Superior Control of Spacecraft Re-Entry Trajectory

Abstract

This paper focuses on the re-entry phase of lunar return spacecraft and addresses the design optimization of their re-entry trajectories in real-world conditions. Considering various constraints of re-entry flights, this study introduces a refined superior control theory, drawing from Xuesen Qian’s descriptions in engineering control theory, and presents a specific superior control algorithm. The designed superior control algorithm and the traditional weighted optimal control algorithm were employed to simulate the lunar return and re-entry processes. Two representative trajectories were selected for a comparative analysis to obtain various parameters. Results indicate that the trajectory optimized using the weighted optimal control algorithm can only ensure that multiple performance indexes are optimized according to preset weights but cannot achieve superior performance in all metrics. In contrast, trajectories optimized using the superior control algorithm effectively leverage the permissible floating range of performance indexes without exceeding the maximum limit, thereby ensuring superior performance in all metrics. This paper is the first to refine the superior control theory proposed by Xuesen Qian, to design a specific algorithm theory for superior control, and to apply it to aerospace re-entry trajectory optimization—providing a theoretical foundation for future non-weighted control algorithm developments.

1. Introduction

In the design of control systems, traditional control methods evaluate the quality based on a given single index. If multiple aspects are required to be good in this process, it is impossible to achieve this using this single index control method. The most common method is to clarify the main and secondary optimization requirements of the system, assign corresponding weights to each index, and obtain a comprehensive index after weighting. By optimizing only this single comprehensive index, the optimization result can be obtained. However, in a huge system engineering, the complexity of optimization problems increases exponentially. It is not applicable to design control methods based on a single index, and assigning weights based on experience often ignores secondary contradictions. When the index with a large weight reaches an optimal value, the secondary indexes cannot be satisfied at all. However, secondary needs also need to be met in large systems [1].

Some control algorithms are also trying to reduce the presence of weighting factors, such as PID control, which does not directly use weighting factors but achieves control by adjusting the three parameters of proportional integral and differential [2]. Although the Linear Quadratic Regulator (LQR) involves the selection of weight matrices, once the design is completed, the control law itself does not involve the weighting process [3]. Model predictive control (MPC) may use weights to balance different performance metrics during the design process, but its core algorithm does not directly rely on weighting factors but instead calculates control inputs by optimizing a performance metric within a prediction horizon [4]. Sliding mode control designs a sliding surface and corresponding control law to ensure that the system state can reach and remain on the sliding surface without deviation, without involving weighting factors and relying on the invariance and robustness of the system [5,6]. Adaptive control can automatically adjust its parameters to adapt to dynamic changes in the system, usually without using weighting factors, but relies on an adaptive law to adjust the control action [7]. However, adaptive laws also select one or some indexes for adaptive design, which has the same problem as weighted control, that is, it cannot simultaneously meet multiple index requirements. A robust controller is designed to deal with model uncertainty. By modeling the uncertainty of the system, it usually does not use weighting factors [8]. It designs a controller by ensuring that all allowed uncertainties are modeled, making the closed-loop system stable for all allowed uncertainties and meeting preset performance metrics. H∞ control theory designs a controller to minimize the maximum gain from external disturbances to the system output [9]. Although weights may be involved in the design process, the final control law does not contain weighting factors.

When it is desired to optimize the performance indexes of each subsystem and its subsystems in the system simultaneously, superior control without weight coefficients is required. Initially, Xuesen Qian proposed the idea and theory of superior control in Engineering Cybernetics [1] but did not provide specific simulation modeling methods. This article first theoretically refines the theoretical content of superior control and then proposes a simulation algorithm for superior control.

