# Extension of the Rigid-Constraint Method for the Heuristic Suboptimal Parameter Tuning to Ten Sensor Fusion Algorithms Using Inertial and Magnetic Sensing

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Optimal vs Suboptimal Working Conditions

#### RCM Description

#### 2.2. Selected SFAs

- Mahony et al., 2008 [14] (MAH), with 2 parameters;
- Madgwick et al., 2011 [12] (MAD), with 1 parameter;
- Valenti et al., 2015 [15] (VAC), with 9 parameters;
- Seel et al., 2017 [16] (SEL), with 4 parameters;
- MATLAB complementary filter R2020a (MCF), the implementation of VAC by the MathWorks with only two parameters.

- Sabatini 2011 [17] (SAB), with 6 parameters;
- Ligorio and Sabatini 2015 [18] (LIG), with 6 parameters;
- Valenti et al., 2016 [19] (VAK), with 3 parameters;
- Guo et al., 2017 [20] (GUO), with 3 parameters;
- MATLAB Kalman filter R2020a (MKF), the implementation by MathWorks of the filter by Roetenberg et al., 2005 [21], with 8 parameters.

#### 2.3. Experimental Setup and Protocol

^{3}of volume; the maximum variation of the magnetic norm was limited to 1 µT.

#### 2.4. Data Processing

#### Orientation Estimation and Error Computation under Optimal and Suboptimal Conditions

- ${\mathit{q}}_{\mathit{S}\mathit{P}}$ is the ground-truth orientation expressed in the quaternion form. It describes the orientation of the LCS of SP referred to its initial orientation and it was obtained by using the SVD technique [22]. From trigonometry considerations, as described in Section 2.5.1. of [10], the errors which affect the ground-truth orientation are lower than 0.5 deg;
- ${\mathit{p}}_{\mathbf{1}vec}$ and ${\mathit{p}}_{\mathbf{2}vec}$ are the two vectors which contain, for each SFA, the values of the two parameters ranging from $0$ to $uppe{r}_{1}$ and from 0 to $uppe{r}_{2}$, respectively. In general, the two upper limits were chosen large enough to ensure that all the relevant search space was explored. The values of $uppe{r}_{1}$ and $uppe{r}_{2}$ can be observed in the figures of Appendix A. The lower limit for all the SFAs was set to zero but for a
_{th2}of VAC which was set to the value of the first threshold for the accelerometer measurements (a lower value would be meaningless since for the constraint is a_{th2}≥ a_{th1}). The average number of combinations explored was not the same for all the SFAs since it was a trade-off between computational costs and the search space size (on average it amounts to 360 combinations).

#### 2.5. Data Analysis

#### 2.5.1. Identification of the Optimal Region for Each Scenario and the Corresponding Optimal Absolute Error

- Optimal absolute orientation error: ${e}_{opt}=\mathrm{min}\left(\mathit{e}\right)$. In other words, ${e}_{opt}$ is the lowest error achievable when both parameter values are optimally tuned.
- The optimal region correspond to the range of the parameter values whose combinations provide errors within [${e}_{opt}$, ${e}_{opt}$ + 0.5 deg], where 0.5 is the uncertainty related to the ground-truth errors: $\{{\mathit{p}}_{op{t}_{1}},{\mathit{p}}_{op{t}_{2}}\}=\{\left({\mathit{p}}_{\mathbf{1}vec},{\mathit{p}}_{\mathbf{2}vec}\right)|\mathit{e}\le {e}_{opt}+0.5deg\}$.

#### 2.5.2. Identification of the Suboptimal Parameter Values for Each Scenario and the Corresponding Suboptimal Absolute Error

- Minimum relative orientation difference: ${\delta}_{sub}=\mathrm{min}\left(\mathit{\delta}\right)$.
- The suboptimal region is defined by the values of ${\mathit{p}}_{\mathbf{1}\mathrm{vec}}$ and ${\mathit{p}}_{\mathbf{2}vec}$ corresponding to ${\delta}_{sub}$: $\left\{{\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{1}}},{\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{2}}}\right\}=\{\left({\mathit{p}}_{\mathbf{1}vec},{\mathit{p}}_{\mathbf{2}vec}\right)|\mathit{\delta}={\delta}_{sub}\}.$ When the region, $\left\{{\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{1}}},{\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{2}}}\right\}$ was formed by two or more separated sub-regions, only the largest was considered.
- The suboptimal parameter values (${p}_{1\mathrm{c}}$ and ${p}_{2\mathrm{c}}$) are the values of ${\mathit{p}}_{\mathbf{1}vec}$ and ${\mathit{p}}_{\mathbf{2}vec}$ corresponding to the centroid of the suboptimal region: $\left\{{p}_{1\mathrm{c}},{p}_{2\mathrm{c}}\right\}=\mathrm{centroid}\left({\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{1}}},{\mathit{p}}_{\mathit{s}\mathit{u}{\mathit{b}}_{\mathbf{2}}}\right)$.
- The suboptimal absolute orientation error is the absolute orientation error corresponding to ${p}_{1c}$ and ${p}_{2c}$: ${e}_{sub}=$ $\mathit{e}\left({p}_{1\mathrm{c}},{p}_{2\mathrm{c}}\right)$.

#### 2.5.3. RCM Validation Metric

## 3. Results

#### 3.1. Optimal and Suboptimal Errors

#### 3.2. Optimal and Suboptimal Regions

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CF | Complementary Filter |

GCS | Global Coordinate System |

KF | Kalman Filter |

LCS | Local Coordinate System |

MIMU | Magneto-Inertial Measurement Unit |

RCM | Rigid Constraint Method |

rms | Root Mean Square |

SFA | Sensor Fusion Algorithm |

SP | Stereophotogrammetric System |

STD | Standard Deviation |

Absolute orientation | the orientation of the LCS of a system with respect to its GCS |

Absolute orientation error | the difference between the orientation of the LCS of a MIMU computed by a SFA and its actual orientation computed by the optical reference (SP) and expressed by the angle given by the axis-angle convention |

${e}_{opt}$ | minimum absolute orientation error which corresponds to the selection of the optimal parameter values |

${e}_{sub}$ | absolute orientation error which corresponds to the selection of the suboptimal parameter values |

Optimal parameter region | the range of parameter values for which the orientation errors are equal to ${e}_{opt}$ plus 0.5 deg |

Relative orientation difference | the difference between the LCSs of two MIMUs both computed by a SFA and expressed by the angle given by the axis-angle convention |

Suboptimal parameter region | the range of parameter values corresponding to the minimum of the relative orientation difference |

Suboptimal parameter values | Parameter values corresponding to the centroid of the suboptimal parameter region |

## Appendix A

**Figure A1.**This appendix provides the optimal and the suboptimal regions for each SFA. Mono-dimensional intervals were represented instead of two-dimensional regions for those SFAs with only one parameter value tuned (MAD; MCF, GUO, MKF).

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**Figure 1.**The experimental setup employed. Three pairs of MIMUs were aligned (from the bottom: Xsens—MTx, APDM—Opal and Shimmer—Shimmer3). The LCSs of the MIMUs are represented in blue. The three central markers define the LCS of the SP (in green). The MIMUs and SP systems were arranged so that their axes were aligned with the axes of the board (dashed red arrows). Figure adapted from [9].

**Figure 2.**Exemplificative description of the movements of the board in terms of Euler angles for the intermediate trial. As evident, from the graph, the first three rotations were performed around one axis at a time, while the last part of the movement is a combination of the movement around the three axes.

**Figure 3.**The grid-search approach followed to compute the absolute orientation error and the relative orientation difference for a given combination of the two parameter values. This process has been applied to each SFA for each of the nine experimental scenarios. Red and green arrows are related to the computation of the absolute error and relative difference, respectively.

**Figure 4.**Boxplot of the distribution of the 90 residuals (∆e). Outliers are also reported(red cross). The limit of 0.5 deg chosen to consider the suboptimal errors equivalent to the optimal error is also highlighted.

**Figure 5.**On the left: the optimal regions (one for each experimental scenario) obtained for LIG. On the right: the suboptimal regions (one for each experimental scenario) and their centroids obtained for LIG.

**Table 1.**This table reports the details of the two parameters selected for optimal and suboptimal tuning along with their default values. Adapted from [10].

CF | # Params | ${\mathit{p}}_{1}$ | Default | ${\mathit{p}}_{2}$ | Default | ||
---|---|---|---|---|---|---|---|

MAH | 2 | k_{p}—inverse gyroscope weight | 1 | rad/s | k_{i}—weight for online bias estimation | 0.3 | rad/s |

MAD | 1 | β—inverse gyroscope weight | 0.1 | rad/s | / | / | |

VAC | 9 | g_{mag}—magnetometer weight | 0.01 | a.u. | a_{th2}—threshold for accelerometer vector selection | 0.2 | a.u. |

SEL | 4 | τ_{acc}—accelerometer time constant | 1 | s | τ_{mag}—magnetometer time constant | 3 | s |

MCF | 2 | g_{mag}—magnetometer weight | 0.01 | a.u. | / | / | |

KF | # Params | ${p}_{1}$ | Default | ${p}_{2}$ | Default | ||

SAB | 6 | σ_{gyr}—inverse gyroscope weight | 0.007 | rad/s | a_{th}—threshold for accelerometer vector selection | 40 | mg |

LIG | 6 | σ_{gyr}—inverse gyroscope weight | 1 | rad/s | c_{b}—Gauss-Markov parameter of the prediction model to set the variance of external acceleration and ferromagnetic disturbances | 1 | a.u. |

VAK | 3 | σ_{gyr}—inverse gyroscope weight | 0.004 | rad/s | σ_{acc}—inverse accelerometer weight | 0.014 | m/s^{2} |

GUO | 3 | σ_{gyr}—inverse gyroscope weight | 0.001 | rad/s | / | / | |

MKF | 8 | σ^{2}_{gyr}—inverse gyroscope weight | 9.14 × 10^{−5} | (rad/s)^{2} | / | / |

System | Software | Sampling Frequency |
---|---|---|

Xsens—MTx | MT Manager Version 1.7 | 100 Hz |

APDM—Opal | Motion Studio Version 1.0.0.201712300 | 128 Hz (resampled at 100 Hz) |

Shimmer—Shimmer3 | Consensys v.1.5.0 | 100 Hz |

Vicon—T20 | Nexus v2.7 | 100 Hz |

**Table 3.**The STD of the three sensors computed for each of the six MIMUs during one minute of static acquisition. Taken from [10].

STD | Accelerometer (mg) | Gyroscope (deg/s) | Magnetometer (µT) | ||||||
---|---|---|---|---|---|---|---|---|---|

x | y | z | x | y | z | x | y | z | |

Xsens-MTx #1 | 0.86 | 0.80 | 0.85 | 0.38 | 0.39 | 0.37 | 0.06 | 0.04 | 0.04 |

Xsens-MTX #2 | 0.82 | 0.86 | 0.80 | 0.44 | 0.40 | 0.40 | 0.05 | 0.06 | 0.06 |

APDM-OPAL #1 | 0.38 | 0.33 | 0.38 | 0.16 | 0.23 | 0.11 | 0.26 | 0.23 | 0.20 |

APDM-OPAL #2 | 0.34 | 0.32 | 0.35 | 0.16 | 0.27 | 0.19 | 0.26 | 0.25 | 0.20 |

Shimmer-Shimmer 3 #1 | 1.06 | 0.97 | 1.26 | 0.09 | 0.08 | 0.09 | 0.84 | 0.84 | 0.69 |

Shimmer-Shimmer 3 #2 | 1.12 | 1.09 | 1.29 | 0.06 | 0.06 | 0.06 | 0.97 | 0.97 | 0.58 |

**Table 4.**The optimal and suboptimal errors are reported together with their residual for each SFA and for each experimental scenario. The residual values higher than 0.5 deg are highlighted in bold.

CF | ${\mathit{e}}_{\mathit{o}\mathit{p}\mathit{t}}$ | ${\mathit{e}}_{\mathit{s}\mathit{u}\mathit{b}}$ | $\mathbf{\u2206}\mathit{e}$ | KF | ${\mathit{e}}_{\mathit{o}\mathit{p}\mathit{t}}$ | ${\mathit{e}}_{\mathit{s}\mathit{u}\mathit{b}}$ | $\mathbf{\u2206}\mathit{e}$ | ||
---|---|---|---|---|---|---|---|---|---|

Xsens | Slow | MAH | 2.5 | 2.5 | 0 | SAB | 2.2 | 2.2 | 0 |

Intermediate | 2.4 | 3.8 | 1.4 | 2.1 | 2.1 | 0 | |||

Fast | 4.0 | 4.2 | 0.2 | 2.4 | 2.4 | 0 | |||

APDM | Slow | 3.8 | 5.6 | 1.8 | 5.0 | 5.1 | 0.1 | ||

Intermediate | 4.8 | 4.9 | 0.1 | 5.7 | 5.8 | 0.1 | |||

Fast | 8.2 | 9.2 | 1 | 8.3 | 10.0 | 1.7 | |||

Shimmer | Slow | 3.4 | 3.7 | 0.3 | 4.5 | 4.5 | 0 | ||

Intermediate | 4.6 | 5.3 | 0.7 | 4.9 | 4.9 | 0 | |||

Fast | 7.6 | 10.6 | 3 | 8.5 | 9.6 | 1.1 | |||

Xsens | Slow | MAD | 2.7 | 2.7 | 0 | LIG | 1.9 | 2.4 | 0.5 |

Intermediate | 2.5 | 4.0 | 1.5 | 2.0 | 3.8 | 1.8 | |||

Fast | 4.0 | 4.0 | 0 | 2.9 | 3.4 | 0.5 | |||

APDM | Slow | 3.8 | 3.8 | 0 | 3.6 | 3.9 | 0.3 | ||

Intermediate | 4.6 | 4.8 | 0.2 | 4.9 | 5.1 | 0.2 | |||

Fast | 8.1 | 8.2 | 0.1 | 4.6 | 4.9 | 0.3 | |||

Shimmer | Slow | 3.9 | 4.1 | 0.2 | 4.4 | 4.6 | 0.2 | ||

Intermediate | 4.9 | 5.1 | 0.2 | 4.0 | 4.2 | 0.2 | |||

Fast | 8.8 | 10.8 | 2 | 6.3 | 6.5 | 0.2 | |||

Xsens | Slow | VAC | 4.0 | 4.0 | 0 | VAK | 1.2 | 1.5 | 0.3 |

Intermediate | 5.0 | 5.1 | 0.1 | 1.6 | 1.7 | 0.1 | |||

Fast | 7.2 | 7.2 | 0 | 2.5 | 2.5 | 0 | |||

APDM | Slow | 3.5 | 4.4 | 0.9 | 3.6 | 4.1 | 0.5 | ||

Intermediate | 6.1 | 6.4 | 0.3 | 6.0 | 6.9 | 0.9 | |||

Fast | 8.3 | 11.3 | 3 | 9.2 | 10.4 | 1.2 | |||

Shimmer | Slow | 3.8 | 3.8 | 0 | 4.0 | 4.6 | 0.6 | ||

Intermediate | 10.2 | 10.8 | 0.6 | 4.4 | 5.7 | 1.3 | |||

Fast | 11.5 | 15.2 | 3.7 | 8.2 | 10.6 | 2.4 | |||

Xsens | Slow | SEL | 3.1 | 3.5 | 0.4 | GUO | 2.3 | 2.3 | 0 |

Intermediate | 2.5 | 3.9 | 1.4 | 2.3 | 2.3 | 0 | |||

Fast | 5.1 | 5.1 | 0 | 5.7 | 5.7 | 0 | |||

APDM | Slow | 3.7 | 3.8 | 0.1 | 4.2 | 4.5 | 0.3 | ||

Intermediate | 7.1 | 7.1 | 0 | 5.1 | 5.7 | 0.6 | |||

Fast | 8.0 | 10.0 | 2 | 9.4 | 9.4 | 0 | |||

Shimmer | Slow | 3.4 | 3.5 | 0.1 | 4.0 | 4.2 | 0.2 | ||

Intermediate | 5.0 | 6.3 | 1.3 | 5.1 | 5.1 | 0 | |||

Fast | 9.4 | 10.8 | 1.4 | 13.7 | 14.4 | 0.7 | |||

Xsens | Slow | MCF | 3.3 | 3.4 | 0.1 | MKF | 4.2 | 4.3 | 0.1 |

Intermediate | 6.1 | 6.1 | 0 | 4.8 | 4.8 | 0 | |||

Fast | 6.6 | 7.5 | 0.9 | 6.7 | 6.9 | 0.2 | |||

APDM | Slow | 3.8 | 4.8 | 1 | 3.6 | 3.8 | 0.2 | ||

Intermediate | 12.3 | 12.5 | 0.2 | 5.3 | 5.3 | 0 | |||

Fast | 7.9 | 9.6 | 1.7 | 7.2 | 7.2 | 0 | |||

Shimmer | Slow | 5.0 | 5.2 | 0.2 | 3.9 | 4.2 | 0.3 | ||

Intermediate | 10.0 | 13.3 | 3.3 | 8.4 | 9.8 | 1.4 | |||

Fast | 8.6 | 8.8 | 0.2 | 9.9 | 10.0 | 0.1 |

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**MDPI and ACS Style**

Caruso, M.; Sabatini, A.M.; Knaflitz, M.; Della Croce, U.; Cereatti, A. Extension of the Rigid-Constraint Method for the Heuristic Suboptimal Parameter Tuning to Ten Sensor Fusion Algorithms Using Inertial and Magnetic Sensing. *Sensors* **2021**, *21*, 6307.
https://doi.org/10.3390/s21186307

**AMA Style**

Caruso M, Sabatini AM, Knaflitz M, Della Croce U, Cereatti A. Extension of the Rigid-Constraint Method for the Heuristic Suboptimal Parameter Tuning to Ten Sensor Fusion Algorithms Using Inertial and Magnetic Sensing. *Sensors*. 2021; 21(18):6307.
https://doi.org/10.3390/s21186307

**Chicago/Turabian Style**

Caruso, Marco, Angelo Maria Sabatini, Marco Knaflitz, Ugo Della Croce, and Andrea Cereatti. 2021. "Extension of the Rigid-Constraint Method for the Heuristic Suboptimal Parameter Tuning to Ten Sensor Fusion Algorithms Using Inertial and Magnetic Sensing" *Sensors* 21, no. 18: 6307.
https://doi.org/10.3390/s21186307