antenna configuration method for rf measurement based on

Research ArticleAntenna Configuration Method for RF Measurement Based onDOPs in Satellite Formation Flying

Weiqing Mu ,1,2 Rongke Liu ,1 Zijie Wang,1 and Yukai Liu1

1School of Electronic and Information Engineering, Beihang University, Beijing, China2China Electronics Technology Group No. 22 Research Institute, Qingdao, China

Correspondence should be addressed to Rongke Liu; [email protected]

Received 13 January 2018; Accepted 31 May 2018; Published 30 August 2018

Academic Editor: Paolo Gasbarri

Copyright © 2018 Weiqing Mu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Radio frequency (RF) measurement technology provides a relative navigation solution that can be of great importance to havesignificant potential for application to the satellite groups. With the development of the RF relative measurement sensors, it isfound that the antenna configuration of the sensors affects the precision of relative position and attitude measurement. Thisstudy proposes improvements to the precision of the sensors by the virtue of the optimal antenna configuration. Furthermore,the concept of dilution of precision (DOP) is extended to the RF relative measurement sensors, and a new dilution called IDOPis proposed as a benchmark to determine the precision of relative position and attitude measurement in engineering applicationsof the space mission. In order to select the optimal antenna configuration in real time in a scene where the intersatellite positionand attitude change dynamically, this study presents an optimal antenna configuration selection strategy and models the antennaconfiguration selection as a combinatorial optimization problem. Furthermore, the genetic algorithm (GA) with two encodingmechanisms is proposed to solve this problem. Finally, numerical results are presented to verify the robustness.

1. Introduction

Satellite formation flying (SFF) has brought several advan-tages and privileges to space missions. However, some newchallenges have engendered in maintaining the configurationof a set of satellites in formation flying [1, 2]. SFF requires ahigh level of accuracy of relative navigation and control [3].RF metrology is generally considered as the most suitedmethod for relative navigation and extremely precise controlfor SFF, and it also provides a solution for intersatellite com-munication and network [4]. RF-based accurate relative nav-igation can be executed by means of onboard embeddedsystems which afford relative measurements. The onboardembedded systems can be of advantage in a way it enhancesthe precision achieved by Global Navigation Satellite System(GNSS), and more importantly, it can be used in the deepspace where GNSS is not available [5]. Numerous missionswith high-accuracy demands on the intersatellite positionhave been flown or proposed. The National Aeronauticsand Space Administration has proposed the autonomousformation flying (AFF) technology for many missions

(Star-Light/DS3/ST3) [6–8]. Formation flying radio fre-quency (FFRF) sensor is another RF-based technology devel-oped by the National Center for Space Studies in thePRISMA mission [9, 10].

So far, the RF technology is of high importance both forcurrent and future space missions. Based on the RF measure-ment model, the positions and attitudes of intersatellite canbe estimated from a batch of observations collected at a singleepoch by the use of a least-squares estimator. This studyanalyzed the antenna configuration and found that antennaconfiguration affects the estimation precision of the relativepositions and attitudes in RF measurement to a large extent.For this reason, a new set of DOPs is presented as a bench-mark for the optimal antenna configuration selection,because of their capability of predicting the precision of rela-tive position and attitude measurement.

DOPs have been used in satellite navigation, for example,geometric dilution of precision (GDOP) and attitude dilutionof precision (ADOP) in GNSS technology. GDOP is a valuethat indicates the effectiveness of the GNSS satellite geometrydistribution on the navigation, by checking the values of

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 4216971, 14 pageshttps://doi.org/10.1155/2018/4216971

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https://doi.org/10.1155/2018/4216971

which a receiver can determine the most appropriate setof GNSS satellites for a particular positioning precision[11, 12]. The same thought is applied to the RF relativemetrology. However, GDOP only analyzes the estimationprecision of position parameters and excludes the precisionof attitude parameters [11]. Similarly, ADOP focuses on pre-diction of the attitude parameter estimation precision, and itis defined by the single-difference carrier-phase measure-ment model, which is different from commonly used RFmeasurement model [12]. In addition, [13, 14] are for theimprovement of the ADOP when the GNSS is employed insome special scenarios. Both GDOP and ADOP are not suit-able for RFmethodology. As a further development, a new setof DOPs is presented to predict and quantify the precision ofRF relative measurement. The distance dilution of precision(DISDOP) is defined, whose definition is the same as AZU-DOP, ELEDOP, EUDOP, and TDOP. These DOPs start witha new approach to the intersatellite position and attitude esti-mation that produces the distance, azimuth angles, elevationangles, and Euler angles as the solution form. However, in adynamically relative moving scene, if DOPs are employedto obtain the optimal antenna configuration, the resultsobtained by different DOPs are different. For example, in theestimation of intersatellite attitude, EUDOP (ωDOP, θDOP,and φDOP) can be employed to obtain the optimal antennaconfiguration; moreover, the order of magnitude of these threeDOP values is different. Therefore, this study proposed a newdilution called IDOP and a new matrix of weighted coeffi-cients which represents the proportion of each DOP in theIDOP. In this way, the different requirements of the estimatedprecision of intersatellite position and attitude parametersin different stages of the space mission can be satisfied.

