Advertisement

Rigid Body Dynamics

  • Pål Johan From
  • Jan Tommy Gravdahl
  • Kristin Ytterstad Pettersen
Chapter
Part of the Advances in Industrial Control book series (AIC)

Abstract

Dynamics is the study of how forces affect the motion of rigid bodies. In this chapter we introduce the fundamental topics required to derive the dynamic equations for rigid bodies with the results obtained in the previous chapters on rigid body kinematics as a starting point. In this way we obtain a well-defined formulation of the dynamics without singularities and other artifacts. The formulation can be used to derive the dynamics of bodies with different configuration spaces, i.e., both flat Euclidean spaces and non-Euclidean configuration spaces on manifolds. The equations are well suited for simulation and controller design.

Keywords

Rigid Body Velocity Variable Configuration Space Lagrange Equation Position Variable 
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.

References

  1. Arnold, V. I. (1989). Mathematical methods of classical mechanics. Berlin: Springer. CrossRefGoogle Scholar
  2. Bremer, H. (1988). Über eine zentralgleichung in der Dynamik (Vol. 68, pp. 307–311). Google Scholar
  3. Bullo, F., & Lewis, A. D. (2000). Geometric control of mechanical systems: modeling, analysis, and design for simple mechanical control systems. New York: Springer. Google Scholar
  4. Cameron, J. M., & Book, W. J. (1997). Modeling mechanisms with nonholonomic joints using the Boltzmann-Hamel equations. The International Journal of Robotics Research, 16(1), 47–59. CrossRefGoogle Scholar
  5. Duindam, V., & Stramigioli, S. (2007). Lagrangian dynamics of open multibody systems with generalized holonomic and nonholonomic joints. In Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, San Diego, CA, USA (pp. 3342–3347). Google Scholar
  6. Duindam, V., & Stramigioli, S. (2008). Singularity-free dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints. IEEE Transactions on Robotics, 24(3), 517–526. CrossRefGoogle Scholar
  7. Egeland, O., & Gravdahl, J. T. (2003). Modeling and simulation for automatic control. Trondheim: Marine Cybernetics AS. Google Scholar
  8. Fossen, T. I. (2002). Marine control systems. Trondheim: Marine Cybernetics AS. 3rd printing. Google Scholar
  9. Fossen, T. I., & Fjellstad, O. E. (1995). Nonlinear modelling of marine vehicles in 6 degrees of freedom. International Journal of Mathematical Modelling Systems, 1(1), 17–28. Google Scholar
  10. From, P. J. (2012a). An explicit formulation of singularity-free dynamic equations of mechanical systems in Lagrangian form—part one: single rigid bodies. Modeling, Identification and Control, 33(2), 45–60. CrossRefGoogle Scholar
  11. From, P. J. (2012b). An explicit formulation of singularity-free dynamic equations of mechanical systems in Lagrangian form—part two: multibody systems. Modeling, Identification and Control, 33(2), 61–68. CrossRefGoogle Scholar
  12. From, P. J., Duindam, V., & Stramigioli, S. (2012). Corrections to “Singularity-free dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints”. IEEE Transactions on Robotics, 28(6), 1431–1432. CrossRefGoogle Scholar
  13. Gibbs, J. W. (1879). On the fundamental formulae of dynamics. American Journal of Mathematics, 2(1), 49–64. MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hamel, G. (1949). Theoretische Mechanik. Berlin: Springer. CrossRefzbMATHGoogle Scholar
  15. Herman, P., & Kozlowski, K. (2006). A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators. Archive of Applied Mechanics, 76(9–10), 579–614. CrossRefzbMATHGoogle Scholar
  16. Jarzbowska, E. (2008). Quasi-coordinates based dynamics modeling and control design for nonholonomic systems. Nonlinear Analysis, 71(12), 118–131. CrossRefGoogle Scholar
  17. Kane, T. R., & Levinson, D. A. (1985). Dynamics: theory and applications. New York: McGraw Hill. Google Scholar
  18. Kane, T. R., Likins, P. W., & Levinson, D. A. (1983). Spacecraft dynamics. New York: McGraw Hill. Google Scholar
  19. Kozlowski, K., & Herman, P. (2008). Control of robot manipulators in terms of quasi-velocities. Journal of Intelligent & Robotic Systems, 53(3), 205–221. CrossRefGoogle Scholar
  20. Kwatny, H. G., & Blankenship, G. (2000). Nonlinear control and analytical mechanics a computational approach. Boston: Birkhäuser. CrossRefzbMATHGoogle Scholar
  21. Lagrange, J.-L. (1788). Mécanique analytique. Chez la Veuve Desaint. Google Scholar
  22. Lesser, M. (1992). A geometrical interpretation of Kanes equations. Journal of Mathematical and Physical Sciences, 436(1896), 69–87. MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lewis, A. D. (1996). The geometry of the Gibbs-Appel equations and Gauss’s principle of least constraint. Reports on Mathematical Physics, 38(1), 11–28. MathSciNetCrossRefzbMATHGoogle Scholar
  24. Marsden, J. E., & Ratiu, T. S. (1999). Texts in applied mathematics. Introduction to mechanics and symmetry (2nd ed.). New York: Springer. CrossRefzbMATHGoogle Scholar
  25. Maruskin, J. M., & Bloch, A. M. (2007). The Boltzmann-Hamel equations for optimal control. In IEEE conference on decision and control, San Diego, CA, USA (pp. 554–559). Google Scholar
  26. Murray, R. M., Li, Z., & Sastry, S. S. (1994). A mathematical introduction to robotic manipulation. Boca Raton: CRC Press. zbMATHGoogle Scholar
  27. Park, F. C., Bobrow, J. E., & Ploen, S. R. (1995). A Lie group formulation of robot dynamics. The International Journal of Robotics Research, 14(6), 609–618. CrossRefGoogle Scholar
  28. Poincaré, H. (1901). Sur une forme nouvelle des équations de la mécanique. Bull Astron. Google Scholar
  29. Rao, A. (2006). Dynamics of particles and rigid bodies—a systematic approach. Cambridge: Cambridge University Press. Google Scholar
  30. Rossmann, W. (2002). Lie groups—an introduction through linear algebra. Oxford: Oxford science publications. Google Scholar
  31. Sagatun, S. I., & Fossen, T. I. (1992). Lagrangian formulation of underwater vehicles. In Conference of systems, man and cybernetics, Charlottesville, VA, USA (pp. 1029–1034). Google Scholar
  32. Selig, J. M. (2000). Geometric fundamentals of robotics. New York: Springer. CrossRefGoogle Scholar
  33. Tanner, H. G., & Kyriakopoulos, K. J. (2001). Mobile manipulator modeling with Kane’s approach. Robotica, 19, 675–690. CrossRefGoogle Scholar
  34. Zefran, M., & Bullo, F. (2004). Lagrangian dynamics, robotics and automation handbook. Boca Raton: CRC Press. Google Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Pål Johan From
    • 1
  • Jan Tommy Gravdahl
    • 2
  • Kristin Ytterstad Pettersen
    • 2
  1. 1.Department of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway
  2. 2.Department of Engineering CyberneticsNorwegian Univ. of Science & TechnologyTrondheimNorway

Personalised recommendations