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 developed. 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.

手眼标定是将传感器中心的测量映射到机器人工作空间以及使机器人能够精确移动传感器的关键问题之一，本文提出了两种手眼标定问题的解决方案，并针对线性和非线性方法进行了稳定性分析，结果表明非线性优化方法对噪声和测量误差更具鲁棒性。