Hand - Eye Calibration 1

{ 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 the hand-eye calibration problem. Hand-eye calibration is important in at least two types of tasks: (i) map sensor centered measurements into the robot workspace frame and (ii) allow the robot to precisely move the sensor. In the past some solutions were proposed in the particular case of the sensor being a TV camera. With almost no exception, all existing solutions attempt to solve a homogeneous matrix equation of the form AX = XB. This paper has the following main contributions. 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 0 Y B. The advantage of the latter formulation is that the extrinsic and intrinsic parameters of the camera need not be made explicit. Indeed, this formulation directly uses the 34 perspective matrices (M and M 0) associated with 2 positions of the camera with respect to the calibration frame. 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: single scan-line cameras, stereo heads, range nders, etc. Second, we develop a common mathematical framework to solve for the hand-eye calibration problem using either of the two formulations. We represent rotation by a unit quaternion. We present two methods, (i) a closed-form solution for solving for rotation using unit quaternions and then solving for translation and (ii) a non-linear technique for simultaneously solving for rotation and translation. Third, we perform a stability analysis both for our two methods and for the classical linear method developed by Tsai & Lenz TL89]. This analysis allows the comparison of the three methods. 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.