# Estimation of Foot Trajectory and Stride Length during Level Ground Running Using Foot-Mounted Inertial Measurement Units

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Collection and Processing

#### 2.2. Foot Trajectory Estimation

#### 2.2.1. Spatial Error Correcting (SEC) Method

_{i}and MS

_{i}

_{+1}). The process for estimation of foot trajectory from IMU data is shown in Figure 4. Estimation of the foot trajectory starts with first estimating the orientation of the IMU. In this study, we used a quaternion to represent the IMU orientation. Quaternions are defined using four parameters, one defining the angle of rotation and three defining the axis of rotation. The differential of a quaternion is obtained as follows:

_{x}, ω

_{y}, and ω

_{z}are the angular velocity of the IMU in the global coordinate system. Because an IMU measures the acceleration and angular velocity data in the local coordinate system fixed to the IMU, the data must be transformed into the global coordinate system to calculate foot trajectory. For this transformation, the roll (ϕ) and a temporary pitch angles (θ) of the IMU at MS

_{i}were calculated as follows:

_{x}, a

_{y}, and a

_{z}are three components of the measured acceleration at MS

_{i}. Because the yaw angle (ψ) cannot be estimated from the acceleration data, it was temporarily set to zero at this stage. After the MS

_{i}, the orientation of the IMU was calculated using the measured angular velocity until the next MS (MS

_{i}

_{+1}). The measured acceleration was transformed into the global coordinate system using the calculated quaternion, and then the gravitational acceleration was removed by subtracting 9.8 m/s

^{2}from the vertical component of the transformed acceleration. The foot trajectory was then calculated by integration of the gravity removed acceleration as follows (Figure 5a):

_{g}, v

_{g}, and p

_{g}are the gravity removed acceleration, velocity, and position of the foot in the global coordinate system, and the initial velocity (v

_{0}) and position (p

_{0}) are set as zero. Assuming that the runners run straight on the level ground in one cycle, the mediolateral and vertical displacements of the foot at MS

_{i+1}should also be zero. Therefore, the yaw angle was calculated from the position of the foot at MS

_{i+1}as follows:

_{x}and p

_{y}are the mediolateral and anteroposterior positions of the foot at MS

_{i}

_{+1}. Similarly, the pitch angle was recalculated from the position of the foot at MS

_{i+}

_{1}. Because the relationship between the pitch angle and vertical position at MS

_{i}

_{+1}is nonlinear, a gradient descent method was performed to calculate the pitch angle that makes the vertical position at MS

_{i+1}zero with the initial pitch angle defined as follows:

_{x}) and vertical positions (p′

_{x}) at MS

_{i+1}are equal to zero from the SEC calculation method, the stride length was defined as anteroposterior position (p′

_{y}) at MS

_{i+1}.

#### 2.2.2. Linear Dedrifting (LD) Based on the Velocity Method

_{i}was estimated from the measured acceleration (Equations (2) and (3)), and gravity was removed from the acceleration in the global coordinate system. A linear function in all three components.

#### 2.3. Evaluation

## 3. Results

^{2}in the SEC method compared to the LD method. We used Bland–Altman plots to compare estimated stride length and reference stride length (Figure 8). The offsets were −0.14 m with [−0.32, 0.04] 95% limits of agreement for the LD method, and −0.01 m with [−0.13, 0.11] 95% limits of agreement for the SEC method.

## 4. Discussion

_{i+1}) velocity zero. This reduction in the velocity resulted in underestimation of the stride length, especially for higher running speeds.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Visualization of detected initial contact (IC), midstance (MS), and toe-off (TO) events using the acceleration data.

**Figure 4.**Flow chart of the data postprocessing steps for the estimation of foot trajectory from foot-mounted IMU.

**Figure 5.**Example of original and corrected foot trajectory obtained from foot mounted IMU of the velocity was used to make both velocities of foot at MS

_{i}and MS

_{i}

_{+1}zero. The foot trajectory was then calculated from the dedrifted velocity, and the stride length was calculated from the absolute value of the foot position at MS

_{i}

_{+1}.

**Figure 6.**Example of foot trajectories obtained by linear dedrifting (LD) and spatial error correcting (SEC) methods (solid line) with reference (broken line).

