Unveiling $SO(3)$ Parallel Robot Variants: Application of the Optimal Robot to a Humanoid Eye

This study presents a systematic motion analysis and classification of $SO(3)$-type parallel robot variants using an analytical Lie algebra approach. These robots are known for their ability to perform arbitrary rotations around a fixed point, making them suitable for various applications. Despite their architectural diversity, existing research has largely treated them on a case-by-case basis, limiting the exploration of all potential variants and the benefits derived from this diversity. By applying a generalized analytical approach through the reciprocal screw method, we systematically examine the kinematic conditions for limbs that generate $SO(3)$ motion. As a result, we identify 73 distinct non-redundant limb types capable of producing the desired $SO(3)$ motion. Our approach includes an in-depth algebraic motion-constraint analysis, uncovering common characteristics across different variants. This leads us to identify 73 symmetric and 5,256 asymmetric variants, for a total of 5,329, each with unique capabilities. Finally, we selected a computationally optimized, miniaturized robot from this set for use in a humanoid eye system.