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Graphics II 91.547

Animation 2

Articulated Figures &

Deformation

Session 7

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Animating Articulated Structures

0

Articulated structure

- Figure that consists of a series of rigid links

connected at joints.

- Joints may be:

=Revolute (or rotary): only angles can vary
=Prismatic: sliding joint where length of link can

vary

0

Kinematics

- Specification or study of motion independent of the

underlying forces that produce the motion. It

includes all geometrical and time related properties

of the motion

0

Degrees of freedom (DOF)

- Number of independent variables necessary to

specify the state of the structure

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Animating Articulated Structures

0

State vector

- Vector of all possible configurations of an

articulated structure.

- A set of independent parameters defining the

positions and orientations, and rotations of all

joints constituting the figure forms a basis of

the state space.

- The dimension of the state space = DOF of

structure

0

End effector

- Free end of the chain of links



 



( , ,..., )

1

2

N

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Forward vs. Inverse Kinematics

Forward Kinematics:

X  f ( )



The motion of all

joints

is specified explicitly by the

animator. The motion of the end effector is the
accumulation of all motions that lead to the end effector.

Inverse Kinematics:

 



f

1

( )

X

The animator defines the position of end effectors only.
Inverse kinematics solves for the position and the orientation
of all joints in the link hierarchy that lead to the end effector.

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A Simple 2-Link Articulated Structure

X( , )

x y



1



2

l

1

l

2

X 































( cos

cos(

), sin

sin(

))

cos

( sin )

(

cos )

( sin )

(

cos )

l

l

l

l

x

y

l

l

l l

l

x

l

l

y

l

y

l

l

x

1

1

22

1

2

1

1

2

1

2

2

1

2

2

1

2

2

2

1 2

1

2

2

1

2

2

2

2

1

2

2

2

























Forward:

Inverse:

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A Simple 2-Link Articulated

Structure:

Two Solutions

X( , )

x y



1



2

l

1

l

2

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Representing Articulated Figures:

Denavit-Hartenberg (DH) Notation














a

s

z

x

z

z

x

d

x

x

z

x

x

z

i

i

i

i

i

i

i

i

i

i

i

i

i

i

is the distance from to measured along (length of the link)

is the angle between and measured about (twist of the link)

is the distance between the

and axes, measured along (distance between links)

is the angle between

and measured about

i

i

1

1

1

1





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Representing Articulated Figures:

Denavit-Hartenberg (DH) Notation

Rotation of angle about the

axis aligning the

axis with the axis

R

z

x

x

z

i

i

i

i









1

1

cos

sin

sin

cos









i

i

i

i



























0 0
0 0

0

0

1 0

0

0

0 1

Translation along the axis of a distance

to make the x - axes coincident

T

z

d

zd

i

i

 1

1 0 0 0
0 1 0 0
0 0 1
0 0 0 1

d

i

























Translation along the axes of a distance

making the two origins coincident

T

x

a

xi

i

i 1

1 0 0
0 1 0

0

0 0 1

0

0 0 0

1

1

a

i

























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Representing Articulated Figures:

Denavit-Hartenberg (DH) Notation

Rotation

of an angle

about finally

to make the coordinate frames coincident

i-1

R

x

x

i





1

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

cos

sin

sin

cos









i

i

i

i



































Composing these four transformation into
a single transformation from frame to :

i

i-1

(

)

cos

sin

sin cos

cos cos

sin

sin

sin sin

cos sin

cos

cos

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

d

d





















































1

1

1

1

1

1

1

1

1

1

0

0

0

0

1

T































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Representing Articulated Figures:

Denavit-Hartenberg (DH) Notation

All frame-to-frame transformations can be concatenated to
form a single transformation that links frame 0 to frame N:

0

0

1

1

2

2

3

1

T

T T T

T

N

N

N





. . .

(

)

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The Jacobian

Given a function:

where is of dimension and is of dimension ,
the Jacobian is the

matrix of partial derivatives:

Therefore:
d

Dividing by incremental time element gives:

ij

X

X

J

J

X J

X J















f

n

m

n m

f

d

i

j

( )

( )

( )









 

 

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The Jacobian:

Why is it Useful for Inverse

Kinematics?

The inverse kinematics problem is stated:

 



f

1

( )

X

For complex articulated figures, this is too complex to solve
for analytically. The problem can be solved incrementally by

localizing about the current state vector and inverting the Jacobian
to give:

d

d

 



J

X

1

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The Jacobian:

Why is it Useful for Inverse

Kinematics?

