Spring Rigid Body Simulation

Spring Contents Unconstrained Collision Contact Resting Contact

Spring Review Particle Dynamics State vector for a single particle: System of n particles: Equation of Motion

Spring Rigid Body Concepts

Spring Rotational Matrix Direction of the x, y, and z axes of the rigid body in world space at time t.

Spring Velocity Linear velocity Angular veclocity Spin:  (t) How are R(t) and  (t) related? Columns of dR(t)/dt: describe the velocity with which the x, y, and z axes are being transformed

Spring Rotate a Vector

Spring = = Change of R(t)

Spring Rigid Body as N particles Coordinate in body space

Spring Center of Mass World space coordinate Body space coord.

Spring Force and Torque Total force Total torque

Spring Linear Momentum Single particle Rigid body as particles

Spring Angular Momentum I(t) — inertia tensor, a 3  3 matrix, describes how the mass in a body is distributed relative to the center of mass I(t) depends on the orientation of the body, but not the translation.

Spring Inertia Tensor

Spring Inertia Tensor

Spring [Moment of Inertia (ref)]ref Moment of inertia

Spring Table: Moment of Inertia

Spring Equation of Motion (3x3)

Spring Implementation (3x3)

Spring Equation of Motion (quaternion) 3×3 matrix quaternion

Spring Implementation (quaternion)

Spring Non-Penetration Constraints

Spring Collision Detection

Spring Colliding Contact

Spring Collision Relative velocity Only consider v rel < 0 Impulse J : J

Spring Impulse Calculation [See notes for details]

Spring Impulse Calculation For things don ’ t move (wall, floor):

Spring Uniform Force Field Such as gravity acting on center of mass No effect on angular momentum

Spring Resting Contact: See Notes

Spring Exercise Implement a rigid block falling on a floor under gravity x y 5 3 thickness: 2 M = 6 Moments of inertia Ixx = ( )M/12 Iyy = ( )M/12 Izz = ( )M/12 Inertia tensor

Spring x y 5 3 Three walls