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 Ld qk 1 qk t D1 Ld qk qk 1 t 0 Discrete Euler Lagrange Equations 14 Variational Integrator Solve for qk 1 D2 Ld qk 1 qk h D1 Ld qk qk 1 h 0 qk qk qk 1 h L q k 1 h qk qk 1 qk h 0 L q k h L qk qk 1 L qk qk 1 L qk 1 qk L qk 1 qk q h q q h q k 1 k 1 k k 0 q h q h h q h q 15 Solution Nonlinear Root Finder f qk 1 D2 Ld qk 1 qk h D1 Ld qk qk 1 h 0 q i 1 k 1 q i k 1 i k 1 i k 1 f q Df q 16 Simple Example Spring Mass Continuous Lagrangian 1 2 1 2 L q q mx kx 2 2 L d L Euler Lagrange Equations kx m x 0 q dt q Simple Integration Scheme 1 2 k xk 1 xk hx k h xk 2 m k x k 1 x k h xk m 17 Simple Example Spring Mass Discrete Lagrangian 2 1 xk 1 xk 1 2 Ld xk xk 1 h m kxk 2 h 2 Discrete Euler Lagrange Equations D2 Ld xk 1 xk h D1 Ld xk xk 1 h 0 m 2 xk 1 2 xk xk 1 kxk 0 h Integration h2k xk xk 1 xk 1 2 m 18 Comparison 3 Types of Integrators Euler easiest least accurate Runge Kutta more complicated more accurate Variational EASY ACCURATE 19 Euler h 0 001 Runge Kutta ode45 Variational h 0 001 0 04 0 02 velocity 0 0 02 0 04 0 06 0 08 0 97 0 98 0 99 1 1 01 1 02 position 1 03 1 04 1 05 1 06 20 0 51 Euler h 0 001 Runge Kutta ode45 Variational h 0 001 0 508 Energy 0 506 0 504 0 502 0 5 0 498 Notice 0 10 20 30 40 50 time s 60 70 80 90 100 Energy does not dissipate over time Energy error is bounded 21 Variational Integrators are Symplectic Simple explanation area of the cat head remains constant over time Stein A Desbrun M Discrete geometric mechanics for variational time integrators 22in Discrete Differential Geometry ACM SIGGRAPH Course Notes 2006 Forcing Functions Discretization of Lagrange d Alembert principle 23 Constraints D2 Ld qk 1 qk h D1 Ld qk qk 1 h g qk T k f z k 1 0 g qk 1 qk 1 z k 1 k i f z i 1 i k 1 z k 1 z k 1 i Df z k 1 24 Example Constrained Double Pendulum w Damping 2 x g q 0 y x y q 1 2 1 F q 0 0 K 1 K 2 x y 25 Example Constrained Double Pendulum w Damping Constraints strictly enforced h 0 1 No stabilization heuristics required 26 Complex Examples From Literature E Johnson T Murphey Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates IEEE Transactions on Robotics 2009 a k a Beware of ODE 27 Complex Examples From Literature Variational Integrator ODE 28 Complex Examples From Literature 29 log Complex Examples From Literature Timestep was decreased until error was below threshold leading to longer runtimes 30 Applications Marionette Robots E Johnson and T Murphey Discrete and Continuous Mechanics for Tree Represenatations of Mechanical Systems ICRA 2008 31 Applications Hand modeling E Johnson K Morris and T Murphey A Variational Approach to Stand Based Modeling 32 of the Human Hand Algorithmic Foundations of Robotics VII 2009 Applications Non smooth dynamics Fetecau R C and Marsden J E and Ortiz M and West M 2003 Nonsmooth Lagrangian mechanics and variational collision integrators SIAM Journal on Applied Dynamical Systems 33 Applications Structural Mechanics Kedar G Kale and Adrian J Lew Parallel asynchronous variational integrators International Journal for Numerical Methods in Engineering 2007 34 Applications Trajectory optimization O Junge J E Marsden S Ober Bl baum Discrete Mechanics and Optimal Control in Proccedings of the 16th IFAC World Congress 2005 35 Summary Discretization of the variational principle results in symplectic discrete equations of motion Variational integrators perform better than almost all other integrators This work is being applied to the analysis of robotic systems 36 Discussion What else can this idea be applied to Optimal Control is also derived from a variational principle Pontryagin s Minimum Principle This idea should be taught in calculus and or dynamics courses We don t need accurate simulation because real systems never agree 37