POSITION & ORIENTATION ANALYSIS. This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a.

POSITION &

ORIENTATION ANALYSIS

This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3-D space

After this lecture, the student should be able to:•Appreciate the distinction between direction of rotation and axis of rotation•Express the motion of a rigid body in terms of the motion of a point of the body and the rotation of the body

Position & Orientation Analysis

General Motion += Translation Rotation

Summary of Previous Lecture

•Translation: defined by the position between the origins of frame w.r.t. frame {X,Y,Z}. Translation will not change the orientation between the frames:

}ˆ,ˆ,ˆ{ 321 eee

•Orientation: The changes in the orientation of the can be viewed as a result of rotations of frame w.r.t. frame {X, Y, Z}.

}ˆ,ˆ,ˆ{ 321 eee}ˆ,ˆ,ˆ{ 321 eee

)]([)()()()( 0tPQtRtPQtPtQ

R(t) times vector before rotation

=Vector after rotation

Rotation between two configurations

In general, given any arbitrary orthonormal set attached to the rigid body

)()()()()()(

)()()()()()(

111

2022

111

0011

tFtRtRtFtRtF

tFtRtFtFtRtF

The above shows that if two configurations F(t2) and F(t1) are known relative to F(t0), then we can find the configuration of F(t2) relative to F(t1) using R, where )()( 1

12 tFtFR

Therefore:

)()(

)()()()(

12

11

211

2

tRFtF

tFtFtRtRR

Properties of Rotational tensor

100

010

001

IRRT

TRR 1

1)det( RR is proper orthogonal:

For an orthonormal set F:1

1

)1)(1(

SSF

FF T

T

xy

xz

yz

S

SS

SS

SS

S

0

0

0

Where S is a skew-symmetric matrix:

Pure Rotation

So far, we use R (a matrix) to represent the rotation. Is there another way we can represent the rotation? Consider a clock. To define the rotation of the hand represented by vector “AB”, we need at least 2 things:•the axis of rotation <a> (unit vector) and •the angle of rotation about axis <a>

12

6

39 A

B

“AB(t0)” before rotation

B* “AB(t1)” after rotationP

Axis of rotation <a>

positive

Axis of rotation

(angle of rotation)

Axis of Rotation for Pure Rotation

Notice that regardless of how the clock is orientated, once we know the axis of rotation and the angle of rotation, we can define the orientation of final vector “AB(t1)” if given the initial vector “AB(t0)”. To find the axis of rotation, we need a point P so that

12

6

39 APAxis of rotation <a>

Note: if you rotate AP any angle about AP, you will still end up with AP

APAPR

tAPtAPtAPR

)(

)()()( 010

Let A be an nn matrix. If there exists a and a nonzero n1 vector such that

xxA

x

then is called an eigenvalue of A and is called an eigenvector of A corresponding to the eigenvalue

x

Refresh:

APAPRNow:

R is a 33 matrix and is a 31 vector. Relate =1, AR,

AP

APx

Therefore, is the eigenvector of R corresponding to =1

AP

AP

APa

Unit vector for axis of rotation is

Axis of Rotation for Pure Rotation

Example: Axis of Rotation for Pure Rotation

Determine the direction of rotation given by

36329

4948

33364

491

R

The direction of rotation is the same as the axis of rotation. Given R, we can solve for the eigenvector of R corresponding to =1 to get

AP

Solution:

0

0

0

13329

44048

333645

491

0

0

0

)4936(329

4)499(48

3336)494(

491

0

0

0

100

010

001

1

36329

4948

33364

491

z

y

x

z

y

x

z

y

x

AP

AP

AP

AP

AP

AP

AP

AP

AP

APIR

Example: Axis of Rotation for Pure Rotation

Example: Axis of Rotation for Pure Rotation

013329

044048

0333645

zyx

zyx

zyx

APAPAP

APAPAP

APAPAP

2

09819606516045

0333645

y

z

zyzyx

zyx

APAP

APAPAPAPAP

APAPAP

Multiply 3rd eqn by -5 and add it to 1st eqn to eliminate xAP

Example: Axis of Rotation for Pure Rotation

Divide 2nd eqn by and simplify using the known result:yAP

32

4832

0)2(44048

044048

y

x

y

x

y

z

y

x

AP

AP

AP

AP

APAP

AP

AP

013329

044048

0333645

zyx

zyx

zyx

APAPAP

APAPAP

APAPAP

Example: Axis of Rotation for Pure Rotation

T

yT

zyx

y

x

y

z

APAPAPAPAP

AP

AP

APAP

2132

32

,2

TT

y

y

a

AP

AP

AP

APa

63271

2132

73

ˆ

2132

2132

ˆ22

2

(Remember this axis of rotation. We need it later in another e.g.)

