multi-uavs cooperative localization algorithms with

Research ArticleMulti-UAVs Cooperative Localization Algorithms withCommunication Constraints

Xiaowei Fu,1 Haiyang Bi,2 and Xiaoguang Gao1

1School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, China2Shenyang Aircraft Design and Research Institute, Shenyang, Liaoning 110035, China

Correspondence should be addressed to Xiaowei Fu; [email protected]

Received 4 June 2017; Revised 31 July 2017; Accepted 3 August 2017; Published 7 September 2017

Academic Editor: Filippo Cacace

Copyright © 2017 Xiaowei Fu 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.

Communication is the basis of multi-UAVs cooperative localization. There will be some communication delay and packet losswhen one UAV communicates with others, and these communication restrictions may have negative influences on multi-UAVcooperative localization. In this paper, the communication among UAVs is described as Bernoulli random variables, and twokinds of cooperative localization algorithms are proposed. One is centralized algorithm, where a kind of one-step predictionstrategy is designed.The other is distributed algorithm, where a kind of prediction-compensation strategy is proposed andweightedexpectation state estimation method is designed. Simulation results show the effectiveness of the proposed algorithms.

1. Introduction

In the future, air-to-ground attack and target indicationwill be important tasks for UAVs, and targets localizationaccuracy is the key factor to influence whether UAVs canaccomplish these tasks or not. Currently, the single stationtargeting method is often used by UAVs, but multi-UAVscooperative positioning can improve localization accuracy.Communication is the basis of multi-UAVs cooperativepositioning. However, there always are some communicationlatency and information packet loss when UAVs communi-cate with each other, and these communication restrictionsmay have negative effect on multi-UAVs cooperative local-ization.

In order to eliminate the negative effects of commu-nication delay and packet loss, it is necessary to presentsome information fusion mechanisms to improve multi-UAVs cooperative positioning accuracy. A kind of subopti-mal fusion estimation method is proposed in [1] to resolvethe random packet loss problem of centralized multiplesensors information fusion. Literature [2] proposed amethodof optimal estimation to resolve the problem of randomcommunication latency packet loss. Literature [3] presents a

three-period distributed Kalman fusion estimation strategyfor wireless sensor networks random packet losses problem.Literature [4] is concerned with the distributed fusion esti-mation problem for discrete-time stochastic linear systemwith multiple delay. Literature [5] uses a predicted modal todescribe communication latency, and a kind of suboptimalH-filter is designed. A kind of centralized H-filter is designedto resolve fusion problem with communication latency, dataout-of-order, and packet loss in [6].

All this work is based on the fact that there was priorinformation about communication delay and packet loss.But in most situations, it is hard to get this information inadvance. Two methods are proposed to describe communi-cation latency and packet loss. One is Markov chain in [7];the other is random variable with Bernoulli distribution in[8–10].

In this paper, the communication latency and packet lossamong UAVs are described as Bernoulli random variables,and two kinds of cooperative localization algorithms areproposed. One is centralized algorithm, where a kind of one-step prediction strategy is designed. The other is distributedalgorithm, where a kind of prediction-compensation strat-egy is proposed and weighted expectation state estimation

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 1943539, 8 pageshttps://doi.org/10.1155/2017/1943539

https://doi.org/10.1155/2017/1943539

2 Mathematical Problems in Engineering

p1T(x, y, z)

p3T(x, y, z)

p2T(x, y, z)

Figure 1: Multi-UAVs cooperative Localization.

method is designed. Simulation results show the effectivenessof the proposed algorithms.

2. Description of Multi-UAVs CooperativeLocalization Problem

In Figure 1, multiple UAVs are detecting the target indepen-dently. One UAV is defined as the fusion center, which isthe leader of multi-UAVs; other UAVs are followers. EachUAV gets target location information by passive sensor; thenthe positioning information is transmitted to the fusioncenter. The leader fuses this information to obtain accuratepositioning information.

Packet loss and communication latency in the processof communication have negative influence on the precisionof UAV’s cooperative localization. Packet loss leads to thefact that fusion center cannot receive complete data packetfrom followers, and communication delay leads to the factthat the leader cannot receive data packet from followerssynchronously. All these result in the fact that the leadercannot obtain accurate target position.

