Joint Angle vs. Orientation Angles
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Although joint angle is commonly computed from the coordinates of the joint centers as a user-angle (see the User-Angle Issues section for the details), it can be also computed from the relative orientation angles of the joint. In 1-DOF joints such as the hinge joint, the joint angle corresponds to one of the orientation angles and no additional computation is required. But in joints with more than 1 DOF, additional computation must be performed to obtain the joint angle.

From [6] of Computation of the Orientation Angles:

,    [1]

where TD/P = the transformation matrix from the proximal reference frame to the distal reference frame at a joint, and = the relative orientation angles between the segments at the joint. Now, let's assume that the Z axes of the segmental reference frames coincide with their respective longitudinal axes of the segments:

,    [2]


,    [3]

where s = the unit vector of a segment's longitudinal axis.

As one can see in [3], the longitudinal axis vector of the distal segment described in the proximal reference frame has nothing to do with angle because it is the rotation angle about the Z axis. The angle between the longitudinal axes of the proximal and distal segments forming a joint () can be computed from the scalar product of the two s vectors: from [2] and [3],

.    [4]

Thus, the joint angle is

.    [5]


Young-Hoo Kwon, 1998-