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Page 10 -- Joint Coordinate System

The global positions of targets on a rigid segment are used in conjunction with anatomical calibrations to determine the body-fixed axes about which segment rotation is described. The relative motion of two body segments can be defined by a **Joint Coordinate System** described by two segment-fixed axes and a mutually orthogonal floating axis. The **Joint Coordinate System (JCS)** was proposed by Grood and Suntay (1983) to eliminate the temporal sequence dependency of Euler angle techniques and to encourage the use of clinically relevant models. Unfortunately, the drawback to the JCS is that an orthogonal coordinate system is not guaranteed.

The joint coordinate system is defined by two independent body-fixed axes and the common perpendicular. Angular rotation of the bodies is about one or more ot these spatial axes. Grood and Suntay (1983) indicate that it is necessary to specify:

- the bone coordinate system
- the body-fixed axes and the reference axes, and
- the translation reference point

Essentially we must define the anatomical axes of interest from bony markers, the clinical axes of rotation, and the origin of the joint coordinate system for a complete analysis of motion.

An interactive demonstration is provided to illustrate the orientation of two body fixed axes and the relative rotation. Using the mouse, the user can rotate the view. Once axis rotations have been entered, rotation of the femur relative to the pelvis can be seen.

For best results, try zero rotations first. (If initial configuration has not been reset, try emptying your network cache file.)

The axes are defined such that the origin is located approximately at the hip joint center.

Pelvis Axis: approximate right to left ASIS

Femur Axis: Greater Trochanter to Intracondylar point

Floating Axis: mutual perpendicular

EULER ANGLES

Euler angles are a set of three rotations about the local coordinate system of a moving body. The current axis of rotation depends on the resultant axis position due to previous rotations. Therefore these angles are sequence dependent, and the order of rotations must be specified, to accurately determine the final body orientation.

ALTERNATIVE METHODS

Other investigators (Veldpaus et al., 1988, Spoor & Veldpaus, 1980, Challis, 1995) support the use of a least squares method to calculate a helical axis and rotation angle for one body in a sequence of poses. This approach utilizes numerous markers in an attempt to reduce error through redundancy and risks increased marker crossover.

ANGLE DECOMPOSITION

Recall that the rotation submatrix of the transformation is a multiplication matrix of the dot products of the unit vectors of the two body coordinate systems, and therefore includes trigonometric functions of the three angles of rotation, denoting flexion, abduction, and external rotation.

The transformation matrix is calculated as a function of local body coordinates relative to global or relative body coordinates. The rotation submatrix is comprised of trigonometric functions which describe the angles of flexion, abduction, and external rotation between the two coordinate systems. It is necessary to decompose this matrix to solve for these angles of rotation. This is done by simultaneously inverting and substituting three trigonometric relationships to solve for each angle:

The rotation matrix selected for decomposition is dependent on the JCS axis definition. For more details, refer to Grood and Suntay, 1983, and Craig, 1986. Simplifying and substituting, we find the angles corresponding to flexion, external rotation, and abduction, respectively: