# Continuous Motion Estimation of Knee Joint Based on a Parameter Self-Updating Mechanism Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A self-adaptive optimized DBN, depending on the original sEMG signals of different subjects, was built to complete the reconstruction of sEMG sequences.
- An adaptive regression model fused with BPNN was established to achieve the optimal estimation of continuous joint angle.
- A parameter self-updating mechanism was applied to update the model parameters using a small amount of data from new subjects to satisfy personalized demand.

## 2. Materials and Methods

#### 2.1. Data Acquisition and Pre-Processing

^{TM}, Delsys Corporation, Natick, MA, USA) were used to acquire raw sEMG signals with a frequency of 1928 Hz. The layout scheme is shown in Figure 2. The optimal location for the sensors is on the midline of the muscular abdomen between the nearest innervated area and the tendon junction [30,31,32,33]. At this position, the sEMG signals with the greatest amplitude are obtained [33,34].

_{A}= [x

_{A}, y

_{A}, z

_{A}], P

_{B}= [x

_{B}, y

_{B}, z

_{B}], P

_{C}= [x

_{C}, y

_{C}, z

_{C}]. Therefore, the formula for calculating the angle of the knee joint is:

_{knee}is the angle between the extension line of the thigh chain l

_{1}and the shank chain l

_{2}, l

_{1}= [x

_{A}− x

_{B}, y

_{A}− y

_{B}, z

_{A}− z

_{B}], l

_{2}= [x

_{C}− x

_{B}, y

_{C}− y

_{B}, z

_{C}− z

_{B}].

_{i}is the angle value of the ith sampling, and N signifies the number of sampling points, here N = 10.

#### 2.2. Feature Reconstruction by DBN

_{1}, v

_{2}, …, v

_{m})

^{T}represents the normalized multi-dimensional vector of the input layer. m is the number of neurons in the input layer. w

_{ij}is the weight matrix. h = (h

_{1}, h

_{2}, …, h

_{n})

^{T}denotes the multi-dimensional vector of the hidden layer. n is the number of neurons in the hidden layer. The concept and training process of RBM are presented below.

#### 2.3. DBN Adaptive Optimization Fused with the PSO Algorithm

_{1}and c

_{2}denote learning factors, and r is a random number between 0 and 1. V

_{i}

^{k}, X

_{i}

^{k}, pbest

_{i}

^{k}, and gbest

_{i}

^{k}are the velocity, position, local optimum, and global optimum of particle i at the kth iteration, respectively. The linear decreasing weight method was applied to ensure that the PSO algorithm had a good ability to balance search and development.

_{max}and w

_{max}refer to the maximum and minimum values of inertia weights, k is the current number of iterations, and k

_{max}is the maximum number of iterations.

#### 2.4. Construction of the Adaptive Regression Model

_{in}represents the weight matrix of the hidden layer, the W

_{out}denotes the weight matrix of the hidden layer, and b

_{in}and b

_{out}are threshold vectors. We used the test sets to verify the results when all the parameters were updated.

#### 2.5. Result Evaluation Indicators

_{knee}could be estimated using the sEMG signals. This paper specified three metrics to evaluate the quantitative difference between actual and estimated values in all regression models: root mean square error (RMSE), correlation coefficient (CC), and R

^{2}score.

## 3. Experiments and Results

#### 3.1. Subjects

#### 3.2. Experimental Procedure

#### 3.3. Model Training

_{1}, v

_{2}, v

_{3}, v

_{4}]

^{T}and the “real” angle data θ

_{knee}.

_{max}= 10, the learning factors C

_{1}and C

_{2}were 0.9 and 0.5, the random number was 1, w

_{max}and w

_{min}were 0.9 and 0.5, respectively. The number of hidden layer nodes in DBN had to be considered for the performance and the computational cost of network training. The minimum error and the lowest computational load parameters were chosen. We explored the impact of the range of hidden layer nodes on the estimation results. Figure 7 shows the RMSEs under different ranges of hidden layer nodes. The number of hidden layer nodes was between 1 and the maximum number of hidden layer nodes (hmax). The best performance was achieved when hmax was set to 50. However, when selecting other hmax values, relatively large errors would present due to the network not achieving optimal performance.

