  Whenever a sensor is mounted on a robot hand it is important to know the
relationship between the sensor and the hand. The problem of determining this
relationship is referred to as hand-eye calibration, which is important in at
least two types of tasks: (i) map sensor centered measurements into the robot
workspace and (ii) allow the robot to precisely move the sensor. In the past
some solutions were proposed in the particular case of a camera. With almost no
exception, all existing solutions attempt to solve the homogeneous matrix
equation AX=XB. First we show that there are two possible formulations of the
hand-eye calibration problem. One formulation is the classical one that we just
mentioned. A second formulation takes the form of the following homogeneous
matrix equation: MY=M'YB. The advantage of the latter is that the extrinsic and
intrinsic camera parameters need not be made explicit. Indeed, this formulation
directly uses the 3 by 4 perspective matrices (M and M') associated with two
positions of the camera. Moreover, this formulation together with the classical
one cover a wider range of camera-based sensors to be calibrated with respect
to the robot hand. Second, we develop a common mathematical framework to solve
for the hand-eye calibration problem using either of the two formulations. We
present two methods, (i) a rotation then translation and (ii) a non-linear
solver for rotation and translation. Third, we perform a stability analysis
both for our two methods and for the classical linear method of Tsai and Lenz
(1989). In the light of this comparison, the non-linear optimization method,
that solves for rotation and translation simultaneously, seems to be the most
robust one with respect to noise and to measurement errors.
