mt411 robotic engineering
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MT411 Robotic Engineering. Chapter 3 Coordinate Transform. Asian Institution of Technology (AIT). Narong Aphiratsakun, D.Eng. Cartesian Coordinate. - PowerPoint PPT PresentationTRANSCRIPT
MT411Robotic Engineering
Asian Institution of Technology (AIT)
Chapter 3
Coordinate Transform
Narong Aphiratsakun, D.Eng
Cartesian Coordinate
The position of an object in space is described with Cartesian coordinates. Coordinate axes for 2-3 dimensional system (2D and 3D) are normally labeled as shown in the figure below. There is no standard in computer graphics and axis labeling varies from books to books. But consistence of direction is important in the analysis. In this class, we would follow a Roll-Pitch-Yaw axis (R-P-Y).
X
Z
Y
Y
X
X (Roll)
Z (Yaw)
Y (Pitch)
a
b g
Cartesian Coordinate
Here the chosen coordinate frame forms a right hand set vectors. The positive direction of rotation is chosen in accordance with the right hand as illustrated.
Hold your right hand open, with your thumb pointing in the direction of the axis of rotation, and your fingers pointing in the direction of a second axis. Curl your fingers through 90 degree until they point in the direction of the third axis. This is a positive direction.
(for this class)Take CW : -ve and CCW : +ve
Position
Coordinate of point p can be specify with respect to frame 0 or frame 1.
For example0 15 2.8
,6 4.2
p p
Rotation
This notation specify rotation of coordinates vectors for the axes of frame 1 with respect to coordinate frame 0.
01R
11 1201
21 22
a aHOW TOOBTAIN THIS ROTATION MATRIX R
a a
Rotation in the plane
11 1201
21 22
a aR
a a
Rotation in the plane
1 001
1 0
01 1
.cos
sin .
: 0
x xx
x y
x x with referenceto axis
Rotation in the plane
01
01 1
cos
sin
: 0
y
y y with referenceto axis
01
1 001
1 0
cos(90 ) cos90 cos sin 90 sincos
sin sin(90 ) sin 90 cos cos90 sin
.sin
cos .
y
y xy
y y
Rotation in the plane
1 0 1 001
1 0 1 0
. . cos sin
. . sin cos
x x y xR
x y y y
x1relative to frame x0y0.
This notation specify rotation of coordinates vectors for the axes of frame 1 with respect to coordinate frame 0.
01R
Rotation in the plane
0 1 0 110
0 1 0 1
. .
. .
x x y xR
x y y y
x0 relative to frame x1y1.
This notation specify rotation of coordinates vectors for the axes of frame 0 with respect to coordinate frame 1.
10R
Rotation in the plane
dot product is commutative : . .i j j ix y y x
0 1 0 110
0 1 0 1
. .
. .
x x y xR
x y y y
1 0 1 001
1 0 1 0
. .
. .
x x y xR
x y y y
0 11 0
TR R
Rotation in 3D
The projection technique before scale to the 3 dimensional case. The matrix is given as
1 0 1 0 1 0
01 1 0 1 0 1 0
1 0 1 0 1 0
. . .
. . .
. . .
x x y x z x
R x y y y z y
x z y z z z
Rotation in 3D : Rotation about z-axis
,
cos sin 0
sin cos 0
0 0 1zR
0 1
1 0
1 0
1 0
1 0
. 1
. 0
. 0
. 0
. 0
z z
and
x z
y z
z x
z y
Rotation in 3D : Rotation about x-axis and y-axis
,
1 0 0
0 cos sin
0 sin cosxR
,
cos 0 sin
0 1 0
sin 0 cosyR
0 1. 1x x
0 1. 1y y
Rotational Transformation : 2D
Many times, we would like to know the new coordinates from the rotation from old coordinates.
new oldP A P
Rotational Transformation : 2D
If L =1,
By normal convention, we know that
1 3 1,
2 2
T
P
1 230 , 30
2 3 1,
2 2
T
P
Let’s apply the formula: 2 1P A P
12
1 3cos( 60) sin( 60) 2 2sin( 60) cos( 60) 3 1
2 2
A
2
1 3 3 32 2 2 2
1 13 12 22 2
P
Rotational Transformation : 2D
EX: If L =1, 1 3 1,
2 2
T
P
1 2 80
12A
2P
Rotational Transformation : 3D
In a similar way,
1 0 1 0 1 0
01 1 0 1 0 1 0
1 0 1 0 1 0
. . .
. . .
