# Assessing the Performance of Sensor Fusion Methods: Application to Magnetic-Inertial-Based Human Body Tracking

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Proposed Benchmarking Method: Overview

#### 2.2. The Timed up and Go Dataset

^{2}, ±1500°/s and ±600 μT of full-range scale, respectively), was secured to the participants’ lower back (lumbar region of the spine, approximately at L3–L4 vertebrae level, Figure 2), using an elastic belt. MIMU data were collected at 128 samples/s. A plastic plaque equipped (Figure 2) with a cluster of four infrared reflective markers was rigidly attached to the MIMU case for ground-truth data acquisition using a nine-camera motion capture system (Vicon MX3, Oxford, UK) at 100 sample/s.

**Figure 2.**(

**Left**) Sensor location on the participants’ body, axes orientation and TUG scheme, and (

**Right**) Opal MIMU with the infrared reflective marker cluster.

^{−3}rad/s, 5 × 10

^{−3}m/s

^{2}and 0.15 µT). At this point in time, both the experimental and the ideal data required to devise the different scenarios for the SFA evaluation were available.

#### 2.3. Scenario Selection

**Table 1.**The six evaluation scenarios considered in this work, listed theoretically from the best to the worst case.

Scenario | Gyroscope | Accelerometer | Magnetometer |
---|---|---|---|

MOD | simulated | gravity-only | simulated |

SIM | simulated | simulated | simulated |

GYR | measured | simulated | simulated |

ACC | simulated | measured | simulated |

MAG | simulated | simulated | measured |

MEAS | measured | measured | measured |

#### 2.4. Sensor Fusion Algorithms

**Figure 3.**Overview of the Kalman-based SFAs considered in this work: (

**Left**) Algorithm 1 and (

**Right**) Algorithm 2.

#### 2.5. Measure of Performance

#### 2.6. Statistical Analysis

## 3. Results

**Figure 4.**Heading (

**a**) and attitude (

**b**) ground-truth curves; heading (

**c**) and attitude (

**d**) error angles obtained for all the six scenarios considered. The colored bands in the upper row denote activities of sit-to-stand and stand-to-sit (

**yellow**), walking (

**blue**), and 180° turns around the cranio-caudal axis.

**Figure 5.**RMS

_{head}and RMS

_{att}median and inter quartile ranges obtained for the two considered SFAs and all the six tested scenarios.

**Table 2.**Results of the one-way repeated measures ANOVA for both the attitude and heading errors and for each tested algorithm. Degrees of freedom for the effect (df

_{scenario}) and for the error term (df

_{error}) are reported together with F values, p values and partial eta squared (η

^{2}).

Algorithm | MOP | df_{scenario} | df_{error} | F | p | η^{2} |
---|---|---|---|---|---|---|

1 | RMS_{att} | 1.13 | 25.93 | 193.95 | <0.001 | 0.85 |

RMS_{head} | 3.09 | 71.21 | 661.30 | <0.001 | 0.97 | |

2 | RMS_{att} | 1.05 | 24.26 | 113.60 | <0.001 | 0.83 |

RMS_{head} | 3.37 | 77.67 | 285.84 | <0.001 | 0.92 |

**Table 3.**Post-hoc analysis: marginal differences between the scenarios indicated in the first and second column for both algorithms. Significant differences are indicated with an asterisk.

Tested Scenario | Testbed Scenario | Algorithm 1 | Algorithm 2 | ||||||
---|---|---|---|---|---|---|---|---|---|

Attitude | Heading | Attitude | Heading | ||||||

GYR | SIM | 2.11 | * | 3.82 | * | 1.03 | * | 2.61 | * |

MEAS | −0.9 | * | −0.19 | −1.22 | * | −0.98 | * | ||

ACC | SIM | 0.25 | * | 0.54 | * | 0.15 | * | 0.61 | * |

MEAS | −1.95 | * | −3.47 | * | −2.10 | * | −2.99 | * | |

MAG | SIM | 0.00 | 1.93 | * | 1.76 | * | 1.66 | * | |

MEAS | −2.20 | * | −2.08 | * | −0.49 | * | −1.94 | * | |

MOD | SIM | −0.46 | * | −0.22 | * | −0.25 | * | −0.64 | * |

**Table 4.**Results (Z and p-values) of the comparison between Algorithms 1 and 2 for the SIM and MEAS scenarios. To improve the table readability/clarity, significant differences are also indicated with an asterisk.

Considered Scenario | Attitude | Heading | ||||
---|---|---|---|---|---|---|

Z | p | Z | p | |||

SIM | −4.286 | <0.001 | * | −4.286 | <0.001 | * |

MEAS | −2.143 | 0.032 | * | −0.829 | 0.407 |

## 4. Discussions

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- ${x}_{k}$ and ${\widehat{x}}_{k}$ are, respectively, the real and estimated value of the state vector $x$;
- ${{}^{-}\widehat{x}}_{k}$ and ${{}^{+}\widehat{x}}_{k}$ are, respectively, the a priori and a posteriori estimate of the state vector $x$ at the k-th time instant;
- ${h}^{n}$ and ${h}^{b}$ are the representation of the Earth’s magnetic field in the $\left\{n\right\}$ and $\left\{b\right\}$ reference frames, respectively;
- ${g}^{n}$ and ${g}^{b}$ are the representation of the Earth’s magnetic field in the $\left\{n\right\}$ and $\left\{b\right\}$ reference frames, respectively;
- ${}^{g}\mathit{\sigma}$, ${}^{a}\mathit{\sigma}$ and ${}^{m}\mathit{\sigma}$ are the standard deviations of the gyroscope, accelerometer and the magnetometer measurement, respectively;
- ${}^{b}\mathit{\sigma}$ is a tuning parameter encoding the expected severity of the magnetic disturbances in method B;
- ${c}_{a}$ and ${c}_{b}$ are the Gauss-Markov parameters of the prediction models in method A;
- $H$, $Q$ and $R$ are, respectively, the extraction matrix, the process noise matrix and the measurement noise matrix in the Kalman filter;
- ${T}_{s}$ is the sampling time;
- $\times $ represents the cross product operator;
- $\otimes $ represents the quaternion product operator;
- ${q}^{bn}$ and ${R}^{bn}$ represents, respectively, the quaternion and the rotation matrix encoding the orientation from the $\left\{n\right\}$ reference frame to the $\left\{b\right\}$ reference frame.
- ${I}_{n}$ and ${0}_{n}$ are the $n\times n$ identity and null matrix;
- $[p\text{\hspace{0.17em}}\times ]$ is the cross product matrix for the vector $p={\left[\begin{array}{ccc}{p}_{1}& {p}_{2}& {p}_{3}\end{array}\right]}^{T}$, $[p\text{\hspace{0.17em}}\times ]=\left[\begin{array}{ccc}0& -{p}_{3}& {p}_{2}\\ {p}_{3}& 0& -{p}_{1}\\ -{p}_{2}& {p}_{1}& 0\end{array}\right]$;
- $\Omega (p)=\left[\begin{array}{cc}-\left[p\times \right]& p\\ -{p}^{T}& 0\end{array}\right]$;
- $\Xi \left(q\right)=\left[\begin{array}{c}\left[{}^{v}q\times \right]+{q}_{4}\cdot {I}_{3}\\ {}^{v}q^{T}\end{array}\right]$ with $q={\left[\begin{array}{cccc}{q}_{1}& {q}_{2}& {q}_{3}& {q}_{4}\end{array}\right]}^{T}={\left[\begin{array}{cc}{}^{v}q& {q}_{4}\end{array}\right]}^{T}$ being a generic unit quaternion;
- $\Psi \left(q,p\right)=\frac{\partial \left(q\otimes \left[\begin{array}{c}p\\ 0\end{array}\right]\otimes {q}^{-1}\right)}{\partial \left(q\right)}$

Algorithm A1 |

for each measured sample from the sensors do |

${\mathit{\omega}}_{k}^{b}\leftarrow gyroscope\text{\hspace{0.17em}}sample$ |

Prediction |

Predict gravity and acceleration (KF_{acc}) |

${}_{acc}{}^{-}x_{k}=\left[\begin{array}{c}{}^{-}\widehat{g}_{k}^{b}\\ {}^{-}\widehat{a}_{k}^{b}\end{array}\right]=\left[\begin{array}{cc}\mathrm{exp}(-[{\mathit{\omega}}_{k-1}^{b}\times ]\cdot {T}_{s})\text{\hspace{0.17em}}& {0}_{3}\\ {0}_{3}& {c}_{a}\cdot {I}_{3}\end{array}\right]\left[\begin{array}{c}{}^{+}\widehat{g}_{k-1}^{b}\\ {}^{+}\widehat{a}_{k-1}^{b}\end{array}\right]$ |

Process noise covariance matrix (KF_{acc}) |

${}_{acc}Q_{k-1}=\left[\begin{array}{cc}\left[{}^{-}\widehat{g}_{k}^{b}\times \right]& {0}_{3}\\ {0}_{3}& {c}_{b}\cdot {I}_{3}\end{array}\right]\left[\begin{array}{cc}{I}_{3}\cdot {\left({}^{g}\mathit{\sigma}{T}_{s}\right)}^{2}& {0}_{3}\\ {0}_{3}& {I}_{3}\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cc}\left[{}^{-}\widehat{g}_{k}^{b}\times \right]& {0}_{3}\\ {0}_{3}& {c}_{b}\cdot {I}_{3}\end{array}\right]}^{T}$ |

Predict Earth’s magnetic field and disturbances (KF_{mag}) |

${}_{mag}{}^{-}x_{k}=\left[\begin{array}{c}{}^{-}\widehat{h}_{k}^{b}\\ {}^{-}\widehat{d}_{k}^{b}\end{array}\right]=\left[\begin{array}{cc}\mathrm{exp}(-[{\mathit{\omega}}_{k-1}^{b}\times ]\cdot {T}_{s})\text{\hspace{0.17em}}& {0}_{3}\\ {0}_{3}& {c}_{a}\cdot {I}_{3}\end{array}\right]\left[\begin{array}{c}{}^{+}\widehat{h}_{k-1}^{b}\\ {}^{+}\widehat{d}_{k-1}^{b}\end{array}\right]$ |

Process noise covariance matrix (KF_{mag}) |

${}_{mag}Q_{k-1}=\left[\begin{array}{cc}\left[{}^{-}\widehat{h}_{k}^{b}\times \right]& {0}_{3}\\ {0}_{3}& {c}_{b}\cdot {I}_{3}\end{array}\right]\left[\begin{array}{cc}{I}_{3}\cdot {\left({}^{g}\mathit{\sigma}{T}_{s}\right)}^{2}& {0}_{3}\\ {0}_{3}& {I}_{3}\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cc}\left[{}^{-}\widehat{h}_{k}^{b}\times \right]& {0}_{3}\\ {0}_{3}& {c}_{b}\cdot {I}_{3}\end{array}\right]}^{T}$ Update: |

$H={}_{mag}H={}_{acc}H=\left[\begin{array}{cc}{I}_{3}& {I}_{3}\end{array}\right]$ |

Measurement update (KF_{acc}) |

${}_{acc}y_{k}\leftarrow accelerometer\text{\hspace{0.17em}}sample$ |

${}_{acc}R={I}_{3}\cdot {}^{a}\mathit{\sigma}^{2}$ |

${}_{acc}{}^{+}\widehat{x}_{k}\leftarrow $ KalmanUpdate (${}_{acc}{}^{-}x_{k}$, ${}_{acc}Q_{k-1}$, ${}_{acc}y_{k}$, $H$, ${}_{acc}R$) |

Measurement update (KF_{mag}) |

${}_{mag}y_{k}\leftarrow magnetometer\text{\hspace{0.17em}}sample$ |

${}_{mag}R={I}_{3}\cdot {}^{m}\mathit{\sigma}^{2}$ |

${}_{mag}{}^{+}\widehat{x}_{k}\leftarrow $ KalmanUpdate (${}_{acc}{}^{-}x_{k}$, ${}_{mag}Q_{k-1}$, ${}_{mag}y_{k}$, $H$, ${}_{mag}R$) |

Orientation computation (TRIAD algorithm) |

Normalize ${}^{+}\widehat{g}_{k}^{b}$ and ${}^{+}\widehat{h}_{k}^{b}$ to one |

${M}_{obs}=\left[\begin{array}{ccc}{}^{+}\widehat{g}_{k}^{b}& {}^{+}\widehat{g}_{k}^{b}\times {}^{+}\widehat{h}_{k}^{b}& {}^{+}\widehat{g}_{k}^{b}\times \left({}^{+}\widehat{g}_{k}^{b}\times {}^{+}\widehat{h}_{k}^{b}\right)\end{array}\right]$ |

${M}_{ref}=\left[\begin{array}{ccc}{g}^{n}& {g}^{n}\times {h}^{n}& {g}^{n}\times \left({g}^{n}\times {h}^{n}\right)\end{array}\right]$ |

${\widehat{q}}_{k}^{bn}=matrixToQuaternion\left({M}_{ref}{M}_{obs}^{T}\right)$ |

end for |

Algorithm A2 |

for each measured sample from the sensors do |

${\mathit{\omega}}_{k}^{b}\leftarrow gyroscope\text{\hspace{0.17em}}sample$ |

Prediction |

Predict the quaternion and the magnetic disturbance |

${}^{-}\widehat{x}_{k}=\left[\begin{array}{c}{}^{-}\widehat{q}_{k}^{bn}\\ {}_{m}{}^{-}\widehat{b}_{k}^{n}\end{array}\right]=\left[\begin{array}{cc}\mathrm{exp}\left(\Omega \left({\mathit{\omega}}_{k-1}^{b}\right){T}_{s}\right)& {0}_{3}\\ {0}_{3}& {I}_{3}\end{array}\right]\left[\begin{array}{c}{}^{+}\widehat{q}_{k-1}^{bn}\\ {}_{m}{}^{+}\widehat{b}_{k-1}^{n}\end{array}\right]$ |

Process noise covariance matrix |

${Q}_{k-1}=\left[\begin{array}{cc}{\left(\frac{{}^{g}\mathit{\sigma}{T}_{s}}{2}\right)}^{2}\Xi \left({}^{+}\widehat{q}_{k-1}^{bn}\right){I}_{3}\Xi {\left({}^{+}\widehat{q}_{k-1}^{bn}\right)}^{T}& {0}_{4\times 3}\\ {0}_{3\times 4}& {}^{b}\mathit{\sigma}^{2}\cdot {I}_{3}\end{array}\right]$ |

Update: |

compute ${}^{-}\widehat{R}^{bn}=quaternionToMatrix({}^{-}\widehat{q}^{bn})$ |

${y}_{k}=\left[\begin{array}{c}{}_{mag}y_{k}\\ {}_{acc}y_{k}\end{array}\right],\begin{array}{c}{}_{mag}y_{k}\leftarrow magnetometer\text{\hspace{0.17em}}sample\\ {}_{acc}y_{k}\leftarrow accelerometer\text{\hspace{0.17em}}sample\end{array}$ |

Sensor data validation |

${}_{acc}R=\{\begin{array}{c}{I}_{3}\cdot {}^{a}\mathit{\sigma}^{2},\text{\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left|{}_{acc}y-{}^{-}\widehat{R}^{bn}{g}^{n}\right|<{}_{acc}\mathit{\epsilon}\\ \infty ,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}otherwise\end{array}$ |

${}_{mag}R=\{\begin{array}{c}{I}_{3}\cdot {}^{m}\mathit{\sigma}^{2},\text{\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left|{}_{mag}y-{}^{-}\widehat{R}^{bn}\left({h}^{n}-{}_{m}{}^{-}\widehat{b}_{k}^{n}\right)\right|<{}_{mag}\mathit{\epsilon}\\ \infty ,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}otherwise\end{array}$ |

$R=\left[\begin{array}{c}{}_{mag}R\\ {}_{acc}R\end{array}\right]$ |

Jacobian calculation |

$H=\left[\begin{array}{cc}\Psi \left({}^{-}\widehat{q}_{k}^{bn},{h}^{n}+{}_{m}{}^{-}\widehat{b}_{k}^{n}\right)& {R}^{bn}\\ \Psi \left({}^{-}\widehat{q}_{k}^{bn},{g}^{n}\right)& {0}_{3}\end{array}\right]$ |

${}^{+}\widehat{x}\leftarrow $KalmanUpdate (${}^{-}\widehat{x}$, ${Q}_{k-1}$, ${y}_{k}$, $H$, $R$) |

end for |

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## Share and Cite

**MDPI and ACS Style**

Ligorio, G.; Bergamini, E.; Pasciuto, I.; Vannozzi, G.; Cappozzo, A.; Sabatini, A.M.
Assessing the Performance of Sensor Fusion Methods: Application to Magnetic-Inertial-Based Human Body Tracking. *Sensors* **2016**, *16*, 153.
https://doi.org/10.3390/s16020153

**AMA Style**

Ligorio G, Bergamini E, Pasciuto I, Vannozzi G, Cappozzo A, Sabatini AM.
Assessing the Performance of Sensor Fusion Methods: Application to Magnetic-Inertial-Based Human Body Tracking. *Sensors*. 2016; 16(2):153.
https://doi.org/10.3390/s16020153

**Chicago/Turabian Style**

Ligorio, Gabriele, Elena Bergamini, Ilaria Pasciuto, Giuseppe Vannozzi, Aurelio Cappozzo, and Angelo Maria Sabatini.
2016. "Assessing the Performance of Sensor Fusion Methods: Application to Magnetic-Inertial-Based Human Body Tracking" *Sensors* 16, no. 2: 153.
https://doi.org/10.3390/s16020153