LI Jian，ZHANG Gang

Research Center of Satellite Technology，Harbin Institute of Technology，Harbin 150080，P.R.China

Abstract: This paper proposes an intelligent low?thrust orbit phasing control method for multiple spacecraft by simultaneously considering fuel optimization and collision avoidance.Firstly，the minimum?fuel orbit phasing control database is generated by the indirect method associated with the homotopy technique.Then，a deep network representing the minimum-fuel solution is trained.To avoid collision for multiple spacecraft，an artificial potential function is introduced in the collision-avoidance controller.Finally，an intelligent orbit phasing control method by combining the minimum-fuel neural network controller and the collision-avoidance controller is proposed.Numerical results show that the proposed intelligent orbit phasing control is valid for the multi-satellite constellation initialization without collision.

Key words：orbit phasing control；low thrust；deep neural networks；collision avoidance

0 Introduction

In recent years，the number of on orbit space?craft increases rapidly，thus it is important to devel?op the intelligent autonomous control scheme of spacecraft.Due to high specific impulse and high precision，low-thrust thrusters such as electric pro?pulsion are important ways for orbit control.There?fore，the low-thrust trajectory optimization problem has attracted many attentions.The low-thrust trajec?tory optimization methods mainly include the direct method［1］，the indirect method［2］，and the shapebased method［3］.Compared with the other two methods，the indirect method can guarantee the first-order optimality condition，but it is sensitive to initial values.Generally speaking，the shape-based method and the direct method can be used to pro?vide the initial guess［4］，and the homotopy tech?nique［5-6］can be used to reduce the difficulty for the initial guess.However，traditional methods need complex numerical calculations，and they cannot be used on board.

Recently，the intelligent technology represent?ed by the deep neural network and the machine learning provides a new way to solve the low-thrust trajectory optimization problem.The powerful fit?ting ability of the deep neural network is used to pro?vide an initial guess for the costate variable in the in?direct method［7-8］，but the exact value of the initial costate variable also needs to be solved by numeri?cal iterations.In addition，the neural network can be directly used to obtain the optimal control［9-11］.For the training data in neural networks，Cheng et al.［10］used an actor-indirect method to employ a network learning architecture，and Izzo et al.［11］proposed a new general methodology called“backward genera?tion of optimal examples”to create the database.Compared with the traditional methods，the preced?ing intelligent studies do not need complex numeri?cal calculations，but they are not valid for multiple spacecraft.

For the multi-spacecraft control，two impor?tant objectives are fuel optimization and collision avoidance.For the fuel optimal control problems，the model predictive control method and the convex optimization method are used for multiple space?craft［12］.However，the computational burden of this method increases significantly for large number of spacecraft and long transfer time.Therefore，it is not suitable for on-board applications.For the colli?sion avoidance problem，each spacecraft need to have the autonomous control capability，thus the ar?tificial potential function（APF）is widely used in the control design［13-14］.In addition，another idea is to combine the sliding mode control with the APF control to realize the closed-loop multi-satellite con?trol under the collision constraint［15-17］.However，all preceding methods are based on the relative dynamic model.For the absolute dynamic model，Yu et al.［18］proposed a quadratic APF controller to achieve the spacecraft autonomous cluster without collision，but they did not consider the fuel optimality in the con?trol process.In summary，there is no control meth?od considering both the fuel optimality and the colli?sion constraint.

The main contribution of this paper is to pro?vide an intelligent autonomous orbit phasing control method by simultaneously considering the fuel opti?mization and the collision avoidance.This paper is structured as follows：In Section 1，the dynamic model is introduced and the indirect optimal control method is given；in Section 2，the database genera?tion method with the homotopy technique，and the design and training process of neural network are proposed；then，the minimum-fuel neural network controller is combined with the collision-avoidance controller in Section 3；in Section 4，two numerical examples are provided；finally，the conclusions are given in Section 5.

1 Low?Thrust Trajectory Optimi?zation

1.1 Dynamic model

The cylindrical coordinate system is shown in Fig.1，wherer，θandzare the radial distance，the azimuth angle，and the altitude of the spacecraft，re?spectively.

The motion of spacecraft can be described as the following differential equation［19］

where

andx=[r，θ，z，vr，vθ，vz]T，herevr，vθandvzare the derivatives ofr，θandz，respectively；α=[αr，αθ，αz] is the thrust direction；mis the mass；Ispis the specific impulse of thruster；μ=398 600.441 5 km3/s2is the gravitational constant；g0=9.806 65 m/s2is the gravitational acceleration at sea level；u=T/Tmax，andu∈[0，1] is the en?gine thrust ratio，hereTandTmaxare the current and maximum thrust magnitudes，respectively；R=is the distance from the center of the space?craft to the Earth.

