Initial trajectory design of low-thrust spacecraft considering attitude constraints

Abstract

The fuel carried by deep space exploration spacecraft is crucial for the completion of their exploration missions, and the fuel for attitude control engines is even more precious. In order to reduce the control requirements for attitude control systems, this paper proposes a shape-based trajectory optimization algorithm that considers attitude constraints for low-thrust spacecraft. This method obtains a more accurate transfer trajectory by considering the change rate and change range constraints of the propulsion acceleration direction of spacecraft. By comparing the simulation results without considering spacecraft attitude constraints, it is confirmed that the proposed algorithm considering attitude constraints is very important for the initial design of transfer trajectories. This is of great significance for high-precision initial trajectory optimization of deep space exploration missions.

Introduction

Early deep-space exploration used pulse rocket engines, but due to their low specific impulse, the cost of deep-space exploration was too high, and many missions were not technically achievable or economically affordable. Continuous low-thrust spacecraft, due to its high specific impulse, are very suitable for deep-space exploration¹. Under the same fuel consumption, they can generate larger velocity increments, greatly improving the payload ratio of the spacecraft, saving fuel consumption for the mission², and also improving the flexibility of the flight trajectory³.

According to the working principle of a continuous low-thrust propulsion system, the thrust generated is very small, and the propulsion system needs to operate stably for a long time to provide sufficient speed increment for the spacecraft⁴. This makes the corresponding trajectory design and optimization very challenging⁵. For trajectory optimization of continuous low-thrust spacecraft, direct or indirect methods can generally be used. The direct methods transform continuous trajectory optimization problems into nonlinear programming problems. The indirect method converts the optimal control problem in continuous trajectory optimization into a boundary value problem of two or more points, and then solves the nonlinear equations⁶. However, both the direct method⁷ and the indirect method⁸ have problems such as high computational complexity and difficulty in obtaining initial solutions.

In order to carry out initial mission design more effectively, designers need to balance requirements accuracy, optimality level, and computational load. Some intelligent algorithms have also been applied to solve trajectory optimization problems for continuous low-thrust spacecraft. Rauwolf and Coverstone-Carroll⁹ used genetic algorithms to design an low-thrust transfer trajectory. They divided the transfer trajectory into multiple segments and then optimized each segment trajectory. Ghosh and Conway¹⁰ applied particle swarm optimization technology to obtain approximate optimal solutions for various dynamic optimization problems. Sentinella and Casalino¹¹ used genetic algorithms to obtain initial solutions for indirect optimization methods. Pontani and Conway¹² used particle swarm optimization method to study the optimization problem of space transfer trajectories for continuous low-thrust spacecraft, providing initial guesses for indirect optimization algorithms. These intelligent algorithms are computationally complex and only suitable for situations with low precision requirements.

Therefore, some scholars have proposed the shape-based method^13,14 to quickly generate continuous low-thrust transfer trajectories¹⁵. These methods assume that the trajectory shape of the spacecraft satisfies certain functional forms^16,17, and then optimize the unknown variables of the function to meet various constraint requirements¹⁸. The advantage of shape-based methods is that they can efficiently and flexibly provide initial trajectories that meet the constraints of motion equations, boundary conditions, and maximum propulsion acceleration, while minimizing the objective function. The finite Fourier series (FFS) method proposed by Taheri and Abdelkhalik^19,20 also avoids solving any form of nonlinear algebraic equations when optimizing three-dimensional transfer trajectories, further reducing computational load. Fan et al.²¹ proposed the Bezier shape-based method, which, under the condition of only considering the propulsive acceleration constraint^22,23, can obtain transfer trajectories with better performance indicators in shorter computation time compared to the FFS shape-based method¹⁹.

The various shape-based methods currently proposed assume that the thrust direction of low-thrust spacecraft is arbitrary^24,25, only limiting the maximum thrust amplitude. However, in the practical application of low-thrust spacecraft, the distribution of thrusters is fixed, and the direction of thrust cannot change arbitrarily in a short period of time. The spacecraft requires a certain amount of time for attitude maneuver to provide thrust in the corresponding direction. Spacecraft can use momentum wheels and attitude control engines to adjust attitude, or use attitude control thrusters for momentum wheel unloading. The momentum wheel adopts electric energy control, and the attitude control engine needs to consume fuel. When the fuel is depleted, the spacecraft cannot use the attitude control engine to control its own attitude. Taking the Dawn spacecraft²⁶ with a total mass of 1217.8 kg as an example, it only carried 45.5 kg of hydrazine propellant for partial attitude adjustment, while the fuel used for orbit control was 425.3 kg. Therefore, it is necessary to constrain the attitude control of the spacecraft within a certain range to utilize momentum wheels as much as possible for attitude control, thereby reducing the use of attitude control engines. Therefore, this paper proposes a shape-based trajectory optimization algorithm that considers attitude constraints of low-thrust spacecraft to achieve more accurate trajectory optimization, which is of great significance for high-precision initial trajectory optimization of deep-space exploration missions.

