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Discrete Variational Mechanics Benjamin Stephens J E Marsden and M West Discrete mechanics and variational integrators Acta Numerica No 10 pp 357 514 2001 M West Variational Integrators PhD Thesis Caltech 2004 1 About My Research Humanoid balance using simple models Compliant floating body force control Dynamic push recovery planning by trajectory optimization C C t F PL PL t PR http www cs cmu edu bstephe1 2 http www cs cmu edu bstephe1 3 But this talk is not about that The Principle of Least Action The spectacle of the universe seems all the more grand and beautiful and worthy of its Author when one considers that it is all derived from a small number of laws laid down most wisely Maupertuis 1746 5 The Main Idea Equations of motion are derived from a variational principle Traditional integrators discretize the equations of motion Variational integrators discretize the variational principle 6 Motivation Physically meaningful dynamics simulation Stein A Desbrun M Discrete geometric mechanics for variational time integrators 7in Discrete Differential Geometry ACM SIGGRAPH Course Notes 2006 Goals for the Talk Fundamentals and a little History Simple Examples Comparisons Related Work and Applications Discussion 8 The Continuous Lagrangian Q configuration space TQ tangent velocity space L TQ R L q q T q q U q Lagrangian Kinetic Energy Potential Energy 9 Variation of the Lagrangian Principle of Least Action the function q t minimizes the integral of the Lagrangian T T L q t q t dt L q t q t q t q t dt 0 0 Calculus of Variations Lagrange 1760 T Variation of trajectory with endpoints fixed L q t q t dt 0 0 Hamilton s Principle 1835 10 Continuous Lagrangian L d L 0 q dt q Euler Lagrange Equations 11 Continuous Mechanics L q t q t T q q U q L d L d T U q dt q q dt q T U d T q q dt q T U 2T 2T q q q q q q q q M q q C q q G q 0 12 The Discrete Lagrangian T h L QxQ R L q t q t dt L q q d k k 1 h T qk 1 qk Ld qk qk 1 h L qk h h L qk h qk 1 13 Variation of Discrete Lagrangian D2



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