# A Deep Learning Approach for Foot Trajectory Estimation in Gait Analysis Using Inertial Sensors

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.2. Reference Trajectories and Gait Parameters

#### 2.3. Data Preparation

#### 2.4. Network Architecture

#### 2.5. Training and Hyperparameters Optimization

^{®}V100 GPU with 16GB of memory.

#### 2.6. Model Performance Evaluation

#### 2.7. Gait Parameters from Predictions

#### 2.8. Instrument Comparison and Validation

#### 2.9. Comparison with the Conventional Gait Analysis Approach

## 3. Results

#### 3.1. Network Training and Trajectories Estimation

#### 3.2. Gait Parameters Using the Deep Learning Approach

#### 3.3. Robustness to Changes in Orientation

#### 3.4. Comparison with the Conventional Gait Analysis Approach

## 4. Discussion

#### 4.1. Problem Formulation

#### 4.2. Deep Learning-Based Gait Analysis Performance

#### 4.3. Comparison with the Conventional Gait Analysis Approach

#### 4.4. Framing our Method within the State-of-the-Art

#### 4.5. Limitations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Definition of spatial gait parameters based on heel and toe trajectories: (

**a**) Stride length and swing width. (

**b**) Minimum toe clearance (MTC).

**Figure 4.**Data analysis process diagram: data preparation, network training, and model performance evaluation.

**Figure 6.**Data analysis process diagram: estimating gait parameters from predicted heel and toe trajectories.

**Figure 7.**Events detection from acceleration magnitude (obtained from the inertial sensor data) and vertical acceleration (obtained from predicted toe trajectories). FO—foot off; FC—foot contact; MS—mid-stance; Ref-FO—reference FO; Ref-FC—reference FC.

**Figure 8.**Mean squared error (loss) achieved while training the model with the best selection of hyperparameters. Loss was calculated from normalized trajectories.

**Figure 9.**An example of predicted (full lines) and reference (dashed lines) trajectories, obtained from a sample in the test set. Heel and toe trajectories are shown in a different color.

Partition | Number of Subjects | Number of Strides |
---|---|---|

Training set (60%) | 14 | 31,470 |

Validation set (20%) | 6 | 13,872 |

Test set (20%) | 6 | 12,474 |

Parameter | Configuration Space | Optimal Configuration |
---|---|---|

Number of Units | 32–160 | 160 |

Dropout | 0–0.30 | 0.1 |

Learning Rate | $1\times {10}^{-6}$–$1\times {10}^{-2}$ | $1\times {10}^{-3}$ |

Batch Size | 100–400 | 100 |

Metric | Validation Set | Test Set | ||
---|---|---|---|---|

Heel Traj. | Toe Traj. | Heel Traj. | Toe Traj. | |

MSE (mm^{2}) | $538.0$ | $534.6$ | $1043.7$ | $1087.7$ |

MAE (mm) | $7.6$ | $7.0$ | $8.6$ | $8.2$ |

RMSE (mm) | $14.2$ | $13.3$ | $17.2$ | $16.8$ |

Eucl. Dist. (mm) | $17.9$ | $16.6$ | $20.2$ | $19.5$ |

**Table 4.**Performance on the test set, including turns (n = 2068 strides). Shown are mean values (standard deviation), limits of agreement, RMSE, correlation, and equivalence interval (p-value).

^{†}All correlations were based on Spearman and have $p<0.01$.

^{‡}Equivalence tests were based on Wilcoxon signed-rank test.

Parameter | IMU | VICON | Rel. Error | Abs. Error | Lim. Agr. | RMSE | Corr. ^{†} | Equival. ^{‡} |
---|---|---|---|---|---|---|---|---|

Stride dur. (s) | $1.15\phantom{\rule{4pt}{0ex}}\left(0.20\right)$ | $1.15\phantom{\rule{4pt}{0ex}}\left(0.20\right)$ | $0.00\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $[-0.08,0.08]$ | $0.04$ | $0.99$ | $\pm 0.06\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Swing dur. (s) | $0.40\phantom{\rule{4pt}{0ex}}\left(0.07\right)$ | $0.39\phantom{\rule{4pt}{0ex}}\left(0.06\right)$ | $0.00\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.03\right)$ | $[-0.07,0.08]$ | $0.04$ | $0.88$ | $\pm 0.02\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Stance dur. (s) | $0.75\phantom{\rule{4pt}{0ex}}\left(0.15\right)$ | $0.76\phantom{\rule{4pt}{0ex}}\left(0.16\right)$ | $-0.01\phantom{\rule{4pt}{0ex}}\left(0.03\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $[-0.06,0.05]$ | $0.03$ | $0.98$ | $\pm 0.04\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Cad. (st/min) | $107.5\phantom{\rule{4pt}{0ex}}\left(17.4\right)$ | $107.5\phantom{\rule{4pt}{0ex}}\left(17.8\right)$ | $0.1\phantom{\rule{4pt}{0ex}}\left(4.7\right)$ | $1.8\phantom{\rule{4pt}{0ex}}\left(4.4\right)$ | $[-9.2,9.4]$ | $4.7$ | $0.99$ | $\pm 5.37\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