Angle of Rotation for Pure Rotation

For the clock, the hand is always perpendicular to <a>, the axis of rotation. Generally, given a point C on the rigid body, the line “AC” may not be perpendicular to axis <a>. Nevertheless, once the axis of rotation <a> is obtained from R, we can always resolve “AC” into 2 components: the first component ACn is normal and the second component ACp is parallel to the axis <a>:

aaACACACACAC

ACaaACACACAC

aaACAC

pn

nnp

p

ˆˆ

ˆˆ

ˆˆ

A

PAxis of rotation <a>

C

ACp

ACn

Time t0

At time t1, AC has rotated an angle of about axis <a> to AC* :

ACp

A

PAxis of rotation <a>

C

ACn

C*

aaACtACtRACtR

ACtRaaACtACtR

n

n

ˆ)ˆ()()()(

)(ˆ)ˆ()()(

0

0

aaACACACtR pP ˆˆ)(

But

npnp ACtRACtRACACtRtACtRtACAC )()(])[()]()[()(* 01

Therefore

Angle of Rotation for Pure Rotation

In other word, their cross product will be a vector in the direction of axis <a>.

We can find once we determine axis <a>.

nACtR )(ACp

A

P*Axis of rotation <a>

C

ACn

C*

aaACtACtRACtR n ˆ)ˆ()()()( 0

nACtR )(Notice that axis <a> is normal to both and

nAC

Therefore )sin()(2

nnn ACACtRAC

Where is the angle of rotation

Angle of Rotation for Pure Rotation

)sin(baba But

Angle of Rotation for Pure Rotation

Therefore, given R and , we can find the axis of rotation )( 0tAC

a

aatACACp ˆˆ)( 0

pn ACtACAC )( 0

)(

nACR

)()(

nn ACRAC

2

nAC

)()(

nn ACRAC

2

)(

)()()sin(

n

nn

AC

ACRAC

Example: Angle of Rotation for Pure Rotation

Determine the angle of rotation of a point C of the rigid body if

36329

4948

33364

491

111)( 0

R

tAC T

Solution:

The axis of rotation for R has been found from the previous example to be

Ta 63271

ˆ

Example: Angle of Rotation for Pure Rotation

Tn

TTpn

TTp

p

AC

ACACAC

AC

aaaACAC

171627491

6324911

111

6324911

63271

711

ˆ76

73

72

111ˆˆ

TT atAC 63271

ˆ,111)( 0

With

4926

24011274

4917

4916

4927

2

2222

n

n

AC

AC

Example: Angle of Rotation for Pure Rotation

36329

4948

33364

491

,171627491

RAC TnWith

1

4

3

71

343

1372

1029

491

)(

17

16

27

36329

4948

33364

491

491

)(

2

n

n

ACR

ACR

Example: Angle of Rotation for Pure Rotation

4926

156785271

491

)(

156

78

52

71

491

71

74

73

4917

4916

4927

ˆˆˆ

)(

14371

)(,171627491

222

nn

nn

nT

n

ACRAC

kji

ACRAC

ACRAC

Example: Angle of Rotation for Pure Rotation

,2,1,0,22

1

)(

)()()sin( 2

nn

AC

ACRAC

n

nn

49262

nACStory so far ….

4926

)(

nn ACRAC

Therefore

Explicit Expression of R given axis and angle of Rotation

Given R we can obtain the axis of rotation and the angle of rotation. If we are given the axis of rotation and the angle of rotation, then we should be able to derive the rotational matrix R as ),ˆ( aRR

ccaasacaasacaa

sacaaccaasacaa

sacaasacaaccaa

R

zzxzyyzx

xyzyyzyx

yxzzxyxx

)1()1()1(

)1()1()1(

)1()1()1(

Given and kajaiaa zyxˆˆˆˆ

where

)cos(

)sin(

c

s

Explicit Expression of R given axis and angle of Rotation

Example 1: Find R for a rotation of radians about [1, 0, 0]T

cs

sc

csa

sac

cca

R

ccaasacaasacaa

sacaaccaasacaa

sacaasacaaccaa

R

x

x

x

zzxzyyzx

xyzyyzyx

yxzzxyxx

0

0

001

0

0

00)1(

)1()1()1(

)1()1()1(

)1()1()1(

2

,001ˆ0ˆ0ˆ1ˆ Tkjia Solution:

where

)cos(

)sin(

c

s This is a rotation about the X-axis

Explicit Expression of R given axis and angle of Rotation

Example 2: Find R for a rotation of radians about [0, 1, 0]T

cs

sc

csa

cca

sac

R

ccaasacaasacaa

sacaaccaasacaa

sacaasacaaccaa

R

y

y

y

zzxzyyzx

xyzyyzyx

yxzzxyxx

0

010

0

0

0)1(0

0

)1()1()1(

)1()1()1(

)1()1()1(

2

,010ˆ0ˆ1ˆ0ˆ Tkjia Solution:

where

)cos(

)sin(

c

s This is a rotation about the Y-axis

Explicit Expression of R given axis and angle of Rotation

Example 3: Find R for a rotation of radians about [0, 0, 1]T

100

0

0

)1(00

0

0

)1()1()1(

)1()1()1(

)1()1()1(

2

cs

sc

cca

csa

sac

R

ccaasacaasacaa

sacaaccaasacaa

sacaasacaaccaa

R

z

z

z

zzxzyyzx

xyzyyzyx

yxzzxyxx

,100ˆ1ˆ0ˆ0ˆ Tkjia Solution:

where

)cos(

)sin(

c

s This is a rotation about the Z-axis

This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3-D space

The following were covered:•The distinction between direction of rotation and axis of rotation•The motion of a rigid body in terms of the motion of a point of the body and the rotation of the body

Summary