Preliminary Mathematical Concepts

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


This chapter presents the main mathematical tools used throughout the book. The necessary mathematical background is presented, which in turn will be used to develop mathematical models that accurately describe the mechanical behavior of our systems, which is the main goal of mathematical modeling. The models are derived so that they are well suited for analysis, simulation, and control of the mechanical system.

In this chapter the most important tools from Lie theory and differential geometry are presented in a brief but concise manner. The theory is presented in such a way that it can serve as an introduction to differential geometry for roboticists not familiar with the topic. We focus on the tools and properties required for modeling mechanical systems with focus on presenting the reader with a well-defined framework well suited to derive a singularity-free formulation of the kinematics and dynamics of complex mechanical systems. Even though the main mathematical tools are presented in this chapter, the readers that are not interested in a deep mathematical analysis of these systems, but rather a more practical approach, can go straight to Chap.  3 on rigid body modeling.


Reference Frame Rigid Body Multibody System Configuration Space Inertial Frame 
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.


  1. Absil, P.-A., Mahony, R., & Sepulchre, R. (2008). Optimization algorithms on matrix manifolds. Princeton: Princeton University Press. zbMATHGoogle Scholar
  2. Antonelli, G. (2006). Underwater robots. Motion and force control of vehicle-manipulator systems. Berlin: Springer. Google Scholar
  3. Brockett, R. W. (1984). Robotic manipulators and the product of exponentials formula. In Proceedings of mathematical theory of networks and systems, Beer Sheva, Israel (pp. 120–129). CrossRefGoogle Scholar
  4. 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
  5. Cartan, E., & Adam, P. (2000). Sur la structure des groupes de transformations finis et continus. Nony. Google Scholar
  6. Cayley, A. (1854). On the theory of groups, as depending on the symbolic equation θ n=1. Philosophical Magazine, 7(47), 40–47. Google Scholar
  7. Crouch, P. E., & Grossman, R. (1993). Numerical integration of ordinary differential equations on manifolds. Nonlinear Science, 3(1), 1–33. MathSciNetCrossRefzbMATHGoogle Scholar
  8. Duindam, V. (2006). Port-based modeling and control for efficient bipedal walking robots. Ph.D. thesis, University of Twente. Google Scholar
  9. Engø, K., & Marthinsen, A. (1997). Application of geometric integration to some mechanical problems. Multibody System Dynamics, 2(1), 71–88. CrossRefGoogle Scholar
  10. Fossen, T. I. (2002). Marine control systems. Trondheim: Marine Cybernetics AS. 3rd printing. Google Scholar
  11. Gingsberg, J. (2007). Engineering dynamics. New York: Cambridge University Press. CrossRefGoogle Scholar
  12. Hairer, E. (2001). Geometric integration of ordinary differential equations on manifolds. BIT Numerical Mathematics, 41(2), 996–1007. MathSciNetCrossRefGoogle Scholar
  13. Hughes, P. C. (2002). Spacecraft attitude dynamics. Mineola: Dover Publications. Google Scholar
  14. Iserles, A. (1984). Solving linear ordinary differential equations by exponentials of iterated commutators. Numerische Mathematik, 45(2), 183–199. MathSciNetCrossRefzbMATHGoogle Scholar
  15. Jordan, D. W., & Smith, P. (2004). Nonlinear ordinary differential equations—an introduction to dynamical systems. Oxford: Oxford University Press. Google Scholar
  16. Killing, W. (1888). Mathematische Annalen: Vol. 31. Die Zusammensetzung der stetigen/endlichen Transformationsgruppen Google Scholar
  17. Kwatny, H. G., & Blankenship, G. (2000). Nonlinear control and analytical mechanics a computational approach. Boston: Birkhäuser. CrossRefzbMATHGoogle Scholar
  18. Lee, T., Leok, M., & McClamroch, N. (2011). Geometric numerical integration for complex dynamics of tethered spacecraft. In Proceedings of American control conference, San Francisco, California, USA (pp. 1885–1891). Google Scholar
  19. Lee, T., McClamroch, N., & Leok, M. (2005). A lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3d pendulum. In Proceedings of IEEE conference on control applications, Toronto, Canada (pp. 962–967). Google Scholar
  20. Lie, S. (1888). Encyclopædia Britannica. Theorie der Transformationsgruppen I. Google Scholar
  21. Lie, S. (1890). Encyclopædia Britannica. Theorie der Transformationsgruppen II. Google Scholar
  22. Lie, S. (1893). Encyclopædia Britannica. Theorie der Transformationsgruppen III. Google Scholar
  23. McLachlan, R. I., & Quispel, G. R. W. (2006). Geometric integrators for ODEs. Journal of Physics. A, Mathematical and General, 39(19), 5251–5285. MathSciNetCrossRefzbMATHGoogle Scholar
  24. Meng, J., Liu, G., & Li, Z. (2005). A geometric theory for synthesis and analysis of sub-6 DoF parallel manipulators. In Proceedings of the IEEE international conference on robotics and automation, Barcelona, Spain (pp. 3342–3347). Google Scholar
  25. 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
  26. Munthe-Kaas, H. (1998). Runge-Kutta methods on lie groups. BIT Numerical Mathematics, 38(1), 92–111. MathSciNetCrossRefzbMATHGoogle Scholar
  27. Murray, R. M., Li, Z., & Sastry, S. S. (1994). A mathematical introduction to robotic manipulation. Boca Raton: CRC Press. zbMATHGoogle Scholar
  28. Park, J., & Chung, W.-K. (2005). Geometric integration on Euclidean group with application to articulated multibody systems. IEEE Transactions on Robotics, 21(5), 850–863. CrossRefGoogle 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. Study, E. (1903). Geometrie der Dynamen. Die Zusammensetzung von kräften und verwandte gegenstände der Geometrie. Ithaca: Cornell University Library. zbMATHGoogle Scholar
  32. Tapp, K. (2005). Matrix groups for undergraduates. New York: Am. Math. Soc. zbMATHGoogle Scholar
  33. Tu, L. W. (2008). An introduction to manifolds. New York: Springer. zbMATHGoogle 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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