## 3 Adhesion Coefficient Fusion Estimation Based on Improved Singular Value Decomposition UKF  
### State-Space Model  
The four longitudinal forces \(F_{xi}\) on the front and rear wheels of the vehicle are selected as the state variables \(x_{k|0}\), and the vehicle's longitudinal acceleration \(a_{x}\) and yaw acceleration \(\dot{\omega}_{z}\) form the observations \(y\). The estimation system model is:  
\[x_{k+1}=f_{k}x_{k|0}+\omega_{k} \tag{11}\]  
\[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]  
Figure 3: Tire dynamics model. Calculate the next step prediction and covariance matrix for the state variables:  
\[\hat{x}_{k+1|k}=\sum_{i=0}^{2n}\omega_{m}^{(i)}x_{k+1|k}^{(i)} \tag{19}\]  
\[P_{k+1|k}=\sum_{i=0}^{2n}\omega_{c}^{(i)}\Big{(}x_{k+1|k}^{i}-\hat{x}_{k+1|k} \Big{)}\Big{(}x_{k+1|k}^{i}-\hat{x}_{k+1|k}\Big{)}^{T}+\mathbf{Q}_{k} \tag{20}\]  
where \(\mathbf{Q}_{k}\) is the state noise matrix, and its value affects the confidence level in the state values; a smaller value indicates greater trust in the state values. Based on the time-updated \(\hat{x}_{k+1|k}\) and \(\mathbf{P}_{k+1|k}\), and the 2n+1 Sigma points and their weights obtained through the sampling strategy, the measurement function yields:  
\[y_{k+1|k}^{i}=\mathbf{H}_{k}x_{k+1|k}^{i} \tag{21}\]  
Similar to Equations (19) and (20), the measurement means \(\hat{y}_{k+1|k^{\prime}}^{i}\) variance \(\mathbf{P}_{y,k+1}\), and covariance \(\mathbf{P}_{xy,k+1}\) can be obtained. \[\hat{y}_{k+1|k}=\sum_{i=0}^{2n}\omega_{m}^{(i)}y_{k+1|k}^{(i)} \tag{22}\]  
\[\mathbf{P}_{y,k+1}=\sum_{i=0}^{2n}\omega_{c}^{(i)}\Big{(}y_{k+1|k}^{i}-\hat{y}_{k+ 1|k}\Big{)}\Big{(}y_{k+1|k}^{i}-\hat{y}_{k+1|k}\Big{)}^{T}+\mathbf{Q}_{k+1} \tag{23}\]  
\[\mathbf{P}_{xy,k+1}=\sum_{i=0}^{2n}\omega_{c}^{(i)}\Big{(}x_{k+1|k}^{i}-\hat{x}_{ k+1|k}\Big{)}\Big{(}y_{k+1|k}^{i}-\hat{y}_{k+1|k}\Big{)}^{T} \tag{24}\]  
where \(\mathbf{R}_{k+1}\) is the observation noise matrix, and its value affects the confidence level in the observations; a smaller value indicates greater trust in the observations. The traditional UKF process assumes that the noise covariance matrix is fixed, but this can lead to significant estimation errors. To account for the impact of varying road noise on the estimation results, we introduce the Sage-Husa noise-adaptive filtering method. This method adjusts the observation noise covariance for changing noise conditions. By adaptively adjusting the noise covariance in real-time, this filtering method improves the estimation accuracy and reduces the impact of bumpy road noise on the estimation. The Sage-Husa adaptive filtering algorithm is as follows:  
The estimated covariance value of the system observation noise is calculated as:  
\[\mathbf{R}_{k+1}=(1-d_{k})\mathbf{R}_{k}+d_{k}\Big{(}e_{k+1}e_{k+1}^{T}-\mathbf{P}_{y,k+1 }\mathbf{P}_{y,k+1}^{T}\Big{)} \tag{25}\]  
where \(d_{k}\) is the forgetting factor, and b is the forgetting factor, usually ranging between 0.95 and 1. \(e_{k}\) represents the difference between the true observation value and the estimated observation value at time k, i.e., the observation residual. It is challenging to ensure that the process noise covariance matrix and the observation noise covariance matrix remain non-negative definite during vehicle operation. Therefore, the improvements to \(\mathbf{Q}_{k}\) and \(\mathbf{R}_{k}\) are as follows [33]:  
\[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]0 \[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]1The Kalman filter gain matrix is:  
\[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]2  
The system state is updated as:  
\[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]3  
The system covariance is updated as:  
\[y_{k+1}=h_{k}x_{k|0}+v_{k} \tag{12}\]4  
The above calculation process constitutes the complete AUKF algorithm.