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Rigid Body Dynamics (unconstrained)

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Rigid Body Dynamics(unconstrained)

State vector of a single particle

Change of Y(t) over time

Solved by any ODE solver (Euler, Runge-Kutta, etc.)

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)

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

How are R(t) and w(t) related?

Single particle

No effect on the angular momentum

Gravity does not exert torque

Angular momentum stays the same

quaternion

multiplication

Unit quaternion

as rotation

quaternion derivative

Equation of

motion