However, when the RF relative measurement sensors areemployed in the relative motion scene of satellites, it is foundthat the optimal antenna configuration changes with thechange of intersatellite position and attitude. Reference [6]proposed the rule that the antennas should be placed as farapart as possible; based on this empirical rule, some antennaconfigurations can be obtained, but it is not optimal in adynamic scene. Reference [15] also presented this issue inGNSS-based attitude measurement, but the optimal antennaconfiguration is not obtained, and its model is differentfrom this study. For these reasons, an antenna configura-tion selection strategy is proposed in this study. This studymodels the antenna configuration selection in a dynamicscene as a combinatorial optimization problem. Further-more, the binary-coded and the real-coded GAs are pro-posed, respectively, to solve this combinatorial optimizationproblem. The experimental results show that the proposedGA-based strategy is very effective and reliable. The measure-ment precision corresponding to the optimal antenna config-uration obtained by the proposed GA-based strategy is muchhigher than that of the normal antenna configuration.

The organization of this paper is as follows. Section 2describes the RF relative measurement model and the DOPmodel. Section 3 presents the GA-based strategy which isused to obtain the optimal antenna configuration. The exper-imental results and conclusion of this paper are shown inSections 4 and 5, respectively.

2. RF Relative Measurement and DOPModeling

2.1. Estimation Model of RF Relative Measurement. To esti-mate the total seven parameters including the relative posi-tions, attitudes, and clock error, enough observables need tobe obtained in the measurement epoch. Moreover, a rangemeasured between any transmitting antenna and receivingantenna on different satellites is considered as observable.Thus, the antenna configuration in the RF measurementaffects the estimation model and the RF measurement accu-racy (the proof is done in Section 2.2). The antenna configu-ration includes two aspects: the number of antennas and thefixed locations of antennas. We firstly consider the number ofantennas to determine the estimation model, and the fixedlocations of antennas will be analyzed in Section 3. Multipletransmitting and receiving antenna configuration is shownin Figure 1.

Two aspects need to be considered to determine thenumber of antennas: the number of observables and the sep-arate number of transmitting and receiving antennas. First ofall, the observation obtained by antennas must be more thanthe number of the parameters to be estimated. Secondly, sup-pose there is 1 transmitting antenna on one of the satellitesand there are three receiving antennas on another satellite,the shape formed by the antennas is a regular triangular pyr-amid in which the base is defined by receiving antennas, andthe summit is the transmitting antenna, and the two satellitesare at the two ends of the centerline of the regular triangularpyramid as illustrated in the right part of Figure 1. In thiscase, the attitude of the intersatellite changes when both thesatellites rotate by the centerline; however, the observablesare always consistent, so that the estimation matrix is irre-versible. For this reason, it is necessary to increase the trans-mitting antenna to change the shape of antennas andincrease the number of observables. Therefore, the antennanumber of 5 which includes 2 transmitting antennas and 3receiving antennas on each satellite is determined. Hence,12 ranging observables are obtained in measurement epochs.

In practice, the ranging observables is obtained by thecarrier phase. It is assumed that the carrier-phase integerambiguity has been solved by using the integer ambiguity res-olution algorithms. The commonly used algorithms areLAMBDA and LAMBDA improved algorithms [16]. This

z

y

x

Receiver 1Receiver 1

Transmitter

Transmitter

Receiver 3

Receiver 2 Receiver 2 Receiver 3

Figure 1: Multiple transmitting and receiving antennaconfiguration.

2 International Journal of Aerospace Engineering

ranging observables are affected by the clock error betweentwo satellites; then, the measurement model is given by [17]

ρ = ρG + ρτ + ε, 1

where ρ is the observed range, ρG is the geometric range,and ε is the noise of the ranging; and

ρτ = c τR − τT = cΔτ, 2

where τR is the clock error at the receiving satellite and τT isthe clock error at the transmitting satellite; and

ρG = rrT , 3

where r is the ranging vector. Thus, the measurementmodel is

ρi = rrT + cΔτ + ε 4

Consider the position of transmitting satellite A is pT,with orientation angles ωT, θT, and φT. The local offset coor-dinates of the transmitting antenna on satellite A are uT. Thesame description applies to the receiving satellite with sub-script R replacing T. Thence, the geometric range model isgiven by

r = PR +Q φR, θR, υR uR − pT +Q φT, θT, υT uT 5

Note that any orientation can be reached by a rotationthrough ω angle around the x-axis, followed by a rotationthrough θ angle around the rotated y-axis, followed by a rota-tion through φ angle around the rotated z-axis. The rotationscan be represented by the rotation matrix Q.

Δx mi ≅ d m

i − d mi−1 , α

mi − α

mi−1 , γ

mi − γ

mi−1 ,

w mi −w m

i−1 , θmi − θ

mi−1 , φ

mi − φ

mi−1 ,

Δτ mi − Δτ m

i−1

6

In addition, a reference point associated with satellite A isat the origin of the Cartesian reference frame. The rangingmodel can be expressed as

r = d cos α cos γ, d cos α sin γ, d sin α

+ Q φR, θR, υR uR −Q φT, θT, υT uT ,7

where d is the distance between the transmitting andreceiving antennas, α is the azimuth angle, and γ is theelevation angle.

