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)

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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 w(t) related?

Linear and Angular Velocity


R t and w t

R(t) and w(t)


R t and w t1

R(t) and w(t)


Velocity of a particle

Velocity of a Particle


Force and torque

Force and Torque


Linear momemtum

Single particle

Linear Momemtum


Center of mass

Center of Mass


Angular momemtum

Angular Momemtum


Inertia tensor

Inertia Tensor


Inertia tensor1

Inertia Tensor


Equation of motion

Equation of Motion


Inertia tensor of a block

Inertia Tensor of a Block


Inertia tensor table ref

Inertia Tensor Table (ref)


Uniform force field

Uniform Force Field

No effect on the angular momentum


The football in flight ref

The Football in Flight (ref)

Gravity does not exert torque

Angular momentum stays the same


Using quaternion

Using Quaternion

quaternion

multiplication

Unit quaternion

as rotation

quaternion derivative

Equation of

motion


Computing qdot

Computing Qdot


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