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Dynamics and control of a class of ground based mobile flexible manipulators Ries, Thomas Charles 1993

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DYNAMICS AND CONTROL OF A CLASS OF GROUND BASED MOBILE FLEXIBLE MANIPULATORS By Thomas Charles Ries B. Sc. (Physics), University of British Columbia, 1976 B. A.Sc. (Mechanical Engineering), University of British Columbia, 1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED ScIENcE  in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING  We  accept to  this  the  thesis as conforming  required standard  THE UNIVERSITY OF BRITISH COLUMBIA November  0  Thomas  1993  Charles  Ries,  1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely availaNe for refer ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Mechanical Engineering The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5  Date:  ABSTRACT  The thesis studies dynamics and control of a two-link flexible manipulator, free to undergo planar translational and slewing maneuvers, through numerical simulation. To begin with. kinematics and kinetics of the system are investigeted, for a general N-link system undergoing planar motion, leading to the nonlinear, nonautononious and coupled equations of motion. obtained using the Lagrangian procedure. A parametric study follows to assess the influenc.e of system variables on the dynamical response and particularly on the positioning error of the payload. A joint based non-collocated algorithm, using the Feedback Linearization Technique (FLT) that accounts for the complete nonlinear dynamics of the system. is proposed for the tip control. The results are useful in the design of the ground based flexible, mobile servicing systems which are gradually appearing on the industrial scene and are likely to be more common in the future.  11  Table of Contents  ABSTRACT  ii  List of Figures  vi  LIST OF SYMBOLS  xiii  ACKNOWLEDGEMENT 1  2  INTRODUCTION  1  1.1  Preliminary Remarks  1  1.2  A Brief Review of the Relevant Literature  2  1.3  Scope of the Investigation  3  FORMULATION OF THE PROBLEM  5  2.1  System Model  7  2.2  Reference Frames, Generalized Coordinates and Maneuvers  7  2.3  Natural Modes of a Bar in Bending Vibration  10  2.3.1  Modal functions  10  2.3.2  Modal integrals  12  2.4  Kinematics  14  2.5  Kinetics  19  2.5.1 2.6  Potential Energy 2.6.1  2.7  Kinetic Energy for the N-Link System  22 26  Potential Energy for the N-Link System  Governing Equations of Motion  27 29  in  2.7.1 3  4  30  DYNAMICAL RESPONSE  33  3.1  The Numerical Approach  3.2  Definition of the Program Variables  35  3.3  Numerical Data used in the Simulation  36  3.4  Simulation Results and Analysis: Tip Trajectory Tracking  37  3  CONTROLLED DYNAMICAL RESPONSE  61  4.1  Preliminary Remarks  61  4.2  Feedback Linearization  63  4.3  Control Implementation  65  4.3.1  Quasi-Open Loop Control  68  4.3.2  Quasi-Closed Loop Control  69  4.4  5  Equations of Motion for the N-Link System  Application of the Quasi-Closed Loop Control to the Mobile Flexible Two-link Manipulator  71  4.4.1  Stationary Manipulator in Zero Gravity Field Without Payload  71  4.4.2  Mobile Manipulator in Zero Gravity Field Without Payload  74  4.4.3  Stationary Manipulator in Non-Zero Gravity Field Without Payload  4.4.4  Mobile Manipulator in Non-Zero Gravity Field Without Payload  95  4.4.5  Mobile Manipulator in the Gravity Field With Payload  96  .  95  CONCLUDING REMARKS  112  5.1  Summary of Results and Conclusions  112  5.2  Comments About the Numerical Simulation  114  5.3  Recommendations for Future Work  116  Bibliography  118  APPENDICES  121  iv  A KINETIC ENERGY EQUATIONS  121 123  B POTENTIAL ENERGY  123 124  C EQUATIONS OF MOTION  124  V  List of Figures  2.1  A commercial general purpose six degrees of freedom robotic manipulator  2.2  A schematic diagram of the mobile, flexible two-link manipulator  2.3  Reference frame and generalized coordinates used in the mathematical modeling  .  .  of the mobile flexible two-link manipulator system 2.4  6 8  16  Schematic of the open chain of coordinate systems representing infinitesimal flexible mass elements dm 1 in the local and the inertial reference coordinate frames  3.5  20  Flowchart for the earth based translating flexible two-link manipulator dynamics simulation program  3.6  34  (A): Time strobe position plot of the two-link flexible manipulator with station ary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 4s  3.7  (B): Comparison between specified  40  and /3) and actual (p and p3k)  positions of the manipulator tip during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 4s 3.8  (C): Comparison between the specified (/3k and  /32)  41 and actual (8 and 82) po  sitions of the joint angles during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 4s 3.9  42  (D): Comparison between the specified (/3 and /3) and actual (p , and ps,) 2 positions of the joint angles as well as the flexible generalized coordinated during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 4s  43 vi  3.10 (A): Time strobe position plot of the two-link flexible manipulator with station ary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 2s 3.11 (B): Comparison between specified  (/3  and  8 P )  44  and actual (p and p3k)  positions of the manipulator tip during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 2s 3.12 (C): Comparison between specified  (/3,  ,2, /3px  45  and /3) and actual (6, 82, px  and ps,) positions of the base and joint angles during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 2s  46  3.13 (D): Comparison between the flexible generalized coordinates 6 3 for each link during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 2s  47  3.14 (A): Time strobe position plot of the two-link flexible manipulator with station ary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in is  48  3.15 (B): Comparison between specified (3 and i3) and actual (p and p3k) positions of the manipulator tip during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in is  49  3.16 (C): Comparison between specified (j3 and /32) and actual (6 and 62) positions of the joint angles during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in is  50  3.17 (D): Comparison between specified (/37’ and /3,,,) and actual (p , and p) posi 2 tions of the joint angles as well as the flexible generalized coordinated during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in is  51 vii  3.18 (A): Time strobe position plot of the two-link flexible manipulator with station ary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)rn in 4s 3.19 (B): Comparison between specified (j3 and  fl)  53  and actual (p and p3k)  positions of the manipulator tip during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 4s 3.20 (C): Comparison between specified  (th  54  and /32) and actual (0 and 02) positions  of the joint angles during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 4s 3.21 (D): Comparison between specified  55 (/ p 3 x  and 3) and actual (p and ps,) posi  tions of the joint angles as well as the flexible generalized coordinated during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 4s  56  3.22 (A): Time strobe position plot of the two-link flexible manipulator with station ary base, using two modes for each link, during a straightilne tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 2s 3.23 (B): Comparison between specified  (/ p 3 3x  and  fl)  57  and actual (P3w and p3)  positions of the manipulator tip during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 2s 3.24 (C): Comparison between specified (i3 and  132)  58  and actual (Oi and 02) positions  of the joint angles during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 2s  59  3.25 (D): Comparison between specified (/3 and /3) and actual (p,, and ps,,) posi tions of the joint angles as well as the flexible generalized coordinated during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 2s  60 vi”  4.26 A block diagram explaining control through the Feedback Linearization Tech nique as applied to the mobile, flexible, two-link, ground based manipulator.  .  64  4.27 Block diagram for the quasi-open ioop control of the two-link flexible manipula tor  70  4.28 Block diagram for the quasi-closed loop control of the mobile manipulator under study  72  4.29 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the PD control with the base held fixed 4.30 (B) Graphs of (from top left figure clockwise) /3 & p3w,  /3,3  75  & P3y, I 3 &  1, /32  & 02, VS time [seconds] for the stationary flexible two-link manipulator under the PD control  76  4.31 (C) Graphs of (from top left figure clockwise)  Pi, 3 /  & p, /3 & py,  6 j j  &  621,  612  & 622, VS time [secondsi for the stationary flexible two-link manipulator under the PD control  77  4.32 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the FLT control with the base held fixed 4.33 (B) Graphs of (from top left figure clockwise) 3, & p,  73y 3 /  &  P3y,  78  /3 &  61, /32  & 02, VS time [seconds] for the stationary flexible two-link manipulator under the FLT control  79  4.34 (C) Graphs of (from top left figure clockwise)  /3 & p, /3 & p,,,  & 621,  ‘512  & 622, VS time [seconds] for the stationary flexible two-link manipulator under the FLT control  80  4.35 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the PD control with the base translating as shown 4.36 (B) Graphs of (from top left figure clockwise) /32  & p3w, /3 & P3y, /3 &  81  Si,  & 62, VS time [seconds] for the moving flexible two-link manipulator under  the PD control  82 ix  4.37 (C) Graphs of (from top left figure clockwise) /3 & p, /3 & &  ‘522,  &  ps,,  521, 612  VS time [seconds] for the moving flexible two-link manipulator under the  PD control  83  4.38 (A) Time history response of the manipulator, showing the tip and the link postions in the presence of the FLT control with the base translating as shown  85  4.39 (B) Graphs of (from top left figure clockwise) /3, & p3w, 3 & 8, p/ 3y & P3y, /32 & 92, VS time [seconds] for the moving flexible two-link manipulator under the FLT control  86  4.40 (C) Graphs of (from top left figure clockwise) /3 & p, /3 & p, &  622,  &  621,  612  V5 time [seconds] for the moving flexible two-link manipulator under the  FLT control  87  4.41 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the PD control with the base held fixed during a full three quadrant slewing maneuver  89  4.42 (B) Graphs of (from top left figure clockwise) &  2,  ‘ P 3 3x  &  J3 /3p  & P3y,  &  9i,  /3  VS time [seconds] for the stationary flexible two-link manipulator under  the PD control, during a three quadrant slewing maneuver  90  4.43 (C) Graphs of (from top left figure clockwise) /3 & p, 3 / p y &J &  622,  ‘5fl  &  ‘521 612  VS time [seconds] for the stationary flexible two-link manipulator under  the PD control, during a three quadrant slewing maneuver  91  4.44 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the FLT control with the base held fixed during a full three quadrant slewing maneuver  92  4.45 (B) Graphs of (from top left figure clockwise)  /1  & p3,,  & pay,  & 0, 132  & 92, VS time [seconds] for the stationary flexible two-link manipulator under the FLT control, during a three quadrant slewing maneuver x  93  4.46 (C) Graphs of (from top left figure clockwise) /3 & ps,, fl & &  622,  ps,, 6 i  & 621,  612  VS time [seconds] for the stationary flexible two-link manipulator under  the FLT control, during a three quadrant slewing maneuver  94  4.47 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the PD control with the base held fixed, during a full three quadrant slewing maneuver in a nonzero gravity field 4.48 (B) Graphs of (from top left figure clockwise) / p3x & P3z, 3 &  2,  97  P3y 3 /  & P3y, i3 &  01, /32  VS time [seconds) for the stationary flexible two-link manipulator under  the PD control in nonzero gravity field 4.49 (C) Graphs of (from top left figure clockwise)  98 & p, /3 & pa,,  611  &  621, 612  & 622, VS time [seconds] for the stationary flexible two-link manipulator under the PD control in nonzero gravity field  99  4.50 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the FLT control with the base held fixed, during a full three quadrant slewing maneuver in a nonzero gravity field 4.51 (B) Graphs of (from top left figure clockwise) / 3 & p3,,  3y 3 /  100  & P3y,  &  i, /32 0  & 02, VS time [seconds] for the stationary flexible two-link manipulator under the FLT control in nonzero gravity field  101  4.52 (C) Graphs of (from top left figure clockwise) /3 & i, / py & 3  py,  &  621, 612  & 622, VS time [seconds] for the stationary flexible two-link manipulator under the FLT control in nonzero gravity field  102  4.53 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the PD control with the base translating as shown, during a full three quadrant slewing maneuver in a nonzero gravity field 4.54 (B) Graphs of (from top left figure clockwise)  13 &  /37,3  & P3y,  i 3 /  .  &  .  .  .  103  91,  /32 & 02, VS time [seconds] for the moving flexible two-link manipulator under the PD control in nonzero gravity field  xi  104  4.55 (C) Graphs of (from top left figure clockwise) &  622,  Px 3 I  & J2,  py 3 /  &  Jy  &  621, 612  VS time [seconds] for the moving flexible two-link manipulator under the  PD control in nonzero gravity field  105  4.56 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the FLT control with the base translating as shown, during a full three quadrant slewing maneuver in a nonzero gravity field 4.57 (B) Graphs of (from top left figure clockwise) /32  & p3 ,, 2  P3y 3 /  .  .  .  .  106  & P3y, th & 01,  & 02, VS time [seconds] for the moving flexible two-link manipulator under  the FLT control in nonzero gravity field 4.58 (C) Graphs of (from top left figure clockwise)  107 /37i,  &, 2, p  /3 & ps,, 611 & 621,  612  & 622, VS time [seconds] for the moving flexible two-link manipulator under the FLT control in nonzero gravity field  108  4.59 (A) Time history response of the manipulator, showing the tip and the link positions in the presence of the FLT control with the base translating as shown during a full three quadrant slewing maneuver in a nonzero gravity field with a payload  109  4.60 (B) Giaphs of (from top left figure clockwise) /32  p3 /3 & , /3 & 2,  P3y,  & Oi,  & 02, VS time [seconds] for the moving flexible two-link manipulator under  the FLT control in nonzero gravity field with a payload 4.61 (C) Graphs of (from top left figure clockwise)  /3 &  110  ry & p, 3 /  & 621,  612  & 622, VS time [seconds] for the moving flexible two-link manipulator under the FLT control in nonzero gravity field with a payload  xli  111  LIST OF SYMBOLS  angle between the reference frames F:_i and F crk  sum of angles c + a-1+  + a,  position vector to the origin of the frame F: w.r.t. frame F 1 _ 2 projected on to the inertial frame i 3 1  specified joint angle for link 1  /32  specified joint angle for link 2 specified position vector to the tip (payload) w.r.t. the inertial frame x-component of /9 w.r.t. the inertial frame y-component of  I P 3 x  /3 w.r.t. the inertial frame  x-component of the specified base or joint 1 position vector in the inertial frame y-component of the specified base or joint 1 position vector in the inertial frame cos(a) 2 cos(c-i ÷ 1+...+ak) -a generaiized co-ordinate associated with mode  j of link  i  m  number of modes used in the simulation  0 F  inertial frame  1 F  base and link 1 local frame at joint I  2 F  link 2 local frame at joint 2  3 F  payload local frame at its CM  ID  K.E. part of the differential equation derived from the time Xlii  derivative segment of the Lagrangian equation but contains only the acceleration terms fTD  K.E. part of the differential equation that is derived from the time derivative segment of the Lagrangian equation but does not contain any acceleration terms  FT  the K.E. part of the differential equation that is not derived from the time derivative segment of the Lagrangian equation and does not contain any acceleration terms  fu  the P.E. part of the differential equation that is derived from the Lagrangian equation  Ji  joint 1 joint 2  Ha,b  homogeneous transformation matrix between the frames Fb and Fa  3 k  eigenvalue for the mode function  3 (kl)  spatial frequency for the mode function q5  1 L  link 1  cj  link2 angular velocity vector of the frame F w.r.t. the frame F:_i projected onto the inertial frame used only in the general formulation p3  position vector to the tip (payload) w.r.t. the inertial frame  , 2 p3  X-component of p  p3y  Y-component of p mode function for link i, mode  j  modal integral for the link i with modes  j and k  normalized modal integral with i = 1 position vector to the origin of the frame F 1 w.r.t. the inertial frame p.o.t.  projected onto xiv  p  X-component of the base (joint 1) response in the inertial frame  2 p  Y-component of the base (joint 1) response in the inertial frame  D  time varying system mass matrix acceleration of the generalized coordinates vector  ‘7  generalized control force vector control force acting on the base at the origin of F 1 in the X 0 direction  Q  control force acting on the base at the origin of F 1 in the Y 0 direction  1 Qe  control torque acting at joint 1 control torque acting at joint 2  Ra,b  rotation transformation matrix between frames Fb and Fa position vector to the elemental mass dm:  w.r.t.  the local frame F  position vector to the elemental mass dm 2 w.r.t. the inertial frame sin sin(cr+a+1+..