The state vector which consists of 7 parameters that needto be estimated is defined:

x = d, α, γ, ω, θ, φ, τ 8

Equations (4) and (5) present that the measurementequation is nonlinear. So, a linearization using a first-orderTaylor series expansion is performed; nextly, a least squaremethod is applied to solve the measurement equations:

u mi ≅

∂ρmi∂d

,∂ρmi∂α

,∂ρmi∂γ

,∂ρmi∂w

,∂ρmi∂θ

,∂ρmi∂φ

, 1 , 9

wherem is themth iteration, i is the ith measurement observ-

able, u mi is the mth iteration measurement vector of the ith

measurement observable, and ρi is the ith measurementobservable which is obtained by ranging a pair of transmit-ting and receiving antennas on different satellites.

Furthermore, define the following vector for the iteration:

Δx mi ≅ d m

i − d m‐1i , α

mi − α

m−1i ,

γmi − γ

m‐1i , w m

i −w m−1i ,

θmi − θ

m−1i , φ

mi − φ

m−1i ,

Δτ mi − Δτ m−1

i ,

10

where Δxi is the state vector which is equal to the differencebetween the two successive iterations, and i ∈ 1, 12 .

So the measurement equation can be shown with amatrix form as

Δy m =H m ⋅ Δx mi + ε, 11

where

Δy m = Δρ m1 , Δρ m

2 ,… , Δρ m12

T,

H m = u m1 , u m

2 ,… , u m12

T,

ε = εΔρ 1i, εΔρ 2

i,… , εΔρ 12

i

T

12

Δρi is the remaining error of the measurementobservable, which is equal to the difference between twosuccessive iterations.

Furthermore, the iterative process can be given by [17]

Δx m−1 = v H m−1 TWH m−1 −1H m−1 TWΔy m−1 , 13

x m = x m−1 + Δx m−1 , 14

where H is a matrix of 12 rows and 7 columns and W is theweight matrix:

W =

w1

w2

⋯

wN

, wi =1σρi

, 15

where σρi is the ranging error in the ith ranging channel.If it is assumed that the ranging precision of each rangingchannel is the same, W becomes an identity matrix.

When ∥Δxk∥< threshold, the iterative process stops.

3International Journal of Aerospace Engineering

2.2. DOP Modeling for Basis of Optimal AntennaConfiguration Selection. The approach to derive the AZU-DOP, ELEDOP, and EUDOPs is similar to the derivation ofGDOP described in [17]. The state vector is defined as in(8). The covariance of Δx is given as

cov Δx = E ΔxΔxT 16

If all the errors of observables in Δy have the samevariance σρ with zero mean and are uncorrelated with eachother, then

E ΔyΔyT = σ2ρW−1, 17

where W is a weight matrix. By (13), (15), and (16), the fol-lowing equation is obtained:

cov Δx = E ΔxΔxT

= E HTWH−1HTW ΔyΔyT

=σ2ρW−1

WTH HTWH−1

= HTWH−1HTWH HTWH

−1σ2ρ

= HTWH−1σ2ρ

18

The covariance of each parameter needs to be estimatedin (17), which is the product of the diagonal elements ofmatrix D defined in (18).

E Δd2 =D11 σ2p

E Δα2 =D22 σ2p

E Δγ2 =D33 σ2p

E Δw2 =D44 σ2p

E Δθ2 =D55 σ2p

E Δφ2 =D66 σ2p

E Δτ2 =D77 σ2p

⟹

σd = D11 σp,

σα = D22 σp,

σγ = D33 σp,

σw = D44 σp,

σθ = D55 σp,

σφ = D66 σp,

στ = D77 σp,

D = diag HTWH−1 ,

19

where Dii refers to the ith row and the ith column element ofmatrix D.

The following dilutions are used to indicate the effect ofintersatellite antenna geometry on the estimation accuracyof the relative position and attitude parameters: AZUDOP,ELEDOP, EUDOPs (including ωDOP, θDOP, and φDOP),and TDOP. Define these new DOPs as follows:

DISDOP, AZUDOP, ELEDOP,ωDOP, θDOP, φDOP, TDOP

= D = diag HTWH−1

20

In the space missions, the requirements of measurementaccuracy are different in different stages of flight. For exam-ple, when both the satellites are in the long-distance phase,the accuracy of attitude measurement is reduced due to thelimitation of measurement accuracy of distance, so the atti-tude measurement is abandoned; however, in the short-distance stage, the more precise attitude measurement canbe obtained. Therefore, the level of attention to the precisionof intersatellite relative attitudes and positions can be set todifferent in RF measurement. As a result, it is necessary tointroduce a new matrix called interest matrix Imatix to rep-resent the level of attention to the estimation precision of rel-ative attitude and position parameters. The new definedinterest matrix is as follows:

Imatrix = id, iα, iγ, iω, iθ, iφ, iT , 21

id + iα + iγ + iω + iθ + iφ + iT = 1 22

So IDOP is defined as follows:

IDOP = DISDOP, AZUDOP, ELEDOP, ωDOP,θDOP, φDOP, TDOP ITmatrix

23

Since a smaller IDOP corresponds to a higher estimationaccuracy, the objective function of the optimal antenna con-figuration optimization model is shown in a new form, whichis as follows:

min IDOP 24

3. Antenna Configuration

3.1. Problems of Optimal Antenna Configuration Selection.IDOP provides an effective tool and basis to get the optimalantenna configuration. When the RF measurement sensorsare employed in space missions, the relative position andattitude change dynamically; and IDOP contains the coeffi-cients corresponding to the relative position and attitudeparameters which exist in matrix H. As a result, it will bea serious problem that the value of IDOP changes withthe change of intersatellite position and attitude. Moreover,in a dynamic scenario, the optimal antenna configurationbased on IDOP is also changed with time. Furthermore,to verify the extent of its impact, a traversal simulationexperiment is presented.

