A Study Of Inverse Kinematics And Trajectory Planning For The Permanent Magnet Spherical Motor

The inverse kinematics equations of the PMSM are nonlinear multivariable equations with strong coupling and difficult to solve.In order to avoid the difficulty of solving analytical problems,the solution of inverse kinematics are changed from nonlinear multivariable equations into constrained optimal problems,the rotation angles are obtained by using optimization algorithm.The solution of the PMSM's inverse kinematics model is the basis of dynamics control,motion analysis,off-line programming and trajectory planning for PMSM.Based on the experience of the robot inverse kinematics solution,the gradient projection method was used to solve the inverse kinematics problem of the PMSM.The simulation results showed that when the initial value of Euler angle was not zero and the trajectory of the output shaft of the rotor was complex,the gradient projection method had high accuracy.In this paper,the gradient projection method was used to solve the inverse kinematics problem but the amount of calculation of the gradient projection method was large.To solve the problems,an improved inverse kinematics model was proposed based on the ideas that Euler angles' sine and cosine can be linearized when they tend to be infinitesimal.The improved inverse kinematics model simplified largely the complexity of the coupling degree of inverse kinematics equations.The effectiveness of improved model was verified by comparing the improved inverse kinematics model with the original in accuracy and speed of the trajectory of nutational motion.The trajectory of Euler angle solved by the inverse kinematics model was not satisfactory in smoothness,operation time,dynamic constraints and obstacle environment of PMSM.In practical applications,we wanted to shorten the operation time of the PMSM to improve the working efficiency.Additionally,in order to minimize the energy consumption and ensure the obstacle avoidance,the results solved by inverse kinematics model are needed to generate the trajectory planning in the Euler angle space.With the resolving results of the improved inverse kinematics model,the Euler angles space trajectory was planned.The trajectory of Euler angles was generated by quintic B-spline and the optimal time nodes were obtained by Particle Swarm Optimization based on simulate anneal(SA-PSO)algorithm.The shortest operation time was obtained on condition that the output shaft of the rotor avoids obstacles and dynamics constraints were satisfied.