+crk) i 8  joint angle response for link 1  82  joint angle response for link 2  3 T  total specified excursion time of slew trajectory at a constant velocity  w.r.t.  with respect to  Xo (i)  initial specified X-coordinate at the tip during slew maneuver  Xo (f)  final specified X-coordinate at the tip during slew maneuver  0 X  X-coordinate axis of the inertial frame  1 X  X-coordinate axis of the frame F 1  2 X  X-coordinate axis of the frame F 2  xl  X-coordinate of the elemental mass dm 1 of link 1 w.r.t. the frame F 1  2 x  X-coordinate of the elemental mass 2 dm of link 2 w.r.t. the frame F 2  0 (i) Y  initial specified Y-coordinate at the tip during slew maneuver  0 (f) Y  final specified Y-coordinate at the tip during slew maneuver xv  0 Y  Y-coordinate axis of the inertial frame  1 Y  Y-coordinate axis of the frame F 1 Y-coordinate axis of the frame F 2  Yl  Y-coordiuate of the elemental mass dm 1 of link 1 w.r.t. the frame F 1  Y2  Y-coordinate of the elemental mass dm 2 of link 2 w.r.t. the frame F 2  1 yi  deflection of the tip of link 1 w.r.t. to the frame F 1  Ci  position vector of the flexible elemental mass dm: w.r.t. the frame F used only in the development of the generalized equations of motion damping constant of the base in the x-direction  c,  damping constant of the base in the y-direction  1 c  damping constant of joint 1 damping constant of joint 2  1 (El)  flexural rigidity of link 1  2 (El)  flexural rigidity of link 2  g  gravitational acceleration  k  stiffness of the base in the x-direction  k  stiffness of the base in the y-direction  1 k  stiffness of joint 1  2 k  stiffness of joint 2  11  lenght of link 1  12  lenght of link 2  1 m  mass of link 1  2 m  mass of link 2  rn  mass of the manipulator base  mp  mass of the manipulator payload  xvi  ACKNOWLEDGEMENT  The author wishes to express his sincere thanks to Prof. V.J. Modi and Dr. F. Karray for their guidance throughout the preparation of this thesis. The project was supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A-2181, and the Networks of Centers of Excellence Program, Grant No. IRIS/C-8, 5-55380. Special thanks are extended to my wife, Louise, for her patience, encouragement, and tireless assistance.  xvii  Chapter 1  INTRODUCTION  1.1  Preliminary Remarks  The continuously evolving technology of robotic joint actuators coupled with the introduction of direct drive systems have led to a rapid increase in speed and payload capacity of ground based manipulators. As a result, the flexibility of previously considered rigid links have become increasingly significant. Due to rigidity requirements, present generation of robotic manipula tors [1], [2] are normally restricted in their load carrying capacity to approximately 5-10 % of their own weight. For example. the Cincinatti-Milicron T3R3 robot weighs about 2000 Kg but can carry a payload of only about 20 Kg. If a controller is designed to account for the: • flexibility of the links: • flexibility of the joint actuator systems; • flexibility of the base structure of the manipulator; • effects of gravity; then, in principle, it may be possible to increase the payload/robot weight ratio as well as the manipulator speed for a given mass and positioning requirements. There are other obvious advantages in accounting for the manipulator’s flexible character: • Cheaper drive components can be used since less power is required to move the lighter structural components, assuming equivalent accelerations and payload capacity to the rigid link design. 1  • Safer operation in case of collision, since lighter components are used. • Added weight reduction due to simpler actuator and structural design. This is a result of the reduced power requirement. Now the direct drive actuators can easily be adapted thereby eliminating the need for gears. This also results in reduced backlash and improved actuator linearity. • Increased structural compliance due to flexibility in the links. End effector compliance is very important in standard pick and place maneuvers involving delicate assembly op erations. Flexible links can provide this compliance in a very simple and cost effective manner. Suprising as it may seem, interest in the ground based mobile, flexible manipulators. ac counting for the gravitational effects, is relatively recent and the associated literature rather scarce. With this as background and the above mentioned benefits as motivation, a reasonably general model of the ground based mobile flexible manipulator is considered for study to gaimi some insight into its dynamical performance and control.  1.2  A Brief Review of the Relevant Literature  A challenge introduced by the flexibility consideration is so immense that, as pointed out before. research in this area was initiated only in the past decade and the accumulated literature is quite limited, particularly with respect to the ground based systems. This, of course, is understandable as for most industrial applications, links and joints were essentially rigid; hence flexibility was never an issue, at least for the intended objectives. On the other hand, for space applications, where weight is always a major governing factor, flexible structural members in the forms of solar panels, antennas and beams were introduced relatively earlier. However, now the structures, such as the Canada Arm aboard the Space Shuttle [3], operate in the microgravity environment with an entirely different class of problems. Modi [4], Chan [5] and Mah [6] have reviewed this literature at a considerable length. 2  Studies aimed at dynamics and control of ground based mobile, flexible. multi-link manip ulators, accounting for gravity, are indeed rare. However, there have been some investigations concerning tip control of a single flexible link [7], [8]. Almost all commercial robots use cob cated control (the position sensor located at the joint actuator where the generalized force is applied), as then it is easier to maintain stability of the controlled system. With noncolocated position sensing, stable control is extremely difficult to chieve. In that sense Binford and Can non’s contribution is noteworthy [9]. They successfully demonstrated tip-position control of a single flexible link manipulator (fixed at one position, i.e. not mobile) using noncolocated (po sition sensing at the tip rather than at the joints where generalized forces are applied) control strategy.  Several different procedures have been studied to control single and multi-link manipulators. They include: Linear Quadratic Regulator [LQR, [5]]; variable structure or sliding mode control [10]; model following adaptive techniques [11]; ard pole placement, selftuning control [12]. In most cases, their applications are limited to stationary manipulators in general, with two links if rigid and a single link when flexible. R.ovner and Franklin [13] incorporated variation of the payload mass in their formulatioi and applied a selftuning regulator approach to adaptive control. of a flexible single link manipulator. Although accurate noncobocated tip control was achieved it was found to be quite sensitive to modeling errors. Modeling and control of robots with rigid links but elastic joints have also received some attention [14]. The thesis takes a small step in this neglected area of dynamics and control of earth-based mobile flexible manipulators accounting for gravitational forces.  1.3  Scope of the Investigation  In general earth-based mobile, flexible, multilink manipulators, with flexible joints and sup ported by a flexible base, present a formidable problem in dynamics and control. Rapid and  accurate control of the payload using non-colocated position sensing add to the challenge. The thesis considers a simplified planar model of such a system to gain insight into the complicated 3  dynamics involved, effect of parametric changes, and develop an effective control strategy. To begin with a mobile manipulator, with an arbitrary number of Euler-Bernoulli beamtype links and joints with torsional flexibility, supported by a. mobile flexible base is considered. The manipulator carries an arbitrary point pay]oad, and is free to undergo translational as well as slewing maneuvers in the vertical plane. The continuum system deformations, also in the vertical plane, are discretized using admissible functions in conjunction with time dependent generalized coordinates. The governing nonlinear, nonautonomous and coupled equations of motion are obtained using the Lagrangian procedure. As can be expected, the governing equations are not amenable to any closed-form solution. A special case of two-link manipulator is considered and a numerical code written for its dynamical analysis and control. A parametric study follows to help appreciate the effects of system variables and slew maneu vers on the dynamical response. It also suggests a need for control under critical combinations of system parameters and initial conditions. Finally, a noncolocated control strategy is devel oped for the payload trajectory tracking and its effectiveness is assessed over a range of slewiiig time histories. The thesis ends with some concluding comments and recommendations for possible extension of the study.  4  Chapter 2  FORMULATION OF THE PROBLEM  Most general purpose commercial robots, such as the one shown in Figure 2.1, have 4  -  6  degrees of freedom. Normally the base, joints and links are treated as rigid distributed mass structures. In reality, this is an approximation. To make the positioning and tracking of the payload accurate to within a prescribed tolerance, designers are required to considerably over design the structural and drive components to make the end-tip position sufficienly predictable. This results in a prohibitively excessive robot weight combined with meager payload capacity and slow slewing response. One approach to overcome these undesireable elements would be to reduce, significantly, the total mass of each element but retain its structural integrity through an acceptable factor of safety with respect to mechanical stresses based on maximum loads. This would result in a lighter and faster manipulator. If the control strategies are modified to include the flexibility effects of the links, the base and the joints in the dynamical equations, then the manipulator can be made to position the payload as accurately as before. With this as background, the development of the formulation starts with a mathematical model for an earth based flexible, mobile, manipulator. The model aims at realistic measures of time and effort, yet remains sufficiently comprehensive to capture essential features of the system. Reference frames and generalised coordinates are selected to describe the state of the system. Deformations of continuous flexible members are presented by discrete modal functions. The kinetic and potential energy expressions are derived and the governing equations of motion obtained using the Lagrangian procedure. 5  Elbow extcnsOfl Arm  Shoulder swivel  Yaw Ar in  W fist  SW CC p  Roll Pitch Base  Figure 2.1: A commercial general purpose six degrees of freedom robotic manipulator  6  2.1  System Model  Consider a mobile manipulator with an arbitrary number of Eulerian beam-type members, interconnected to form an open chain geometry, carrying a point payload  nip.  The two-link  version of this general configuration is schematically presented in Figure 2.2. The joints J are considered to have torsional rigidity as well as damping. Translation motion of the mobile base. treated as a point mass, is also represented realistically through inclusion of the linear stiffness and viscous damping. The system is free to undergo translational and slewing maneuvers as well as link deformations in the vertical plane (plane of the paper). The model is carefully selected so as to retain essential features of a robotic manipulator, particularly with reference to the elastic character of the base, joints and links. The emphasis is on the flexibility effects on the payload trajectory and its control.  2.2  Reference Frames, Generalized Coordinates and Maneuvers  The manipulator shown in Figure 2.2 is assigned local reference base coordinates and generalized coordinates as described here in more detail. The elasticit of the base and the mechanisms at the joints will affect the performance and accuracy of the manipulator by permitting oscillations about the nominal specified slew angles. As a result the base support flexibilities are modeled by straight linear springs and velocity (viscous) dampers. The position of the base C.M., and hence joint 1, is located w.r.t. and projected onto the inertial reference frame F 0 by the vector defined by p and  whose x and y components are  p.  The joint flexibility is modeled by a linear torsional spring with a velocity damper. Flexible rotations of the links are described by the  joint  angle, 8 , and the specified rotations designated  as, j, where i is the link number; cr,, the angle that the local reference frame F, associated with the link i, makes with the previous reference frame F , in this case about the Z_ 1 _ 2 1 axis; and / 3 is the specified slew at the joint i measured between the link i and the X_ 1 axis at the 7  Figure 2.2: A schematic diagram of the mobile, flexible two-link manipulator.  S  location of the joint i. The actual joint angle at i is 8 and is measured the same way as the t3 with definitions as follows: = =  where  62+cl;  is the angle of link 1 at the tip of link 1 measured in the local frame F . For the 1  Lagrangian formulation the generalized coordinates are 81, 82, p, and  Py.  The links are assumed to be Euler-Bernoulli beams and their deformations are modeled by spatial modal functions together with time dependent generalized coordinates, t)  where k, represents the associated with the  th  th  (x)6(t). 1 Ø  =  mode for the ii” link while  3 j 5  (I) is the  j generalized coordinate  link. n is the number of modes used in the discretization process.  Theoretically, the summation should extend to infinity, however, in practice only a small number of modes are generally used, depending upon the situation. In fact, in the reported literature, most investigators consider the effect of only the first mode since it normally dominates the response. As can be expected, for rapid convergence to the true response, the selected modal functions jj(x) should satisfy all geometric and natural boundary conditions. This, of course, is often  difficult and for complex structures virtually impossible. Fortunately, the modes satisfying the following conditions do present an acceptable set [14], [15], [16], [17] of admissible functions as: • they are able to describe any link profile; • they are differentiable over the system domain and to the degree of the differential equa tions modeling the dynamics; • they satisfy at least the geometric boundary conditions of the slope and deflection at the ends of the link. 9  In general. functions that satisfy the above conditions are orthogonal with respect to one an other. This property may be used to advantage in deriving the equations of motion. This is illustrated in the following secion.  Natural Modes of a Bar in Bending Vibration  2.3  2.3.1  Modal functions  The translating flexible manipulator under consideration has infinite degrees of freedom. The method of assumed modes effectively transforms the governing partial differential equations into a set of ordinary differential equations. The choice of available modal functions is also large. Modes for beams with various combinations of end conditions are possible caiiclidates. One might consider the use of unconstrained modes incorporating time varying boundary con ditions [7]. In this study, to minimize the complexity, cantilever modes of an Eulerian beam are used as admissible functions for the manipulator links. The effects of rotary inertia and shear deformation, which become significant for large deflecions and high frequencies [141. are neglected here. For a cantilever beam with free end (no tip-mass), 02  (El (x)  02y(x,t)’\ 8z2  ,i  —  dy(x,t) —m(a  where: El (x)  =  flexural ridgidity of the beam  m(x)  =  mass per unit length at any point x;  y(x,t)  =  transverse displacement solution subject to given bounday conditions reflecting the manner in which the ends are supported.  