In this paper, a differentiable friction model is proposed to estimate the longitudinal tire-road friction force of vehicle systems. A novel adaptive nonlinear observer-based parameter estimation scheme has been developed to estimate the parameters of friction model, which requires the signals from the existing sensors signals such as wheel rotational speed and vehicle speed. Different from conventional gradient and recursive least square (RLS) methods, the filtered regression parameter estimation error is introduced in the novel adaptive laws, which can guarantee the observer error convergence to zero and the estimated parameter also convergence to their real value. The Lyapunov method is used to prove the stability of the proposed methods. The robustness of the developing method against bounded disturbances is also proved. Simulation results illustrate that the proposed method can realize relatively accurate estimation of the friction with variations in speed and road gradient.

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