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27( 00 61787
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This is to certify that the

dissertation entitled

DESIGN AND CONTROL OF CONTRAINED ROBOTIC SYSTEMS FOR
ENHANCED DEXTERITY AND MOBILITY

presented by

Mark Mew Minor

has been accepted towards fulfillment
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PhD. Mechanical Engineering

 

 

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DATE DUE | DATE DUE DATE DUE
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11/00 C‘JCiRCIDateDuth-p.“

Design and Control of Constrained Robotic Systems for
Enhanced Dexterity and Mobility

By

Mark Andrew Minor

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

2000

ABSTRACT

Design and Control of Constrained Robotic Mechanisms
for Enhanced Dexterity and Mobility

By

Mark Andrew Minor

With continued developments in the field of robotics, there has been considerable
research on manipulation, mobility, and control. Research in these areas has led to the
development of sophisticated mechanisms and systems with enhanced capabilities. While
innovations in actuator and sensor technology has significantly contributed towards
meeting new challenges, the design problems have become increasingly complex due to
greater number of constraints imposed by limitations in space, weight, power, and
computational capability. Innovative design and control is often the key to successful
implementation of these systems and the subject of this dissertation. Presented herein is
the design of three separate systems, each with its own unique challenges in dexterity,
mobility, and control. Specifically, we present the design of a dexterous mechanism for
minimally invasive surgery, a miniature climbing robot for reconnaissance operation, and

the control of a spherical robot which offers to provide maximum mobility and stability.

A surgical instrument was designed for minimally invasive surgery with optimized load
capacity and dexterity. The instrument fits through a standard 10 mm port and is
optimized for bi-directional 180° articulation and unlimited rotation of a pair of forceps at
the tip of the instrument. The state-of-the-art in surgical instrumentation does not have
similar capabilities. A miniature climbing-robot for inspection and reconnaissance in
urban environments is also developed. Similar to the surgical instrument design, which is
constrained by space limitations, the climbing-robot design is constrained by weight, size,
and power usage. A biped kinematic structure was chosen for the climbing-robot for its

inherent advantages. This kinematic structure allows the robot to traverse horizontal and

vertical surfaces, and transition between surfaces of different inclination. Similar to the
climbing-robot, the spherical robot is intended for reconnaissance missions, but its
control problem is more challenging due to the presence of nonholonomic constraints. A
unique coordinate system is defined on the sphere for the purpose of control — this results
in simple linear and curvilinear motion on the plane for independent control action.
These motions are exploited to design trajectories between initial and final configurations
of the sphere described by the two Cartesian coordinates of its point of contact and three
orientation coordinates. The solution of these trajectories are straightforward and require
minimal computation. Among the three problems considered, experimental work was
done with the climbing-robot. The robot was designed, built, and tested under computer

control.

Copyright by
Mark Andrew Minor
2000

To my loving wife Kristin...

...we finally get to move on to the next stage of life.

ACKNOWLEDGMENTS

I would like to take this opportunity to thank and acknowledge those who have helped me
complete this milestone in my life. It is impossible mention everyone since this has been
a lifelong process beginning in elementary school in and proceeding through my graduate
degrees. My parents, of course, come first since they have been so crucial to nurturing my
work etiquette, passion to help people, and mechanical inclinations. They have given me

so many advantages in life, I cannot thank them enough.

I must also state my appreciation to Northwestern Michigan College. The faculty at
NMC (Dick Minor, Jim Coughlin, Terry Sievert, Ray Niergarth, Jay Berry, Lesslie
Spencer, Jerry Williams, Bob Rudd, Adam Gahn, Ken Stepnitz, Erik Wildman, and Steve
Balance, to name a few) have played a pivotal role in helping me realize my passion for
both education and engineering. They provided me with an unsurpassed foundation in
math, science, machining, and manufacturing techniques that I use as a daily, and I would

have certainly taken a different path in life if not for these individuals.

At the University of Michigan, I developed my passion for design, modeling, and control
of dynamic systems. The teachings of Drs. Bruce Karnopp, Robert Keller, Noboru
Kikuchi and Alan Ward played significant roles in developing this passion and luring me

to graduate school, for which I am thankful.

Of course, I must thank Dr. Craig Somerton for attracting me to Michigan State
University. MSU has offered an excellent education and opportunities to refine my
teaching skills. For these teaching opportunities, I am thankful to Dr. Ronald Rosenberg.

He has provided me many opportunities at MSU and has been a very inspiring educator

vi

and advisor. Likewise I must thank Drs. Clarke Radcliffe, Steve Shaw, Brian Feeny,
Alejandro Diaz, Alan Haddow, Hassan Khalil, and Craig Gunn for their active roles in
helping me expand my understanding of dynamics systems, design methodologies, and
teaching. I must also extend thanks to my PhD advisor, Ranjan Mukherjee. He has
provided me with many great opportunities to work on very exciting research projects

and in turn develop the skills to profit on these with publications, patents, and proposals.

Lastly, I must thank all of my friends at MSU. The dynamic systems group, Brookes
Byam, Charles Birdsong, Joe DeRose, Joga Setiawan, Tuhin Das, Gary Gosciak, Brennan
Sicks, and Sally Starr have provided countless hours of stress relief, lunch time follies,
and stimulating discussions. Without their support, this would have been a much more
difficult process. I must also thank the Mechanical Engineering staff, which includes
Carol Bishop, Aida Rodriguez, Nancy Chism, Kathy Chomas, and Roy Bailiff, who have

always been there to help me and lend a hand in times of need.

More than anything, though, all of these fine people I have acknowledged have been like

family, and family is what makes life so enjoyable.

vii

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. xi
LIST OF FIGURES ........................................................................................................... xii
LIST OF ABBREVIATIONS .......................................................................................... xvi
1. Introduction ................................................................................................................. 1
1.1 A New Era ............................................................................................................... l
1.2 Objectives ................................................................................................................ 1
1.3 Document Structure ................................................................................................. 2

2. A Surgical Mechanism with Enhanced Dexterity and Reachability .......................... 4
2.1 Motivation ............................................................................................................... 4
2.2 Related Research ..................................................................................................... 6
2.3 Design of an Articulated Mechanism ...................................................................... 8
2.3.1 Kinematic Structure ......................................................................................... 8
2.3.2 Input-Output Force Relationship ..................................................................... 9
2.3.3 Optimization of the AMMIS Link Configuration ......................................... 14

2.4 Dexterous Articulated Linkage for Surgical Applications .................................... 17
2.4.1 Methodologies Explored ...................... 1 ........................................................ 17
2.4.2 Description of Functionality .......................................................................... 18
2.4.3 Force Analysis ............................................................................................... 20
2.4.4 Load Capacity Maximization ........................................................................ 24

2.5 Manufacturing, Performance, and Practical Considerations ................................. 27
2.6 Other Considerations ............................................................................................. 29

3. Design and Implementation of a Miniature Climbing Robot .................................... 30
3.1 Motivation ............................................................................................................. 3O

viii

3.2 Basic Structure ...................................................................................................... 31

3.3 Existing Wall Climbing Robots ............................................................................ 32
3.4 Design Constraints ................................................................................................ 35
3.5 Robot Kinematics .................................................................................................. 36
3.5.1 General Form ................................................................................................. 36
3.5.2 Performance Criteria ..................................................................................... 37
3.5.3 Joint Structure: The Simple Biped ............................................................... 38
3.5.4 Joint Structure: The Two-Legged Biped (revolute hip joint) ....................... 40
3.5.5 Joint Structure: The Two-Legged Biped (prismatic hip joint) ..................... 42
3.5.6 Joint Structure: the Two-Legged Biped (hybrid hip joint) ........................... 43
3.5.7 Selection of Kinematics Structure ................................................................. 47
3.5.8 Coupled DOF and the Two-Legged Biped ................................................... 49
3.6 Robot Design ......................................................................................................... 52
3.6.1 Biped Structure .............................................................................................. 52
3.6.2 Force Analysis: climbing a vertical surface supported by foot 1 ................. 57
3.6.3 Force Analysis: climbing a vertical surface supported by Foot 2 ................ 62
3.6.4 Force Analysis: Steering on a vertical surface while supported by Foot 2 .. 64
3.6.5 Actuator Selection Criteria ............................................................................ 65
3.6.6 Actuator Selection ......................................................................................... 69
3.6.7 Structural Optimization ................................................................................. 73
3.7 Testing ................................................................................................................... 78
3.8 Future Work .......................................................................................................... 80
3.9 Other Considerations ............................................................................................. 81
4. Control of a Spherical Robot for Enhanced Mobility ............................................... 83
4.1 Motivation ............................................................................................................. 83

ix

4.2 Kinematics Model ................................................................................................. 87

4.3 Reachability of Arbitrary Configurations ............................................................. 92
4.4 Kinematics Models in the Literature ..................................................................... 96
4.5 Traditional Nonholonomic Control Techniques ................................................... 99
4.6 Simple Geometric Motion Planning .................................................................... 101
4.6.1 Partial Reconfiguration Algorithm .............................................................. 102
4.6.2 Simulation of Partial Reconfiguration Algorithm ....................................... 104
4.6.3 Complete Reconfiguration Algorithm ......................................................... 106
4.6.4 Simulation of Complete Reconfiguration Algorithm .................................. 110
4.6.5 Reachability of All orientations .................................................................. 112

4.7 Other Comments ................................................................................................. 116

5. Concluding Remarks ............................................................................................... 118
Bibliography .................................................................................................................... 121
Appendix A ..................................................................................................................... 128
Appendix B ..................................................................................................................... 155

LIST OF TABLES

Table 1. Mmax for different values of n and Ur using long links (3:0, y=90 degrees) 10
Table 2. Mm. for configurations of common lengths using either short or long links. 13
Table 3. Mmax for configurations of equal length using permutations of .......................... 13
Table 4. Results of AMMIS link optimization, sorted in decreasing optimality. ............. 16
Table 5. Gear tooth forces and effective torque for a typical instrument configuration
(L=14.5mm, r3=2.0mm, D=6.35mm, LB=2.7mm, r—m>=10mm, Ftip=5N, Diametral
Pitch = 64). ................................................................................................................ 23
Table 6. Optimum gear thicknesses. ................................................................................. 27

Table 7. Data indicating space requirements and mobility of various robot joint
structures and strides. ................................................................................................ 47

Table 8. Weighted performance evaluations of the robot configurations operating in

open, confined, and mixed environments .................................................................. 48
Table 9. Maxon motor performance (10mm, .75w, plastic planetary gearbox). ............... 70
Table 10. Maxon motor performance (13mm, 1.2w, metal planetary gearbox). ............. 71
Table 11. Portescap motor performance (13mm, 2.5w, metal planetary gearbox). ......... 71
Table 12. Portescap motor performance (17mm, 3.2w, metal planetary gearbox). .......... 71

xi

LIST OF FIGURES

Figure l - AMMIS Kinematic Structure ............................................................................. 8
Figure 2 - Long and short link diagrams ........................................................................... 11
Figure 3. Reachable and Dexterous Workspaces ............................................................. 14
Figure 4. The DALSA Mechanism. .................................................................................. 18
Figure 5. Exploded view of DALSA linkage. ................................................................... 19
Figure 6. DALSA mechanism Free-Body-Diagrams. ...................................................... 20

Figure 7. The BIped Climbing Adaptable Robot (RAMRl) scaling a vertical surface. . 31

Figure 8. Space requirement typified by cross sectional area required for walking. ....... 37
Figure 9. The Simple Biped robot. ................................................................................... 38
Figure 10. Mobility of evaluation of the simple biped structure. ..................................... 39

Figure 11. Space requirements of the simple biped structure during flipping and
crawling strides. ........................................................................................................ 39

Figure 12. Simple biped and two-legged biped attempting to traverse a 270° comer. 40
Figure 13. Mobility evaluation of the revolute hip two legged biped. ............................. 41

Figure 14. Space requirements of the revolute hip two-legged biped during flipping and
crawling strides. ........................................................................................................ 41

Figure 15. Locomotion of a Two-Legged Biped robot using a prismatic hip joint. ........ 42
Figure 16. Mobility evaluation of the prismatic hip two-legged biped robot. ................. 43

Figure 17. Space requirements of the prismatic hip two-legged biped during flipping and
crawling strides. ........................................................................................................ 43

Figure 18. A hybrid revolute and prismatic joint shown assembled and in exploded view.44
Figure 19. The hybrid revolute and prismatic joint shown in several positions. ............. 45
Figure 20. Joint and link structure of the Two-Legged Biped with coupled DOF. ......... 50

Figure 21. Illustrated pictures of the Biped Climbing robot with a revolute hip joint ..... 52

xii

Figure 22. An exploded diagram of the Biped Climbing Robot with Smart Robotic Feet

(Dangi, Stam, et al. 2000). ........................................................................................ 53
Figure 23. Belt drive system coupling articulation of Ankle #1 and the hip joint. .......... 54
Figure 24. Ankle #1 and the differential. .......................................................................... 55
Figure 25. The worm drive at joint 4. .............................................................................. 56
Figure 26. Free body diagrams of the climbing robot supported by Foot #1 on a vertical

surface (some forces omitted for clarity). ................................................................. 58
Figure 27. Free body diagrams of the climbing robot supported by Foot #2 on a vertical

surface (some forces omitted for clarity). ................................................................. 61
Figure 28. Diagram of the climbing robot steering via Foot #1 on a vertical surface. 64
Figure 29. Diagram of robot Boundary Conditions used for Structural Analysis ............ 73
Figure 30. FEM analysis of Link 3 subjected to maximum loading. ............................... 75
Figure 31. FEM analysis of Link 4 subjected to maximum loading. ................................ 77
Figure 32. FEM analysis of Link 5 subjected to maximum loading. ................................ 77

Figure 33. Biped climbing a vertical surface. Start position and beginning of motion. .. 79

Figure 34. Biped climbing a vertical surface, halfway through step ................................. 79
Figure 35. Biped climbing a vertical surface. Near end of step and final position. ........ 79
Figure 36. Spherobot Mobile Robot ................................................................................. 84
Figure 37. Nonholonomic constraints: parallel parking. ................................................. 85
Figure 38. Spherobot Coordinate System. ....................................................................... 87
Figure 39. Spherobot Orientation Variables. ................................................................... 88
Figure 40. Partial Reconfiguration of the sphere. .......................................................... 102
Figure 41. Partial reconfiguration simulation: top View. .............................................. 104
Figure 42. Partial reconfiguration simulation: chart view ............................................. 105
Figure 43. Complete reconfiguration process, top view. ............................................... 107
Figure 44. Complete reconfiguration process, perspective view. .................................. 108

xiii

Figure 45. Complete reconfiguration simulation: top view. .......................................... 111

Figure 46. Complete reconfiguration simulation: chart view. ...................................... 112

Figure 47. Paths of point P as A05 varies.from Error! Objects cannot be created from
editing field codes. to Error! Objects cannot be created from editing field
codes ........................................................................................................................ 1 13

Figure 48. Net sphere re-orientation A¢G as a function of Error! Objects cannot be
created from editing field codes. for Error! Objects cannot be created from

editing field codes. and r = 0.25, 0.5, 1.0, 2.0, 5.0, 10.0. ....................................... 114
Figure 49. Counter Clock-Wise (CCW) and alternate Clock-Wise (CW) and paths for

complete reconfiguration ......................................................................................... 115
Figure 50. Assembled view of DALSA surgical mechanism (sheet 1). ........................ 129
Figure 51. Exploded view of DALSA surgical mechanism (sheet 2). ........................... 130
Figure 52. DALSA Component: Bevel Gear Pinion (sheet 3). ..................................... 131
Figure 53. DALSA Component: Bevel Gear (sheet 4). ................................................ 132
Figure 54. DALSA Component: Forceps (sheet 5). ...................................................... 133
Figure 55. DALSA Component: Spur Gear with 48DP and 12 teeth (sheet 6). ........... 134
Figure 56. DALSA Component: Base Link (sheet 7). .................................................. 135
Figure 57. DALSA Component: Drive Shaft (sheet 8). ................................................ 136
Figure 58. DALSA Component: Tip Link (sheet 9). .................................................... 137
Figure 59. DALSA Component: Tip Retainer Plate (sheet 10). ................................... 138
Figure 60. DALSA Component: drive link (sheet 11). ................................................. 139
Figure 61. DALSA Component: Idler Pulley (sheet 12) ............................................... 140
Figure 62. DALSA component: driveshaft retaining plate (sheet 13). ......................... 141
Figure 63. DALSA Component: Spur Gear 64DP, 16 Teeth (sheet 14). ...................... 142
Figure 64. DALSA Component: Pin (sheet 15). ........................................................... 143
Figure 65. DALSA Componenent: Alignment Pin (sheet 16). ..................................... 144
Figure 66. DALSA Component: Drag Link (sheet 17). ................................................ 145

xiv

Figure 67.
Figure 68.
Figure 69.
Figure 70.
Figure 71.
Figure 72.
Figure 73.
Figure 74.
Figure 75.
Figure 76.
Figure 77.
Figure 78.
Figure 79.
Figure 80.
Figure 81.
Figure 82.
Figure 83.
Figure 84.
Figure 85.
Figure 86.
Figure 87.
Figure 88.

Figure 89.

DALSA Component: Swivel (sheet 18). ..................................................... 146
DALSA Component: Spur Gear 64DP, 12 Tooth (sheet 19). ..................... 147
DALSA Component: Spur Gear 64DP, 16 Tooth, .07SID (sheet 20). ........ 148
DALSA Component: Forceps Jaw (sheet 21) .............................................. 149
DALSA Component: Tip Retainer Spacer (sheet 22). ................................ 150
DALSA Component: Drive Shaft Spacer (sheet 23). .................................. 151
DALSA Component: Bevel Gear Spacer (sheet 24). .................................. 152
DALSA Component: Drive Shaft Extension (sheet 25). ............................. 153
DALSA Component: Drive Shaft Coupler (sheet 26). ................................ 154
Exploded View of the RAMRl climbing robot. ........................................... 156
RAMRl Component: Differential Housing ................................................. 157
RAMRl Component: Bracket for Foot #1. ................................................. 158
RAMRl Component: Bracket for Motor #1. ............................................... 159
RAMRl Component: Link #3. .................................................................... 160
RAMRl Component: Link #5 ..................................................................... 161
RAMRl Component: Link 4 Bracket for Motors #2 and #3. ...................... 162
RAMRl Component: Link 4 Motor Bracket Coupling Plate ...................... 163
RAMRl Component: Pin for Joint #3. ........................................................ 164
RAMRl Component: Pin for Joint #4. ......................................................... 165
RAMRl Component: Link 3 Support Strut. ................................................ 166

RAMRl Component:
RAMRl Component:

RAMRl Component:

Final SRF Suction Block Design. ............................ 167
Early SRF Suction Block Design. ............................ 168

Modified Worm. ....................................................... 169

XV

10.
11.
12.
13.
14.
15.
l6.
17.

18.

LIST OF ABBREVIATIONS

. AMMIS — Articulated Manipulator for Minimally Invasive Surgery

CCW - Counter Clock Wise

CW — Clock Wise

DALSA - Dexterous Articulated Linkage for Surgical Applications
DOF - Degree Of Freedom

DP — Diametral Pitch of gear

. FBD — Free Body Diagram

FEA — Finite Element Analysis

MIS — Minimally Invasive Surgery

mNm — Torque unit of rrrilli Newton meter

MPC - Maximum Power Consumption of DC motors
OD - Outside Diameter

PA — Pressure Angle of gear tooth profile

PLC — Package Length Constraint of DC motor
RAMR — Reconfigurable Adaptable Micro Robot
SEA - Super Elastic Alloy

SLSF — Shaft Load Safety Factor of DC motor

TSF — Torque Safety Factor of DC motor

xvi

1. Introduction

1.1 A New Era

During the twentieth century, an era of robotics was born from dreams and necessity. In
their earliest conception, robots were imagined as humanoid devices that would serve
people and relieve them of laborious activities (Asimov 1940; Capek 1923). Despite this
creative view of robotics, the technology to implement them did not yet exist. It was
necessity that actually spurred the innovation leading to the first robotics system.
Designed to manipulate radioactive materials at Argonne National Laboratories during
World War H (Goertz and Thompson 1954), this first master-slave system integrated
mechanisms with parallel kinematics, DC motors, and analog controllers to create
position control and force reflection. This project helped motivate numerous researchers

to pursue dreams of robotic systems with improved dexterity, control, and repeatability.

1.2 Objectives

Expected productivity gains and improved work environments are primary motivations
for implementing robots in manufacturing. Robots have been able to achieve these
improvements because they are designed to perform a range of repetitive actions with
greater precision, repeatability, and speed than humans. These capabilities are the result
of several decades of research on manipulation, mobility, and control of robotic systems.
Further expansion of robotics is likewise dependent on continued research in these areas.
Since all robots are constrained by available space, weight, power, and computational
resources, achieving an optimal balance of these constraints and desired performance is a
significant challenge. This is especially true when new applications are becoming

increasingly constrained.

In this dissertation, the objective is to achieve new and innovative solutions in design and
control for state-of-the-art robotic applications where constraints on space, weight,
power, and computational ability are both substantial and critical. This is achieved
through the design and control of constrained robotic mechanisms for enhanced dexterity
and superior mobility. Several cutting-edge applications of robotics are especially

selected to emphasize these issues.

To best address dexterity in a constrained environment, a dexterous instrument for
Minimally Invasive Surgery (MIS) is designed to provide dexterity, reachability, and load
capacity not yet achieved by other instruments. Space constraints are again addressed in
conjunction with mobility in the design of a miniature climbing-robot for urban
environments. The resulting robot is the smallest of its kind to date and provides very
formidable design constraints on space, weight, and power usage. Mobility is further
examined from a controls approach in the design of a spherical reconnaissance robot.
Unlike the climbing robot, which possesses relatively straightforward kinematics, the
spherical robot has non-linear nonholonorrric rolling constraints, which prohibit the
application of traditional control techniques. Unlike existing techniques in the literature
that are not computationally effective, a very efficient approach is devised for planning
the motion of the spherical robot by exploiting fundamental motions created by a unique
coordinate system defined on the surface of the sphere. A summary of the document

structure follows.

