Algebraically, the problem is known to lead to a quadratic homogeneous equation in the unknown pose matrix, which prevents any direct solution. The solution first proposed in the literature exploited the properties of proper orthogonal matrices, while decoupling the orientation problem from the position problem. This solution, relying on perfect measurements, led, for two sets of measurements, to a system of nine linear equations in four unknowns. This procedure was later on streamlined by means of quaternions, but these required the introduction of an iterative procedure relying on the singular-value decomposition of a matrix at each iteration. If we consider that the singular-value decomposition is itself iterative, the new procedure did not lead to a computational gain. Moreover, this procedure neither considered measurement noise.Given two frames, and , attached to the gripper of a robotic end-effector (EE) and to a sensor rigidly mounted on the EE, respectively, find the relative pose--position and orientation--of the two frames, from measurements of the EE pose by means of sensor and joint-encoder readouts.

The issue of noisy measurements was addressed for the first time as recently as 1995. The solutions proposed so far, however, are not yet implementable online. In this project we developed an algorithm to solve the hand-eye problem that relies on recursive linear least squares and is, hence, applicable online. To this end, we resort to the invariant formulation of rotations developed at CIM's Robotic Mechanical Systems Laboratory. (Figure 6.17).

Figure 6.17: Layout of the hand-eye calibration problem

The solution that we propose is based on two nested recursive least-square processes, the external one, for the estimation of the the relative orientation, the internal one, depending on the former, for the estimation of the relative position of the origins of the two frames and . Each process is based on the standard recursive least-square algorithm.

Upon convergence, the external process yields a rotation matrix that is not necessarily proper orthogonal, and hence, not acceptable. However, the proper-orthogonal component of this matrix is computed by post-processing, using polar-decomposition filtering. The latter is based on Higham's algorithm.

Tests run so far on a laser range finder mounted on a six-axis robot in the Computer Vision Laboratory have been successful in the implementation of the algorithm developed here.

*J. Angeles, G. Soucy, F. Ferrie*