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Spring 2006 1
Rigid Body Simulation
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Spring 2006 2
Contents
Unconstrained Collision ContactResting Contact
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Spring 2006 3
Review Particle Dynamics
State vector for a single particle:
System of n particles:
Equation of Motion
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Spring 2006 4
Rigid Body Concepts
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Spring 2006 5
Rotational Matrix
Direction of the x, y, and z axes of the rigid body in world space at time t.
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Spring 2006 6
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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Spring 2006 7
Rotate a Vector
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Spring 2006 8
= =
Change of R(t)
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Spring 2006 9
Rigid Body as N particlesCoordinate in body space
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Spring 2006 10
Center of Mass
World space coordinate
Body space coord.
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Spring 2006 11
Force and Torque
Total force
Total torque
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Spring 2006 12
Linear MomentumSingle particle
Rigid body as particles
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Spring 2006 13
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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Spring 2006 14
Inertia Tensor
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Spring 2006 15
Inertia Tensor
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Spring 2006 16
[Moment of Inertia (ref)]
zzzyzx
yzyyyx
xzxyxx
III
III
III
I
Moment of inertia
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Spring 2006 17
Table: Moment of Inertia
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Spring 2006 18
Equation of Motion (3x3)
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Spring 2006 19
Implementation (3x3)
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Spring 2006 20
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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Spring 2006 21
Implementation (quaternion)
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Spring 2006 22
Non-Penetration Constraints
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Spring 2006 23
Collision Detection
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Spring 2006 24
Colliding Contact
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Spring 2006 25
Collision
Relative velocityOnly consider vrel < 0
Impulse J:
J
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Spring 2006 26
Impulse Calculation
[See notes for details]
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Spring 2006 27
Impulse Calculation
For things don’t move (wall, floor):
000
000
000
011 1I
M
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Spring 2006 28
Uniform Force Field
Such as gravity
acting on center of mass
No effect on angular momentum
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Spring 2006 29
Resting Contact: See Notes
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Spring 2006 30
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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Spring 2006 31
xy
5
3
Three walls