1.3 Document Structure

The structure of the dissertation is now presented. Design and control of constrained
robotic mechanisms for enhanced dexterity and mobility is addressed via several state-of-

the-art applications of robotics. To best address dexterity and reachability in a

constrained environment, a dexterous instrument for Minimally Invasive Surgery (MIS)
is designed in Chapter 2. Space constraints are again addressed in conjunction with
enhanced mobility in the design of a miniature climbing-robot for urban environments in
Chapter 3. Mobility is further address from a controls approach in the design of a control
strategy for a spherical reconnaissance robot with extremely non-linear nonholonomic

rolling constraints in Chapter 4.

In each of the aforementioned chapters, a motivational statement and review of
specifically relevant literature is presented for each of these systems. The design process,
associated analysis, optimization, and proof of concept are also presented. At the end of
each chapter, a brief statement about the accomplishments in dexterity and mobility is
made. Final concluding remarks summarizing these accomplishments are presented in
Chapter 5. Appendix A contains design drawings for the surgical mechanism, Appendix
B contains design drawings for the climbing robot, and Appendix C contains details of

the load analysis conducted on the climbing robot in Chapter 3.

2. A Surgical Mechanism with Enhanced
Dexterity and Reachability

2.1 Motivation

Since the late 19805, minimally invasive surgery has gained widespread acceptance
because of the significant advantages realized by the patient in the form of reduced
trauma, recovery time, and procedural costs. Despite growing demand, however, fewer
new Minimally Invasive Surgical (MIS) procedures are being developed. This can be
attributed to the fact that the skill required to perform advanced procedures is beyond the
ability of most surgeons, given the current level of technology available. Dexterous

instruments can help reverse this trend.

Current state-of-the-art instruments in MIS provide single joint articulation of the tip up
to 90 degrees (Ethicon Endo Surgeries 1998; Pivotal Medical Innovations 1996). Despite
tip articulation, they are incapable of placing sutures with arbitrary orientation at surgical
sites. We propose to alleviate dexterity and reachability inadequacies of current MIS
instruments through the development of the Dexterous Articulated Linkage for Surgical

Applications (DALSA).

The DALSA mechanism is a geared serial link mechanism that can be coupled to form
either a manual or robotic instrument. Design of DALSA highlights the impact of space
and operational constraints on robotic systems that must achieve a desired workspace,
load capacity, and dexterity. DALSA can be appreciated in light of the following

stringent constraints imposed by MIS:

1. Instrument Size: A port on the patient body, which typically varies between 2 and
15mm diameter (Rosen and Carlton 1997 ; Schurr, Melzer, et al. 1993; Yuan, Lee, et al.
1997), determines the size of the mechanism. Smaller ports reduce scarring, but larger
ports allow instrument designs with greater load capacity and dexterity. The common

10mm port was selected for DALSA design.

2. Articulation and Tip Rotation: Difficult tasks in MIS, such as suturing, require that the
tip of the instrument approach tissue at arbitrary orientation and perform surgical tasks.
Given the four DOF inherent to a port, dexterity necessitates that two additional DOF are
provided. In DALSA, these two DOF are provided via 180° bi-directional tip articulation
and unlimited rotation about the articulated longitudinal axis. Articulation is divided

among several links, to provide encircling capability and improved reachability.

3. Load Capacity. The desired load capacity of the instrument is dictated by the intended
surgical procedures. For suturing, the instrument will drive a suture through tissue,
displace the tissue for alignment purposes, and pull the suture material for knot tying. The
pull-out strength of a suture in tissue varies between 2N and 15N (Bunt and Moore 1993;
Trowbridge, Lawford, et al. 1989). The driving force of a needle through soft tissue is
much less, on the order of 3N (Seki 1988). A driving force of 5N at the tip of a 1cm

needle was targeted for DALSA design.

4. Durability and Reliability. The necessity for reliable instruments during surgery is

obvious.

This chapter is organized as follows. In section 2.2, we exanrine state-of-the-art surgical
instrumentation. In section 2.3, we present and optimize the articulation component of

DALSA. In section 2.4, we integrate tip rotation and actuation with the articulated

linkage to complete DALSA design. In the same section we evaluate its performance and
optimize its structure. In section 2.5 we address manufacturability and cost
considerations. Final observations about the design of dexterous robotic mechanisms are

provided in section 2.6.

2.2 Related Research

MIS instruments are comprised of a slender rigid tube and an end-effector. The
instruments are inserted through a port in the human body that constrains the motion of
the end-effector by allowing it to move with four DOF. These DOF are insertion depth
and the three axis of rotation about the insertion point. Numerous approaches have been
adopted to provide greater freedom of movement to the end-effector. The simplest of
these consist of a single revolute joint supporting the end-effector. Such mechanisms are
typically actuated by push-pull rods and, or, four bar linkages (Faraz and Payandeh 1997;
Hassler, Murray, et al. 1994; Jaeger 1987; Wales, Paraschac, et al. 1997; Zvenyatsky,
Aranyi, et a1. 1995). A mechanism using bevel gears for articulation has also been
developed (Knoepfler 1993). Including the four DOF inherent to a port, these
mechanism allow the end-effector to have five DOF, which is inadequate for full

dexterity.

Other approaches to articulation use a serial chain of links with revolute joints.
Distributing articulation among several joints decreases sharp bends in the instrument and
is less harmful to cables or wires routed throughout the mechanism. One such device
(Huitema 1998) uses a flexible neck formed by five vertebrae and integral actuator bands
for steering. This device only provides one articulation DOF and is inadequate for full
dexterity. Other serial link devices with similar limitations use lead screws (Faraz and

Payandeh 1997) or five bar linkages (Guo, Qian, et al. 1992; Zhang, Guo, et al. 1992). A

steerable channel (Schurr, Melzer, et al. 1993), actuated by opposing tendons routed
along the channel, provides articulation in two directions as well as rotation of the end-
effector. Though the instrument provides the necessary DOF for complete dexterity, it
lacks sufficient rigidity. In an improved design, a chain of five four bar linkages was
used to develop a trunk mechanism with a central passage for end effector operation
(Mueglitz, Kunad, et al. 1993; Schurr, Melzer, et al. 1993). The trunk provides excellent
dexterity via 180° articulation and unlimited tip rotation for full dexterity. However, it
suffers from two drawbacks: the long chain of links creates large actuation forces
reducing controllability, and the Super Elastic Alloy (SEA) tube used for driving tip
rotation is unreliable when subjected to large cyclic strains in articulated configurations
(Plietsch 1997). A device using four serial links, AMMIS, implements planetary gearing
with unity ratio to drive successive links for i180° articulation distributed equally
amongst the links (Mukherjee, Minor, et al. 1998). Starting with the AMMIS structure,
DALSA adds additional DOF for full dexterity and optimizes the mechanical structure. It
provides the same dexterity as the trunk mechanism but has improved rigidity and
reduced force magnification. The gearing in DALSA introduces backlash which must

restricted during manufacturing.

Among other instruments, tendon actuated Stuart-platforms are used for end-effector
dexterity (Cohn, Crawford, et al. 1995; Schenker, Das, et al. 1995). The first of these
lacks repeatability and provides less than 90° articulation, which makes it unsuitable for
dexterous tip placement. The second device is larger (¢25mm) and uses a serial chain of
tendon actuated links supporting a wrist. This device provides high accuracy and
minimal backlash, but its large size prohibits MIS applications. A number of dexterous
mechanisms can be found in industrial applications, but it is doubtful if they can be
effectively miniaturized for MIS (Lande and David 1981; Milenkovic 1987; Rosheim
1987; Stanisic and Duta 1990; Susnjara and Fleck 1982).

2.3 Design of an Articulated Mechanism

2.3.1 Kinematic Structure

In this section, we discuss the articulation of DALSA, which is provided by the
Articulated Manipulator for Minimally Invasive Surgery (AMMIS) (Mukherjee, Minor,
et al. 1998). AMMIS is a serial concatenation of geared linkages where every link,
except the last, is both a driven-link and a driving-link. The last link is only a driven-
link. This idea is used to transfer power from a single actuator to simultaneously
articulate each link. The idea is realized by the four-link AMMIS in Figure 1, where the

driving gear is located at the base; Link-0.

Describing AMMIS in terms of planetary gearing terminology, it suffices to say that each
link in AMMIS functions as a carrier arm and as a sun. Link-l is a carrier for planetary

Gears 4 and 5, and its motion driven by Gear-1 produces rotation of the planets since

r Link 3 Gear 10
f / /—Gear 9
L g.) Q Q j; . /Gear 8
/1 Irv-arrayIlium-1"» 1"?"
1 II

/ //Gear12-j v" I '
5 Gear 13 Gear 11 ‘1'

Gear 14 Link 2

 

 

'Y Link 4 (end link) Link 1

5‘ 4‘1
Gear 1 (driving mad/fl

I‘.\
Link 0 (base link)\’/" \
\ f

Figure 1 - AMMIS Kinematic Structure

'11
“a

 

Gear 3, acting as a sun, is stationary. The successive link, Link-2, is then driven by the
rotation of planet Gear 5 and sun Gear-7 integral to Link-1. Each successive set of link
and gears functions similarly. Unity gear ratio between planets and suns, and even
number of planets, produces identical relative motion between each link. A net rotation

of 4B of the fourth link occurs for B rotation of Gear-1.

2.3.2 Input-Output Force Relationship

2.3.2.1 AMMIS with arbitrary articulation and load orientation

We study the input-output force relationship to determine the magnitude of the forces
required to drive AMMIS. If these forces are large, they will lead to failure of the
miniature components. In addition, actuation of the mechanism will not be

ergonomically suitable

Consider the free-body diagrams of the last three links of an n-link AMMIS for an
articulation angle B. Let F denote the tip force exerted by AMMIS acting at an arbitrary
angle 7. For i=1,2,..n, let P,, acting at an angle of 0:, denote the pin force at the point of
articulation and f,- denote the driving force of the i-th link. Ignore gear pressure angles
and assume common gear pitch radii, r. The static equilibrium conditions for the last

three links are,

P" cosan -Fcosy—fn sin/3 =0
P" sinan—Fsiny—fn cosfl=0 (2.1)
fur—FLsiny=0

Pn_l cos a“ — P" cos(,6 + an )- f,,_l sin ,3 = 0
1r>,,_l sin a“ — P" sin(fl + a, )— f,,_l cos 3 = 0 (2.2)
6Pnrsin(,B + an)— 4fnr - Hr = 0

P,,_2 cos arm2 — P,,_l cos(,6 + an_1)- f,,_2 sin [3 - fn sin ,6 = 0
Pit-2 Sin arr—2 — Pn—l Sin(fl + arr-l ) — fn-2 COS/3 - fn COS/3 = O
6Pn_lrsin(fl + arm1 )- f" r(6cos ,6 + 1) - 4fn_,r — fn_2r = 0

(2.3)

Equations (2.1) and (2.2) are used to solve for f,,, an, Pu, and then fH, a“, PH.

Subsequently Eq. (2.3) can be used recursively to solve for the unknown forces of the k-

th link, k=n-2, n-3, 2, I, in the following manner:

ft : —4fk+l +6Pr+1 Sin(.3 +ak+l )_ fk+2 (6COSIB +1)

a _ tan-1|:fk C05fl+PE+1Sin(fl+ak+r)_fk+2cosfl:|
k - . .
fr Slnfl‘I' Pk+l c05(16‘l’a'lm)“fr+2 Sin/3

Pk = iJsz + Pk2+l + fk2+2 + szPkH Sing/9 + am )" 2fkfk+2 COS Zfl — 2Pk+lfk+2 Sing/(+1

(2.4)

 

 

Force magnification in AMMIS is denoted by M = (f1 / F). Numerical simulations were

used to compute M for different values of ,6, y, n, and Ur ratio. The following

observations were made:

1. M is maximum at ,B=0 and 7:190“,
2. The maximum force magnification, Mmax, varies nonlinearly with n;
3. Mmax is independent of T, but linearly proportional to l/r.

The first and third observations are intuitive, but the second is not. To quantify the
nonlinear relationship between Mmax and n, we construct Table 1. From the data therein,
we deduce Mmax=3n2+2n when l/r=5 and Mm=3n2+3n when l/r=6. The nonlinear
relationship can be attributed to the fact that Mmax increases as the number of articulated
segments increase, even when the total length of AMMIS is constant. This is

investigated in the next section.

 

 

 

 

 

 

 

 

 

Table 1. Mmax for different values of n and Ur usin long links ([3=0, y=90 degrees)
n=2 n=3 n=4 N=5 n=6
Ur=5 16 33 56 85 120
1.11:6 18 36 6O 90 126

 

10

 

2.3.2.2 AMMIS with fixed length

To change the number of links of AMMIS without changing its total length, we introduce
the "short link", shown in Figure 2 adjacent to a normal link. The short link eliminates the
intermediate gears, maintains the direction of articulation, and reduces the effective
length from 6 to 2 pitch radii. Every normal link can be replaced with three short links
without changing the total length of AMMIS.

AMMIS with 2, 3, and 4 links were studied. All links, except the last, were replaced by
short links, resulting in 4, 7 and 10 link configurations. The last link was not replaced
since it should have sufficient length to support an end-effector. Force magnification for
these configurations can be seen in Table 2 with two variations of the last link; L=5r and
L=6r. The results indicate that Mm is linearly proportional to the number of links when
the total length of AMMIS is kept constant; irrespective of the last link length. In the
case where L=5r, Mmax=33 for 3 links and 77 for 7 links. For L=6r, Mm=60 for 4 links

and 150 for 10 links. This result is not intuitive.

Also consider force magnification when the normal links are partially replaced with short

links. This will determine whether Mmax depends upon the relative location of the short

fllntermediate gear locations

0 Ms W

 

 

 

 

 

 

 

 

 

 

——P.R.—--—P.R.—--—P.R.—- P.R.—--
f 6 PR. * RR. = Pitch Radius
samurai EMS—UM

Figure 2 - Long and short link diagrams

ll

links in the mechanism. We consider variations of an AMMIS initially with three normal
links and an end link. Nineteen variations are possible and are shown in Table 3, for
L=6r. The data, sorted in the order of increasing Mmax, reveals:

S

19. Mmax is the least for the four-link configuration, followed by the six-link
configurations, and then by eight and ten-link configurations.

20. Data in Table 2 indicates Mm is not quite proportional to the number of links - it
is dependent on the relative position of the short and normal links. For example,
for n=6, the ten variations in the link configuration produce Mm varying between
78 and 102.

21. For a given n, Mmax is lower for link configurations with normal links closer to the
base. The ten configurations for n=6 and the seven configurations for n=8
support this conclusion.

12

Table 2. Mia, for configurations of common lengths using either short or long links.

 

 

 

 

 

 

 

 

 

 

Link Configuration i”; £1116;
6r, L (n=2) 16 1 8
2r,2r,2r,L (n=4) 32 36
6r,6r,L (n=3) 33 36
2r,2r,2r,2r,2r,2r,L (n=7) 77 84
6r,6r,6r,L (n=4) 56 60
2r,2r,2r,2r,2r,2r,2r,2r,L (n=10) 140 150

 

Table 3. Mnrm for configurations of equal length using permutations of

short and long links.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . Mum
Lrnk Configuration 1: 6r
6r,6r,6r,L (n=4) 60
6r,6r,2r,2r,2r,L (n=6) 78
6r,2r,6r,2r,2r,L (n=6) 82
6r,2r,2r,6r,2r,L (n=6) 86
2r,6r,6r,2r,2r,L (n=6) 86
6r,2r,2r,2r,6r,L (n=6) 90
2r,6r,2r,6r,2r,L (n=6) 90
2r,6r,2r,2r,6r,L (n=6) 94
2r,2r,6r,6r,2r,L (n=6) 94
2r,2r,6r,2r,6r,L (n=6) 98
2r,2r,2r,6r,6r,L (n=6) 102
6r,2r,2r,2r,2r,2r,2r,L (n=8) 108
2r,6r,2r,2r,2r,2r,2r,L (n=8) 1 12
2r,2r,6r,2r,2r,2r,2r,L (n=8) 1 16
2r,2r,2r,6r,2r,2r,2r,L (n=8) 120
2r,2r,2r,2r,6r,2r,2r,L (n=8) 124
2r,2r,2r,2r,2r,6r,2r,L (n=8) 128
2r,2r,2r,2r,2r,2r,6r,L (n=8) 132
2r,2r,2r,2r,2r,2r,2r,2r,2r,L (n=10) 150

 

 

13

 

2.3.3 Optimization of the AMMIS Link Configuration

We select several criteria for optimizing the AMMIS link configuration using the
perspectives of a design engineer and a surgeon. These criteria are quantified using cost

functions (Arora 1989) described below:

1. Workspace and Dexterous Workspace: Workspace is the region that the end-
effector can reach and dexterous workspace is the sub-space that can be reached with
all orientations (Craig 1989). An MIS instrument should have a large and dexterous
workspace. For simplicity, we consider lo, 6, and B in Figure 3 to compute both
workspaces by area instead of volume. Examination of Figure 3 and the restriction
imposed on B obviates that the tip will not be able to reach any point in the workspace

with all orientations. For this reason the definition of dexterous workspace is relaxed

 

 

 

 

 

 

ALI-Y (mm)
1 20 — I I I I I \l I l I I I _
‘I 00 — ‘
80 F ‘
80 r “
4O
9
20 r ‘
-1 oo -80 ~40 -2o 20 4o 60 so 1 00 +me)
10
Workspace Constraints
-7: II
R habl W ksp A so 5 A
\S§§§§§§ eac e or ace _ 1|: 1!
vwvv v.v O A S B S A
’ .’::': :o’o°::: Dexterous- Reachable Works ace 0 5 I S 80
0.3:: ’0’:’o’o’:‘« P 0

Figure 3. Reachable and Dexterous Workspaces

14

to include regions that provide 2700 range of end-effector orientation. Reachable and
dexterous workspaces are denoted by AT and Ad, respectively; and their cost functions

are C1=Ar and C2=Ad. A similar criteria was used by (Faraz and Payandeh 1997).

. Force Magnification: The negative effects of large force magnification were
outlined in the previous section. The cost function for force magnification will be

C3=Mmax.

. Backlash: Backlash occurs in mechanisms due to clearance allowances that assure

smooth operation. It can degrade tip placement, accuracy, and repeatability.

Backlash depends on the number of meshing gears and for AMMIS the cost function

will be C4 =ns +312", where n, and nn are the number of short and long links,

respectively.

. Average tip path curvature: A dexterous surgical instrument is expected to
articulate with a small radius of curvature. The average radius of curvature of the

path trace by the tip is therefor a good indicator of dexterity. It can be expressed as

7: 0

where n is the number of links, p is the instantaneous tip path curvature, and B is the

relative articulation between successive links. The cost function is chosen as C5 = p.

. Maximum articulation angle per link: Fewer links will require greater articulation

per link for a net 180° articulation and adversely increases stresses in cables routed

along the mechanism. The cost function is represented by C6 = filmx = 7r / n.

15

We limit our search among configurations with two, three, and four links where the last
link is a normal link supporting a surgical tool. The optimal configuration is obtained by
finding the minimum weighted sum of the normalized cost functions. The choice of
weights depend upon the desired characteristics of the surgical instrument, but are

somewhat subjective. The weighted sum is described as
CF = —w,C,” — wZCzN + W3C3N + w4C,” + WSCSN + W6C: (2.6)

where w, and Cl." , i=1,...,6 are the weights and normalized cost functions, respectively,

corresponding to the elementary cost function Ci, i=1,...,6. The idea behind the
optimization process is to choose the link configuration that has large values of C,” and

C 5’ , and small values of C3” , Cf , C5” and C: . For the task of suturing, the weights are

chosen as
WI :80 W2 :60 W3 :40 w4 =3.0 w5 =1.0 W6 :10 (2.7)
The maximum weights were assigned to dexterous workspace and force magnification.

Table 4. Results of AMMIS link optimization, sorted in decreasin optimality.

Link Individual Cost functions

Configuration C,” Cf C 3" Cf' C.” C: Net Cost
6r,2r,L 0.30 0.24 0.18 0.20 0.21 0.28 -1.98
2r,L 0.25 0.17 0.09 0.05 0.15 0.42 -1.93
6r,2r,2r,L 0.32 0.26 0.26 0.25 0.23 0.21 -1.92
2r,2r,L 0.26 0.19 0.16 0.10 0.18 0.28 -1.83
6r,L 0.26 0.22 0.12 0.15 0.21 0.42 -1.79
6r,6r,2r,L 0.33 0.31 0.32 0.35 0.30 0.21 -1.65
2r,2r,2r,L 0.26 0.22 0.24 0.15 0.20 0.21 -1.57
6r,6r,L 0.27 0.29 0.24 0.30 0.30 0.28 -1.46
2r,6r,L 0.23 0.24 0.21 0.20 0.26 0.28 -l.30
2r,6r,2r,L 0.26 0.26 0.29 0.25 0.27 0.21 -1.30
6r,2r,6r,L 0.27 0.32 0.34 0.35 0.31 0.21 -1 . 14
6r,6r,6r,L 0.27 0.36 0.39 0.45 0.38 0.21 -0.86
2r,2r,6r,L 0.22 0.27 0.32 0.25 0.29 0.21 -0.82
2r,6r,6r,L 0.22 0.31 0.37 0.35 0.35 0.21 -0.55

 

 

Rank

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

335:8xoooxlmm-bwmu—

 

l6

Average tip path curvature had the least weight since suturing is typically performed in a
small region without gross steering. Optimization results are shown in Table 4, with data
sorted in the order of decreasing optimality. The optimum AMMIS is comprised of three
links among which the first and second links are normal and short links, respectively. In
terms of actual (not normalized) data, this configuration has a workspace of 206cm2,
dexterous workspace of 72cm2, force magnification of 28, backlash resulting from 4
mating gears, average path curvature of 29.4mm, and 60° maximum articulation angle

per link.

2.4 Dexterous Articulated Linkage for Surgical Applications

2.4.1 Methodologies Explored

Tip rotation as well as articulation is necessary to position a suture needle in arbitrary
orientations. Towards this objective, several methods for tip rotation in articulated
configurations were explored (Minor and Mukherjee 1999). These include flexible
shafts, miniature motors, hydraulic systems, tendons, and geared mechanisms. Flexible
shafts encounter large cyclic stresses: a straight wire core wrapped with multiple layers
of wires is too compliant and SEA tubes are prone to failure. Miniature motors of 3 and
5mm diameter are available, but their length and load capacity prohibits their usage.
Hydraulic actuators were discounted because of leakage potential, additional hardware
requirements, and control complexity. Tendon actuation was considered, but not used

due to large actuation and pre-load forces.

