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© Möller/Parent/Machiraju

Rigid Body Dynamics

CMPT 466

Computer Animation

Torsten Möller


Advanced Algorithms

• Hierarchical Kinematic Modeling

– Forward Kinematics

– Inverse Kinematics

• Rigid Body Constraints

– Basic Particle Forces

– Collision

• Controlling Groups of Objects

– particle systems

– flocking behaviour


Reading

• Chapter 4 of Parents book

• Foley, van Dam, etc.; Angel; Watt;

Glassner; Hill; Baker + Hearns


Rigid Body Dynamics

• Keyframing can be tedious - especially to

get ‘realism’

• Simulate physics by

– programming equations of motions

– setting initial conditions

• Animator gives up control

• Animator gets ‘realistic’ motion

automatically.


Simulation Update Cycle

Object properties

Position, orientation

Linear and angular velocity

Linear and angular momentum

mass

Calculate forces

Calculate accelerations

Using mass, momenta

Update object properties


Particle Under Forces

• Forces

– Gravity

– Wind

– Springs

– Collision avoidance

– Soft constraints

• Calculate acceleration due to forces

• Calculate update to object velocity &position


Forces - Basic Physics

• Friction - static and kinematic

• Gravity

• Viscosity

• Springs (Hooke’s law)

• Damping


F

Fs

!

Fs

= usFN

Friction

• Supporting object

• Resting contact

• Normal force

• Static friction


!

Fs

= usFN

!

Fk

= ukFN

FN

v

F

Fk

Friction (2)

• Supporting object

• Resting contact

• Normal force

• Static friction

• Kinetic friction


!

Fgravity =Gm

1m2

r2

!

aobject =F

mobject

=Gmearth

rearth2

= 32 ft /s2

= 9.8m /s2

Gravity


• kv - depends on shape of object

• n – viscosity depends on properties of liquid

• v- velocity (should be low)

• For spherical object:

• Terminal velocity - viscosity and gravitybalance

• E.g., for sphere:

!

Fv

= "kvnv

!

kv

= 6"r

!

mg = 6"rnv

Viscosity


!

Fspring = kspring (x " xrest )

xrest

x

x

F

F

Springs (Hooke’s Law)

• Spring’s rest length: exerts zero force


Spring Mesh

• Edges => springs

• Internal springs to stabilize shape


!

Fdamping = "kdampingv

!

F = (x " xrest )kspring " vkdamping

Damping

• Calm down spring oscillations


Define point masses

postion

velocity

mass

force

fixed?

Define springs

point 1

point 2

rest length

kspring

kdamper

Multiple time samples per frame?

Mass-Spring-Damper System


For each point

Initialize force with wind

For each spring

Calculate spring-damper force

spring.point1.force += force

spring.point2.force -= force

For each point

acc = gravity

acc += force/mass

newVel = velocity + acc*dt

position += dt*(velocity+newVelocity)/2

Velocity = newVelocity

Mass-Spring-Damper System


Examples

• 1, 4, 16 time samples per frame


Examples (2)

• Multiple connectivity; fixing a node adding

gravity and wind


Examples (3)

• Adding damping; making a flag

• Use controlled random numbers to add

interest


Randomize

• Controlled randomness adds more interest

– To initial values (positions, velocities)

– To force fields (wind direction, wind speed)

– To spring constants, masses

– To joint angles

• Proximal joints: lower amplitude

• Distal joints: higher amplitude

• Coordinate frequency and phase


Angular Springs

• Use angles (cosine) between normals

• Place a linear spring between ends of

triangles


Constrain Forces (soft

constraints)

F

F

F

Non-penetration Fix to surface


Particle Under Forces

• Forces

– Gravity

– Wind

– Springs

– Collision avoidance

– Soft constraints

• Calculate acceleration due to forces

• Calculate update to object velocity &position

© Möller/Parent/Machiraju!

x(t + "t) = x(t) + v(t)"t

v(t + "t) = v(t) + a(t)"t

x(t + "t) = x(t) +(v(t) + v(t + "t))

2"t

x(t + "t) = x(t) + v(t)"t +1

2a(t)"t 2

Particle under Linear Force

• Given: a(t)

• Point: x(t)

v(t)

• Uses acceleration at start of interval for

entire interval!


