Enhanced Northern Goshawk Optimization–Elman neural network-based tire-road friction coefficient estimation for vehicle ABS control

The instability of weight updates in conventional Elman neural networks (ENNs) compromises the accuracy of tire-road friction coefficient estimation. To overcome this limitation, an enhanced Northern Goshawk Optimization (NGO)–ENN framework is proposed for tire-road friction coefficient estimation. A neural network dataset was constructed based on a seven-degree-of-freedom vehicle dynamics model, and an enhanced NGO algorithm was introduced to optimize the weight matrix of the ENN. The optimization process incorporated a reverse learning strategy, a Cauchy–Gauss mutation mechanism, and a dynamic neighborhood search strategy to enhance convergence and reduce absolute training errors. In addition, a Fourier function method was used to fit the ideal slip ratio – tire-road friction coefficient relationship. Theoretical analyses, simulations, and experimental validation were performed to evaluate the proposed approach. The results demonstrate that the proposed enhanced NGO–ENN method achieves a 32.92% and 34.28% reduction in the root mean square error (RMSE) compared to the conventional ENN method under constant and variable friction coefficient conditions, respectively. Experimental implementations on roads with constant friction coefficients further reveal an average RMSE reduction of 59.43%, significantly improving anti-locking braking control efficacy. The results validate the superiority of the proposed enhanced NGO–ENN framework in enhancing tire-road friction coefficient estimation accuracy and robustness, thereby advancing the practical applicability of anti-locking braking control.