# Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem
## 1 Introduction  
Maintaining, detecting, or intercepting targets in space has become a vital area of space technology research, providing countries with a more significant space information edge. Meanwhile, satellite constellations, which are made up of a large number of satellites orbiting in the same orbit [1, 2], such as the GPS Navigation System [3] and the Beidou Navigation System [4, 5], or satellite communication systems, such as OneWeb and StarLink [6], are playing an indispensable role in society, as well as in the field of national defense. Therefore, detecting or maintaining the satellites in these constellations has emerged as an essential research topic. The flyby multi-target problem [7, 8], particularly the flyby non-coplanar multi-target problem [9, 10], was investigated to some extent, but most solutions require numerous orbit maneuvers and are incredibly dependent on the ground station [11]. Minimizing the number of orbital maneuvers can effectively decrease the mission constraint, and thus, enhance the efficacy of each orbital maneuver; therefore, it is crucial to study the single-impulse flyby co-orbital multi-target problem. The Lambert problem, defined as the problem of finding the impulse time and value with two given positions and the transfer time, is the fundamental problem in the single-target flyby/interception problem. A typical way of solving the Lambert problem is to establish a connection between the transfer time and a Kepler element [12, 13, 14]. It is also common to convert the Lambert problem into an optimization problem by adding constraints [15, 16, 17] to achieve the optimal solution to the interception problem [18, 19]. Unfortunately, these methods are sensitive to an initial value for iteration, and thus, theb-spline interpolation function is introduced to provide the selection strategy of the initial value [20]. The Lancaster method is also combined to access initial values quickly [21]. Furthermore, heuristic algorithms, such as particle swarm optimization [21; 22], brainstorm optimization [23], the simulated annealing method [24], deep neural networks [25], and deterministic artificial intelligence [26], can also be utilized to solve the Lambert problem. The multi-target flyby/interception problem was previously investigated based on single-target flyby/interception research. For cosmic navigation [27], there are no cases of multiple planets in the same orbit; therefore, research is focused on the problem of multiple planets in the flyby hetero-orbit plane. For example, [PERSON] [28] examined the trajectory of flybys of Venus and Mars on a single interplanetary trip and conducted actual flight tests. In the area of artificial earth satellites, [PERSON] [29] studied the non-coplanar multiple-target interception problem with two or three impulses, proposed a general solution method, and improved the NSGA-II algorithm to seek the optimal solution. [PERSON] [30] suggested a traversing points-based method for intercepting multiple targets on the Walker constellation, which was able to quickly determine the orbit of non-coplanar multi-target interception. The feasibility of the method in a circular orbit was demonstrated by [PERSON] [31], who utilized a random greedy adaptive search procedure to optimize a sequence of rendezvous maneuvers by a spacecraft with multiple targets. [PERSON] [32] explored non-coplanar two-target interception with a single-impulse problem, establishing nonlinear equations that are solved using the Newton iteration method, predicated on the relationship between angular momentum and position vector between interplanar orbits. [PERSON] [33] simultaneously studied the problem of coplanar multi-target interception but did not conduct further studies on co-orbital targets. For a satellite constellation consisting of multiple orbital planes, a nonplanar multi-target flyby/intercept can be important as a way to traverse all satellites. However, at the same time, co-orbital multi-target flyby/interception, as another perspective, also has some importance.