Euler Integration

!

g(ti+1) = g(ti) + " g (ti)#t

!

" g (ti)

!

g(ti)


Euler Integration (2)

t

Derivative field


Numeric Approximation

Underlying fuction: Sine

Cos(x) is velocity function


Numeric Approximation (2)

Step size = 0.2Euler method



Step size = 5.0Euler method



Euler method Step size = 2.0


Midpoint Method

• A.k.a. 2nd order Runga Kutta

!

g(ti+h

2

)

!

" g (ti)

!

" g (ti+

h

2

)

!

g(ti)

!

g(ti+1) = g(ti) + " g (ti+

h

2

)#t


Numeric Approximation

Midpoint method Step size = 2.0


4th Order Runge Kutta

g (ti)

!

g(ti+1) =k1

6+k2

3+k3

3+k4

6

g(ti+1)

k1

k2

k3 k4


Conserved in a

closed system

Equations of Motion

• Linear force

• Angular force: torque

• Linear momentum

• Angular momentum

• Resistance to linear force: mass

• Resistance to angular force: inertia tensor.


Rotational Movement

• Represent orientation with rotation matrix:

R(t)

• Represent angular velocity with vector: !(t)

– direction is axis of rotation

– magnitude is speed of rotation

• Angular velocity insensitive to distance

from center or rotation


Rotational Movement (2)

b

r (t)

"

!(t)

!

˙ r (t) = "(t) r(t) sin(#)

!

˙ r (t) ="(t) # r(t)!

a = b + r(t)



!

R(t) = R1(t) R

2(t) R

3(t)[ ]

!

˙ R (t) = "(t) # R1(t) "(t) # R

2(t) "(t) # R

3(t)[ ]

( ) BA

B

B

B

AA

AA

AA

BABA

BABA

BABA

BA

z

y

x

xy

xz

yz

xyyx

zxxz

yzzy

*

0

0

0

=

!!!

"

#

$$$

%

&

!!!

"

#

$$$

%

&

'

'

'

=

!!!

"

#

$$$

%

&

('(

('('

('(

=)

Define the following for notational convenience:

It is the dual notation !


!

˙ R (t) ="(t)*R(t)

q(t) = R(t)q + x(t)

˙ q (t) ="(t) # (R(t)q) + v(t)

˙ q (t) ="(t) # (q(t) $ x(t)) + v(t)


• q is on object and x(t), R(t) is position,

orientation of object


• Points are often used in CG for masses and

objects

• Integration of differential mass times

position in object

!

M = mi"

!

x(t) =miqi(t)"M

Center of Mass


Force and Torque

!

F = ma

a = F /m

F(t) = f i(t)"# i(t) = (q(t) $ x(t)) % f i(t)

#(t) = # i(t)"F

F


Linear Momentum

!

p = mv

P(t) = mi˙ q i(t)"

• If the center of mass is at the origin:

!

P(t) = Mv(t)

˙ P (t) = M ˙ v (t) = F(t)


Angular Momentum

!

L(t) = ((q(t) " x(t)) #mi($ ˙ q (t) " v(t))

= (R(t)q #mi(%(t) # (q(t) " x(t))))$

= (mi(R(t)q # (%(t) # R(t)q)))$

!

L(t) = I(t)"(t)

˙ L (t) = # (t)


Inertia Tensor

!

Iobject =

Ixx Ixy Ixz

Ixy Iyy Iyz

Ixz Iyz Izz

"

#

$ $ $

%

&

' ' '

!

I(t) = R(t)IobjectR(t)T

!

Ixx = "(q)(qy2### + qz

2)dxdydz


Inertia Tensor (2)

!

Ixx = "(q)(qy2### + qz

2)dxdydz

!

Ixx = mi(yi2

+ zi2)"

Iyy = mi(xi2

+ zi2)"

Izz = mi(xi2

+ yi2)"

!

Ixy = mixiyi"

Ixz = mixizi"

Iyz = miyizi"


Inertia Tensor for a Cuboid

!