Every follower communicates with the leader directly.There will be packet loss and time delay in the processof communication, and there was no prior informationabout information delay and packet loss. There only was thepossibility of packet loss and time delay.

3. Multi-UAVs CooperativePassive Localization

Based on the description ofmulti-UAVs cooperative localiza-tion problem, this section presents two kinds of multi-UAVscooperative passive localization algorithms under commu-nication constraints, including centralized and distributedalgorithm.

3.1. Centralized Multi-UAVs Cooperative Localization Algo-rithm. In this algorithm, every follower sends its sensed

information to the leader without processing, and the leaderfuses all this received information to obtain target position.

We use the following state-space model to describetarget’s motion and communication constraints

x𝑘+1 = Φ𝑘x𝑘 + Γ𝑘w𝑘, (1)

z𝑖𝑘 = H𝑖𝑘𝑥𝑘 + k𝑖𝑘, (2)

y𝑖𝑘 = 𝜉𝑖1 (𝑘) z𝑖𝑘 + 𝜉𝑖2 (𝑘) z𝑖𝑘−1+ (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘)) y𝑖𝑘−1,

(3)

where x𝑘 represents the target position state at time 𝑘, z𝑖𝑘represents target position information that is sensed by the 𝑖thUAV at time 𝑘, y𝑖𝑘 represents target position information thatis received by the leader from 𝑖thUAVat time 𝑘,Φ𝑘 representssystem state transition matrix, Γ𝑘 represents system noisematrix, H𝑖𝑘 represents the 𝑖th UAV’s measurement matrix attime 𝑘, k𝑖𝑘 represents measurement noise of 𝑖th UAV at timek, and w𝑘 represents system noise.

In formula (3), variable 𝜉𝑖1(𝑘) and 𝜉𝑖2(𝑘) are used todescribe the communication situation between fusion centerand other UAVs, and these variables obey the Bernoullidistribution. If the 𝑖th UAV is the leader, then 𝜉𝑖1(𝑘) = 1represents the fusion center received from the leader whichhas no packet loss or time delay. If the 𝑖th UAV is a follower,𝜉𝑖2(𝑘) depends on whether the fusion center receives messagefrom the follower. If the fusion center receives target positionwithout time delay, 𝜉𝑖1(𝑘) = 1 and 𝑦𝑖𝑘 = 𝑧𝑖𝑘; if the fusioncenter receives target position with time delay, 𝜉𝑖2(𝑘) = 1 and𝑦𝑖𝑘 = 𝑧𝑖𝑘−1; if the fusion center does not receive target positionat time 𝑘, which means packet is lost, then 𝜉𝑖1(𝑘) = 0 and𝜉𝑖2(𝑘) = 0.

From formulas (2) and (3), we can get the result as follows:

y𝑖𝑘 = 𝜉𝑖1 (𝑘) (H𝑖𝑘x𝑘 + k𝑖𝑘) + 𝜉𝑖2 (𝑘) z𝑖𝑘−1+ (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘)) y𝑖𝑘−1

= 𝜉𝑖1 (𝑘)H𝑖𝑘x𝑘 + 𝜉𝑖2 (𝑘) z𝑖𝑘−1+ (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘)) y𝑖𝑘−1 + 𝜉𝑖1 (𝑘) k𝑖𝑘.

(4)

A new positioning system is constructed by usingextended Kalman filter:

x𝑘+1 = Φ𝑘𝑥𝑘 + Γ𝑘w𝑘,y𝑘 = H𝑘x𝑘 + 𝜉1 (𝑘) k𝑘.

(5)

Mathematical Problems in Engineering 3

In formula (5)

x𝑘 = [[[

x𝑘z𝑘−1y𝑘−1

]]],

W𝑘 = [w𝑘V𝑘] ,

V𝑘 = [[[[[

k1𝑘...k𝐿𝑘

]]]]],

z𝑘 = [[[[[

z1𝑘...z𝐿𝑘

]]]]],

y𝑘 = [[[[[

y1𝑘...y𝐿𝑘

]]]]],

H𝑘 = [[[[[

H1𝑘...H𝐿𝑘

]]]]],

Φ𝑘 = [[[[

Φ𝑘 0 0H𝑘 0 0𝜉1 (𝑘)H𝑘 𝜉2 (𝑘) I𝑚 − 𝜉1 (𝑘) − 𝜉2 (𝑘)

