Skip to main content
SpringerLink
Log in

Kinematics

  • Chapter
  • First Online:
Springer Handbook of Robotics

Part of the book series: Springer Handbooks ((SHB))

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.

figure 1

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 269.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 349.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

3-D:

three-dimensional

6-D:

six-dimensional

C:

cylindrical joint

DOF:

degree of freedom

H:

helical joint

P:

prismatic joint

R:

revolute joint

U:

universal joint

References

  1. W. R. Hamilton: On quaternions, or on a new system of imaginaries in algebra, Philos. Mag. 18 (2000)

    Google Scholar 

  2. E.B. Wilson: Vector Analysis (Dover, New York 1960), based upon the lectures of J.W. Gibbs (reprint of the 2nd edn. published by Charles Scribner's Sons, 1909)

    Google Scholar 

  3. H. Graßmann: Die Wissenschaft der extensiven Größe oder die Ausdehnungslehre (Wigand, Leipzig 1844)

    MATH  Google Scholar 

  4. J.M. McCarthy: Introduction to Theoretical Kinematics (MIT Press, Cambridge 1990)

    Google Scholar 

  5. W.K. Clifford: Preliminary sketch of bi-quarternions, Proc. Lond. Math. Soc. 4, 381–395 (1873)

    MATH  Google Scholar 

  6. A.P. Kotelnikov: Screw calculus and some applications to geometry and mechanics (Annal. Imp. Univ., Kazan 1895)

    Google Scholar 

  7. E. Study: Geometrie der Dynamen (Teubner, Leipzig 1903)

    MATH  Google Scholar 

  8. G.S. Chirikjian, A.B. Kyatkin: Engineering Applications of Noncommutative Harmonic Analysis (CRC, Boca Raton 2001)

    MATH  Google Scholar 

  9. R. von Mises: Anwendungen der Motorrechnung, Z. Angew. Math. Mech. 4(3), 193–213 (1924)

    Article  MATH  Google Scholar 

  10. J.E. Baker, I.A. Parkin: Fundamentals of Screw Motion: Seminal Papers by Michel Chasles and Olinde Rodrigues, School of Information Technologies (University of Sydney, Sydney 2003), translated from O. Rodrigues: Des lois géométriques qui régissent les déplacements d'un système dans l'espace, J. Math. Pures Applicqu. Liouville 5, 380–440 (1840)

    Google Scholar 

  11. R.S. Ball: A Treatise on the Theory of Screws (Cambridge Univ. Press, Cambridge 1998)

    MATH  Google Scholar 

  12. J.K. Davidson, K.H. Hunt: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics (Oxford Univ. Press, Oxford 2004)

    MATH  Google Scholar 

  13. K.H. Hunt: Kinematic Geometry of Mechanisms (Clarendon, Oxford 1978)

    MATH  Google Scholar 

  14. J.R. Phillips: Freedom in Machinery. Vol 1. Introducing Screw Theory (Cambridge Univ. Press, Cambridge 1984)

    Google Scholar 

  15. J.R. Phillips: Freedom in Machinery. Vol 2. Screw Theory Exemplified (Cambridge Univ. Press, Cambridge 1990)

    Google Scholar 

  16. G.S. Chirikjian: Rigid-body kinematics. In: Robotics and Automation Handbook, ed. by T. Kurfess (CRC, Boca Raton 2005), Chap. 2

