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

Holonomic and Non-holonomic constraints Bilateral and Unilateral constraints

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

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)

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

Constrained Dynamics

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

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

Lagrangian Dynamics Suitable for systems with fixed configuration

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

In Lagrange’s Equation