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Design Optimization of Long-Reach Flexible M/m
Robots Using Integrated Design Method
Sasan Raghibizadeh
A thesis submitted in conformity with
the requirements for the degree of
DOCTOR OF PHILOSOPHY
Department of Mechanical and Industrial Engineering
University of Toronto
@Copyright by Sasan Raghibizadeh 1998
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Design Optimization of Long-Reach Flexible M/m
Robots Using Integrated Design Method
Sasan Raghibizadeh
A thesis submitted in conformity with
the requirements for the degree of
DOCTOR OF PHILOSOPHY
Department of Mechanical and Industrial Engineering
University of Toronto
1998
Abstract
Traditionally, the mechanical and control design of a robot are performed separately
based on various objectives. This research is aimed at enhancing (optimizing) the design
of long-reach flexible macro/rnicro robots by integrating the mechanical and control
design stages. The scope of the work is general in the sense that it covers a wide range of
applications of such robots.
A framework is provided for the integrated design of flexible macro/rnicro robots.
Nonlinear programming methods are used for generating the optimal values of
mechanical and control design variables for given objective functions (performance
measures) and design constraints. Formulation of a performance measure for integrated
design purposes requires a knowledge of the task requirements, and dynamics and control
of the robot. In this context, a modular systematic procedure is proposed for the automatic
generation of finite element model of a robot with arbitrary number of flexiblehgid links.
A comparative analysis is performed to establish the advantage of using the
Integrated Design Method (DM) over Traditional Design Method (TDM). The analysis is
based on the results of three design case studies. The first case study represents TDM,
where the control design is initiated after the mechanical design is completed; and the
second and third case studies use IDM to minimize the total mass and tracking error of an
unconstrained flexible M/m robot, respectively. The comparative analysis clearly shows
the superiority of IDM over TDM.
To facilitate the usage of IDM, perfomance measures are proposed for different
applications of long-reach M/m robots. The applications of such robots are identifled and
classified into three general groups: non-contact, contact, and bracing; and a performance
measure is proposed for each group of applications. The performance measures are
general in the sense that they can be applied to robots with various configurations and
degrees of freedom.
A type of contact tasks is proposed with a new micro robot configuration. It is
shown that, the flexible macro can be designed separately from the rigid micro, and a
performance measure is proposed for minimizing the cycle time of the flexible bracing
macro by using DM. To reduce the dynamic forces generated by the micro on the bracing
point, a novel conceptuaUconfiguration design of the micro is proposed. The new design
of the micro reduces the weight, dynamic interactions between the micro and macro, and
simplifies the control design.
This research is a first step towards the integration of specific design processes in
order to achieve better products. Some limitations of this work are: i) the effect of axial
load (due to external and centrifugal forces) on the stiffness matrix has been ignored; the
validity of this assumption has been numerically verified for the case studies; ii) the
proposed performance measures are formulated based on the linearized closed-loop
model of a flexible Mlm robot with a simple joint variable PD controller; new
formulations of performance measures are required for more complicated non-linear
control methods in which the closed-loop system behavior cannot be expressed
adequately by the linearized model. The work can be extended by considering a wider
variety of tasks, other robot configurations, other control methods, and by integrating
other design processes which are part of flexible automation systems.
iii
Acknowledgments
I would like to extend my sincere gratitude to my supervisor professor Andrew
Goldenberg, director of the Robotics and Automation Laboratory (RAL), for his
thoughdul guidance, advice, and support. I wish to thank professors: M. Kircanski, W.L.
Cleghom, A.N. Sinclair, R.G. Fenton, Y. Stepaneko, B.A. Francis, J.W. Zu, and C.B.
Park for their constructive comments and suggestions on this work.
Special thanks to S. Raghibi, V. Safavi, K. Sadeghy, Y. H. Tse, and other friends Kamran
Fariborz, Kambiz, Amir, Madjid, Vahid, Masod, and Reza for their support,
encouragement, and understanding.
Financial assistance from the Ministry of Culture and Higher Education of Iran and the
University of Toronto is also gratefully acknowledged.
TO My Family
Contents
Abstract
Acknowledgments
List of Tables
List of Figures
Nomenclature
1 Introduction 1
1.1 Preliminary Remarks .......................................................................... 1
1.1. I Statement of the Thesis ................................................................................................................. 4
1.2 Literature Survey ............................................................................... 4
1.2. I Kinematics Design ........................................................................................................................ 5
1.2.2 Dynamics Design ..................................... .- ............................................ 5
1.2.3 Integrated design .......................................................................................................................... 7
....................................... 1.3 A Framework for Integrated Design Optimization 9
1.4 Contributions. .............. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Organization of the Thesis ................................................................. -15
2 Modeling of Flexible Robots for Design Optimization 17
2.1 Review of Flexible-Link Kinematics and FEM .......................................... 18
2.1.1 Related Works ............................ ,, .., ....................................................................................... 19
2.1.2 Kinematics ................................................................................................................................... 21 . . 2.1.3 Frnrte Element Method ................. ... ........................................................................................ 24
2.1.4 Dynamics ..................................................................................................................................... 27
2.2 Expansion of Dynamics Equations ........................................................ 29
2.2. I mansion of Inertia Matrix ........................................ .............................................................. -30
2.2.2 Erpamion of Gravity Vector ........................... ,. .................................................................... -35
2.2.3 Expansion of S m e s s Matrir ........................... ,., ....................................................................... -36
2.3 Finite-Element Modeling of Planar Flexible Robots .................................... 38
.......................................................................... 2.3.1 Equations of Motion of Planar Flexible Robots 38
2.3.2 A Procedure for Automatic Generation of Equations of Motion ................................................ 47
2.4 Summary ..................................................................................... 50
3 Integrated Design of A Planar Mlm Robot 51
3.1 Mechanical Configuration and Control of the M/m Robot ........................... -52
3.1. I Mechanical Ann .................... .. ............,....................................................... ............................. 52
3 . I . 2 Selection of a Control Algorithm ...................................... .,, ............................. 53
3.2 Design Case Studies ....................................................................... ..58 3.2.1 Case I: Design based on TDM .................................................................................................. -59
3.2.2 Objective Functions and Design Constraints. ...... .....,. .............................................................. -65
3.2.3 Case 2: Design based on [DM for Minimizing the Total Mass ................................................... 75
3.2.4 Case 3: Design based on lDM for Minimizing the Tracking Error ......................................... 78
3.3 Comparative Analysis of the Design Case Studies ..................................... 79
3.3.1 Simulation ................................................................................................................................... 79
3.3.2 Distribution of the Closed-loop Poles ......................................................................................... 84
......................................... 3.3.3 Comparative Analysis of TDM and IDM ............................... .... 86
3.4 Conclusions ................................................................................... 88
vii
4 Integrated Design of Constrained M/m Robots
4.1 Applications of Long-Reach M/m Robots .............................................. -90
4.2 Background on the Control of Constrained Robots ..................................... 92
4.3 Closed-Loop Model of a Constrained Flexible Wm Robot ........................... 95
4.3.1 Dynamics Model of a Constrained Flen'ble W m Ann ............................................................. ...96 4.3.2 Control Algorithm ..................................................................................................................... 101
4.3.3 Closed-Loop System .................................................................................................................. I02
4.3.4 Linearized Closed-loop Model ................................................................................................ 105
.................................................... 4.4 Study of the Closed-Loop Response 106
.......................................... 4.4.1 PositiodForce Tracking Petfiormance uftite Constrained Robot 106
4.4.2 Stability Analysis of the Closed-Loop System ......................................................................... 1 0 9
................................................. 4.5 Formulation of the Objective Function 110
................................................................................................ 4.5.1 Position Error .............. ..... 111
4.5.2 Force Error ............................................................................................................................... 123
................................................................................... 4.5.3 Total Perjonnance ...................... .. 1 14
.................................................................................... 4.6 Summary 114
5 Design of A Bracing 1Wm Robot
5.1 Performance Measure of a Bracing Flexible Robot ................................... 117
5.1.1 Task Plan .................................................................................................................................. 118
.............................. ........................................ 5.1.2 Control Algorithm and Pet$omnce Measure ... I21
.......................................................... 5.2 Design of a Novel Micro Robot 124
.......................................................................................... 5.2.1 Mechartical Configuration Design 1 2 4
......................................................................... 5.3 Kinematics Model -126
.................................................................................................................. 5.3.1 Fonvard kinematics 126
................................................................................................................... 5.3.2 Inverse kinematics 1 2 9
................................................................................................ 5.3.3 Derivative Terms and Jacobian 131
........................................................................... 5.4 Dynamics Model 132
................................................................................ 5.4.1 Kinetic and Potential Energies of Link I 132
.................................................................................. 5.4.2 Kinetic and Potential Energies of Link 2 133
5.4.3 Total Kinetic and Potential Energies ........................................................................................ 135
.................................................................................... 5.5 Summary 136
viii
6 Conclusions and Recommendations 138
................................................................ 1.1 Summary and Conclusions 138
.................................................... 1.2 Recommendations for Future Work. 140
Bibliography
List of Tables
3.1. Mechanical Parameters and Their Nominal Values .................................... 60
3.2. Moment of Inertia and Lowest Natural Frequency of the M/m Arm ................. 62
3.3. Initial Control Gains. Natural Frequencies. and Damping Ratios .................... 63
3.4. Designed Values of the Control Gains of the M/m Arm ............................... 63
3.5. The Nominal Values of Control Variables ............................................. -76 3.6. Lower and Upper Bounds on Design Variables and Beams' Cross-sections ....... 76
3.7. The Final Values of MI Design Variables (Normalized Values) .................... 77
3.8. Maximum and RMS values of the Tracking Error in the Task and Joint Space .... 82
3.9. Closed-Loop System Poles at the Configuration 8 = [O.O.O. a /20] ................ 83
...................... 3.10. The Normalized Influence Factors of the Closed-Loop Poles 85
3.1 1: Final Values of the Objective Function and Other Important Characteristics of the
.......................................................................................... M/m robot 86
List of Figures
2.1 : Coordinate Frames of the FIexibIe Link i and DH Parameters ........................ 22
2.2. A Simple Beam Element with Two Nodes and Four Parameters .................... 25
2.3. A Beam with Three Elements and Four Nodes ......................................... 26
2.4. Displacements u and Rotations v of Node k of Link i in a 3D Space ................ 27
2.5: Adding Link i to the Previous Links and Expanding the Generalized Coordinate
Vector and Dynamics Matrices ................................................................. 30
2.6. Coordinate Frames for a Planar Flexible Robot ......................................... 39
3.1. A CLink Planar FlexiblelRigid M/m Robot ............................................ 53
................ 3.2. Desired Trajectories of the Macro and the Micro in the Task Space 56
3.3. Desired and Actual Positions of the Mano and Micro in the Task Space ........... 58
3.4: (a) Robot Configuration at 8 = [On a /3. -lr /6 . -n /3] , and (b) Configurations of
the M/m Robot for Stability Analysis .......................................................... 64
3.5. The Real Part of the Rightmost Poles at Different Configurations ................... 64
3.6. Cross-section of the Flexible links ....................................................... 73
3.7. Final Values of Control Design Variables (from Table 3.7) ......................... 77
3.8. Final Values of Mechanical Design Variables (from Table 3.7) .................... 78
3.9. Desired Trajectories .................................. .. ................................. 80
3.10. Tracking Errors of the Macro and Micro (Case 1) ................................... 81
3.11. Tracking Errors of the Macro and Micro (Case 2) ................................... 81
3.12. Tracking Errors of the Macro and Micro (Case 3) ................................... 81
3.13: Maximum and RMS values of the Tracking Error in the Task and Joint Space
...................................................................................................... 82
3.14. Distribution of the Closed-Loop Poles in the Complex Plane ...................... 84
3.15. The Normalized Muence Factors (from Table 3.10) ............................... -85
4.1. Desired and Actual Trajectories of the Macro and Micro ............................ 108
4.2. PositiodForce Tracking Errors of the Endeffector ................................... 108
4.3. FlexturaI Deflections/Rotations at the Distal Ends of Links I and 2 ............... 108
4.4: (a) Robot Configuration at 8 = [Q n /3. -z /6.- n /3] . and (b) Configurations of
the M/m Robot for Stability Analysis ....................................................... 110
4.5. The Real Part of the Rightmost Poles of the Closed-Loop System ................. 110
5.1. A Bracing Macro/rnicro Robot ........................................................... 119
5.2. Free-Body Diagram of the Bracing Device ............................................ 119
5.3. One Motion Cycle of the Macro ........................................................ 120
5.4: Design Concept of a Robot With Flexible Links Moving on a Curved
Surface .............. .. ........................................................................ 125
5.5. One Possible Configuration Design of ROCS ......................................... 126
5.6. A Schematic Diagram of ROCS ........................................................ I27
5.7. One link of ROCS and the Intersection Ellipse ........................................ 127
5.8. First Link of ROCS and its Angular Velocity .......................................... 132
5.9. Second Link of ROCS and its Angular Velocity ..................................... 134
xii
Nomenclature
Roman
macro's end-effector acceleration
Denavit-Hartenberg parameter of link i (link length)
real constants
NM x N, input matrix
a positive scale factor
damping matrix
structural damping of the flexible links
constant damping matrix of the linearized model
damping matrix in the task space
reduced damping matrix in the task space
length of the motor axis
Denavit-Hartenberg parameter of link i Oink offset)
total mass of link i
total mass of lumped mass i
number of flexural coordinates of the macro
flexural generalized coordinate vector
macro's joint error vector
static deflections due to the gravity
dynamics deflections due to inertial forces
end-effector position error in the task space
end-effector position and force errors
Young's modulus
partitions of the identity matrix I,&
partitions of the identity matrix I,
bending stiffness of flexible link i
bending stiffness of element j about the Z axis of link i
bending stiffness about the respective axes of the ch link
a constant matrix
objective function
micro's dynamics force on the bracing device
friction force of the contact surface
macro's clamping force on the bracing device
normal contact surface
gravity-force vectors added to Fgi-, due to link i
P component of the gravity-force vector due to link i
contribution of the rh lumped mass to the total gravity-force vector Fg,
vector of Lagrange multipliers
desired force vector
vector of centrifugal and Coriolis forces
desired force
configuration-dependen t gravity-force vector
gravity-force vector of links 1 to i- l
gravity acceleration
gravity-acceleration vector
xiv
constraint function
torsional stiffness about the xi axis of the link
gravity force vector in the task space
reduced gravity force vector in the task space
inertia tensor (about the center-of-mass) of link i
inertia tensor (about the center-of-mass) of lumped mass i
additive terms to the inertia matrix Mi-, due to Link i
inertia-tensor of links 1 and 2
influence factors of real (r) and complex (c) modes
influence factors of real (r) and complex (c) modes
the second moment of inertia of the rotor of Motor 1 about its axis
the second moment of inertia of the rotor of Motor 2 about its axis
the second moment of inertia of Motor 2 about Z-axis
the second moment of inertia of the payload
local objective function
global objective function
Jp4 J m =% micro's Jacobian matrix
J M Jacobian matrix of the macro
JP4 J~ =a4 Jacobian matrix of a flexible M/m arm
Jz-, , J,-, , Jii Jacobian matrices
Jrci 9 J R ~ Jacobian matrices of rigid link i
O J ~ , O J~~~ Jacobian matrices expressed in the base b e
A Jacobian matrix
J Y - J , macro and micro parts of the end-effector Jacobian matrix J ,
k a constant scale factor
stiffness matrix
spring constant
proportional gain of the force controller
structural stiffness of the flexible links
global stiffness matrix of link i
stiffness matrix of element j on link i
constant stiffhess matrix of the linearized model
diagonal matrices of proportional and derivative gains
proportional and integral force-control gains
stiffness and damping matrices of the endeffector's impedance
proportional and derivative gains of the macro's PD controller
stiffhess matrix in the task space
reduced stiffness matrix in the task space
element length
length of element j on link i
link length
i'h link length
initial spring length
spring length
mass per unit length of flexible link i
link masses
payload mass
inertia matrix
inertia matrix of links 1 to i- I
components of the inertia matrix
constant inertia matrix of the Linearized model
xvi
macro's inertia matrix corresponding to the rigid-body motions of the links
contribution of rigid link i to the inertia matrix
inertia matrix in the task space
reduced inertia matrix in the task space
number of degrees of freedom of the end-effector in the task space
number of constraints
total number of degrees of freedom
number of elements of the beam
number of elements of a flexibIe Iink i
number of nodes of the beam
number of joints (or rigid-body motions)
number of real and complex poles of the closed-loop system
end position-vector of the macro
end position-vector of the micro
position vector of the link center-of-mass w.r.t. the base frame.
position vector of the origin of frame i- 1 w.r.t. to the base frame
3x 1 position vector
3x 1 position vector
position vector of a point xi on link i
kh modal coordinate
generalized coordinates vector
generalized coordinate vector of links 1 to i- 1
a nominal c o d iguration
P element of the generalized force vector
number of degrees of freedom of the micro
relative position of a point on link i w.r.t. the proximal end of the link
number of joints of the macro
xvii
3x3 rotation matrix
3x3 rotation matrix
gross traveling time between two points
kinetic energy
kinetic energy
total kinetic energy
tracking error measure
total mass
displacement of the f' node of an element
vector of nodal displacements and rotations of link i
flexural displacement and rotation of node k on link i
rotation of the $ node of an element
center-of-mass velocity
potential energy
potential energy
velocity vector of the center-of-mass of link i
potential energy due to elastic deflections of flexible elements
potential energy due to the gravity force
linear and angular velocity vectors of frame (Xi-,. Y&L~.Z-~) w.r.t. the base
lower bounds on design variables
total potential energy
upper bounds on design variables
velocity vector of a point xi on the link i w.r.t. the base frame
centrifugaUCoriolis force vector in the task space
reduced centrifugaVCoriolis force vector in the task space
link cross-section variables
link angular velocity
xviii
weighting factor for configuration i
a diagonal weighting matrix
work of non-conservative forces
spatial coordinate along the length of link i
x coordinate of the fmt node of an element
x coordinate of the fmt node of element j on link i
vector of design variables
coordinate frame attached to the distal end of link i
vector of task coordinates
reduced vector of task coordinates
element displacement at a point x and a time t
flexural deflection in & direction
force control subspace of the hybrid task
y, E %("-*)X' position control subspace of the hybrid task
zi (xis t ) flexural deflection in direction
Greek
a the angle between the link and motor axis
a i Denavit-Hartenberg parameter of link i (link twist)
Pi absolute rotation angle of the link coordinate frame
vector of generalized coordinates of link i
6 a small virtual variation from the actual value
6i flexural displacement
A denotes the small deviations from the nominal values
Ai = (6qP 6,, 6, IT
small flexural deflections of the flexible link i dong axes (Xi. LZi )
xix
end-eff'tor position error
error in the joint space comsponding to the real mode r
element local variable
damping ratio of all modes
damping ratio of dominant modes
damping ratio of the most effective modes
closed-loop damping ratio
rotation angle of joint i
vectors of joint displacements and velocities
desired trajectories of the macro and micro in the joint space
complex eigenvalues
real component of the dominant poles
real eigenvalues
mass per unit volume (density)
torque of joint i
macro and micro vectors of joint input torques
endeffector polar coordinates
constraint equations
torsion angle about the xi axis of the rh link
kfi mode shape
#= =a + j q complex eigenvalues and eigenvectors
r @ i = (Px, * Pyf* Pq )
small flexural rotations of the flexible link i along axes (X* Yi,Zi)
3 real eigenvectors
~ ( 4 ) a vector of mutually independent functions of the generalized coordinates
q = ( u v v;)
Acronyms
AMM
CDV
DH
DOF
FEM
FMRM
IDM
M/m
MaED
MDV
NPM
PD
PID
RMS
ROCS
TDM
flexural displacements and rotations of link i at node k
lowest clamped nahual frequency of the arm
closed-loop servo bandwidth
angular velocity of link i
minimum closed-loop frequency
closed-loop natural frequency
Assumed Mode Method
Control Design Variable
Denavi t-Hartenberg
Degree-Of-Freedom
Finite Element Method
Flexible MacrolRigid Micro
Integrated Design Method
Macro/micro
Maximum End-point Deflection
Mechanical Design Variable
Nonlinear Programming Method
Proportional-Derivative
Proportional-Integral-Derivative
Root-Mean-Square
Robot for Curved Surfaces
Traditional Design Method
Chapter 1
Introduction
1.1 Preliminary Remarks
Long-reach robots have potential applications in different industries including:
1 ) construction: for finishing tasks (e.g., painting, plastering, jointing, welding), concrete
application, inspection, and repair works mar901 [Gro89]; 2) highway: for paint removal
and surface preparation of steel bridges, visual inspection, painting, etc.
~ o o 9 5 ] ~ 9 5 ] ; 3) aircrafi and shipbuilding: for spray painting, paint removal,
cleaning, and inspection (visual and ultrasonic) [SDH93] [SW90]; 4) nuclear a d waste
management: for waste handling, structural demolition, restoration of storage tanks,
inspection, etc. [Red95][CJ92]; and 5) space: for visual inspection, servicing, and
assembly.
This project is aimed at enhancing the performance1 of long-reach macro/micro
(M/m) robots for a wide range of applications using optimization techniques. Such a
robot consists of a long-reach manipulator (the macro) which carries a small and light
manipulator (the micro) at the end. The macrohnicro configuration is used because it
provides superior performance over the ordinary configuration (i.e., a long-reach robot
without the micro). Previous research [CKC91][SHH89] showed that W m robots can
Definition of "performance" is task dependent. The following chapters provide performance measures for
different classes of tasks.
easily achieve bandwidths of up to several times the fmt natural frequency of the macro
arm using a simple joint variable PD control method. On the other hand, the control
bandwidth of the same long-reach robot without the micro cannot exceed even half of its
fmt natural frequency if a satisfa~tory closed-loop response is required wM74]. A long-
reach manipulator generally exhibits a large structural flexibility with a very low natural
frequency. This limitation of the control bandwidth to half of the fmt natural frequency
(for the arm without the micro) severely degrades the control performaoce. Therefore, the
W m configuration is used to provide a fast and precise response with a simple and robust
joint variable control algorithm. The macro provides a large workspace while the micro
furnishes dexterity and precision.
Long-reach M/m robots are complicated electromechanical systems with
structural flexibility and kinematics redundancy. Designing such robots requires a special
attention to be dedicated to the mutud interactions between the mechanical arm and its
controller. This integration of the mechanical and control designs has not been given
enough attention in robotics research. Works on mechanical design were mostly based on
kinematics andlor dynamics characteristics of the mechanical arm only, and seldom
considered the control design. Similarly, works on control design seldom attempted to
modify the mechanical design, and mostly focused on designing a "better" controller for a
"given" mechanical arm. This work is based on a new design approach, called Integrated
Design Method (ZDM), where a mechanical a m and its controller are designed
concurrently.
D M directly incorporates control objectives into the mechanical design stage, and
combines the mechanical and control design parameters. Since the final behavior of a
robot depends on both mechanical and control design, it would be reasonable to design
both parts concurrently to satisfy the task objectives. This integrated approach is specially
useful for more complex systems (e.g., flexible arms) where the interactions between the
mechanical structure and control are more complicated and cannot be perceived through
intuition or stated explicitly in terms of simple design rules.
IDM can be compared with Traditional Design Method (TDM), in which, the arm
linkage is designed based on some kinematics and dynamics considerations only (e.g.,
payload, workspace, structural deflections, and natural frequency) and without
considering control objectives. For example, ''tracking error", which is an important
performance measure for some applications of robots, is only considered during the
control design, when the mechanical design is completed. This separation of the
mechanical and control design usually leads to a more conservative design which
generates heavier and stiffer arms than the one based on DM. Previous reported works
demonstrated substantial improvements achieved by integrating the mechanical and
control design stages [SHH89] [PA94].
