Simultaneous Hand-Eye and Robot-World Calibration by Solving the $AX=YB$ Problem Without Correspondence

Calibration is often an important and necessary step in the use of image-guided systems. In the case of the$AX=YB$problem, the relative hand-eye (X) and robot-world (Y) transformations must be determined to provide accurate data for use in control. As an added difficulty, the exact correspondence between the streams of sensor data (A’s andB’s) is typically unknown due to asynchrony in sampling rates and processing time. One common scenario is a constant shift between the two data streams. Therefore, in this paper, we present a probabilistic method to simultaneously solve forXandYwithouta prioriknowledge of the correspondence between the streams ofA’s andB’s. We begin by discussing probability density functions on$SE(3)$and then use Euclidean-group invariants to obtain an exact solution forXandY. We then present a method to simultaneously recoverXandYand the correspondence between temporally shifted data sets using a correlation method. Following this, we show how to solve the problem in the case when the data are completely scrambled, corresponding to a complete loss of temporal information. Finally, we numerically simulated the proposed method with asynchronous data and noise added to the stream ofB’s to verify its efficiency and robustness.