Robust Estimation and Experimental Evaluation of Longitudinal Friction Forces in Ground Vehicles

Abstract

A longitudinal force estimation based on wheel dynamics and unscented Kalman filter is proposed in this report to address the difficulties in the conventional tire-based approaches. Although it seems that implementation of a tire model in the estimation procedure should result in more accurate results, especially for non-linear regions, complexities in identifying the tire parameters due to the variation of the road and tire conditions leads to inaccurate results for harsh maneuvers on slippery roads. Moreover, the estimation process requires reliable measurements and this necessitates utilizing dynamic models with feasible measurements. Consequently, wheel dynamics is employed to extend the fidelity of the algorithm. For such a model, wheel speeds as reliable and feasible measurements are available. In this strategy, the complex tire-road interaction can be discarded since the wheel speeds are being observed by wheel sensors and the values of the effective torques are provided by motor drives then the longitudinal forces at each individual corner of the vehicle can be estimated independently. Experimental and simulation results confirm the validity of the algorithm in slippery road conditions as well as normal conditions. The newly developed structure has a strong potential to be integrated with other state estimation, such as longitudinal/lateral velocities and lateral forces.

Distributed robust vehicle state estimation

“…in which B c = [0 1 1] T , the estimation input is u = R eω −v xt , the output y = µ x is the normalized longitudinal force, which can be obtained from road friction-independent approaches using nonlinear and sliding mode observers [22]- [24], Kalman-based estimation [25], [26], and unknown input observers [27]- [29]. The measurement and process noises are denoted by w m and w p = [w 1 w 2w1 ]…”

Estimation of longitudinal speed robust to road conditions for ground vehicles

“…Hence, estimation of tire forces independent of road conditions would be a remedy. Longitudinal force estimation independent of the road friction may be classified on the basis of wheel dynamics into the nonlinear and sliding mode observers [22]- [24], Kalman-based estimation [25], [26], and unknown input observers [27], [28]. This section provides two force estimation approaches, an unknown input observer and a Kalman-based method.…”

“…Proper capturing of nonlinearities contributes to the unscented transformation that defines the Sigma vectors X ∈ R N ×2N +1 , (N is the length of the state vector), which are supposed to propagate through the nonlinear system. With some minor changes, UKF can also be employed for parameter estimation instead of state estimation for the vehicle parameter identification [29], [30] and for the longitudinal force estimation [26]. For the force estimation with UKF, the effective torque T t provides input u k ; the wheel speed is assumed to be the state x k , and the estimated longitudinal forceF x is denoted by the estimated parameterf .…”

“…In summary, the following Lemma presents two tests for observability of the velocity estimator with stochastic initial conditions. Lemma 1: [42] The Kalman estimator (21) on the system (20) with an error covariance matrix (22) and stochastic initial stateP 0|−1 = ψI, ψ ∈ R + is stochastically observable if the condition λ max (M f ) = 0 (test 1) holds for a finite time t f and λ max (N k+1 ) < λ b for t k ≥ t f (test 2) with a predefined bound λ b , where M k+1 is obtained from the modified Riccati equation (26) and the procedure provided in (A6), (A7) in the Appendix. Employing the condition λ max (M f ) = 0 for a finite time t f < ∞, the modified Riccati equation (26) changes toP k+1|k = N k+1 + S k+1 which leads to a simplified form of N k+1 as in:…”

“…These two tests have been performed on the proposed observer withQ = 2.5e − 3 andR = 1e − 3 and results are depicted as follows. Figure 2 exhibits λ max (M f ), where M f is obtained from (26) and the procedure provided in the Appendix. Different experiments such as DLC, brake-in-turn (BiT), sharp turn, and steering on dry and slippery (snow/ice) roads have been performed and results are illustrated in Fig.…”