PDF Ch. 2: Spatial Descriptions and Transformations

Ch. 2: Spatial Descriptions and Transformations

2.1 Coordinate System

• Cartesian coordinate system

• three mutually orthogonal unit vector

• origin point

• Cartesian coordinate frame, e.g.

• usually fixed or attached to the moving body

{ }

^A

Ch. 2: Spatial Descriptions and Transformations

2.2 Representing Position

• attach points to the locations of interest

• the point is the point itself; we know where it is; it is a geometric entity, i.e. no numerical values involved

• we can describe the point by position vector represented in any coordinate system

0

1

5 6

2.8 4.2 p

p

= ⎢ ⎥⎡ ⎤

⎣ ⎦

⎡− ⎤

= ⎢ ⎥

⎣ ⎦

[ ]

[ ]

0 1

1 0 1

0 1 0

10 5

10.6 3.5 ,

T

T

O

O O O

=

= − ∴ ≠

Ch. 2: Spatial Descriptions and Transformations

2.3 Representing Orientation

• attach frames to the bodies of interest

• Rotation is the rotation itself. There are several ways to describe.

• One way is to describe the unit vectors of one frame with respect to the other frame

0 0 0

1 1 1

1 0 1 0

1 0 1 0

1 0 1

0

T T

R

c s

s c

θ θ

θ θ

⎡ ⎤

= ⎣ ⎦

⋅ ⋅ −

⎡ ⎤ ⎡ ⎤

= ⎢⎣ ⋅ ⋅ ⎥ ⎢⎦ = ⎣ ⎥⎦

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

x y

x x y x x y y y

x y

Ch. 2: Spatial Descriptions and Transformations

Basic Rotation Matrix

,

,

,

1 0 0

0 0

0

0 1 0

0

0 0

0 0 1

x

y

z

R c s

s c

c s

R

s c

c s

R s c

θ

θ

θ

θ θ

θ θ

θ θ

θ θ

θ θ

θ θ

⎡ ⎤

⎢ ⎥

= ⎢ − ⎥

⎢ ⎥

⎣ ⎦

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢− ⎥

⎣ ⎦

⎡ − ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

Rotation matrix property

•

•

• the columns (rows) are mutually orthogonal

• each column (row) is a unit vector

•

det R =1

1

RT = R⁻

( )

^{10R −1 = 01R}

Ch. 2: Spatial Descriptions and Transformations

Ex.

0 1

1/ 2 1/ 2 0

0 0 1

1/ 2 1/ 2 0

R

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ − ⎥

⎣ ⎦

1 0

0 1

1/ 2 0 1/ 2

1/ 2 0 1/ 2

0 1 0

R RT

⎡ ⎤

⎢ ⎥

= ⎢ − ⎥ =

⎢ ⎥

⎢ ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

2.4 Rotational Transformation

•

• the coordinates of a point in one frame is transformed to the coordinates of the same point in another frame by the rotational matrix

0 0 1

p = 1R p

coordinate transformation

Ch. 2: Spatial Descriptions and Transformations

•

coordinate transformation

0

1 orientation of transformed coordinate frame with respect to fixed coordinate frame

R =

Ch. 2: Spatial Descriptions and Transformations

•

• a point is rotated according to the rotation matrix to a new location; both of which are described in the same coordinate frame

0 0

b a

p = R p

operator

Ch. 2: Spatial Descriptions and Transformations

Similarity Transformation

• is the transformation of same operator from one frame into another frame

• is the operator described in

• is the same operator described in

• is the coordinate transformation

• then,

• first, transform the coordinates from to

• second, operate in by the operator

• third, display the result back from to by

• same as operate in by the operator

A { }

⁰

B { }

¹

0 1T

( )

^{01 1 01}

B = T ⁻ A T

{ }

¹

{ }

⁰

{ }

⁰

A

{ }

⁰

{ }

¹

( )

^{01T −1}

{ }

¹

B

Ch. 2: Spatial Descriptions and Transformations

Ex.

