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.
Similar content being viewed by others
References
Mujika I, Padilla S (2001) Physiological and performance characteristics of male professional road cyclists. Sports Med 31(7):479–487
Too D (1990) Biomechanics of cycling and factors affecting performance. Sports Med 10(5):286–302
Foley J, Bird S, White J (1989) Anthropometric comparison of cyclists from different events. Br J Sports Med 23(1):30–33
Abbiss CR, Laursen PB (2008) Describing and understanding pacing strategies during athletic competition. Sports Med 38(3):239–252
Lamberts RP, Lambert MI, Swart J, Noakes TD (2012) Allometric scaling of peak power output accurately predicts time trial performance and maximal oxygen consumption in trained cyclists. Br J Sports Med 46(1):36–41
Smith J, Dangelmaier B, Hill D (1999) Critical power is related to cycling time trial performance. Int J Sports Med 20(06):374–378
Lucia A, Hoyos J, Perez M, Santalla A, Earnest CP, Chicharro J (2004) Which laboratory variable is related with time trial performance time in the tour de france? Br J Sports Med 38(5):636–640
Atkinson G, Brunskill A (2000) Pacing strategies during a cycling time trial with simulated headwinds and tailwinds. Ergonomics 43(10):1449–1460
Cangley P, Passfield L, Carter H, Bailey M (2011) The effect of variable gradients on pacing in cycling time-trials. Int J Sports Med 32(02):132–136
Foster C, Snyder A, Thompson NN, Green MA, Foley M, Schrager M (1993) Effect of pacing strategy on cycle time trial performance. Med Sci Sports Exerc 25(3):383–388
Earnest CP, Foster C, Hoyos J, Muniesa C, Santalla A, Lucia A (2009) Time trial exertion traits of cycling’s grand tours. Int J Sports Med 30(04):240–244
Olds T (2001) Modelling human locomotion. Sports Med 31(7):497–509
Faria EW, Parker DL, Faria IE (2005) The science of cycling: factors affecting performance—part 2. Sports Med 35(4):313–338
Jeukendrup AE, Martin J (2001) Improving cycling performance. Sports Med 31(7):559–569
Maroński R (1994) On optimal velocity during cycling. J Biomech 27(2):205–213
Gordon S (2005) Optimising distribution of power during a cycling time trial. Sports Eng 8(2):81–90
Dahmen T, Wolf S, Saupe D (2012) Applications of mathematical models of road cycling. IFAC Proc Vol 45(2):804–809
De Jong J, Fokkink R, Olsder GJ, Schwab A (2017) The individual time trial as an optimal control problem. Proc Inst Mech Eng Part P J Sports Eng Technol 231(3):200–206
Yamamoto S (2018) Optimal pacing in road cycling using a nonlinear power constraint. Sports Eng 21(3):199–206
Wolf S, Biral F, Saupe D (2019) Adaptive feedback system for optimal pacing strategies in road cycling. Sports Eng 22(1):6
Morton RH (1996) A 3-parameter critical power model. Ergonomics 39:611–9
Sundström D, Carlsson P, Tinnsten M (2014) Comparing bioenergetic models for the optimisation of pacing strategy in road cycling. Sports Eng 17(4):207–215
Di Prampero P, Cortili G, Mognoni P, Saibene F (1979) Equation of motion of a cyclist. J Appl Physiol 47(1):201–206
Olds TS, Norton K, Lowe E, Olive S, Reay F, Ly S (1995) Modeling road-cycling performance. J Appl Physiol 78(4):1596–1611
Martin JC, Milliken DL, Cobb JE, McFadden KL, Coggan AR (1998) Validation of a mathematical model for road cycling power. J Appl Biomech 14(3):276–291
Muller S, Uchanski M, Hedrick K (2003) Estimation of the maximum tire-road friction coefficient. J Dyn Syst Meas Contr 125(4):607–617
de Jong J, Fokkink R, Olsder GJ, Schwab A (2016) A variational approach to determine the optimal power distribution for cycling in a time trial. Procedia Eng 147:907–911
Sundström D, Carlsson P, Tinnsten M (2014) The influence of course bends on pacing strategy in road cycling. Procedia Eng 72:835–840
Biral F, Da Lio M, Lot R, Sartori R (2010) An intelligent curve warning system for powered two wheel vehicles. Eur Transport Res Rev 2(3):147–156
Rice RS (1973) Measuring car-driver interaction with the g-g diagram. In: SAE Technical Paper, SAE International, p 2
Biral F, Bertolazzi E, Mauro DL (2014) The optimal manoeuvre. Modell Simul Control Two-Wheeled Vehic 2014:119–154
Tavernini D, Massaro M, Velenis E, Katzourakis DI, Lot R (2013) Minimum time cornering: the effect of road surface and car transmission layout. Vehic Syst Dyn 51(10):1533–1547
Kegelman JC, Harbott LK, Gerdes JC (2017) Insights into vehicle trajectories at the handling limits: analysing open data from race car drivers. Vehic Syst Dyn 55(2):191–207
Sharp RS (2008) On the stability and control of the bicycle. Appl Mech Rev 61(6):060803
Astrom KJ, Klein RE, Lennartsson A (2005) Bicycle dynamics and control: adapted bicycles for education and research. IEEE Control Syst Mag 25:26–47
Cangley P (2012) Aspects of modelling performance in competitive cycling. PhD thesis, University of Brighton
Lot R, Biral F (2014) A curvilinear abscissa approach for the lap time optimization of racing vehicles. IFAC Proc Vol 47(3):7559–7565
Biral F, Lot R (2009) An interpretative model of gg diagrams of racing motorcycle. In: Proceedings of the 3rd ICMEM international conference on mechanical engineering and mechanics. Beijing, Repubblica Popolare Cinese, Ottobre, pp 21–23
