Page - 138 - in Maximum Tire-Road Friction Coefficient Estimation

C Vehicle model structure for sensitivity analysis The structure of the differential equation for time step k can then be represented as z˙(k) =M−1 ·(q(z(k),z(k−1),δi(k),MD,i(k),µmaxi (k))−k(z(k))). (C.1) with k = f(z(k)) = f (vx,vy)(k) and the inputs for q which are the wheels’ steering angles δi(k), the wheels’ driving and braking torquesMD,i(k) and the friction potential µmaxi (k). The dependency on q and z˙(k−1) originates from the simplifications in the calculation of the tire load variation and the effective tire radius, see Section 4.3. A systematic overview of the dependencies is given in Table C.1 and C.2. Table C.2.: Relation between state variables and the inputs as well as the friction po- tentials in the vehicle model adapted for the sensitivity analysis Inputs Parameterµmax v˙x x x x x x x v˙y x x x x x x bω˙z x x x x x x ω˙fl x x x ω˙fr x x x ω˙rl x x ω˙rr x x F˙Dy,fl x x F˙Dy,fr x x F˙Dy,rl x F˙Dy,rr x C.1. Numerical implementation of vehicle model for automatic differentiation To calculate the Jacobian J and the derivative fc using AD, the vehicle model has to be at least one time differentiable. To achieve at least weak derivatives, numerical approximations were necessary for the following two unsteady functions. 138