Joint n-1
) 0-, Link n-1
Link n-2
Joint n
8n
Joint n+1
on.l
Link n
Figure 3.1. Kinematic link frames as attached to the links of a manipulator (Paul, 1981).
A translational homogeneous transformation of one coordinate frame with respect to another by
a vector, ai + bj + ck, is described by
S10 0 a
Trans(a,b,c) = 1 c
.0 0 0 1. (3-4)
The general relationship between two successive frames (the coordinate frame of link n
and the coordinate frame of link n-1, see Figure 3.1) is expressed as the product of two rotation
transformations and two translation transformations. This relationship is referred to as a
general A matrix (Paul, 1981)
"-'An = Rot(zn-l,On) Trans(0,0,dn) Trans(a,,0,0) Rot (xn,an) (3-5)
where n-An = relationship between frame n-1 and n,
Rot(zn.i,9) = rotation transformation about zn.-, an angle On,
Trans(0,0,dn) translation along zn-, a distance dn,