Research ArticlePseudosatellite Dynamic Positioning of UAV Pod Based onan Improved SR-UKF Algorithm

Qinghua Zhang,1,2,3 Fengjuan Rong,2 Yangyang Sun ,2 Lei Gao,2 and Naishu Zhu2

1College of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China2Army Engineering University of PLA, Nanjing, Jiangsu 210007, China3State Key Laboratory of Geo-Information Engineering, Xi’an, Shaanxi 710054, China

Correspondence should be addressed to Yangyang Sun; bryant8011@163.com

Received 31 October 2018; Accepted 6 December 2018; Published 27 December 2018

Academic Editor: Krzysztof S. Kulpa

Copyright © 2018 Qinghua Zhang 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.

In view of the practical engineering problem of dynamic positioning of rotor UAV pod and the strong nonlinear problem indynamic operation of UAV, a real-time estimation of the dynamic position of the rotor UAV pod is proposed by using the SR-UKF algorithm.This algorithm uses the nonlinear propagation of UT transform to generate the point set to maintain the mean andcovariance information and thus achieves higher precision. Moreover, it uses the square root of covariance instead of covariance toparticipate in the recursive operation, thus improving the numerical stability of the filter and reducing the amount of computation.In this work, a pseudosatellite positioning platform was constructed in a field site in Nanjing. Based on evaluation of the spacegeometry of pseudosatellite base station, the accuracy of several nonlinear filtering algorithms was analyzed and evaluated usingGPS RTK positioning results. It was found that the SR-UKF algorithmwas themost accurate and efficient algorithm. It canmeet therequirements of dynamic positioning of the rotor-wing UAV pod.The experiment results of this algorithm provide a more efficientpositioning algorithm and implementation means for the actual engineering of UAV pod positioning using pseudosatellite system,which has high application value.

1. Introduction

In recent years, more and more scientific research andengineering projects have focused on carriers for the mul-tirotor UAV technology [1–4]. When detecting unexplodedordnance that is below ground, investigators usually travel byplane and use noncontact detection to ensure safety. [5–7].However, in order to improve the resolution of the detection,the use of low altitude flight, which requires a high level ofdriver skills, the rotor UAV has very high controllability andstability of flight [8, 9] and is the ideal carrier for unexplodedbomb detection. In order to improve the detection accuracyand reliability, a flexible connection between the sensor andthe UAV body is fixed. Based on this, the study target was theUAV’s pod and not the UAV itself.

Generally, the UAV is equipped with a low precisionsingle-frequency GPS positioning antenna [10, 11], but thesensor in the case is attached through a soft connection. Asa result, it cannot maintain accurate relative position with the

UAV due to maneuvering of the UAV and the influence ofwind. Thus, it is necessary to carry the sensor’s pod for a cer-tain accuracy of the individual positioning. In order to ensurethe accuracy of unexploded bomb detection, the positioningaccuracy of the sensor pod should be better than 10 cm.GPS/Beidou RTK is a common real-time high-precisionpositioning device, which has a three-dimensional accuracyof up to 1-2 cm [12–15]. However, the satellite positioningis heavily dependent on the number of visible satellites.In complex terrain conditions, such as deep valleys, denseforests, and high slope environment, its positioning accuracyis seriously affected, and it cannot even locate the target [16].Pseudosatellite is a kind of positioning equipment which canflexibly lay out the transmitting base station, which can dealwith the complex environment well. This paper adopted a2.4GHz wireless ranging device as pseudosatellite locationequipment, and a specific position estimation algorithm wasdesigned to locate theUAV suspended pod by using its outputdistance value. In the maneuvering process, the trajectory of

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 4568746, 12 pageshttps://doi.org/10.1155/2018/4568746

2 Mathematical Problems in Engineering

the UAV is usually nonlinear [17], which requires a suitablenonlinear filtering algorithm for its location.

Nonlinear filtering algorithm is widely used in real engi-neering problems, such as precision orbit determination andtime synchronization [18], satellite positioning [19], vehicledynamic navigation [20], and wireless positioning [21]. Withthe development of theoretical mathematics and the tractionof control engineering, nonlinear filtering algorithms suitablefor different needs are developing very quickly. The commonnonlinear filtering algorithms can be divided into three cate-gories according to the mathematical principle involved. Thefirst class is the analytic function approximation method, ofwhich the most representative example is the EKF (ExtendedKalman Filter) [22]. The EKF algorithm is based on Taylorseries expansion processing nonlinearmethod. As it is simpleand easy to achieve the characteristics, this algorithm iswidely used by the engineering community. However, thereare many complicated problems in linearization error andhigh dimensional solution Jacobian matrix [23, 24]. Thesecond class is based on deterministic sampling, the mostrepresentative of which is the UKF (unscented Kalman filter).This algorithm obtains the mean and covariance of thestate space equation using the UT (unscented transform),and substitutes into the standard Kalman filter to solve thecalculation [25]. This method can obtain a higher accuracythan EKF, and does not solve the Jacobian matrix. The thirdclass is based on the uncertainty sampling method. Themost representative example is PF (particle filter) withMonteCarlo sampling method instead of the UT-based samplingmethod, which has better resistance and adaptive abilityto deal with strong nonlinear/nongaussian problems. Whilethis algorithm is still in rapid development, there are someproblems such as filtering divergence from random sampling[26], large amount of particles, [27], particle degradation[28] and sample exhaustion [29], and so on. Hence, it isnot widely used in practical engineering. In conclusion, theUKF algorithm has high-precision and low computationalcomplexity, but it is time-consuming. It is proposed touse this method to dynamically locate the rotor-wing UAVpod in this paper. At the same time, according to therequirement of nonlinearity and higher calculation efficiencyof UAV’s motion process, the traditional UKF algorithmis improved so as to obtain the positioning result withhigher precision and better stability in practical engineer-ing.