If the beam is assumed uniform, LI is a constant and the governing differential equation reduces to y 2 0 04 2 y +a —i=O, 10  where a2  =  El m  —  For a simple cantilever beam, the boundary conditions are zero displacement and slope at the fixed end and zero shear force and moment at the free end. More specifically: dy  y(O)=O;  I=o=O;  0 !x=L  y 3 d xLO  y 2 d  The equation of motion can now be solved by the method of separation of variables. y  =  (x).6(t). where (x) depends only on the spatial coordinate x and 6 (t) depends only on time. The new set of equations to be solved has the form: and where: [34  —  m .2 4 El  ,2 2  The general solution to these equations is given by: (x)  =  4 c 2 Cjsint3x+C s 3 c inh5x+C os8x+C oshi3x  6(t)  =  coswt 6 sint ± C 5 C  Now the modal function 0(x) has the form (x)  =  (sin 3z  —  sinh i3x) —  sin 131 + sinh 31 (cos /3x cos 31 + cosh/31  where cos3lcoshi3l Here: —  —  El  a2’  11  =  —1  —  cosh 3x),  a  2  El =  —;  El  =  flexural rigidity of the beam  m.  =  mass of the beam per unit length  =  circular frequency  =  length of the beam  1  in  The transcendental characteristic equation yields infinite set of eigenvalues  Inserting these  values into the equation for (x) gives the desired modes. which are orthogonal and self-adjoiiit. The presence of a payload at the tip of the terminal arm would introduce shear bouiidarv condition. The characteristic frequency equation cosi3lcosh 3i + I  now  modifies to  !id (sin /3lcosh dl in  —  cosj3lsinh :31),  with the form of the modal function (x) remaining the same as before. The first six roots of the new characteristic equation are summarized in the table below for four payload conditions:  2.3.2  0  1  2  5  (3i)  1.88  1.25  1.08  0.87  (/31)2  4.70  4.03  3.98  3.95  (,8l)3  7.86  7.13  7.10  7.08  (/31)4  11.0  10.3  10.2  10.2  (/31)5  14.1  13.4  13.4  13.3  (/31)6  17.3  16.5  16.5  16.5  Modal integrals  The modal integrals are associated with the kinetic and potential energy evaluation of the system. They represent integration of dynamical quantities over the out before,  system domain.  As pointed  orthogonal character of the modes simplify calculation of the energies and hence  derivation of the equations of motion. 12  The modal integrals are defined as follows: fij(x)m(x)dx; = =  f  j(x)dm  =  jj(x)m(x)dx; xm(x)dx;  Xij =  Dij  =  dx 2 (EI)j (d2&x)  Normalizing with respect to the length of the beam (i.e. length of the manipulator arm) gives: =  =  Xij 4 Dij 4  =  =  (EI) 2l  where: i  =  link number  j  =  mode number;  x  =  lx’;  dx  =  ldx’;  x O  x’  <1;  m (x)  =  rn  M  =  total mass of the link i;  =  j =  j  =  uniformly distributed mass of the link i;  (x’) dx’; dj (x’) dx’;  =  j  x’q (x’) dx’; ‘3  =  —  —  —  (kl)  j  ((sin (kl) x’  +  sinh (kl) x’) +  3 x’ + cosh (kl) ‘)) di’: c (cos (kl)  3 + sinh (k1) sin (kl) +cosh(kl) 3 cos(kl)  These norrnlized integrals may be evaluated by numerical integration and may be used for a manipulator link with constant flexural rigidity and mass distribution along its length. Values of the integrals as affected by the number of modes used in the discretization process, for in  =  0,  are summarized in the table below: No. of MODES  MODAL INTEGRALS 4  /  F  F  ij  Sj  Xj  3 D2  1  0.78299  1.00000  0.568830  12.3620  2  0.43393  1.00000  0.090762  485.510  3  0.25442  1.00000  0.032425  3806.30  4  0.18172  1.00000  0.016450  14606.0  5  0.14147  1.00000  0.009985  39956.0  6  0.11569  1.00000  0.006669  89096.0  Note, the integrals involving products of different modes are not presented here as they vanish due to the orthogonality condition. However, this is not always true for ni  0 because  the modal functions for a cantilever beam with tip mass are not necessarily orthogonal.  2.4  Kinematics  To derive the governing equations of motion it is necessary to evaluate the kinetic and potential energies of the system. For this we require the position and velocity vectors for the elemental  mass, drn , in Link i. 1 The fundamental kinematic expressions for displacement and velocity are developed in this section. Two approaches are reviewed to aid in choosing a technique best suited for the intended purpose. The first approach utilizes homogeneous transformation matricese [18), [19] to obtain 14  the position and velocity of dm with respect to and projected onto the inertial reference frame. The second approach [5], [6] employs vector methods to arrive at the same end result. By direct experimentation, it became appearant that for simpler systems such as the present two link manipulator, derivation of the equations of motion is considerably easier through the use of homogeneous transformation matrices. This is because the vector method creates a large number of redundant terms which eventually cancel out in the final set of the differential equations. However, it should be noted that the vector method provides a relatively simpler form of the equations of motion for a general system of N flexible links that are mobile and operate under the influence of gravity. The various reference frames, and generalized coordinates are selected to specify orientation of the system as shown in Figures 2.2 and 2.3. Note, the two diagrams are similar except that the latter is a bit simpler; and while using the homogeneous transformations the local coordinate vector is changed to 5. The various reference coordinate systems F , F 0 . F 1 2 aiid 3 are local frames describing the kinematic specifications of the elemental mass segments of F each link with respect to (w.r.t.) and projected onto the respective local frame. F 0 is the base or inertial reference frame. F 1 represents the local coordinate frame for link 1 whose origin is coincident with the C.M. of the mobile base as well as the centroid of joint 1, and is fixed to link 1 so that the Link is aligned with the X -axis of the local frame. F 1 2 is the local coordinate frame for link 2 with the origin coincident with the centroid of joint 2 and the same conditions apply as for the previous link. F 3 is the local coordinate frame for the payload mass M, which is considered to be a point (i.e. negligible rotational inertia). In the local coordinate frame the position vector  fl  locates the mass element dm which is defined in terms of its components T= [xj,yj] 1  Note that the choice of reference coordinate frames located at the joints facilitates the use of cantilever modal functions. This makes the derivation and computation simpler. Since the X axis is aligned with the rigid body neutral axis of the link, the transverse 15  Xe  ye XI  Figure 2.3: Reference frame and generalized coordinates used in the mathematical modeling of the mobile flexible two link manipulator system.  16  deflection and rotation of a mass element drn in the link y(x,t)  =  cj ( 2 xj,t)  =  =  are defined as:  Ej(x)bjj(t)  As an example, joint. 2, located at the tip of link 1, is displaced by an amount y 1 and rotated through  1 a1  w.r.t. the frame F . 1  Before determining the kinetic and potential energies, the displacement and velocity of the elemental mass dm must be established w.r.t. the inertial reference frame F . The displacement 0 of dm, is first determined w.r.t. and projected onto the local frame, by the vector  .  Through  a series of homogeneous transformations (i.e. H ), 0 1 it is transferred to the inertia) frame. The position vector of the mass element dm 1 on the link i with respect to the inertial frame is given byi.  where: = 1 , 0 H  , 0 R  =  (  1 , 0 R  0  1 p. 0  1)  rotation matrix (3 x 3) for the transformation from the local frame F 3 to the inertial frame F 2  =  position vector to the origin of the local frame F w.r.t. the inertial frame.  Determination of the transformation matrix H 3 is accomplished by postmultiplying the local , 0 transformation matrices, 3 H , 1 _ ,, in the consecutive order from the inertial frame to the frame 2 in question. The position vectors to the mass element drn F 3 are defined as follows:  —  —  =  p 1 Ho,  Cl  1 —S  0 p  1 S  1 C  0  p,  Yi  0  0  1  0  0  0  001  1  =  17  — -.  2 r  =  jH 0 H p 2 , 1  1 C  —Si  0 p  2 —S C 2  1 S  Ci  0 py  52  2 C  0  0  10  0  0  0  01  0  0  ‘1  0  1 Y1  Yi  0  10  0  0  01  =  where:  i  =  link or reference frame number cosc  =  sinaj  =  length of link i =  =  =  xi  =  distance along the link i (or the X-axis) from joint  yj(zj)  =  transverse deflection of link i at position x  i  Inserting tile required variables, differentiating w.r.t. time and multiplying the appropriate matrix elements, the absolute velocities are obtained for the mass elements din , din 1 , as well 2  as lnp: S 9 1 —x  —  1 thS-  8C + yiC’i x 1  -.  —  —  C’ ±j3 1 yj6 2 5 +J)y 1 6 Y1  TI =  0 0  —  12 5 Y2  212 x 1 C 2+2 C’i ‘  —  —  à 2 y 1 C 2 Y212SI2  —  S Ô 1  S 6 +1  r2  0 0 18  —  —  Si  —  —  11 + y C 1 Ô 11 + j y S 1 Ô  2  —  )  12 2 S Y1  12 + 12 ã 2 +(1 )C  2 Y1  2  à 2 yj 1 S 2  —  —  1161 SI  + 116151  —  —  51  —  1 C 11  Tp =  —  1 yi  1 + j3 61 C  12 + y S 1 6  ,  0 0 In the next section the use of vector method is demonstrated to identify its strength and wealnesses. Although it is a somewhat complex approach on one hand, it also offers some intuitive insight into the various kinematic terms contributing to the overall system equations. For the present study of the moving two link manipulator. homogeneous transformations are used directly, on a term by term basis. as it is far more efficient.  2.5  Kinetics  Alternatively, in the vector form, the generalized position and velocity of the elemental mass dm in an open chain of coordinate systems may be presented as: =  =  It should be noted that all the above vector variables must be projected onto the inertial reference frame F . This implies that all vectors stated in terms of the local reference frame F 0 must be transformed by a series of consecutive rotational transformation matrices. Consider Figure 2.4 for a representation of the open chain of coordinate systems with flexible links. With the position and velocity of each elemental mass dm determined, the kinetic energy T and potential energy U of the system can be derived by integrating over the total mass M of the system as a whole. The kinetic energy [20], [21] for the mobile flexible manipulator can be represented as:  T =  =  21M 2JM,2  19  N  N1  Figure 2.4: Schematic of the open chain of coordinate systems representing infinitessmal flexible mass elements dm in the local and the inertial reference coordinate frames  20  =  =  5Jdin 1 x)• 2 TCMT+TCMR+TMv+THH  =  where: ‘7-’  ICMT  = =  TCMR  I  -  -  7)jU?flj  total translational K.E. of the C.M. of the system;  (’ 1 f  x 5) (c x )din,  =  1  =  total rotational K.E. of the C.M. of the system w.r.t. the local frames; N  = =  THH  r  i= JM 7)2  =  ‘p  ‘MV  N  1  Jji, P P  total K.E. w.r.t. the local frames;  =  =  angular momenta terms for the system:  =  instantaneous inertia matrix w.r.t. the local frames.  The kinetic energy TCMT describes a nominal value associated with the C.M. of each frame w.r.t. and p.o.t. the inertial frame. The energy TCMR is the rotational kinetic energy associated with the instantaneous inertia measured in each frame. Kinetic energy arising from modal vibration in the flexible links is described by TMV. Energy THH represents the coupling between the various degrees of freedom within the manipulator system as a whole. it should be noted that, when evaluating T for the Lagrangian procedure, all vector pa 0 through a series of rotational raineters must ultimately be projected onto the inertial frame F transformations. The vector method offers good intuitive insight into the nature of the various contributors to the total kinetic energy. However, it is more labour intensive compared to the homogeneous transformation procedure. It leads to numerous redundant cross-terms in the 21  differential equations of motion which ultimately cancel. Hence the homogeneous transform method was used to derive the kinetic energy for the specific case in study (Appendix A). How ever, in the following sections vector method is used to determine the form of the generalized equations of motion for an open chain of N flexible links. 2.5.1  Kinetic Energy for the N-Link System  The nature of the vector method and its extension to a multi-degree of freedom system of links is demonstrated in this section. The expression obtained  iii  the previous section for the  kinetic energy, T, can be further transformed into a general expression for an N link manipulator system as follows: T  2JM  =  =  =  =  +( (c x fli) .  +  ( x ) jdm.  If the local frame mass element position vector is changed from  to ( to indicate that the  body being integrated over is a flexible cantilever beam, then the kinetic energy expression takes the following form; N  ; 1 T=T  =  +  J  +  1+ çxdm  2  IM  cdmi]  This can be rewritten in pure summation form by making the following substitutions: O,i  =  1  +  2  +  +  j;  22  =  , +R 0 R 1 + , 0  =  + 1 ao+a + 2 ; •+a a  ; 1 , 0 + R  02 + R +R 1 +a , 0  =  .  .  .  +  =  j=1  where:  Ro•o  =  Identity matrix I  1 , 0 R  =  Rotation transformation matrix from the inertial frame to the local link frame. i  =  Position vector describing the origin of the frame i w.r.t. and p.o.t. the previous frame i  =  —  1 in the chain  Position vector describing the origin of the frame i w.r.t. the previous frame but p.o.t. the inertial frame  =  Position vector of the origin of frame i w.r.t. and p.o.t. the inertial frame  =  The position vector of the elemental mass dm, w.r.t and p.o.t. local frame i  =  The position vector of the elemental mass dm w.r.t. and p.o.t. the inertial frame  =  Angular velocity vector of frame i w.r.t. and p.o.t. the previous local frame i  =  —  1;  Angular velocity vector of frame i w.r.t. and p.o.t. the inertial frame.  The total kinetic energy of the system then becomes:  T  =  i=1  i=1  j=1  j=1  N  [roi_i_i.  + 23  J 1 Ro, ‘ui)]  [()T  •  ()]  f 1 Ro,  +  Carrying out the required algebraic manipulations the generalized kinetic energy equation can be further reduced to the following form: T=  !T  ] + 2 [ivw+2  {I:+2L  +3]  where: 1 in  rv1  —  inN  Global rigid body mass matrix for the total system;  N t=1  i  =  =  i  N t=2  1  11  i=2’i  :=2’i  Ifl  Ill  In  •..  In  Global rigid body inertia matrix for the total system  ,_ R x 1 , 0 t 1 R 1 +_  + xR + L _ Ro, 1 a , 20 Ei=  +  x 1 , 0 R 1 4 1 11 1R 111 , 0 _ +_ =  A time dependent inertia vector 24  1 Ro,_12  1 , 0 R _ 2n Global momentum vector  i=2  [  ) 3 RQ_j  _ 1 , 0 R  Global angular momentum vector  0 Ito,  v=  R 1 , 0 2 I’  Ro,nj Global velocity vector for local reference frames  L)  =  2 , 0 R p4 •  Global angular velocity vector for local reference frames  25  =  IM, cdm  =  fM’  2i  =  4j  = =  J  drn =  j  dm;  1 M  Instantaneous inertia matrix of the link subsystem i w.r.t. the local reference frame i  In the global kinetic energy equation, T, the bracketted part of the first expression contains tlw linear momentum terms. The bracketted part in the second expression contains the angular inomentnuin terms. The third expression contains the kinetic energies as measured in the local reference frames.  2.6  Potential Energy  The following three sources are the sole contributors to the potential energy of the u1oi)il flexible two link system: • gravitational field • strain energy of the flexible links • linear elastic strain energy of the joints resulting in: U=UG±UL+UJ.  The gravitational potential energy for the twolink flexible manipulator is given by =  where =  gravitational acceleration;  =  position vector of the element mass dm 1 w.r.t. the inertial frame. 26  The strain energy due to bending is given by UL  -  j(EI)(4)2dx  =  where: (EI)  =  bending stiffness of the link i  =  curvature of the link i  The potential energy stored within the flexible joints is  Uj  =  _8)2 2 kC13  where: k,  =  torsional stiffness of the joint i  =  specified slew angle of the joint i  =  response argle at joint i.  Details of the potential energy expressions are given in Appendix B. 2.6.1  Potential Energy for the N-Link System  The required potential energy terms  G, 1 t L  Uj clescibed above can be generalized to N links  utilizing the orthogonality properties of the modal functions: 0 U  = —  =  —•  J  jdm  1 _ 0 R  mj + R_ 11 , 1 j=i  UL =  =  Uj  =  27  where:  i11  i12  13 ‘I  i1rn 4  21 (P  i22 4  i23  i2m  em1  irn2  :m3 4  1 K  =  Global stiffness matrix in bending associated with link i  i1 6 i2 6 =  6777, =  (x)(x)dx; 1 j(EI)  ijk  l. 3 1 4 (x)  Global generalized coordinate vector associated with link i  =  generalized coordinate associated with the jth mode of link i ; i  =  spatial admissible function associated with mode j of link i  —  —  =  ô (x) k 2 2 Ox  gravitational acceleration field vector;  28  =  1,  ...,  m  total mass of link associated with local frame  =  j  torsional spring rate of joint i  (i3,  —  6)  angular deflection between the joint i and link i, and is dependent on joint stiffness.  =  The remaining parameters were defined in the previous sections.  