Variational integration for articulated body dynamics

1
Variational integration for articulated body
dynamics

• Rahul Narain
• COMP790-072 final project presentation
• Dec. 13, 2006

2
Variational integration

• Instead of
• Mq'' F
• use variational principles
• min ? F(q,q') dt
• e.g. F Lagrangian L k.e. - p.e.
• More physically valid behaviour energy
  momentum conservation, etc.

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Variational principles

• Trajectory should be local min of functional
• S(q,q') ? F(q,q') dt
• S(qdq, q'dq) - S(q,q') ? 0

q(t)
q(t)dq(t)
t
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Variational integration

• Discretize the integral
• S(q) ? Fdk,k1
• S(qdq) - S(q) ? 0

qk
qkdqk
t
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Variational integration

• Functional does not vary with small changes in qk
• ?/?qk S(q) 0
• ?/?qk (Fdk-1,k Fdk,k1) 0
• Discrete Euler-Lagrange equation
• Given qk-1 and qk, find qk1

qk-1
qk
Fdk-1,k
qk1
Fdk,k1
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Dynamics meta-algorithm

• Formulate Fdk,k1 (usually the Lagrangian) and
  derivatives with respect to qk, vk1
• Substitute into DEL equation
• Simplify to get a time-stepping
  scheme(preferably an explicit one)

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Articulated bodies

• Tree structure
• Reduced coords xj, ?j, vj, ?j relative to parent
  link
• Maximal coords xj, Rj, vj, ?j in world space

uj
xj,vj or ?j, ?j
Link j
xj, Rj
Link i
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Formulating the Lagrangian

• Maximal coordinates easy!
• L T - U
• Kinetic energy T ? (½miviTvi ½?iTIi?i)
• Potential energy U ? U(xi,Ri)
• Derivatives are ridiculously straightforward

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Formulating the Lagrangian

• Reduced coordinates only via maximal
  coordinates
• Finding the derivatives is a mess
• Prismatic ?/?xi L -uiT?i-1piA
• ?/?vi L uiTpiA
• Revolute ?/??i L -uiT?iLiA - uiTvipiA
• ?/??i L uiTLiA
• where piA, LiA linear, angular momentum of
  subtree at joint i

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Maximal coordinates ho!

• DEL equations become really simple
• But have to use constraints to enforce joint
  connectivity
• Nonlinear constraints!
• Could simplify by linearizing the constraints
  locally like everybody else does

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Linearizing constraints

• Example simple pendulum
• (x-o)T(x-o) r2
• Linearize at x x0
• (x-o)T(x0-o) r2
• Turns out, variationalintegration doesnt like
  that

o
r
x0
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Linearizing constraints

• Pendulum started at rest at 90
• Time step of 0.1 sec
• Explicit Euler
• Variational, exact constraint
• Variational, linearized constraint

10
50
60
?
0
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Not linearizing constraints

• Linearization loses one of the main reasons for
  using variational integration
• Not easy to satisfy non-linear constraints

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Reduced coordinates then

• Prismatic ?/?xi L -uiT?i-1piA
• ?/?vi L uiTpiA
• Revolute ?/??i L -uiT?iLiA - uiTvipiA
• ?/??i L uiTLiA
• Plugging into DEL gives equations in terms of piA
  and LiA
• which depend on state variables of all joints

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Sparsification?

• Weve got a big linear system, O(n3) to solve
• Baraff 1996 faced a similar problem, used tree
  structure to allow Gaussian elimination in O(n)
• Seems promising
• piA pi ? pjA
• Similarly for LiA
• pi, Li depend only on state of ancestral joints

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Sparsification?

• If by manipulation, we could eliminate ? pjA, we
  would have a linear time algorithm
• But multiplied by different terms in different
  places
• Couldnt find a solution
• People generally dont apply variational
  approaches to non-linear coordinate systems

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Back to maximal coordinates

• SHAKE algorithm for constrained particle dynamics
  (e.g. molecular dynamics simulation)
• Equivalent to variational integration with
  constraints
• Stage 1 Move particles, ignoring constraints
• Stage 2 Find correction forces to satisfy
  constraints (requires iterative solve)

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Applying SHAKE

• SHAKE works on systems of particles with
  constrained distances (atoms with bonds)
• Articulated bodies are more complex
• Links have mass and rotational inertia
• Joints have different degrees of freedom
• SHAKE would have to be modified
• Havent worked out the details yet
• Would probably be slower than existing methods

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Conclusion

• Variational integration is hard to apply to
  articulated body dynamics
• Reduced coordinates non-linear underlying domain
  ? coupled system of equations
• Maximal coordinates non-linear constraints ?
  iterative solve required

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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.
• D. Baraff. Linear-Time Dynamics using Lagrange
  Multipliers. SIGGRAPH 1996.

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Thanks

• Questions?

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Thanks

• Questions?
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