The structure of this paper is as follows: The secondsection introduces the principle of pseudosatellite rang-ing equipment including hardware composition, static ren-dezvous and positioning principles, pseudosatellite spaceconstellation distribution, and evaluation of three aspects.The third section introduces the traditional UKF algorithmand the SR-UKF algorithm for improving it. The fourth sec-tion describes the field test analysis, including the design ofthe test site, the evaluation results of the spatial constellationdistribution, and the evaluation of accuracy of the SR-UKFalgorithm results. The fifth section summarizes and con-cludes the work of this paper from the aspects of theoreticalimprovement, technical method, and engineering test effect.Further, this final section analyzes the shortcomings of the

theory and technology and the problems that require furtherimprovement.

2. Pseudosatellite Positioning Equipment andIts Positioning Principle

2.1. Pseudosatellite Ranging System. Apseudosatellite rangingsystem provided by Beijing Digital Space Technology Co.,Ltd., was used in the experiment, which can provide ahigh-precision observation of the distance between the basestations and the observation stations. This paper intends touse these distance observations to design a specific dynamicfiltering algorithm tomeet the needs of UAVpod positioning.

The pseudosatellite system used in this paper uses thesignal of 2.4GHz band and provides distance observationat the same time. As shown in Figure 1, the experimentalsetup includes 1 observation station and 5 base stations, andtwo-way ranging can be obtained between the base stationand the observation station. At the same time, the systemcontains a special base station located on the top of themobilemeasuring vehicle (blue base station in Figure 1). The basestation can measure the distance between the observationstation and the collected distance between the observationstations to the base stations, which is used as the next dynamicpositioning input.

The hardware devices used in the pseudosatellite basestations and observation stations in this experiment aresimilar, as shown in Figure 2. A mobile power supply, usually10000mA power supply under full power, can continue topower the setup for more than 2 hours, which can ensurecompletion of the experiment.

2.2. Pseudosatellite Observation Equation and Least-SquaresSolution. The principle of pseudosatellite positioning isbased ondistancemeasurement of the space rear intersection,which is similar to the principle of satellite positioning. Thepseudosatellite single point positioning is based on observa-tion of the distance between the pseudosatellite transmittingbase station and the user receiver antenna.The coordinates ofthe observation point corresponding to the observatory aredetermined on the basis of the known accurate pseudosatel-lite base station coordinates.

It is known that the base station coordinates of pseu-dosatellites are (𝑋1, 𝑌1, 𝑍1), (𝑋2, 𝑌2, 𝑍2), ⋅ ⋅ ⋅ , (𝑋𝑛, 𝑌𝑛, 𝑍𝑛), inwhich 𝑛 is the number of pseudosatellite base stations. For the𝑖th pseudosatellite base station, the distance between it and theobservation station antenna is measured to be

𝑆𝑖 = √(𝑋𝑖 − 𝑥)2 + (𝑌𝑖 − 𝑦)2 + (𝑍𝑖 − 𝑧)2 + 𝛿𝑖 (1)

where 𝑆𝑖 is the distance between the pseudosatellite basestation and the observatory and the 𝑖 pseudosatellite basestation coordinates (𝑋𝑖, 𝑌𝑖, 𝑍𝑖) are precisely known. Then,to solve the observatory coordinates (𝑥, 𝑦, 𝑧) would requireat least three pseudosatellite base stations. Assuming thatthere are more than three pseudosatellite base stations and

Mathematical Problems in Engineering 3

Base station Base station

Base stationBase station

Base station

1

21

2

1

2

1

21

2

3

USB

Workstation

Observation station

Figure 1: EPS pseudosatellite system.

Figure 2: PS base station/observation station.

observation stations (𝑛 ≥ 3), there are the followingequations:

𝑆1 = √(𝑋1 − 𝑥)2 + (𝑌1 − 𝑦)2 + (𝑍1 − 𝑧)2 + 𝛿1𝑆2 = √(𝑋2 − 𝑥)2 + (𝑌2 − 𝑦)2 + (𝑍2 − 𝑧)2 + 𝛿2

⋅ ⋅ ⋅𝑆𝑛 = √(𝑋𝑛 − 𝑥)2 + (𝑌𝑛 − 𝑦)2 + (𝑍𝑛 − 𝑧)2 + 𝛿𝑛

(2)

The nonlinear equations (2) are linearized, that is, thefirst-order Taylor formula is expanded. Assuming that theinitial approximate position of the observatory is (𝑥0, 𝑦0, 𝑧0),then the ranging equation can have the following expression:

𝑆𝑖 = √(𝑋𝑖 − 𝑥0)2 + (𝑌𝑖 − 𝑦0)2 + (𝑍𝑖 − 𝑧0)2+ 𝜕𝑆𝑖𝜕𝑥 (𝑥 − 𝑥0) + 𝜕𝑆𝑖𝜕𝑦 (𝑦 − 𝑦0) + 𝜕𝑆𝑖𝜕𝑧 (𝑧 − 𝑧0) + 𝛿𝑖