Iobject =1

12

M b2 + c 2( ) 0 0

0 M a2 + c 2( ) 0

0 0 M a2 + b2( )

"

#

$ $ $

%

&

' ' '

• With sides a, b, and c


Inertia Tensor for a Sphere

!

Iobject =2MR

2

5

1 0 0

0 1 0

0 0 1

"

#

$ $ $

%

&

' ' '


The Equations - State Vector

S (t) =

x (t)

R (t)

P (t)

L (t)

Update with velocity

Update with angular velocity

Update with force

Update with torque

!

v(t) = P(t) / M

"(t) = I(t)#1

L(t)

˙ P (t) = M ˙ v (t) = F(t)

˙ L (t) = $ (t)

!

I(t) = R(t)IobjectR(t)T

!

˙ R (t) ="(t)*R(t)


Update State Vector

!

d

dtS(t) =

d

dt

x(t)

R(t)

P(t)

L(t)

"

#

$ $ $ $

%

&

' ' ' '

=

v(t)

((t)*R(t)

F(t)

) (t)

"

#

$ $ $ $

%

&

' ' ' '


Update Orientation

• ! (t) * R(t) and renormalize

• Extract axis-angle from ! (t) and rotate

columns of R(t)

• Use quaternions


Baraff 1991

Collisions

• Now, what about collisions?

– Detecting them (kinematics)

– Finding response (dynamics)


Collisions

• Collision Avoidance

– force fields, steer to avoid, etc.

– as in flocking

• Collision Detection

– calculating when objects overlap

– during a time interval

• Collision Response

– Distribution of mass important

– Localized forces

– computation v. accuracy

– back up time v. after-the-fact


d1

d2

!

tcollision

= ti+ "t

d1

d1

+ d2

Collision Detection

• Either back time up to point of collision and

go from there

OR

• Respond to collision only at time steps

forward (penalty method)

• Multiple collisions?


Collision Resp. Strategies

• Strictly kinematical response

– quick and easy

– good visual results for (spherical) particles

• Penalty method

– used when no backing up in time is enforced

– easy to use, however, forces can be unnatural

• Impulse forces

– Usually used when time is backed up


!

E p( ) = ax + by + cz + d < 0

Kinematic Response

• Detect Collision - Point-Plane

• Every time step, see if point is one “wrong”

side of plane

• Detect collision by planar equation


Kinematic Response (2)

• Negate velocity component in direction of

normal

• Using damping:

v

N

!

" v = v # 2(v $ N)N

!

(v " N)N

!

" v = v # 1+ k( )(v $ N)N


Kinematic Response (3)

• The role of k:

– K=0.8 – get bounces

– Not physically based, but reasonable visual

response


Penalty Method

• Associate closest point of object with spring

• Penalize when interpenetration occurs

• Spring force F = k*d

• Apply –F to point

• Compute acceleration and effect velocity along

normal

d© Möller/Parent/Machiraju

Testing Planar Polyhedra

• Point inside a convex/concave polyhedron

test

• Edge-plane intersection

• Temporal aliasing can occur


Swept Volumes

• Compute motion of one relative to other

• Compute swept volume

• Determine if other object intersects swept

volume

• Tractable only for simple motion


After collision is detected

• Consider applying a force to both bodies

– nature “simulates” collisions like this

– Interpenetrating objects will remain so for at

least one more time step

• Cannot instantaneously change velocity

• Forces need time to be integrated to accelerations

and velocities


After collision is detected

• Still, we need instantaneous change in

velocity => Impulse

– A hack for impenetrable bodies exist

– Generalization of subtle surface properties

– Simulates a large force for a small time step

– Will change velocity like we need


Calculating impulse

• Duration of impulse is “no time”

– Actually a small amount of time

– Other forces ignored during this period

• No friction too !


