Fast cooperative trajectory optimization and test verification for close-range satellite formation using Finite Fourier Series method

Mingying HUO, Zichen FAN, Niming QI, Zhiguo SONG, Xin SHI

a School of Astronautics, Harbin Institute of Technology, Harbin 150001, China

b China Academy of Launch Vehicle Technology, Beijing 100076, China

KEYWORDS Collision-avoidance;Finite Fourier Series (FFS);Satellite formation reconfiguration;Satellite simulators;Shape-Based (SB) method

Abstract The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites. To this end, this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series (FFS). The FFS method can rapidly generate the collision-avoidance threedimensional trajectory. The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed. Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration. The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites. This fast trajectory generation method has great significance for on-line, constantly satellite formation reconfiguration.

1. Introduction

With the higher requirements for the function of satellites,more and more payloads are carried by traditional satellites.Therefore, the development cycle of traditional satellites is long, the cost is high and the fault tolerance ability is low.Compared with a single large satellite, the formation satellite system has the advantages of high efficiency, low cost, high survivability et al.1-3Therefore, satellite formation flying has become an attractive topic, and extensive effort has been put on this novel technique.4-8However,there are many technologies need to be solved9and be tested on the ground to minimize the risk of flying for formation satellites. Therefore,designing a testbed of formation satellites is of great significance for the physical simulation and performance verification of the key technologies in the satellite formation.

The main focus of this paper is on the formation reconfiguration of satellite formation. Robertson et al.10studied the path planning of the satellite formation reconfiguration.Mbede et al.11planned the flight path with the virtual potential function method.Izzo and Pettazzi12presented a satellite pathplanning technique able to make a set of identical spacecraft acquire a given configuration. Richards et al.13introduced a method for finding fuel-optimal trajectories for spacecraft subjected to avoidance requirements in satellite formation reconfiguration. Zhang and Duan14used an improved pigeoninspired optimization algorithm for solving the optimal formation reconfiguration problems. This paper uses the Shape-Based (SB) method to rapidly generate low-thrust trajectories in the formation reconfiguration and considers at the same time collision avoidance between the satellites.

Low-thrust satellites need a method to approximate flight paths and mission cost.For the generation of flight paths,indirect and direct optimization methods all require a reasonable initial solution. Therefore, the rapid initial trajectory design is very important, and because of the fast calculation speed of the SB method, the SB method has been proposed for the initial low-thrust trajectory design. Petropoulos and Longuski15proposed the first SB method, who used the exponential sinusoid function to describe the trajectory.After that,many people put forward many SB methods, such as Pascale and Vasile,16Wall and Conway,17Xie et al.,18Novak and Vasile,19Zeng et al.20and Pelonia et al.21Recently, Taheri and Abdelkhalik22,23used the Finite Fourier Series (FFS)approximation to generate the low-thrust trajectories. These methods provide a new idea for fast generation of low-thrust trajectories.

This work generates the flight trajectories by the FFS method with free time at low computational time and verifies the effectiveness by test verification. Different from Refs.,22,23which sets the flight time of the transfer trajectory as a fixed value, the flight time in this paper is an optimization variable,not a fixed value.At the same time,this paper does not use the FFS method to generate a flight path,but to generate multiple flight paths at the same time, which adds more optimization variables. And this paper should consider the collision avoidance problem among multiple satellites, which increases the constraints and complexity of the problem. The Gauss Pseudospectral Method (GPM) discretizes state variables and control variables on a series of Gauss points, and constructs Lagrange interpolation polynomials with discrete points as nodes to approximate state variables and control variables.By deriving the global interpolation polynomials, the derivative of state variables to time is approximated, and the constraints of differential equations are transformed into a set of algebraic constraints,which can transform the optimal control problem into a parameter optimization problem with a series of algebraic constraints. In this paper, the results obtained by the FFS method are used as an initial guess in the GPM solver to verify the applicability of the results.Fast generation of flight trajectories is very important for satellite formation reconfiguration,which enables satellites to calculate the trajectories of reconstructed formation everywhen.

This paper is organized as follows. In Section 2, the problem description is presented. In Section 3, the FFS method with free time is briefly described. And unknown coefficients initialization is presented in Section 3. Finally, in Section 4,simulation analysis and physical experiment are carried out respectively. The effectiveness of FFS method is verified by comparing FFS method with GPM method in simulation.Section 5 concludes the paper.

2. Problem description

Because all satellites in satellite formation are fast maneuvering in short distance, Hill equation is not considered in this paper. (In fact, the method in this paper is also applicable to the case of long-distance satellite formation, that is, considering orbit dynamics.) The reference coordinate is the inertial system, and the Equations of Motion (EoM) in the reference coordinate are

where ΔV1, ΔV2, ..., ΔVncorrespond to the propellant consumed by satellites in flying, and a1, a2, ..., anare the corresponding satellite thrust acceleration. Because all satellites start and complete maneuvers at the same time, their total flight time is the same.

