Prediction of pacing and cornering strategies during cycling individual time trials with optimal control

Abstract

A new dynamic model for predicting road cycling individual time trials with optimal control was created. The model included both lateral and longitudinal bicycle dynamics, 3D road geometry, and anaerobic source depletion. The prediction of the individual time trial performance was formulated as an optimal control problem and solved with an indirect approach to find the pacing and cornering strategies in the respect of the physical/physiological limits of the system. The model was tested against the velocity and power output data collected by professional cyclists in two individual time trial Giro d’Italia data sets: the first data set (Rovereto, n = 15) was used to adjust the parameters of the model and the second data set (Verona, n = 13) was used to test the predictive ability of the model. The simulated velocity fell in the $\mathrm{CI}_{95\%}$ of the experimental data for 32 and 18% of the duration of the course for Rovereto and Verona stages, respectively. The simulated power output fell in the $\mathrm{CI}_{95\%}$ of the experimental data for 50 and 25% of the duration of the course for Rovereto and Verona stages respectively. This framework can be used to input rider’s physical/physiological characteristics, 3D road geometry, and conditions to generate realistic velocity and power output predictions in individual time trials. It, therefore, constitutes a tool that could be used by coaches and athletes to plan the pacing and cornering strategies before the race.

6. Appendix

6.1 Vehicle dynamics and bioenergetics

6.2 Numerical values

• m = 77 kg, cyclist’s body mass + bicycle mass

• $I_X$ = 77 $\mathrm{kgm}^2$,

• h = 1.2 m, height of the centre of mass from road surface

• $\rho$ = 1.23 $\mathrm{kg}/\mathrm{m}^3$

• $A_\mathrm{f}$ = 0.35 $\mathrm{m}^2$, frontal area

• kv = $1/2 A_\mathrm{f} C_{\rm D}\rho$ = 0.13 kg/m, air drag force coefficient

• $c_\mathrm{rr}$= 0.0035, rolling friction coefficient

• g= 9.81 $\mathrm{m/s}^2$, constant of gravity

• L= 1.4 m, bicycle length

• $\delta _{\rm max}$= 0.52 rad, maximal steering angle

• $vW_{\mathrm{n}_{\rm max}}$ = 50 W/s, maximal power output variation

• $\mathrm{ay}_{\max}=\mathrm{ax}_{\max}$= 9.81 $\mathrm{m/s}^2$, maximal acceleration $({\mu_{\mathrm y}=\mu_{\mathrm x}=1})$

• $W_{{\rm J}_1}$ = 0.01, steering angle weight in the objective function

• $W_{{\rm J}_2}$ =0.01, power output variation weight in the objective function

• $n_0$ = 0 m, initial lateral displacement

• $\alpha _0$ = 0 rad, initial attitude

• $\phi _0$ = 0 rad, initial roll angle

• $\phi _\mathrm{dot0}$ = 0 rad/s, initial roll rotational velocity

• $\delta _\mathrm{n0}$ = 0, initial normalised steering angle

• $W_\mathrm{n0}$ = $W_{\rm C}/W_\mathrm{max}$, initial normalised power output

• $EAn_0$ = 22.000 J, initial anaerobic sources

• $W_\mathrm{C}$ = 440 W, critical power output

• $W_\mathrm{max}$ = 1870 W, maximal power output

• $\mathrm{EAnzero}$= 22.000 J, maximal anaerobic sources

• $L_\mathrm{width}$ = 4 m, left road width

• $R_\mathrm{width}$ = 4 m, right road width

• $Vw_0$ = 2.6 and 2.7 $\mathrm{m/s}^2$, wind velocity (Rovereto and Verona ITT, respectively)

• wD = $-\pi /2$ and $- 3/4\pi$, wind direction (Rovereto and Verona ITT, respectively)