Page 179 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 165
that of 3 with respect to 2. The position coordinate vector of point A can be expressed with
respect to coordinates 2 and 3 as follows (Figure 3.17),

r = T ⋅ r (3.200)
3A 34 4A
r = T ⋅ r = T ⋅ T ⋅ r (3.201)
2A 23 3A 23 34 4A
r = T ⋅ r = T ⋅ T ⋅ T ⋅ r (3.202)
1A 12 2A 12 23 34 4A
r = T ⋅ r = T ⋅ T ⋅ T ⋅ T ⋅ r (3.203)
0A 01 1A 01 12 23 34 4A
T
where, r = [ x y z 1] .The r , r , r , r are similarly defined. Notice that
4A 4A 4A 4A 3A 2A 1A 0A
T is the description (position coordinates of the origin and the orientation of the axes
12
of coordinate system 2 with respect to coordinate system 1) of coordinate system 2 with
respect to coordinate system 1. Then the reverse description, that is the description of
coordinate system 1 with respect to coordinate system 2, is the inverse of the previous
transformation matrix,
T = T −1 (3.204)
21 12
The (4x4) transformation matrix has a special form and the inversion of the matrix also has
a special result. Let

⎡ R 12 p ⎤
A
⎢ ⎥
T 12 = −−−−−− ⎥ (3.205)
⎢
⎢ 000 1 ⎥
⎣ ⎦
Then, the inverse of this matrix can be shown as
T
⎡ R T −R ⋅ p ⎤
A
12 12
⎢
T −1 = −−−−−− ⎥ (3.206)
12 ⎢ ⎥
000 1
⎢ ⎥
⎣ ⎦
Also notice that the order of transformations is important (multiplication of matrices are
dependent on the order),
T ⋅ T ≠ T ⋅ T (3.207)
12 23 23 12
For a general purpose multi degrees of freedom mechanism, such as a robotic manipulator,
the relationship between a coordinate frame attached to the tool (the coordinates of its
origin and orientation) with respect to a fixed reference frame at the base can be expressed
as a sequence of transformation matrices where each transformation matrix is a function
of one of the axis position variables. For instance, for a four degrees of freedom robotic
manipulator, the coordinate system at the wrist joint can be described with respect to base
as (Figure 3.17)
T = T ( ) ⋅ T ( ) ⋅ T ( ) ⋅ T ( ) (3.208)
04 01 1 12 2 23 3 34 4
where , , , are the positions of axes driven by motors. Any given position vector
2
1
4
3
relative to the fourth coordinate frame (r ) can be expressed with respect to the base as
4A
follows,
r 0A = T ⋅ r 4A (3.209)
04
= T ( ) ⋅ T ( ) ⋅ T ( ) ⋅ T ( ) ⋅ r 4A (3.210)
2
23
34
3
4
12
1
01
T
where r 4A = [ x , y , z ,1 ] the three coordinates of the point A with respect to the
4A
4A
4A
coordinate frame 4.