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

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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
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http//www.cs.cmu.edu/bstephe1
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But this talk is not about that

• ?

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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
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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

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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
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Goals for the Talk

• Fundamentals (and a little History)
• Simple Examples/Comparisons
• Related Work and Applications
• Discussion

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The Continuous Lagrangian

• Q configuration space
• TQ tangent (velocity) space
• LTQ?R

Kinetic Energy
Potential Energy
Lagrangian
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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
Hamiltons Principle 1835
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Continuous Lagrangian
Euler-Lagrange Equations
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Continuous Mechanics
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The Discrete Lagrangian

• LQxQ?R

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Variation of Discrete Lagrangian
Discrete Euler-Lagrange Equations
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Variational Integrator

• Solve for

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Solution Nonlinear Root Finder
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Simple Example Spring-Mass

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

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Simple Example Spring-Mass

• Discrete Lagrangian
• Discrete Euler-Lagrange Equations
• Integration

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Comparison 3 Types of Integrators

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

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• Notice
• Energy does not dissipate over time
• Energy error is bounded

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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
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Forcing Functions

• Discretization of LagrangedAlembert principle

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Constraints
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Example Constrained Double Pendulum w/ Damping
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Example Constrained Double Pendulum w/ Damping

• Constraints strictly enforced, h0.1

No stabilization heuristics required!
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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

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Complex Examples From Literature
Variational Integrator
ODE
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Complex Examples From Literature
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Complex Examples From Literature
log
Timestep was decreased until error was below
threshold, leading to longer runtimes.
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Applications

• Marionette Robots

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

• Structural Mechanics

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

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Discussion

• What else can this idea be applied to?
• Optimal Control is also derived from a
  variational principle (Pontryagins Minimum
  Principle).
• This idea should be taught in calculus and/or
  dynamics courses.
• We dont need accurate simulation because real
  systems never agree.

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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, 1834