2.7  Governing Equations of Motion  The nonlinear differential equations of motion for the mobile two-link flexible manipulator were derived using the Lagrange equation, d IOT’\ dt \ aqk J)  —  where k  =  l,...,nq  ;  -——-  ÔT —  —  dqk  +  öU —  dqk  = Qk  T arid U are the total kinetic and potential energies, respectively qk and  Q k are the generalized coordinates and the generalized forces, respectively; and  flq  number of generalized coordinates. Specifically, the generalized coordinates are defined as follows: p  =  x-comnponent of the base position vector  py  =  y-component of base position vector ;  =  joint angle of link 1;  =  joint angle of link 2  =  generalized coordinate associated with the  j 1 s  wherej=l =  th  mode of link 1,  rn;  generalized coordinate associated with the jt1 mode of link 2, wherej=l  m.  29  is the total  The specified time varying slewing coordinates are: =  mobile base translation vector  =  slew angle at joint 1;  =  slew angle at joint 2.  The governing equations of motion may be written as: d /OT\ T 0(1 —j-)—--+---d oqj oqk oqk  fD+fTD-fT+h’  =Q,;  =  =  Q  =  Here D is the symmetric positive definite time varying inertia matrix for the system: and  c  represents the non-linear vector containing all the cross influence force terms that are generally associated with centripetal and coriolis accelerations. They are functions of velocity and not acceleration of the generalized coordinate vector  containing the  input  denotes the generalized force vector  joint control torque values. For the generalized modal coordinates the  associated generalized forces in the vector  must be equal to zero since it is assumed that  the individual modes of each link cannot be controlled externally. A summary of the nonlinear differential equations is presented in Appendix C. 2.7.1  Equations of Motion for the N-Link System  The kinetic and potential energy expressions for a general system with N flexible links were presented earlier:  T  =  N  =  N  + R;_ ] + A 1  30  N  +  —  9)2  Using the Lagrangian procedure, the governing equations of motion can be obtained, as shown before, from the following equation: ÔT  d IOT\ at \oqk  )  I  —  +  oqk  OU oqk  =  where T and U are the global kinetic and potential energies of the system; with  qi  and Qk as  the generalized coordinates (i.e. joint angles, flexural modes) and the associated generalized forces (i.e. forces, torques). Making a substitution for the bracketted expressions to simplify the equations gives:  where  1 fr  =  2 b  =  ; 2 M+2  Performing the required algebra leads to the generalized equations of motion for the exible iV-linld system:  1 Qk  (ôcT’\  (d  b -  =  + V—T  Id  (o  \  j--  -  o -  aT\ (d (&T’\ (d (2” 2 0b 2 — 1 + + ----i-—————,b w—T iIi— —  di  \  ôqk  /  T. 0  +  qk 9 ‘  rô  1 + ——b b 2+v qk 9 l  di  qk) 8  qk 9 l  31  +  2 •T0b Oqk  q 9  /  dq  +TK1)  1 , 0 +R  [ORti;t_i  (i7nk+Li)  (+]  +k°’  This can be further expanded and then simplified into the  more  recognizable form. FOr  the present, it is sufficient to say that since all the terms are clearly defined, an algebraic algorithm may be formulated for the N-link. flexible, mobile manipulator. This algorithm may then be integrated with a symbolic algebraic manipulation software. such as MATHCAD or MATHEMATICA [22], to automatically produce the complete system of differential equations. which can be numerically solved with graphical output through a single command operation.  32  Chapter 3  DYNAMICAL RESPONSE  The dynamic equations of motion derived in the previous chapter are nonlinear, nonau tonomous, and coupled. They are not amenable to any known closed form solutions without considerable simplification. To retain the subtleties of the response characteristics, the equa tions are integrated numerically over a range of system parameters and prescribed maneuvers.  3.1  The Numerical Approach  The numerical code is in VS FORTAN77 language and is so written as to isolate the effects of various system parameters such as the number of modes used, inertia and flexibility properties of the links, end-tip trajectory tracking, maneuvers, etc. The numerical solution is obtained using the IMSL Version.lO [37] differential equation solver subroutine called DIVPAG and the MTS(; system subroutine GEARB. Both subroutines utilize Gear’s method [36] to solve initial value problems of ordinary differential equations with greatly varying time constants (i.e. stiff systems). DIVPAG appears to be a relatively slower program with limited capability. On the other hand, the GEARB routine is more versatile and can tackle more difficult situations while giving reliable results. For the mobile flexible two-link manipulator under study, the structural frequencies are in the range of 0.5-100 Hz depending upon the link and joint parameters as well as the number of modes used. A flow chart for the translating flexible two-link manipulator simulation code is presented in Figure 3.5. 33  !j  I  oq o  .  3.2  Definition of the Program Variables  The following is a summary of the program state variables, output data variables, and their corresponding symbolic notations.  This list should be referred to in conjunction with the  simulation plots that follow:  BETA 1  =  THETA1  =  BETA2  =  specified joint angle for link 1  0  =  joint angle response for link I  =  132  =  specified joint angle for link 2  THETA2  =  62  =  joint angle response for link 2  BP3  =  ,‘3  =  specified position vector for the manipulator tip or payload w.r.t. the inertial frame  P3  =  J3  BP3X  =  Px = 3 ‘  X-component of  p3  =  X-component of 3  =  Y-component of 3  P3X BP3Y  =  ;3  P3Y  =  p  BPX  =  Z’ 3 ‘  =  position vector to the manipulator tip (payload) w.r.t. the inertial frame /33  Y-cornponent of 3 =  X-component of the specified base (joint 1) position vector in the inertial frame  PX  =  =  X-component of the base (joint 1) position vector response in the inertial frame  BPY  =  =  Y-component of the specified base (joint 1) position vector in the inertial frame  PY  =  py  =  Y-component of the base (joint 1) position vector response in the inertial frame  DLi  (j)  =  jj 6  =  generalized co-ordinate associated with the mode 35  j  of the link i.  3.3  Numerical Data used in the Simulation  As a first step many simulation runs were carried out with different parameters and under vary ing slewing conditions until an average parametric configuration was established about which certain parameters can be varied so that the differences in the resultant response characteristics may be observed. In the following simulation. the physical system that is modelled is constructed from 0.O5mx 0.05m x 5.Om long square aluminum links. When not specified explicitly, the link cross-section and mass distribution are considered uniform, with the joints assumed massless. The payload is considered as a point mass. The following parameter values represent the nominal configuration mentioned above:  1 (El)  =  4 3.0 x i0  [iv  2 (El)  =  3.0 x  [N  =  20.0  [Kg]  =  total mass of link 1  =  20.0  [Kg]  =  total mass of link 2  .  ] = 2 7fl  flexural stiffness of link 1  ] = 2 in  flexural stiffness of link 2  11  =5.0  [in)  =  total lenght of link 1  12  =5.0  [in]  =  total lenght of link 2  mB  =  00.0  [Kg]  =  total mass of the manipulator base  =  00.0  [Kg]  =  total mass of the manipulator payload  =  stiffness of the base in the x-direction  =  stiffness of the base in the y-direction  [N.,i j rod  =  stiffness of joint 1  rod  =  stiffness of joint 2  =  damping constant in the x-direction  [N  k  4 =1.0x10  k  4 =1.0x10  1 K  =  2 k  4 =1.0x10  , 2 C  =1.0  [N  , 3 C  =1.0  I7  =  damping constant in the y-direction  1 c  4 =1.0x10  lrad/s  =  damping constant of joint I  2 C  4 =1.0x10  =  damping constant of joint 2.  4 1.0 x i0  I [N  I  [  I ?fl/S IN  [N.m I rod/s  36  g  =  gravitational acceleration.  starting specified inertial X-coordinate of slew trajectory at the manipulator tip, 6.9m  0 (i) X 0 (i) Y  0.0  =  starting specified inertial Y-coordinate of slew trajectory at the manipulator tip, 6.9m  0 X  (f)=  final specified inertial X-coordinate of slew trajectory at the manipulator tip, 6.9iii  Yo  (f)  =  final specified inertial Y-coordinate of slew trajectory at the manipulator tip, 6.9m  8 T  =  total specified excursion time of slew trajectory at a constant velocity  Nm  =  number of modes used in the simulation to estimate flexural dynamics of the links, 2  In the initial simulation runs, most of these values were kept constant. The parameters with different values are identified in the captions under the plots.  Simulation Results and Analysis: Tip Trajectory Tracking  3.4  For a planned system simulation with a wide variation in parameter values and initial conditions would result in, literally, enormous amount of information. Therefore, a more concise yet rational approach is adopted. Only the typical results useful in establishing trends are presented here. The following group of simulation response plots are intended to establish characteristics of the time dependent variables in terms of the manipulator tip and joint trajectory tracking for the basic parametric configuration mentioned above. Information concerning the effect of the number of modes used in the simulation model is also explored. In the following example, the manipulator payload, i.e. the end effector (tip of link 2), is required to follow a prescribed trajectory. The trajectory is taken to be a straight line and the payload as well as the base mass are considered zero. The tip starts at the global coordinate [Xo (i) Yo (i) 0] and is required to follow a straight line at a constant velocity to reach the global coordinates [X 0 (f) . 0 V  (f)  0 J in precisely  is required to be stationary at the origin. 37  seconds. The manipulator base  Kinematicaliy, the global slewing coordinates of the manipulator are calculated in prescribed increments of the distance, at each time-step, along the slew trajectory, in global coordinates. Using the inverse kinematic equations of the manipulator, the joint angles are calculated at each time-step and are used to provide the joint control torques,  Q, Q, Qe 1  and  , 2 Q  via  the following PD type control equations: =  (/3—p)k+ (&_i3)c  =  (/3—p],)k+ (/v—Jy)’  1 Qe  =  (i3  2 Qe  =  _Ô (i ) c — (i3 ) 2 + k 6 2  —  1 + 9)k  (  —  6]) c : 1  The torques are then used to drive the system equations which, when solved numerically, yield generalized coordinate solutions for the time interval in question. For each simulation run, there are 4 frames within each group. and they are identified by the suffix letters (A). (B), ((‘) and (D). The data used in the first simulation run are specified in Figures 3.6, 3.7. 3.8. Figure numbers with suffix letter (A) represent time strobed figures of the manipulator at specific time intervals (Figure 3.6). The slew trajectory is shown as the solid line while the actual path of the end tip is indicated by the dashed line. The simulation time interval T 3 is set to 4 seconds so that the lead or lag of the tip relative to the desired trajectory can be clearly observed. Figure numbers with suffix letter (B) represent the slew trajectory superimposed onto the actual end-tip path vs time, but separated into the X and Y components in global coordinates as shown in Figure 3.7. Figure numbers with suffix (C) represent the slew joint angles superimposed onto the actual joint angle trajectories vs time, plotted for each joint. The slew angles are calculated from the inverse kinematic relations mentioned before which in turn are based on the required slew trajectory of the manipulator tip in global coordinates. 38  The top two plots of figure numbers with suffix (D) represent the slew trajectory superim posed onto the actual path or motion of the base but separated into the X and Y components in global coordinates as indicated on the plots (Figure 3.9). The bottom two plots of the same figure represents the generalised modal coordinates vs time. Each graph represents the super position of the same modal coordinate from each link, so that the dynamic coupling may be easier to observe. This first run can be perceived as basic test of the dynamic response of the flexible 2 link manipulator without payload. The traverse time from start to finish, from [6.9, 6.9, 0.0] to [1.0, 1.0. 0.0] is 4 seconds at a constant velocity. There is no ramping of speed at either ends leading to lag at the start and overshoot at the end (Figure 3.6). This is mostly due to the phase mismatching between the two joint angles. Note. the maximum deviation of the tip occurs at the start of the slew maneuver. This is due to the fact that the joint 2 slew rate is twice that of joint 1. Link 2 also has an advantage of greater effective inertial leverage over link 1. The manipulator tip then settles into the slew trajectory with progressively smal]er oscillations about the norm. This is due to the viscous damping at the joints and the base. Since the slew rate is relatively small, the overshoot and excursion at the end are essential]y negligible. The starting excursion of the tip can be more clearly seen in Figure 3.7. Figure 3.8 shows that eventhough the manipulator tip may deviate from the required tra jectory, the joint angles follow the commanded profile quite closely. This is the first hint of the problem involved in controlling the tip by judicious application of the joint torques. In the next two sets of simulation runs all conditions are maintained the same as before but the traverse period is reduced to 2 and 1 second respectively (Figures 3.10, 3.11, 3.12, 3.13 and Figures 3.14, 3.15, 3.16, 3.17). As expected, an increase in the slew velocity by two fold increases the tip excursions significantly. With the I second traverse period the tip excursions become very large. Even the joint angles are starting to show significant departure from the calculated slewing angles. Obviously, a sophisticated control strategy would be required to maintain the desired trajectory. 39  0 0 ‘—4  0 C  0  0  0 -Co  0  I 0  0 CO  I  0 C\2  I 0 CD  ‘-4  C?I 0  Figure 3.6: (A): Time strobe position plot of 2 link flexible manipulator with stationary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in 4s. 40  C -  I 7  -  // F I  I  C  C  a  /7  -  o C  I  I  Q  CD  ó  I C1  0  -  4 I  / / I /2  I  I  o  o  o  1’  -  I  I  I  0 CD  0  0  C  0 0  Figure 3.7: (B): Comparison between specified and and actual (p and p3k) position of the manipulator tip during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in 4s. 41  0  C-) CD  -  -  -  Cs2  I,  I  ‘  / /  II I I II  -  C2  1/ /  cz -  -  _CD 0  ‘4  r  0  I  I  o  o  1  c\  —  0  a  a  CD  c I  d I  -c  \\  -  -  ci.  \\\ SE C%  C%  ‘  I I I  a  I I I I  -  0  I 0  I  C’  I  I  CD  CD  —  a  —  ó  0  a  Figure 3.8: (C): Comparison between specified (3 and /32) and actual (9 and 82) position of the joint ang’es during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in 4s. 42  .  .-.  Cl)  0-  -S  —.-.  (DO  (D  clq  —  j. 0-  (D  .  O  -$  ;  o o,o-S  —0.16  —0.08  0.00  (xHF’  0.08  0.16  0.24  —0.08  —0.04  0.00  0.04  0.08  (D  •  ,—  q  (D  —  0.12  (D  O—. (D  o  0.0 0.8  0.8  I  I  1.6 2.4 TIME . s  1.6 2.4 TIME , s  I  ,mI  4.0  I  3.2  4.0  6221J  612  3.2  I  AAñAflARA FJy\j\,/yyyyyyyl  0.0  -  -  -  -  0.0  -  -  -  -  —0.10  —0.05  0.00  0.05  0.0  —  0.10-  0.15-  —0.08  —0.04  0.00  0.04  0.08  0.12  0.8  I  0.8  I  I  I  I  2.4 1.8 TJME,s  .  ,m] 2 ã 1  2.4 1.6 TIME,s  I  3.2  I  3.2  I 4.0  4.0  An A A. v’Ivv 4 ‘w.v.i’1y  3.90  C  a a a CD  a C\2  a a C C  0 CD  0 CD  0  C C  SI  Figure 3.10: (A): Time strobe position plot of 2 link flexible manipulator with stationary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in 2s. 44  C  SE  rd-S  E0  0 0 C  C 0  q CD  0 C  CD  0 0  SE 4 4’  Cc? ‘I  0 0 0  o o  o CD  0 CD  0  0  0 0  1  Figure 3.11: (B): Comparison between specified (/ p3x and /3,,,3) and actual (p and p3k) position 3 of the manipulator tip during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in 2s. 45  0  0  ‘-1  ----aq  cj  (_)  —  (0  (0  (0  (D,  0  (0  U)  p clq —  I1.  ---  (0  c..  (0  ,  0  0.0  0.0  0.4  0.4  2  0.8 1.2 TIME , s  0.8 1.2 TIME,s  m ----fl . 2  1.6  2.0  0.0  0.0  0.4  0.4  0.8 1.2 TIME,s  0.8 1.2 TIME.s  1.6  2.0  ----fl m 71 O ,m  3.12C  -1  —  ;•  CD  C)  CD  II  a)  CDCD  CD  o  r)  CD  CD  -f  CD CD  -f  (DCD  I,  CD  clq  cn  CD  —.  -  -  -  —0.16  —0.08  0.00  :  0.0  -  -  ) 1 (x10  0.08  0.16  0.24  I  0.4  I  I\ !r I  1.2 0.8 TIME,s  I  .  I  0.00  I  2.0 1.6  —0.16  —0.08  I  YYYYr  0.08  0.16  0.24  0.0  0.4  0.8 1.2 TIME,s  3.13D  1.6  2.0  0 0  0  0 C\2 0 C\2  0 Co  0 0  o Co  p  p  p  01  C\2  CD  0  Figure 3.14: (A): Time strobe position plot of 2 link flexible manipulator with stationary base, using two modes for each link, during a straightline tip maneuver from’ the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in Is. 48  Q if  I / I  _CQc  /  F  _4  -  c I  o o  C  I  I  I  o  o  o  o C  1”  / —  I  -  OE  0  I$  0  Figure 3.15: (B): the (1, 1, O)m O)mn of  I  I  I  I  I  o o  o  0 CD  0  0 C  Comparison between specified  manipulator tip  to  in  during  a  straightline  S and 3 (fi  tip maneuver from  is.  49  0 0  and actual the inertial  (p3  and  p3) position  coordinate  (6.9, 6.9,  C C-)  (0  1’ ,44  C 0 CO  0 E  A  C  I  I  I  I  I  C  C  C 0  0  C C CO 0  ‘-4  0 CO 0-  Cs  Cs  C2 0 0 0 ‘-4  0  0  Figure 3.16: (C): Comparison between specified (3 and /32) and actual (6 and 62) position of the joint angles during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, 0)m to (1, 1, 0)m in is. 50  N  C  q  1-4  ‘-4  0  C  N 1’  S  o  0  *  0  0 N (.  