During the re-entry flight of the lunar landing, due to the proportional relationship between the aerodynamic heating rate and the flight speed raised to the power of 3.15, when the re-entry point speed is 10.8 km/s, the aerodynamic heating, overload, and dynamic pressure are much more severe than when the speed is 7.8 km/s [10]. Therefore, in order to avoid out-of-control and landing point deviation caused by severe aerodynamic ablation [11] and to reduce aerodynamic heating, overload, and dynamic pressure [12,13,14]—in response to the severe aerodynamic heating environment caused by a re-entry speed of 10.8 km/s, in order to reduce the cost of thermal protection systems and other overall design schemes [15,16,17], such as more fuel for braking and orbit-changing engines, and reduce the payload—it is necessary to optimize the design of the return re-entry trajectory [18,19,20]. For the lunar landing return re-entry scenario with a re-entry speed of 10.8 km/s [21], this article uses C++ to perform programming simulation and optimizes several lunar landing return re-entry trajectories and their parameters using two algorithms. The performance indexes of the optimized design are maximum payload. They also reduce the mass of the thermal protection system and reduce fuel consumption of the orbit maneuvering engine.

This article organizes the re-entry trajectories obtained from the simulations of the two algorithms and conducts a detailed analysis using a representative trajectory for each. The analysis results show that both algorithms can optimize the available re-entry trajectory, but their performance in terms of optimization performance indexes is different. The innovative aspects of this article are as follows:

(1)

For the first time, Xuesen Qian’s superior control theory was refined, and the theoretical model of superior control was written in detail;

(2)

A simulation algorithm of Xuesen Qian’s superior control theory was designed for the first time;

(3)

For the first time, superior control was applied to the field of aerospace re-entry trajectory optimization for analysis.

2. Optimization Mathematical Model for Re-Entry Trajectory and Superior Control Algorithms

2.1. Performance Indexes for Re-Entry Trajectory

There are three performance indexes for the optimized re-entry trajectory system in this article: heating rate ${\dot{Q}}_S$, overload $n_Y$, and dynamic pressure q. Minimize the following performance indexes:

$$J_{min}=\underset{\overrightarrow{U}\left(t\right)}{min}{\int }_{t_0}^{t_f}\left[\begin{array}{c}{\dot{Q}}_S\hfill \\ n_Y\hfill \\ q\hfill \end{array}\right],$$

(1)

Each performance index in the above equation is a real-valued function of the system state $x\left(t\right)$ and control variables $u\left(t\right)$. Traditional performance indexes require weight coefficients, while superior control algorithms do not have weight coefficients in their performance indexes.

2.2. Constraints on Re-Entry Trajectory

The differential equation of flight mechanics during re-entry flight is as follows:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=f[t,\mathbf{x}\left(t\right),\mathbf{u}\left(t\right)],$$

(2)

$\mathbf{x}=[V,\gamma ,\psi ,h,\theta ,\varphi ]$ is the status parameter; $\mathbf{u}=[\alpha ,\sigma ]$ is the control parameter; t is time. The differential equations of flight motion can be found in [4,5,6,7,8].

Heating rate ${\dot{Q}}_S$ and overload $n_Y$ are subject to the following constraints:

$${\dot{Q}}_S\le {\dot{Q}}_{SMax},$$

(3)

$$n_Y\le n_{YMax},$$

(4)

$${\dot{Q}}_S\left(t\right)={\displaystyle \frac{17600}{\sqrt{R_N}}}\sqrt{{\displaystyle \frac{\rho }{{\rho }_0}}}{\left({\displaystyle \frac{V}{V_0}}\right)}^{3.15},$$

(5)

In the above equation, $V_0$ = 10.8 km/s; $\rho$ and ${\rho }_0$ are the atmospheric densities at local and sea levels, respectively; and the subscript “Max” is the maximum allowable value. The control variables are constrained as follows:

$${\alpha }_{Min}\le \alpha \le {\alpha }_{Max},$$

(6)

$${\sigma }_{Min}\le \sigma \le {\sigma }_{Max},$$

(7)

The subscript ‘Min’ represents the minimum allowable value. The subscript ‘Max’ represents the maximum allowable value. From starting point 0 to ending point F,

$$\begin{array}{c}V\left(t_0\right)=V_0,\gamma \left(t_0\right)={\gamma }_0,\psi \left(t_0\right)={\psi }_0,\hfill \\ \theta \left(t_0\right)={\theta }_0,h\left(t_0\right)=h_0,\varphi \left(t_0\right)={\varphi }_0;\hfill \end{array},$$

(8)

$$\begin{array}{c}V\left(t_f\right)=V_f,\gamma \left(t_f\right)={\gamma }_f,\psi \left(t_f\right)={\psi }_f,\hfill \\ \theta \left(t_f\right)={\theta }_f,h\left(t_f\right)=h_f,\varphi \left(t_f\right)={\varphi }_f;\hfill \end{array},$$

(9)

The subscript ‘0’ represents the initial time. The subscript ‘f’ represents the terminal time.

2.3. Mathematical Model for Re-Entry Trajectory Design

The mathematical model for re-entry trajectory design can be found in papers [11,12,13,14,15,16]. This article considers the six degree of freedom variables during the re-entry flight of spacecraft, models its re-entry process, and obtains differential equations:

$$\begin{array}{c}\dot{V}={\displaystyle \frac{Pcos\alpha cos\beta -c_{x}qS}{m}}-gsin\gamma +\\ {\omega }_d^{2}rcos\varphi (sin\gamma cos\varphi -cos\gamma sin\psi sin\varphi )+a_{Wx}\end{array},$$

(10)

$$\begin{array}{c}\dot{\gamma }={\displaystyle \frac{P(sin\alpha cos\sigma +cos\alpha sin\beta sin\sigma )}{mV}}+{\displaystyle \frac{c_{y}qScos\sigma }{mV}}-\\ {\displaystyle \frac{c_{x}qSsin\sigma }{mV}}+{\omega }_d^{2}rcos\varphi {\displaystyle \frac{cos\gamma cos\varphi +sin\gamma sin\psi sin\varphi }{V}}+\\ ({\displaystyle \frac{V}{r}}-{\displaystyle \frac{g}{V}})cos\gamma +2{\omega }_{d}cos\psi cos\varphi +{\displaystyle \frac{a_{Wy}}{V}}\end{array},$$

(11)

$$\begin{array}{c}\dot{\psi }={\displaystyle \frac{P(sin\alpha cos\sigma +cos\alpha sin\beta cos\sigma )}{mV}}-{\displaystyle \frac{qS}{mVcos\gamma }}(C_{y}sin\sigma +\hfill \\ C_{z}cos\sigma )+2{\omega }_d(tg\gamma sin\psi cos\varphi -sin\varphi )-\hfill \\ {\displaystyle \frac{{\omega }_d^{2}r}{Vcos\gamma }}cos\psi sin\varphi cos\varphi -{\displaystyle \frac{V}{r}}cos\gamma cos\psi tg\varphi +{\displaystyle \frac{a_{Wz}}{Vcos\gamma }}\hfill \end{array},$$

(12)

$$\dot{h}=Vsin\gamma +V_{Wy},$$

(13)

$$\dot{\theta }=Vcos\gamma cos\psi +V_{Wx},$$

(14)

$$\dot{\varphi }=-Vcos\gamma sin\psi +V_{Wz},$$

(15)

In the above equation, V, $\gamma$, and $\psi$, respectively, represent the speed, path angle, and heading angle of the re-entry vehicle; $\alpha$, $\beta$, and $\sigma$ correspond to the angle of attack, sideslip angle, and roll angle, respectively; S and L, respectively, represent the reference area and reference length; $C_x$, $C_y$, and $C_z$ are the drag coefficient, lift coefficient, and side force coefficient, respectively; h, $\theta$, and $\phi$ are the altitudes at which the spacecraft is located and the distance traveled in the longitude and latitude directions, respectively; P is the engine thrust; $V_{W_y}$, $V_{W_x}$, and $V_{W_z}$ are the wind speeds in the vertical, longitude, and latitude directions of the spacecraft in the ground coordinate system, respectively; $a_{W_y}$, $a_{W_x}$, and $a_{W_z}$ are the wind shears in the y, x, and z directions in the ballistic coordinate system; R is the Earth’s radius; and $g_0$ is the gravitational acceleration at sea level.

