Variational integration for articulated body dynamics

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Variational integration for articulated body
dynamics

• Rahul Narain
• COMP790 project progress report
• Nov. 16, 2006

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Background

• Variational principles in classical mechanics
• e.g. Principle of least action
• Path of system minimizes action ? L(q,v) dt
• where L(q,v) T(v) - U(q)
• Discretize integral ? time-stepping scheme

Kinetic energy
Potential energy
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The discrete equations

• qk1 - qk hkvk1
• pk1 - pk ?Ld(qk,vk1)/?vk1 Fd(qk,vk1)
  hk?k?g(qk)
• hkpk1 ?Ld(qk,vk1)/?qk
• g(qk1) 0
• Need to plug in external forces,derivatives of
  Lagrangian

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Current status

• Worked out the derivatives of the Lagrangian
• for both reduced coordinate and maximal
  coordinate formulations
• Ready to put into code

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Remaining work

• Implement!
• I might need (constrained?) non-linear
  optimization code
• Practical comparison of speed, accuracy, etc.
  with other integration methods

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Long-term goals

• More general animation scenarios
• Contact handling
• Collision response
• Adaptive dynamics
• in the context of variational integration

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References

• L. Kharevych, Weiwei, Y. Tong, E. Kanso, J. E.
  Marsden, P. Schröder, and M. Desbrun. Geometric,
  Variational Integrators for Computer Animation.
  Eurographics/ACM SIGGRAPH Symposium on Computer
  Animation, 2006.
• A. Stern and M. Desbrun. Discrete Geometric
  Mechanics for Variational Time Integrators. In
  course notes of Discrete Differential Geometry
  An Applied Introduction, course at ACM SIGGRAPH
  2006.
• J.E. Marsden and M. West. Discrete mechanics and
  variational integrators. Acta Numerica 10, 2001.