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:
where
where S = the current sensor, and T = the transmitter. The transformation matrix from the transmitter frame to the sensor frame can be described as
Comparison of [3] above and [3] in the Orientation Angles page reveals the the following relationship:
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 - 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
where T = the transmitter reference frame, and G = the global frame. Also from [5]:
The transformation matrix from the sensor reference frame to the global reference frame is then
Likewise, the transformation from the global frame to the sensor reference frame is
Transformation from one sensor reference frame (A) to another (B) can be written as
where matrices
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:
where rS = the global position of
the sensor,
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- |
,
[1]
, 
,
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]
.
[3]
. [4]
Fig.
1
, [5]
. [6]
. [7]
. [8]
, [9]
and 
are the direct output of
the system. Pay attention to the direction of the axes of the sensor reference
frame.
, [10]
= 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.