Rigid Body Transformations

Rigid Body Kinematics

University of Pennsylvania

1

Rigid Body Transformations

Vijay Kumar



2

Remark about Notation

Vectors x, y, a, … Ax u, v, p, q, …

Matrices A, B, C, … AAB

g, h, …

Potential for Confusion!



3

The 3vector a and its 3skew symmetric matrix counterpart a

For any vector b

ab = A b

3

2

1

12

13

23

2112

3113

2332

3

2

1

3

2

1

0

0

0

b

b

b

aa

aa

aa

baba

baba

baba

b

b

b

a

a

a

a

a = A



4

Rigid Body Transformation

Rigid Body Displacement

Rigid Body Motion

3: ROtg

O

R33: ROg t



5

Coordinate Transformations and Displacements

Transformations of points Transformation (g) of points induces an action (g*) on vectors

What are rigid body transformations? Displacements? g preserves lengths g* preserves cross products

p

qg (p)

g (q)

v

g* (v)



6

Rigid Body Transformations in R3

Can show that the most general coordinate transformation from {B} to {A} has the following form

position vector of P in {B} is transformed to position vector of P in {A}

description of {B} as seen from an observer in {A}

OAPBB

APA rrRr

x

y

z

ArP

O

BrP

ArO’

z'

y'

x'

{A}

O'

A

BP

Rotation of {B} with respect to

{A}

Translation of the origin of {B} with respect to origin of {A}



7

Rotational transformations in R3

Properties of rotation matrices Transpose is the inverse Determinant is +1

Rotations preserve cross products

R u R v = R (u v)

Rotation of skew symmetric matrices

For any rotation matrix R:

R w RT = (R w)



8

Example: Rotation Rotation about the x-axis through

cossin0

sincos0

001

,xRot

x

y

z

Displacement

x

y

z



9

Example: RotationRotation about the y-axis through

y

z

x

z'x'

cos0sin

010

sin0cos

,yRot

Rotation about the z-axis through

100

0cossin

0sincos

,zRot

y

z

x

y'

x'



10

Rigid Motion in R3

The same equation can have two interpretations: It transforms the position vector of any point in {B} to the position vector in {A} It transforms the position vector of any point in the first position/orientation (described

by {A}) to its new position vector in the second position orientation (described by{B}).

x

y

z

ArP’

O

ArO’

z'

y'

x'

{A}

O'

{B}

ArP P

P’

BrP’

x

y

z

ArP

O

BrP

ArO’

z'

y'

x'

{A}

O'

A

BP

OAPBB

APA rrRr OAPAB

APA rrRrCoordinate transformation from {B} to {A} Displacement of a body-fixed frame from {A} to {B}



11

Mobile Robots

x

y

100

cossin

sincos

y

x

gWRW

R

WAR



12

The Lie group SE(3)

IRRRRrR

0

rRAA TTRRSE ,,,

13 333

31

http://www.seas.upenn.edu/~meam520/notes02/RigidBodyMotion3.pdf



13

SE(3) is a Lie groupSE(3) satisfies the four axioms that must be satisfied by the elements of an algebraic group:

The set is closed under the binary operation. In other words, if A and B are any two matrices in SE(3), AB SE(3).

The binary operation is associative. In other words, if A, B, and C are any three matrices SE(3), then (AB) C = A (BC).

For every element A SE(3), there is an identity element given by the 44 identity matrix, ISE(3), such that AI = A.

For every element A SE(3), there is an identity inverse, A-1 SE(3), such that A A-1 = I.

SE(3) is a continuous group. the binary operation above is a continuous operation the product of any two

elements in SE(3) is a continuous function of the two elements the inverse of any element in SE(3) is a continuous function of that element.

In other words, SE(3) is a differentiable manifold. A group that is a differentiable manifold is called a Lie group[ Sophus Lie (1842-1899)].



14

Composition of DisplacementsDisplacement from {A} to {B}

Displacement from {B} to {C}

Displacement from {A} to {C}x

y

z

O

z'

y'

x'

{A}

O'

z''

x''

y''

O''

{B}

POSITION 1

POSITION 2

POSITION 3

{C}

,131

'

0

rRA

OAB

A

BA

,131

0

rRA

OBC

B

CB

1

11

1

0

rrRRR

0

rR

0

rR

0

rRA

OAOBB

AC

BB

A

OBC

BOAB

A

OAC

A

CA

Note XAY describes the displacement of the body-fixed frame from {X} to {Y} in reference frame {X}



15

Composition (continued)

x

y

z

O

z'

y'

x'

{A}

O'

z''

x''

y''

O''

{B}

POSITION 1

POSITION 2

POSITION 3

{C}

Note XAY describes the displacement of the body-fixed frame from {X} to {Y} in reference frame {X}

Composition of displacements Displacements are generally described

in a body-fixed frame Example: BAC is the displacement of a

rigid body from B to C relative to the axes of the “first frame” B.

