CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
Description: specify attributes of objects with which a manipulation
system deals. Objects are parts, tools, and the manipulator itself.
1. Description of position
Once a coordinate system is established, locate any point in the
universe with a 3×1 position vector.
px
P = p y = px p y pz P = px Xˆ A p yYˆA pz Zˆ A
A T A
,
pz
CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
There are also other 3-tuple descriptions of position of points.
r = px2 p 2y pz2 r = px2 p 2y
arctan( p y / px ) z pz
arctan( pz / px2 p 2y ) arctan( p y / px )
px = r cos cos p x = r cos
p y r cos sin p y r sin
pz r sin pz z
Spherical Cylindrical
CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
2. Description of orientation
Attach a coordinate system, {B}, to the body, give a description of this
coordinate system relative to the reference system, {A}.
One way: write unit vectors of three principal axes of {B} in terms of the {A}.
Rotation matrix: Stack three unit vectors together as the columns of a 3×3
matrix.
r11 r12 r13
A ˆ A ˆ A ˆ
B R X B YB Z B r21
A
r22 r23
r31 r32 r33
rij : components of any vector, the projections of
that vector onto the unit directions of its
reference frame.
CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
A
Each component of B R can be written as dot product of a pair of unit
vectors:
Xˆ B Xˆ A YˆB Xˆ A Zˆ B Xˆ A cos( Xˆ B , Xˆ A ) cos(YˆB , Xˆ A ) cos( Zˆ B , Xˆ A )
A ˆ A ˆ A ˆ
Xˆ B YˆA YˆB YˆA Zˆ B YˆA cos( Xˆ B ,YˆA ) cos(YˆB ,YˆA ) cos( Zˆ B ,YˆA )
A
B R X
B B Y Z B
ˆ ˆ ˆ ˆ
X B Z A YB Z A Zˆ B Zˆ A cos( Xˆ B , Zˆ A ) cos(YˆB , Zˆ A ) cos( Zˆ B , Zˆ A )
Dot product of two unit vectors yields the cosine of the angle between
them, rotation matrices are often referred to as direction cosines.
E.g.: Basic rotation matrix (Elementary Rotations)
1 0 0 cos 0 sin cos sin 0
R( X , ) 0 cos sin , R(Y , ) 0 1 0 , R( Z , ) sin cos 0
0 sin cos sin 0 cos 0 0 1
CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
A
Rows of the B R are the unit vectors of {A} expressed in {B}. Hence, AB R ,
description of {A} relative to {B}, is given by transpose of rotation matrix.
B Xˆ AT
A ˆ A ˆ A ˆ
B T T
ˆ B ˆ B ˆ B ˆ
B R X B YB Z B YA X A YA Z A A R R BA RT
A B T B
, A
B ˆT
Z A
From linear algebra, inverse of a matrix with orthonormal columns is
equal to its transpose:
A Xˆ BT
A T A
YˆB Xˆ B YˆB Zˆ B I 3 R AB R 1 AB RT
A T A A A A
B R B R B
A ˆT
Z B
SO(3) {R R 33 | RRT I ,det( R) 1}
CHAPTER 1: Spatial description
§1.1 Description: position, orientation, and frames
3. Description of frame
Information needed to completely specify whereabouts of manipulator
hand is a position and an orientation. Situation of a position and an
orientation pair arises so often that define an entity called a frame.
E.g.: {B} is described relative to {A} {B} { BA R , A PBORG }
Graphical representation: An arrow is drawn from one origin to another,
represents the position of origin at head of arrow in terms of frame at tail of
arrow.
CHAPTER 1: Spatial description
§1.2 Mappings: changing description
Change descriptions from frame to frame: express same quantity in
terms of various reference coordinate systems.
1. Mappings involving translated frames
A position defined by a vector B P . We wish to
express the point in space in terms of {A}, A P ,
when {A} has the same orientation as {B}.
A
P B P A PBORG
2. Mappings involving rotated frames
Origins of two frame {A}, {B} are coincident.
Know definition of a vector w.r.t {B}, B P , and
rotation matrix BA R , its definition w.r.t {A}, A P :
A
P BA R B P
CHAPTER 1: Spatial description
§1.2 Mappings: changing description
Computation is possible when description of orientation of {B} is known
A
relative to {A}, B R . Components of A P may be calculated as:
A
px = B Xˆ A B P B Xˆ AT B P A px B Xˆ A B P B Xˆ AT
A B ˆ B B ˆT B
p = BYˆ B P BYˆ T B P P py = YA P YA P BA R B P
A A
y A A
A B ˆ B B ˆT
pz = B Zˆ A B P B Zˆ AT B P pz Z A P Z A
A
E.g.: {B} is rotated relative to {A} about Ẑ by 30 degrees.
cos sin 0 0.866 0.500 0.000
0 0.500 0.866 0.000
B R sin cos
A
0 0 1 0.000 0.000 1.000
0.0 1.000
B
P 2.0 A
P BA R B P 1.732
0.0 0.000
CHAPTER 1: Spatial description
§1.2 Mappings: changing description
3. Mappings involving general frames
– Origin of {B} is not coincident with that of {A}, has a general vector
A
offset, PBORG .
– {B} is rotated w.r.t {A}, BA R .
B
First change P to its description relative to an intermediate frame that
has same orientation as {A}, but whose origin in coincident with the origin
of {B}. Then account for the translation between origins by simple vector
addition. A
P BA R B P A PBORG
4. Homogeneous transform
Define a 4×4 matrix operator and use
4×1 position vectors:
A P BA R A
PBORG B P
P ABT B P
A
,
1 0 0 0 1 1
CHAPTER 1: Spatial description
§1.2 Mappings: changing description
This representation is conceptually clear, but is mainly for convenience
when thinking and writing equations down on paper, because of time
wasted multiplying ones and zeros.
R t 44
3
SE (3) T T R | R SO(3), t R
0 1
E.g.: {B} is rotated relative to {A} about Zˆ A by 30 degrees, translated 10
units in Xˆ A , and 5 unit in YˆA . Find A P , where B P [3.0 7.0 0.0]T .
0.866 0.500 0.000 10.0
0.500 9.098
0.866 0.000 5.0
BT
P ABT B P 12.562
A A
0.000 0.000 1.000 0.0
0.000
0 0 0 1
CHAPTER 1: Spatial description
§1.3 Transformation arithmetic
1. Compound transformation
C
{C} is known relative to {B}, and {B} is known relative to {A}, we have P
A
and wish to find P .
General transform: B
P BRCP BP C CORG
A
P BA R B P A PBORG BA R( CB R C P B PCORG ) A PBORG
( BA R CB R) C P ( BA R B PCORG A PBORG )
Homogeneous transform:
B
P CBT C P
A
P ABT B P ABT CBT C P CAT C P
T ABT CBT
A
C
BA R CB R A
R B PCORG A PBORG
A
CT B
0 0 0 1
CHAPTER 1: Spatial description
§1.3 Transformation arithmetic
2. Inverting transform
A
{B} is known with respect to a {A}, B T , inverting transform to get a
B
description of {A} relative to {B}, A T .
AB R B
PAORG
B
AT
0 0 0 1
– First, recall from discussion of rotation matrices: B
A R BA RT
A
– Next, change the description of PBORG into {B}:
( PBORG ) AB R A PBORG B PAORG
B A
B
PAORG AB R A PBORG BART APBORG
BA RT BA RT A PBORG
B
AT
0 0 0 1
Note that, with our notation: BAT ABT 1
But the orthogonal property does not hold good: B
AT 1 BAT T
CHAPTER 1: Spatial description
§1.3 Transformation arithmetic
E.g.: {B} is rotated relative to {A} about Ẑ by 30 degrees and translated
four units in Xˆ and three units in Yˆ . Find Τ .
B A
A A A BT
0.866 0.500 0.000 4.0
0.500 0.866 0.000 3.0
BT
A
0.000 0.000 1.000 0.0
0 0 0 1
BA RT BA RT A PBORG
B
AT
0 0 0 1
0.866 0.500 0.000 4.964
0.500 0.866 0.000 0.598
0.000 0.000 1.000 0.0
0 0 0 1
CHAPTER 1: Spatial description
§1.3 Transformation arithmetic
3. Transform equations
{D} can be expressed as products of transformations in two different
ways:
U
T UAT DAT
D
T DAT UBT CBT CDT
U
A
U
DT UBT CBT CDT
In case that all transforms are known except CBT : have one transform
equation and one unknown transform.
B
CT UBT 1 UAT DAT CDT 1
Transform equations can be used to
solve for transforms in the case of n
unknown transforms and n transform
equations.
CHAPTER 1: Spatial description
§1.3 Transformation arithmetic
E.g.: know the transform TB T , and where the tabletop is located in space
B
relative to the manipulator’s base, S T , and the location of frame attached
S
to the bolt lying on the table relative to the table frame, G T . Calculate
position and orientation of the bolt relative to hand, GT T .
B
TT GTT BST GST T BTT 1 BST GST
T
G
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
Nine elements of rotation matrix are not all independent.