The six degrees of freedom equations of motion used in this paper include the wind field term—wind shear—for two reasons: Firstly, with the improvement of re-entry accuracy requirements, more influencing factors should be considered. Secondly, when the spacecraft enters other planets with a high wind field magnitude, the influence of the wind field must be considered

Resistance of spacecraft during aerodynamic deceleration re-entry into the atmosphere, D, is as follows:

$$D={\displaystyle \frac{1}{2}}\rho V^2{C}_{D}S,$$

(16)

Among them, $\rho$ is the density of air of the current position; $C_D$ is the drag coefficient.

2.4. Performance Indexes of Theoretical Model for Superior Control

The concept of superior control refers to the process of optimizing performance indicators without using weighting coefficients, that is, without artificially imposing importance restrictions on each performance indicator, and optimizing all performance indicators equally.

In the theory of superior control, the performance index matrix is $J=\left[\begin{array}{c}{J}_1,J_2,J_3,\dots \dots ,J_n\hfill \end{array}\right]$, and each performance index has its own constraints: $J_{1Min}\le J_1\le J_{1Max}$; $J_{2Min}\le J_2\le J_{2Max}$; $J_{3Min}\le J_3\le J_{3Max}$; ……; $J_{nMin}\le J_n\le J_{nMax}$. The performance indexes of more specific optimization control problems in spacecraft re-entry trajectory optimization can be expressed as follows:

$$J_{min}=\underset{U_{DU},U_{DP}}{\overset{{†}_{\forall A}}{opt}}\left\{\begin{array}{c}{P}_w\left(t\right)\\ {\mathsf{\Omega }}_T\left(t\right)\\ {\overrightarrow{W}}_{PI}\left(t\right)\\ \mathsf{\Phi }[\overrightarrow{x}\left(t_f\right),t_f]\\ {\int }_{t_0}^{t_f}\mathsf{\Theta }[t,\overrightarrow{x},\mathsf{\Lambda },\mathsf{\Xi },\overrightarrow{U}\left(t\right),{\overrightarrow{U}}_s,S_X]dt\\ \cdots \cdots \end{array}\right\},$$

(17)

In the above equation, ${†}_{\forall A}$ represents all detected information, including internal structural information of the spacecraft and external environmental information; $U_{DU}$ refers to variables that spacecraft users can control and specify, including static and dynamic variables; $U_{DP}$ is a variable that spacecraft designers can control; $P_w\left(t\right)$ is the flight mission that the spacecraft needs to complete, including waypoints, mission points, and the impact of the mission on the spacecraft; ${\mathsf{\Omega }}_T\left(t\right)$ refers to the flight’s restricted area that the spacecraft needs to avoid, specifically expressed as ${\mathsf{\Omega }}_{Threat}(t,x,h,z)$; ${\overrightarrow{W}}_{PI}\left(t\right)$ represents the performance indexes of the spacecraft itself, including structural indexes and the index requirements of each subsystem, including weight, cost, volume, load, etc; $\mathsf{\Phi }[\overrightarrow{x}\left(t_f\right),t_f]$ represents terminal technical indexes, namely the terminal error; ${\int }_{t_0}^{t_f}\mathsf{\Theta }[\dots \dots ]$ refers to the conceptual performance indexes that may be obtained during the flight process, where $\overrightarrow{x}$ refers to the flight state of the spacecraft; $\mathsf{\Lambda }$ refers to the overall mathematical model of the spacecraft; $\mathsf{\Xi }$ refers to the mathematical model of the environment in which the spacecraft is located. The system motion equation is Equation (2), where $x\left(t_0\right)=x_0$, $\mathbf{f}=f_1,f_2,\dots \dots ,f_k$, that is, the state space is k-dimensional. In this paper, $k=3$. The index function (1) can also be written in vector form:

$${\displaystyle \frac{d\mathbf{z}}{dt}}=\varphi [\mathbf{x},\mathbf{u},t],\mathbf{z}\left(t_0\right)=0,$$

