Analysis of a Constant Jacobian Nonholonomic Constraint 2R1T Parallel Mechanism Based on Plane-Sphere Pure Rolling

To address the difficulty of establishing unified displacement-level forward and inverse kinematics and effectively characterizing the orientation evolution of spherical-platform parallel mechanisms under pure rolling constraints, a 3-P(S) parallel mechanism based on a plane–sphere pure rolling nonholonomic constraint is proposed. The mechanism outputs a 2R1T motion with the rotation axis always parallel to the xy-plane of the fixed coordinate system. By combining the geometric constraints of the mechanism with the no-slip rolling condition, an analytical mapping between input velocities and the instantaneous platform velocity is established, yielding the forward velocity kinematic model. In view of the kinematic characteristics of the spherical output platform, a characteristic-particle method is developed to reconstruct the platform pose via particles rigidly attached to the spherical shell, and an inverse velocity kinematic model is established for a prescribed characteristic-particle trajectory. Performance analysis shows that the Jacobian matrix is constant, and the condition number, local conditioning index, and manipulability are globally constant after normalization, exhibiting an orthogonal modal decomposition. Numerical examples, numerical accuracy analysis, and workspace analysis verify the correctness of the model. A physical prototype was fabricated and experimentally tested. Gyroscope measurements agree with theoretical predictions. The results provide a theoretical basis and experimental evidence for the kinematic analysis and performance optimization of symmetric parallel mechanisms with rolling contact.