Download The Concepts of Orientation/Rotation Transformations

The Concepts of Orientation/Rotation
‘Transformations’
ME 4135 -- Lecture Series 2
Fall 2011, Dr. R. Lindeke
1
Overview Outline
• Defining Terminology
• Defining the Orientation Matrix
– By Axial Rotation
– By Physical Meaning
• Defining Orientation Effects
– RPY Orientation Control
– Euler and Robotic Orientation Control
2
Symbol Interpretation
• Lower Case BOLD letters
indicate vectors
• Upper Case BOLD letters
indicate a Matrix
• Letters w/ “Hat” indicate
Unit Vectors (Unbolded)
• Arrow Above 2 Uppercase
Letters indication Start/Stop
Points of a vector
r,s,d,a,b,c
p,q,v,w,y,z
ω,α,θ,δ,φ
F,M,J
iˆ, ˆj , kˆ, Iˆ, Jˆ , Kˆ
ON
3
Symbol Interpretation
• Typically the Capitol Letters: A,T,Q & R indicate
translational &/or rotation Transformation
Matrices
• Transformation are often (Always?) between
different Frames of Reference, thus:
• Here, B is a “Body” Frame and G is the “Ground,”
“Global” or “Inertial” Frame – later is often labeled
the 0 (zero) frame
G
B
G
B
G
B
TB (or alternatively) T
TG (or alternatively) T
4
Symbol Interpretation
• Left “Superscript” on a vector
indicates frame vector is
expressed in
G
r
G
rP
– Here a vector ‘r’ expressed in
ground frame
• Right “Subscript” on a vector
indicates ‘tip’ point that the
vector points to
– Here a vector ‘r’ pointing to
point ‘P’ expressed in frame G –
and implied it is also measured
in G frame
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Symbol Interpretation
• Left Subscript in a
vector indicates the
Frame of
measurement
– Here we are
indicating a vector
ending at point P as
measured in the B
Frame but Expressed
in the G Frame
G
B P
r
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Symbol Interpretation
• Right Subscript on angular velocity
indicates a Moving Frame while
left subscript indicates frame of
measurement
– Here angular velocity of Frame B
w.r.t. G Frame
• Right Superscript on angular
velocity vector indicates frame in
which velocity is expressed
– Here angular velocity of Frame B1
measured in G Frame but expressed
into B2 Frame
For more symbolic interpretations see the text Preface and lists at the end of each chapter!
G
B
B2
G
B
1
7
Dealing With Rotational Kinematics
Note: Vector r positioning point P can be defined in either Frame G
(Xp,Yp,Zp) or in Frame B (xp,yp.zp)
8
Dealing With Rotational Kinematics
• Typically we then need a
“Quick” tool to relate
these two definitions of
the same point!
• Since we are talking
about rotations only, we G
r
will develop an
Orientation Transform to
do this (Q)
• It is based in the “Inner
Product” approach to
geometric relationships
 Qrot ,angle r
B
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Simple (single axial) Rotational Transformations
• About Z axis (by  or rad):
• About Y axis (by  or rad):
• About X axis (by  or rad):
QZ ,
Cos
  Sin
 0
Sin 0
Cos 0 ,  about ZG
0
1 
QY ,
 Cos
  0
  Sin
Sin 
1
0  , about YG
0 Cos 
QX ,
0
0
1
 0 Cos
0 Sin

 Sin  , about XG
Cos 
0
10
A Plan View of a “Pure” Z-Rotation
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By Inner Products Meaning:
ri  xi iˆ  yi ˆj  zi kˆ
G
r  X Iˆ  Y Jˆ  Z Kˆ
B
i
i
i
i
by Inner Products these two definitions are related:
X i  Iˆ  ri  Iˆ  xi iˆ  Iˆ  yi ˆj  Iˆ  zi kˆ
Y  Jˆ  r  Jˆ  x iˆ  Jˆ  y ˆj  Jˆ  z kˆ
i
i
i
i
i
Z i  Kˆ  ri  Kˆ  xi iˆ  Kˆ  yi ˆj  Kˆ  zi kˆ
12
Leading to matrix Form:
Iˆ  iˆ
Iˆ  ˆj
 Xi  
Y   
ˆ  iˆ
ˆ  ˆj
J
J

i

 