Assume that there are two satellites whose relative atti-tudes and positions need to be estimated, and the side lengthof each cube satellite is normalized to 1. Figure 1 shows thecoordinate frame and body frame of the RF relative measure-ment. There are 2 transmitting and 3 receiving antennas,which are, respectively, installed in the one face of each satel-lite. The coordinate values of all the antennas in each bodyframe are summarized in Table 1.


The results in Figure 2 are consistent with our analysis,and with the changes of the relative attitude of intersatellite,the value of IDOP corresponds to a consistent change. As aresult, the optimal antenna configuration obtained based onIDOP will change in real time, when the RF relative mea-surement sensors are applied in a dynamic scene. Therefore,as the relative attitude of the intersatellite changes, the opti-mal antenna configuration corresponds to a consistentchange. Furthermore, for the purpose of obtaining the opti-mal antenna configuration in dynamic scenes, it is necessaryto change installation locations of the antennas on satelliteplatforms. The easiest way is to traverse all the reliableinstallation locations, but in applications, this way is notachieved due to its high computational complexity. There-fore, it is necessary to propose a method reasonably in accor-dance with the application requirements to get the optimalantenna configuration.

3.2. Antenna Configuration Selection Strategy Based onGenetic Algorithm

3.2.1. Antenna Configuration Selection Strategy. An effectiveantenna configuration strategy is proposed to maintain highestimation precision of the RF relative measurement, whichis to select some of the antennas from the reliable positionsin real time to form the optimal antenna configuration basedon IDOP. According to [6], antennas should be placed asfar apart as possible. Thus, in the application, assume thatthere are 8 installed locations of antennas, which are the8 corners of each cube satellite. Thence, the antenna combi-nation corresponding to the minimum value of the IDOP isselected as the optimal antenna configuration. According tothis antenna configuration strategy, it is necessary to select2 antennas as transmitting antennas and 3 antennas asreceiving antennas out of 14 antennas regardless of antennavisibility and orientation. Therefore, there are totally C2

14C312 = 20020 kinds of combinations. Due to this large number

of combinations, if the traversing method is used to calculateall the IDOPs corresponding to the 20020 kinds of antennaconfigurations, the real-time performance of the traversing

method is not high due to the high computational complex-ity. Therefore, this study models the antenna configurationselection as a combinatorial optimization problem, and thisproblem is described as (24)

f it si =min IDOP si , ∀si ∈Ω, 25

where Ω = s1, s1, s1,… , s20020 and Ω is a solution set com-posed of 20020 kinds of antenna configuration.

3.2.2. GAs for the Proposed Strategy. Among the methods ofsolving combinatorial optimization problems, GA is an effec-tive method [18]. In turn, the GA-based method is proposedfor obtaining the optimal antenna configuration in this study.In the use of GA to solve this optimization problem, the pur-pose is to obtain the optimal antenna configuration corre-sponding to the smallest value of IDOP. Thence, the IDOPvalue corresponding to the antenna configuration is consid-ered as the fitness in GA. The objective function, the functionto be optimized, provides the mechanism for evaluating eachstring [19], and the fitness function is a special type of objec-tive function that is shown as

f it si =max1

IDOP

=max1

diag HTH−1 ITmatrix

,26

where si is the ith kind of antenna configuration and Bcodeiand Qcodei, corresponding to the binary-coded GA and thereal-coded GA, respectively.

The encoding mechanism is a point of attention in theapplication of GA. In this study, two encoding mechanismsare proposed, which are used in the proposed GAs and areknown as the binary-coded and real-coded mechanisms,respectively. Therefore, the binary-coded and the real-coded mechanisms are shown in Figure 3.

Case 1 (the binary-coded mechanism). It is proposed toencode a chromosome of antenna configuration, which isshown in Figure 3. A chromosome in the GA is representedby a 14-bit binary codeword, and each coded bit representsthe antenna at a location on the satellite. If the ith bit is 1, thismeans that the antenna at the ith location is selected andused; whereas, if it is 0, it means that it is unused. Since theantenna selection strategy of selecting 5 antennas from 14antennas is adopted, the constraint is shown in (26) withthe binary-coded mechanism.

〠14

n=1Bcodei bitn = 5, 27

where codei is the coded value of the gene in the ith chromo-some and bitn is the nth genes of the chromosome; a chromo-some represents an antenna configuration and a generepresents an antenna position in an antenna configuration.

Table 1: Antenna coordinates of the tandem satellite.

Antenna coordinates

Satellite A

Transmitter 1 (0.5 0 0)

Transmitter 2 (0.5–0.5 0.5)

Receiver 1(0.5 0.5 0.5)

Receiver 2 (0.5 0.5–0.5)

Receiver 3 (0.5–0.5 -0.5)

Satellite B

Transmitter 1 (0.5 0 0)

Transmitter 2 (0.5–0.5 -0.5)

Receiver 1 (0.5–0.5 0.5)

Receiver 2 (0.5 0.5 0.5)

Receiver 3 (0.5–0.5 -0.5)


This binary-coded mechanism is useful to select 5 of the14 antennas; however, it is impossible to select 2 of the 5selected antennas as the transmitting antennas and theremaining 3 as the receiving antennas. In fact, for the purposeof selecting 2 antennas among the 5 antennas as the transmit-ting antenna, there are C2

5 = 10 kinds of combination. In thismethod, the IDOP values corresponding to the above 10

combinations are traversed and compared, and the smallestIDOP is selected as the individual fitness (28).

