spacetime constraints andrew witkin michael kass presenter: fayun luo
TRANSCRIPT
Spacetime Constraints
Andrew WitkinMichael Kass
Presenter: Fayun Luo
Outline
• Introduction and Motivation
• A Particle Example
• SQP Method
• Extension to Complex Models
• Discussion
• A Tiny Movie Demo
Early Computer Simulations
Question: can we generate those motion automatically?
The first computer generated simulation—Pixar’s Luxo, Jr. 1986
Yes, by adding physics in the simulation.
Physically-based Approaches
Solving Initial Value Problems
V0
x0 x1
Physically-based ApproachesConstraints force methods
A man executing a kickA man executing a kick
The same kick on a frictionless floor
Physically-based Approaches
Spacetime Constraints
Basic idea is solve for the character’s motion and varying force over the entire time.
Newtonian physics
Object Function
Roadmap
• Introduction and MotivationIntroduction and Motivation
• A Particle Example
• SQP MethodSQP Method
• Extension to Complex ModelsExtension to Complex Models
• DiscussionDiscussion
• A Tiny Movie DemoA Tiny Movie Demo
Problem Statement
0 mgfxm
dttfRT
0
2|)(|
bTxax )(,)0(
Governing Equation (Motion Equation):
Object Function (Energy Consumption):
Boundary Conditions:
gf(t)
Discretize continuous function
1n
i
Discretize unknown function x(t) and f(t) as:
x1, x2, …xi, … xn-1, xn
f1, f2, …fi, … fn-1, fn
Our goal is to solve these discretized 2n values.
Next step is to dicretize our motion equation and object equation.
Difference Formula
h
xx
h
xx
h
xxx iiiiiii 2
1111
xi - 1 xi xi + 1
xi - 0.5 xi + 0.5
h h
211
11
5.05.0 2
h
xxx
hh
xx
h
xx
h
xxx iii
iiii
iii
Backward Forward Middle
Middle
Discretized Function
3,2,0)2(
0
112
imgfxxxh
m
mgfxm
iiii
bxbTx
axax
4
1
)(
)0(
t
x
x1, f1
x2, f2
x3, f3x4, f4
Motion equation:
Boundary Conditions:
24
1
0
2
||
|)(|
ii
T
fR
dttfR
Object Function: When does R have minimum value?
0000
0
4321
f
R
f
R
f
R
f
R
f
R
i
Roadmap
• Introduction and MotivationIntroduction and Motivation
• A Particle ExampleA Particle Example
• SQP Method
• Extension to Complex ModelsExtension to Complex Models
• DiscussionDiscussion
• A Tiny Movie DemoA Tiny Movie Demo
Generalize Our Notation
0S
R
3,2,0)2( 112 imgfxxx
h
miiii
bxax 41
t
x
x1, f1
x2, f2
x3, f3
x4, f4
Unknown vector:
S = (S1, S2, …Sn)
Constraint Functions:
Ci(S) = 0
Object Function R(S):
0000
0000
4321
4321
f
R
f
R
f
R
f
R
x
R
x
R
x
R
x
R S = (x1, x2, x3, x4, f1, f2, f3, f4)
j
iij S
CJ
jiij SS
RH
2
SQP Step One
S
R
0S
R
))(())((!2
1))(()()( 22''' axOaxafaxafafxf
Pick a guess S0, evaluate
Taylor series expansion of function f(x) at point a is:
))(()( 2002
2
00
SSOSSS
R
S
R
S
R
SSSS
'100
00 ))((0
SSSSorSSS
SSSHS
Rij
SS
Most likely
Similarly, we have:
= 0 Omit
SQP Step Two
))(()()()( 2'1
'1
'1
'1
SSOSSS
CSCSC
SS
SSSSSSorSSS
SSSJSC ij
0'1
'1
'1
'1
'1 ))(()(
Now we got S1’, evaluate our constraints Ci(S1
’), if equal to 0, we are done but most likely it will not evaluate to 0 in the first several steps.
So, let’s say Ci(S1’) ≠ 0, let’s apply Taylor series expansion on
the constraint function Ci(S) at point S1’ :
1001 SSsoSSSS
Then we will continue with step one and step two until we got a solution Sn which minimize our object function and also satisfy our constraints.
= 0 Omit
S0 S1’ S1 S2
’ S2 … Sn
Graphical Explanation of SQP
S0 S1’
S1S2’ S2
C(S)S
R
S
Roadmap
• Introduction and MotivationIntroduction and Motivation
• A Particle ExampleA Particle Example
• SQP MethodSQP Method
• Extension to Complex Models
• DiscussionDiscussion
• A Tiny Movie DemoA Tiny Movie Demo
Difficulties
• Set up the motion equations
• Define the objective equation
• Evaluate the derivatives
• Fit them into our SQP solver
Derive Motion EquationUse Lagrangian Dynamics to derive our motion equations dynamically:
0)(
T
q
T
dt
d
T – Kinetic Energy
q – Generalized Coordinates
Q – Generalized Forces
The Authors’ Automatic System
Graphical User Interface
Function Boxes Dynamic System
SQP Solver
T, Q, q J
R H
Roadmap
• Introduction and MotivationIntroduction and Motivation
• A Particle ExampleA Particle Example
• SQP MethodSQP Method
• Extension to Complex ModelsExtension to Complex Models
• Discussion
• A Tiny Movie DemoA Tiny Movie Demo
Lagrangian Dynamics
0)(
T
q
T
dt
d
Define T for complex system is still too much work.
Define Objective Functions
• Walking on hot coals
• Walking on eggs
• Carrying a bowl of hot soup
• Pursued by a bear
Define appropriate objective functions may be extremely difficult:
The Author’s Automatic System
Symbolic Analysis is really complex, especially for complex system.
The state of art symbolic analysis tool is Matlab, Maple.
The author’s automatic system may work for some relative simple system.
Local Optimization vs Global Optimization
S
R
S0 S* S*
Roadmap
• Introduction and MotivationIntroduction and Motivation
• A Particle ExampleA Particle Example
• SQP MethodSQP Method
• Extension to Complex ModelsExtension to Complex Models
• DiscussionDiscussion
• A Tiny Movie Demo
References
• Pixar, Luxo, Jr. 1986• Ronen Barzel, et al. Dynamic Constraints.
Siggraph 1987• Paul Issacs and Michael Cohen, Controlling
Dynamic Simulation with Kinematic Constraints, Proc. Siggraph 1987
• David C. Brogan, et al. Spacetime Constraints for Biomechanical Movements. Applied Modeling and Simulation, 2002
• Phillip Gill, et al. Practical Optimization, Academic Press, New York, NY, 1981