LIU Qingguo,LIU Xinxue,WU Jian,and LI Yaxiong

Xi’an High-tech Institute,Xi’an 710025,China

Abstract:Fast computation of the landing footprint of a space-toground vehicle is a basic requirement for the deployment of parking orbits,as well as for enabling decision makers to develop real-time programs of transfer trajectories.In order to address the usually slow computational time for the determination of the landing footprint of a space-to-ground vehicle under finit thrust,this work proposes a method that uses polynomial equations to describe the boundaries of the landing footprint and uses back propagation(BP)neural networks to quickly determine the landing footprint of the space-to-ground vehicle.First,given orbital parameters and a manoeuvre moment,the solution model of the landing footprint of a space-to-ground vehicle under finit thrust is established.Second,given arbitrary orbital parameters and an arbitrary manoeuvre moment,a fast computational model for the landing footprint of a space-to-ground vehicle based on BP neural networks is provided.Finally,the simulation results demonstrate that under the premise of ensuring accuracy,the proposed method can quickly determine the landing footprint of a space-to-ground vehicle with arbitrary orbital parameters and arbitrary manoeuvre moments.The proposed fast computational method for determining a landing footprint lays a foundation for the parking-orbit configu ation and supports the design of real-time transfer trajectories.

Keywords:space-to-ground vehicle,landing footprint,back propagation(BP)neural network,fast computational method,Pontryagin’s minimum principle.

1.Introduction

Spacecraft such as return satellites,manned spacecraft,space shuttles,and space-to-ground kinetic weapons,manoeuvring from their orbits to the earth surface are uniformly referred to as space-to-ground vehicles[1–3].The landing footprint of the vehicle is an important indicator for assessing the ability of the space-to-ground vehicles.The fast computation of the landing footprint can lay a solid foundation for the work of a large number of repeated and real-time computations for the landing footprint,such as the parking-orbit configuration of the space-to-ground vehicle and the real-time programs of transfer trajectories developed by decision-makers.A typical space-to-ground transfer trajectory consists of a transition trajectory segment and a re-entry trajectory segment.Depending on the drag and lift of the spacecraft in the re-entry segment,the re-entry trajectories are classified as ballistic,semiballistic,or gliding[4,5].

The landing footprint consists of landing points on the boundaries.Each of the landing points is the result obtained by solving the transfer trajectory.Therefore,the optimization of the transfer trajectory is the basis of determination of the landing footprint.In the existing literature,direct methods[6,7],indirect methods[8,9],and hybrid methods[10–12]are used to study the optimization of transfer trajectories of different types of space-to-ground vehicles.The hybrid methods,one of which is adopted in this paper to optimize the transfer trajectory,combine the advantages of the direct and indirect methods.

Some methods have been proposed to study the landing footprint of a space-to-ground vehicle in the existing literature.Saraf et al.[13]proposed an analytical method for calculating the landing footprint based on the guidance mode of the space shuttle.Li et al.[14]proposed a method that uses scheduling of the drag pro file to the normalized energy between the upper and lower bounds.This leads to finding the near and far edges of the landing zone.Hu et al.[15]used a genetic algorithm(GA)to compute the landing footprint.The existing literature on the landing footprint consists of solving for the plurality of points on the boundaries of the landing footprint,and then connecting these points on the boundary to determine the landing footprint. However, such methods are too longtime consuming to efficiently solve the transfer trajectory of the space-toground vehicle under finite thrust,and to determine the landing footprint.In order to shorten the computational time,this work departs from the traditional methods of determining the landing footprint by connecting boundary points,and proposes a fast method,which uses polynomial equations to describe the boundaries of the landing footprint.Moreover,back propagation(BP)neural networks are used to quickly achieve a nonlinear mapping of the boundary of the landing footprint with arbitrary orbital parameters and arbitrary manoeuvre moments.