Gearing was examined in two scenarios. In the first, a geared reduction unit at the tip of
the instrument would reduce the impact of shaft compliance. While load capacity could
be increased substantially, the effects of compliance were still appreciable. Alternatively,

by placing a third layer of gears along the AMMIS linkage, the exiting compact structure

17

 

Figure 4. The DALSA Mechanism.

could also be used to transmit power to the tip. Viewed as the best alternative, this
method uses very little additional space and allows routing of cables and accessories to
the end—effector. The serial chain of gears introduces some backlash, but that can be

minimized via tolerance control.

2.4.2 Description of Functionality

To describe our approach for tip rotation, refer to Figure 4. As shown in this figure,
DALSA DOF include articulation A, and A2, tip rotation B, and end-effector actuation,
C. Power for these DOF are transmitted through tendons a1 and a2, a rotary shaft b, and a
push—pull cable c, respectively. An exploded view of DALSA is shown in Figure 5.
Rotation of the forceps at the tip of the instrument is accomplished via rotation of the
drive shaft 17, which drives bevel gears 6 and 5. Bevel gear 5 is coupled to Gear 35 and
drives gears 34 through 28. This ultimately drives bevel gear 4 and forceps 9. Actuation
of the forceps is achieved by push-pull cable 1. Push-pull actuation of the cable pivots
the upper jaw about the axis of screw 8. The cable rotates with the forceps and swivels
freely via the cylinder 43. Articulation is provided by tendons 23 and 24, which wrap
around the idler pulley 22 and around the end of the drive-link 21. They then pass

through the holes in the drive-link where they are restrained. Tension applied to tendon

23 displaces the tendon and hence rotates the drive-link about the axis of pin 41 in a

clockwise direction.

Wrapping the tendon about the pulley and then the end of the drive-link results in a
constant linear relationship between tendon displacement and link rotation, which aids
controllability. Identical to the AMMIS actuation, rotation of the drive-link 21 by some
angle B causes the subsequent links to articulate by angle B relative to the previous link.
Since this is a three-link mechanism, rotation B of the drive-link results in 3B net
articulation of the tip-link. In an actual surgical instrument, the base-link 16 will be
supported by a 10mm diameter tube, which will be coupled to a handle or motor drive at
the exterior end. The tube will be aligned with the base-link by pin 42 and retained by

the screws 36.

 

Figure 5. Exploded View of DALSA linkage.

l9

2.4.3 Force Analysis

In addition to the forces discussed in Section 2.3.2, we consider the forces exerted by the
bevel gears and additional gear pack for tip rotation. We examine DALSA in a straight
configuration for the sake of simplicity, which is justifiable in light of the results of
Section 3.2. Considering the free-body-diagrams of DALSA, in Figure 6, adopt the

following notation,

Raj force acting between the i‘11 pin and the ju‘ gear
B” force acting between the ith pin and the jm link
G. force acting on the i‘” gear tooth.

I

The load, F up, is applied to a needle tip with radius rnp orientated at angle 0 relative to

the horizontal plane of the mechanism. The bevel gear force Br and the torque TB

 

A. Tip and Needle M. Pin 7
B. Bevel Gear N. Pin 6
C. Tip Link (link 3) 0. Pin 5
D. Drag Link (link 2) P. Pin 4
E. Drive Link (link 1)

F. Gear 8

G. Gear 7

H. Gear 6

1. Gear 5

J. Gear 4

K. Gear 12

L. Gear 13

Figure 6. DALSA mechanism Free-Body-Diagrams.

20

balance the load. Taking moments about the x-axis of the Tip (labeled A), we have,

3, = 'T—”’ Fm, (2.8)
r a
where r3 is the bevel gear pitch radius. We can determine the torque TB acting on the

Bevel Gear (labeled B) as,
T8 = Br r3 : rTlP FTIP (2'9)

Neglecting axial and radial separation forces created by the bevel gears, it can be shown

that the relevant forces supporting the tip are described by F y and T z where,

F), = FT,P(sin(9)+:TE-], T2 = FT,P[Lsin(0) — LB 2i] (2.10)

r3 rs
and L and LB are defined in Figure 6. Neglect the contribution of gear pressure angle
effects and bending moment and assume that D denotes the pitch diameter of all spur

gears, except for gear 8, which is 1%6D. The bevel gear is coupled to Gear 8 (labeled F)

and the torque T3 produces the reaction forces G9 and 13,68 described by,

Gs = P868 22512:}?

Gear 8 acts on Gear 7 (labeled G), producing reactions along the lower gear-pack

(labeled H, I, J) of the mechanism. This results in forces G,- and P,“ for i=1...7.

Neglecting friction and pressure angles, these forces can be determined as

§r_F_ Po.- =EEEFEE

G.— ,,.
3D

— 2.11
.30 ()

21

Considering the previously described forces acting on the Tip Link (labeled C), we can

show that the forces 09 and P7"3 acting on the Tip Link are,

 

F 11 . 4
09 :2 3P[[L+LB+rB+§D)81n(0)-§rm,] (2.12)
2F
P7“ =__DT£_IIL+LB +rB +—18—1D]sin(6)—§rm] (2.13)

To determine the forces on the Drag Link (labeled D), we balance the forces on Pin 7

(labeled M) yielding, '

P7L2 = P7L3 +P7G7

2.14)
:2-€%&[[L+LB +r3 +_18—1'D]Sin(6)+_.1;rTIP:I (

Applying force Pf2 to the Drag Link (labeled D), we obtain the driving and support

forces Gu and If.” as,

 

011:4 1:3” [L+LB+rB+%D)sin(6) (2.15)
L2 FTIP 11 -
P6 =6—B— L+Ls+ra+§D srn(0) (2.16)

The force R,“ acting on Pin 6 (labeled N) and the Drive Link (labeled E) is

Pa“ : [2L2 _ PGGG = 6%[(L+ LB + r3 +lgl-D)Sin(6)-i—:- r1119] (2.17)

Since Gll = G12 = G13 , we can show that the forces Pfl and P:1 acting on the Drive Link
are,

F
P4“ = P5“ = P5012 —P565 =8—;1[[L+LB +rB +%D)sin(9)—§rm] (2.18)

22

Balancing moments applied to the Drive Link about the axis of TIN, the driving torque is

obtained as,

TIN : -3‘5DGIO +3DI’6Ll _2DI)5L1+ 0P4“

(2.19)
: FTIP|:[3L+3LB 'I' 37'}; +%D)Sin(0) _§_ rTIP:|

Forces pertinent to gear design are Gg, Gg, G; 1, and TIN, and are studied at 6 = —%

where the maximum amplitudes occur. Table 5 shows the maximum of these forces and
their equivalent torque for typical system parameters. As expected, driving forces are
larger near the base of the instrument. The tooth force representing TIN is fictitious since
the drive link does not have gear teeth on its driven end, but that force is shown for
comparison to the force magnification, Mm, introduced in Section 2.3.2.1. Recalling the

definition Mmax=(fI/F), here we have,

Mmax = fl = 37.6 (2.20)
5N

which is larger than Mmax = 28, obtained previously. This can be attributed to the

increased tip length resulting from the forceps and gearing for tip rotation.

Table 5. Gear tooth forces and effective torque for a typical instrument configuration
(L=14.5mm, rB=2.0mm, D=6.35mm, LB=2.7mm, r1~1p=10mm,
Ftip=5N, Diametral Pitch = 64).

 

 

 

 

 

Force Tangential Effective Initial
Component Tooth Force Torque Thickness (mm)
03 21.0 N 66.7 N-mm 0.97 mm
69 -60.0 N -190 N-mm 2.76 m
G” -87.9 N -279 N-mm 4.05 mm
TIN -188 N ~597 N-mm 8.66 mm

 

 

 

 

 

 

23

2.4.4 Load Capacity Maximization

Decreased force magnification and optimized gear designs maximize load capacity.
Reducing tip-link and forceps length decreases force magnification to the level
mentioned above. Efforts to accomplish this include a compact forceps with integrated
upper bevel gear, as shown in Figure 6 (labeled A), and a smaller diameter for Gear 8,
labeled F. Nonetheless, force magnification is excessive and the gear designs must be

optimized to maximize load capacity.

In order to understand how gear design parameters affect tooth capacity, we examine
stress characteristics of an involute tooth profile. Based upon AGMA stress formulae,
(Shigley and Mischke 1989) the maximum bending stress, 0, occurring at the root of the
tooth is described in US units by,

a = Miflfi’fa = 3.71% (2.21)
K, F J F

where W, is the load, Pd is the Diametral Pitch (DP), F (tooth thickness), , Ka=1.0
(application factor), Kv=0.95 (dynamic loading factor), KS=I.0 (size factor), Km=I.3
(load distribution factor), and J=0.22 (geometry factor). To determine tooth capacity,

we must first calculate allowable bending stress, which is determined by,

(2.22)

 

where S, is the endurable bending strength, KL is the life factor, K1 is the temperature
factor, and KR is the reliability factor. We use high strength UNS 842000 stainless steel
tempered at 400°F with an ultimate strength of 250ksi (ASM International Handbook
Committee 1996) which has an endurance strength of Se=69ksi, assuming 107 cycles

(Shigley and Mischke 1989). Based on Eq. (2.22), the allowable stress is equivalent to

24

the endurance limit when we use KL=1.0 (107 load cycles), KT=I.0 (operating
environments up to 250°F), and K,=I.0 (reliability factor 0.99). Solving Eq.(2.21) for

tooth load, W,, and using a = 69ksi we have,

W, s 7924; (2.23)

d
Using the original AMMIS tooth profile, 64 Diametral Pitch (DP) and 20° Pressure
Angle (PA), the load capacity is W,=21.7F (N per mm tooth thickness, F). Tooth
thickness is calculated to support the corresponding loads, both shown in Table 5. Notice
that the face width for T [N is much larger than that required by 09 at the opposite end of
the same link. Application of a tendon drive at Tm eliminates the need for the thicker link
at TIN and permits more effective use of the space. Tensile forces on the tendons
approach 199N, which is safely within the limits of a 1mm Vectran (Cortland Cable

Company 1997) cable with a 800N capacity.

As equation (2.23) indicates, gear load capacity is determined by thickness, F, and
diametral pitch, Pd, which are both limited. The instrument must be encased in a flexible
tube, passing through a 10mm port, and hence the net gear thickness is limited by the
9mm annular space. Considering the major diameter of the gear profile to be 7.14mm
(corresponding to a 0.25in pitch diameter), the maximum gear pack thickness fitting
within the tube is 5.5mm, which is insufficient for the net 7.78mm thickness of Ga, 09,

and G“ indicated by Table 5.

The net 5.5mm thickness must be distributed such that load capacity is maximized. Prior
to optimizing the thickness of the gears, we modify the tooth PA and DP to maximize
capacity. A smaller DP implies larger teeth and increased load capacity whereas PA

determines the minimum DP a gear of a particular diameter may possess. Standard 20°

25

and 25° PA prefer at least 18 and 12 teeth, respectively. Our largest feasible tooth profile
with 48 DP and 25° PA has a tooth capacity of 28.9 (N/mm thickness) as compared to
21.7 (N/mm thickness) for a smaller 64 DP 25° PA tooth, as seen from Eq.(2.23). The
former profile is used to strengthen the top two layers of DALSA, which is where the
largest forces occur. The later is used in the bottom layer, which supports less load, to
provide a smaller 12-tooth gear on the tip link (Figure 6, labeled F) to decrease tip length

and force magnification.

Based on the above profiles, we distribute the 5.5mm thickness of material amongst the

three gear layers optimally using standard gradient based techniques (Arora 1989).
Given 6 = —% , the gear life parameters previously stated, and the linkage configuration

as described, our optimization problem is to:

Find {t1 t2 :3} that maximizes {Fm} subject to,

ELIM— s 0 where a, = 1486M = 2340-174
am, Dt, t,
M4130 where 02:83 LB+rB+L+lD+firnP —"5’--=5012ffl
amax 8 3 Dt2 r2
MSG where 0'3 =1672[LB+rB+L+%D)=73SOEE
O'max I3

11 H2 +t3 35.5

where am = 69000 , and t1, t2, and t3 are the bottom through top gear pack thicknesses.

Matlab results of the optimization are shown in Table 6. Using the optimal values, we
are assured that in the worst case scenario the maximum stresses occur simultaneously in

all three layers. In this case, the maximum infinite-life load supported by the tip of a 1cm

26

Table 6. Optimum gear thicknesses.

 

 

 

 

 

 

 

 

Variable Items Effected Optimum
(Figure 6) Value

t1 F,G,H,I,J 0.88 mm

12 C, E 1.88 mm

t3 D, K, L 2.75 mm

 

suture needle is evaluated to be 4.54 N. If an infinite life is not required, the instrument

can support intermittent loads approaching 8.7N.

We must ensure that the bevel gears are also capable of supporting the loads in the worse
case. From Eq.(2.8) , the tangential load, Br, transmitted by the bevel gears is 23N for a
4.5N needle load and 44N for the intermittent 8.7N needle load. Based on bevel gear
strength formulae found in the Machinery’s Handbook (Oberg and Jones 1973), the

maximum tangential tooth load, W,, in pounds is,

W = M (2.24)

’ KM P075
where 0;,” = 69ksi is the maximum allowable bending stress, F =0.037in is the tooth
width, J: 0.15 is the geometry factor, KM=I.0 is the mounting factor, and P=64 is the
DP. W, is computed to be approximately 17 pounds (575N), which is sufficient to

provide a safety factor of 1.7 even for the 8.7N needle load.

2.5 Manufacturing, Performance, and Practical Considerations

DALSA is made of hardened stainless steels that possess high strength and corrosion
resistance, but that are difficult and expensive to machine. To achieve high precision in
gear profiles, a gear is typically hobbed, heat treated, and ground. This eliminates
residual heat treatment deformation and assures a precise tooth profile. Due to the small

size of the gears and unusual geared links, precision grinding is quite difficult. This

27

leaves the machinist with the alternatives of shaping hardened materials or using methods

such as wire EDM and lapping.

Stainless steel gears of like materials and equal hardness also present difficulties during
operation. Specifically, micro-welding occurs at the pitch line when the gears mate under
high pressure. During micro-welding, small amounts of material are typically removed,
resulting in pitting. Pitting decreases smoothness of operation and induces premature
tooth failure. To eliminate these problems, our design alternates material types (UNS

842000 and UNS 844003) and/or hardness between mating gears and pins.

Due to the number of meshing gears in DALSA, great care is paid to backlash
minimization. If backlash is poor, accuracy and repeatability of instrument tip placement
will suffer. Backlash is reduced by precise control of manufacturing tolerances, which
increase costs. Towards this end, clearance providing a slip fit between the gears, links,
and pins is allowed to vary between 0.0001” and 0.0007”. Gear shapes are controlled to
AGMA 12 standards with class B backlash allowance. As such, the backlash per mating
gear is intended to be less than 0.0005” at the pitch diameter. Considering these
allowances, we anticipate backlash of the articulation DOF at the tip to be 0.017”
(0.4mm). Allowing 0.001” backlash in the bevel gears and considering backlash in the
gear train, the backlash at the needle tip due to rotation is 0.028” (0.7mm). The
composite backlash for articulation and rotation at the needle tip is 0.036” (0.9mm).
Needless to state, backlash will be further diminished by friction between mating

components and compliance of the flexible tube covering the mechanism.

28

2.6 Other Considerations

A key to advancing and improving the application of robotics is the design of
mechanisms that provide sufficient dexterity, workspace, and load capacity in lieu of
constraints on space and operability. This has been addressed via the design of a robotic
arm for enhanced dexterity in minimally invasive surgery. It was found that innovative
kinematics structures can assist in providing workspace, but engineering optimization is
required to achieve the configuration that provides satisfactory performance. Additional
DOF were also required to achieve necessary dexterity in placing the end-effector. In
this case, dexterity was accomplished by augmenting the articulation structure with an
additional gear arrangement. It was also found that this augmented structure required
further optimization to achieve necessary load capacity. The augmented structure also
resulted in coupled degrees of freedom where articulation also results in tip rotation,
unless compensated. Compensation is not a problem since the kinematic structure is well
known and proper control can counteract coupled DOF. The resulting instrument,
DALSA, can thus be integrated as part of a surgical robotics system or to form a manual
instrument. In summary, while innovative kinematics are extremely beneficial for
achieving dexterity in lieu of constraints, engineering practice is mandatory to assure that
proper load capacity, dexterity, workspace, and durability can be achieved. In the next

chapter, mobility is also considered.

29

3. Design and Implementation of a Miniature Climbing
Robot

3.1 Motivation

In this chapter we address the problem of designing an autonomous climbing robot that is
a fraction of the size of any existing climbing robot in the literature. The robot must be
sufficiently small to travel through confined spaces, such as ventilation ducts, and to
minimize detection while traveling along the outside of a building. These environments
require that the robot travel on surfaces with varying inclinations, such as floors, walls,
and ceilings, and be able to traverse between such surfaces. Thus, the robot must be
capable of adapting and reconfiguring for various environments and modes of operation,
be self-contained, and be capable of carrying wireless sensors, such as a camera or
microphone and their transmitters. The purpose of deploying such a robot would be for
inspection, isolating the source of a biological hazard, or for gathering information about

a hostile situation within a building (T ummala, Mukherjee, et al. 1999).

All of the above requirements pose a challenging problem in robot design. The system is
substantially constrained by available space, weight, and power, and yet the robot must
provide sufficient mobility to climb and traverse between a variety of surfaces. This
problem is similar to the surgical mechanism of the previous chapter in that space
constraints are crucial; both devices must pass through and operate in confined
environments. The climbing robot, however, encounters weight and power limitations
not considered during the design of the surgical instrument. Consequently, a

conservative approach must be used for minimizing weight and maximizing mobility,

3O

which is just the opposite of the surgical instrument where increased weight usually

implies greater strength and load capacity.

3.2 Basic Structure

In response to aforementioned requirements, a biped Reconfigurable Adaptable Miniature
Robot (RAMRl) has been designed, constructed, and evaluated [Dulimarta, 2000 #33;
Minor, 2000 #30. During this first stage of the development process, the design focus has
been on selection, development, and testing of a kinematics structure. RAMRl, shown in
Figure 7, uses a biped format with four joints and five links. A revolute hip joint
supports two legs and articulated ankles at the ends of the legs. A Smart Robotic Foot
(SRF) (Dangi, Stam, et al. 2000) at the ends of the ankles provides the robot with the
capability to grip climbing surfaces. Steering is provided by one of these ankles. An
under actuated structure allows three motors to drive the robot and permits weight

reduction, space savings, and power conservation. To expedite the prototype process,

 

Figure 7. The BIped Crnbig Apbdatale Robot (RAMRl) scaling a vertical surface.

31

RAMRI has been built twice as large as its desired size. The prototype, shown in Figure
7, measures approximately 45mm x 45mm x 248m in its longest configuation, weighs
335grams, operates under tethered control, and provides video surveillance. The larger
prototype has emphasized scale—related issues of the kinematics that result in increased
actuator loads and power consumption. During the next stage, power consumption will
be much less due to smaller size, but packaging of custom on-board electronics and

actuators will provide greater challenges.

3.3 Existing Wall Climbing Robots

Numerous efforts have been undertaken to design mobile robots for climbing inclined
surfaces such as walls or ceilings. Common motivations are inspection or maintenance of
tall buildings, ship hulls, nuclear facilities, or pipelines where the presence of human life
would be a risk, infeasible, or simply cost prohibitive. Due to the intended functionality
of these robots, they are typically much larger than RAMRl, as we shall point out in the
following survey. Like RAMRl, however, most of them also rely on suction to climb the
inclined surface. The majority employ crawling or walking for locomotion, with the
suction device placed at the end of a leg. Biped, quadruped, and eight legged robots are

typical. A few climbing robots are also known to be wheel driven or thruster propelled.

Biped robots are most suitable to wall climbing in situations where appreciable external
loads are not expected. Since a single foot supports the robot during walking, safety is a
concern: slippage of the supporting foot may cause the robot to fall. For this reason,
many biped robots are intended for inspection purposes only, where the payload consists

of lightweight sensors, such as cameras.

32

Aircraft inspection is the intended application of the biped ROSTAM IV (Bahr, Li, et al.
1996). ROSTAM IV has ankle joints similar to RAMRl, allowing articulation at one
ankle and both steering and articulation at the other ankle. The middle joint of ROSTAM
IV is prismatic, which limits the robot to traversing between inclined surfaces with acute
angles, such as the internal comer of a room. To its benefit, ROSTAM IV ’s center-of-
gravity remains close to the climbing surface during walking. This reduces the bending
moment applied to the feet and consequently increases the robot’s load capacity and

safety.

The biped robot ROBIN (Pack, Christopher, et al. 1997) features similar ankle joints to
ROSTAM IV, but its middle joint is revolute rather than prismatic, which is actually
stereotypical of biped robots. ROBIN uses opposing pneumatic actuators to control its
link angles, but the current design is only capable of traversing surfaces with an
inclination up to 60° from horizontal. The structure of the biped created by Nishi (Nishi
1992) for building inspection is very similar to ROBIN, but it features steering capability
at both ankles and its joints are actuated by electric motors. Nishi’s robot is capable of

climbing vertical surfaces and traversing between planes with different inclination.

The joint structures of ROBIN and Nishi’s robot are most similar to RAMRl, which uses
revolute hip and ankle joins and a differential in one ankle to permit steering. RAMRl is
capable of climbing surfaces with any inclination and traversing between inclined planes.
Steering capability permits RAMR] to move to any point on a surface and also follow a
prescribed path. Currently, RAMRl operates in a tethered fashion, but future revisions
will provide tether-less autonomous operation. The greatest difference between RAMRl
and the aforementioned robots is its size and weight. In its longest configuration, when

the links are parallel, RAMRl occupies an envelope 0.25m long by 0.05m wide by 0.05m

33

high with a mass of 335gr. The next version of RAMRl will be less than half the current

size.

