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.
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