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Rotation Matrix Joint Center

Rotation Matrix

Electromagnetic motion sensors typically provide with their positions and orientations at a given instant in real time. Assuming each sensor represents a rigid body, the location of the origin and the orientation of the local reference frame each sensor represents are readily available. From [41] of the Helical (Screw) Axis page:

Fig. 1

,    [1]

where P_i = the reference frame of the linked proximal segment (fixed body) in frame i, D_i = the reference frame of the linked distal segment (moving body) in frame i, = the rotation matrix for the relative rotation of the distal body to the proximal body from frame i to frame i+1, and T_A/B = the rotational transformation matrix from frame B to frame A (Fig. 1). The transformation matrix from the proximal segment reference frame to the distal segment reference frame can be obtained as follows:

,    [2]

where T_A/G = the transformation matrix from the global reference frame to frame A. The transformation matrices from the global (inertial) frame to the local (segmental) reference frames are the immediate output of the motion sensors. From [1] and [2]:

    [3]

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Joint Center

For a multi-DOF joint formed by two segments, proximal and distal, let vector & be the positions of the sensors on the proximal and distal segments (proximal and distal sensors), respectively, in frame i. The relative positions of the distal sensor to the proximal sensor can be described as

,    [4]

where = the relative position of the distal sensor to the proximal sensor in frame i, etc. The relative positions observed in the proximal body reference frame become

,    [5]

where = position vector described in reference frame A.

The relative positions of the distal sensor to the proximal sensor in frames i and i+1 suffice the following relationship:

,    [6]

where = the position of the joint center observed in the proximal reference frame, and

.    [7]

In [6] it was assumed that there is no relative translation between the proximal and distal segments. [6] actually provides with 3 equations and by expanding [6] to N - 1 intervals a total of 3 * (N - 1) linear equations of can be obtained:

.    [8]

The position of the center of the multi-DOF joint observed in the proximal reference frame () can be obtained by solving [8] through the least square approach.

Once is known, the global position of the joint center in each frame can be computed:

,    [9]

where = the global position of the joint center in frame i.

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© Young-Hoo Kwon, 1998-