# A Novel Approach of Modelling and Predicting Track Cycling
## 2 Materials and Methods
### Implementation  
The explicit-forward iteration of all time-dependent variables is conducted in the distance domain at equidistant points to facilitate the direct comparison of computed values to observations taken at fixed positions in the velodrome (start line, half-round, round, finish line, etc.). A suitably small increment \(\Delta s\) has to be selected to account for the unsteady characteristics of all variables. Within each distance increment \(\Delta s\), a non-uniform linear motion with constant acceleration is assumed.  
The sequence of computational steps is outlined in Figure 1. Starting from a standstill, with initial values \(t=0\), \(s=0\), \(v=0\), \(\varphi=0\), the resulting drag forces \(F_{D}\) and \(F_{RR}\) can be determined using Equations (7) and (8). The cadence \(PR\) results from Equation (10). With these status variables known, the pedal force applied by the athlete is returned from the physiological model and the propulsive force at this position is estimated using Equation (16). The momentary acceleration of the system can now be calculated from the force balance in Equation (6). The time increment \(\Delta t\) elapsed for moving one increment is computed by solving the differential equation of motion for the time interval \(t\), taking into account that due to the lean angle, the centre of gravity travels a distance shorter than \(\Delta s\):  
\[\Delta t_{1,2}=-\frac{v}{a}\pm\sqrt{\frac{v^{2}}{a^{2}}+\frac{2\Delta s\cdot(1 -\sin\varphi/H)}{a}} \tag{17}\]  
The result of the square root is positive for \(a>0\), negative for \(a<0\) and the case \(a=0\), which is numerically most unlikely to appear, has to be excluded. Now, the initial velocity for the next distance increment can be calculated:  
\[v_{1}=v_{0}+a_{0}\cdot\Delta t_{0} \tag{18}\]  
The absolute position of the system relative to the start and the total run-time until this position is reached results from a summation of the respective increments:  
\[s=\sum\Delta s;\quad t=\sum\Delta t \tag{19}\]  
In this contribution, a generic track 250 m in length with circular corners of radius \(R=25\) m shall be assumed. This results in a 46.46 m straight, followed by a 78.54 m long curve, with a period of 125 m. With the absolute distance known, the position \(s^{*}\) within half a lap can be computed:  
\[s^{*}=\text{mod}\left(\frac{d}{125\,\text{m}}\right) \tag{20}\]  
From \(s^{*}\), the local corner radius can be determined, and the corresponding lean angle computed using Equation (11), assuming that the transition between cycling upright in the straights and inclined in the curves or vice versa occurs in a linear fashion over a distance of ca. 20 m centred around the geometric origin of the bend. Thus, the lean angle is multiplied with a form function that is defined to be zero in the straights, \"1\" in the bends and a linear function of \(s\) during the transition from one to the other. From this adjusted lean angle, the local load factor from Equation (12) and all derivatives thereof can be calculated.  
Usually, at the start, the athlete accelerates the bike in a standing position (_STAND_) for the initial 140 to 170 m. This affects the aerodynamic characteristics, as well as the physiological capabilities. To account for this, a different set of parameters is used for \(c_{D}\cdot A\), \(A\), \(\tau\) and \(c\) for this phase. The iteration cycle is repeated until the system reaches the desired finish position.