XU Xiaoyi,WANG Chunhui,and JIN Zhonghe

Micro-Satellite Research Center,Zhejiang University,Hangzhou 310027,China

Abstract:In this paper,a random access inter-satellite ranging(RAISR)system is designed.The ranging accuracy is optimized by an algorithm to greatly improve the ranging accuracy.This paper verifie the feasibility of the RAISR system through a series of theoretical analysis,numerical simulation,hardware system design and testing.The research work brings the solution to the design and accuracy optimization problem of the RAISR system,which eliminates the main error caused by the satellite dynamic characteristics and frequency source drift of the RAISR system.The accuracy of the measurement system has been significantl improved.

Keywords:satellite ranging,random access,distributed network.

1.Introduction

Since the beginning of the 21st century,the application results of satellite communication measurement technology have brought about major changes in communication,navigation and positioning,etc.,but a single spacecraft has been unable to meet the requirements of low cost,high reliability,fastness and efficiency for modern aerospace technology.In order to adapt to a variety of new user needs,distributed satellite systems came into being[1].With the concepts of distributed satellite systems such as formation flying,separation module spacecraft,and constellation,the cooperation of multiple satellites and space missions has become an important trend in space technology development[2–4].Most of the satellite systems that have been in-orbit experiments are networks of multiple homogeneous satellites[5,6],such as the National Aeronautics and Space Administration(NASA)Edison Demonstration of Smallsat Networks(EDSN)Project[7,8],the Swift Gamma-Ray Burst Mission Project[9],the European Space Agency(ESA)Project for On-board Autonomy(PROBA-3)[10],and the Chinese Academy of Sciences Space Ultra-Low Frequency Radio Observatory(SULFRO)Project[11].In order to extend the capabilities of individual systems and further reduce development costs,the concept of heterogeneous networks was proposed and applied to satellite networks,including federated satellite systems(FSS)proposed by Selva et al.[12],the heterogeneous spacecraft network(HSN)[13]proposed by Faber et al.,and the Internet of Satellites(IoS)proposed by Ruiz et al.[14].These systems have different mission objectives.Satellites from different research institutes can communicate with each other,and integrate heterogeneous networks to share functions such as shared payload data,forwarding remote command and telemetry data,and on-board hardware resource.Alessandro[15]pointed out that heterogeneous networks would become one of the three major development trends of distributed satellite systems in the future.At present,many research institutions and scholars have proposed to adopt a random peer-to-peer network architecture[16–29].Due to the randomness and irregularity of satellite distribution in heterogeneous networks[30],regular and orderly formation flying in homogeneous networks is no longer suitable for heterogeneous networks,so random networking has become an area we need to focus on.For stochastic networking technology,inter-satellite relative measurement is the premise to ensure the coordinated implementation of space system tasks.Inter-satellite communication measurements can reduce the dependence on ground stations,such as distributed satellite systems,small satellite formation flying,and satellite communication systems,reduce signal transmission delays,and improve the invulnerability and maneuverability of the entire system.Thus the satellite needs to have the capability of inter-satellite ranging.

Cooperative network refers to a network where both anchor nodes and other nodes need to communicate with the located node[31].The location of anchor nodes generally depends on global poisoning system(GPS).GPS costs are relatively high,and it may be unavailable in some cases.It becomes very important for the topological relationship between nodes to realize the co-location between nodes.The scheme proposed in this paper can be applied to responsive dynamic wireless sensor networks(WSNs),that is,data information is transmitted only when a specific situation occurs in the network.Each node independently completes the positioning process through ranging information and is not limited to GPS[32–35].

According to the characteristics and requirements of inter-satellite communication and ranging of distributed satellite systems,this paper designs a random access intersatellite ranging(RAISR)system scheme,and eliminates the main error through the precision optimization approach,which greatly optimizes the ranging accuracy.This is very important for the future collaborative space mission, filling the gap in the RAISR system.This paper studies the ultra high frequency(UHF)band,but the scheme is applicable to communication satellites in each band.