(3)

Let 𝑆0 = √(𝑋𝑖 − 𝑥0)2 + (𝑌𝑖 − 𝑦0)2 + (𝑍𝑖 − 𝑧0)2. The aboveformula can be rewritten as

𝑆𝑖 = 𝑆0 + 𝑥0 − 𝑋𝑖𝑆0 (𝑥 − 𝑥0) + 𝑦0 − 𝑌𝑖𝑆0 (𝑦 − 𝑦0)+ 𝑧0 − 𝑍𝑖𝑆0 (𝑧 − 𝑧0) + 𝛿𝑖

(4)

Taking 𝐿 = [𝑆1, 𝑆2, ⋅ ⋅ ⋅ , 𝑆𝑛]𝑇 as the observation and𝐻 = [𝐻1, 𝐻2, ⋅ ⋅ ⋅ , 𝐻𝑛]𝑇 as the design matrix, wherein 𝐻𝑖 =[(𝑥0 − 𝑋𝑖)/𝑆0, (𝑦0 − 𝑌𝑖)/𝑆0, (𝑧0 − 𝑍𝑖)/𝑆0], 𝐿 = [(𝑥 − 𝑥0), (𝑦 −𝑦0), (𝑧−𝑧0)]𝑇 is the unknown that needs to be solved, and 𝛿 =[𝛿1, 𝛿2, ⋅ ⋅ ⋅ , 𝛿𝑛]𝑇 is the error vector, the following observationequation can be obtained:

𝐿 = 𝐻𝑥 + 𝛿 (5)

Since (5) is a linear equation, the least-squares solutioncan be obtained as

𝑥 = (𝐻𝑇𝑃𝐻)−1𝐻𝑇𝐿 (6)

2.3. Estimation of Spatial Distribution of Pseudosatellite BaseStation (DOP). DOP (dilution of precision) is an importantindex to measure the performance of satellite navigationsystem, which reflects the degree of contribution of observa-tional information to the solution of unknown parameters.DOP has the dual meanings of measurement and matrixmathematics: the ratio of pseudodistance location errorcaused by the geometrical structure of GNSS satellites inthe measurement and the trace characteristics of the inversematrix of least-squares adjustment in algebra. In pseudosatel-lite location, the spatial distribution of pseudosatellite basestation has great influence on the location result. Thus, thispaper attempts to use DOP value to evaluate the spatialdistribution of pseudosatellite independent base station.

4 Mathematical Problems in Engineering

In formula (5) of the previous section, the design matrixof the observation equation is

𝐻 =[[[[[[[[

𝑎𝑥1 𝑎𝑦1 𝑎𝑧1𝑎𝑥2 𝑎𝑦2 𝑎𝑧2... ... ...𝑎𝑥𝑛 𝑎𝑦𝑛 𝑎𝑧𝑛

]]]]]]]]

(7)

where 𝑎𝑥𝑖, 𝑎𝑦𝑖, 𝑎𝑧𝑖 (i = 1, 2, . . ., n) represent the cosineof the direction vector between the receiver position and thepseudosatellite base station position, i.e.,

𝑎𝑥𝑖 = 𝑥 − 𝑋𝑖𝑆𝑖𝑎𝑦𝑖 = 𝑦 − 𝑌𝑖𝑆𝑖𝑎𝑧𝑖 = 𝑧 − 𝑍𝑖𝑆𝑖

(8)

𝑋𝑖 is the component of the location of pseudosatellite basestation in the 𝑋-direction, 𝑥 indicates the component of theobservatory position in 𝑋-direction, and 𝑆𝑖 is the directionvector of the station and the position of pseudosatellite basestations. It is assumed that the error covariance matrix of thepseudodistance observation error value is𝜎2 (pseudodistanceobservation precision). According to the law of error propa-gation, the covariance matrix of 𝑥 is as shown as follows:

𝐶𝑂𝑉 (𝑥) = (𝐻𝑇𝑃𝐻)−1𝐻𝑇𝑃𝜎2𝑃𝐻(𝐻𝑇𝑃𝐻)−1 (9)

Assuming P is the unit matrix, formula (9) has thefollowing form:

𝐶𝑂𝑉 (𝑥) = (𝐻𝑇𝐻)−1 𝜎2 (10)

Then, the indexes PDOP, HDOP, and VDOP can berepresented by the following formulas:

𝐷𝑖𝑎𝑔 (𝐻𝑇𝐻)−1 = 𝐷𝑖𝑎𝑔 (𝑎11, 𝑎22, 𝑎33) (11)

𝑃𝐷𝑂𝑃 = √𝑎11 + 𝑎22 + 𝑎33 (12)

𝐻𝐷𝑂𝑃 = √𝑎11 + 𝑎22 (13)

𝑉𝐷𝑂𝑃 = √𝑎33 (14)

The above is the DOP value of several of the mostcommon expressions. By analysis of formulas (10) and (11)-(14), the following conclusion can be obtained: the smaller theDOPvalue, the smaller the covariancematrix of the estimatedvalue X. That is, a higher parameter estimation accuracy canbe achieved.

3. Research on Pseudosatellite DynamicLocation Algorithm

In order to eliminate systematic or random errors in theobservation, considering the dynamic characteristics of the

UAV pod, the method of filtering is adopted in the processof locating rather than the least-squares batch processingmentioned earlier. Filtering is one of the commonly useddata processing methods in navigation, positioning, andmeasurement engineering. The purpose of filtering is toeliminate or weaken the interference of error by specialprocessing of a series of measurement (observation) datawith errors, so as to recover a position parameter estimationproblem which is disturbed by noise as much as possible.

