In this paper we solve the problem of estimating the relative pose between a robot's gripper and a camera mounted rigidly on the grip-per in situations where the rotation of the gripper w.r.t. the robot global coordinate system is not known. It is a variation of the so called hand-eye calibration problem. We formulate it as a problem of seven equations in seven unknowns and solve it using the Grobner basis method for solving systems of polynomial equations. This enables us to calibrate from the minimal number of two relative movements and to provide the first exact algebraic solution to the problem. Further, we describe a method for selecting the geometrically correct solution among the algebraically correct ones computed by the solver. In contrast to the previous iterative methods, our solution works without any initial estimate and has no problems with error accumulation. Finally, by evaluating our algorithm on both synthetic and real scene data we demonstrate that it is fast, noise resistant, and numerically stable.
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