Calibration means building the transformation matrices between the robot end effector and the camera (call it Z), and between the robot base and the reference system (call it X). The mathematical expression is the matrix equation, in homogeneous coordinates, AX=ZB. The known matrices are A, camera calibration, and B, the end effector position in the robot base system. Despite the availability of various methods, the authors seek to improve the solution for high-quality reconstruction, for multiple cameras, and for outdoor robots.

Calibration is an optimization problem, iteratively solved with a nonlinear least squares method. Two classes of cost functions are considered: one uses n robot positions, while the other takes the camera re-projection error instead of A. Moreover, three different angle representations, Euler, axis, and quaternions, are used. In total, 18 methods are defined and six error measures considered.

The performance of the 18 methods is evaluated on eight real datasets, together with seven literature algorithms. The study concludes that the performance is dataset dependent, the second-class cost function consistently gives lower errors in re-projection but greater rotation and translation errors, and the angle representation has a minor effect. The reconstruction error is consistently low with some of the 18 methods. It is hard to find which algorithm is always the best, but comparative results help users make the choice. To this end, the methods are provided as open-source code.

The paper is easy to follow; however, a better graphical representation of the tables is missing. A warning about computation time is also missing. On the same dataset, some algorithms take about one second, while the best results in the new methods may require hundreds of seconds, making them unfeasible for real-time operations.