IDM has been under development for active flexible space structures during the
past decade w84]m92]. However, its application to the robotics field is new (see the
literature survey in Section 1.2.3). Unlike space structures, robots are nonlinear systems
with changing configurations. The applications and task requirements of robots are also
different from those of space structures. New approaches are required to apply D M to
robots. This project is aimed at advancing the design of long-reach flexible M/m robots
using Integrated Design Method (DM).
In this thesis, IDM means simultaneous design of a mechanical arm and its
controller, and Traditional Design Method (TDM) means the current design practice,
where the control design is initiated after the mechanical design is nearly completed. A
design process have the following stages ~ix95]~ym94]: i) specifications stage;
ii) conceptual stage; iii) configuration stage; iv) parametric stage; and v) detailed design
stage. The conceptual and configuration stages of design are more abstract, intuition-
based, and less formal, and are very difficult to automate. This thesis focuses on
developing IDM for the parametric stage of design, which is more suitable for applying
mathematical programming tools.
In the parametric stage, specific values are assigned to design parameters (e.g.,
link lengths, link cross-sections, mass distribution, etc.). Nonlinear programming
methods are used for computing the optimal values of mechanical and control design
parameters. In order to obtain an optimal design, task requirements must be translated
into a suitable objective function (performance measure) and a set of constraints.
Formulation of performance measures is task and system dependent. This research
provides general purpose performance measures for different applications of long-reach
Wm robots.
1.1.1 Statement of the Thesis
This work is the first attempt to apply IDM to the design of long-reach flexible M/m
arms. The scope of the work is general as it considers a wide range of applications with
different task requirements. A general framework is provided in Section 1.3 to be used in
subsequent chapters for the integrated design of long-reach Mlm robots. Then, a
comparative analysis is carried out to highlight the advantages of IDM over TDM, as
applied to long-reach flexible M/m robots. The comparative analysis is performed based
on the numerical results of three representative design cases studies on non-contact
applications of such robots (i.e., when the robot is unconstrained).
Once the superiority of the IDM over TDM is established for non-contact
applications of long-reach Mlm robots, the work is continued by extending D M to a
wider range of applications of long-reach flexible M/m robots (e.g., applications in which
the robot is in contact with the environment). Since one major step in the development of
IDM is the formulation of suitable performance measures (as is evident in the integrated
design case studies), IDM is extended by formuiating general purpose performance
measures for different applications of long-reach M/m robots. Formulation of the
performance measures and design constraints for integrated design purposes requires a
knowledge of the task requirements, dynamics and control issues of the system. These
aspects are addressed in this thesis for flexible long-reach M/m robots.
1.2 Literature Survey
A large number of papers have been published on the design of robotic manipulators,
which can be categorized into three main groups: kinematics, dynamics, and integrated
design. Most of the papers in the robotics area consider only the kinematics design, some
address the dynamics design, and only a few discuss the integrated design approach.
121 Kinematics Design
Works in this group consider only kinematics task requirements, and performance
measures are all kinematics quantities used for obtaining unknown kinematics parameters
such as Denavit-Hartenberg (DH) parameters. Because only the kinematics model of the
arm (e.g., the Jacobian matrix) is required, this approach is the simplest one.
Many kinematics performance measures are defined based on the Jacobian matrix
or its properties. Gonzalez et al. [GAR931 designed a robot based on a prescribed
Jacobian matrix at a specified configuration. Kinematic manipulability considers the
determinant of the Jacobian matrix Wos851, and provides a measure of the maximum
achievable endeffector velocities in the task space when the joint rates are bounded.
Isotropy is defined based on the condition number of the Jacobian matrix, and is a
measure of the end-effector positioning (or force) accuracy and uniformity in different
directions [AL92] [SC82] [GL93]. S toughton and Arai [SA93] used the average value of
the condition number over a central region of the workspace to define a global dexterity
measure.
Other performance measures were proposed to represent general kinematic
characteristics of an arm. A manipulator may be designed to be free from internal
singularities [RS93]mo185], to have an analytical closed-form solution for its inverse
kinematics Bo1891, to provide the maximum reachable workspace for specified link
lengths [ P S 8 8 ] m 8 6 ] , or to satisfy some kinematics task specifications (e.g., avoiding
specific obstacles or reaching certain points) pK93].
1.2.2 Dynamics Design
Since the dynamics behavior of a mechanical arm is affected by kinematics parameters
(e.g., DH parameters) as well as dynamics parameters (e.g., links masses and inertias),
dynamics design is more general than kinematics design. Many dynamics performance
measures are defined based on the properties of inertia matrix, represented either in the
joint or task space.
It is desirable to design a manipulator with a diagonal andor configuration-
invariant inertia matrix. With a configuration-invariant inertia matrix, all nonlinear
inertial forces in dynamics equations vanish and linear control methods can be applied. A
diagonal inertia matrix is also desirable, because it decouples the dynamics equations.
Yang and Tzeng [YT86] designed a four-DOF robot with an invariant inertia matrix.
Dynamics measures such as generalized inerria ellipsoid (GIE) [Asa82] and d y ~ m i c s
isotropy m 9 3 ] were used to minimize the coupling terms and variation of a robot
inertia matrix. Singh and Rastegar [SF2921 used the statistical variations of the condition
number of the inertia matrix as a global measure for minimizing nonlinear inertial terms.
Unfortunately, complete decoupled and invariant inertia matrix cannot be achieved except
for some simple configurations (up to two-DOF for serial manipulators) [Asa89].
Other dynamics performance measures were proposed to specify the dynamics
manipulability (i.e., the maximum achievable end-effector accelerations when the joint
torques are bounded) [Yos85], load carrying capacity, or stiffness-to-mass ratio of robots
FMP961. A good summary of kinematics and dynamics performance measures can be
found in [RB89].
Several problems can be recognized while surveying the works on kinematics and
dynamics design of robots. First, almost all of them consider the design of rigid
manipulators only. Although designing a manipulator to be flexible is not desirable
(except for some special applications), in some cases (e-g., designing long-reach
manipulators or ultra-fast robots) structural flexibility is inevitable and must be
considered in the design process. Second, kinematics and dynamics performance
measures reflect the behavior of the mechanical arm only, while the final behavior of a
robot in a specified environment is determined by its mechanical structure as well as its
control system. In the above works, mutud interactions between the mechanical arm and
control were totally or partially ignored during the mechanical design process.
As mentioned earlier, an integrated design approach, which measures the whole
system behavior against the final task requirements, leads to a more efficient and reliable
design. On the other hand, ignoring control requirements in the mechanical design stage
may impose severe limitations on the practical implementation of the controller.
1.2.3 Integrated design
Unlike the traditional design method, IDM directly incorporates control objectives into
the mechanical design process. In IDM, performance measures reflect the find task
requirements (e.g., the endeffector tracking error), and they contain kinematics,
dynamics, and control design parameters. This approach to mechanical and control design
was demonstrated on some simple linear systems with few design parameters in [SHH89]
and [Jac89].
So far, D M has found its main applications in design optimization of advanced
flexible space structures. Onoda and Haftka [OH871 used IDM to minimize the initial and
operational costs of a beam-like spacecraft system subjected to external disturbances. To
suppress the vibrations, they used a linear state feedback control whose gains were
concurrently designed with mechanical parameters to reduce the total (i.e., initial plus
operational) cost by up to 45%. Using IDM, Belvin and Park [BP88] reduced the total
energy of a cantilever truss beam by 56%. The total energy was defined as the kinetic and
potential energy due to structural vibrations, plus the control energy spent by the
actuators. Canfeld and Meirovitch [CM94] solved a multiobjective optimization problem
for minimizing the total energy and total mass of a simply supported beam with three
actuators. Their results indicated a 29% reduction in the peak vibration amplitude without
reducing the closed-loop damping or exceeding the actuators limits.
For realistically large structures, the number of design variables can be very high
such that it practically prohibits the implementation of the integrated design optimization.
Suitable formulation of performance measures and constraint functions is critical for the
computational feasibility of DM. To improve the computational efficiency, Messac and
Malek m 9 2 ] provided the closed-form solution of a performance measure, which was
successfully applied to the integrated design of a 15%rder system with 50 mechanical
and 7 control design variables. Jin and Schrnit [JS92] utilized approximation concepts
such as linking mechanical and/or control design variables to reduce the number of design
variables and improve the computational efficiency.
Space structures are linear systems with fixed configurations, and their design
performance measures are usually the total mass, control effort, or disturbance-rejection
error, and the constraints are generally the size, fundamental frequency, actuator forces,
and closed-loop stability of the system Optimal state variable feedback control method is
commonly used for suppressing the structural vibrations due to internal or external
disturbances. On the other hand, robots are nonlinear systems with variable
configurations. Their task requirements and control methods are. usually different from
those of space structures. New performance measures and constraint hnctions must be
developed for different applications of robots. So far, only a few number of papers were
published on the integrated design of robots.
Rai and Asada W93J used IDM to design an ultra-high speed single-link flexible
arm for unconstrained point-to-point applications. The experimental results indicated a
69.6% reduction in the total mass, 83.2% reduction in the link inertia, and 34.6%
reduction in the cycle time. They proposed an interactive design method in which, the
user modifies the design variables based on the sensitivity information provided by the
computer in each iteration [RA91].
Park and Asada [PA941 developed an ultra-fast flexible two-DOF robot for pick-
and-placeapplications using D M . Settling time of the linearized closed-loop model was
minimized, subject to the constraints on the total mass, link inertias, maximum vertical
deflection of the arm, and the workspace. The experimental results indicated that the
speed of point-to-point motion increased to twice that of the initial design.
All these works on D M demonstrate a significant improvement achieved by
concurrently designing the mechanical structure and control. However, the applications of
IDM in the robotics field are far from being developed. To date. there has been no
publication on the integrated design of robots for applications other than high speed
point-to-point motion or for more complicated configurations such as M/m robots. This
project is an attempt to illustrate and extend the applications of IDM to long-reach M/m
robots.
Integrated design of long-reach robots requires knowledge of the dynamics and
control of flexible robots. The finite element or assumed mode model is commonly used
for the simulation, control, and design of flexible arms rG95]. Due to the complexity of
the equations, hand derivation of the closed-form dynamics model of a flexible M/m arm
8
is nearly impossible. Several methods were proposed for automatic generation of
dynamics equations using commercial symbolic calculation softwares. Chapter 2 presents
a summary of the related works, followed by the formulation of a new procedure which is
modular and more efficient than the previous works on the symbolic computation of the
finite-element model.
Position and force control of robots has been the subject of numerous research
works, some of which pertain to flexible M/m robots. The related literature is reviewed in
Chapters 3 and 4. For integrated design purposes, a simple non-model-based control
method is preferable because: i) it does not conceal the dynamic behavior of the
mechanical arm in the closed-loop model (this allows for better tailoring of the
mechanical arm for its task requirements); and ii) it improves the computational
efficiency during an optimization process. In Chapter 3, a joint variable PD control is
used for non-contact applications of flexible Mlm robots. The closed-loop response is
studied, followed by the formulation of a new performance measure for non-contact
applications. Chapter 4 uses an impedance control to formulate a new performance
measure for the arm in contact with an environment. For the contact motion, a hybrid task
representation, originally proposed by Yoshikawa (Yos871 for rigid robots, is extended to
flexible M/m robots.
1.3 A Framework for Integrated Design Optimization
This section provides a general framework used in other chapters for the integrated design
of longreach flexible M/m robots. In this thesis, DM means concurrent design of
mechanical and control parameters of the system, and TDM means the current
engineering practice in which, mechanical and control design are performed separately. A
survey of literature (presented in Section 1.2) indicated that a large number of papers
have been published on IDM since the past decade; the majority of them addressed the
integrated design of flexible space structures and ody a few applied IDM to robots. In all
cases, the results demonstrated a significant improvement of performance achieved using
D M rather than TDM,
An example of the inefficiency of TDM is the robot-to-payload mass ratio of
current industrial robots, which is generally over 50 [CACgl]. Compared with a human
a m , current robots are very inefficient, and inefficiency means lower productivity, higher
power consumption, and higher cost. One main reason for designing such heavy robots is
the separation between the mechanical and control design stages. The mechanical arm is
usually designed based on the desired payload, workspace, arm deflection, lowest
structural frequency, and some other kinematics and static considerations. The actuators
and transmission systems are then selected to provide the desired velocities and
accelerations. When the mechanical design is completed, the control design is initiated to
fulfill the final task requirements (e.g., the required tracking accuracy and speed). This
separation of the mechanical and control design may lead to a very heavy and stiff arm
structure (and therefore, heavy actuators and transmission systems) as it requires that the
elastic deflections of the mechanical arm remain smaller than the desired tracking
accuracy.
Although this traditional approach may be adequate for designing low
performance robots, it cannot provide satisfactory results when more stringent constraints
on performance, achievable speed, total mass, or size of workspace are applied. In such
cases TDM leads to an undesirably heavy structure with oversized actuators and
transmission systems. Design of lightweight, high speed, or long-reach robots requires
that the arm flexibility and mutual interactions between the flexible arm and its control
system are taken into account. The design case studies in Chapter 3 show that D M can
substantially reduce the total mass without deteriorating the tracking performance.
IDM may be used in different stages of design. Generally, A design process has
the following stages [Dix95] pym94] : i) specifications; ii) conceptual; iii) configuration;
iv) parametric; and v) detailed design. In the first stage, functional requirements are
elaborated. For example, in this research, we start from the basic need of accurate
positioning/manipulation of an object in a large free space or in contact with a far surface,
and try to define the functional requirements. In the conceptual stage, a long-reach robot
(i.e., a serial chain of several links and actuators) is chosen, and in the configuration stage
the number, type, and location of joints, links, actuators and sensors are determined. The
robot control algorithm can be determined at this stage, as well. Then, in parametric
stage, specific values are assigned to design parameters such as link lengths, link masses,
and control gains. Finally, the design is completed by selection of all components, and
preparation of documents for manufacturing. The whole process is iterative, and in some
stages we may need to return to previous stages if a satisfactory result is not achieved.
The conceptual and configuration stages of design are more abstract, intuition-
based, less formal, and are very difficult to formulate analytically. Some researchers tried
to formalize the earlier stages of design by providing general guidelines and rules (or
axioms) [Suh90][NG97J; however, intuition is still an important and inseparable part of
these earlier stages of design.
Mathematical tools are more easily applied to the parametric stage of design. This
research focuses on the integrated parametric design of flexible M/m robots. Nonlinear
Programming Methods (NPMs) provide powefil tools for the automation of the
parametric stage. They allow for designing a complicated nonlinear system, with a large
number of design variables and conflicting constraints on the system characteristics.
NPMs have been actively used for the integrated design of large space structures
[MT84]m92], and more recently for optimal design of robots w89]pA94]. NPMs
are specially useful for DM, as they provide means for simultaneous (concurrent)
optimization of mechanical and control design variables, subject to objectives and
constraints which are nonlinear functions of both mechanical and control system
characteristics. A general nonlinear optimization problem is mathematically stated as
follows:
minimize the objective function (or pe@onnance measure): f ( X ) ,
subject to
the inequulity construintfunctions: Gi ( X ) S 0 for i = 1.. . . . NC , and
the lower and upper bounds on design variables: Vlb S X < Vub , where the objective and constraint functions, f (XI and Gi(X) , are nonlinear functions
of the nxl vector of design variables X, NC is the number of design constraints; and VZb
and Vub are vectors containing the lower and upper bounds on the values of design
variables. A more general problem may contain some equality constraints, as well.
Sometimes, an equality constraint can be used to eliminate a design variable.
Performance measures and constraint functions are formulated based on task
requirements and the desired behavior of the closed-loop system. The general framework
(or procedure) used in subsequent chapters for the integrated design of robots is stated as
foltows:
Identi& the task objective(s) and design constraints based on the application. For
example, some applications require the end-effector to carry a payload on a specified
path in a given workspace. In this case, the tracking performance is a task objective,
and the size of the workspace is a design constraint. Chapter 4 presents a
comprehensive List of applications for longreach robots and identifies their task
objectives.
Propose a suitable configuration for the robot, and identifi the Mechanical (i.e..
kinematics and dynamics) Design Variables (MDVs). The robot configuration may be
determined based on some kinematics considerations (egg., shape and size of
workspace, and obstacle avoidance requirements), the required accuracy,
manufacture-ability and cost (for exampie, precise linear joints are more costly than
rotary joints). The final configuration may be selected after several design iterations.
After choosing the arm configuration, MDVs (which represent the unknown
dimensions, masses, etc.) are identified. For example, we may choose a SCARA
configuration for the robot, and consider the link lengths as two unknown MDVs.
Derive the d y ~ m i c s model of the arm as afwction of MDVs. After determining the
robot configuration, it is possible to derive a parametric dynamics model of the
mechanical arm. For example, the parametric dynamics model of a SCARA robot can
be obtained as a function of its unknown link lengths. The unknown link lengths may
appear as symbols L, and L, in the parametric (or symbolic) model.
Propose or select a suitable control algorithm for the robot, and identifr the Control
Design Variables (CDVs). The control algorithm must be simple, yet stable, robust,
and it must satisfy task requirements. The following chapters provide some control
algorithms for non-contact, contact, and bracing applications of flexible M/m robots.
Although the control algorithm is known in this stage, the numerical values of the
control gains may not be known. The unknown control gains constitute the CDVs.
5. Using the djnamics model and feedback control algorithm, derive the closed-loop
model of the robot as a Mction of MDVs und CDVs. The closed-loop model shows
the relationship between the input commands to the controller (eg., the desired
trajectory) and the robot motion (e.g., the actual trajectory). In other words, the
closed-loop model represents the dynamic behavior of the robot. Both MDVs and
CDVs appear in this model.
6. Funnulate the objective and constraint functionrr Based on the closed-loop model
and task requirements, formulate the objective (performance measure) and constraint
functions to quantitatively represent the task objective and design constraints
identified in Step 1. Also, determine the upper and lower bounds on the design
variables. The objective and design constraints are functions of both MDVs and
CDVs. Formulation of the objective and constraint functions is illustrated in Chapters
3,4, and 5.
7 . Find the optimal values of MDVs and CDVs using a suitable nonlinear programming
method.
The above steps are illustrated in the following chapters through case studies and
by formulating several performance measures and constraint functions for different
applications of long-reach M/m robots.
1.4 Contributions
This work is the fmt attempt to apply D M to flexible long-reach manipulators. The
scope of the work is general in the sense that it covers non-contact, contact, and bracing
applications of long-reach M/m arms. The main contributions of the thesis are as follows:
1. Development of a new procedure for automatic generation of the closed-fonn finite
element model ofwble/rgid a m . The procedure is applicable to serial arms with
arbitrary numbers of flexible and rigid links. Compared with the previous related
works, this procedure has the following advantages: i) it is based on a modular
formulation of dynamics equations, in which the dynamic effects of each link is
explicitly computed; and ii) all spatial variable dependent integrals of shape functions
are separated from the other terms in dynamics equations, and each integral is
replaced by a variable. As a result, the integrals are computed only once for each
element type, and it is quite easy to use different element types for the modeling.
2. Illustration a d comparative analysis of IDM and TDM, based on case studies. The
closed-form finiteelement model of a 4-DOF planar Flexible-MacrolRigid-Micro
(FMRM) is generated and used for three design case studies in Chapter 3. The first
case study represents TDM, in which the mechanical design precedes the control
design. The second and third design case studies use IDM to minimize the total mass
(Case 2) and tracking error (Case 3) of the M/m robot, respectively. Comparison of
the results of the first design case study with those of the second and third case studies
is evident of a significant improvement of performance and reduction of mass
achieved by integrating the mechanical and control design stages.
3. Closed-form solution of a pe$ormance measure for non-contact applications of a
flexible M/m arm. A performance measure is provided to quantify the tracking error of
a flexible W m robot during a non-contact motion. The favorable numerical properties
of the performance measure are illustrated by two integrated design case studies.
4. Closed-fonn solution of a pegonname measure for contact applications of a flerible
M/m arm. A performance measure is provided to quantify the position and force
tracking errors of a flexible Mlm robot during a contact motion. To formulate the
closed-loop model, the hybrid task representation based on curvilinear coordinates
[Yos85] is extended to flexible Wm robots.
5 . Design of a Bracing M& robot, which includes: i) closed-fonn solution of a
performance measure for the integrated design of a bracing flexible robot (the
macro); and ii) conceprual and configuration design and modeling of a novel micro
robot for the bracing macro robot. A performance measure is provided to quantify the
cycle time of a bracing flexible robot, and a novel micro robot (called Robot for
Curved Surfaces or "ROCS") is designed to reduce dynamic interactions between the
bracing macro and micro. ROCS is a specidconfiguration robot used, with a bracing
long-reach robot, for scanning (or positioning on) a c w e d surface, when an active
control of contact forces is not required. Compared with a spatial-configuration (3D)
robot, ROCS requires less degrees of k d o m and a simpler control algorithm to
cover a curved surface. Deployment of ROCS at the end of a long-reach bracing mbot
provides an effective solution for scanning a curved surface in a large workspace
(e.g., a large storage tank).
1.5 Organization of the Thesis
The thesis is organized in six chapters. Chapter 1 briefly presents the objectives and the
rationales behind them, reviews the related literature and the state-of-the-art in robotics
design, introduces the general methodology and framework used in subsequent chapters
for integrated design optimization, and summarizes the main contributions of the thesis.
Chapter 2 focuses on the kinematics and dynamics modeling of flexible M/m
robots. A modular systematic procedure is proposed for the automatic expansion of the
closed-form kinematics and dynamics equations of such robots. Closed-form finite-
element equations of motion are required for the integrated design case studies in
Chapter 3, and for simulations and stability analyses in Chapters 3 and 4.
Chapter 3 provides the details of three representative design case studies on non-
contact applications of a 4-DOF flexible M/m robot. In the first case study (which
represents the traditional design method), the control gains are derived for a nominal
mechanical design. In the second and third case studies, integrated design optimization
method is used; a new performance measure for non-contact applications of long-reach
Mlm arms is formulated, and the design constraints are provided. Finally, a comparative
analysis is performed between DM and TDM based on the numerical results of the case
studies.
Chapter 4 extends IDM to other applications of long-reach M/m robots. A
comprehensive list of the potential applications of these robots is compiled from which,
the task requirements are identified and classified into three main groups of non-contact,
contact, and bracing tasks. As the development of DM for non-contact tasks was the
focus of Chapter 3, Chapter 4 and Chapter 5 are devoted to the development of IDM for
contact and bracing tasks, respectively. Chapter 4 is continued by formulating a new
performance measure for the integrated design of constrained flexible Wm robots.
Chapter 5 focuses on bracing applications of long-reach Mlm robots. The first part
of the chapter provides a task plan for surface scanning applications of bracing Mlm
robots (egg., ultrasonic or visual inspection, laser cutting, etc.), and based on that,
introduces a performance measure for the integrated design of such robots. The second
part of the chapter focuses on the conceptual and configuration design and modeling of a
novel micro robot (called "ROCS") for scanning applications of bracing M/m robots.
Finally, Chapter 6 summarizes the work and conclusions, and offers
recornmendations for fhture work.
Chapter 2
Modeling of Flexible Robots for Design Optimization
Chapter 1 provided a general methodology and framework used in the subsequent
chapters for the integrated design optimization of flexible Mlm robots. Integrated design
of robots at the parametric stage was formulated as a mathematical non-linear
programming problem. The proposed framework requires the closed-form analytical
(symbolic) solution of the equations of motion. Therefore, this chapter is devoted to the
kinematics and dynamics (finiteelement) modeling of flexible M/m robots.