The research content of this paper is as follows:First,a solution model for the landing footprint of a space-toground vehicle under finite thrust is established.Second,given arbitrary orbital parameters and an arbitrary manoeuvre moment,a fast computational model for determining the landing footprint of the space-based vehicle based on BP neural networks is provided.Finally,the effectiveness of the method is demonstrated through simulation.

2.Computation of the landing footprints

In this work,the equations of motion are described in the coordinate system fixed to the earth.The determination of the landingfootprintis performedwith the longitudeas thex-coordinateandthelatitudeas they-coordinate.Givenorbital parameters and a manoeuvre moment,the outlines of the solution model are as follows:

(i)Determine the minimum longitude or latitude of all landing points;

(ii)Determine the maximum longitude or latitude of all landing points;

(iii)Choose many points between the minimum and the maximum longitude or latitude in(i)and(ii),and determine the boundary points corresponding to the chosen points;

(iv)Connect all of the points obtained in(i),(ii)and(iii)to determine the landing footprint of the space-to-ground vehicle.

The transfer trajectories are required to compute the minimum longitude or latitude and the maximum longitude or latitude of all of the landing points, ae well as the boundary points between the minimum and the maximum longitude or latitude.The procedure for using a mixed method to solve the transfer trajectory problem is described by the following:Take the longitudes or latitudes of the landing points of(i),(ii)and(iii)as the optimization index.The transfer trajectory optimization problem is then converted into a two-point boundary value problem using Pontryagin’s minimum principle[16– 18].The initial values of the adjoint variables and the values of the partial state variables are adjusted by means of a GA[19–21],and the transfer trajectory and the landing footprint are then obtained.

2.1 Motion differential equations in the earth- fixed coordinate system

The differential equations[22,23]of motion are plane and the latitude tangent),is the dimensionless geocentric distance,Θis the longitude,Φis the latitude,is the engine thrust,is the gas jet velocity,is the dimen-

whereis the dimensionless velocity,γis the velocity inclination angle,ψis the course angle(the angle between the projection of the velocity vector on the local horizontal sionless spacecraft quality,tis the dimensionless time,is the dimensionless drag,is the dimensionless lift,is the dimensionless lateral force,αis the angle of attack,βis the sideslip angle,is the dimensionless earth rotation angular rate,andare the dimensionless gravitational components when only the first three terms of the spherical harmonic expansion are considered.In the transition trajectory segment,the values of,andare zero.In the re-entry trajectory segment,the value ofis zero and the atmospheric model is the US standard atmosphere(1976).

The equations for the dimensionless parameters are as follows:

whereJ2is the coefficient of the second order principal spherical harmonic function and the value ofJ2is 1.082 63e–3.rref,mref,Vref,trefandgrefare given by the following equations:

whereμis the gravitation constant and the value ofμis 3.986 005e+14 m3/s2.m0is the initial mass of the vehicle andREis the radius of the earth[24]:

The parameters used to describe the orbit of a space-toground vehicle in space are the orbital radiusa,the flatteninge,the orbital inclinationi,the ascending node right ascensionΩ,the perigee angleωand the true anomalyf.The relationship between the orbit parameters and the motion parameters in the absolute coordinate system is

whereVI,γI,ΨI,rI,ΘIandΦIare the velocity,the velocity inclination angle,the course angle,the geocentric distance,the longitude and the latitude in the absolute coordinate system,respectively.

The relation between the motion parameters in the absolute coordinate system and the motion parameters in the earth- fixed coordinate system are

2.2 Computational steps of the landing footprints

The optimization indexes of research ideas(i),(ii)and(iii)are given by

whereJ1,J2andJ3are optimization indexes corresponding to research ideas(i),(ii)and(iii),respectively.ΘmandΦmare the longitude and the latitude of the manoeuvre point.The variablesΘbandΦbare the longitude and the latitude of the landing point.

