The estimation of the orientation in space of a body using a sensor fusion algorithm (SFA) applied to the recordings of a wearable Magnetic and Inertial Measurement Unit (MIMU) requires the proper setting of the values of its parameters. [1
]. A common method to set the parameter values consists in exploiting the ground-truth information to minimize the overall difference between the estimated and the true orientation for a given recording (optimal working condition).
However, the latter approach may be unfeasible since gold standard such as stereophotogrammetric system (SP) are rarely available out of specialized human movement laboratories. Recent works have confirmed the crucial role played by the criteria used to select appropriate parameter values [8
]. It has also been shown that several intrinsic and extrinsic factors should be considered when performing this selection. Among them, the most critical ones are sensor noise characteristics, amplitude of motion, intensity of ferromagnetic disturbances and time required for the algorithm to converge. In light of these considerations, alternative strategies for selecting parameter values without orientation references are needed. To address this problem, we have recently presented in [9
] a rigid-constraint method (RCM) for a sub-optimal estimation of the values of the single parameter (β) of the sensor fusion algorithm by Madgwick et al., [12
] using a heuristic procedure which does not rely on an orientation reference. The assumption on which the method relies, is that two MIMUs aligned on a rigid body must have a null orientation difference during the movement. The orientation of the two MIMUs was computed for several values of β, separately. The main finding was that the value of β which minimizes the relative orientation difference between the MIMUs also guarantees small absolute orientation errors: the difference between the obtained and the optimal error (i.e., minimum) was 1.5 deg on average and 2.5 deg at most. This can be justified by considering that the error and disturbance characteristics affecting the sensors embedded in the two MIMUs are, in general, independent, as discussed more in detail in Section 2.1
“RCM description”. To the best of our knowledge, the RCM is the only method in the literature that specifically addressed the problem of estimating the most suitable parameter values (suboptimal) without relying on the use of a gold standard.
In this framework, this paper aims at verifying the generalizability of the RCM presented in [9
] to different sensor fusion algorithms including five complementary filters (CFs) and five Kalman filters (KFs). To this end, nine experimental scenarios were considered including six MIMUs from three different manufacturers and three rotation rate magnitudes. The motions consisted in a mix of 2D and 3D rotations. For each scenario, the absolute errors corresponding to the selected suboptimal values were compared with those obtained under optimal working conditions. When this difference was lower or equal than 0.5 deg, the RCM was considered completely successful in estimating the suboptimal parameter values corresponding to the optimal orientation error for that specific SFA and experimental scenario.
One of the most important steps when using a sensor fusion algorithm is the choice of the value(s) of filter parameter(s) [2
] since they heavily affect the orientation accuracy. As discussed in [10
], a possible choice is to perform a fine-tuning of these values specifically for a given experimental scenario using the ground-truth orientation. The errors obtained this way are representative of the best performance achievable with a given SFA. Furthermore, it has been observed that there is not a common intersection among the optimal regions when varying the experimental scenario (see Appendix A
“Optimal regions (intervals)”). This stresses the importance of tuning the parameter values of each SFA according to the specific scenario under analysis (device model and rotation rate).
In this work, suitable parameter values were estimated using the RCM method, which does not rely on any ground-truth orientation to reflect a more common situation during the everyday use of the MIMUs. Table 4
shows that the residuals between the optimal and suboptimal errors were lower or equal to 0.5 deg in 60 cases out of 90, between 0.5 and 1.0 deg in 10 cases, and higher than 1.0 deg in the remaining 20 cases. Thus, the RCM allowed to estimate the orientation with errors equivalent to the optimal approach in 67% of the cases. In the remaining 33%, the maximum residual amounted to 3.7 deg. Overall, the median residual was equal to 0.2 deg and the computed mean to 0.6 deg. These results corroborate the findings obtained by Caruso et al., 2020 [9
] for a single SFA and suggest the possibility to properly tune a generic SFA on different scenarios without using any orientation reference.
Among the 30 cases in which
was higher than 0.5 deg, an increase of the rotation rate magnitudes led to inferior performance of RCM. In fact, 14/30 cases were at fast rotation rate vs 12/30 at intermediate and only 4/30 at slow. This is in line with the unfavorable effect of the rotation rate in the orientation estimation accuracy, as widely recognized in the literature [2
]. Moreover, also the specific device model had an influence on the accuracy of the RCM. In the 30 cases in which
was higher than 0.5 deg, 14 were associated with Shimmer, 11 with APDM while 5 with Xsens. Finally, the LIG was the SFA for which the RCM performed the best. In fact, only 1 out of 9 residuals was higher than 0.5 deg while 5 out of 9 residuals were higher than 0.5 deg for MAH and VAK. The four residuals marked as outliers in Figure 4
were equal to 2.4 deg, 3 deg, 3.3 deg and 3.7 deg and were obtained with Shimmer at intermediate and fast rotation rates.
The figures shown in Appendix A
“Suboptimal regions (intervals)” suggest that the suboptimal regions do not consist of a single point. This is because multiple combinations of the two parameter values provide the same minimum of the relative orientation difference. However, it should be highlighted that the absolute errors corresponding to these combinations may be different.
Some limitations must be considered when using the RCM. Since the method relies on the differences between the errors affecting the two different accelerometers and magnetometers, when their mutual distance approaches zero also the differences tend to be less evident. In this case, the relative orientation difference may be very small, but it does not guarantee low absolute errors, especially if the orientations of the two MIMUs are estimated giving a high weight to the accelerometer and magnetometer readings. The authors suggest placing the two MIMUs with a mutual distance of at least a few centimeters, compatibly with the size of the rigid body support.
Some applications can benefit from this approach. In fact, as described in [28
] and detailed in [9
] a miniaturized plastic case may be designed for each specific application to host two MIMUs and to rigidly attach them to the body segment of interest. If only one MIMU is necessary for the data collection, the miniaturized case may host both the MIMU to be employed and an additional MIMU and a preliminary movement acquisition which mimics the gesture under the same experimental scenarios (similar rotation rate magnitude and device model) may be performed to estimate the suboptimal parameter(s) of the selected SFA.