Case 2 (the real-coded mechanism). It is also shown inFigure 3, which is described as follows. Firstly, 14 candidatelocations where the antennas are installed are encoded inthe quaternary; each of the coded bits represents an

1

0

−1

−1 −1

1 10 0y

z

x

TransmitterReceiver

(a)

1

0

−1

−1 −1

1 10 0y

z

x

TransmitterReceiver

(b)

10.5

−0.50

Relat

ive a

ttitu

de an

gle (

rad)

0 10 20 30 40 50 60 70 80 90 100t (s)

PsiThetaPhi

(c)

10

2015

50 10 20 30 40 50 60 70 80 90 100

t (s)

IDOP corresponds to antenna configuration 1IDOP corresponds to antenna configuration 2

The v

alue

of I

DO

P

(d)

Figure 2: IDOP with the change of relative attitude of the intersatellite: (a) the optimal antenna configuration 1 when satellite relativeattitudes psi = 0.174 (rad), theta = 0.174 (rad), phi = 0.174 (rad); (b) the optimal antenna configuration 2 when satellite relative attitudespsi = 0.042 (rad), theta = 0.236 (rad), phi =−0.277 (rad); (c) trajectory of satellite relative attitude; (d) IDOPs in the case of satellite relativeattitude changing.

zy

x

1

1

8 9 10 11 12 13 141

1

Tx1 Tx2 Rx1 Rx1 Rx2

1 114 14 14 1 14 1 14

Bcodei

Qcodei

1 1

2

2

2

2

5

4

3

3 4

8

7

0 0 0 0

6

0

7

0 0 0 0

12

10

14

11

13

9

6

5

Figure 3: Encoded method of chromosomes.


installation location on the satellite; then, five antennas to beinstalled are determined, which include two transmittingantennas (Tx1 and Tx2) and three receiving antennas (Rx1,Rx2, and Rx3); and finally, a location (a total of 14 locations)for each of the five antennas is selected as the mounting loca-tion. In addition, the locations chosen for each antenna needto be different from each other, so constraints are as follows:

〠5

j=1〠5

k=1k≠j

Qcodei bit j ⋅Qcodei bitk = 0 28

Furthermore, the steps of solving the combinatorial opti-mization problem of optimal antenna configuration selectionwith the real-coded and binary-coded GAs are as follows:

(i) Initialization: generate an initial population Ω0. Ini-tialize the probability of crossover pc0 and mutationpm0, respectively.

(ii) Selection: evaluate the fitness function f it si . Theroulette wheel selection method based on a pro-portional selection mechanism is used to obtainthe current population Ωt [18]. If the size of thegroup Ωt is N and the fitness of the individual siis f it si , the selection probability of the individualsi is

P si =f it si

〠Nj=1 f it sj

29

The roulette selection method is implemented as follows:

(1) Generate a random number r ∈ 0, 1 .

(2) If r ≤ q1, chromosome s1 is selected, where qi isthe cumulative probability of the chromosome si(i = 1, 2,… ,N), which is calculated as

qi = 〠i

j=1P sj 30

(3) If qk−1 ≤ r ≤ qk 2 ≤ k ≤N , chromosome sk is selected.

(iii) Crossover: due to the two proposed encodingmechanisms, there are two corresponding rulesto constrain individuals which are used in thecrossover.

Case 1. Choose three genes (coded bits) randomly in eachparent chromosome for crossover, and the three genes needto meet the following conditions:

〠3

n=1Bcodep1 bitn = 〠

3

n=1Bcodep2 bitn , 31

where Bcodep1and Bcodep2are the binary-code values of thegenes used for crossover in the two parents; bitn is the nthselected genes of the two parents.

Case 2. Due to the use of real-coded mechanism, the cross-over operator is done according to the following rules shownin (31) [20]. The kth chromosome Qcodek and the lth chro-mosome Qcodel are selected to undergo crossover at the jthgene bitj.

Qcodek bitj = Qcodek bitj 1 − b + Qcodel bit j b,

Qcodel bitj = Qcodel bit j 1 − b + Qcodek bitj b,32

where b is a random number and b ∈ 0, 1 .

According to the above methods, the selection ofgenes in the parent chromosomes is completed. Then,choose the parent chromosomes based on the probabilityof crossover pc. First of all, associate a random number from0, 1 with each chromosome in Ωt , and add the chromo-some to the parent pool set if the associated number isless than pc.

For pc selection, [21] recommends the use of adaptiveprobabilities of crossover pc to realize the twin goals ofmaintaining diversity in the population and sustaining theconvergence capacity of the GA, and this method is calledthe AGA algorithm. The probability of crossover can auto-matically vary with the fitness. If pc is appropriate, it cankeep the diversity of the population while ensuring theconvergence of genetic algorithm. Therefore, the design ofadaptive pc is improved to avoid the following problems:the superior individuals are in a nearly invariant state inthe initial stage, resulting in GA converging to a locallyoptimal solution.

pc =pc1 −

pc1 − pc2 fmax − f itfmax − f avg

, f it ≥ f avg,

pc1, f it < f avg,33

where f avg is the average fitness of the population, fmax is themaximum fitness of the population, and pc1 and pc2 are theconstraint parameters of pc.