The remainder of this paper is organized as follows.The RAISR system design scheme and the capture method are presented in Section 2.Section 3 includes system ranging accuracy optimization and simulation.Experimental testing and verification are introduced in Section 4.This paper is concluded in Section 5.

2.Scheme of RAISR system

2.1 RAISR ranging scheme design

The principle of dual one-way ranging is that the two satellites perform the ranging measure each other to transmit measurement signals to the other party for distance measurement,and obtain a pair of distance measurement values with opposite magnitudes and opposite directions.And the two measured values are added to effectively separate the clock influences,thereby obtaining more accurate distance measurements.Under the random access system,each satellite transmits a signal,and other visible satellites can complete a one-way distance measurement with respect to the satellite of the transmitted signal.Taking four satellites as an example,the principle of the RAISR system is shown in Fig.1 below.

Fig.1 Four-satellite mutual distance measurement method

Every satellite can transmit signals,and other visible satellites can complete one-way distance measurement with respect to this transmitting signal satellite.Taking four satellites as an example,the multi-satellite mutual ranging method is shown in Fig.1.The time for each satellite to transmit measurement signals is random.Satellite A(SA),Satellite B(SB),Satellite C(SC),and Satellite D(SD)in the figure send signals atandrespectively,When SA emits a signal,the pseudoranges of the SA measured by SB,SC,and SD are,,and,respectively.When SC emits a signal,the pseudoranges of the SC measured by SA,SB,and SD are,and,respectively.When SD emits a signal,the pseudoranges of the SD measured by SA,SB,and SC are,and,respectively.When SB emits a signal,the pseudoranges of the SB measured by SA,SC,and SD are,and,respectively.Therefore,each satellite only needs to transmit a signal once,and it can complete the bidirectional distance measurement between any two satellites.

2.2 Error analysis

Under the RAISR system,because there is no strict time synchronization,the measurement accuracy of the dualone-way system is affected by many factors.After analysis,it is found that the error caused by the dynamic motion of the satellite and the drift of the frequency source is the main systematic error.This section will qualitatively and quantitatively analyze the main errors of the dual one-way inter-satellite ranging system to obtain measurement system errors.

2.2.1 Satellite dynamic motion error

Fig.2 shows the effect of dynamic motion on the RAISR system.

Fig.2 Effect of satellite dynamic motion on RAISR

A(t0),A(t1),A(t2)andA(t3)represent SA at timet0,t1,t2andt3respectively.B(t0),B(t1),B(t2)andB(t3)represent SB at timet0,t1,t2andt3respectively.The distance of the signal propagation within the propagation delay is not equal to the instantaneous distance between SA and SB.The relationship between the signal propagation distance and the instantaneous distance between the satellites is shown in Fig.2.The nominal time of SA emission time,SB reception time,SB emission time,and SA reception time are respectively recorded ast0,t1,t2andt3.is the propagation distance vector of the pseudo-code measurement signal indicating SA emission and SB reception,the value of which is equal to c,where c is the speed of light in vacuum,andis the transmission time of the signal through the measurement link SA.Sis the propagation distance vector of the pseudo-code measurement signal indicating SB emission and SA reception,the value of which is equal to candis the transmission time of the signal passing through the measurement link.ΔA(t1)indicates the distance vector of SA flight during.ΔB(t3)indicates the distance vector of the SB flight during.s(t0),s(t1),s(t2)ands(t3)respectively represent the inter-satellite distance(ISD)vectors at timet0,t1,t2andt3,and the expression of the inter-satellite pseudorange value is

where ΔtB(t1)is the difference between the clock face time and the nominal time of SB att1,ΔtA(t0)is the difference between the clock face time and the nominal time of SA att0,ΔtB(t2)is the difference between the clock face time and the nominal time of SB att2,and ΔtA(t3)is the difference between the clock face time and the nominal time of SA att3.andare the distance values respectively corresponding to the ionospheric delay ofand.andare the distance values respectively corresponding to the hardware transmission delay ofandandare the distance values respectively corresponding to the measurement noise delay ofand.

