Page 9c -- Relative Motion of Two Rigid Bodies

A common use of 4x4 transformation matrices is for the description of relative motion between two rigid bodies. The position and attitude of each body is defined by an the orientation of an embedded coordinate system relative to the global system.

For example, to describe rotation at the hip, we may want to relate motion of the thigh to the pelvis in three dimensions. The hip joint centers can be defined by a relationship between the left and right ASIS, from which an axis of rotation can be defined, forming the basis of a joint coordinate system to be described later.

For a given segment, the position matrix for the local to global coordinate transformation has been described previously (Relative Coordinate Systems). Likewise, the transformation from anatomical to local coordinates was calculated earlier (Anatomical Calibration). The product of these matrices is equivalent to a transformation from anatomical to global coordinates, and must be renewed at each instant in time.

[T_AG]=[T_AL][T_LG]

Similarly, a transformation between two anatomical coordinate systems (for example, the pelvis and the femur to define hip rotation) can be calculated from previously defined relationships between global and each anatomical coordinate system. Essentially, we are comparing the position and attitude of the coordinate systems of two body segments at each moment to determine the relative motion.