Adaptive Kinematic Mapping Based on Chebyshev Interpolation: Application to Flexure-Jointed Micromanipulator Control

For flexure-jointed mechanisms and manipulators, accurate kinematic models typically involve complex nonlinear descriptions of elastic behavior and are ill-suited for real-time control computations. An alternative approach is to use multidimensional function approximation methods, with optimization via online kinematic identification and calibration. In this article, a function interpolation approach is proposed based on Chebyshev approximation theory for near-optimal minimization of maximum positioning errors. The method allows fast calibration procedures using a small number of data points. The implementation involves a correction mapping that operates on command input variables before an approximate inverse kinematic model is applied. An adaptation algorithm is further proposed that can be used to update and refine mappings: 1) in a localized space for improved precision for the current task, or 2) globally by using calibration points chosen to match the Chebyshev nodes of the overall workspace. Results are shown for simulation of a flexure-jointed X-Y motion stage and for experiments on a X-Y-Z micromanipulation platform with Delta-type parallel kinematics. For the experiments, direct measurement of platform position was achieved using a microscope vision system. The proposed method gave order-of-magnitude improvements in positioning accuracy compared with the pseudo-rigid-body modeling approach and was found to out-perform direct visual servoing when operating with similar image capture rates.