Skip to main content
SpringerLink
Log in

Dynamics

  • Chapter
  • First Online:
Springer Handbook of Robotics

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

Abstract

The dynamic equations of motion provide the relationships between actuation and contact forces acting on robot mechanisms, and the acceleration and motion trajectories that result. Dynamics is important for mechanical design, control, and simulation. A number of algorithms are important in these applications, and include computation of the following: inverse dynamics, forward dynamics, the joint-space inertia matrix, and the operational-space inertia matrix. This chapter provides efficient algorithms to perform each of these calculations on a rigid-body model of a robot mechanism. The algorithms are presented in their most general form and are applicable to robot mechanisms with general connectivity, geometry, and joint types. Such mechanisms include fixed-base robots, mobile robots, and parallel robot mechanisms.

In addition to the need for computational efficiency, algorithms should be formulated with a compact set of equations for ease of development and implementation. The use of spatial notation has been very effective in this regard, and is used in presenting the dynamics algorithms. Spatial vector algebra is a concise vector notation for describing rigid-body velocity, acceleration, inertia, etc., using six-dimensional (GlossaryTerm

6-D

) vectors and

The goal of this chapter is to introduce the reader to the subject of robot dynamics and to provide the reader with a rich set of algorithms, in a compact form, that they may apply to their particular robot mechanism. These algorithms are presented in tables for ready

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

ABA:

articulated-body algorithm

CRBA:

composite-rigid-body algorithm

DOF:

degree of freedom

JPL:

Jet Propulsion Laboratory

JSIM:

joint-space inertia matrix

OSIM:

operational-space inertia matrix

RNEA:

recursive Newton–Euler algorithm

References

  1. R. Featherstone: The calculation of robot dynamics using articulated-body inertias, Int. J. Robotics Res. 2(1), 13–30 (1983)

    Article  Google Scholar 

  2. J.J. Craig: Introduction to Robotics: Mechanics and Control, 3rd edn. (Prentice Hall, Upper Saddle River 2005)

    Google Scholar 

  3. R.E. Roberson, R. Schwertassek: Dynamics of Multibody Systems (Springer, Berlin, Heidelberg 1988)

    Book  MATH  Google Scholar 

  4. J.Y.S. Luh, M.W. Walker, R.P.C. Paul: On-line computational scheme for mechanical manipulators, Trans. ASME J. Dyn. Syst. Meas. Control 102(2), 69–76 (1980)

    Article  MathSciNet  Google Scholar 

  5. M.W. Walker, D.E. Orin: Efficient dynamic computer simulation of robotic mechanisms, Trans. ASME J. Dyn. Syst. Meas. Control 104, 205–211 (1982)

    Article  MATH  Google Scholar 

  6. D. Baraff: Linear-time dynamics using lagrange multipliers, Proc. 23rd Annu. Conf. Comp. Graph. Interact. Tech., New Orleans (1996) pp. 137–146

    Google Scholar 

  7. J. Baumgarte: Stabilization of constraints and integrals of motion in dynamical systems, Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Featherstone: Rigid Body Dynamics Algorithms (Springer, New York 2008)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. J. Angeles: Fundamentals of Robotic Mechanical Systems, 2nd edn. (Springer, New York 2003)

    MATH  Google Scholar 

  11. R.S. Ball: A Treatise on the Theory of Screws (Cambridge Univ. Press, London 1900), Republished (1998)

    MATH  Google Scholar 

  12. J.M. Selig: Geometrical Methods in Robotics (Springer, New York 1996)

    Book  MATH  Google Scholar 

  13. D.T. Greenwood: Principles of Dynamics (Prentice-Hall, Englewood Cliffs 1988)

    Google Scholar 

  14. F.C. Moon: Applied Dynamics (Wiley, New York 1998)

    Book  MATH  Google Scholar 

  15. R. Featherstone: Robot Dynamics Algorithms (Kluwer, Boston 1987)

    Book  Google Scholar 

  16. R. Featherstone: Spatial v2, http://royfeatherstone.org/spatial/v2 (2012)

  17. S. McMillan, D.E. Orin: Efficient computation of articulated-body inertias using successive axial screws, IEEE Trans. Robotics Autom. 11, 606–611 (1995)