According to the Pontryagin’s minimum principle,the Hamiltonian function is given by

From(10),the covariate variables satisfy the differential equations in(11).It should be noted that only the transition trajectory with the thrust control is considered and the thrust is constant in the computational model of the landing footprints.

From the sufficient conditions of optimality and(8),the optimal thrust directions are obtained by

and

where sgn()is the sign function.

The transition trajectory satisfies the initial boundary conditions:

The transition trajectory satisfies the terminal boundary constraints:

whereVa,γaandare the dimensionless speed,the velocity inclination angle and the geocentric distance at the terminal of the transition trajectory respectively;is the dimensionless time at the terminal point of the transition trajectory.

Therefore,the cross-section conditions are given by

When the optimization indexJ3is calculated,the reentry trajectory satisfies the constraints:

whereΘbandΦbare the longitude and latitude of the landing point at the dimensionless time,respectively.

Because the optimization indexes do not include time,

In order to achieve the optimization indexes under the given constraints,11 or 12 parameters are taken as optimization variables including the initial values of sevenadjoint variablesandthe values of four state variablesVa,γa,andor the values of five state variables,,,andΦb(Θb).Since the initial value of an adjoint variable can be obtained from(10)and(19)andλΘ()≡0,only nine or ten variables need to be optimized.A fourth order Runge-Kutta method[25,26]is used to compute the starts from(1)and(6)and the Adams predictor-corrector method is adopted to compute the remaining integral equations.The implementation steps of the GA are as follows:

Step 1When optimization indexesJ1andJ2are calculatedandtaare encoded.When optimization indexJ3is calculated,andΦb(Θb)are encoded.Twenty chromosomes are generated randomly as the initial population.The crossover probabilityPc,the mutation probabilityPmand the maximum number of iterationsNare set.

Step 2The fitness function is shown in(7),(8)and(9).When a chromosome is determined,(11)is integrated to get the optimal thrust directions under the constraints(15)to(19).And(1)is then integrated to obtain the transfer trajectory.The fitness values of all the chromosomes in the current generation are computed.

Step 3The chromosomes are selected using roulette wheel selection.Some genes on two different chromosomes reciprocally cross according to the crossover probability and others mutate according to the mutation probability.The execution of selection,crossover and mutation leads to the next generation population.

Step 4Steps 2 and 3 are executed until the GA converges withε=10?6or the maximum number of iterationsNis reached.

The hybrid method gives the transfer trajectories and landing footprints at a given manoeuvre point.

3.The proposed fast computational method of landing footprints

BP neural networks have the advantages of strong learning ability,good nonlinear mapping ability and good fault tolerance[27–29].They consist of two sub-networks:the signal forward propagation network and the error BP network.The signal forward propagation network operates in such a way that the output results are obtained after the input parameters are processed layer by layer in the neural network.The error BP network operates in such a way that the output values are transmitted to the network in the opposite direction to modify the weight and threshold values between the neurons of the entire network,until the requirements of the output results are met.The signal forward propagation and error BP are called the BP neural networks training process.

In this section, the input and output parameters of the BP neural network are determined and then used to provide a model based on BP neural networks for the fast computation of landing footprints.Finally,a computational model of relative errors is given.

3.1 Determination of input and output parameters

The determination of the input and output parameters is the premise of the BP neural networks training.Polynomial equations( fitting curves)are adopted to describe the boundaries of the landing footprint,transforming the problem of determining the landing footprint into a problem of finding the coefficients of the polynomial equations.Given a manoeuvre moment of the space-to-ground vehicle,a polynomial equation can be used to approximately express the landing footprint.However,the landing footprint varies with manoeuvre moments.Thus,the BP neural networks can effectively solve the problem of an irregular change in landing footprints.

Fig.1 shows the boundary of the landing footprints under the conditions 0?i45?or 135?i180?,whereiis the orbital inclination.Fig.2 shows the boundary of the landing footprints under the conditions 45?

Fig.1 Sketch diagram of the landing footprints when 0?i45?or 135?i180?