The structure of this paper is as follows. Firstly, the coordinate systems and constraint conditions are described. Then, the Bezier shape-based trajectory optimization algorithm considering attitude constraints is introduced and compared through simulation. Finally, the conclusion is provided.

Problem description

Firstly, establish the heliocentric ecliptic inertial system $o_s{x}_s{y}_s{z}_s$, as shown in Fig. 1. The origin $o_s$ is the center of mass of the sun, with the positive $x_s$-axis pointing towards the equinox, the positive $z_s$-axis perpendicular to the ecliptic plane and pointing towards the north pole of the ecliptic, and the $y_s$-axis forming the right-hand system. The flight trajectory of the low-thrust spacecraft is described using the heliocentric cylindrical coordinate system, where $\rho$ is the radial distance, $\theta$ is the azimuth angle, and z is the altitude of the spacecraft.

Figure 1.

The heliocentric ecliptic inertial system and the heliocentric cylindrical coordinate system.

The equations of motion in the cylindrical coordinate system are

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \ddot{\rho }-\rho {\dot{\theta }}^2+{\mu }_{\odot }\rho /r^3=a_{\rho }\hfill \\ \hfill & \rho \ddot{\theta }+2\dot{\rho }\dot{\theta }=a_{\theta }\hfill \\ \hfill & \ddot{z}+{\mu }_{\odot }z/r^3=a_z\hfill \end{array}\right)\end{array}$$ 1

where $r=\sqrt{{\rho }^2+z^2}$ is the distance between the spacecraft and the sun, ${\mu }_{\odot }$ is the gravitational constant of the sun, and $a_{\rho }$, $a_{\theta }$ and $a_z$ are the propulsion acceleration components in three directions, respectively.

The required total propulsion acceleration a is

$$\begin{array}{c}\hfill a=\sqrt{a_{\rho }^2+a_{\theta }^2+a_z^2}\end{array}$$ 2

The speed increment $\mathrm{\Delta }V$ required for the low-thrust spacecraft to complete the transfer process is

$$\begin{array}{cc}& \mathrm{\Delta }V={\int }_0^{T}adt\hfill \end{array}$$ 3

where T is the total flight time.

Constraint conditions

Boundary constraints

When a spacecraft departs from Earth or other celestial bodies and intersects with an asteroid, the following boundary constraints need to be met at the position and speed of departure and arrival.

$$\begin{array}{c}\hfill \begin{array}{c}\rho (\tau =0)={\rho }_i,\rho (\tau =1)={\rho }_f,{\rho }^{\prime }(\tau =0)=T{\dot{\rho }}_i,{\rho }^{\prime }(\tau =1)=T{\dot{\rho }}_f\\ \theta (\tau =0)={\theta }_i,\theta (\tau =1)={\theta }_f,{\theta }^{\prime }(\tau =0)=T{\dot{\theta }}_i,{\theta }^{\prime }(\tau =1)=T{\dot{\theta }}_f\\ z(\tau =0)=z_i,z(\tau =1)=z_f,z^{\prime }(\tau =0)=T{\dot{z}}_i,z^{\prime }(\tau =1)=T{\dot{z}}_f\end{array}\end{array}$$ 4

where $\tau$ represents dimensionless time ($\tau =t/T\in [0,1]$), the subscripts “i” and “f” represent the departure and arrival conditions, respectively, and the superscripts $_{}^{·}$ and $_{}^{\prime }$ represent the derivatives of t and $\tau$, respectively.

Propulsion acceleration constraints

For low-thrust spacecraft, the maximum propulsion acceleration amplitude needs to be constrained as follows.

$$\begin{array}{c}\hfill a=\sqrt{a_{\rho }^2+a_{\theta }^2+a_z^2}\le a_{\max }\end{array}$$ 5

where $a_{\max }$ is the maximum propulsion acceleration amplitude of the low-thrust spacecraft.