A third style of biped developed by Yano (Yano, Suwa, et al. 1997) uses a central body
and feet supported by two DOF ankles. One DOF is rotation of the foot about an axis
orthogonal to the climbing surface and the other DOF is prismatic such that the foot
raises and lowers. To walk, this robot swings its body about one foot and then the other.
Yano’s robot is not capable of traversing between surfaces of different inclination. It is
the closest in size to RAMRl since it measures 0.38m x 0.25m x 0.17m, but its 8kg mass

is much greater. ROSTAM IV is the closest in mass, at 4kg.

Quadruped and higher legged robots are used more frequently when greater load capacity
and stability are required. ROSTAM (Bahr, Li, et al. 1996), NINJA-1 (Hirose,
Nagakubo, et al. 1991), and ROBUG II (Luk, Collie, et al. 1991) are all quadruped robots
featuring multi-DOF legs protruding from a central body and supporting feet with suction
pads. ROBUG III (Luk, Collie, et al. 1996) and MSIV (Ikeda, Nozaki, et al. 1992) are
eight legged robots. While these greater numbers of legs provide greater safety and load
capacity, they use more complicated mechanisms, which are more difficult to miniaturize

and control.

A number of robots have been designed to crawl on walls. These robots usually possess
bodies with several movable segment and suction cups on each segment. Like a
caterpillar, these robots create locomotion by moving these segments relative to each
other and applying suction to the corresponding segments, as in the robot by Gradetsky
(Wilson 1990). Similar strategies have been used for underwater welding robots (Bach,
Haferkamp, et al. 1996), inspecting hazardous ducts (White, Hewer, et al. 1998), and

inspection in nuclear power plants (Briones, Bustamante, et al. 1994). The robot used in

34

the inspection of nuclear plants, known as ROBICEN, features an articulated segment for

traversing between horizontal and inclined planes.

Another class of wall climbing robots, or more appropriately wall driving robots, features
drive wheels for propulsion and a method of keeping the vehicle in contact with the
inclined surface. Robots by Nishi (Nishi 1996) use a single large sucker, like an inverse
hovercraft, or a series of thrusters to maintain contact with the wall. The robot Disk
Rover (Hirose and Tsutsumitake 1992) uses magnetic disks for wheels to hold the robot

on ferro-magnetic surfaces and provide propulsion.

3.4 Design Constraints

Factors governing the design of RAMR] are determined by its intended application,
namely, untethered gathering of sensory data in an environment where space will be
limited and detection undesirable. As mentioned earlier, the robot may be sent through
the ventilation ducts of a building, into the rooms of the building, or along the outside of
a building or structure. In all of these environments, the robot will need to send wireless
sensory data’ about that environment to a remote location and follow navigational
instructions. To accomplish this goal, the robot will need to operate for extended periods
with limited power supply. For these reasons, initial design efforts have focused on
miniaturization and limiting weight to minimize power usage. In order to validate design
concepts more efficiently, space constraints have been relaxed slightly to permit the use
of standard miniature components wherever possible. Design and selection of the robot
structure, actuators, and sensors have all been completed after considering all of the

above factors. The following design constraints have thus been established:

1. Mobility. The robot must be capable of (a) walking and climbing on horizontal
and vertical surfaces, (b) traversing between surfaces with varying inclinations,
and (c) navigating between specified locations.

35

2. Size. In its largest configuration, the robot should occupy a space less than 20cm
x 5cm x 5cm.

3. Space Requirements. The space required above the surface to maneuver the robot
should be minimized to improve performance in confined locations.

4. Weight. The holding capacity of the suction feet, termed the Smart Robotic Foot
(SRF), governs the maximum weight of the robot. Therefore weight
minimization is critical for safe operation.

5. Sensors. Due to space requirements, sensors critical for operation are to be used.
These include wireless video for reconnaissance, pressure sensors for sensing
secure foot placement, and binary joint position sensors for calibrating motor
encoders.

6. Control. Backlash in the joints and minimal sensing for foot placement require a
controller capable of providing adaptability to a broad range of environmental
conditions.

3.5 Robot Kinematics

3.5.1 General Form

Several candidate designs were considered prior to constructing RAMRl. As mentioned
previously, the objective was small size, extended operation, and adaptability. Several
variations of biped, quadruped, and wall driving robots were considered. While the
quadrupeds, and higher legged robots, possess safety advantages and increased load
capacity, they also require more actuators and increasingly complicated mechanisms.
This translates to increased weight, power usage, and control complexity, which are not
desirable. Thus, given limited package space and power supply, more complicated
designs were considered to be infeasible and design efforts focused on simpler structural

formats, such as the biped.

A biped robot can use a structure with fewer components and actuators. Thus, power

usage is decreased, weight is reduced, and the design can be miniaturized more easily.

36

Reliability is also increased since there are fewer parts that can fail. Variations on the
biped robot were primarily considered for these reasons. These include structures with a
simple central body supporting two feet (the simple biped) and structures consisting of
two legs joined by various types of hips (the revolute, prismatic, and hybrid-hip bipeds).

Suction cup feet are used to support all of these formats.

3.5.2 Performance Criteria

To compare each of these configurations objectively, they are each evaluated based on
their ability to provide mobility and operate in confined environments. Mobility is
determined by the capability of the robot format to move between planes with relative
inclination of -135°, -90°, -45°, 45°, 90°, and 135°. For each angle of inclination that the

robot can cross, one point is awarded. These points are then summed together to

/
5.0cm/

<3

1/ ..

/- Space required, A=5.0 x 20.0 = 100

 

20.0cm

 

 

 

 

<9”
V

Figure 8. Space requirement typified by cross sectional area required for walking.

37

establish the mobility rating, M, for the particular kinematics structure. A larger mobility

rating, M, indicates that a structure can traverse a broader range of obstacles.

Operability of a robot structure in confined environments is determined by the space that
the robot requires for walking. This can be typified by considering cross sectional area
that the robot passes through while completing a step, as indicated in Figure 8. Strides
characterizing minimum and maximum space requirements for each format are evaluated
to determine which strides are suited to a particular environment. Less required space

indicates better operability in confined spaces.

3.5.3 Joint Structure: The Simple Biped

The simple biped structure is shown in Figure 9. The robot has a rigid central body with
ankle joints at each end of the body that support suction cup feet. In order to traverse
between inclined planes, as shown in Figure 9, it is necessary that each ankle provide
articulation to the feet. Steering must be also be provided to at least one of the feet to
permit the robot to move between arbitrary points on a plane and also to change its angle
of approach to an inclined surface. Steering at both feet would simplify these matters,
but would significantly increase weight. Thus, weight savings are balanced by increased

motion planning.

x‘ ’.' I: “f

:/ Revolute Ankle Joint

T F _q/ FOOt

Figure 9. The Simple Biped robot.

 

 

Jl

 

*-

38

The mobility evaluation of the simple biped is shown in Figure 10. As the figure
indicates, it is unlikely that the simple biped would be capable of crossing between planes
with relative inclinations of —135° , —90° , or —45°. This limitation is attributed to the
rigid central body that prevents the structure from placing its foot around comers. Thus,
the structure possesses limited dexterity. In situations where dexterity is less important,
such as while crossing between planes with inclinations of 45° , 90° , and 135° , the robot

performs satisfactorily. Thus the net mobility of the simple biped is M = 3.

The space requirement of the simple biped during flipping and crawling strides is
illustrated in Figure 11. As the robot flips its entire body end-over end about one foot, as
indicated, the flipping stride is achieved. For uniformity, it is assumed that the net length
of the robot is the maximum 20cm and that its width is also the maximum Scm. Thus
during the flipping stride, the robot requires a space of 100cm2 for locomotion. A
crawling stride is also shown in Figure 11. In this stride, the robot lifts one foot slightly
off the ground and spins its body about the other foot. Here it is also assumed that the

robot occupies the maximum 5cm height, but since the foot is elevated slightly, it is

Relative Inclination of Surfaces
~135° -90° -45° 45° 90° 135°

Fail Fail Fail Pass t Pass Pass

Figure 10. Mobility of evaluation of the simple biped structure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flipping Crawling
A= 20.0cm x 5.0cm =C1_970cm2 I A: 5.5cm x 20.0cm = 110mm2
20.0 cm
7‘ 5.5 cm
"‘— 20.0 cm T.
CE: ----- 1’ ........... 1 """ . -— l
A ' r'\ ' [K r A ‘ i r

 

 

 

 

 

 

 

 

 

Figure 11. Space requirements of the simple biped structure during
flipping and crawling strides.

39

 

 

\

......
......

I

.......

 

 

 

 

 

 

Simple Biped '- Two-legged Biped
(cannot traverse 270° corner) (capable of traversing 270° corner)

Figure 12. Simple biped and two-legged biped attempting to traverse a 270° comer.

assumed that a 5.5cm height is required. Thus, the net space required during the crawling

stride is 110cm2.

3.5.4 Joint Structure: The Two-Legged Biped (revolute hip joint)

The two legged biped with revolute hip joint is shown in Figure 12 adjacent to the simple
biped. Similar to the simple biped, this two-legged robot requires articulation at both
ankles to cross between inclined surfaces and a minimum of steering at one ankle to
move between specified locations. Thus, the balance of weight reduction and motion
planning complexity can be argued here as well. Enhanced dexterity is the dominant
feature provided by the revolute hip joint. This permits the robot to reach around

obstacles and place its foot.

The enhanced dexterity of the revolute hip format is evident when we examine its
mobility, as illustrated in Figure 13. Unlike the simple biped, the revolute hip biped is
capable of ascending all six of the inclined surfaces. Thus, the mobility of the revolute

hip two legged biped is M = 6.

40

Depending on stride, the revolute hip also effects space requirements during walking. If
we again consider a robot that requires the maximum cross sectional area, Figure 14
indicates that the space required by the flipping locomotion of the revolute hip robot is
identical to the simple biped, which is 100cm2. During crawling, however, the revolute
hip robot is able to substantially reduce its space requirement to 47.5cm2 by folding its

body in half.

Crawling and Flipping strides each have advantages and disadvantages. The crawling
stride, for example, is slower and requires more complex joint movements, which also
requires more actuators. However, it requires less space above the robot during
locomotion and helps maintain a lower profile. The flipping stride requires twice as
much space above the robot during locomotion and creates a greater moment arm when
the robot “flips” while moving up a vertically inclined surface. Increased actuator loads
and foot holding requirements are a direct result. Crawling, on the other hand, maintains
a lower profile to the surface, which helps balances the weight added by increased

Relative Inclination of Surfaces
-l35° -90° -45° 45° 90° 135°

Figure 13. Mobility evaluation of the revolute hip two legged biped.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flipping Crawling
A= 20.0cm x 5.0cm =v10,0cm2 A= 9.5cm x 5.0cm = 47.5cm2
A. T HQ“.
\‘~\ Ir’l” 20.0 cm

 

 

 

 

 

 

 

Figure 14. Space requirements of the revolute hip two-legged biped during flipping and
crawling strides.

41

 

numbers of actuators. Hence, the crawling locomotion has a marginally detrimental
effect on stability on an inclined surface and permits the robot to travel through confined

spaces.

3.5.5 Joint Structure: The Two-Legged Biped (prismatic hip joint)

The two-legged biped robot with the prismatic hip joint is shown in Figure 15. As
indicated, collinear portions of the body sliding relative to each other form the two legs of
the robot. At the end of each leg are suction cup feet supported by articulated ankles.
They permit the robot to walk between planes with varying inclinations. Steering is
provided by at least one of these ankles to allow the robot to walk between arbitrary

locations.

The prismatic hip impacts mobility while walking between inclined surfaces.
Unfortunately, the prismatic hip joint possesses many of the same limitations as the
simple biped since the robot cannot reach around obstructions, as indicated in Figure 16.
Thus the robot is likely to not be useful on inclinations of —135° , -90° , or —45°. Using

the mobility rating outlined earlier, the net mobility of the prismatic hip biped is then

M=3.

:5} Sliding Locomotion

...over a flat surface ...across a 270° corner

.\:\ :

(b') (b) (a)
I?

Figure 15. Locomotion of a Two-Legged Biped robot using a prismatic hip joint.

     

42

The benefits of the prismatic hip are evident when we examine the space requirements for
walking, shown in Figure 17. Assuming that the robot can decrease its length from 20cm
to 13.5cm, which is somewhat arbitrary, the flipping stride only requires 67.5cm2. While
walking in a crawling stride, the space requirement is decreased substantially to 27.5cm2.
Thus, it is apparent that the prismatic hip joint is well adapted to operation in confined

spaces, but at the cost of decreased mobility.

3.5.6 Joint Structure: the Two-Legged Biped (hybrid hip joint)

A re-configurable middle joint that is a hybrid between the prismatic and revolute joints
should provide the robot with superior mobility. Such a joint would permit the robot to
reconfigure itself such that it could behave as a revolute or prismatic joint as required by
the environment and terrain. As indicated earlier, the revolute hip is best for dexterous
foot placement whereas the prismatic joint is ideal for confined locations. Thus, if the
hybrid joint can behave as both a revolute and prismatic joint, mobility and space

requirements will improve. Discrete regimes of revolute or prismatic behavior would

Relative Inclination of Surfaces
-l35° -90° -45° 45° 90° 135°

Figure 16. Mobility evaluation of the prismatic hip two-legged biped robot.

 

 

 

 

 

 

 

 

 

Flipping Crawling

 

A= 13.5cm x 5.0cm = 67.5cm2 A= 5.5cm x 5.00m = 27.5cm2

 

r--v
P 1 '
I l
l h “é I” i I

' 13.5 cm _ 5-5 cm
A ' A ' A .A. m it
Figure 17. Space requirements of the prismatic hip two-legged biped during flipping and
crawling strides.

 

 

 

 

 

43

also make control of the joint easier and permit it to be driven by a single actuator for

space and weight savings.

Numerous versions of a hybrid prismatic-revolute joint can be conceived. One
realization is shown in Figure 18. The hybrid mechanism consists of a movable link that
uses pins to guide its motion via slotted guide links. These pins, Guide Pins 2 & 3 in
Figure 18, travel in Guide Slots 2 & 1 on the Guide Links, respectively. The Moving
Link is driven via the Thrust Link, which travels in Guide Slots 1 & 2 via Guide Pins 1 &
2, respectively. The motion of the Thrust Link, and hence the Moving Link, is driven via

the Drive Block and Drive Rod, which are essentially a lead screw and mating tackle.

Notice that Guide Slot 2 possesses a straight segment that allows the mechanism to act as

a prismatic joint and a curved portion that permits the mechanism to function as a

  
   
  

. ssembled Mechanism
(upper guide not shown for clarity)

Upper guide link

1 ' uide Pin 3
Moving Link

Drive Block\ ‘6

 

Guide Pin 1—/ l Gmd" Pm 2
‘ Drive Rod Thrust Link
(threaded lead screw)

Guide Slot 2
(revolute portion)

 
  

Guide Slot 2
(prismatic portion)

 

Lower guide link
Figure 18. A hybrid revolute and prismatic joint shown assembled and in exploded view.

44

revolute joint. The transition between prismatic and revolute behavior occurs when
Guide Pin 3 reaches the limit of Guide Slot 1, as shown in the exploded view. As the
Thrust link continues to drive the Moving link, Guide Pin 2 is force to travel in the
revolute portion of Guide Slot 2. Hence, via a single actuator, such as a coreless DC
motor, the hybrid mechanism is capable of behaving as either a prismatic and revolute

joint. Examples of these distinct modes of operation are shown in Figure 19.

As Figure 19 indicates, the hybrid mechanism is capable of exhibiting prismatic or
revolute behavior. The particular design shown is capable of expanding from a net length
of 100mm in the closed position to 150mm in the open position. As the thrust link
continues to drive the mechanism, revolute behavior ensues and the link can actuate from
the straight configuration, denoted as 0° articulation, to a 120° counter-clockwise

articulation. A 30° articulation is shown in Figure 19. Thus, the hybrid joint is fully

    

Prismatic Mode (closed)

/
‘/_‘=

‘-},"_ 34’:

    
  

corresponds to prismatic
link behavior

      
  
 
 

Revolute Mode
(partially articulated)

Pin 2 in curved track
corresponds to revolute

joint behavior

Figure 19. The hybrid revolute and prismatic joint shown in several positions.

45

capable of behaving as either a prismatic joint, or to a limited degree as a revolute joint.

The limited articulation provided by hybrid joint must be debated since it constrains the
motion of the robot. The articulation provided by the joint is certainly sufficient to allow
the robot to perform a crawling stride similar to that of the revolute hip joint, shown in
Figure 14. The flipping stride, also shown in that figure, is not achievable since that
stride requires at least i90° hip articulation, which the hybrid cannot provide in the
design presented. However, the design can be modified to provide bi-directional

articulation in addition to prismatic motion.

The hybrid is also limited while crossing planes of various inclinations. This is because
the robot must first approach the inclined surface in a manner such that its limited
articulation can be exploited. If the robot can maneuver into such an orientation, it is
then limited in its capability to cross between the surfaces since it is forced to use the
crawling stride. This stride predisposes the robot to place its trailing foot behind the
leading foot, and thus the ability of the robot to cross between planes of -l35° , —90° , or
—45° is entirely dependent upon the leading foot being placed sufficiently far onto the
inclined surface such that there is room for the trailing foot behind it. As a result, the
limited articulation of the hybrid joint does not entirely realize the benefits of revolute
hip. Whereas the revolute hip received a mobility score of 6, the hybrid joint only
receives a rating of M = 3+ 3(0.5) = 4.5. Full marks are awarded for crossing between
inclinations of 45°, 90°, and 135° where the hybrid can behave as a prismatic joint.
Half points are awarded for crossing between planes of —135° , —90° , or —45° due to

the limited capability of the robot in these situations.

Despite the mobility limitations of the hybrid joint, it capable of minimizing space usage

during locomotion. When the joint is in its prismatic regime, the robot uses the very

46

minimal 27.5cm2 as in the case crawling with the prismatic joint. In the revolute
crawling mode, the robot only requires 47.5cm2. So despite mobility limitations realized
by the current hybrid design, it does capture the best space requirements of both the

revolute and prismatic hips.

3.5.7 Selection of Kinematics Structure

The mobility and space requirement data derived in the previous sections is presented in
Table 7. The data is indexed and identified by hip joint style and walking stride. For
analysis purposes, the data is normalized relative to the average performance of the

designs. Normalized mobility is determined by,

M_N=_‘___
' M

avg

M.—Ma, 1 "
3 where Mavg=;21M,.

and the normalized space requirements are calculated by,
[4.1-N = A — Aavg

avg

1 n
where Am = Z2, A, .

Table 7. Data indicating space requirements and mobility of various robot joint
structures and strides.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

47

Configuration Space Requirement Mobility
index, i HIP Walkin N N
Structure Strideg A‘ A" M‘ M"

1 Simple Flip 100 0.52 3 -O.27

2 Simple Crawl 1 10 0.67 3 -0.27

3 Revolute Flip 100 0.52 6 0.45

4 Revolute Crawl 47.5 -0.28 6 0.45

5 Prismatic Flip 67.5 -0.02 3 -0.27

6 Prismatic Crawl 27.5 -0.58 3 -0.27

7 Hybrid Revolute 47.5 -O.28 4.5 0.09
crawl

8 Hybrid Prismatic 27.5 -O.58 4.5 0.09
crawl

Avg: 68.4 Avg: 4.13

 

These normalized expressions are designed such that positive values indicate better than
average performance and negative values indicate lower than average performance. The
prismatic (i = 6) and hybrid (i = 8) hips possess the best normalized space requirement

rating of 0.58. The revolute hip formats (2' = 3, 4) have the best mobility rating of 0.45.

To evaluate each joint structure and walking stride in different environments, a weighted
sum of the normalized mobility and space requirement data is formed. The weighted

value of a configuration is thus calculated from,
a, = —A,."wA + M,”wM

where three sets of weightings were selected to emphasize operation in unconfined
(wA=0.5, wM=I.5), confined (wA=I.5, wM=0.5), and mixed (wA=I.0, wM=I.0)
environments. These weighted calculations are shown in Table 8. The data indicates that
operation in an open environment is best achieved by #4, which is the revolute hip with
crawling stride, which received the highest rating of 0.59. Operation in a confined
environment is best achieved by #8, the hybrid hip with prismatic crawling stride, with a
rating of 0.92. This is followed very closely by #6, which received a rating of 0.74. In an
environment where equal emphasis is placed on mobility and space requirement, the

revolute hip with crawling stride (i = 4) performs best with a rating of 0.73. This is

Table 8. Weighted performance evaluations of the robot configurations operating in
open, confined, and mixed environments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Configuration Environment

index, i Structure Stride Open Confined Mixed
1 Simple Flip —0.53 -0.91 -0.79
2 Simple Crawl -0.61 -1.14 -0.94
3 Revolute Flip 0.20 -0.55 -0.06
4 Revolute Crawl 0.59 0.65 0.73
5 Prismatic Flip -0.28 -0.17 -0.30
6 Prismatic Crawl 0.02 0.74 0.31
7 Hybrid Revolute 0.23 0.46 0.37
8 Hybrid Prismatic 0.38 0.92 0.67

 

4s

 

followed distantly by the hybrid hip joint with prismatic stride (i = 6) with a rating of

0.67.

The revolute middle joint is selected for several reasons. First, the capability of the
hybrid joint is not entirely satisfactory in its current form, and it is believed that
improvements can be made to achieve bi-directional articulation. Since this robot is in its
first phase of construction, and a prototype must be constructed within the first year,
these improvements are left as the subject of later research. Secondly, the revolute joint
is a much simpler mechanism, which inevitably will be more reliable since it has fewer
parts and mechanisms that could fail. Reliability was deemed paramount since success of
this first prototype would be largely responsible for renewed funding, and hence the
revolute hip joint was selected based on its superior performance and anticipated
reliability. The development of an improved hybrid joint is the subject of the later

research. Details of the revolute hip design will now be presented.