Fig.2 Sketch diagram of landing footprints when 45?

and

wherexis the longitude andyis the latitude;the valuesare the fitting coefficients of the boundary curve on the high latitude side;and the valuesare the fitting coefficients of the boundary curve on the low latitude side.

Similarly,when 45?

and

Once(20),(21),(22)and(23)are determined,the coefficients ofx4,x3,x2,xandx0are taken as the output parameters of the BP neural network under the conditions 0?i45?or 135?i180?.We defineas the output parameters of the boundary curve on the high latitude side,andandas the output parameters of the boundary curve on the low latitude side.Similarly the coefficients ofy4,y3,y2,yandy0are taken as the output parameters under the conditions of 45?

When 0?i45?or 135?i180?,

and

When 45?

and

The factors affecting the landing footprint are the orbital parameters and the manoeuvre moment. Viewing the earth as a homogeneous ellipsoid and considering the periodicity characteristic of the space-to-ground vehicle, the determination of the landing footprint in one period can represent the landing footprint at an arbitrary manoeuvre moment. The period T is given by

The manoeuvre momenttmis in the rangewherekis an arbitrary non-negative integer.The input parameters of the BP neural network are the orbital radiusa,the flatteninge,the orbital inclinationi,the ascending node right ascensionΩ,the perigee angleω,the true anomalyfand the manoeuvre momenttm.

The input parameters need to be normalized as follows:

3.2 Construction of the BP neural network model

We provide four BP neural network models,each of which consists of a three-layer network:input layer,output layer and hidden layer as shown in Fig.3–Fig.6.

Fig.3 No.1 BP neural network model

Fig.4 No.2 BP neural network model

Fig.5 No.3 BP neural network model

Fig.6 No.4 BP neural network model

The differences of these four BP neural network models are the output parameters and the training data.The input layer of the four BP neural network models has seven nodes.The input parameters of the BP neural network are the orbital radiusa,the flatteninge,the orbital inclinationi,the ascending node right ascensionΩ,the perigee angleω,the true anomalyfand the manoeuvre momenttm.The four BP neural network models have five nodes in the output layer.The output parameters of the No.1 BP neural network model areand1;the output parameters of the No.2 BP neural network model are;the output parameters of the No.3 BP neural network model are;the output parameters of the No.4 BP neural network model are.The number of hidden layer nodes of the four neural network models is

wherepis the number of nodes in the input layer,qis the number of nodes in the output layer,andsis a natural number between 0 and 10.The training data of the four BP neural network model are the actual footprints successively obtained under conditions of(24),(25),(26)and(27).

The signal forward propagation means that the signal received by the input layer is transmitted layer by layer until the output layer results are generated.The relevant mathematical expressions for signal forward propagation are shown in(31)–(34).

The input signal netiof the numberinode of the hidden layer is given by

wherewijis the weight from the numberjnode of the input layer to the numberinode of the hidden layer,xjis the input parameter of the numberjnode of the input layer,andθiis the threshold of the numberinode of the hidden layer.

The output signaloiof the numberinode of the hidden layer is given by

whereφ()is the Sigmoid function,an activation function commonly used in BP neural networks.

The input signal netkof the numberknode of the output layer is given by

wheretkiis the weight from the numberinode of the hidden layer to the numberknode of the output layer,andukis the threshold of the numberknode of the output layer.

The output signalokof the numberknode of the output layer is given by

whereψ()is the Purelin function[32],another activation function commonly used in BP neural networks.

BP of the error means that the error computed from the output layer is back propagated to the hidden layer.The weight valueswijandtkiand threshold valuesθiandukare then adjusted using the gradient descent method related to the errors of the nodes in the hidden layer and the output layer.This process is iterated until the output of the modified network is close to the expected value. The mathematical expressions related to error BP are shown in

ForPtraining samples,the quadratic error criterion functionEPof the BP neural network is

whereis the true value of thepth sample at thekth node in the output layer;andis the corresponding value as computed by the BP neural network model.

