Computer Animation Algorithms and Techniques Physically Based Animation

Computer Animation Algorithms and Techniques Physically Based Animation Rick Parent Computer Animation

Physics Review force, mass, acceleration, velocity, position a m f v’ v a vave Rick Parent Computer Animation

Physics Review: Gravity Rick Parent Computer Animation

Physics Review: Spring-damper Rick Parent Computer Animation

Physics Review: Momentum conservation of momentum (mv) Rick Parent Computer Animation

Physics Review: Linear v. angular Linear postion velocity acceleration mass force momentum Rick Parent Angular rotational velocity rotational acceleration moment of inertia torque angular momentum Computer Animation

Spring-mass-damper system Topological springs: each edge of object Cross springs: keep stable Angular springs: maintain angle Virtual springs (‘soft’ constraints) Proportional derivative controllers (PDCs) Rick Parent Computer Animation

Useful in modeling: Flexible objects Cloth, flags, clothes Enforcing ‘soft’ constraints while reacting to outside forces, try to maintain a distance or keep in contact Rick Parent Computer Animation

Spring-mass-damper system Rick Parent Computer Animation

Spring-mass-damper system Rick Parent Computer Animation

Spring-mass-damper system Rick Parent Computer Animation

Angular spring Rick Parent Computer Animation

Particle systems http: //www. vecpix. com/tutorials/3 dstudio/sm 003. php Rick Parent Computer Animation

Particle systems Lots of small particles - local rules of behavior Create ‘emergent’ element Particles: Do not collision with other particles Don’t cast shadows on other particles Might cast shadows on environment Do not reflect light - usually emit it Rick Parent Computer Animation

Particle system Rick Parent Computer Animation

Particle system implementation add new particles assign attributes to particles deallocate any dead particles animate existing particles, modify attributes render particles Rick Parent Computer Animation

Particle systems http: //www. videotutorialsrock. com/opengl_tutorial/particle_system/home. php Rick Parent Computer Animation

Particle systems http: //www. cs. sun. ac. za/~lvzijl/courses/rw 778/grafika/Open. GLtuts/Big/graphicsnotes 018. html Rick Parent Computer Animation

Rigid body simulation Bodies in free fall and under collisions Orientation and rotational movement Collision detection & impulse force of collision Collision response Numerical approximation Equations of motion Center of mass & Mass distribution Forces & torques Momentum & Inertia Tensor Rick Parent Computer Animation

Rigid body simulation Rick Parent Computer Animation

Integration Rick Parent Computer Animation

Gravity Rick Parent Computer Animation

(Explicit/Forward) Euler integration Rick Parent Computer Animation

Step size explicit Euler integration dx = 0. 2 dx = 5. 0 Rick Parent Computer Animation

Step size explicit Euler integration dx = 2. 0 midpoint method dx = 2. 0 Rick Parent Computer Animation

Rotational velocity identical rotational velocity a) on-axis b) off-axis also has (instantaneous) linear velocity Rick Parent Computer Animation

Axis of rotation Rick Parent Computer Animation

Handling contact Collision detection Particle-plane collision Testing polyhedra Collision response kinematic response penalty method Impulse force of collision Constant contact Friction Resting Contact Rick Parent Computer Animation

Collision detection: point-plane Rick Parent Computer Animation

Collision response: kinematic Rick Parent Computer Animation

Collision response: damped Rick Parent Computer Animation

Collision response: penalty Rick Parent Computer Animation

Collision response: penalty Rick Parent Computer Animation

Collision detection: polyhedra Rick Parent Computer Animation

Collision detection: swept volume Rick Parent Computer Animation

Collision detection: time of impact Rick Parent Computer Animation

Coefficient of restitution Rick Parent Computer Animation

Impulse response Rick Parent Computer Animation

Friction Rick Parent Computer Animation

Resting contact Rick Parent Computer Animation

Constrained dynamics Forces applied to linked appendages external forces applied at joints Forward dynamics apply (linear or rotational) force at joints propagate force (and reaction to it) throughout linkage Inverse dynamics from desired motion, determine joint forces Rick Parent Computer Animation

Reaction to external force component reacts to force applied force from component outward, re-establish distance constraints Rick Parent Computer Animation

Dynamics of Linked Hierarchies Constrained dynamics The Featherstone equations propagate forces along linkages Rick Parent Computer Animation

Forward dynamics Torque at ui Rick Parent Computer Animation

Cloth http: //cg. in. tu-clausthal. de/publications. shtml From flags to curtains to clothes Flexible, thin material Self collisions as well as collisions with environment Rick Parent Computer Animation

Cloth High-level: model folds directly catenary curve between support points Low-level: thin sheet of geometry collision detection & response Lower-level: model internal structure warp and weft threads Rick Parent Computer Animation

Cloth - model physics Sheet of geometric elements (triangles) mass-spring-damper system finite element method Collision detection and response Method of numerical integration: accuracy v. efficiency Super-elasticity: how ‘stretchy’ Rick Parent Computer Animation

Cloth Baraff, D. and Witkin, A. 1998. Large steps in cloth simulation. In Proceedings of the 25 th Annual Conference on Computer Graphics and interactive Techniques SIGGRAPH '98. ACM, New York, NY, 43 -54. Rick Parent Computer Animation

Cloth Huamin Wang, Ravi Ramamoorthi, James O'Brien Data-Driven Elastic Models for Cloth: Modeling and Measurement SIGGRAPH 2011 Rick Parent Computer Animation

Enforcing Soft and Hard Constraints Energy minimization Space-time constraints Rick Parent Computer Animation

Energy Minimization Useful functions Position Normal Implicit surface Rick Parent Computer Animation

Energy Minimization: useful functions Position to fixed point Position matching Abutting Floating attachment Rick Parent Computer Animation

Energy minimization Rick Parent Computer Animation

Spacetime Constraints - example f - function to be minimized over the interval, e. g. , applied force Constraints: e. g. equations of motion, initial conditions Rick Parent Computer Animation

Spacetime Constraints - example constraints Rick Parent Computer Animation

Spacetime Constraints Numerical solution Rick Parent Computer Animation