**Figure 7.**Relationships of stride lengths between estimated by linear dedrifting (LD) and spatial error correcting (SEC) methods and reference.

**Figure 8.**Bland–Altman plot with mean and difference in stride lengths estimated by linear dedrifting (LD) and spatial error correcting (SEC) methods and reference. Limits of agreement are shown as average difference (dashed line) ± 1.96 standard deviation of the difference (dotted line).

**Figure 9.**Mean and standard deviation of stride length estimated by linear dedrifting (LD) and spatial error correcting methods (SEC) with measured stride length at different speed ranges.

N | Age (Years) | Height (m) | Mass (kg) | |
---|---|---|---|---|

Female | 22 | 40.7 ± 12.6 | 1.59 ± 0.05 | 49.9 ± 4.7 |

Male | 57 | 35.1 ± 13.9 | 1.70 ± 13.9 | 61.3 ± 6.8 |

Running Speed (m/s) | Number of Trials |
---|---|

<2.7 m/s | 87 |

2.7~3.3 m/s | 102 |

3.3~3.9 m/s | 95 |

3.9~4.5 m/s | 101 |

>4.5 m/s | 26 |

**Table 3.**Mean and standard deviation of correlation coefficient and root mean square error between foot trajectories estimated with linear dedrifting (LD) and spatial error correcting (SEC) methods and the reference trajectory.

X | Y | Z | |
---|---|---|---|

Correlation coefficient | |||

LD | 0.06 ± 0.48 | 0.99 ± 0.00 | 0.56 ± 0.27 |

SEC | 0.48 ± 0.44 | 1.00 ± 0.00 | 0.97 ± 0.08 |

Root mean square error (m) | |||

LD | 0.26 ± 0.17 | 0.17 ± 0.11 | 0.24 ± 0.20 |

SEC | 0.06 ± 0.04 | 0.05 ± 0.02 | 0.03 ± 0.01 |

**Table 4.**Mean and standard deviation of error and absolute error of estimated stride length by linear dedrifting (LD) and spatial error correcting method compared to the reference.

LD Method | SEC Method | |||
---|---|---|---|---|

Error | Abs. Error | Error | Abs. Error | |

<3 m/s | −0.05 ± 0.03 * | 0.05 ± 0.03 * | 0.00 ± 0.03 | 0.03 ± 0.02 |

3–4 m/s | −0.06 ± 0.04 * | 0.06 ± 0.04 * | −0.01 ± 0.03 | 0.02 ± 0.02 |

>4 m/s | −0.08 ± 0.04 * | 0.08 ± 0.04 * | −0.01 ± 0.02 | 0.02 ± 0.01 |

**Table 5.**Intraclass correlation coefficient (ICC) and confidence interval (CI) of ICC for stride length estimated by linear dedrifting (LD) method and spatial error correcting (SEC) method at different speed ranges.

LD Method | SEC Method | |||
---|---|---|---|---|

ICC (3, 1) | CI of ICC | ICC (3, 1) | CI of ICC | |

<3 m/s | 0.957 | [0.949–0.963] | 0.952 | [0.944–0.959] |

3–4 m/s | 0.912 | [0.897–0.925] | 0.961 | [0.954–0.967] |

>4 m/s | 0.878 | [0.847–0.903] | 0.968 | [0.959–0.975] |

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**MDPI and ACS Style**

Suzuki, Y.; Hahn, M.E.; Enomoto, Y.
Estimation of Foot Trajectory and Stride Length during Level Ground Running Using Foot-Mounted Inertial Measurement Units. *Sensors* **2022**, *22*, 7129.
https://doi.org/10.3390/s22197129

**AMA Style**

Suzuki Y, Hahn ME, Enomoto Y.
Estimation of Foot Trajectory and Stride Length during Level Ground Running Using Foot-Mounted Inertial Measurement Units. *Sensors*. 2022; 22(19):7129.
https://doi.org/10.3390/s22197129

**Chicago/Turabian Style**

Suzuki, Yuta, Michael E. Hahn, and Yasushi Enomoto.
2022. "Estimation of Foot Trajectory and Stride Length during Level Ground Running Using Foot-Mounted Inertial Measurement Units" *Sensors* 22, no. 19: 7129.
https://doi.org/10.3390/s22197129