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Chapter 2: Description of position and orientation
1
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOutline:
• Introduction.• Descriptions: positions, orientations and frames.• Mappings: changing description from frame to frame• Operators: translations, rotations, and transformations
Chapter 2: Description of position and orientation
2
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSIntroduction:
• Location of an object in 3D space?
• Robot links• Robot tool• Parts in the Workspace (WS )• Environment (obstacles, walls…)
• Position & orientation
Attach frame (coordinate system) to each object rigidly.
How to represent these quantities mathematically?
Find the transformation between these frames
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSIntroduction
• Frames attachment. Don’t forget to define the universe frame
Mapping between these frames
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSDescriptions: positions, orientations and frames
• Position {A} ≡ frame A
≡ the 3D position vector of point P calculated in {A}
,
xAOA P y
z
P
P P
P
,
AOA PP
,AOA PP
OA
P
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSDescriptions: positions, orientations and frames
• Orientation?– Position in 3D-space is not enough,
the orientation must be also described.How to describe the orientation of {B} relative to {A}?
– One possible solution:Decompose the unit vector directions of {B} ( ) in {A}, as follows:
similarly for and
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSDescriptions: positions, orientations and frames
• Orientation?if we put in one matrix
= 3x3 Rotation matrix between{A} and {B}, {B} expressedin {A}
complete description:{B} =
,{ , }A AB OA BR P
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSDescriptions: positions, orientations and frames
• Orientation?Note that,
and is the dot-product which is equivalent to the cosine of the angle between these two vectors. Also,
Hence,
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSDescriptions: positions, orientations and frames
• Rotation matrix characteristics:– All columns have unit magnitude – And they are orthogonal Orthonormal matrix
ˆ 1ABX
ˆ ˆ ˆ ...A A AB B BX Y Z
1
Recall that
TA AB B
TA BB A
R R
R R
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
• Translation without rotationAssume {A} and {B}, and . What is if {A} and {B} have the same orientation?
,BOB PP
,AOA PP
,BOB PP
,AOA PP
,AOA OBP
, , ,
, , ,
A A AOA P OA OB OB P
A A BOA P OA OB OB P
P P P
P P P
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
• Rotation without translation ≡ Rotation matrix between {A} and {B}Note that
In matrix form
ABR
,BOB PP
,
,
,
ˆ
ˆ
ˆ
A B Bx A O P
A B By A O P
A B Bz A O P
P X P
P Y P
P Z P
O
, ,
ˆ
ˆ
ˆ
TB
AAx
TA A B BO P y A O P
ATz B
A
XP
P P Y P
PZ
, ,
1
, , ,
, ,
Or,
A A BO P B O P
TB A A A AO P B O P B O P
B B AO P A O P
P R P
P R P R P
P R P
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
, ,
0
Given, 2 , What is ?
0
B AOB P OA PP P
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
, ,
0
Given, 2 , What is ?
0
B AOB P OA PP P
Solution
Note that the vector P is not changing in space, but we are calculating its description in different frame
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
• Translation and rotation In more compact form
• Or,
,AOA OBP
,BOB PP
, , ,
, , ,
A A AOA P OA OB OB P
A A A BOA P OA OB B OB P
P P P
P P R P
, , ,
1 0 0 0 1 1
A A A BOA P B OA OB OB PP R P P
, ,A A BOA P B OB PP T P
(4x1) vectors
,AOA PP
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
• Translation and rotation In more compact form
• Or,
,AOA OBP
,BOB PP
, , ,
, , ,
A A AOA P OA OB OB P
A A A BOA P OA OB B OB P
P P P
P P R P
, , ,
1 0 0 0 1 1
A A A BOA P B OA OB OB PP R P P
, ,A A BOA P B OB PP T P
(4x1) vectors
ABT Homogeneous transform ≡ trans.+rot. in a single matrix
,AOA PP
≡ Complete description of {B} relative to {A}
More computations (disadvantage)
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMapping (changing description from frame to frame)
,Find .AOA PP
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
, ,
,
9.1
12.6
0
1
9.1
12.6
0
A A BOA P B OB P
AOA P
P T P
P
Mapping (changing description from frame to frame)
Solution ,Find .A
OA PP
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSTransformation operations: (Multiplication and inversion)
• Multiplication (Compound transformation)
, ,
, ,
, ,
, ,
, ,It can be proven that0 0 0 1
B B COB P C OC P
A A BOA P B OB P
A A B COA P B C OC P
A A COA P C OC P
A A BC B C
A B A A BA B C OA OB B OB OCC
P T P
P T P
P T T P
P T P
T T T
R R P R PT
AC R ,
AOA OCP
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSTransformation operations: (Multiplication and inversion)
• Inverse
1
,
,
, , , ,
Given , what is ?
From we can extract both and .