C  C I  I  I  I  0  0  0  0  N  N 0  —  0  0  0 0  o  0  —  I  I  0  0  -  C  p31) 0  C  N 0  N 0  I C 0  I  I  p3  p3 0 0  r4  0  I  I  —  o  I  I  4  0  I  01 0  I  I  I  I  0 0  q  0  —-—  c 0  0  0  I  I  oi  o  0  d  0 oz 0  Figure 3.17: (D): Comparison between specified (j3 and /3) and actual (p and py) position of the joint angles as well as the flexible generalized coordinated during a straightline tip maneuver from the inertial coordinate (6.9, 6.9, O)m to (1, 1, O)m in is. 51  Keeping all the parameters constant, the above simulations were repeated but with the direction of the translational maneuver reversed: the manipulator tip moving from (1.0.1.0) to (6.9, 6.9)m. The traverse time is 4 seconds. Note, the manipulator tip trajectory signature is quite different (Figures 3.18, 3.19, 3.20, and 3.21). The main feature appears to be the dominance of low frequency excursions which are further emphasized in Figures 3.23, 3.24, and 3.25 where the maneuver time is reduced to 2s. Again it is evident that a more sophisticated control strategy is required to achieve the desired performance. Response results were also obtained by systematically varying the trajectory length, orien tation, speed and direction. In general, they showed similar trends with the main conclusion as above. With an increasing speed of maneuvers, a sophisticated control procedure will 1w required to track the trajectory precisely.  52  C C  C CD  C C\2 C C\2 C’,  C CD  C C C C  C CD  ci C\2  C\2  CD  I  I  C  Figure 3.18: (A): Time strobe position plot of 2 link flexible manipulator with stationary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (1, 1, O)m to (6.9, 6.9, O)m in 4s. 53  C 6  a)  \  -  \  -‘-  1J1  C,,  ‘S  c’i  E E l4  -ó C  I  I  I  I  I  o  o  0  0  0 C2  o  -  0 0  ‘-4  Figure 3.19: (B): Comparison between specified (3 and /37,3k) and actual (P3 and p3k) position of the manipulator tip during a straightline tip maneuver from the inertial coordinate (1. 1, O)m to (6.9, 6.9, O)m in 4s. 54  C 0  \  _o C  -a d  o  o  o C  .7  7  -  I  U  /  / /  U 4 _  /  /  / /  .1  J  /  -à  /  I 0  C\2  I CO  ci  .1  0  I  —  0 0  0  Figure 3.20: (C): Comparison between specified and p82) and actual (6 and 62) position of the joint angles during a straightline tip maneuver from the inertial coordinate (1, 1, 0)m to (6.9, 6.9, 0)m in 4s. 55  Cl)  —  -  ,  .  -.--  0(D  -$  Cl)  cJq  -‘  CDCl  OH  N 0 CD  O  ,—‘  C)  CD  +  ‘•—  o  O  Ci)  CD  -  —  (D  O  0.8 1.6 2.4 TIME,s 3.2 4.0  0.0 0.8  1.6 2.4 TIME,s  3.2  4.0  —0.12  —0.07  —0.07  —0.12  —0.02  0.03  0.08  0.13  —0.02  (x  0.0  —0.032  —0.022  —0.01  —0.02  —0.012  0.0  —0.002  0.008  0.02 0.01  0.018  0.03  0.0  0.0  0.8  0.8  1.6 2.4 ThAE.s  1.6 2.4 TJME.s  3.2  3.2  4.0  4.0  __  3.21D  ___  0  a a  ‘—4  a cc a CQ  a -cc a I  I  I  I  I  a a  a cc  a  a  a cc  I  I  -  ‘-4  a  Figure 3.22: (A): Time strobe position plot of 2 link flexible manipulator with stationary base, using two modes for each link, during a straightline tip maneuver from the inertial coordinate (1, 1, O)rn to (6.9, 6.9, O)m in 2s. 57  0 C2 rD C f)  ‘I  C  0  o o  o CD  C CD  C  C CQ  C 0 0 C’2  4’ l’  a 0 0 0  a  0  0 CD  0  0 C’2  0 0  ‘-4  Figure 3.23: (B): Comparison between specified (3 and f pi,) and actual (p and p3k) position 3 of the manipulator tip during a straightline tip maneuver from the inertial coordinate (1, 1, O)m to (6.9, 6.9, O)m in 2s. 58  C-) CN  r’) —  —‘-4  \  Eq  aD  -CE  cx;::  C’2  I  I  a  a  -à  I C  I  ‘0  -ci  ..  ®  /  I  7 —  ‘—4  A’  I I  CQ  / /  u2  —CE  4.  I  —a  I I  /  I  I  Cl,  ‘-  -o  I  0 0  C 0  Figure 3.24: (C): Comparison between specified (/3 and /32) and actual (8 and 82) position of the joint angles during a straightline tip maneuver from the inertial coordinate (1, 1, O)rn to (6.9, 6.9, O)m in 2s. 59  CD  .,  CD  CD CD  -  CD  ci,  ,—  -  p  CD  —,  .  0-  ci,  0..  ‘—  oq  0 CD  °rD  “-s  cq  0-’  o o  —  CD  o  (x  0.0 0.4 0.8 1.2 TIME,s 1.6 2.0  0.0 0.4  0.8 1.2 TIME . s  1.6  2.0  —0.04  —0.02  —0.24  —0.14  —0.04  0.06  0.16  0.26  —0.06  —0.02  0.0  —0.04  0.0  0.02  0.04  0.02  0.04  0.06  0.0  0.0  0.4  0.4  0.8 1.2 TIME,s  0.8 1.2 TJME,s  2.0  1.6  2.0  621,mI  1.6  ,mJ  3.250  Chapter 4  CONTROLLED DYNAMICAL RESPONSE  4.1  Preliminary Remarks  A relatively general formulation for studying the behaviour of a mobile two link manipulator was presented earlier in Chapter 2. Its dynamical response characteristics were studied during  tracking of a straight line trajectory at predetermined speeds using the classical colocated proportional-derivative (PD) linear feedback technique applied to the rigid body generalized coordinates. The results suggested a need for a more effective control procedure that would account for the system nonlinearity and coupling better. The next logical step would be to implement a nonlinear control on the system. One approach to the problem would be application of the sliding mode proceedure [23] If the right hand side of a differential equation is discontinuous around a hypersurface and if the trajectory of the solution points toward the discontinuity, then it is reasonable that the trajectory eventually slides along the hypersurface. Control laws can be derived that will force the manipulator to follow a specified trajectory defined by these surfaces. This is accomplished by an optimal choice of sliding surfaces. A major problem with this approach pertains to unmodelled dynamics which usually results in high frequency oscillations of the manipulator as it slides along the surface. This is a direct consequence of the switching nature of the controller. Furthermore, the manipulator joints are subjected to high reversing torque impulses which are required for this method to work properly. This is not always practical with many electro-mechanical actuators. Some researchers have improved the controller performance using a filtering process with a high bandwidth for the sliding variable [24] [25]. In an alternate approach, the sliding mode approach is incorporated into an adaptive PD feedback controller 61  leading to the zero velocity error. The sliding mode controller elliminates the nonzero position errors. Of course, as is often done, one may proceed to linearize the governing equations.This is accomplished by considering an incrementa] deviation of the state vector from its operating point at each sampled instant of time. The equations of motion are modified to reflect this change. Re-arranging, one has two sets of equations: • The first group resembles the original set of non-linear equations describing the large motion of the manipulator. • The second group of equations describe the incremental deviation of the state vector from the large motion at each sampled point in time. From the second group we get a linearized model by neglecting the second and higher degree terms of the delta state vector. This can be rearranged to form a set of state equations thereby obtaining the linearized system state matrix about a given operating point for flexible manipulators, control strategy based on linearized  system  139],  [40]. However.  nioclels has often proved  to be inadeciuate. Actual dynamics of the robot arms often deviated significantly from that predicted by a linearized approach. A solution, proposed by Freund [27], uses the state feedback to decouple the nonlinear system in such a way as to make arbitrary pole placement possible. However, this method is difficult to apply to systems with more than three degrees of freedom. Iii a subsequent modification, by first simplifying the equations of motion aiid then strategically partitioning them, Freund [28] showed that the method could be extended to systems with more than three degrees of freedom. A promising technique that may provide adequate control for both rigid as well as flexi ble manipulators is a form of inverse method based on the Feedback Linearizations Technique (FLT). It was first investigated by Beijczy [30]. Singh and Schy [31] as well as Spong and Vidyasagar [32] [33] used the FLT to formulate robust control proceedures for rigid manipula  tors. The method was later applied to the control of robots with flexible joints by Spong [29]. 62  Modi and Karray [34] [35] extended the FLT to include structural flexibility and applied it to the proposed space station based mobile manipulator modeled by Chan [5]. The FLT is sometimes referred to as the computed torque technique, which in effect is a special case of the FLT. This is because the technique feeds back the rigid body state vector to calculate the foreward rigid body dynamics of the system and uses the resultant generalized forces to drive the manipulator according to the required trajectory. The effect of this is to linearize the system such that only a linear compensator is required to obtain the required system output. There are certain advantages to this approach: the control algorithm is quite simple and the linear compensator is easy to implement. The proceedure is schematically described through a block diagram in Figure 4.26. As can be expected, the method has some limitations just as other control proceedures. It relies on the accurate knowledge of the system model and structural parameters for robustness in the rigid degrees of freedom. Furthermore, another disadvantage is, knowledge of the flexible generalized coordinates is necessary for determination of the control torques and forces. In practice, the problem can be readily overcome by introducing strain gauges to gain information concerning the first few modes.  4.2  Feedback Linearization  The Feedback Linearization Technique (FLT), as applied to a nonlinear rigid body system. is reviewed here. Consider a system governed by  D(r,t) +  where  Q  U  (kr,r,t)  (r,r,t),  represents the generalized coordinate vector for the rigid system and  (4.1)  (q,.,.,t) is  the nonlinear generalized force/torque control vector. Taking the control torque time history as:  63  +  H  I  0  1  CD  —0-  q  (D  II.  HZ  0  Linear PD Control j Applied to the Rigid Degrees of Freedom  Desired Trajectories  r  Inverse Dynamics Based On Non Linear Feedback Control  Q  ->  Moving Flexible Two Link Ground Based Manipulator Dynamics  qqq  Output State Variables  = + K (ir  where  ,  and  J,.  (4.2)  ; —  +K  (  —  r) ;  (4.3)  denote the trajectory related desired acceleration, velocity and displace  ment, respectively, leads to the linear closed ioop control equation as  (4.4)  This can be rewritten as,  €±K±K= 0, where  =  —  .  ë.=  L  .. —  and ë=  ,. —  .  (4.5)  are the acceleration, velocity, and displacement  errors, respectively. Note, the error is driven towards zero in an asymptotic manner given the correct choices for the position and velocity gain matrices. If we choose tO decouple the system such that each individual degree of freedom can be tuned separately then a suitable choice would a diagonal matrix. The feedback gain matrix K will have diagonal elements equal to  .  This is directly proportional to the speed of response and  will thus determine the response time of the generalized coordinate qj. Similarly, the feedback gain matrix K may have diagonal elements equal to 2w, which will make the jth generalized coordinate response critically damped. In general, therefore, the larger the value of w 3 the more robust will be the response.  4.3  Control Implementation  hnplementation of the FLT begins with the appropriately partitioned system mass matrix D. The equations of motion are arranged into the groups of rigid and flexible degrees of freedom, 65  the former related to Px’ coordinates  jj 6  p, 6 i,  6 generalized coordinates while the latter to the flexible  , t). Rearranging these into a matrix format gives 1 (x  Dif  Fr  -IDif  Dfr where: Drr  ()  •..  =  ...  (4.6)  PI  qf  is the 4 x 4 system mass submatrix representing base translational and joint  rotational degrees of freedom: Drf(rif) is a  4  x 2m submatrix that represents the coupling  between the rigid and the flexible generalized coordinates; Dfr = D is a 2m x 4 submatrix: and Dif  (flj)  is a 2m x 2m submatrix that represents only the flexible degrees of freedom.  Here in is the number of modes used to represent the link deflections. F,.  and  are 4 x 1 and 2in x I vectors representing the remaining coupled nonlinear  j  forcing terms.  ,.  and  j  are the control forces supplied by the actuators. In general it would  be impractical to control the modal degrees of freedom associated with the manipulator links. In that case  Q  is set to zero. Thus  is required to control the rigid degrees of freedom  directly and the flexible degrees of freedom indirectly. To design the controller, the next step is to rewrite the submatrices in equation (4.6) in such a way as to determine  aid to linearize the closed-loop system. This can be accomplished  qiite readily as follows. Putting  j  = 0, equation (4.6) becomes:  Drr.+Dri+r  =  Dfr + Dffj + Ff  =  Qr; =0.  (4.7) (4.8)  From Eq. (4.8),  = —D D 1 66  —  1f. 1 D  (4.9)  Substituting from Eq. (4.9) into Eq. (4.7) gives  D1c) +  Drrr + Drf (_D1Dfrr  (Drr  —  DrfD’Dfr)  +  (Pr  —  =  DrfD1f) =  (4.10)  .  Putting:  D  =  DrrDrfD’Dfr  (4.11)  =  Pr—DrfDf;  (4.12)  gives  (4.13)  D+FQr.  This represents the rigid body generalized force vector similar to the original differential equa tions of motion and is a function of all the generalized coordinates. Since it is a complete equation. it includes the effects of all the generalized coordinates. Applying the FLT Eq. (4.13) gives the control effort as  =  (,j,t)+  ,  (4.14)  where:  (4.15) =  r+Kv(L  =  —D’D  =  &r;  (4.18)  = ji.—.  (4.19)  r)+lCp($rr) ; —  67  (4.16) (4.17)  The control effort thus becomes  =  D  r+  (4.20)  + D(Kvr+Kpr).  Note, the control effort consists of two components:  Qr,pd  =  D13r+F ;  (4.21)  D(Kvër+Kpë’r).  (4.22)  The first one is the nonlinear feedback component  Qr.ni.  It. cancels the effects of the nonlinear  torque/force disturbances on the joint and base actuators resulting from the system dynamics. The effectiveness of this part of the controller is strictly dependent on the closeness of the dynamical equations to the actual manipulator dynamics being modeled. The second the linear proportional/derivative feedback component  respect to the errors E,. and  .  ‘,pd•  one  is  It ensures robust behaviour with  which may arise as a result of imprecise modeling of the actual  manipulator dynamics. Of course, in practice this is always the case and the linear controller is therefore always included to improve performance. In order to determine the control effort, evaluation of the FLT variable arrays D and F is necessary, which in turn requires the knowledge of  and  j.  With this in mind, two different  proceedures, Quasi-Open Loop Control (QOLC) and Quasi-Closed Loop Control (QCLC), are suggested by Modi et ai. [34].  4.3.1  Quasi-Open Loop Control  If the situation is such that most or all of the generalized variables are not amenable to direct measurement, then these variables must be determined using the nearest approximation to the real system being controlled. In this scheme, the central idea is to evaluate the generalized modal coordinates in an off-line parallel integration procedure, i.e. the dynamics of 68  4 is computed  without the use of Q.. However, öj is governed by the specified trajectory,  •8  /r  and the  dynamics may be obtained from Eq. (4.17):  =  L  =  (4.23)  ;  D (/‘r. f )Dfr(/r, qf)v  —  D ( 1 /r, f )Fç(/3r,/3r, qf, )  (4.24)  .  Solution to Eq. (4.24) results in two rewards. Firstly, it allows the observation of the ap proxi?nated behaviour of the modal coordinates and its derivatives, i.e. j, j, and j, off-line without the need for direct measurement. Furthermore, it allows for the off-line computation of the control effort 7 r for the rigid degrees of freedom. Figure 4.27 presents a block diagram for the Quasi-Open Loop Control scheme. The procedure would demand considerable amount of computational effort to solve the equations. Also, the method does not fully linearize the control system since it utilizes estimated values. 4.3.2  Quasi-Closed Loop Control  On the other hand. if the situation is such that most of the generalized coordinates are available for measurement than this approach is probably a better one. Here, the rigid and flexible degrees of freedom are both utilized and calculated simultaneously at each control interval, as shown by the following equations: (4.25) qf(qf)  =  Note, Ff is now a function of  .  