Unified formulas of constrained Jacobian and Hessian matrices for 3T1R overconstrained parallel mechanisms

The article proposes a general approach for solving the unified formulas of the constrained Jacobian matrix and constrained Hessian matrix of three translational and one rotational (3T1R) parallel manipulators (PMs). First, based on the constraint properties of PMs, the constrained couples of the limbs are divided into two categories: invariable and variable constrained couples. The unified 2 × 6 constrained Jacobian matrix and 2 × 6 × 6 constrained Hessian matrix of the 3T1R PM are established. Then, a typical 2-(UPU)2R PM is selected, and the 6 × 6 Jacobian matrix and 6 × 6 × 6 Hessian matrix of the mechanism are deduced. The singularity of the mechanism is analyzed based on the 6 × 6 Jacobian matrix. Finally, the kinematics are derived, and results demonstrating the proposed method were provided.