CH 1
CH 1
CH 1
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 )
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.
Dot product of two unit vectors yields the cosine of the angle between
them, rotation matrices are often referred to as direction cosines.
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
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
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
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
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
BA RT BA RT A PBORG
B
AT
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
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
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
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
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 .
Extracting:
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
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.: 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
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?