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Axis Alignment

In motion analysis, it is advantageous to align a global axis (an axis of the global reference frame) with the direction of motion because it will simplify the data analysis substantially. The global reference frame is initially fixed to the calibration frame. In other words, one can align a horizontal axis of the global reference frame with the direction of motion through careful placement of the calibration frame. If the calibration frame has cubic or hexahedral shape, this should be fairly easy.

In the survey method, on the other hand, one can place two range poles along the X axis and use one as the origin pole and the other as the X-axis pole. The real-life coordinates of the control points based on this reference frame setup can be easily obtained.

Careful placement of the calibration frame is sufficient in most cases but there are some circumstances in which one must take additional measures to accurately define the global reference frame. One simple but flexible approach is to use the so-called additional points.

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The Additional Point Strategies in 3-D Analysis

One can place up to 3 additional markers in the field of view in addition to the control points during the camera calibration to define the orientation of the global reference frame. There are three possible strategies in the 3-D analysis:

 No. of Additional Points Axis Transformation Strategy 1 2 3 Translation only 1-Axis Rotation + Translation 2-Axis Rotation + Translation

1-Point Method

This method can be used when one needs to move the origin of the reference frame (to translate the reference frame) (Figure 1).

Figure 1

The XCFYCFZCF system shown in Figure 1 is the reference frame fixed to the calibration frame while the XYZ system is the new (translated) frame. Marker A1 is the additional point. Only the origin of the reference frame moves in this method while the unit vectors of the axes remain intact.

2-Point Method

This method can be used when the vertical axis is aligned properly but the horizontal axes are not (Figure 2). Two additional points are used in this method: A1 and A2. A1 serves as the origin while A2 determines the direction of one horizontal axis (the Y axis in the example shown in Figure 2). As a result, the Y axis of the global reference frame will be aligned with the direction of motion in Figure 2.

Figure 2

The unit vectors of the new reference frame can be obtained as

[1]

where i, j, k = the unit vectors of the global reference frame (k is known already), and A = the vector drawn from A1 to A2. In other words, A2 does not have to be exactly on the Y axis. As long as the vector drawn from A1 to A2 is in the YZ plane, both X and Y axes will be defined correctly. (Remember it was initially assumed that the vertical axis was properly aligned.)

3-Point Method

This method can be used when none of the axes of the reference frame fixed to the calibration frame is aligned properly (Figure 3). Three additional points must be placed on the flat surface: A1, A2, and A3. Two vectors can be defined from the three additional points: A drawn from A1 to A2 and B drawn from A1 to A3. A accurately defines one horizontal axis (the X axis in Figure 3) while B serves as the temporary second horizontal axis (the Y axis).

Figure 3

The unit vectors of the new reference frame can be obtained as

[2]

In other words, the first horizontal axis must be precisely defined by A1 and A2 but A3 does not have to be on the 2nd horizontal axis. It will be fine as long as both vectors A and B are perpendicular to the 3rd axis (the Z axis in Figure 3) thus defines the plane that is perpendicular to the Z axis.

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The Additional Point Strategies in 2-D Analysis

Two additional point strategies can be used in the 2-D analyses with up to 2 additional points.

 No. of Additional Points Axis Transformation Strategy 1 2 Translation only 1-Axis Rotation + Translation

These are similar to those of the 3-D analysis strategies.

The 1-point method is for translation only. By placing one additional point in the field, one can move the origin of the global reference frame (translate the reference frame) (Figure 4).

Figure 4

The 2-point method can be used when the axes are not properly aligned with the direction of motion. Two additional points, A1 and A2 shown in Figure 5, defines one axis (X axis in this example).

Figure 5

The unit vectors of the new global frame can be obtained as follows:

[3]

where A = the vector drawn from A1 to A2, and i, j = unit vectors of the global axes. i, j and A are all 3-D vectors with their third components being 0.

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Application

The axis alignment based on the additional points is extremely useful when one collects the ground reaction force data in conjunction with motion analysis. In order to combine both data sets for, perhaps, inverse dynamics one must unify the reference frames. Figure 4 illustrates the reference frames involved in this situation.

Figure 6

The plate reference frame (XPYPZP system in Figure 4) is fixed to the center of the surface of the plate. By using three corners of the plate as additional points, one can define the global reference frame at one corner (A1) of the plate. Therefore, the axes of the new global reference frame become parallel to those of the plate reference frame. By translating the force plate reference frame to the global reference frame located at the corner, one can unify the reference frames used in the motion analysis and in the ground reaction force analysis. In reality, one needs to translate the center of pressure data only because the axes of the two systems are already parallel. The dimension of the plate is known, so this should be a simple matter.

Placing markers at the corners of the plate may cause problems in some cases and it is often advantageous to use a set of specially-designed markers for this purpose. Figure 7 shows an example.

Figure 7

The additional markers in Figure 7 are not on the surface but above the plate. The height of the marker is known so the origin of the reference frame can be easily translated back to the surface of the plate.

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