112


where u45 is a sign ambiguity and Trig(x) = .(1- x2).

 Equation (9.45) and (9.46) yield


 C23 = ltx/S45 (9.50)

 S23 = tz /S45 (9.51)


respectively, when S45 is not zero. In the special case where S45 has an extremely small absolute value, 845 = 0 or 7 and the current value of el will allow closed form solving of the remaining joint variables and f(81) can still be computed. Next, S4 and C4 are computed from equation (9.42),

 S4 = (9pz d3)/a4 (9.52)


 C4 = u4 Trig(S4), (9.53)


where u2 is another sign ambiguity, and C2 and S2 are obtained from (9.47) and (9.48) respectively,

 C2 = ( 1px a4 C4 C23 al)/a2 (9.54)


 S2 = (Pz a4 C4 S23)/a2. (9.55)

Finally the values of e3 and e5 are computed as

 83 = atan2(S23,C23) atan2(S2,C2) and


e5 = atan2(S45,C45) atan2(S4,C4)

..