Geometric Algorithms for Kinematic Calibration of Robots Containing Closed Loops

We present a coordinate-invariant, differential geometric formulation of the kinematic calibration problem for a general class of mechanisms. The mechanisms considered may have multiple closed loops, be redundantly actuated, and include an arbitrary number of passive joints that may or may not be equipped with joint encoders. Some form of measurement information on the position and orientation of the tool frame may also be available. Our approach rests on viewing the joint configuration space of the mechanism as an embedded submanifold of an ambient manifold, and formulating error measures in terms of the Riemannian metric specified in the ambient manifold. Based on this geometric framework, we pose the kinematic calibration problem as one of determining a parametrized multidimensional surface that is a best fit (in the sense of the chosen metric) to a given set of measured points in both the ambient and task space manifolds. Several optimization algorithms that address the various possibilities with respect to available measurement data and choice of error measures are given. Experimental and simulation results are given for the Eclipse, a six degree-of-freedom redundantly actuated parallel mechanism. The geometric framework and algorithms presented in this article have the desirable feature of being invariant with respect to the local coordinate representation of the forward and inverse kinematics and of the loop closure equations, and also provide a high-level framework in which to classify existing approaches to kinematic calibration.