Spring 20131 Rigid Body Simulation. Spring 20132 Contents Unconstrained Collision Contact Resting Contact.

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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

Solved by ODE Solvers (Euler, RK4, 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)

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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

Uniform Force Field

No effect on the angular momentum

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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

Inertia Tensor Table (ref)Solid sphere

Hollow sphere

Solid ellipsoid

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The Football in Flight (ref)Gravity does not exert torqueAngular momentum stays the same

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Equation of Motion (3x3)

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Implementation (3x3)

Using Quaternion

quaternion

multiplication

Unit quaternionas rotation

Equation of motion

quaternion derivative

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Later …

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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)

Computing Qdot

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Incremental rotation represented in quaternion:

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dt

dtq

dt

dq

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Non-Penetration Constraints

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Collision Detection (Particle)

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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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Resting Contact

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Solve the contact forces fi so that for

Relative displacement at contact point i:

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In contact at t0:

and we want: We need:

Similar logic, we want:

Solving Contact Forces

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First constraint on contact forces fi:

Second constraint on fi (repulsive):

Third constraint (no force when contact breaks)

Quadric Programming(or Linear Complementarity Program to be exact)

Details (appendix D)

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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