# Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem
## 6 Conclusions  
In this study, the single-impulse flyby co-orbital spacecraft problem and the Lambert problem were studied, and a derivative-free method was provided. Focusing on the Lambert problem, the vertical component \(e_{T}\) of a single variable was selected, the relationship between \(e_{T}\) and the reciprocal of the transfer time \(\Delta t^{-1}\) was used to propose an initial value calculation method based on quadratic function fitting, and then this was combined with the secant method. This generated a non-derivative method to solve the problem. For the single-impulse flyby co-orbital spacecraft problem, using the relationship between the two intersection points of the target orbit and the transfer orbit, the problem was transformed into a nonlinear equation of the time \(t_{1}\) for a given impulse time \(t_{0}\), and the iterative formulas of the linear interpolation search method for initial value selection and the iterative formula of the chord secant method were given. The simulation results showed that the proposed method could effectively improve the solution efficiency of the Lambert problem and demonstrate the effectiveness of the transfer orbit calculation results for a given impulse time. In addition, the general conditions for the existence of a solution were presented based on simulation data, providing a foundation for future research on this problem.  
Of course, this research needs to be continued in further depth and focused on the following aspects. First, the details of the model should be further optimized. In this study, only a solution for the two-body condition was considered, and the next step could be to continue exploring the solution to this problem under orbital uptake. The second aspect is to analyze the access efficiency of a single pulse for the same orbit multi-target problem, and further investigate how to optimally use fuel by studying the access of a single target to the satellite constellation using different pulses to achieve it. Third, an engineering-based implementation of the Lambert algorithm is possible. In this study, we only used the MATLAB environment to compare the computational efficiency of different algorithms. In engineering, low-level languages are typically used; therefore, it will be more meaningful to engineer the algorithm using C language or Fortran. At this stage, vehicle trajectory optimization is becoming more and more intelligent, but compared with other intelligent  
Figure 10: The eccentricity and average \(\Delta\
u\) of the solutions.  
fields, whether it is stochastic AI or deterministic AI, the crucial prerequisite for intelligent vehicle trajectory is its mathematical and physical basis. Therefore, the next step in our research is determining how to make intelligent spacecraft learn to use these formulas.  
Conceptualization, H.S. and Z.D.; methodology, H.S.; software, H.S.; validation, H.S.; formal analysis, H.S.; investigation, H.S.; resources, H.S.; data curation, H.S.; writing--original draft preparation, H.S.; writing--review and editing, L.L.; visualization, H.S.; supervision, H.S.; project administration, H.S.; funding acquisition, Z.D. and L.X. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.