Transformations and Kinematics

Transformations and

Kinematics

Jehee Lee

Seoul National University

Transformations

• Linear transformations

• Rigid transformations

• Affine transformations

• Projective transformations

T

Global reference frame

Local moving frame

Linear Transformations

• A linear transformation T is a mapping between

vector spaces

– T maps vectors to vectors

– linear combination is invariant under T

• In 3-spaces, T can be represented by a 3x3 matrix

)()()()( 1100

0

NN

N

i

ii TcTcTccT vvvv

major) (Row

major)(Column )(

3331

1333

Nv

vMvT

Examples of Linear Transformations

• 2D rotation

• 2D scaling

y

x

y

x

cossin

sincos

ys

xs

y

x

s

s

y

x

y

x

y

x

0

0

yx, yx ,


• 2D shear

– Along X-axis

– Along Y-axis

y

dyx

y

xd

y

x

10

1

dxy

x

y

x

dy

x

1

01


• 2D reflection

– Along X-axis

– Along Y-axis

y

x

y

x

y

x

10

01

y

x

y

x

y

x

10

01

Properties of Linear Transformations

• Any linear transformation between 3D spaces

can be represented by a 3x3 matrix

• Any linear transformation between 3D spaces

can be represented as a combination of rotation,

shear, and scaling

• Rotation can be represented as a combination

of scaling and shear

Affine Transformations

• An affine transformation T is an mapping

between affine spaces

– T maps vectors to vectors, and points to points

– T is a linear transformation on vectors

– affine combination is invariant under T

• In 3-spaces, T can be represented by a 3x3

matrix together with a 3x1 translation vector

)()()()( 1100

0

NN

N

i

ii TcTcTccT pppp

131333)( TpMpT

Homogeneous Coordinates

• Any affine transformation between 3D spaces

can be represented by a 4x4 matrix

• Affine transformation is linear in homogeneous

coordinates

110)(

131333 pTMpT

Examples of Affine Transformations

• 2D rotation

• 2D scaling

1100

0cossin

0sincos

1

y

x

y

x

yx, yx ,

11100

00

00

1

ys

xs

y

x

s

s

y

x

y

x

y

x


• 2D shear

• 2D reflection

11100

010

01

1

y

dyx

y

xd

y

x

11100

010

001

1

y

x

y

x

y

x


• 2D translation

11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x


• 2D transformation for vectors

– Translation is simply ignored

00100

10

01

0

y

x

y

x

t

t

y

x

y

x

Properties of Affine Transformations

• Any affine transformation between 3D spaces can be

represented as a combination of a linear transformation

followed by translation

• An affine transf. maps lines to lines

• An affine transf. maps parallel lines to parallel lines

• An affine transf. preserves ratios of distance along a line

• An affine transf. does not preserve absolute distances

and angles

Rigid Transformations

• A rigid transformation T is a mapping between

affine spaces

– T maps vectors to vectors, and points to points

– T preserves distances between all points

– T preserves cross product for all vectors (to avoid

reflection)

• In 3-spaces, T can be represented as

1det and

where,)( 131333

RIRRRR

TpRp

TT

T

Rigid Body Rotation

• Rigid body transformations allow only rotation

and translation

• Rotation matrices form SO(3)

– Special orthogonal group

IRRRR TT

1det R

(Distance preserving)

(No reflection)

Rigid Body Rotation

• R is normalized – The squares of the elements in any row or column

sum to 1

• R is orthogonal – The dot product of any pair of rows or any pair

columns is 0

• The rows (columns) of R correspond to the vectors of the principle axes of the rotated coordinate frame

IRRRR TT

Taxonomy of Transformations

• Linear transformations – 3x3 matrix

– Rotation + scaling + shear

• Rigid transformations – SO(3) for rotation

– 3D vector for translation

• Affine transformation – 3x3 matrix + 3D vector or 4x4 homogenous matrix

– Linear transformation + translation

• Projective transformation – 4x4 matrix

– Affine transformation + perspective projection

Taxonomy of Transformations

Projective

Affine

Rigid

Kinematics

• How to animate skeletons

(articulated figures)

• Kinematics is the study of

motion without regard to

the forces that caused it

Hierarchical Models

• Tree structure of joints and links

– The root link can be chosen arbitrarily

• Joints – Revolute (hinge) joint allows rotation about a fixed axis

– Prismatic joint allows translation along a line

– Ball-and-socket joint allows rotation about an arbitrary axis

Human Joints

Forward and Inverse Kinematics

1

2

)F(),( iqp ),(F 1qp

iForward Kinematics Inverse Kinematics

1

2

Forward Kinematics: A Simple Example

• A simple robot arm in 2-dimensional space

– 2 revolute joints

– Joint angles are known

– Compute the position of the end-effector

)sin(sin

)cos(cos

21211

21211

lly

llx

e

e

2

1

1l

2l

),( ee yx

Forward Kinematics: A Simple Example

• Forward kinematics map as a coordinate transformation

– The body local coordinate system of the end-effector was initially

coincide with the global coordinate system

– Forward kinematics map transforms the position and orientation

of the end-effector according to joint angles

2

1

1l

X Y

1

0

0

1

Ty

x

e

e2l

X

Y

A Chain of Transformations

1

0

0

1

Ty

x

e

e

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

2

1

1l

X Y

2l

X

Y

Thinking of Transformations

• In a view of body-attached coordinate system

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

X

Y

X

Y

X

Y



1X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT



1

1LX

Y

X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

2

1

1L

X

Y



X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

2

1

1L

X Y

2L



X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT


• In a view of global coordinate system

X

Y

X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT



X

Y

2L

X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

2

X Y

2L



X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT



21L

X Y

2L

X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT



2

1

1L

X Y

2L

X

Y

100

010

01

100

0cossin

0sincos

100

010

01

100

0cossin

0sincos 2

22

221

11

11

2211

ll

translrottranslrotT

How to Handle Ball-and-Socket Joints ?