SL (cm) | $126.7\phantom{\rule{4pt}{0ex}}\left(27.3\right)$ | $129.4\phantom{\rule{4pt}{0ex}}\left(28.4\right)$ | $-2.6\phantom{\rule{4pt}{0ex}}\left(10.0\right)$ | $5.8\phantom{\rule{4pt}{0ex}}\left(8.6\right)$ | $[-22.3,17.0]$ | $10.4$ | $0.94$ | $\pm 6.47\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Speed (cm/s) | $115.9\phantom{\rule{4pt}{0ex}}\left(39.3\right)$ | $118.0\phantom{\rule{4pt}{0ex}}\left(40.0\right)$ | $-2.2\phantom{\rule{4pt}{0ex}}\left(10.5\right)$ | $5.8\phantom{\rule{4pt}{0ex}}\left(9.0\right)$ | $[-22.7,18.4]$ | $10.7$ | $0.97$ | $\pm 5.90\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

SW (cm) | $9.4\phantom{\rule{4pt}{0ex}}\left(10.3\right)$ | $10.4\phantom{\rule{4pt}{0ex}}\left(10.8\right)$ | $-1.0\phantom{\rule{4pt}{0ex}}\left(5.8\right)$ | $2.6\phantom{\rule{4pt}{0ex}}\left(5.3\right)$ | $[-12.3,10.3]$ | $5.9$ | $0.88$ | $\pm 0.52\phantom{\rule{0.277778em}{0ex}}\left(1.0\right)$ |

MTC (cm) | $1.7\phantom{\rule{4pt}{0ex}}\left(0.4\right)$ | $1.9\phantom{\rule{4pt}{0ex}}\left(0.8\right)$ | $-0.2\phantom{\rule{4pt}{0ex}}\left(0.8\right)$ | $0.6\phantom{\rule{4pt}{0ex}}\left(0.6\right)$ | $[-1.9,1.5]$ | $0.9$ | $0.20$ | $\pm 0.10\phantom{\rule{0.277778em}{0ex}}\left(0.3\right)$ |

**Table 5.**Performance on the test set, excluding turns (n = 1108 strides). Shown are mean values (standard deviation), limits of agreement, RMSE, correlation, and equivalence interval (p-value).

^{†}All correlations were based on Spearman and have $p<0.05$.

^{‡}Equivalence tests were based on Wilcoxon signed-rank test.

Parameter | IMU | VICON | Rel. Error | Abs. Error | Lim. Agr. | RMSE | Corr. ^{†} | Equival. ^{‡} |
---|---|---|---|---|---|---|---|---|

Stride dur. (s) | $1.14\phantom{\rule{4pt}{0ex}}\left(0.21\right)$ | $1.15\phantom{\rule{4pt}{0ex}}\left(0.21\right)$ | $0.00\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.01\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $[-0.08,0.08]$ | $0.04$ | $0.99$ | $\pm 0.06\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Swing dur. (s) | $0.39\phantom{\rule{4pt}{0ex}}\left(0.07\right)$ | $0.39\phantom{\rule{4pt}{0ex}}\left(0.05\right)$ | $0.00\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $[-0.08,0.08]$ | $0.04$ | $0.92$ | $\pm 0.02\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Stance dur. (s) | $0.75\phantom{\rule{4pt}{0ex}}\left(0.16\right)$ | $0.76\phantom{\rule{4pt}{0ex}}\left(0.16\right)$ | $-0.01\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $[-0.05,0.04]$ | $0.02$ | $0.99$ | $\pm 0.04\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Cad. (st/min) | $107.9\phantom{\rule{4pt}{0ex}}\left(17.5\right)$ | $107.7\phantom{\rule{4pt}{0ex}}\left(17.4\right)$ | $0.2\phantom{\rule{4pt}{0ex}}\left(2.8\right)$ | $1.3\phantom{\rule{4pt}{0ex}}\left(2.5\right)$ | $[-5.3,5.6]$ | $2.8$ | $0.99$ | $\pm 5.39\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

