Rotations and Translations

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Mathematical terms The inner product of 2 vectors a,b is defined as:

The cross product of 2 vectors is defined as:

A unit vector will be marked as:

, , , , ,x y z x y z x x y y z za a a a b b b b a b a b a b a b

X

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Representing a Point 3D A three-dimensional point

A is a reference coordinate system

here

z

y

xA

p

p

p

P

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Representing a Point 3D (cont.)

Once a coordinate system is fixed, we can locate any point in the universe with a 3x1 position vector.

The components of P in {A} have numerical values which indicate distances along the axes of {A}.

To describe the orientation of a body we will attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system.

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Example

0

2

1

PB ?PA

134.05.02866.0)30sin()30cos( 00 YB

XB

XA PPP

232.2YAP

0.866 0.500 0.000 1 0.134

0.500 0.866 0.000 * 2 2.232 ``

0.000 0.000 1.000 0 0.000

AP

000.0ZAP

AX

BX

AYBY

PB

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Description of Orientation

ˆˆˆˆˆˆ

ˆˆ ˆˆ ˆˆ

ˆˆ ˆˆ ˆˆ

ˆˆˆ

ABABAB

ABABAB

ABABAB

BA

BA

BAA

B

ZZZYZX

YZYYYX

XZXYXX

ZYXR

BX

AX

BX

AYBY

is a unit vector in B

is a coordinate of a unit vector of B in coordinates system A (i.e. the projection of onto the unit direction of its reference)

BA X

BX

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Example Rotating B relative to A around Z by30

000.1000.0000.0

000.0866.0500.0

000.0500.0866.0

RAB

AX

BX

AYBY

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Example In general:

AX

BX

AYBY

100

0cossin

0sincos

ZAB R

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Using Rotation Matrices

PRP BAB

A 9

Translation

AX

AY

AZ

BX

BY

BZ

PB

BORGAP

BORGABA PPP

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Combining Rotation and Translation

BORGABA

BA PPRP

AX

BX

PB

BORGAP

PAAY

BY

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What is a Frame ? A set of four vectors giving position and

orientation information. The description of the frame can be

thought as a position vector and a rotation matrix.

Frame is a coordinate system, where in addition to the orientation we give a position vector which locates its origin relative to some other embedding frame.

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Arrows Convention An Arrow - represents a vector drawn

from one origin to another which shows the position of the origin at the head of the arrow in terms of the frame at the tail of the arrow. The direction of this locating arrow tells us that {B} is known relative to {A} and not vice versa.

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Rotating a frame B relative to a frame A about Z axis by degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0]T?

Example

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000.0

562.12

098.9

000.0

000.5

000.10

0.000

7.562

0.902-

000.0

000.5

000.10

000.0

000.7

000.3

000.1000.0000.0

000.0866.0500.0

000.0500.0866.0

PA

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Extension to 4x4

110001

PPRP BBORG

AAB

A

We can define a 4x4 matrix operator and use a 4x1 position vector

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Example

If we use the above example we can see that:

1000

000.0000.1000.0000.0

000.5000.0866.0500.0

000.10000.0500.0866.0

TAB

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P in the coordinate system A

1

000.0

562.12

098.9

1

000.0

000.7

000.3

1000

000.0000.1000.0000.0

000.5000.0866.0500.0

000.10000.0500.0866.0

PA

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Formula

PTP BAB

A

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Compound Transformation

PTP BAB

A PTP CBC

B

PTTP CBC

AB

A TTT BC

AB

AC

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Several Combinations

TTT

TTTT

TT

TTTTTT

BC

UB

UC

DC

DA

UA

UC

AU

UA

DC

BC

UB

DA

UA

UD

1

1

11

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Example Rotating a frame B relative to a frame A about

Z axis by degrees and moving it 1 units in direction of X and 2 units in the direction of Y. What will be

?ABT

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Example cont - 2 We create a homogeneous transformation

using the function se2: T1 = se2(1, 2, 30*pi/180)T1 = 0.8660 -0.5000 1.0000 0.5000 0.8660 2.0000 0 0 1.0000Note that this is a 2D matrix, we eliminate

the z axis

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Example cont - 3

We can also plot this, relative to the world coordinate frame, by:

>> axis([0 5 0 5]);>>trplot2(T1, 'frame', '1', 'color', 'b')

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Example cont - 4

We can plot a point by:>> P = [3; 2];>> plot_point(P, ‘*’);

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Example cont - 5 Now we got something like this: The point P is

known with respectto {0}

We want todetermine the coordination of point P with respect to {1}

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Example cont - 6

What do we know ?

But…

>> P1 = inv(T1)* [P; 1]

>> P1 = h2e( inv(T1) * e2h(P) );h2e: homogenous to euclidean

0 0 11P T P

1 1 0 0 10 1( )P T P T P

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Example cont - 7

This also can be achieved by:>> homtrans(inv(T1), P)

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Example Rotating a frame B relative to a frame A about Z axis by

degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0]T?

>> T = se2(10,5,30*pi/180) >> P = [3;7;1] >> P2 = T*P >> P = inv(T)*P2

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Notes

Homogeneous transforms are useful in writing compact equations; a computer program would not use them because of the time wasted multiplying ones and zeros. This representation is mainly for our convenience.

For the details turn to chapter 2.

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