    Article  Google Scholar 

  18. L. Sciavicco, B. Siciliano: Modeling and Control of Robot Manipulators, 2nd edn. (Springer, London 2000)

    Book  MATH  Google Scholar 

  19. J. Slotine, W. Li: On the adaptive control of robot manipulators, Int. J. Robotics Res. 6(3), 49–59 (1987)

    Article  Google Scholar 

  20. K.S. Chang, O. Khatib: Operational space dynamics: Efficient algorithms for modeling and control of branching mechanisms, Proc. IEEE Int. Conf. Robotics Autom., San Francisco (2000) pp. 850–856

    Google Scholar 

  21. O. Khatib: A unified approach to motion and force control of robot manipulators: The operational space formulation, IEEE J. Robotics Autom. 3(1), 43–53 (1987)

    Article  Google Scholar 

  22. Y.F. Zheng, H. Hemami: Mathematical modeling of a robot collision with its environment, J. Robotics Syst. 2(3), 289–307 (1985)

    Article  Google Scholar 

  23. W. Khalil, E. Dombre: Modeling, Identification and Control of Robots (Kogan Page Sci., London 2002)

    MATH  Google Scholar 

  24. 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 

  25. H. Brandl, R. Johanni, M. Otter: A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix, Proc. IFAC/IFIP/IMACS Int. Symp. Theory Robots, Vienna (1986)

    Google Scholar 

  26. R. Featherstone: Efficient factorization of the joint space inertia matrix for branched kinematic trees, Int. J. Robotics Res. 24(6), 487–500 (2005)

    Article  Google Scholar 

  27. R. Featherstone: An empirical study of the joint space inertia matrix, Int. J. Robotics Res. 23(9), 859–871 (2004)

    Article  Google Scholar 

  28. K. Kreutz-Delgado, A. Jain, G. Rodriguez: Recursive formulation of operational space control, Proc. IEEE Int. Conf. Robotics Autom., Sacramento (1991) pp. 1750–1753

    Google Scholar 

  29. K.W. Lilly: Efficient Dynamic Simulation of Robotic Mechanisms (Kluwer, Boston 1993)

    Book  MATH  Google Scholar 

  30. K.W. Lilly, D.E. Orin: Efficient O(N) recursive computation of the operational space inertia matrix, IEEE Trans. Syst. Man Cybern. 23(5), 1384–1391 (1993)

    Article  Google Scholar 

  31. G. Rodriguez, A. Jain, K. Kreutz-Delgado: Spatial operator algebra for multibody system dynamics, J. Astronaut. Sci. 40(1), 27–50 (1992)

    MathSciNet  Google Scholar 

  32. R. Featherstone: Exploiting sparsity in operational-space dynamics, Int. J. Robotics Res. 29(10), 1353–1368 (2010)

    Article  Google Scholar 

  33. P. Wensing, R. Featherstone, D.E. Orin: A reduced-order recursive algorithm for the computation of the operational-space inertia matrix, Proc. IEEE Int. Conf. Robotics Autom., St. Paul (2012) pp. 4911–4917

    Google Scholar 

  34. R.E. Ellis, S.L. Ricker: Two numerical issues in simulating constrained robot dynamics, IEEE Trans. Syst. Man Cybern. 24(1), 19–27 (1994)

    Article  Google Scholar 

  35. J. Wittenburg: Dynamics of Systems of Rigid Bodies (Teubner, Stuttgart 1977)

    Book  MATH  Google Scholar 

  36. R. Featherstone, D.E. Orin: Robot dynamics: Equations and algorithms, Proc. IEEE Int. Conf. Robotics Autom., San Francisco (2000) pp. 826–834