(iv) Mutation: similar to crossover, there are two rulesof mutation corresponding to the two codedmechanisms.

Case 1. Associate a random number from 0, 1 with eachgene in each chromosome in the population Ωt and mutatethis gene if the associated number is less than pm. Moreover,if the child meets the sum of the coded values of the child’sgenes that is the same as that of the parent, it is added tothe children pool set.


Case 2. The operation of selecting the jth gene (bit j) in the ithchromosome (Qcodei) for mutation is as follows:

where Qcodei bitmin = 1, Qcodei bitmax = 14,

f g = r21 − gGmax

2, 35

where r2 and r are random numbers; r ∈ 0, 1 , r2 ∈ 0, 1 , andr determines the trend of changes in gene values; f g is adecreasing function, and g is the current generation; Gmaxis the maximum generation. The design of adaptive pm issimilar to pc, which meets the following conditions:

pm =pm1 −

pm1 − pm2 fmax − f itfmax − f avg

, f it ≥ f avg,

pm1, f it < f avg,36

where pm1 and pm2 are the constraint parameters of pm.

(v) Stopping condition: the difference between the aver-age fitness of the population and the best fitness ofthe population is the condition to stop iterationin these proposed GAs. If stopping conditions aresatisfied, then terminate. Otherwise, iterations keepundergoing.

The proposed GAs are experimentally validated. Initial-ize Ω0 = 30, pc0 = 0 4, pm0 = 0 4 and set pc1 = 0 9, pc2 = 0 6,pm1 = 0 1, pm2 = 0 001, and Gmax = 200. The convergenceeffect of the GAs with both the encoding mechanisms thatis shown in Figure 4 is obtained.

The results in Figure 4 show that both the average fitnessof the population and best fitness of the population converge.As for the binary-coded GA, after 20 iterations, the averagefitness of the population and best fitness of the populationhave converged. Moreover, the IDOP corresponding tothe optimal antenna configuration obtained by geneticalgorithm is 8.167, which is greatly improved comparedwith IDOP = 52 28 in the absence of underdoing the pro-posed GA. As for the real-coded GA, the average fitnessand best fitness of the population converge after 65 itera-tions, and the IDOP corresponding to the optimal antennaconfiguration obtained by the GA is 8.167, which is greatlyimproved compared with IDOP = 107 1 in the absence ofunderdoing the proposed GA. All of these show that theGAs proposed in this study can indeed optimize the antennaconfiguration. The advantage of using GA to get the optimalantenna configuration is to reduce computational complexity

compared to the traversal algorithm. From the previousanalysis, it shows that traversing the 20020 antenna con-figuration combinations requires a total of 20020 times ofIDOP calculation.

In contrast, in the proposed binary-coded GA, there are20 iterations, 30 individual fitness is calculated for each iter-ation, and in each individual fitness calculation, there areC25 = 10 times of IDOP calculation which are obtained by

the traversal model. Therefore, after 20∗30∗10 times ofIDOP calculation, the optimal antenna configuration isobtained. The real-coded GA converges after 65 iterations,so it contains 65∗30 times of IDOP calculations.

3.3. Verification of Proposed GA-Based Strategy. For thepurpose of verifying the correctness of the optimal antennaconfiguration obtained by the GAs proposed in this study,the following experiment is done. Assuming that the relativeattitude of intersatellite is changing, the history of changingattitudes is as shown in Figure 2(c). The optimal antennaconfiguration of each attitude angle is solved by usingthe traversal method and the proposed genetic algorithm,respectively, and the results are shown in Figure 5.

The results in Figure 5 show that the optimal antennaconfiguration obtained by using the GAs is consistent withthe results obtained from the traversal method in the caseof relative attitude changes between satellites. Therefore,the GAs proposed in this study are effective in obtainingthe optimal antenna configuration with low complexityand high real time.

4. Experimental Work and Results

4.1. Equipment Setup. In this study, the self-developed proto-type of the RF relative measurement sensors and the systemconfiguration and analysis platform are applied as the verifi-cation devices of the proposed algorithm, based on which theexperimental scenarios are built. The prototype of the RFrelative measurement sensor is shown in Figure 6, whichincludes a baseband signal processing unit, a radio frontend, and antennas. In the case of satellite relative motion,for the purpose of verifying the influence of the antenna con-figuration on the estimation accuracy of the relative positionand attitude of intersatellites, we have independently devel-oped the simulation software, which is used as the verifica-tion platform of this study. In this software, we can set theantenna configurations and the relative position and attitudeof intersatellites, as well as the estimation method whichincludes LS and EKF algorithm. Finally, the results are com-pared with the actual results to verify the effect of the

Qcodei bit j =Qcodei bitj + Qcodei bitj −Qcodei bitmax ∗ f g , r ≥ 0 5,

Qcodei biti + Qcodei bitmin −Qcodei bitj ∗ f g , r < 0 5,34


proposed methods on the measurement accuracy. The soft-ware interface of the RF relative measurement sensor config-uration and analysis system is shown as Figure 7.