Crouch TN, Burton D, LaBry ZA, Blair KB (2017) Riding against the wind: a review of competition cycling aerodynamics. Sports Eng 20(2):81–110
Burke E (2003) High-tech cycling. Human Kinetics, Champaign
Jones AM, Vanhatalo A (2017) The ‘critical power’concept: applications to sports performance with a focus on intermittent high-intensity exercise. Sports Med 47(1):65–78
Shearman S, Dwyer D, Skiba P, Townsend N (2016) Modeling intermittent cycling performance in hypoxia using the critical power concept. Med Sci Sports Exerc 48(3):527–535
Ashtiani F, Sreedhara VSM, Vahidi A, Hutchison R, Mocko G (2019) Experimental modeling of cyclists fatigue and recovery dynamics enabling optimal pacing in a time trial. In: 2019 American Control Conference (ACC), IEEE, pp 5083–5088
Biral F, Bertolazzi E, Bosetti P (2016) Notes on numerical methods for solving optimal control problems. IEEJ J Industry Appl 5(2):154–166
Rao AV (2009) A survey of numerical methods for optimal control. Adv Astronaut Sci 135(1):497–528
Bertolazzi E, Biral F, Da Lio M (2005) Symbolic-numeric indirect method for solving optimal control problems for large multibody systems. Multibody Sys Dyn 13(2):233–252
Bays PM, Wolpert DM (2007) Computational principles of sensorimotor control that minimize uncertainty and variability. J Physiol 578(2):387–396
Todorov E (2004) Optimality principles in sensorimotor control. Nat Neurosci 7(9):907–915
Viviani P, Flash T (1995) Minimum-jerk, two-thirds power law, and isochrony: converging approaches to movement planning. J Exp Psychol Hum Percept Perform 21(1):32
Harris CM, Wolpert DM (1998) Signal-dependent noise determines motor planning. Nature 394(6695):780–784
Flash T, Meirovitch Y, Barliya A (2013) Models of human movement: trajectory planning and inverse kinematics studies. Robot Autonom Syst 61(4):330–339
Harris CM (2009) Biomimetics of human movement: functional or aesthetic? Bioinspiration Biomimetics 4(3):033001
Jeukendrup AE, Craig NP, Hawley JA (2000) The bioenergetics of world class cycling. J Sci Med Sport 3(4):414–433
Housh TJ, Devries HA, Housh DJ, Tichy MW, Smyth KD, Tichy A (1991) The relationship between critical power and the onset of blood lactate accumulation. J Sports Med Phys Fitness 31(1):31–36
Padilla S, Mujika I, Orbananos J, Angulo F (2000) Exercise intensity during competition time trials in professional road cycling. Med Sci Sports Exerc 32(4):850–856
FRIERE SKIBAP, Chidnok W, Vanhatalo A, Jones AM (2012) Modeling the expenditure and reconstitution of work capacity above critical power. Med Sci Sports Exerc 44(8):1526–1532
Fintelman D, Sterling M, Hemida H, Li F (2015) The effect of time trial cycling position on physiological and aerodynamic variables. J Sports Sci 33(16):1730–1737
Reiser RF, Maines JM, Eisenmann JC, Wilkinson JG (2002) Standing and seated wingate protocols in human cycling. a comparison of standard parameters. Eur J Appl Physiol 88(1–2):152–157
Bouillod A, Pinot J, Valade A, Cassirame J, Soto-Romero G, Grappe F (2018) Influence of standing position on mechanical and energy costs in uphill cycling. J Biomech 72:99–105
Dal Bianco N, Bertolazzi E, Biral F, Massaro M (2019) Comparison of direct and indirect methods for minimum lap time optimal control problems. Vehic Syst Dyn 57(5):665–696
Swain DP (1997) A model for optimizing cycling performance by varying power on hills and in wind. Med Sci Sports Exerc 29(8):1104–1108
Fayazi SA, Wan N, Lucich S, Vahidi A, Mocko G (2013) Optimal pacing in a cycling time-trial considering cyclist’s fatigue dynamics. In: 2013 American Control Conference, IEEE, pp 6442–6447
Sundström D, Bäckström M, Carlsson P, Tinnsten M (2015) Optimal distribution of power output and braking for corners in road cycling in Science and Cycling. Utrecht 1–2:2015
Fitton B, Symons D (2018) A mathematical model for simulating cycling: applied to track cycling. Sports Eng 21(4):409–418
Zignoli A, Biral F, Pellegrini B, Jinha A, Herzog W, Schena F (2017) An optimal control solution to the predictive dynamics of cycling. Sport Sci Health 13(2):381–393
Acknowledgements
We are deeply thankful to A. Giorgi, M. Quad, and D. Sanders and all the professional riders for providing the racing data. We thank P. Menaspà for the fruitful discussions on models and modelling methodologies. We thank E. Bertolazzi for providing valuable assistance during the optimal control problem formulation.
Funding
Partially funded by the Fondazione Cassa di Risparmio di Trento e Rovereto (CARITRO) (Grant Number: 2017.379).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
6. Appendix
6. Appendix
1.1 6.1 Vehicle dynamics and bioenergetics
1.2 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)
Rights and permissions
About this article
Cite this article
Zignoli, A., Biral, F. Prediction of pacing and cornering strategies during cycling individual time trials with optimal control. Sports Eng 23, 13 (2020). https://doi.org/10.1007/s12283-020-00326-x
Published:
DOI: https://doi.org/10.1007/s12283-020-00326-x