SL (cm) | $136.1\phantom{\rule{4pt}{0ex}}\left(23.3\right)$ | $138.8\phantom{\rule{4pt}{0ex}}\left(24.6\right)$ | $-2.6\phantom{\rule{4pt}{0ex}}\left(5.4\right)$ | $3.8\phantom{\rule{4pt}{0ex}}\left(4.7\right)$ | $[-13.3,8.0]$ | $6.1$ | $0.98$ | $\pm 6.94\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

Speed (cm/s) | $124.8\phantom{\rule{4pt}{0ex}}\left(38.2\right)$ | $127.1\phantom{\rule{4pt}{0ex}}\left(39.3\right)$ | $-2.2\phantom{\rule{4pt}{0ex}}\left(7.0\right)$ | $3.9\phantom{\rule{4pt}{0ex}}\left(6.2\right)$ | $[-15.9,11.4]$ | $7.3$ | $0.99$ | $\pm 6.35\phantom{\rule{0.277778em}{0ex}}\left(0.0\right)$ |

SW (cm) | $2.7\phantom{\rule{4pt}{0ex}}\left(2.2\right)$ | $3.5\phantom{\rule{4pt}{0ex}}\left(2.1\right)$ | $-0.8\phantom{\rule{4pt}{0ex}}\left(1.8\right)$ | $1.3\phantom{\rule{4pt}{0ex}}\left(1.5\right)$ | $[-4.3,2.7]$ | $2.0$ | $0.76$ | $\pm 0.17\phantom{\rule{0.277778em}{0ex}}\left(1.0\right)$ |

MTC (cm) | $1.6\phantom{\rule{4pt}{0ex}}\left(0.3\right)$ | $1.8\phantom{\rule{4pt}{0ex}}\left(0.7\right)$ | $-0.2\phantom{\rule{4pt}{0ex}}\left(0.8\right)$ | $0.5\phantom{\rule{4pt}{0ex}}\left(0.6\right)$ | $[-1.7,1.3]$ | $0.8$ | $0.08$ | $\pm 0.09\phantom{\rule{0.277778em}{0ex}}\left(0.2\right)$ |

**Table 6.**Random orientation simulation results (n = 2139 strides). Shown are mean values (standard deviation), RMSD, Correlation—with Spearman or Pearson—, and equivalence interval (p-value).

^{†}All correlations have p < 0.01.

^{‡}All equivalence tests are based on Wilcoxon signed-rank test.

Parameter | Original Orientation | Random Orientation | RMSD | Correlation ^{†} | Equivalence ^{‡} |
---|---|---|---|---|---|

Stride dur. (s) | $1.17\phantom{\rule{4pt}{0ex}}\left(0.27\right)$ | $1.17\phantom{\rule{4pt}{0ex}}\left(0.27\right)$ | $0.03$ | ${r}_{s}=0.99$ | $\pm 0.06\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

Swing dur. (s) | $0.40\phantom{\rule{4pt}{0ex}}\left(0.07\right)$ | $0.40\phantom{\rule{4pt}{0ex}}\left(0.06\right)$ | $0.03$ | ${r}_{s}=0.92$ | $\pm 0.02\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

Stance dur. (s) | $0.77\phantom{\rule{4pt}{0ex}}\left(0.23\right)$ | $0.77\phantom{\rule{4pt}{0ex}}\left(0.23\right)$ | $0.02$ | ${r}_{s}=0.99$ | $\pm 0.04\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

Cad. (st/min) | $106.4\phantom{\rule{4pt}{0ex}}\left(18.5\right)$ | $106.4\phantom{\rule{4pt}{0ex}}\left(18.4\right)$ | $2.5$ | ${r}_{s}=0.99$ | $\pm 5.32\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

SL (cm) | $126.1\phantom{\rule{4pt}{0ex}}\left(27.4\right)$ | $126.8\phantom{\rule{4pt}{0ex}}\left(26.3\right)$ | $5.7$ | ${r}_{s}=0.98$ | $\pm 6.34\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

Speed (cm/s) | $114.3\phantom{\rule{4pt}{0ex}}\left(40.0\right)$ | $114.7\phantom{\rule{4pt}{0ex}}\left(38.5\right)$ | $6.5$ | ${r}_{s}=0.99$ | $\pm 5.73\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

SW (cm) | $9.3\phantom{\rule{4pt}{0ex}}\left(10.3\right)$ | $9.5\phantom{\rule{4pt}{0ex}}\left(10.2\right)$ | $2.3$ | ${r}_{s}=0.94$ | $\pm 0.47\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

MTC (cm) | $1.7\phantom{\rule{4pt}{0ex}}\left(0.4\right)$ | $1.6\phantom{\rule{4pt}{0ex}}\left(0.4\right)$ | $0.3$ | ${r}_{s}=0.78$ | $\pm 0.08\phantom{\rule{4pt}{0ex}}\left(0.0\right)$ |

**Table 7.**Comparison of the performance achieved with the deep learning approach and the conventional approach. Shown are mean absolute errors (standard deviation), reported for the samples in the test set.

