Hybrid geometric calibration and model-free pose compensation of a 5R1P spherical parallel manipulator using forward kinematics

This study presents a hybrid calibration and model-free error compensation framework to enhance the positioning accuracy of a 5R1P (R: revolute, P: prismatic) spherical parallel manipulator. Calibrating such mechanisms is challenging due to the analytical intractability of their inverse kinematics. To address this, a forward-kinematics-only approach is developed that eliminates reliance on inverse models while preserving high precision. The method integrates geometric calibration based on Denavit–Hartenberg (D-H) modeling with a two-stage optimization combining Particle Swarm Optimization (PSO) and Levenberg–Marquardt (LM) refinement. To correct residual errors, a Positioning Error Compensation Method (PECM) is introduced, which iteratively updates joint commands using a numerically estimated Jacobian. The PECM operates in two modes: one utilizing direct end-effector measurements and another employing an Adaptive Neuro-Fuzzy Inference System (ANFIS) to predict positioning errors. Experimental validation on a 5R1P prototype demonstrates a 97% reduction in Cartesian error using measured feedback and a 73% reduction using ANFIS-based predictions. The proposed framework provides a generalizable, inverse-kinematics-free solution suitable for real-time implementation in robotic systems requiring compactness, accuracy, and reliability.