The main difference between pseudosatellite positioningsystem and satellite location is that the pseudosatellite emit-ter is relatively close to the user receiver. This may bringsome favorable aspects in the positioning process, such asthe influence of ionosphere on pseudosatellite observation.However, it may also cause some unfavorable effects, suchas near-far effect and linearization error. In this section,we first introduce the most commonly used Kalman filteralgorithm in satellite positioning. Then, according to theactual nonlinear problem of locating model in this research,the commonly used EKF algorithm and the UKF algorithmwith higher theoretical accuracy are introduced. To accountfor the uncertainty of the trajectory of UAV pod and theinstability of the distance observation of pseudosatellitesystem in this study, a more suitable SR-UKF (Square RootUnscented Kalman Filter) algorithm was designed by theauthors.

3.1. Standard Kalman Filter and EKF Algorithm. Kalman fil-ter was first proposed in 1960. It is a filtering algorithmwhichobtains the required parameters by estimating algorithmfrom the noise-containing observations, and is a real-timerecursive method which can be realized by computer. Unlikethe least-squares position estimation algorithm mentionedearlier, Kalman filter can realize the dynamic position estima-tion of motion vectors and has higher parameter estimationefficiency. The basic principle of standard Kalman filter hasbeen covered in previous literature cited in the introductionand explained in detail. The unified description of the statespace equation and the observation and updates under theBayesian framework are given here, and the formula is notdeduced.

The state equation and the measurement equations of thepseudosatellite positioning system are as follows:

𝑋𝑘 = 𝑓 (𝑋𝑘−1) + 𝑊𝑘 (15)

𝐿𝑘 = ℎ (𝑋𝑘) + 𝑒𝑘 (16)

where 𝑋𝑘, 𝐿𝑘 are the state vector and an observation vector,f(⋅), h(⋅) are the state equation and an observation equation,and 𝑊𝑘, 𝑒𝑘 are the state noise and the measurement noise.The Kalman filter can be described by recursive Bayesianestimator.The state equation and observation equation are asfollows:

𝑃 (𝑋𝑘 | 𝐿𝑘−1)= ∫𝑃 (𝑋𝑘 | 𝑋𝑘−1) 𝑃 (𝑋𝑘−1 | 𝐿𝑘−1) 𝑑𝑋𝑘−1 (17)

Mathematical Problems in Engineering 5

Measurement updates are as follows:

𝑃 (𝑋𝑘 | 𝐿𝑘) = 𝑃 (𝐿𝑘 | 𝑋𝑘) 𝑃 (𝑋𝑘 | 𝐿𝑘−1)𝑃 (𝐿𝑘 | 𝐿𝑘−1) (18)

In the above formula, 𝐿𝑘 = [𝐿1𝐿2 ⋅ ⋅ ⋅ 𝐿𝑘]𝑇 is the observationvector.

The real situation is that, due to the irregular motionof UAV, there is nonlinearity of the state equation and theobserved equation A. EKF algorithm is the most commonalgorithm used to solve the nonlinearity of the filteringprocess, and it deals with the first-order linearization of thestate equation and the observation equation, as shown in (19)and (20).

𝐹𝑘−1 = 𝜕𝑓𝜕𝑋𝑘−1 |𝑋𝑘−1 = 𝑋𝑘−1 (19)

𝐻𝑘 = 𝜕ℎ𝜕𝑋𝑘 |𝑋𝑘−1 = 𝑋𝑘 (20)

After linearization of the state equation and the observa-tion equation, the standard Kalman filter can be used to dealwith it, which is not discussed here. The following sectionswill focus on themore accurate nonlinear filtering algorithm,UKF, and its improved algorithm.

3.2. UT Transform and UKF Algorithm. Because the EKFalgorithm adopts first-order linearization, its mean valueand covariance have higher first-order accuracy, and it alsocontains large truncation error.TheUKF algorithm proposedby Julier in 1997 uses theUT transform to represent the distri-bution of the state equation using the Sigma point set, whichmaintains the mean and covariance information throughnonlinear propagation of this point set. In general, EKF canachieve second to third order accuracy. The following is abrief introduction to the UKF algorithm.

The first step is to generate a Sigma point set. As shown inthe following equation, 𝑛𝑥 is the dimension of the parameter,𝜆 = 𝑥2(𝑛𝑥 + 𝑘) − 𝑛𝑥 is a first scaling parameter, 𝛼 determinesthe degree of dispersion of the Sigma point and is usuallyset to a small value (10−3 < 𝛼 < 1), 𝑘 is the second scalingparameter and is usually set to 0 or 3−𝑛𝑥, 𝛽 is used to reducethe covariance approximation to the Taylor expansion of fourtimes the error generated, and G(√(𝑛𝑥 + 𝜆)∑𝑘−1 )𝑖 is the i-line vector or column vector of the ∑𝑘−1 square root of aweighted covariance matrix.

