FIGURE $3\mathrm{.}20$: Schematic of a $3R$ wrist in which all three axes intersect at a point and

are mutually orthogonal. This design is used in the PUMA $560$ manipulator and many

other industrial robots$\mathrm{.}4\mathrm{.}23[38]$ Examine a specific situation in the $3R$ wrist $($Fig$.3\mathrm{.}20)$ in which the estimation

of ${}_4$ becomes undefined in programming. The inverse kinematic model is given as

${}_4=$Atan$2(a_y,a_x)$

${}_5=$Atan$2(C_4{a}_x+S_4{a}_y,a_z)$

${}_6=$Atan$2(-S_4{n}_x+C_4{n}_y,-S_4{o}_x+C_4{o}_y)$

where

$n_x=C_4{C}_5{C}_6-S_4{S}_6$

$n_y=S_4{C}_5{C}_6+C_4{S}_6$

$o_x=-C_4{C}_5{S}_6-S_4{C}_6$

$o_y=-S_4{C}_5{S}_6+C_4{C}_6$

$a_x=C_4{S}_5$

$a_y=S_4{S}_5$

$a_z=C_5$