Attitude constraints

The thruster layout of low thrust spacecraft is fixed, and the direction of propulsion acceleration cannot be arbitrarily changed in a short period of time. The spacecraft requires a certain amount of time for attitude maneuver to provide thrust in the corresponding direction, so the direction of propulsion acceleration cannot change instantaneously. Therefore, the following constraint on the change rate of propulsion acceleration direction for low-thrust spacecraft can be obtained.

$$\begin{array}{cc}& \left|\frac{\text{d}{\mathit{e}}_a}{\text{d}t}\right|\le {\omega }_{\max }\hfill \end{array}$$ 6

where ${\mathit{e}}_a$ is the unit vector of the propulsion direction of the low-thrust spacecraft, and ${\omega }_{\max }$ is the maximum angular rate generated by the attitude control system.

Like the Dawn spacecraft²⁶, the low-thrust spacecraft have very little fuel for attitude control engines, so in order to reduce the working time of attitude control engines, it is necessary to constrain the thrust adjustment angle of the spacecraft within a certain range.

$$\begin{array}{cc}& \left|\mathrm{\Delta },{\beta }_{{\mathit{e}}_a}\right|\le {\beta }_{\max }\hfill \end{array}$$ 7

where $\mathrm{\Delta }{\beta }_{{\mathit{e}}_a}$ is the angle between the thrust direction at each moment and the initial thrust direction, and ${\beta }_{\max }$ is the maximum angle constraint. The initial thrust direction is defined as the direction of the propulsion acceleration of the low-thrust spacecraft at the departure time. As the propulsion acceleration of the spacecraft also changes with the optimization of the flight trajectory during the optimization process, the initial thrust direction is not determined in advance, but varies according to the optimization situation.

Bezier shape-based method with attitude constraints

$\rho$, $\theta$, and z are expressed as Bezier curve equations in the following form.

$$\begin{array}{cc}\hfill c(\tau )& =\sum _{j=0}^{n_c}{B}_{c,j}(\tau )P_{c,j}c=\{\rho ,\theta ,z\}\hfill \end{array}$$ 8

where $n_c$ are the orders of Bezier curve equations, $P_{c,j}$ are the unknown coefficients, and $B_{c,j}(\tau )$ are the Bezier basis functions of the state variables.

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill B_{c,j}(\tau )& =\frac{n_c!}{j!(n_c-j)!}{\tau }^j{(1-\tau )}^{n_c-j}\hfill & \hfill j\in [0,n_c]\end{array}\end{array}$$ 9

The first and second derivatives of $c(\tau )$ with respect to $\tau$ can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill c^{\prime }(\tau )& =\sum _{j=0}^{n_c}{B}_{c,j}^{\prime }(\tau )P_{c,j}\hfill \\ \hfill c^{\prime \prime }(\tau )& =\sum _{j=0}^{n_c}{B}_{c,j}^{\prime \prime }(\tau )P_{c,j}\hfill \end{array}\right)\end{array}$$ 10

Therefore, when $\tau =0$ and $\tau =1$,

$$\begin{array}{c}\hfill B_{c,j}(\tau =0)=\left\{\begin{array}{cc}1& j=0\\ 0& j\in [1,n_c]\end{array}\right)B_{c,j}(\tau =1)=\left\{\begin{array}{cc}0& j\in [0,n_c-1]\\ 1& j=n_c\end{array}\right)\\ \hfill & B_{c,j}^{\prime }(\tau =0)=\left\{\begin{array}{c}-n_{c}j=0\\ n_{c}j=1\\ 0j\in [2,n_c]\end{array}\right)B_{c,j}^{\prime }(\tau =1)=\left\{\begin{array}{cc}0& j\in [0,n_c-2]\\ -n_c& j=n_c-1\\ n_c& j=n_c\\ \end{array}\right)\hfill \\ \hfill \end{array}$$ 11

By applying the boundary constraints of Eq. (4), the following results can be obtained:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill c_i& =c(\tau =0)=P_{c,0}\hfill \\ \hfill c_f& =c(\tau =1)=P_{c,n_c}\hfill \\ \hfill T{\dot{c}}_i& =c^{\prime }(\tau =0)=n_c(P_{c,1}-P_{c,0})\hfill \\ \hfill T{\dot{c}}_f& =c^{\prime }(\tau =1)=n_c(P_{c,n_c}-P_{c,n_c-1})\hfill \end{array}\end{array}$$ 12

