in the
x-axis). Shown here (a) and (b) are two different view angles of the resulting workspace.
(a)
(b)
Figure 72: The constant orientation workspace of the HD2 haptic further reduced to about 0.51m3
when the handle is held at an rotated angle of -60 degree along the x-axis. Shown here are (a)
isometric view and (b) front view of the resulting workspace. More views are presented in the
Appendix.
118
6.5 Manipulability Measure
First we study the manipulability ellipsoid of HD2 haptic using only the two
chains (upper and lower chain) described by DH parameters given in Table 5. The
manipulability ellipsoid at a given end-effector position is then plotted and shown in
Figure 73 (a)-(d). Using this formulation, the manipulability ellipsoid indicates that the
device can move along the y direction with least effort, and can resist forces with
minimum effort along the x and z-direction [134].
(a)
(b)
(c)
(d)
Figure 73: Scaled manipulability ellipsoid of (HD)2 shown in (a) isometric view; (b) front view; (c)
top view; and (d) side view.
119
We then look at the manipulability of the HD2 haptic device, formulated with the
parallelogram structure in the system. The combined in-parallel system Jacobian now has
four serial chains. In addition to the two chains formulate using DH parameters given in
Table 5, additional two chains are modeled using the DH parameters given in Table 6.
The inclusion of the parallelogram structure through these two additional chains, allow L3
to be actuated through the parallelogram located at the same location as the 2 nd actuator.
As expected, this changes the device manipulability ellipsoid considerably, as shown in
Figure 74, with four other views showing in Figure 75 (a)-(d).
Figure 74: With the inclusion of parallelogram structure, the manipulability ellipsoid changes
considerably for the HD2 haptic device.
120
(a)
(b)
(c)
(d)
Figure 75: Comparison of manipulability ellipsoids of HD2 haptic device with end-effector
positioned at (2.5, 0, 3.44), modeled with (in blue color) and without (in yellow color) the
parallelogram structure at various view (a) front view; (b) side view; (c) top view; and (d)
Isometric view.
From Figure 75 (a)-(d), with the addition of the parallelogram system and moving
the actuator from the base of the 3rd link, the device now cannot move as easily as it used
to in the x-y plane. And the z-direction becomes the direction where it can move most
easily. At the same time, the device can now resist force more easily in the x-y plane. The
manipulability ellipsoid at another end-effector position and angle is given in Figure 76
(a)-(d).
121
(a)
(b)
(c)
(d)
Figure 76: Comparison of manipulability ellipsoids of HD2 haptic device with end-effector
positioned at (2.8, 2, 3.44) and tilted 60 deg along the x-axis, modeled with (in blue color) and
without (in yellow color) the parallelogram structure at various view (a) front view; (b) side view;
(c) top view; and (d) Isometric view.
6.6 Isotropy Index
We choose to study the manipulability index over the entire workspace on x-y
plane (
1
1'
0 ) with the handle held vertically. The isotropy index/ inverse of
condition number contour plot of the HD2 device (with and without parallelogram sub-
122
system) are given in Figure 77(a) and (b). Figure 77(c) shows compares the isotropy
index of the HD2 device with and without the parallelogram sub-system.
(a)
(b)
(c)
Figure 77: Inverse condition number/ Isotropy index of HD2 haptic (a) with and (b) without
including the parallelogram structure in the Jacobian formulation, evaluated at plane z=0.344m
and the handle is held vertically. (c) Although the condition number without the parallelogram
structure is better (higher number), the isotropy index with the addition of parallelogram
structure is more uniform throughout the workspace.
123
6.7 Stiffness Measures
Similar to manipulability study, we compute the stiffness indices map of the HD2
haptic device with and without the parallelogram structure in the formulation.
Specifically, we look at the x-y plane where
1
0 . The resulting stiffness map of
1'
HD2 formulated without the parallelogram structure is given in Figure 78 (a)-(f). The
stiffness map of HD2 formulated with the inclusion of parallelogram structure is given in
Figure 79 (a)-(f). Note that the x-, y-, z- and
,
,
directions are corresponding
to frame {0} , or frame {W } .
The first thing to notice is that the stiffness in the
-direction in both Figure 78(b)
and Figure 79(b) is almost non-exist. This is because it is the joint along the axis of the
handle, which is a passive joint. Similar to the case of PHANToM 1.5 system, where one
can see the contour of the stiffness in the x-, y-, and z-direction here similar to those of
PHANToM 1.5. With the inclusion of parallelogram, the stiffness in the x- and zdirection decreases. The results were similar to the result obtained for the PHANToM 1.5.
On the other hand, the stiffness in the y-direction didn‟t change much with inclusion of
the parallelogram structure. This is expected as the main contribution to this stiffness is
from the 1st joint. The stiffness in
direction increase considerably with the adding of
the parallelogram since rotation on that axis mostly related to the 2nd joint and involves
moving the parallelogram structure. The stiffness in
direction on the other hand,
reduces considerably since rotation along this axis did not involve the 2nd joint much.
124
2
Stiffness Map of HD without Parallelogram Sub-system
Stiffness in  -direction
Stiffness in  -direction
4
4
Stiffness in  -direction
4
4.502 5e-00
7
0.954691.432
9.00
5 1e
-0 07
2
2
0.35801
2
-2
-2
-2
2
50
4.
4
0.119 3
07
-0
5e
-4
-4
0.4
0.6
84
17
05
2
0.8 1
47
29
1.33
15
y-position
4
19 3
0.1
08
42
0.2
0
y-position
0.3
58
01
y-position
67
38
0.2
31 5
13.33
68
0.9
7
00
e25
50
4.
0
0
4
10
12
0.
0.47
7 34
0.363
12
0.59668
-4
0.23867
0.35801
0.95469
-6
-6
1.1934 1.79
0
1
2
3
x-position
4
5
0
6
-6
4.5025e-0 07
1
2
(a)
3
x-position
4
5
6
0
1
2
3
x-position
(b)
Stiffness in X-direction
5
6
(c)
Stiffness in Y-direction
4
4
Stiffness in Z-direction
4
4
1.2314
1.231
4
2
2
0.61569
2
0.46176
(d)
4
1.
45
36
y-position
1.03
8
3
2.076
6
y-position
5
6
0
1
2
0.37639
0.60
2 22
1.2044
y-position
0.6
22
98
3
x-position
9
56
61
0.
-6
2
0.15392
-4
0.30784
-6
1
84
07
0.3
-2
-4
3
61
1.6
0
0
0.225 83
-2
0.15055
-4
0.376 39
0.45166
-2
0.52694
68 9
1.8
0.41532
0
0.30111
64
30
0.8
0
3
x-position
69
0.615
0.769 61
-6
4
(e)
5
6
1.3853
0
1
2
3
x-position
4
5
6
(f)
Figure 78: Stiffness of (HD)2 in each directions, with formulation without considering the
parallelogram structure in the upper and lower linkages.
125
2
Stiffness Map of HD with Parallelogram Sub-system
Stiffness in  -direction
Stiffness in  -direction
4
4
0.4
28 5
7
2
0
8
23
95
0.0
0.38095
0.19048
38
52
09
0.
0.0
47
61
9
71
0.285
0.
57
14
3
2
0
y-position
y-position
0.047619
76
04
0.9
-2
0.476 19
19
76
0.
0.66667
-2
6
47
1.0
0.857
14
0.9
52 3
8
-4
0.952 38
-6
1
2
3
x-position
(a)
0.
61
90
5
y-position
0.23
81
33
33
0.3
0.2
85 7
1
0.23
81
0.71
0.6419
290
5
0.428
57
0.33333
0.28
5 71
-4
1
2
3
x-position
4
5
6
0.28571
52
09
0.3
-6
0
3
x-position
4
5
6
0
1
(d)
2
0.142 86
-2
1
85 7
0.2
-6
2
0.095 238
0.238 1
0.28571
0.38095
7
85 1 9
8
42 0.4761
0. .523
0
19
0.26
1
0.19048
0.
07
14
29
9
61
47
0.
0
0.21429
0.23
81
0
3
x-position
(e)
0.14286
8
23
0.11905 66 67
0.1
0.142 86
0 48
0.19
9
0.261
6
(c)
0.19048
y-position
2
0.333 33
0.38095
9 095
0.047 61 0.
-4
5
0.2381
0.523 81
0.42857
19
76
04
0.
81
0.33333
4
3
0.02
9
42
1.1
Stiffness in Y-direction
4
-2
4
(b)
Stiffness in X-direction
0
1
-6
0
2
8
47 6 23
1.0 1.
76
1.04
-4
4
5
6
(f)
Figure 79: Stiffness of (HD)2 in each directions, with formulation including the parallelogram
structure in the upper and lower linkages.
126
6.8 Trajectory Tracking
The Jacobian of the HD2 system is formulated using the method described in
Section 6.2. We verify the ability to simulate the system together with a trajectory
following task [176]. A task-space controller was developed to allow the end-effector tip
(position only) to track a moving spatial trajectory describe as x d
and z d
cos(2t ) , as t
0
2
R sin(t ); yd
R cos(t ) ;
, without constraining the end-effector angles. The
controller takes the following form:
Jv
1
xd
K xd
x
(86)
with an initial offset position of the end-effector (where frame {7} locate) by 0.25 in the
z -direction of frame {0} . In Figure 80(a), the desired and the actual trajectories of the
end-effector are plotted and we see the convergence to the desired trajectory. Figure 80(c)
shows the angle profile of all the joints during the trajectory tracking and Figure 80(b)
shows the trajectory tracking of the HD2 in the inertial frame. Figure 80(d) show the
length of the handle throughout the trajectory tracking task, shows that the position and
angular constraints were both satisfied.
127
Desired vs. Actual Trajectory
Desired
Actual
3.8
z-position
3.6
3.4
3.2
Actual Trajectory
3
3
Desired Trajectory
2.5
3.5
3
2
2.5
1.5
y-position
2
x-position
 1 (rad)
(a)
(b)
0.05
0
-0.05
-0.1