    Google Scholar 

  37. C.A. Balafoutis, R.V. Patel: Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach (Kluwer, Boston 1991)

    Book  MATH  Google Scholar 

  38. A. Jain: Robot and Multibody Dynamics: Analysis and Algorithms (Springer, New York 2011)

    Book  MATH  Google Scholar 

  39. L.W. Tsai: Robot Analysis and Design: The Mechanics of Serial and Parallel Manipulators (Wiley, New York 1999)

    Google Scholar 

  40. K. Yamane: Simulating and Generating Motions of Human Figures (Springer, Berlin, Heidelberg 2004)

    MATH  Google Scholar 

  41. F.M.L. Amirouche: Fundamentals of Multibody Dynamics: Theory and Applications (Birkhäuser, Boston 2006)

    MATH  Google Scholar 

  42. M.G. Coutinho: Dynamic Simulations of Multibody Systems (Springer, New York 2001)

    Book  MATH  Google Scholar 

  43. E.J. Haug: Computer Aided Kinematics and Dynamics of Mechanical Systems (Allyn and Bacon, Boston 1989)

    Google Scholar 

  44. R.L. Huston: Multibody Dynamics (Butterworths, Boston 1990)

    MATH  Google Scholar 

  45. A.A. Shabana: Computational Dynamics, 2nd edn. (Wiley, New York 2001)

    MATH  Google Scholar 

  46. V. Stejskal, M. Valášek: Kinematics and Dynamics of Machinery (Marcel Dekker, New York 1996)

    MATH  Google Scholar 

  47. L. Brand: Vector and Tensor Analysis, 4th edn. (Wiley/Chapman Hall, New York/London 1953)

    MATH  Google Scholar 

  48. F.C. Park, J.E. Bobrow, S.R. Ploen: A lie group formulation of robot dynamics, Int. J. Robotics Res. 14(6), 609–618 (1995)

    Article  Google Scholar 

  49. M.E. Kahn, B. Roth: The near minimum-time control of open-loop articulated kinematic chains, J. Dyn. Syst. Meas. Control 93, 164–172 (1971)

    Article  Google Scholar 

  50. J.J. Uicker: Dynamic force analysis of spatial linkages, Trans. ASME J. Appl. Mech. 34, 418–424 (1967)

    Article  Google Scholar 

  51. A. Jain: Unified formulation of dynamics for serial rigid multibody systems, J. Guid. Control Dyn. 14(3), 531–542 (1991)

    Article  MATH  Google Scholar 

  52. G. Rodriguez: Kalman filtering, smoothing, and recursive robot arm forward and inverse dynamics, IEEE J. Robotics Autom. RA-3(6), 624–639 (1987)

    Article  Google Scholar 

  53. G. Rodriguez, A. Jain, K. Kreutz-Delgado: A spatial operator algebra for manipulator modelling and control, Int. J. Robotics Res. 10(4), 371–381 (1991)

    Article  Google Scholar 

  54. J.M. Hollerbach: A recursive lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity, IEEE Trans. Syst. Man Cybern. SMC-10(11), 730–736 (1980)

    Article  MathSciNet  Google Scholar 

  55. M.W. Spong, S. Hutchinson, M. Vidyasagar: Robot Modeling and Control (Wiley, Hoboken 2006)

    Google Scholar 

  56. K.W. Buffinton: Kane's Method in Robotics. In: Robotics and Automation Handbook, ed. by T.R. Kurfess (CRC, Boca Raton 2005), 6-1--6-31

    Google Scholar 

  57. T.R. Kane, D.A. Levinson: The use of kane's dynamical equations in robotics, Int. J. Robotics Res. 2(3), 3–21 (1983)

    Article  Google Scholar 

  58. C.A. Balafoutis, R.V. Patel, P. Misra: Efficient modeling and computation of manipulator dynamics using orthogonal cartesian tensors, IEEE J. Robotics Autom. 4, 665–676 (1988)

    Article  Google Scholar 

  59. X. He, A.A. Goldenberg: An algorithm for efficient computation of dynamics of robotic manipulators, Proc. 4th Int. Conf. Adv. Robotics, Columbus (1989) pp. 175–188

    Google Scholar 

  60. W. Hu, D.W. Marhefka, D.E. Orin: Hybrid kinematic and dynamic simulation of running machines, IEEE Trans. Robotics 21(3), 490–497 (2005)

    Article  Google Scholar 

  61. C.A. Balafoutis, R.V. Patel: Efficient computation of manipulator inertia matrices and the direct dynamics problem, IEEE Trans. Syst. Man Cybern. 19, 1313–1321 (1989)

    Article  Google Scholar 

  62. K.W. Lilly, D.E. Orin: Alternate formulations for the manipulator inertia matrix, Int. J. Robotics Res. 10, 64–74 (1991)