3.5.8 Coupled DOF and the Two-Legged Biped

The revolute hip structure is used as a platform for the robot. To reduce the number of
actuators, and hence weight, consider the concept of coupling the DOF of the robot. As
shown in Figure 20, the Two-Legged Biped, with a revolute middle joint, has four DOF.
At one ankle, which consists of Link 1, Joint 1, Link 2, and Joint 2, there is one DOF for
steering (Joint 1), described by angle y, and one DOF for pivoting (Joint 2), described by
angle a. At the other ankle, Link 5, there is only a single DOF for articulation (Joint 4),
described by angle B. At the hip joint, Joint 3, there is a revolute DOF for controlling the
angle between the legs, described by angle A. Each DOF powered individually would

require four actuators. Coupling the motion of two of these DOF would reduce the

49

number of actuators by 25% and hence decrease the weight of the robot significantly

since the actuators are one of largest contributors to the robot’s net weight.

Obviously, the steering DOF at Joint 1 must remain uncoupled. However, coupling
between Joints 2 and Joint 3 is practical because the hip and ankle joints tend to function
in unison. A belt drive coupling the joints, conceptually illustrated in Figure 20, can
accomplish this. Assuming symmetric foot and link sizes, as illustrated in the figure, the
change in link angles during a flip motion will be M = 180° and Ad = 90°. Selecting
the pulley diameter at Joint 2 twice as large as the pulley diameter at Joint 3 guarantees
that this relationship will be maintained. Hence, by driving the middle joint, Joint 3, the

ankle joint, Joint 2, will be driven appropriately.

Coupled DOF also produces side effects that may not be desirable. Once the joints are
coupled, they then function as one DOF and capabilities that were based upon their
independent actuation no longer exist. For example, the coupling designed above allows
the robot to perform the flipping stride, Figure 14, but the crawling stride is no longer
possible because the relative rotations of the hip and ankle joints are not the same for
both strides. Another complication is encountered when aligning Foot 1 with the surface

[thwm

/ x
/

/

  

 

Figure 20. Joint and link structure of the Two-Legged Biped with coupled DOF.

50

because independent control of Joint 2 is not feasible. Rather, Joints 3, 4, and 5 must all
be actuated simultaneously to align the foot with the surface. Hence, slightly increased
control complexity and decreased operability are compromises that must be made with
coupled DOF. As mentioned earlier, though, if these compromises are acceptable, a few

lines of computer code weigh much less than an additional actuator!

51

3.6 Robot Design

3.6.1 Biped Structure

Figure 21 contains illustrated pictures of the current climbing robot. This figure is a little
cluttered by wiring on the robot and an exploded diagram of the robot has been included
in Figure 22. Full detail drawings of the components shown in Figure 22 are available in
Appendix B. As described in the previous sections, the current format of the biped
climbing—robot consists of a two-legged structure using a revolute hip joint to support two
legs of equal length. At the end of each of these legs are ankle joints supporting suction
cup feet. Ankle #1 consists of Joint 1 and Joint 2 which provide DOF for steering and

articulation. Ankle #2 provides only articulation via Joint 4.

 
 
 
 
 

Joint3
(hip) 7% Motor3 \

Link 4
“(leg 2)

Ankle #2

  
 
 
  

Pulley &
Timing belt
Joint 4

L'"k 2 Link 5
(foot 2)

Link 1
(foot 1)

Figure 21. Illustrated pictures of the Biped Climbing robot with a revolute hip joint.

52

Coupling between the hip, Joint 3, and Joint 2 is accomplished via belt driven pulleys,
shown in Figure 22 and Figure 21. This reduces the DOF of the robot and three motors
are used to drive the four joints. Motor #2 drives the Hip Joint by a worm gear system
and rotation of the Hip Joint is coupled to articulation of Joint #2. Motor #2 is supported
by Link #3, Figure 22, and the shaft of the geared motor unit supports the worm. The
worm is made of hardened steel and has a Diametral Pitch (DP) of 48, a single 14.5° lead,
and a .38” Outside Diameter (OD). As indicated in Figure 22 the worm gear (48 DP, 16
tooth) is pressed into Links #1 and #5 with a filleted raised shoulder to provide stability
and improved coupling between the gear and link. The revolute hip joint under
consideration is created by Shaft #3, which is firmly retained by the motor bracket
integral to Link #4. The shaft provides a bearing surface for the worm gear and thus

relative rotation between Link #3 and Link #4 is achieved when the worm rotates.

Motor 1 :—

   

Shafi 2\
Differential Housing

B vcl Gears
Position Sensor
(Joint l) .
. ’ \ l l ! Motor 2
\ i Shah 3—/ g j
‘ Q " ‘ RulonJ bushing : __
. - Link 4 '
A ' ' Link 3 ,
L \nn. .1 If! 3.....- @
Link 1 @ ] \Fillcted Shoulder i x
i ‘ Link 5

SRF , :
(SmartRobotic Foot) Ankle #1 j 8h 0 ‘ Ankle #2

 

 
      
    

  

xii
Belt drive systern/ .
(coupmg ms l & 3) / Position Sensors
Figure 22. An exploded diagram of the Biped Climbing Robot
with Smart Robotic Feet (Dangi, Stam, et al. 2000).

53

Coupling between articulation of the Hip joint and Ankle #1 is achieved via the belt—
pulley system, shown in Figure 22 and Figure 23. A pulley clamped to the end of Shaft 3
drives the coupled DOF and retains Link #3. Rotation of Link #3 causes the relative
motion between the fixed pulley and the link, and this drives the belt motion. At Ankle 1,
there is a mating pulley clamped to the Differential Housing and the belt motion causes it
to rotate. The amount of rotation between the hip joint is half as much as the rotation of
the differential joint by design of the 2:1 sizing of the pulleys. Timed belts and pulley
were used to guarantee consistent relative position of the joints and a slight preload was

used to prevent slipping.

As mentioned before, as the hip joint articulates it simultaneously causes articulation of
Joint #2 via the belt drive system, indicated in Figure 23. This motion is also coupled to
the rotation of Joint #1 via the bevel gears indicated in Figure 22 and Figure 24. To help
understand this phenomenon, assume that Motor #1 is not allowed to rotate and realize
that this motor controls the rotation of the bevel gears. Also note that the bottom bevel

gear rotation is not coupled to articulation of the differential joint. Hence, the bottom

llliil .
u \lilllilllillllll llilll i

ll llllilllllllllllliliil llllli
lllllllllllllllllllllllllllllllu
IillIlillllllillilllll‘lillll ‘ .
illll‘lllllI llllllllllll .m’yi‘

illlllllill “Mural“ .l ‘41
lllllllll lllllj .

p 1 l llljl
Hildl‘lliii lll

'l‘ll ll ll

1 llli'l ‘ln‘li ll‘l‘llll

- will“ Him
mill n“lllllml ll

 

n . “
: llllmlllllllln H‘lll'l.l.llllillul lllillil'lilll ln’l l l:

, . . p M l a
Figure 23. Belt drive system coupling articulation of Ankle #1 and the hip joint.

54

bevel gear in the differential will not rotate as the differential articulates. However, as
the differential articulates it does displace Foot #1, and this displacement causes the foot
and its bevel gear to circle travel in a circle about the stationary bevel gears. Because of
the mating bevel gears, this motion is necessarily coupled to the rotation of the foot about
Shaft l, and thus the coupled articulation DOF are actually coupled to the steering DOF,
too. Motor #1 is responsible for driving the bevel gears of the differential, and hence

must compensate for this coupling during actuation of the hip joint.

The differential is supported via a RulonJ bearing in Link #3, indicated in Figure 22, and
also by Shaft #2, which is clamped into the bracket supporting Motor #1. RulonJ is a
Teflon embedded polymer that provides the lubricant qualities of Teflon without the
undesirable malleability of Teflon. Shaft #2 is responsible for supporting the upper
portion of the differential housing as well as the bottom bevel gear of the differential. A
brass bushing is used in the upper portion of the differential to reduce friction. At

assembly, spacers must be placed below the bottom bevel gear to control its mating

III. I IIIIIIIIIIIIIIIIIIIII m'l‘llll‘l

IIIIIII‘. I lIlIlllllllllIlllIIllIl IIIIIIIIIIIIIIIIIIIIIII ml
I ' IIIII
WmmwmmwwWMWMNMl

llllllllilllllllllllll II ,lIIIIlllIIIII
IIlIIIllIIIIIIIIII'I II' ‘IIIIII
lIllll llllIll IIIIII II'IIIII

‘ I.” I l:llllIIlIIIlIl ; MI I I' I I I “I
lulllllllilllllllh‘l- I" “mil: . 5‘" ': III]:lI‘I‘ulIII‘III‘I’I‘IIIIIIIIIIIIIIIIIIIIIIIIllIIlIIIIIIIIIlIlIlIlIII }
MWWWMI ‘ww HNMWMWMWW

mnmu lWM
fim$nmwwi ,W ,3.H.nw\“ 'iIWIWWWMWWWWWW

‘ "I l,IIIIIII‘IIIlIlIlIlIlllllllllll
MnIIIIIIIIIIIIIIIIIIIIII
"I Ijl I III IIIIIIIIIIIIIIIIIIII
NLIIIIIIIIIIIIIIII

"I. ..I III IIIIIIIIIIII I I II
N, ..I 1III lIIIlll IIIIIIIIIIII
j” IIm
,lL‘I

HMWWWW
‘I, III IIIIIII, III“ M
meMW
mlllfi
III.

25::-
3::
- :gi
ML?

1‘.

i;

l l I I lllll‘ Il l

IE‘I'IIilIIIIIIIIIlH
I I I

l Illlllllnllm ill

I

A .Er'“ A

 

l'l "IIIIIIIII
, WW IIIIIIII'IIIIIIIIIIIIIIIIII
, Ml” IIIIIIIIIIIIIIIIIIIIIIIIIIlIII
.. lll IIIIIIIIIIIIIII‘IIIII
‘ lllll'lll lIllIIllimW
I I IIIIIIIIIII
IIIIIIIIIIIIIIIIIIIIIII II

I; M IIIlIIII IIIIIIIIIII
IIIIIIlI
, lIIIlI IIIIIIIIIIIIIIIIIII

.IIIIIIIIIIIIIII
\l II‘I‘IIIIII
l‘illlllllIIlIll

, ”‘I‘IIIIIl
Figure 24. Ankle #1 and the differential.

55

relationship with the bevel gear driving the foot. The bevel gear is brass, so its friction
with the shaft and steel spacers is minimal. Similarly, spacers placed between the motor
bracket and Link #3 control the interaction between the lower bevel gear and the bevel
gear mounted to Motor #1. A double set of miniature ball bearings support Shaft #1 and
the foot. They are pressed and retained in the differential housing. Such a durable and
low-friction support for Shaft #l is especially important since the entire load of the robot

is supported as a cantilever by this joint.

Motor #3 drives articulation of Ankle #2 via a worm drive identical to the one used in the
hip joint. There are three motivators for selecting a worm drive at these joints. The first
is that the worm gear compactly provides additional gear reduction to allow for smaller
gear boxes. As important, the worm drive allows the actuator to be mounted transversely
to the joint axis and reduced the size of the robot. Hence, the actuators become integral
parts of Link #4, as shown in the figures, rather than a large object protruding out the side
of the robot. The last motivator is the self-locking characteristic present in most worm
drives. In the situation that the robot may have to maintain a position for an extended
period, the actuators may sit idle. Unfortunately though, self-locking is a product of

friction, and friction is synonymous with inefficiency and wasted power.

IIIIIHI l "I‘IIII
M . II’I‘I'IIIIIIIIII‘II

,. I'IIllI‘I “ ‘
IIIIIIIIII‘.I'll‘.W IWMM
.. II I“ .

 

56

To counteract friction effects, a combined Msz and Graphite coating was applied to the
worm, worm gear, and shafts. These components, which all featured polished steel
rubbing against brass, typically possess a coefficient of friction of about 0.3. The
application of the Msz and Graphite coating reduces the friction to a rated value of
about 0.07 (Dow-Corning 1999). The coating is quite durable, with a rated life of 71
hours under IOOOlb per the federal pin and vee-block test method 791a-3807, so it
provides suitable dry lubrication for the prototype. Liquid lubricants were not used

because of the open joint structure and undesirable messiness.

3.6.2 Force Analysis: climbing a vertical surface supported by foot 1

The purpose of performing this force analysis is to determine the feasibility of different
actuators for Motor #2. A static force analysis is performed on the robot in the
configurations that apply the largest forces on the actuators. This will help determine the
ideal actuator among numerous alternatives. All given actuators under consideration are
geared DC motor units with position encoders. Each model weighs a different amount,
and has its own geometric properties and load capacities. These variations change the
load requirements from unit to unit, and thus each unit must thus be considered
independently to determine feasibility of its application. Of course, the selection of

Motors #1 and #3 both have direct consequence on Motor #2, and vice versa.

The robot will be analyzed while climbing a vertical surface in a straight, cantilevered,
orientation where the maximum moments and loads are encountered. Feasibility of an
actuator will be confirmed by a safety factor of at least 2.0 at this operating point. Such a
safety factor should be sufficient to accommodate dynamic forces as well as unexpected
friction. Figure 26 shows the robot supported by Foot 1. The case of the robot supported

by Foot 2 will be considered later in the selection of Actuator #3.

57

The robot supported by Foot #1 is shown assembled and as Free-Body-Diagrams (FBD)

in Figure 26. The following notation is used in the figure:

 

 

///////7/

FB = Belt force acting on the pulley

Fm = Tangential worm load

me = Radial worm load

L, = Length of the i"' link between joints i and i+l

PX, = Force on the i"' joint in the x — direction

P. = Force on the 1"” joint in the y - direction

:5

TF. = Static friction torque on the i’” joint

I

W. = Total weight of the 1'”1 body

d, 2 Distance to the center of mass of the 1"” body from its support point.

rs, = Pitch radius of worm gear

. o m
0 o O po©0@o@o@oo O

 

If

 

 

 

 

 

 

 

    
  

 

 

 

 

 

 

 

 

 

Differential
(Link #2)
P
Y1 TF2 FB
Mm sz
P Px4
x2
F901 #1 TF4 Foot #2
(Link #1) W1) Pvz 3 (Link #5)
P Leg #1 Leg #2 p
(Lmk #3) (Link #4) F Tr Y“
P w2
x3 D
Px4
W p Mm W with
1
«1.4 Y' d2——>i +dz’l

Figure 26. Free body diagrams of the climbing robot supported by Foot #1 on a vertical

surface (some forces omitted for clarity).

58

r51. = Radius of shaft supporting the 1"” joint
a = angle of belt between pulleys

xi = worm lead angle

a! = worm gear pressure angle

,u = Static coefficient of friction

Using the above notation while examining Foot #2, the tangential worm force, er2’ and

pin forces, PX4 and PM, are determined to be:

1

erz =r—(VV4d4-l-TF4) (3.1)
3

PM = R3, (3.2)

PY4 = W: + FWRZ (3-3)

The radial force on the worm drive, me , is generically determined by (Meritt 1971),

Fm. = FM. tan w (3.4)

where l]! = 14° is the pressure angle of the worm wheel. The friction torque on the 1"”

joint, T; , is then generically determined by,

TH = rsan erz + Pnz (35)

Equations (3.1) through (3.5) are then used to form a quadratic expression that is solved

for the tangential worm wheel force, FWD:

 

2 2 4 2 2 2
—rGW4d4-W4tan(\|l)r32u -W4v¢2d4r6tan(\v)r52u -rs4u +rs2p. d42+tan(\u)2r52p d42+r62r52u

Fm‘

 

2 2 2
'32“. +WW) r52“. _r02 (3.6)

59

Similarly, the forces supporting Link #3 are Fm, PX3, and P”. They are calculated as

follows,
F -1Wd T—T F F F4—F2 37
er—T( 3 3+ F3 F4+ prpl+ BrP1+( Sy WR)L3) ( : )
G
PX3=FW1_FBX_PX4+FW2 (3.8)
= Fwn " Fax
R’3z‘V3+R’4_FWR2+FWRl_FBY (39)
=W3+m+me—pr
where
FBX = F3 cosa (3.10)
F” = F3 sina (3.11)
and Otis the angle of the belt between the pulleys, determined by,
tana- (3.12)

Before an expression for FW“ can be formed, the pulley force F; must be evaluated via
analysis of Link. Keep in mind that the pulley force acts as a torque on the Differential,
which then creates relative rotation between Link #2 and the Differential. This torque is
essentially the input required to rotate Link #2 and will be determined from this link. The

forces supporting Link #2 are determined as,

Px2=FW1-Px3 (313)
=st
Rzzwz'l'Pn-me (314)
=VVZ+‘V3+“I4—FBY -
1
F3 =r—[“/2d2+TF2_TF3+FWTer +(PY3—FWR1)L2] (3°15)
P2

60

Equations (3.7) and (3.15) each form non-linear equations in their fully expanded form

which must be solved numerically for F3 and thus Fwn. Once these equations have been

solved, sufficient information exists for determining the load applied to Actuator #2.

Since the force Fm is created by a worm drive system, we first realize that the force is

equivalent to a torque applied to the worm gear, denoted by MW]:

chr = err0 (3-16)

and that a worm torque, MW1 is required to create the worm gear torque. Considering

that torque loss due to friction in worm drives can be substantial, we first consider the

efficiency of the worm drive (Meritt 1971):

EFF = tan ’1 = 0.38 (3.17)

 

tan(). + arctan ,u)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Differential L3
(Link #2) *—d§—~
‘Q’ ._., H r
. 3"sz
__Q_ (Po ? rd ()1 - Oo®o o®o S/r PX4
i
1U W Wzi L W H P Y4 F4 /
Foot #1 ‘___ I 3 /
(Link #1) d' D d2 L93 #2 r Pa 1 I ’
I'D L2 (Link #4) F 3‘ a /
Leg #1 D) /
(Link #3) Px4 (
w4 :
Foot #2 *d-J /
(Link #5)

 

Figure 27. Free body diagrams of the climbing robot supported by Foot #2 on a vertical
surface (some forces omitted for clarity).

61

Given that the reduction ratio of the worm drive is 20:1, the required worm torque is then,

M
MW! ___ WGl
20EFF

 

(3.18)

Selection of Actuator #1 is made based upon the capability of the actuator to provide this
torque while operating near a preferred joint speed. This will be considered further in

Section 3.6.6 after the worst-case loads for the other actuators have been determined.

3.6.3 Force Analysis: climbing a vertical surface supported by Foot 2

In this section the robot is again considered while climbing a vertical surface, Figure 27,
but now Foot #2 supports it. The robot is again in a straight, cantilevered, configuration
where the maximum load on Foot #2 occurs. In this situation, however, the forces due to
the other bodies can be considered cumulatively since the coupling created by the belt
drive does not affect Actuator #3. Essentially, those forces are internal to the rest of the

robot. The forces acting on Foot #2 and affecting the actuator are determined to be,

P“ = PW,2 (3.19)
PM=W,+W3+W2+WD+WI+FWR2 (3.20)
TF4 : I'M/1‘] Px‘:2 + Py42 ‘ (3.21)

FWR2 = Fpm tan 1]! (3.22)

___L W3(L3"'da)+Wz(L3+Lz_d2)+

sz _ (3.23)
r0 +Wr(LJ+Lz+LD+dr)+WD(La+l’z+dD)+TF4

Equations (3.19) thru (2.3) are then used to form a quadratic equation for the driving

force sz, which is not shown here because of its extreme length. Once sz is

Calculated, then the effective torque on worm gear is determined by,

chz = Fwan (3.24)

62

The worm torque can then be determined by,

_ MWGl

M _ 3.25
W‘ 20EFF ( )

where the efficiency, EFF, is determined by Equation (2.22). This worm torque will then

be used in Section 3.6.6 for selecting Actuator #3.

63

3 .6.4 Force Analysis: Steering on a vertical surface while supported by Foot 2

The maximum loads for steering the climbing robot will now be examined. These forces
occur when the robot is steering on a vertical surface. If independent articulation of
Joints #2 and #3 were possible, this maximum would occur when the hip joint and ankle
joint #2 were straight, ankle joint #1 was bent 90°, and the orientation was such that the
robot was perpendicular to the gravitational field. Rather, articulation of Ankle #1 is
directly coupled to the hip joint, and the aforementioned configuration cannot be
achieved. Instead, as the hip joint bends, so does Ankle #1, and the effective moment
created by steering the robot changes. The maximum of that moment, denoted by T1,

occurs when the robot is in the configuration shown in Figure 28, and is calculated as,
7; = Wzd2 cos 45°+W3(L, + d3)cos 45°+W4(d4 + (L2 + L3)cos45°) (3.26)

This equation will also be used in Section 3.6.6 for actuator selection.

\\\\\\\\\\\\\\\\\\\\\\\\

T02 VICW FOOt #1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d g (9 (Link #1)
4
7‘ ’ W4 - Differential w
Foot #2 (Lmk #2) . D
(Link #5)
1 W3
w2
T1
(12
Leg #2 12 d L2 Leg #1
(Lmk #4) 3 (Link #3)
g I] =
l ’ Front View
0
1:1 \1

 

213* I

Figure 28. Diagram of the climbing robot steering via Foot #1 on a vertical surface.

64

3 -6.5 Actuator Selection Criteria

We now consider the force analysis conducted in the previous sections to determine the

viability of various actuators applied to the climbing robot. There are a number of

criteria that the optimal actuator must satisfy. They are summarized in the order that they

were examined during the design process:

1.

Torque Safety Factor (TSF). The actuators must be able to provide a safety factor
of at least 2.5 to compensate for unanticipated friction affects and for dynamic
forces, which are anticipated to be quite small due to small joint velocities. The
TSF is calculated based on the worst-case joint torques, described in the previous
sections, and the maximum continuous torque that the motor unit can supply to
the joint. A generic expression for TSF is,

Z. Te

TSE = = ‘ ”L (3.27)
nT / (5,l nT

r (on: .i e! i can! .i

 

where T,- is the worst case joint torque, n,- is the gear reduction, 8,.“ is the net
efficiency of the gear reduction, and Tm“- is the maximum continuous torque of

the motor.

Packing Length Constraint (PLC). The actuators must fit within the allotted
space, where net length is the primary concern. Variations in the design could
accommodate excess length to some degree, but as rule the length of Actuator #1
should be less than 55mm and the length of Actuators #2 and #3 should be less
than 52mm. The PLC is formulated as the amount of available space, or
unavailable as the case may be, when the actuator is incorporated in to the robot.