1

1*
 2 (rad)
0
1
2
3
4
5
6
7
8
3.4

3.2

2
2*
0
1
2
3
4
5
6
7
8
 3 (rad)
3.2

3
3

2.8
3*
0
1
2
3
4
5
6
7
8
 4 (rad)
-1

4

-1.5
4*
 5 (rad)
0
1
2
3
4
5
6
7
-1.2
8

5
-1.4

5*
-1.6
 6 (rad)
0
1
2
3
4
5
6
7
20
15
10
5
8

6

6*
0
1
2
3
4
5
6
7
8
Time (sec)
(c)
(d)
Figure 80: (a) The trajectory tracking result shows the desired path and the traced path; (b)
show the HD2 tracking the trajectory; (c) angle profile of all the joints in the system; and (d) the
length of the handle throughout the trajectory tracking show the position and angular constraints
were satisfied.
128
6.9 SimMechanics / MapleSim Model
Our goal is to be able to design, analysis, prototyping, controller design and
ultimately validation using a combination of virtual prototyping and hardware-in-the-loop
testing. Our paradigm for rapid development, refinement and implementation of system
design emphasizes: (i) development of the control scheme in a user-friendly, graphical,
high-level block diagrammatic language (using MATLAB/Simulink) (ii) simulation,
testing and refinement of the control system by virtual prototyping (using COTS
simulation tools); (iii) rapid conversion of the refined control system into a form suitable
for real-time execution (using Real-Time-Workshop/QuaRC) for hardware-in-the-loop
testing, as shown in Figure 81. In this work, we developed a SimMechanics model as
well as a MapleSim model that can be used as a plant model for HIL simulation [177].
Figure 81: Hardware-in-the-loop (HIL) design and testing framework of the HD2 haptic device.
129
Figure 82: SimMechanics model of HD2 haptic device with its corresponding VRML model for
visualization.
(a)
(b)
Figure 83: (a) MapleSim model of the HD2 haptic device with its corresponding 3D visualization
model.
6.10 Chapter Summary
In this chapter, we study the workspace, manipulability, and stiffness indices of
an in-parallel haptic device – HD2. The HD2 is an example of a parallel haptic device
created by piecing together two PHANToM 1.5 type serial manipulators that we studied
in the previous chapter. We use the modular formulation that we present in previous
130
chapters, to formulate the Jacobian of the HD2 device. In particular, we study how the
inclusion of the parallelogram sub-system changes the manipulability ellipsoid and their
stiffness indices.
With these performance measures, we can then perform optimization similar to
those shown with PHANToM 1.5 device in the previous chapter, based on the specific
application where we want to deploy the system.
131
7 Musculoskeletal Analysis-based Performance
Measures
Thus far, we have looked at various performance measures can be use to study a
haptic device, focusing mainly on the device. However, when in use, a haptic device is
always attached to a human user. This close coupling between a human user and the
device suggests that we should include the human user in studying the performance of
haptic devices. This is a challenging task: a human user holding a haptic device creates
an additional closed loop constraint. This new system that consists of a human and a
device, and the bi-directional transmission of force and moment affecting the
performance of the overall system.
Figure 84: When a human user holding on a haptic device, a geometric loop-constraint between
them and the bi-directional transmission of forces and moments create challenges to evaluate the
performance of the combined human-device interactions.
132
Often times, when designing a haptic device, the human itself is usually the last
piece of the entire system to be put in the loop for evaluation. Human-in-the-loop studies
tend to be set up as human interacting with physically prototyped devices coupled with
human performance evaluation by statistical analysis [178-181]. Alternately, the human
can be modeled as a mechanical admittance connected to the haptic device through spring
and damper and subsequently a compensator maybe design to improve the performance
of the device [182-184]. In a recent paper by Low, et al. [185] who proposed a
framework through the use of musculoskeletal models, coupled with the dynamics of
haptic devices for analysis of haptic rehabilitation device.
In recent work, researchers realized the importance of incorporating human
factors in the design of haptic devices, and start to incorporate human factors in the
design process of haptic device. In previous chapter, we listed several researcher
guidelines [78-80] that suggested the inclusion human in the evaluation of haptic devices.
The goal is to find performance measures to allow a designer quantitatively evaluate the
effect of the coupling between human-machine interactions.
The coupling between human and machine can have different levels. In the
position level, it is the study of common reachable workspace between a human and the
haptic device. As seen in previous studies, haptic devices have limited operating
workspace. For devices that have limited workspace but the task required a larger
workspace, proper scaling of distance or repositioning of the device is used. Peer et al.
[174] use a simplified human model (planar) to study common workspace that is
reachable by both the human user and two haptic interfaces (Shown in Figure 85(a)), and
133
perform optimization to find out the ideal placement of the devices and human user for
their application. Komoguchi et al. [173] use simplified human kinematic model to study
human-machine collision detection as well as avoidance, as shown in Figure 85(b). In
recent time, many Computer Aided Design (CAD) package offers accurate CAD
mannequin with accurate anthropometric data and similar degree-of-freedom as a human
user, e.g. NX Human from Siemens, and Creo Elements/Pro (formerly Pro/Engineer)
Manikin Extension from PTC. This allows a designer to study design considerations such
as workspace and ergonomic, within a CAD package where the device is modeled.
(a)
(b)
Figure 85: (a) A simple planar human kinematic model created to include human user in
optimizing the haptic device workspace, given in [174-175]; and (b) Collision avoidance and
motion planning by modeling both the human and haptic device as linkages, given in [173].
In the velocity level alone, it is the determination of the best possible position
between human and haptic device, such that both human (forearm) and haptic device can
have maximum manipulability in the operation space. The challenge here is the human
134
model, which typically has much higher degree-of-freedom than a haptic device, is
usually redundantly actuated. We have yet to see any work performed in this area even
with a simplified human arm model. Human arm is known to have similar stiffness
ellipsoid as a manipulator that is determined experimentally [186-188]. Similar study
could be performed by studying the stiffness in both the human arm and the device, and
their combined stiffness. In the force level, the interaction between human user and
device is typically modeled as a spring-damper system, and study their dynamic
interactions [182-184], and the stability of the haptic devices.
7.1 Haptic-based Virtual Rehabilitation Systems
In the past decade, researchers [189-194] including our group [195-202]
examined the application of robotic devices and automation technology to assist, enhance,
quantify, and document physical rehabilitation tasks which now offers a useful template
for sensorimotor training tasks. Functional evaluation methods tend to be limited to
observation-based (and subjective) assessment of quantities such as dexterity, speed,
coordination of one or both hands; joint range-of-motion, and strength using semiquantitative tests (such as Fugl-Meyer Index, Rivermead Motor Assessment Score).
Quantitative (and objective) functional assessments are vital both from the viewpoint of
initial evaluation as well as subsequent creation, progress monitoring and adaptation of
an effective training regimen.
Recently, much work has been done in developing haptic-based virtual
rehabilitation systems [178, 181, 192, 203-206]. The haptic enabled virtual reality
systems not only allowed patient to do physical therapy exercises and functional
rehabilitation exercises, it also allow automatic and transparent patient data collection.
135
This allows a therapist to monitor progress, and change the exercise level of difficulty,
and repeat evaluations over time. Example of haptic rehabilitation devices are shown in
Figure 86. Similarly, adjustable and reconfigurable rehabilitation exercise machines are
also being developed (see Figure 87 for some examples).
Examples of Haptic Rehabilitation Devices
(a)
(b)
(c)
Figure 86: Example of haptic rehabilitation devices:(a) the MIT-MANUS for upper-body function
rehabilitation after stroke, developed in MIT [207]; (b) the ‚Rutgers Ankle‛ Rehabilitation
Interface for ankle rehabilitation developed in Rutgers University [208]; and (c) The haptic
rehabilitation device for upper-limb motor function rehabilitation developed in Shibaura Institute
of Technology [209].
Examples of Commercial Rehabilitation Devices
(a)
(b)
(c)
Figure 87: Examples of adjustable, reconfigurable rehabilitation exercise machines from BIODEX:
(a) Upper Body Cycle; (b) System 4, and (c) BioStep Clinical Pro.
Hence, it is beneficial to be able to create, test, evaluate and refine the haptic
rehabilitation program to be deployed on such rehabilitation devices following a Virtual
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Prototyping (VP) methodology [38, 210]. Virtual prototyping refers to functional
simulation, quantitative performance analysis and iterative refinement of suitable
products and processes in a virtual environment. By permitting realistic, accurate and
quantitative testing of multiple intermediate models within a virtual environment, virtual
prototyping, also known as Simulation-Based Design (SBD), has rapidly gained
popularity and become a crucial part of most engineering design processes.
Here, we examine the application of this VP methodology to study and refine the
interactions of a patient with the rehabilitation equipment and regimen completely in a
virtual environment. This is realized by the integration of engineering support tools (such
as visualization, musculoskeletal analysis, and optimization) and structured therapist
involvement within a Virtual Design Environment (VDE) to facilitate the evaluation and
the subsequent refinement of the of rehabilitation regimen in the presence of variability.
Such framework leveraging computational, visualization, and decision support tools has
immense value for rapidly performing what-if type analyses, or parameter-sensitivitystudies. We use a case-study of a driving simulator [211-212] to illustrate these three
aspects in more concrete terms.
7.2 Critical Challenges
The adoption of a computational-analysis paradigm is beneficial from the
viewpoint of allowing us to think of a studying a rehabilitation regimen in the form of a
„design problem‟ [213]. In such a setting, there is a clear emphasis on systematic
generation, evaluation and elimination of design choices. SBD tools in engineering have
capitalized on setting-up and solving such design problems by coupling parametric
models with functional simulation tools and optimization methods.
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7.2.1 Computational Musculoskeletal Analysis Tools
Many computational tools have been developed for kinematic and dynamic
analysis of vertebrate musculoskeletal systems, building on an Articulated Multi-Body
Systems
(AMBS)
framework
[214]. Constrained
musculoskeletal
system-level
computational models can be constructed modularly by placing physiologic and
behavioral constraints on anatomical components (e.g., bone, muscle, and tendon). Such
musculoskeletal analysis tools allow monitoring the internal human variables - a wide
variety of biologically relevant data (from lengths, forces, reactions of muscles, tendons,
joints to metabolic power consumption and mechanical work) can be accessed.
Alternatively, other higher-level abstracted performance measures may be developed,
allowing a therapist to directly monitor the effects of a regimen on target muscle groups
and allow grasping of functional relationship of muscles.
(a)
(b)
Figure 88: (a) AnyBody musculoskeletal modeling environment [202]; and (b) OpenSim open
source software system interface [215].
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The ability to perform functional evaluations depends on the availability of
functional simulation tools. However, the effectiveness of the functional simulation tools
is limited by the extent of capture of the underlying physics, the modeling and analysis
fidelities and ultimately computational power. In developing rehabilitation regimen, the
effective simulation of interactions between the human user, the device and the
environment becomes critical. Varying levels of such simulations are possible – e.g.
several digital-human modeling tools such as Jack allowed kinematic modeling of
human-device-environment interactions. The ability to monitor internal human variables,
such as joint angles and torques, from the virtual avatar formed the basis of a usercustomized development of human-worn products [38, 210]. In recent times, a number of
computational
musculoskeletal-analysis
tools
such
as
SIMM
[216]
from
MusculoGraphics Inc., AnyBody [217] from Anybody Technology A/S, and Human
Body Model (HBM) from Motek Medical have become available. Open source software
like OpenSim [215] also allow user to develop musculoskeletal models and perform
dynamic simulation of movement. Musculoskeletal systems consist of numerous bones
connected together at joints, activated by muscles and thus can be treated within
framework of Articulated Multi-Body Systems (AMBS) [218]. Unlike traditional
engineering
systems,
musculoskeletal
systems
inherently possess
considerable
redundancies, which can be resolved typically leveraging an optimization approach [218].
Several packages such as SIMM focus on the forward dynamic simulation problem while
others such as AnyBody implement an inverse dynamic approach [217-219]. Such tools
can now allow monitoring the dynamic of internal human variables (such as muscleforces and muscle activities).
139
Among these, the AnyBody Modeling System offered a convenient tool for
modeling and analyzing various vertebrate musculoskeletal systems. The AnyBody
musculoskeletal model is built up as a constrained articulated multibody system with
rigid skeletal bones overlaid with multiple muscles that serve to both constrain and
actuate the system. The governing equations can be obtained as the constrained dynamic
equations of this articulated multibody system. However, the significant actuation
redundancy creates indeterminacy for resolving muscle-actuator forces via inverse
dynamic analyses. The indeterminacy in muscle force-distribution is resolved using an
optimization approach. For AnyBody, it is the minimization of the maximal muscle
activity subject to equality constraints (multibody dynamic equations) and non-negative
muscle-force constraints [220]:
f
min G  fmuscle   max  muscle,i
f
 Ni

