Abstract
In this work, we began to take forward kinematics of the Gough–Stewart (G-S) platform as an unconstrained optimization problem on the Lie group-structured manifold SE(3) instead of simply relaxing its intrinsic orthogonal constraint when algorithms are updated on six-dimensional local flat Euclidean space or adding extra unit norm constraint when orientation parts are parametrized by a unit quaternion. With this thought in mind, we construct two kinds of iterative problem-solving algorithms (Gauss–Newton (G-N) and Levenberg–Marquardt (L-M)) with mathematical tools from the Lie group and Lie algebra. Finally, a case study for a general G-S platform was carried out to compare these two kinds of algorithms on SE(3) with corresponding algorithms that updated on six-dimensional flat Euclidean space or seven-dimensional quaternion-based parametrization Euclidean space. Experiment results demonstrate that those algorithms on SE(3) behave better than others in convergence performance especially when the initial guess selection is near to branch solutions.
Highlights
The closed loop kinematic relations between the moving platform and the fixed base platform in 3-D space made forward kinematic problems a challenging issue among all kinematic research of parallel manipulators during the past few decades
We address the forward kinematic as an unconstrained optimization problem on a Lie group-structured manifold SE(3)
This section is devoted to formulating the forward kinematic problem of a general type G-S platform as a minimum optimization problem
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. There exist three main research directions to address the forward kinematics problem, which are the analytic method, numerical iterative method, and auxiliary-sensor-based method. As there exist multiple direct solutions, research attempting to solve the forward kinematic problem is further expected to determine a unique actual pose (position and orientation) of the G-S platform. An experimental comparison is expected to show that the ways of updating the iteration on the local parameter space make the forward kinematic optimization algorithms more susceptible to being stuck in branch solutions.
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