# Vehicle Pose Estimation Method Based on Maximum Correntropy Square Root Unscented Kalman Filter
## 2 Problem Description and Existing Methods
### Square Root Unscented Kalman Filter SLAM  
In general, it is difficult to directly integrate the probability expression in Equation (3). In SLAM, Bayesian filtering methods in probability statistics are usually used for the approximate calculation [13]. Furthermore, since the motion model and observation model of the mobile robot are the nonlinear system, nonlinear Bayesian filtering methods are usually used. Among them, the Square Root Unscented Kalman Filter SLAM (SRUKF-SLAM) algorithm [23] is a classic algorithm for solving the SLAM problem of the mobile robot. The algorithm uses Unscented Transform (UT) to approximate the posterior probability density of the system state and uses the square root form to ensure the positive definiteness of the covariance matrix, reduce the calculation error of the state estimation, and improve the calculation efficiency of the algorithm, so as to improve the real-time performance of the algorithm. The algorithm employs Unscented Transform (UT) to approximate the posterior probability density of the system state. By employing Cholesky decomposition to obtain the square-root matrix of the covariance, it ensures positive definiteness of the covariance matrix, reduces computational errors in state estimation, and enhances algorithmic efficiency. The SRUKF-SLAM algorithm employs the Unscented Transform (UT) technique. The UT operates by strategically selecting sigma points that encapsulate the complete statistical properties (mean and covariance) of the state vector distribution. These sigma points are then propagated through the nonlinear system function, generating a transformed set of sigma points. The posterior statistics are subsequently computed through weighted averaging of these propagated sigma points. Furthermore, the algorithm utilizes the square-root matrix \(\mathbf{S}\) of the covariance instead of the original covariance matrix \(\mathbf{P}\) to guarantee the positive definiteness of the covariance matrix. The selection of sigma points in the SRUKF-SLAM algorithm can be determined by  
\[\mathbf{\xi}_{i}=\left\{ \begin{array}{ll}\hat{\mathbf{x}},&i=0\\ \hat{\mathbf{x}}+\left(\sqrt{(n+\lambda)\mathbf{S}}\right)_{i},&i=1,2,\cdots,n\\ \hat{\mathbf{x}}-\left(\sqrt{(n+\lambda)\mathbf{S}}\right)_{i-n},&i=n+1,\cdots,2n\end{array}  \right., \tag{4}\]  
where \(\hat{\mathbf{x}}\) is the mean vector of the known state vector \(\mathbf{x}\), and \(\mathbf{S}\) is the square root matrix of the covariance matrix \(\mathbf{P}\) of the state vector \(\mathbf{\(\left(\sqrt{(n+\lambda)S}\right)_{i^{\prime}}\) is the \(i\)-th row of the square root matrix of \((n+\lambda)S\), which can be obtained by Cholesky decomposition. The weight corresponding to each sigma point in Equation (4) is  
\[\left\{ \begin{array}{l}\omega_{0}^{m}=\frac{\lambda}{n+\lambda}\\ \omega_{0}^{c}=\frac{\lambda}{n+\lambda}+\left(1-\alpha^{2}+\beta\right)\\ \omega_{i}^{m}=\omega_{i}^{c}=\frac{1}{2(n+\lambda)},\;\;i=1,\cdot\cdot\cdot,2n\end{array} \right., \tag{5}\]  
where \(\omega_{i}^{m}\) denote the weights of the mean vector, and \(\omega_{i}^{c}\) denotes the weights of the covariance matrix. The superscripts m and c represent the mean and covariance, respectively. The subscripts 0 and \(i\) represent the initial value and \(i\)-th sigma point, respectively. \(\alpha\) determines the spread of the sigma points, the range of values for \(\alpha\) is \(0\leq\alpha\leq 1\). \(\beta\) is a parameter related to the prior knowledge of the distribution of the state vector and is usually taken as 2. \(\lambda\) is a composite scaling factor, given by  
\[\lambda=\alpha^{2}(n+\varphi)-n, \tag{6}\]  
where the value of parameter \(\varphi\) is usually \(3n\)[43]. The following is the specific process of the SRUKF-SLAM algorithm.