    Google Scholar 

  17. R.M. Murray, Z. Li, S.S. Sastry: A Mathematical Introduction to Robotic Manipulation (CRC, Boca Raton 1994)

    MATH  Google Scholar 

  18. A. Karger, J. Novak: Space Kinematics and Lie Groups (Routledge, New York 1985)

    Google Scholar 

  19. R. von Mises: Motorrechnung, ein neues Hilfsmittel in der Mechanik, Z. Angew. Math. Mech. 2(2), 155–181 (1924)

    Article  MATH  Google Scholar 

  20. J.D. Everett: On a new method in statics and kinematics, Mess. Math. 45, 36–37 (1875)

    Google Scholar 

  21. F. Reuleaux: Kinematics of Machinery (Dover, New York 1963), reprint of Theoretische Kinematik, 1875, in German

    Google Scholar 

  22. R. Featherstone: Rigid Body Dynamics Algorithms (Kluwer, Boston 2007)

    MATH  Google Scholar 

  23. K.J. Waldron: A method of studying joint geometry, Mechan. Mach. Theory 7, 347–353 (1972)

    Article  Google Scholar 

  24. T.R. Kane, D.A. Levinson: Dynamics, Theory and Applications (McGraw-Hill, New York 1985)

    Google Scholar 

  25. J.L. Lagrange: Oeuvres de Lagrange (Gauthier-Villars, Paris 1867)

    MATH  Google Scholar 

  26. J. Denavit, R.S. Hartenberg: A kinematic notation for lower-pair mechanisms based on matrices, J. Appl. Mech. 22, 215–221 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Khalil, E. Dombre: Modeling, Identification and Control of Robots (Taylor Francis, New York 2002)

    MATH  Google Scholar 

  28. K.J. Waldron: A study of overconstrained linkage geometry by solution of closure equations, Part I: A method of study, Mech. Mach. Theory 8(1), 95–104 (1973)

    Article  Google Scholar 

  29. R. Paul: Robot Manipulators: Mathematics, Programming and Control (MIT Press, Cambridge 1982)

    Google Scholar 

  30. J.J. Craig: Introduction to Robotics: Mechanics and Control (Addison-Wesley, Reading 1986)

    Google Scholar 

  31. K.J. Waldron, A. Kumar: The dextrous workspace, ASME Mech. Conf., Los Angeles (1980), ASME paper No. 80-DETC-108

    Google Scholar 

  32. R. Vijaykumar, K.J. Waldron, M.J. Tsai: Geometric optimization of manipulator structures for working volume and dexterity, Int. J. Robotics Res. 5(2), 91–103 (1986)

    Article  Google Scholar 

  33. J. Duffy: Analysis of Mechanisms and Robot Manipulators (Wiley, New York 1980)

    Google Scholar 

  34. D. Pieper: The Kinematics of Manipulators Under Computer Control, Ph.D. Thesis (Stanford University, Stanford 1968)

    Google Scholar 

  35. C.S.G. Lee: Robot arm kinematics, dynamics, and control, Computer 15(12), 62–80 (1982)

    Article  Google Scholar 

  36. M.T. Mason: Mechanics of Robotic Manipulation (MIT Press, Cambridge 2001)

    Book  Google Scholar 

  37. H.Y. Lee, C.G. Liang: A new vector theory for the analysis of spatial mechanisms, Mech. Mach. Theory 23(3), 209–217 (1988)

    Article  Google Scholar 

  38. R. Manseur, K.L. Doty: A robot manipulator with 16 real inverse kinematic solutions, Int. J. Robotics Res. 8(5), 75–79 (1989)

    Article  Google Scholar 

  39. M. Raghavan, B. Roth: Kinematic analysis of the 6R manipulator of general geometry, 5th Int. Symp. Robotics Res. (1990)

    Google Scholar 

  40. D. Manocha, J. Canny: Real Time Inverse Kinematics for General 6R Manipulators, Tech. Rep. (University of California, Berkeley 1992)

    Book  Google Scholar 

  41. B. Buchberger: Applications of Gröbner bases in non-linear computational geometry, Lect. Notes Comput. Sci. 296, 52–80 (1989)

    Article  MathSciNet  Google Scholar 

  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. 326–334

    Chapter  Google Scholar 

  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. Mech. Transm. Autom. Des. 107, 189–195 (1985)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  45. R. Manseur, K.L. Doty: Fast inverse kinematics of 5-revolute-axis robot manipulators, Mechan. Mach. Theory 27(5), 587–597 (1992)

