## 2 Materials and Methods
### Fitting the Data  
[PERSON]'s Magic Formula was fit to the experimental force and moment data [6] in a two-step process. First the data was smoothed using a 1D weighted, Blaise filter. Then, coefficients for [PERSON]'s Magic Formula, whose general form is depicted in Equation (2), were found to best fit the smoothed datasets for lateral force due to slip and camber, as well as to the self-aligning moment due to slip. Specifically, constraints from [PERSON]'s Motorcycle Magic Formula Tire model, which more equally emphasizes the camber and slip angles, were applied. The utility of the [PERSON]'s Motorcycle Magic Formula for this comparison were two-fold in that, given the specified constraints, each of the coefficients, and particular combinations thereof, had some physical significance, as explained below. Secondly, the fit data could be compared graphically and even extrapolated slightly to supplement the sometimes limited range of angles achieved during the physical test. \[y=D\sin[\text{Carctan}[Bx-E(Bx-\text{arctan}Bx)]], \tag{2}\]  
\[Y(X)=y(x)+S_{V}, \tag{3}\]  
\[x=X+S_{H}, \tag{4}\]  
where  
* \(Y\): output variable. In this case, either lateral force, \(F_{y}\), or self-aligning moment, \(M_{Z_{SA}}\);
* \(X\): input variable; here, either slip or camber angle in radians; and
* \(B\): stiffness factor, which determined the slope at the origin;
* C: shape factor, which controlled the limits of the range of the sine function;
* \(D\): peak value (when C was constrained as specified);
* \(E\): curvature factor for the peak, controlling its horizontal position;
* \(S_{H}\): horizontal shift;
* \(S_{V}\): vertical shift. The product BCD, obtained by multiplying the respective Pacejka coefficients, corresponded to the slope at the origin, that is, linear stiffness of the data, which was normalized by applied vertical load, and represented the cornering stiffness coefficient for slip, camber stiffness coefficient, and self-aligning moment coefficient, respectively. Twisting torque due to camber, influenced predominantly by the difference in peripheral velocities across the bicycle tire's toroidal shape [7], was modelled using the nonlinear formula from the FastBike multibody simulation software depicted in Equation (5). \[M_{Z_{TW}}(\varphi)=m_{r}\varphi N\Big{(}1+t_{w}\varphi^{2}\Big{)}, \tag{5}\]where  
* \(M_{Z_{TW}}\): twisting torque due to camber, which was then normalized by
* \(N\): the tire normal load in Newtons; and
* \(\varphi\): camber angle in radians;
* \(m_{r}\): linear twisting torque coefficient;
* \(t_{w}\): non-linear twisting torque coefficient. Figure 3 shows the results of the curve fitting for the \(29\times 2.3\)\" knobby tire. The yellow dots represent the cloud of data collected during the fixture's separate sweeps of slip and camber. The thin blue line depicts the smoothed curve from the Blaise filter. Finally, the red dashed line represents the respective fit curve for that data, whereas the red plus symbol represents the origin of the fit curve given any offsets. In this case, the Magic Formula curves for lateral force for both slip and camber incorporated a small vertical offset, whereas that for self-aligning moment incorporated both a small vertical and horizontal offset. The twisting torque formulation also contained a vertical offset term. Figure 3: Force and moment raw data, smoothed lines, and fit curves for \(29\times 2.3\)\" knobby on 25 mm rim at 25 psi (1.7 bar) with (**a**) normalized lateral force vs. slip angle, (**b**) normalized lateral force vs. camber angle, (**c**) normalized self-aligning moment vs. slip angle and (**d**) normalized twisting torque vs. camber angle. As can be seen, the lateral force versus slip (Figure 3a) and, to an even greater extent, the self-aligning moment versus slip (Figure 3c) exhibited the characteristic shape of the Magic Formula, whereas the lateral force versus camber (Figure 3b) showed less curvature, as did the twisting torque (Figure 3d), with some negative (downward) curvature in this example.