The Modeling and Numerical Solving Method of the Spatial Mechanism with Lie Group and Lie Algebra

An efficient and accurate kinematics modeling and computational approach is proposed for the direct and inverse kinematics solving of the Stewart parallel mechanism. The problem is formulated directly as a pose -attitude modeling and differential calculation problems using Lie group and Lie algebra. The velocity and acceleration equations are derived out by using the Lie group differential on the displacement and attitude model. The Lie group equation is translated into the Lie algebraic type according to the mapping relation of Lie group and algebraic. The Newton iterative is used to solve the equation, and the Jacobi matrix based on the Lie algebraic is derived. The rotation matrixes of the push rods are obtained with Newton iterative based on the attitude responses, the angular velocity and acceleration are solved by the linear equation solving method. Based on a group parameters, the initial and end lengths of the 6 push rods are obtained by the inverse solution method, then the motions of the rods are planned. The equations are simulated with MATLAB. The simulation results indicate that the numerical method based on Lie algebraic can solve the parallel mechanism system.