𝜒(0)𝑘−1 = 𝑋𝑘−1 (21)

𝜒(𝑖)𝑘−1 = 𝑋𝑘−1 + (√(𝑛𝑥 + 𝜆) ∑𝑘−1

)𝑖

𝑖 = 1, ⋅ ⋅ ⋅ 𝑛𝑥 (22)

𝜒(𝑖)𝑘−1 = 𝑋𝑘−1 − (√(𝑛𝑥 + 𝜆) ∑𝑘−1

)𝑖

𝑖 = 𝑛 + 1, ⋅ ⋅ ⋅ 2𝑛𝑥(23)

𝑤(𝑚)0 = 𝜆(𝑛𝑥 + 𝜆) , 𝑖 = 0 (24)

𝑤(𝑐)0 = 𝜆(𝑛𝑥 + 𝜆) + (1 − 𝛼2 + 𝛽) , 𝑖 = 0 (25)

𝑤(𝑚)𝑖 = 𝑤(𝑐)𝑖 = 𝜆(2 (𝑛𝑥 + 𝜆)) , 𝑖 = 1, ⋅ ⋅ ⋅ , 2𝑛 (26)

UKF time update is given by the following formulas (27)-(29):

𝜒(𝑖)𝑘−1 = 𝑓 (𝑋(𝑖)𝑘−1) (27)

𝑋𝑘 = 2𝑛𝑥∑𝑖=0

𝑤(𝑚)𝑖 𝜒(𝑖)𝑘−1 (28)

∑𝑋𝑖

= 2𝑛𝑥∑𝑖=0

𝑤(𝑐)𝑖 (𝜒(𝑖)𝑘−1 − 𝑋𝑘) (𝜒(𝑖)𝑘−1 − 𝑋𝑘)𝑇 +∑𝑊𝑘

(29)

The status update is given by formulas (30)-(36)

𝐿(𝑖)𝑘 = ℎ (𝜒(𝑖)𝑘−1) (30)

𝐿𝑘 = 2𝑛𝑥∑𝑖=0

𝑤(𝑚)𝑖 𝐿(𝑖)𝑘 (31)

𝐾𝑘 = ∑𝐿𝑘𝑋𝑘

−1∑𝐿𝑘𝐿𝑘

(32)

𝑋𝑘 = 𝑋𝑘 + 𝐾𝑘 (𝐿𝑘 − 𝐿𝑘) (33)

∑𝑋𝑘

= ∑𝑋𝑘

−𝐾𝑘 ∑𝐿𝑘𝐿𝑘

𝐾𝑇𝑘 (34)

∑𝐿𝑘𝑋𝑘

= 2𝑛𝑥∑𝑖=0

𝑤(𝑐)𝑖 (𝜒(𝑖)𝑘−1 − 𝑋𝑘) (𝐿(𝑖)𝑘 − 𝐿𝑘)𝑇 (35)

∑𝐿𝑘𝐿𝑘

= 2𝑛𝑥∑𝑖=0

𝑤(𝑐)𝑖 (𝐿(𝑖)𝑘 − 𝐿𝑘) (𝐿(𝑖)𝑘 − 𝐿𝑘)𝑇 +∑𝑘

(36)

3.3. An Improved UKF Algorithm. Although the accuracyof UKF algorithm is much higher than that of EKF, it canonly reach the three-dimensional precision of 10 cm forthe engineering example of this paper (see the results ofcalculation example). Moreover, the UKF algorithm is moretime-consuming than the EKF algorithm. Considering thepractical requirements of the project, this paper proposesto use the square root nontrace Kalman filter algorithm(square root unscented Kalman filter, SR-UKF) to carry onthe dynamic positioning of the UAV pod. This algorithmuses the square root of covariance instead of covarianceto participate in the recursive operation. It is expected tofurther improve the numerical accuracy and stability of thefiltering algorithm, and reduce the computational time tosome extent. This section briefly describes the three parts of

6 Mathematical Problems in Engineering

the SR-UKF algorithm, which are initialization, time update,and observation update.

The first part is initialization. As shown in formula (19),X𝑖0 is the initial state of the UAV pod, whose dimension factoris 𝑛𝑥, and S𝑋,𝑖0 is the Cholesky factor of the codefense initialmatrix P𝑖0. In the state space equation, it is generally assumedthat the state noise and the measurement noise are Gaussianwhite noises.Then, the Cholesky factor dimension expansionform of the state vector and its covariancematrix can be givenby the following equations:

X𝑎,𝑖0 = 𝐸 [X𝑖0, 𝜔𝑖0] (37)

S𝑎,𝑖0 = [[S𝑋,𝑖0

S𝜔𝑖

0,𝑖

0

]] (38)

in which S𝜔𝑖

0,𝑖

0 = cholesky{Q𝜔𝑖0

} and Q𝜔𝑖0

= diag{𝜎2𝐷, 𝜎2𝑎}.The second part is the time update. For state vectors

and covariance matrices, their Cholesky factor dimensionexpansions have the following operations: At t = 0 time, X𝑎,𝑖𝑡and Sa,it can be obtained by formulas (37) and (38), and atother times of t > 0, X𝑎,𝑖0 and S𝑎,𝑖0 can be obtained by formulas(39) and (40).

X𝑎,𝑖𝑡 = 𝐸 [X𝑖𝑡, 𝜔𝑖𝑡] (39)

S𝑎,𝑖𝑡 = [[S𝑋,𝑖𝑡

S𝜔𝑖

𝑡,𝑖𝑡

]] (40)

in which S𝜔𝑖

𝑡,𝑖𝑡 = cholesky{Q𝜔𝑖

𝑡} and Q𝜔𝑖

𝑡= diag{𝜎2𝐷, 𝜎2𝑎}.

