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.

翻译：本研究采用解析速度级方法，系统枚举了球形（$SO(3)$）类并联机器人的所有变体。这类机器人以能绕固定点实现任意旋转而著称，适用于众多应用场景。尽管其构型多样，现有研究大多以个案方式探讨，这阻碍了对所有可能变体的探索，从而限制了构型多样性带来的优势。通过采用互易螺旋法的广义解析方法，我们系统探索了产生$SO(3)$运动的所有支链运动学条件，进而识别出所有73种适用于生成目标$SO(3)$运动的非冗余支链类型。该方法涉及深入的代数运动约束分析，并识别不同变体间的共同特征，由此系统探索了所有73种对称变体及5256种非对称变体，合计5329种，每种变体可能具有不同的工作空间能力、刚度性能和动力学特性。因此，掌握所有变体有助于促进新型球形机器人的创新，并帮助我们针对特定应用轻松找到最优方案。