Page 183 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 169
where T 12 is described by the orientation of the second coordinate frame and position
coordinates of its origin with respect to the first coordinate frame.

⎡ e 11 e 12 e 13 x 1A ⎤
e 21 e 22 e 23 y 1A
⎢ ⎥
T 12 = ⎢ ⎥ (3.226)
⎢ e e e z ⎥
31 32 33 1A
⎢ ⎥
⎣ 0 0 0 1 ⎦
⎡ 0.0 −1.00.0 −0.5 ⎤
⎢ ⎥
1.0 0.00.0 0.5
= ⎢ ⎥ (3.227)
⎢ 0.0 0.01.0 0.0 ⎥
⎢ ⎥
⎣ 0 0 0 1 ⎦
Notice that the orientation portion of the matrix is the coefficients of the relationships
between the unit vectors,
⃗ i = cos ⋅ ⃗ e + cos ⋅ ⃗ e + cos ⋅ ⃗ e (3.228)
11 1 12 2 13 3
= e ⋅ ⃗ e + e ⋅ ⃗ e + e ⋅ ⃗ e 3 (3.229)
12
1
11
13
2
= 0.0 ⋅ ⃗ e + (−1.0) ⋅ ⃗ e + 0.0 ⋅ ⃗ e (3.230)
1 2 3
⃗ j = cos ⋅ ⃗ e + cos ⋅ ⃗ e + cos ⋅ ⃗ e (3.231)
21 1 22 2 23 3
= e ⋅ ⃗ e + e ⋅ ⃗ e + e ⋅ ⃗ e 3 (3.232)
23
1
22
2
21
= 1.0 ⋅ ⃗ e + (0.0) ⋅ ⃗ e + (0.0) ⋅ ⃗ e 3 (3.233)
1
2
⃗ k = cos ⋅ ⃗ e + cos ⋅ ⃗ e + cos ⋅ ⃗ e 3 (3.234)
32
2
1
31
33
= e ⋅ ⃗ e + e ⋅ ⃗ e + e ⋅ ⃗ e (3.235)
31 1 32 2 33 3
= (0.0) ⋅ ⃗ e + (0.0) ⋅ ⃗ e + (1.0) ⋅ ⃗ e (3.236)
1 2 3
Hence,
r 1P = T ⋅ r 2P (3.237)
12
= [−1.51.50.01 ] T (3.238)
Example The purpose of this example is to illustrate graphically that the order of a
sequence of finite rotations is important. If we change the order of rotations, the final
orientation is different (Figure 3.19). In other words, T ⋅ T ≠ T ⋅ T .Let T represent a
2
2
1
1
1
◦
◦
rotation about the x-axis by 90 , and T represent a rotation about the y-axis by 90 . Figure
2
3.19 shows the sequence of both T followed by T and T followed by T . The resulting
2
2
1
1
final orientations are different since the order of rotations are different. Finite rotations are
not commutative. We can show this algebraically for this case.
⎡ 1.0 0.00.00.0 ⎤
⎢ ⎥
0.0 0.01.00.0
T = ⎢ ⎥ (3.239)
1
⎢ 0.0 −1.00.00.0 ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
⎡ 0.00.0 −1.00.0 ⎤
0.01.0 0.00.0
⎢ ⎥
T = ⎢ ⎥ (3.240)
2
⎢ 1.00.0 0.00.0 ⎥
⎢ ⎥
⎣0 0 0 1 ⎦
Clearly, T ⋅ T ≠ T ⋅ T .
2
1
1
2