1. Introduction Constraint: important features in computer animation
Simulate rigid structure (length, angle,…)
IK application
Solutions:
Penalty method
Constrained dynamics

2. Holonomic and Non-holonomic constraints
Bilateral and Unilateral constraints

3. Constraint Formulation
C(q) = 0 equality (bilateral) constraints
Behavior function: the function value reaches ZERO when we are happy
Examples x r x1 x2 x1 x2 l

4. Penalty Method Convert behavior functions to force laws
The constraint force competes with all other forces (thus, not precisely satisfied)
Can have numerical problems (stiff eq)

5. Example
(x,y) r
(fx,fy)

6. Constrained Dynamics

7. Example (x,y) r (fx,fy) How is this used:
Compute all other forces as fa
Then compute the constraint force fc
Continue with the ODE solver

8. Semi-Implicit Euler for ODE
Not guaranteed to be stable, but usually is

10. Lagrangian Dynamics
Suitable for systems with fixed configuration

11. Example
(x,y) u
(fx,fy)

12. In Lagrange’s Equation