Rigid Body Dynamics (unconstrained)

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Presentation transcript:

Rigid Body Dynamics (unconstrained)

Simulation Basics State vector of a single particle Change of Y(t) over time Solved by any ODE solver (Euler, Runge-Kutta, etc.)

Rigid Body Concepts Body space Origin: center of mass p0: an arbitrary point on the rigid body, in body space. Its world space location p(t) Spatial variables of the rigid body: 3-by-3 rotation matrix R(t) and x(t)

The Rotation Matrix Three columns of R(t) correspond to the axes of the body-space in the world space

Linear and Angular Velocity How are R(t) and w(t) related?

R(t) and w(t)

R(t) and w(t)

Velocity of a Particle

Force and Torque

Single particle Linear Momemtum

Center of Mass

Angular Momemtum

Inertia Tensor

Inertia Tensor

Equation of Motion

Inertia Tensor of a Block

Inertia Tensor Table (ref)

Uniform Force Field No effect on the angular momentum

The Football in Flight (ref) Gravity does not exert torque Angular momentum stays the same

Using Quaternion quaternion multiplication Unit quaternion as rotation quaternion derivative Equation of motion

Computing Qdot

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