The proof of a better performance by one-stage than two-stage estimation in robot calibration

Calibration of a robot manipulator requires an external instrument to measure its end-effector locations. If the parameters are estimated directly from the measurements, it is called one-stage estimation. Otherwise, it is called two-stage estimation. It was previously observed that two-stage estimation has the drawback of error propagation. Theoretically, it is known that for linear system, the minimum variance is achieved using the best linear unbiased estimator in one-stage estimation. However, the two-stage estimation can also achieve the minimum variance under certain conditions. In this paper, the statistical properties of both are explored in detail under a newly established mathematical fact. The result can be extended to the nonlinear estimation problem via approximation. Simulations of robot calibration by a coordinate measuring machine and eye-in-hand camera are conducted and the results confirm the theoretical analysis.