AMERICAN SOCIETY OF BIOMECHANICS
Presented at the Twenty-First Annual Meeting
of the American Society of Biomechanics
Clemson University, South Carolina
September 24-27, 1997
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| AMERICAN SOCIETY OF BIOMECHANICS
Presented at the Twenty-First Annual Meeting |
Although helical angles are the method of choice for representing rigid body angular orientation in a research setting, they are nonintuitive for representing spinal angles, especially in a clinical setting. Clinicians communicate about spinal motion using the well-established anatomical plane angles: flexion/ extension, lateral bending, and axial rotation. These angles are based on the projections seen in x-ray and computed tomography images. A scientifically sound method is needed to convey three-dimensional coupled spinal motion in terms of these familiar planar angular components; for example, "X degrees of flexion/extension was coupled with Y degrees of lateral bending and Z degrees of axial rotation."
The two most common techniques for extracting planar angles from three-dimensional rigid body motion are the Euler method and the projection method. The validity of both techniques can be questioned because both techniques give significantly varying results depending on the rotation sequence or vectors used in their application to spinal motion. In addition, both methods have been shown to be unstable at some angles approaching 90°. A new method of computing spinal rotation angles based on a cylindrical representation of vertebral geometry is presented that overcomes the problems of the Euler and projection techniques. This method gives values of flexion/extension, lateral bending, and axial rotation that are consistent with intuitive expectations over ranges of both small and large angles and are stable approaching 180°. This alternative technique, the method of "tilt and twist", is easily understood in geometrical terms using an analogy of two stacked, overlappable cylinders to represent a lower fixed vertebra and an upper moving vertebra, but may also be implemented as a classical, non-Cardanic Euler rotation sequence (Ry, Rx, Ry'). This method is proposed as a standard in three-dimensional spinal motion studies.
Currently, two techniques are chiefly used for determining spinal coupling angles from marker coordinate data: the Euler method and the projection method. In the Euler method, an equivalent sequence of three rotations (one about each coordinate axis) is calculated that would rotate the vertebra from its initial orientation to the orientation observed. In the projection method, vectors associated with a vertebra are projected in each plane and the angles between the vector projections in the vertebra's initial and final orientation are measured.
Both the Euler and projection techniques are ambiguous in their application to the spine: there are six possible rotation sequences when using the Euler method (Smith and Fernie, 1991) and there are two possible angles in each plane when using the projection method (Crawford et al., 1996). Although resolution of ambiguities is possible for angles less than about 30° (Crawford et al., 1996), it is unclear which Euler sequence or projection angle set best applies when studying larger spinal motion.
In addition to their ambiguity, a major difficulty with both the Euler and projection methods is that a singularity is approached in cases where a vector for projection nears orthogonality to the projection plane and in cases where the second Euler angle approaches 90° (gimbal lock—Woltring, 1991). At the singularity, the angles are undefined. Near the singularity, small changes in orientation result in large changes in the angles calculated (Woltring, 1991).
Because a representation of spinal angles capable of passing 90° is desirable in some cases (e.g., measuring head-neck motion) and because it is unclear which, if any, of the current methods provides intuitively correct spinal coupling angles over a large range, a new technique for determining spinal angles was developed: the method of tilt and twist.
In the tilt/twist method, the vertebrae of a motion segment are represented by two stacked, overlap-pable cylinders, as shown in Figure 1. Three angles are measured geometrically from vectors associated with the cylinders in their initial and final orientations: the tilt magnitude (phi), tilt azimuth (theta), and twist (tau) angles.
Figure 1: Stacked cylinder analogy showing tilt and twist.
The angle phi is the magnitude of bending of the upper cylinder with respect to the lower cylinder. The angle theta is the direction in the transverse plane in which this bending occurs. The angle tau is the twisting of the upper cylinder on the lower cylinder.
Twist corresponds to the axial rotation spinal angle. Flexion/extension (F) and lateral bending (L) are the anteroposterior and lateral components (respectively) of tilt, derived from tilt azimuth and tilt magnitude using the following relations:
F = phi x cos(theta) L = phi x sin(theta)
Although described above as angles measured geometrically from spinal vectors (similar to the projection method), the tilt magnitude, tilt direction, and twist angles may instead be obtained from a sequence of three rotations (similar to the Euler method). This sequence is non-Cardanic, i.e., rotation occurs about the same axis twice. The motion is represented as first a rotation theta about the y-axis, then a rotation phi about the new x-axis, then a rotation eta about the new y-axis. The angle tau is the sum of theta and eta. Representation as a non-Cardanic Euler sequence is advantageous because matrix techniques may then be used to invoke the resulting rotations (Paul, 1982).
To compare the stability of the tilt/twist angles, the tilt/twist-derived spinal angles, projection angles and Euler angles, a stability test was devised where a one-degree conical test movement was applied to a mathematical model of a vertebra in a specified starting angular orientation, and the range of resulting angles reported as determined by each method. This stability test was performed for rotations about each coordinate axis.
For small movements (<30°), the tilt/twist-derived spinal angles behave similarly in flexion/extension and lateral bending to the projection angles derived from the inferosuperior vector (the preferred projection angles for describing spinal motion—Crawford et al., 1996). In axial rotation, they approximate the average of the projection angles derived from the anteroposterior and lateral vectors. In terms of Euler angles, the tilt/twist-derived spinal angles behave similarly in flexion/extension and lateral bending (<30°) to Rx and Rz from the Ry->Rz->Rx Euler sequence (the preferred Euler sequence for spinal motion—Crawford et al., 1996). In axial rotation, they behave approximately the same as the average of Ry determined from all six Euler sequence permutations. For larger rotations, the tilt/twist-derived spinal angles do not follow any of the Euler or projection angles.
As expected, the Euler and projection methods showed instabilities as they approached 90° rotation about certain axes. Although the tilt azimuth angle showed instability near 0° and 180°, the tilt magnitude, twist, and tilt/twist-derived flexion/extension and lateral bending angles were stable from 0° to nearly 180°, twice the range of spinal angles derived from the Euler or projection methods.
Because they are based on an intuitively straightforward geometrical representation and are stable over twice the range of Euler and projection angles, the tilt/twist-derived spinal angles are recommended as the standard for representation of spinal coupling.
Crawford, N.R. et al. Hum. Mvmt. Sci., 15, 55-78, 1996.
Paul, R.P. Robot Manipulators: Mathematics, Programming, and Control. (pp.45-71), MIT Press, 1982.
Smith, T.J., Fernie, G.R. Spine, 16, 1197-1203, 1991.
Woltring, H.J. Hum. Mvmt. Sci., 10, 603-16, 1991.
This work was funded in part by a grant from the National Science Foundation (grant #BCS 9257395-01)
