Rigid body dynamics II Solving the dynamics problems UNC Chapel Hill S Redon M C Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions Discussion UNC Chapel Hill S Redon M C Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions Discussion UNC Chapel Hill S Redon M C Lin Algorithm Overview UNC Chapel Hill S Redon M C Lin Algorithm Overview Two modules Collision detection Dynamics Calculator Two sub modules for the dynamics calculator Constrained motion computation accelerations forces Collision response computation velocities impulses UNC Chapel Hill S Redon M C Lin Algorithm Overview Two modules Collision detection Dynamics Calculator Two sub modules for the dynamics calculator Constrained motion computation accelerations forces Collision response computation velocities impulses Two kinds of constraints Unilateral constraints non penetration constraints Bilateral constraints hinges joints UNC Chapel Hill S Redon M C Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions Discussion UNC Chapel Hill S Redon M C Lin Constrained accelerations Solving unilateral constraints is enough When vrel 0 have resting contact All resting contact forces must be computed and applied together because they can influence one another UNC Chapel Hill S Redon M C Lin Constrained accelerations UNC Chapel Hill S Redon M C Lin Constrained accelerations Here we only deal with frictionless problems Two different approaches Contact space the unknowns are located at the contact points Motion space the unknowns are the object motions UNC Chapel Hill S Redon M C Lin Constrained accelerations Contact space approach Inter penetration must be prevented Forces can only be repulsive Forces should become zero when the bodies start to separate Normal accelerations depend linearly on normal forces This is a Linear Complementarity Problem UNC Chapel Hill S Redon M C Lin Constrained accelerations Motion space approach The unknowns are the objects accelerations Gauss principle of least contraints The objects constrained accelerations are the closest possible accelerations to their unconstrained ones UNC Chapel Hill S Redon M C Lin Constrained accelerations Formally the accelerations minimize the distance over the set of possible accelerations a is the acceleration of the system M is the mass matrix of the system UNC Chapel Hill S Redon M C Lin Constrained accelerations The set of possible accelerations is obtained from the non penetration constraints This is a Projection problem UNC Chapel Hill S Redon M C Lin Constrained accelerations Example with a particle The particle s unconstrained acceleration is projected on the set of possible accelerations above the ground UNC Chapel Hill S Redon M C Lin Constrained accelerations Both formulations are mathematically equivalent The motion space approach has several algorithmic advantages J is better conditionned than A J is always sparse A can be dense less storage required no redundant computations UNC Chapel Hill S Redon M C Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions Discussion UNC Chapel Hill S Redon M C Lin Computing a frictional impulse We consider one contact point only The problem is formulated in the collision coordinate system j j UNC Chapel Hill S Redon M C Lin Computing a frictional impulse Notations v the contact point velocity of body 1 relative to the contact point velocity of body 2 vz the normal component of v vt the tangential component of v a unit vector in the direction of vt fv ft the normal and tangential frictional z and t components of force exerted by body 2 on body 1 respectively UNC Chapel Hill S Redon M C Lin Computing a frictional impulse When two real bodies collide there is a period of deformation during which elastic energy is stored in the bodies followed by a period of restitution during which some of this energy is returned as kinetic energy and the bodies rebound of each other UNC Chapel Hill S Redon M C Lin Computing a frictional impulse The collision occurs over a very small period of time 0 tmc tf tmc is the time of maximum compression vz UNC Chapel Hill vz is the relative normal velocity We used vrel before S Redon M C Lin Computing a frictional impulse jz UNC Chapel Hill jz is the impulse magnitude in the normal direction Wz is the work done in the normal direction S Redon M C Lin Computing a frictional impulse v v 0 v0 v tmc v v tf vrel vz Empirical Impact Law v z v z Poisson s Hypothesis j z j0z j0z j z 1 j0z Newton s Stronge s Hypothesis Wz Wz0 2 Wz0 Wz 1 2 Wz0 Energy of the bodies does not increase when friction present UNC Chapel Hill S Redon M C Lin Computing a frictional impulse Sliding Dry dynamic friction v t 0 ft fn v t static friction v t 0 ft fn Assume UNC Chapel Hill no rolling friction S Redon M C Lin Computing a frictional impulse 1 1 1 1 v t 1 r1I1 r1 r2I2 r2 j t Kj t m1 m2 where r p x is the vector from the center of mass to the contact point 0 rz ry r r 0 r z x ry rx 0 UNC Chapel Hill S Redon M C Lin The K Matrix K is constant over the course of the collision symmetric and positive definite UNC Chapel Hill S Redon M C Lin Collision Functions Change variables from t to something else that is monotonically increasing during the collision v Kj Let the duration of the collision 0 The functions v j W all evolve over the compression and the restitution phases with respect to UNC Chapel Hill S Redon M C Lin Collision Functions We only need to evolve vx vy vz and Wz directly The other variables can be computed from the results for example j can be obtained by inverting K UNC Chapel Hill S Redon M C Lin Sliding or Sticking Sliding occurs when the relative tangential velocity v t 0 Use the friction equation to formulate v t 0 ft fn v t Sticking occurs otherwise Is it stable or instable Which direction does the instability get resolved UNC Chapel Hill S Redon M C Lin Sliding Formulation For the compression phase use v z v z is the relative normal velocity at the start of the collision we know this At the end of the compression phase v 0z 0 For the restitution phase use Wz Wz0 is the amount of work that has been done in the compression phase From Stronge s hypothesis we know that Wz 1 2 Wz0 UNC Chapel Hill S Redon M C Lin Sliding Formulation Compression phase equations are vx k x d 1 vy k y dv z k z Wz vz