  This study presents a systematic enumeration of spherical ($SO(3)$) type
parallel robots' variants using an analytical velocity-level approach. These
robots are known for their ability to perform arbitrary rotations around a
fixed point, making them suitable for numerous applications. Despite their
architectural diversity, existing research has predominantly approached them on
a case-by-case basis. This approach hinders the exploration of all possible
variants, thereby limiting the benefits derived from architectural diversity.
By employing a generalized analytical approach through the reciprocal screw
method, we systematically explore all the kinematic conditions for limbs
yielding $SO(3)$ motion.Consequently, all 73 possible types of non-redundant
limbs suitable for generating the target $SO(3)$ motion are identified. The
approach involves performing an in-depth algebraic motion-constraint analysis
and identifying common characteristics among different variants. This leads us
to systematically explore all 73 symmetric and 5256 asymmetric variants, which
in turn become a total of 5329, each potentially having different workspace
capability, stiffness performance, and dynamics. Hence, having all these
variants can facilitate the innovation of novel spherical robots and help us
easily find the best and optimal ones for our specific applications.