The adjustment of the weight value Δwijin the hidden layer is performed by using

The adjustment of the threshold value Δθiin the hidden layer is performed by using

The adjustment of the weight value Δtkiin the output layer is implemented by using

The adjustment of the threshold value Δukin the output layer is implemented by using

In(36)– (39),the parameterηis the learning rate.

The training error of the BP neural network propagates forward through the signal,and the BP of the error becomes iteratively smaller and smaller until stable weights and thresholds are obtained.

3.3 Computation of relative error

The relative error refers to the error between the landing footprint obtained by the fast computational method based on the BP neural network and the landing footprint obtained by the least squares method.

Fig.7 shows the fitting boundary curves and the boundary curves obtained by the BP neural network.Fig.8shows the area of the landing footprints enclosed by the fitted curve.The regionsS1,S2,S3andS4are the areas of the non-overlapping landing footprints enclosed by the fitting curves and the BP neural network.

Fig.7 Fitting boundary curves and the boundary curves obtained by the BP neural network

Fig.8 Area of the landing footprints enclosed by the fitted curve

The relative error ΔSis calculated by using the following formula:

When0?i45?or 135?i180?,Srefers to the area enclosed by the fitting curves of the(20)and(21),and the range ofxis obtained by using(7)and(8).After combining(20),(21),(24)and(25)to obtain the intersections of four curves,S1,S2,S3andS4are obtained by computing the area enclosed by four curves with the longitudinal range between any two intersections.

When 45?

In Fig.7,four regionsS1,S2,S3andS4are taken as the numerators of(40),and there areKregions that can be used as molecules,S1,S2,...,Sk.

4.Simulations and analyses of results

4.1 Simulations

The values of the parameters of the space-to-ground vehicle simulations are as follows:the mass of the vehicle is 500 kg,the engine thrust is 100 N,the gas jet velocity is 3000m/s,the shape of the vehicle is an axisymmetric coneand the characteristic area of the vehicle is 0.02m2.The orbital radiusais in the range[6 700 km,11 000 km],the flatteningeis in the range[0,1),the value of the orbital inclinationiis in the range[0?,180?],the right ascension of ascending nodeΩis in the range[0?,360?],the argument of the perigeeωis in the range[0?,360?],the true anomalyfis in the range[0?,360?].Each of the six parametersa,e,Ω,ω,fandtmtakes three random values within their respective range.The parameteritakes four random values,two in the range of 0?i45?or 135?i180?and the other two in the range of 45?

Table 1 Ten experiments

The learning rate of the four BP neural network models is 0.01,the maximum number of iterations is 2 000,the training error is 0.001,and the number of hidden layer nodes is ten.If the relative error is smaller than 2%,it is considered that the usage requirements are met.

Twenty-four blade servers are used for the simulations,and each of the blade servers has 10 blades.The program is performed using visual studio 2012.The main steps of the program are as follows:

Step 1The training data of each space-to-ground vehicle is computed.At first,the minimum longitude or latitude and the maximum longitude or latitude of all of the landing points at one manoeuvre moment are computed by using two blades.Then,58 blades are used to compute 58 landing points between the minimum longitude or latitude landing point and the maximum longitude or latitude landing point.Then,60 manoeuvre moments are uniformly selected in one orbit period and the landing footprints at these manoeuvre moments are computed.This step corresponds to the theory in Section 2.

Step 2The boundary curve and its coefficients are obtained by least squares fitting.This step is described in de-tail in Section 3.1.

Step 3BP neural network training.This step is described in Sections 3.2–3.3.

Step 4Test the effectiveness of the proposed method.

Methods proposed by Li et al.[14]and Hu et al.[15]are used to compare to the proposed method.The parameters of the simulation are taken from[14,15].

