106
In practical situations, the derivative of f can be estimated by'
df/de1 = f(e1 + 6e1) f(E1) ]/ SE1, where Se1 is a small increment of a1.
When the value of G1 is close enough to a solution, the complexity can be reduced by computing df/de1 numerically at any iteration using the values of 81 and f(81) in the preceding iteration,
(df/dEl)i = (fi-1 fi)/(8 i-l E1i )
where the superscript represents the iteration number at which the variable is computed. This saves the
computational cost of Eqs. (9.31) through (9.36) and avoids the problem of special cases that occur when division by a number close to zero is needed in any of those equations.
The procedure just described was programmed to compute the joint variables for 10 equidistant points on a linear trajectory with constant orientation that will move the endeffector from the initial pose
,/2/2 0 /2/2 1 ./2/2 0 -/2/2 -1/2
0 1 0 -1/2
0 0 0 1
..