## 3 Adhesion Coefficient Fusion Estimation Based on Improved Singular Value Decomposition UKF
### Longitudinal Force Fusion Estimation of Adhesion Coefficient  
The Dugoff tire model requires fewer parameters to be fitted, but it cannot reflect the change in friction force with slip ratio when the tire enters the nonlinear region, leading to significant errors in the calculated longitudinal tire force. Similarly, when the longitudinal force is known, back-calculating the adhesion coefficient also results in substantial errors. Therefore, this paper will fuse the longitudinal force calculated from tire dynamics with the longitudinal force estimated from vehicle dynamics to calibrate the deviation, thereby making the final obtained adhesion coefficient more accurate. Considering that the vehicle dynamics model has errors, while the tire dynamics formula is highly reliable on high-adhesion roads when the vehicle is not slipping, but its reliability decreases significantly when slipping occurs, the estimated results will be fused with the tire dynamics. Let the estimated longitudinal force be \(F_{e}\), and the longitudinal force calculated by the tire dynamics model be \(F_{m}\). Then, the fused longitudinal force \(F_{x}\) is:  
\[F_{x}=uF_{m}+(1-u)F_{e} \tag{30}\]  
where u is the confidence level, ranging from [0,1], with higher values indicating greater trust. On high-adhesion roads (non-slip conditions), u should be set to a higher value, relying more on the tire dynamics calculation results. On low-adhesion, easily slidable roads, u should be set to a lower value, relying more on the vehicle dynamics estimation results. According to the Dugoff tire model, the normalized longitudinal force \(F_{x}^{0}\) can be calculated, and the adhesion coefficient can be obtained by the following equation:  
\[\mu_{l}=\frac{F_{x}}{F_{x}^{0}} \tag{31}\]  
According to the Burckhardt model, the optimal slip ratio typically varies between 3% and 17%, depending on different road adhesion conditions. To reduce the impact of road tire adhesion estimation accuracy on ASR performance, [PERSON] [21] proposed a fixed slip ratio method based on the Burckhardt model. The fixed slip ratio can be optimized through the following method:  
\[\left\{ \begin{array}{c}\frac{\mu_{l}^{l}\left(\lambda_{l}^{q}\right)}{\mu_ {m}^{\max}\left(\lambda_{l}^{q\prime\prime}\right)}\geq 95\%\\ f(\lambda_{i})=\sum\limits(1-\frac{\mu_{l}^{l}\left(\lambda_{l}^{q}\right)}{ \mu_{m}^{\max}\left(\lambda_{l}^{q\prime\prime}\right)}\right)\\ f(\lambda_{i}^{o})=\min f(\lambda_{i}),\lambda_{i}\in[0,1]\end{array} \right. \tag{32}\]\(\lambda_{i}^{o}\) can be obtained by solving the nonlinear programming problem:  
\[ \begin{split}&\min f\big{(}\lambda_{i}^{o}\big{)}=\sum\biggl{(}1- \frac{\mu_{i}^{i}}{\lambda_{\text{max}}^{i}}\biggr{)}\\ &\text{S.t.}\ \left\{\begin{array}{c}\frac{\mu_{i}^{i}\big{(} \lambda_{i}^{o}\big{)}}{\mu_{\text{max}}^{i+1}}\geq 95\%\\ \lambda_{i}\in[0,1]\end{array}\right.\\ \end{split}  \tag{33}\]  
The pseudocode of the algorithm is shown as Algorithm 1. First, estimate the longitudinal force \(F_{e}\) based on system input, concurrently calculate \(F_{m}\) using the tire dynamics model, and fuse them together considering the confidence level to obtain \(F_{x}\). Subsequently, calculate the adhesion coefficient based on the Duggoff tire model, and finally incorporate it into the Burckhardt model to determine the optimal slip ratio. ```
Input: measurement: \(a_{x},\ \ \omega_{x}\), \(T\); confidence level: \(u\), Sampling times: M. Initialization: \(F\), \(P\), \(Q\), \(R\) for k = 1: M do \(Fe\) is calculated by estimation model; \(Fm\) was calculated by tire dynamics model;  Calculate the fusion longitudinal force by \(F_{x}=uF_{m}+(1-u)F_{e}\)  Calculate the adhesion coefficient by \(\mu_{i}=\frac{F_{x}}{F_{x}^{2}}\) \(\min f\big{(}\lambda_{i}^{o}\big{)}=\sum\biggl{(}1-\frac{\mu_{i}^{i}}{\lambda_ {\text{max}}^{i+1}}\biggr{)}\)  Calculate the optimal slip rate by \(\text{S.t. }\ \left\{ \begin{array}{c}\frac{\mu_{i}^{i}\big{(}\lambda_{i}^{o} \big{)}}{\mu_{\text{min}}^{i+1}}\geq 95\%\\ \lambda_{i}\in[0,1]\end{array}