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# Rigid Body Dynamics (unconstrained) - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Rigid Body Dynamics (unconstrained)' - weylin

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Presentation Transcript

### 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.)

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 body-space in the world spacew(t) related?

Linear and Angular Velocity

R(t) and body-space in the world spacew(t)

R(t) and body-space in the world spacew(t)

Velocity of a Particle body-space in the world space

Force and Torque body-space in the world space

Single particle body-space in the world space

Linear Momemtum

Center of Mass body-space in the world space

Angular Momemtum body-space in the world space

Inertia Tensor body-space in the world space

Inertia Tensor body-space in the world space

Equation of Motion body-space in the world space

Inertia Tensor of a Block body-space in the world space

Inertia Tensor Table ( body-space in the world spaceref)

Uniform Force Field body-space in the world space

No effect on the angular momentum

The Football in Flight ( body-space in the world spaceref)

Gravity does not exert torque

Angular momentum stays the same

Using Quaternion body-space in the world space

quaternion

multiplication

Unit quaternion

as rotation

quaternion derivative

Equation of

motion

Computing Qdot body-space in the world space