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As an alternative to robot calibration, a method based on attitude estimation from measurements of Cartesian coordinates of a set of more than three non-collinear landmark points of each robot link is proposed here. From these data, a set of rotation matrices with almost identical axes of rotation and varying angles of rotation are estimated by a simple least-square fit. Since no constraint is imposed on the proper orthogonality of these matrices, their least-square estimates most likely will fail to be orthogonal. A further filtering based on the Polar-Decomposition Theorem, provides the most likely orthogonal matrix. The most likely unit vector defining the direction of the axis of rotation is chosen as the normalized mean of the estimated set of axes and the coordinates of the point of the axis lying closest to the origin of coordinates is computed using a least-square fit. The method proposed here was implemented using actual measurements taken from an industrial robot.