Page - 146 - in Maximum Tire-Road Friction Coefficient Estimation

E Longitudinal velocity estimation E.2. Observer-based estimation of the longitudinal velocity Several methods using observers to estimate the longitudinal velocity vx or the wheels’ slips sx,i directly are described in the literature. Daiß and Kiencke compared two dif- ferent observers for estimating the wheel slip sx with an accuracy of 1 % using sensors for the four wheel speeds, the longitudinal body acceleration and the vehicle’s yaw rate, [DK95]. The use of a Kalman filter applied to a two-track vehicle model and Fuzzy logic showed similar results in terms of the accuracy of calculating sx. Concerning computa- tionaland implementational effort, theFuzzy logicapproachseemedtobeadvantageous. Nevertheless, this accuracy is not sufficient for the present application. Klomp et al. estimated both vx and the road slope from measurements of the wheels’ torques, the wheels’ speeds and bax using a Kalman filter, [KGB14]. Only one wheel’s velocity (usually the one with the lowest wheel slip) is used as an input. Therefore, it is necessary to detect high values of sx robustly and early, which is a crucial part of the proposed algorithm. Experimental results on ice showed that the achieved estimate was between 5 % of the real vx, except when the test started on a road slope of 10 %. Imsland et al. propose a non-linear observer to estimate vx and vy based on signals of bax,ωz andωi, [IJF +05]. In a first step, vx is estimated using mainly the wheel speeds, but also bax and ωz. Then, vy is estimated also using bay and δS. The basis is a two- track vehicle model used to describe the motion in the horizontal plane. Experimental results show some shortcomings on surfaces with low µmax. Nevertheless, results were only shown for highly dynamic manoeuvres and on a circle, which makes the estimation of accuracy in normal driving states difficult. Tanelli only uses measurements of bax andωi to estimate vx within a non-linear es- timator, [TSC06]. It is assumed that the effective tire radii re,i is known and bax is compensated by pitch effects. Road slope is not considered. At low vehicle speeds, con- stant vehicle speeds or soft accelerating or braking (e.g. bax<−0.8 m/s), the mean of the four wheel speeds multiplied by their re,i is used for vx. When accelerating, only the wheel speedsof thenon-drivenwheelsareused. Braking is themostcriticaldrivingstate for estimation. In this state, the filtered bax is integrated using a discrete time integra- tor. Nevertheless, for a braking manoeuvre starting at about 30 kph and bax≈3 m/s2, the maximum error in estimating vxwas shown to be below 0.4 m/s. 146