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Rigid Body Simulation Spring 2006 1 Contents Unconstrained Collision Contact Resting Contact Spring 2006 2 Review Particle Dynamics State vector for a single particle: System of n particles: Equation of Motion Spring 2006 3 Rigid Body Concepts Spring 2006 4 Rotational Matrix Direction of the x, y, and z axes of the rigid body in world space at time t. Spring 2006 5 Velocity Linear velocity Angular veclocity Spin: w(t) How are R(t) and w(t) related? Columns of dR(t)/dt: describe the velocity with which the x, y, and z axes are being transformed Spring 2006 6 Rotate a Vector Spring 2006 7 Change of R(t) = Spring 2006 = 8 Rigid Body as N particles Coordinate in body space Spring 2006 9 Center of Mass World space coordinate Body space coord. Spring 2006 10 Force and Torque Total force Total torque Spring 2006 11 Linear Momentum Single particle Rigid body as particles Spring 2006 12 Angular Momentum I(t) — inertia tensor, a 33 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 2006 13 Inertia Tensor Spring 2006 14 Inertia Tensor Spring 2006 15 [Moment of Inertia (ref)] I xx I I yx I zx Spring 2006 I xy I yy I zy I xz I yz I zz Moment of inertia 16 Table: Moment of Inertia Spring 2006 17 Equation of Motion (3x3) Spring 2006 18 Implementation (3x3) Spring 2006 19 Equation of Motion (quaternion) quaternion x(t ) v(t ) 1 d d q(t ) 2 w (t )q(t ) Y (t ) F (t ) dt dt P(t ) L(t ) (t ) 3×3 matrix Spring 2006 20 Implementation (quaternion) Spring 2006 21 Non-Penetration Constraints Spring 2006 22 Collision Detection Spring 2006 23 Colliding Contact Spring 2006 24 Collision J Relative velocity Only consider vrel < 0 Impulse J: Spring 2006 25 Impulse Calculation [See notes for details] Spring 2006 26 Impulse Calculation For things don’t move (wall, floor): 1 1 0 M Spring 2006 0 0 0 I 1 0 0 0 0 0 0 27 Uniform Force Field Such as gravity acting on center of mass No effect on angular momentum Spring 2006 28 Resting Contact: See Notes Spring 2006 29 Exercise Implement a rigid block falling on a floor under gravity Moments of inertia Ixx = (32+22)M/12 Iyy = (52+22)M/12 Izz = (32+52)M/12 y 3 x 5 thickness: 2 Spring 2006 M = 6 Inertia tensor 132 I body 0 0 0 29 2 0 0 132 1 0 I body 0 34 0 2 0 0 2 34 0 2 29 0 30 Three walls Spring 2006 31