Therefore,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& P_{c,0}=c_i\hfill \\ \hfill & P_{c,1}=c_i+T{\dot{c}}_i/n_c\hfill \\ \hfill & P_{c,n_c-1}=c_f-T{\dot{c}}_f/n_c\hfill \\ \hfill & P_{c,n_c}=c_f\hfill \end{array}\right)\end{array}$$ 13

Then, the dimensionless time is discretized using mth-degree Legendre-Gauss distribution.

$$\begin{array}{c}\hfill {\tau }_1=0<{\tau }_2<\cdots <{\tau }_{m-1}<{\tau }_m=1\end{array}$$ 14

Therefore, Eqs. (8) and (10) can be written in the following matrix form.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[c\right]}_{m\times 1}={\left[B_c\right]}_{m\times (n_c+1)}{\left[P_c\right]}_{(n_c+1)\times 1}\\ {\left[c^{\prime }\right]}_{m\times 1}={\left[B_c^{\prime }\right]}_{m\times (n_c+1)}{\left[P_c\right]}_{(n_c+1)\times 1}\\ {\left[c^{\prime \prime }\right]}_{m\times 1}={\left[B_c^{\prime \prime }\right]}_{m\times (n_c+1)}{\left[P_c\right]}_{(n_c+1)\times 1}\\ \end{array}\right)\end{array}$$ 15

where $\left[P_c\right]={\left[\begin{array}{ccccc}{P}_{c,0}& P_{c,1}& {\left[X_c\right]}_{(n_c-3)\times 1}^{\text{T}}& P_{c,n_c-1}& P_{c,n_c}\end{array}\right]}^{\text{T}}$. ${\left[X_c\right]}_{(n_c-3)\times 1}={\left[\begin{array}{ccc}{P}_{c,2}& \cdots & P_{c,n_c-2}\end{array}\right]}^{\text{T}}$ are the unknown coefficients.

Equation (5) can also be written in the following form.

$$[a]=\sqrt{{[a_{\rho }]}^2+{[a_{\theta }]}^2+{[a_z]}^2}\le a_{\text{max}}$$ 16

Therefore, the continuous low-thrust trajectory optimization problem can be written as the nonlinear programming problem as follows.

$$\begin{array}{c}\hfill \underset{{\left[X_c\right]}_{(n_c-3)\times 1},T}{min}\mathrm{\Delta }V\\ \hfill & \mathrm{s}.\mathrm{t}.\left[a\right]\le a_{\text{max}},\left|\frac{\mathrm{\Delta }{\mathit{e}}_a}{\mathrm{\Delta }t}\right|\le {\omega }_{\max },\left|\mathrm{\Delta },{\beta }_{{\mathit{e}}_a}\right|\le {\beta }_{\max }\hfill \end{array}$$ 17

where the unknown Bezier coefficients ${\left[X_c\right]}_{(n_c-3)\times 1}$ and the total flight time T are variables that need to be optimized, the speed increment $\mathrm{\Delta }V$ required for the transfer process of the low-thrust spacecraft shown in Eq. (3) is the performance indicator during the flight process, and in the process of optimizing unknown variables, the constraints shown in Eqs. (5)–(7) need to be always satisfied.

Through Eqs. (11) and (13), it can be seen that when $n_r=n_{\theta }=n_z=3$, $\left[P_c\right]$ can be directly determined by the boundary conditions of Eq. (4). Therefore, the third-order Bezier curve can be used for initialization. Since the initial conditions only need to be calculated once, the number of discrete points selected for calculating the initial conditions is $n_{\text{APP}}$ ($n_{\text{APP}}>m$).

Numerical simulations

Selecting the generation of transfer trajectories for detecting asteroid Dionysus as a simulation example, and comparing it with examples that do not consider attitude constraints, confirms the importance of the algorithm proposed in this paper.

The initial orbital parameters are taken from Ref. 21. The orders of Bezier curve equations and discrete points are set as $n_r=16,n_{\theta }=18,n_z=16$, $m=120$, and $n_{\text{APP}}=600$. $a_{max}$ is $8.0\times {10}^{-5}$ $\mathrm{m}/s^2$²¹, and the number of revolutions is 5²¹. ${\omega }_{\max }=8^{\circ }/\text{s}$ and ${\beta }_{\max }={50}^{\circ }$. In order to obtain a suitable launch date, all dates within the range of the Modified Julian Date (MJD) 52000 and 59000 with 5-day intervals are calculated. The calculation results of the launch date are shown in Fig. 2, where the blue circle represents all feasible solutions, and the red point (MJD=52940) represents the optimal result within the current launch date range. Therefore, the selected launch date for the subsequent simulation is MJD=52940.

