AnnouncementsDisclaimerA Bead on a WirePenalty ConstraintsBasic IdeasGeometric InterpretationBasic Formulation (F = ma)Implicit Representation of ConstraintsMaintaining ConstraintsConstraint GradientConstraint ForcesDerivationExample: A Bead on a WireDrift and FeedbackConstrained Particle SystemsCompact NotationParticle System Constraint EquationsImplementationsMathematical FormulationTake a Closer LookConstraint StructureSlide 22Other ModificationConstraint Force EvaluationParametric Representation of ConstraintsParametric ConstraintsParametric bead-on-wire (F = mv)Some Simplification……Lagrangian DynamicsUNC Chapel HillM. C. LinAnnouncementsHomework #1 is due on Tuesday, 2/14/06Reading: SIGGRAPH 2001 Course Notes on Physically-based ModelingUNC Chapel HillM. C. LinDisclaimerThe following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright 2001 by Andrew Witkin at Pixar).UNC Chapel HillM. C. LinA Bead on a WireDesired 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 HillM. C. LinPenalty ConstraintsHow about using a spring to hold the bead to the wire?Problem–Weak springs sloppy constraints–Strong springs instabilityUNC Chapel HillM. C. LinBasic IdeasConvert each constraint into a force imposed on a particle (system)Use principle of virtual work – constraint forces do not add or remove energySolve the constraints using Lagrange multipliers, ’s–For particle systems, need to use the derivative matrix, J, or the Jacobian Matrix.UNC Chapel HillM. C. LinGeometric InterpretationUNC Chapel HillM. C. LinBasic Formulation (F = ma)Curvature(k) has to matchk depends on both a & v:–The faster the bead is going, the faster it has to turnCalculate fc to yield a legal combination of a & vUNC Chapel HillM. C. LinImplicit Representation of ConstraintsUNC Chapel HillM. C. LinMaintaining ConstraintsUNC Chapel HillM. C. LinConstraint GradientUNC Chapel HillM. C. LinConstraint ForcesUNC Chapel HillM. C. LinDerivationUNC Chapel HillM. C. LinExample: A Bead on a WireUNC Chapel HillM. C. LinDrift and FeedbackUNC Chapel HillM. C. LinConstrained Particle SystemsUNC Chapel HillM. C. LinCompact NotationUNC Chapel HillM. C. LinParticle System Constraint EquationsUNC Chapel HillM. C. LinImplementationsA global matrix equationMatrix block structure with sparsityEach constraint adds its own piece to the equationUNC Chapel HillM. C. LinMathematical FormulationUNC Chapel HillM. C. LinTake a Closer LookUNC Chapel HillM. C. LinConstraint StructureUNC Chapel HillM. C. LinConstrained Particle SystemsUNC Chapel HillM. C. LinOther ModificationUNC Chapel HillM. C. LinConstraint Force EvaluationAfter computing ordinary forces:UNC Chapel HillM. C. LinParametric Representation of ConstraintsUNC Chapel HillM. C. LinParametric ConstraintsUNC Chapel HillM. C. LinParametric bead-on-wire (F = mv)UNC Chapel HillM. C. LinSome Simplification……UNC Chapel HillM. C. LinLagrangian DynamicsAdvantages–Fewer DOF’s–Constraints are always metDisadvantages–Difficult to formulate constraints–Hard to combine
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