Solution of the Forward Kinematic Problem of 3UPS-PU Parallel Manipulators based on Constraint Curves

Adrián Peidró, Luis Payá, Sergio Cebollada, Vicente Román, Óscar Reinoso

Abstract

Algebraic elimination methods for solving the forward kinematic problem of parallel manipulators are fast and obtain all solutions, but they require eliminating all unknowns except one, and solving a high-degree univariate polynomial whose coefficients often have expressions too complex to be obtained symbolically. This prevents parameterizing these coefficients in terms of all the kinematic parameters involved, which requires repeating the elimination process again whenever these kinematic parameters change. To avoid this, this paper presents an new method to solve the forward kinematics of 3UPS-PU parallel manipulators by eliminating only one unknown, reducing the system to an easily parameterizable set of planar constraint curves in the space of the remaining unknowns, which contain all real solutions of the forward kinematics. By sampling points from these curves densely, and sorting the sampled points using k-d trees, the proposed method manages to search all real solutions along these curves. The proposed method is compared to previous methods that obtain all solutions and is shown to perform about 100 times faster than these methods, but is less general than them.

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