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 <inline formula> <tex math notation="LaTeX"$AX=YB$/tex math>/inline formula> problem, the relative hand eye <inline formula> <tex math notation="LaTeX"$X$/tex math>/inline formula> and robot world <inline formula> <tex math notation="LaTeX"$Y$/tex math>/inline formula> 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 <inline formula> <tex math notation="LaTeX"$A$/tex math>/inline formula>'s and <inline formula> <tex math notation="LaTeX"$B$/tex math>/inline formula>'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 for <inline formula> <tex math notation="LaTeX"$X$/tex math>/inline formula> and <inline formula> <tex math notation="LaTeX"$Y$/tex math>/inline formula> without <italic>a priori</italic> knowledge of the correspondence between the streams of <inline formula> <tex math notation="LaTeX"$A$/tex math>/inline formula>'s and <inline formula> <tex math notation="LaTeX"$B$/tex math>/inline formula>'s. We begin by discussing probability density functions on <inline formula> <tex math notation="LaTeX"$SE(3)/tex math>/inline formula> and then use Euclidean group invariants to obtain an exact solution for <inline formula> <tex math notation="LaTeX"$X$/tex math>/inline formula> and <inline formula> <tex math notation="LaTeX"$Y$/tex math>/inline formula> We then present a method to simultaneously recover <inline formula> <tex math notation="LaTeX"$X$/tex math>/inline formula> and <inline formula> <tex math notation="LaTeX"$Y$/tex math>/inline formula> and the correspondence between temporally shifted data sets using a correlation method. Following this, we show how to solve the problem in the case when the data are completely scrambled, corresponding to a complete loss of temporal information. Finally, we numerically simulated the proposed method with asynchronous data and noise added to the stream of <inline formula> <tex math notation="LaTeX"$B$/tex math>/inline formula>'s to verify its efficiency and robustness.
IEEE Robotics and Automation Letters