 , i  1,

, nmuscle
(87)
Subject to : Cf = r
(88)
f muscle,i  0
(89)
Many contemporary studies [221-223] have examined validation of Anybody
forward- and inverse-dynamics simulations in various application-contexts. Additionally,
the ability to resolve muscle activities/forces (beyond the more traditional joint forces and
moments), and relate these directly to Electromyography data (EMG) was also extremely
attractive from the view point of validation [221-222].
Hence, we use this „insight‟ into the human to help study the effects of variability.
For example, a rehabilitation routine might aim to target specific muscle groups of a
patient. A wide variety of musculoskeletal analysis results (ranging from lengths, forces,
140
reactions of muscles, tendons, joints, etc), can be accessed. However, this raw data needs
to be further processed to create performance measures that can be of utility to the
therapist. Hence, in this thesis, (i) peak muscle forces; (ii) average muscle forces; and (iii)
muscle force fluctuation (difference between maximum/minimum muscle forces) will
serve as our performance measure. Such performance measures allows a therapist to
directly monitor the effects of a regimen on target muscle groups and allow changes to
rehabilitation program based on patient condition. Alternatively, other higher level
abstracted measures may be developed from the physical measurements of kinematic and
dynamic quantities available from the model.
Parametric Study of Bipedal Walking
Figure 89: Systematic and parametric musculoskeletal analysis based study can be used to better
understand bipedal walking frequency [200].
As an example, we have used the musculoskeletal analysis framework to
parametrically study bipedal walking frequency [200]. Researchers have hypothesized
that animal locomotory patterns seen are consistent with the resonant frequencies
endowed by their musculoskeletal structures [224-225]. Further it is posited that systems
succeed in minimizing their energy expenditure by moving at this resonant frequency.
We systematically study this hypothesis in the specific context of bipedal locomotion.
Researchers have sought to correlate the preferred strike frequency with the resonant
frequencies of the model or used indirect measurement such as oxygen consumption,
electromyography (EMG) to assess expended effort. In our study, we employed virtual
141
prototyping with a capable musculoskeletal simulation model to study the same
hypothesis. We benchmark against the available literature and demonstrate that valuable
insights can be obtained that can complement the current knowledge-base in biped
locomotion [200].
Parametric Study of Bicep Curl
Biceps Long Muscle Force Plot
Biceps Short Muscle Force Plot
160
D.Mass
D.Mass
D.Mass
D.Mass
D.Mass
D.Mass
140
D.Mass = 10 Kg
D.Mass = 8 Kg
0 Kg
2 Kg
4 Kg
6 Kg
8 Kg
10 Kg
100
D.Mass = 6 Kg
80
D.Mass = 4 Kg
60
300
D.Mass = 8 Kg
250
D.Mass = 6 Kg
0 Kg
2 Kg
4 Kg
6 Kg
8 Kg
10 Kg
D.Mass = 4 Kg
150
D.Mass = 2 Kg
100
D.Mass = 0 Kg
20
(a)
=
=
=
=
=
=
200
D.Mass = 2 Kg
40
0
D.Mass
D.Mass
D.Mass
D.Mass
D.Mass
D.Mass
D.Mass = 10 Kg
350
Muscle Forces, (N)
Muscle Forces, (N)
120
=
=
=
=
=
=
D.Mass = 0 Kg
50
0
0.1
0.2
0.3
0.4
0.5
0.6
Time, (sec)
0.7
(b)
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time, (sec)
0.7
0.8
0.9
1
(c)
Figure 90: (a) Parametric bicep curl study on a simplified upper-arm/shoulder musculoskeletal
model and muscle force profiles for both (b) biceps short and (c) biceps long muscles.
In another effort, a parametric bicep-curl study on the Upper Arm/ Shoulder
musculoskeletal model was created to allow healthcare students (including OT/PT) in a
Gross Anatomy class to systematically study relationships between form and function
[202, 226]. Pedagogically, this example (Shown in Figure 90) is highly relevant in its
ability to illustrate system-level effects of various components. But more importantly, we
were able to extend it easily to explore the effects of variability of geometry, dynamics,
and regimen on musculoskeletal performance. Specifically, they could study variability
of bicep force-performance due to parameter variations in (i) initial elbow joint position
(ergonomic); (ii) the mass of the dumbbell (device); and (c) the elbow-joint velocity
(regimen). Fig. 3(b) and (c) depict the variation in Bicep short and Bicep long muscles
for a parametric variation of dumbbell mass between 0 kg to 10 kg (while keeping the
142
initial elbow-angle at 90 and elbow-velocity at 45 rad / sec ). In addition to the
quantitative results, such virtual experimentation can help develop the students‟ intuition
about musculoskeletal performance trends.
Parametric designs simplify the process of systematic generation of choices
especially in a computer-based implementation. However, the appropriate selection of
design variables which are both biomechanically relevant and are of significant to the
therapist poses challenges. Design variables can encompass geometry as well as regimen
parameters for the user-device interaction. In the case of rehabilitation device refinement,
geometric parameters could include individual user geometries such as limb length, etc
[227-228], device properties for customizable devices [38, 210], as well as properties
determined by the ergonomics of placement of the user with respect to the device.
Regimen parameters such as dynamically adjustable springiness, damping, as well as
frequency, amplitude of a desired motion can also serve as design variables. These design
variables can be used to systematically explore the space of feasible alternatives. Coupled
with selection of a suitable performance measure, an optimization problem can be setup
to determine the „best‟ set of values for a rehabilitation regimen, form a potentially large
set of choices thereby serving as a decision-support for a therapist tool.
7.3 Performance Measures
A wide variety of musculoskeletal analysis results (ranging from lengths, forces,
reactions of muscles, tendons, joints, etc), can be accessed. However, this raw data needs
to be further processed to create performance measures that can be of utility to the
designer (to improve the design of haptic devices) or therapist (to design patient
rehabilitation progress). It is important to note that one should use the performance
143
measure that is being validated with the computational package. For example, in
AnyBody, the quantity „muscle-activity‟ has shown to correlate well with EMG
measurement while in HMB by Motek, the computed muscle forces have shown good
correlation with EMG data.
7.3.1 Muscle Forces
In the study of a haptic motor rehabilitation program [201-202], (i) peak muscle
forces; (ii) average muscle forces; and (iii) muscle force fluctuation (difference between
maximum/minimum muscle forces) serve as our performance measure. Such
performance measures allows a therapist to directly monitor the effects of a regimen on
target muscle groups and allow changes to rehabilitation program based on patient
condition. Alternatively, other higher level abstracted measures may be developed from
the physical measurements of kinematic and dynamic quantities available from the model.
7.3.2 Muscle Activity
In the absense of direct measures, the amounts of energy expenditure, is
correlated and estimated through EMG and oxygen consumption measures or inverse
dynamics techniques. The electrical signal associated with the contraction of a muscle is
a commonly-used measure of muscle activity [229-230] and can be captured as an
Electromyogram (EMG). Similarly, the localized measurement of oxygen consumption,
measured by blood-gas measurement, reflects the energy consumption in that muscle.
Recent studies have shown good correlations of EMG and local oxygen consumption to
external load as well as to muscle forces [231]. In another study [200], we use a measure
called “muscle-activity” as a fraction (between 0 and 1) of maximum voluntary muscle144
contraction-force. In other words, the activity at a given point in the gait cycle indicates
the percentage of maximum muscle force that can be exerted. Muscle activity profiles
obtained from AnyBody have been shown to correlate very well with EMG data [232].
We can further post process this “activity level” to determine (i) peak values indicative of
the range of muscle force required (since all begin at activity level 0); (ii) average value
over a gait cycle (used to develop metabolic energy consumption estimates).
As noted previously, the muscle activity profile is comparable to the EMG data,
which in turn relates to muscle contraction forces and energy cost and hence can be used
as a metric for our study. For example, Figure 91 shows the muscle activity profile of the
„right muscle‟, a muscle modeled in our simple pendulum study [200] and its
corresponding average muscle activity (shown in next section) at frequencies 0.2Hz and
0.4Hz.
Right Muscle Activity in One Swing Cycle
0.09
0.08
Muscle Activity @ Freq = 0.2Hz
0.07
0.06
Muscle Activity @ Freq = 0.4Hz
Activity
0.05
Average Muscle Activity @
Freq = 0.2Hz
0.04
0.03
0.02
Average Muscle Activity @
Freq = 0.4Hz
0.01
0
-0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
Scaled Time, sec
0.7
0.8
0.9
1
Figure 91: Right muscle activity profile (solid) and average muscle activity (dashed) for one
pendulum swing cycle used in [200].
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7.4 Case Study: Haptic Motor Rehabilitation Programs
As an example to show the suggested integrated musculoskeletal simulation
framework that allow one to study human-device interaction, we use a motorrehabilitative Haptic Virtual Driving Environment (hVDE)
[211-212] to illustrate
various aspects of the study of variability in a rehabilitation regimen. This hVDE was
developed by integrating a commercial-of-the-shelf (COTS) force-feedback steering
wheel with parameterized rehabilitation therapies to serve as a home-based inexpensive
personal-movement-trainer (as shown in Figure 92(a)).
(a)
(b)
(c)
Figure 92: (a) the actual setup that is being modeled and studied[212]; (b) Front view of the user
and driving wheel (rehabilitation device) arrangement in the case study; and (c) the
corresponding top view of the same arrangement.
Specifically, users are instructed to perform structured-rehabilitation-exercises in
the form of driving-tasks along prescribed parametric paths while holding the driving
wheel with one- or both-hands. However, a very important question arises in such a
setting – what is the effect of variability in terms of (i) how the equipment is setup
(ergonomic); and (ii) how the exercise is performed on the actual functional rehabilitation
146
(regimen). These questions are critical to evaluate the effectiveness of a rehabilitation
device.
Our framework allows the therapist to answer these questions quantitatively using
a range of (i) parametric sweep; and (ii) optimization studies to be discussed next.
Ultimately, such quantitative evaluation is intended to help the therapist to determine the
„best‟ combination of the two for a specific user. This framework brings together several
tools within a VDE as shown in Figure 93. The VDE consists of a Central Therapist
Interface with which additional human musculoskeletal modeling and optimization
modules interact.
7.4.1 Central Interface
However, the interface for interaction with the AnyBody system remain very
inaccessible to a non-computational user due to the lack of both programming knowledge
and numerical-analysis background to take advantage of the „computational testing
paradigm‟ (see Figure 88(a)). In this respect, Anybody system offers a convenient
Application Programming Interface (API) to facilitate access to the underlying
computational engine from external programs which we exploit by creating a Graphical
User Interface (GUI) which allows for visualization and provides a restricted means for
user interaction with AnyBody settings (e.g. using sliders). Further, using the GUI, the
therapist can select the rehabilitation regimen, the objective function, as well as
customize patient geometry information. The therapist can then perform parametric
studies by varying the appropriate design variables.
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Figure 93: Paradigm of our framework for VP of rehabilitation device: A MATLAB GUI that
serves as the Center Interface that allows the therapist to examine the effects of different regimen
and determines the ‘best’ regimen based on user’s geometric information. The AnyBody engine is
responsible for the computation of the muscle forces, while the optimization routine is handle
using MATLAB’s optimization toolbox.
7.5 Musculoskeletal Models
The rehabilitation environment model (user and device) is setup using the
AnyBody Modeling System. Parametric models of the human patient can be extracted