Parameter | Including Turns | Excluding Turns | ||
---|---|---|---|---|

Deep Learning Approach | Conventional Approach | Deep Learning Approach | Conventional Approach | |

Trajectories (mm) | $8.7\phantom{\rule{4pt}{0ex}}\left(14.7\right)$ (heel) $8.7\phantom{\rule{4pt}{0ex}}\left(13.2\right)$ (toe) | $9.8\phantom{\rule{4pt}{0ex}}\left(12.7\right)$ (sensor) | $7.4\phantom{\rule{4pt}{0ex}}\left(16.3\right)$ (heel) $7.5\phantom{\rule{4pt}{0ex}}\left(14.8\right)$ (toe) | $8.6\phantom{\rule{4pt}{0ex}}\left(13.0\right)$ (sensor) |

Stride dur. (s) | $0.02\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.05\right)$ | $0.01\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.05\right)$ |

Swing dur. (s) | $0.02\phantom{\rule{4pt}{0ex}}\left(0.03\right)$ | $0.03\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.04\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.05\right)$ |

Stance dur. (s) | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ | $0.02\phantom{\rule{4pt}{0ex}}\left(0.02\right)$ |

Cad. (st/min) | $1.8\phantom{\rule{4pt}{0ex}}\left(4.4\right)$ | $1.9\phantom{\rule{4pt}{0ex}}\left(4.8\right)$ | $1.3\phantom{\rule{4pt}{0ex}}\left(2.5\right)$ | $1.4\phantom{\rule{4pt}{0ex}}\left(2.9\right)$ |

SL (cm) | $5.8\phantom{\rule{4pt}{0ex}}\left(8.6\right)$ | $10.1\phantom{\rule{4pt}{0ex}}\left(10.9\right)$ | $3.8\phantom{\rule{4pt}{0ex}}\left(4.7\right)$ | $8.1\phantom{\rule{4pt}{0ex}}\left(8.1\right)$ |

Speed (cm/s) | $5.8\phantom{\rule{4pt}{0ex}}\left(9.0\right)$ | $9.4\phantom{\rule{4pt}{0ex}}\left(11.2\right)$ | $3.9\phantom{\rule{4pt}{0ex}}\left(6.2\right)$ | $7.6\phantom{\rule{4pt}{0ex}}\left(9.6\right)$ |

SW (cm) | $2.6\phantom{\rule{4pt}{0ex}}\left(5.3\right)$ | $3.5\phantom{\rule{4pt}{0ex}}\left(6.2\right)$ | $1.3\phantom{\rule{4pt}{0ex}}\left(1.5\right)$ | $1.9\phantom{\rule{4pt}{0ex}}\left(4.2\right)$ |

MTC (cm) | $0.6\phantom{\rule{4pt}{0ex}}\left(0.6\right)$ | $1.5\phantom{\rule{4pt}{0ex}}\left(1.0\right)$ | $0.5\phantom{\rule{4pt}{0ex}}\left(0.6\right)$ | $1.4\phantom{\rule{4pt}{0ex}}\left(0.8\right)$ |

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**MDPI and ACS Style**

Guimarães, V.; Sousa, I.; Correia, M.V. A Deep Learning Approach for Foot Trajectory Estimation in Gait Analysis Using Inertial Sensors. *Sensors* **2021**, *21*, 7517.
https://doi.org/10.3390/s21227517

**AMA Style**

Guimarães V, Sousa I, Correia MV. A Deep Learning Approach for Foot Trajectory Estimation in Gait Analysis Using Inertial Sensors. *Sensors*. 2021; 21(22):7517.
https://doi.org/10.3390/s21227517

**Chicago/Turabian Style**

Guimarães, Vânia, Inês Sousa, and Miguel Velhote Correia. 2021. "A Deep Learning Approach for Foot Trajectory Estimation in Gait Analysis Using Inertial Sensors" *Sensors* 21, no. 22: 7517.
https://doi.org/10.3390/s21227517