kinematic modeling of wheeled mobile .mobile robot, kinematics, wheeled robot, forward solution,

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  • - -

    I

    Patrick F. Muir and Charles P. Neuman

    Department of Electrical and Computer Enginewing The Robotics Institute

    CamegieMellon University Pittsburgh, Pennsylvania 15213

    June 1986

    Copyright @ 1986 CarnegieMellonUnhrersity

    Kinematic Modeling of Wheeled Mobile Robots

    This research has been supported by the Office of Naval Research under Contract NOOO14-81 -Ko0503 and the Department of Electrical and Computer Engineering, Carnegie-Mellon Unhrenity.

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  • Unclassified SECURITY CIASSIFICATIOK OF THIS PACE I When Doto EntcrcA

    ~ --. REPORT DOCUMENTATION PAGE

    1. REPORT NUMBER 2. COVT ACCESSION NO.

    CMU-RETR-8612 4. TITLE (and Subtde)

    KINEMATIC MODELING OF WEEELED MOBILE ROBOTS

    7. AUTHOR (a)

    Patrick F. Muir and Charles P. Neuman

    D. PERFORMlFiC ORGANIZATION NAME AND ADDRESS The Robotics htitute, Mobile Robot Lab; and The Department of Electrical and Computer Engineering, Carnegie-Meloon University, Schenley Park, Pittsburgh, PA 15213

    Office of Naval Research, Dr. Alan R. M e y r d t z , Code 433,800 I f . CONTROLLING OFFICE NAME AND ADDRESS

    N. Quincy St., Arlington, VA 22217

    14. MONlTORlNG AGENCY NAME & ADDRESS (if different / o m Contrdkng Offsee)

    6. DISTRIBUTION STATEMENT (of thu Report)

    READ INSTRUCTIONS BEFORE COMPLETING FORM

    3. RECIPIENT'S CATALOG NUMBER

    6. TYPE OF REPORT & PERIOD COVEREI Te~hnicai Report, 5/84 - 5/86

    6. PERFORMING ORG. REPORT NUMBER

    8. CONTRACT OR GRANT NUMBER (a) N00014-81-K-0503

    10. PROGRAM ELEMENT, PROJECT, TASK AREA I WORK UNIT NUMBERS

    Work Unit NR 610-001

    12. REPORT DATE June 1986

    126

    Unclassified

    IS. NUMBER OF PAGES

    ,IS. SECURITY CLASS. (of thu report)

    IS.. DECLASSlPlCATlON/DOW1JCRADlNG SCHEDULE

    None

    17. DlSTlBUTlON STATEMENT (of the obrtroct entered in Block to; if different from Beport) unlimited

    8. SUPPLEMENTARY NOTES

    IO. KEY WORDS (Cunbnue on rewerae 8de if aeeeaaory and identih b# block nuder) . Mobile Robot, Kinematics, Wheeled Robot, Forward Solution, Inverse Solution, Mobility Charscteristics, Actuation Characteristics, Smsing Characteristics, Bobot Design, Dead Reckoning, Kinematic Control, slip Detcctiw uranua

    ). ABSTRACT (Cophnnc on rewerae d e if nmum ond tdentih by block number) We farmulate the kinematic equatid-motion of wheeled mob& robots incorporating conventid, omnidirectional, and ball wheels. While our approach parallels the Linematic modeling of stationary ma- nipulators, we extend the methoddo& to accomodate such specid characteristics of wheeled mobile roboh 1u multiple dosed-link chains, higher-pair contact points between a wheel and a d a c e , and unactuated and unaamd wheel degreed-fredam. We sarrrey atistiug wheeled mobile robots to motivate our d d - opmcnt. To communicate the kinematic featuxea of wheeled mobile robots, we introduce a diagrammatic convention and nomenclature. We apply the Seth-Uicker convention to assign cooodizmte a m and develop,