(18)

Combine the state equation and performance index equation, denoted as

$$\mathbf{y}\left(t\right)=\mathbf{x}\left(t\right),\mathbf{z}\left(t\right),g=\{f,\varphi \},$$

(19)

The entire system can be abbreviated as

$${\displaystyle \frac{d\mathbf{y}}{dt}}\left(t\right)=g[\mathbf{x},\mathbf{u},t],y\left(t_0\right)=\{{\mathbf{x}}_0,0\},$$

(20)

2.5. Optimization Algorithm for Superior Control of Re-Entry Flight Trajectory

The design of the optimized control algorithm for the entire re-entry flight trajectory is as follows:

• Step 1:

Set a time step $\Delta t$, calculate only one performance index at each step, such as the first performance index in the first step and the second performance index in the second step, taking turns calculating each performance index. After calculating the last performance index, recalculate the first performance index.

If $\Delta t=1$,

${\mathbf{J}}_c=[J_1,J_2,J_3,\dots ,J_n]\times {[1,\underset{(n-1)copiesof0}{\underbrace{0,0,\dots ,0}}]}^T$;

If $\Delta t=2$,

${\mathbf{J}}_c=[J_1,J_2,J_3,\dots ,J_n]\times {[0,1,\underset{(n-2)copiesof0}{\underbrace{0,\dots ,0}}]}^T$;…;

If $\Delta t=i$,

${\mathbf{J}}_c=[J_1,J_2,J_3,\dots ,J_n]\times {[\underset{(i-1)copiesof0}{\underbrace{0,\dots ,0}},1,\underset{(n-i)copiesof0}{\underbrace{0,\dots ,0}}]}^T$.

Among them, ${\mathbf{J}}_c$ is the performance metric that needs to be optimized, n is the number of performance metrics, and ${[\underset{(i-1)copiesof0}{\underbrace{0,\dots ,0}},1,\underset{(n-i)copiesof0}{\underbrace{0,\dots ,0}}]}^T$ is the superior optimization matrix, $i=1,2,\dots ,n$.

If $\Delta t>n$, ${\mathbf{J}}_c$ starts the loop from $\Delta t=1$ to $\Delta t=n$ again, which continues until the calculation is completed.

• Step 2:

Definition: The maximum allowable limit refers to a value designed for performance indicators; this value is closer to the maximum value of this performance indicator and is used to determine whether the current value of a performance indicator needs to be optimized. When the value of a performance metric exceeds its maximum allowable limit, optimization of that performance metric begins. When the value of a performance indicator does not exceed its maximum allowable limit, the performance indicator is allowed to change freely without optimization.

When a certain performance index $J_{k_1}$ exceeds the maximum allowable limit, optimization is carried out for it, and the current rotation performance index and the performance index that need to be optimized are calculated. The maximum allowable limit designed in this article is 0.8 $J_{k_{1}Max}$, where $J_{k_{1}Max}$ is the maximum value constrained by performance metric $0.8J_{k_1}$. Let the current number of performance indexes exceeding the maximum permitted limit be m and the set of performance indexes exceeding the maximum permitted limit be $\overrightarrow{k}=[k_1,k_2,\dots ,k_m]$.

