Spring 20061 Rigid Body Simulation. Spring 20062 Contents Unconstrained Collision Contact Resting...

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Rigid Body Simulation

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Contents

Unconstrained Collision ContactResting Contact

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Review Particle Dynamics

State vector for a single particle:

System of n particles:

Equation of Motion

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Rigid Body Concepts

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Rotational Matrix

Direction of the x, y, and z axes of the rigid body in world space at time t.

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Velocity

Linear velocity Angular veclocity Spin: (t)

How are R(t) and (t) related?Columns of dR(t)/dt: describe the velocity with which the x, y, and z axes are being transformed

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Rotate a Vector

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= =

Change of R(t)

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Rigid Body as N particlesCoordinate in body space

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Center of Mass

World space coordinate

Body space coord.

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Force and Torque

Total force

Total torque

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Linear MomentumSingle particle

Rigid body as particles

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Angular Momentum

I(t) — inertia tensor, a 33 matrix, describes how the mass in a body is distributed relative to the center of mass

I(t) — inertia tensor, a 33 matrix, describes how the mass in a body is distributed relative to the center of mass

I(t) depends on the orientation of the body, but not the translation.

I(t) depends on the orientation of the body, but not the translation.

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Inertia Tensor

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Inertia Tensor

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[Moment of Inertia (ref)]

zzzyzx

yzyyyx

xzxyxx

III

III

III

I

Moment of inertia

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Table: Moment of Inertia

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Equation of Motion (3x3)

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Implementation (3x3)

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Equation of Motion (quaternion)

)(

)(

)()(

)(

)(

)(

)(

)(

)( 21

t

tF

tqt

tv

tL

tP

tq

tx

dt

dtY

dt

d

3×3 matrix

quaternion

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Implementation (quaternion)

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Non-Penetration Constraints

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Collision Detection

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Colliding Contact

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Collision

Relative velocityOnly consider vrel < 0

Impulse J:

J

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Impulse Calculation

[See notes for details]

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Impulse Calculation

For things don’t move (wall, floor):

000

000

000

011 1I

M

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Uniform Force Field

Such as gravity

acting on center of mass

No effect on angular momentum

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Resting Contact: See Notes

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Exercise

Implement a rigid block falling on a floor under gravity

x

y

5

3

thickness: 2M = 6

Moments of inertiaIxx = (32+22)M/12Iyy = (52+22)M/12Izz = (32+52)M/12

342

292

132

1

234

229

213

00

00

00

00

00

00

bodybody II

Inertia tensor

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xy

5

3

Three walls