## 3 Results
### Analytical Corroboration of the Longitudinal Braking Model  
A comparison of the simulated stopping distance against [PERSON] [PERSON]'s stopping distance formula was performed (Equation (1)). Simulations were run in initial velocity increments of one meter per second. The simulated stopping distances are slightly shorter than the theoretical formula due to aerodynamic drag, which the formula neglects. This formula also assumes that both tires have a constant friction coefficient, that both brakes are applied at tire friction limits, and that the bike and rider have zero pitch displacement [3]. It will be shown in the simulation results that this formula drastically underestimates stopping distances because it essentially assumes braking with both wheels locked. This is an extremely unstable condition. In actuality, most riders increase the pressure until the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Value** & **Units** & **Notes** \\ \hline \(m\) & 100 \({}^{*}\) & kg & Bike and rider combined mass \\ \(a_{cm}\) & 0.7 \({}^{*}\) & m & Distance from center of mass to front tire contact patch \\ \(b_{cm}\) & 0.4 \({}^{*}\) & m & Distance from center of mass to rear tire contact patch \\ \(l_{cm}\) & 1.1 \({}^{*}\) & m & Height of center of mass from ground plane \\ \(C_{d}\) & 0.3 & & Coefficient of drag for bike with rider \\ \(A_{f}\) & 1.1 \({}^{*}\) & m\({}^{2}\) & Frontal area of bike with rider \\ \(I_{yy}\) & 0.16 & kg\(\cdot\)m\({}^{2}\) & Mass-moment of inertia of each wheel \\ \(L_{rel}\) & 0.6 & m & Tire relaxation length (tire response time) \\ \(R_{e}\) & 0.35 & m & Loaded tire radius \\ \(H_{lientic}\) & 0.5 & & Brake pad and rotor coefficient of friction \\ \(d_{horc}\) & 0.03 & m & Brake caliper piston bore diameter \\ \(R_{m}\) & 0.085 & m & Mean rotor radius \\ \(V_{0}\) & 24 \({}^{*}\) & km/h & Velocity at initiation of braking \\ \hline \hline \end{tabular} \({}^{*}\)These values were altered for each corroboration test.  
\end{table} 
Table 2: Values selected for each variable used in the bicycle longitudinal dynamics and bicycle tire simulation blocks during corroboration tests. Figure 3: Locations of the dimensions described in Table 2 relative to the center of mass and ground plane. rear wheel locks and then decrease pressure slightly, thus, only reaching around half of the front tire friction limit. [PERSON] derived formulations that include tire, bearing, and aerodynamic drag forces [22]. The velocity profile during a braking event is compared to the simulation model for front braking, rear braking, and simultaneous braking. The simulated stopping times are about one-tenth of a second longer, resulting in approximately five percent error. When the simulation output was initially compared against [PERSON]'s results, there were large discrepancies between results for the front and rear wheel braking simulations but not the simultaneous braking simulation; this was crucial in discovering that Simulink references the vehicle center of mass to the axles and not the ground plane. Figure 4 shows the comparison between the simulated and analytical results. Tip-over initiation can be easily calculated by summing the moments about the front tire contact patch when the rear wheel begins to lift off the ground. The result shows that the maximum acceleration before tip-over is only a function of the distance from the front tire contact patch (\(a_{cm}\)) and the height of the center of mass from the ground plane (\(h_{cm}\)) as shown in Equation (5) [23]. \[\ddot{x}_{crit}=\frac{a_{cm}}{h_{cm}} \tag{5}\]  
The model results in an average of three percent higher deceleration (measured in \(g\)'s) at 40 km/h due to the aerodynamic drag force in the model. At low speeds, the simulation results are within 0.4% of the calculation (Table 3).