_{knee}measured using Vicon is used as the training output. The maximum number of iterations is 1000, and the learning rate is 0.01.

_{knee}were the output. The number of BPNN input layer nodes was 3. Figure 9 illustrates the trained BPNN structure. Y is the input sEMG feature vectors, θ signifies the output estimate angle, and W, b represents the weight and bias, respectively. The numbers of neurons are 3, 12, and 1. Figure 10 shows the trained normal BPNN. The input of the normal BPNN is the RMS features of four-channel sEMG signals and the actual angle values θ

_{knee}denote the output. The numbers of neurons are 4, 12, and 1.

#### 3.4. Comparison of Estimated Results

^{2}through (16), (17), and (18). Table 2 lists the means and standard deviations computed by fifteen tests. As can be seen in Table 2, our method presented significant advantages over the other two models. The most accurate results were obtained for squatting in S1 and knee flexion/extension in S3, representing the optimal performance achieved by our method under the same conditions. The average RMSEs of our method between the estimated results and the actual values were 9.42 ± 0.31° and 7.36 ± 0.25°, indicating that the error between the actual and estimated values was tiny. And the RMSEs of our method were significantly lower than 10.54 ± 1.16° (p = 0.001), 8.64 ± 0.61° (p < 0.005) of DBN-BP and 14.14 ± 1.56° (p < 10

^{−4}), 9.60 ± 0.86° (p < 10

^{−7}) of BPNN. In addition, the average CCs of our method were 0.96 ± 0.01 and 0.94 ± 0.01, which were numerically closer to 1. The average R

^{2}scores are 0.92 ± 0.01 and 0.90 ± 0.01, indicating a good match between the curves generated by the estimation results and the measured curves. However, the CCs and R

^{2}scores of the other two networks both indicated that their errors were much larger than our method.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**(

**a**) The RMSEs of squat with respect to different ranges of hidden nodes; (

**b**) the RMSEs of knee flex/ext with respect to different ranges of hidden nodes.

**Figure 8.**(

**a**) The RMSEs for PSO-DBN trained by different epochs in squat; (

**b**) the RMSEs for PSO-DBN trained by different epochs in knee flex/ext.

**Figure 11.**(

**a**,

**c**,

**e**) Estimated results of the squat based on PSO-DBN-BP, DBN-BP, and BPNN. (

**b**,

**d**,

**f**) Estimated results of the knee flex/ext based on PSO-DBN-BP, DBN-BP, and BPNN.

**Figure 12.**(

**a**) The new five-dimensional reconstructed features for four-channel sEMG signals by using PSO-DBN. (

**b**) The three-dimensional features by using DBN for dimensionality reduction.

Subject | Squat | Knee Flex/Ext | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Number of Neurons in Each Layer | Time/s | Number of Neurons in Each Layer | Time/s | |||||||