The exact form of dynamics equations of a flexible M/m robot is expressed in
terms of a hybrid partiaVordinary set of differential equations. However, the exact form is
not suitable for design, simulation, and control purposes. Approximation techniques such
as Finite Element Method (FEM) or Assumed Mode Method (AMM) must be used to
obtain a finite dimensional model. Due to the ~ i ~ c a n t complexity of the intermediate
steps and final dynamics equations, hand derivation of them is nearly impossible and is
prone to errors. Moreover, any future modification of the robot configuration would
require a significant development time. As a result, a systematic procedure was required
for automatic generation of the closed-form analytical solution of equations of motion.
This issue is addressed in this chapter.
Section 2.1 provides a basis for Sections 2.2 and 2.3 by reviewing background
materials and the terminology on kinematics, finite element method, and dynamics of a
multi-link flexible arm. Formulation of the kinematics equations in this section is based
on the Link coordinate frames and homogenous transformation matrices presented in
[TG957 and Foo841.
Section 2.2 proposes a new approach for the expansion of inertia, gravity, and
stiffness matrices of a flexible-link robot. Compared with previous related works, the
procedure has the following features: i) a modular formulation of the dynamics equations
is used, which allows for the explicit computation of the dynamic effect of each link, and
provides more physical insights to the structure of the equations; ii) all spatial variable
integrals over the link lengths are separated from the other terms, which allows for a more
efficient computation of the integrals; and iii) the inertia, gravity, and stiffness matrices
are calculated directly (with less computations) as compared with some previous methods
that fmd the dynamics matrices indirectly form kinetics and potential energies.
The formulations in Section 2.2 are general and can be applied to both planar and
spatial robots. Section 2.3 simplifies the formulation in Section 2.2 to provide a
procedure for automatic generation of the finite element model of a planar robot with
arbitrary numbers of flexible and rigid links. The procedure can be easily extended to
spatial robots using the formulations in Section 2.2.
2.1 Review of Flexible-Link Kinematics and FEM
The purpose of this section is to providelreview background material and the terminology
used in Sections 2.2 and 2.3 for the formulation of an automatic modeling procedure.
This review includes a procedure for kinematics modeling of flexible-link robots
[Boo84], the finite element model of a flexible link using beam elements [TG95], and the
general closed-form solution of dynamics equations Wos901.
The motion of a flexible robot is represented by flexural deflections of flexible
links super-imposed on rigid-body motions of the arm. Rigid-body motions are usually
represented by joint variables, and flexural deflections of flexible links are represented by
functions that show the deflections at each point along the links. These continuous
functions can be approximated (or discretized) by functions with a finite number of
parame ten. The flexural and rigid-body parameters together constitute the total number
of deg& of freedom of a discretized dynamics model.
Complexity of flexible systems practically inhibits the manual expansion of
closed-form equations of motion. A systematic method (procedure) is required for
automatic derivation of equations of motion using computer packages for symbolic
computations. The main purpose of Sections 2.1, 2.2, and 2.3 is to formulate a recursive,
modular, and efficient procedure for automatic computer generation of the closed-form
finite element model of robots with arbitrary numbers of flexible and rigid links. The
following assumptions are made for the derivation of the kinematics and dynamics
models:
1. The robot consist of several flexible andlor rigid links, serially connected together
using revolute joints.
2. In their undeformed configurations, links are straight with zero offset.
3. Flexural deflections are small in comparison with the arm dimensions.
4. Euler-Bernoulli beam theory is used for the modeling of flexible links.
5. The effect of axial loads on the stiffness matrix is found negligible and is ignored.
2.1.1 Related Works
A flexible M/m robot has an infinite number of degrees of fieedom, and its dynamics is
represented by a set of hybrid partial/ordinary differential equations. Hamilton's principle
can be used to obtain the equations of motion and boundary conditions Wei701,
f ~ d r + r ~ w ~ d t = 0 , 11 (2.1)
where
L = T - V (2-2)
is the Lagrangian, W,is the work of non-conservative forces, and 6 is a small virtual
variation from the actual value. In (2.2). T and V are the total kinetic and potential energy
of the arm, respectively.
The hybrid partiaVordinary differential form of equations provides an accurate
(exact) representation of a continuos system behavior. However, this form of dynamics
equations is not suitable for simulation, control, and design purposes. Approximation
techniques must be used to discretize the exact form of the equations into a discrete
system model with a finite number of degrees of freedom.
19
The two most common discretization methods are Finite Element Method (FEM)
and Assumed Mode Method (AMM). Each method has its own advantages and
disadvantages. FEM is more versatile than AMM and can be applied to more complicated
configurations (e.g., a rob0 t with non-uniform cross-section links). Therefore, for design
purposes, FEM is more suitable than AMM.
Based on homogenous transformation matrices, Book Po0841 proposed a method
for the systematic derivation of assumed mode model of flexible arms using the Lagrange
equation. Transformation matrices are used to provide the position of a desired point on a
flexible or rigid link with respect to (w.r.t.) a fixed base h e .
Based on Book's approach, Centinkunt et al. [CB87] developed a procedure for
automatic expansion of the closed-form equations of motion using a commercial
symbolic calculation software. To improve the efficiency of the algorithm, they extracted
all possible combinations of spatial-variable dependent terms and replaced them with
symbolic names, so that the integrals w.r.t. spatial variables can be performed only once.
Korayem et al. m93] followed the same approach to develop a software on a PC-based
symbolic language Mathematica.
Lin et al. [LL94] proposed a more efficient formulation of the assumed mode
model by removing the "trace" operator from the equations, and using 3x 1 translation and
3x3 rotation matrices instead of 4x4 homogeneous transformation matrices. Furthermore,
they directly obtained the inertia matrix by separating non-velocity terms in the kinetics
energy. However, in their approach, they did not compute spatial-variable dependent
integrals explicitly.
The above papers used AMM for model discretization. In general, FEM is more
versatile than AMM. It can be used for modeling of links with a complex geometry or
non-uniform cross-section, and can also be extended to closed-loop manipulators.
However, for beams with simple geometry and boundary conditions, AMM usually
provides a lower order model for the same accuracy [JK!93].
Theodore et al. [TG95] provided a procedure for automatic generation of finite
element model of flexible robots with rotary and prismatic joints. However, they do not
extract the shape function integrals from the dynamics equations, and the computation of
the integrals is repeated for every element. The formulation of the kinetics and potential
energy is similar to that in [LL94]. They compared AMM with FEM in terms of
computational efficiency, accuracy, and versatility. FEM was used by other researchers as
well [SD8 11, -861, [NS87, [BT89], [Jon90], and [CACgl]. Most of these papers,
however, focused only on formulations and illustration of them by simple examples, and
majority of them consider only planar robots. They did not aim at automating the
modeling process. Therefore, the formulations are not suitable for the systematic
computer generation of the closed-form model.
It is desirable to provide a modular formulation of the dynamics equations, so that
the effect of each link on the dynamics model is explicitly expressed. Li &87] proposed
a modular formulation of dynamics equations for a Mlm robot mounted on a base
(M/m+B), and based on that, he studied the contribution of each part (i.e., the macro,
micro, and base) in the dynamics model and the interactions among the three parts. His
formulation, however, contains only three partitions corresponding to the three
subsystems, and it does not provide a systematic procedure for the derivation of the
dynamics model of each subsystem, or a system with more than three subsystems.
Due to the lack of a general-purpose, modular, and efficient procedure for
automatic generation of the finiteelement model of flexible multi-link robots, Section 2.2
proposes an iterative modular method for automatic modeling of such robots with an
arbitrary number of subsystems. Each separate link can be considered as one subsystem.
By starting from the first link and adding a new link in each iteration, we can
systematically expand the dynamics equations. Section 2.3 simplifies this procedure for
the finiteelement modeling of planar multibody systems.
2.1.2 Kinematics
Figure 2.1 shows three links of a flexible robot. Links are numbered consecutively from 1
to n, where link 1 is directly attached to the base frame (01, and link n is the end effector.
We use the approach in [TG95] to assign the frames. Frame (XbYbZi) is attached to the
distal end (farthest from the base) of link i, oriented such that the Zi axis is aligned with
the axis of joint i+ 1 and Xi axis is aligned with link i. The frame (&&a is defined in
such a way that when link i is in its undeformed configuration, the two frames (XbY-.Zi)
and (&La are coincident. In Figure 2.1, Bi denotes the rotation of joint i about
axis, Pi.1 represents the position vector of the origin of frame (Xi-l,Yi-r,Zi-l) w.r.t. to the
base frame, and ri denotes the relative position vector of an arbitrary point on link i w.r.t.
the proximal end of this link.
Figure 2.1 : Coordinate Frames of the Flexible Link i and DH Parameters
The 4x4 homogeneous transformation matrix from frame (X*YhZi) to frame
(Xi-l, Yi- l ,z- l) is given by [TG95],
"'q ="'Ai Ei , (2.3)
where
COS& -sin8,cosai sineisinai aicosei
sine, cosO,cosai -c0s8~sina, aisinei i - ' ~ , =
O sin ai cosai 4 0 0 0 I
is the 4x4 transformation matrix fiom frame (& & z) to frame Yi-l.Zi-l); ei , ai , di ,
and ai are the Denavit-Hartenberg parameters of link i; and
is the transformation matrix from frame (XbYbZi) to frame (&&a, caused by small
flexural rotations Oi =(cp,,p,,p,) T and deflections Ai = (6,, 6,. 6, )' of the flexible
link i along coordinate axes (&&a. The 4x4 homogeneous transformation matrices
from frames (Xi, Yi.2') and (Xi, YbZ) to the base frame (Xa Ya&) are given by
respectively. In (2.6) and (2.7), Og and 'Iti are 3x3 rotation matrices, and O F and '4
are 3x1 position vectors. The position vector of any point ( x i ) on link i w.r.t. the base
frame is given by
C i "q=Op-,+ R, q,
where
is the relative position vector expressed in frame (X;.Y;.Zi), xi is the spatial coordinate
along the link length, and yi (x i . t ) and zi (x i , t ) are flexural deflections in B and
directions, respectively.
Note that the dependence of yi and q on the spatial coordinate xi makes the
system infinite dimensional, leading to coupled ordinary and partial differential equations
of motion. In order to obtain a finite dimensional model, the deflection functions yi and
ti must be discretized on the spatial coordinate xi. In other words, the dependence of yi
and 4 on the spatial variable xi must be replaced by its dependence on a finite number of
time-dependent parameters. Discretization of the model is the subject of the next part.
2.1.3 Finite Element Method
This part reviews the finite element model of a flexible link. Finite Element Method
(EM) and Assumed Mode Method (AMM) are the most common discretization methods
used in robotics. The main idea is to approximate yi and zi, which are unknown
functions of the spatial variable x i , with a combination of several known functions of xi .
In AMM, a flexural displacement y (or z ) is approximated by a truncated modal series
where p,(x) is the k* mode shape and qJt) is the modal coordinate. In this method,
the dependence of y on the spatial variable x is replaced by its dependence on a finite
(m) number of parameters q,(t). The main difficulty in this method is to find a suitable
set of mode shapes.
In FEM, instead of using mode shapes that are defined over the entire length of
the link, we consider the link as an assemblage of a finite number of elements and define
the mode shapes (or in this case, "shape functions") over an element span. Therefore, we
can use simple functions (usually low order polynomials) to define shape functions of an
element. These shape hctions are then multiplied by the displacements at finite number
of points (called "nodes") to express the displacement at any point of a continuous
element.
Figure 2.2 shows a simple beam element that can be used for modeling of a
flexible link. The link deflection y(x, t ) is approximated as
y(x, t ) = ye (x. t ) for x i < x < x; ,
where ye(x, t ) is the element displacement at a point x and a time t. The simplest beam
element has two nodes and four parameters. Therefore, the element displacement at any
point x of the element is given by
where
is the element local variable, and I, and x i are the length and x coordinate of the first
node of the element, respectively;
w e ) = ( ~ ; ( t ) N35) W ( t ) NX)) (2.14)
is a 1x4 matrix of shape functions, and
W e ( ) = ( ( t u; ( t ) v; (t$ (2.15)
is a 4x 1 vector of nodal parameters. In (2.1 S), ute ( t ) and v , ~ ( t ) denote the displacement
and rotation of the f' node of the element, respectively (Figure 2.2).
Figure 2.2: A Simple Beam Element with Two Nodes and Four Parameters
The beam shape functions are third order polynomials defined by
N; = I - 35' + 2e3,
N: = 1,&5 - I)', N; = 5'(3 - 25),
N; = I, {'(C - I). Adjacent elements are connected together at the nodal points. Continuity of the
displacement function y(x, r ) and its first derivative (w .r.t. x) are assured over the entire
span of the beam by using the nodal displacements and rotations as the element
parameters. Figure 2.3 shows a flexible link with three beam elements. At each node we
have two parameters and at node zero (i.e., the left node of the fvst element), both
parameters are zero:
U ~ ( ~ ) = U ~ ( ~ ) = ~ ( O . ~ ) = O and v , ( t ) = v , ' ( t ) = ~ ~ I , , o = O
Therefore, the total number of parameters that completely define the displacement y(x, t)
at any point x of the continuous beam is six.
Figure 2.3: A Beam with Three Elements and Four Nodes
In general, the total number of parameters N, for describing a beam deflections in
one plane is given by
N, = N e x 2 = ( N , - 1 ) x 2
where N, and N, are the number of elements and number of nodes of the beam,
respectively. The displacement at any point of the beam is obtained by knowing the nodal
displacements and rotations.
In a three-dimensional (3D) space, each node of a simple beam element has six
parameters, i.e., two bending and one axial displacements, two bending rotations, and one
torsion (Figure 2.4). However, usually the beam has more rigidity in the axial direction x,
and the axial displacement ux can be ignored. Therefore, the vector of parameters that
represents the flexural displacements and rotations of link i at node k is given by
Y : = ( u { ~ v& ui; vi:, vif,). (2- 19)
For a flexible link i with Nei elements and a joint rotation 4. the vector of parameters is
given by
= ( q 'Y: - - - Y,i&)*. (2.20)
This vector completely defines the relative motion of link i w.r.t to the previous links.
Figure 2.4: Displacements u and Rotations v of Node k of Link i in a 3D Space
2-14 Dynamics
This section reviews the general structure of a flexible arm dynamics model, and
introduces the terminology used in subsequent sections. The discretized dynamics model
of a flexible robot can be obtained by substituting the total kinetics energy T and potential
energy Vof the robot into the Lagrange equations of motion
dT dV - '(") - -- +-=Q, for j = ~ 2 . . . , ~ ~ ~ , dt * j dq, &j
where qj is the jh element of the generalized coordinates vector q, and Qj is the
corresponding generalized force, Nw is the total number of degrees of freedom, and dot
" -" means derivative w.r.t. time. For a flexible robot approximated by finite elements, the
total kinetics energy T and potential energy V is expressed in terms of nodal and joint
positions and their derivatives w.r.t. time. The total kinetics and potential energies are
given by
and
V=V,+V', (2.23)
where V, is the conf~guration-dependent potential energy due to the gravity force, and
is the total potential energy due to elastic deflections of flexible elements. In (2.22) and
(2.24), M and K are the inertia and stiffness matrices. respectively. The generalized
force/torque Qj corresponding to generalized coordinate q,
where ei and r i are the rotation angle and torque of joint
total number of joints (or rigid-body motions).
is given by
(2.25)
i, respectively, and N, is the
Substituting (2.22) to (2.25) into (2.21), yields the closed-form equations of
motion
where q is the vector of generalized coordinates; Fg(q) is the configuration-dependent
gravity-force vector given by
Fcc(q, 4) is the vector of centrifugal and coriolis forces given by
N N
Fccj = z f h j k d 1 4 k for j = I, 2,...,N,
and
JMjl I JM@ hjk = -- -- for j , I ,k=I ,2 ,..., Nb,, a 2 %
where Mi, and M, are the components of the inertia matrix; B is the N,, x N, input
matrix given by
and r is the N, x 1 vector of joint torques. Subsequent sections provide a procedure for
automatic generation of all the matrices and vectors in (2.26).
2.2 Expansion of Dynamics Equations
This section presents a new approach for the iterative expansion of the inertia matrix,
gravity-force vector, and stiffness matrix of a flexible robot. Compared with the previous
related works, this approach is more efficient because of its modular structure, the
explicit expression of the spatial-variable integrals, and direct computation of dynamics
matrices (without computing potential and kinetics energies). Besides, the contribution of
each link in the dynamics model is clearly shown in the modular formulation.
Consider a robot with n flexiblelrigid Links, serially connected together using
revolute joints. The idea is to start from link 1 (closest to the base) and expand the vector
of generalized coordinates and dynamics matrices (i-e., inertia, gravity-force, and
stiffness) as links 2, 3, ..., n are added to link 1 (see Figure 2.5). In other words, if qi-,
denotes the generalized coordinates vector of links 1 to i- 1, and ri represents the vector
of generalized coordinates of link i given by (2.20), then the generalized coordinates
vector of links 1 to i can be defined as
Correspondingly, the inertia matrix, gravity-force vector, and stiffness matrix of links 1 to
i-1 will be expanded to accommodate the dynamic contribution of the last link i. This
approach provides a means for iterative modular expansion of dynamics matrices. When a
new link is added to the arm, only the dynamic effects of that link on the rest of the arm is
computed and added to the dynamics model.
2.2.1 Expansion of Inertia Matrix
When link i is added to links 1 to i-1, corresponding to the expansion of the generalized
coordinate vector (2.3 I ) , the inertia matrix is also expanded as
Inertia Matrix:
Gravity Vector:
Figure 2.5: Adding Link i to the Previous Links and Expanding the Generalized
Coordinate Vector and Dynamics Matrices
where Mi-, is the inertia matrix of links 1 to i- 1 ; and I,, , Ii+, and Iii are, respectively
the increase of the inertia matrix Mi-, due to link i, coupling inertia between link i and
the previous links, and the inertia matrix of link i when the previous links are fixed. The
30
I i - I i inertia matrix [I;-' Iu ] represents the total contribution (or effkct) of the last link i to ' i - I i
the robot inertia matrix. The new inertial components I,, , and I, are obtained in
the next part fiom the formulation of the kinetic energy for link i.
Flexible Links
From (2.32). the kinetics energy of link i is given by
W e use the velocity transformation formula between relative coordinate frames
V* = yI +mi-, xr, +i,, (2.34)
to obtain the total kinetics energy of link i. In (2.34). V, denotes the velocity vector of a
point xi on the link i w.r.t. the base frame (see Equation 2.9 for xi), the notation x denotes
the cross product of two vectors, r;- represents the relative position of point xi w.r.t. the
origin of frame (Xi-lrYi-IIZi-I) (see Figure 2.1), and x-, and ti+-, are the linear and
angular velocity vectors of frame (Xi.ll Yi.l, Zi.1) w .r. t. the base frame, respectively. The
total kinetics energy of flexible link i is given by
where the notation "." denotes the inner (scalar) product, defined for two arbitrary column
vectors a and b as
a - b = a T b . (2.3 6)
From (2.34), we have
and using Jacobian matrices, we can express the velocity terms in (2.37) in terms of qi-,
and ri as
K 4 = Jn-1 4 - 1 9
= JK-l @i-1 ,
and
6 = Jii ri , where Jn-, , J,+ and Jii are the corresponding Jacobian matrices.
(2.38), we can rewrite the link kinetics energy as
I; =ql+q2+q3 +qr +Ts+q6,
where
and
(2.3 8c)
Using (2.35) to
is a 3x3 matrix, defined by
and the notation x] denotes, for an arbitrary 3x 1 vector a = (a, a, aZ)' ,
Substituting (2.40) into (2.39) and comparing the result to (2.33) yields
and
where the scalar
is the total mass of link i; the vector
represents the position of the link center-of-mass multiplied by its total mass; the matrix
is the inertia tensor of link i; and the matrices Di, and Di5 are defined by
and
Note that (2.43) explicitly gives the contribution of link i to the robot inertia matrix, and
all spatial-variable integrals in this equation are separated from the other terms.
Rinid Links and Lnm~ed Masses
The kinetics energy of a rigid body is the sum of two terms: the energy due to the
translation of its center-of-mass, and the energy due to rotation of the Link about its center
of mass. Each rigid link increases the number of generalized coordinates by one
ri =e i , (2.45)
and increases the kinetics energy of the arm by
where Di and HG are the total mass and inertia tensor (about the center-of-mass) of link
i, respectively; V, is the velocity vector of the center-of-mass; and mi is the angular
velocity of link i. The velocity terms in (2.46) can be expressed as
= JTCi Qi (2.47)
and
mi = J,&, (2.48)
where J,, and J, are the corresponding Iacobian matrices. Substituting (2.47) and
(2.48) into (2.46) and extracting non-velocity terms yields
where M, is the contribution of rigid link i to the inertia matrix.
Similarly, the kinetics energy of lumped mass i with its center-of-mass located at
the joint i+ l (attached to the distal end of the link i ) is given by
where D, and R, are the mass and inertia tensor (about the center-of-mass) of lumped
mass i. Rewriting (2.38) for link i and substituting the results into (2.50) gives
which readily provides the contribution of the lumped mass i to the inertia matrix Mi as
M , = D , J ~ J , + J : H , J , . (2.52)
22.2 Expansion of Gravity Vector
When link i is added to Links 1 to i-1, corresponding to the expansion of the generalized
coordinate vector (2.3 I), the gravity-force vector is also expanded as
where FG-, is the gravity-force vector of links 1 to i-1; and fgi-, and fgii a,
respectively, the gravity-force added to Fgi-, due to link i and the gravity-force of the
link i when the other links are fixed. The new gravity terms fgi-, and fgii are obtained
from the formulation of the gravitational potential energy for link i.
Flexible Links
The gravitational potential energy of flexible link i is given by
where G is the gravity-acceleration vector, and P, is the position vector of a point xi on
link i. Differentiating (2.54) w.r.t. helm generalized coordinate yields
where fg, is the 7 component of the gravity-force vector due to link i. By substituting
the equations
and
into (2.55), we obtain
Rieid Links and L m d M8sse5
The gravitational potential energy of rigid link i is given by
VGi = Di G - P,, (2.59)
where Di is the total mass of the link, and P, is the position vector of the link center-of-
mass w .r. t. the base frame. Differentiating (2.59) w .r. t. the f' generalized coordinate
yields
%i fg, =-- *a *, dpfi is the where fg, is thel' component of the gravity-force vector due to link i, and - a4,
column of the Jacobian matrix J , . Using the Jacobian matrix, we can rewrite (2.60) as
Similarly, the gravitational potential energy of lumped mass i with its center-of-mass
located at joint i+ 1 (attached to the distal end of link i ) is given by
V' = D,G*Pi, (2.62)
where D, is the mass and 4 is the position vector (w.r.t the base frame) of lumped
mass i. Differentiating (2.62) w.r.t. the generalized coordinates yields
~ ~ , = D , J ; G . (2.63)
where J: is the Jacobian matrix, and fg, is the contribution of the P lumped mass to
the total gravity-force vector Fgi .
2.2.3 Expansion of Stiffness Matrix
Stiffness matrix of a flexible robot is due to the elastic potential energy stored in flexible
links, and it is only a function of the flexural parameters of flexible links. For flexible
link i, the elastic potential energy is given by
where Ui is the vector of nodal displacements and rotations of the link i, defined by
In (2.65), Y: is the vector of nodal parameters defined by (2.19). Because the elastic
potential energy of a flexible link is only a function of the link flexural parameters, the
global s t f iess matrix of a robot with multiple flexiblehigid links has a block diagonal
where the ?' diagonal block corresponds to the zh flexible link. The elastic energy of a
robot is not affected by its rigid-body motions ( O,, 8,. ,em ); therefore, the diagonal
terms of the stiffness matrix corresponding to the rigid-body parameters are zero.
