## 3 Methodology
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbols/Variables** & **Definition** \\ \hline \(\Delta t_{n,n-1}\) & The time interval between neighboring nodes \(n-1\) and \(n\). \\  & The point cloud poses of the \(n\)th laser frame, \(\mathbf{p}_{n}=\{p_{k}|k=1,2,3,\cdots\}\), \(p_{k}\) represents the coordinate of a point, \(p_{k}=(x_{k},y_{k},z_{k})^{\mathrm{T}}\). \\ \(x_{n}=[\mathbf{R}_{n},\ \mathbf{t}_{n}]\) & The state of the \(n\)th laser historical key frame, that is, the \(n\)th vehicle state node. \(\mathbf{R}_{n}\) is the rotation matrix of the pose relative to the origin coordinate, \(\mathbf{t}_{n}\) is the translation vector, \\ \(\mathbf{p}_{n}=\mathbf{R}_{n}\mathbf{p}_{n-1}+\mathbf{t}_{n}\). \(\mathbf{R}\in\mathrm{SO}(3)\), \(\mathbf{t}\in\mathbb{R}^{3}\). \(\mathbf{p}_{n}\), \(x_{n}\) and \([\mathbf{R}_{n},\ \mathbf{t}_{n}]\) can represent one another. \\ \(\tilde{x}_{n}=\left[\mathbf{R}_{n}^{\mathrm{init}},\ \mathbf{t}_{n}^{ \mathrm{init}}\right]\) & The initial value of the \(n\)th frame pose, and it is calculated based on the pose-transformation matrix provided by the previous node and the IMU. The final pose state of the vehicle after factor graph optimization is \(\hat{x}_{n}\). \\ \(fan_{i}\) & The fan-shaped sector sequence number in a frame of point cloud data. The fan-shaped sector is divided into equal angles. \\ \(\mathbf{p}_{fan_{i}}^{n}\) & The collected pose data of the \(i\)th fan-shaped sector in the \(n\)th frame; \(\hat{\mathbf{p}}_{fan_{i}}^{n}\) represents the initial prior pose, \\ \(\Delta t_{fan_{i}}\) & The data-collection time interval between the \(i\)th fan-shaped sector and the previous. \\ \(row\), \(col\) & Represent the rows and columns of the matrix, respectively. \(I(row,col)\) represents the matrix values corresponding to the rows and columns of the polarized matrix of point cloud. \\ \(res_{h}\) & The horizontal angular resolution of the Lidar scan. \\ \(\theta\) & Angle. \(\theta_{k}^{\mathrm{h}}\) represents the horizontal angle of point \(k\). \(\theta_{k}^{\mathrm{v}}\) represents the vertical angle of point \(k\). \(\theta_{ab}^{\mathrm{h}}\) represents the vertical angle between point \(p_{a}\) and \(p_{b}\). \\  & The \(n\)th Lidar historical key frame containing features, the frame coordinates are in the Lidar body \\ \(\mathbb{F}_{n}\) & coordinate system. \(\mathbb{F}_{n}=\left\{\mathbb{F}_{n}^{\mathrm{e}},\mathbb{F}_{n}^{\mathrm{p}}\right\}\), \(\mathbb{F}_{n}^{\mathrm{e}}\) represents the extracted polarized edge feature; \(\mathbb{F}_{n}^{\mathrm{p}}\) represents the extracted polarized planar feature. \\ \(\mu\mathbb{F}_{n}\) & The \(n\)th Lidar historical keyframe in the global coordinate system. \\ \(c_{k}\) & The curvature of the point \(p_{k}\). \\ \(\mathbf{S}\) & Point sets distributed on the left and right sides of point \(k\), \(|\mathbf{S}|\) represents the number of point sets distributed. \\ \(r\) & Point depth value. \(r_{j}\) is the depth value of the \(j\)th point in \(\mathbf{S}\) in the polarized matrix of point cloud, and \(r_{k}\) is the depth value of the \(k\)th point in \(\mathbf{S}\) in the polarized matrix. \\  & Threshold. \(\varepsilon^{\mathrm{e}}\) represents the curvature threshold for points with edge features; \(\varepsilon^{\mathrm{P}}\) represents the curvature threshold for points with planar features; \(\varepsilon_{dis}\) represents the Euclidean distance threshold; \(\varepsilon_{ab}^{\mathrm{h}}\) represents the vertical angle difference threshold between point clouds; \(\varepsilon_{l}\) represents the range threshold for the parallel parking fitting line; \(\varepsilon_{uv}\) represents the range threshold for the perpendicular parking fitting line. \\ \(\mathbf{M}_{\varepsilon}\) & The local map within the range of \(\varepsilon_{dis}\) is composed of historical key frames with edge features and planar features. \\ \(\Delta\mathbf{T}_{n-1,n}\) & Odometry factor for vehicle state \(\hat{x}_{n-1}\) to \(\hat{x}_{n}\). \\ \(\Delta\mathbf{T}_{n}^{\mathrm{ICP}}\) & Closed-loop detection factor calculated by ICP. \\ \(\mathbf{M}_{\mathrm{ROI}}\) & The ROI in the map where parking space detection is required. \\ \(\mathbf{p}_{cluster_{i}}\) & Point clouds of different clusters after clustering. \(cluster_{i}\) represents the category of the point cloud cluster. \\ \(\mathbf{p}_{cluster_{i}}\) & The number of categories is positively correlated with the number of vehicles contained in the ROI. \\  & Obstacle vehicle point cloud, \(vehicle_{i}\), represents the category of the retained obstacle vehicle. Rules: The sides of the vehicle door are the long side, represented by the character \(l\); the sides of the front and rear windshield of the vehicle are the wide side, represented by the character \(w\). \\ \(\mathbf{p}_{\mathrm{CH}}\) & Convex hull point cloud of the obstacle vehicle obtained after drawing the convex hull of the obstacle vehicle. \(\mathrm{CH}_{i}\) represents different convex hull categories drawn for different obstacle vehicles. \\ \(size_{l}\), \(size_{w}\) & \(size_{l}\) represents the vehicle length size constraint; \(size_{w}\) represents the vehicle width size constraint. \\ \hline \hline \end{tabular}
\end{table}