Rigid Body Kinematics

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

Abstract

Rigid body kinematics is the study of the displacement, velocity, and acceleration of a rigid body with respect to a reference. We introduce the notion of reference frames and associate each reference frame with a rigid body. We thus achieve a mathematical framework for describing how rigid bodies move with respect to each other and with respect to the inertial reference frame.

Single rigid body motion serves as the basis for multibody motion. In addition to describe the configuration and motion space of vehicles and mobile robots, single rigid body kinematics is also the basis of robotics, i.e., multibody motion with additional kinematic constraints imposed on the motion space. Based on the concepts introduced in Chap.  1, the location, velocity, and acceleration of single rigid bodies are described in terms of well-defined mathematical entities. This chapter thus serves as an introduction to rigid body modeling as a part of an introductory robotics course.

Keywords

Manifold Torque Expense 

References

  1. 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
  2. Duindam, V. (2006). Port-based modeling and control for efficient bipedal walking robots. Ph.D. Thesis, University of Twente. Google Scholar
  3. 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
  4. 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
  5. Egeland, O., & Gravdahl, J. T. (2003). Modeling and simulation for automatic control. Trondheim: Marine Cybernetics AS Google Scholar
  6. Fossen, T. I. (1994). Guidance and control of Ocean vehicles. Chichester: Wiley. Google Scholar
  7. Fossen, T. I. (2002). Marine control systems. Trondheim: Marine Cybernetics AS. 3rd printing. Google Scholar
  8. 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
  9. 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
  10. From, P. J., Duindam, V., Pettersen, K. Y., Gravdahl, J. T., & Sastry, S. (2010). Singularity-free dynamic equations of vehicle-manipulator systems. Simulation Modelling Practice and Theory, 18(6), 712–731. CrossRefGoogle Scholar
  11. Gallier, J. (2001). Texts in applied mathematics series. Geometric methods and applications: for computer science and engineering. Berlin: Springer. CrossRefGoogle Scholar
  12. Hanson, A. J. (2006). Visualizing quaternions. San Francisco: Morgan Kaufmann. Google Scholar
  13. Hervé, J. M. (1978). Analyse structurelle des mécanismes par groupe des déplacements. Mechanism and Machine Theory, 13(4), 435–437. CrossRefGoogle Scholar
  14. Kuipers, J. B. (2002). Quaternions and rotation sequences. Princeton: Princeton University Press. Google Scholar
  15. Marsden, J. E., & Ratiu, T. S. (1999). Texts in applied mathematics. Introduction to mechanics and symmetry (2nd ed.). New York: Springer. CrossRefMATHGoogle Scholar
  16. Meng, J., Liu, G., & Li, Z. (2007). A geometric theory for analysis and synthesis of sub-6 DoF parallel manipulators. IEEE Transactions on Robotics, 23(4), 625–649. CrossRefGoogle Scholar
  17. Murray, R. M., Li, Z., & Sastry, S. S. (1994). A mathematical introduction to robotic manipulation. Boca Raton: CRC Press. MATHGoogle Scholar
  18. Rao, A. (2006). Dynamics of particles and rigid bodies—a systematic approach. Cambridge: Cambridge University Press. Google Scholar
  19. Rossmann, W. (2002). Lie groups—an introduction through linear algebra. Oxford: Oxford Science Publications. Google Scholar
  20. Selig, J. M. (2000). Geometric fundamentals of robotics. New York: Springer. CrossRefGoogle Scholar
  21. Siciliano, B., Sciavicco, L., Villani, L., & Oriolo, G. (2011). Advanced textbooks in control and signal processing. Robotics: modelling, planning and control. Berlin: Springer. Google Scholar
  22. Tu, L. W. (2008). An introduction to manifolds. New York: Springer. MATHGoogle Scholar
  23. Wen, J. T.-Y., & Kreutz-Delgado, K. (1991). The attitude control problem. IEEE Transactions on Automatic Control, 30(10), 1148–1162. MathSciNetCrossRefGoogle 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

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