Measuring the 3D motion space of the human ankle

2.1Measuring instrument

In this study, we used a motion-capturing system (Motive) produced by NaturalPoint, Inc. This system is able to acquire accurate data about the actual motion space of the ankle [17]. As shown in Fig. 2, this system consists of 12 cameras, a supporting camera frame, a computer, a rotary head with three degrees of freedom, special clothing, shoes, and marks. These cameras indirectly capture the 3D coordinates of the marked points on the human foot and feed them back to the computer for continuous recordkeeping.

Figure 1.

Motion capturing system.

Figure 2.

The coordinate system after calibration.

Sixty graduate students (30 males and 30 females) were recruited to participate in this study. Participant demographic information is shown in Table 1.

2.2Methodology

The motion capturing system is not able to directly measure rotation angles. Therefore, we used vectors and trigonometric functions to translate the measured three-dimensional coordinates of the marked points into rotation angles in three-dimensional space. In order to follow trigonometric function principles, three markers were needed for data measurement. These markers were named Mark1, Mark2, and Mark3. In order to confirm the relative positions of the cameras in space, and to confirm the horizontal plane, we used the rotary head with three degrees of freedom and the calibration bar to recalibrate the cameras. We did this to avoid any errors that could be caused by slight equipment movements between measurements. Figure 2 depicts the mark positions and the coordinate system that were produced after the calibration of the motion capturing equipment.

If we fixed marks directly on the foot, the data and results would be inaccurate. Some marks might be blocked and the corresponding coordinates would not be able to be obtained through reflection during measurement. Toe movement could also cause measurement errors. We used bandages to horizontally fix a stiff, thin board with three marks on the sole of the foot, through the principle of translation and extension. On the board, the line from Mark1 to Mark2 was perpendicular to the line from Mark2 to Mark3. The positions of the marks are shown in Fig. 4. Using the board prevented metrical errors due to uneven soles, and greatly improved measurement accuracy.

Figure 3.

Mark positions on the board.

Figure 4.

The measurement process.

Rotation of the leg following ankle movements can generate falsely inflated data. To address this, a medical bed with an adjustable rack was used during the measurement process. The rack could move both horizontally and vertically. We fixed the rack on the middle of the bed. The leg was fixed on the rack with bandages in order to prevent it from rotating after ankle movement, and the rack was adjusted to an appropriate position relative to the subject’s height to ensure that the rack did not influence the data. This method helped us to obtain more accurate data of the ankle’s 3D motion space. Since the rotation angle is a relative value, we defined a reference point (referred to as the initial state) in which the subject’s leg was perpendicular to the ground, the horizontal plane for the sole of foot was perpendicular to leg, and the toes were in front of the ankle (see Fig. 4b). In this state, the rotation angles of the ankle in all directions were zero. We had the subjects sit on the bed with the stiff, thin board fixed on the sole of the foot while the leg was fixed on the rack in the initial state. Then, a model was built to measure the data. After the equipment started simultaneously recording the mark coordinates, participants let the ankle rotate to reach any point it could achieve in three-dimensional space. After several minutes, we stopped collecting data and recorded the data in a spreadsheet. When measuring the data of the left foot, the participant sat to the left of the rack, as this made it easier for the left foot to be in its initial state, resulting in more accurate data. Likewise, when measuring the data of the right foot, the subjects sat to the right of the rack. The measurement process is shown in Fig. 4.

Through data analysis, we calculated the standard deviation (σ) of the data. The standard deviation is a measure of the reliability of the data we obtained. We did this to confirm whether there was a significant difference between study groups, and whether that difference could be applied widely. We used a t-test to analyze the obtained data during data analysis. sig< 0.05 is considered to be a significant difference, otherwise, there is no difference.

3.1Basic principles of data analysis

The data of the left and the right foot were separated after all subjects were measured. Table 2 shows the data of one foot. Columns C, D, and E in the table represent the coordinates (x1,y1,z1) of Mark1; columns F, G, and H in the table represent the coordinates (x2,y2,z2) of Mark2; and, columns I, J, and K in the spreadsheet represent the coordinates (x3,y3,z3) of Mark3. Although we could not obtain the exact location of the origin, we had the mark coordinates, which were relative to the origin. Vector and trigonometric functions were used to translate the coordinates into rotational angles through analyzing the data of the three marks. This didn’t affect the measurement of the ankle’s 3D motion space.