In the case of Actuator #1, PLC is formulated as,

65

PLC, = 55 - l, (3.28)

where L; is the net length of the actuator. Similarly, the PLC for Actuators #2 and

#3 are formulated as,

PLC” = 52 — 1,, (3.29)

Obviously, a feasible actuator has a positive PLC.

. Shaft Load Safety Factor (SLSF). The compactness of the robot necessitates that
the gear boxes attached to the motors support the anticipated gear forces. Gear
forces applied to the output shaft of the transmission include radial force and
subsequent torque. In either case, the SLSF should be at least 1.5 if an infinite
life rating is provided by the transmission manufacturer, or 2.5 if only an
intermittent transmission rating is provided. In the case of Actuator #1, which

drives steering, the SLSF is calculated by,

SLSFl = minimum(-AlB-, -F—’) (3.30)
MR FR

where MB and F, are the moment and normal loads created by the bevel gear, and
MR and F R are the rated moment and radial loads of the motor unit. The moment

load is calculated as,

MB =dBFr

(3.31)
= 7.0F; [mNm]

where dB=7.0mm is the distance from the bevel gear pitch circle to the gearbox

face. The force Fn is determined by,

66

 

F,=F,tan1//= 2M“ 2C tam/I
d 2C— f (3.32)
= 0.11M51 [N]

Ms) (mN m) is the shaft load applied by the motor, d: 7.9mm is the pitch diameter
of the gear, f=1.8mm is the face width of the gear, and C=6.5mm is the distance

from the apex of the bevel gear pair to the far edge of the gear tooth.

Similarly, the shaft load safety factor for Actuators #2 and #3 are determined by
the normal forces created by the worm drives. The expression for SLSF for the
middle joint is,

Fm

SLSF2 = minimum[ MBW‘ , J (3.33)
MR FR

 

Again, MR and F R are the rated moment and radial loads of the motor unit, and
Maw) and F,w1 are the bending moment and normal load created by the worm

drive. They are calculated by,
Mam = FrW2dw2 (3'34)

Em = 3%; tan 1,1 (3.35)

where dw2=1 1 mm is the axial distance along the worm from the face of the motor
unit perpendicular to the center of the worm, W=I4.5 ° is the pressure angle of the
worm, M52 is the shaft torque created by the motor, and d=8.5mm is the worm
pitch diameter. This expression can also be expressed as a function of the
required joint torque, sz, if we also consider the efficiency of the worm drive,

8 2.3 , and its reduction ratio n=20. The resulting expression is,

67

2M
F =——W2 tan
"“2 den V’ (3.36)

=.010Mw2 [N]

This expression also applies to ankle joint 2.

. Maximum Power Consumption (MPC). Assuming that several actuators satisfy
the above specifications, final selection of an actuator will be based on a robot
with minimal MPC. Power consumption is based on the power requirement
necessary for the worst case load conditions at the prescribed JAV. Since DC

motors are considered, power consumption, P, is calculated by,
P = VI

where V is the necessary voltage to achieve the desired output speed. It is
determined by required joint speed, and the Speed Constant and gear reduction of
the motor under consideration. The current, I, necessary to achieve the joint
torque is determined by the efficiency and reduction ratio at the joint, and by the

Torque Constant of the motor.

The aforementioned criteria are evaluated with common joint velocities. The joint

velocities are determined by the desired walking speed of the robot. During this initial

prototype stage where investigative value has overshadowed performance, it has been

established that the robot should be capable of completing a complete flipping step in at

least than 7.5 seconds. This corresponds to the hip joint moving at 4 RPM, the steering

joint moving at 2 RPM, and Ankle #3 moving at 2RPM. Since most actuators are

capable of these performance levels, they will serve as an operating point where power

consumption is evaluated. The net weight is also evaluated to assure adequate holding

capacity of the suction cup feet. It is calculated by summing the weight of all of the robot

68

components, including the feet, actuators, structural components, and fasteners. Wiring
has not been considered. The above criteria will now be applied during the evaluation

and selection of actuator units for the climbing robot.

3.6.6 Actuator Selection

Several groups of actuators will now be examined. The analysis will start with the
smallest actuators and proceed to larger devices in order to determine a range of feasible
units. Groups of actuators with similar size amongst each group will be examined to
minimize the number of permutations that must be considered. Units that were totally
unrealistic, that is, they could not satisfy the TSF criterion or were far too large, are not

discussed.

The size of motors considered range from 10mm diameter to 17mm diameter, and
0.75watts to 3.2watts, respectively. Motor manufacturers with a broad product range,
including motor selection, gear heads, and encoders, were the primary concern. These
include API Portescap (API-Portescap 1998), Maxon Precision Motors Inc (Maxon-
Precision-Motors 1999), and Micro M0 Electronics (Micro-Mo-Electronics 1997). All of
these companies provide thorough product data, which is critical during the design stage.
Units available through API and Maxon were primarily considered since the units offered
by Micro M0 were quite similar to those from Maxon, and Micro Mo’s engineering units

used throughout their catalog were mixed.

The smallest units examined were the Maxon RE010. They are 10mm diameter, 0.75
watt, planetary-geared (plastic housing) DC motors with digital magnetic encoders.

Since joint level feedback was not incorporated into this first prototype, the encoders

Table 9. Maxon motor performance (10mm, .75w, plastic planetary gearbox).
Error! Objects cannot be created from editing field codes.

69

were necessary to control joint position indirectly. This allowed the use of packaged
motor controllers ready to interface with a computer serial port. Using these motors, the
net mass of the robot is 235gr, and as Table 9 indicates, they motors satisfy the TSF and

PLC. Unfortunately, the plastic housing is unable to support the shaft loads.

The next units examined were the Maxon RE013. They are 13mm diameter DC motors,
also with magnetic encoders, and planetary gearboxes with metal housings for higher
load capacity. This characteristic is indicated by Table 10 which shows a that the unit
has satisfactory TSF and SLSF. In this case, however, the motor unit is too long, as
indicated by the negative PLC values, and is not feasible. The 13mm diameter Portescap
units shown in Table 11 also have similar capability and also suffer from excess length

demonstrated by their PLC values.

70

Table 10. Maxon motogaerformance (13mm, 1.2w, metal planetary gearbox).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% Motor #1 Motor #2 Motor #3

Model Information RE013 6V 1.2W RE013 6V 1.2W RE013 6V 1.2W
Weight, gr 38 35 35
Diameter, mm 13 13 13
Net Length, mm 58.4 55 55
Gear Ratio 1119 275 275
Max Radial Load, N 40 40 40
Max Shaft Moment, mNm 240 240 240
Torque Constant, mNm/A 3.4 3.4 3.4
S ed Constant, RPM/volt 2820 2820 2820
Torque Safety Factor (TSF) 4.5 5.5 6.2
Package lflgth Constraint (PLC) -3.4 -3 -3
Shaft Load Safety Factor (SLSF) 1.48 12.92 14.44
Max. Power Consumption (MPC) 0.07 0.57 0.26

 

Table 11. Portescap motor performance (13mm, 2.5w, metal planetary gearbox).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9__a_t_§ Motor #1 Motor #2 Motor #3 ‘
Model Information 13N88 12V 2.5W 13N88 12V 2.5W 13N88 12V 2.5W
Weight, gr 33 30 30
Diameter, mm 13 13 13
Net Lethh, mm 67.6 63.5 63.5
Gear Ratio 352 88 88
Max Radial Load, N 20 5 5
Max Shaft Moment, mNm 40 40 40
Torque Constant, mNm/A 9.1 9.1 9.1
Speed Constant, RPM/volt 1053 1053 1053
Torque Safety Factor (TSF) 3.2 4.4 5.8
Package Length Constraint (PLC) -12.6 -11.5 -11.5
Shaft Load Safety Factor (SLSF) 0.90 1.96 2.21
Max. Power Consumption (MPC) 0.08 0.55 0.21
Table 12. Portescap motor performance (17mm, 3.2w, metal planetary gearbox).
_D_a_L_a Motor #1 Motgg_#_2_ Motor #3
Model Information 17N88-208P 6V 17N88-208P 6V 17N88-208P 6V
Weight, gr 43 43 43
Diameter, mm 17 17 17
Net Length, mm 53.8 49.7 49.7
Gear Ratio 166 30.2 30.2
Max Radial Load, N 20 20 5
Max Shaft Moment, mNm 160 160 40
Torque Constant, mNm/A 7.2 7.2 7.2
S ed Constant, RPM/volt 1333 1333 1333
Torque Safety Factor (TSF) 2.8 2.8 3.1
Package Length Constraint (PLC) 1.2 2.3 2.3
Shaft Load Safety Factor (SLSF) 0.81 7.10 8.00
Max. Power Consumption (MPC) 0.07 0.52 0.23

 

 

 

 

 

71

 

 

 

Since the larger units available from Maxon continue to violate the PLC constraint, larger
units from Portescap were examined. The next units evaluated were 17mm Portescap
17N88 geared DC motors with metal planetary gearing and optional ball bearing or
bushing output shaft support. As Table 12 indicates, these units satisfy the TSF and PLC
requirements very well. The optional ball bearing shaft support is used on Motor #2 for a
SLSF of 7.1 and the bushing SLSF is used on Motor #3 for a SLSF of approximately 2.0.
The ball bearing support is also used on Motor #1, but due to the high loads on the bevel
gearing, the SLSF is only 0.8, which is risky for short term operation even though these
ratings are based on infinite life. The only motor that had a reasonable SLSF=1.5 on this

joint was the Maxon RE013 with metal gearbox, but it was 4mm too long.

A compromise had to be made for the sake of producing a prototype rapidly, and the
Portescap 17N88 actuators were selected. The weakness of the steering actuator was
deemed acceptable for short term operation, and the design could be perfected during the

next stage of research.

72

3.6.7 Structural Optimization

Now that the final actuators have been established, the structure of the robot can be
examined to assure that it can support the given loads and to determine if its weight can
be further minimized. The holes that have been shown on the structure until this point
are actually a product of these efforts. Prior to the analysis, however, the joint loads
must be examined one last time for the purpose of applying Finite Element Analysis

(FEA) to the structure.

The robot is once again considered climbing a vertical surface in a straight, cantilevered,
orientation as shown in Figure 29. Unlike the case of the actuator investigations, shown
in Figure 26, the robot is turned 90° about its longitudinal axis such that the bending
moments are applied to the links to exploit their weakest bending strength. This
configuration and corresponding boundary conditions are shown in Figure 29. The

boundary conditions consist of the Joint loads and appropriate anchor points created by

  

Foot 1 supports robot in Link 2 analysis

 

  

   
         
 

‘, \- , 1'; Anchor points
Anchor points .. j 7 ’ . ~ 1- ~ Eng/—
[ F _ _ .- h../ Anchor Points —
Link3 L2 'I M
M
L2 Link 5

Figure 29. Diagram of robot Boundary Conditions used for Structural Analysis

73

the structure. For the sake of worst-case load conditions, it is assumed during the
analysis of Link #2 that the robot is supported by Foot #1, that the strut provides no
support, all resistance to bending is provided by the motor mount locations, and that the
preload of the belt has no effect. Supporting the robot by Foot #1 creates the largest

bending moments in the link.

Based on structural components manufactured from Aluminum 6061 with an un-
tempered yield strength of 8 ksi, the effective force, F L2, and moment, M12, applied to

Link 3 are determined to be,

FL, = W, + W, = 1.94 N (3.37)

ML2 = W3d3 + W;(L, + d4)=136 mNm , (3.38)

In the analys1s of Links 4 and 5, it is assumed that Foot #2 supports the robot. This
creates the largest forces on these members. Contributions of the motors to the loading of

Link 4 have been depicted as F M), Fm, MM), MM; where,

FM1 = FM'2 = 0.42 N (3.39)

MM2 2 MM = 11.3 mNm (3.40)
The external force and moments applied to Link #4 are,

F,3 =W, +w,, +W2 = 1.41 N (3.41)

M3 = W,(1Q - 21,) + W,(z.~ + d!) + WD(L, + d0) = 99.7 mNm ' (3.42)
and the external forces and moments applied to Link #5 are,

FL4 =Wl +WD+W2+W3 =2.63N (3.43)

74.

M..=%(L.+L.-d.)+

‘ (3.44)
+W.(Iq+lg+d.)+WD(I.+I.+dD)+W.(l.—d.)=263mNm

The final analysis of Link #3 is shown in Figure 30. As the image indicates, the largest
stresses occur near the motor bracket mount location, which is anticipated because that is
the anchor location, indicated in , for the link. There are five 7.9mm diameter holes in
the link to reduce the weight of the link by 1.4gr, a 19% decrease in mass of the link.

The maximum stress is 1.9 ksi, which provides a safety factor of 4.2.

The final analysis of Link #4 is shown in Figure 31. Eight 7.9mm diameter holes have
been added to the upper and lower plates to decrease the weight by 1.1gr, an 8% decrease
in structural weight of the link. The maximum stress of 2.7 ksi occurs in the motor

brackets near the joint supports. This provides a safety factor of 3.0 with Aluminum

Stress (psi)
RESULTS: 4- ac. 1,STRESS_4,LOAD Star 2

STRESS - VON MISES MIN: 1.20E-01 MAX: 1.85E+03
DEFORMATION: 2- ac. 1,DISPLACEMENT_2,LOAD SET 2 1.66E+03
DISPLACEMENT - MAG MIN: 0.00E+00 MAX: 3. 73E-03

1 8551-03

1.48803
1.29E+03

‘ (r. .. . '_'I: . j ".
" . . .. , ' ‘ ‘ ' t.~ - -
, -./ _ ''''' . h,
" ' ’ a ' . ' ' ‘ ' '1- ~‘ .‘VA’45;1‘A:‘0' ' +

 

7.40E+02

5.55E+02

3.70E+02

1.85E+02

 

1205-01
Figure 30. FEM analysis of Link 3 subjected to maximum loading.

75

6061, or a safety factor of 5.6 using stronger Aluminum 7075 with an un-tempered yield

strength of 15ksi.

The final analysis of Link 5 is shown in Figure 32. It has a maximum stress of 7.6 ksi for
a safety factor of 1.97 using Aluminum 7075, which is necessary with this part. Some
large stresses are shown near the worm gear mounting pocket, but those can be attributed
to the application of point loads. Maximum stresses occur at the opposite end of the link,
near the foot. These are due to large bending moments. These could be averted by
webbed side structures added to the link, if the aforementioned stresses result in

deformation. Deflection in all of these results was determined to be minimal, with at

most .004”.

76

911969991-

RESULTS: 2- ac. 1,STRESS_2,LOAD SET 1
STRESS - VON MISES MIN: 5.11E+00 MAX: 2.63E+03 263903
DEFORMATION: 1- BC. 1,OISPLACEMENT_1,LOAD SEr 2375,03

DISPLACEMENT - MAG MIN: 0.00E+00 MAX: 2.87E-03
2.11E-I-03

  

1

5.31 E+02

2.68E+02

 

5.11E+00

Figure 31. FEM analysis of Link 4 subjected to maximum loading.

  

 
             
 

             
 

    
  

     

 

Stress (psi)
RESULTS: 2- 13.0. 1,STRESS_2.LOAD 31:11 4 “E+03
STRESS - VON MISES MIN: 8.04E+01 MAX: 7.56E+03
FRAME OF REF: PART “65,03
”-41 ‘0 3.97E+03
M“ 5,350} 1‘3“!)
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. . .1 44"
”11%? 2.51 E+03
mar-"‘7' 7' ‘49 “
'{fillfi’we 153::
1511.111. , _ my 2.03E+03
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4'1‘éfififiwyfi‘lstéht‘fi. 4¢3$¢&.:‘vp‘5§1¢9m'
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199115139315 4,19
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44159939,. 53;; 1:"
5 35mm ‘ 5.67E+02
8.04E+01

Figure 32. FEM analysis of Link 5 subjected to maximum loading.

77

3.7 Testing

The robot was tested climbing on a vertical surface, walking on a horizontal surface,
walking between a horizontal and vertical surfaces (crossing a 90° comer), and walking
between specified locations on a vertical surface (Minor et a1 2000, Dulimarta et al.
2000). During the walking and climbing tests on horizontal and vertical surfaces,
respectively, the robot started from an initial position and walked a specified number of
steps. A sequence of pictures of the robot climbing a vertical surface is shown in Figure
33, Figure 34, and Figure 35. The robot performed these tasks with no difficulties. Only
a small amount of uncertainty was encountered when placing the foot at the end of the
step, and varying the joint angle slightly until the foot was secured on the surface

counteracted this.

One of the most difficult tasks was walking between specified locations on a vertical
surface. While walking between points, the robot was to follow a path on the surface to a
final destination. This task involved walking vertically up the surface, turning 90°,
walking across the surface several steps,‘turning 90°, and walking up the surface another
two steps. Uncertainty of the steering angle played a major while the robot was walking
across the surface since gravity tended to turn towards the ground. A joint level

potentiometer was incorporated to offset this effect.

78

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11“

111|Il| ‘.1‘1‘":"I‘Z”i"’ 1 II-IIIIIIIIIIII "1 H1||||I|l||111 ||1I1|l|1I|1I1||I1IIIIIIIIIIIIIIIII
‘ 111111
1111))111. 1. 1111111 11) 111IIIIIIIII|I|I|I|IIII..I11|I1IIII||||

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‘ H’ .

.._.
#51:, ... __
‘— .4: _,
- 'L.
— f—
, ..
- v,

4:17 f_’ 1111‘ “"1th \
11) E1111 Wfiuh‘lh‘nn“ V 11111 "W". 1111511)) I 1111111111111
1‘1 “1“”; ”1” 1111' 11' ”1‘ 1,1 Ill \ III

Figure 33. Biped climbing a vertical surface. Start position and beginning of motion.

111 '11'11“

 

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11H "(‘1’
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Figure 34. Biped climbing a vertical surface, halfway through Step.

‘ ’ 1111111” 11‘ .11111111

11 '11 '. 141W 111'1' ' ‘
"I“ 1:11-11:11) 11111111111111“) 1111111111" )1“ )II11I)1\11))1“ 1111 ._

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Figure 35. Biped climbing a vertical SUrface. Near end of step and fnal position.

79

The most challenging task was crossing the 90° corner between the vertical and
horizontal surfaces. Again, uncertainty caused this difficulty because the robot does not
have adequate sensors for determining the attitude of its feet with respect to the climbing
surfaces. Our test was based on the hypothetical situation that if the robot was placed in a
known position, it could move its joints to a determined set of angles and the foot should
be very close to perpendicular to the next surface. With slight reorientation of the
angles, the foot will make contact and the robot can cross to the inclined surface. Again,
however, since joint level control is not used, backlash affects the joint angles and the
final position of the foot will not quite be perpendicular. The affects of gravity and
compliance in the feet also add to this problem. Hence, until sensors such as feelers or
proximity sensors, are incorporated into the robot, climbing between surfaces will be

difficult.

3.8 Future Work

Research during the next stage of this project will proceed in two directions. One focus
will be the further miniaturization of the aforementioned climbing robot to about half its
current size. This will provide formidable design challenges due to miniaturization and
restricted package space. Benefits will be reaped in the form of weight savings and
extended operability. This second prototype will also permit further exploration into the
design of the structural components for improved rigidity and efficiency, as well as joint

level sensors for improved accuracy.

The second, and more significant, research focus will be on the development of the
hybrid joint. While the proposed joint permits prismatic and revolute behavior, its
revolute behavior is somewhat limited in range of motion. In fact, the hybrid joint only

permits articulation in one direction, albeit between zero and 120°. This limited

8O

articulation allows the robot to cross 90° and 270° corners satisfactorily and perform the
crawling and sliding walking strides. It does not, however, permit the robot to perform
the flipping stride effectively. Incorporation of joint level sensors into this design will be

very challenging.

3.9 Other Considerations

In this chapter, design of a climbing robot for superior mobility under significant
limitations on available space and power has been studied. The robot developed is
capable of adapting to walking or climbing on vertical or horizontal surfaces, traversing
between surfaces with various inclinations, and also steering to any point on a surface.
The robot designed is motivated towards being the smallest robot of its nature and
possessing an adaptable structure to allow the robot to traverse confined spaces as well as
rapidly across surfaces. A conceptual hybrid prismatic-revolute joint was presented for
this purpose. The initial robot, however, for the sake of rapid exploration, was built two
times scale and with only a revolute hip joint. This allowed a prototype to built rapidly
and reliably, and to permit the exploration of coupled DOF for reduced weight. An
analysis of the force in the robot was then conducted to determine the set of feasible
actuators, and then a FEM analysis of the structure was performed to assure adequate

strength and provide further weight reductions.

Design of the robot emphasized many important points, only some of which were
encountered during design of the surgical instrument. Like the surgical instrument
design, the climbing robot design required design innovations combined with engineering
analysis and optimization to determine the most adaptable and trustworthy kinematics
structure. Unlike the surgical instrument, however, weight, actuation, and power usage

are major issues for the climbing robot. This is partially attributed to limited holding

81

capacity of the suction feet, which in turn limit space and weight available for the power
supply and actuators. Also unlike the surgical instrument, the actuators themselves play a
major role in the design; both in terms of the weight that they contribute and the loads
that they can support. As such, a balance between actuator weight and capacity had to be
achieved, which ultimately resulted in the actuators contributing to almost 50% of the
robot weight, whereas the robot frame only contributed to 10% of the weight. While this
may not be the case with less constrained robots, it certainly was an issue here. This
application of mobile robotics makes it very clear that actuator weight is one of the major
issues in designing mobile robots and we pursued designs based on under actuation.
Future advances in mobile robot design will be accomplished with further research on
under-actuated mechanisms and development of actuators with higher power to weight

ratios.

82

4. Control of a Spherical Robot for Enhanced Mobility

4.1 Motivation

In the previous chapters, the focus was design for dexterity and mobility given constraints
on space, weight, and power. In this chapter, an equally important limitation is addressed
during the design of a mobile robot: control of a robot for maximized mobility given
limited computational ability. Since the wall-climbing robot possesses straightforward
kinematics, computational usage for control and motion planning is not an issue. To
address computational usage directly, a particular system is chosen with substantial
nonlinear constraints that do not readily present a solution. The task at hand is point-to-
point reconfiguration of a spherical robot. Such a robot has highly nonlinear
nonholonomic rolling constraints, and traditional non-linear control techniques are not
applicable. Several techniques have been presented in the literature, but in this chapter
we develop a simpler and less computationally demanding algorithm for controlling the

sphere. We now examine the problem closely.