    Article  Google Scholar 

  63. S. McMillan, D.E. Orin: Forward dynamics of multilegged vehicles using the composite rigid body method, Proc. IEEE Int. Conf. Robotics Autom. (1998) pp. 464–470

    Google Scholar 

  64. U.M. Ascher, D.K. Pai, B.P. Cloutier: Forward dynamics: Elimination methods, and formulation stiffness in robot simulation, Int. J. Robotics Res. 16(6), 749–758 (1997)

    Article  Google Scholar 

  65. R. Featherstone: A divide-and-conquer articulated-body algorithm for parallel O(log(n)) calculation of rigid-body dynamics. Part 2: Trees, loops and accuracy, Int. J. Robotics Res. 18(9), 876–892 (1999)

    Article  Google Scholar 

  66. MSC Software Corporation: Adams, http://www.mscsoftware.com/

  67. T. Kane, D. Levinson: Autolev user's manual (OnLine Dynamics Inc., Sunnyvale 2005)

    Google Scholar 

  68. Real-Time Physics Simulation: Bullet, http://bulletphysics.org/wordpress (2015)

  69. Georgia Tech Graphics Lab and Humanoid Robotics Lab: DART, http://dartsim.github.io (2011)

  70. S. McMillan, D.E. Orin, R.B. McGhee: DynaMechs: An object oriented software package for efficient dynamic simulation of underwater robotic vehicles. In: Underwater Robotic Vehicles: Design and Control, ed. by J. Yuh (TSI Press, Albuquerque 1995) pp. 73–98

    Google Scholar 

  71. Open Source Robotics Foundation: Gazebo, http://gazebosim.org (2002)

  72. R. Smith: Open Dynamics Engine User Guide, http://opende.sourceforge.net (2006)

  73. Microsoft Corporation: Robotics Developer Studio, http://www.microsoft.com/robotics (2010)

  74. P.I. Corke: A robotics toolbox for MATLAB, IEEE Robotics Autom. Mag. 3(1), 24–32 (1996)

    Article  Google Scholar 

  75. Robotran: http://www.robotran.be (Center for Research in Mechatronics, Université catholique de Louvain 2015)

  76. J.C. Samin, P. Fisette: Symbolic Modeling of Multibody Systems (Kluwer, Dordrecht 2003)

    Book  MATH  Google Scholar 

  77. M.G. Hollars, D.E. Rosenthal, M.A. Sherman: SD/FAST User’s Manual (Symbolic Dynamics Inc., Mountain View 1994)

    Google Scholar 

  78. M. Sherman, P. Eastman: Simbody, https://simtk.org/home/simbody (2015)

  79. G.D. Wood, D.C. Kennedy: Simulating Mechanical Systems in Simulink with SimMechanics (MathWorks Inc., Natick 2003)

    Google Scholar 

  80. W. Khalil, D. Creusot: SYMORO+: A system for the symbolic modeling of robots, Robotica 15, 153–161 (1997)

    Article  Google Scholar 

  81. W. Khalil, A. Vijayalingam, B. Khomutenko, I. Mukhanov, P. Lemoine, G. Ecorchard: OpenSYMORO: An open-source software package for symbolic modelling of robots, Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatron. (2014) pp. 126–1211

    Google Scholar 

  82. Cyberbotics Ltd.: Webots User Guide, http://www.cyberbotics.com (2015)

  83. I.C. Brown, P.J. Larcombe: A survey of customised computer algebra programs for multibody dynamic modelling. In: Symbolic Methods in Control System Analysis and Design, ed. by N. Munro (Inst. Electr. Eng., London 1999) pp. 53–77

    Google Scholar 

  84. J.J. Murray, C.P. Neuman: ARM: An algebraic robot dynamic modeling program, Proc. IEEE Int. Conf. Robotics Autom., Atlanta (1984) pp. 103–114

    Google Scholar 

  85. J.J. Murray, C.P. Neuman: Organizing customized robot dynamic algorithms for efficient numerical evaluation, IEEE Trans. Syst. Man Cybern. 18(1), 115–125 (1988)