– Each is a unit vector Xˆ 1 , Yˆ 1 , Zˆ 1
– All three must be mutually perpendicular Xˆ Yˆ 0 , Xˆ Zˆ 0 , Yˆ Zˆ 0
Problem: whether representations of orientation can be conveniently
specified with three parameters?
Cayley’s formula: for any proper orthonormal matrix, there exists a skew-
symmetric matrix ( S S ) such that:
T
0 Sz Sy
R ( I3 S )1 ( I 3 S ) S Sz 0 Sx
S y Sx 0
An immediate consequence is that any 3×3 rotation matrix can be
specified by just three parameters.
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
1. X-Y-Z fixed angles
Orientation of a frame {B} as follow: Start with the frame coincident with
a known reference frame {A}. Rotate {B} first about Xˆ A by an angle , then
about YˆA by an angle , and, finally, about Zˆ A by an angle .
This convention is referred to as roll, pitch, yaw angles.
c s 0 c 0 s 1 0 0
c 0 0 1 0 0 c s
B RXYZ ( , , ) s
A
0 0 1 s 0 c 0 s c
c c c s c s c c s c s s
s c s s s c c s s c c s
s c s c c
Keep in mind that the order of the
three rotations.
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
E.g.: rotations don’t generally commute.
1.000 0.000 0.000 0.866 0.500 0.000
RX (30.0) 0.000 0.866 0.500 RZ (30.0) 0.500 0.866 0.000
0.000 0.500 0.866 0.000 0.000 1.000
0.87 0.43 0.25
RZ (30.0) RX (30.0) 0.50 0.75 0.43
0.00 0.50 0.87
RX RZ RZ RX
0.87 0.50 0.00
RX (30.0) RZ (30.0) 0.43 0.75 0.5
0.25 0.43 0.87
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
Inverse problem: extracting equivalent three fixed angles from a rotation
matrix.
Transcendental equations: there are nine equations and three unknowns if
equation is equated to a given rotation matrix.
c c c s c s c c s c s s r11 r12 r13
A
R ( , , ) s c s s s c c s s c c s r21 r22 r23
B XYZ
s c s c c r31 r32 r33
A tan 2(r31 , r112 r212 )
A tan 2(r21 / c , r11 / c )
A tan 2(r32 / c , r33 / c )
Atan2(x,y) use the signs of both x and y to identify the quadrant in which
the resulting angle lies.
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
2. Z-Y-X Euler angles
Orientation of a frame {B} as follow: Start with the frame coincident with
a known reference frame {A}. Rotate {B} first about Zˆ B by an angle , then
about YˆB by an angle , finally, about Xˆ B by an angle .
c s 0 c 0 s 1 0 0
c 0 0 1 0 0 c s
B RZ'Y'X' RZ ( ) RY ( ) RX ( ) s
A
0 0 1 s 0 c 0 s c
c c c s c s c c s c s s
s c s s s c c s s c c s
s c s c c
B RXYZ ( , , ) B RZ'Y'X' ( , , )
A A
Duality:
3 rotations taken about fixed frame
yield same orientation as 3 rotations in
opposite order about moving frame.
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
E.g.: Z-Y-Z Euler angles ( is commonly used )
Orientation of a frame {B} as follow: Start with the frame coincident with
a known reference frame {A}. Rotate {B} first about Zˆ B by an angle , then
about YˆA by an angle , finally, about Zˆ B by an angle .
c c c s s c c s s c c s r11 r12 r13
A
R ( , , ) s c c c s s c s c c s s r21 r22 r23
B Z'Y'Z'
s c s s c r31 r32 r33
Extracting:
A tan 2( r312 r322 , r33 )
A tan 2(r23 / s , r13 / s )
A tan 2(r32 / s , r31 / s )
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
We have seen three conventions for specifying orientation: X-Y-Z fixed
angles, Z-Y-X Euler angles and Z-Y-Z Euler angles. These conventions are
examples of a set of 24 conventions that we will call angle-set conventions.
Of these, 12 conventions are for fixed-angle sets, and 12 are for Euler-angle
sets.
Because of duality of fixed-angle sets with Euler-angle sets, there are
really only 12 unique parameterizations of a rotation matrix by using
successive rotations about principal axes.
3. Equivalent angle-axis representation
Describing orientation of a frame {B} as follow:
Start with the frame coincident with a known
reference frame {A}. Then rotate {B} about the
vector A Kˆ by an angle according to the right-
hand rule.
