ECE 450 Introduction to Robotics

1
ECE 450 Introduction to Robotics

• Section 50883
• Instructor Linda A. Gee
• 9/14/99
• Lecture 05

2
Relative Transformations in Robot Workspace

• Assign a separate coordinate system to each part
  of the assembly
• O4 Xt(4), Yt(4) Zt(4)
• Pt(0) Xt(0), Yt(0) Zt(0)
• Hi,n
• ? Hi,i1 Hi,i1 Hi1,i2 Hn-1,n

n-1
i1
3
Relative Transformations contd

• Pt(i) H1,n Pt(n)

Xt(n) Yt(n) Zt(n) 1
Xt(i) Yt(i) Zt(i) 1
H1,n
Pt(0) H0,4 Pt(4)
4
Transformations in the Workspace
Xt(4) Yt(4) Zt(4) 1
Xt(0) Yt(0) Zt(0) 1
H0,1 H1,2 H2,3 H3,4
To move a robot gripper tool to point Pt, obtain
its coordinates by using homogeneous
transformations.
5
Assembly Workspace
Pt
Not drawn to scale
z4
y4
z3
Top View
x4
O4
O3
1
5
y3
x3
z2
10
3
O2
y2
1

1
z0
x2
z1
3
2
2
y1
4
O0
O1
x1
5
y0
x0
6
Workspace Example

• Given
• O0 (0,0,0) and O1 (-5,10,10)
• Pt in O4 xt(4) 5
• yt(4) 3
• zt(4) 6
• Find Pt in O0

7
Method

• Calculate the homogeneous transformation matrices
  from O0 to O4
• Define H0,1 , H1,2 , H2,3 , H3,4 such that
• H0,4 H0,1 H1,2 H2,3 H3,4

8
Homogeneous Transformation Matrices

H 0,1
0 -1 0 -5
angle ? 90 degrees rotated about z-axis
1 0 0 10
0 0 1 10
0 0 0 1
0 1 0 4

H 1,2
angle ? -90 degrees rotated about z-axis
-1 0 0 2
0 0 1 2
0 0 0 1
9
Homogeneous Transformation Matrices contd
1 0 0 0
no rotation translation only
H 2,3

0 1 0 0
0 0 1 10
0 0 0 1
0 -1 0 1
H3,4

angle ? 90 degrees rotated about z-axis
1 0 0 -2
0 0 1 1
0 0 0 1
10
Solving for the Composite Transformation Matrix
H0,4

Xt(4) Yt(4) Zt(4) 1
Xt(0) Yt(0) Zt(0) 1

11
Solution
5 3 6 1
Xt(0) Yt(0) Zt(0) 1

Xt(0) -9 Yt(0) 17 Zt(0) 29
Pt(0) Xt(0), Yt(0) Zt(0)