Announcements l Homework 1 is due on Tuesday

Announcements l Homework #1 is due on Tuesday, 2/14/06 l Reading: SIGGRAPH 2001 Course Notes on Physically-based Modeling UNC Chapel Hill M. C. Lin

Disclaimer l The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001 by Andrew Witkin at Pixar). UNC Chapel Hill M. C. Lin

A Bead on a Wire l Desired Behavior – The bead can slide freely along the circle. – It can never come off no matter how hard we pull. So, how do we make this happen? UNC Chapel Hill M. C. Lin

Penalty Constraints l How about using a spring to hold the bead to the wire? l Problem – Weak springs sloppy constraints – Strong springs instability UNC Chapel Hill M. C. Lin

Basic Ideas l Convert each constraint into a force imposed on a particle (system) l Use principle of virtual work – constraint forces do not add or remove energy l Solve the constraints using Lagrange multipliers, ’s – For particle systems, need to use the derivative matrix, J, or the Jacobian Matrix. UNC Chapel Hill M. C. Lin

Geometric Interpretation UNC Chapel Hill M. C. Lin

Basic Formulation (F = ma) Curvature(k) has to match l k depends on both a & v: l – The faster the bead is going, the faster it has to turn l UNC Chapel Hill Calculate fc to yield a legal combination of a & v M. C. Lin

Implicit Representation of Constraints UNC Chapel Hill M. C. Lin

Maintaining Constraints UNC Chapel Hill M. C. Lin

Constraint Gradient UNC Chapel Hill M. C. Lin

Constraint Forces UNC Chapel Hill M. C. Lin

Derivation UNC Chapel Hill M. C. Lin

Example: A Bead on a Wire UNC Chapel Hill M. C. Lin

Drift and Feedback UNC Chapel Hill M. C. Lin

Constrained Particle Systems UNC Chapel Hill M. C. Lin

Compact Notation UNC Chapel Hill M. C. Lin

Particle System Constraint Equations UNC Chapel Hill M. C. Lin

Implementations l. A global matrix equation l Matrix block structure with sparsity l Each constraint adds its own piece to the equation UNC Chapel Hill M. C. Lin

Mathematical Formulation UNC Chapel Hill M. C. Lin

Take a Closer Look UNC Chapel Hill M. C. Lin

Constraint Structure UNC Chapel Hill M. C. Lin

Constrained Particle Systems UNC Chapel Hill M. C. Lin

Other Modification UNC Chapel Hill M. C. Lin

Constraint Force Evaluation After computing ordinary forces: UNC Chapel Hill M. C. Lin

Parametric Representation of Constraints UNC Chapel Hill M. C. Lin

Parametric Constraints UNC Chapel Hill M. C. Lin

Parametric bead-on-wire (F = mv) UNC Chapel Hill M. C. Lin

Some Simplification…… UNC Chapel Hill M. C. Lin

Lagrangian Dynamics l Advantages – Fewer DOF’s – Constraints are always met l Disadvantages – Difficult to formulate constraints – Hard to combine constraints UNC Chapel Hill M. C. Lin