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DESCRIPTION
Variational Methods for solid mechanicsTRANSCRIPT
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
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2http://www.cs.cmu.edu/~bstephe1
3http://www.cs.cmu.edu/~bstephe1
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, 1746
5
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, 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• L:TQ→R
)(),(),( qUqqTqqL
Kinetic Energy Potential EnergyLagrangian
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Variation of the Lagrangian
• Principle of Least Action = the function, q*(t), minimizes the integral of the Lagrangian
TT
dttqtqtqtqLdttqtqL00
))()(*),()(*())(*),(*(
0))(),((0
T
dttqtqL Variation of trajectory with endpoints fixed
“Hamilton’s Principle” ~183510
“Calculus of Variations” ~ Lagrange, 1760
Continuous Lagrangian
0
q
L
dt
d
q
L
“Euler-Lagrange Equations”11
Continuous Mechanics)(),())(),(( qUqqTtqtqL
)( UTqdt
d
L
dt
d
q
L
q
T
dt
d
q
U
q
T
qqq
Tqqq
T
q
U
q
T
22
0)(),()( qGqqCqqM 12
The Discrete Lagrangian
• L:QxQ→R
13
hqqLdttqtqL kkd
hT
T
,,)(),( 1
L
kqh
1kq
hh
qqqLhqqL kkkkkd
11 ,,,
Variation of Discrete Lagrangian
0,,,, 1112 tqqLDtqqLD kkdkkd
“Discrete Euler-Lagrange Equations” 14
Variational Integrator
• Solve for :
0,,,, 1112 hqqLDhqqLD kkdkkd
0,, 111
hh
qqqL
qh
h
qqqL
qkk
kk
kkk
k
0,,,, 1111
11
h
qqq
q
Lh
h
qqq
q
L
h
qqq
q
Lh
h
qqq
q
L kkk
kkk
kkk
kkk
1kq
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Solution: Nonlinear Root Finder
)(
)(
1
11
11 i
k
iki
kik qDf
qfqq
0,,,,)( 11121 hqqLDhqqLDqf kkdkkdk
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Simple Example: Spring-Mass
• Continuous Lagrangian:
• Euler-Lagrange Equations:
• Simple Integration Scheme:
22
2
1
2
1, kxxmqqL
0
xmkxq
L
dt
d
q
L
kkk
kkkk
xm
khxx
xm
khxhxx
1
21 2
1
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Simple Example: Spring-Mass
• Discrete Lagrangian:
• Discrete Euler-Lagrange Equations:
• Integration:
18
22
11 2
1
2
1,, k
kkkkd kx
h
xxmhxxL
0,,,, 1112 hxxLDhxxLD kkdkkd
02 112 kkkk kxxxx
h
m
1
2
1 2
kkk xxm
khx
Comparison: 3 Types of Integrators
• Euler – easiest, least accurate
• Runge-Kutta – more complicated, more accurate
• Variational – EASY & ACCURATE!
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0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
position
velo
city
Euler h=0.001
Runge-Kutta (ode45)Variational h=0.001
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Notice:
•Energy does not dissipate over time
•Energy error is bounded
0 10 20 30 40 50 60 70 80 90 100
0.498
0.5
0.502
0.504
0.506
0.508
0.51
time (s)
Ene
rgy
Euler h=0.001
Runge-Kutta (ode45)Variational h=0.001
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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
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Forcing Functions
• Discretization of Lagrange–d’Alembert principle
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Constraints
)(
)(
1
11
11 i
k
iki
kik zDf
zfzz
0
)(
)(,,,,)(
1
11121
k
kT
kkkdkkdk
qg
qghqqLDhqqLDzf
k
kk
qz
1
1
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Example: Constrained Double Pendulum w/ Damping
1
2
),( yx
0)(
y
xqg
2
1
0
0
)(
K
KqF
2
1
y
x
q
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Example: Constrained Double Pendulum w/ Damping
• Constraints strictly enforced, h=0.1
26No 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”
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Complex Examples From Literature
Variational Integrator
ODE
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Complex Examples From Literature
29
Complex Examples From Literaturelo
g
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 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 33
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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• Trajectory optimization
Applications
O. Junge, J.E. Marsden, S. Ober-Blöbaum, “Discrete Mechanics and Optimal Control”, in Proccedings of the 16th IFAC World Congress, 2005 35
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 (“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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