# A Novel Approach of Modelling and Predicting Track Cycling
## 3 Results  
Mean pedal force and the pedalling rate with corresponding power output for one athlete during both 750 m-time trials is illustrated in Figure 2. The gear ratio of the first run (\(i=3.6\)) corresponds to a distance of 7.56 m per pedal revolution. Here, the mean power output was 651 W at a mean cadence of 117 rpm. In the second run, a higher gear ratio \(i=4.14\), resulting in the development of 8.7 m was chosen. In the second race, the mean power was 691 W at a mean cadence of 104 rpm. Figure 2: Mean pedal force \(F_{T}\), pedalling rate \(PR\) and power output P during two 750 m time trials. (**a**) First run at the development of 7.56 m per pedal revolution, (**b**) second run at the development of 8.7 m.  
In Figure 3, the maximum pedal force \(F_{max}\) attained at \(PR=0\) was calculated from the data measured in the first run and the physiological model fitted, returning high to excellent agreement. The physiological model parameters were then used to approximate the data measured in the consecutive run with different gearing, and the fit quality was assessed again. It was established that the physiological model can accurately reproduce the decline in \(F_{max}\) with time and that this athlete-specific fatigue characteristic can be generalised and reused to forecast following runs, even in a different setup. While this worked sufficiently well for five athletes, the physiological parameters had to be adjusted for athlete no. 6 to fit the data captured in the second run. Model parameters found for all six athletes in the two different phases (_STAND_, _SIT_) of the race with the associated model quality, measured by the determination coefficient \(R^{2}\), are detailed in Table 1. When comparing the \(R^{2}\) of the different runs using a paired \(t\)-test, no statistically significant differences were found (\(p>0.05\); \(d<0.1\)). Figure 3: Time-dependent maximal mean pedal force \(F_{max}\) with model function (prediction) calculated by Equation (5) for (**a**) the first and (**b**) the second run. The physiological model parameters were fitted to the data of the first run. In the next step, the physiological model was tested for correct reproduction of the dependency between cadence \(PR\) and mean pedal force \(F_{T}\) by comparing the measured pedal forces to projected values. Figure 4 illustrates for all six athletes, each in two runs, the deviation between the modelled and the measured data, plotted against the distance travelled. With a bias of 2.05 N at a standard deviation of 14.77 N, the overall agreement was found to be satisfactory. Applying the physical model to the pedal force returned by the physiological model as an input, the expected instantaneous velocity of the system and the mechanical power required by the athlete can be computed. In Figure 5, the modelled velocity at the wheels is compared to the measurements, and a very good agreement was observed. A similar comparison is displayed in Figure 6 for the mechanical power, where both quantities are highly correlated despite the data returned from the power meter showing significant fluctuations. A linear regression analysis was employed to the modelled and measured velocity and power output, respectively. A high degree of agreement was returned for both quantities. Table 2 tabulates the \(R^{2}\) found in the two runs and the proportionality factor \(k\) between the modelled and measured data.