from libraries and potentially customized to reflect the specific patient characteristics (the
bones maybe re-dimensioned or the peak muscle forces can be customized). Similarly,
148
the model of the rehabilitation device can be created within the same framework. The
model of the user and the model of the device can thus be co-simulated and the obtained
results can be used for refinement.
(a)
(b)
Figure 94: The human musculoskeletal model, the rehabilitation device, and their interaction used
in the case study were modeled using AnyBody Modeling System. Shown here is the (a) Side view
and (b) Front view of the model.
The musculoskeletal model, shown in Figure 94(a) and (b), can help a therapist
model, analyze and visualize various aspects the patient‟s interaction with the
surrounding environment. If desired, such as model can also be customized with specific
patient geometry and muscular characteristics. However, for computational efficiency,
we restricted the human model to the upper-body bone-tendon-muscle from the pelvis up
much like others [233].
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7.5.1 Parametric Studies
The five geometry parameters involved in this study, xw , yw , zw , Rw ,w , pertain to
the ergonomics of placement of the wheel with respect to the patient. These are spatial
location of the center of the steering wheel ( xw , yw , zw ) ; the radius of the steering wheel
( Rw ) ; and the tilt angle of the wheel with respect to the global y-axis ( w ) , as shown in
Figure 92(b) and (c). We can then perform parametric study for each of selected
parameters within the reachable region of the user. This process allows one to analyze the
effect of one parameter while others were held fixed. While any of these five parameters
can be used as design variables, we choose to study only two of them ( xw , w ) in this
study.
The speed and amplitude at which the patient turns the wheel serve to
parameterize the regimen of the rehabilitation. Increasing the speed of the turning allows
a quicker completion of the regimen but requires greater muscle forces for completion of
the task. In this paper, the rehabilitation regimen is such that the patient turns the wheel at
an angular velocity of 30 / sec for amplitude of 330 turn, thus it takes 11.0 seconds to
perform such task (shown in Figure 95(a)-(d)).
(a) t=0sec
(b) t=4.5sec
(c) t=8.0sec
(d) t=11.0sec
Figure 95: (a)-(d) The hand placement on the steering wheel at different time instant in the
parametric study, view from the top of the virtual model. (a) The initial position; (b)-(c) the
intermediate positions; and (d) the final position.
150
In particular, we perform parametric sweep of xw (corresponding to the distance of
the steering wheel in front of the user) from 0.15 m to 0.45 m, at a constant angular
velocity of 30 / sec for a total of 330 turn. The results are shown in Figure 96(a) and (b).
On the other hand, to study how the muscle force changes with respect to the steering tilt
angle, we fixed the steering wheel at (0.25, 0.1,0.25) and vary the tilt angle from 0 to 60 .
The results are shown in Figure 97(a) and (b).
(a)
(b)
Figure 96: (a) Surface and (b) 2D plot of the maximum combine muscles force for a patient to
turn a wheel at a constant angular velocity of 30deg/sec (with 330 deg total movement), for a 0
to 60 degree tilted wheel angle, with wheel center located at (0.25,
-0.1, 0.25), measured in
meters.
From Figure 96(a) and (b), we see that the peak of the combine muscle force
decreases from about 0.92N to 0.72N as the steering tilt angle increases from 0 to 60 (in
region where time = 3 sec to time t = 6 sec). This corresponds to the motion where the
hand is at the lower region of the wheel, as shown in Figure 95(b)-(c). On the other hand,
the combined muscle force profile corresponding to different xw positions is given in
Figure 97(a) and (b). In this study, we varied the xw location from 0.15 m to 0.45m while
the yw and zw are located at -0.1m and 0.25m respectively. The wheel tilt angle is
151
maintained at 0 deg (no tilt) for this study. From Figure 97(d), we can see that as the
wheel being placed farther away from the user, the maximum muscle force drops from
1.43N (at x = 0.15m) to 0.96N (at x = 0.41m).
(a)
(b)
Figure 97: (a) Surface and (b) 2D plot of the maximum combine doing the same motion, this time
with 0 deg tilted angle (no tilt) , for wheel x position varied from 0.1m to 0.45m (far or close in
front of the user).
7.5.2 Optimization Studies
An optimization study was created to help the therapist to determine the best
possible placement of the device (within the feasible region) and the regimen for specific
objective. Two different objective functions were studied: (a) Maximum combined muscle
force fluctuation; and (b) Average combined muscle force. For simplicity, we choose only
two design variables in this problem, the tilt angle  w and wheel‟s x-position xw . Thus,
both objective functions are function of the x-position and tilt angle, f ( xw ,w ) . Hence, the
general form of the optimization problem may be written as:
Min fi  xw , w 
xw ,  w
152
(90)
s.t :
g1 : 0.15  xw  0.25
(91)
g 2 : 0   w  60
Since only two design variables are being studied, the function space of these two
objective functions can be explicitly plotted. Figure 98(a) and (b) are contour plots of the
function space with relatively few data points used (13 data points in the tilt angle
direction and 6 data points at the x direction).
(a)
(b)
Figure 98: (a) Function space of maximum combined muscle forces fluctuation as the objective
function and steering wheel’s x location and tilt angle as the design variables; and (b) Function
space of average muscle force as the objective function and steering wheel’s x location and wheel’s
tilt angle as the design variables.
Although the resulting contour of the function space does not look very smooth,
these contour plots give us pretty good estimate of the function space and a reasonable
good estimate for the location of the design variables. From these two function plots, we
can obtain the general rules: (i) to minimize the muscle force fluctuation, one would like
to place the wheel closer to the user with a higher tilt angle; (ii) to minimize the
153
combined average force used in performing the rehabilitation routine, one would place
the wheel farther away from the user and with small or no tilt angle.
Thus, the setup of the optimization allows creation of a decision-support tool for
the therapist. For larger problems, with more design variables, the visualization based
solution process finding the „best‟ solution may not be viable. However, one can
capitalize on the availability of powerful numerical optimization techniques to achieve
similar outcomes.
7.5.3 Discussion
We presented an integrated musculoskeletal simulation based design framework
that facilitates the rapid design and refinement of the ergonomic and regimen of a haptic
motor rehabilitation device. The integrated Virtual Design Environment serves as a
decision support tool for a therapist to determine the „best‟ rehabilitation program for a
particular user leveraging design and computational tools from musculoskeletal analysis,
optimization and visualization. Our preliminary result, demonstrated with a case study of
a motor-rehabilitative Haptic Virtual Driving Environment (hVDE). In particular, we
present our design framework using a case study which demonstrates the practical
implementation of our framework with preliminary result. Much work needs to be done
in terms of (i) creating a more seamless VDE interface for the therapist; (ii) developing
quantitative performance measures that are well balanced computationally and are of
significance/ relevance to the therapist; and (iii) applying the framework to a wider range
of settings. We are actively pursuing all these aspects in our current work.
154
Although the case study used to demonstrate our framework is a driving
rehabilitation device, the basic ideas and the design framework in this paper are
applicable to the design of variety of other devices that are working closely with human
user (such as exoskeletons, assistive devices, haptic devices and wearable devices), where
musculoskeletal measure of the user is critical. Such a framework could potentially be
used to (i) quantitatively study the effectiveness of existing rehabilitation devices, and
subsequently refinement of such devices; and (ii) quantitatively study the effect of
variability in the user and/or device to the effectiveness of the rehabilitation regimen, i.e.
the robustness of a particular rehabilitation regimen.
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8 Discussion and Future Work
The performance of the Haptic-User-Interface (HUI) that allows a human user to
interact with virtual environment to realize a typical haptic simulation, plays a very
critical role in determining the „realism‟ of a haptic rendering (referring to Figure 2). In
this thesis, we studied and analyzed various haptic device design-architectures with the
goal of improving the haptic rendering realism.
We focused on formalizing a design methodology, emphasizing modularly
piecing together serial-type haptic devices to create in-parallel haptic devices, that even
today is being employed (albeit informally) to realize various commercial- and researchin-parallel haptic devices. This methodology builds on the framework used to design and
analyze cooperation in multiple articulated robotic systems (as seen in legged walkers,
multifingered hands and multiple-arm grasping) to realize robust and flexible composite
systems.
From an engineering design standpoint, this approach is also attractive by
allowing the reuse of part (individual module), and new device design can be created by
piecing these individual modules in various way (shown in Figure 6 and Figure 45).
Finally, from an analysis standpoint, it is attractive that we can systematically study
system performance on the basis of individual chain/module performance characteristics.
This is the approach that we systematically espouse in this dissertation.
156
However, the overall system performance depends on design choice, the nature of
the individual arms, as well as their interactions. Using PHANToM 1.5 and HD2 as our
examples, we study the workspace, manipulability, and stiffness of these two systems
systematically. At the same time, we were able to quantify the effects of this added
parallelogram sub-system to performance improvements (manipulability and stiffness) as
well as the entailed drawbacks (such as reduction of overall workspace) in both
PHANToM 1.5 and HD2 systems. With these performance measures, a designer can then
systematically and quantitatively analyze, design, and evaluate haptic devices based on
specifications required by the intended haptic applications.
It is also important to note that we only discussed three performance measures
(workspace, manipulability, and stiffness) in this work. In defining these performance
measures, we also make several assumptions such as the actuators are perfect, joint
stiffness are linear and perfectly constant, electronics components are perfect and
manufacturing tolerance is perfect. Other design considerations such as manufacturing
cost, ease of control, interference of linkages, are likely important but are not pursued in
our work.
Traditionally, the study of haptic-user-interfaces focuses only on the individual
device performance (workspace, manipulability, and stiffness) alone. For devices
working closely with human user (rehabilitation devices, exoskeletons, haptic devices,
assistive device, etc.), it is important to be able to characterize the performance-effects of
the close coupling to a human, early in the design phase. „Human-in-the-Loop‟ (HIL)
testing typically entails physical human-factors testing of human user responses to
various experimental situations. In our effort, we examined use of a virtual
157
musculoskeletal analysis framework in which to co-simulate the human and the device,
and extract quantitative performance measures from the human musculoskeletal model.
These quantitative performance measures can then be used for performing optimization
on human-device interactions as demonstrated using a haptic motor-rehabilitation
regimen refinement scenario. However, a more careful study that includes the forces and
moment exerted by the haptic device and how it affects the muscle forces should be set
up to study the haptic interactions between the human and the haptic device.
8.1 List of Contributions
The contribution from this work can be categorized broadly in two parts: (i)
quantitative and systematic performance measures formulation for haptic devices
evaluation; and (ii) quantitative performance measures to study human-machine
interactions using musculoskeletal analysis framework. Specifically, the work can be
summarized as follow:

