A survey of formal methods for determining the centre of rotation of ball joints.

The determination of an accurate centre of rotation (CoR) from segment marker positions is of interest across a wide range of applications, but particularly for clinical gait analysis and for estimating the hip joint centre during surgical intervention of the knee, for limb alignment purposes. For the first time in this survey of formal methods, we classify, analyse and compare different methods (geometric, algebraic, bias compensated algebraic, and Pratt sphere fit methods, as well as the centre transformation technique, the Holzreiter approach, the helical pivot technique, the Schwartz transformation techniques, the minimal amplitude point method and the Stoddart approach) for the determination of spherical joint centres from marker position data. In addition, we propose a new method, the symmetrical CoR estimation or SCoRE, in which the coordinates of the joint centre must only remain constant relative to each segment, thus not requiring the assumption that one segment should remain at rest. For each method, 1000 CoR estimations were analysed with the application of isotropic, independent and identically distributed Gaussian noise (standard deviation 0.1cm) to each of the marker positions, to all markers on the segment simultaneously and the two in combination. For the test conditions used here, most techniques were capable of determining the CoR to within 0.3 cm, as long as the spherical range of motion (RoM) of the joint was 45 degrees or more. Under the most stringent conditions tested, however, the SCoRE was capable of best determining the CoR, to within approximately 1.2mm with a RoM of 20 degrees . The correct selection and application of these methodologies should help improve the accuracy of surgical navigation and clinical kinematic measurement.