## 1 Introduction
[PERSON] presented a robust hierarchical estimation scheme for the vehicle driving state based on the maximum correntropy square-root cubature Kalman filter (MCSCKF), and achieved good results [17]. A longitudinal-lateral cooperative estimation algorithm based on an adaptive square-root cubature Kalman-filter (ASRCKF) was proposed by [PERSON], this algorithm can accurately estimate the vehicle status under different driving conditions [18]. The CKF relies on the determined cubature point to calculate the posterior probability density function. Compared with the EKF, it does not need to calculate the complex Jacobian matrix, and the filtering is more accurate and avoids the truncation error. Compared with the UKF, the cubature points of the CKF appear symmetrically in a lower one-dimensional subspace, and the weights of each cubature point are equal. It is not necessary to set the parameters in advance like the UKF. The selection method is simpler, the adaptability is stronger, and the calculation speed is faster. However, the filtering accuracy of CKF can only reach the third order, and its positive cubature point weight will cause the estimation point to cross the boundary or produce a complex form of estimation point, and when n is large enough, the cubature point may exceed the integral region [19]. Therefore, many scholars have deduced the high-order spherical-radial cubature rule based on the third-order spherical-radial cubature rule, proposed the high-order cubature Kalman filter (HCKF) and applied it [20; 21; 22]. HCKF needs to increase the order of both the volume criterion and the Gaussian Laguerre criterion when expanding the spherical radial integration to higher order, and calculate the area fraction of the n-dimensional hypersphere. The calculation process is complex, resulting in poor high-order scalability of the CKF algorithm. In order to overcome the above problems, GHCKF has been proposed and applied by relevant scholars. GHCKF applies the theory of numerical integration, which is simple in form, efficient in calculation and better in scalability [23; 24; 25]. In this paper, we will establish a state equation based on the seven degrees of freedom and the tire model of this vehicle. At the same time, we will establish a measurement equation based on measurement sensor data; in this way, a new research framework for vehicle state estimation and road adhesion coefficient estimation will be constructed. We propose a GHCKF estimator, which can estimate the vehicle state and road adhesion coefficient, respectively, and use the adaptive theory to upgrade the GHCKF to an AGHCKF estimator with a stronger robustness and effectiveness. In this paper, the CKF, GHCKF, and AGHCKF algorithms will be developed, and will also be compared through simulation and experiments to verify the superiority of the AGHCKF algorithm. A high-order Kalman estimator will provide a new and reliable method for vehicle state estimation and road adhesion coefficient estimation. As shown in Figure 1, state equations and measurement equations will be established, and Kalman and adaptive theory will be applied to build a CKF/GHCKF/AGHCKF algorithm, which will jointly estimate the vehicle state and road adhesion coefficient. The vehicle sensor measures the steering wheel angle, wheel speed, yaw rate, lateral/longitudinal acceleration, and other parameters, then inputs them into the seven degrees of freedom nonlinear vehicle model, the vehicle state estimator, and the road adhesion coefficient estimator. The vehicle model uses the measurement parameters to calculate the tire longitudinal force and lateral force, and the calculated values are provided to the vehicle state estimator, and then the longitudinal/lateral velocity and yaw rate are estimated via the vehicle state estimator. The vehicle state estimator feeds the estimated values back to the nonlinear vehicle model, calculating parameters such as the sideslip angle and the tire slip rate.