Advertisement

Kinematics

  • Kenneth Waldron
  • James Schmiedeler

Abstract

Kinematics pertains to the motion of bodies in a robotic mechanism without regard to the forces/torques that cause the motion. Since robotic mechanisms are by their very essence designed for motion, kinematics is the most fundamental aspect of robot design, analysis, control, and simulation. The robotics community has focused on efficiently applying different representations of position and orientation and their derivatives with respect to time to solve foundational kinematics problems.

This chapter will present the most useful representations of the position and orientation of a body in space, the kinematics of the joints most commonly found in robotic mechanisms, and a convenient convention for representing the geometry of robotic mechanisms. These representational tools will be applied to compute the workspace, the forward and inverse kinematics, the forward and inverse instantaneous kinematics, and the static wrench transmission of a robotic mechanism. For brevity, the focus will be on algorithms applicable to open-chain mechanisms.

The goal of this chapter is to provide the reader with general tools in tabulated form and a broader overview of algorithms that can be applied together to solve kinematics problems pertaining to a particular robotic mechanism.

Keywords

Coordinate Frame Euler Angle Joint Model Rolling Contact Spherical Joint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abbreviations

DOF

degree of freedom

References

  1. 1.1.
    W. R. Hamilton: On quaternions, or on a new system of imaginaries in algebra, Philos. Mag. 18, installments July 1844 - April 1850, ed. by D. E. Wilkins (2000)Google Scholar
  2. 1.2.
    E.B. Wilson: Vector Analysis (Dover, New York 1960), based upon the lectures of J. W. Gibbs, (reprint of the second edn. published by Charles Scribnerʼs Sons, 1909)Google Scholar
  3. 1.3.
    H. Grassman: Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre (Wigand, Leipzig 1844)Google Scholar
  4. 1.4.
    J.M. McCarthy: Introduction to Theoretical Kinematics (MIT Press, Cambridge 1990)Google Scholar
  5. 1.5.
    W.K. Clifford: Preliminary sketch of bi-quarternions, Proc. London Math. Soc., Vol. 4 (1873) pp. 381–395Google Scholar
  6. 1.6.
    A. P. Kotelnikov: Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan (1895)Google Scholar
  7. 1.7.
    E. Study: Geometrie der Dynamen (Teubner, Leipzig 1901)Google Scholar
  8. 1.8.
    G.S. Chirikjian, A.B. Kyatkin: Engineering Applications of Noncommutative Harmonic Analysis (CRC, Boca Raton 2001)zbMATHGoogle Scholar
  9. 1.9.
    R. von Mises: Anwendungen der Motorrechnung, Z. Angew. Math. Mech. 4(3), 193–213 (1924)CrossRefGoogle Scholar
  10. 1.10.
    J.E. Baker, I.A. Parkin: Fundamentals of Screw Motion: Seminal Papers by Michel Chasles and Olinde Rodrigues (School of Information Technologies, The University of Sydney, Sydney 2003), translated from O. Rodrigues: Des lois géométriques qui régissent les déplacements dʼun système dans lʼespace, J. Math. Pures Applicqu. Liouville 5, 380–440 (1840)Google Scholar
  11. 1.11.
    R.S. Ball: A Treatise on the Theory of Screws (Cambridge Univ Press, Cambridge 1998)zbMATHGoogle Scholar
  12. 1.12.
    J.K. Davidson, K.H. Hunt: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics (Oxford Univ Press, Oxford 2004)zbMATHGoogle Scholar
  13. 1.13.
    K.H. Hunt: Kinematic Geometry of Mechanisms (Clarendon, Oxford 1978)zbMATHGoogle Scholar
  14. 1.14.
    J.R. Phillips: Freedom in Machinery: Volume 1. Introducing Screw Theory (Cambridge Univ Press, Cambridge 1984)Google Scholar
  15. 1.15.
    J.R. Phillips: Freedom in Machinery: Volume 2. Screw Theory Exemplified (Cambridge Univ Press, Cambridge 1990)Google Scholar
