Research on fast forward kinematics of an offset-type flexible micro-motion Delta parallel mechanism

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In this paper, a fast numerical iterative algorithm is proposed based on the Pseudo-Rigid-Body-Model and geometric method to deal with the problem of the offset-type flexible Delta mechanism forward kinematics with multiple solutions and without analytical solutions. In forward kinematics, the quartic kinematic equations are reduced to quadratic ones by variable substitution. As the simplest nonlinear equations, quadratic equations can be represented by the coefficient matrices which is suitable for computer calculation. And due to the operational properties of the matrix, the updating and iterative process in the steps of Newton’s method can be simplified and improved to solve the forward kinematics more efficiently. The convergence and singularity of the proposed iterative algorithm are also analysed. By controlling the offset-type flexible micro-motion Delta parallel mechanism to move along three different expected spatial trace curves and measuring the displacements, the experiment results show that the Root Mean Square Error between the measured values and the expected values of linear positioning is 0.9177 μm for 37.4166 μm. Moreover, it is proved by numerical examples that the proposed iterative algorithm takes only 0.53 ms on average to solve the forward kinematic problems. The calculation time is reduced by 90.3% on average compared to the traditional Newton’s method, which provides a feasible solution for real-time control based on forward kinematics.