The reduced system of equations (4.9)-(4.12) determines
candidate solutions for joint variables 2, 3, 4, and 5. Once this system of equations is solved, the remaining two variables can be found by using more equations from (4.1) and then tested for consistency. The power of this approach
will become apparent for specific manipulators as further simplification using Eqs. (4.5)-(4.8) becomes obvious. Furthermore, simplification by use of rotation inner-product invariance is computationally economical and provides
greater insight into the structure and properties of the inverse kinematic equations.
Additional Inverse Kinematics Equations
Equations (4.9)-(4.12) are necessary, but not sufficient. Although they are satisfied by all solution sets of Eq. (4.1), they are also, in general, satisfied by extraneous solutions. This problem was reported by Tsai and
Morgan (1984) as
Another problem with considering Eqs. (4.9)-(4.12) alone is the presence of sign ambiguities. In many
practical situations, one of the equations will allow a closed-form solution for either the sine or the cosine function of a resolute variable 8. The other function needs
to be computed using the Pythagorean identity, which offers two values opposite in sign. 'Although both. signs can be tried in the search for a solution, in some cases the number
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