]]]],

Γ𝑘 = [[[Γ𝑘 00 I𝑚0 𝜉1 (𝑘)

]]],

𝑚 = 𝐿∑𝑖=1

𝑚(𝑖),H𝑘 = [𝜉1 (𝑘)H𝑘 𝜉2 (𝑘) I𝑚 − 𝜉1 (𝑘) − 𝜉2 (𝑘)] ,

𝜉1 (𝑘) = diag [𝜉𝑖1 (𝑘) I𝑚(1) , . . . , 𝜉𝑖1 (𝑘) I𝑚(𝐿)] ,𝜉2 (𝑘) = diag [𝜉𝑖2 (𝑘) I𝑚(1) , . . . , 𝜉𝐿2 (𝑘) I𝑚(𝐿)] ,

(6)

where 𝑖 represents the identification number of UAV, theleader UAV’s identification number is 1, 𝐿 represents thequantity of UAVs, and diag(∙) represents the diagonal matrixwhich diagonal variable is ∙, while I𝑚 is𝑚 × 𝑚 unit matrix.

The system which formula (5) represented has the noisestatistics information as follows:

Q𝑤 = 𝐸 [W (𝑡)WT (𝐾)]= 𝐸{[w (𝑡)

V (𝑡)] [wT (𝑘) ,VT (𝑘)]} = [Q𝑘 00 QV

] ,QV = 𝐸 [V (𝑡)VT (𝐾)] = (𝑅)𝑚 ,S = 𝐸 [W (𝑡)VT (𝐾)] = 𝐸{[w (𝑡)

V (𝑡)]VT (𝑡)}

= [ 0QV] ,

(7)

where 𝜉𝑖1(𝑘) and 𝜉𝑖2(𝑘) obey the Bernoulli distribution,prob(𝜉𝑖1(𝑘) = 1) = 𝛼𝑖1, prob(𝜉𝑖2(𝑘) = 1) = 𝛼𝑖2, 0 ≤ 𝛼𝑖1 ≤ 1,and 0 ≤ 𝛼𝑖2 ≤ 1, 𝑖 = 1, 2, . . . , 𝐿, while 𝛼𝑖1 represents theprobability of fusion center receiving position without timedelay. 𝛼𝑖2 represents the probability of fusion center receivingposition with time delay.𝜉𝑖(𝑘) obeys the Bernoulli distribution, so that 𝐸[𝜉𝑖(𝑘)] =𝛼𝑖 and 𝐸[(𝜉𝑖)2(𝑘)] = 𝛼𝑖, 𝑖, 𝑗 = 1, 2, . . . , 𝐿. 𝐸[∙] represents themathematical expectation of ∙.

Based on the feature of the Bernoulli distribution, forabove system, we can get the conclusion as follows:

Φ𝑘 = 𝐸 [Φ𝑘] = [[[Φ𝑘 0 0H𝑘 0 0𝛼1H𝑘 𝛼2 I𝑚 − 𝛼1 − 𝛼2

]]],

H𝑘 = 𝐸 [H𝑘] = [𝛼1H𝑘 𝛼2 I𝑚 − 𝛼1 − 𝛼2] ,

Γ𝑘 = 𝐸 [Γ𝑘] = [[[Γ𝑘 00 I𝑚0 𝛼1

]]],

(8)

where

𝛼1 = 𝐸 [𝜉1 (𝑘)] = diag (𝛼11I𝑚1 , . . . , 𝛼𝐿1 I𝑚𝐿) ,𝛼2 = 𝐸 [𝜉2 (𝑘)] = diag (𝛼12I𝑚1 , . . . , 𝛼𝐿2 I𝑚𝐿) .

(9)

And then

ΔΦ𝑘 = Φ𝑘 −Φ𝑘

= [[[0 0 (𝜉1 (𝑘) − 𝛼1)H𝑘0 0 𝜉2 (𝑘) − 𝛼20 0 (𝛼1 − 𝜉1 (𝑘)) + (𝛼2 − 𝜉2 (𝑘))

]]]

T

,

ΔH𝑘 = H𝑘 −H𝑘 = [[[

(𝜉1 (𝑘) − 𝛼1)H𝑘𝜉2 (𝑘) − 𝛼2(𝛼1 − 𝜉1 (𝑘)) + (𝛼2 − 𝜉2 (𝑘))

]]]

T

.