Traditionally, mobile robots have employed multiple wheels for locomotion and stability.
In contrast, the spherical robot discussed here relies on its outer shell for mobility and
support. The robot, named Spherobot, is shown in a sectional view in Figure 36. Notice
that the robot uses a central body with reciprocating masses supported on radial spokes in
a tetrahedral arrangement. These masses cause the robot to roll when their position and
accelerations impart a reactive moment on the robot body. In this conception, the robot
also provides a retractable camera for reconnaissance purposes and retractable limbs that
provide stability while the camera is in use. Additional end-effectors and sensors can be

incorporated to provide functionality for retrieving specimens or detecting

83

 

 

 

 

 

Figure 36. Spherobot Mobirefobot.

chemical/nuclear threats. The sturdy exoskeleton provides protection for the robot in

hostile conditions.

Spherobot is not the first spherical robot to be conceived. To date in the literature, this
honor belongs to (Halme, Schonberg, et al. 1996), whose spherical robot contains a
single-wheeled device that drives around inside the sphere to create a rolling moment by
offsetting the sphere’s center of gravity. A similar concept was soon after exploited by
the Sphericle (Bicchi, Balluchi, et al. 1997), which uses a car driving around inside the
sphere to create the propelling moments. Because of the rolling nature of the
mechanisms used to drive the counterweights, both robots have additional nonholonomic
constraints that linrit mobility and complicate control strategies. The reciprocating
masses used in Spherobot can shift their cumulative center of gravity and acceleration in

any direction at any time, which provides superior mobility.

In this chapter, our endeavor is to create simple control strategies for moving Spherobot
to a specified location and orientation. The challenge of this task is created by the non-

slip rolling constraints, referred to as nonholonomic constraints, which prevent direct

84

 

 

Start
E: [:1 E Q
r: [:3 [[25 Q
l:]( D
1:]

Finish

Figure 37. Nonholonomic constraints: parallel parking.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

linear motion to the prescribed final position and orientation. The parallel parking of an
automobile, shown in Figure 37, is a simple example of nonholonomic constraints. The
tires of the automobile prevent the vehicle from moving directly sideways into the
parking space. Rather, the vehicle must steer along a complex curve to reach its parallel
parking space. While rolling Spherobot to its final destination, similar, but more
complicated, constraints are encountered. This subject has even gained a classical name

in the literature: the ball and plate problem.

In recent years, numerous researchers have studied the ball and plate problem. In 1990,
Li and Canny (Li and Canny 1990) considered the problem in the context of rolling
surfaces. They showed that the system was controllable and they also defined a three
step algorithm for completing the task. The first step converges the position coordinates.
The second step converges two of the three orientation coordinates using Lie Bracket
motions, which create a spherical triangle on the surface of the sphere. The last step uses
a circular trajectory to converge the third orientation variable. More recently, Bicchi
(Bicchi, Prattichizzo, et al. 1995) investigated a technique, with limited success, that
applied a series of piecewise independent controls and then integrated the state equations
to solve for the magnitude of each of the pieces. Mukherjee (Mukherjee and Minor 1999;
Mukherjee, Minor, et al. 1999) later presented two algorithms based on spherical
trigonometry and on the geometry created by the motion of the sphere on the plane to

achieve controllability. The later will be discussed further in this chapter. Optimal

85

control, in the sense of control effort minimization, of the open loop problem has been

addressed by Jurdjevic (J urdjevic 1993).

As the parallel parking example points out, direct motion between initial and final
configurations is not likely, and the first question that must be answered is whether a path
exists between the specified orientations. This will be one of the first subjects that are
addressed. A proof of existence of a path requires a mathematical model of the sphere’s
kinematics, which will first be developed in Section 4.2. The existence of a path between
configurations will be discussed in Section 4.3. In section 4.4, other kinematics models
in the literature are reviewed. Then the application of traditional nonholonomic control
techniques are discussed in Section 4.5. We present our motion planner in Section 4.6.
Simulation results are presented in to demonstrate the efficacy of the motion planner in

Section 4.6.4.

86

4.2 Kinematics Model

Consider Spherobot on the x-y plane, shown in Figure 38. The position of the sphere is
determined by the point Q, at the center of the sphere, relative to a globally fixed
Cartesian coordinate system with its origin at point 0. It is assumed that the sphere rolls
on the (x, y} plane without slipping or spinning. Orientation of the sphere is determined
relative to point P on the surface of the sphere. A new Cartesian system {x’, y’} is
established by rotation about the z axis by angle or, —7r S a S It , such that P lies in the X’-
2 plane directly above the x’ axis. The elevation of point P is described by the angle 9,
O S 6 S 7: , measured about the y’ axis relative to the z-axis. Control inputs consisting of

angular velocities, a); and a); , are aligned with the x’ and y’ axis, respectively. The

position of the sphere is described in the x-y plane by the vector, R00,
Rog = {L y} (4.1)

As indicated by Figure 39 the sphere contacts the plane at point R, and the velocity of

W \

R=(rx,ry)

 

Side View

 

 

a»
O X
Figure 38. Spherobot Coordinate System.

87

point Q relative to the plane can be determined by,

R00 = {8‘} = RQR = ”X RRQ ={ 60, } (4-2)

y __wx

where {(x, y) e 932}. The control inputs (0, and my can then be expressed relative to

the x’ and y’ axis by the transformation:

60, cos a — sin a a);
= . , (4.3)
my srn a cos a my

Applying this transformation to Equation (4.2) yields,

5: sina , cosa ,
[H ]w[ l.., (4.4)
y —cosa srna

State equations for aand Hare derived from the position vector RQp,
RPQ = {sin 0cos a sin 0cosa cos 0}T (4.5)

Taking the time derivative of the position vector yields the relative velocity vector,

 

 

 

Figure 39. Spherobot Orientation Variables.

88

6cosBcosa — dsin (93in a

R

PQ = boos 63in a — dsin 0cos a (4.6)

—9sin6
Note, however, that these velocities are created by the control inputs (0; and w’y. The

velocity vector can then be expressed as a function of the control inputs as,

a); sin acos 6 + (0’, cos acos (9
RPQ = a) x RPQ = (0; cos acos 6 + a); sin acos 6 (4.7)

—a)’y sin 6
Equating terms of Equations (4.6) and (4.7) yields,
('9 = a); (4.8)

d = —a); cot 6 (4.9)

where {66 9i‘ I-fl < 6 < 71,61: 0} and a e 9“. The input to; also causes the sphere to

rotate about the axis QP, which results in a change of the orientation variable (0

determined by,

I

é>=w.

 

= a); csc (9 (4.10)

8111

Together, the position and orientation variables then form a state model, which

completely describes the sphere rolling on a plane:

Vi- F sina ' Pcosaq

)3 —cosa sina

9 = O w§+ 1 a); (4.11)
d —cot6 O

_¢j _ cscH _ _ O _

 

 

 

 

 

 

I I
Xl cox + X2 coy.

89

Note that as 6—> 0 or in, three things occur. First, the rate of change of a and (I)

become extremely sensitive to input in the a); direction. This is demonstrated by,

Xr(9=0)=[sina —cosa' 0 —oo oo]

To maintain boundedness of a and ¢ near this point, control u, = a); must become very

small and actually go to zero at 6:0 or in. This could be accomplished by a

coordinate transformation such as,
a); = (bx sin 6 (4.12)

Secondly, note that the orientation of a); is defined by the angle a formed by the

position of point P above the x-y plane, Figure 39. The angle or becomes undefined
when 6:0 since P is directly above, or below, the center of the sphere. As a
mathematical state variable, however, a will retain its value when 6:0 as long as
a); = 0. Hence, in a practical application, there is no reason why (0'y could not be used
to drive 6—> O or i 7:. In fact, since a is not clearly defined when 6 = O, a could just
as conveniently be assigned to a new value that might define a more desirable orientation
of a); and w’y. This point will be exploited in Section 4.6.3 for the definition of a full
reconfiguration motion-planning algorithm. Thus, the limitations of set M will actually

be exploited rather advantageously.

A third consideration, and probably most important, regarding 6 = O or i 7! must also be
discussed. Consider the point P on the surface of the sphere. Normally, when
6 at O or i 71', both variables a and 6 are required to describe the longitude and latitude
of point P. However, when 6—> O or i 7: , the point P is at the topmost, or bottommost,
position of the sphere, and or becomes poorly defined. Note, however, that when P is at

these extreme positions a is not necessary for describing the point’s position. Hence, at

90

these two extreme locations, the a state variable is unnecessary, and the system could be
reduced to a four state model! This aspect is key in the complete reconfiguration
algorithm discussed in Section 4.6.3 where the point P is brought to the topmost position

during the algorithm.

The sphere’s path curvature on the plane is determined by

[XWT

pit = 2
( ) [xy—yx] (413)
[(2);2 +w;2]3 '

2
I3 I - I I ' I I2 I
[—01, cot6+coxwy —a)ya)x —a)y a), cot 6]

 

If a); at 0 and a); = 0 , then the curvature is infinite which is the case of linear motion in

the x’ direction. Conversely, if a); $0 and a)’y =0, it can be shown that curvilinear

motion is created, indicated in Figure 38, and is described by,
p = tan 6 (4.14)

Motion planning strategies presented later exploit these curved and linear paths to derive
methods for converging the sphere to a specified final orientation and position. When
applying such independent controls, the a and ¢ states can be expressed as dependent

functions,
é): —dsec6 (4.15)

and the state model can be reduced for simplicity to,

91

5c = a); cosa+w; sina

. I - I

y = a) srna-a) cosa

. f ‘ (4.16)
6—wy

a = —a); cot6

where a desired change in ¢can be determined as a function of a and 6 during the motion

planning stages.

4.3 Reachability of Arbitrary Configurations

Since we assume that no slip is present in the sphere’s motion, the state model of the
sphere is of the form of a drift free system. Reachability of arbitrary configurations of
such a system is determined by first considering a generic drift free system with multiple

inputs of the form,

p:ZuiXi(p)=ulX1(p)+u2X2(p)+...+ume(p) (4.17)

i=1

Herman and Krener (Halme, Schonberg, et al. 1996; Hermann and Krener 1977) discuss

many valuable concepts regarding controllability of such a system. Ins the context of a

system in the form of Equation (4.17) defined on a set M ; 9i" , define A(po) as the set
of points that are accessible from an initial point, pO 6 M . If A(po) = M , it is said that

the system is controllable at x0, and if A( p) = M V p e M , it is said that the system is

controllable over the set. Controllability is determined by Hermann and Krener (Halme,
Schonberg, et al. 1996; Hermann and Krener 1977) and elsewhere in the literature using

Chow's Theorem (Chow 1939), with some modifications (Li and Canny 1990).

The theorem begins with a drift free system of the form of Equation (4.17) where
pe M c; R" E R5, U =[u1 um]e N g R'" E R2. Then, based on the control vectors

92

Xi, establish the lie distribution V( p). In the case of a two input five state model, we

have,

v(p).—.span[Xl X2 X3 X4 X5]

3X 3X
X3 =[X1’X2]=3p‘2'81 __ap_182

X, =[X,,X,] (4.18)
X5 = [X29 X3]

If the rank(V(p)) == an e M , then the controllability rank condition is satisfied and the

system is weakly controllable over the set M. Loosely speaking, weak controllability
over a set will guarantee that a path exists between points in the set, but that the time
required for completing that path may not be finite. Of course it is also required that the
system (4.17) be well defined over M such that a solution to the differential equations

exists.

Recall from Equation (4.11) that the model of Spherobot is controlled by the vectors

X,=[sina —cosa 0 —cot6 csc6]

X2=[cosa sina l O 0]
where

M = [{x,y,a,6,¢} e 9i5|—7r< 6<n,6¢ o]. (4.19)

Note that M is defined in a way such that X; is bounded over the set and its integral is
well defined. In practicality, however, this restriction does not mean that 6 = 0 or in
cannot be reached by the system. Clearly the control vector X2 permits full control of 6 ,
regardless of its value, but a few subtle points must be considered. First, as pointed out in

Section 4.4, the primary purpose in limiting M is to assure that the state model, in

93

particular or, is well defined. This is easily managed by assuring that (1); =0 when

6 = 0 or in, as prescribed in Section 4.4 via Equation (4.12). Secondly, the limitation

that a); = 0 does not necessarily mean that the system can only be steered in a single

direction. Recall that a is not well defined in this situation, implying that it could take
on any value, and hence it can be arbitrarily assigned such that a)’y has a desirable
direction. This property is exploited later in the full reconfiguration algorithm, Section
4.6.3, to steer the sphere in a new direction when 6 = 0. It can thus be argued that the
application of Chow’s theorem, will provide proof of controllability over M, but that set

M will not necessarily be the only controllable region.

Per application of Chow’s Theorem, controllability over M can be proven by examining
the dimension of the lie distribution described by Equation (4.18). First calculate X3 via

the Lie bracket of X 1 and X2,

X3 2 [X,, X2]
T (4.20)
= [sin acot 6 —cos acot 6 0 —1 — cot2 6 csc 6cot 6]
then X4 is calculated similarly by the Lie bracket of X1 and X 3,
X, s[X,,X,]=[cosa sina o o 0]T (4.21)
and then finally X5,
X5 5 [X2’ X3]
= [—2 sin a(1 — cot2 6), 2 cos a(l + cot2 6), (4.22)

0, 2cot 6(1 + cot2 6), — 2csc 6(1+ cot2 6)]T

Using these vectors, the distribution V is formed:

94

sin 61 cos a sin acot 6 cos a —2 sin a(1+ cot2 6)—
— cos a sin a — cos acot 6 sin a 2cos a(1 + cot2 6)
V = O 1 0 0 O (4.23)
- cot 6 0 —(l + cot2 6) 0 2cot6(1+ cot2 6)

csc 6 0 csc 6cot 6 0 2csc 6(1 + cot2 6) j

 

 

and the determinant of the distribution is,

det V = -— csc 6 (4.24)

Since (4.24) is well defined and non-zero over M, it is concluded that V maintains full
rank. According to Chow’s theorem, the system is controllable between any prescribed
configurations in M. A similar conclusion follows for the reduced order system described

by Equation (4.16). The Lie distribution for this system is,

P sin a cos a sin acot 6 cos a-
—cosa sin a —cosacot 6 sin a
Vredur‘ed : [X1 X2 X3 X4] = O 1 O 0 (425)
_—cot6 0 —l—cot26 0 _

 

 

which has full rank over M, as the determinant indicates:

det V - 1 (4.26)

reduced —

Thus, the reduced order system is also controllable over M. It can be argued further that

the reduced order system can also be used for complete reconfiguration of the system.

Recall Equation (4.15), which demonstrates coupling between a and ¢ when the system

is subjected to (I); alone:

(i): —c'tsec6 (4.15)

Later in this chapter, section 4.6.3, an algorithm will be specified that exploits this

property to drive the system from some random initial configuration to one with 6 = 0

95

where the value of a is irrelevant for describing the sphere’s orientation. Since a is
unnecessary for describing this final position, the variable a can be used with Equation

(4.15) to yield a desired final (0. Thus, the system can be completely reconfigured via

the reduced order system. Reconfiguration to final positions with 6¢ 0 can be

accommodated via a coordinate transformation in 6 and ¢ that would place a different

point, say P’, to the topmost position on the sphere.

4.4 Kinematics Models in the Literature

The rolling constraints imposed above are identical to those in the controls literature
referring to the classical ball and plate problem (Bloch and Crouch 1995; Brockett and
Dai 1993; Jurdjevic 1993), and the more general rolling contact of rigid bodies problem
(Bicchi, Prattichizzo, et al. 1995; Li and Canny 1990). In the ball and plate problem,
locomotion is achieved by relative motion of two parallel plates on opposite sides of the
sphere in non-slip contact. The rolling contact problem typically does not discuss
actuation. In the case of Spherobot, locomotion is achieved by positioning internal
masses such that an unbalance is generated, which creates a driving moment and rolling

motion on the plane is achieved.

In these other works, different methods have been used to describe the orientation of the
sphere. As the reader will recall from Section 4.2, the definition of the kinematics model
can yield very beneficial behavior that can be exploited for control design. The
formulation used by Li and Canny (Li and Canny 1990) tracks the contact point between
the surfaces of the sphere and plane in R4 and establishes a fifth variable I]! for contact
angle. Spherical coordinates [u], v1} and Cartesian coordinates lug, vz} determine the
contact point on the surface of the sphere and plane, respectively. Angle wmeasures the

angle between the x—axis of the spherical and Cartesian systems. This formulation is

96

quite similar to our own, with a few exceptions. Whereas we track the point P, which we

wish to drive to a final orientation, they track the contact point between the sphere and

ground. They also measure the inclination of the point from the x-y plane, where we

measure the inclination from the z-axis. The unit sphere formulation of the kinematics

model is (Li and Canny 1990):

 

 

P - r -

O
secul 0
—sint// a) + —cost// a)
—cost// sinw

 

 

 

 

_— tan u1 _ 0

(4.27)

where, similar to above, the inputs or), and a)y are oriented to the moving frame of the

sphere. Similar controllability of this realization is demonstrated by its Lie distribution,

 

F 0 -1
sec u1 0
VLi,Canny :
— cos w sin y!
— tan u1 O

0 0 0

secultanuI 0 —secu,(1+2tan2u,)
-sint// -cost// —sint//tanu1 —cosw 23int1/(l+tan2ul

—cost//tan ul sin r/I 2cos l/f(l + tan2 ul
—(l+tan2u,) 0 2tan ul(1+tan2u,

 

)
)
)_

which has full rank over M =[{ul,vl,u2,v2,t//}E9i5| —7r/2 < ul <7r/2, -7t<u2 <7t]

as demonstrated by the determinant,

det V Mam”, = sec u1

(4.28)

and hence controllability is proven over M. Since M does not cover the complete surface

of the sphere, Li and Canny define a second model, which is supposedly controllable at

the remaining points not contained in M, but this fact has not been able to be confirmed.

97

The kinematics formulation of (Bicchi, Prattichizzo, et al. 1995) is fundamentally similar

to the (Li and Canny 1990) model, with the exception that the inputs a), and a)y are

relative to the fixed x-y axis, respectively. Such a formulation is,

 

 

g. g. g. g. a.
ll

sint/I/cosv

 

tan v sin l]!

0
—1
(1)
cos l/l

 

+

r 1
0
cost/I/cosv
—-sin l/l

 

 

- tan vcosr/I j

(4.29)

where {x, y} is the location of the sphere on the plane, (a, v} is the location the point on

the surface of the sphere, and W is the contact angle. This model is then reformulated by

transforming the inputs according to,

to generate the system model,

 

cos v sin l]!
a) =
cosv cos w

I- -1

l
0
sin v

cos t]! cos v

 

 

 

“a R' s- <. :0

—sin r/Icosv

cos w

, é)
—s1nr//

—sinr//

 

 

—cosr//

(4.30)

(4.31)

which, similar to models (2), (4), and (5), permits direct control over the sphere

orientation variables. More importantly, Bicchi (Bicchi, Prattichizzo, et al. 1995) uses

this formulation to solve the state equations analytically given a series of piecewise

independently applied constant controls, ii), and 502 , which are then solved for

numerically to drive the system to a desired trajectory. These equations are very

computationally intensive to solve and not well suited to real time control.

98

4.5 Traditional Nonholonomic Control Techniques

To date, the best understood class of nonholonomic systems is the canonical chained

form. First and second order examples of such systems are, respectively:

5‘1 = “1
X1 = “1 5C2 = “2
X2 = u, and x2, = x214,
J.‘3 = xzur 55211 = xzrur
55212 = xzruz

where u1 and u2 are control- inputs. Extensive research in the literature has been

dedicated to controlling systems in this form, and to converting systems to this form for
that very purpose. The method of steering such systems by sinusoidal inputs was studied
initially by Brockett (Brockett 1981) and in more detail by Murray and Sastry (Murray
and Sastry 1993). Steering these systems using piecewise constant inputs was
investigated by Bicchi (Bicchi, Prattichizzo, et al. 1995), Jacob (Jacob 1992), and
Lafferriere (Lafferriere and Sussmann 1991). Exponential feedback stabilization has
been accomplished by Sordalen (Samson 1995) using a non-smooth time varying
feedback law, by Morin (Morin and Samson 1997) using backstepping techniques, and by
Tayebi (T ayebi, Tadjine, et al. 1997) via an invariant manifold approach, to only name a

few.

Feedback techniques for converting noncanonical systems to chained form have been
presented by Murray (Murray and Sastry 1993) and Sordalen (Sordalen 1993). Both use
a constructive proof based on the differential properties of a canonical system to
determine a feedback transformation to convert the system to chained form. Thus, the

techniques prescribe a set of sufficiency conditions that such a function should satisfy as

99

guidelines for selecting a transformation. Hence, the process requires intuition and trial

and error, on the behalf of the user.

Fortunately, a test is known for determining if a transformation to chained canonical form

exists (Murray, Zexiang, et al. 1994). This test is based upon the controllability filtration

of the system, I“, ,

F0 = span(Xl, X2)
1‘, = r, +[1"0,I‘0]
1“2 = 1“l +[r,,r,] (4.32)
Pk = Flt-l +[Fk—1’FO]
The growth vector, y , formed by the controllability filtration is,

y = [dim 1“,, dim 1“,, dim 1“,] (433)

Which is then used to define the relative growth vector, 0'. For five-state two-input

systems, the relative growth vector must be of the form,
a = (2,1,1,1)

for a transformation to chained form to exist (Murray, Zexiang, et al. 1994). For the

sphere, the corresponding growth vector is,
y = [2, 3, 5, 5] (4.34)
which corresponds to a relative growth vector,

0 = (2,120)

100

Clearly, the system cannot be converted to chained form and the aforementioned

techniques are not applicable.

4.6 Simple Geometric Motion Planning

In this section we define geometric algorithms that exploit independent actuation of a);

and (0’y for establishing a path between initial and final configurations. These control

actions are defined as,

a) (02:0 and (0;;t0

b) w1¢0, a); =0,and 6¢0,i7z.

Control action (a) results in linear motion along the x’ axis and varies 6. Define point F,
as shown in Figure 40, as the point along the x’ axis where the sphere stops when 6—-> 0.

