# Voltage-Based Braking Controls for Electric Vehicles Considering Weather Condition and Road Slope
## 2 Definitions and Assumptions  
One presents the definitions and assumptions used in this article. For convenience, \(c(\theta)\) denotes \(\cos(\theta)\), and \(s(\theta)\) denotes \(\sin(\theta)\). Let \(w\) define the angular velocity of vehicle wheels. We utilize the vehicle longitudinal motion modeling in [13]. Consider the case where the brake is initiated to make the vehicle stop. According to [13], the forces acting on the vehicle in straight motion are as follows.  
\[m\psi=-F_{x}-F_{a}-F_{g} \tag{1}\]  
where \(m\) is the vehicle mass, and \(v\) denotes the longitudinal speed. \(F_{x}\) is the braking force, which is the controlled force produced by the brake. \(F_{a}\) is the aerodynamic force given by:  
\[F_{a}=0.5*\rho*C*\
u^{2} \tag{2}\]  
where \(\rho\) denotes the air density, and \(C\) denotes the drag coefficient. In (1), \(F_{g}\) is the gravitational force due to the road slope:  
\[F_{g}=m*g*s(\theta). \tag{3}\]  
Here, \(\theta\) denotes the road slope, and \(g\) denotes the gravitational acceleration. See that the road slope \(\theta\) is considered in our vehicle motion model.  
Note that this gravitational force did not appear in [16], since [16] considered the case where \(\theta=0\). In other words, [16] assumed a flat road, not a sloped road. Assuming that the vehicle's inclination relative to the road is zero, the road slope \(\theta\) can be measured by installing the IMU on the vehicle.  
Let \(S\) denote the wheel slip ratio given by  
\[S=\frac{v-R*w}{v}=1-\frac{R*w}{v}. \tag{4}\]  
Here, \(R\) is the wheel radius, according to [19]. The slip ratio shows the wheel slip according to the longitudinal velocity of the vehicle and the angular velocity of the wheel [19].  
Note that \(v-R*w\leq v\) in (4). Moreover, \(v\geq R*w\), since the vehicle's longitudinal velocity cannot be smaller than \(R*w\), considering the wheel slip. Thus, \(0\leq\|S\|\leq 1\) follows [19]. When \(w=0\), then \(\|S\|=1\) and the wheel lock up will happen [19].  
Let \(\mu(S)\) denote the adhesion coefficient, which changes depending on seasonal road conditions (e.g., dry, ice, snow, summer, winter, etc.), and the graph of \(\mu(S)\) with respect to \(S\) is addressed in [19]. The authors of [19] showed that the Pacejka Formula has the most appropriate results for the full range of the wheel slip ratio than other tire models, such as the polynomial model. Note that [19] derived \(\mu(S)\) by doing experiments in various environmental conditions.  
Note that \(\mu(S)\) is available using [19]. According to [19], \(\mu(S)\) in wet or dry road conditions is plotted in Figure 1.  
This \(\mu(S)\) is used in our MATLAB simulations.  
Using \(\mu(S)\) in [19], \(F_{x}\) in (1) is the braking force given by:  
\[F_{x}=\mu(S)*m*g*c(\theta). \tag{5}\]  
Note that this braking force is distinct from the braking force in [16], since [16] considered the case where \(\theta=0\). In other words, [16] assumed a flat road, not a sloped road. Note that the braking force in (5) considers \(\mu(S)\), which depends on seasonal road conditions (e.g., dry, ice, snow, summer, winter, etc.).  
Suppose the vehicle initiates the stop maneuver at time 0. Let \(d\) define the _braking distance_, given as  
\[d=\int_{0}^{T_{f}}v(t). \tag{6}\]  
Here, \(T_{f}\) is the time when \(v(t)\) becomes zero by applying the brake.  
Obstacles, such as pedestrians or stop signs, can be detected using on-board sensors [2; 3; 4]. One assumes that the sensing range of on-board sensors is longer than the braking distance \(d\). For avoiding collision with an object, the vehicle initiates a braking system automatically when there is an obstacle which is within \(d\) distance from the vehicle.