For problems involving one satellite transfer of two locations, the following 12 Boundary Conditions (BCs) need to be satisfied

where 0 ≤τ = t/T ≤1 is the scaled time. And each satellite has the above 12 BCs.

3. FFS method with free time

3.1. Fourier approximation

According to Ref.23, x is approximated with FFS for each satellite as follows:

3.2. Unknown coefficient initializations

Initialization of unknown Fourier coefficients is to provide an approximation of the coordinates(x,y,z)at m set of Legendre Gauss discretization points. And initialization of unknown Fourier coefficients can be written as

Although there is a big gap between the initial estimation of flight time under such assumptions and the actual value, the simulation results show that these deviations have little effect on the simulation results.After many iterations,the flight time tends to be a reasonable solution.

4. Simulation example and test verification

4.1. Simulation example

A three-dimensional simulation example of three satellites is given to verify the effectiveness of the proposed method. In this work, the performance index asked to be minimized here is the propellant consumed by all satellites. The maximum

Fig. 2 is relative distance between three satellites obtained by using FFS.As can be seen from the figure,every two satellites can be kept at a safe distance, and there will be no collision between the satellites. Fig. 3 shows the thrust acceleration of three satellites. It can be seen from the figures that the thrust acceleration of each component and the total thrust acceleration of each satellite are below the maximum thrust acceleration value,which can realize the reconfiguration of satellite formation under the existing conditions.

Fig.1 Flight paths of three satellites obtained by using GPM in 3D.

Fig. 2 Relative distance between three satellites obtained by using FFS in 3D.

Fig. 3 Thrust acceleration of three satellites in 3D.

In this scenario,the simulation results are shown in Table 1.The flight time obtained by using FFS is 603.0702 s, and that obtained by using GPM is 603.0696 s, and the performance index obtained by using FFS and GPM are 0.1023 m/s and 0.1019 m/s respectively. The difference between the performance index of the two methods is only approximately 0.39%. However, the difference in computational time between the two methods is large. The computational time of generating the initial flight paths by using FFS is 141.7790 s.This is only 1.14%of the computational time used to generate the further optimized flight paths by using GPM which is 12413.6049 s,even if the FFS provided an initial guess for the GPM.Therefore,from this simulation example,we can see that FFS method can obtain a good initial solution with very short calculation time.

4.2. Test verification

To verify the effectiveness, the FFS method is tested and validated by a formation control testbed. Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration. The control testbed for satellite formation is shown in Fig. 4. The satellite simulatorscan carry out two degrees of freedom translations on the granite platform by using air suspension technology and have the function of trajectory control.

Table 1 Three-dimensional simulation results.

Fig. 4 Control testbed for satellite formation.

Table 2 Three-dimensional test results.

Fig.5 Flight paths of three satellites obtained by using GPM in 2D.

Fig. 6 Relative distance between three satellites obtained by using FFS in 2D.

The initial flight paths designed by using FFS and the further optimized flight paths obtained by using the GPM solver are illustrated in Fig.5.Fig.6 is relative distance between three satellites obtained by using FFS.Fig.7 shows the thrust acceleration of three satellites. It can be seen from the figures that every two satellites can be kept at a safe distance and the thrust acceleration of each component and the total thrust acceleration of each satellite are below the maximum thrust acceleration value, which can realize the collision-avoidance reconfiguration of satellite formation under the existing conditions.

Physical experiments are carried out through the simulation results. The experimental results show that the FFS method can effectively reduce the risk of collision between satellites to rapidly generate trajectories. The results of physical experiments and simulation are basically the same.

5. Conclusions

(1) This paper presents a method to rapidly generate lowthrust collision-avoidance 3D trajectories in the satellite formation reconfiguration using FFS method.

(2) Compared with the GPM method, for 3D trajectories the FFS method only takes 1.14% of the computation time to get the result that the performance index is only 0.39% different. This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.

(3) The proposed method is tested and validated by simulation and a formation control testbed to verify the effectiveness. The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.

(4) The method proposed in this paper is applicable to the formation of short-range satellites and to the formation of long-range satellites (considering orbital dynamics).Because the anti-collision requirements between longdistance satellites are lower,it is easier to solve the flight paths.However,in order to meet the requirements of the ground test,this paper takes the short-range satellite formation without considering the orbit dynamics as an example, conducts simulation experiments, and then provides physical verification.

Fig. 7 Thrust acceleration of three satellites in 2D.

Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (Nos. 11702072 and 11672093).