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Discrete Variational Mechanics

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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. About My Research. Humanoid balance using simple models

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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

About My Research

- Humanoid balance using simple models
- Compliant floating body force control
- Dynamic push recovery planning by trajectory optimization

http://www.cs.cmu.edu/~bstephe1

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

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

Motivation

- Physically meaningful dynamics simulation

Stein A., Desbrun M. “Discrete geometric mechanics for variational time integrators” in 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

The Continuous Lagrangian

- Q – configuration space
- TQ – tangent (velocity) space
- L:TQ→R

Lagrangian

Kinetic Energy

Potential Energy

Variation of the Lagrangian

- Principle of Least Action = the function, q*(t), minimizes the integral of the Lagrangian

“Calculus of Variations” ~ Lagrange, 1760

Variation of trajectory with endpoints fixed

“Hamilton’s Principle” ~1835

Continuous Lagrangian

“Euler-Lagrange Equations”

The Discrete Lagrangian

- L:QxQ→R

Variation of Discrete Lagrangian

“Discrete Euler-Lagrange Equations”

Variational Integrator

- Solve for :

Simple Example: Spring-Mass

- Continuous Lagrangian:
- Euler-Lagrange Equations:
- Simple Integration Scheme:

Simple Example: Spring-Mass

- Discrete Lagrangian:
- Discrete Euler-Lagrange Equations:
- Integration:

Comparison: 3 Types of Integrators

- Euler – easiest, least accurate
- Runge-Kutta – more complicated, more accurate
- Variational – EASY & ACCURATE!

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

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” in Discrete Differential Geometry. ACM SIGGRAPH Course Notes, 2006

Forcing Functions

- Discretization of Lagrange–d’Alembert principle

Example: Constrained Double Pendulum w/ Damping

- Constraints strictly enforced, h=0.1

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”

Complex Examples From Literature

log

Timestep was decreased until error was below threshold, leading to longer runtimes.

Applications

- Marionette Robots

E. Johnson and T. Murphey, “Discrete and Continuous Mechanics for Tree Represenatations of Mechanical Systems,” ICRA 2008

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, 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

Applications

- Structural Mechanics

Kedar G. Kale and Adrian J. Lew, “Parallel asynchronous variational integrators,” International Journal for Numerical Methods in Engineering, 2007

Applications

- Trajectory optimization

O. Junge, J.E. Marsden, S. Ober-Blöbaum, “Discrete Mechanics and Optimal Control”, in Proccedings of the 16th IFAC World Congress, 2005

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

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.

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