Rotational operator in is

in where the relative rotation between two frames is

,

Rz _θ

{ }

⁰

1 0 0

0 0

c s

s c

θ θ

θ θ

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ − ⎥

⎣ ⎦

{ }

¹

0 1

0 0 1

0 1 0

1 0 0 R

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢− ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

2.5 Composition of Rotations

• Rotation w.r.t. current frame Æ POST-multiply

• Rotation w.r.t. fixed frame Æ PRE-multiply

• Rotational transformation is not commute

Ch. 2: Spatial Descriptions and Transformations

2.6 Parameterization of Rotations

• A 3x3 orthogonal matrix is one of the ways to describe the rotation

• 3 degrees of freedom for arbitrary rotation Æ 3 free variables

• Euler-angles, fixed-angles, angle-axis representation

Ch. 2: Spatial Descriptions and Transformations

Euler-angles representation

• any rotation can be created by three rotations about an axis of the moving frame

• 12 possibilities

•

φ θ ψ

, , ZYZ Euler-angles

( )

' ' ' , , , , ,

Z Y Z

Z Y Z

R R R R

c c c s s c c s s c c s s c c c s s c s c c s s

s c s s c

φ θ ψ

φ θ ψ

φ θ ψ φ ψ φ θ ψ φ ψ φ θ φ θ ψ φ ψ φ θ ψ φ ψ φ θ

θ ψ θ ψ θ

=

− − −

⎡ ⎤

⎢ ⎥

= ⎢ + − + ⎥

⎢ − ⎥

⎣ ⎦

see Craig’s Appendix B

Ch. 2: Spatial Descriptions and Transformations

Euler-angles representation

( )

sol #1, s

θ

> 0

( )

( )

( )

2

33 33

13 23 31 32

atan2 , 1 atan2 ,

atan2 ,

r r

r r r r θ

φ ψ

= −

=

= −

13 0 || 23 0

r ≠ r ≠

( )

sol #2, s

θ

< 0

( )

( )

( )

2

33 33

13 23

31 32

atan2 , 1 atan2 ,

atan2 ,

r r

r r r r θ

φ ψ

= − −

= − −

= −

Yes

Ch. 2: Spatial Descriptions and Transformations

Euler-angles representation

θ

= 0

(

11 21

)

atan2 r r, φ ψ+ =

13 0 || 23 0

r ≠ r ≠

θ π

=

(

11 12

)

atan2 r , r φ ψ− = − −

No

r33

= 1 = -1

representational singularity

representational singularity

Ch. 2: Spatial Descriptions and Transformations

Fixed-angles representation

• any rotation can be created by three rotations about an axis of the fixed frame

• 12 possibilities

•

φ θ ψ

, , XYZ (RPY) fixed-angles

(

, ,

)

, , ,

XYZ Z Y X

R R R R

c c s c c s s s s c s c s c c c s s s c s s s c

s c s c c

φ θ ψ

ψ θ φ

φ θ φ ψ φ θ ψ φ ψ φ θ ψ φ θ φ ψ φ θ ψ φ ψ φ θ ψ

θ θ ψ θ ψ

=

− + +

⎡ ⎤

⎢ ⎥

= ⎢ + − + ⎥

⎢ − ⎥

⎣ ⎦

see Craig’s Appendix B

Ch. 2: Spatial Descriptions and Transformations

Fixed-angles representation

sol #1 sol #2

Yes

Ch. 2: Spatial Descriptions and Transformations

Fixed-angles representation

No

representational singularity

representational singularity

Ch. 2: Spatial Descriptions and Transformations

Angle-axis representation

• A rotation can be obtained by rotating about the specified axis by the particular angle