Temporarily ignoring the influence of other factors such as ionospheric delay and hardware delay,and temporarily thinking that the clock difference remains unchanged,the expression of the inter-satellite pseudo-range value can be expressed as

whereandare the propagation distances of the measurement signals in the measurement linksand,respectively,and are also the measurement targets of the ranging process.

wherevABindicates the radial relative motion velocity vector between satellites,τd=t2?t1.ISD vectors(t1)can be expressed as

Use the velocity vector and the propagation delay product to represent the satellite flight vector as

where˙rA(t1)is the velocity of SA att1and˙rB(t3)is the velocity of SB att3,(ti)represents the ISD unit vector of the no-minal timeti.

ηA=andηB=are,respectively,the velocity component ofs(t1)and the velocity component ofs(t3).Therefore,ρcan be written as

Therefore,the ISD calculated by the dual one-way ranging formula settlement is equal to the instantaneous ISD minus the systematic error introduced by the dynamic motion of the satellite.In the expression of systematic error,it can be seen that the magnitude of the error term depends on the orbit of the satellite andτd,and the larger theτd,the larger the error.

When the maximum ISDSmax=4000km,τ=13.3ms,vAB=3 km/s,τd=1 s,the system error caused by the dynamic motion of the satellite is 1520m with quantitative analysis.

2.2.2 Frequency source drift error

When analyzing the influence of frequency source drift on measurement accuracy,the effects of ionospheric delay and hardware transmission delay are temporarily ignored,and it is assumed that the propagation delays of the measurement linksandare equal,that is,==τ.The nominal moments of SA emission time,SB reception time,SB transmission time,and SA reception time are defined as time nodes 0,1,2,and 3,and the time node 0 is set tot=0.

In the measurement process,each satellite obtains the transmission time of the signal by making the local pseudocode phase different from the pseudo-code phase received by the received pseudo-code phase.Considering the influence of frequency source drift,the instantaneous frequencies of SA and SB are

θA(t)andθB(t)are the local pseudo code phase signals of SA and SB,fcis the nominal pseudo code rate of SA and SBdτare the frequency drift functions of SA and SB respectively.

De fine the phases of the A and B pseudo codes of the nodenin the random access network areθAnandθBnrespectively,and then the pseudo-code phase of SA at node 3 is

The pseudo-code phases of the same SB at node 1 and node 2 are

SA receives the pseudo-code signal sent by SB at node2 and node 3,extracts the pseudo-code phase of the received signal,and then compares it with the local pseudo-code phase to obtain the pseudo-code phase difference.

The signal propagation delay does not exceed 17 ms at most,and the drift functionsα1(t)andβ1(t)are functions that change very slowly.Therefore,the drift functionα1(t)andβ1(t)can be considered constants in the time of one measurement process.

The ISD solved by the dual one-way ranging formula can be expressed as

The measurement error is quantified without applying the interpolation algorithm.Taking the 40 MHz and 1 ppm crystal as an example,the frequency drift parameter has a value range of

Whenτd=1 s,α0=40 Hz,β0=?40 Hz,α1=1 Hz/s,β1=1 Hz/s,the ISD measurement error caused by frequency source drift is 303.75 m.

3.Ranging accuracy optimization and simulation

3.1 Ranging precision principle

In the RAISR system,due to different characteristics of transmission and reception,the error caused by the dynamic motion of the satellite and the drift of the frequency source is amplified.This paper proposes to use the interpolation algorithm to correct the time.The interpolation algorithm is based on pseudo-range.The observation itself calculates different moments of the measurement,and corrects the measured value at a certain moment to another time,effectively eliminating the systematic error caused by the dynamic motion of the satellite and the drift of the frequency source.

For two satellites that are ranging from each other,the polynomial interpolation method is shown in Fig.3.

