Conformal Geometric Algebra-based Forward Kinematics Analysis Method for the (2-SPR+RPS)+(3-SPR) Serial-Parallel Hybrid Mechanism

Abstract Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years. However, notable theoretical gaps persist, particularly in the field of analytical solutions for forward kinematics. To address this, this paper proposes an innovative forward kinematics analysis method based on Conformal Geometric Algebra (CGA) for complex hybrid mechanisms formed by serial concatenation of such parallel mechanisms. The method efficiently represents geometric elements and their operational relationships by defining appropriate unknown parameters. It constructs fundamental geometric objects such as spheres and planes, derives vertex expressions through intersection and dual operations, and establishes univariate high-order equations via inner product operations, ultimately obtaining complete analytical solutions for the forward kinematics of hybrid mechanisms. Using the (2-SPR+RPS) + (3-SPR) serial-parallel hybrid mechanism as a validation case, three configuration tests implemented in Mathematica demonstrate that: for each configuration, the upper 3-SPR mechanism yields 15 mathematical solutions, while the lower 2-SPR+RPS mechanism yields 4 mathematical solutions. After geometric constraint filtering, a unique physically valid solution is obtained for each mechanism. SolidWorks simulations further verify the correctness and reliability of the model. This research provides a reliable analytical method for forward kinematics of hybrid mechanisms, holding significant implications for advancing their applications in high-precision scenarios.