This paper mainly stuides the orbit phasing con?trol in the phase initialization and phase reconfigura?tion missions for the multi-satellite constellation.For different satellites in the same orbit（with the same semi-major axis，eccentricity and inclination），the right ascension of the ascending node（RAAN）change rates caused by J2 perturbation are almost the same.Thus，the J2 perturbation is ignored in the orbit phasing control here.

1.2 Optimal control problem

For the fuel optimal orbit phasing control prob?lem with free transfer timetf，the performance index can be expressed as

To overcome the difficulty arising from solving the bang?bang control，the homotopic approach is adopted in the performance index，so we have

whereε∈[0，1] is the homotopy parameter.As theεdecreases from 1 to 0，the problem changes from the easily solved energy optimal problem into the fu?el optimal control problem.Whenε=0，the prob?lem is the fuel optimal problem.

For the performance index in Eq.（4），the fol?lowing Hamiltonian function is constructed by intro?ducing the costate variableλ.

and the optimal thrust magnitude ratiou?is

whereρis the switching function，shown as

The differential equation of the costate variableλis

where

Since the transfer time of the spacecraft is free，there are two additional boundary conditions，i.e.

Then，the two-point boundary value problem in the cylindrical coordinate system with the homoto?py parameter is obtained，and the corresponding shooting equation is

whereΛε=[λri，λθi，λzi，λvri，λvθi，λvzi，λmi，tf] is the vari?able needs to be solved，andxcandxtare the states of the chaser and the target，respectively.

2 Network Training

This section describes the methods for the data generation，the neural network architecture and the training method.The purpose is to obtain a mini?mum-fuel neural network controller for the autono?mous fuel-optimal control in real time.

2.1 Training data generation

In order to realize the minimum-fuel neural net?work controller，the input of the network includes the current statexci，the current massmciand the ex?pected statexti.The output is the optimal control variableU*=(u，θr，θz)，whereθr∈[-π，π]，θz∈[-π/2，π/2] are the thrust direction angles.The thrust direction angles can be expressed as

In this way，we can combine the spacecraft state and the control to construct a set of training da?ta(xci，mci，xti，U*).

Based on the optimal solution by the homotopy technique，a database can be generated.By perturb?ing the initial state of the nominal trajectory，the un?disturbed state can provide a good initial guess for the disturbed state.For the nominal optimal trajecto?ry，the initial and terminal states are，re?spectively，and the optimal costate variable isΛ*.A new set of values of the initial stateand the ex?pected stateis

In the above process，firstly the true anomalyφin the range［0，360°）is divided into 500 equal values，and the homotopy method is used to solve the optimal orbits at different phases，which are viewed as the nominal orbits with different phases.Then，for these nominal orbits，the other five orbit elements are disturbed to obtain many optimal solu?tions.Finally，all the obtained optimal orbits are combined to establish the optimal control database.

2.2 Network structure selection

In this paper，two kinds of neural networks are established.The first one is the thrust-ratio net?work，which predicts the thrust ratiou.Note thatuof the fuel optimal control is only 0 or 1，and we es?tablish a classified neural network to fit the thrust magnitude term in the control.The second one is the thrust-direction-angle network，which is to pre?dict the thrust direction anglesθrandθz.

Due to the strong nonlinearity of the optimal control，the network needs to have a certain com?plexity to capture the relationship between the opti?mal control and the state.Therefore，to avoid under fitting and over fitting for the thrust ratio，a neural network is designed with three hidden layers and 128 neurons in each layer for the control magnitude network.To realize the fitting of the thrust direction angles，a neural network is designed with nine hid?den layers and 128 neurons in each layer.The specif?ic structure of the network is Fig.2.

Fig.2 Minimum-fuel neural network architecture

2.3 Selection of activation function and loss function

This subsection mainly introduces the architec?ture design of neural network and the selection of ac?tivation-function.

Firstly，the fully connected feedforward net?work is selected.The activation function of the out?put layer is determined according to the range of the output.For the thrust-ratio network，the Sigmoid function is used as the activation function of the out?put layer.For the thrust-direction-angle network，the Tanh function is selected as the activation func?tion of the output layer.For the hidden layer of the network，the ReLU function is selected as the acti?vation function.