4.2. Experiments and Results

4.2.1. DOP Verification. To verify the exactness of all theDOPs defined in this study, three experimental scenarioswere presented, which are by changing the intersatellite dis-tance, the azimuth angle, and one of the attitude angles,respectively, to get the changing rules of DOPs. The satelliteB platform investigated in these three cases is assumed tobe oscillated about intersatellite distance, azimuth angle,and yaw angle with a periodic velocity, respectively. The his-tory of the intersatellite distance, azimuth angle, and yawangle is shown in Figures 8(a), 9(a), and 10(a), respectively.

In three experimental scenarios, the coordinate values ofall the antennas in each body frame are summarized inTable 1. The ranging precision is verified by the ground-based experiment with the prototype of the RF relative mea-surement sensors; the experimental scenario is set up asshown in Figure 11. In the experiment, the transmitter sendsthe DSBPSK modulated signal, and the signal is received bythe receiver, and it is down-converted to a baseband signal,

then the baseband signal is then processed. In the basebandsignal processing unit, the carrier phase of the signal isobtained after the signal capture and tracking module. Theranging value is obtained according to the obtained carrierphase. In addition, the ChipScope software records and dis-plays the phase of the signal, which is processed by FPGAsin the baseband processing unit. Thence, by the ChipScopesoftware, the ranging variance is observed, which is as shownin Figure 12. The ranging precision σρ ≈ 0 38mm, which is inaccordance with the regulation of the GNSS technology thatthe carrier range error is approximately 1% of carrier wave-length due to the carrier frequency of the ranging signal thatis approximately 20GHz [6, 22].

Figures 8–10(b) show the changes of DOPs, as functionsof time; the change of the DOPs is exactly correlated with therelative motion of the intersatellite.

Figures 8(c), 9(c), 9(d), and 10(c) compare the measure-ment errors in the parameter estimation against the analyti-cally calculated variances; the variances of the parametererrors change with a fluctuation due to the variance of theDOPs. In the figures, the parameter errors are well within

0.14

0.12

0.08

Aver

age fi

tnes

s of p

opul

atio

n

0.06

0.04

0.02

00 50

Iterations150100 200

0.1

Binary-coded mechanismReal-coded mechanism

(a)

0.125

0.12

0.115

0.11

0.105

0.1

0.095

0.090 10 20 30 40

Best

fitne

ss o

f pop

ulat

ion

Iterations

Binary-coded mechanismReal-coded mechanism

(b)

Figure 4: Convergence of the genetic algorithm: (a) the history of average fitness of population; (b) the history of best fitness of population.

IDOP of the best antenna configuration obtained by GAsIDOP of the best antenna configuration obtained by traversal method

5.56

6.57

7.58

8.5

IDO

P

10 20 30 40 50 60 70 80 90 1000

t (s)

Figure 5: IDOP with the change of relative attitude of theintersatellite.

Baseband signalprocessing unit

RF front end

Antennas

Figure 6: The prototype of the RF relative measurement sensors.


the analytical 2σρ bounds calculated on the basis of DOPs. Itis worth noting that the variances of the ωDOP change onlywith a small fluctuation; however, its trend is consistent withthe other DOPs.

Experimental data show that the DOPs can predict theestimation errors; thus, for the purpose of the improvementof the estimation precision, the DOPs can be utilized to ana-lyze the antenna configuration of the RF relative measure-ment sensors in the space mission.

4.2.2. IDOP Verification. To verify the reliability of IDOP,assume that there are two satellites whose relative attitudesand positions need to be estimated. Figure 1 shows the coor-dinate frame, and there are 2 transmitting and 3 receivingantennas, which are, respectively, installed in the one faceof each satellite. The coordinate values of all the antennasin each body frame are summarized in Table 1.

As shown in Table 2, the transmitting antenna 1 on satel-lite A and the transmitting antenna 1 on satellite B synchro-nously move on the one face of both satellite platforms, andthe other antenna locations are fixed. All simplifications areto reduce the computational complexity. The transmittingantenna moves on the viewing surface, the movement ofwhich is shown in Figure 13(a), and the values of DOPs arecalculated and recorded. Based on the assumptions, the con-straint conditions are as follows:

−0 5 ≤ yT1 ≤ 0 5, yT1 = yT2,

−0 5 ≤ zT1 ≤ 0 5, zT1 = zT237

All values of IDOP are obtained by the traversing simula-tion model as shown in Figure 13(b).

Figure 7: The RF relative measurement sensor configuration and analysis system.

10864

Dist

ance

(m)

20 20 40 60 80 100

Time (s)120 140 160 180 200

(a) History of the intersatellite distance

20 40 60 80 100Time (s)

120 140 160 180 200

DO

P

0.45

0.35

0.25

0.4

0.3

DISDOP

(b) History of the DISDOP

0 20 40 60 80 100Time (s)

120 140 160 180 200Dist

ance

erro

r (m

)

1

0

−1

×10−4

Analytical 2𝜎 boundError

(c) Comparisons of distance estimation error against analytically

calculated variances

Figure 8: History of scenario 1.