In Fig.3,SA emits a signal at its clock face timeT0.SB receives the signal and measures the one-way ranging resultρB1.SB transmits a signal at its clock face timeT2.SA receives a signal and measures the one-way ranging resultρA1.With the number of signals transmitted by SA and SB increasing,SA and SB respectively obtain the one-way ranging results(ρA1,ρA2,ρA3,...,ρAM)and(ρB1,ρB2,ρB3,...,ρBN).The moments at which SA and SB transmit signals are random,so the obtained ranging results are not equally spaced.In Fig.3(b),the polynomial interpolation method refers to fitting the pseudo-range value corresponding to the clock time of SB receiving time by using the actually measured pseudorange(ρA1,ρA2,ρA3,...,ρAM),for example,ρA(T1)corresponding to timeT1of SA clock face time,as shown by the red dotted line in the figure,the bidirectional pseudo-range observation(ρA(T1),ρB1)that has been subjected to polynomial interpolation is used to solve the ISD.Similarly,SB uses the ranging result(ρB1,ρB2,ρB3,...,ρBN)to fit the pseudo-range(ρB(T3),ρB(T5),ρB(T11),ρB(T15),ρB(T17),...),which corresponds to the clock face time of SA.We can calculate the fitted values and measured values.That is to say,a bidirectional distance measurement can be completed every time a satellite sends a signal by polynomial interpolation.

Fig.3 Algorithm based on polynomial interpolation

3.2 Capture method

The fast acquisition method of direct spread spectrum signals by the physical layer is delay multiplication.The influence of Doppler frequency offset can be eliminated by multiplying the received signal with the conjugate signal delayed by itself.The received pseudo-code phase can be calculated quickly by fast Fourier transform(FFT).The principle block diagram is shown in Fig.4.

Fig.4 Delay and multiplication method block diagram

The digital intermediate frequency signal output by the analog-to-digital converter(ADC)is multiplied by its delayed complex conjugate after being down-mixed and lowpass filtered.LPF is short for low pass filter.NCO is short for numerically controlled oscillator.IFFT is short for inverse fast Fourier transform.The local pseudo-code is multiplied by its delayed local pseudo-code signal,and then the two are related to the frequency domain code phase search.The received intermediate frequency signal output by the ADC can be expressed as

wherePis the received signal power,Tsis the sampling period,nTsis the sampling time,c(nTs?τ)is the pseudocode sequence,τis the unknown delay,andfIF,fdandθ0are the nominal intermediate frequency of the carrier,the Doppler frequency and the initial phase.w(nTs)represents Gaussian white noise and obeys the normal distributionN(0,σ2).The received intermediate frequency signal is digitally down-mixed with the local carrier,and then the high-frequency component is filtered through a low-pass filter.The output I and Q low-frequency components are expressed as

wherewI(nTs)andwQ(nTs)represent noise terms.The two signals are delayed by an integer number of chip times.The delay time is expressed askTs,kis a fixed constant,and then the delay amount is conjugated and multiplied by the undelayed signal component, because the delay amountkis a fixed constant. After the delay multiplication process,the Doppler frequency in the obtained signal is eliminated,leaving only the pseudo-code signal and noise.Then,the phase of the received pseudo-code is directly obtained by the FFT.After obtaining the pseudo-code phase and compensating the local pseudo-code signal,the local pseudocode is used to despread the received direct-spread spectrum(DS)signal to obtain a single carrier,and then the Doppler frequency of the carrier can be directly calculated to complete the capture of the received signal.

3.3 Simulation

The interpolation algorithm eliminates the error caused by the dynamic motion of the satellite,greatly reduces the error caused by the drift of the frequency source,but introduces the interpolation error.The magnitude of the interpolation error is related to the sampling interval and the polynomial order.This article uses the Satellite Tool Kits(STK)to generate ISD simulation data for numerical analysis of interpolation errors.The satellite orbit is set to J4 perturbation and low earth orbit(LEO).The two satellites are located in different orbits.The duration of the simulated data is 1 h,and the interval is 0.05 s.Using this ISD simulation data,the interpolation precisions for different sample sampling intervals and different polynomial orders are calculated separately.In this paper,the difference accuracy is evaluated by the maximum error and the root mean square error(MSE).The distance between the lines is taken as the true value,the interpolation valuePat the time of the difference is compared with the corresponding ISD analog valuePtrue,and the interpolation error ΔPis obtained.