On the basis of this, the probability density of the statevectors is approximated by selecting a few deterministicsampling points for the calculation of UT transform sam-pling points. The weights of these sampling points can bedetermined by formulas (41)-(45) using proportion correctedsymmetry sampling.

𝜆𝑎 = 𝛼22 (𝑛𝑎 + 𝑘𝑘)𝑎 (41)

𝜔𝑎𝑚𝑜 = 𝜆𝑎(𝑛𝑎 + 𝜆𝑎) (42)

𝜔𝑎𝑐𝑜 = 𝜆𝑎(𝑛𝑎 + 𝜆𝑎) + (1 − 𝛼22 + 𝛽) (43)

𝜔𝑎𝑚𝑢𝑖 = 𝜔𝑎𝑐𝑢𝑖 = 𝜆𝑎(2𝑛𝑎 + 2𝜆𝑎) (44)

𝜂𝑎 = √(𝑛𝑎 + 𝜆𝑎) (45)

On the basis of the UT transformation, the sample point 𝜒𝑖𝑡 ata time t can be represented as

𝜒𝑖𝑡 = [𝑋𝑎,𝑖𝑡 𝑋𝑎,𝑖𝑡 + 𝜂𝑎𝑆𝑎,𝑖𝑡 𝑋𝑎,𝑖𝑡 − 𝜂𝑎𝑆𝑎,𝑖𝑡 ] (46)

A sampling point for the position of the rotor drone pod wasgenerated by the upper equation. Here, 𝜅 and 𝛼2 are scale

scaling factor and positive scale factor, respectively, 𝛽 is apriori information parameter to minimize the loss of high-order term precision, and𝜔ac𝑢𝑖 and𝜔amui are the weights of thesample points 𝑢𝑖.

One-step prediction of time update is performed using𝜒𝑖𝑡’s nonlinear function as 𝑓(∙). By propagating 𝜒∗ 𝑖𝑡+1/𝑡, thestate prediction value (one-step prediction) of the position ofthe rotor UAV pod is obtained by weighting𝑋𝑖𝑡+1/𝑡.

𝜒∗ 𝑖𝑡+1/𝑡 = 𝑓 (𝜒𝑖𝑡, 𝑢𝑖𝑡) (47)

𝑋𝑖𝑡+1/𝑡 = 2𝑛𝑎∑𝑢𝑖=0

𝜔𝑎𝑚𝑢𝑖𝜒∗ 𝑖𝑢𝑖,𝑡+1/𝑡 (48)

The predicted values for the covariance matrix (one-stepprediction) are shown in formulas (49) and (50)

𝑆𝑖𝑡+1/𝑡 = 𝑞𝑟 {√𝑤𝑎𝑐1:2𝑛𝑎 (𝜒∗ 𝑖1:2𝑛𝑎 ,𝑡+1/𝑡 − 𝑋𝑖𝑡+1/𝑡)} (49)

𝑆𝑖𝑡+1/𝑡 = choleskyupdate {𝑆𝑖𝑡+1/𝑡, 𝜒∗ 𝑖0,𝑡+1/𝑡 − 𝑋𝑖𝑡+1/𝑡, 𝑤𝑎𝑐0} (50)

where 𝑞𝑟{∙} returns the matrix of Cholesky factor, 𝑃 = 𝑆𝑆𝑇,and update factor choleskyupdate{∙} returns the matrix 𝑃 ±√]𝐵𝐵𝑇 of the upper triangular Cholesky factor. This factorupdate can reduce the nonpositive qualitative probability ofthe covariance matrix.

The third part is the measurement update. For the SR-UKF algorithm, the generation process of the new samplingpoint in the nonlinear problem is as follows: in the pseu-dosatellite positioning process of UAV pod, themeasurementnoise ]𝑖 is assumed to be Gaussian white noise, the newsampling point dimension is 𝑛𝑏 = 𝑛𝑥, the number is 𝐿𝑏 =2𝑛𝑏 + 1, and 𝑛a is used instead of 𝑛b. Then, using the bindingformula (23), 𝜆𝑏, 𝜂𝑏, and weight parameters 𝜔𝑏𝑚0 , 𝜔𝑏𝑐0 , 𝜔𝑏c𝑢𝑖,and 𝜔𝑏mui can be calculated. A new sampling point with𝑋𝑖𝑡+1/𝑡and 𝑆𝑖𝑡+1/𝑡 can be recalculated to represent the location ofUAVpod using the second part of the time update.

𝜒𝑖𝑡+1/𝑡 = [𝑋𝑖𝑡+1/𝑡 𝑋𝑖𝑡+1/𝑡 + 𝜂𝑏𝑆𝑖𝑡+1/𝑡 𝑋𝑖𝑡+1/𝑡 − 𝜂𝑏𝑆𝑖𝑡+1/𝑡] (51)

Here, 𝜂𝑏 = √𝑛𝑏 + 𝜆𝑏. A new sampling point is used in thepositioning process of the UAV pod. Based on this, we canget the one-step predictive value 𝛾𝑖𝑢𝑖,𝑡+1/𝑡 of the variable, andafter considering the weight factor, we can get the predictedvalue ��𝑖𝑡+1/𝑡.