We demonstrated a framework that allow systematically formulate the Jacobian of in-
parallel system by combining contribution from individual chain. This formulation also
allows symbolic computation of the Jacobian.

We demonstrated our formulation and compared our results with the current literature,
and subsequently use it to study the workspace, manipulability, and stiffness of
PHANToM 1.5 and HD2 haptic devices.

In particular, we study the effect of an added parallelogram sub-system, which is
commonly found in various haptic devices, to the overall manipulability and stiffness of
the entire system.
158

We summarized various possible variants of haptic device by modularly piecing a
modular haptic device. It can be seen, from Figure 45, many candidate haptic devices can
be synthesized using the PHANToM 1.5 as a modular building block.

We examined a musculoskeletal analysis framework to allow a designer to
quantitatively study and evaluate devices that are to be used in intimate contact with a
human user. Quantitative performance measures obtained from musculoskeletal models
can be used to study design parameters that relates human-machine interactions and
showed the feasibility and effectiveness of this framework.
8.2 Future Work
As with any research, we have unearthed a host of research questions requiring
further investigations.

The presence of the offset link between the universal joint and the handle for the HD2
haptic device complicates (and prevents) development of the analytical forward and
inverse-kinematics (as noted in Appendix A.7). Elimination of this offset offers potential
for analytical kinematic solution and could be pursued.

Formulation of the analytical workspace expression would also greatly aid the
optimization of device workspace.

Our results matches very well with Park and Kim [119], who study the manipulability
of in-parallel system in a differential geometry framework. Bicchi and Prattichizzo [26,
115, 121] studied manipulability of parallel systems with a finely partitioned Jacobian for
a nuanced study of various types of singularities and could help provide further insight
into the system performance.
159

We shown the systematic and quantitative performance evaluation for various in-
parallel haptic devices – however it is necessary to set up multi-objective optimization to
simultaneously optimize several performance measures (that some time can be
conflicting to each other). This can be implemented easily using the quantitative
performance measures that we defined here but was not attempted here due to penalty of
time.