    Article  Google Scholar 

  46. S.C.A. Thomopoulos, R.Y.J. Tam: An iterative solution to the inverse kinematics of robotic manipulators, Mechan. Mach. Theory 26(4), 359–373 (1991)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  48. J. Zhao, N. Badler: Inverse kinematics positioning using nonlinear programming for highly articulated figures, Trans. Comput. Graph. 13(4), 313–336 (1994)

    Article  Google Scholar 

  49. D.E. Whitney: Resolved motion rate control of manipulators and human prostheses, IEEE Trans. Man Mach. Syst. 10, 47–63 (1969)

    Article  Google Scholar 

  50. H. Cheng, K. Gupta: A study of robot inverse kinematics based upon the solution of differential equations, J. Robotic Syst. 8(2), 115–175 (1991)

    Article  Google Scholar 

  51. L. Sciavicco, B. Siciliano: Modeling and Control of Robot Manipulators (Springer, London 2000)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  54. D.E. Orin, W.W. Schrader: Efficient computation of the jacobian for robot manipulators, Int. J. Robotics Res. 3(4), 66–75 (1984)

    Article  Google Scholar 

  55. D.E. Whitney: The mathematics of coordinated control of prosthetic arms and manipulators, J. Dynamic Sys. Meas. Control 122, 303–309 (1972)

    Article  Google Scholar 

  56. R.P. Paul, B.E. Shimano, G. Mayer: Kinematic control equations for simple manipulators, IEEE Trans. Syst. Man Cybern. 11(6), 339–455 (1981)

    Google Scholar 

  57. R.P. Paul, C.N. Stephenson: Kinematics of robot wrists, Int. J. Robotics Res. 20(1), 31–38 (1983)

    Article  Google Scholar 

  58. R.P. Paul, H. Zhang: Computationally efficient kinematics for manipulators with spherical wrists based on the homogeneous transformation representation, Int. J. Robotics Res. 5(2), 32–44 (1986)

    Article  Google Scholar 

  59. K.J. Waldron, K.H. Hunt: Series-parallel dualities in actively coordinated mechanisms, Int. J. Robotics Res. 10, 473–480 (1991)

    Article  Google Scholar 

  60. H. Asada, J.J.E. Slotine: Robot Analysis and Control (Wiley, New York 1986)

    Google Scholar 

  61. F.L. Lewis, C.T. Abdallah, D.M. Dawson: Control of Robot Manipulators (Macmillan, New York 1993)

    Google Scholar 

  62. R.J. Schilling: Fundamentals of Robotics: Analysis and Control (Prentice Hall, Englewood Cliffs 1990)

    Google Scholar 

  63. M.W. Spong, M. Vidyasagar: Robot Dynamics and Control (Wiley, New York 1989)

    Google Scholar 

  64. T. Yoshikawa: Foundations of Robotics (MIT Press, Cambridge 1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre of Mechatronics and Intelligent Systems, University of Technology Sydney, City Campus, 15 Broadway, NSW 2001, Ultimo, Australia

    Kenneth J. Waldron

  2. Department of Aerospace and Mechanical Engineering, University of Notre Dame, IN 46556, Notre Dame, USA

    James Schmiedeler

Authors
  1. Kenneth J. Waldron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Schmiedeler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenneth J. Waldron .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Naples Federico II, Via Claudio 21, 80125, Naples, Italy

    Bruno Siciliano

  2. Department of Computer Sciences, Artificial Intelligence Laboratory, Stanford University, 450 Serra Mall, CA 94305, Stanford, USA

    Oussama Khatib

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Waldron, K.J., Schmiedeler, J. (2016). Kinematics. In: Siciliano, B., Khatib, O. (eds) Springer Handbook of Robotics. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-319-32552-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-32552-1_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32550-7

  • Online ISBN: 978-3-319-32552-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation