# Using Accelerometer Data to Tune the Parameters of an Extended Kalman Filter for Optical Motion Capture: Preliminary Application to Gait Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preliminary Test

#### 2.1.1. Experimental Data Collection

#### 2.1.2. Sensor Orientation and Geomagnetic Frame of Reference

^{2}) was added to the vertical component of the resulting acceleration; the presence of the marker attached at the point made it unnecessary the use of any motion reconstruction method, thus eliminating a source of error for the optical system. Second, the acceleration provided by the IMU at point 4 was expressed in frame O by multiplying it by matrix ${R}_{\mathrm{OI}}^{5}$ (orientation provided by the IMU #5, attached to that point). Third, the acceleration provided by the IMU at point 4 was expressed in frame O by multiplying it by matrix ${R}_{\mathrm{OI}}$ (orientation provided by the optical system after the mentioned filtering of the marker trajectory). The resulting accelerations and their comparison are shown in Section 3.

#### 2.2. Gait Analysis

#### 2.2.1. Experimental Data Collection

#### 2.2.2. Skeletal Model and Kinematics

#### 2.2.3. Motion Reconstruction from Motion Capture Data

#### 2.2.4. Extended Kalman Filter for Motion Reconstruction

**F**is the state propagation matrix, ${a}_{k}$ is the process noise vector, and $\Gamma $ is the noise gain matrix. The DWNA is a second-order kinematic model, so the state vector contains the 52 degrees of freedom, $q$, along with their first time derivatives, $\dot{q}$. Accelerations are introduced in the system through the process noise vector

**a**. This vector contains the 52 independent accelerations, being each of them a discrete-time zero-mean white sequence. Therefore, they are assumed to be constant along every time step, and their values are random variables with a zero-mean normal distribution of variance ${\sigma}_{a}^{2}$. This variance has dimensions of squared acceleration for the translational DOFs, and squared angular acceleration for the angular ones. In order to reduce the number of parameters, in this work the same numerical value will be used for all of them.

**F**and

**Γ**for the whole system are the result of assembling these individual matrices, following the structure of the state vector

**x**.

**Q**matrix for the whole system is the result of assembling these individual matrices, as done for the state transition and noise gain matrices.

**z**, which in this case contains the absolute x, y and z coordinates of the 36 optical markers, as a function of the state vector

**x**,

**R**is a diagonal matrix, whose diagonal elements are all equal to the sensor noise variance ${\sigma}_{s}^{2}$, which has dimensions of squared length.

**q**and, consequently, in

**x**.

**k**and the skeletal degrees of freedom

**q**, being the objective function the quadratic error between measured and estimated marker positions,

**F**, leading to the so-called a priori estimate ${\widehat{x}}_{k+1}^{-}$. The estimate covariance matrix

**P**is updated accordingly, by using matrices

**F**and

**Q**,

**D**is built, such that

**z**, and the same is done to their corresponding rows in

**H**, so they do not affect the correction.

#### 2.2.5. Calculation of the Accelerations

**g**the gravity vector (9.81 m/s

^{2}in the positive vertical direction, as it would be perceived by the IMU). To get the virtual acceleration, vector ${a}_{i}$ must be expressed in the local frame of the IMU, I,

## 3. Results

#### 3.1. Preliminary Test and Calibration

^{2}when using the orientation provided by the IMU. Moreover, the accelerometer shows some peaks that are not captured by the optical system, for instance when the plate touches the ground after the 30 s mark. Due to the low sampling rate of the optical system, it cannot capture high-frequency events such as impacts, regardless of the filter cutoff frequency.

#### 3.2. Gait Analysis

#### 3.2.1. Vaughan’s Method

^{2}. Conversely, for low cutoff frequencies (below 8 Hz), the accelerations were too smooth, not reaching the experimental peak measurements of the inertial sensors. As opposed to the preliminary test, some acceleration peaks can be captured by the optical system at high cutoff frequencies, due to the softer contacting materials involved in this case, but at the cost of very noisy accelerations along the whole capture. The lowest errors were obtained for a cutoff frequency of 12 Hz, as highlighted in Table 1.

#### 3.2.2. Extended Kalman Filter

^{2}whereas for rotational DOFs it is in rad/s

^{2}.

^{2}(or rad/s

^{2}, depending on the corresponding coordinate), and cutoff frequencies ranging between 6 and 30 Hz, with respect to those directly measured by the inertial system.

^{2}combined with a 20 Hz Butterworth filter.

## 4. Discussion and Limitations of the Study

^{2}(or rad/s

^{2}, depending on the corresponding coordinate) and a cutoff frequency of 20 Hz. Noise was eliminated, peaks measured by the IMUs were almost reached, and the resulting RMSEs were better than those incurred by Vaughan’s method. Moreover, the EKF offered consistent kinematics by providing constant lengths of the body segments along the motion. Vaughan’s method is similar to those proposed in [19] and, like them, does not impose the kinematic constraints to compute the joint kinematics from the marker trajectories. Therefore, it would require an additional step to correct these inconsistencies before dynamic analysis.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The three reference frames involved in the calibration: fixed global reference frame of the optical system (subscript O); Earth-fixed global reference frame of each IMU (in grey, subscript E and superscript i); moving local reference frame of all the IMUs and the wooden plate (subscript I).

**Figure 4.**3D human model: (

**a**) graphical output; (

**b**) multibody model showing the segments: joints (black dots), and marker locations (white dots).

**Figure 6.**Kinematics at segment level to get the virtual acceleration (acceleration of the point of the body where the IMU is attached) from the optical motion capture recordings. The black dots represent joints connecting the segment with its neighbors. The white dots represent the markers attached to the segment.

**Figure 7.**Orientation errors in roll, pitch and yaw incurred by the nine IMUs with respect to the optical system (reference).

**Figure 8.**Comparison of the global acceleration of point 4 of the wooden plate, obtained by three methods: from the optical system (red); from the inertial system with the orientation provided by the inertial system (blue); from the inertial system with the orientation provided by the optical system (black).

**Figure 9.**Accelerations obtained from the optical system with Vaughan’s method, for cutoff frequencies of 6 Hz (black), 12 Hz (blue) and 40 Hz (cyan), respectively, vs. accelerations measured by the IMUs (red).

**Figure 10.**Detail of accelerations at the left foot obtained from the optical system with Vaughan’s method, for cutoff frequencies of 6 Hz (black), 12 Hz (blue) and 40 Hz (cyan), respectively, vs. accelerations measured by the IMUs (red).

**Figure 11.**Accelerations obtained from the optical system with the EKF for combined process noise variances and cutoff frequencies of 0.1/30 Hz (black), 1/20 Hz (blue) and 50/6 Hz (cyan), respectively, vs. accelerations measured by the IMUs (red).

**Figure 12.**Detail of accelerations at the left foot obtained from the optical system with the EKF-based method for combined process noise standard deviations and cutoff frequencies of 0.1/20 Hz (black), 1/15 Hz (blue) and 50/6 Hz (cyan), respectively, vs. accelerations measured by the IMUs (red).

**Table 1.**RMSE of the accelerations obtained from the optical system through Vaughan’s method with different cutoff frequencies, with respect to the accelerations measured by the IMUs, taken as reference. The row with the lowest mean RMSE is highlighted in red.

Cutoff Freq. (Hz) | RMSE (m/s^{2}) | |||||||
---|---|---|---|---|---|---|---|---|

Pelvis | R Thigh | L Thigh | R Tibia | L Tibia | R Foot | L Foot | Mean | |

6 | 0.626 | 1.034 | 1.123 | 1.583 | 1.502 | 2.743 | 2.485 | 1.585 |

8 | 0.578 | 1.005 | 1.071 | 1.538 | 1.448 | 2.640 | 2.405 | 1.336 |

10 | 0.559 | 0.997 | 1.043 | 1.515 | 1.428 | 2.571 | 2.362 | 1.309 |

12 | 0.559 | 1.003 | 1.031 | 1.508 | 1.426 | 2.529 | 2.339 | 1.299 |

15 | 0.583 | 1.034 | 1.036 | 1.521 | 1.445 | 2.504 | 2.328 | 1.306 |

20 | 0.680 | 1.138 | 1.086 | 1.607 | 1.510 | 2.526 | 2.354 | 1.363 |

25 | 0.840 | 1.303 | 1.181 | 1.771 | 1.602 | 2.590 | 2.422 | 1.464 |

30 | 1.045 | 1.517 | 1.314 | 2.002 | 1.706 | 2.679 | 2.526 | 1.599 |

40 | 1.531 | 2.035 | 1.654 | 2.602 | 1.931 | 2.897 | 2.815 | 1.933 |

**Table 2.**RMSE of the accelerations obtained from the optical system through the EKF-based method with different combinations of process noise standard deviations and cutoff frequencies, with respect to the accelerations measured by the IMUs, taken as reference. The row with the lowest mean RMSE is highlighted in red.

Acc. Std. (m/s^{2} or rad/s^{2}) | Cutoff Freq. (Hz) | RMSE (m/s^{2}) | |||||||
---|---|---|---|---|---|---|---|---|---|

Pelvis | R Thigh | L Thigh | R Tibia | L Tibia | R Foot | L Foot | Mean | ||

0.1 | 6 | 0.717 | 1.047 | 1.195 | 1.663 | 1.624 | 2.649 | 2.430 | 1.618 |

0.1 | 10 | 0.679 | 1.020 | 1.155 | 1.679 | 1.618 | 2.530 | 2.338 | 1.377 |

0.1 | 15 | 0.676 | 1.018 | 1.136 | 1.678 | 1.614 | 2.433 | 2.274 | 1.354 |

0.1 | 20 | 0.682 | 1.022 | 1.133 | 1.674 | 1.611 | 2.377 | 2.229 | 1.341 |

0.1 | 25 | 0.690 | 1.028 | 1.135 | 1.672 | 1.612 | 2.341 | 2.198 | 1.335 |

0.1 | 30 | 0.698 | 1.035 | 1.140 | 1.672 | 1.615 | 2.318 | 2.176 | 1.332 |

0.5 | 6 | 0.606 | 1.011 | 1.108 | 1.513 | 1.442 | 2.553 | 2.274 | 1.313 |

0.5 | 10 | 0.545 | 0.969 | 1.030 | 1.514 | 1.412 | 2.340 | 2.105 | 1.239 |

0.5 | 15 | 0.557 | 0.965 | 1.017 | 1.524 | 1.415 | 2.183 | 2.004 | 1.208 |

0.5 | 20 | 0.582 | 0.977 | 1.034 | 1.530 | 1.424 | 2.098 | 1.943 | 1.198 |

0.5 | 25 | 0.608 | 0.996 | 1.058 | 1.538 | 1.439 | 2.049 | 1.907 | 1.199 |

0.5 | 30 | 0.634 | 1.018 | 1.083 | 1.553 | 1.458 | 2.018 | 1.887 | 1.207 |

1 | 6 | 0.604 | 1.006 | 1.099 | 1.484 | 1.433 | 2.565 | 2.267 | 1.307 |

1 | 10 | 0.538 | 0.963 | 1.026 | 1.454 | 1.383 | 2.330 | 2.072 | 1.221 |

1 | 15 | 0.562 | 0.961 | 1.033 | 1.457 | 1.379 | 2.161 | 1.955 | 1.188 |

1 | 20 | 0.601 | 0.978 | 1.067 | 1.466 | 1.393 | 2.074 | 1.890 | 1.183 |

1 | 25 | 0.639 | 1.004 | 1.105 | 1.483 | 1.417 | 2.026 | 1.857 | 1.191 |

1 | 30 | 0.679 | 1.036 | 1.143 | 1.511 | 1.446 | 1.999 | 1.843 | 1.207 |

10 | 6 | 0.632 | 0.987 | 1.132 | 1.473 | 1.439 | 2.580 | 2.324 | 1.321 |

10 | 10 | 0.576 | 0.926 | 1.086 | 1.393 | 1.343 | 2.344 | 2.131 | 1.225 |

10 | 15 | 0.606 | 0.928 | 1.115 | 1.374 | 1.300 | 2.174 | 2.002 | 1.187 |

10 | 20 | 0.665 | 0.973 | 1.168 | 1.397 | 1.307 | 2.084 | 1.934 | 1.191 |

10 | 25 | 0.739 | 1.041 | 1.228 | 1.452 | 1.342 | 2.037 | 1.909 | 1.218 |

10 | 30 | 0.823 | 1.124 | 1.292 | 1.538 | 1.391 | 2.014 | 1.913 | 1.262 |

50 | 6 | 0.633 | 0.989 | 1.134 | 1.478 | 1.442 | 2.578 | 2.330 | 1.323 |

50 | 10 | 0.574 | 0.933 | 1.088 | 1.401 | 1.348 | 2.343 | 2.142 | 1.229 |

50 | 15 | 0.603 | 0.943 | 1.119 | 1.386 | 1.307 | 2.177 | 2.018 | 1.194 |

50 | 20 | 0.666 | 1.000 | 1.179 | 1.424 | 1.319 | 2.091 | 1.955 | 1.204 |

50 | 25 | 0.753 | 1.085 | 1.253 | 1.507 | 1.359 | 2.047 | 1.937 | 1.243 |

50 | 30 | 0.859 | 1.190 | 1.336 | 1.632 | 1.416 | 2.032 | 1.953 | 1.302 |

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**MDPI and ACS Style**

Cuadrado, J.; Michaud, F.; Lugrís, U.; Pérez Soto, M.
Using Accelerometer Data to Tune the Parameters of an Extended Kalman Filter for Optical Motion Capture: Preliminary Application to Gait Analysis. *Sensors* **2021**, *21*, 427.
https://doi.org/10.3390/s21020427

**AMA Style**

Cuadrado J, Michaud F, Lugrís U, Pérez Soto M.
Using Accelerometer Data to Tune the Parameters of an Extended Kalman Filter for Optical Motion Capture: Preliminary Application to Gait Analysis. *Sensors*. 2021; 21(2):427.
https://doi.org/10.3390/s21020427

**Chicago/Turabian Style**

Cuadrado, Javier, Florian Michaud, Urbano Lugrís, and Manuel Pérez Soto.
2021. "Using Accelerometer Data to Tune the Parameters of an Extended Kalman Filter for Optical Motion Capture: Preliminary Application to Gait Analysis" *Sensors* 21, no. 2: 427.
https://doi.org/10.3390/s21020427