At this point $t<n\Delta t$, when $t=k_1$ and when the $k_1$-th performance metric has exceeded the maximum allowable limit $J_{k_1}$≥ 0.8 $J_{k_{1}Max}$, in addition to the performance metric being calculated alternately, $J_{k_1}$ is optimized and calculated simultaneously until $J_{k_1}$ < 0.8 $J_{k_{1}Max}$. The mathematical expression is

${\mathbf{J}}_c=[J_1,J_2,J_3,\dots ,J_n]\times {[\underset{(k-1)copiesof0}{\underbrace{0,\dots ,0}},\underset{k-th}{\underbrace{1}},\underset{(i-1-k)copiesof0}{\underbrace{0,\dots ,0}},\underset{i-th}{\underbrace{1}},\underset{(n-i)copiesof0}{\underbrace{0,\dots 0}}]}^T$.

• Step 3:

Calculate all performance metrics that exceed the maximum permitted limit and optimize them until they are optimized to within the permitted limit. If a certain index always exceeds the maximum allowable limit during the optimization process, the optimization will be maintained until the calculation of trajectory optimization is completed.

3. Numerical Simulation Results and Analysis

3.1. Numerical Simulation Background

The research object of this article is the trajectory of the re-entry flight phase of the lunar return spacecraft, which will undergo re-entry flight in the airspace with wind fields. Use superior control algorithms and weighted optimal control algorithms to optimize the trajectory of the lunar return spacecraft to reach the determined destination. The starting point for trajectory optimization is located at an altitude of 120 km above the ground, with an initial flight speed of 10,800 m/s. The lunar return spacecraft is required to meet the minimum requirements of multiple indexes during flight in airspace with wind fields.

The simulation involves comparing the various parameters of trajectories optimized by two control algorithms. One of them is the superior control algorithm designed in this article, and the other is to use traditional optimal control algorithms for trajectory optimization. The performance index calculation method is shown in Equation (21).

$$J_0=\underset{\overrightarrow{U}\left(t\right)}{min}{\int }_{t_0}^{t_f}(a{\dot{Q}}_S+b{n}_Y+cq)dt,$$

(21)

Among them, a, b, and c are the weight coefficients of their corresponding performance indicators, which are manually set and can be modified according to the situation.

During the simulation process, various parameters of the lunar return spacecraft were inputted, and numerical simulations were conducted using C++ programming in the WIN11 system to obtain different re-entry trajectory data. The basic parameter settings for the mathematical model of the lunar return spacecraft are as follows: lift-to-drag ratio $\lambda$ = 0.35, aerodynamic reference area S = 95 m², and mass M = 20,000 kg. The heating rate is required to at least meet the condition ${\dot{Q}}_S\le {\dot{Q}}_{SMax}$ = 450 kw/m². Overload should at least meet the requirement $n_Y\le n_{YMax}$ = 2 g.

The horizontal wind field is shown in Figure 1. The starting point of each blue small arrow represents the coordinates of the point, the length of the arrow indicates the wind speed at the point, and the direction of the arrow is the direction of the wind at the point. This article is a fixed wind field simulation, and the specific wind speed can be set and modified as needed in the simulation. In practice, the trajectory can be re-optimized based on the real-time detected changes in the wind field situation.

3.2. Simulation Results and Analysis

Several trajectories were obtained through a simulation, and each algorithm took one trajectory to serve as an example in this article.

Figure 2 shows the trajectory diagrams of two optimized lunar return spacecraft’s re-entry and landing at the same target location. The blue trajectory represents the result obtained by the superior control algorithm, and the green trajectory represents the result obtained by the weighted optimal control algorithm. It can be seen that the trajectories obtained by both algorithms have achieved the goal of reaching the target destination. Relatively speaking, the green trajectory is smoother, while the blue trajectory has more small turns. This is because the blue trajectory will change the currently optimized index in real time based on whether the current performance index value reaches 80% of the maximum allowable limit. Therefore, the current speed direction in the optimized trajectory will also change accordingly. Figure 3 is a 2D diagram of the trajectories optimized by the two algorithms, including a top view and a side view.