The musculoskeletal analysis framework provides a good starting point for
quantitative coupled-performance analysis of human-haptic device interactions but needs
to be better explored with more detailed case studies.
160
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Appendix A
A.1 Jacobian Formulation Example: Stanford Manipulator
A Stanford manipulator is shown in Figure 99.
Figure 99: Stanford Manipulator, a R-R-P-R-R-R serial manipulator and its D-H frame
assignment.
179
The DH frames and parameter assignment, according to the convention used in
[Robotic Handbook] is shown in Table 8, where terms with an * denote joint variables.
i
i
ai
di
1
0
0
2
/2
3
4
i
i ,initial
0
*
1
/2
0
d2
*
1
/2
/2
0
d 3*
/2
0
0
*
1
0
0
0
*
1
0
0
0
*
1
0
/2
5
6
0
/2
-
Table 8: DH Frame assignment for Stanford manipulator.
A.1.1 Spatial Vector Formulation
Using the frame assignment given in Table 8, the Homogenous Transformation
matrices from frame {0} to frame {6} (note that in this case frame {6} is also the endeffector frame) are given as follow:
0
A1
3
A4
cos
sin
0
0
cos
0
sin
0
1
1
4
4
sin 1
cos 1
0
0
sin 4
0
cos 4
0
0
0
1
0
0
0
0
1
0
1
0
0
1
A2
0
0
0
1
where the initial angle of
4
A5
1
and
sin 2
0
cos 2
0
cos
0
sin
0
cos 5
0
sin 5
0
sin
0
cos
0
2
2
2
5
5
0
1
0
0
0
1
0
0
0
d2
0
1
0
0
0
1
2
0
0
1
0
5
cos
sin
0
0
A3
A6
1
0
0
0
6
6
0
1
0
0
sin 6
cos 6
0
0
0
d3 0
0
1
0
0
1
0
0
0
0
1
are carried out in forming the above homogeneous
transformation matrix. Now we compute the direction of each joint axis (as expressed in
the base frame), which are given by:
180
0
z1
0
z4
0
z6
0
0
0
0 , z2
1
R1 0
1
0
sin
R4 0
1
cos
0
R2 0
1
sin
1
0
cos
1
R6 0
sin
sin
1
2
sin
1
sin
1
1
, z3
R3 0
1
cos
1
sin
sin
1
4
sin
1
4
cos
cos
cos
1
sin
cos
2
cos
4
cos
2
4
2
1
cos
cos 1 sin
sin 2
0
0
, z5
sin
0
R5 0
1
2
cos 1 sin
cos 2
0
1
sin
4
sin
1
2
2
cos
cos 1 cos
4
sin 2 cos 4
2
cos
4
2
cos
4
,
(92)
4
The vector i pn* , which is the vector defined from the reference frame of i th link
to the origin of the end-effector frame are given by (Note that frame {6} itself is also the
end-effector frame):
6
p
*
6
0
4
p*6
0
R4 4 r5
p*6
0
1
*
6
0
0
0,
0
0
R6 r6
2
p
0
R6 0
0
6
5
p*6
0
R5 0
0
0
0
0,
0
0
0
5
R5 r6
3
6
p*6
R2
R1
0
d2
0
d3 sin 1cos 2
d3 cos 1cos 2
d3 sin 2
p*6
0
1
2
*
6
0
0
R5 0
0
*
6
0
0
0
0
0
0
0
R3 0
0
0
0
0
0
0
0
0
p
0
R3 3 r4
0
d3 sin 1cos 2
d3 cos 1cos 2
d3 sin 2
0
0
0
3
p
*
6
0
d3
0
R2 2 r3
R1 r2
p
0
0
5
d3 sin 1cos 2 d2 cos 1
d3 cos 1cos 2 d2 sin 1
d3 sin 2
Note that J 3 is a prismatic joint and the rest are revolute joints, from Eqn. (6):
181
(93)
-d3 cos 1 cos 2 - d2 sin
-d 3 sin 1 cos 2 d2 cos
z × 1 p6*
J1 = 1
=
z1
d3 sin 1 sin
-d3 cos 1 sin
1
1
z × 2 p6*
, J2 = 2
=
z2
0
0
0
1
- sin 1 cos
cos 1 cos
3 *
sin 2
z × p6
J3 = 3
=
0
z3
0
0
2
2
d3 cos 2
cos 1
sin 1
0
0
0
4 *
0
z × p6
J4 = 4
=
sin 1 sin 2
z4
cos 1 sin 2
cos 2
2
2
,
0
0
z5 × 5 p6*
=
- cos 1 sin
z5
- sin 1 sin
J5
0
sin 1 cos
4
cos
cos
4
1
sin 2 cos 4
0
0
6 *
0
z × p6
J6 = 6
=
- cos 1 sin 4 sin 1 cos
z6
- sin 1 sin 4 cos 1 cos
sin 2 cos 4
cos
cos
2
2
,
4
4
(94)
cos
cos
2
2
4
4
As such, we obtain the Jacobian of the Stanford manipulator as follow:
J
J 1, J 2 ,
-d3c 1c
-d3s 1c
2
2
, Jn
- d2s
d2c
1
1
d3s 1s
-d 3c 1s
2
2
-s 1c
c 1c
0
0
0
2
0
0
0
0
0
0
0
d3c
0
c
1
0
0
s
1
0
c 1s
0
c
1
2
0
s
2
2
s 1s
2
2
(95)
-c 1s
4
s 1c 2c
4
-c 1s
4
s 1c 2c
4
-s 1s
4
c 1c 2c
4
-s 1s
4
c 1c 2c
4
2
s 2c
4
s 2c
4
A.1.2 Twist-based Jacobian Formulation
The twist contribution from each joints, expressed in frame {0} are given as
182
0
t1
0
0
1
0
0
0
0
t2
- cos 1 sin
- sin 1 sin
0
t5
cos
sin
0
0
0
0
sin
sin
1
cos
2
0
t3
- sin 1 cos 2 cos
cos 1 cos 2 cos
4
sin 2 cos 4
2
sin
1
cos
1
4
d2 cos
d2 cos
2
d2 cos
4
d3 sin
4
d3 sin
w
sin 1 cos 2
cos 1 cos 2
sin 2
- cos 1 sin
- sin 1 sin
4
4
0
t6
sin
4
d3 sin
4
0
0
0
1
1
cos
4
sin
4
2
t4
- sin 1 cos 2 cos
cos 1 cos 2 cos
4
sin 2 cos 4
4
d2 cos
2
sin
1
cos
sin 1 sin 2
cos 1 sin 2
cos 2
d2 sin 1 cos 2 d3 cos
-d2 cos 1 cos 2 d3 sin
-d2 sin 2
2
d2 cos
d2 cos
4
d3 sin
4
4
1
4
4
4
d3 sin
d3 sin
1
4
4
The Jacobian matrix, expressed in frame {0} is thus given as:
J
J1 J 2
0 c
0 s
Jn
s 1s
1
0
1
0
c 1s
c
1
0
0
0
0
s 1c
0
0
0
0
c 1c 2
s 2
2
d2s 1c
-c 1s
2
-s 1s
2
- s 1c 2c
4
c 1c 2c
s 2c
2
d 3c
2
4
-d2c 1c 2 d 3s
-d2s 2
s 1s 2 (d2c
1
c 1s 2 (d2c
c 2 (d2c 4
1
-c 1s
4
-s 1s
4
- s 1c 2c
4
d 3s 4 )
4
s 1s 2 (d2c
d 3s 4 )
d3s 4 )
c 1s 2 (d2c
c 2 (d2c 4
4
c 1c 2c
s 2c
4
4
4
4
4
d3s 4 )
4
4
d3s 4 )
d3s 4 )
The twist contribution from each joints, expressed in frame {W} where frame {W}
have the same orientation as frame {0}, but locate at frame {6} are given as
w
t1
0
0
1
-d3 cos 1 cos 2 - d2 sin
-d3 sin 1 cos 2 d2 cos
0
- cos 1 sin
- sin 1 sin
w
t5
w
t2
1
1
- sin 1 cos
- cos 1 cos
4
sin 2 cos 4
0
0
0
4
cos 1
sin 1
0
d3 sin 1 sin
-d 3 cos 1 sin
d 3 cos 2
cos
cos
2
2
w
0
0
0
t3
2
2
- cos 1 sin
- sin 1 sin
4
4
w
183
w
sin 1 cos 2
cos 1 cos 2
sin 2
t6
t4
sin 1 sin 2
sin 1 cos 2
cos 2
0
0
0
- sin 1 cos
- cos 1 cos
4
sin 2 cos 4
0
0
0
4
cos
cos
2
2
4
4
The Jacobian matrix, expressed in frame {W} is thus given as:
J
J1 J 2
Jn
0
c
0
s
1
-d3c 1c
-d3s 1c
2
2
0
0
1
0
s 1c
0
c
0
- d2s
d2c
1
1
s 1s
1
d3s 1s
-d 3c 1s
d3c 2
2
2
s 1c
c 1c 2
s 2
2
-c 1s
2
2
-s 1s
2
4
- s 1c 2c
4
- c 1c 2c
s 2c
4
4
-c 1s
4
-s 1s
4
- s 1c 2c
4
4
- c 1c 2c
4
s 2c
0
0
0
0
0
0
0
0
0
4
(96)
Which is the same as the one formulated using the spatial vector formulation.
A.2 Numerical Workspace Computation
Workspace computation of parallel manipulators can be difficult due to the
irregular shape of the workspace volume. Since workspace of a parallel manipulator is
often used as a performance metric in designing parallel manipulators, it is important to
develop a computational efficient way of computing the area enclosed by the workspace.
Feasible workspace of parallel manipulators is usually computed numerically, leaving the
user with large data clouds that represent the feasible points. These data points are often
scattered. For planar mechanism, these data clouds are x and y points. Traditionally, one
would evaluate if these data points lies within a 2D „box‟ define by x-y grid size: if it lie
inside this box, the area of the box will be counted. Adding up these „boxes‟ will give an
184
estimate of the total area. For spatial volume estimation, a 3D cube will be used instead
of a box. Here two ways of numerically compute an area enclosed by the workspace are
presented. The polyarea algorithm can be used to computes the area enclosed by a set of
points efficiently, as long as the point do not intersect each other and the area that it
enclosed is closed. Hence to use this algorithm, we basically need to find the points that
define the boundary of the workspace. There are two ways to get these boundary points.
Using an example, where the feasible workspace of a parallel manipulator is a square, we
show how these two algorithms can be used to obtain the boundary points. Subsequently,
a case study with a circular workspace is studied, as well as its application in computation
of the constant orientation workspace area for the HD2 haptic device at a specific plane.
Algorithm 1: Set up x-, y-grid. For each x and y point, check if the point is
reachable. If it is reachable, store the point. You end up getting points shown in Figure
100(a). To get the boundary points, one needs an algorithm to filter out the interior points,
leaving only the points on the boundary (Figure 100(b)). However, these boundary points
are usually not organized, if one plotted them sequentially (Figure 100(c)). Hence, the
second algorithm needs to order these points sequentially around the boundary, the endresult is shown in Figure 100(d). With these points, the area can be computed using
polyarea algorithm.
Algorithm 1
(a)
(b)
(c)
185
(d)
Figure 100: (a) raw data points; (b) interior points are filtered; (c) the boundary points are not
arranged sequentially around the boundary; (d) re-ordered points sequentially envelop the
boundary.
The accuracy of this method depends on the grid size, and the specified grid
spacing. In general, the smaller the grid size, more accurate representation of the
boundary can be generated. The disadvantage of this algorithm is that the algorithm that
removes the interior points and subsequently re-organizing the point sequentially around
the workspace boundary can be time consuming.
Algorithm 2: Instead of recording all the points on the grid that in the feasible
workspace, this algorithm create a grid z(x,y), where interior points are mark as 1 and
exterior points are mark as 0. Through this grid, one can generate a surface plot (Figure
101(a)), and subsequently a contour plot of this surface (Figure 101(b)). The points that
specify these contour plot can then be extracted, and use as boundary points, for
workspace area computation, as shown in Figure 101(c).
Algorithm 2
Figure 101: (a) Surface plot that represent the workspace; (b) Contour at specific level can be
generated; (c) the workspace boundary point extracted from the contour plot.
The accuracy of this method depends, in addition to grid size and grid spacing, is
the selection of the level at which the contour is specified. Using the square workspace as
186
an example, select the contour at level 1.0 give us the exact value of the workspace area,
which is 16 unit^2. Choosing other contour levels result in a workspace area larger than
16. However, for workspace that is more irregular shaped, choosing contour at level 0.5
will give us a more accurate result. Nonetheless, choosing contour at level 1.0 will
always give us a more conservative estimation of the workspace. The advantage of this
algorithm is that the computation is much faster than the previous algorithm as there is no
need for remove interior points as well as re-organizing of points.
A.2.1 Case Study: Circular Workspace
Workspaces of planar parallel manipulators are usually irregularly shaped. Hence
a circular workspace study will show us the accuracy of the above two algorithms under