Composition of transformations Same basic idea AAC = AAB BAC

Note that our description of transformations (e.g., BAC) is relative to the “first frame” (B, the frame with the leading superscript).



16

S u b g r o u p N o t a t i o n D e f i n i t i o n S i g n i f i c a n c e

T h e g r o u p o fr o t a t i o n s i n

t h r e ed i m e n s i o n s

S O ( 3 ) T h e s e t o f a l l p r o p e r o r t h o g o n a lm a t r i c e s .

S O R T T3 3 3 R R R R R R I,

A l l s p h e r i c a l d i s p l a c e m e n t s .O r t h e s e t o f a l l d i s p l a c e m e n t s

t h a t c a n b e g e n e r a t e d b y as p h e r i c a l j o i n t ( S - p a i r ) .

S p e c i a lE u c l i d e a n

g r o u p i n t w od i m e n s i o n s

S E ( 2 ) T h e s e t o f a l l 3 3 m a t r i c e s w i t h t h es t r u c t u r e :

c o s s i n

s i n c o s

r

rx

y

0 0 1

w h e r e , r x , a n d r y a r e r e a l n u m b e r s .

A l l p l a n a r d i s p l a c e m e n t s . O rt h e s e t o f d i s p l a c e m e n t s t h a tc a n b e g e n e r a t e d b y a p l a n a r

p a i r ( E - p a i r ) .

T h e g r o u p o fr o t a t i o n s i n

t w od i m e n s i o n s

S O ( 2 ) T h e s e t o f a l l 2 2 p r o p e r o r t h o g o n a lm a t r i c e s . T h e y h a v e t h e s t r u c t u r e :

c o s s i n

s i n c o s

,

w h e r e i s a r e a l n u m b e r .

A l l r o t a t i o n s i n t h e p l a n e , o rt h e s e t o f a l l d i s p l a c e m e n t s

t h a t c a n b e g e n e r a t e d b y as i n g l e r e v o l u t e j o i n t ( R - p a i r ) .

T h e g r o u p o ft r a n s l a t i o n s i nn d i m e n s i o n s .

T ( n ) T h e s e t o f a l l n 1 r e a l v e c t o r s w i t hv e c t o r a d d i t i o n a s t h e b i n a r y

o p e r a t i o n .

A l l t r a n s l a t i o n s i n nd i m e n s i o n s . n = 2 i n d i c a t e s

p l a n a r , n = 3 i n d i c a t e s s p a t i a ld i s p l a c e m e n t s .

T h e g r o u p o ft r a n s l a t i o n s i n

o n ed i m e n s i o n .

T ( 1 ) T h e s e t o f a l l r e a l n u m b e r s w i t ha d d i t i o n a s t h e b i n a r y o p e r a t i o n .

A l l t r a n s l a t i o n s p a r a l l e l t o o n ea x i s , o r t h e s e t o f a l l

d i s p l a c e m e n t s t h a t c a n b eg e n e r a t e d b y a s i n g l e

p r i s m a t i c j o i n t ( P - p a i r ) .

T h e g r o u p o fc y l i n d r i c a l

d i s p l a c e m e n t s

S O ( 2 ) T ( 1 ) T h e C a r t e s i a n p r o d u c t o f S O ( 2 ) a n dT ( 1 )

A l l r o t a t i o n s i n t h e p l a n e a n dt r a n s l a t i o n s a l o n g a n a x i s

p e r p e n d i c u l a r t o t h e p l a n e , o rt h e s e t o f a l l d i s p l a c e m e n t s

t h a t c a n b e g e n e r a t e d b y ac y l i n d r i c a l j o i n t ( C - p a i r ) .

T h e g r o u p o fs c r e w

d i s p l a c e m e n t s

H ( 1 ) A o n e - p a r a m e t e r s u b g r o u p o f S E ( 3 ) A l l d i s p l a c e m e n t s t h a t c a n b eg e n e r a t e d b y a h e l i c a l j o i n t

( H - p a i r ) .

Subgroups of SE(3)

Rigid Body Transformations

Documents