# Neural Networks for Muscle Forces Prediction in Cycling

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Techniques

- Definition of a kinematic model to evaluate the position of every segment of the leg involved in the gesture;
- Definition of the inverse dynamics to evaluate the muscular torque for every joint;
- Calculation of the muscular forces through the data obtained with the two previous steps.

#### 2.1. Biomechanical Model Identification

_{c}, and θ

_{p}and calculating the other angles as reported in [15] (θ

_{s}(θ

_{c}, θ

_{p}) and θ

_{g}(θ

_{s}))—ankle, knee and hip joint moments are calculated using the inverse dynamics, using inertial parameters given by literature [34]. Afterwards, these data are used to implement the three equilibrium equations at each joint, involving the following muscles, that represent the minimum set to be involved in the model: (1) Tibialis anterior (TA); (2) Soleus (SO); (3) Gastrocnemius (GA); (4) Vastii (VA); (5) Rectus femoris (RF); (6) Short head of biceps femoris (BFs); (7) Long head of Biceps Femoris (BFl); (8) Iliacus (IL); (9) Gluteus Maximum (GLM). The relation between the muscular moments and the muscular forces at each joint j is given by the equation:

_{j}represents the muscular moment at the j-th joint, N

_{j}is the number of muscles acting on the j-th joint, F

_{i}is the muscular force exerted by the i-th muscle and d

_{ij}is the effective moment arm of the i-th muscle from the j-th joint. The values of muscular moment arms were calculated as a second order function of the joint angle in % of the length of the segment on which muscle belly is located, based on the equation reported in [35]. As the number of equations is not sufficient to calculate muscular force values, these were calculated by minimizing the cost function:

_{i}and F

_{i}

_{max}respectively the physiological cross sectional area and the maximum force value for the i-th muscle, obtained by the literature [34]. The cubic exponent used in the equation (3), guarantees the best tradeoff between the muscular contractile force and the maximum duration of the contraction. This cost function is widely used in literature [36,37] as it relies on the co-activation of all the muscles involved in the gesture.

#### 2.2. Neural Network Design

- Calculation of the relative rotational angle between the frame of the bicycle and the thigh, θ
_{S}; - Estimation of muscle forces.

_{S}is defined in an implicit transcendental equation:

_{B}+ X

_{D}and Y = Y

_{B}− Y

_{D}, assuming that:

_{S}using numerical methods (e.g. optimization algorithms implemented in Matlab, MathWorks, Natick, MA, USA), but even if this is an assessed solution that gives accurate results [15], it is iterative and consequently slow. To solve this problem, for the calculation of the angle θ

_{S}, a Multiple Input Single Output (MISO) NN with two inputs, four hidden neurons and a single output, has been used. The inputs were the angle between the frame of the bicycle and the crank (θ

_{C}) and the angle between the frame of the bicycle and the pedal (θ

_{P}), which can be expressed as a function of X and Y. Indeed, even if the X and Y displacements could be used as direct inputs to the Neural Network, this would require an additional computational step, since the direct outputs of the measurement system are θ

_{C}and θ

_{P}. To reduce the computational cost of the algorithm, the calculation of X and Y was included in the neural network tasks, using as input the raw data from the crank and pedal angle sensors. For this first step, training data set was composed of 7050 samples from three different subjects, and 118,000 samples from a different new subject composed the testing dataset. It is important to highlight that since the problem is analytical it is not necessary to use measured input data to train the NN, as long as it belongs to a sensible function domain [40]. In this work being the measured data set oversampled with respect to its frequency content and covering the whole angular domain, it was more practical to use it instead of synthetic data. The comparison between the old method and the new one was done through direct error estimation. This is justified considering in this first phase the NN approach as an approximation of the quasi-analytical solution.

**Figure 3.**Root Mean Square Error (solid) and Standard Deviation (dashed) Error plot function of the number of neurons (from 1 to 10).

#### 2.3. Bland-Altman Plots

- The 1.96 σ
_{diff}boundary for the difference distribution, pointing out how much the two methods spread. - The regularity of the distribution along the mean axis, to identify variable-related error patterns.
- The symmetry of the distribution around the zero, addressing systematic bias of the measurements.

