E2-03 - spatial descriptions - transformer.pdf

 C.B. Pham 3-1
Chapter 3: Spatial descriptions &
transformations
- Object location in space specified by location of a
selected point on it and orientation of the object.
- A coordinate system
(frame) is attached
rigidly to the object.
Then proceed to
describe the position and
orientation of this frame
with respect to some
reference coordinate
system.

 C.B. Pham 3-2
3.1. Descriptions: positions, orientations,
and frames
Description of a position
Once a coordinate system is established, any point in the
universe can be located with a 3  1 position vector.

 C.B. Pham 3-3
and frames

 C.B. Pham 3-4
Description of an orientation
-The complete location of the hand is still not specified until
its orientation is also given.
and frames
- A coordinate system (B) has
been attached to the body in a
known way. A description of {B}
relative to (A) now suffices to give
the orientation of the body.

 C.B. Pham 3-5
Note: Rotation matrix are simply the projections of a frame
onto another frame.
• Projections of
{B} onto {A}:
and frames
• Projections of
{A} onto {B}:

 C.B. Pham 3-6
Example:
and frames

 C.B. Pham 3-7
Note: In a rotation matrix, rows / columns are orthogornal
unit vectors.
and frames

 C.B. Pham 3-8
Example: Determine missing elements in the following
rotation matrix.
From properties of rotation matrix:
and frames

 C.B. Pham 3-9
and frames

 C.B. Pham 3-10
Description of a frame
A 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.
and frames

 C.B. Pham 3-11
3.2. Mappings from frame to frame
- It is concerned how to express the same quantity in terms
of various reference coordinate systems.
• Mappings involving translated frames

 C.B. Pham 3-12
• Mappings involving rotated frames

 C.B. Pham 3-13
Example: Determine AP. Given that a frame {B} is rotated
relative to frame {A} about 𝑍 by 30 degrees, with:
Rotation matrix

 C.B. Pham 3-14
• Mappings involving general frames

 C.B. Pham 3-15
Generally, it is desired to think of a mapping from one
frame to another as an operator in matrix form.





































1
1
0
0
0
1
P
P
R
P B
BORG
A
A
B
A
(4  1) (4  4) (4  1)
The 4 x 4 matrix above is called a homogeneous transform.

 C.B. Pham 3-16
Example: A frame {B} is rotated relative to frame {A} about
𝑍 by 300, translated 10 units in 𝑋𝐴, 5 units in 𝑌
𝐴 . Determine
AP, given BP = [3.0 7.0 0.0]T

 C.B. Pham 3-17
3.3. Operators: translation, rotation, &
transformation
The same mathematical forms used to map points between
frames can also be interpreted as operators that translate
points, rotate vectors, or do both.
• Translational operators
A translation moves a
point in space a finite
distance along a given
vector direction.

 C.B. Pham 3-18
To write this translation operation as a matrix operator:
The DQ operator may be thought
of as a homogeneous transform
of a special simple form.
where qx, qy, and qz are the components of the translation
vector 𝑄.
transformation
where q is the signed magnitude of the translation along
the vector direction 𝑄.

 C.B. Pham 3-19
• Rotational operators
Another interpretation of a rotation matrix is as a rotational
operator that operates on a vector AP1 and changes that
vector to a new vector, AP2, by means of a rotation, R.
transformation
In this notation, "RK()" is a rotational operator that
performs a rotation about the axis direction 𝐾 by  degrees.
This operator can be written as a homogeneous transform
whose position-vector part is zero.