and  (4.26)  rather than 3,. and /r, and that the control action  of the PD controller is now on a linearized system. This is true if the system model is exact but in practice this is never the case. Robust behaviour, about the desired trajectoy, of the contolled output is mostly dependent on the linear controller part of Eq. (4.16) and the arrays  K and K. This approach is represented in a block diagram form in Figure 4.28. A bit more 69  0  S  CD  C-  CD  0  CD  0  0  0 0  CD  Cl)  CD  0  -1  CD  Manipulator Dynamics  Offline Computational Estimation of Flexible Modal Dynamics  Nonlinear Controller  PD Controller  computation is involved here than in the previous scheme but this disadvantage is more than offset by the increased precision and robustness in the controlled degrees of freedom. Also the QCLC is less sensitive to system uncertainties than the QOLC.  Application of the Quasi-Closed Loop Control to the Mobile Flexible Two-link  4.4  Manipulator The implementation of the QOLC as opposed to the QCLC using the existing program is radically different. The QOLC requires major modifications to the program since it involves a secondary double integration scheme. On the other hand, the QCLC involves minimal additions and revisions. Now the program remains much the same as before but with the addition of a modified control subroutine called FRCFC2, which calculates the required control effort based on the desired trajectory generated by the subroutine INVKB2. The main purpose of FRCFC2 is to rearrange and then partition the mass matrix D, to evaluate D and the control effort  a,..  , and then  calculate  Since the rigid degrees of freedom can be made to follow the desired  trajectories precisely, the main interest now is to track the flexible degrees of freedom trajectory with accuracy. The present study is limited to the QCLC of a moving, ground based, flexible, two-link manipul.ator. 4.4.1  Stationary Manipulator in Zero Gravity Field Without Payload  All the physical parameters and dynamic constants of the manipulator are kept essentially the same as in the earlier study to facilitate a meaningful comparison between the two control schemes. The joint stiffnesses are purposely reduced here by an order of magnitude to illustrate the effects of a highly flexible joint drive system. The duration of the maneuver is in the range 1.0  -  2.0 seconds. Numerical values used in the simulation are listed below:  c  =  1000 N/rn/s  c  =  1000 N/rn/s  =  1000 N-m/rad/s 71  I  0  0 0  Ci)  0  Ci)  C  Nonlinear Controller  PD Controller Manipulator Dynamics  1000 N.m/Tad/s 1 (El)  =  30000 N-rn 2  2 (El)  =  30000 N-rn 2  =  1000 N/rn  =  1000 N/rn  =  1000 N-m/rad  2 k  =  1000 N-rn/rad  ‘1  =  5m  ‘2  =  5m  =  20 Kg  =  20 Kg  =  0  7722  mp  0  Note, the gravitational field is purposely set to zero initially to observe the system dynamics without any outside influence. The joint, base, and link flexural stiffnesses are set fairly low in order to emphasize their effects. In a practical system. the above parameters  may  correspond  to a manipulator constructed from standard .05m x .05m x 5m square solid aluminum bar sections. Only the first two cantilever modes are used to save simulation time and cost. A preliminary study showed that the use of additional modes does not appreciably change the payload tracking characteristics. In this set of simulations results (Figures 4.29% 4.30, and 4.31), the base is held stationary and the payload is required to follow a rather complex trajectory described by the solid line curve in Figure 4.29. The actual trajectory traced by the payload is indicated by the dash line. The frozen positions of the links at successive time-steps are also included (time-strobe overlays). As shown in Figure 4.29, the manipulator starts the trajectory from the positive X axis. It is appearant that the PD control is rather inadequate. The tracking errors are indeed  quite large. This can be seen, more clearly, from Figures 4.30 and 4.31 where p, p3y, 73  ,  and  62 are plotted against their desired trajectory counterpart. There is as much as 6 meters of following error between p and  ‘ P 3 3x  4 meters between P3y and p,; 0.5 radians between  and  8; and 0.3 radians between 132 and 62. The combined effect of these leads to a large l)hase lag between the desired and actual payload load positions as observed in Figure 4.29. Having observed the dynamic response of the manipulator to sinusoidal trajectory tracking in the presence of the PD control, the next. step was to assess effectiveness of the FLT. This is shown in Figures 4.32, 4.33, and 4.34. Note, there is a major improvement in the tracking behaviour of the manipulator tip. The maximum deviation from the desired trajectory is about 0.5 meter. It occurs at the beginning and at the end of the travel where the peak acceleration takes place. Of course. these perturbations are primarily due to the flexible character of the cantilever links. This can be verified by analyzing Figures 4.33 and 4.34. Plots of 6, 82. p . 3 and p vs time clearly show that there is no detectable deviation from the desired trajectories, suggesting that the only source of the discrepancy must be the flexible degrees of freedom over which there is no direct control at this level.  4.4.2  Mobile Manipulator in Zero Gravity Field Without Payload  Figures 4.35, 4.36, and 4.37 show the tracking characteristics of the manipulator under the PD control with the base now free to move. All other parameters have the same values as before. The base is made to follow a sinusoidal desired trajectory of amplitude 5 meters in both the -X and +Y directions at a driving frequency of 1.0 radians/second. The first thing to note is that the tracking error is not as great as when the base was stationary under the same control scheme. Furthermore, the tip phase lag is considerably less. This suggests that the nature of the base motion will have considerable effect on the tip tracking efficiency. Figures 4.38, 4.39, and 4.40 show the tracking characteristics of the mobile manipulator under the FLT control. 74  0 0  Q  CD  0 C\2  S  C\2  S.’  Co  0 0 0 CD  0  0 C\2  0 CD  I  I  C C  Figure 4.29: (A) Time history response of the manipulator showing the tip and the link postions in the presence of PD control with the base held fixed 75  .  —,  C)  cq  0.8  1.4  2.2  3.0  00  2.0  6.0  8.0  100  /  0.4  -‘  I 0.4  7,  /  I,  I,  0.0  -  -  0.0  —  -  -  -  ,  ‘S 5,  -  \  Jx ,m  I J 1.6 2.0  ‘•,  -  SS  ‘S  \ S  1.6  ‘S S  \  \\  \  \\  ,m  2.0  02 ,m  2  S__________  S  0.8 1.2 s 1 TIME  TIME  0.8  ‘S  I 1.2  ‘S  -  0.0  -  -  -  0.0  —  —  -  -  ::::  —0.8  —0.4  0.0  0.4  —12.0  —8.0  —  0.0  4.0  80  ‘S  ‘\ ‘S  S  \  ‘5’  S’,  ‘\  -  1.6  /  °1  I 1.6  “-S  -  ----  1.2  \  s  I 1.2  S  ,  5S  \  TIMERs  0.8  i”  ‘.  \  S  ‘.  TIME  I 0.8  ‘S S  ‘\  0.4  “S  I 0.4  ‘S  -  2.0  ,mI  ,m]  I 2.0  _____ _____ _____ _________ _____ __________ _____ __ __  4.30B  -.4  0.4 0.8 1.2 TIME • S 1.6  2.0  0-’  0-  ;• CD  0.0 0.4  0.8 1.2 TIME • s  1.6  2.0  —0.2  —0.16  0.2  0.4  0.6  0.8  0.0  0.00  612 m 1522 ,m  —0.08  (x  0.08  0.16  0.24  —0.04  —0.15 0.0  0.0  —0.10  (DO  —  0.04  —0.05  (D  Cl)  0.12  0.16  0.06  0.05  0.10  0.0  (DC)  CJD  Tj  0.0  0.0  ,m  0.8 1.2 TIME • S  0.4  0.8 1.2 s 1 TIME  621 ,m  c5,,  0.4  1.6  1.6  2.0  2.0  4.31C  0 0 ‘I  0  S.’  C C\2  S C)  C Co  b  C CD  CQ  C C\2  0 CD  I  I  CI C  Figure 4.32: (A) Time history response of the manipulator showing the tip and the link positions in the presence of FLT control with the base held fixed  78  (0  -.,  I  -  0-  ,  ,;-  Ca  -  c;;.  tth’  -  U  Tj  ,  \  2.0  / 0.8  1.2  1.6  2.0  —1.6  —0.8  —0.4  —1.6 0.4  \  0.0  —1.2  0.0  -  2 fl  0.4  —0.8  0.0  0.8  1.6  2.4  —12.0  —4.0 1.6  —8.0  0.0  0.8 1.2 TIME s  —4.0  4.0  4.0  0.4  7?zj  0.0  0.0  I  F,mI  8.0  12.0  16.0  0.0  0.0  0.4  0.4  0.8  1.2  0.8 1.2 TIME,s  2.0  ,rn  ,m  1.6  2.0  9, ,m  fl  1.6  3y  ----a  _____  4.34B  00  0  C,,  I’)  9-I’)  -  0  C.,—  0-  CD  0q  (DO  -  C., oi  CD  C,,  (Do  +  CD  —0.06  —0.04  —0.02  0.0  0.02  0.04  —0.05  —0.04  —0.03  —0.02  —0.01  0.0  0.0  0.0  -  -  -  -  0.4  0.4  I I  0.8 1.2 TIME,s  0.8 1.2 TIMEs  I  1.6  1.6  2.0  2.0  .mI  —0.8  —0.4  0.0  0.4  0.8  1.2  0.0  0.01  0.02  0.03  0.04  0.05  0.0  0.0  0.4  0.4  0.8 1.2 TIME.s  1.6  2.0  1.6  2.0  ,m -1 6 -1  0.8 1.2 TIME,s  ----m  4.35C  S  E  C C  cD C cc  C C\2  C C\2  C Co  C  C  C  C  cc  C  C  C  C\2  C  Figure 4.35: (A) Time history response of the manipulator showing the tip and the link positions in the presence of PD control with the base translating as shown.  81  (.3  U)  (V  (‘3  C.,  0  -  ‘1 (0  11  U)  0-,  r)  —1.0  —0.5  0.0  0.5  1.0  1.5  8.0  12.0  fi  0.0  —  -  -  -  -  I  I  I  0.8 1.2 TIME,s  I  ,e 2 m  2 , m  0.4  ly’  014  *\  ‘\  “S  \  1.6  \  \\\  2.0  I  “S.  \  m ?x 1  ----n.m  —1.6  —1.2  —0.8  —0.4  0.0  0.4  —6.0  —4.0  —2.0  0.0  2.0  4.0  0.0  —  —  -  0.0  0.4  I  0.4  I  --  0.8 1.2 TJME,s  I  -  0.8 1.2 TIME,s  .-  t-_  1.6  I  -----  m  2.0  2.0  I  --  ,m  3y  1 e  1.6  ____  4.388  .-  _4,  0  -1  -  Cl)  9:  -  —  C,  DCD  q  (0  -  -..)  I’-  I I I I I  0.4  0.6  0.8  —0.2 2.0  2.0  —0.15 1.6  1.6  0.0  0.8 1.2 TIME.s  0.8 1.2 TIME • s  —0.10  0.4  0.4  0.0  2.0  0.2  0.0  0.0  -  —0.05  ) 1 (xlO  0.0  0.05  0.10  —6.0  —4.0  4.0  -  —2.0  8.0  10.0  6.0  -  -  0.0  2.0  4.0  0.0  —  -  -  -  -  // 0.4  I  ,/7  0.4  62,  ----6,1  0.0  -  -  -  -  I  —-  0.8 1.2 TIME.s  j___  “  0.8 1.2 TIME •  1.6  N  1.6  2.0  2.0  4.39C  As before, considerable improvement in the tracking efficiency of the manipulator is ap parent. The rigid degrees of freedom have virtually no detectable tracking error as observed earlier. One point of interest is the vibrational characteristics of the first two flexible degrees of freedom of the modal coordinates. Comparing Figures 4.31, 4.37 with Figures 4.34, 4.40. considerable difference in the frequency content of the wavefonus is noticeable. With the PD control the high frequency oscillations of the first and second modal coordinates fade rapidly leaving only the deflection brought about by the nominal motions. On the other hand, with the FLT control, the high frequency contributions dominate the response. This situation would be greatly improved by structural damping. The set of Figures 4.41. 4.42, and 4.43 as well as Figures 4.44, 4.45, and 4.46 show respons of the system as it performs a full three quadrant slewing maneuver. The intent here is to explore a different maneuver profile while keeping the system parameters unchanged.  The  desired trajectories are chosen to best observe the effects of a constant gravitational field as well as the inclusion of a payload at. the tip of the manipulator. as described in the following section. Figure 4.41 shows the inadequacy of the simple PD controller in the presentce of relatively low joint stiffiiesses. However, the FLT controller proves its superiority even though the spring and damping constants are left unchanged (Figure 4.44). Note that the tip follows the required trajectory quite closely save for the deviations that occur due to the link vibrations, which cannot be directly controlled. Furthermore, a large torque required for the maneuver through three quadrants completed in 2s adds to the discrepancy. Again the slow damping oscillatory motion of the first and second cantilever modes is noted in Figure 4.46, which appears to be assosciated with the FLT control. Consider the fundamental cantilever mode of both the links. Since the coefficient of stiffness and the mass per unit length are equal for both links, the natural frequencies are also equal. As a result both links are locked into synchronous oscillation but out of phase by 180 deg. This effect appears to be independent of the manipulator configuration but remains superimposed on the deflection created by the 84  0 0 C)  0 CD  o  0  o  0  cc  C\2  0 CD  I  I  Figure 4.38: (A) Time history response of the manipulator showing the tip and the link postions in the presence of FLT control with the base translating as shown 85  N  N  -I  -I  N  0  “a’  N  0  0  0  0  0 0  N  0  0  0  I  0  I  ‘-  I  ‘-I  I  0  0  N  ‘-4  ‘-4  N12  0  0  q 0 0 C’)  ‘-I  0  0 C’)  0 N  I  0 1-  I  0  0 N  0  ó  -  Figure 4.39: (B) Graphs of (from top left figure clockwise) / p & P3y, /3 & 3 / P3x & 3 & 82, VS time [seconds) for the moving flexible 2 link manipulator under FLT control  86  81, /32  —1  -.,  .+  ci  CD  -  o  t)  —  0  CD  —.-  CD s 1-.  CDO  —,  0  cl  —.%  CD  -  -  —  I-  .-  l)  0.4 0.8 1.2 TIME •  1.6 2.0  0.4  •  0.8 1.2 TIME  1.6  2.0  —0.4  —0.04  0.0  0.0  —0.02  —0.8  0.4  0.0  —0.06  0.8  1.2  0.02  0.04  0.0  —.0 0.0  1.0  2.0  —4.0  —3.0  3.0  4.0  —1.0 —2.0  6.0  0.0  kfL  0.0  -  -  0.0  0.4  I  A  0.4  1.6  .  I  1.2 0.5 TIME s  I  1.6  I  rN  0.5 1.2 TIME  2.0  I  2.0  S.m  4.43C  motions of the rigid degrees of freedom. These oscillations are a consequence of the fundamental mode coupling between the links. Second mode effects are not observed because they are small, at least by an order of magnitude. This suggests that to minimize modal response ol the links. they should be designed to avoid the frequency coupling. This can be achieved quite readily by selecting the flexural stiffness appropriately.  88  C  C 0 ‘I  C Co  C C\2  C 0 0  0 CO  0  ‘I  0 C\2  0  I  I  CI C  Figure 4.41: (A) Time history response of the manipulator showing the tip and the link positions in the presence of PD control with the base held fixed doing a full three quadrant slewing maneuver 89  CD  o  /V  c  1  -—  -  —  S  S  EE  C  -e  ‘5•__. S .•_S’  I  I  o  I  c  o  I  o 0  I 0  I  0  ‘N)  I  I  0  c  I  c  0 C  0  C  I  o /Ci  H  ¶  -  /  I  -  ‘.‘\  “\  _,_;  —  —  —I  —  “ ‘\  S 5  “1  /  -oE  C..’  /  H  I,  /  Figure 4.42: (B) Graphs of (from top left figure clockwise) /3 & p3,, & p3y, /3 & 9J /32 & 82, VS time [seconds] for the stationary flexible 2 link manipulator under PD control during a three quadrant slewing maneuver 90  —.  -.  cq  •  2’  If  00  (  0-  —  2.0  0.6  0.0 —0.2 —0.4  —0.12 —0.20  ,  1.6  2.0  —0.04  ) 1 (xlO  0.8 1.2 TIME s  1.6  —0.2  0.2  0.8 1.2 TIME,s  ,m  /1  0.0  0.0  I  0.4  0.4  ,  I  0.8 1.2 TIME,s  0.8 1.2 TIME s  I  ---,  1.6  1.6  I  I  2.0  2.0  0.0_———-t”””  U.G_  fi  0.4  0.04  0.4  0.4  -  0.6_  0.4  0.0  0.0  /\  0.8  0.12  0.20  —0.08  —0.03  0.02  0.07  0.12  0.17  _______  4.47C  ___-  0  0 C C  cc C C\2 0 C\2  0  cc 0 C C  0 CO  C  C CQ  I  C  cc I  CI 0  Figure 4.44: (A) Time history response of the manipulator showing the tip and the link positions in the presence of FLT control with the base held fixed doing a full three quadrant slewing maneuver 92  C  H q  0 0  0  C c’2  ci  -4  -4  C’2 -4  0  0  0 0  0 C  0  C’2  .  C2  0  0 0  0  C’2  C’2  -  0 C,,  -4  0  0  I  0  0  I  -r  0  0  0  C 0 -  0  C 0  0  -  Figure 4.45: (B) Graphs of (from top left figure clockwise) / /37,3k & p3y, i3 & 81. /32 P3x £ 3 & 82, VS time [seconds] for the stationary flexible 2 link manipulator under FLT control during a three quadrant slewing maneuver 93  (0  (D  —---‘  T  3  q  -...  Cl)  -I..  ()  j. ;:  -$  0  0-  0  .  —  00  *  0  -  C)  -  p,  )  —0.12  —0.08  —0.04  0.0  0.04  0.08  —0.05  —0.04  —0.03  —0.02  —0.01  0.00  0.0  0.0  0.4  0.4  ,  0.8 1.2 TIME s  0.8 1.2 TIMERs  1.6  1.6  2.0  2.0  •1I1  —1.0  —0.5  0.0  0.5  1.0  1.5  0.00  0.01  0.02  0.03  0.04  0.05  0.0  0.0  0.4  0.4  0.8 1.2 TIME ,s  ,  0.8 1.2 TIME s  1.6  1.6  2.0  2.0  4.51C  4.4.3  Stationary Manipulator in Non-Zero Gravity Field Without Payload  Figures 4.47, 4.48, and 4.49 show the response of the stationary manipulator using PD control and under the influence of a gravitational acceleration of 9.8m/s . The results may be directly 2 compared with those shown in Figures 4.41. 4.42, and 4.43. It is evident that the joint stiffnesses are insufficient to carry the two links through the full maneuver range when gravity is present (Figure 4.47). Note, in Figures 4.47 and 4.49, the effect of gravity offsets the base and the entire manipulator down as far as 0.3 meters below the origin. The effect is better seen in Figures 4.41 and 4.43. Next, the FLT control is applied in the same situation as above. Figures 4.50. 4.51, and 4.52 show, as before, that the rigid degrees of freedom are instantly brought into line with the desired trajectories. Of course. this happens since we have full knowledge of all the structural parameters as well as all of the degrees of freedoms. The base deflection is no longer a dominant factor in the manipulator tip deviation from the desired trajectory due to gravity. The major contribution comes from the flexible degrees of freedom. The plots of the flexible coordinates and 6 in Figure 4.52 indicate the level of the contribution. Note. where the initial peak deflections occur, they combine in such a manner as to create the greatest deviation from the desired trajectory. It may be pointed out that the overshoot at the end of the slewing maneuver was absent when the gravitational acceleration was neglected.  