• Three revolute joints whose axes intersect at a

point (equivalent to Euler angles), or

• 3D rotation about an arbitrary axis

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

21

zz

zz

yy

yy

xx

xx

zyx translrotrotrottranslT

Floating Base

• The position and orientation of the root segment

are added

1

2

3

221100 rottranslrottranslrottransltranslrotT rr

X

Y

Joint & Link Transformations

• Each segment has its own coordinate frame

• Forward kinematics map is an alternating multiple of

– Joint transformations : represents joint movement

– Link transformations : defines a frame relative to its parent

1L

3L

2L

3322110 JLJLJLJT

The position and orientation of

the root segment

1st link transformation

1st joint transformation

Joint & Link Transformations

• Both are rigid transformations in general

– Joint transformations may include translation

• Human joints are not ideal hinges

– Link transformations may include rotation

• Some links are twisted

Denavit-Hartenberg Notation

),(),(),(),( 11 iiiii ZRotdZTransaXTransXRotL

iiii

iiii

iiii

iiii

ZXX

ZXXd

XZZ

XZZa

about measured and between angle the

along measured to from distance the

about measured and between angle the

along measured to from distance the

1

1

1

1

Link Transformations

• How do you compute the link transform for the ith joint if

you know the position and orientation of the ith joint as

well as its parent’s position and orientation at the neutral

pose ?

1iT

iL

iT

iii

iii

TTL

TLT

1

1

1

Representing Hierarchical Models

• A tree structure

– A node contains a joint transformation

– A arc contains a link transformation

0J

1J

2J

3J

1L

2L

3L

Forward and Inverse Kinematics

1

2

)F(),( iqp ),(F 1qp

iForward Kinematics Inverse Kinematics

1

2

Why Inverse Kinematics ?

• Environmental interactions

– Pick up an object or place feet on the ground

– Hard to do with forward kinematics

• The pose of the character is described in the joint angle space

• The environmental interaction is described in the work (Cartesian)

space

Inverse Kinematics: A Simple Example

• A simple robot arm in 2-dimensional space

– Two revolute joints

– The position of the end-effector is known

– Compute joint angles

2

1

1l

2l

X

Y

),( ee yx

Analytic Solution for A Simple Example

2

1

1l

2l

X

Y

21

222

2

2

11

2

21

222

2

2

12

22

1

2

2

222

11

1

22

1

2

2

222

11

22

1

22

2cos

2)cos(

2cos

2)cos(

cos

)cos(

ll

yxll

ll

yxll

yxl

lyxl

yxl

lyxl

yx

x

yx

x

ee

ee

ee

ee

ee

ee

ee

e

ee

e

),( ee yx

Redundancy in Human Arms

Why so difficult to get a closed-form solution ?

• Redundancies – Multiple solutions (# of unknowns > # of equations)

• Joint limit – The range of each unknown is bounded

• Reachable workspace – No solution, or

– A unique solution, or

– Multiple solutions

• Multiple goals – Four limbs may have constraints simultaneously

– Intermediate links can also be constrained

Iterative Methods

• Iteratively refine joint angles to the goal

– Consider infinitesimal changes

),(from),,(compute 321 yx

),(from),,(compute 321 yx

),( yx

3

1 2

Jacobians of Forward Kinematics Map

• Forward Kinematics Map

• Jacobian

3

2

1

321

321

3

2

1

yyy

xxx

FFF

FFF

y

xJ

),,(),(),( 321 FFFyx yx ),( yx

3

1 2

Goal

Jacobian of Forward Kinematics Map

• If the inverse of the Jacobian can be computed, …

),( yx

3

1 2

Goal

t

t

ttt

y

xt

t

1

3

2

1

3

2

1

3

2

1

13

2

1

J

A System of Linear Equations

• # of unkowns = the dimension of = n

• # of equations = the dimension of x = m

• J is a (m x n) matrix

• The solution maybe unique if m=n

• The linear system is under-specified if m<n

• The linear system is over-specified if m>n

xθJ

the pseudoinverse of Jacobian

is required

Redundancy

• A single solution must be chosen from multiple solutions

– “Closest” to the current configuration

• Pseudo inverse minimizes joint angle rates (locally)

– Move outermost links the most

• The outermost link sweeps a smallest region (visual change)

– Minimum time and effort

• Dynamics involves

– Secondary goal

• Additional constraints

– Natural looking

• Biomechanical experiments

Summary

• Very simple structure allows an analytic solution

• Most of complex articulated figures requires a

numerical solution

• May not always get the “right” answer

– Need to tweak the solution later

Transformations and Kinematics

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