# An optimal control solution to the predictive dynamics of cycling

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## Abstract

### Purpose

Pure predictive dynamics aims at predicting the set of driving inputs in the absence of any a priori data and can be applied in movement science to generate biomechanical variables in many different what-if scenarios. The objective of this research was to solve the problem of the predictive dynamics of sub-maximal cycling by means of an optimal control computational algorithm that makes use of an indirect method.

### Methods

To this, a 2D two-legged seven bodies three degrees of freedom model of the lower limbs of a cyclist has been developed and validated against the average behaviour of eight well-trained cyclists pedalling at different sub-maximal intensities (100, 220, 300 W) at constant cadence (90 rpm). Experimental data adopted in model validation consists of the hip, knee, ankle joint centre and crank kinematics and the right/left crank torques.

### Results

It has been found that the model can replicate the major features of pedalling biomechanics and the ability of a cyclist to deliver a larger torque if a larger power output is required and the cadence is kept constant. The reported mismatches with experimental data get smaller as the power output increases.

### Conclusions

It is suggested that: (1) an optimal control based on an indirect method approach can provide a solution to the predictive dynamics of sub-maximal cycling, (2) predictive dynamics adapts accordingly to real data for changes in power output.

## Keywords

Cycling biomechanics Optimal control Indirect method Predictive simulation## Notes

### Acknowledgements

The authors would like to thank Prof E. Bertolazzi for the valuable help in developing the most convenient mathematical formulations of the models.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflicts of interest.

### Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

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