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Dynamics

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Book cover Springer Handbook of Robotics

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 (6-D) vectors and tensors.

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 access.

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Abbreviations

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. Robot. Res. 2(1), 13–30 (1983)

    Article  Google Scholar 

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

    Google Scholar 

  3. R.E. Roberson, R. Schwertassek: Dynamics of Multibody Systems (Springer-Verlag, Berlin/Heidelberg/New York 1988)

    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. Measur. 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. Measur. Control 104, 205–211 (1982)

    Article  MATH  Google Scholar 

  6. D. Baraff: Linear-Time Dynamics using Lagrange Multipliers, Proc. SIGGRAPH ʼ96 (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  MATH  MathSciNet  Google Scholar 

  8. R. Featherstone: Rigid Body Dynamics Algorithms (Springer, Berlin, Heidelberg 2007)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

  16. S. McMillan, D.E. Orin: Efficient Computation of Articulated-Body Inertias Using Successive Axial Screws, IEEE Trans. Robot. Autom. 11, 606–611 (1995)

    Article  Google Scholar 

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

    Google Scholar 

  18. J. Slotine, W. Li: On the Adaptive Control of Robot Manipulators, Int. J. Robot. Res. 6(3), 49–59 (1987)

    Article  Google Scholar 

  19. K.S. Chang, O. Khatib: Operational Space Dynamics: Efficient Algorithms for Modeling and Control of Branching Mechanisms. In: Proc. of IEEE International Conference on Robotics and Automation (San Francisco 2000) pp. 850–856

    Google Scholar 

  20. O. Khatib: A Unified Approach to Motion and Force Control of Robot Manipulators: The Operational Space Formulation, IEEE J. Robot. Autom. 3(1), 43–53 (1987)

    Article  Google Scholar 

  21. Y.F. Zheng, H. Hemami: Mathematical Modeling of a Robot Collision with its Environment, J. Robot. Syst. 2(3), 289–307 (1985)

    Article  Google Scholar 

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

    Google Scholar 

  23. J. Denavit, R.S. Hartenberg: A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices, J. Appl. Mech. 22, 215–221 (1955)

    MATH  MathSciNet  Google Scholar 

  24. 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. In: Proc. of IFAC/IFIP/IMACS International Symposium on Theory of Robots, (Vienna 1986)

    Google Scholar 

  25. R. Featherstone: Efficient Factorization of the Joint Space Inertia Matrix for Branched Kinematic Trees, Int. J. Robot. Res. 24(6), 487–500 (2005)

    Article  Google Scholar 

  26. R. Featherstone: An Empirical Study of the Joint Space Inertia Matrix, Int. J. Robot. Res. 23(9), 859–871 (2004)

    Article  Google Scholar 

  27. K. Kreutz-Delgado, A. Jain, G. Rodriguez: Recursive Formulation of Operational Space Control, Proc. of IEEE International Conference on Robotics and Automation (Sacramento, CA April 1991) pp. 1750–1753

    Chapter  Google Scholar 

  28. K.W. Lilly: Efficient Dynamic Simulation of Robotic Mechanisms (Kluwer Academic, Norwell, MA 1993)

    MATH  Google Scholar 

  29. 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 

  30. 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 

  31. J. Wittenburg: Dynamics of Systems of Rigid Bodies (B.G. Teubner, Stuttgart 1977)

    MATH  Google Scholar 

  32. R. Featherstone, D.E. Orin: Robot Dynamics: Equations and Algorithms. In: Proc. of IEEE International Conference on Robotics and Automation, (San Francisco, April 2000) pp. 826–834

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  43. F.C. Park, J.E. Bobrow, S.R. Ploen: A Lie Group Formulation of Robot Dynamics, Int. J. Robot. Res. 14(6), 609–618 (1995)

    Article  Google Scholar 

  44. M.E. Kahn, B. Roth: The Near Minimum-time Control of Open-loop Articulated Kinematic Chains, J. Dyn. Syst. Measur. Control 93, 164–172 (1971)

    Article  Google Scholar 

  45. J.J. Uicker: Dynamic Force Analysis of Spatial Linkages, Trans. ASME J. Appl. Mech. 34, 418–424 (1967)

    Google Scholar 

  46. A. Jain: Unified Formulation of Dynamics for Serial Rigid Multibody Systems, J. Guid. Control Dyn. 14(3), 531–542 (1991)

