24 Electric and Hybrid Vehicles define a tangential co-ordinate system that moves along with the vehicle with respect to a fixed two- dimensional system. The roadway description will be utilized to calculate the distance traversed by a vehicle along the roadway. The fixed co-ordinate system is attached to the earth such that the force of gravity is perpen- dicular to the unit vector i F . Let us consider a straight roadway, i.e., the steering wheel is locked straight along the x F -direction. The roadway is then on the x F y F plane of the fixed co-ordinate system (Figure 2.1). The two-dimensional roadway can be described as y f F F = ( ) x .
The roadway position vector r x ( ) F between two points a and b along the horizontal direction is r x ( ) F F = + x i F F f x ( ) j a F F for . ≤ ≤ x b The direction of motion and distance traversed by the vehicle is easier to express in terms of the tangent vector of the roadway position vector given as ( ) = = dr df T x F i F + j F dx F dx F The distance norm of the tangent vector T x ( ) F is 2 df T x ( ) F = + 1 dx F The tangential roadway length s is the distance traversed along the roadway. Mathematically, s is the arc length of y f F F = ( ) x over a ≤ ≤ x b F . Therefore, s = ∫ b T x ( ) F F dx .
a The roadway percent grade can be described as a function of the roadway as − 1 df ( ) x β ( ) x F = tan F . dx F (a) (b) FIGURE 2.1 (a) Fixed co-ordinate system. (b) Roadway on the fixed co-ordinate system. yF yF j F r(x ) f(xF0) F0 xF T( x ) i j F0 zF F F k F xF0 i xF F yF yF j F f(xF0) xF yF r(x ) F0 T( x ) F0 xF j F i F xF0 xF