## 3 Algorithm for Multi-Station Satellite Orbit Optimization in Common-View
### Simulation Annealing Algorithm and Optimization  
The six parameters that characterize a satellite's Keplerian orbit are important in determining the most efficient trajectory design. This design challenge can be approached as a static parametric optimization problem with a finite set of choice variables [26; 27]. However, it is important to note that numerical gradient-based optimization algorithms may only converge to local optima, while heuristic methods have the potential to reach global optimal results. Therefore, there is a significant interest in utilizing specialized optimization algorithms to solve constrained trajectory optimization problems [28]. Additionally, heuristic methods can bypass computational obstacles associated with Hessian and Jacobian matrices by eliminating the need for derivative information. The general form of a constrained optimal control problem is as follows [29]:  
\[\dot{x}(t)=f(x(t),u(t))\, \tag{10}\]  
where \(x(t)\in\mathbb{R}^{n_{x}}\) and \(y(t)\in\mathbb{R}^{n_{u}}\), defining the system state and control variables over the time interval \(t\in[t_{0},t_{f}]\), with \(n_{x}\) and \(n_{u}\) being the dimensions of the system state and control variables, respectively. The form of the boundaries and constraints are as follows:  
\[ \begin{array}{c}g(x(t),u(t))\leq 0\\ x(t_{0})=x_{0}\\ x(t_{f})=x_{f}\end{array} \, \tag{11}\]  
The general form of the objective function is:  
\[\underset{u(t)}{min}\,J_{1}=\int_{t_{0}}^{t_{f}}L(x(t),u(t))dt+\Phi(x(t_{f}),t _{f})\, \tag{12}\]  
where \(L(x(t),u(t))\) and \(\Phi(x(t_{f}),t_{f})\) represent the process and terminal performance indicators, respectively. The simulated annealing algorithm is a probabilistic metaheuristic algorithm used for global optimization. It operates on the principle of gradually reducing a system's energy to reach a low-energy state, akin to the process of cooling a solid from high temperatures [21]. This method is particularly effective in handling nonlinear, discontinuous, and stochastic objective functions. It proves advantageous in scenarios with a high-dimensional state space and substantial memory requirements [29]. The quality of the solution for combinatorial minimization problems hinges on several factors: the length of the Markov chain, the halting criterion, the temperature cooling rate, and the initial temperature. The generating principle of the domain aligns with the interference mechanism of the Metropolis algorithm, and the annealing method can be adjusted as necessary. Given an extremely low temperature and an extremely slow decay rate, the simulated annealing algorithm can theoretically approach the global optimal solution with high precision. To ensure reasonable computation time and manage the instability of the simulated annealing algorithm caused by the probability requirement, we propose the following annealing strategy:  
1. Initial parameter settings: The initial guesses are selected randomly, while the upper and lower parameter boundaries are set at \(i\in[0,180]\) and \(\Omega\in[0,360]\), correspondingly. The length of the Markov chain is set at 200. 2. Cooling strategy: We adopt the linear annealing schedule, given by \(T=T_{init}-C*I\), where \(T_{init}\) is the initial temperature and \(C\) is the cooling coefficient [30]. To expand the search space, the initial temperature is set at 500. Following this, linear cooling is performed to set the cooling coefficient to 0.8. The Metropolis criterion is a criterion for accepting new solutions [30]. This will aid in quick convergence, reducing randomness, and avoiding local optimum solutions as confirmed by the Metropolis acceptance criterion. 3. Perturbation strategy: With the introduction of control strategies and by focusing perturbation strategies, various perturbation techniques are established, aligning with distinct Markov chain period durations. A Markov chain is a type of stochastic process in which the future state of the system is determined solely by its current state, without any influence from its past evolution [29].