## 2 Methods
### Point Cloud Alignment and Parameter Optimization Point Cloud Pose Estimation and Optimization
#### 2.3.2 Fine Registration by J-ICP  
After the coarse alignment, the two point clouds have been roughly aligned, but it is not sufficient to meet the accuracy requirements. Therefore, fine registration after coarse registration is a necessary step in the process of acquiring precise registration results of the given point cloud set. The ICP algorithm [21] has become a classic algorithm in point cloud fine alignment due to its wide range of applications. This algorithm is based on matching point pairs to align surfaces. It involves finding nearest-neighbor point pairs between a source and target point cloud, calculating transformation parameters and an objective function, applying the transformation to a subset of the source cloud, and iteratively optimizing the objective function to achieve convergence and obtain the optimal transformation matrix. In essence, the ICP algorithm is employed to determine the transformation matrix between two frames by minimizing a specific objective function:  
After the coarse alignment, the two sets of point clouds have been roughly overlapped, but they do not meet the required accuracy. Therefore, fine alignment is necessary to improve the accuracy of the point cloud alignment results. The ICP algorithm [21] is a well-known method for accurate point cloud alignment, which involves surface fitting based on aligning pairs of points. This algorithm identifies nearest-neighbor point pairs between the target and source point, calculates transformation parameters and an objective function, and then operates on a subset of the source point cloud. Through continuous iterations, the algorithm optimizes the objective function to achieve convergence conditions and determine the optimal transformation matrix. In essence, the ICP algorithm is utilized to estimate the transformation matrix between two frames in order to minimize the objective function defined in Equation (6). \[\underset{R,t}{\text{min}}E(R,t)=\sum_{N}^{i=1}\parallel Rp_{i}+t-p_{closet} \parallel_{2}^{2} \tag{6}\]  
\[p_{closet}=\underset{q_{i}\in Q}{\text{argmin}}\parallel p_{i}-q_{i} \parallel_{2}^{2} \tag{7}\]  
where \(p_{i}\) represents a point in the point cloud set of the current frame, while \(q_{i}\) is the corresponding point in the next frame, and \(Q\) denotes the set of all points in the point cloud of the next frame. The relationship between them is expressed in Equation (7). When aligning point cloud data acquired during small angle tumbling maneuver for targets with symmetry features, it has been observed that the traditional ICP algorithm often becomes stuck in local optimal solutions, resulting in a notable increase in alignment error. To address this issue, this paper introduces a novel approach called Joint Iterative Closest Point (J-ICP), which aims to minimize the distance between point clouds from multiple frames by taking into account the point cloud data obtained at various time points during small angle transformations. Assuming that the point clouds \(P_{1},P_{2},\ldots,P_{n}\) are acquired separately at \(t_{1},t_{2},\ldots,t_{n}\) time, the goal of J-ICP is to find a set of transformations \(\{T_{1},T_{2},\ldots,T_{n-1}\}\) to align each point cloud \(P_{i}\) with reference \(P_{r}\) and minimize the total alignment error of multiple frames by iterating. The target function can be expressed as:  
\[E(T_{1},T_{2},\ldots,T_{n-1})=\sum_{i=1}^{n-1}\sum_{p\in P_{i}}\parallel T_{i }(p)-NN_{P_{r}}(T_{i}(p))\parallel^{2} \tag{8}\]where \(NN_{P_{t}}(p)\) denotes that the point \(p\) is the nearest neighbor in the reference point cloud \(P_{r}\), and \(\|\cdot\|^{2}\) denotes the squared Euclidean distance. The transformation \(T_{i}\) is a rigid-body transformation with rotational and translational components that could align each of the \(P_{i}\) with the \(P_{r}\). For the transformation matrix under different frames obtained by the iterative nearest point algorithm \(T_{ji}= \begin{bmatrix}R&t\\ 0&1\end{bmatrix} \), which consists of a rotation matrix \(R\) and translation matrix \(t\).