Note that control action (b) causes the sphere to travel along a curve orthogonal to the x

axis with radius p as determined by Equation (4.14), repeated here for convenience,
p = tan 6

As 6—> 0 , note that the radius of curvature approaches zero, but that as 6—> in/ 2 , the
curvature becomes infinite, which corresponds to linear motion. Control action (a) is
used to modulate the radius of the path, since 6 = w’y. This moves point C along the x’
axis. When control action (b) is applied, the sphere then travels along the curved path, as
indicated in Figure 40. Note that during action (b) the variation of aand ¢ are dependent
on 6 (see Equation (4.11)), but that 6 is constant. Since 6 is unchanged the distance CF
remains constant, and F travels in a circular path concentric with the paths of the points Q

and P.

101

4.6.1 Partial Reconfiguration Algorithm

These properties will now be exploited to design an algorithm for moving the sphere to a
specified point with a desired orientation. For simplicity, let that point be the origin of
the coordinate system and let the orientation be such that 6 = 0. Reconfiguration with
the final values of a and ¢ also considered will be discussed in the next section.
Alternate final positions, and values of 6, can be accomplished by coordinate

transformations.

The steps of the algorithm are now defined with reference to Figure 40. Assume the

sphere is in the initial position {x0 y0 a0 60} with fixed point P and pivot point C as
indicated in Figure 40. The partial reconfiguration the steps are as follows:
1. Define the unique circle passing through the points 0 and F, and with its center on

the line CF at point C’. Assuming that the sphere is initially at point Q with
6 = 60. Apply control action (a) to drive 6 —-> 6’ and C —> C’ .

L
’

 

 

 

 

 

 

 

 

 

Figure 40. Partial Reconfiguration of the sphere.

102

2. Apply control action (b) such that the sphere travels along the circle and F ——> 0 ,
Q’—>Q”,and P’—> P”.

3. Apply control action (a) such that 6—9 0 , Q” —> 0 , and P” returns to its topmost
position. This completes the reconfiguration maneuver.

Several calculations must be completed prior to executing this procedure. The equation

of the circle passing through points 0 5 (0,0) and

F E (xF, yF) = (x0 —6cosa', y0 — 6sin a)

must be determined. The center of the circle, C’ 5 (a,b) , lies along the line with slope a

passing through point (x0, yo) , which is defined by,

b=atanao+(yF—thana0)=(a—xF)tanao+yF (4.35)
Equating the distances 0C ’ and C’F, we have
(12+!)2 =(a-xF)2+(b— y,,)2 (4.36)
Equations (4.35) and (4.36) are then solved for a,

2 2
a : xF - yF +2xFyF tan an (4.38)
2(xF + yF tan a0)

 

and then Equation (4.35) is used to solve for b. The angle 6’ , 0<6’SIt/2, is then

determined from,

tan6’-6’ = Jaz +1)2 (4.39)

The angle of segment CO, corresponding to the final angle a! of the x’ axis, is,

(”l
af = arctan —
-a

103

4.6.2 Simulation of Partial Reconfiguration Algorithm

Simulation of the partial reconfiguration process will now be demonstrated. Let the

sphere the sphere begin at the initial condition:

{x y a 19 ¢G}’={433 2.50 ”/3 ”/6 o}’
The final conditions are required to be,

{x y 6}={0 0 0}

where a and ¢are allowed to vary freely. Figure 41 shows the reconfiguration process

viewed from directly above and Figure 42 shows the variables as functions of time.

The sphere starts at point Q with the camera opening at point P and the point q on the

equatorial circle orthogonal to QP. The fixed point, F, is determined to be
F =(xF, yF) = (4.07, 2.05). Point C, the center of the concentric paths traveled by

points Q and P, is calculated via sequential application of Equations (4.38) and then

 

 

 

 

 
 

1 1 1 l

- I
-3 -2 -1 0 1 2 3 4 5 6

 

Figure 41. Partial reconfiguration simulation: top view.

104

(4.35) to be C=(a, b)=(2.7l, ~31). Angle 6’ , which defines the radius of the

concentric paths about point C, is calculated via Equation (4.39) to be 6’ = 1.33 = 76.1°.

During Step 1 of the partial reorientation process, the sphere rolls in the ac direction, via

control action (a), until 6’ = 1.33 as indicated in Figure 42. Control action (b) is then
applied during Step 2 point F reaches point 0 (Figure 41), which occurs when
a = a, = 3.03 = 173° (Figure 42). At this location, the center of the sphere is 1.33units
away from the final point 0. Application of control action (a), Step 3, completes the
process by driving 6—>0, Q-> 0, and q”—>q”’. Notice that coincidentally the

reorientation of point q is A¢G = —27r, which means that point q maintains the same

orientation in the x-y plane.

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 1 1 Step 2 1 Step 3
f v I T I .l j I I
. I . j . . L 1 ; .
. k. ‘ j I ; .
xo~ I . 1 f; ;
1 - 1 . ,
1 l 1 1 l 1 1 l I 1 l
o l 50 1 100
T I ! I I I I I I I I
’ I ‘ l ; ‘
l : I ; ;
o l l I L 1 ; l 1
0 l 50 | 100
2 r r ' I I r r I I r I,
. I ' - 1 Z
1 i i 3 ¥
0 1 1 1 1 1 r 1 g 1 1
0 j 50 I 100
I f | 1 I I 1 I I I 1
i g 3 ' 1 i i F
a2). . ; . r ;. j . .;. . ,. . _.
i 1 _ l i f
1 1 1 1 1 1 1 r J 1 1
0 j 50 l 100
r r l I 1 i T I
i E ' . ' i :
¢ . I I l '
_1 1 1 l 1 1 1 1 1 ' 1 1
O 10 20 I 30 4O 50 60 70 I 80 90 100
<— Step 1 :1? Step 2 444 Step 3—>
Tlme.sec

Figure 42. Partial reconfiguration simulation: chart view.

105

4.6.3 Complete Reconfiguration Algorithm

The previous algorithm can be modified to also move the angle (2), defined in Figure 39,

to a Specified final value as well as x, y, and 6. This modified algorithm will drive
6—> 0 when the sphere reaches its final position and the point P to its topmost position
on the sphere. At this location, P is directly above the center of the sphere and a

becomes undefined. Hence, a is not mentioned as a specified output of this algorithm.

The variable ¢, however, defines the location of a point q on the equatororial circle of

the sphere perpendicular to line QP. Thus, specification of (I) is actually required to

complete the description of the sphere’s final orientation. The variable a does play a

role in defining the algorithm for complete reconfiguration Since a and (I) are coupled by

Control Action (b), as indicated by Equation (4.15), shown below for convenience.
(l): —c'tsec6 (4.15)

Recall that control action (b) does not affect 6 , but actually causes the Sphere to travel on

the x-y plane along a curved path with radius p=tan6. While traveling this path,

control action (b) integrates the value of a if 6 ¢ in/ 2 and results in a net change of a

denoted as Aa. Equation (4.15) can then be reduced to,
ma = —Aa sec 6’ (4.40)

where 6’ is the value 6 of while control action (b) is applied. This relationship indicates

that a desired A¢ can be accomplished by an appropriate selection of 6’ and Ad. First
recall, however, that (I) is defined relative to the x’ axis, as indicated in Figure 39, which

rotates by Aa during this algorithm. If we define A¢G as the change in (0 relative to the

globally fixed x-y coordinate frame resulting from the orientation process, we then have:

A¢G = A0!(l - sec 6’) (4.41)

106

The complete reconfiguration algorithm will now be defined so as to provide a basis for

discussing Equations (4.40) and (4.41) further. Referring to Figure 43 and Figure 44, let
the initial and final values of the states (x, y, 0, or, (1)) be (x0_ yo, 00, 010, (pa) and (0, 0, 0, 0,
(pi ), respectively.

1. The Sphere starts at the initial position (x0, yo) with point P oriented by 60 and
a0, which also defines the x’ axis. Apply control action (a) to make 6 —) 0 ,
which takes P to its topmost position and the center of the sphere from
(x0, yo) —> F E (xF, yF) = (x0 -— 6o cosao, y0 - 60 sin (10).

2. Since a is undefined when 6 = O, a can be specified such that sphere will roll in
a new direction a’ when control action (a) is applied. Apply control action (a)
such that 6 a 6’ and Q —> Q’ , P —) P’ and C —> C’. Note that this step defines
a unique circle passing through points 0 and F with its center at C’. Variables a’
and 6’ , which determine the location of C’ , and hence the circle, are calculated
in part by either Equations (4.40) or (4.41), such that Act = a’ —- af and

A¢ = (1),, — (1),, will be satisfied.

Side View

 

y' ,y .
V
Figure 43. Complete reconfiguration process, top view.

107

3. Apply control action (b) to drive F —-> 0 along the circular segment designed in
the previous step. Note that Q’ and P’ travel along concentric segments such

that Q’-—)Q”, P’—> P”, a’—>af,and ¢o—>¢f. Notethat (0 and a have
converged to their final values!

4. Now apply control action (a) to drive 6 —> 0 which brings Q” —9 0 and hence
(x, y) —> (0,0). Note that control action (a) neither affects a or (D , and complete
reconfiguration is complete.

The reconfiguration algorithm hinges on the calculation of the circle created during step

2. Since points F and O are pre-determined, the circle is dependent on the coordinates of

C’ E(a,b), which are calculated based on the choice of Aa = a f —a’ and 6’ which
create a desired A¢. The angle A0: is traversed by the robot during step 3 when

F —> 0. By inspection of Figure 43 and the application of trigonometry, the locations of

points 0, C’, and F reveal,

. 1 x2 + 2
Aa a a, — a" = 2msm[§‘/fi) (4.42)

This equation can be expressed as a function of 6’ . To do this, recall that the radius of

the circular path through 0 and F is expressed as p= tan6’ . Inspection of Figure 43

then yields the following expression since the sphere has unit radius:

p = C’Q’= tan 6’ = \laz + b2 + 6’ (4.43)

 

 

Figure 44. Complete reconfiguration process, perspective view.

108

Terms in 6’ are then moved to the right hand side to provide,

tan6’ — 6’ = \(a2 +1;2 (4.44)

Substituting this expression into the denominator of Equation (4.42) reveals,

2+ 2
Aa=2arcsin[ V“ y” ) (4.45)

2(tan6’—6’)

which can then be substituted into Equation (4.41) to solve numerically for 6’ given a

desired A¢G :

 

(tan6’—6’)

I 2 2
A¢G = —-2 arcsin[2 xF + y; ](l — sec 6’) (4.46)

The calculated 6’ is then substituted back into (4.41) to determine Aa. Based on the

angle of line OF, we can then determine (If and 61’ according to the following

relationships that can be confirmed by inspection of the geometric properties of Figure

43,
y, 7! Ad
a’=arctan — +——— (4.47)
xF 2 2
a, = arctan(-yi) + 5 + 93 (4.48)
x,r 2 2

While not necessary for the algorithm, the location of C’ 2 (a,b) can be determined by

considering the line passing through 0 with slope at],

b=atanaf (4.49)

109

The center of the circular path lies on the segments OC’ and FC’, and their lengths can be

equated to Show,

2

a2 + b2 = (xF — a) +(yF —b)2 (4.50)

which is then used in conjunction with Equation (4.49) to solve for a and b,

2 2
xF+yF

2(x, + y, tan oz!) (451)

 

a:

2 2
xF+yF

b: 2(xF + yF tanaf)

 

tan a, (4.52)

4.6.4 Simulation of Complete Reconfiguration Algorithm

Simulation of the full reconfiguration algorithm will now be demonstrated. For
comparison purposes, let the initial conditions be identical to those used in the simulation

of the partial reconfiguration algorithm. That is, let the initial conditions be:

{x y a 0 ¢,}’={433 2.50 m3 7r/6 o}’

The final conditions will also be similar, with the exception that (DC is also Specified:

{x y 19 ¢G}’={o 0 0 —157z}’

As in the previous example, point Q represents the initial position of the sphere, and
points P and q are used to orient the sphere’s surface, as shown in Figure 45. During
Step 1, control action (a) is applied and the sphere rolls such that Q —) F and P —-> F ,
which corresponds to 6 —> O as shown in Figure 46. Since P is directly above the center
of the sphere, the angle a is poorly defined. A new angle a’ =1.90=109° is then
determined by Equation (4.47) such that a net Aa = 0.266 = 153° is achieved during Step
2, as illustrated by Figure 45 and Figure 46. The desired Aa is calculated by Equation
(4.45), which is based on a desired 6’ =152 = 86.9° as calculated by Equation (4.46) to

110

achieve the initially Specified 13% = —37z/ 2 = —270°. Recall that A¢G measures the net
rotation of the point q in the equatorial plane J. to line QP when QP is aligned with the
z-axis. Hence, specification of A¢G is equivalent to specifying the position of point q on

the equator of the sphere.

AS Figure 45 and Figure 46 indicate, step 2 results in a f = 2.17 = 124°, via Equation
(4.48), and Q’-—> Q” , P’ —-> P”, and q’—) q”. Angle a f is also used to calculate the
center of the concentric paths, C = (9.68,—14.l7) via Equations (2.9) and (2.8). Step 3
rolls the sphere such that Q” —> 0, P” —> 0, q”—) q’” , and thus (x, y,6)—> (0, O, 0).
Note that point q has rotated a net of A¢G = -37z/ 2 = —270° between point F with q, and

point 0 with q’”. Hence, complete reconfiguration has been completed, and by only

using a few simple initial calculations.

 

 

 

 

 

5 r 1 l I r 1 I H
4 )—
3 .—
>~ 2 l-
i
1 L . . k).
0 L
"‘\ 2 _/ . :note: C=.(9.68,-14.17) not shown
-1 1 1 “X414 ,. 1 1 1 1 1
-3 -2 -1 0 1 2 3 4 5 6

X
Figure 45 . Complete reconfiguration simulation: top view.

111

 

 

 

Step 4

 

 

 

 

 

 

 

 

 

>1
1
I I
I t
f ..l 1 .
[__—
1 .
' .
1 1 I I
1 200
I l l
; 1 .
. ; I :
I I 1 l L l l P—L“
O

 

I

100 1 200
I

 

 

 

 

 

 

 

 

  

 

 

 

 

I
I
O I 200
I I I I I r I | I
. | . p |
' ‘ I : I .
1 1 1 t L 1 1 1 1 1 I 1
O I I 100 I 200
0 T I I ‘ I I I I I
‘ I 1 I : I I I
4 . I 3 I ; - ; 1
, 1 : . - I
_5 1 I J J 1 1 1 1 x i I 2
o 20: 40 L 60 so too 120 140 160 I180 200
Step 1 Step 2 r Step 3 =‘. Step 4

 

 

Time. see.
Figure 46. Complete reconfiguration simulation: chart view.

4.6.5 Reachability of All orientations

In Section 4.3, Chow’s theorem was applied to the system model to prove that a path
exists between any initial and final configurations of the sphere. The question that
remains is whether the complete reconfiguration algorithm presented in 4.6.3 is capable
of this. Based on the partial reconfiguration algorithm, it is obvious that the Sphere can

be brought to the point 0 with the point P at the topmost position. In that example, the

result was (x, y,6) —) (0,0,0). Full reconfiguration is accomplished when the point q also

moves to a specified final location on the equator of the sphere. The ability to place point
q is determined by the achievable A¢G calculated by Equation (4.41) from Section 4.6.3

for the sphere traveling in a Counter-Clock-Wise (CW) path along the arc:
A¢G = Aa(l — sec 6’) (4.41)

This equation indicates that the achievable A¢G are dependent upon Aa and 6’ . As

indicated in Figure 47, the range of Ad that can be attained is dependent on whether the

112

center of the path is above or below OF. If the center of the path is below the line, as
indicated by point C2, the range of Aa is 0°S Aa $180°. The lower limit of Ad = 0°

occurs when the radius of the path is infinite and 6’ = 90°. This corresponds to the

minimum A¢G achieved, A¢G = —r , shown in Figure 48, where r is the length of OF

determined by,

r = 1’xi+ y: (4.53)

The upper limit of Aa =180° occurs when the point is at the midpoint of OF. As the

center of the path moves above the line OF, as indicated by Path 3 with center C3 in

\ Centers of path circles"
\, lie on this line

\

\

\

 

Figure 47. Paths of point F as A“ varies.from Error! Objects cannot be created from
editing field codes. to Error! Objects cannot be created from editing field codes.

113

Figure 47, the range of Aa increases to include 180°S Aas360°. Again, the lower
limit of Ad =180° occurs when the center of the path is at the midpoint of OF. The
upper limit of Ad = 360° occurs as radius of the path becomes infinite. This case is
impractical because the Sphere is traveling along a circle with infinite radius away from
the point 0. Theoretically, it would need to travel an infinite distance to the right until it

returns back to point 0 from the left side of Figure 47.

The net reorientation A¢G of the sphere that occurs as a result of 0°S Ad 5 360° can be

determined numerically by Equation (4.46), calculated previously:

 

61c

    
 

41c

21:-

 

 

/
r=1.0
r=0.5
'21P r=0.25

- ccw
paths
-61c ' ‘

-21I —1r 0 1r 2n 1
-c -d A“ a b

Figure 48. Net sphere re-orientation A¢G as a function of Error! Objects cannot be

created from editing field codes. for Error! Objects cannot be created from editing
field codes. and r = 0.25, 0.5, 1.0, 2.0, 5.0, 10.0.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

114

l 2 2
+
A¢G = 2arcsin 1" y;

2 (tan 6’— 6’) (l—sec 6’) (4.46)

The results of this equation are shown in Figure 48 for locations of point P with r = 0.25,

0.5, 1.0, 2.0, 5.0, and 10. As indicated above, the smallest reorientation A¢G of the

sphere is determined by 11% = —r, and as Aa —) 360° then A¢G —> —<>o. The converse

of these ranges holds true when the sphere travels in the opposite Clock-Wise direction

along the circular paths. An example of this path is indicated in Figure 50, and the

reorientation A¢G in this case is determined by

’ 2 2
+
A¢G = —27z + 2 arcsin xF yF

2 (tan 0, _ 6’) (l — sec 6’) (4.54)

The reorientation A¢G results as determined by Equation (2.1) for the sphere traveling

  
 

v’,——_‘—
I, _ _ \

Figure 50. Counter Clock-Wise (CCW) and alternate Clock-Wise (CW) and paths for
complete reconfiguration.

115

along the CW paths are also shown in Figure 48 for —360°S Aa S 0°.

It is indicated by Figure 48 that complete reorientation of A¢G can be achieved by
selecting different values of Aa , regardless of the distance of the sphere, r, from its final
destination. For example, Figure 48 illustrates that for r = 2, a range -47r S A¢G S -27r
can be achieved by selecting a S Aa S b. Thus, the actual range of A¢G is offset by 4n

such that the net reorientation is 0 S A¢G S 27:. Similar conclusions can be made for the

sphere traveling in the CW paths.

4.7 Other Comments

In this chapter, the spherical mobile robot, Spherobot was considered. Spherobot
encounters computational constraints that are caused by its nonlinear nonholonorrric
rolling constraints. These constraints are such that traditional control techniques do not
apply, and techniques proposed in the literature are not adequate for efficient control. To
resolve this problem, a unique moving coordinate system is defined on the sphere. This
causes the robot to follow linear and curvilinear paths on the plane while each control
input is driven independently. The properties of these paths are then examined more
closely and used to derive intuitive geometric motion planning strategies for moving the
sphere to a prescribed location and orientation on the plane. The benefit of these
strategies is their Simplicity and limited calculations required, which suits them especially

to real—time implementation.

116

CONCLUDING REMARKS

117

5. Concluding Remarks

Design and control of constrained robotics systems for enhanced dexterity and mobility
has been examined via three case studies. In the first, the design of surgical mechanism
for improved dexterity and reachability in MIS was examined. This system is
constrained in size and in allowable motion by a port on the patient abdomen. The design
achieves a delicate balance between load capacity, dexterous workspace, and DOF using
an optimized geared linkage with tendon actuation. The resulting mechanism, which can
be coupled to form a robotic or manual instrument, provides forceps at the tip of a
linkage capable of bidirectional 180° articulation. The forceps are provided with
unlimited rotation, actuation, and the capability to support a 4.5N load at the tip of a 1cm

suture needle.

The second system considered is a miniature climbing robot for urban reconnaissance.
Space constraints are also important for this system, as it must maneuver ventilation ducts
and the exterior of buildings while minimizing detection. The resulting robot, which is
the smallest of its kind, also satisfies mobility requirements and weight limitations using
an innovative under actuated revolute hip biped structure. The system is capable of
walking, climbing surfaces with any angle, traversing between surfaces with various

inclinations, and moving between specified locations.

In the third case study, a spherical mobile robot for reconnaissance is examined. The
robot provides excellent mobility, since it can roll in any direction, but control of the
robot is complicated by its nonlinear nonholonorrric motion constraints. The sphere

reorientation problem represents a fundamental motion control dilemma that is not solved

118

via traditional approaches, which are more complicated and computationally intense. We
achieve efficient point-to-point motion planning with a desired reorientation using an
intuitive geometric motion planner. The motion planner is based upon a unique
coordinate system defined on the sphere such that independent actuation of the control
inputs causes the sphere to travel along intuitive linear and curved paths. Straightforward
and efficient geometric planning techniques are then used to achieve point-to-point
motion and desired reorientation of the sphere. Thus, superior mobility of the sphere can

be achieved with very minimal computation.

In each of these state-of-the-art systems, innovative design and control strategies are key
for providing enhanced dexterity and mobility. Each system has provides unique
constraints in space, power, weight, and computational limitations that could not be
solved using traditional methods. AS robotics applications continue to expand,
innovative solutions, such as those presented here, will be crucial to successful system

realization.

119

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120

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126

APPENDICES

127

Appendix A

DALSA Design Drawings

128

 

 

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Figure 51. Assembled view of DALSA surgical mechanism (sheet 1).

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Figure 76. DALSA Component: Drive Shaft Coupler (sheet 26).

 

 

154

Appendix B

Climbing Robot Design Drawings

155

 

 

 

 

 

 

 

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Figure 77. Exploded View of the RAMRl climbing robot.

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Figure 80. RAMRI Component: Bracket for Motor #1.

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