S1 | 37 | 26 | 16 | 5 | 31.7 | 8 | 48 | 11 | 44 | 40.4 |

S2 | 34 | 32 | 29 | 35 | 43.4 | 34 | 41 | 14 | 17 | 29.3 |

S3 | 2 | 41 | 48 | 32 | 36.6 | 38 | 2 | 42 | 49 | 42.0 |

S4 | 5 | 40 | 25 | 34 | 39.0 | 18 | 14 | 40 | 19 | 34.1 |

S5 | 2 | 49 | 46 | 42 | 41.1 | 17 | 28 | 37 | 32 | 41.4 |

Subject | Model | Squat | Knee Flex/Ext | ||||
---|---|---|---|---|---|---|---|

RMSE | CC | R^{2} | RMSE | CC | R^{2} | ||

1 | PSO-DBN-BP | 5.57 ± 0.44 | 0.99 ± 0.01 | 0.97 ± 0.01 | 8.71 ± 0.16 | 0.93 ± 0.01 | 0.88 ± 0.01 |

DBN-BP | 6.37 ± 1.04 | 0.98 ± 0.01 | 0.96 ± 0.01 | 9.09 ± 0.36 | 0.92 ± 0.01 | 0.85 ± 0.01 | |

BPNN | 8.79 ± 2.71 | 0.94 ± 0.04 | 0.89 ± 0.07 | 9.84 ± 1.22 | 0.89 ± 0.03 | 0.79 ± 0.05 | |

2 | PSO-DBN-BP | 9.96 ± 0.51 | 0.95 ± 0.01 | 0.90 ± 0.01 | 8.34 ± 0.17 | 0.93 ± 0.01 | 0.88 ± 0.01 |

DBN-BP | 11.72 ± 1.28 | 0.93 ± 0.01 | 0.86 ± 0.02 | 12.06 ± 0.41 | 0.90 ± 0.01 | 0.84 ± 0.02 | |

BPNN | 12.90 ± 1.45 | 0.90 ± 0.01 | 0.81 ± 0.04 | 12.64 ± 0.47 | 0.88 ± 0.02 | 0.81 ± 0.02 | |

3 | PSO-DBN-BP | 11.82 ± 0.10 | 0.95 ± 0.01 | 0.90 ± 0.01 | 6.09 ± 0.34 | 0.96 ± 0.01 | 0.93 ± 0.01 |

DBN-BP | 13.53 ± 2.61 | 0.93 ± 0.03 | 0.85 ± 0.06 | 7.21 ± 1.18 | 0.95 ± 0.02 | 0.89 ± 0.04 | |

BPNN | 21.71 ± 1.40 | 0.81 ± 0.02 | 0.66 ± 0.03 | 8.77 ± 0.99 | 0.92 ± 0.02 | 0.85 ± 0.04 | |

4 | PSO-DBN-BP | 10.49 ± 0.15 | 0.95 ± 0.01 | 0.90 ± 0.01 | 5.87 ± 0.34 | 0.97 ± 0.01 | 0.95 ± 0.01 |

DBN-BP | 10.86 ± 0.24 | 0.94 ± 0.01 | 0.89 ± 0.01 | 6.75 ± 1.30 | 0.95 ± 0.02 | 0.91 ± 0.04 | |

BPNN | 12.32 ± 0.99 | 0.93 ± 0.01 | 0.87 ± 0.01 | 8.12 ± 0.86 | 0.93 ± 0.02 | 0.87 ± 0.03 | |

5 | PSO-DBN-BP | 9.29 ± 0.33 | 0.96 ± 0.01 | 0.93 ± 0.01 | 7.77 ± 0.23 | 0.93 ± 0.01 | 0.86 ± 0.01 |

DBN-BP | 10.22 ± 0.62 | 0.95 ± 0.01 | 0.91 ± 0.02 | 8.07 ± 0.39 | 0.91 ± 0.01 | 0.84 ± 0.01 | |

BPNN | 14.97 ± 1.24 | 0.88 ± 0.02 | 0.78 ± 0.04 | 8.64 ± 0.75 | 0.90 ± 0.02 | 0.82 ± 0.04 | |

Overall | PSO-DBN-BP | 9.42 ± 0.31 | 0.96 ± 0.01 | 0.92 ± 0.01 | 7.36 ± 0.25 | 0.94 ± 0.01 | 0.90 ± 0.01 |

DBN-BP | 10.54 ± 1.16 | 0.95 ± 0.01 | 0.89 ± 0.02 | 8.64 ± 0.61 | 0.93 ± 0.01 | 0.87 ± 0.02 | |

BPNN | 14.14 ± 1.56 | 0.89 ± 0.02 | 0.80 ± 0.04 | 9.60 ± 0.86 | 0.90 ± 0.02 | 0.83 ± 0.04 |

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Li, K.; Zhang, J.; Cao, J.
Continuous Motion Estimation of Knee Joint Based on a Parameter Self-Updating Mechanism Model. *Bioengineering* **2023**, *10*, 1028.
https://doi.org/10.3390/bioengineering10091028

**AMA Style**

Li J, Li K, Zhang J, Cao J.
Continuous Motion Estimation of Knee Joint Based on a Parameter Self-Updating Mechanism Model. *Bioengineering*. 2023; 10(9):1028.
https://doi.org/10.3390/bioengineering10091028

**Chicago/Turabian Style**

Li, Jiayi, Kexiang Li, Jianhua Zhang, and Jian Cao.
2023. "Continuous Motion Estimation of Knee Joint Based on a Parameter Self-Updating Mechanism Model" *Bioengineering* 10, no. 9: 1028.
https://doi.org/10.3390/bioengineering10091028