The commonly used indicators for evaluating the accuracy of the difference are the maximum error and the root MSE.The formula for calculating the root MSE is

The ISD simulation values are sampled at an average sampling interval of 0.5 s,0.6 s,0.7 s,4.9 s,and 5 s,and the 2nd-order to the 8th-order Lagrange polynomial interpolation(LPI)precision is calculated.The maximum error of statistical interpolation is root MSE.In each case,100 times value calculation and statistical results,the uniform interval sampling and non-uniform interval sampling calculation results are shown in Fig.5–Fig.11.

Through statistical comparison analysis,under the same conditions,the interpolation error of equally spaced samples is smaller than that of non-uniform interval sampling.When the average sampling interval of non-uniform interval samples is less than 2 s,the interpolation polynomial order should select a smaller value,and the 2nd-or 3rdorder is more suitable.The average sampling interval is in the range of 2 s to 5 s,and the interpolation polynomial order is the4th-order.That is to say,when the satellite system measurement task is frequent,the interpolation algorithm adopts the 2nd-or 3rd-order to bring a smaller error.When the system has fewer measurement tasks,the 4th-order interpolation algorithm has a smaller error.Using the results of the analysis to perform polynomial interpolation on the system allows the interpolation error to be controlled in the order of millimeter.

Fig.5 The 2nd-order LPI

Fig.6 The 3rd-order LPI

Fig.7 The 4th-order LPI

Fig.8 The 5th-order LPI

Fig.9 The 6th-order LPI

Fig.10 The 7th-order LPI

Fig.11 The 8th-order LPI

4.System demonstration

The physical map of the ranging platform of the RAISR system is shown in Fig.12,and its connection relationship is shown in Fig.13.The current test platform contains a total of three satellites,and the satellites are connected by radio frequency(RF)cables and RF connectors.The output signal of each satellite is split into two paths by a splitter,which are connected to the combiner of the other two satellites,thus ensuring the connection between any two satellites.Each satellite transmits the distance measurement value to the personal computer(PC)through the universal asynchronous receiver/transmitter(UART)interface,and the ISD is solved by the PC.

Fig.12 Test bench

Fig.13 Connection relationship

The hardware platform of the inter-satellite communication measurement system designed in this paper is a new generation of miniature measurement and control transponder developed by the research group.Its structure is shown in Fig.14.The RF transmit front end and the RF receive front end have only amplifiers and RF filters,which function to filter and amplify the RF signal.The functions of downmixing,analog wideband automatic gain control(AGC)processing,analog-to-digital conversion,digital-toanalog conversion,and up-conversion are all performed in the baseband single chip.The digital processing module is responsible for signal synchronization,relative ranging,performing multiple access protocols,protocol data packets,etc.,and is implemented in a Xilinx field programmable gate array(FPGA).Fig.15 is the work flow of the inter-satellite ranging system.

Fig.14 Hardware platform structure functional block diagram

Fig.15 Work flow of inter-satellite ranging system

Step 1The application layer sends the measurement signal command pulse.

Step 2The media access control(MAC)layer receives the measurement command pulse.

Step 3The MAC layer successfully contends to the channel and sends a complete measurement frame.

Step 4After receiving the measurement signal,the satellite samples the measurement signal and transmits it to the PC.

Step 5The distance is calculated by PC.