4.4.4  Mobile Manipulator in Non-Zero Gravity Field Without Payload  Keeping all the parameters the same as in the previous sections, the effect of a moving base is observed in a nonzero gravity field. Figures 4.53, 4.54, and 4.55 illustrate effectiveness of the PD control as the mobile manipulator undergoes the previous slew maneuver. Corresponding results for the FLT control are presented in Figures 4.56, 4.57, and 4.58. The payload in both the cases is absent. Note, the base motion is almost enough to carry the two links over the top dead centre during the PD control. Also, the acceleration of the base delays the tip and stretches out the track 95  path which is further away from the desired trajectory then in the previous simulation (Figure 4.47). Comparison of the mobi]e manipulator response results with those for the stationary case in the previous section show favourable agreement. The exceptions are the modal coordinates. They tend to exhibit greater excursions for the mobile manipulator case. This trend is also observed in the FLT control data. 4.4.5  Mobile Manipulator in the Gravity Field With Payload  Finally, the attention is turned to the most general case that accounts for the mobile character of the manipulator, operating in the gravitational field, and carrying a payload (Figures 4.56. 4.57, and 4.58). The gravity field is taken as 9. n/s ; the FLT control is active; the base starts 2 in the fourth quadrant and is moving in the direction as shown in Figure 4.59. The payload is 3.0 kg. The response is basically as expected. In Figure 4.61 the response plots of  i 6  and 62  show a measure of damping effect on the fundamental mode. The higher order vibrations are suppressed somewhat (compared to those observed in Figure 4.58). although the peak deviation from the desired trajectory is slightly higher in the presence of a payload. As well, the frequency of the vibration of the first mode of link 2 has decreased by about a factor of 1.5 while the vibration frequency in link 1 has gone up by about the same amount. A larger overshoot is noted for this case, at the end of the slewing maneuver, then with any other simulation results obtained so far (Figure 4.59). The response of the second modal coordinate for both the links is somewhat surprising and  22  (512  in Figures 4.58 and 4.61). Firstly, the frequency of vibration of link 2 is slightly higher  than that of link 1. Secondly. the maximum deviation is less than that observed in the previous section where the payload was zero.  96  0 0  0 C  0 C\2 0 C\2  0 Cc  0  0  0  C  c\i  O  01  C  I  ‘I  Figure 4.47: (A) Time history response of the manipulator showing the tip and the link positions in the presence of PD control with the base held fixed doing a full three quadrant slewing maneuver in a nonzero gravity field 97  e ci  / j/’  (I ‘N  —  —  /  -  -  \\  -o I  I  0  0 C  N  I 0  I 0 0  0  —---  N  I C2  0  I C  0 0  C 0  -  C  ( 1—;  ‘s  -  ,,  —ó  I C 0  :  S.  31  -  T  Figure 4.48: (B) Graphs of (from top left figure clockwise) & p, / j & 8, 3 P3y & P3y, / 3 I2 & 82, VS time [seconds] for the stationary flexible 2 link manipulator under PD control in nonzero gravity field. 98  ,(_)  (D  0  — -  0  ,)  -.  —  0-  -  0-  o  0  (D  ,-  -  cn  9-8  <  •-‘  —0.24  —0.16  —0.08  ) 1 (x10  0.0  0.08  0.16  —0.12  —0.08  —0.04  0.0  0.04  0.08  jI)  0.0  -  -  -  -  -  0.0  0.4  I  0.4  I  0.8 1.2 TIME , s  I  0.8 1.2 TIME • s  1.6  I  2.0  2.0  I  612 ,rn 622 ,m  1.6  —0.5  —0.3  —0.1  0.1  0.3  0.5  —0.32  —0.24  —0.16  —0.08  0.0  0.08  \  0.0  -  -  —  -  -  0.0  -  \  I  0.4  I  ,  I  0.8 1.2 TIME s  1.6  /  j# ,1  s  -  2.0  -  -  2.0  rn 1 6 ---11 621 rn  1.6  -  0.40.8 TJME.s  ----,rn  4.55C  C C  C CO  C C\2 C  c\i C CD  C C C  C  C  CD  o  C  C\2  C  Figure 4.50: (A) Time history response of the manipulator showing the tip and the link positions in the presence of FLT control with the base held fixed doing a full three quadrant slewing maneuver in a nonzero gravity field 100  ES  C C2  C’2  -4  -  -  0  ci  0  c  cD  cr  C  C  0  0  a  a 0  0  0  ‘  0  SE  C C,,  C C3  -‘  0  0 0  0  0  0  -4  -4  C’ ‘-I  o  o  0  C’)  I  0  0  0  0 0  a  0  0 —  17  o  0  0  -  Figure 4.51: (B) Graphs of (from top left figure clockwise) / i & 6, 3 P3y & P3y, 1 8 3 & p, ‘ 82, VS time [seconds] for the stationary flexible 2 link manipulator under FLT control in nonzero gravity field. 132 &  101  1q  —  C  ,-..‘  C  —  O  .-  (DO  o  * -  -  a  (D  Nb)  2.0  1.6  2.0  6,2 ,m 622 ,m  1.6  0.0  0.5  1.0  1.5  —1.0 0.8 1.2 TIME • s  0.8 1.2 TIME,s  —0.12 0.4  0.4  0.0  0.01  0.02  —0.5  0.0  0.0  P,m  —0.07  —0.02  0.03  0.08  0.13  —0.05  —0.04  —0.03  0.03  —0.02  ,xn  0.04  —0.01  ----i x  0.05  0.0  0.0  -  0.0  0.4  AA  0.4  I  0.8 1.2 TIME,s  I  0.8 1.2 s 1 TIME  1.6  I  2.0  2.0  I  621 ,m  1.6  V  4.58C  C C C  cc  C CQ  cc C C C  cc  CQ  C\2  I  cc  C  I  Figure 4.53: (A) Time history response of the manipulator showing the tip and the link positions in the presence of PD control with the base translating as shown doing a full three quadrant slewing maneuver in a nonzero gravity field 103  p)  .  H  -S.--  (4  .  04  -  CD  0  -e 0-  .+  C.) c1.. i  0  —  0  o.f  rji  , lyq <b)  —2.4  —1.6  —0.8  0.0  0.8  1.6  —4.0  0.0  4.0  8.0  12.0  16.0  \\  0.0  0.0  -  -  -  -  -  0.4  0.4  I  S  I  0.8 1.2 TIME • s  0.8 1.2 TIME • s  I  S  “S  1.6  1.6  2.0  2.0  -  —0.8  0.0  0.8  1.6  2.4  3.2  —15.0  —10.0  —5.0  0.0  ‘I  I,,’.  5’  --  __I  iI”  0.4  0.4  :-0.0  —  —  -  -  0.0  -  —  5.0_  10.0  /  s  //  ,  I 1.2  /  1.6  —  I  —--S  2.0  I 2.0  JI, 3y ,m 14  I 1.6  ____  /  0.8 1.2 TIME • s  _//  TIME  F 0.8  /  /  /  /‘s  ----S.  4.60B  0’  t..)  —  oP  0.  0  -,  -  0  r  (D  cyq  0  0  g-  o&)  0  rJ)  —  (D  ‘  I I I  1.6  1.8  I  2.0  2.0 0.4  —5.0  4.0  —3.0  t,,’  —0.6  —0.4  —0.16 —0.24  —0.2  —0.08  ) 1 (x10  0.0  0.8 1.2 TIME,s  0.8 1.2 TIME,s  —  -  0.0  —  “I  1’ 1\  I  /  0.4  ‘,g  I  0.4  I  /  ,,/  ‘-I  ,m  ,111  0.8 1.2 T!ME.s  021  017  1.8  I  :/  1.6  ----  ,__i  0.8 1.2 TIME,s  gig,’,?  /7  i/i  ‘.\  1  ,,l,i  0.0  —  -  -  —1.0_  0.0  0.0  0.4  0.4  I  P  lix  L_—_.  ----  0.2  0.0  0.0  -  -  -  -  0.08  0.16  —2.0  0.0  2.0  4.0  6.0  8.0  \  2.0  2.0  ,1T1  ,m  4.61C  0 0 ‘—4  C CD C C\2  C\2  CD  0 0  o O  0 CD  0 C\2  0 CQ  0 CD  I  I  Of 0  Figure 4.56: (A) Time history response of the manipulator showing the tip and the link positions in the presence of FLT control with the base translating as shown doing a full three quadrant slewing maneuver in a nonzero gravity field 106  rw.)  0  0  SE  c’i C  -I  0  C  0  0  C  C  o  0  c  cr  I  rI  0  c -  ci  0 —  0  0  II  LI  II Ii  C  II Ii  —  II  II  C2 ‘-I  0  0 0 0 0 It)  C42  ‘  O  0 —  0  0  t-  It)  0 0  I  0 0  0 I  -‘  Figure 4.57: (B) Graphs of (from top left figure clockwise) /3 & P3x, / p3y 3 P3y, i3 & 6, 132 & 62, VS time [seconds] for the moving flexible 2 link manipulator under FLT control in nonzero gravity field. 107  :  <  N  o  -  -  —  0-  0--S  -  0  D  cq  0 0  -+,  0C1)  -  0  <-s  —0.15  —0.07  0.01  0.09  0.17  0.25  —1.0  0.0  0.0  0.0  -  1.0..  2.0  3.0..  4.0\  0.4  0.4  \ \  \  ,  0.8 1.2 TIME . s  0.8 1.2 TIME s  I  \ 2.0  -  1.6  6 Z 2  2.0  ,m  ó 1 2 ,m  1.6  I  —1.6  —0.8  0.0  0.8  1.6  2.4  —5.0  —4.0  —3.0  —2.0  —1.0  0.0  0.0  0.0  0.4  0.4  0.8 1.2 TIME,s  0.8 1.2 TIME,s  1.6  1.6  2.0  2.0  4.64C  0 0  C CD  0 CQ  C\2  CC  0  o  o C  o c\i  o  o  0  C\2 SI  Figure 4.59: (A) Time history response of the manipulator showing the tip and the link positions in the presence of FLT control with the base translating as shown doing a full three quadrant slewing maneuver in a nonzero gravity field with payload 109  -  I El  I  II  I  I) Ii  I  II I I  I’  N  -4  -4  0  0  0  0  0  o  N  N  -4  t  cS  N  ci  -e  o  o  o  0  IE El  -c’i  LI II I Ii  I’ II  ,,j)  I  “7  II I’  ‘V  I  -4  ‘7 I  0 in  N  I  0 t  -e  I  o  / I  -;  I  0  0  V..;  -  -  aD  -4  --4  N  \\  //  H I ‘  -  -o 0  I  I  CD D2  _-4  C  I  C 0  -ó  CD 0  Figure 4.60: (B) Graphs of (from top left figure clockwise) / & p3 , /3 & 61, 11 3 & p, /32 & 62, VS time [seconds] for the moving flexible 2 link manipulator under FLT control in nonzero gravity field with payload. 110  0.0 0.4 0.8 1.2 s 1 TIME 1.6  2.0  1.6  2.0  —1.6  —0.12 0.8 1.2 TIME,s  —0.8  —0.07  0.4  0.0  —0.02  0.0  0.8  0.03  2.4 1.6  0.13  —  —5.0  0.08  —1.0  .-.,  0  C  —.  (D  aq  —4.0  0.0  o  —2.0  —1.0  0.0  —3.0  2.0  3.0  4.0  1.0  -  —‘‘  Cl)  o -., 9-o  3  (D  -<  0.0  -  -  _•\  0.4  I  -  I  ,‘  ,m  I  1.2 0.8 TIME , s  I  \1  1.6  I  1.6  ----a  /—  0.8 1.2 TIME,s  / \ I--”  ó 21 ô  0.4  I  I  //  /  / 0.0  -  -  -  2.0  2.0  4.67C  Chapter 5  CONCLUDING REMARKS  5.1  Summary of Results and Conclusions  Developed in this thesis is a general formulation for investigating planar dynamics and control of a ground based mobile flexible two-link manipulator. This formulation is also applicable to the space station based systems involving flexible slewing appendages that can be modelled as cantilever beams. Two methods were considered for the development of the dynamical equations of motion. The first approach uses homogeneous traimforinations while the second employs vector analysis to arrive at the kinematic relations for the manipulator, and then the kinetic and potential energies of the system. The use of homogeneous transformations resulted in considerably less effort and proved to be efficient. A computer code is developed for integration of the highly nonlinear, nonautonomous and coupled equations of motion representing a stiff system. Validity of the formulation and the program was assessed through analysis of simple cases. This involved graphical representations of the manipulator in stick figure form showing the configuration and location of the manipulator at discrete time intervals. Although the amount of information that can be obtained through a systematic parametric study is enormous, the focus is on results that may help in establishing trends. In general, the approach is to select a set of realistic parameters representing a highly flexible physical system and subject it to various slew maneuvers and control strategies. The objecctive was to gain some insight into the effectiveness of the control procedures. During the simulation, it became evident that the system can become unstable with certain parameter values and manipulator configurations. This was also affected by the damping of the rigid degrees of freedom. Of course, to avoid numerical instability it was essential to set the 112  error tolerance below the critical value (10—3 in the present study). The system stiffness also proved to be an important parameter in governing its stability. Under certain combinations of joint and structural flexibilties and slew maneuvers the tip trajectory error was found to be excessive even in the presence of the FLT control. The manipulator flexibility influences the end tip payload position by superimposing a high frequency component over the rigid response. A major influence on the response of the manipulator is the time history of the slew maneu ver. Maneuvers involving high acceleration rates led to greater vibration of the flexible degrees of freedom. This effect was more pronounced due to abrupt changes in velocity at the beginning or the end of a slew maneuver. However, the effect was greatly minimized through introduction of a smooth sinusoidal maneuver profile. The choice of two modes was found to be adequate in representing the tip dynamics. As expected, the presence of payload tended to reduce the vibratory response in the flexib’e degrees of freedom. A promising new approach in the nonlinear control of systems is the Eeedback Linearization lechnique (FLT). The method is straightforeward and the control algorithm in its basic form is simple. For control of a highly nonlinear system such as the mobile flexible manipulator considered here, this approach looks very promising. If all of the state variables are known and used in the control algorithm (Quasi-closed Loop Lontrol (QOLC)), the rigid degrees  of freedom, which are colocally controlled, can be made to follow the desired trajectory with considerable accuracy. The effect of the FLT is to decouple the rigid degrees of freedom from one another and from the flexible ones such that the coupling effects on the rigid body dynamics  are eliminated. The converse, of course, is not true since we cannot directly control the flexible degrees of freedom. The flexible body dynamics is thus susceptible to dynamic influence from the rigid degrees of freedom. Hence, the manipulator tip tracking error is strictly due to the combined stuctural flexibility of the links. This fact implies that we require a secondary level of control to deal with the flexible modes. There are certain aspects which need attention for implementation of the FLT in a practical situation.  Demand on computational power is the major one. 113  Fortunately, the computer  technology has made over years, and continues to make, spectacular advances. With parallel processing architecture, real time implementation of control, even for a complex system. may be within reach. Control of the rigid degrees of freedom requires precise knowledge of thc correct structural parameters of the manipulator so that an accurate model may be derived for the controller. Furthermore, sensor information for the flexible degrees of freedom must be obtained for the controller. This would require additional filter circuits and other additional hardware. One way to circumvent this problem is to use Quasi-open Loop Lontrol (QOL(’ which utilizes sensor feedback information of the rigid degrees of freedom and calculates th predicted flexible degrees of freedom to be used by the main controller algorithm. This, however. would require additional computational power. Thus success of the FLT is largely dependent on the advances in the computer and software technologies.  5.2  Comments About the Numerical Simulation  There are a number features regarding the simulation program that are worth commenting on, particularly for its future use in studying dynamics and control of mobile flexible mamlipu lators with payload and operating in the gravity field enviornment: • The program can be used to simulate the links with as many as six modes. Additional modes can be accomodated simply by making the appropriate changes in the r&evant subroutines. Only two modes were used in the present study because it offered desireable trade off between the simulation speed and faithful representation of the flexible system dynamics. The mode numbers used can be changed through the PARAMETER state ments, using the global ALTER editor command, as described at the beginning of the MAIN routine. • The program can be used in any system envoirnment that supports the ANSI standard FORTRAN77 assembler language. For use in the UBC-MTSG enviornment, just issue 114  the commands stated at the start of the MAIN routine, but first be sure to CREATE the required files as stated. To run in the UBC-UNIXG enviornment, the basic assembler and link commands must be modified.  • The program can be joined with the dynamical models of other systems and then simulated simultaneously as one single unified system. The dynamic model of the manipulator, contained in the subroutine FCN and used by the MTSG’s integra.tion subroutine GEAR B (or IMSL’s DIVPAG), may be easily modified by adding the extra terms to the mass matrix D. the forcing vector  and the over all system state vector  .  Subsequently.  the manipulator must be dynamically synthesized with the connecting system via the generalized force vector  Q  • A useful feature for a highly flexible systems under the influence of externally induced steady state force fields (such as the gravitational field), which has the effect of preloading the system to assume a state of ‘deflected’ static equilibrium, is the MTSG subroutine HYBRID I (or the IMSL library subroutine DNEQNF). In general, its purpose is to find the zero of a system in N variables by a modification of the Powell Hybrid Method. Objective here is to evaluate the initial conditions of all the generalized coordinates of the system in static equilibrium under the influence of the gravity. The reason for this is to start the slewing maneuver of the manipulator from a ‘true’ static equilibrium condition. In other words, now the velocity and acceleration of the generalized coordinates, both flexible and rigid, are zero. Note, however, that the IMSL subroutine DNEQNF works flawlesly with this program. On the other hand, HYBRID1 has not been shown to work properly with this code and requires a certain amount of debugging before use. As a result it has been COMMENTED out and deactivated. This, however, does not seriously effect the results because the stiffness coefficients are relatively high and the slew maneuvers are fast and therefore the generalized forces are high enrnigh to dominate any effects of nonzero initial conditions. 