• this operation can be described easily in the coordinate frame having a principle axis aligned with the axis of

rotation

• what is the representation of the operation in the reference coordinate frame

• similarity transformation

Ch. 2: Spatial Descriptions and Transformations

Angle-axis representation

{ }

⁰ : reference coordinate frame

{ }

¹ : coordinate frame having axis aligned with the axis of rotation

z

0 1

, 1 , 0

k z

R _θ = R R⋅ _θ ⋅ R

0

1 , ,

1

0 , ,

z y

y z

R R R

R R R

α β

β α

− −

= ⋅

= ⋅

2

2 ,

2

x x y z x z y

k x y z y y z x

x z y y z x z

k v c k k v k s k k v k s

R k k v k s k v c k k v k s

k k v k s k k v k s k v c

θ

θ θ θ θ θ θ

θ θ θ θ θ θ

θ θ θ θ θ θ

⎡ + − + ⎤

⎢ ⎥

= ⎢ + + − ⎥

⎢ − + + ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

Angle-axis representation

11 22 33 1

acos 2

r r r

θ = ^⎛⎜⎝ ^{+ + − ⎞}⎟⎠

32 23

13 31

21 12

1 2 sin

r r r r r r θ

⎡ − ⎤

⎢ ⎥

= ⎢ − ⎥

⎢ − ⎥

⎣ ⎦

k

Two solutions are and(^k,θ ) (^{− −k, θ})

Representational singularity when orθ = 0 θ π= , k would be undefined

Ch. 2: Spatial Descriptions and Transformations

2.7 Homogeneous Transformation

• general rigid body motion involves both translation and rotation

• affix the coordinate frame to the rigid body; body fixed frame

• motion of the body = motion of the moving frame relative to the fixed frame

• combine the orientation of the moving coordinate frame and the position of its origin Æ homogeneous transformation matrix

0 0

0 1 1

1

0 1

R p

T ⎡ ⎤

= ⎢ ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

Some properties and meanings

•

• three interpretations: relative pose of two frames, coordinate transformation (mapping), and operator

• motion performed relative to the current frame Æ post- multiplication

• motion performed relative to the fixed frame Æ pre- multiplication

•

1 1 0 0 0

( )

1 0 0 1 1 1 0 1

0 1

0 1 0 1

T T

R p R R p

T ⎡ ⎤ ⎡ − ⎤ T

⁻

= ⎢ ⎥ ⎢ = ⎥ =

⎣ ⎦ ⎣ ⎦

0 1

A B A A B

A B C B B C

C

R R p R p

T ⎡ + ⎤

= ⎢ ⎥

⎣ ⎦

Ch. 2: Spatial Descriptions and Transformations

Ex.

A frame is described as initially coincident with . We then rotate about the vector

passing through the point by an amount . Give the frame description of .

{ }

^B

{ }

^A

{ }

^{B Ak =}

[

^{0.707 0.707 0}

]

^T

[

^{1 2 3}

]

^T

A p =

θ = 30^D

{ }

^B

Ch. 2: Spatial Descriptions and Transformations

Sol.

Using the similarity transformation, set up as the frame where its origin is at , but has the same

orientation as . The given operation represented in this frame is

The coordinate transformation between and is

{ }

^C

A p

0.9330 0.0670 0.3536 0 0.0670 0.9330 0.3536 0 0.3536 0.3536 0.8660 0

0 0 0 1

CT

⎡ ⎤

⎢ − ⎥

⎢ ⎥

= ⎢− ⎥

⎢ ⎥

⎣ ⎦

{ }

^A

{ }

^C

{ }

^A

Ch. 2: Spatial Descriptions and Transformations

Sol.

Therefore, the given operation expressed in is

which, after multiplying of gives .

( )

¹

0.9330 0.0670 0.3536 1.1276 0.0670 0.9330 0.3536 1.1276 0.3536 0.3536 0.8660 0.0484

0 0 0 1

A C C C

A A

T = H ⁻ T H

⎡ − ⎤

⎢ − ⎥

⎢ ⎥

= ⎢− ⎥

⎢ ⎥

⎣ ⎦

1 0 0 1

0 1 0 2

0 0 1 3

0 0 0 1

C AH

⎡ − ⎤

⎢ − ⎥

⎢ ⎥

= ⎢ − ⎥

⎢ ⎥

⎣ ⎦

{ }

^A

I4

{ }

^A

{ }

^B