(10)


In the system of multi-UAVs cooperative localization,one-step state predicted error is as follows:

x𝑘+1,𝑘 = x𝑘+1 − x𝑘+1,𝑘= (Φ𝑘 − L𝑘H𝑘) x𝑘,𝑘−1 + ΔΦ𝑘x𝑘 − L𝑘ΔH𝑘x𝑘+ Γ𝑘W𝑘 − L𝑘𝜉1 (𝑘)V𝑘.

(11)

From this formula, we know there was system statein one-step predicted error. So the system state covariancematrix is introduced. We use q(𝑡) to represent system statecovariance, q𝑘 = 𝐸[x𝑘xT𝑘 ]. The next step can be representedas follows:

q𝑘+1 = Φ𝑘𝑞𝑘ΦT𝑘 + [[[0 0 00 0 00 0 A1

]]]⊗ [Φ1𝑘q𝑘Φ1T𝑘 ]

+ [[[0 0 00 0 00 0 A2

]]]⊗ [Φ2𝑘q𝑘Φ2𝑇𝑘 ] + [[

[0 0 00 0 00 0 A3

]]]

⊗ q𝑘 +Q,

(12)

where

A1 = diag (𝛼11 (1 − 𝛼11) 1𝑚1 , . . . , 𝛼𝐿1 (1 − 𝛼𝐿1) 1𝑚𝐿) ,A2 = diag (𝛼12 (1 − 𝛼12) 1𝑚1 , . . . , 𝛼𝐿2 (1 − 𝛼𝐿2) 1𝑚𝐿) ,A3 = diag ((𝛼11 (1 − 𝛼11) + 𝛼12 (1 − 𝛼12))⋅ 1𝑚1 , . . . , (𝛼𝐿1 (1 − 𝛼𝐿1) + 𝛼𝐿2 (1 − 𝛼𝐿2)) 1𝑚𝐿) ,

Φ1𝑘 = [[

[0 0 00 0 0H𝑘 0 0

]]],

Φ2𝑘 = [[

[0 0 00 0 00 1 0

]]],

(13)

where ⊗ represents Hadamard Product; we define B ⊗ C =(𝑏𝑖𝑗 × 𝑐𝑖𝑗)𝑟×𝑠, and also

[[[[

𝑎1I𝑚1d

𝑎𝑘I𝑚𝑘]]]]𝑟×𝑟

(𝑏𝑖𝑗)𝑟×𝑠 [[[[

𝑐1I𝑛1d

𝑐𝑙𝑏𝑛𝑘]]]]𝑠×𝑠

= [[[[[

𝑎1𝑐11𝑚1𝑛1 ⋅ ⋅ ⋅ 𝑎1𝑐𝑙1𝑚1𝑛𝑙... d...

𝑎𝑘𝑐11𝑚𝑘𝑛1 ⋅ ⋅ ⋅ 𝑎𝑘𝑐𝑙1𝑚𝑘𝑛𝑙]]]]]𝑠×𝑠

⊗ (𝑏𝑖𝑗)𝑟×𝑠 ,

Q = 𝐸 [Γ𝑘w𝑘wT𝑘 Γ

T𝑘] = [[[

[

Γ𝑘Q𝑘ΓT𝑘 0 00 QV QV𝛼10 𝛼1QV 𝛼1QV𝛼1

]]]].(14)

Based on above fundamental theory, multi-UAVs cen-tralized cooperative localization algorithm based on Kalmanfilter is presented as follows.

(1) One-step state prediction:x𝑘+1,𝑘 = Φ𝑘x𝑘,𝑘. (15)

(2) Mean square error of one-step prediction:

P𝑘+1,𝑘 = [Φ𝑘 − L𝑘H𝑘]P𝑘,𝑘−1 [Φ𝑘 − L𝑘H𝑘]T +Q

− L𝑘𝛼1ST𝑘 Γ

T𝑘 − Γ𝑘S𝑘𝛼1LT𝑘 + ΔΦ𝑘q𝑘ΔΦT𝑘

+ L𝑘ΔH𝑘q𝑘ΔHT𝑘L

T𝑘 − ΔΦ𝑘q𝑘HT

𝑘LT𝑘

− L𝑘ΔH𝑘q𝑘ΔΦT𝑘 .