4.2 Analyses of results

Fig.9–Fig.18 show the boundaries of,the fitting boundary curves of and the boundary curves of each landing footprint as obtained by the BP neural networks in the ten sets of testing experiments.As can be seen from Fig.9–Fig.18,the boundary curves obtained by the BP neural network,the fitting boundary curves and the boundaries of the landing footprints are basically the same.

Fig.9 Landing footprint of No.1 experiment

Fig.10 Landing footprint of No.2 experiment

Fig.11 Landing footprint of No.3 experiment

Fig.12 Landing footprint of No.4 experiment

Fig.13 Landing footprint of No.5 experiment

Fig.14 Landing footprint of No.6 experiment

Fig.15 Landing footprint of No.7 experiment

Fig.16 Landing footprint of No.8 experiment

Fig.17 Landing footprint of No.9 experiment

Fig.18 Landing footprint of No.10 experiment

Table 2 and Table 3 give the coefficients of the fitting boundary curves obtained by using the least squares method,including(boundary curves on the high latitude side);(boundary curves on the low latitude side);(boundary curves on the high longitude side);(boundary curves on the low longitude side).Table 4 andtained by the BP neural network,including(boundary curves on the high latitude side);and(boundary curves on the low latitude side);(boundary curves on the high longitude side);(boundary curves on the low longitude side).From Table 2 to Table 5,we can see that the errors between coefficients of the fitting boundary curves obtained by the least squares method and those obtained by the BP neural network are smaller than the training error of 0.001.

Table 2 Coefficients of fitting curves(high latitude or longitude side)

Table 3 Coefficients of fitting curves(low latitude or longitude side)

Table 4 Results(high latitude or longitude side)obtained by the BP neural networks

Table 5 Results(low latitude or longitude side)obtained by the BP neural networks

Table 6 shows that the relative errors ΔSare smaller than 2%,which meet the usage requirements.

Table 7 is a comparison of the different computational times.The variablesta,tbandtcare the computational time required for computing all of the landing points of each set of experiments,the average computational time for one landing point and the computational time for the landing footprint based on the BP neural network,respectively.As can be seen from Table 7,tcis less than 0.10/000ofta,andtcis less than 0.20/000oftb.

Table 6 Relative errors in ten experiments

Table 7 Comparison of the computational time

Table 8 shows the relative errors of the proposed method,the method in[14]and the method in[15].ΔS1and ΔS2are the relative errors obtained by the methods proposed in[14]and in[15],respectively.We can see from Table 8 that the relative errors in each experiment obtained by methods proposed in[14]and in[15]are bigger than the proposed method in this paper.

Table 8 Relative errors of three methods

Table 9 shows the computational time of the proposed method,the method in[14]and the method in[15].The parameterstdandteare the computational time for the landing footprints based on methods proposed in[14]and[15],respectively.As can be seen from Table 9,the computational time in each experiment obtained by methods proposed in[14]and[15]are longer than the proposed method.

Table 9 Computational time of three methods

Above all,the simulation results demonstrate that the proposed method can efficiently determine the landing footprints of space-to-ground vehicles with arbitrary orbital parameters and arbitrary manoeuver moment under the premise of ensuring accuracy.

5.Conclusions and future research

A BP neural network for determining the coefficients of the fitting boundary curves,which realizes fast computation of the landing footprint of a space-to-ground vehicle is proposed.The simulation results demonstrate that the proposed method can determine landing footprints in 0.01 s while ensuring a relative error within 2%.The proposed method lays a foundation for the deployment of parking orbits and for decision makers to develop real-time programs of transfer trajectories,which enriches the relevant theories of space engineering.

There are still some shortcomings in this paper.For example,it is important to try to decrease the number of hardware devices used to compute the training data.Additionally, finding a way to solve the transfer trajectories under finite thrust should be further studied.Moreover,the polynomial equations used to describe the boundaries of landing footprints by BP neural networks can be further generalized by using alternatives such as piecewise polynomial functions(splines)as fitting curves.