Geometric algorithms for closed chain kinematic calibration

We develop a high-level, unified framework for the calibration of kinematic chains containing closed loops, passive joints, and an arbitrary number of actuators. Our approach rests on viewing the configuration space of the kinematic chain as an embedded sub-manifold of an ambient manifold, and formulating error measures based on the natural metric in this ambient manifold. Both joint encoder readings and end-effector position and orientation measurements can be uniformly included into this framework. Kinematic calibration is now formulated in a coordinate-invariant way (i.e., independent of the local representation of the forward and inverse kinematics, and of the loop closure constraints) as an optimal multidimensional surface-fitting problem to a given set of data points. We present algorithms that directly solve this nonlinear constrained optimization problem, in contrast to the existing coordinate-dependent approaches that are based on a linearization of the kinematic equations. Experimental and simulation results are presented for the Eclipse, a novel 6 DOF overactuated parallel mechanism designed for rapid machining applications