60

40

20

Azi

mut

h (d

eg)

0

−200 20 40 60 80 100

Time (s)120 140 160 180 200

(a) History of the intersatellite azimuth angle

DO

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AZUDOPELEDOP

(b) History of AZUDOP and ELEDOP

Azi

mut

h er

ror (

deg)

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0

−0.01

−0.020 20 40 60 80 100 120 140 160 180 200

Time (s)


(c) Comparisons of azimuth angle estimation errors

against analytically calculated variances

Elev

atio

n er

ror (

deg)

−0.020 20 40 60 80 100 120 140 160 180 200

−0.01

0

0.01

0.02

Time (s)


(d) Comparisons of elevation angle estimation errors

against analytically calculated variances

Figure 9: History of scenario 2.

0 20 40 60 80 100 120 140 160 180 200Time (s)

−0.050

0.05

Yaw

erro

r (de

g)

0.50

−0.05

Pitc

h er

ror (

deg)

0 20 40 60 80 100 120 140 160 180 200Time (s)

0.50

−0.05

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erro

r (de

g)

0 20 40 60 80 100 120 140 160 180 200Time (s)


0 50

807060

100 150 200

0500

1000

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P

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100 120 140 160 180 200

θDOPφDOP

𝜔DOP

0 20 40 60 80 100 120 140 160 180 200Time (s)

−200

204060

Yaw

(deg

)

(a)

(b)

(c)

Figure 10: History of scenario 3: (a) history of yaw angle of satellite B; (b) history of the EUDOPs; (c) comparisons of elevation angleestimation errors against analytically calculated variances.


The results of the simulation are as follows Figure 13(b).As locations of the antennas change, the correspondingvalue of IDOP also changes accordingly, and the value ofIDOP is the smallest when the transmitting antenna is atthe location of (0.5, −0.5, and −0.5), and by the field test,this location is the transmitting antenna location of theoptimal antenna configuration.

4.2.3. Performance Analysis of Proposed GA-Based Strategy.To compare the proposed method with the performance ofthe existing static antenna selection method, the followingexperiment is designed. Assume that the satellites’ attitudechanges constantly at a certain distance, and the history ofthe changing attitudes is as shown in Figure 2(c). In thisdynamic scenario, the proposed method is used to optimizethe antenna configuration in real time and record its IDOPvalues in various attitudes. Then, the recorded IDOP valuesare compared with the IDOP values corresponding to the

Figure 12: Results of ranging precision by ChipScope.

Transmitter Receiver

FPGA on line simulator

Baseband signalprocessing unit

Figure 11: Relative measurement sensor ranging precision verification experiment.

Table 2: Antenna coordinates of both satellites.

Antenna coordinates

Satellite A

Transmitter 1 (0.5 yT1zT1)

Transmitter 2 (0.5 −0.5 0.5)Receiver 1 (0.5 0.5 0.5)

Receiver 2 (0.5 0.5 −0.5)Receiver 3 (0.5 −0.5 −0.5)

Satellite B

Transmitter 1 (0.5 yT2zT2)

Transmitter 2 (0.5 −0.5 -0.5)Receiver 1 (0.5 −0.5 0.5)Receiver 2 (0.5 0.5 0.5)

Receiver 3 (0.5 −0.5 −0.5)


antenna configuration mentioned based on the rule of [6];the antenna configurations are shown in Table 1. The exper-imental results are shown in Figure 14. The data in Figure 14show that the IDOP obtained in the proposed method has alarger degree of reduction than the one proposed in [6] inthe dynamic scenario, and the measurement accuracy hasbeen improved. Furthermore, with the proposed antennaconfiguration mentioned in [6], as the attitude changes, themeasurement accuracy also fluctuates greatly, which affectsthe robustness of the relative measurement sensor. There-fore, the dynamic antenna configuration selection strategybased on GA improves not only the measurement accuracyof the RF relative measurement sensor but also its robustnessin the dynamic scene.

5. Conclusions

(i) Based on the RF metrology model, IDOP is proposedas a basis of obtaining the optimal antenna configura-tion, due to fact that it can predict the estimationaccuracy with respect to the distance, azimuth angle,elevation angle, and attitude of the intersatellite.

(ii) In the implementation of space missions, thedynamic optimal antenna selection strategy is mod-eled as a combinatorial optimization problem, and

GAs with two encoding mechanisms are proposedto solve this combinatorial optimization problem.

(iii) In dynamic scenarios, the proposed GA-based strat-egy of the optimal antenna configuration selection isverified, and the measurement accuracy of the RFrelative measurement sensor has been improvedwith the proposed strategy.

(iv) In the future work, more factors can be considered inthe combinatorial optimization problem, such asantenna visibility and orientation.

Data Availability

The programs analyzed during the current study are not pub-licly available, due to a part of simulation programs in use,but are available from the corresponding author on reason-able request. Most of the data generated or analyzed duringthis study are included in this published article.

Conflicts of Interest

Authors declare that there is no conflict of interest regardingthe publication of this paper.

4035302520

IDO

P

1510

50

10 20 30 40 50t (s)

60 70 80 90 100

IDOP of the stable antenna configurationIDOP of the optimal antenna configuration obtained by proposed strategy

Figure 14: IDOP with the change of relative attitude of the intersatellite.

z

y

(a)

0.5

IDO

P

−0.500

zy−0.5

0.5

5040302010

0

(b)

Figure 13: The traversal model and history of the IDOP: (a) the traversal model; (b) IDOP obtained by traversal model.


Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no. 91438116) and the Program forNew Century Excellent Talents (no. NCET-12-0030).

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