Figure 4 shows a comparison of moving the starting points of the two trajectories to the same location. It can be clearly be seen from the middle that the trajectories are very similar during the initial period of re-entry, indicating that the selection of weighted design during the initial period is easy to ensure that multiple indicators are within the ideal range.

Figure 5 shows the curves of the velocity of the trajectories optimized by the two algorithms over time. It can be seen that the trends of the two curves are similar, and the landing time is relatively shorter compared to the superior control algorithm.

Figure 6 shows a comparison of the heating rates of trajectories optimized by the two algorithms. Figure 7 is a comparison of the overload of trajectories optimized by the two algorithms. Combining the two figures, it can be seen that in Figure 6—when the heating rate value of the superior control exceeds the maximum allowable limit of 80%, indicated by the red dashed line in the figure—priority is given to optimizing only the heating rate performance indicator. At this time, the overload has not reached 80% of the maximum allowable limit, so the overload is allowed to continue to rise. In Figure 7, it can be seen that the maximum overload value obtained by superior control is nearly twice as large as that of the weighted optimal control, but its value is still within the allowable range. This is because the weighted class control algorithm calculates the weights of each performance indicator as fixed values, which is an empirical calculation method that cannot adapt to the state of each performance indicator itself in practice. In practical applications, it cannot adapt to every sudden or unknown state in natural situations.

Figure 8 shows the temperature changes over time for the trajectories optimized by the two algorithms. It can be seen that under the same external environment, the temperature trends of both algorithms are similar, with a sharp increase at first, fluctuations at high temperatures, and then a decrease. After a period of time, they rise again and reach a maximum before decreasing. The maximum value is basically the same, and the minimum and maximum values are slightly higher trajectories for superior control algorithms. Figure 9 shows the variation of path and heading angle over time for the two optimized algorithms.

Figure 10 shows the variation of control variables of trajectories optimized by the two algorithms over time. It can be seen that in terms of the angle of attack control, the two are relatively close, while in terms of roll angle control, superior control has more operations than weighted optimal control. This also reflects that superior control is more sensitive to changes in body parameters and requires more control actions to achieve control objectives.

4. Conclusions and Discussion

In the past, research on control algorithms did not intentionally reduce the role of weights. This article innovatively refines Xuesen Qian’s concept of superior control without weights into a specific theory and designs corresponding algorithms for programming and simulations. This article applies the designed superior control algorithm to the trajectory optimization of lunar return and re-entry, and optimizes the re-entry point velocity of 10.8 km/s using both the superior control algorithm and the weighted optimal control algorithm. Research has shown that both algorithms have optimized feasible re-entry trajectories, with the maximum heating rate of the superior control being smaller than that of the weighted optimal control. In terms of overload, the maximum overload of superior control slightly exceeds 80% of the maximum allowable limit, which slightly greater than the maximum overload value of weighted optimal control.

From the simulation results in this article, it can be seen that both the superior control algorithm and the weighted class algorithm can obtain usable re-entry trajectories. However, the performance indexes in the superior control algorithm will be optimized more evenly and will not be prioritized for important indexes, nor will their limits be ignored because they are not the most important indexes. On the other hand, superior control better utilizes the floating range of performance indexes that do not exceed 80% of the maximum allowable limit, which provides greater optimization space for performance indexes that exceed 80% of the maximum allowable limit.

With the rapid development of contemporary computer computing power, there will be more and more performance indexes in the field of aerospace re-entry trajectory optimization, and there will be more refined requirements for optimization constraints. We believe that researchers will gradually stop using known weights to limit performance indexes. Moreover, we suggest that more simulations should be conducted for experimentation. In addition, superior control can be applied in the field of aerospace re-entry trajectory optimization and in the optimization calculation of other multi-optimization objectives using superior control algorithms. This is also worthy of further research.