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 impor tant in at least two types of tasks: (1) map sensor centered measurements into the robot workspace frame and (2) tasks allowing the robot to precisely move the sensor. In the past some solutions were proposed, particularly in the case of the sensor being a television camera. With almost no exception, all existing solutions attempt to solve a homogeneous matrix equation of the form AX = X B. This article has the following main contributions. First we show that there are two possible formulations of the hand-eye calibration problem. One formu lation is the classic one just mentioned. A second formulation takes the form of the following homogeneous matrix equation: MY = M'YB. 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 3 x4 perspective matrices (M andM' ) associated with two positions of the camera with respect to the calibration frame. Moreover, this formulation together with the classic one covers a wider range of camera-based sensors to be calibrated with respect to the robot hand: single scan-line cameras, stereo heads, range finders, 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 and present two methods: (1) a closed-form solution for solving for rotation using unit quaternions and then solving for translation and (2) a nonlinear technique for simultane ously solving for rotation and translation. Third, we perform a stability analysis both for our two methods and for the lin ear method developed by Tsai and Lenz (1989). This analysis allows the comparison of the three methods. In light of this comparison, the nonlinear optimization method, which solves for rotation and translation simultaneously, seems to be the most robust one with respect to noise and measurement errors.

Recently, Zhuang et al.[1] proposed a method that allows simultaneous computation of the rigid transformations from world frame to robot base frame and from hand frame to camera frame. Their method attempts to solve a homogeneous matrix equation of the form AX = ZB. They use quaternions to derive explicit linear solutions for X and Z. In this short paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above:1) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation; 2) a method based on nonlinear constrained minimization and which simultaneously solves for rotations and translations. These results may be useful to other problems that can be formulated in the same mathematical form. We perform a sensitivity analysis for both our two methods and the linear method developed by Zhuang et al. [1]. This analysis allows the comparison of the three methods. In the light of this comparison, the nonlinear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.Index Terms-Hand/eye calibration, quaternion algebra, robot/world calibration.

No abstract