    Article  Google Scholar 

  86. F.C. Park, J. Choi, S.R. Ploen: Symbolic formulation of closed chain dynamics in independent coordinates, Mech. Mach. Theory 34, 731–751 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  87. M. Vukobratovic, N. Kircanski: Real-Time Dynamics of Manipulation Robots (Springer, New York 1985)

    Book  MATH  Google Scholar 

  88. J. Wittenburg, U. Wolz: Mesa Verde: A symbolic program for nonlinear articulated-rigid-body dynamics, ASME Des. Eng. Div. Conf., Cincinnati (1985) pp. 1–8, ASME Paper No. 85-DET-151

    Google Scholar 

  89. J.Y.S. Luh, C.S. Lin: Scheduling of parallel computation for a computer-controlled mechanical manipulator, IEEE Trans. Syst. Man Cybern. 12(2), 214–234 (1982)

    Article  Google Scholar 

  90. D.E. Orin: Pipelined approach to inverse plant plus jacobian control of robot manipulators, Proc. IEEE Int. Conf. Robotics Autom., Atlanta (1984) pp. 169–175

    Google Scholar 

  91. R.H. Lathrop: Parallelism in manipulator dynamics, Int. J. Robotics Res. 4(2), 80–102 (1985)

    Article  Google Scholar 

  92. C.S.G. Lee, P.R. Chang: Efficient parallel algorithm for robot inverse dynamics computation, IEEE Trans. Syst. Man Cybern. 16(4), 532–542 (1986)

    Article  Google Scholar 

  93. M. Amin-Javaheri, D.E. Orin: Systolic architectures for the manipulator inertia matrix, IEEE Trans. Syst. Man Cybern. 18(6), 939–951 (1988)

    Article  MATH  Google Scholar 

  94. C.S.G. Lee, P.R. Chang: Efficient parallel algorithms for robot forward dynamics computation, IEEE Trans. Syst. Man Cybern. 18(2), 238–251 (1988)

    Article  MathSciNet  Google Scholar 

  95. M. Amin-Javaheri, D.E. Orin: Parallel algorithms for computation of the manipulator inertia matrix, Int. J. Robotics Res. 10(2), 162–170 (1991)

    Article  Google Scholar 

  96. A. Fijany, A.K. Bejczy: A class of parallel algorithms for computation of the manipulator inertia matrix, IEEE Trans. Robotics Autom. 5(5), 600–615 (1989)

    Article  Google Scholar 

  97. S. McMillan, P. Sadayappan, D.E. Orin: Parallel dynamic simulation of multiple manipulator systems: Temporal versus spatial methods, IEEE Trans. Syst. Man Cybern. 24(7), 982–990 (1994)

    Article  Google Scholar 

  98. A. Fijany, I. Sharf, G.M.T. D'Eleuterio: Parallel O(logN) algorithms for computation of manipulator forward dynamics, IEEE Trans. Robotics Autom. 11(3), 389–400 (1995)

    Article  Google Scholar 

  99. R. Featherstone: A divide-and-conquer articulated-body algorithm for parallel O(log(n)) calculation of rigid-body dynamics. Part 1: Basic algorithm, Int. J. Robotics Res. 18(9), 867–875 (1999)

    Article  Google Scholar 

  100. R. Featherstone, A. Fijany: A technique for analyzing constrained rigid-body systems and its application to the constraint force algorithm, IEEE Trans. Robotics Autom. 15(6), 1140–1144 (1999)

    Article  Google Scholar 

  101. P.S. Freeman, D.E. Orin: Efficient dynamic simulation of a quadruped using a decoupled tree-structured approach, Int. J. Robotics Res. 10, 619–627 (1991)

    Article  Google Scholar 

  102. Y. Nakamura, K. Yamane: Dynamics computation of structure-varying kinematic chains and its application to human figures, IEEE Trans. Robotics Autom. 16(2), 124–134 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering, The Australian National University, RSISE Building 115, ACT 0200, Canberra, Australia

    Roy Featherstone

  2. Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Avenue, OH 43210-1272, Columbus, USA

    David E. Orin

Authors
  1. Roy Featherstone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David E. Orin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roy Featherstone .

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

Featherstone, R., Orin, D.E. (2016). Dynamics. In: Siciliano, B., Khatib, O. (eds) Springer Handbook of Robotics. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-319-32552-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-32552-1_3

  • 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