The elastic energy of flexible link i is given by
where E?: and EZ,? are bending stiffness about the respective axes of the link. and
GJ; and cpf are the torsional stiffness and torsion angle about the xi axis of the $ link,
respectively.
This section provided a modular formulation for an iterative expansion of the
inertia matrix, gravity-force vector, and stifmess matrix of a flexible robot. The modular
formulation of the dynamics matrices clearly shows the structure of dynamics equations
and the contribution of each link to the dynamics of the arm. All spatial-variable
dependent integrals were separated from the other terms and replaced by symbolic names.
The equations in this section are used in the next section to produce a procedure for
automatic generation of the finite element model of planar flexible robots.
2.3 Finite-Element Modeling of Planar Flexible Robots
Using the formulations in Sections 2.1 and 2.2, this section presents a modular systematic
procedure for automatic generation of the finite element model of a planar robot with
arbitrary numbers of flexible and rigid links. First, the equations of motion are obtained
for a planar (2D) robot by simplifying the equations in Sections 2.1 and 2.2. Second, the
spatial-variable (xi) dependent integrals in dynamics equations are calculated using FEM.
Explicit computation of the finite element integrals increases the efficiency of the
procedure, as each integral is computed only once. Finally, a procedure for automatic
generation of the symbolic closed-form equations of motion is presented. Although the
procedure is proposed for planar manipulators, the approach is general and can be easily
extended to spatial cases.
2.3.1 Equations of Motion of Planar Flexible Robots
Sections 2.1 and 2.2 presented the equations of motion for spatial flexible robots. The
new features of the procedure were: i) modularity of the formulation, in which the
dynamics matrices are first obtained for the first link and then, expanded to accommodate
the effect of the other links (2.32). (2.53), and (2.66); and ii) separation of spatial-variable
dependent terms (2.43~) and (2.44) from the other terms.
This section simplifies the spatial form of the equations to obtain a planar (2D)
version of the kinematics and dynamics equations. The explicit calculation of the spatial-
variable dependent integrals distinguishes the proposed procedure from previous ones.
Kinematics
Kinematics equations provide the position and orientation of link coordinate frames and
the center-of-mass position of each rigid link. Jacobian matrices are then obtained by
differentiating the position vectors w .r. t. the generalized coordinates.
Figure 2.6 depicts the ? link of a planar flexible robot with revolute joints. In this
figure, pi is the absolute rotation angle of frame (&E) w.r.t. the base frame (&,Yo), and
denotes the flexural rotation angle at the distal end of flexible Link i.
The position of a point xi on link i w.r.t. the origin of frame (Xiel, E-l) is given by
The rotation matrix between frame (Xi, and the base frame (Xo, Yo) is given by
and the rotation matrix between frame (Xi, Yi) and the base frame (X6 Yo) is given by
where the rotation angle pi is obtained from the following recursive formula:
pi = pi-I + Q),-~ + ei for i = 1.2, ..., n , (2.72)
starting from
P o =cP,o = O = (2.73)
Using the rotation matrices, we can obtain the position of the origin of coordinate frames
from the following recursive formulas:
and
starting from the base
represents the flexural displacement at the distal end of flexible link i. For rigid links, the
flexural displacement 6,, and rotation 9, are zero, and frames (Xi,&) and (XbYi) are
coincident. Therefore, (2.70) and (2.74) can be used to obtain the rotation matrix and the
position vector between the link frame (Xb Yi) and the base frame (Xh YO). The position of
the $ link center-of-mass is given by 0- '
O pci + Ri 'ra , (2.78)
where 'r, is the position vector between the link center-of-mass and the origin of frame
(Xi-r, Yi.r), expressed in the P coordinate h e .
Differentiating the position vectors '4 and OP, w.r.t. the generalized coordinate
vector qi of links 1 to i, yields
where O J , and ' J , are the Jacobian matrices expressed in the base frame, and ' J , is
the Jacobian matrix expressed in the f' frame.
F i t e Element Model
Rewriting (2.1 1) and (2.12) for link i and element j, we have
yi(xi , t)=y~(xi. t) for x&<x, < x i ,
and
y i ; . ( x i ~ t ) = ~ ' ( ~ ) U , ' ( t ) ,
where
defines a local (normalized) variable, Ne({) is the vector of shape functions,
is the vector of nodal parameters, % denotes the length, and x& denotes the x coordinate
of the first node (see Figure 2.2) of element j on link i. Equation (2.19) defines the
parameters for a simple beamelement in the three-dimensional space. For a planar robot,
this vector has only two components
y c = ( u L vif , ) , (2.85)
where ui:, and v;, denote the flexural displacement and rotation of node k on link i in the
y and z directions, respectively. The link generalized-coordinate vector is then given by
ri +, y; y;JT (2.86)
where the components of the last term represent the flexural displacement
6, =u&
and rotation
- 2 U. - ' i . ~ e i
at the distal end of link i. Finally, the generalized-coordinate vector of links 1 to i is given
by
Using (2.71) to (2.73), we can show that the rotation angle of frame (Xr. Yi) wxt. the base i
frame is equal to Z(B, +q,). Differentiating this angle w.r.t. the generalized k=l
coordinates (2.89) yields
where the non-zero terms in the Jacobian matrix J , correspond to 8, and v:,,, in the
generalized-coordinate vector (2.89).
Using FEM, this section provides formulations for calculating all spatial-variable (4)
dependent integrals in the dynamics equations presented in Section 2.2. Explicit
computation of the spatial-variable and shape-function integrals increases the
computational efficiency and reduces the memory usage. Only planar robots are
considered here.
For each flexible link, the integrals in (2.44) can be computed by: fint, calculating the
integrals for each element of the link, and then, adding up the results to obtain the
integrals over the link span. For instance, the total mass of link i is obtained by fint,
calculating the mass of each element j of the link, and then, adding up the element masses
4 Nei Nei Nei
Di = Irnidxi = dri =c D; =&til,', j=r 0 j=I j=l
where the superscript e on a parameter shows that the parameter belongs to the element.
In the same fashion, we can obtain the link moment-of-inertia For a planar robot, only
the last component of the inertia tensor (i.e., the moment-of-inertia about the Z axis) is
required. Substituting (2.82) and (2.83) in (2.44~) yields
42
Similarly, substituting (2.69) into (2.44b) and carrying out the integrals give Nei
ci = Cc;
To obtain Di,, we use the following equation:
By dividing link i into several elements, we obtain
Nei
]?qi,dri = ~ l ~ ; ( ; ~ . where
is the time derivative of 'r;. defined by (2.69). and
is the Jacobian matrix for element j. We define the generalized-coordinate vector for
element j as
From (2.98) and (2.99)- we obtain
where
Finally, using (2.96), (2.97), and (2.101), we obtain
Nei
Equation (2.103) gives the global matrix Di4 in terms of the element matrices Di4. In the
same way, we can obtain Di5 from
where the subscript z means the z-component of the vector. Substituting (2.98) and (2.99)
into (2.104) and simplifying the terms yield
where
Equation (2.105) gives the global matrix Dis in terms of the element matrices I)&.
Similarly, by substituting (2.98) and (2.99) into
and simplifying the t e r n , we obtain
where
(2.109)
Equation (2.108) gives the global inertia matrix I, in terms of the element matrices I:j.
Rewriting (2.43a) for planar robots yields
where
and C, and C, are the components of
Finally, the inertia matrix I,-,, is obtained by substituting Did and Di, in (2.43b)
b) Gravily-Force Vector
Rewriting (2.58a) for planar robots yields
fgi-1 = Qi-1 G * (2.1 13)
where Q,, is given by (2.11 1). The gravity-force vector fg, is obtained by substituting
Dir in (2.58b)
The equation
I Nei I V, = -u: K~ ui = E-CJiT K; u;,
2 j=l 2
presents the global stiffness matrix Ki in terms of the element stiffness matrices Ki; . The
elastic potential energy for element j is given by
where Hi; is the bending stiffness of the element about the Z axis. From (2.1 15). we find
Evaluation of EIement Intenrals
Thus far, we transfomed link integrals (i.e., the integrals over the entire link span) into
element integrals (i.e., integrals over the element span). The final form of dynamics
equations contains the following definite integrals:
and
AU these integrals are calculated only once for each element type. For example, for a
simple beam element, substituting (2.16) into (2.1 17) to (2.120) and evaluating the
integrals yield
and
r12 61, -12 61J
where 1, denotes the element length.
S:=
2.3.2 A Procedure for Automatic Generation of Equations of Motion
Based on the formulations given in Section 2.3.1, this section presents a procedure for
automatic generation of the finiteelement model of planar flexiblehigid robots. The
procedure can be readily coded in any symbolic calculation language, such as
Mathernatica, to produce the symbolic closed-form equations of motion.
62, 41,' - 6 1 , -2Ze2 -12 d l , 12 dl,
Algorithm 1: Inaut Data
'
Step 1. Input number of links: n.
Step 2. Input gravity-acceleration vector: G . Step 3. Input Link type and parameters:
For i= 1 to n
If Rigid
LinkType[i] = 0.
Input: L,, Di, 'r,. Ha.
Else If Flexible
LinkType[i] = 1.
Input number of elements: N, . For j= l to N,
Input: I ; , mi;, EI,; .
Next j.
End If.
Next i.
Step 4. Input lumped-mass parameters:
For i=l ton
Input: D,, H , . Next i.
Aleorithm 2: Kinematics
Step I. Initialize variables, using (2.73) and (2.76).
Step 2. Calculate position vectors, rotation matrices, and Jacobians:
For i=l ton
If Rigid
Define: Ti = Oi . Set: (p, =a, = O .
Calculate: p i , O$, '4, and OP, , using (2.71) to (2.78).
i Calculate: J , , JTi , and J~ , using (2.79) and (2.80).
Else If Flexible
Defme: ri , p, , and 6, , using (2.86) to (2.88).
Calculate: pi, O$, '4 , and '8, using (2.70) to (2.75).
Calculate: O J , , and ' J , , using (2.79).
End If.
Define: qi , using (2.3 1).
Calculate: J , , using (2.90).
Next i.
Aleorithm 3: Dvnamicus
Step 1. Initialize variables.
Step 2. Calculate inertia matrix, gravity-force vector, and stiffness matrix:
For i=l ton
If Rigid
Calculate mass due to Link i, using (2.49).
Calculate gravity-force due to link i, using (2.61).
Else If Flexible
For j= l to N,
Define: U; in terms of generalized coordinates, using (2.84) to (2.86).
Calculate: Di , Hi , C; , using (2.9 1 ), (2.93), and (2.95).
Add to: Di , Hi , Ci , using (2.9 1 ), (2.92), and (2.94).
Calculate: Di4 , D& , and 1; , using (2.102), (2.106), and (2.109).
Add to: Did, Di, , and I , , using (2.103), (2.105), and (2.108).
Calculate: Q,, , I,, , and I,, , using (2.1 1 1 ), (2.1 1 O), and (2.43b).
Calculate: fgi-, , and fg,, , using (2.1 13) and (2.58b).
Calculate: Ki , using (2.116).
Add to: Ki , using (2.1 14).
Calculate: xii j+I = x& + 1;.
Next j.
End If.
Expand the inertia matrix, gravity-force vector, and stiffness matrices using
(2.32), (2.53), and (2.66), respectively.
Calculate the mass due to lumped-mass i, using (2.52).
Calculate the gravity-force due to lumped-mass i, using (2.63).
Add M, to the robot inertia matrix, and fg, to the gravity-force vector.
Next i.
Step 3. Calculate hi, by substituting the components of inertia matrix into (2.29).
Step 4. Calculate centrifugal and coriolis forces, using (2.28).
2.4 Summary
This chapter provided a new procedure for automatic generation of the symbolic closed-
form equations of motion for robots with arbitrarily numbers of rigid and flexible links.
The new features that distinguish the procedure from the previous ones are: i) modularity
of the formulation (2.32), (2.53), and (2.66); ii) explicit computation of spatial-variable
integrals of the shape functions (2.43c, 2.44, 2.93, 2.95, 2.99, 2.106, 2.109, and 2.1 16);
and iii) direct derivation of the matrices in the dynamics equations (procedure in Section
2.3.2).
Section 2.1 presented the background material on kinematics and finite element
modeling of serial flexiblelrigid robots. Kinematics equations were obtained using links
coordinate frames and transformation matrices, and the shape functions, nodal parameters
of a simple beam element, and the general form of dynamics equations were reviewed.
Section 2.2 proposed a method for systematic expansion of dynamics matrices (i.e.,
inertia, stiffness, and gravity-force) of flexiblehgid robots. All spatial-variable dependent
integrals were extracted from the equations (2.43~) and (2.44). This provided means for
efficient calculation of the integrals using finite element matrices in Section 2.3. Finally,
a modular iterative procedure was provided for the automatic computer generation of the
equations of motion for planar flexibldrigid robots.
Chapter 3
Integrated Design of A Planar M/m Robot
Section 1.3 presented a framework for Integrated Design Method (IDM) and explained
the Traditional Design Method (TDM). This chapter performs a comparative analysis
between IDM and TDM based on three representative design case studies. A Clink planar
Flexible-MacroRigid-Micro (FMRM) robot is used in the studies. The kinematics
configuration and control algorithm of the robot are provided fmt, and the mechanical
and control design variables are introduced. The optimal values of the design variables
that provide a good tracking performance in a free space (i.e., when the robot is not
constrained by the environment) are then obtained for each case study.
The first case study represents TDM, in which the control gains are designed for
the nominal values of mechanical parameters. The second case study uses D M to
minimize the total mass, and the third case study uses IDM to minimize the end-effector
tracking error, subject to constraints on some of the robot characteristics. The results of
the first case study are compared with those of the second and third case in order to
establish the advantage of using D M over TDM.
Section 3.1 introduces the mechanical configuration, provides the control
algorithm, and discuss the trajectory planning of the FMRM robot. Section 3.2 presents
the details of the three design case studies. The design constraints are provided, and a new
performance measure is proposed for the integrated design optimization of unconstrained
long-reach Mlm robots. Finally, the results of the case studies are used in Section 3.3 for
a comparative analysis between D M and TDM. The comparative analysis clearly shows
the superiority of IDM over TDM.
3.1 Mechanical Configuration and Control of the M/m Robot
This section introduces the Mlm robot that is used for the design case studies. First, the
mechanical configuration and control algorithm of the robot is presented, and the
mechanical and control design variables are introduced. Then, the trajectory planning of
the arm is discussed based on the method proposed in [YHM96]. The material in this
section is used in subsequent sections for the design case studies.
3.1.1 Mechanical Arm
Figure 3.1 shows the robot configuration which consists of a 2-DoF flexible macro arm
carrying a 2-DoF rigid micro arm. The macro provides a large work-space using two long
flexible links (links 1 and 2), and the micro performs dexterous manipulations using two
short and light rigid links (links 3 and 4). The coordinate frames 1 to 4 are attached to the
distal ends of links 1 to 4, respectively. In Figure 3.1, P, and P, denote the end position-
vector of the m a m and the micro, respectively; and Oi denotes the rotation angle of the
f' joint.
The robot is characterized by twenty-one mechanical parameters, as follows:
I ) Kinematics Parameters:
Link lengths: L, for i = I, 2,3,4.
2) Dynamics Parameters:
Mass per unit length of flexible links: mi for i = 1.2.
Bending stiffness of flexible links: Eli for i = L2.
Mass of lumped masses: D, for i = 1.2.
Inertia of lumped masses: H, for i = 442.
Mass and inertia (about the mass-center of) rigid links: Di, tr', for i = 3.4.
Location of the mass-center of rigid links: L, for i = 3,4.
Mass and inertia of the payload: DL,, H,, .
The next section presents a control algorithm for the M h arm, and introduces the
control design parameters.
w DU,HL, * - - - - .
Link 4: D3, Hcjr LC,, 4 4 - - - - - - - -
Link 1:
Figure 3.1 : A CLink Planar Flexible/Rigid Mlm Robot
3.1.2 Selection of a Control Algorithm
A joint variable PD control plus feedfornard compensation of gravity forces is considered
for the M/m robot. This controller is simple and provides satisfactory results (as will be
shown in simulations). Except for the joint gravity torques (which can be calculated off-
line based on the desired trajectory) the controller does not require the dynamics model of
the flexible arm. In the sequel, the control algorithm and the rationales behind its
selection are provided.
Control AIporitbm
Practically, every manipulator has inherent structural flexibility. However, the arm
flexibility does not affect the controller performance if the robot moves slow enough such
that the arm's natural modes are not excited. For fine motions of a flexible arm with joint
variable feedback control, Book et al. DM741 showed that the arm flexibility can be
ignored (and the arm can be considered rigid) if the lowest clamped natural frequency of
the arm od, ' exceeds three times the closed-loop servo bandwidth m, . However, the
arm flexibility must be taken into account if the servo bandwidth o, exceeds half of the
clamped natural frequency ok [CBgO].
Long-reach manipulators have very low clamped natural frequencies, and the
rigidity constraint (which indicates that oh must be smaller than o&, ) imposes a severe
limitation on the servo bandwidth if a satisfactory control response is required with a joint
variable feedback controller. To increase the servo bandwidth, one can increase the arm
stiffness and/or use a more sophisticated control algorithm. Increasing the arm stiffness
leads to undesirably heavy links and actuators, and using a noncollocated model based
control algorithm [CS84] significantly reduces the robustness and increases the
complexity of the control system.
On the other hand, joint-variable (collocated) feedback controllers have a simple
structure (which makes it suitable for numerical optimization), good stability and
robustness, and are widely used in practice [CB90]. The main disadvantage of collocated
controllers is their low performance in terms of the closed-loop bandwidth and tip
positioning accuracy. A better solution is to use a joint variable feedback controller for
the long-reach flexible manipulator, and to employ a fast rigid micro manipulator at the
tip of the long-reach macro for achieving the desired servo bandwidth and positioning
accuracy. Bandwidths up to several times the first clamped natural frequency of the arm
are easily attainable using the macro-micro structure and a simple control algorithm
' The lowest clamped natural frequency of the arm is the ftrst natural frequency of the a m when all joints
are Iocked and the arm is fully extended.
(CKC9 11 [SHH89]. Therefore, a joint variable controller can provide satisfactory results
if a macro/micro configuration is used.
Based on the above discussion, we use a joint variable Proportional-Derivative
(PD) controller for the Wm arm in Figure 3.1. The control algorithm of the macro is
given by
and the control algorithm of the micro is given by
7, =-~, (8, - 8 d , ) - ~ ~ ( & , - 6 d , ) + ~ ~ ~ ,
where the subscripts M and m refer to the macro and micro, respectively; Kp and KD
denote the diagonal matrices of proportional and derivative gains, respectively; .r denotes
the vector of joint input torques; 8 and 6 represent the vectors of joint displacements
and velocities, respectively; Fg denotes the vector of joint gravity torques; and the
superscript d denotes the desired values of the joint variables.
Equations (3.1) and (3.2) can be combined together and written as one equation
=-llP(e-ed)- K , ( & - ~ ~ ) + F ~ , (3.3a)
where
are the vectors of input joint torques, joint displacements, joint velocities, and gravity
torques, respectively; and
are the proportional and derivative gain matrices.
Due to the redundancy and flexibility of long-reach Mlm robots, their trajectory
planning is complex. Different methods may be used to obtain the desired trajectories of
the macro and micro in the joint space (i.e., edy , e:, and their derivatives w.r.t. time).
The following part presents a trajectory planning method for such robots, which, except
for some minor changes, closely follows the method proposed in -961.
Tniectorv Planning
A macro-micro is a redundant arm. The macro is used to locate the micro in the vicinity
of the area in which the micro performs the task. Due to its elastic deflections, the macro
does not provide a good positioning accuracy, and the micro must compensate the errors
caused by the macro. Therefore, we require a rough positioning accuracy at the macro's
tip and an accurate tracking capability at the endeffector of the micro.
The desired trajectories of the macro and micro are usually known in the task
space. For example, to scan a surface, we may want the macro to move the micro on a
straight line while the micro moves up and down (Figure 3.2).
The end positions of the macro and micro are given by forward kinematics
equations
4 =P'(eB,) (3 -4)
and
P4 = F',(e, e,, em), (3-5)
where e , B, , and 8, are the components of the generalized coordinate vector
and represent the flexural
micro, respectively . ates, joint angles of the macro, and joint angles of the
Desired trajectory
of micro
Desired trajectory
of macro - -
Figure 3.2: Desired Trajectories of the Macro and the Micro in the Task Space
The following procedure can be used to transform the desired trajectories from the
task space into the joint space:
S t e ~ 1: We assume that the macro is rigid (i-e.. we set e = 0 ) . Substituting e = 0 into
(3.4), gives
p," = P*(o, 0.8d,), (3 97)
where 0d, represents the vector of desired joint angles of the rigid macro, and the
superscript d denotes the desired value of the variables. The vector of desired joint angles
8d, is obtained by inverting the forward kinematics equation (3.7).
Step 2: Due to the elastic deflections, the actual tip position of the macro in the task
space P, will be different from the desired one P: even if the joint angles coincide with
the desired values (see Figure 3.3). The elastic deflections of the macro can be divided
into two parts: \
e=e,+e,, (3.8)
where e, denotes the static deflections due to the gravity, and e, denotes the dynamics
deflections due to inertial forces. The static deflections depend only on the arm
configuration, and can be calculated off-line. The desired trajectory of the macro in the
joint space 8; can then be obtained by modifying the desired trajectory of the rigid
macro 0d, in order to compensate the static deflections -961. This gives
ed,=ed,+sed,, (3.9)
where 68$ modifies the rigid macro's trajectory so as to compensate the error due to
the static deflections.
Steo 3: Finally, the desired joint position and velocity of the micro are obtained so as to
compensate the end-effector position and velocity errors in the task space. The micro's
trajectory error in the task space is given by (P, - P:) and in the joint space is given by
(8, - 8:) , where P, and P: are the actual and desired end-effector positions in the task
space, respectively; and 8, and 8: are the actual and desired joint positions of the
micro, respectively. Transforming the error from the task space into the joint space using
the micro's Jacobian matrix J , = - E 'XM, yields ae,
Desired Position of Micro,
Desired Position of Macro. Pt
Deflected Position of Macro, ' 2
Figure 3.3: Desired and Actual Positions of the Macro and Micro in the Task Space
Substituting (3.10) and (3.1 1) into (3.2) and using the following equation
p4 = JMmQ (3.12)
yields the control algorithm of the micro as a function of the desired end-effector
trajectory
r, = -R, J:' [p4 (q) - p,d] - K ~ ~ J ; ' ( 3 ~ ~ 4 - pt) + Fg, , (3.13)
as, where J , = - E !ItDLNy denotes the Jacobian matrix for the flexible M/m ann. 47
3.2 Design Case Studies
This section presents the details of three representative case studies of the parametric
design of the M/m arm introduced in Section 3.1. In the parametric design stage, the arm
58
configuration and its control algorithm are known, and the unknowns are the mechanical
and control design variables. The robot introduced in Section 3.1 is characterized by
twenty-one mechanical and eight control parameters. Control parameters consist of the
proportional and derivative gains of the four joint-variable PD controllers, and
mechanical parameters consist of kinematics and dynamics parameters of the arm.
The fmt case study represents TDM, and the second and third case studies use
DM to minimize the total mass and the tracking error of the robot, respectively. The
objective is to show, by comparing Cases 2 and 3 with Case 1, the advantage of using
IDM (rather than TDM) for designing long-reach M/m robots. Section 3.2.1 presents the
results of TDM (Case I), in which the control gains are designed for the nominal values
of mechanical parameters. The objective functions and design constraints are formulated
in Section 3.2.2 to be used for the integrated design case studies. Sections 3.2.3 and 3.2.4
provide the optimum values of mechanical and control design variables based on DM
(Cases 2 and 3), respectively. A new performance measure is proposed in Section 3.2.2
for the integrated design of unconstrained flexible M/m robots.
3.2.1 Case 1: Design based on TDM
Traditionally, mechanical design of the arm precedes its control design. This approach is
illustrated in this case study. First, the nominal values of the mechanical parameters are
obtained based on some kinematics and statics considerations and without considering
control objectives. Then, the control gains are obtained for the designed mechanical arm
so as to achieve the best possible performance.
Nominal Values of Mechanical Design Parameters
Table 3.1 shows the nominal values of mechanical design parameters. The nominal
values have been obtained based on the following considerations:
1) The required payload mass and moment of inertia are
DL, = 20 Kg,
HL,=O.Z lCg-m2.
Table 3.1 : Mechanical Parameters and Their Nominal Values
2) The required maximum reach of the macro and the micro are 20 (m) and 2 (m),
respectively. Therefore, we have
L,+L, =20 m , (3.14)
L 3 + L 4 = 2 rn. (3.15)
The workspace of a robot consists of all the points in the task space that the end-effector
can reach. For the macro and micro, the maximum workspace is obtained when the link
lengths are equal. This gives
L , = L 2 = 1 0 m,
L 3 = L 4 = 1 m .
3) The maximum static deflection of the M/m arm at the end-effector (due to gravity) is
required to be less than 1 (m). The maximum deflection is obtained when the arm is fully
extended in horizontal position with aU the joints locked. Besides, the static deflections of
both flexible links are required to be of the same order of magnitude. With the parameters
in Table 3.1, we have
Max. Static Defection at the End-Effector = 0.625 m,
Static Deflection at the End of Link I = 0.16 rn,
Static Deflection at the End of Link 2 = 0.12 m
4) The micro robot is designed to be rigid, and the maximum static deflection at the end-
effector of the micro is required to be lo4 (m).
5) The links are made of steel with Young's modulus E = 2 x 10" Pa , and density
p = 7800 Kg/rn3.
The next part provides the control gains for the nominal values of the mechanical
parameters in Table 3.1.
Desinn of the Control Gains
Previous investigations on the joint variable PD control of flexible arms proved that, the
closed-loop bandwidth of a flexible arm must be less than two thirds of the lowest
clamped natural frequency of the arm w: in order to achieve satisfactory damping ratio
for the dominant poles [CB90]. On the other hand, it is shown that by deploying a fast
micro at the tip of the flexible macro, it is possible to achieve closed-loop bandwidths
several times higher than the lowest clamped natural frequency [CKC9 l][SHH89]. These
results suggest the following procedure for designing the PD control gains of a flexible
M/m robot:
The proportional gains of the macro are determined so as to provide the closed-loop
bandwidth of less than two thirds of the lowest structural fnquency of the arm when
all joints are locked &,, . The derivative gains of the macro are obtained so as to provide enough damping for
the dominant closed-loop poles of the flexible arm.
Without considering the macro, the proportional gains of the micro are determined so
as to achieve the highest possible closed-loop bandwidth of the micro, which is
normally several times higher than ok . The derivative gains of the micro are obtained so as to provide enough damping for
the micro controller (usually critical damping).
Finally, the initial values of the PD gains, which are obtained in steps 1 to 4, must be
tuned to their final values by inspecting the closed-loop response of the M/m robot.
61
The lowest clamped natural frequency of the M/m arm is computed when the arm
is fully extended (i-e., all joint angles are zero). Table 3.2 shows the na- frequency
06. of the arm and the moment of inertia about the joint axes when the arm is fully
extended. In the table, Mii represents the second moment of inertia of the distal part of the
arm (consists of the links between the i'A joint and the end-effector) about the joint,
when all the joints are locked. If all the joints except joint i are locked, and the arm
flexibility is ignored, then the closed-loop model of the arm with a PD control of joint i is
given by the following second order differential equation
~~~8~ + K,& + K,Bi = 0 , (3.16)
where K,. and K, are the proportional and derivative gains of the iLh joint PD controller,
respectively. The standard form of a second order dynamic system is given by
ei + 2{,0,8~ + o:Oi = 0 , (3.17)
Table 3.2: Moment of Inertia and Lowest Natural Frequency of the M/m Arm
which can be compared to (3.17) to represent the PD control gains in terms of the closed-
loop natural frequency o, and damping ratio 5, . From (3.16) and (3.17), we have
Variable
%"
2 Kpi = M i p & (3.18a)
and
K, = 2 5 , 0 ~ M i i , (3.18b)
which specifies the PD gains in terms of the natural frequency arrd damping ratio of the
closed-loop system. Having the clamped natural fkquency of the mechanical arm wk in
Table 3.2, one can estimate the desired closed-loop natural frequency o, and damping
Value 5.15
Unit r d s e c
ratio ti of different joints, using the procedure provided earlier in this subsection for
designing the control gains. The estimated values are then substituted into (3.18) to obtain
the control gains. Table 3.3 shows the estimated control gains and the comsponding
closed-loop frequencies and damping ratios for all the joints.
Table 3.3: Initial Control Gains, Natural Frequencies, and Damping Ratios
Note that the natural frequencies and damping ratios obtained from (3.18) are only a
rough estimation of the actual closed-loop frequencies and damping ratios of the Mlm
arm, as the arm flexibility was ignored in the above formulations. Due to structural
flexibility, the actual closed-loop frequencies and damping ratios of the macro are usually
lower than those listed in Table 3.3. Therefore, it is necessary to tune the estimated values
of control gains by inspecting the closed-loop response of the flexible M/m robot. The
final (tuned) values of the gains are listed in Table 3.4.
Joint 1 (macro) Joint 2 (macro) Joint 3 (micro) Joint 4 (micro)
Table 3.4: Designed Values of the Control Gains of the Mlm Arm
1 Joint 1 (macro) 1 9.214E+5 I 9.648ErS 1
Kpi (N.m) 2.71E6 4.1 OE+5 2E+5 5Ei-4
1 Joint 4 (micro) 1 SE+4 1 2.03E+3 1
Z (N.m.sec)
1.26E+6 1.90E+5 1 .05W 2.03E-3
Joint 2 (macro) Joint 3 (micro)
Numerical examination of the closed-loop poles at a large number of robot
conf@uations (i.e., at different combinations of joint angles) showed the local stability of
the designed controller. Figure 3.4a shows some of the configurations at which the local
stability has been shown. The second joint angle 8, is iterated from zero, incremented
0,
(radlsec) 3.45 3.45 38.2 49.3
1.829EsS 2E+5
ti #
0.8 0.8 1 .O 1 -0
1.601E+5 1.05Ei-4
each time by n /6, and ends to 5z /6 . For each value of 8, , the micro joint angles 8,
and 8, are incremented by x / 6 and a/M, respectively, to cover the workspace of the
micro,
Figure 3.4: (a) Robot Configuration at 8 = [O, n/3,-~c /6,-~r /3] , and
(b) Configurations of the M/m Robot for Stability Analysis
Figure 3.5: The Real Part of the Rightmost Poles at Different Configurations
Figure 3.5 shows the real part of the rightmost poles (dominant poles) of the
closed-loop system at each robot configuration. The maximum value is -1.0235, which
indicates the local stability of the robot at different configurations.
3 3 3 Objective Functions and Design Constraints
This section defines the objective and constraint hnctions that are used in the following
sections for integrated design optimizations (see Section 1.3 for the definition of a general
optimization problem). A new performance measure' is proposed to be used as a measure
of the tracking error (TI?) of an unconstrained flexible Mlm robot. In case studies 2 and 3,
the total mass (XM) and tracing error (TE) are the objective functions, respectively. The
objective functions and design constraints are formulated in Subsections 1 and 2,
respectively.
1. Objective Functions
Case studies 2 and 3 use IDM to minimize the total mass (234) and tracking error (TE) of
the Mlm robot, respectively. Definitions of these two objective functions are provided in
this subsection,
a) Total M a s (TM)
The total mass of the arm is simply given by
lM= LI mI+ L,m,+D,, +DL,+ D, +D,. (3.19)
In case study 2, ZU has been used as an objective function to minimize the total mass,
and in case study 3 it has been used as a constraint function to set an upper bound on the
total mass of the arm.
b) Tracking Error (TE)
A performance measure is proposed that represents the tracking error of an unconstrained
flexible M/m robot in terms of its mechanical and control design variables. This function
can be used for the design optimization of a robot either as an objective function to
minimize the tracking error or as a constraint to set an upper bound on the tracking error.
The simplest way of defining such a hnction is to simulate the real time motion
of the robot on a specific trajectory and use the sum-of-square or the maximum absolute
' In an optimization process, a performance measure can be used as an objective hnction to minimize a
quantity or as a design constraint to defme a bound on the quantity.
65
value of error as a measure of the tracking accwacy. This method, however, has two main
drawbacks. Fit, simulating a flexible macro-micro robot is computationally demanding
and is not suitable for optimization. Second, the function represents the tracking error for
the specified path only. A suitable performance measure must be computationally
efficient (usually a closed-form solution is required), and should represent the global
behavior of the robot. To address these two issues, a new tracking performance measure
is proposed. The formulation is based on the linearized closed-loop model of the flexible
macro-micro arm. The linearized closed-loop model of a flexible M/m robot about a
nominal configuration q, is obtained by substituting the vector of generalized
coordinates
9 = 4 0 + 4
and the vector of joint torques
.r=r,+A.r
into the dynamics model (2.26), which yields
M O M + C& + KO@ = BAr ,
where the notation A denotes the small deviations from the nominal values, and the
matrices M,, C,, and KO are constant N, x N,, inertia, damping, and stiffness
matrices, respectively, given by
Mo = W 4 0 ) *
c, = Ck, ) ,
The vector A7 E %"I contains the small deviations of the joint torques From their
nominal values ro , the vector
contains small deviations of the generalized coordinates from their nominal values, and
the matrix
is the constant input matrix. In (3.24)- Ae E 9tN'"* represents the flexural displacements
and rotations at the nodal points, A e , E Sh' represents the rigid body motions of the
macro (i.e., macro's joint positions), and A e , E 'Xal represents the micro's joint
positions. For the Wm robot in Figure 3.1, we have
In general, the total number of flexural parameters and the total number of degrees of
freedom of the arm are given by
Nf = Z(nl +nJ (3.27)
and
Nbf = Nf +N, = 2(nI + n , ) + 4 , (3.28)
respectively; where ni is the number of finite beam elements of the ich link and N, = 4 is
the number of rigid body motions of the arm (i.e., the number of joints).
Rewriting the PD control algorithm in Section 3.1.2 for small deviations from the
base configuration yields
AZ, = - K , , ( A ~ , -A@:) - K ~ ~ ( A & -A&:) (3.29)
and
Arm = -K, J ; ; ' ( J , ~ ~ - m,d) - K, J;'(J,,* - bf) , (3.30)
where the Jacobian matrices J,,, tzIN and J , E%"*w are calculated at the base
configuration 4,. Substituting (3.29) and (3.30) into the linearized dynamics model
(3.22) yields the Linearized closed-loop model
~ ~ e + ~ , + + K , d p = B [ K , A ~ ' + K , A ~ ~ ] ,
where
and
are the desired joint positions and velocities, and
and
are the closed-loop stifmess and damping matrices, respectively.
The end-effector position error in the task space is given by
ex =dp,d-dp4 =AP,d-.J,,Aq (3.36)
where ex E SZX1 is the end-effector position error in the task space (in x and y directions),
and dP,d and dP, are, respectively, the desired and actual displacements of the end-
effector in the task space. Solving the closed-loop model (3.31) for zero input, i.e.
d e d = A Q ~ = 0 , and substituting the solution into (3.36) yields
where a,, b,, and cc are real constants, -1, and a, are real eigenvalues and
eigenvectors, -Ac + j wc and 0, = A, + jnc are complex eigenvalues and eigenvecton,
and N, and N, are the number of red and complex poles of the closed-loop system,
respectively. The eigenvalues
problem:
and eigenvectors are the solution of the following eigen
(3.38)
From (3.37), we can see that the contribution of each mode to the end-effector position
error is determined by the eigenvector of the mode premultiphed by the arm's Jacobian
matrix. We try to obtain a numerical factor that shows the influence of each mode on the
endeffector position error. The influence factor of different modes can be obtained by
activating one mode at a time and by measuring the end-effector error corresponding to
that mode.
When only one real mode is active, (3.37) yields - I , r Aqr =ar@,e (3.39)
where A q, is the error in the joint space corresponding to the real mode r. To obtain the
constant a,, we assume that the weighted norm of the error A q, at t = 0 is bounded to
one. This gives
1 1 ~ qr (o)llw = a r ll@rllw = 1 9 (3.40)
which. subsequently, yields
The notation l~ll,,, denotes the weighted norm, and is defined, for a vector0 , as
ll@II: = @ T ~ r ~ @ , (3.42)
where W is a diagonal weighting matrix. The endeffector position error is then given by
The performance measure is defined by the integral of the square error:
Substituting (3.43) into (3.44) and carrying out the integration w.r.t. time yields
where
is the influence factor of the real mode r, which shows the effect of this mode on the end-
effector tracking error.
Similarly, for a complex mode c, (3.37) yields
4, = [A, cos(wC t -cc)-f2, sin(@, t -c,)] . (3.47)
To obtain the constant bc , we assume that the weighted norm of the error Aq, at t = 0 is
bounded to one. This gives
I I A qc (0)1Iw = bc IFc COS(C, ) + sin@, )[Iw = 1 (3.48)
which, subsequently, yields
where
It can be seen that R, and the mode amplitude b, are functions of the initial phase c,.
We can use the maximum value of R, to make sure that the weighted norm 11 qc(t)llw in
(3.47) is always less than or equal to one, or we may simply use the mean value of R, .
Numerical examinations have shown that both mean and maximum values lead to similar
results. Taking the mean value of R, and substituting into (3.49) yields
1
2
Substituting the endeffector position error
e,(t)=-J&q, =-bcJh [ ~ ~ c o ~ ~ , , t - c ~ ) - R ~ s i n ( ~ ~ t - ~ ~ ) ] . ~ ~ ~
into the performance measure (3.44, carrying
the mean value of the result w.r.t. c, , yields
where
out the integration w.r.t. time, and taking
(3.53)
is the influence factor of a complex mode c. Finally, the local objective hnction2 is
defined as
where
is the mean value of all the influence factors. Influence factors in (3.55) are divided by
their mean-value (3.56) to obtain dimensionless factors that represent the relative
importance of different system poles.
The local objective function (3.55) represents the tracking error at one
configuration of the arm. For design optimization purposes, we need an objective
function that represents the global tracking capability of the robot at different
configurations. A global objective function can be obtained by taking the average (or
weighted average) value of the local measures in several representative arm
configurations:
where n is the number of configurations. Note that we may use a weighted sum of the
local objective functions
to represent the relative importance of different configurations of the robot. In (3.58), wi
is the weighting factor for configuration i.
' The local performance measure specifies the robot performance in one configuration.
71
a) Link Cross Section
A rectangular cross section has been used for the flexible links (Figure 3.6). The mass per
unit length (m) and the bending stiffness (EI) of the arm are expressed in terms of the Link
cross-section variables ( w, t,,,, h, t, ), as follows:
and
where p and E are the mass per unit volume (density) and Young's modulus of the link
material, respectively. All or some of the cross-section parameters (w, t,, h, t, ) can be
used as design variables in the optimization process. However, dynamic behavior of the
arm depends only on rn and EI, and it does not depend directly on the cross-section
parameters. This means that two (out of four) cross-section parameters are redundant.
Since only m and EI directly affect the arm's dynamic behavior, we use these two
parameters (instead of the cross-section parameters) as design variables in the case
studies. The constraints on m and EI are obtained from the constraints on the cross-
section parameters. In the case studies, we assume that w and tw are constant, and th and
h are variable parameters bounded by the following inequalities:
and
Solving (3.59) and (3.60) for t, and h, and substituting the results into (3.61) and (3.62)
yields the following constraints
and
where
and
Figure 3.6: Cross-section of the Flexible links
b) Mmornum End-point Defection (MoED)
Another constraint on the mechanical design variables is obtained by setting an upper
bound on the static deflection of the arm (due to the gravity) when it is fully extended and
all the joints are locked. This constraint prevents from obtaining a very flexible arm
during the optimization process.
c) Real Component of the Dominunt Poles (A,,,)
In order to diminish the residual vibrations of the flexible arm as fast as possible, we have
to decrease the settling time of the dominant poles of the closed-loop system. This is
achieved by pushing the dominant poles of the dosed-loop system further to the left in
the complex plane (i.e., by decreasing the real component of the dominant poles). In the
design case studies, the real components of dominant poles are constrained to be less than
a certain upper bound. The upper bound must be a negative number to ensure the local
stability of the closed-loop system.
ti) Minimum Closed-Loop Frequency (m,)
The lowest undamped natural frequency of the closed-loop system can be used as a
measure of the links flexibility and/or the size of the proportional control gains. A very
low undamped natural frequency indicates very flexible links andfor very low control
gains. Therefore, the lowest undamped natural frequency of the closed-loop system is
constrained to be more than a certain lower bound,
e) Damping Ratio of the Most Effective Modes (cgKni*) The most effective modes of the closed-loop system are those that have the highest effect
on the endeffector position error. The influence factors defined by (3.46) and (3.54) can
be used to determine the most effective closed-loop poles. To minimize the endeffector
overshoot, we constrain the damping ratios of the most effective poles to be more than a
certain lower bound.
fl Damping Ratio of D o m i ~ n t Modes (g,,, )
The dominant closed-loop modes are those modes that have the largest settling time (i.e.,
the rightmost poles in the complex plane). These modes usually correspond to the low
frequency modes of the macro. In order to control the overshoot of these modes, we
constrain the damping ratios of the dominant poles to be more than a certain lower bound.
g) Damping Ratio of All Modes (6,)
Finally, in order to improve the overall response of the closed-loop system, we constrain
the damping ratio of all significant poles to be more than a certain lower bound. The
significant poles are those poles whose absolute values are less than a specified number.
3.23 Case 2: Design based on IDM for M' ' ' ' g the Total Mass
In this case study, control and mechanical design variables are used concurrently to
minimize the total mass (TM) of the M/m arm, subject to fifteen inequality constraints.
The fifteen design constraints are as follows:
The first four constraints (GI to G4) are given by (3.63) to (3.66), and represent the
bounds on the cross-section variables of the first flexible link (see Table 3.6).
The constraints Gs to G8 are similar to GI to G4, and represent the bounds on the
cross-section variables of the second flexible link (see Table 3.6).
The other seven constraints Gg to G I ~ are bounds on the:
Maximum End-point Deflection: MaED S 1.2 (m),
Tracking Enor TE 5 27 (see),
Real component of the most dominant poles: A,,, S -1.2 (I/sec),
Minimum closed-loop frequency: o, 2 1.5 ( r d s e c ) .
Damping ratio of the most effective modes: 5 , . 2 0.8,
Damping ratio of the poles whose real components are greater than -5 l/sec:
g,,, 2 0.6.
Damping ratio of the poles whose absolute values are less than 100 l/sec: 6 , 2 0.15.
The constraints Glr to Gls are configuration dependent. In such cases, the
constraint function is evaluated at different arm configurations, and the worst case will be
considered as the constraint value. In addition to the inequality constraints, the equality
constraint given by (3.14) is also applied.
All design variables are scaled by their nominal values, and the scaled
(normalized) variables are used in the optimization process. The nominal values of
mechanical parameters are shown in Table 3.1, and the nominal values of control gains
are shown in Table 3.5. AU design variables are constrained by their lower and upper
bounds. Table 3.6 shows the lower and upper bounds on the nomalized values of design
variables and on the cross section parameters of the fI exible links.
TabIe 3.5: The Nominal Values of Control Variables
Table 3.6: Lower and Upper Bounds on Design Variables and Beams' Cross-sections
Table 3.7 lists all the design variables, and summarizes their final (optimal) values
for the three design case studies. The values in Table 3.7 are the normalized values.
Figures 3.7 and 3.8 graphically represent the optimal values of control and mechanical
design variables in Table 3.7. The final values of design variables in Case 1 have been
used as the initial (starting) values in the optimization process of Case 2 and Case 3. The
initial value of the objective function TM is 1.67 (ton) and the final (optimal) value is
1.25 (ton), which indicates 25% percent reduction of the total mass of the arm fkom the
initial value.
Table 3.7: The Final Values of All Design Variables (Normalized Values)
Scaled Values of Control Design Variables
Figure 3.7: Final Values of Control Design Variables (from Table 3.7)
I Scaled Values of Mechanical Design Variables
Figure 3.8: Final Values of Mechanical Design Variables (from Table 3.7)
3.2.4 Case 3: Design based on IDM for Minimizing the Tracking Error
This case study is identical to Case 2 except that in Case 3 the tracking error (TE) is the
objective function and the total mass (TM) is constrained, while in Case 2 the total mass
(TM) is the objective function and the tracking error (TE) is constrained. All other
constraints and the lower and upper bounds on the design variables are identical in both
case studies. The constraint Glo in Case 2 (TE) is replaced by
Totd Mass: TM 5 1.55 (ton).
Similar to Case 2, all design variables are scaled by their nominal values, and the
scaled (normalized) variables are used in the optimization process. Tables 3.1 and 3.5
show the nominal values of mechanical and control parameters, respectively. Table 3.6
shows the lower and upper bounds on the normalized values of design variables and on
the cross section parameters of the flexible links.
The final (optimal) values of all design variables are listed Table 3.7, and are
graphically shown in Figures 3.7 and 3.8. The values in Table 3.7 are the normalized
values. The fmal values of variables in Case 1 have been used as the initial (starting)
values in the optimization process. The initial value of the objective function TE is 2.9 1
(llsec) and the final (optimal) value is 1.89 (I/s~c), which indicates 35% percent
reduction of the objective function TE its the initial value.
3.3 Comparative Analysis of the Design Case Studies
Section 3.2 detailed the three design case studies. The final (optimal) values of all design
variables are listed in Table 3.7. Base on the results of the case studies, this section
performs two comparisons between the Traditional Design Method D M ) and Integrated
Design Method (DM). To show the robot performance, the tracking capability of the
M/m robot has been simulated for each set of parameter values in Table 3.7, and the
results are presented in Section 3.3.1. Distribution of the closed-loop poles and their
influence factors are discussed in Section 3.3.2. The simulation results and closed-loop
system characteristics are then used in Section 3.3.3 for the comparative analysis.
3.3.1 Simulation
The unconstrained motion of the robot is simulated for each design case study. The
simulation results demonstrate the tracking performance of the robot for each design case
study. The desired trajectories of the macro and micro are shown in Figure 3.9. The
macro moves on a straight line, while the micro oscillates around the straight path. The
oscillations produce enough excitation of the arm so as to demonstrate its tracking
capability. A structural damping of the form Ce = 004 M e + 0.001 K G was considered for
each finite element, where Ce, Me, and Ke are the element damping, inertia, and
s tiffhess matrices, respectively.
Figures 3.10, 3.1 1, and 3.12 show the end-point tracking errors of the macro and
micro for Case 1,2, and 3, respectively. Here, the tracking error is defined as the absolute
value of the distance between the desired and actual trajectories.
Table 3.8 shows the Maximum and Root-Mean-Square (RMS) values of the
tracking errors for the macro and micro in the task and Joint spaces1. The RMS of the
tracking error e(t) is defined by
' The tracking error in the joint space is the difference beomen the desired and actual position of the joints.
79
where T is the simulation time. The end-points tracking errors of the macro and micro (in
the task space) are denoted by MacroError and MicroError, and the tracking errors of
joints 1 to 4 are denoted by ThlError to Th4Error, respectively.
Figure 3.9: Desired Trajectories
Figure 3.10: Tracking Errors of the Macro and Micro (Case 1)
Figure 3.1 1 : Tracking Errors of the Macro and Micro (Case 2)
Figure 3.12: Tracking Errors of the Macro and Micro (Case 3)
Table 3.8: Maximum and RMS values of the Tracking Error in the Task and Joint Space
I I Unit
Th3Error I rad par~h4~rror I rad Msqth4~rror I rad
Case 1 Case 2 Case 3 7,008E-03 4.444E-03 2.66 1E-03
(a) E n d a c t o r Tracking Error
(c) Maximum Joint Errors
1 2 3 4
Joint Wm bet
(b) Macro's Tracking Error
(d) RMSQ of Joint Enom
1 2 3 4
Joint klum ber
Figure 3.13: Maximum and RMS values of the Tracking Error in the Task and Joint
Space
Note that the tracking error of the macro is considerably large; however, the tracking
performance of the micro (i.e., tracking at the end-effector of the M h arrn) is quite good.
3.3.2 Distribution of the Closed-loop Poles
The closed-loop poles are obtained for the three case studies when the robot is at the
configuration 8 = [0, 0,0, z /20] in which, the macro is fully extended. Table 3.9
summarizes the numerical results, and Figures 3.14a to 3.14~ show the distribution of the
poles in the complex plane for Case 1 to Case 3, respectively. The poles that have the
largest effect on the end-effector position are depicted with mows. The influence factors
of the closed-loop poles, defined by (3.46) and (3.54), are listed in Table 3.10, and are
graphically shown in Figure 3.15. The influence factors are normalized by dividing them
by their mean values (3.56).
Table 3.9: Closed-Loop System Poles at the Configuration 8 = [O, 0.0, ir /20]
I Case 1 I cas : 2 I Cas I Real I Imaginary I Real
Pole 1 -4403.2 0 -40 19.7 Pole 2 -1491.4 0 - 1263.6 Pole 3 I -653.91 01 -1001.6 Pole 4 -27.544 191.64 - 19.54 Pole 5 -27.544 -191.64 - 19.54 Pole 6 - 1 1.626 88.33 1 - 1 1.765 Pole 7 - 1 1.626 -88.33 1 - 1 1.765
Pole 10 1 -7.64391 23.0461 -5.93451 19.4061 -32.789) 0 Pole 1 1 I -7.6439 1 -23.0461 -5.9345 1 - 19.4061 -7.72761 2 1.485
Pole 13 1 -8.81951 01 -9.7327) 01 -12.6741 0 Pole 14 1 -1.67561 1.07531 - 1.3241 1 -47771 -1.3711 1.4275
I
Pole 15 1 - 1.67561 - 1.0753 1 - 1.3241 - 1.47771 -1.3711 - 1.4275
(a) Case I
(b) Case 2
100
lmag ( W s e c )
(c) Case 3
Closed-Loop Macro-micro Poles: W=O
100
lmag (radsec)
Real (1 fsec)
Closed-Loop Macro-micro Poles: Th2=0 I I
-40 -30 -20 -1 0 0 Real (I/sec)
Figure 3.14: Distribution of the Closed-Loop Poles in the Complex Plane
Table 3.10: The Normalized InfIuence Factors of the Closed-Loop Poles
I I Case 1 I Case 2 1 Case 3 1
Pole 2 Pole 3
I
Pole 12 1 2.01E-01 7.89E-04
Pole 4 Pole 5
8 -43E-05 1.26E-01 3.69E-04 3.69E-04
Pole 13 Pole 14
Normalized Influence Factors of ClorreWoop Poles
4.26E-05 1.38E-02
Pole 15 Pole 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pole Wmber
2.0 1 E-02 8.11E-02
8.38E-05 8.38E-05
5.9 1 E-05 1.37E-07
Figure 3.15: The Normalized Influence Factors (from Table 3.10)
1.28E-03 1.28E-03
1.37E-07 3.32E-09
1 .TOE45 4.44E-08
1.39E-04 1.88E-07
4.44E-08 8.27E-09
1.88E-07 2.37E-08
33.3 Comparative Analysis of TDM and IDM
Sections 3.3.1 and 3.3.2 presented the simulation results and the distribution of closed-
loop poles for the three case studies. This section performs two comparative analyses
between the design case studies to show the advantages of IDM over TDM. The results of
Case 1 (representing TDM) are compared with those of Cases 2 and 3 (representing
IDM)*
The comparative analyses are based on: i) the fmal vaiues of some important robot
characteristics (e.g., total mass TM and tracking error 279, and ii) simulation results of
Sections 3.3.1 and 3.3.2. Table 3.11 summarizes the final (optimal) values of the
objective function and other important characteristics of the M/m robot. The variables in
Table 3.1 1 are defined in Section 3.2.2.
Table 3.1 1 : Final Values of the Objective Function and Other Important Characteristics of
the Mlm robot
Case 1 I Case 2 Case 3 - 1.2254
1,6506 0.8082
0.6548
0.15
1.8939 1 .S5
h
Comparative Analvsis Between Case 1 and Case 2
The main objective of Case 2 is to minimize the total mass of the arm (TM), without
increasing the tracking error (which is considered as the main task requirement in the case
studies) and without compromising other characteristics defined by the constraint
functions.
Considering Case 1 as a basis for the comparison, we can see from Table 3.1 1
that, in Case 2 the total mass has been decreased by 255, while the other properties
maintained their previous values (or even improved). For example, the tracking
performance measure TE has been improved by 7%. Only the damping ratio of the
dominant mode has been decreased, but the settling time of this mode have been
improved. The dominant modes correspond to the low fRquency modes of the macro, and
they do not have an strong effect on the end-effector performance.
The simulation results show that the Maximum and M S values of the end-
effector tracking error (MaxMcroError and RMSqMicroError) have been reduced
(improved) by 37% in Case 2. This can be explained by comparing the most effective
poles of Case 1 and Case 2 shown in Table 3.9 and Figure 3.14. It can be seen that in
Case 2 these poles have been pushed further to the left side of the complex plane (which
means a smaller settling time), and the damping ratios have been increased (which means
less overshoot).
The comparative study of Case 1 and Case 2 clearly demonstrate better
performance achieved by using IDM rather than TDM. The results indicate a considerable
reduction of the total mass and, at the same time, a noticeable improvement of the
tracking capability of the M/m robot.
Com~arative Analvsis Between Case 1 and Case 3
The main objective of Case 3 is to minimize the tracking error of the arm (TE), without
increasing the total mass and without compromising the other system characteristics
defined by the constraint functions.
Considering Case 1 as a basis for the comparison, we can see from Table 3.1 1
that, in Case 3 the tracking performance measure TE has been decreased by 35%, while
the other properties maintained their previous values (or even improved). For example,
the total mass TM has been improved by 7%. Similar to Case 2, the damping ratio of the
dominant mode crwl, has been decreased, but the settling time of this mode have been
improved.
The simulation results show that the Maximum and RMS values of the end-
effector tracking error (MarMcroError and RMSqMicroError) have been reduced
(improved) by 62% and 58% in Case 3, respectively. It can be seen in Table 3.9 and
Figure 3.14 that in Case 3 the most effective poles have been pushed further to the left
87
side of the complex plane (which means a smaller settling time), and the damping ratios
have been increased (which means less overshoot).
The comparative study of Case 1 and Case 3 clearly demonstrate a better
performance achieved by using DM rather than TDM. The results indicate a considerable
reduction of the tracking error (i-e.. the objective function). and at the same time. a
noticeable reduction of the total mass of the W m robot.
3.4 Conclusions
Integrated design method (DM) and Traditional Design Method Crr,M) were illustrated
and compared through three representative case studies. A new performance measure
(3.55 and 3.57) was formulated to represent the tracking error of an unconstrained
flexible M/m arm, and the constraint functions were introduced for the design case
studies,
Two comparative analyses were performed between IDM and TDM based on
simulation results and other robot characteristics. Both comparative analyses clearly
confirmed the superiority of D M as compared with TDM. It was demonstrated that, by
concurrently designing the mechanical arm and its controller, it is possible to reduce the
total mass of the arm considerably without sacrificing the tracking performance, or to
improve the tracking capability significantly without increasing the total mass.
Chapter 4
Integrated Design of Constrained M/m Robots
In Chapter 3, we applied Integrated Design Method (DM) to a planar, U o f , flexible
M/m arm. The results of the three case studies proved the effectiveness of D M in
comparison with the Traditional Method of Design (TDM). Details were presented in
Chapter 3, and a new objective function was formulated to be used as a measure of the
tracking performance of unconstrained flexible M/m robots.
This chapter extends the results of the previous chapter to other applications of
M/m robots, where the robot is in contact with the environment. Section 4.1 presents a
list of potential applications of long-reach M/m manipulators and classifies them into
several groups based on their task requirements. One major group of tasks requires the
robot to be in contact with the environment. This motivates us to formulate a performance
measure for the integrated design of constrained flexible M/m robots.
In this chapter, a performance measure is formulated based o n the linearized
closed-loop model of a constrained flexible M/m robot. To perform a contact task, the
macro cames the micro to the place of interest in the workspace, and the micro controls
the position and contact forces of the endeffector. The related literature on the
positiodforce control of constrained M/m robots is reviewed in Section 4.2. Derivation of
the linearized closed-loop model is the focus of Section 4.3. In Section 4.3.1, the
dynamics model of a constrained flexible Mlm robot is obtained using a combined
jointltask space specification of the generalized coordinates, in which, the dynamics
model of the flexible macro is specified in the joint space, while that of the micro is
specified in the task space. This is an extension of the curvilinear coordinate
representation of the hybrid task [Yos87] to flexible M/m robots. Section 4.3.2 introduces
a control algorithm for a constrained flexible M/m robot A joint variable PD control is
used for the macro, and an impedance plus force control is used for the micro. The results
of Sections 4.3.1 and 43.2 are then combined to obtain the closed-loop model in Section
4.3.3. Similar to the formulation of the performance measure (3.55) in Chapter 3, the
performance measure in this chapter is formulated based on the linearized closed-loop
model of the robot. Section 4.3.4 presents the linearized closed-loop model of a
constrained flexible M h robot.
The stability and tracking performance of the proposed controller was extensively
studied using numerical methods and computer simulation. Some of the results are
presented in Section 4.4. Although, an analytical proof is not provided for the global
stability of the proposed controller, a large number of simulations and numerical stability
analyses provide a high level of confidence that the proposed controller stabilizes a
constrained flexible robot with a satisfactory performance.
Finally, based on the linearized closed-loop model, a new performance measure is
formulated in Section 4.5. The proposed performance measure can be used for the
integrated design optimization of constrained long-reach M/m robots, in a similar way
that the tracking performance measure (3.55) was used in the previous chapter for the
integrated design of unconstrained M/m robots (see Case Study 3 in Section 3.2.4).
The contribution of this chapter is to provide the closed-form solution of a
performance measure for the integrated design of constrained long-reach M/m robots,
which is an extension of IDM to contact applications of such robots. The contribution
includes the extension of the curvilinear coordinate representation of the hybrid task
Pos871 to flexible M/m robots, applying impedance control method to such robots, and
studying the stability and performance of the controller.
4.1 Applications of Long-Reach M/m Robots
Detailed implementation of IDM on a planar 4-dof M/m robot (in Chapter 3) revealed
that one major difficulty in applying D M to the optimal design of robots is the
formulation of a suitable objective function (performance measure) for a given
tasWapplication. Developing a Library of general purpose performance measures for
different applications facilitates the application of IDM to design of flexible Macro/miao
robots. A designer can then select a suitable performance measure to design the robot for
a given application. Formulation of such performance measures helps automate the
parametric design process of M/m robots for a wide range of applications.
As a first step, a survey was performed to prepare a list of applications of M/m
robots. The survey indicates that long-reach robots have possible applications in the
following areas:
Construction Indusny: hauling and positioning of large building components (e.g.,
steel beams, precast members, etc.); automatic concrete distribution; interior finishing
tasks (e.g., painting, plastering, jointing, welding, etc.); inspection and repair works
with large structures war901 [Gr089].
Highway Industry: paint removal and surface preparation of steel girder bridges (e.g.,
sand-blasting); visual inspection, painting, and maintenance of bridges which includes
the underwater inspection of abutment and pier scour woo95][LMR95].
Aircraft Industry: spray painting; paint removal (using plas tic-media- blasting);
inspection (visual and ultrasonic); cleaning [SDH93][SW90].
Shipbuilding Industry: paint removal, painting, and inspection; precise movements of
frames and rigs.
Nuclear Zndurry: process equipment removal (e.g., piping, vessels, etc.); waste
handling; structural demolition; walllfloor decontamination; restoration of storage
tanks Bed951.
Space: visual inspection; grabbing satellites; servicing; assembly (holding, fitting,
inserting, etc.).
Detailed study of the applications and their task requirements indicates that a M/m
manipulator may be designed to perform one (or some) of the following basic tasks:
Task1 Continuous path-tracking in a free space (laser or plasma-torch cutting, etc.). -
Task2 Continuous path-tracking in a free space while an external (or a reaction) force is - exerted on the end-effector (spray painting. sand-blasting, etc.).
Task3 Pick-and-place tasks (handling, positioning, etc.). - Task4 Precise control of the contact forces and position of the end-effector when the -
robot is in contact with a surface (ultrasonic inspection, cleaning, assembly, etc.).
Task5 Working with vibrating devices while the robot is in contact with the environment - (circular saw, jack-hammer, drill, etc.).
Task6 Bracing the macro to a fixed point while the micro is working (for tasks which - require a high accuracy in position andlor force control).
In a more general sense, the above tasks can be categorized into three main p u p s
of: i) non-contact, ii) contact, and iii) bracing tasks. Tasks 1 to 3 are non-contact tasks.
Tasks 4 and 5 are contact tasks, and Task 6 requires the bracing of the flexible part
against the environment.
A general purpose performance measure (3.55) was formulated in the previous
chapter for non-contact applications of flexible M/m robots. This chapter formulates a
performance measure for contact tasks, and the next chapter focuses on the bracing
applications of flexible M/m robots.
In the sequel, modeling and control issues of constrained flexible M/m robots are
addressed, and a performance measure is proposed that quantifies the endeffector
position/force tracking performance. The performance measure is computationally
efficient as the closed-form solution is provided, and it is generic since it can be applied
to M/m robots of different configurations and degrees of freedom.
4.2 Background on the Control of Constrained Robots
Hybrid positiodforce control of constrained robots has been the subject of numerous
research works. Raibert and Craig w81] first proposed a generic framework for the
hybrid control of rigid robots. The underlying assumption of their work is that the
subspaces of constraint forces and unconstrained motions are mutually orthogonal. This
assumption, however, is not always true [Go195]. They specified the hybrid task w.r.t. a
Cartesian "task framey', and they used selection matrices to decouple the position and
force subspaces. The decoupled variables are transformed into the joint space, in which
they are multiplied by joint control gains to produce the input forces/torques to the robot
joints.
Hogan [Ho85] proposed the impedance control method in which the robot
behaves as an impedance with predetermined stiffness, damping, and inertia, in its
dynamic interactions with the environment. He stated that for most manipulatory tasks,
the environment can be suitably modeled as an admittance; therefore, the robot must
assume the behavior of an impedance to be complement with the environment. He
proposed a feeback linearization method, similar to the resolved acceleration by Luh, et
al. LWP801 and Shin, et al. [SL85], to implement the impedance control.
Using a similar structure to that in [RC81], Khatib F;h87] proposed the
operational space hybrid control method. The idea is to formulate the dynamic model of
the end-effector in the task frame, and to use the model for compensating the nonlinear
behavior of the robot at the end-effector. An and Hollerbach [AH891 stated that the
hybrid control of [Kh87] and [SLSS], and the impedance control of mo85] are essentially
identical, and all belong to the class of dynamic based control methods. They compared
performance of the resolved acceleration method with that of the hybrid control of
[RC81] and the modified stiffness control of Salisbury [Sal80] to show the importance of
using the dynamic model in force control algorithms.
To use curvilinear coordinates for specifying a hybrid task was fmt proposed by
Yoshikawa [Yos87]. For rigid holonomic constraints, he used the algebraic equations of
constraints for defining the curvilinear coordinates (herein called the "task coordinates"),
and proposed a feedback linearization algorithm based on the task coordinates. Similarly,
McClamroch and Wang m 8 8 ] utilized a specific set of curvilinear coordinates to
formulate their model-based hybrid control algorithm.
Goldsmith [Go1953 provided a theoretical justification for using the task
coordinates (proposed in [Yos87]) to specify a hybrid task. He utilized the task
coordinates as a unified theoretical framework for representing and comparing different
hybrid control methods. He proved that for any holonomic constraint, it is possible to find
a set of generalized task coordinates in which position and force variables are statically
decoupled.
Although hybrid control of rigid robots has been a very active area of research
wS93], there have been only a few works on control and modeling of constrained
flexible manipulators. Using the Singular Value Decomposition (SVD) of the constraint
equations, Lew and Book [LB93] extended the hybrid control method proposed in
[RC81] to flexibldrigid robots that make multiple contacts with the environment. The
SVD reduction of constrained dynamics equations can be effectively applied only when
the constraint equations are either linear or linearized around a reference position. Rocco
and Book [RB96] reduced the order of constrained dynamics equations of a flexible robot
by assuming that the algebraic constraint equations can be explicitly solved for the same
number of rigid variables. Their assumption can be justified only if the robot kinematics
and constraint surfaces are represented by simple algebraic equations.
Kim, et. al. KSK96] proposed a simple PD-based hybrid control method for
spatial flexible robots. The force error acts in the normal direction to the constraint
surface, and the position error is mapped into a subspace which is orthogonal to the force
error (similar to the hybrid control of FCSl]). They used a lumped parameter model of
the arm in their analysis.
Yoshikawa, et. al. [YHH94] and [YHM96] proposed a model-based dynamic
hybrid positionlforce control method for flexible-macrojrigid-micro robots. They
discussed the trajectory planning of a flexible M/m robot, and used a simple PD-
controller to move the macro on the desired trajectory. The macro was used for a rough
positioning of the micro in the workspace, and the accurate positiodforce control of the
end-effector was achieved by the micro, which used a model-based dynamic hybrid
control method to compensate for the errors induced by the macro. They provided the
experimental results to verify the effectiveness of their contml algorithm.
To date, a large number of papers were published on the hybrid positiodforce
control of robots, some of which were reviewed above. The research in this field mostly
focused on rigid robots, and a majority of the proposed methods are model-based
controllers that require the dynamics model of the robot [SL85]~085][Kh87]~0~87]
m 8 8 ] . For flexible robots, model-based controllers tend to be very complicated and
sensitive to dynamic parameters. In Section 4.3.2 a non model-based impedance control
algorithm is introduced and used for the formulation of a performance measure for
constrained flexible M/m robots.
Works on hybrid positiodforce control of flexible Mim robots did not provide a
suitable framework for specifying the task coordinates and formulating the closed-loop
model of a constrained flexible M/m robot. For rigid robots several methods were
proposed [RC81]~87]~os87]FT(K88], among which, the hybrid task specification
based on cunilinear coordinates [Yos87] was shown to be more general [Go195].
Section 4.3 extends the curvilinear coordinate specification of a hybrid task to the flexible
M/m robot, using a combined joint/task space specification of the generalized
coordinates,
4.3 Closed-Loop Model of a Constrained Flexible M/m Robot
In this section, a feedback control algorithm is provided for a constrained flexible Mlm
robot, and the closed-loop model is obtained. The closed-loop model is then linearized to
be used for the formulation of a performance me- in Section 4.5. A combined
joinvtask space specification of the generalized coordinates (herein called "task
coordinates") is proposed for specifying the hybrid task and deriving the closed-loop
model of constrained flexible M/rn robots.
Section 4.3.1 provides the dynamics model of a constrained flexible Mlm in task
coordinates, and Section 4.3.2 introduces a control algorithm for the integrated design of
such robots. A simple PD control is used for the macro, and a conventional impedance
control algorithm is used for micro arms. The closed-loop model is then formulated in
Section 4.4.3. The model is partitioned into two parts representing the position and force
control subspaces. Finally, a linear closed-loop model of the robot at a nominal
configuration is derived in Section 4.4.4.
43.1 Dynamics Model of a Constrained Flexible M/m Arm
Let the tip (end-effector) of the robot make contact with a very stiff environment, and the
constraint forces acting along the normals to the constraint surfaces. The constraint
equations are easily written in terms of the Cartesian coordinates of the end-effector in a
suitable reference frame. However, using the forward kinematics equations of the robot,
we can always represent the constraint equations in terms of the joint coordinates of the
robot as:
8(4)=0 (4.1)
where 8 E sex' represent constraint equations in terms of the generdkd coordinates
where e E W' , E$, E SM, and 8, E %W are the flexural coordinates, joint
displacements of the macro, and joint displacements of the micro, respectively. The
number of elements of q is equal to the total number of degrees of freedom of the robot,
i.e., e + R + r = Nh, . The dynamics model of a robot in contact with a holonomic
where M E 9tNa "*M is the inertia matrix, FCC( q, q ) E 'R Nw XI is the centrifugal and
Coriolis force vector, Fg(q ) E % NM Xl N X N is the gravity force vector, C E % and
N xN K E% are
mu1 tiphers, and
damping and stiff'ness matrices, F, E %"c"' is the vector of Lagrange
(4.4)
is the vector of joint torques corresponding to the generalized coordinates vector (4.3),
respectively. The notation
denotes the gradient vector. The gradient of a scalar is a I x Nw row vector. We assume
that the number of degrees of fkeedom of the micro is equal to the dimension of the task
space, i.e., r = n; therefore, the micro has enough degrees of freedom
hybrid task. The damping and stifmess matrices (C and K ) are given by
and
where C, E Ye and KF E %me are the structural damping and stiffness
flexible links.
to perfom the
(4.7)
matrices of the
To specify the hybrid task of the micro, we use the curvilinear coordinate
representation of Yoshikawa [Yos87]
where y , E %("-"c and y, E %'@ are, respectively, the position and force control
subspaces of the hybrid task, given by
Y, =Y(Q)*
Y, =WlL
where y(q) is a vector of ( n - n,) mutually independent functions of the generalized
coordinates, and p(q) is a vector of n, independent constraints. When the end-effector
is in contact with the environment, from (4.1) and (4.9b), we have
y, = O . (4-10)
Similar to -941, we assume that the macro is used only for the rough
positioning of the micro, and the precise hybrid positioo/force control of the end-effector
is performed by the micro. Therefore, we specify the dynamics of the micro in the task
space using (4.8), which is suitable for hybrid control, and that of the macro in the joint
97
space. This combined joint/task space specification of the dynamics model is achieved by
proposing a new vector of generalized coordinates (herein called the "task coordinates")
which is used instead of q to specify the dynamics equations. Note that (4.1 1) is an
extension of the curvilinear coordinate rep~sentation of Yoshikawa (4.8) to a flexible
M/m robot. The time derivative of the task coordinates is given by
j t = J y & (4.12)
where
is the Jacobian matrix, and J , E I" Ny"' and J,, E ~'""' are, respectively, the
macro (M) and micro (m) parts of the endeffector Jacobian matrix ( J , ), given by
The micro part J,, is a square matrix, and is invertable if the micro is not in a singular
configuration. Inverting (4.12) and differentiating the result w.r.t. time gives
Q = J ; ' ~ (4-14)
and
where J;' is the inverse Jacobian matrix, given by
Rewriting the dynamics equations (4.3) in the task space, using (4.1 1) to (4.15) and the
foilowing identity
V ; P = J ; E ; , (4- 171,
yields
where
are the inertia., damping, stiffness, gravity force, and centrifugaVCoriolis force matrices in
the task space, respectively; and
and
are the partitions of the unit matrix
Equation (4.17) can be verified by direct substitution of (4.13) and (4.21). Substituting
(4.6), (4.7) and (4.16) into (4.19) gives
C , = C , (4.23a)
K y = K . (4.23b)
Equation (4.18) specifies the dynamic behavior of the M/m robot, even if the end-effector
moves in a free space. When the end-effector is in contact with the environment, we can
use (4.10) to reduce the number of the task coordinates and the order of dynamics
equations. Using (4. lo), we write the task coordinates vector as
Y = E : Y I ~ (4.24)
where
is the reduced vector of the task coordinates, which defines the position subspace of the
hybrid controller. Substituting (4.23) and (4.24) into (4.18) and premultiplying the
equation by El yields
MYIjiI+Cyi j , +K,,y ,+V, ,+G,, =E,J;*L (4.26)
where
M,E:. (4.27a)
C,, = El c E: (4.2%)
K,, = E, K E: , (4.27~)
vy, = E, vY(g Y,& Y , ), (4.27e)
are the reduced inertia, damping, stiffness, gravity, and centrifbgaVCoriolis matrices,
respectively. Note that the contact forces F2 do not appear in (4.26), and this equation
specifies the unconstrained subspace of the dynamics model in terms of position variables
y, and input joint torquedforces r . Substituting (4.23) and (4.24) into (4.18) and
premultiplying the equation by E, yields the force variables
specified in terms of position variables y, and input torquedforces Z . The following
identities
E,E: = o , (4.29a)
E, E: = I , , (4.29b)
E ~ C ~ = O , (4.29~)
E2 KE: = O , (4.29d)
have been used for deriving (4.26) and (4.28).
In summary, we obtained the dynamics model of a flexible Mlm robot in contact
with rigid holonomic constraints. A new combined joinutask space representation of the
generalized coordinates (4.1 1) was used to specify the macro's dynamics in the joint
space and the micro's dynamics in the task space. The curvilinear coordinates proposed
by Yoshikawa (4.8) were used to represent the micro's task space coordinates. Finally,
the new vector of generalized coordinates (4.11) was used to obtain the reduced form of
the dynamics model (4.26)- which does not include the contact forces and is only a
function of position variables.
4.3.2 Control Algorithm
This section introduces a control algorithm for constrained long-reach M/m robots to be
used for integrated design purposes. Such robots consists of a long-reach flexible macro
which is used for rough positioning of the micro in the workspace, and a small rigid
micro that performs the precise hybrid positiodforce control of the endeffector. Similar
to the previous chapter, we use a simple joint variable PD control plus a feedforward
gravity compensation for the macro. This position control algorithm is best represented in
the joint space.
For the micro, we use a conventional impedance control plus a feedforward
gravity compensation. This non model-based impedance control was shown to be
effective and stable for rigid-link robots in the presence of joint and/or contact
compliance [Go195]. Although model-based hybrid control methods such as the dynamic
hybrid control of -941 may be more accurate than non model-based methods
[AH89], they are generally more complicated and less robust, and are not suitable for
integrated design purposes as they cover the effect of mechanical parameters in the
closed-loop response.
The impedance plus force control algorithm of the micro is given by
7, = - J L [ E , z ( K , ~ , + R , ~ , ) - E : ( F : - K ~ ~ ) ] + F ~ , (4.30)
and the joint variable PD control of the macro is given by
ry =-KPM e, - K,, e, +Fg,,
where K, and KDm are the stiffness and damping matrices of the end-effector's
impedance, respectively; Kf is the proportional gain of the force controller and F: is
the desired force vector, K, and KDM are the proportional and derivative gains of the
macro's PD controller, respectively; the vectors
e, =E$, -ei, (4.32a)
d eF=Yp-Ypr (4.32b)
e, =F, -F;, (4.32~)
are the macro's joint, and the end-effector position and force errors, respectively; and
and
are the partitions of the unit matrix
I,, = [E; E; ] .
In (4.30) and (4.31), Fg, E 3"' and Fg, E %"Y are, respectively, the macro and micro
parts of the gravity vector
and FgF E Rn' is the flexible part. Note that the micro's control algorithm is represented
in terms of the task coordinates (4.8)' and that of the micro is represented in terms of the
joint coordinates.
4.3.3 Closed-Loop System
Using the results of Sections 4.3.1 and 4.3.2, this section formulates the closed-loop
model of a constrained flexible W m robot. Using definitions (4.4)' (4.21), (4.34)- and
(4.36) and equations (4.16)' (4. Wd), and (4.30), we obtain
where the following identities were used:
E, E;=I,,
Substituting (4.37) into (4.28) yields the force error vector
Similarly, from (4.4), (4.16), (4.20), and (4.33), we fmd
where
Substituting control equations (4.30) and (4.3 1) into (4.40), and substituting the resulted
equation into the dynamics model (4.26) yields the closed-loop model
where
c, =
and
are the closed-loop damping and stifmess matrices, respectively; the vector
is the effect of the gravity force on the flexible structure; and the vector
contains the flexturd displacements e, and the position control errors of the macro and
micro; and
is the desired value of the position vector y, . In deriving (4.42), the equations (4.6) and
(4.7). and the identity
T * p Ep = bC) 9 (4.48)
were used. Note that the force control error ef appears in the closed loop model.
Substituting (4.39) into (4.42) gives the closed-loop model in terms of the position error
vector e ,, , as
where
and
are the new centrifugaVCoriolis force vector and inertia matrix, respectively. Using
(4.38b), we can write the right hand side of (4.49) as
We can see that the last row of (4.52), which corresponds to the endeffector position
error, is zero. This means that the desired contact force F: does not affect the steady-
state position error of the endeffector, and the steady-state position error is zero.
However, it affects the static position error of the macro's joint controller.
4.3.4 Linearized Closed-loop Model
The linearized closed-loop model of a constrained flexible Mlm robot is obtained to be
used for the formulation of a performance measure in Section 4.5. The linearized closed-
loop model of the robot about a nominal configuration yo is obtained by substituting the
following identities:
Y,=YO+AY,? Y,=AY,, Y , = A Y ~ (4.53) d d
Y: = Y O +A!/ 3 j : = ~ , , d , jif=djif, (4.54)
e,, = Y , - Y : = ~ ~ , + ~ ~ , , (4.55)
where
into the non-linear closed loop model (4.49) and (4.39, and canceling out the static and
second order terms. The subscript denotes the nominal (or static) value of the
variable, and the notation "A " denotes a smdl variation of the variable fiom its nominal
(or static) value. The linearized closed-loop model of the robot is given by
M, by, + Co A,, + 111, &,, = -M, AY:, (4.58)
e, = -(I, + K,)-' & M, E; (di,, + AJ:), (4.59)
where Mo , Co , and My, are, respectively, the inertia matrix Me, damping matrix C, ,
and inertia matrix My evaluated at the nominal configuration yo, and
is the stiffness matrix at the nominal configuration.
The results of this section are used in the next section for studying the closed-loop
tracking performance and local stability of a constrained flexible M/m robot with the
proposed control algorithm (4.30) and (4.3 1).
4.4 Study of the Closed-Loop Response
The tracking performance, robustness, and stability of the proposed control algorithm
(4.30) and (4.31) have beem extensively studied using computer simulations and
numerical analyses. The stability and performance have been studied for both rigid and
flexible environments and for a variety of gain values. Although, such a numerical study
does not prove the global stability of the closed-loop system, it provides a high level of
confidence that the proposed control method is stable, as the robots perform satisfactorily.
This section provides some of the numerical results. The Clink planar flexible M/m arm
described in Chapter 3 is used for the study. F i t , based on the non-linear closed-loop
equations (4.39) and (4.49), the position/force tracking performance of the robot is
simulated in Section 4.4. I. The end-effector is assumed to be in contact with a circular
surface. Then, the local stability of the closed-loop system is studied in Section 4.4.2,
using the linearized closed-loop equations (4.58) and (4.59). The numerical results
demonstrate a satisfactory performance of the proposed control algorithm.
4.4.1 PositiodForce Tracking Performance of the Constrained Robot
The 4-link planar flexible M.m arm described in Chapter 3 is used to demonstrate the
positiodforce tracking performance of the control algorithm proposed in Section 4.3.2.
Figure 3.1 in Chapter 3 shows the mechanical configuration of the flexible Mhn arm. The
nominal values of the mechanical parameters (listed in Table 3.1) are used for simulation.
The endeffector of the micro is assumed to follow a circle in the arm's plane of motion.
The micro's task coordinates (4.8) are easily defined in terms of the endeffector
Cartesian coordinates x and y , as
Y , = v ( * . y ) = y .
where xc and y, are the coordinates of the center, and R is the radius of the constraint
circle. The forward kinematics equations of the arm (3.5) express the end-effector
Cartesian coordinates in terms of the generalized coordinates q.
The following parameter values were used in the simulation:
Constraint circle
xc=25 m , y ,=IO m,and R = I O m .
Proportional control gains of the macro are obtained from Table 3.4 in Chapter 3, as
Derivative control gains of the macro are obtained from Table 3.4 in Chapter 3, as
Gains of the micro's hybrid controller
K h = 1 0 E + 4 N / m ,
K h = 5 E + 3 N.sedrn,
K, = l E + 3 .
Desired value of end-effector force variable
F,d = 2 0 N / m .
The endeffector trajectory starts from y = I2 m on the circle and ends at y = 8 rn
after two seconds. While the micro is performing the hybrid task in contact with the
circular surface, the macro carries the base of the micro on a path near the circle. We
assume that the desired trajectory of the macro in the Cartesian space is a straight line
from point ( x , y ) = (14.12 ) m to point (14.8) m .
Figure 4.1 : Desired and Actual Trajectories of the Macro and Micro
Figure 4.2: PositionForce Tracking Erron of the End-effector
Figure 4.3: Flexural Deflections/Rotations at the Distal Ends of Links 1 and 2
Figure 4.1 shows the desired and actual trajectories of the macro and micro. The
desired trajectories are denoted by "+", and the actual trajectories by "o". me simulation
results are shown in Figures 4.2 and 4.3. Figure 4.2 shows the position and force errors of
the end-effector versus time. The maximum position error is 2.2 mm, and the force error
is less than 0.05 percent of the desired value. Figure 4.3 shows the flexural deflections
and rotations at the end of the flexible links I and 2.
4.43 Stability Analysis of the Closed-Loop System
The linearized model of the closed-loop system (4.58) has been used for local stability
analysis of the closed-loop system. The controller gains and mechanical parameters of
Section 4.4.1 have been used in the analysis. It is assumed that the endeffector of the
robot is constrained to move on a vertical line (parallel to the Y axis).
The local stability of the closed-loop system has been studied at a large number of
robot configurations, some of which are shown in Figure 4.4b. The second joint angle 8,
starts from zero, incremented each time by n /6 , and ends to 5x /6 . For each value of
0, , the micro joint angles 0, and 8, are incremented by x / 5 and / 90, respectively,
to cover the micro's workspace. The configurations for which - z / 6 < 8, c n / 6 were
ignored so as to avoid the micro's singularity.
At each robot configuration. the real part of the rightmost poles of the closed-loop
system has been obtained. The results are shown in Figure 4.5. From Fieme 4.5, we can
see that all the poles are located in the left hand side of the complex plane, and therefore,
the closed-loop system is locally stable in all configurations. Figure 4.5 shows the results
of the local stability analysis when the robot is in contact with a very stiff environment.
Stability of the robot were also studied when the endeffector was in contact with a
flexible surface, and stable results were obtained.
In summary, this section presented some simulation resuIts to study the closed-
loop response of a constrained flexible Wm arm with the proposed control algorithm in
Section 4.3.2. The results demonstrated a satisfactory response of the closed-loop system
and verifi~ed its local stability.
Figure 4.4: (a) Robot Configuration at 8 = [O, r /3, -x /6, -a /3] , and
(b) Configurations of the M/m Robot for Stability Analysis.
Figure 4.5: The Real Part of the Rightmost Poles of the Closed-Loop System
4.5 Formulation of the Objective Function
Using the linearized closed-loop model of a constrained flexible 1Wm robot presented in
section 4.3.4, this section formulates a new objective function for integrated design of
such robots. The proposed objective function is a measure of positiodforce tracking
performance of the endeffector. The procedure is similar to the one presented in Section
3.2.2 for the formulation of Tracking Error (TE).
Section 4.5.1 provides an objective function that measures the position tracking
performance of a constrained flexible Mlm robot, and Section 4.5.2 presents an objective
function for the force tracking performance of such a robot. The results are then
combined to provide an objective function for integrated design purposes.
4.5.1 Position Error
For zero acceleration set-point dyf = 0 , equation (4.58) gives
Mo AEy, + C, by, + KO &,, = 0
Solving the closed-loop model (4.61) for the error yields
where q, b,, and c, are real constants; -Ar and 0, are real eigenvalues and
eigenvecton; -A, + j a, and @, = A, + j a, are complex eigenvalues and
eigenvectors; and N, and NIm are the number of real and complex poles of the closed-
loop system, respectively. The eigenvalues and eigenvectors are the solution of the
following eigen problem:
The end-effector position error Ae, corresponds to the micro part of the error vector
A,, (see 4.46). Therefore, we have
de, = Em h,, ,
where Em is a selection matrix defined by
Substituting (4.62) into (4.64) yields
which gives the end-effector position error in terms of the eigenvalues and eigenvectors
of the closed-Imp system We define the performance measure as the integral of the
squared error w .r.t. time:
Comparison of (4.66) and (4.67) to (3.37) and (3.44) in Chapter 3, shows that the end-
effector error and objective function defined in this section are similar to those defined by
(3.37) and (3.44). In fact, by replacing Em with - J , in (4.65), we obtain (3.37).
Therefore, we can use the procedure of Section 3.2.2 to formulate the objective function.
Using (3.41) and (3.51), we obtain the modal amplitudes a, and 6, as
where the notation IlII, denotes the weighted norm defined by (3.42). Replacing J ,
with - Em in (3.46), (3.54), and (3.55), yields the local objective function of the end-
effector position error
where 1: and I: are, respectively, the influence factors of real (r) and complex (c)
modes, given by
and mean( I: , 1: ) is the mean value of all the influence factors defined by (3.56).
4 3.2 Force Error
For zero acceleration set-point dy,d = 0 , equation (4.59) gives
=a,+
where
is a constant matrix. Differentiating (4.62) w.r.t. time and substituting the result in (4.73)
yields
where the constants a, and 6, are given by (4.68) and (4.69), and
=a:@r, (4.76a)
We define the performance measure as the integral of the squared error w.r.t. time:
W e can obtain the local objective function by using the analogy between the performance
measures (4.77) and (4.67) and the error functions (4.75) and (4.66). Using (4.68) to
(4.72), we obtain the local objective function of force error as
where if, and I:, are, respectively, the influence factors of real (r) and complex (c)
modes, given by
and mean( I; , I: ) is the mean value of all the influence factors defined by (3.56).
4.5.3 Total Performance
For the integrated design of a constrained robot, we need an objective function that
measures both position and force tracking performances of the robot. This can be
achieved by combining the position and force perfonnance measures
Jw = Jparition + Jfome t (4.8 1)
where c, 2 0 is a scale factor, and J,, is an objective function that represents the local
perfonnance of the robot about a nominal configuration. To obtain a global measure of
performance, we use the average value of the local performance measures at different
representative arm configurations:
where n is the number of configurations. Representative arm configurations can be
selected based on the probability of the arm working at different configurations. A
weighted sum of the local objective functions can also be used to define the global
performance measure. For example, if the probability of the arm working at one
configuration is higher than the other configurations, the local objective function at that
configuration can be weighted higher before substituting it in (4.82).
4.6 Summary
A new performance measure (4.70. 4.78, 4.81, and 4.82) was formulated for the
integrated design optimization of constrained long-reach M/m robots. The performance
measure represents both position (4.70) and force (4.78) tracking capability of such
robots. Formulation of the performance measure was based on a suitable form of the
linearized closed-loop model, where the position (4.58) and force (4.59) subspaces were
separated from each other. This separation of subspaces was achieved by a combined
jointltask space specification (4.1 1) of the dynamics model (4.26, 4.28) and control
algorithm (4.30, 4.3 1). The satisfactory response of the closed-loop system was
demonstrated in Section 4.4.
The contribution of this chapter was to provide the closed-form solution of a
performance measure (4.70, 4.78,4.8 1, and 4.82) for the integrated design of constrained
long-reach M/m robots. This extends IDM to the contact applications of such robots. The
contribution includes an extension of the curvilinear coordinate representation of the
hybrid task mos87] to flexible M/m robots (4.8, 4.9, and 4.1 I), which allows for the
separation of the position (4.49) and force (4.39) subspaces; application of the impedance
control method (4.30) to constrained flexible W m robots; and study of the closed-loop
performance (see Section 4.4).
Chapter 5
Design of A Bracing M/m Robot
The first section of Chapter 4 classified the applications of long-reach M/m robots into
the three groups: non-contact, contact, and bracing applications. This chapter focuses on
the design of a long-reach M/m robot for bracing applications. Such a robot can be used
for fast and accurate scanning of a surface in a large workspace. To enhance the design of
such a robot, IDM is used to minimize the cycle time of the bracing flexible macro
(Section 5.1), and a specialconfiguration micro robot is designed to reduce the dynamics
interactions between the micro and macro (Section 5.2).
Bracing a flexible M/m arm against a stationary frame can significantly improve
the structural stiffness, speed, and positioning accuracy. It provides an effective means for
overcoming structural vibrations, which is one of the main drawbacks of long-reach,
lightweight manipulators. Task requirements, and hence, performance measure(s) of
bracing W m robots are different from those of the non-bracing M/m robots studied in
Chapters 3 and 4.
Section 5.1 presents a task plan for bracing Mlm robots and identifies their task
requirements. The macro performs a point-to-point motion and braces itself against the
environment; then, the micro performs the fine manipulations. The task plan allows for
the separate design of the macro and micro, as it reduces the dynamics interactions
between the two parts and separates their operating times'. According to the task plan, the
main task requirement of the macro is to have a short cycle time for a point-to-point
motion, and the micro is required to have a minimum dynamic interaction with the macro.
To minimize the cycle time of the bracing macro, a performance measure is
proposed in Section 5.1 to be used for the integrated design of such a robot. Only the
formulation of the performance measure is presented here (without any case study), as it
will be used in a similar way that (3.55) was used in Case study 3 (see Chapter 3).
Bracing the flexible macro against the environment reduces the effect of the
macro's structural vibration on the operation of the micro. However, the bracing
performance of the macm can be affected by the dynamic forces generated by the
operation of the micro (herein called the "coupling forces"). TO reduce the coupling
forces, a novel special-~o~guration micro robot (called a Robot for Curved Surfaces or
"ROCS") is proposed. Section 5.2 presents the conceptuaUconfiguration design of ROCS.
The new design reduces the weight and dynamic interactions between the micro and
macro, and simplifies the control design of the micro. Kinematics and dynamics models
of ROCS are formulated in Sections 5.3 and 5.4, respectively.
The contribution of this chapter is to enhance the design of bracing long-reach
M/m robots. The contribution involves: i) closed-form solution of a performance measure
for the integrated design of a bracing long-reach robot; and ii) conceptuaVconfiguration
design and modeling of a novel micro robot for bracing applications (ROCS).
5.1 Performance Measure of a Bracing Flexible Robot
This section discusses the integrated design of a bracing flexible robot for tasks that
require a fast and accurate scanning of a surface in a large workspace (e.g., inspection of a
large storage tank). First, a task plan is presented and task requirements are identifiled in
Section 5.1.1. The task plan describes the actions that a bracing Mlm may perform to
' According to the task plan, the macro and micro do not operate simultaneously, and dynamics interactions
between the macro and micro are reduced by bracing the macro against a fix point while the micro is
operating.
complete a typical operational cycle of a scanning task. The macro performs a point-to-
point motion to locate the micro in a specified position, then the micro performs the task
while the macro braces itself against the contact surface.
It is shown that, the bracing task plan reduces the dynamic interactions (or
dynamic coupling) between the rigid micro and the flexible structure of the macro, and it
simplifies the mechanical and control design. Section 5.1.2 presents a control algorithm
for a bracing flexible arm, and formulates a new performance measure to be used for the
integrated design of the robot.
5.1.1 Task Plan
Figure 5.1 shows a M/m robot in contact with a stationary surface. The macro arm is a
flexible long-reach manipulator that carries the micro to the place of interest. When the
desired destination is reached, the macro locates the base of the micro and braces itself
against the contact surface. Bracing the macro against the environment improves the
stiffness and accuracy of the system [WA85], and partialIy (or fully) eliminates the
dynamic effects of the flexible macro on the operation of the micro. The micro can then
be used for fast and accurate positioning of the endeffector on the surface.
Figure 5.2 shows the forces applied to the bracing device (the gravity force, and
the moments are not shown). The macro applies a clamping force f, in the direction
normal to the contact surface, the micro exerts a dynamics force f,, and the contact
surface applies a normal f, and a tangential friction force fp to the clamping device. If
the normal force applied by the macro f, is large enough (in comparison with the
micro's dynamic force f,), the friction force f, will be sufficient to suppress the
dynamic force of the micro fD in tangential direction and fix the base of the micro on the
contact surface.
Figure 5.1 : A Bracing Macro/micro Robot
- Contact Surface
- -p
Figure 5.2: Free-Body Diagram of the Bracing Device
This bracing plan reduces the dynamic interactions (or coupling) between the
micro and the flexible structure of the macro, and therefore, enhances the performance
and simplifies the micro's control algorithm. To minimize the coupling effects, the micro
is required to be as light as possible so as not to exert large dynamic forces on the
clamping device. For delicate environments, where excessive forces may damage the
contact surface, vacuum or magnetic pads may be deployed in the bracing device for
increasing the friction force without increasing the nonnal force applied by the macro.
Figure 5.3 depicts a typical motion cycle of the macro from one position
(position 1) to another position (position 4) on the contact sUTface. The macro can be
moved by an operator as a telerobot, or it may operate in an autonomous mode and follow
a computer generated trajectory. When the bracing device comes in contact with the
environment, a force regulator is activated to regulate the force applied by the macn, to
the bracing device.
Figure 5.3: One Motion Cycle of the Macro
One operational cycle of the macro has the following stages (see Figure 5.3):
Staee 1 Force controller is deactivated at position I : The force controller is deactivated
so that the robot can leave the surface.
Stage 2 Slow separation from the contact surface: The macro moves away the contact
surface with a low velocity, until it reaches position 2 which is far enough from the
surface to start a fast motion.
Stage 3 Fast motion from position 2 to position 3: Because the robot is away From the
contact surface, it is safe to move it with the highest possible speed to a new position
(shown as position 3) which is close to the destination point 4.
Staee 4 Stop at position 3: Fast motion of the macro to position 3 will induce vibrations
on the flexible arm. The vibrations must be settled down at position 3 before the robot
can proceed to contact the environment.
Stage 5 Slow motion towards the contact suface: The macro moves towards the surface
with a low velocity until it contacts the environment. The maximum approach velocity (in
the direction normal to the surface) depends on the maximum allowable impact force that
can be applied to the surface. The magnitude of the impact force is proportional to the
endeffector approach velocity Wal941. Another factor that affects the impact force is the
effective mass of the macro at the contact point, which depends on the robot
configuration, inertia matrix, joints friction, and control gains m 9 6 ] .
Stage 6 Force controller is activated at position 4: When the bracing device comes in
contact with the environment. the force controller is activated in order to regulate the
normal force applied to the surface.
The above-mentioned task plan indicates that the main task requirements of the
macro are: i) fast and reliable point-to-point motion in the free space; and ii) regulation of
the normal force f, while the micro is performing the task. The micro arm must be
designed to be rigid, yet as light as possible so as to minimize the dynamic force f,.