𝛾𝑖𝑢𝑖,𝑡+1/𝑡 = ℎ (𝜒𝑖𝑡+1/𝑡) + ]𝑖 (52)

��𝑖𝑡+1/𝑡 = 2𝑛𝑏∑𝑢𝑖=0

𝜔𝑏𝑚𝑢𝑖𝛾𝑖𝑢𝑖,𝑡+1/𝑡 (53)

The predicted value of the square root factor of the covariancematrix is as follows:

𝑆𝑌,𝑖𝑡+1/𝑡 = 𝑞𝑟 {√𝑤𝑏𝑐1:2𝑛𝑏 (𝛾𝑖1:2𝑛𝑏 ,𝑡+1/𝑡 − ��𝑖𝑡+1/𝑡)√𝑅𝑖} (54)

Mathematical Problems in Engineering 7

1611

1614

1616

1615 1542

1420

15 m Field measuring vehicle.

250

240

230

220

210

200

Y (m

)

X (m)110 120 130 140 150 160

BoundaryBase station of PS

Figure 3: Experimental field.

Six rotor UAV

GPS receiver Base station of PS

Observation station of PS

Figure 4: Pseudosatellite observation station, GPS RTK, and UAVin dynamic experiment.

𝑆𝑌,𝑖𝑡+1/𝑡 = choleskyupdate {𝑆𝑌,𝑖𝑡+1/𝑡, 𝛾𝑖0,𝑡+1/𝑡 − ��𝑖𝑡+1/𝑡, 𝑤𝑏𝑐0} (55)

where 𝑅𝑖 is the measurement noise covariance matrix of theI moment.

4. Results Analysis

4.1. Design of the Experiment. In order to analyze thedynamic location of the soft-connected pod of rotor-wingUAV, a flat area in Nanjing was selected for the experiment(Figure 3). At the experimental site, 6 towers with a heightof about 15 meters were installed to fix the pseudosatellitebase stations. Five of the base stations were located near thetops of the towers, and the 6th base stations were fixed atthe top of a field measuring vehicle for data transmissionwith workstations. The initial position of the base stationswere measured by the total station. The geometrical positionis shown in the lower right corner of Figure 3, and thecoordinate values are listed in Table 1.

In order to analyze the accuracy of several pseudosatellitepositioning algorithms, and make the positioning results ofGPS RTK technology as the truth value (accuracy of 1-2 cm),the pseudosatellite observation station andGPSRTK antennawere bundled together in the experiment, and a nylon tape

Table 1: Coordinates of the pseudosatellite base station.

Number x y z1420 129.816 203.625 80.7951542 119.455 235.326 76.421611 158.838 227.548 78.9621614 148.813 237.06 78.0731615 132.364 243.764 79.7681616 114.442 204.804 62.123

was used for soft connection with the UAV. Figure 4 showsan ongoing dynamic positioning experiment in which thedrone’s range is located inside a polygonal area surroundedby the towers.

4.2. Analysis of the DOP Value of Pseudosatellite Base Station.According to the location of base station, the space distribu-tion of base station in the test field was evaluated by usingthe DOP value formula given in Section 2.3. The red line inFigure 5(a) is the flight trajectory of the UAV. Figures 5(b),5(c), and 5(d) show the distributions of PDOP, HDOP, andVDOP, respectively. Through analysis, it was found that theflight area of the UAV was in the ideal area of DOP value,which is very advantageous to the experiment.

4.3. Accuracy Analysis of Dynamic Positioning Algorithmfor SR-UKF Algorithm. In this paper, the dynamic three-dimensional position of UAV pod in space was calculatedby using EKF, UKF, and SR-UKF algorithms, respectively.Figures 6(b), 6(c), and 6(d) present the values of X, Y, andZ computed by the three algorithms, respectively. From theanalysis of four figures, it can be found that the results of UKFand SR-UKF algorithms are very close most of the time, andthe results of EKF calculation are poor, especially at the startand end of UAV flight.

In order to analyze the accuracy of the three differentalgorithms, the results obtained by GPS RTK were usedas the real value (accuracy is better than 2 cm), and thecalculation errors of the three methods in X-, Y-, and Z-directions were obtained, respectively, as shown in Figures7(a)–7(c). Analysis of Figure 7 shows that the positioningerror of EKF algorithm in the X-, Y-, and Z-directions isgreater than that of UKF and SR-UKF algorithms, especiallyin the X-direction. Sometimes, the error reached the meterlevel, and far exceeded the maximum positioning error limitof the UAV pod. Figure 8 presents an error box diagram ofthe three algorithms, which shows that the error of the SR-UKF algorithm is small and concentrated and contains lessoutliers.

Finally, the accuracy of the three algorithms was analyzedstatistically. Table 2 gives the statistical results of the EKF,UKF, and SR-UKF algorithms in X-, Y-, and Z-directions,and it can be observed that the SR-UKF algorithm hasthe highest parameter estimation accuracy. The standarddeviation of three-dimensional estimation was calculated tobe 0.4261m for EKF, 0.1066m for UKF, and 0.0717m forSR-UKF. At the same time, the calculation time statistics

8 Mathematical Problems in Engineering

Flight pathBoundaryBase station of PS

1616 1420

1611

1542

1615

1614

Flight path of UAV in X−Y plane

200

210

220

230

240

250Y

(m)

120 130 140 150 160110

X (m)

(a) Measurement area

110 120 130 140 150 160X (m)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1616 1420

1615

1614

1611

1542

200

210

220

230

240

250

Y (m

)

(b) Distribution of PDOP value

110 120 130 140 150 160200

210

220

230

240

250

X (m)

Y (m

)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1616

1615

1614

1611

1420

1542

(c) Distribution of HDOP Value

110 120 130 140 150 160200

210

220

230

240

250

X (m)

Y (m

)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1616 1420

1615

1614

1611

1542

(d) Distribution of VDOP value

Figure 5

Table 2: Statistics of positional errors of the three algorithms.