    Article  MATH  Google Scholar 

  47. G. Rodriguez: Kalman Filtering, Smoothing, and Recursive Robot Arm Forward and Inverse Dynamics, IEEE J. Robot. Autom. RA-3(6), 624–639 (1987)

    Article  Google Scholar 

  48. G. Rodriguez, A. Jain, K. Kreutz-Delgado: A Spatial Operator Algebra for Manipulator Modelling and Control, Int. J. Robot. Res. 10(4), 371–381 (1991)

    Article  Google Scholar 

  49. 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 

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

    Google Scholar 

  51. K.W. Buffinton: Kaneʼs Method in Robotics. In: Robotics and Automation Handbook, ed. by T.R. Kurfess (CRC, Boca Raton, FL 2005), 6-1 to 6-31

    Google Scholar 

  52. T.R. Kane, D.A. Levinson: The Use of Kaneʼs Dynamical Equations in Robotics, Int. J. Robot. Res. 2(3), 3–21 (1983)

    Article  Google Scholar 

  53. C.A. Balafoutis, R.V. Patel, P. Misra: Efficient Modeling and Computation of Manipulator Dynamics Using Orthogonal Cartesian Tensors, IEEE J. Robot. Autom. 4, 665–676 (1988)

    Article  Google Scholar 

  54. X. He, A.A. Goldenberg: An Algorithm for Efficient Computation of Dynamics of Robotic Manipulators. In: Proc. of Fourth International Conference on Advanced Robotics, (Columbus, OH, 1989) pp. 175–188

    Google Scholar 

  55. W. Hu, D.W. Marhefka, D.E. Orin: Hybrid Kinematic and Dynamic Simulation of Running Machines, IEEE Trans. Robot. 21(3), 490–497 (2005)

    Article  Google Scholar 

  56. 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 

  57. K.W. Lilly, D.E. Orin: Alternate Formulations for the Manipulator Inertia Matrix, Int. J. Robot. Res. 10, 64–74 (1991)

    Article  Google Scholar 

  58. S. McMillan, D.E. Orin: Forward dynamics of multilegged vehicles using the composite rigid body method, Proc. IEEE International Conference on Robotics and Automation (1998) pp. 464–470

    Google Scholar 

  59. U.M. Ascher, D.K. Pai, B.P. Cloutier: Forward Dynamics: Elimination Methods, and Formulation Stiffness in Robot Simulation, Int. J. Robot. Res. 16(6), 749–758 (1997)

    Article  Google Scholar 

  60. 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. Robot. Res. 18(9), 876–892 (1999)

    Article  Google Scholar 

  61. MSC Software Corporation: Adams, [On-line] http://www.mscsoftware.com/ (Nov. 12 2007)

    Google Scholar 

  62. T. Kane, D. Levinson: Autolev Userʼs Manual (OnLine Dynamics Inc., 2005)

    Google Scholar 

  63. 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, NM 1995) pp. 73–98

    Google Scholar 

  64. R. Smith: Open Dynamics Engine User Guide, Available online: http://www.ode.org (Nov. 12 2007)

    Google Scholar 

  65. Microsoft Corporation: Robotics Studio [On-line] http:www.microsoft.com/robotics (Nov. 12 2007)

    Google Scholar 

  66. P.I. Corke: A Robotics Toolbox for MATLAB, IEEE Robot. Autom. Mag. 3(1), 24–32 (1996)

    Article  Google Scholar 

  67. M.G. Hollars, D.E. Rosenthal, M.A. Sherman: SD/FAST Userʼs Manual (Symbolic Dynamics Inc., 1994)

    Google Scholar 

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

    Google Scholar 

  69. Cyberbotics Ltd.: Webots User Guide, Available online: http://www.cyberbotics.com (Nov. 8 2007)

    Google Scholar 

  70. I.C. Brown, P.J. Larcombe: A Survey of Customised Computer Algebra Programs for Multibody Dynamic Modelling. In: The Use of Symbolic Methods in Control System Analysis and Design, ed. by N. Munro (The Institute of Engineering and Technology, London 1999) pp. 53–77

