Abstract
The Laboratory of Robotics Systems (LSRO) has developed two high precision robots of 3 and 6 degrees of freedom. In order to calibrate the systems, a large amount of inputoutput pairs have been measured. Input: position of the motors, Output: position of the endeffector. At these "training points" the inputoutput relation is now known exactly. At all other points the relation is found by interpolation (or, in the case of noisy measurements, by approximation).
We approached the problem from a theoretical point of view and considered the general case of nonuniform sampling in m dimensions, where the solution can be expressed through radial basis functions (RBF). For the case of uniform sampling we derived a solution for the approximation and the interpolation problem expressed in the basis of cubic Bsplines. In contrast to radial basis functions, Bspines have the advantage of a finite support which allows fast reconstruction.
Three algorithms have been implemented in MATLAB:
 multidimensional interpolation based on the existing MATLAB function "interpn"
 threedimensional interpolation and approximation for nonuniform sampling based on radial basis functions
 multidimensional interpolation and approximation based on cubic Bsplines
The LSRO has provided two experimental data sets of the "Delta Cube" robot. The first set of 15^3 training points was used to compute the RBF and Bspline coefficients. The second set of 10^3 reference points was then used to validate the proposed algorithms. All three algorithms produced good results. The goal of errors below 100 nm has been surpassed with errors around 17 nm. It has also been shown, that better results can be achieved if interpolation is replaced by approximation (accuracy improvement of roughly 10%). A graphical user interface has been created in order to facilitate further experiments.
