Simulation of holonomic mechanical systems by means of automatic differentiation

We discuss both the theoretical framework and implementation issues for the simulation of holonomic mechanical systems using automatic differentiation. Our approach allows the direct simulation of such systems based on its Lagrangian. The Lagrangian may be given as explicit equations or as an algorithm containing control structures such as loops. The method is illustrated by the well-known ball and beam system.

[1]  Klaus Röbenack,et al.  Controller design for nonlinear multi-input – multi-output systems based on an algorithmic plant description , 2007 .

[2]  R. Abraham,et al.  Manifolds, tensor analysis, and applications: 2nd edition , 1988 .

[3]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[4]  Andreas Griewank,et al.  ADOL‐C: Automatic Differentiation Using Operator Overloading in C++ , 2003 .

[5]  Darryl D. Holm,et al.  Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions , 2009 .

[6]  Martin Berz,et al.  Computational differentiation : techniques, applications, and tools , 1996 .

[7]  K. Röbenack,et al.  Simulation of Nonholonomic Mechanical Systems Using Algorithmic Differentiation , 2012 .

[8]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

[9]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[10]  Andreas Griewank,et al.  Automatic Differentiation of Algorithms: From Simulation to Optimization , 2000, Springer New York.

[11]  Andreas Griewank,et al.  Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++ , 1996, TOMS.

[12]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[13]  Klaus Röbenack,et al.  Direct simulation of mechanical control systems using algorithmic differentiation , 2011 .

[14]  P. Kokotovic,et al.  Nonlinear control via approximate input-output linearization: the ball and beam example , 1992 .