Efficient Kinematic Calibration for Parallel Manipulators Based on Unit Dual Quaternion

The unit dual quaternion (UDQ)-based product-of-exponential (POE) formula has achieved efficient kinematic calibration for serial manipulators. However, due to the presence of unknown passive joint displacements, it is difficult to directly establish explicit forward kinematic models for parallel manipulators (PMs). This forms a barrier to subsequent error modeling and compensation. This work establishes a novel UDQ-based forward kinematic model for a PM by utilizing constraints on the identical pose of the moving platform across all its chains. Furthermore, the adjoint transformation of UDQ's twist is derived for PM's error modeling. Notably, an index matrix is introduced to achieve a unified representation of active or passive joint displacement. Thereby, this UDQ-based kinematic error modeling method is applicable to general PMs. In addition, an error compensation method is proposed for a PM using the UDQ-based local POE formula, which incorporates the developed forward kinematic model to adjust active joint displacements. The proposed method offers significant runtime savings compared to the traditional homogeneous-transformation-matrix-based POE formula due to the compact data structure and reduced arithmetic operations.