Section 5.1.2 presents a control law and a performance measure that satisfy the macro's
task requirements, and Section 5.2 introduces a novel design of a lightweight micro
manipulator.
5.1.2 Control Algorithm and Performance Measure
This section presents a control algorithm and a performance measure to be used for the
integrated design of the flexible macro. When the robot moves in a free space, a joint
variable PD control law plus gravity compensation (similar to the one in Section 3.1) is
used to control the flexible arm. The control law for the unconstrained motion of the
macro is given by
Z, =-K, (6, - 8 d , ) - ~ ~ ~ ( 6 ~ - Q $ ) + F ~ , , (5. I a)
where z, is the input torque vector to the joints; K, and KDM are the proportional and
derivative gain matrices, respectively; 8, and 6, are the joints position and velocity
vectors, respectively; and Fg, is the macro's gravity vector. The notation dot "." denotes
the derivative w.r.t. time, and the subscript 'W denotes the desired value.
When the bracing device comes in contact with the environment, a force regulator
is activated to regulate the force applied by the macro to the bracing device. The parallel
control algorithm proposed by [ChS93] provides a simple, yet effective way for
combining the position control law with a force regulator, when the accuracy is not the
121
main objective. The parallel control algorithm combines the PD position control law with
a PI force control law as follows:
where J , is the Jacobian matrix of the macro; Kn and are the proportional and
integral force control gains, respectively; F, is the desired force; and e, is the force
control error (i.e., the difference between the actual and desired values of the force). One
advantage of this control algorithm is that the position control law remains intact even
after the contact. Moreover, the integral action force control law with velocity feedback
used in this control algorithm improves the force response and reduces the impact force
The operational cycle of the macro in Section 5.1.1 indicates that the cycle time is
mainly affected by: i) the settling time of the closed-loop system; and ii) the time for the
gross motion of the macro in Stage 3. Reducing the settling time directly affects the time
for Stage 4, and it also improves vibration damping and tracking performance of the
macro. With a better tracking performance and less vibration, one can move the points 2
and 3 closer to the surface, and thereby, reduce the time for Stages 2 and 5 as well.
In order to reduce the time for Stage 3 (i.e., the gross motion time to the point of
interest), the links must be designed to be as light a s possible, and powerful actuators
must be used to increase the acceleration of the arm. Lightweight links also reduces the
impact force, and thereby, reduces the time for Stages 2 and 5 by increasing the allowable
speed Unfortunately, reducing the mass increases the flexibility, and thereby, the settling
time. Therefore, it is important to find a correct balance between the settling time and
gross motion time.
The settling time of the macro in each configuration is inversely proportional to
the real part of the dominant poles (Ah3.,,,) of the linearized closed-loop system about
that configuration wango], and the gross traveling time ( t , - ) between two points is
inversely proportional to the square-root of the macro's end-effector acceleration ( a, ) in
the task space, i.e., t , =(&)-I. A good measure of the maximum achievable end-
effector acceleration in the task space (when the joint torques are bounded) is obtained
based on the Dynamic Manipulability Measure @MM) proposed by Yoshikawa
Iyos85]. Using DMM, we have I
where J v is the Jacobian matrix relating the macro's end-effector translational velocities
to the macro's joint velocities, M, is the macro's inertia matrix corresponding to the
rigid-body motions of the links (i.e., when all flextural deflections are zero), and R is the
number of rows of the Jacobian matrix J , . Note that DMM provides a measure which is
R proportional to (a,) ; therefore, in (5.2) the exponent f is used to obtain a measure
which is proportional to the end-effector acceleration.
A local performance measure (i.e., a measure for one configuration of the robot) is
obtained by combining the settling and gross motion times using a suitable scale factor. A
global performance measure JgW of the cycle time is defined based on the average
value of the local measures in several representative configurations. Computing the gross
motion time using (5.2), combining it with the settling time, and taking the average for
several representative robot configurations, yield
where the subscript i indicates that the Jacobian matrix, inertia matrix, and the real part of
the dominant poles are calculated in configuration i; n is the number of representative
configurations; and k is a balancing scale factor between the settling time and gross
motion time.
Only the formulation of the performance measure (5.3) is provided in this section
(without any case study). The proposed performance measure can be used for the
integrated design optimization of bracing long-reach Mlm robots, in a sirnilar way that
the tracking performance measure (3.55) was used in Chapter 3 for the integrated design
of unconstrained M/m robots (see Case Study 3 in Section 3.2.4). Designing a
Lightweight micro robot for the bracing macro arm is the subject of the next section.
5.2 Design of a Novel Micro Robot
The previous section presented a task plan for bracing long-reach M/m manipulators, and
provided a performance measure for the integrated design of the bracing macro robot.
Bracing was introduced as a mean for reducing the mutual dynamic interactions between
the flexible macro and micro. This section focuses on the design of a novel micro robot
(called a Robot for Curved Surfaces or bbROCS") for bracing applications. The new
design concept reduces the weight and the dynamic force generated by the micro (f, in
Figure 5.2), and simplifies the control design. Deployment of ROCS as a micro mounted
at the tip of a long-reach bracing macro, provides an effective solution for those
applications that require accurate high speed scanning of a surface in a large workspace.
5.2.1 Mechanical Configuration Design
Theoretically, two Degrees Of Freedom (DOF) are sufficient to locate a point on a
surface. In practice, however, a spatial (3D) robot is used to cover a curved surface. The
extra DOF(s) is(are) used for modulating contact forces. In many practical situations the
accurate control of normal forces is not required, and one can simplify the mechanical
design and eliminate the need for a hybrid controller by using a two-DOF robot for
locating the endeffector on a curved surface. ROCS uses two-DOF to cover a curved
surface. This reduces the number of actuators and weight, compared with a 3D robot.
The main idea is to design a robot with enough flexibility to conform itself to the
shape of the curved surface on which it moves. This allows to use the minimum number
of degrees of freedom to cover the surface. Figure 5.4 illustrates the design concept for a
2-DOF ROCS. Each DOF consists of a motor, a conforming link, a passive pushing
mechanism (e.g., a spring), and an end-part that fkely moves on the surface using the
rollers.
As the robot moves, the two pushing mechanisms keep the links on the surface,
and generate a normal force to the surface at the end-effector. Since the number of
degrees of freedom in the task space is equal to the number of actuators, there is no need
for a hybrid controller. A simple joint variable PID control method is sufficient to control
the end-effector position on the surface.
Curved Surface, e.g., a Storage Tank
Long-Reach Arm End-Effec tor and Rollers
Flexible Link
Pushing Mechanism
Figure 5.4: Design Concept of a Robot With Flexible Links Moving on a Curved Surface
Based on this idea, different designs are conceivable. One practical realization of
this concept is shown in Figure 5.5. In this design, instead of using a flexible link, a
revolute joint and a cardan joint have been used to give the link the required flexibility to
conform itself to the curvature of the surface. Each DOF uses a spring to keep the link in
contact with the surface and to adjust the normal force to the surface.
Kinematics and dynamics models of ROCS depend not only on the mechanical
configuration of the arm, but on the geometry of the surface, as well. Kinematics and
dynamics of ROCS for cylindrical surfaces are discussed in the sequel.
I Long-rwch manipulator
M u t e Joint 2
Figure 5.5: One Possible Configuration Design of ROCS
5.3 Kinematics Model
Figure 5.6 shows a schematic diagram of a 2-DOF ROCS on a cylindrical surface, where
the dark lines represent the links and motor axes. Polar coordinates ( q j ) are used to
specify the endeffector position on the surface. Kinematics model of ROCS on a
cylindrical surface is formulated in this section. Forward kinematics is discussed in
Section 5.3.1, followed by formulation of inverse kinematics in Section 5.3.2. Finally,
Section 5.3.3 presents the derivative terms and the Jacobian matrix, which are used for
obtaining the dynamics model.
5.3.1 Forward kinematics
The forward kinematics model expresses the endeffector coordinates ( c p 2) in terms of
the joint rotation angles 8, and 8,. Figure 5.6 shows that, the end-effector polar
coordinates (q , 2) can be obtained by, k t , calculating the polar angles p, and p2 and
elevations z, and z2 of Links 1 and 2 separately, and then adding the corresponding
terms together.
Figure 5.7 shows the schematic diagram of a link of ROCS on a cylindrical
surface. The motor axis (AB) and the Link (BC) constitute a plane (herein called "link-
motor plane") whose intersection with the cylindrical surface outlines an ellipse. In this
figure, R is the cylinder radius, L is the link length, d is the length of the motor axis (from
the surface), a is the angle between the link and motor axis, cp and z are the polar
126
coordinates of the end-point of the link (C). and 8 is the inclination angle of the Link-
motor plane measured from a horizontal plane. We want to obtain the end-point
coordinates (z and q) in terms of the inclination angle 0 .
Figure 5.6: A Schematic Diagram of ROCS
Figure 5.7: One link of ROCS and the Intersection Ellipse
Based on the triangles in Figure 5.7, five equations
z = R sinp tan@, (5.8)
are obtained, where (5.8) represents the intersection ellipse in the polar coordinate
system, and w and p are temporary variables shown in Figure 5.7. Solving (5.4) to (5.8)
for five unlcnowns 2. q, p, w, and a and simplifying the results give
which represents cp in terms of the inclination angle 8. When 0 approaches zero or i~ the
right-hand-side of (5.9) approaches 010, which is an undefined number. To remove the
ambiguity, we use the binomial expansion
to expand the second term in the numerator of (5.9). When 0 approaches zero or ir, the
condition
holds, and (5.9) reduces to
which provides g , when 8 approaches zero or z Having found the polar angle tp (from
5.9 or 5-12), we can compute the elevation z from the following formula
For a 2-DOF arm, the same procedure can be followed. However in this case, the
inclination angle 8 of the link-motor plane is expressed in terms of joint rotation
angles 0, and 0,. For Link 1, thc inclination angle 0 is equal to the first joint rotation
angle 8,, and for the second link the inclination angle @ is equal to 8,, = 0, + 6,. The
endeffector coordinates are then given by (see Figure 5.6)
(PC@,& 1 = V J ~ I ) + P # L ), (5.14)
d@,& 1 = z;W; )+ zJ@,2 ) 9 (5.15)
where q, and p2 are obtained by substituting 8, and 6, into (5.9), respectively; and z,
and 2, are obtained by substituting 8, and 8, into (5.13), respectively.
A program has been written to test the forward kinematics equations. The
procedure is as follows:
Read joint rotation angles 8, and 0,.
If condition (5.1 1) holds use (5.12), otherwise, use (5.9) to obtain q, ( OI ) and q2 (8, )
with positive signs.
IfcosO, < Othen 9, =-cp,.
Kcos~,, < 0 then tp2 =-q2.
Use (5.13) to obtain z, (6, ) and 2, (GI, ) with positive signs.
If sine, < 0 then z , = -2, . I€ sin@,, < 0 then 2, = -2, . Use (5.14) and (5.15) to obtain the endeffector coordinates cp and z .
5.3.2 Inverse kinematics
The inverse kinematics model expresses the joint rotation angles 9 , and 8, in terms of
the given endeffector coordinates cp and 2 . Rewriting (5.14) and (5.15) and writing
(5.13) for Links 1 and 2, we found four equations
~ p C w ' 2 1 = OJe; )+92(012 ), (5.14)
z C ~ A!, ) = zJ@I )+ zJe, ) 9 (5.15)
which can be solved for four unknowns, 2, , q , qI , and q2. Substituting
2 Q)I y = sin (-) 2
into (5.14) to (5.17) and solving the equations for the new variable y gives the fourth-
order polynomial
(A' + D ' ) ~ ' + ( Z A B + ~ D E - D ' ) ~ ' + (B' + ~ A C - ~ D E + E ' ) ~ ' +(~Bc- E ' ) ~ + C ~ =o '
where the coefficients are given by
A =(b, - r ~ , ) ~ -k:,
The polar angle p, is obtained by solving the polynomial (5.19) and substituting
the result in (5.18). Ferrari's method provides an analytical solution to the 4th order
polynomials.
Having found q, , the variables (p,, 2, , and z, are then obtained substituting the
result into (5. M), (S.l6), and (S.l7), respectively. Equation (5.15) is used to select the
correct sign of these variables among several possibilities. Finally, the joint rotation
angles are computed from the following equations:
9, = atan2( z, , R sinq, ) ,
O2 = a t u z , , Rsintp2 )-0, .
Based on the above procedure, a program was written to verify the validity of the
equations.
5.3.3 Derivative Terms and Jacobian
Derivative terms and Jacobian matrix are important for static force and velocity analyses,
and for obtaining the dynamics model using Lagrange's method. From (5.6). we obtain
and from (5.1 6) and (5.17)' we get
Using (5.8) and (5.25), we obtain
&, Rsin Q:
where
el, = el, and 61, = 8, + 0,.
Combining (5.25) and (5.26) yields
Similarly, from (5.24) and (5.26), we obtain
Equations (5.26) and (5.28) represent the derivatives of the endeffector coordinates w.r.t.
the joint coordinates. Therefore, they define the elements of the Jacobian matrix.
5.4 Dynamics Model
This section formulates the kinetics and potential energies of a 2-DOF ROCS on a
cylindrical surface. Kinetics and potential energies of the links and payload are calculated
separately, and the results are added to yield the total values.
5.4.1 Kinetic and Potential Energies of Link 1
Figure 5.8 shows a schematic diagram of the fmt link on a cylinder. Without lack of
generality, we can assume that the center of mass C is in the middle of the link, and the
spring is connected to C from one end, and to the middle of the motor axis from the other
end.
Figure 5.8: First Link of ROCS and its Angular Velocity
The angular velocity of the link w, has two components: one is 8, along the
motor axis, and the other is dr, dong the revolute joint axis (normal to the link-motor
plane). A coordinate frame Cx,y,z, is attached to the link's center-of-mass C to be used
as a reference frame. The Y-axis is parallel to the motor axis, and the YZplane is parallel
to the link-motor plane (Figwe 5.8). The center-of-mass velocity v,, and link angular
velocity w, are given by
w,=[ -&I 6, o]', (5.30)
which can be substituted into
to give the total kinetic energy of the f i t DOF. Note that the vector components are
written in the reference frame Cx,yczc . In (5.34), Ihmr, is the second moment of inertia
of the rotor of Motor 1 about its axis, and m,, and [ I , , ] are the mass and inertia-tensor
of Link 1, respectively. For a uniform slender link, the last term will be simplified to
The total potential energy of the fiat DOF is composed of the gravitational and
elastic potential energies, and is given by
I I Vl =-m,, gz, ++ u:,,
2
where
is the change of the spring length L,, from its initial value &, g is the gravity
acceleration, and K, is the spring constant.
5.4.2 Kinetic and Potential Energies of Link 2
Figure 5.9 shows a schematic diagram of the second link. Without lack of generality, we
can assume that the center of mass C is in the middle of the link, and the spring is
connected to C from one end, and to the middle of the motor axis From the other end.
Similar to the reference frame of Link 1, a reference coordinate frame Cxcy,z, is attached
to the link's center of mass C.
The angular velocity of the link w, consists of one vector along the cylinder axis,
qi,, another vector along the motor axis, 8,, , and the last one along the revolute joint
axis, dr,. The angular velocity, represented in the centersf-mass coordinate frame, is
and the center-of-mass velocity v,, is given by
where
qi,k = qi, [cos~,, o sine,Jr ,
7 v , =[i,co~9~~-R@,sin0,, 0 i ls in0,+R~,c~s8, , ] .
Substituting (5.40) to (5.42) into (5.39) yields the center-of-mass velocity
The total kinetic energy of the second degree-of-freedom T, is then obtained by
substituting (5.38) and (5 -43) into
where mwur2, Ihur2, and IMr2 are the mass, second moment of inertia of Motor 2
about Z-axis, and second moment of inertia of the rotor of Motor 2, respectively; and mu
and [I,] are the mass and inertia-tensor of Link 2, respectively.
The total potential energy of the second DOF is composed of the gravitational and
elastic potential energies, and is given by
where
is the change of the spring length L,, from its initial value &, g is the gravity
acceleration, and K2 is the spring constant.
5.4.3 Total Kinetic and Potential Energies
The payload kinetic (T, ) and potential (V, ) energies are given by
where m, , I,, and I, are the payload mass, second moment of inertia of the load
about the cylinder vertical axis, and second moment of inertia of the load about Motor 2
axis, respectively.
The total kinetic and potential energy of the robot are simply obtained by adding
the kinetic and potential energies of all the components
In practice, R is usually much larger than L,, L, , d l , and d, , and those terms
that include ti and @ are negligible in comparison with those with e and z. For
instance, if
R = S m ; L , = L , = I r n ; d , = d , = O J r n ,
we have
&' r 006 and (;)_ r 0 J 3
Disregarding the small terms in the kinetics and potential energy formulas will greatly
simplifies the equations.
Finally, the dynamics model is obtained by taking the partial derivatives of T and
V with respect to el, e2, 6 , and e2, and substituting the terms into Lagrange's equation.
Equations of Section 5.3.3 are used to calculate the time derivatives of other variables
such as qi and i in terms of the joint velocities 8, and e2.
5.5 S u m m a r y
The design of a bracing long-reach M/m robot was the focus of this chapter. To enhance
the performance of such a robot, a performance measure was proposed (5.3) to minimize
the cycle time of the bracing macro robot using DM, and a special-configuration micro
robot (called a Robot for Curved Surfaces or "ROCS") was designed to reduce the
dynamic interactions between the micro and the bracing macro.
Section 5.1 presented a task plan for bracing M h robots, and formulated the
performance measure (5.3) for the integrated design of such robots. M y the formulation
of the performance measure was provided (without case study). Section 5.2 presented the
conceptual and ~ o ~ g u r a t i o n design of ROCS. The special configuration design of ROCS
reduces the weight and dynamic forces generated by the micro. Finally, the kinematics
and dynamics models of ROCS were fonnuiated in Sections 5.3 and 5.4, respectively.
The contribution of this chapter was to enhance the design of bracing long-reach
Mlm robots. The contribution involved: i) closed-form solution of a performance measure
(5.3) for the integrated design of a bracing long-reach robot; and ii) conceptual/
configuration design (Figures 5.4 and 5.5) and kinematics (Section 5.3) and dynamics
(Section 5.4) modeling of ROCS .
Chapter 6
Conclusions and Recommendations
1.1 Summary and Conclusions
The main objective of this research was to enhance the design of long-reach M/m robots
for a wide range of applications using integrated design method (DM). IDM was
introduced and compared with traditional design method (TDM). Traditionally. the
mechanical and control design of robots are performed separately based on different
objectives. In DM, the mechanical and control design stages an performed concurrently
in order to achieve a performance which is otherwise impossible using TDM. A
comparative analysis, based on three design case studies, illustrated the advantage of
using D M over TDM.
Chapter 3 detailed the three design case studies of noncontact applications of a
planar 4DOF flexible M/m robot. The first case study represented TDM and the second
and third ones represented DM. A new modular systematic procedure for automatic
generation of the closed-form finite element model of flexible M/m arms was proposed in
Chapter 2. The finite element model, along with the control algorithm of the robot, was
used for the formulation of design objectives/constraints and for simulations. A new
performance measure (3.55 to 3.57) was formulated and used in the design cases studies
as a global measure of the endeffector tracking-error.
In Case 2, IDM was used to minimize the total mass without increasing the
tracking error. As a result, the total mass was reduced by 25% and the tracking-error
measure by 7% in comparison with Case 1 which represented TDM. In Case 3, D M was
used to minimize the measure of the tracking-error (TE) without increasing the total mass
of the robot. As a result, the TE measure and total mass were decreased (compared with
Case 1) by 35% and 7%, respectively. Simulation of the closed-loop system showed that
the maximum tracking error of the endeffector on a specified path was decreased by 37%
in Case 2 and by 62% in Case 3, as compared with Case I. Comparative analyses of the
numerical results illustrated the advantage of using IDM over TDM.
The detailed implementation of the integrated design case studies in Chapter 3
revealed that one major difficulty in applying IDM to long-reach flexible M/m robots was
the formulation of a suitable performance measure for a given set of task requirements. A
performance measure must clearly reflect the main task requirements, and it must be
computationally well behaved and efficient (closed-form solution is usually required). To
facilitate the application of D M to long-reach M h robots, the thesis provided the
closed-form solution of general-purpose performance measures for different applications
of such robots. Chapter 3 presented the closed-form solution of a performance measure
(3.55)-(3.57) for non-contact applications d long-reach M/m robots.
To extend IDM to other applications, a comprehensive list of applications was
compiled in Chapter 4 indicating that such robots have potential applications in variety of
industries such as: construction, highway, aircraft and shipbuilding, space, nuclear, and
waste management. Based on their task requirements, different applications were grouped
into three major classes of noncontact, contact, and bracing applications. Then, closed-
form solutions of performance measures for contact (4.70, 4.78, 4.81, and 4.82) and
bracing (5.3) applications were formulated in Chapters 4 and 5, respectively.
Design of bracing long-reach flexible robots was addressed in Chapter 5. First, a
performance measure (5.3) was proposed for the integrated design of bracing flexible
robots; then, the conceptuaVconfiguration design and modeling of a novel micro Robot
for Curved Surfaces (ROCS) were presented in Sections 5.2 to 5.4. ROCS attaches to a
bracing long-reach robot for fast and accurate scanning of c w e d surfaces. The new
design reduces the dynamic interactions between the bracing arm and micro, and
simplifies the control design.
1.2 Recommendations for Future Work
The following topics are suggested for future investigations:
Experimental study of IDM for long-reach Mh robots: To date, there is no report on
the experimental study of DM for long-reach M/m robots. Experimental study is
required to identify all design constraints imposed by standard components (e.g.,
actuators, transmission systems, and sensors) and other practical considerations (e.g.,
manufacture-ability and robustness), and to obtain an accurate dynamics model of a
complicated spatial (3D) flexible M/m robot. Using standard (catalogue) components
may lead to an integer (or discrete) optimization problem. Efficient techniques must
be developed for solving such optimization problem with discrete and continuous
design variables.
Development of a computer sofhvare for IDM: Integrated design of complicated
electromechanical systems requires an integrated software environment DGK951,
which combines dynamics modeling, control, formuiation of objective and constraint
functions, and suitable optimization methods. The software must support both
symbolic and numerical computations.
Study of the non-minimum phase behavior of flexible-macro/rigid-micro robots:
Further study is required for better understanding the dynamic characteristics of
flexible arms and their implications on control design. One important dynamic
characteristic of flexible robots, which has profound effects on control design, is their
non-minimum phase behavior (caused by the noncollocation of actuators and
sensors). To date, research in this area mostly focused on linear structures or a single
flexible link with a rotary joint [SF90]~u91][PA91]. Study of the non-minimum
phase behavior of more complicated non-linear systems (e-g., a flexible M/m robot)
can provide valuable guidelines for design purposes [CKC9 I].
Fuuy-crisp formulation of objective and constraint fimctions for integrated design
purposes: In practical design problems, it is not always convenient (or possible) to
specify design objectives and constraints precisely using crisp mathematical
functions. This is specially hue in the earlier stages of design (i.e., conceptual and
configuration design stages). where the design objectiveslconstraints may be better
presented by imprecise words than by crisp mathematical functions. Knowledge-
based systems or fuzzy set theory can be used to deal with imprecision and
uncertainty Dix95]mao87] [RCh96]. Using hybrid fhy/crisp representation of
design objectives and constraints (for integrated design purposes) is an open area
5. Development of D M for other systems: This thesis addressed the integrated
parametric design of flexible W m robots. Design of many other electromechanical
systems (e-g., active vibration damping or active suspension systems) can be equally
benefited from DM. Furthermore, IDM can be effectively used in conceptual and
configuration design stages, which leads to mechatronics design approaches me961
[AP961*
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