Figure 2.

The calculation results of Earth-Dionysus launch date.

The three-dimensional transfer trajectories from Earth to Dionysus, with and without attitude constraints, is shown in Fig. 3. Figures 4 and 5 show the variation curves of propulsive acceleration with flight time under two simulation conditions, and Figs. 6 and 7 show the variation curves of thrust direction angle with time, respectively. From the three-dimensional transfer trajectories shown in Fig. 3, it can be seen that considering attitude constraints has a significant impact on the transfer trajectory of spacecraft, which also confirms that considering attitude constraints can obtain more accurate flight trajectories. The comparison between Figs. 4 and 5 also shows that both simulation conditions can meet the constraint of propulsion acceleration. When considering attitude constraints, the total propulsion acceleration of the spacecraft needs to be maintained at a high amplitude for a long time, which can also be seen by comparing the required velocity increment during the transfer process.

Figure 3.

Transfer trajectories of Earth-Dionysus, with and without attitude constraints(AC).

Figure 4.

The propulsion acceleration of the Earth-Dionysus transfer trajectory without considering attitude constraints.

Figure 5.

The propulsion acceleration of the Earth-Dionysus transfer trajectory considering attitude constraints.

Figure 6.

Thrust direction angle of the Earth-Dionysus transfer trajectory without considering attitude constraints.

Figure 7.

Thrust direction angle of the Earth-Dionysus transfer trajectory considering attitude constraints.

By comparing Figs. 6 and 7, it can be seen that without considering attitude constraints, the maximum change in thrust direction angle of the spacecraft will be close to ${180}^{\circ }$. By applying attitude constraints, the change in thrust direction angle of the spacecraft can be constrained within ${50}^{\circ }$. From the simulation results in Figs. 6 and 7, it can also be seen that without considering attitude constraints, the low-thrust spacecraft not only need to perform large-scale attitude maneuvers, but also have a very high frequency of attitude maneuvers; after applying attitude constraints, the low-thrust spacecraft not only reduced the range of attitude maneuvers, but also significantly reduced the frequency of attitude maneuvers, which greatly reduced the control requirements of the attitude control system and proved the effectiveness of the algorithm proposed in this paper.

The detailed comparison of the two simulation conditions is shown in Table 1. When not considering attitude constraints, the required speed increment $\mathrm{\Delta }V$ for the transfer process is 16.99 km/s, the flight time is 3520.28 days, and the calculation time is 2.12 s; when considering attitude constraints, the required speed increment $\mathrm{\Delta }V$ for the transfer process is 21.67 km/s, the flight time is 3352.63 days, and the calculation time is 7.95 s. Through the comparison, it can be seen that due to the constraint of attitude, the required $\mathrm{\Delta }V$ for the transfer process has increased, but the flight time has decreased by 167.65 days. Due to the consideration of attitude constraints, the calculation becomes more complex, and the required computation time also increases.

Table 1.

Results comparison.

Simulation conditions	$\mathrm{\Delta }V$ (km/s)	Flight time (day)	Computation time (s)
With attitude constraints	21.67	3352.63	7.95
Without attitude constraints	16.99	3520.28	2.12

Conclusions

This paper proposes a shape-based trajectory optimization algorithm that considers attitude constraints for low-thrust spacecraft in order to reduce the control requirements for attitude control systems. By considering the change rate and change range constraints of the propulsion acceleration direction of spacecraft, the Bezier shape-based method was used to generate more accurate transfer trajectories. Compared with the situation without considering attitude onstraints, applying attitude constraints reduces the maximum change in spacecraft thrust direction angle from close to ${180}^{\circ }$ to within ${50}^{\circ }$, while also lowering the frequency of attitude maneuvers. This confirms that the method proposed in this paper can effectively reduce the attitude maneuver range of spacecraft, thereby reducing the fuel consumption of attitude control engines. This is crucial for the long-distance flight of spacecraft.

Author contributions

Z.F., F.C. and W.L. conceived the experiment(s), Z.F., W.K. and J.Q. conducted the experiment(s), Z.F., M.H., N.Q. and S.X. analysed the results. All authors reviewed the manuscript.

Data availability

All data generated or analysed during this study are included in this published article [and its supplementary information files].

Competing interests

The authors declare no competing interests.