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Politecnico di Torino - Dipartimento di Meccanica Stefano Pastorelli

POSITION and ORIENTATION of COORDINATE SYSTEMS

Frame of reference 0 : origin O

axes x y z

unit vectors k ji

Frame of reference i : origin P

axes u v w

unit vectors

Position of origin P with respect to the reference frame O-xyz

Position vector written in terms of its components p x, p y, p z :

k p j pi p p z y x

z

y

x

p

p

p

p0

Orientation of the reference frame P-uvw with respect to the reference frame O-xyz

Units vectors pointing along axes u, v, w written in terms of their components:

k c jcic uz uyux

k c jcic vz vyvx

k c jcic wz wywx

cux, ... cwz = direction cosines of the axes u, v, w with respect to the frame O-xyz

conditions: 1 ( 1222 uz uyux ccc ); 1 ; 1

0 ( 0 vz uz vyuyvxux cccccc ); 0 ; 0

Rotation matrix of reference frame i (P-uvw) with respect to the reference frame 0 (O-xyz):

wz vz uz

wyvyuy

wxvxux

i

cccccc

ccc

A 0000

Oy

z

u

vw

P

x

p





Orientation of the reference frame O-xyz with respect to the reference frame P-uvw

Rotation matrix of reference frame 0 (O-xyz) with respect to the reference frame i (P-uvw):

100

0

0

0

0

i

T

i

wz wywx

vz vyvx

uz uyux

T

T

T

iiii A A

ccc

ccc

ccc

k ji A

Position vector of a point R in two different coordinate systems with coincident origins

position vector of point R in the frame 0 (O-xyz): T z y x r r r r 0

position vector of point R in the frame i (P-uvw): T wvu

ir r r r

k c jcicr k c jcicr k c jcicr r r r r wz wywxwvz vyvxvuz uyuxuwvu

r Ar i

i 00

r Ar ii 0

0

i A0 = 3 3 transformation matrix of a vector from reference frame i to reference frame 0.

Oy

z

u

vw

P

x

p

OPy

z

u

vw

x

R

r





Position vector of a point R in two different coordinate systems

with homogenous coordinates notation

position vector of point R in the frame 0 (O-xyz): T z y x r r r r 0

position vector of point R in the frame i (P-uvw): T wvu

ir r r r

PROP OR

r A p r i

i 000

wwz vvz uuz z z

wwyvvyuuy y y

wwxvvxuux x x

r cr cr c pr

r cr cr c pr

r cr cr c pr

Homogenous coordinates notation:

110110001

00

w

v

u

T

i

w

v

u

z wz vz uz

ywyvyuy

xwxvxux

z

y

x

r

r

r

p A

r

r

r

pccc

pccc

pccc

r

r

r

r Ar i

i ˆˆ ˆ

00

i A0 = 4 4 homogeneous transformation matrix; it denotes translation and rotation of

reference frame i with respect to reference frame 0; it transforms a position vector from

reference frame i to reference frame 0.

Oy

z

u

vw

P

x

p

R

r i

r 0





Inverse of the transformation matrixi A0

i A0 = transformation matrix of reference frame i with respect to reference frame 0

r Ar i

i ˆˆˆ

00 r Ar Ar i

i

iˆˆˆˆˆ

0

0

010

0

ˆ Ai = transformation matrix of reference frame 0 with respect to reference frame i

T

ii

i A A A ˆˆˆ 010

0

10

ˆˆ

000

10

0

T

T

i

T

i

i

i p A A

A A

Evaluation of the inverse matrix 10 ˆ i A from the matrix

i A0 :

10

0

1010

ˆˆ 21

00

100

T T T

i

ii

U x X p A A A

U X Ai 1

0 T

ii A A X 010

1

0 p x Ai 0

2

0 p A p A xT

ii

00010

2

O

z

u

vw

P

x

p

R

r

i

r 0





Composition of homogeneous transformation matrixes in a multiframe system

r A... A Ar n

n

nˆˆˆˆˆ

1

2

1

1

00

ni

i

i

i

n A A1

10 ˆˆ

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