Then，the mean square error（MSE）between the training data and the network prediction results is adopted as the loss function，shown as

where the Net function is the neural network to be trained，andNrepresents the total number of sam?ples used for training.The input vectorXiis com?posed of the states，the target vectorcontains the optimal control to be learned，and the symbolsωandbdenote the weight and bias of network，re?spectively.In addition，to avoid system memory ex?plosion caused by all data input into the network，the batch size is set to be 500.

3 Collision Avoidance

When multiple spacecraft are controlled simul?taneously，the collision risk increases.To avoid col?lision，this section proposes a collision-avoidance method based on the APF.

3.1 Control process

In order to avoid collision，the repulsive field is introduced to prevent spacecraft collision during ma?neuvers.Assume that the collision between space?craft is given in the form of distance.In this way，when there is no collision risk between spacecraft，it uses the optimal control obtained by neural net?work；however，when there is collision risk，the APF is used to avoid collision.The control process of spacecraft is shown in Fig.3，and the steps are given as follows.

Step 1Judge whether the current state of the spacecraft meets the terminal constraints.If no，use the intelligent controller generated in Section 2.

Step 2If the spacecraft meets the terminal constraints or is being controlled by the artificial in?telligence controller，judge whether there is a colli?sion risk.If yes，it needs to use the APF to avoid collision.

Step 3If the spacecraft has no collision risk or execute the collision-avoidance maneuver，it is necessary to judge whether it meets the terminal constraint again.If yes，the control is completed.If no，return to Step 1 and repeat the above process.Until all spacecraft meet the terminal constraints and there is no collision risk，the control is over.

3.2 Artificial potential function design

In the terminal constraints in Fig.3，the safety constraints of spacecraft are expressed as

whererminrepresents the nearest distance among all spacecraft andLthe allowable minimum distance be?tween spacecraft.The APF of the spacecraft from the nearest spacecraft is

whered0is the radius of the APF.When the dis?tance between spacecraft is less thand0，there is a risk of collision between them.Then，the repulsion force magnitude to the spacecraft is

wherekis the gain coefficient.The influence of APF on energy consumption in the whole control process can be changed by adjusting the gain coeffi?cientk.Because the spacecraft will fly by other spacecraft during the whole period，the safe dis?tance between spacecraft can be ensured by chang?ing the semi-major axis.

Note that the semi-major axis does not change under the acceleration component perpendicular to the tangential direction in the orbit plane.Then，this paper mainly uses the tangential acceleration compo?nent to change the semi-major axis to achieve the purpose of collision avoidance.Assume that the tan?gential acceleration direction of spacecraftSiisαui.Thus，αuican be expressed as

whereaiis the semi-major axis of the spacecraftSi，andda0represents the deviation limit of the semimajor axis to prevent the thrust direction oscillation caused by the small difference of semi-major axis.The tangential acceleration vector can be written in the coordinate system［S，T，W］，which denotes the radial，transverse and normal directions，respec?tively.Then，we have

where

Then，the next step is to transform the frame[S，T，W]Tinto the geocentric inertial coordinate system，shown as

whereR1(?) andR3(?) represent rotation matrices that rotate vectors by the angle?about theX-andZaxes，respectively［20］.The acceleration direction in the cylindrical coordinate systemαO-rθz=[αr，αθ，αz]Tcan be obtained as

The outputs of the neural network are[unet，θr，θz].Thus，the acceleration direction vector of the spacecraft isαnet=，and

By combining this acceleration and that by the minimum-fuel neural network controller，the final normalized accelerationto avoid collision is

3.3 State deviation and control limit

To describe whether the spacecraft reaches the expected state，the subsection will give the calcula?tion method of control limit，in which the state devi?ations of the spacecraft are given in the form of cylin?drical coordinates，and its expression is

Eq.（26）indicates the deviation between the current state of the controlled spacecraft and the ex?pected state，the subscript“f”indicates the expect?ed terminal state，and rem(p，q)returns the remain?der after division ofpbyq.

In addition，after the spacecraft meets the ter?minal constraints，it may deviate from the expected state due to further collision avoidance control.In this case，to avoid frequent switching of the control?ler，it needs to design the start and stop control lim?its for each spacecraft，and the stop limit is less than the start limit.If the any one of the state deviations is greater than the starting limit，the controller is switched on.When all state deviations are less than the stop limit，the controller is cut off.