  16. 1.16.
    G.S. Chirikjian: Rigid-body kinematics. In: Robotics and Automation Handbook, ed. by T. Kurfess (CRC, Boca Raton 2005), Chapt. 2Google Scholar
  17. 1.17.
    R.M. Murray, Z. Li, S.S. Sastry: A Mathematical Introduction to Robotic Manipulation (CRC, Boca Raton 1994)zbMATHGoogle Scholar
  18. 1.18.
    A. Karger, J. Novak: Space Kinematics and Lie Groups (Routledge, New York 1985)Google Scholar
  19. 1.19.
    R. von Mises: Motorrechnung, ein neues Hilfsmittel in der Mechanik, Z. Angew. Math. Mech. 2(2), 155–181 (1924), [transl. J. E. Baker, K. Wohlhart, Inst. for Mechanics, T. U. Graz (1996)]CrossRefGoogle Scholar
  20. 1.20.
    J.D. Everett: On a new method in statics and kinematics, Mess. Math. 45, 36–37 (1875)Google Scholar
  21. 1.21.
    R. Featherstone: Rigid Body Dynamics Algorithms (Kluwer Academic, Boston 2007)Google Scholar
  22. 1.22.
    F. Reuleaux: Kinematics of Machinery (Dover, New York 1963), (reprint of Theoretische Kinematik, 1875, in German).Google Scholar
  23. 1.23.
    K.J. Waldron: A method of studying joint geometry, Mechan. Machine Theory 7, 347–353 (1972)CrossRefGoogle Scholar
  24. 1.24.
    T.R. Kane, D.A. Levinson: Dynamics, Theory and Applications (McGraw-Hill, New York 1985)Google Scholar
  25. 1.25.
    J.L. Lagrange: Oeuvres de Lagrange (Gauthier-Villars, Paris 1773)Google Scholar
  26. 1.26.
    J. Denavit, R.S. Hartenberg: A kinematic notation for lower-pair mechanisms based on matrices, J. Appl. Mech. 22, 215–221 (1955)zbMATHMathSciNetGoogle Scholar
  27. 1.27.
    W. Khalil, E. Dombre: Modeling, Identification and Control of Robots (Taylor Francis, New York 2002)Google Scholar
  28. 1.28.
    K.J. Waldron: A study of overconstrained linkage geometry by solution of closure equations, Part I: a method of study, Mech. Machine Theory 8(1), 95–104 (1973)CrossRefGoogle Scholar
  29. 1.29.
    R. Paul: Robot Manipulators: Mathematics, Programming and Control (MIT Press, Cambridge 1982)Google Scholar
  30. 1.30.
    J.J. Craig: Introduction to Robotics: Mechanics and Control (Addison-Wesley, Reading 1986)Google Scholar
  31. 1.31.
    K.J. Waldron, A. Kumar: The Dextrous workspace, ASME Mech. Conf. (Los Angeles 1980), ASME paper No. 80-DETC-108Google Scholar
  32. 1.32.
    R. Vijaykumar, K.J. Waldron, M.J. Tsai: Geometric optimization of manipulator structures for working volume and dexterity, Int. J. Robot. Res. 5(2), 91–103 (1986)CrossRefGoogle Scholar
  33. 1.33.
    J. Duffy: Analysis of Mechanisms and Robot Manipulators (Wiley, New York 1980)Google Scholar
  34. 1.34.
    D. Pieper: The Kinematics of Manipulators Under Computer Control. Ph.D. Thesis (Stanford University, Stanford 1968)Google Scholar
  35. 1.35.
    C.S.G. Lee: Robot arm kinematics, dynamics, and control, Computer 15(12), 62–80 (1982)CrossRefGoogle Scholar
  36. 1.36.
    M.T. Mason: Mechanics of Robotic Manipulation (MIT Press, Cambridge 2001)Google Scholar
  37. 1.37.
    H.Y. Lee, C.G. Liang: A new vector theory for the analysis of spatial mechanisms, Mechan. Machine Theory 23(3), 209–217 (1988)CrossRefGoogle Scholar
  38. 1.38.
    R. Manseur, K.L. Doty: A robot manipulator with 16 real inverse kinematic solutions, Int. J. Robot. Res. 8(5), 75–79 (1989)CrossRefGoogle Scholar
  39. 1.39.
    M. Raghavan, B. Roth: Kinematic analysis of the 6R manipulator of general geometry, 5th Int. Symp. Robot. Res. (1990)Google Scholar
  40. 1.40.
    D. Manocha, J. Canny: Real Time Inverse Kinematics for General 6R Manipulators Tech. rep. (University of California, Berkeley 1992)Google Scholar
  41. 1.41.
    B. Buchberger: Applications of Gröbner bases in non-linear computational geometry. In: Trends in Computer Algebra, Lect. Notes Comput. Sci., Vol. 296, ed. by R. Janen (Springer, Berlin 1989) pp. 52–80Google Scholar
  42. 1.42.