#### 2.4. Experimental Protocol and Validation

_{P}, and the angular displacement of the crank, θ

_{C}. These data were used as input for the biomechanical model described above. The experimental protocol was used for the estimation of the muscular forces using the two different techniques and its validation was:

- Training and validation set of NNs using data obtained previously by a deterministic optimization algorithm.
- Plot analysis between the signals obtained by NNs and signals obtained by the optimization algorithm, and the evaluation of the RMSE and RMSE Standard Deviation.
- Validation of the experimental protocol analyzing Bland-Altman plots extrapolating 1180 random samples from each muscle forces signals.

## 3. Results and Discussion

_{s}(θ

_{c}, θ

_{p}) solving a non linear equation, while the second one was used to estimate muscular forces optimizing the cost function described above.

Output | NN topology | Number of inputs | Number of neurons in the hidden layer | Training set (Samples) | Validation set (Samples) |
---|---|---|---|---|---|

θ_{S} angle | 1 MISO | 2 | 4 | 7,050 | 118,000 |

Muscle Forces | 9 MISO | 3 | 15 | 2,360 | 118,000 |

_{S}angle estimated signals, for one of the subject, obtained using deterministic algorithm optimization and using neural network are both shown in Figure 5.

**Figure 5.**Angle θ

_{S}signals, obtained by the deterministic optimization algorithm (solid gray) and by the neural network (dashed black).

Subject | RMS Error % | Standard Deviation Error |
---|---|---|

1 | 0.085% | 2.68 × 10^{−4} |

2 | 0.134% | 7.35 × 10^{−4} |

3 | 0.077% | 4.53 × 10^{−4} |

^{−3}, showing a good convergence in the research of the optimal solution.

_{diff}boundaries are not negligibly narrow, they can be reasonably attributed to the intrinsic noise affecting the deterministic algorithm (more on this matter will be discussed below). The distribution is acceptably symmetric around the mean axis, excluding the possibility of a systematic measurement error. There is no apparent pattern in the error distribution apart from a border effect, consisting in a clustering of samples on the left side around the axis origin. This is due to boundary conditions used in the model: muscular forces cannot have negative value, because of obvious physiological reasons. In Table 3, both the upper and lower 1.96

_{σdiff}boundaries are shown, together with the width of the boundary divided by the relative muscle mean force (normalized boundary).

Muscle | Upper Boundary | Lower Boundary | Normalized Boundary |
---|---|---|---|

TA | 0.1573 | −0.1577 | 0.6707 |

SO | 0.6296 | −0.6341 | 0.0786 |

GA | 0.5829 | −0.5845 | 0.4660 |

VA | 0.1203 | −0.1212 | 0.0794 |

RF | 0.3996 | −0.3967 | 0.4155 |

BFs | 0.08088 | −0.08076 | 0.6616 |

BFl | 0.7286 | −0.7266 | 0.1354 |

IL | 0.2739 | −0.2748 | 0.1873 |

GLM | 0.9368 | −0.9251 | 0.1311 |

**Figure 7.**Muscle forces of rectus femoris obtained with the neural network (dashed) and with the deterministic algorithm optimization (solid). Sampling frequency 1000 Samples/s.

## 4. Conclusions and Future Developments

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Cecchini, G.; Lozito, G.M.; Schmid, M.; Conforto, S.; Fulginei, F.R.; Bibbo, D. Neural Networks for Muscle Forces Prediction in Cycling. *Algorithms* **2014**, *7*, 621-634.
https://doi.org/10.3390/a7040621

**AMA Style**

Cecchini G, Lozito GM, Schmid M, Conforto S, Fulginei FR, Bibbo D. Neural Networks for Muscle Forces Prediction in Cycling. *Algorithms*. 2014; 7(4):621-634.
https://doi.org/10.3390/a7040621

**Chicago/Turabian Style**

Cecchini, Giulio, Gabriele Maria Lozito, Maurizio Schmid, Silvia Conforto, Francesco Riganti Fulginei, and Daniele Bibbo. 2014. "Neural Networks for Muscle Forces Prediction in Cycling" *Algorithms* 7, no. 4: 621-634.
https://doi.org/10.3390/a7040621