A fast forward kinematics algorithm based on planar quaternion solution for a class of 3-DoF planar parallel mechanisms

This paper presents a algorithm of fast-solving forward kinematics for a class of three-degrees-of-freedom (3-DoF) of planar parallel mechanisms (PPMs) based on the kinematic mapping of constrained planar motions and planar quaternions. Firstly, a kinematic mapping is established based on planar quaternions, which formulates the forward kinematics equations of PPMs into the quadratic equations in the form of planar quaternion. The constrained trajectory equations for the RR-type limbs in the kinematic mapping image space are parameterized. Secondly, an efficient algorithm is proposed to obtain numerical solutions to the equation system. The convergence and singularity issues of the algorithm are discussed, revealing the inherent connection between algorithm singularity and PPMs motion characteristics. Finally, the application of the new algorithm is demonstrated through two examples: a 3-RPR PPM and a 3-PRR PMM. The computational results demonstrate that the proposed algorithm can achieve competitive convergence speed and computational efficiency compared with Newton’s method and quasi-Newton methods. The proposed algorithm provides a possible solution for real-time control based on forward kinematics.