 C.B. Pham 3-20
transformation

 C.B. Pham 3-21
Example: Given AP1 = [0.0 2.0 0.0]T. Compute AP2 obtained
by rotating AP1 about 𝑍 by 30 degrees.
transformation

 C.B. Pham 3-22
• Transformation operators
A frame has another interpretation as a transformation
operator. In this interpretation, only one coordinate system
is involved, and so the symbol T is used without sub- or
superscripts. The operator T rotates and translates a vector
AP1 to compute a new vector AP2.
transformation

 C.B. Pham 3-23
Summary of interpretations
• A description of a frame - describes the frame {B}
relative to the frame {A}:
• A transform mapping:
• A transform operator:












1
0
0
0
BORG
A
A
B P
R
As a general tool to represent
frames, a homogeneous transform
has been introduced. That is a 4 x 4
matrix containing orientation and
position information.
There are three interpretations of
this homogeneous transform:

 C.B. Pham 3-24
3.4. Transformation arithmetic
Compound transformations














1
0
0
0
BORG
A
CORG
B
A
B
B
C
A
B
A
C
P
P
R
R
R
T

 C.B. Pham 3-25
Inverting a transform













1
0
0
0
BORG
A
A
B
A
B
P
R
T
We have:

 C.B. Pham 3-26














1
0
0
0
BORG
A
T
A
B
T
A
B
B
A
P
R
R
T
Example: Frame {B} is
rotated relative to frame
{A} about 𝑍 by 300, and
translated 4 units in 𝑋𝐴
and 3 units in 𝑌
𝐴.

 C.B. Pham 3-27
Determine:
Solution:

 C.B. Pham 3-28
Transform equations

 C.B. Pham 3-29
3.5. More on representation of orientation
• 9 elements
R is conveniently specified with three parameters
• 6 constraints
For a rotation matrix R

 C.B. Pham 3-30
Example: Consider two rotations, one about X by 300, and
one about Z by 300.

 C.B. Pham 3-31
Note:

 C.B. Pham 3-32
roll-pitch-yaw (X - Y - Z fixed angles)
• Rotate {B} first about by an angle  (roll)
• Then, rotate {B} about by an angle  (pitch)
• Finally, rotate {B} about by an angle  (yaw)
A
X̂
A
Ẑ
A
Ŷ
Start with the frame {B} coincident with a known reference
frame {A}.

 C.B. Pham 3-33
roll-pitch-yaw (X - Y - Z fixed angles)

 C.B. Pham 3-34
Z-Y-X Euler angles
• Rotate {B} first about by an angle 
• Then, rotate {B} about by an angle 
• Finally, rotate {B} about by an angle 
B
X̂
B
Ẑ
B
Ŷ
Start with the frame {B} coincident with a known reference
frame {A}.

 C.B. Pham 3-35
Z-Y-X Euler angles

 C.B. Pham 3-36
Solution for  /  / 
If
If

 C.B. Pham 3-37
Equivalent angle - axis
Start with the frame
{B} coincident with a
known frame {A}; then
rotate {B} about the
vector by an angle
 according to the
right-hand rule.
K̂
A

 C.B. Pham 3-38
A general orientation of {B} relative to {A} may be written as
or
Where
If

 C.B. Pham 3-39
Example: A frame {B} is described as initially coincident
with {A}. We then rotate {B} about the vector (passing
through the origin) by an amount 
= 30 degrees. Give the frame description of {B}.
T
A
]
0.0
0.707
7070
.
0
[
ˆ 
K

 C.B. Pham 3-40
A frame {B} is described as
initially coincident with {A).
Then {B} is rotated about the
vector
(passing through the point AP
= [1.0 2.0 3.0]) by an amount
 = 300.
T
A
]
0.0
0.707
7070
.
0
[
ˆ 
K
Give the frame
description of {B}.

 C.B. Pham 3-41
Solution: define two new frames {A’} and {B’} so that their
origins are at AP = [1.0 2.0 3.0]T and

 C.B. Pham 3-42
We have

 C.B. Pham 3-43
3.6. Line vector vs. Free vector
The term line vector refers to a vector that is dependent on
its line of action, along with direction and magnitude, for
causing its effects.

 C.B. Pham 3-44
3.6. Line vector vs. Free vector
A free vector refers to a vector that may be positioned
anywhere in space without loss or change of meaning,
provided that magnitude and direction are preserved.
𝐴
𝑉 = 𝐵
𝐴
𝑅 𝐵
𝑉

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