ˆ  iˆ K
ˆ  ˆj
K
 Zi 
 

and by definition:
Iˆ  kˆ   xi 
 
Jˆ  kˆ   yi 

ˆ
ˆ k 
ki 
K




a  b  a  b  cos   a to b 

Here (example Z rotation):


 Iˆ to iˆ   while Iˆ  iˆ  1
so
Iˆ  iˆ  1 1  Cos ( )  Cos ( )
13
Considering other terms:
Considering Iˆ  ˆj


 Iˆ to ˆj  90   & Iˆ  iˆ  1
Iˆ 
Iˆ 
ˆj  11 Cos (90   )
ˆj  Cos (90)Cos  Sin(90) Sin
Iˆ  ˆj  0  Cos  1 Sin   Sin
And so on …
Until we achieve the original QZ, Matrix
14
These Homogenous Transformation
Matrices – A Physical Interpretation
1. A representation of a Coordinate Transformation
relating the coordinates of a point ‘P’ between 2 likegeometrid (-- ie SO3 --) different coordinate systems
2. A representation of the Position and Orientation
(POSE) of a transformed coordinate frame in the
“space” defined by a fixed coordinate frame
3. An OPERATION that takes a vector rP and rotates
and/or translates it to a new vector rP‘ in the same
coordinate frame
These same concepts also apply equally to just the effect of
simple (one axial) and complex ROTATIONAL operations as well
Try an Example:
Now lets rotate about the G frames axes: 1st Z of 45; then X of 45;
finally Y of 90. The question often asked is where is the P1 point
moved to?
16
What it means (1)
• The point on
the “body” is
(5,30,10)G but
it is also
(5,30,10)B and
the two
Frames are
originally coincident
• Looking at the
first rotation
we can build a
rotational
matrix:
Cos  45   Sin  45  0 


Qz ,45   Sin  45  Cos  45  0 

0
0
1 
.707 .707 0 
Qz ,45  .707 .707 0 
 0
0
1 
Thus :
r  Qz ,45
G
B P1
 5   17.678
5
30    24.749  where 30  is B r
P1
  

 
10   10 
10 
17
What it means (2)
• The body Frame (and the
body point!) have now
moved – as expected,
[except for the Z coordinate]
• Now lets take the second
rotation (about XG) by
45
• Leading to a new
position:
0
0
1



G
QX ,45   0 Cos  45   Sin  45  
 0 Sin  45  Cos  45  
0
0 
1
G
QX ,45   0 0.707 0.707 
 0 0.707 0.707 
we seek:
 17.678  17.678
G '
P1  GQX ,45  24.749    10.429 
 10   24.571 
18
What it means (3)
• Again the
point and
body frame
geometry
have changed
(except for the X
coordinate!)
• Finally lets
take a look at
the effect of
the last Y
rotation
 Cos  90  0 Sin  90  


G
QY ,90  
0
1
0

  Sin  90  0 Cos  90  
 0 0 1
G
QY ,90   0 1 0 
 1 0 0 
"Moving" the Body Point to:
G
P1"  GQY ,90
 17.678  24.571
 10.429   10.429 

 

 24.571  17.678 
But can’t there be an easier way than all of these individual steps – to
get final B Frame orientation and a tool for finding the position?
19
Turns out we can Multiply the 3 “back to back” –
in reverse order to accomplish it!
G
QZ  X Y  GQY ,
G
QZ  X Y  ( is 90,  is 45,  is 45)
 Cos
  0
  Sin
0 0
  0 1
 1 0
.4998
 .4998
 .707
G
QX ,
Sin 
1
0 
0 Cos 
0
1  1
0  0
0  0
.4998
.4998
.707
G
QZ ,
0
1
0 Cos