(16)

(3) State estimation:x𝑘,𝑘 = x𝑘,𝑘−1 + Κ𝑘𝜀𝑘. (17)

(4) Filter gain:

K𝑘 = P𝑘,𝑘−1HT𝑘Q−1𝜀 (𝑘) . (18)

(5) MSE (mean square error) estimation:

P𝑘,𝑘 = P𝑘,𝑘−1 − K𝑘Q𝜀 (𝑘)KT𝑘 , (19)

where

𝜀𝑘 = y𝑘 −H𝑘x𝑘,𝑘−1= H𝑘 (x𝑘 − x𝑘,𝑘−1) + ΔHx𝑘 + 𝜉𝑘1k𝑘,

Q𝜀 (𝑘) = H𝑘P𝑘,𝑘−1HT𝑘 + ΔHq𝑘ΔHT + 𝛼1QV𝛼1.

(20)

L𝑘 means one-step predicted gain matrix:

L𝑘 = Φ𝑘K𝑘. (21)

The initial value of the filter is x0,1 = [x0 0 0]T, p0,1 =[ p0 0 00 0 00 0 0

], and based on the initial variable, the filter iteratesstep by step. The system state x𝑘+1,𝑘+1 could be calculated.And the target state x𝑘+1,𝑘+1 = [I𝑛 0 0] x𝑘+1,𝑘+1 could becalculated.

3.2. Distributed Multi-UAVs Cooperative Localization Algo-rithm. In distributed multi-UAVs cooperative localizationalgorithm, each UAV senses the target localization informa-tion and dose information filter.Then the filter results are sentto the leader. The key problem of the distributed algorithmis how to mitigate the negative aspect of communicationconstraints.

Mitigation strategies of packet loss and time delay aredesigned as follows.


(1) There are no packet loss and no time delay. In thissituation, fusion center uses state estimation andMSEestimation received from followers to fuse directly.

(2) There is no packet loss, but there is time delay. In thissituation, fusion center uses one-step predicted stateand one-step predicted MSE received from followersas the estimated value of the current state and MSE.

(3) There is packet loss. In this situation, fusion centeruses two-step predicted state and two-step predictedMSE received from followers as the estimated value ofthe current state and MSE.

The above strategies can be presented as formula (22),where x𝑖𝑘,𝑘 is target state estimated by 𝑖th UAV at time 𝑘. P𝑖𝑘,𝑘is target state MSE estimated by 𝑖th UAV at time 𝑘. x𝑖𝑘,𝑘−1 isone-step predicted target state estimated by 𝑖th UAV at time𝑘 − 1. P𝑖𝑘,𝑘−1 is one-step target state MSE estimated by 𝑖thUAV at time 𝑘 − 1. x𝑖𝑘,𝑘−2 is two-step predicted target stateestimated by 𝑖th UAV at time 𝑘 − 2. P𝑖𝑘,𝑘−2 is two-step targetstate MSE estimated by 𝑖th UAV at time 𝑘−2. 𝜉𝑖 describes thecommunication between fusion center and 𝑖thUAV. 𝜉𝑖1(𝑘) = 1means fusion center can receive information from 𝑖th UAVwithout time delay at time 𝑘. 𝜉𝑖2(𝑘) = 1 means fusion centercan receive information from 𝑖th UAVwith time delay at time𝑘. 𝜉𝑖1(𝑘) = 0 and 𝜉𝑖2(𝑘) = 0mean fusion center cannot receiveinformation from 𝑖th UAV at time 𝑘

x𝑖𝑘 = 𝜉𝑖1 (𝑘) x𝑖𝑘,𝑘 + 𝜉𝑖2 (𝑘) x𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘))× (𝜉𝑖1 (𝑘 − 1) x𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘 − 1)) x𝑖𝑘,𝑘−2) ,

𝑃𝑖𝑘 = 𝜉𝑖1 (𝑘)P𝑖𝑘,𝑘 + 𝜉𝑖2 (𝑘)P𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘))× (𝜉𝑖1 (𝑘 − 1)P𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘 − 1))P𝑖𝑘,𝑘−2) .