3.2Analysis of dorsal flexure and plantar flexion

The rotation angles of the plantar flexion and dorsal flexure were calculated using the Mark1 and Mark2 coordinates. The formula used is as follows:

If θ was positive, it indicated the rotation angle of the dorsal flexure; if θ was negative, its absolute value indicated the rotation angle of the plantar flexion.

3.3Analysis of adduction and abduction

Converting the coordinates into rotation angles of adduction or abduction was a complicated process. We had two methods of analysis.

Method I:

We simply used Eq. (2) to convert the data and obtain the results. We had to examine two cases to use this method.

Analyzing the data of the left foot with the coordinate system led to the conclusion: if β was positive, it indicated an adduction rotation angle; if β was negative, its absolute value indicated an abduction rotation angle. For the right foot, if β was positive, it indicated an abduction rotation angle; if β was negative, its absolute value indicated an adduction rotation angle.

Method II:

We used different formulas to convert the data from the left foot and the right foot. We chose Eq. (2) to convert the left foot data, and Eq. (3) to convert the right foot data.

When using this method to obtain β, regardless of whether the left or right foot was used, if β was positive, it indicated an adduction rotation angle; and if β was negative, its absolute value indicated an abduction rotation angle.

3.4Analysis of varus and eversion

There was no difference between converting the coordinates to rotation angles of varus or eversion and the coordinates to the rotation angles of adduction or abduction. We could use the same methods for analysis and conversion. The first formula used is:

We used Method I to obtain δ. For the left foot, if δ was positive, it indicated a varus rotation angle; if δ was negative, its absolute value indicated an eversion rotation angle. For the right foot, when δ was positive, it indicated an eversion rotation angle; when δ was negative, its absolute value indicated a varus rotation angle.

The principle of Method II is the same as above. We chose Eq. (4) to convert the coordinates collected from the left feet into rotation angles, and Eq. (5) for the right foot data. If δ was positive, it indicated an eversion rotation angle; if δ was negative, its absolute value indicated a varus rotation angle.

3.5Initial errors

Table 3 displays the results obtained using Method II in three columns named L, M, and N. The data in the three columns are the three rotation angles that correspond to the position of the foot in the coordinate system, measured in degrees. There were small initial errors because the sole was uneven when the ankle was in the initial state with a hard board fixed on the sole of the foot. We obtained more accurate results using rotation angles that were obtained by subtracting the rotation angles that corresponded to the initial position.

3.6Arrangement of data

Throughout this process, we were working to obtain the 3D motion space of the ankle. This refers to the rotation ranges in various directions at different rotation angles of the dorsal flexure or plantar flexion. Therefore, it was necessary to analyze the data of the points on the edge of the motion space. There were some points that were not on the edge or repeated because the ankle rotated to reach any point it could achieve in the three-dimensional space for several minutes during measurement. We are able to create a preliminary screening from these data. Firstly, the data for the points that were not on the edge were deleted, and then we calculated the averages of the data of the points that could be considered repetitions. Table 4 shows that the data in the first 8 columns should be deleted because they are obviously not the data of points on the edge. The data in the last 3 columns represent points that could be considered repetitions, so we obtained the averages of these columns.

After all of the data were screened the first time, we integrated the data for the left foot into one table and the data for the right foot into another table. We placed the rotation angles for the same direction separately into one column.

When we screened the integrated data the second time, individual data that were significantly different from the others were deleted, and then we calculated the averages of the remaining data. For each number in the first column, there is a corresponding positive number, representing the rotation angle in one direction, and a negative number, representing the rotation angle in the opposite direction. We separately average all the positive numbers and the negative numbers. Whichever average has a greater absolute value, it is used to represent the actual rotation angle.

The final data that corresponded to the motion space of the left foot or the right foot were obtained after the data’s arrangement was completed again. These data are shown in Table 5.

Lastly, we integrated the final data for the left foot and the right foot into one table using the same method, and then we obtained the final data for the ankle’s motion space after the averages were calculated. The results are shown in Table 6.