115  • In the subroutine FCN, which contains the dynamical equations of motion of the manip ulator, the forcing vector  is segmented into its basic contributing components and then  summed at the end. This greatly facilitates studying the effects of each of the components on the dynamical response of the manipulator simply by blanking certain terms or combi nations of terms. Amongst other things, one may study what effect inaccurate modelling will have on the stability and robustness of the FLT control of the manipulator. 5.3  Recommendations for Future Work  The intention of this thesis is to present a relatively general formulation, based on a model of contemporary interest, which will provide some insight into the dynamics and control of the complex and highly nonlinear ground based, mobile, flexible, twolink manipultor. The tip tracking of moving flexible manipulators have yet to be studied in sufficient detail, especially under the FLT control. This is a new area of research which promises robust, efficient, control of highly nonlinear systems. To investigate this as well as other areas of interest a few suggestions are listed below: • Extension of the model to three dimension would improve its verstiity through application to a large class of systems. Inpiane and out-of-plane degrees of freedom, including torsion of the manipulator links, would enhance the usefulness of the model significantly. The FLT control of such systems would represent an important advance in the field. • The problem of computational time and cost is a major consideration when performing dynamical simulations of complicated nonlinear systems with large numbers of degrees of freedom. Development of a parallel processing enviorument would go a long way  iii  increasing the speed and the real time implementation of the FLT. • Robust control of the rigid degrees of freedom is strongly dependent on the accuracy of the model. There is considerable scope for parametric studies in this area. As mentioned earlier, the model subroutine FCN is structured to facilitate the investigation of the effects 116  of incorrect modelling of the manipulator being controlled. The effect of inaccurate model on the tip trajectory is an important avenue for future study. • There is a considerable scope for detailed parametric study to assess the influence of; structural properties; —  —  —  —  —  differential equation solver parameters; time histories of slew trajectories; FLT control parameters; payload mass; gravitational acceleration.  • In order to minimize vibration of the flexible links, it is suggested that a secondary con troller be built into the base and the joints. Its purpose would be to superimpose, on the main controller action, a secondary damping action to improve tracking of the manipu]a tor’s tip trajectory. This approach can be easily tested using the existing program. • In the problem formulation chapter. two methods of deriving the dynamical equations of motion were suggested. One employed the use of homogeneous transformation, a niore direct method. The other relies on the vector approach, an appearantly more general, less direct and more labour intensive method. It would be useful to investigate the pros and cons of these two, as well as other approaches. The comparison may be in terms of efficiency of derivation and compatibility with commercially available softwares, to expedite as well as synthesize derivation and simulation processes. • End effector compliance and force control during pick/place operations of industrial robots is a very important area in robotic design [38], particularly if high speed, delicate and precise operations are involved. This would involve introducing constraint relations. Such studies, though challenging, should prove to be equally satisfying.  117  Bibliography  [1] Rivin, Ed., Mechanical Design of Robots, McGraw Hill Book Company. 1988. [2] Andeen, G.B., Robot Design Handbook, McGraw Hill Book Company, 1988. [3] Millar, R.K.A., Graham. W.R., aiid Vigneron. F.R., “Simulation of the Motion of a Shuttle Attached Flexible Manipulator Arm”, Proceedings of the 10th World Congress on Systm Simulation and Scientific Computation, Montreal, Canada, August 1982. Editor: W.F. Ames, Published by Association for Mathematics and Computers in Simulation, pp. 225277. [4] Modi, V.J., “Attitude Dynamics of Satellites with Flexible Appendages A Brief Review”. Journal of Spacecraft and Rocket, Vol.11. No.11, November 1974, pp. 743-751. -  [5] Chan, J.K., Dynamics and Control of an Oriting Space Platform Based Mobilc Fkxibk Manipulator, M.A.Sc. Thesis, University of British Columbia, 1990. [6] Mab, H., On the Nonlinear Dynamics of a Space Platform Based Mobile Flexible Manip ulator, Ph.D. Thesis, University of British Columbia, 1992. [7] Cannon Jr., R.H., and Schmitz E., “Initial Experiments on the End-Point Control of a Flexible One-Link Robot”, The International Journal of Robotics Research, Vol. 3, No. :3, Fall 1984, pp. 62-75. [8] Cannon Jr., R.H., and Rovner, D.M., “Experiments Toward On-Line Identification and Control of A Very Flexible One-Link Manipulator”, The International Journal of Robotics Research, Vol. 6, No. 4, Winter 1987, pp. 3-19. [9] Binford, T.O., and Cannon Jr. R.H., First Annual Report on End Point Control of Flexible Robots, Prepared for Air Force Wright Aeronautical Laboratories, Stanford University, Report No. 13389-01-00, 1983. [10] DeCarlo, R.A., Zak, S.H., and Matthews, G.P., “Variable Structure Control of Nonlinear Multivariable Systems”, Proceedings of the IEEE, Vol. 76, No. 3, March 1988, . 232 212 pp. [11] Balestrino, A., De Maria, G., and Sciavicco, “An Adaptive Model Following Control for Robotic Manipulators”, Journal of Dynamic Systems, Measurements, and Control. Vol. 105, Septeniber 1983, pp. 143-151. [12] Leininger, G.G., and Wang, S., “Pole Placement Self-Tuning Control of Manipulators”, IFAC Symposium on Gomuter Aided Design of Multi-Variable Technological Systems, September 15-17, 1982, G. Leiniuger Symp. Editor, West Lafayette, IN. 118  [13] Rovner. D.M., and Franklin, G.F., “Experiments in Load-adaptive Control of a Very Flex ible One-link Manipulator”, Autornatica, Vol. 24, No. 4, 1988, pp.541-548. [14] Spong. M.W., “Modeling and Control of Elastic Joint Robots”, Journal of Dynamic Sys tems, Measurements, and Control, Vol. 109, December 1987, . 319 310 pp. [15] Meirovitch, L., Elements of Vibration Ananlysis, Mac Graw Hill Book Company, 1986. [16] Meirovitch, L., “Analytical Methods in Vibration”, The MacMillan Co., 1967. [17] Thomson. W.T., Theory of Vibration with Applications, Prentice Hall Incorporated. 1972. [18] Pau], R.P., Robot Manipulators: Mathematics Programming and Control, The MIT Press, 1981. [19] Fu K.S., Gonzalez R.C., Lee C.S.G.. Robotics: Control, Sensing, Vision and Intelligencc, McGraw Hill Book Co., 1987. [20] Goldstein, H., Classical Mechanics, Addison-Wesley Publishing Company, 1965. [21] Meriain, J.L., Dynamics, John Wiley and Sons Inc., 1975. [22] Wolfram, S.. “Mathematica: A System for Doing Mathematics by computer”, 2nd Edition, A ddison-Wesley Publishing Company, 1991. [23] Slotine, J.E. and Sastry, S.S., “Tracking Control Non-linear Systems using Sliding Surfaces with Application to Robot Manipulators”, International Journal of Control, Vol. 38, No. 2, 1983, pp. 465-492. [24] Slotine. J.E., “Sliding Controller Design for Non-linear Systems”. International Journal of Control, Vol. 40, No. 2, 1984, pp. 421-434. [25] Slotine, J.E., “The Robust Control of Robot Manipulators”, International Journal of Robotic Research, Vol. 4, No. 2, 1985, pp. 49-64. [26] Slotine, J.E., and Li, W., “On the Adaptive Control of Robotic Manipulators”, Interna tional Journal of Robotics Research, Vol. 6, No. 3, 1987, pp. 49-59. [27] Freund, E., “The structure of Decoupled Nonlinear Systems”, International Journal of Control, Vol. 21, No. 3, 1975, pp. 443-450. [28] Freund, E., “Fast Nonlinear Control with Arbitrary Pole-Placement for Industrial Robots and Manipulators”, The International Journal of Robotics Research, Vol. 1, No. 1, 1982, pp. 65-78. [29] Spong, M.W., “Modeling and Control of Elastic Joint Robots”, Journal of Dynamic Sys tems, Measurements, and Control, Vol. 109, December 1987, . 319 310 pp. [30] Beijczy, A.K., Robot Arm Dynamics and Control, JPL TM 33-669, California Institute of Technology, Pasadena, California, 1974. 119  [31] Singh, S.N.. and Schy, A.A., “Invertibility and Robust Nonlinear Control of Robotic Sys tem”, Proceedings of 3rd conference on Decision and Control, Las Vegas, Nevada. De cember 1984. pp. 105-1O63. [32] Spong, M.W., and Vidyasagar, M., “Robust Linear Compensator Design for Nonlinear Robotic Control”, Proceedings of IEEE Conference on Robotics and Automation, St. Louis. Missouri, March 1985, PP. 954-959. [33] Spong, M.W., ard Vidyasagar, M., “Robust Nonlinear Control of Robot Manipulator”. Proceedings of the .4th IEEE Gonference on Decision and ‘ontrol, Fort Lauderdah, Florida, December 1985, pp. 1767-1772. [34] Karray, F., Modi, V.J., and Chan, J.K., “Inverse control of Flexible Orbiting Manipula tors”, Proceedings of the American Gontrol Conference, Boston, Mass., June 1991, Editor: A.G. Ulsoy, pp. 1909-1912. [35] Modi, V.J., Karray, F., and Chan, J.K.. “On the Control of a Class of Flexible Manipulators Using Feedback Linearization Approach”. .42ud (‘ongress of the International Astronautical Federation, October, 1991. Montreal, Canada. Paper No. IAF-91-324. [36] Gear. C.W., Numerical Initital Value Problems in Ordinary Diffcrentzal Equations. Preii tice Hall Inc.. Eng]ewood Cliffs, NJ. 1971. [37] Rice, J.R., Numerical Mcthods, Software, and Analysis: IMSL Reference Edition. McGraw Hill Book Company, 1983. [38] Latournell, D.J., Cherchas, D.B., “Force and Motion Control of a Single Flexible Link Ma nipu]ator Link”, GA MROL, Dept. of Mechanical Engineering, University of B.C., Oct.1990. [39] Lee, C.S.G; “On the Control of Robot Manipulators”, Dept. of Electrical and Computer Engineering. University of Michigan, 1984. [40] Ogata, K., Discrete- Time Control Systems. Prentice-Hall Incorporated, 1987.  120  Appendix A  KINETIC ENERGY EQUATIONS  Following are the total kinetic energy expressions for the earth based flexible manipulator links 1 and 2 in terms of their physical parameters and generalized coordinates: LINK 1  !f  =  2  1 vdm  1 jj  =  +91X1?61J  + ?fl —  j3) 1 j+C 1 (S  (S  -  (6 miii +  C’i)  (+ LINK 2 1 T  =  2 JVd7fl  m 2 1 =  12  m  +  2 l  E  •  X2j 4 2 6 j  +  1 S2j 4 2 6 j .  +ri2E ‘Isij6 + m2  (Qii +  ‘i  )  2  + +  +  12 111 ( 1 2 2623 + (La +e 121  +2623)  (ie +  +  —  Ô 1 y  (l2.+)  12 E4)2j62j  —m2  e+ 1 (1  O 1 —m2yj  —  12 (1 a e+ 1  2 s  ) —c 12 (s  3y + i3) 121 (S 12 C  1 ) (S  —  1+C (S j3). 1  Total kinetic energy of the system is given by  T The contribution of payload mass  mp  =  . 2 T+T  and of the mobile base  mB  are included, in the  simulation code. by adding the effect of the point mass to the inertia terms for link 2 and for the base. The joint mass is assumed negligible with respect to the mass of the links.  122  Appendix B  POTENTIAL ENERGY  The total potential energy expressions for the two-link manipulator can be written as: LINK I  +  lThlç  =  1 +gC  j 6 1j  py. 1 + gm  LINK 2  ?fl  2 U  =  I)D2j6L  (1 + yj 2 gm S +1 ) + gm2py C 1  j6 4 gCvi j . +gm 1 S l 2 2 +2 where  U  =  2 Ui+U  =  Total flexural and gravitational potential energy of the system.  123  Appendix C  EQUATIONS OF MOTION  The various contributions to the governing differential equations from the inertia, gyroscopic. centripetal, coriolis and other forces as well as their coupling effects are summarized below: rn  p 1 fTD  =  fTDip  =  6 1 fTD  =  8 (ici  m  —  —  C 1 2Ô  —  (2  -  11j 6 S13  (s;+ C)  1 + (S  —  C)  (ml -  fTDIk  =  —9l(Sl-l-CJ3 ) 1 4lk;  2 O 1 fTD  =  0;  k 2 fTDl6  =  0;  p 2 fTD  =  (l2. —  +12  2 —m  p 2 fTD  =  —  +  ( 1 s 2  E  22i) i2i2  12 C  2j2j  o + 1 c (i 1 (e  —  12 + > (th-  i2i2  22 c (al 2 l+ 12 6 —m2  s 1 (ô  ó+ 1 (Q  2i62i)  sI22o2a)  +  Ci) 1 + yi  124  / 1 S 2 fTD  =  12 S2362i62i) (2a  —  Ô i 1 y 2 2rn  l 2 n  s 2 a  o + 1 (i  —  j=I  771212 +Oi—---  11 + â ( 2 S 11 y ) C 2  (  2 (c  ) 2 + (2 + a  1 2 m  + (12 +  m  m  + l 1 j=1  2j 6 2  j=1  e + 1 (i  1 y1  +  2i2i)  —  j=1  )) C 2 à 1 S+S 2 11 + yi ( 2 E 2i62i) (—lià /  —  )  m  ?fl  C 2 (  li) 2 ai  2 ± S  2j 6 2j  2i2) j=1  111212  +C 1 (S ) 12  —  in  12 +a  2j 6 2j  (SI2J  —  12 C j)  j=1 7712  1 + ((ie  /  slk 2 fTD  =  j + Cij) 1 (S  j 1 i (S 1 yi  —  m  I2  —  Cij)y)); iii  rn  S2i62i62i)  +  6C 2  j=1  2j 6 2j  + Y1 1  2i62i)  (i  kS1a2y1 +j 1  in  771212 (c 2  —  2 2J62) S 2=1  ,  1 2 m  êi 1 kS2 (!I 1 +j  —  +  2 a m  —  12 + ——a  ka2$2 1  2iE52i)  j=1  (n1212  in  e 1 (z  +  +  k(512Py 1  +  12 c )  —  k 1 (  k) 1 a12l  1 2 m ( 1 2+ .7=1  m  ka12 1 +4t  2j 6 2j  j3 12 (S  —  12 C j)  j=1  17Th2 6 k 1 41  1 + Cij); (S  125  ?fl  22 + S22i623) c (a 2 6  ( ( ) ( (  /  rn  fTD2e2  =  212  +  S2j 2 6 j  m  1 Yl  1’2 6 1 +Yl  —  1 2 m ——C 2  2j 6 2j  +  oJ 2  1 Y1 j=1  —  C 2  l1o1+y1i)  E 1  j=1  2 +S  2j 6 2j  j=1  771212  2 +—-—s  e+ 1 (i m  —  l 2 rn ( 2+E 12 1 C (S12y + p)  +12  k 6 2 fTD  =  2j 6 2j  f’ 12 (S  —  ) C 2 â 1 2 + yj S 1 (i  1 2k (a  2i62)  —  S (io + Y1 2 à ) 1  —  12  (S12Jy +  1p fT 1 fT m  s 1 fT  i  (Sj3  4 i 6 j  —  —  1+C (S j) (l1a 1 2  j=1 —  1 fT  0;  k 2 fTl6  0;  p 2 fT  0;  i9 (S j3 + CIbX)41k; 1  p 2 fT  1 e 2 fr  —  12 + Ci (S x) 2  l 2 m ( 1 2+  2i62) j=1  m  + (S12J  —  12 12 C o) j=1  —  j3) (i 1 o + 1 j3 + C 1 (S  2 m 126  E  j=1  i 6 •i  + (S  —  1m2 8 k 1 I  slk 2 fT  11 y 2 C)ni ; 1 8  ê 1 (y  +  kÔ1c2 1 (1  +12 4262  (,‘  3 (Sj  —  —  11 + y 1 ê 2 k (5 1 i  o + 1 (i  j=1  +  +  )  2 + (S C . 1 j 12  —  12 C j)))  2i623) 3=1  k 1 (  11 y 2 (C è  1 2 (m e 2 fT  =  12  +  j + j3) 12 (S 12 C  —  —  y, 2 (c é 1  2i62i)  ô + Y1 1 2 (i S )) + 1 —  11 + 2 (i S  k81S2); 1 I  —  (S12y  +  12 c ))  j=1 in  l2 2j62j  11 y 1 Ô 2 (S  +  ô + 1 2 (1 C  +  (S12J)  —  j)); 2 Ci  j=1  k s 2 fT  2k 6 12S2k  +Q122k  2 C 1 (&yi  fu  0;  1 fu  ; 1 gin  e 1 fu  =  g  o+ 1 2 (1 s  1 m (__c 1  —  j + p)) 12 (S 12 C  1 3=1  fulslk  =  D1kb1k 2  2 e 1 fu  0;  fuIs2k  0:  p 2 fu  0;  p 2 fu  ; 2 gni  + gCI4lk;  f/in =  12 !C  g  —  12 S  ) +1 b 2 (1 2 m C  —  Y1 Si)) 1  j=1  olk 2 fu  =  g  2 fue  =  g  1 2 (m  (‘  (  uk  ?fl2l2C  m  2 Cl  —  12 s  —  j=1  127  22i)  + iikm2C1)  k s 2 fu  =  k 2 gC1  +  D2k 2 2 6 k  The various time dependent terms of the system mass matrix can be deconvoluted from the acceleration/force expressions summarized below:  p 2 fr  =  ) 2 +j3(m  +(O)  1 +o +i  +62  2 fD  =  1 2 m  +  —  (liSj + 2 m  1 2 m  12 (__s12 + C  s 2 (_iirm  (—  12 2j62j) c  2i62i)) j=1  n  1 2 rn  ÷ (__si i 2 i6 22 )); ci  +(O)  ) 2 +j, (rn  1 +o  /  (l2  12 f, c  —  122 s ) 2 b  (1 2 m C +1  —  Si)) 1 Y  \\  2  +61r( +62  12 (—f-c 12 \  1 e 2 fD  =  +‘52r  12 42r); (C’  +x  (—  +j;  —  12 C —  sl22i2i)  12 S  262i)  Ci) 2 +iirm  j=1  m  m212 (sl2  12 +C  2i62)  —  2 (lS + iCi)) m  j=1  \  11m —  12 S  2ab2i)  (1 2 m C +1  2=1  +8  2l  m2  +61(  —  11 Y ) 1 S )  2 + yi, S C 1 )) 2 + 29!! (1  + + i) —2  2j2j  2 S 1 (1  3=1•  128  —  11 Y ) 2 C )  +1r  (  rn  ((711272  I  +  —  S2ii)  2 +i C 1 (1 ) 2 S 1  +  —  Y1 (2)))  j=1  fl12l2  +  (lIr (711211  i (1152 6  —  j=1  2 (_-__c  —  2 s j=1  m  12  2’\  +02 ((9+S236 2j)  1 2 m  +  2+ C 1 (1  yt,  ) 2 S  2j2j  (1152  )+ 2 yj,C  !?212  —  —  yt,  )) 2 C  31  +  +t2r (x2r  2r  2 + Yl, S C 1 (1 )) 2  1 2 (m fD26lk  =  l,k  (_!  +  (+  +o,  k 1 (I  1 +o  k 1  (l2C 12 2  (m2l2CY  ( (Q ’ 1 (‘1r  S12 2i62i) + 1kC17fl2)  —  ( ((+—  (  —  52  4)2i62i)) j=1  + ‘S2J6) —  (%-  +  , + k1,k  fl2 7 ‘P1,k  S2j6j j=1  (rnic  k 1  +  S2’5 31  r (,k4X2r + 4Pj,k42fC2) 2 +b +  +Py  2j 6 2j  )  —  + P1 k 1  S2  52))) 1 Y1  (111212  2 —C  —  S2262J)))  2i2J)))  )  (i 2 —IC  +  —  23 h 2 4 ))  52 j=1  1 2 (rn  12 ——C  —  j=1  12 S  423 623))  3=1  ?fl  +ES2j62  +  m  2+ C 1 (1  yl,  I  lir  (nl2l  m  +  :  3=1  S2j6  )  + ljr  129  — —  ) 2 S  2)  (  2+ C 1 (1  (m2l2SCE6))  11 122 2 m  +1r  —  (1152  31  2 (rn  I  =  1!k51hh12)  m  +1r  o 2 fD  —  j=1  +i, (ir (øik  +62  ) b 2  12 + C S 12  (mi 2 2 C  rn  2 S 1 2j (1 6 E 2j  j=1  rn  E  —  1 Y1  )) 2 C  2  +82  4S2J6)  + +62,. ( x2); 4  =  k S 2 fD  +j.(—2kS12)  +j (+C) +81 (+ x2k 4 +61T  +  2k  k1irC2 2 (4  +  2 + C 1 (1  S))  yi,  X2k) 4 r 1 ‘1  +82 ( x2k) 4  These separate terms are then combined to form a complete set of differential equations as follows:  — — — — —  1 + fD  p 2 fD  + fTD,p + fTD p 2  +  py 2 fD  +  fD,’py e 1 fD  --  1 + e 2 fr  fDl6lk + fD slk + 2  +  p 1 fTD  1 fTD,9  fTDlslk  +  k 2 fDl6  +  k 2 fD2o  +  fTDlS2k  , 2 fTD  9 2 fTD  + fTD 6lk 2  12 + fD 62 + fD e + fTD 2  —  —  + fu p + fu 1 1 p 2  =  Qp;  p 1 fT  p 2 fT  + fU p + fu 1 p 2  =  Qpy  e 1 fT  1 e 2 fT  + fu,e 1 + fu 1 e 2  =  Qe,  + fulolk + fU2Sjk  =  Qslk;  + fu 1 2 + fu e 2  =  ; 2 Qe  + fu2s k 2  =  fTlolk  e 2 fTD  + fTD k s 2  1 p 2 fT  1p fT  —  —  olk 2 fT  12 fT  fTls2k  —  —  —  22 fT  fT2S2k  +  k 2 fuls  The above equations can be represented in a compact form using the vector notation as  i.e.  fD+fTD-fT+fu  =  D+r—ir+?j  =  or  D+f  =  This equation can be rewritten in the following form and solved numerically using a state space based integration algorithm: =  D1(_1).  130  

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