curved boundary. Using a series of grid size of 0.01, 0.02, 0.05, 0.1, we mapped the
workspace of a manipulator that where its reachable workspace is a circle with radius 2,
centered at (0,0). The processes of both algorithms are shown graphically in Figure 102
and Figure 103, for grid size of 0.1 and 0.01; and the results for all cases are tabulated in
Table 9 and Table 10.
Grid Size = Computation Time
0.01
Boundary Points
Grid Size = Computation Time
0.02
Boundary Points
Grid Size = Computation Time
0.05
Boundary Points
Grid Size = Computation Time
0.1
Boundary Points
Algorithm 1
452.55 sec (442.62 sec filter, 9.93 sec sorting)
1128
28.69 sec (27.39 sec filter, 1.29 sec sorting)
564
5.95 sec (4.40 sec filter, 0.55 sec sorting)
224
0.071 sec (0.045 sec filter, 0.026 sec sorting)
112
Algorithm 2
0.036 sec
1605
0.0155 sec
805
0.0021 sec
325
9.5x10-4 sec
165
Table 9: Comparison of computation time and number of boundary points generated in each of
the two algorithms, for various grid size.
187
Grid Size =
0.01
Grid Size =
0.02
Grid Size =
0.05
Grid Size =
0.1
Computed Area
% Difference
Computed Area
% Difference
Computed Area
% Difference
Computed Area
% Difference
Algorithm 1
12.5062 (12.5664)
-4.8x10-1%
12.4528 (12.5664)
-0.9%
12.2750 (12.5664)
-2.32%
11.98 (12.5664)
-4.67%
Algorithm 2
12.5627 (12.5664)
-2.96x10-2%
12.5658 (12.5664)
-4.54x10-3%
12.5562 (12.5664)
-8.05x10-2%
12.5450 (12.5664)
-0.17%
Table 10: Comparison of computed are and % difference with actual area at various grid size, in
each of the two algorithms.
For Algorithm 2, it is obvious that in addition to grid size, the level at which the
contour is evaluated affect the result. For extreme cases, like the square workspace shown
previously, where the square workspace form by four straight lines is also aligned with
the grid, it is necessary to evaluate the contour at level 1.0 to get an accurate estimation
of the actual workspace. Nonetheless, the level at which the contour is evaluated affect
the accuracy of the estimation.
Grid size = 0.1
Algorithm 1
(a)
(b)
188
(c)
Grid size = 0.01
(d)
(e)
(f)
Figure 102: Process of computing area using Algorithm 1. (a) shows the original data points that
are inside the workspace; (b) the filtered and re-organized boundary data points; and (c) the area
computed using these data point using polyarea algorithm, for grid size 0.1. (d)-(f) shows the
same process for data points generated with grid size 0.01.
To study this effect, Figure 104 shows the % differences of the estimated
workspace with actual workspace, at various contour levels, for three different grid sizes
at 0.1, 0.05, and 0.01. From this figure, one can see that for grid size of 0.1, the %
difference could varied from 4.40% (over estimate) to -4.66% (underestimate). At finer
grid size of 0.01, the % difference varies much lesser, from 0.42% to -0.47%. On the
other hand, for both grid sizes, evaluate the workspace at contour level 0.5, give
us %difference of -0.17% (for grid size 0.1) and -0.029% (for grid size 0.01). This is
expected result as the contours are interpolated linearly. Hence, it is advice that one
should evaluate the contour at level 0.5 for workspace area estimation with this algorithm.
However, as grid size becomes finer, the effect of the selection of which level the contour
is evaluated diminished. Still, for more conservative estimation of a given workspace,
one should evaluated the contour at 1.0. Otherwise, evaluate the contour at 0.5 should be
used.
Algorithm 2
189
Grid size = 0.01
(b)
(c)
Grid size = 0.01
(a)
(e)
(f)
(g)
Figure 103: Process of computing area using Algorithm 2. (a) shows the surface plot of points
that are inside the workspace; (b) the level at which each contour can be generated, and
subsequently points can be extracted; and (c) the area computed using these data point using
polyarea algorithm, for grid size 0.1. (d)-(f) shows the same process for data points generated
with grid size 0.01.
190
Figure 104: Percent difference of estimating workspace area from actual workspace at different
contour levels (from 0 to 1.0, with increment of 0.1), for grid size of 0.1, 0.05, and 0.01. The
horizontal line represents perfect estimation.
In previous sections, we verified that both algorithms provide reasonably good
boundary points for subsequent workspace area computation. Algorithm 1 always gave
conservative area estimation, and when the grid size is relatively fine, the error is small.
However, it is not very computational efficiency. Algorithm 2 on the other hand, provides
an adjustable parameter (the level where the contour is to be evaluated), and compute
much faster. In this section, both algorithms will be used to compute the 2D workspace
area of the HD2 Haptic device, basically looking at a slice of the 3D volume workspace at
a particular level. We will look at constant orientation workspace of the HD2 device, at a
horizontal plane where both \theta_1 and \theta_1‟ is zero. Note that this is also the
191
largest attainable horizontal workspace area for the device. Due to the construction of the
HD2 device, there are physical constraints imposed on several joints. Through
measurements, these joint limits are listed in Table 7. By taking these physical limits into
account, using grid size of 0.2, 0.1, 0.05 and 0.02, we compute the workspace area of
HD2 with the two algorithms shown previously. The results are shown graphically in
Figure 105 (for Algorithm 1) and Figure 106 (for Algorithm 2, evaluated at contour level
0.5). With the finest grid size 0.02, the workspace area estimated using Algorithm 1 is
28.4462 unit^2, and is 28.697 unit^2 using Algorithm 2 evaluated at contour level 0.5. To
see how these two algorithms perform, the estimated areas using both algorithms are
plotted in Figure 107. Again, one can see that Algorithm 1 estimation is very close to
Algorithm 2 estimation evaluated at contour level 1.0 (28.4462 unit^2 compare to
28.4458 unit^2) – which is a more conservative estimation of the workspace area. Hence,
similar to the argument that we used in previous case study, a conservative estimation
(underestimate) of the constant orientation workspace of HD2 haptic device is 28.4458
unit^2, where the actual constant orientation workspace of the HD2 haptic device could
be closer to 28.697 unit^2.
Physical (Measured) Joint Limits on (HD)2
Maximum Angle Minimum Angle
1
2
2
3
4
4
Range
30 deg
-30 deg
60 deg
105 deg
-60 deg
165 deg
55 deg
Max is -50 deg, Depends on \theta2
Max is 105 deg
No limit
No limit
NA
0 deg
-180 deg
180 deg
No limit
No limit
NA
Table 11: Measured joint limits for the HD2 haptic device.
192
In this section, we compare two algorithms that numerically compute / estimate
planar workspace of a parallel manipulator. Since Algorithm 2 computes the workspace
much faster than Algorithm 1, in our subsequent work, we will use this algorithm to
estimate the planar workspace of a given parallel manipulator.
Original Data Points
Workspace Computation with Algorithm 1
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Filtered and Re-organized
Data
(a)
Figure 105: (a)-(e) Raw data points representing the constant orientation workspace of HD2
haptic device computed using Algorithm 1 at grid size 0.2, 0.1, 0.05 and 0.02; and (f)-(i)their
corresponding workspace evaluated using the filtered and re-organized boundary points.
193
Generated
Workspace
Algorithm 2
(b)
(c)
(d)
(e)
(f)
(f)
(g)
Contour evaluated at level 0.5
(a)
Figure 106: (a)-(d)Constant orientation workspace of HD2 haptic device computed using
Algorithm 2 at grid size 0.2, 0.1, 0.05 and 0.02; and (e)-(g) their corresponding workspace area
computed with contour evaluated at level 0.5.
194
Figure 107: Workspace area approximation with Algorithm 2 evaluated at various contour levels,
with grid size of 0.02, 0.05 and 0.1. Workspace area approximated with Algorithm 1 also plotted
for the same grid sizes for comparison.
Grid Size = Computation Time
0.02
Boundary Points
Grid Size = Computation Time
0.05
Boundary Points
Grid Size = Computation Time
0.1
Boundary Points
Grid Size = Computation Time
0.2
Boundary Points
Algorithm 1
162.563 sec (144.563 sec filter, 18.90 sec sorting)
1253
4.91 sec (3.71 sec filter, 1.20 sec sorting)
497
4.04x10-1 sec (0.237 sec filter, 0.167 sec sorting)
264
4.61x10-2 sec (0.045 sec filter, 0.026 sec sorting)
120
Algorithm 2
9.5x10-3 sec
1807
2.6x10-3 sec
717
1.8-3 sec
355
1.4x10-3 sec
175
Table 12: Comparison of computation time and number of boundary points generated in each of
the two algorithms, for various grid size used in estimating the constant orientation workspace of
HD2.
195
A.3 Workspace Computation Examples
While adding additional chains to form parallel system improves stiffness of the
system, it also changes the workspace of the system. Here, with a 2-RRR (shown in
Figure 108), 3-RRR (shown in Figure 109), and a 4-RRR (shown in Figure 111) planar
parallel system, we show graphically how that the constant orientation workspace reduces
with the addition of serial chain to the system. The location of the ground link as well as
the orientation of the end-effector changes the workspace as well (shown in Figure 110).
All these factors can be used as design variables if the design goal is to optimize the
workspace of a system.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 108: Constant orientation workspace of a 2-RRR planar manipulator changes as the
distance of their bases varied from (s) L
1.5 to (f) L
196
4.5 with end-effector angle held at zero.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 109: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
Largest workspace exist when L
0.5 to L
1.7321 .
197
4.5 with end-effector angle held at zero.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 110: Constant orientation workspace of a 3-RRR planar manipulator changes as the
distance of their bases varied from L
0.5 to L
Largest workspace no longer exist at L
1.7321 .
198
