# Rigid Body Dynamics (unconstrained) - PowerPoint PPT Presentation

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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.). Body space Origin: center of mass p 0 : an arbitrary point on the rigid body, in body space.

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Rigid Body Dynamics (unconstrained)

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## 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.)

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)

### Rigid Body Concepts

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

### The Rotation Matrix

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

Single particle

### 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