4 Simulation

This section provides two examples to verify the proposed neural network for the fuel optimal so?lution，and the proposed intelligent controller for collision avoidance.Assume that the mass ism=270 kg，the specific impulse of electric thrusters isIsp=3 000 s，and the maximum thrust magnitude isTmax=100m·N.Nominal orbit elements[a，e，i，Ω，ω，φ]=[7 378 km，0.1，0，0，0，0].The maximum transfer time is set to be 4.84×105s.Similar to Ref.［21］，the parameters of APF and control limits applied in this example are listed in Table 1.

Table 1 Parameters of APF and control limits

4.1 Neural network performance

In this subsection，the Monte Carlo tests with 100 satellites are used to verify the proposed neural network controller.The deviations between initial orbit and nominal orbit of each spacecraft are uni?formly randomly distributed as Δa0∈[0，100 km ]，Δe0∈[-0.1，0]，Δω0∈[0，360°]，and Δφ0∈[0，360°].The expected transfer phases are also uni?formly randomly distributed as Δφf(shuō)∈[0，360°].When all state deviations are less than the stop con?straint，the phasing control is stop.The stop limits of the control are listed in Table 1.

The terminal phase error and the fuel error of the proposed minimum-fuel neural network control?ler in the 100 tests are shown in Table 2 and Fig.4.Table 2 shows that the average terminal phase error is 0.347 1°.Compared with the indirect method，the increased fuel consumption of the proposed neural network method is less than 1%.However，the mean computational time of the minimum-fuel neu?ral network controller is 0.009 5 s，which is suitable for on-orbit autonomous control.

Table 2 Terminal phase error and fuel error of the pro?posed neural network controller

Fig.4 Fuel index error distribution

4.2 Constellation initialization simulation

In this subsection，a constellation initialization control is considered for the multi-spacecraft orbit phasing control method.A series of spacecraft are distributed in series on the same orbit plane，and the goal is to make the spacecraft realize phase uniform distribution in a given order in orbit.

The orbital elements of the spacecraft are the same as that of the nominal initial orbit except the initial phase.These spacecraft are initially numbered from large to small according to their initial phaseφ，ranging from 1 to 60 with the interval phase 0.17°.At the initial time，the phase of the 1st space?craft is 354.9°，and that of the 60th spacecraft is 5.1°.

Note that the final distribution order of space?craft is different from its initial order.The spacecraft may collide with each other in the control process.To avoid collision，the boundary of the APF is set to be 20 km.When the distance between two space?craft is less than 1 km，the collision is considered.The parameters of APF and the control limit of spacecraft are listed in Table 1.The expected final order of spacecraft isNumE，i.e.

whereNumEis the row vector arranged from left to right.

When the gain coefficientk=3.2×104，at dif?ferent time instants in the control process，the posi?tions of spacecraft are shown in the Fig.5.After 4.84×105s，the constellation initialization is com?pleted.The curve of the minimum distance among all spacecraft in the whole control process is plotted in Fig.6.The minimum distance is 1.520 km，which is greater than the collision limit（i.e.，1 km）.The fuel consumption of each spacecraft with collision avoidance is 0.025—0.233 kg，and the total fuel consumption is 6.763 kg.

Fig.5 Spacecraft positions at different time instants

Fig.6 Minimum distance among all spacecraft

For different gain coefficientsk，the minimum distance and the total fuel consumption are listed in Table 3.When the gain coefficientk=0，it is equivalent to ignoring the APF in the whole control process.The minimum distance and the fuel con?sumption increases with increasing gain coefficientk.For a smallk，the minimum distance between spacecraft may be less than the collision limit.

Table 3 Minimum distance and total fuel consumption for different k

5 Conclusions

A new intelligent multi-spacecraft orbit phasing control method is developed by simultaneously con?sidering fuel optimization and collision avoidance.Firstly，the indirect method associated with the ho?motopy technique is used to generate the optimal control database.Then，two neural networks are constructed to obtain a minimum-fuel neural net?work controller that can generate the fuel-optimal control according to the current and expected state.In addition，based on the APF，a collision avoid?ance method by adjusting the semi-major axis is pro?posed to handle the collision problem.Finally，the intelligent control is obtained by combining the mini?mum-fuel neural network and the collision-avoid?ance method.

The numerical results show that the proposed minimum-fuel neural network is able to predict the optimal thrust magnitude and thrust direction.For the phasing problem，the fuel consumption increas?es by less than 1% compared with the optimal solu?tion.For the constellation initialization，the increase of fuel consumption of the proposed controller is al?so reasonable compared with the optimal control without the collision constraint.However，the mini?mum distance of the proposed method can meet the collision limit.