(22)

In formula (22), the values of 𝜉𝑖1(𝑘) and 𝜉𝑖2(𝑘) are uncer-tain. So we use Bernoulli random variables to describe them.The probability obeys

prob (𝜉𝑖1 (𝑘) = 1) = 𝛼𝑖1 (𝑘) ,prob (𝜉𝑖2 (𝑘) = 1) = 𝛼𝑖2 (𝑘) ,

(23)

where

0 ≤ 𝛼𝑖1 ≤ 1,0 < 𝛼𝑖2 ≤ 1,𝑖 = 2, . . . , 𝐿.

(24)

𝛼𝑖1(𝑘) is the probability that the fusion center can receiveinformation from 𝑖th UAV without time delay at time 𝑘.

𝛼𝑖2(𝑘) is the probability that the fusion center can receiveinformation from 𝑖th UAV with time delay at time 𝑘

𝐸 [𝜉𝑖1 (𝑘)] = 𝛼𝑖1 (𝑘) ,𝐸 [𝜉𝑖2 (𝑘)] = 𝛼𝑖2 (𝑘) ,

(25)

x𝑖𝑘 = 𝐸 [𝜉𝑖1 (𝑘) x𝑖𝑘,𝑘 + 𝜉𝑖2 (𝑘) x𝑖𝑘,𝑘−1+ (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘))× (𝜉𝑖1 (𝑘 − 1) x𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘 − 1)) x𝑖𝑘,𝑘−2)]= 𝛼𝑖1 (𝑘) x𝑖𝑘,𝑘 + 𝛼𝑖2 (𝑘) x𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1 − 𝛼𝑖2)⋅ (𝛼𝑖1x𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1) x𝑖𝑘,𝑘−2) ,

P𝑖𝑘 = 𝐸 [𝜉𝑖1 (𝑘)P𝑖𝑘,𝑘 + 𝜉𝑖2 (𝑘)P𝑖𝑘,𝑘−1+ (1 − 𝜉𝑖1 (𝑘) − 𝜉𝑖2 (𝑘))× (𝜉𝑖1 (𝑘 − 1)P𝑖𝑘,𝑘−1 + (1 − 𝜉𝑖1 (𝑘 − 1))P𝑖𝑘,𝑘−2)]= 𝛼𝑖1 (𝑘)P𝑖𝑘,𝑘 + 𝛼𝑖2 (𝑘)P𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1 − 𝛼𝑖2)⋅ (𝛼𝑖1𝑃𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1) 𝑃𝑖𝑘,𝑘−2) .

(26)

Based on formula (26), we design distributed multi-UAVs cooperative localization algorithm under unknowncommunication constraints as follows.

(1) Each UAV updates its own state independently:

(a) the leader updates its own state, which includesstate estimation and MSE estimation:

x1𝑘 = x1𝑘,𝑘−1 + K𝑘 (z𝑘 −H𝑘x1𝑘,𝑘−1) ,

P1𝑘 = [I𝑛 − K𝑘H𝑘]P1𝑘,𝑘−1.(27)

(b) followers update their own states, which includestate estimation, MSE estimation, one-step stateprediction, one-step MSE estimation, two-stepstate prediction, and two-step MSE estimation:

x𝑖𝑘 = x𝑖𝑘,𝑘−1 + K𝑘 (z𝑘 −H𝑘x𝑖𝑘,𝑘−1) ,

P𝑖𝑘 = [I𝑛 − K𝑘H𝑘]P𝑖𝑘,𝑘−1,𝑖 = 2, . . . , 𝐿,

x𝑖𝑘,𝑘−1 = Φ𝑘−1x𝑖𝑘−1,P𝑖𝑘,𝑘−1 = Φ𝑘−1P𝑘−1ΦT𝑘−1 + Γ𝑘−1Q𝑘−1ΓT𝑘−1,

𝑖 = 2, . . . , 𝐿,x𝑖𝑘,𝑘−2 = Φ𝑘−1x𝑖𝑘−1,𝑘−2P𝑖𝑘,𝑘−2 = Φ𝑘−2P𝑖𝑘−1,𝑘−2ΦT𝑘−2 + Γ𝑘−2Q𝑘−2ΓT𝑘−2

𝑖 = 2, . . . , 𝐿.