Recall that 0 0 0 1
Also recall that ,
and,
A B AB A B
A A AB B OA OB
B BB A OB OAA
TB AA B
T TB B A A A A AOB OA A OB OA B OB OA B OA OB
B AA B
T T T
T R P
R PT
R R
P R P R P R P
T
1 ,
0 0 0 1
T TA A AB B OA OBR R P
T
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSTransformation operations: (Multiplication and inversion)
• Inverse (example 2.5)– Given
– Find BAT
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSGraphical Representation of the Transform
• Arrow directions indicate transformation between frames:
• i.e.: {A} is defined in {U}, {D} in {A},{C} in {B}… etc.
• Arrow direction:– Same direction
{U} to {A} – Opposite direction
{U} to {A}
UAT
1UAT
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSGraphical Representation of the Transform
• From the figure
• Suppose is unknownfind it!
BCT
Chapter 2: Description of position and orientation
22
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSGraphical Representation of the Transform
• From the figure
• Suppose is unknownfind it!
BCT
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSGraphical Representation of the Transform
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMore on Representation of Orientation
• Orientation Variables:
9 variables: Note that in 3D-space 3 independent variables are required to describe the orientation.
9 variables + 6 constraint equations 3 independent variablesConstraint equations:
– Remark: generally, rotations are not commute,
11 12 13
21 22 23
31 32 33
ˆ ˆ ˆA A A AB B B B
r r r
R X Y Z r r r
r r r
ˆ ˆ ˆ1, 1, 1,
ˆ ˆ ˆ ˆ ˆ ˆ0, 0, 0,
A A AB B B
A A A A A AB B B B B B
X Y Z
X Y Y Z X Z
A B B AB C C BR R R R
Describe the orientation using 3 parameters would be simpler
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMore on Representation of Orientation
• Example:0.866 0.5 0
ˆ( ,30) 0.5 0.866 0 ,
0 0 1zR Rot z
1 0 0
ˆ( ,30) 0 0.866 0.5 ,
0 0.5 0.866xR Rot x
0.87 0.43 0.25 0.87 0.5 0
0.5 0.75 0.43 0.43 0.75 0.5
0 0.5 0.87 0 0.43 0.87z x x zR R R R
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Roll, Pitch, Yaw angles (rotation about fixed axes XYZ){A} and {B} are initially coincident (have the same orientation)– Rotate {B} about by an angle γ– Rotate {B} about by an angle β– Rotate {B} about by an angle α(γ, β, α) ≡ (roll, pitch, yaw) angles
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Roll, Pitch, Yaw angles (rotation about fixed axes XYZ)(,γ), (,β), (, α)
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Roll, Pitch, Yaw angles (rotation about fixed axes XYZ)(,γ), (,β), (, α)
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Roll, Pitch, Yaw angles (rotation about fixed axes XYZ)(,γ), (,β), (, α)
The order is a must!cα = cos(α), cβ = cos(β), c γ = cos(γ),
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Roll, Pitch, Yaw angles (rotation about fixed axes XYZ)
– Given , determine (α, β, γ)? (Inverse problem)
Solution: If cβ ≠ 0, Atan2(y,x)?
11 12 13
21 22 23
31 32 33
AB
r r r
R r r r
r r r
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Z-Y-X Euler Angles: Rotation about moving axes{A} and {B} are initially coincident (have the same orientation)– Rotate {B} about by an angle α– Rotate {B} about by an angle β– Rotate {B} about by an angle γ
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Z-Y-X Euler Angles: Rotation about moving axes{A} and {B} are initially coincident (have the same
orientation)– Rotate {B} about by an angle γ– Rotate {B} about by an angle β– Rotate {B} about by an angle α(γ, β, α) ≡ (roll, pitch, yaw) angles
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Z-Y-X Euler Angles: Rotation about moving axes
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Z-Y-X Euler Angles: Rotation about moving axes{A} and {B} are initially coincident (have the same orientation)– Rotate {B} about by an angle α– Rotate {B} about by an angle β– Rotate {B} about by an angle γ
( , )AB R Rot z
( , )BB R Rot y
( , )B BB BR R Rot x
( , ) ( , ) ( , )A A B BB B B BR R R R Rot z Rot y Rot x
The same as obtained before, however in fixed axes rotations, rotations have opposite order!
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOrientation description using 3 angular parameters:
• Z-Y-Z Euler Angles:The same as Z-Y-X procedure, however the last rotation is around
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSEquivalent Angle-Axis Representation:
• Any relative orientation can be described by a rotation by an angle θ around a given axis (direction)
Angle + direction vector ≡ 3 independent variables
ˆx
y
z
k
K k
k
Axis of rotation Only 2 are independent
2 2 2 1x y zk k k
θ
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSEquivalent Angle-Axis Representation:
• Given θ and , find R
• Given R, find θ and
Chapter 2: Description of position and orientation
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSEquivalent Angle-Axis Representation:
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