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 p 0 : an arbitrary point on the rigid body, in body space.

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Rigid body dynamics unconstrained

Rigid Body Dynamics(unconstrained)


Simulation basics
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)

Rigid Body Concepts


The rotation matrix

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

The Rotation Matrix


Linear and angular velocity

How are R(t) and body-space in the world spacew(t) related?

Linear and Angular Velocity


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


R t and w t1
R(t) and body-space in the world spacew(t)


Velocity of a particle
Velocity of a Particle body-space in the world space


Force and torque
Force and Torque body-space in the world space


Linear momemtum

Single particle body-space in the world space

Linear Momemtum


Center of mass
Center of Mass body-space in the world space


Angular momemtum
Angular Momemtum body-space in the world space


Inertia tensor
Inertia Tensor body-space in the world space


Inertia tensor1
Inertia Tensor body-space in the world space


Equation of motion
Equation of Motion body-space in the world space


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


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


Uniform force field
Uniform Force Field body-space in the world space

No effect on the angular momentum


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

Gravity does not exert torque

Angular momentum stays the same


Using quaternion
Using Quaternion body-space in the world space

quaternion

multiplication

Unit quaternion

as rotation

quaternion derivative

Equation of

motion


Computing qdot
Computing Qdot body-space in the world space


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