Evaluation indicators EKF UKF SR-UKFX Y Z X Y Z X Y Z

Maximum Value 1.340 0.364 0.612 0.261 0.237 0.384 0.153 0.170 0.258Minimum value -0.742 -0.157 0.009 -0.109 -0.089 0.013 -0.035 -0.065 0.015Average 0.277 0.154 0.151 0.079 0.102 0.102 0.055 0.076 0.074Standard deviation 0.402 0.091 0.108 0.063 0.054 0.067 0.039 0.040 0.045

of the three algorithms were analyzed, and the results areshown in Figure 9. It was found that the EKF was muchfaster than theUKF and SR-UKF algorithms, but the accuracywas low. The positioning accuracy of SR-UKF was higherthan that of UKF, and the calculation efficiency was muchimproved.

5. Conclusion

In order to address the dynamic location problem of therotor UAV pod, a pseudosatellite positioning system wasconstructed in the field test site by acquiring the distanceobservation between the pseudosatellite base station and the

Mathematical Problems in Engineering 9

74

73

72

71

70

69

68

67

66120

125

130

135

140

145

X (m)

Z (m

)

223

224

225

226227

228

Y (m)

Start End

EKFUKFSR-UKF

(a) Comparison of three-dimensional positioning results of three algo-rithms

EKFUKFSR-UKF

145

140

135

130

125

120

0 50 100 150 200

Time (s)

X (m

)

(b) Comparison of X-directional positioning results for three algorithms

EKFUKFSR-UKF

Time (s)

Y (m

)

228

227

226

225

224

223

0 50 100 150 200

(c) Comparison of Y-directional positioning results for three algorithms

Z (m

)

EKFUKFSR-UKF

Time (s)0 50 100 150 200

76

74

72

70

68

66

(d) Comparison of Z-directional positioning results for three algorithms

Figure 6

mobile station. Dynamic positioning was studied by using avariety of nonlinear filtering algorithms, including SR-UKF.Furthermore, the results of GPS RTK positioning were usedas the external real value to evaluate these nonlinear filteringalgorithms. The following conclusions were obtained fromthe results:

(1) The spatial configuration of pseudosatellite basestation erected on the test field was first evaluated. The

evaluation indexes PDOP, HDOP, and VDOP were calcu-lated, and it was found that the UAV pod was in a relativelystable DOP value environment during the whole flight, whichis very important for the stability of the positioning result.

(2) As the EKF algorithm usually takes Taylor first-orderapproximation linearization, the higher-order items directlyresulted in a large linearization error, indicating that it wasnot good for the nonlinear flight-time positioning. The EKF

10 Mathematical Problems in Engineering

1

0

−1

dX (m

)

EKFUKFSR-UKF

Time (s)0 50 100 150 200

(a) Error comparison of three algorithms in X-direction

0.4

0.3

0.2

0.1

0.0

−0.1

−0.2

dY (m

)

EKFUKFSR-UKF

Time (s)0 50 100 150 200

(b) Error comparison of three algorithms in Y-direction

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

dZ (m

)

EKFUKFSR-UKF

Time (s)0 50 100 150 200

(c) Error comparison of three algorithms in Z-direction

Figure 7

algorithm was less accurate than the UKF and SR-UKFalgorithms, especially in the X-axis direction. This is becausetheUAV’s flight trajectory in the three-dimensional spacewasmore concentrated in the X-direction, in the X-direction ofthe maneuvering range of more than 20m.The experimentalresults illustrated the disadvantage of EKF in handling highmaneuverability. At the same time, the results of SR-UKFwere better than that of UKF.

(3) Different statistics were used to evaluate the posi-tioning accuracy of the three different nonlinear filtering

algorithms. From the statistical results, it was found thatthe positioning error of EKF was 0.4261m, which does notmeet the positioning requirements of the UAV pod. Thepositioning accuracy of the UKF was 0.1066, which barelymeets the 0.1m positioning accuracy requirement of theUAV pod. The positioning accuracy of the SR-UKF was0.0717m, which has higher positioning accuracy and fullyconforms to the positioning requirements of the UAV pod.The EKF algorithm was more efficient than the UKF andSR-UKF algorithms, but its positioning accuracy was low.

Mathematical Problems in Engineering 11

1.5

1.0

0.5

0.0

−0.5

−1.0

Box

plot

for e

rror

stat

istic

s (m

)

EKF UKF SR-UKF

Figure 8: Boxplot of three algorithms.

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

STD

(m)

0.00

0.02

0.04

0.06

0.08

0.10

Tim

e con

sum

ing

(s)

EKF

UKFSR-UKF

Figure 9: Parameter estimation accuracy and efficiency of the threealgorithms.

Also, SR-UKF algorithmwas significantly faster than theUKFcalculation.Therefore, the SR-UKF algorithmwas found to bean ideal algorithm to locate the UAV pod.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

Thisworkwas carried under funding from theNationalNatu-ral Science Foundation of China (Grant no. 41604024), ChinaPostdoctoral Science Foundation (Grant no. 2016M601815),and the State Key Laboratory of Geo-Information Engineer-ing (SKLGIE2016-Z-1-3).

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