    P. Kovacs: Minimum degree solutions for the inverse kinematics problem by application of the Buchberger algorithm. In: Advances in Robot Kinematics, ed. by S. Stifter, J. Lenarcic (Springer, New York 1991) pp. 326–334Google Scholar
  43. 1.43.
    L.W. Tsai, A.P. Morgan: Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods, ASME J. Mechan. Transmission Autom. Design 107, 189–195 (1985)CrossRefGoogle Scholar
  44. 1.44.
    C.W. Wampler, A.P. Morgan, A.J. Sommese: Numerical continuation methods for solving polynomial systems arising in kinematics, ASME J. Mech. Des. 112, 59–68 (1990)CrossRefGoogle Scholar
  45. 1.45.
    R. Manseur, K.L. Doty: Fast inverse kinematics of 5-revolute-axis robot manipulators, Mechan. Machine Theory 27(5), 587–597 (1992)CrossRefGoogle Scholar
  46. 1.46.
    S.C.A. Thomopoulos, R.Y.J. Tam: An iterative solution to the inverse kinematics of robotic manipulators, Mechan. Machine Theory 26(4), 359–373 (1991)CrossRefGoogle Scholar
  47. 1.47.
    J.J. Uicker Jr., J. Denavit, R.S. Hartenberg: An interactive method for the displacement analysis of spatial mechanisms, J. Appl. Mech. 31, 309–314 (1964)zbMATHGoogle Scholar
  48. 1.48.
    J. Zhao, N. Badler: Inverse kinematics positioning using nonlinear programming for highly articulated figures, Trans. Comput. Graph. 13(4), 313–336 (1994)CrossRefGoogle Scholar
  49. 1.49.
    D.E. Whitney: Resolved motion rate control of manipulators and human prostheses, IEEE Trans. Man Mach. Syst. 10, 47–63 (1969)CrossRefGoogle Scholar
  50. 1.50.
    H. Cheng, K. Gupta: A study of robot inverse kinematics based upon the solution of differential equations, J. Robot. Syst. 8(2), 115–175 (1991)CrossRefGoogle Scholar
  51. 1.51.
    L. Sciavicco, B. Siciliano: Modeling and Control of Robot Manipulators (Springer, London 2000)Google Scholar
  52. 1.52.
    R.S. Rao, A. Asaithambi, S.K. Agrawal: Inverse Kinematic Solution of Robot Manipulators Using Interval Analysis, ASME J. Mech. Des. 120(1), 147–150 (1998)CrossRefGoogle Scholar
  53. 1.53.
    C.W. Wampler: Manipulator inverse kinematic solutions based on vector formulations and damped least squares methods, IEEE Trans. Syst. Man Cybern. 16, 93–101 (1986)CrossRefzbMATHGoogle Scholar
  54. 1.54.
    D.E. Orin, W.W. Schrader: Efficient computation of the jacobian for robot manipulators, Int. J. Robot. Res. 3(4), 66–75 (1984)CrossRefGoogle Scholar
  55. 1.55.
    D. E. Whitney: The mathematics of coordinated control of prosthetic arms and manipulators J. Dynamic Sys. Meas. Contr. 122, 303–309 (1972)Google Scholar
  56. 1.56.
    R.P. Paul, B.E. Shimano, G. Mayer: Kinematic control equations for simple manipulators, IEEE Trans. Syst. Man Cybern. SMC-11(6), 339–455 (1981)Google Scholar
  57. 1.57.
    R.P. Paul, C.N. Stephenson: Kinematics of robot wrists, Int. J. Robot. Res. 20(1), 31–38 (1983)CrossRefGoogle Scholar
  58. 1.58.
    R.P. Paul, H. Zhang: Computationally efficient kinematics for manipulators with spherical wrists based on the homogeneous transformation representation, Int. J. Robot. Res. 5(2), 32–44 (1986)CrossRefGoogle Scholar
  59. 1.59.
    H. Asada, J.J.E. Slotine: Robot Analysis and Control (Wiley, New York 1986)Google Scholar
  60. 1.60.
    F.L. Lewis, C.T. Abdallah, D.M. Dawson: Control of Robot Manipulators (Macmillan, New York 1993)Google Scholar
  61. 1.61.
    R.J. Schilling: Fundamentals of Robotics: Analysis and Control (Prentice-Hall, Englewood Cliffs 1990)Google Scholar
  62. 1.62.
    M.W. Spong, M. Vidyasagar: Robot Dynamics and Control (Wiley, New York 1989)Google Scholar
  63. 1.63.
    T. Yoshikawa: Foundations of Robotics (MIT Press, Cambridge 1990)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Mechanical EngineeringThe Ohio State UniversityColumbusUSA

Personalised recommendations