of sign ambiguities can be reduced by considering more constraints from Eqs. (4.3) and (4.4). Additional equations will also help filter out extraneous solutions and in some cases will ease the solution-finding process rather than complicate it. The x- and y-components of vectors t and p provide convenient additional constraints at the cost of introducing the variable e1. Equations
tx = R1 R2 R3 R4 R5 z x, (4.13)
t = R1 R2 R3 R4 R5 z y, (4.14)
Px = (R1(R2(R3(R415+14)+13)+12)+11) x, (4.15)
and
py = (R1(R2(R3(R415+14)+13)+12)+11) y, (4.16)
where
1 0
x = 0 and y= 1
0 0
are the usual canonical unit vectors, are still independent of 96.
Solving Inverse Kinematic Equations
Once the reduced set of equations (4.9)-(4.12) and the additional equations (4.13)-(4.16) have been expanded, the problem becomes that of extracting the values of the joint angles from the equations which are in terms of the sines
..