Electromagnetic Motion Sensors
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Transformation Matrix
Position of the Sensors
Reference and Related Literature

Transformation Matrix

Electromagnetic motion sensors such as Flock of Bird® (Ascension Technology Corporation, Burlington, VT) generates positions and orientations of the sensors (birds) in real time. According to the manual (Ascension Technology Corporation, 1999), the rotation matrix the system produces is basically the ZYX type:

,    [1]

where , , and = the rotation angles about the X, Y, and Z axis, respectively. This rotation matrix can be applied to a 3-D image to make it follow the rotation of the sensor. In other words, this matrix the transformation matrix from the sensor reference frame to the transmitter reference frame:

,    [2]

where S = the current sensor, and T = the transmitter. The transformation matrix from the transmitter frame to the sensor frame can be described as

.    [3]

Comparison of [3] above and [3] in the Orientation Angles page reveals the the following relationship:

.    [4]

These relationships basically summarize the nature of the rotation angles produced by FOB: the relative orientation angles of the transmitter reference frame to the sensor reference frame. Thus, the transformation matrix from the transmitter reference frame to the sensor reference frame is the XYZ-type with the orientation angles being -, -, and -, respectively.

One problem is the direction of the axes of the transmitter reference frame. The transmitter of the MotionStar® uses a rather unusual axis setup (Fig. 1). Assuming the Y-axis of the transmitter reference frame is aligned with that of the desired global reference frame, the relationship between the transmitter reference frame and the global reference frame can be summarized as

Fig. 1

,    [5]

where T = the transmitter reference frame, and G = the global frame. Also from [5]:

.    [6]

The transformation matrix from the sensor reference frame to the global reference frame is then

.    [7]

Likewise, the transformation from the global frame to the sensor reference frame is

.    [8]

Transformation from one sensor reference frame (A) to another (B) can be written as

,    [9]

where matrices and are the direct output of the system. Pay attention to the direction of the axes of the sensor reference frame.

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Position of the Sensors

The positions of the sensors reported by the electromagnetic system are basically described in the transmitter reference frame. Thus, the global position of a sensor (S) can be described as:

,    [10]

where rS = the global position of the sensor, = the relative position of the sensor to the transmitter described in the transmitter reference frame (the immediate output of the sensor system), and rT = the global position of the transmitter.

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References and Related Literature

Ascension Technology Corporation (1999). MotionStar® Installation and Operation Guide. Burlington, VT: Ascension Technology Corporation.

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© Young-Hoo Kwon, 1998-