One significant disadvantage of the Phase-2 arm-tracking system with the employed UM7-LT IMU built-in EKF was the fact that it tended to drift due to the IMU magnetometer sensor measurement distortion caused by variations of the magnetic field in close proximity to the computing equipment present in the laboratory. This was noticed using a computer visualization model in which, after several seconds of the system operation, the wrist pose estimations and, hence, the humanoid model, were drifting away from the corresponding operator’s real pose. Initially, drift reduction was attempted by recalibrating the IMU sensors each time before the system usage, as outlined in

Section 4.1, and by tuning the process and the measurement noise covariance matrices

$\mathbf{Q}$ and

$\mathbf{R}$ of the IMU built-in EKF.

Figure 10a demonstrates the case when the importance given to magnetometer data in the EKF is relatively low, as compared to that of gyroscope data, which resulted in drifting of the IMU quaternion output. In the second case, shown in

Figure 10b, the EKF “trusts” in magnetometer readings is increased, which resulted in a more stable sensor output, with a well-calibrated magnetometer. In both the cases, the IMU was in a stationary position. However, this setting with high trust in magnetometer data would not be reliable for prolonged arm-tracking trial experiments, of more than 20 s.

Therefore, as an alternative, it was decided to employ another data processing algorithm for achieving more accurate and stable arm motion-tracking, while reducing the need for frequent IMU tuning. Reviewing various IMU data-fusion algorithms, a popular open-source gradient-descent filter (GDF), developed by Madgwick [

19] and available at

http://x-io.co.uk/open-source-imu-and-ahrs-algorithms/, was adopted for processing the IMU raw sensor data, due to its more accurate performance, demonstrated in comparative tests using an XSens MTx sensor with a proprietary Kalman-based algorithm. Multiple advantages of GDF, such as low computational cost, integrated compensation of magnetic field variations and easy tuning, led to the adoption of this algorithm by many researchers for further improvement or comparative analysis with their proposed approaches [

33,

34]. The filter requires only one tuning parameter, i.e., the gain term

$\beta $, representing the gyro measurement error expressed as the magnitude of a quaternion derivative [

19]. While higher values of

$\beta $ (greater than 0.05) result in fast responses of the filter with higher noise amplitude (evaluated using a standard deviation measure for individual IMUs), smaller

$\beta $ gains lead to rapidly increased convergence time, providing more accurate computation, as shown in

Figure 10c,d. Based on trial and error tests, a value of

$\beta =0.04$ was set to ensure accurate and stable tracking performance of the system prototype, compared with the built-in EKF, as demonstrated in the remainder of the paper. However, this led to about 100 s GDF convergence time required for the system prototype initialization, by matching the initial T-pose of the operator (as in

Figure 1a). The filter characteristics corresponding to the selected

$\beta $ gain are indicated as dashed lines in

Figure 10a,d).

The comparative evaluation of the final system prototype with both the IMU built-in EKF and the GDF data processing algorithms was done using the principle of symmetry of geometrical patterns. Consider a sequence of symmetrical up-down and left-to-right movements repeated by a fully stretched human operator’s arm and centered around the user’s fully stretched-to-front arm pose acting as a starting arm pose. Exploiting the fact that the geometrical center of the overall arm’s wrist movements, computed as a mean value of all trajectory points, in the ideal case lies on the motion’s axis of symmetry and closely correlates with the wrist initial position, the most accurate data processing algorithm can be experimentally identified by analyzing their arm trajectory estimations. At the beginning, an operator wearing the arm motion-tracking assumed the initial T-pose shown in

Figure 1a, with the wrist and upper arm IMU modules manually aligned as parallel to each other, for initializing the arm-tracking algorithm after the EKF or GDF convergence. After the tracker initialization procedure was completed, the operator took the starting pose with the user’s wrist positioned at point

$(0.05,-0.5,0.49)$ m, defined in the tracker base frame. In this pose, the fully stretched-to-front user’s arm determined the axis of symmetry of the test motions, aligned parallel to axis

y.

Figure 11 illustrates the estimated time evaluations of the

x,

y and

z positions of the operator’s wrist, i.e.,

${\mathbf{p}}_{\mathbf{wx}}$,

${\mathbf{p}}_{\mathbf{wy}}$ and

${\mathbf{p}}_{\mathbf{wz}}$, respectively, and the 3D wrist motion trajectory, computed with EKF and GDF. In general, as seen from the above figures, both EKF and GDF are able to closely capture the position and orientation patterns of the operator’s arm test motions and demonstrate close overlapping patterns, especially when estimating

${\mathbf{p}}_{\mathbf{wz}}$, as shown in

Figure 11c. This is also confirmed by the approximately equal standard deviations of EKF and GDF estimated trajectories, summarized in

Table 2. However, the gradual increase of deviations of the EKF estimated

${\mathbf{p}}_{\mathbf{wx}}$ and

${\mathbf{p}}_{\mathbf{wy}}$ trajectories, demonstrated in

Figure 11a,b, indicate the effect of the magnetic field variations affecting the magnetometer measurements, causing, in turn, distortion of heading estimations about the yaw axis [

34]. This effect is clearly visible in the 3D wrist trajectory graph in

Figure 11d, where one can observe the dynamic heading distortion of the wrist trajectory estimated with the EKF after the first sequence of operator’s left-right up-down arm movements. On the other hand, the GDF estimated trajectory corresponding to the same arm motions remains stable over time and, as a result, accurately captures the operator’s real 3D movements, as can be also observed in

Figure 11d. This conclusion is also supported by the statistical analysis of the estimated user’s wrist trajectories summarized in

Table 2 (the trajectory mean points are also graphically illustrated in

Figure 11). As can be seen in the table, the geometrical center of the GDF estimated trajectory, defined as the mean point (0.062, −0.342, 0.427), is significantly closer to the test motion’s axis of symmetry and the wrist’s starting position

$(0.05,-0.5,0.49)$ than the EKF mean point

$(0.22,-0.249,0.429)$ with 316%, 18% and −0.4% difference, respectively.