113
and f (81) can be computed. A -few points are noteworthy in this derivation. First, since the above procedure computes f from a guess e1 and not a value of 91that corresponds to
a true solution set, the values of sine and cosine of any angle computed from 2 different equations ( such as (9.45) and (9.46) for 823 or (9.47) and (9.48) for 0)will in general not satisfy the Pythagorean identity (4.17).
An alternate possibility is to use one of the 'two equations to solve for one trigonometric function, either sine or cosine, compute the other from equation (4.17) and use the second equation to avoid a sign ambiguity only. For example, instead of using Eq. (9.46), we can compute S23 from
S2 = Sign(tz/S45) Trig(C45) (9.56)
to insure compatibility with identity (4.17).
As discussed in Chapters 6 and 7, the ability to
compute the function f(E1j) proved sufficient to provide a practical one-dimensional inverse kinematic algorithm for the manipulator of Figure 9.3.
The sign ambiguities u4 and u45 above give four possible combinations that must all be tried in the search for a root of the function f. Once a root of f is found, the values of 911 94 and 95are known along with a value for C3 from the last iteration.
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