4.5 with end-effector angle held at
45 .
(a)
(b)
(c)
(d)
(e)
(f)
Figure 111: Constant orientation workspace of a 4-RRR planar manipulator changes as the
distance of their bases varied from L
2 to L
3.25 with end-effector angle held at zero.
A.4 Manipulability of Parallel Systems
Using Park and Kim work [119] as the benchmark, we compare our formulation
with their work using a five-bar system and a six-bar system and the results matched very
well. The five-bar system with two different actuation schemes is shown in Figure 112.
Similarly, the six-bar system with two actuation schemes is given in Figure 113.
199
Result from [119]
Both base joints are actuated Both 2
nd
joints are actuated
(a)
(b)
(c)
Figure 112: Manipulability ellipsoid of a Five-bar manipulator with two different actuated joints
compared with the result obtained from [119], at three different end effector locations (a), (b) and
(c).
200
Result from [119]
Both base joints and 2nd joint
of left chain actuated
Both base joints and 2nd
joints actuated
(a)
(b)
(c)
Figure 113: Manipulability ellipsoid of a Six-bar manipulator with two different actuation
schemes compared with the result obtained from [119], at three different end effector locations (a),
(b) and (c).
201
Figure 114 shows how the manipulability ellipsoid changes based on inverse
kinematic solution used in the analysis. It is therefore important to maintain a
configuration while evaluating isotropy index, as shown in Figure 115.
Same end-effector position and orientation w/ different inverse solution
Figure 114: For the same end-effector position and angle, manipulability ellipsoid can be different
due to inverse solution of each RRR chain.
202
Isotropy Index
(a)
(b)
(a)
(b)
Figure 115: Inverse of Condition Number map across the workspace of a 3RRR parallel
manipulator changes for different configurations.
203
On the other hand, manipulability ellipsoid also changes corresponding to the
configuration of the chain (for example, spacing between each chain), as shown in Figure
116.
Result from [1]
Both base joints are actuated
Both 2
nd
joints are actuated
Figure 116: Manipulability elliposoid changes as well as workspace as configuration changes.
204
A.5 Stiffness Computation
Stiffness Mapping of 3RPR Manipulator
Stiffness in  -direction
Stiffness in y-direction
0.5
0.45
1.1285
0.4
0.4
1.8309
1.2226
1.7394
98
80
02
00
0.35
0.35
1.3166
1.6478
1.4107
y-position
7
0.25
0.3
y-position
0.
5
19
0.000 56
0.3
7
0.
00
05
61
9
0.0
01
40
49
0.35
0.
00
08
42
9
88
52
9
02
0.0 1 966
0
0.0
49
40
01
0.0
0.0
0.0
02
01
52
96
88
69
0.5
0.45
0.4
y-position
Stiffness in x-direction
0.5
0.45
1.5047
1.5988
0.25
1.5563
0.3
1.4647
0.25
1.3732
1.6928
97
0.000 561
0.2
0.2
0.0011239
0.0014049
0.15
1.2816
0.2
1.7868
1.1901
1.8809
0.15
0.15
1.0986
0.0019669
0.0025288
0.1
0.05
0.25
0.1
0.3
0.35
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.7
0.05
0.25
0.75
0.1
0.3
0.35
0.4
0.45
0.5
x-position
(a)
0.55
0.6
0.65
0.7
0.05
0.25
0.75
0.3
0.35
0.4
(b)
Stiffness in x-direction
0.65
0.7
0.75
0.4
1.7
27
1
1.3
38
6
0.35
1.4
22
3
y-position
y-position
0.6
Stiffness in y-direction
1.2
55
0.35
0.3
1.5
05
9
1.6
40
8
1.5
54
4
0.3
1.4
68
1
1.5
89
6
1.6
73
3
0.25
0.25
0.2
1.3
81
7
1.2
95
4
0.2
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.15
0.35
0.4
(b)
Figure 117: Stiffness maps of the 3-RPR manipulator in the (a)
and (c) y –direction with
0.55
0.45
0.4
0.15
0.35
0.5
x-position
(c)
0.45
(a)
0.45
0.45
0.5
x-position
0.55
0.6
0.65
(c)
–direction; (b) x –direction;
0 , (d), (e), and (f) are stiffness map with end-effector angle
90 . Comparing result with comparing results with [143] (top row).
205
Stiffness Mapping of 3RRR Manipulator
Stiffness in x-direction
Stiffness in y-direction
0.5
0.5
22
35 .1
21
.95
12
16 .0
976
24
.87
8
17
.56
1
14 .63
14
.63
41
24
.87
8
41
17
.56
1
15 .80
65
18.0645
22 .58 06
20.3226
29 .35 48
0.2
29 .35
27.0968
33.871
0.15
0.3
15 .8
0 65
20
.3
22
6
27 .096
8
29.2683
7
70
.1
13
11.70 73
0.2
0.25
61
17 .5
0.25
13
.54
84
0.3
78
24 .8
14
.63
41
20
.3
22
6
0.35
65
.80
15
12
.95
2 44
21
19 .0
y-position
51
.19
32
19 .0
2 44
1
.56
17
0.35
18 .0
6 45
15.80 65
0.4
18
.0
64
5
8 78
20 .4
0.3
87
.83
24
24
.87
8
y-position
0.4
24 .838
22 .5807
6
0.45
27 .8
0 49
49
.80
27
21.9512
26 .3
4 15
0.45
48
0.15
0.35
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.7
0.3
0.35
0.4
0.45
(a)
0.5
x-position
0.55
0.6
0.65
0.7
(b)
Stiffness in x-direction
(c)
Stiffness in y-direction
0.45
0.45
0.4
0.4
27.0968
33.6585
36
.58
54
24
.83
87
y-position
0.3
5
64
.0
18
78
.48
20
49
.80
27
17 .5
61
22 .58
24
.87
20 8
.48
78
5 12
21 .9
16
.09
76
6 83
29 .2
0.25
06
.58
22
54
.58
36
21 .9
512
6
8
1 4 20 .48 7
.4
23
19.0244
0.3
24
.83
87
26
.32
20
0.35
8
.87
24
y-position
0.35
27 .8
0 49
24 .8
3 87
5
58
.6
33
20.3226
06
33 .8
71
0.25
27.0968
31 .61 29
36.129
40.6452
0.2
0.15
0.35
0.2
0.4
0.45
0.5
x-position
0.55
0.6
0.65
0.15
0.35
(d)
0.4
0.45
0.5
x-position
0.55
(e)
0.6
0.65
(f)
Figure 118: Stiffness maps of 3RRR manipulator in the (a) x-direction, (b) y- direction, and (c)
– direction, with end effector angle
angle
0 . (d), (e), and (f) are stiffness map with end-effector
90 . Comparing result with [153] (bottom row).
206
Stiffness Map of 3RRR, 3RRR, 3RRR Manipulator (End-effector angle = 45 deg)
(a)
(b)
(c)
Figure 119: Comparison of the stiffness map of a 3-RRR manipulator in x-direction (2nd row), ydirection (3rd row), and
-direction (4th row) with different joints actuated: (a) 3-RRR; (b) 3-
RRR; and (c) 3-RRR manipulator. End-effector angle
207
45 .
A.6 Workspace of HD2
Additional views of the planar workspace of HD2 device and spatial workspace
with tilted handle are shown in Figure 120 and Figure 121.
(a)
(b)
(c)
Figure 120: Constant orientation workspace of (HD)2 at various heights: (a) isometric view; (b)
side view; and (c) front view.
208
(a)
(b)
(c)
(d)
Figure 121: The constant orientation workspace of the HD2 haptice further reduced to about
0.51m3 when the handle is held at an rotated angle of 60 degree along the x-axi. Shown here are
two different view angles of the resulting workspace.
A.7 Inverse Kinematics of HD2
The upper chain is a spatial RRR manipulator (PHANToM 1.5 manipulator). The
position vector of frame {5} expressed in frame {1} , is given by:
209
1
P5,x
1
L2 cos
L3 sin(
2
1
P5
P5,y
2
3
)
0
1
L2 sin
P5,z
(97)
L3 cos(
2
2
3
)
The position of frame {5} , expressed in frame {F } , is given by:
F
P5
F
P5,x
1
F
P5,y
L1
F
P5,z
P5,x cos
P5,x sin
1
3
z0
1
F
z0)
(99)
P5,x
can be solved substituting Eqn. (97) into first and third row of Eqn. (98):
F
1
P5,x
P5,z
L2 cos
L2 sin
We can first solve for
2
P5,x / cos
( F P5,z
1
L2 sin
2
L3 sin(
2
L3 cos(
2
L3 sin(
F
)
3
2
2
P5,x / cos
1
,N
2
L3 cos(
2
L3 sin( 2
L3 cos( 2
F
1
L1 )
P5,x / cos
)
3
( F P5,z
)
F
(100)
1
L1 )
:
L2 cos
where M
(98)
is given by:
1
( F P5,z
and
P5,z
1
Hence, if the position vector F P5 is known,
2
1
1
)
3
)
3
( F P5,z
3
2
M
N
3
P5,x / cos
F
( P5,z
)
L2 cos
L2 sin
1
L1 )
(101)
2
2
L1 ) .
Square both side of Eqn. (101) and add both equations, we get:
A cos
2
B sin
210
2
C
0
(102)
where A
2ML2, B
M2
2NL2, C
N2
Hence, the two possible solution for
2
2 atan2
3
atan2 M
A2
(103)
:
B2
B
2
L23 . Using half-tan substitution:
B2 C 2
C A
B
t
L22
C2
A2 ,C
A
(104)
Using Eqn. (101):
Now we have shown that
1
,
L2 cos 2, N
2
and
3
L2 sin
2
(105)
2
can be solved once we know the position
of frame {5} expressed in frame {F } , F P5 . This can be determined from using the
position vector of frame {7} expressed in frame {F } , is given by:
F
F
P7
F
P5
R5 5P7
(106)
where F P7 is the end-effector position expressed in frame {F } , which is given as
F
P7
[xe
T
ze ] , and F R5
ye
F
R7 [ 5R7 ] 1 . F R7 is the orientation of the end-effector,
function of
6
.
6
as Euler angle), and [ 5R7 ]
, ,
which is a given quantity (with
1
7
R5 is only a
is a measured quantity, hence F P5 can then be compute from:
F
To compute
4
P5
F
P7
F
R7 7R5 5P7
(107)
, we look at 4P7 and 3P7 :
4
P7,x
4
P7
L7 sin
P7,y
4
P7,z
L6 cos
5
4
5
0
L7 cos
211
5
(108)
L6 sin
5
3
4
P7,x
3
P7,x cos
3
P7
P7,y
4
L3
3
(109)
4
P7,z
P7,x sin
From the first and third row of Eqn. (109),
4
can be obtained by:
4
atan2( 3P7,z , 3P7,x )
4
(110)
To compute the position vector 3P7 , we use the following relation:
F
F
P7
F
3
P7
where
F
R3
1
F
P3
R3
R3 3P7
1
F
F
P7
RF and F P3 is a function of
3
1
,
2
(111)
P3
and
3
only, which we already
solved. F P7 is the end-effector position expressed in frame {F } .
To solve for
5
, instead of using the position vector 4P7 shown in Eqn. (108), we
look at 4P6 :
4
P6,x
4
P6
L6 cos
4
0
4
L6 sin
P6,y
P6,z
From Eqn. (112),
5
5
(112)
5
can be obtained by:
atan2(
5
4
P6,z ,
4
P6,x )
(113)
So now we are left with determining the position vector 4P6 . We can write the
position vector F P6 as:
F
P6
F
P4
212
F
R4 4P6
(114)
where F P4 and F R4 is known since they are function of
1
,
2
and
3
only. We are left
to determine F P6 , which we can obtained from:
F
where F R6
F
R7 [ 6R7 ]
F
1
F
P7
R7 since [ 6R7 ]
F
F
P6
F
P6
R6 6P7
(115)
I . Hence, F P6 can be compute from:
1
F
P7
R7 6P7
(116)
As shown before, F P7 is the end-effector position expressed in frame {F } , which
is given as F P7
[xe
ye
T
ze ] , and F R5
F
R7 [ 5R7 ] 1 . F R7 is the orientation of the
as Euler angle), and vector 6P7 is
end-effector, which is a given quantity (with , ,
only a function of L7 only. Hence, the vector 4P6 :
F
4
P6
Once 4P6 is determined,
5
R4
1
F
F
P6
P4
(117)
can be obtained by Eqn. (113). Till now, we have
shown that once we know F P7 , is the end-effector position expressed in frame {F } ,
which is given as F P7
[xe
ye
T
ze ] , and F R7 , the orientation of the end-effector with
respect to frame {F } , which is a given quantity (with , ,
which is a measured quantity, we can determined
1
,
2
,
3
,
as Euler angles), and
4
, and
5
6
can only be measured, due to the loop constraint.
213
,
.
Note that this inverse solution is not complete, in the sense that we assume
known, whereas
6
6
is