Optimization technique for the calculation of in vitro three-dimensional vertebral motion.

A method for the calculation of translations and Eulerian rotations of an orthogonal axis system with respect to a fixed reference is described with application to the measurement of position in a vertebral motion segment. Kinematic equations were derived to compute the three-dimensional motion of a moving vertebra relative to an adjacent fixed body, without the requirement of a direct physical link between the two bodies. For this calculation, the quadratic error of the lengths of six position vectors was minimized to obtain a mathematically optimal estimate of the translations and rotations. Tests with a rigid model resulted in mean maximum overall system errors of 2.8 percent for the measurement of translation (translations less than 3.5 mm) and 6.1 percent for the measurement of rotations (rotations less than 10 deg) limited by transducer accuracy. The mathematical techniques presented for the quantitative description of rigid body motion, based on the measurement of three reference vectors, may be extended to a broad range of kinematic problems.