## 2 Problem Formulation and Modeling
### Convex Function Construction  
In order to solve for the time-invariant parameter of the desired final state, we construct a convex function using property 1, in which parametric circles intersect at the same point. Firstly, the analytical solution of the intersection of two parameter circles is studied. For two sampling moments \(t_{1}\) and \(t_{2}\), there are two parameter circles. Suppose the centers of the two circles are at \((^{1}x_{c},^{2}x_{c})\) and \((^{2}x_{c},^{2}x_{c})\), and the radii are \({}^{1}r_{c}\) and \({}^{2}r_{c}\), respectively. Thus, the midpoint between the two parameter circles is \(\mathbf{O}=(\frac{1}{2}x_{c}+\frac{2}{2}x_{c},\frac{1}{2}y_{c}+y_{c})\), and we define \(L\) as the distance in between and take the unit basis vector \(\mathbf{e}_{a}=(\frac{2}{L-1}x_{c},\frac{2}{L}y_{c}-\frac{1}{L}y_{c})\) in the direction from one circle's center to another, while the orthogonal unit basis vector is \(\mathbf{e}_{b}=(\frac{2}{y_{c}-1}y_{c},-\frac{2}{L}x_{c}-1}{L})\). Thus, the analytic solution of the two intersections in Equation (12) can be obtained. \[\left\{ \begin{array}{l}\mathbf{J}_{1,2}^{+}=\mathbf{O}+A\cdot\mathbf{e}_{a} +B\cdot\mathbf{e}_{b}\\ \mathbf{J}_{1,2}^{-}=\mathbf{O}+A\cdot\mathbf{e}_{a}-B\cdot\mathbf{e}_{b}\\ \end{array} \right. \tag{12}\]  
where  
\[A=\frac{1}{2}\frac{{}^{1}r_{c}^{2}-^{2}r_{c}^{2}}{2L}\]  
and  
\[B=\frac{1}{2L}\sqrt{-[(^{1}r_{c}+^{2}r_{c})^{2}-L^{2}][(^{1}r_{c}-^{2}r_{c})^{ 2}-L^{2}]}\]  
In terms of the form of the analytical solution, there may be no intersections, i.e., imaginary roots, as shown in Figure 5. Therefore, the search space may not be limited to the real domain space \(\mathbb{R}^{3}\), and in order for the algorithm to run robustly, the problem should be extended to the unitary space \(\mathbb{C}^{3}\) in Section 3. Figure 4: The points on the 1 st, 15,001 st and 30,001 st trajectory lines were taken as sampling points, and the parameter circles were obtained. The process of solving for \(x^{*}_{f^{\prime}},y^{*}_{f^{\prime}}\) and \(\theta^{*}_{f}\) is regarded as the optimal estimation of the parameters. Once these values are obtained, they are substituted into Equation (10), which allows us to obtain the set of three parametric circles \(\mathbb{P}_{c}=\{(i^{x}{}_{c^{\prime}}{}^{i}y_{c^{\prime}}{}^{i}r_{c})|i=1,2,3\}\) for \(t_{i}\in(t_{0},t),i=1,2,3\) with \(t_{i}\in(t_{0},t),i=1,2,3\). There are up to six intersections between the three circles on the \(xOy\) plane, and according to the geometry of the problem, three of them intersect on the trajectory line at the desired end point of the trajectory. If any two parametric circles from them are chosen, say those corresponding to indices \(i\) and \(j\), there will be two intersection points \(I^{+}_{i,j}\) and \(I^{-}_{i,j}\), which constitute the geometric properties of the system. It is preferable to use binary \(s_{i}\) (where the symbol \(+\) represents the positive root and the symbol \(-\) represents the negative root) to represent and define the intersection point combinations simultaneously. It can be observed that there is a \(\frac{1}{2}\) probability of selecting the potential common point (end point of the trajectory) from the intersection points provided by each pair of circles. Consequently, there are eight possible combinations of the intersection points that are selected (\"++\", \"++-\", \"++\", \"+--\", \"-++\", \"-++\", \"--+\", \"--+\", \"--+\", \"----\"). \[\mathbb{C}_{s_{2},s_{1},s_{0}}=\{I^{s_{2}}_{1,2},I^{s_{1}}_{2,3},I^{s_{0}}_{3,1}\},s_{i}\in\{+,-\} \tag{13}\]  
Nevertheless, only a unique combination within the set \(\mathbb{C}_{s_{2},s_{1},s_{0}}\) satisfies the optimization criteria. This is demonstrated as follows.