    Google Scholar 

  71. J.J. Murray, C.P. Neuman: ARM: An algebraic robot dynamic modeling program. In: Proc. of IEEE International Conference on Robotics and Automation, Atlanta, Georgia, March (1984) pp. 103–114

    Google Scholar 

  72. 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 

  73. F.C. Park, J. Choi, S.R. Ploen: Symbolic Formulation of Closed Chain Dynamics in Independent Coordinates, Mech. Machine Theory 34, 731–751 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  74. M. Vukobratovic, N. Kircanski: Real-time Dynamics of Manipulation Robots. In: Scientific Fundamentals of Robotics, Vol. 4 (Springer-Verlag, New York 1985)

    Google Scholar 

  75. J. Wittenburg, U. Wolz: Mesa Verde: A Symbolic Program for Nonlinear Articulated-Rigid-Body Dynamics. In: ASME Design Engineering Division Conference and Exhibit on Mechanical Vibration and Noise, Cincinnati, Ohio, ASME Paper No. 85-DET-151, 1-8, September (1985)

    Google Scholar 

  76. 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 

  77. D.E. Orin: Pipelined Approach to Inverse Plant Plus Jacobian Control of Robot Manipulators. In: Proc. of IEEE International Conference on Robotics and Automation, Atlanta, Georgia, 169–175, March (1984)

    Google Scholar 

  78. R.H. Lathrop: Parallelism in Manipulator Dynamics, Int. J. Robot. Res. 4(2), 80–102 (1985)

    Article  Google Scholar 

  79. 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 

  80. 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 

  81. 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 

  82. M. Amin-Javaheri, D.E. Orin: Parallel Algorithms for Computation of the Manipulator Inertia Matrix, Int. J. Robot. Res. 10(2), 162–170 (1991)

    Article  Google Scholar 

  83. A. Fijany, A.K. Bejczy: A Class of Parallel Algorithms for Computation of the Manipulator Inertia Matrix, IEEE Trans. Robot. Autom. 5(5), 600–615 (1989)

    Article  Google Scholar 

  84. 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 

  85. A. Fijany, I. Sharf, G.M.T. DʼEleuterio: Parallel O(logN) Algorithms for Computation of Manipulator Forward Dynamics, IEEE Trans. Robot. Autom. 11(3), 389–400 (1995)

    Article  Google Scholar 

  86. 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. Robot. Res. 18(9), 867–875 (1999)

    Article  Google Scholar 

  87. R. Featherstone, A. Fijany: A Technique for Analyzing Constrained Rigid-Body Systems and Its Application to the Constraint Force Algorithm, IEEE Trans. Robot. Autom. 15(6), 1140–1144 (1999)

    Article  Google Scholar 

  88. P.S. Freeman, D.E. Orin: Efficient Dynamic Simulation of a Quadruped Using a Decoupled Tree-Structured Approach, Int. J. Robot. Res. 10, 619–627 (1991)

    Article  Google Scholar 

  89. Y. Nakamura, K. Yamane: Dynamics Computation of Structure-Varying Kinematic Chains and Its Application to Human Figures, IEEE Trans. Robot. Autom. 16(2), 124–134 (2000)

    Article  Google Scholar 

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Authors and Affiliations

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

    Roy Featherstone Dr.

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

    David E. Orin Prof

Authors
  1. Roy Featherstone Dr.
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  2. David E. Orin Prof
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Correspondence to Roy Featherstone Dr. or David E. Orin Prof .

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Editors and Affiliations

  1. PRISMA Lab, Dipartimento di Informatica e Sistemistica, Universitá degli Studi di Napoli Federico II, 80125, Napoli, Italy, Via Claudio 21

    Bruno Siciliano Prof.

  2. Artificial Intelligence Laboratory, Department of Computer Science, Stanford University, 94305-9010, Stanford, CA, USA

    Oussama Khatib Prof.

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Featherstone, R., Orin, D.E. (2008). Dynamics. In: Siciliano, B., Khatib, O. (eds) Springer Handbook of Robotics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30301-5_3

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  • DOI: https://doi.org/10.1007/978-3-540-30301-5_3

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