Time of Impact

• Must apply impulse exactly at time ofimpact

• After detecting interpenetration, use binarysearch (or more sophisticated) to fine tune

• Linear time approx

• Beware of “tunneling” when dt is so largecollisions are missed


2D Collisions

• We’re in 2D … collisions are:

– Vertex / edge

– Vertex / vertex

– Edge / edge

• Compute normal to collision

– v/e: perpendicular to edge (pointing towards A by

convention)

– e/e: perpendicular to edge

– v/v: something reasonable (perhaps use an edge-pert)


More Remarks

• Edge/edge collisions are modeled as

point/point in this system

• Only two colliding bodies at a time

• 3D is harder because of variety of collision

types


!

J = F"t = Ma"t = M"v = "(Mv) = "P

!

J = jN

Impulse Force of Collision

• Compute change in momentum

• Impulse is in the direction of the normal:


Equal and

opposite

Calculating an impulse

• Solve for one number, j

– Apply j in direction of N to A

– Apply j in direction of –N to B


Relative Velocity

!

rA t( ) = pA " xA t( )

rB t( ) = pB " xB t( )

!

˙ p A t( ) = vA t( ) +"A t( ) # rA t( )

˙ p B t( ) = vB t( ) +"B t( ) # rB t( )

!

vrel = ˙ p A t( ) " ˙ p B t( )


• Determine contact normal

• Velocity of a point is sum of linear + angular

velocities

• Relative velocity is difference of velocity of 2

points of contact

AxB

A B

!

˙ x A(t) = v

A(t) +"

A(t) # r

A(t)

Relative Velocity (2)


Update Velocities after Collision

!

vA+ = vA

" +jN

MA

vB+

= vB""jN

MB

#A

+ =#A

" + IA"1(t)(rA $ jN)

#B

+=#B

"" IB

"1(t)(rB $ jN)


!

vrel

+= "#v

rel

"

Coefficient of restitution

• Coefficient of resitution

– Models complex compression

& restitution of impacting bodies

– Models dissipation of energy

• # = 1 " superball (perfectly elastic)

• # = 0 " clay (perfectly inelastic)

!

vrel

+" N = #$v

rel

#" N


Computing Impulse

• 1st: Assume objects cannot rotate

• 2nd: Use definition of coeff. of rest. to derive

a second set of v+ equations

• 3rd: Use substitution to solve for j

!

vA" # N +

j

MA

N # N " vB" # N +

j

MB

N # N = "$ vA" " vB

"( ) # N

j1

MA

+1

MB

%

& '

(

) * = " 1+ $( )vrel

" # N

!

˙ q A+

= vA

+= vA

"+

jN

MA

˙ q B+

= vB

+= vB

""

jN

MB

!

vA

+" v

B

+( ) # N = "$ vA

"" v

B

"( ) # N


Computing impulse

• Things to note:

– N doesn’t have to be normalized

– A or B can be fixed by setting mass to infinity

– If MA = 1, MB = inf, vB = 0, #=1

• Computes reflection of vA about N


• Consider velocity of collision point P after

collision

• Derived from two equations

Accounting for rotation

!

vA+

= vA"

+jN

MA

vB+

= vB""jN

MB

#A

+ =#A

" + IA"1(t)(rA $ jN) #B

+=#B

"" IB

"1(t)(rB $ jN)

!

˙ q A (t) = vA (t) +"A (t) # rA˙ q A (t) = vA (t) +"A (t) # rA


Accounting for rotation (2)

• Use definition of coeff. of rest. to derive a

second set of v+ equations

• Use substitution to solve for j

!

N " ˙ q A+# ˙ q B

+( ) = #$N " ˙ q A## ˙ q B

#( )

!

j =" 1+ #( )N $ ˙ q A

"" ˙ q B

"( )1

MA

+1

MB

+ N $ IA

"1(t)(rA % N) % rA + N $ IB

"1(t)(rB % N) % rB


If Vrelative > threshold // actually colliding

Compute velocities of points of contact: va, vb

Compute relative velocities in direction of normal

Given: two points of contact and normal of contact

Compute j

PA += J

PB -= J

LA += rA x J

LB -= rB x J

Else if Vrelative < -threshold, then resting contact

Else they are moving away from each other

Computing Impulse Forces

rigid body dynamics - computing sciencetorsten/teaching/cmpt466/lecturenotes/pdf/09_rigid… ·...

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