X

dX

X

X

 d

X

goal





goal





 d

d

d







J

X

1

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The Jacobian:

How do you Construct it?

V

b

b

b

b

b

b

b

b

b

a

a

a

a

a

a

a

a

a

n

n

x

y

z

xi

yi

zi

x

y

z

x

y

z

xi

yi

zi

x

y

z

x

y

z

xi

yi

zi

x

y

z

n

n

n

n

n

n

n

n

n

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1











 

















































































...

...

...

...

...

...































































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The Jacobian:

How do you Construct it?

a a a

b

a

P P

b

a

P P

b

a

P P

P

xi

yi

zi

xi

n

i

yi

n

i

zi

n

i

i

x y

z

i

i

, ,

(

)

(

)

(

)

are the , , and axes of frame [ ] transformed into frame [ ]

where is the origin of frame [ ]

xi

yi

zi

0



















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Application of Inverse Kinematics to

a Skeleton

Root node

Base node

Open node

End node

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Simple Legged Animation

Using Forward Kinematics

Hip

Upper
Leg

Knee

Lower Leg

Ankle

Foot

Upper leg (hip rotate)

Hip rotate

Lower leg (knee rotate)

Hip rotate + knee rotate

Foot (ankle rotate)

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Simple Legged Animation

Hip Rotation Script

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Simple Legged Animation

Knee Rotation Script

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Simple Legged Animation

Ankle Rotation Script

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Simple Legged Animation

Resulting Animation

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Soft Object Animation

The Basic Approaches

0

Deforming the polygonal representation

- Move the vertices defining the polygons as a

function of time

- Connectivity among vertices must be taken into

account

- Magnitude of deformation must be relatively

small (or “simple”) with respect to vertex spacing

- Can cause polygon “aliasing” problems

=Most noticeable when planar or near planar

regions, represented with sparse vertices, are

deformed

0

Deforming the parametric representation

- Accomplished by deforming the control points
- Object still “well defined”

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Nonlinear Global Deformation

Barr 1984

Barr’s notation:

( , , )

( , , )

X Y Z

F x y z



Deformed vertex

Undeformed vertex

Taper along z axis:

( , , ) ( , , )

( )

X Y Z

rx ry z

r

f z





where

Axial twisting:

( , , ) ( cos

sin , sin

cos , )

( )

X Y Z

x

y

x

y

z

f z



















where

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Nonlinear Global Deformation

An Example

Undeformed

Taper

Twist

Bend

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Free Form Deformation

(Sederburg 1986)

A tricubic Bezier hyperpatch is defined:

Q u v w

p B u B v B w

ijk i

j

k

k

j

i

( , , )

( ) ( ) ( )















0

3

0

3

0

3

An

FFD block

is a rectangular lattice of l x m x n Bezier hyperpatches,

consisting of an array of control points.

(

) (

) (

)

3 1

3

1

3

1

l

m

n

 

 



S

T

U

X

0

X s t u

X

sS tT uU

s t u

( , , )

, ,

[ , ]











0

0 1

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Free Form Deformation

The Algorithm

1. Determine the positions of the vertices in

lattice space

.

2. Deform the FFD block. This is accomplished by moving the
control points. The control points are initially located at:

p

X

i

l

S

j

m

T

k

n

U

i

l

j

m

k

n

ijk



 






 






 






 

 

 

0

3

3

3

0

3

0

3

0

3

,

,

,

3. Determine the deformed position of the vertices
- Given the lattice space coordinates of the vertices, find the
relevant hyperpatch
- convert (s,t,u) to the local parameter coordinate of the hyperpatch,
(u,v,w)
- Evaluate the position using:

Q u v w

p B u B v B w

ijk i

j

k

k

j

i

( , , )

( ) ( ) ( )















0

3

0

3

0

3

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Animating the Deformation:

Factor Curves

0

Extension of the concept of weighted

transformation introduced by Barr

0

Animator breaks down the problem of animating a

deformation into two components:

- A set of transformations that can accomplish the

range of deformation required along with a

parameterization of these transformations

- A set of factor curves in both space and time

that modify the parameters of the deforming

transformation according to where and when

they are applied

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Factor Curves:

An Example

 







0

f w f t

w

t

( ) ( )

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Factor Curves:

An Example: the Animated Spoon