(28)


State �lter and estimation

Error �lter andestimation

Weightedfusion

Weightedfusion

Fusion center Optimal fusion

Leader

Leaderestimation

Informationupdate

k = 1

xik,k−1 = Φk−1 x

ik−1

k = 2

k + 1xik,k−2 = Φk−1 x

ik−1,k−2

Pik,k = [In − KkHk]Pi

k,k−1

Pik,k−1 = Φk−1Pk−1Φ

Tk−1 + Γk−1Qk−1Γ

Tk−1

Pik,k−2 = Φk−2P

ik−1,k−2Φ

Tk−2 + Γk−2Qk−2Γ

Tk−2

x1k = x1

k,k−1 + Kk (zk − Hk x1k,k−1)

P1k = [In − KkHk]P1

k,k−1

· · ·

Follower i

xik,k = xi

k,k−1 + Kk (zk − Hk xik,k−1)

k = 1

k = 2

k + 1

= i1 x

ik,k + i

2 xik,k−1

+ (1 − i1 − i

2)∗ (i

1 xik,k−1 + (1 − i

1) xik,k−2)

· · ·

Pik = i

1Pik,k + i

2Pik,k−1

+ (1 − i1 − i

2)

∗ (i1P

ik,k−1 + (1 − i

1)Pik,k−2)

x0k =

L

∑i=1

aik x

ik x1

k = x0k

P0k =

L

∑i=1

(aik)

2Pik

P1k = P0

k

xik

Figure 2: The algorithm flow chart of distributed multi-UAVs cooperative localization algorithm.

(2) Fusion center fuses all this information:

(a) received information is compensated by fusioncenter as follows:

x𝑖𝑘 = 𝛼𝑖1x𝑖𝑘,𝑘 + 𝛼𝑖2x𝑖𝑘,𝑘−1+ (1 − 𝛼𝑖1 − 𝛼𝑖2) (𝛼𝑖1x𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1) x𝑖𝑘,𝑘−2) ,

P𝑖𝑘 = 𝛼𝑖1P𝑖𝑘,𝑘 + 𝛼𝑖2P𝑖𝑘,𝑘−1+ (1 − 𝛼𝑖1 − 𝛼𝑖2) (𝛼𝑖1P𝑖𝑘,𝑘−1 + (1 − 𝛼𝑖1)P𝑖𝑘,𝑘−2) ;

(29)

(b) compensated information is weighted by scalarsas follows:

x0𝑘 = 𝐿∑𝑖=1

𝑎𝑖𝑘x𝑖𝑘,

𝑎𝑖𝑘 = ( 𝐿∑𝑖=1

1trP𝑖𝑘

)−1 1

trP𝑖𝑘

,

P0𝑘 = 𝐿∑𝑖=1

(𝑎𝑖𝑘)2 P𝑖𝑘.

(30)

(3) The feedback is from fusion center to the leader UAV.

There are no communication constraints between thefusion center and the leader UAV, so we can add the feedback

from the fusion center to the leader UAV to improve stateestimation precision of the leader UAV.

x1𝑘 = x0𝑘,P1𝑘 = P0𝑘. (31)

The algorithm flow chart of distributed multi-UAVscooperative localization algorithm is presented in Figure 2.

4. Simulation Results

Two UAVs are used to localize the moving target. EachUAV localizes the moving target by the single station targetpositioning method. The communicating situation betweenthese two UAVs can be described as follows.

The probability that the fusion center receives targetinformation from follower UAVwithout time delay is 0.4, theprobability that the fusion center receives target informationfrom follower UAV with 1s time delay is 0.3, and the proba-bility that the fusion center cannot receive target informationfrom follower UAV is 0.3.

The targets move with constant velocity alongstraight trajectory, and its initial state is X0 =[100, 10, 100, 10, 100, 0]𝑇, which means the initial targetlocation is (100m, 100m, 100m), and the initial target speed(10m/s, 0m/s, 10m/s).

The interference of the UAV localization obeys Gaussdistribution of which mathematical expectation is 0 andvariance is 10.