 0 Sin
0
0 
.707 .707 
.707 .707 
.707 
.707 
0 
 Cos  Sin
 Sin   Sin Cos
Cos   0
0
.707 .707 0 
.707 .707 0 


 0
0
1 
0
0
0 
1 
20
Computed in MathCad
And note: Using this
Matrix Product “Qeff ” built
in reverse order when
multiplied by the original
vector location of PB we
get the same answer!
(whew!)
21
Physical Focus on the Product Matrix
xB
XG
YG
ZG
yB
zB
.500 .500 .707 
.500 .500 .707 


.707 .707
0 
• It is the physical orientation of the Body frame after the three
rotations taken w.r.t. the Ground frame
• 1st Column is the projection of xB to the Ground Frame’s axes
• And note that each of the columns – or rows – is a unit vector
22
Now, What Happens if we “redefine” the
Rotating Frame between each step?
• This should take us to a new place!
• We can build this overall Orientation Matrix
too – but it must be different if it is ‘moving’
P1 to a new place!?!
• Lets guess that we need just reverse the
individual matrix order before doing the
overall product … well it sound good …
Our Text’s Author calls this “Local Rotation”
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What will it look like?
G
?
QZ  X Y 
 GQZ ,45
Cos
  Sin
 0
.707
 .707
 0
 .500
  .500
 .707
 Sin
Cos
0
.707
.707
0
.500
.500
.707
B1
QX ,45
B2
QY ,45
0  1
0
0   Cos 0 Sin 
0  0 Cos  Sin   0
1
0 
1  0 Sin Cos    Sin 0 Cos 
0  1
0
0   0 0 1
0  0 .707 .707   0 1 0 
1   0 .707 .707   1 0 0 
0.707 
.707 
0 
24
Now using this matrix (from MathCad)
A different final position (as expected) -- so
changing frames of reference between
individual operations can be “easily”
accounted for by multiplying matrices – in
the order of the individual operational
events!
25
So Why do I care in Robotics?
• We will be most interested in Orientation of the End
Effectors
• In the “Real World” orientation is defined in various ways:
– By Roll-Pitch-Yaw (RPY)
– By Euler Angles (EA)
• We find
– RPY defines orientation w.r.t. a fixed frame of reference
– EA defines orientation in a relative sense with a simple rotation
about a base frame and two subsequent re-defined frames (one
axial rotation about a single axis in each frame )
• Zinit followed by Xinterm1 followed by Zinterm2
26
RPY by our Text (I grew up with a different thought as to
axis of Roll wrt Z and Yaw wrt X but it is of no consequence)
Meaning
Globally
As applied
Locally
27
Considering Euler Angles
Initial Z rotation leads
to orientation B’
Final Z rot
leads to the
final
orientation
B‴
Secondary X
rotation Leads to
orientation B”
28
A last thought about these Q matrices
• The inverse of the matrix is simply its
transpose!
 R11 R12 R13 
QB   R21 R22
 R31 R32
then we find:
G
R23 
R33 
 R11 R21
G
QB Inverse   R12 R22
 R13 R23
"transposed" since:
G
QB Inverse  BQG
R31 
R32 
R33 
Effectively the “inverse”
is a mathematical way of
looking at the ground
from the viewpoint of
the body frame so we
must simply reverse
which directions are
‘local’ and which are
‘remote’ as the matrix
was fundamentally
29
defined
Looking at this Idea (From last MathCad effort)
Wow – That worked!
So we can “move” round and round and then back again with
the Orientation matrices
30
Summary
• Orientation effects can be “accounted for” by using a directed series
of Simple Rotations about individual axes when they are multiplied in
an appropriate order
• The Orientation Matrix (NQM) defines the axes (XM, YM, ZM) – of the M
frame – in the space defined by the N Frame
• The inverse of a Q matrix is simply its transpose and it is (REALLY) just
the space being define with the original subscript and superscript
exchanging roles (rows exchanged for columns in the 3x3 matrix)!
– That is (NQM ) -1  MQN
• Using a Q matrix we are able to define geometries known in the M
space into the N space (either way in reality!)
• These Orientations are of interest to robotics in the various ways we
define the orientation problem of robot manipulation
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