When the epoch reduction is not performed,the ISD solution is to add the two mutually opposite but distance measurement values with the shortest bidirectional measurement interval(BMI).As shown in Fig.16,SB receives the measurement signal of the SA emission.The one-way distance values areρB1,ρB2andρB3.SA receives the measurement signals transmitted by SB to obtain the one-way distance valuesρA1andρA2.ForρA1,the inverse ranging value with which the measurement time interval is the smallest isρB1,soρA1andρB1are added to calculate the ISD value.Similarly,ρA2andρB3are combined into a pair of mutually opposite distance measurements.ρB2in Fig.16 is discarded and not used for solving the ISD.

Fig.16 ISD solving method

The nominal frequency of the crystal oscillator used in the system is 40 MHz,and the command average generation rateλ=1.By testing the actual frequency accuracy of the crystal oscillator,the maximum frequency deviation between different crystal oscillators is about 60 Hz.After 2 s,the time deviation of the two crystals can reach 3 μ s.

The crystal oscillator time deviation is converted to a bidirectional distance deviation of 450 m.Fig.17 shows the maximum BMI of 2.65s.The corresponding ISD value is 412.6 m,and the measured ISD is 799.1 m,that is,the maximum ranging error is about 386.5 m.That is to say,the actual ranging error is within the range of the theoretical deviation.

Fig.17 ISDA?Band BMIA?B

Fig.18 shows the relationship between ISDB?Cand the corresponding time interval.Fig.19 shows the relationship between ISDC?Aand the corresponding time interval.

Fig.18 ISDB?Cand BMIB?C

Fig.19 ISDC?Aand BMIC?A

In the case of no optimization of ranging performance,the two-way ranging method based on random access has an MSE of several tens of meters or even more than one hundred meters,which is shown in Table 1.

Table 1 Ranging error

Fig.20 shows the ISD solving method using epoch reduction.

Fig.20 ISD solving method using epoch reduction

After the epoch reduction,the measurement results between the satellites are shown in Fig.21.Fig.21(a)is the ISD value calculated by the combination of the measured distance value of SA and the interpolation distance of SB,denoted asSA?B.Fig.21(b)is the ISD value calculated by the combination of the measured distance value of SB and the interpolation distance of SA,and is recorded asSB?A.As can be seen from comparison with Fig.17(a),after the epoch reduction,the ranging error is significantly reduced.

Fig.21 Ranging value of SA and SB after epoch reduction

Table 2 lists the distance values between any two satellites in the network consisting of three satellites after epoch reduction.It can be seen that when the average time interval between the actual signals transmitted by each satellite is about 1.21 s,the ranging accuracy after the aging of the epochs is in the range of 0.762 m to 1.043 m.

Table 2 Distance error after epoch reduction

It is found that the actual measurement results are deviated from the simulation results.After analysis,it is considered that the time used for the simulation is a more accurate system time.In actual measurement,since the system has no time synchronization,the clock time is different.The interpolation time includes the receiver measurement error,so we carried out the homologous experiment.The three satellites in the network are driven by the same frequency source,and the zero time of the three satellites is kept consistent by resetting.In this state,there is no longer a ranging error caused by the frequency source offset in the system.Fig.22(a)shows the ISD between SA and SB,and Fig.22(b)shows the BMI corresponding to each distance value.

Fig.22 Ranging results of SA and SB in homologous experiments

It is statistically obtained that the ranging MSE at this time is 0.069 m.The interpolated ranging value is basically the same as the ranging variance before unplugged,indicating the interpolate on error in the case of time synchronization.It is less than the thermal noise and the ranging error caused by the internal digital processing of the system.In the absence of time synchronization,the deviation between the actual test result and the theoretical result is mainly caused by the interpolation error.

5.Conclusions

In this paper,we design the RAISR system and analyze the system accuracy by simulation quantitatively.An accuracy optimization approach to the RAISR system is proposed,which eliminates the error caused by the two-way ranging time interval.The physical system test results show that the MSE is reduced from the maximum114.353m to 0.069m,and the accuracy is improved by 99.94%.The ranging accuracy of the RAISR system is demonstrated in the numerical simulation and hardware platform tests.