Discrete Variational Mechanics Benjamin Stephens J.E. Marsden and M. West, “Discrete mechanics and variational integrators,” Acta Numerica, No. 10, pp , 2001 M. West “Variational Integrators,” PhD Thesis, Caltech,

About My Research Humanoid balance using simple models Compliant floating body force control Dynamic push recovery planning by trajectory optimization 2

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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,

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

Physically meaningful dynamics simulation Motivation Stein A., Desbrun M. “Discrete geometric mechanics for variational time integrators” in Discrete Differential Geometry. ACM SIGGRAPH Course Notes,

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 Kinetic EnergyPotential EnergyLagrangian 9

Variation of the Lagrangian Principle of Least Action = the function, q*(t), minimizes the integral of the Lagrangian Variation of trajectory with endpoints fixed “Hamilton’s Principle” ~ “Calculus of Variations” ~ Lagrange, 1760

Continuous Lagrangian “Euler-Lagrange Equations” 11

Continuous Mechanics 12

The Discrete Lagrangian L:QxQ→R 13

Variation of Discrete Lagrangian “Discrete Euler-Lagrange Equations” 14

Variational Integrator Solve for : 15

Solution: Nonlinear Root Finder 16

Simple Example: Spring-Mass Continuous Lagrangian: Euler-Lagrange Equations: Simple Integration Scheme: 17

Simple Example: Spring-Mass Discrete Lagrangian: Discrete Euler-Lagrange Equations: Integration: 18

Comparison: 3 Types of Integrators Euler – easiest, least accurate Runge-Kutta – more complicated, more accurate Variational – EASY & ACCURATE! 19

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Notice: Energy does not dissipate over time Energy error is bounded 21

Stein A., Desbrun M. “Discrete geometric mechanics for variational time integrators” in Discrete Differential Geometry. ACM SIGGRAPH Course Notes, 2006 Variational Integrators are “Symplectic” Simple explanation: area of the cat head remains constant over time 22

Forcing Functions Discretization of Lagrange–d’Alembert principle 23

Constraints 24

Example: Constrained Double Pendulum w/ Damping 25

Example: Constrained Double Pendulum w/ Damping Constraints strictly enforced, h= No stabilization heuristics required!

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

Complex Examples From Literature log 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

Applications Hand modeling E. Johnson, K. Morris and T. Murphey, “A Variational Approach to Stand-Based Modeling of the Human Hand,” Algorithmic Foundations of Robotics VII,

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,

Trajectory optimization Applications O. Junge, J.E. Marsden, S. Ober-Blöbaum, “Discrete Mechanics and Optimal Control”, in Proccedings of the 16th IFAC World Congress,

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

Brief History of Lagrangian Mechanics Principle of Least Action – Liebniz, 1707; Euler, 1744; Maupertuis, 1746 Calculus of Variations – Lagrange, 1760 Méchanique Analytique – Lagrange, 1788 Lagrangian Mechanics – Hamilton,