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
Vector Kˆ is called the equivalent axis of a finite rotation. A Kˆ [k x , k y , k z ]T
A
Consider a reference {C} with Zˆ coincident with Kˆ :
A
C
Xˆ C Xˆ A YˆC Xˆ A Zˆ C Xˆ A nx ox ax ax k x
ˆ ˆ
C R X C YA
A
YˆC YˆA ˆ ˆ
ZC YA n y oy a y , a y k y
ˆ ˆ a k
ˆ ˆ
X C Z A YC Z A Zˆ C Zˆ A nz oz a z z z
A
B RK ( ) CA RRot ( Zˆ C , ) AC R CA RRot ( Zˆ C , ) CA R 1
nx ox a x c s 0 n x ny nz
n y oy a y s c 0 ox oy oz
nz oz a z 0 0 1 a x ay a z
nx ox a x n x c ox s n y c o y s nz c oz s
n y oy a y n x s ox c n y s o y c n z s oz c
nz oz a z ax ay az
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
k x k x v c k x k y v k z s k x k z v k y s
A
R
B K ( ) k x k y v k z s k y k y v c k y k z v k x s v 1 cos
k x k z v k y s k y k z v k x s k z k z v c
Element (1,1): n2c n o s nx ox s ox2c ax2
x x x
k x k x v c (nx2 ox2 ax2 1)
Element (2,1):
nx n y c n yox s nxo y s oxo y c a x a y (nx n y oxo y )c (nxo y n yox )s a x a y
k x k y v k z s ( C Xˆ A CYˆA 0 , A Xˆ C AYˆC AZˆC )
Inverse problem:
r11 r12 r13 r32 r23
r r r 1 1
B RK ( ) r21
A
r22 r23 A cos( 11 22 33 ) , Kˆ r13 r31
2 2sin
r31 r32 r33 r21 r12
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
For any axis-angle pair ( Kˆ , ) , these is another pair, ( Kˆ , ) which
A A
results in same orientation in space, with same rotation matrix describing it.
In converting from a rotation-matrix into a angle-axis representation, we are
faced with choosing between solutions.
For small angular rotations, the axis become ill-defined. If the amount of
rotation goes to zero, the axis of rotation becomes completely undefined.
E.g.: a frame {B} is described as initially coincident with {A}, then be
rotated about the vector Kˆ [0.707,0.70,0] (passing through the origin) by
A T
300 degrees. Give the frame description of {B}.
0.933 0.067 0.354 0.0
k x k x v c k x k y v k z s k x k z v k y s
0.067 0.933 0.354 0.0
k x k y v k z s k y k y v c k y k z v k x s
A
T
0.354 0.354 0.866 0.0
B
k x k z v k y s k y k z v k x s k z k z v c
0 0 0 1
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
E.g.: Frame {B} is described as initially coincident with {A}. Rotate {B} about
the vector A Kˆ [0.707,0.70,0]T (passing through the point P [1,2,3] ) by an
A
amount 30 degrees. Give the frame description of {B}.
Define two new frames, {A’} and {B’}, which are coincident with each
other and have the same orientation as {A} and {B} respectively, but are
translated relative to {A} by an offset that places their origins on the axis of
rotation.
1.0 0.0 0.0 1.0
0.0 1.0 0.0 2.0
AT
A
0.0 0.0 1.0 3.0
0 0 0 1
1.0 0.0 0.0 1.0
0.0 1.0 0.0 2.0
B
BT
0.0 0.0 1.0 3.0
0 0 0 1
CHAPTER 1: Spatial description
§1.4 More on representation of orientation
Now rotate {B’} relative {A’}, this is a rotation about an axis that passes
through the origin:
0.933 0.067 0.354 0.0
0.067 0.933 0.354 0.0
A
BT
0.354 0.354 0.866 0.0
0 0 0 1
Finally, we can write a transform equation to compute the desired frame:
0.933 0.067 0.354 1.13
0.067 0.933 0.354 1.13
A A B
BT AT BT BT
A
0.354 0.354 0.866 0.05
0 0 0 1
CHAPTER 1: Spatial description
§1.5 Transformation of free vectors
Define two basic classes of vector:
– Line vector: refers to a vector that is dependent on its line of action,
along with direction and magnitude, for causing its effects. The effects
of a force vector depend upon its line of (or point of application).
– 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. For example, a pure moment vector is always a
free vector.
If we have a moment vector B N that is known in terms of {B}, we
calculate the same moment in terms of {A} as: N B R N
A A B
V BA R BV
A
Only the rotation matrix relating the two systems is used in transforming.
Relative locations of the origins do not enter into the calculation.
CHAPTER 1: Spatial description
Problems:
1. Definition of rotation matrix and derive elementary rotation matrix.
2. Mappings involving general frames and homogeneous transform.
3. Three kind of representations of rotation matrix specified with three
parameters.
4. Why the order of three rotations of fixed angles and Euler angles are
different when calculating rotation matrix?