  • Undaujfied - I ' SECURITY CLASSIFICATION OP TltlS P A C E (When Data E n t d

    20.) a matrix coordinate transformation algcbra to derive thc cqnationssf-motion. A wheel Jacobian matrix is formulated to rchte thc motions of each whecl to thc motions of the robot. We combine the indi- vidual wheel equations to form the composite robot equation&-motion. We calculate the m c d forward and actuated invcrsc solutions and interpret the conditions which guarauta their Bcistsnce. W c iuterpret the properties of the composite robot equation to charactcrizc the mobility of a whceled mob& robot according to the mobility characterization trec. S-, we apply actuation and sensing characteritstion trccs to dclincatc the robot motions prodaciblc by the wheel actuators and discernable by the wheel sensors, respectively. We apply OUT kinematic model to design, kinanatidmsed control, dead-reckoning, and wheel slip detection. To illustrate the development, we fmulate and interpret the kinematic equatianssf-motion of six prototype wheeled mobile robots.

  • Table of Contents

    1 . Introduction ................................................................................ 1 2 . Survey of Kinematic Configurations ......................................................... 5 3 . Wheel .Types .............................................................................. 11 4 . Kinematic Modeling ....................................................................... 14

    4.1. Introduct.ion. .......................................................... : ................ 14 4.2. Definitions And Assumptions ........................................................... 14 4.3. Coordinate System Assignments ........................................................ 16

    4.3.1. Sheth-Uicker Convention ............................................................ 16 4.3.2. WMR Coordinate Systems .......................................................... 17 4.3.3. Instantaneously Coincident Coordinate Systems ...................................... 19

    4.4. Transformation Matrices ................................................................ 22 4.5. Matrix Coordinate Transformation Algebra ............................................. 29 4.6. Position Kinematics ..................................................................... 31 . 4.7. Velocity Kinematics .................................................................... 33

    4.7.1. Introduction ........................................................................ 33 4.7.2. Point Velocities ...................................................................... 34 4.7.3. Whcel Jacobian Matrix .............................................................. 35 4.7.4. Tranforming Robot Velocities ........................................................ 37

    4.8. Acceleration Kinematics ................................................................ 39 4.9. summary ................................................................................ 40'

    5 . The Composite Robot Equation ............................................................ 42 5.1. Introduction ............................................................................. 42 5.2. Formulation of the Composite Robot Equation .......................................... 43 5.3. Solution of Ax = By ................................................................... 43 5.4. Robot Mobility Characteristics ......................................................... 46 5.5. Actuat.ed Inverse Solution ............................................................ ... 52 5.6. Robot Actuation Characteristics ........................................................ 54 5.7. Sensed Forward Solution ................................................................ 57 5.8. Robot Sensing Characteristics .......................................................... 59 5.9. Conclusions ............................................................................ 62

    6 . Applications ............................................................................... 64 6.1. Introduction ......... : ................................................................. -64 6.2. Design ................................................................................. 64 6.3. Dead Reckoning ......................................................................... 66 6.4. Kinematics-Based Feedback Control .................................................... 68

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    6.5. Whcel Slip Detection ................................................................... 70 6.6. S u m .............................................................................. .71.

    7 . Examples .................................................................................. 73 7.1. Introduction ............................................................................ 73 7.2. Unimation Robot ....................................................................... 73 7.3. Newt ................................................................................... 77 7.4. Uranus ................................................................................. 81 7.5. Neptune ................................................................................. 87 7.6. Rover .................................................................................. 89 7.7. Stanford Cart ............................................. .. ............................ 92 7.8. Conclusions ............................................................................ 95

    8 . Conclusions ................................................................................. 97 9 . Continuing Research ...................................................................... 102 10 . References ............................................................................... 104 A1 . Appendix 1: A Nomenclature and Symbolic Representation of WMRs ................... 110 Al.l. Introduction .......... : .............................................................. 110 A1.2. Symbolic Representation Rules ...................................................... -111 A1.3. Nomenclature Rules .................................................................. 112 A1.4. Examples ............................................................................ 114

    A2 . Appendix 2: Symbol Tables ............................................................. 116 A3 . Appendix 3: Wheel Jacobian Matrice ................................................... 118 A3.1. Introduction ......................................................................... 118 A3.2. Conventional Non-Steered Wheel, ................................

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