. . .
x x y x z x
A x y y y z y
x z y z z z
new oldP A P
Rotational Transformation : 3D
EX: P0 is at [0 2 0]T position of the reference frame. Compute the vector P1 obtained by rotating this vector about Z-axis by 30 degrees CW.
1 0, 30
, 30
1
0.86 0.5 0
0.5 0.86 0
0 0 1
0.86 0.5 0 0 1
0.5 0.86 0 2 1.73
0 0 1 0 0
z
z
p R p
R
p
Rotational Transformation : 3D
EX: If P0 is at [2 0 0]T position of the reference frame. Compute the vector P1 obtained by rotating this vector about Z-axis by 30 degrees CW.
1 0, 30
, 30
1
z
z
p R p
R
p
Rotational Transformation : 3D
EX: If P0 is at [2 0 0]T position of the reference frame. Compute the vector P1 obtained by rotating this vector about Y-axis by 30 degrees CCW.
1 0,30
,30
1
y
y
p R p
R
p
2-Dimensional Transformations with Translation
1
1
cos( ) sin( )
sin( ) cos( )
0 0 1 11
x q x
y r y
Translation can not be produced by assigning values to the elements of a 2x2 matrix.
1
1
( cos( ) sin( ))
( cos( ) sin( ))
x x y q
y y x r
Translation parameter
3-Dimensional Transformations with Translation
1
1
1
cos( ) sin( ) 0
sin( ) cos( ) 0
0 0 1
0 0 0 1 11
x q x
y r y
s zz
For 3D matrix, rotation and translation about z-axis will produced
Homogenous Transformation
3 3
* *
3 1 0 1
1 * 3 1*1
Rotation TranslationH
Homogenous transformation is a matrix representation of a rigid motions : combine of rotation and translation.
For Rotation Homogeneous transformation
Homogenous Transformation
,
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
zR
,
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
xR
,
cos 0 sin 0
0 1 0 0
sin 0 cos 0
0 0 0 1
yR
Homogenous Transformation
Translation in 3-D is more simple as there are three translation components for each direction. The general transformation matrix, corresponding to a translation by a vector q,r,s becomes
1 0 0
0 1 0( , , )
0 0 1
0 0 0 1
q
rD x y z
s
For Translation Homogeneous transformation
Homogenous Transformation
1 0 0
0 1 0 0( , )
0 0 1 0
0 0 0 1
q
D x q
1 0 0 0
0 1 0( , )
0 0 1 0
0 0 0 1
rD y r
1 0 0 0
0 1 0 0( , )
0 0 1
0 0 0 1
D z ss
Homogenous Transformation
EX: Let [x,y,z]G be the global frame and [x,y,z]L be the local frame of the system. Here rotate the local frame by 180 degrees CCW about z-axis and translate along y axis by 3 unit. Find the transformation matrix and test the result whether you have obtained the correct result with pold as [1,0,0].
XG
ZG
YG
XL
ZL
YL
3
0 *Gnew L oldP T P
Homogenous Transformation
0
0
[ ][ ] [ ( ,180 )]*[ ( ,3)]
[1 0 0 1]
1 0 0 0 1
0 1 0 3 0*
0 0 1 0 0
0 0 0 0 1
[ 1 3 0 1]
GL
Told
Gnew L old
Tnew
T R D Rot z D y
P
P T P
P
XG
ZG
YG
XL
ZL
YL
3
Homogenous Transformation
EX: Let [x,y,z]G be the global frame and [x,y,z]L be the local frame of the system. Here translate along y axis by 3 unit and rotate frame by 180 degrees CCW about z-axis. Find the transformation matrix and test the result whether you have obtained the correct result with pold as [1,0,0].
0 *Gnew L oldP T P
XG
ZG
YG
3
XL
ZL
YL
Homogenous Transformation
0
0
[ ][ ] [ ( ,3)]*[ ( ,180 )]
[1 0 0 1]
1 0 0 0 1
0 1 0 3 0*
0 0 1 0 0
0 0 0 0 1
[ 1 3 0 1]
GL
Told
Gnew L old
Tnew
T D R D y Rot z
P
P T P
P
XG
ZG
YG
3
XL
ZL
YL
EX: Perform the below transformations with a = 2 and b = 3. Draw the following frames. Frame 0 is shown.
Given p0 = [0,0,0]T, where is p1 wrt frame 0.
Homogenous Transformation
04 ( ,90 ) ( , ) ( ,90 ) ( , )T Rot y Trans z a Rot x Trans y b
X0
Z0
Y0
0 0 1 2 34 1 2 3 4T R D R Dwhere
Homogenous Transformation
04H
Homogenous Transformation
0 0 04 4
04
p H p
p