MSE of centralized localizationMSE of general Kalman �lter

0

10

20

30

40

50

60

4010 15 20 25 30 35 500 455

Figure 3: The comparison of presented localization algorithmand general Kalman filter algorithm under centralized multi-UAVscooperative localization.

The process noise of the target motion obeys the Gaussdistribution and its mean variance is equal to 5. The covari-ance matrix of the initial target localization is set as follows.

𝑃0 =[[[[[[[[[[[[

100 0 0 0 0 00 5 0 0 0 00 0 100 0 0 00 0 0 5 0 00 0 0 0 100 00 0 0 0 0 5

]]]]]]]]]]]]

. (32)

MSE between the target location and the localizationresults with communication constraints is shown in Figures3 and 4. Figure 3 shows the localization’s MSE calculatedby centralized localization algorithm, while Figure 4 showsthe localization’s MSE calculated by distributed localizationalgorithm.

According to the simulating results, we can figure outthat the general Kalman filter algorithm cannot get desiredlocalization results under communication constraints, andit also has convergent problems. The presented multi-UAVscooperative localization algorithm can get better resultsunder communication constraints.

FromFigure 5 we know that the localization results calcu-lated by centralized multiple UAVs cooperation localizationalgorithm are better than distributed algorithm. But based onthe theory of the algorithm’s complexity, the complexity of thecentralized algorithm is 𝑂((𝑛 + 2𝑚)3 + 𝑚3 + 𝑚(𝑛 + 2𝑚)2 +(𝑛 + 2𝑚)𝑛2), and the complexity of the distributed algorithmis𝑂(𝑚3 + 𝑛3 +𝑚2𝑛 +𝑚𝑛2), where 𝑛 and𝑚 represent system’sstate dimension and observation dimension. So distributedalgorithm is more appropriate for engineering application.

Presented localization algorithmGeneral Kalman �lter algorithm

4010 15 20 25 30 35 500 455Time (s)

0

10

20

30

40

50

60

70

80

Loca

tion

MSE

(m)

Figure 4: The comparison of presented localization algorithmand general Kalman filter algorithm under distributed multi-UAVscooperative localization.

CentralizedDistributed

0

5

10

15

20

25

30

35

40

Loca

tion

MSE

(m)

4010 15 20 25 30 35 500 455Time (s)

Figure 5: The comparison between the distributed and centralizedcooperative localization algorithm.

Both centralized and distributed cooperative localizationalgorithms are IIR filtering approach. Compared to FIRfilter approach, such as hybrid particle/FIR filtering [11],composite particle/FIR filtering [12], and deadbeat dissipativeFIR filtering [13], the presented algorithms have greatermean square error when there are model uncertainty and/ornumerical error in the system, but these proposed algorithmscan still meet the needs of practical positioning. And todesign filters with the same parameters, FIR approaches needmore parameters than these proposed algorithms; this willincrease the amount of computation of the DSP. That is to


say, DSP requires more computation time, and this will havenegative impact on the real-time performance of DSP. Forthe problem of multiple UAVs cooperation localization, thereal-time performance of designed algorithm is critical. Sothe presented algorithms in this paper are a compromisebetween accuracy and real-time, and these algorithms aremore appropriate formultipleUAVs cooperation localization.

5. Conclusion

(1) This paper analyzes the problem of multiple UAVscooperation localization under the condition of com-munication limits and explains the negative influenceof communication latency and information packetloss on the multiple UAVs cooperation localization.

(2) Two kinds of cooperative localization algorithms arepresented. In centralized algorithm, extended dimen-sion information fusion is used to establish the stateequation with communication latency and informa-tion packet loss, and state covariance is consideredbased on projection theory. In distributed algorithm,a kind of predicting and compensating strategy isdesigned to deal with communication constraints.

(3) Simulation verifies the proposed algorithms and anal-yses the complexity and the localization precision ofthe centralized and distributed algorithm.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology onAvionics Integration Laboratory and the Aviation ScienceFoundation under Grant 20155553041.

References

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[11] J. M. Pak, C. K. Ahn, P. Shi, Y. S. Shmaliy, and M. T.Lim, “Distributed hybrid particle/FIR filtering for mitigatingNLOS effects in TOA based localization using wireless sensornetworks,” IEEE Transactions on Industrial Electronics, vol. 64,no. 6, pp. 5182–5191, 2017.

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