Fast Lyapunov Vector Field Guidance for Standoff Target

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2019.2932998, IEEE Access

Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identifier 10.1109/ACCESS.2017.DOI

Fast Lyapunov Vector Field Guidance forStandoff Target Tracking based onOffline SearchSHUN SUN1, HAIPENG WANG1, JUN LIU1, AND YOU HE.1, 21Institute of Information Fusion, Naval Aviation University, Yantai 264001, China (e-mail: [email protected])2College of Electronic Science, National University of Defense Technology, Changsha 410073, China

Corresponding author: Haipeng Wang (e-mail: [email protected]).

This work was supported in part by the National Natural Science Foundation of China under Grant 91538201, Grant 61531020, and Grant61671463.

ABSTRACT To address the problem of the long convergence time of traditional Lyapunov Vector FieldGuidance (LVFG), a fast convergent LVFG designed with guidance function is proposed. The necessarycriteria of a fast convergent vector field are first illustrated through an analysis of the contraction andcirculation components. Then, two types of guidance functions of the distance between aircraft and targetare constructed to replace the original guidance parameter. Considering the saturation constraints from theaircraft performance, the optimal parameter in guidance functions based on offline search is estimated toachieve the best convergence. Meanwhile, standoff tracking for a moving target is solved with analysis ofmotion correction. Simulation results indicate the proposed vector field guidance converges faster to thestandoff circle without traversing the orbit regardless of whether the aircraft’s initial position is inside oroutside the circle, even for a moving target. In addition, selection of parameter in the guidance function isproved to make convergence speed approach the limit.

INDEX TERMS Vector Field Guidance, Lyapunov Function, Standoff Tracking, Fast Convergence,Moving Target, Offline Search.

I. INTRODUCTION

W ITH the development of automatic control and navi-gation technologies, low-cost Unmanned Aerial Vehi-

cles (UAVs) equipped with Flight Control System (FCS) arewidely used in military reconnaissance [1], real-time targettracking [2], geographic mapping [3], forest firefighting [4],and power-line inspection [5].

An important application scenario is continuous trackingand monitoring for individual target, relying on fixed-wingUAV. Fixed-wing UAV, as an inexpensive, high-enduranceautomatic aircraft, maintains economical cruising speed andkeeps the target observable at all times, so the standoff-target tracking approach that makes the UAV maintain a fixedstandoff distance to the target came into being. In extendedapplications, the target may refer to an area of interest orthe geometric centroid of dense targets in the group [6]. Inaddition, standoff distance is designed to protect the aircraftfrom detection and attack threats of hostile targets.

Extensive research works have been performed, includingon “good helmsman” behavior-based approaches for con-

stant line-of-sight orientation relative to the aircraft [7], [8],guidance law using a relative side-bearing angle [9], refer-enced point-based path following guidance methods [10]–[12], Lyapunov Vector Field Guidance (LVFG), and pathplanning based on neural networks [13], [14].

In path-following based on a referenced point, which canalso be called a virtual target, movement control is achievedby reducing the angle between the course of the UAV andthe line of sight to the virtual target to zero. The methodscan be achieved simply and efficiently for stable standofftracking. However, accuracy settling onto the standoff circledepends on relationship between aircraft speed, line of sight,and standoff distance.

LVFG has the advantages of small computational com-plexity and global convergence. The Lyapunov function wasearlier used to design LVFG for standoff target tracking [15].Asymptotically unbiased path-following approaches basedon vector fields were proposed [16], [17]. Arbitrary curvedpath tracking using a vector field of course commands wasfurther discussed in [18]. The above vector fields were stated

VOLUME 4, 2016 1



in two-dimensional space. Sensing and tracking performancecan be improved by changing the altitude of the UAV withthree-dimensional guidance vector fields [19], [20]. Tangentvector field guidance and LVFG were combined to track sev-eral sparse discrete targets effectively and robustly [21], [22].A nondimensional parameter, which is called as a guidanceparameter in the circulation component here, was utilized totransfigure the Lyapunov vector field [23], and strategies forsimultaneous cooperative multiple UAVs standoff trackingwere presented. The above approach was extended to pathfollowing and achieved arrival-angle control [24], [25]. Thecirculation term in traditional LVFG was modified to makeUAV faster settle onto the standoff orbit with a lower max-imum curvature [26]. Standoff target tracking was achievedwith unknown constant airspeed and target motion [27]. Co-operative standoff tracking methods of moving targets werepresented to solve the phase separation problem [28]–[31].Other innovative aspects for circumnavigation can be foundin recent works [32], [33].

Although many significant and constructive research efforthave been performed, long convergence time is not developedto address recently. For the problem, this paper’s principalcontributions are summarized as following:

1) The analysis of curvature problems and three criteriaof fast convergence are illustrated to guide the designof fast LVFG in the Lyapunov vector field guidanceframework.

2) Two types of guidance functions fulfilling the criteriaare constructed to achieve fast convergence, and theirperformance and application scope are analyzed.

3) The optimal parameter estimations in guidance func-tions are proposed based on offline search, and theperformance and the complexity of different solutionswith rough or fine constrains are clarified.

The rest of the paper is organized as follows. In SectionII, we describe the problem, and two guidance functionsare presented and analyzed. The optimal parameter for astationary and moving target was searched offline in SectionIII to design the guidance vector field with the proposedguidance functions. Simulation and analysis for standofftarget tracking are presented in Section IV to demonstratethe advantage of the proposed algorithm, followed by con-clusions in Section V.

II. LYAPUNOV VECTOR FIELD GUIDANCEA. SCENARIO DESCRIPTION

Assuming the position vector of a target in the loiter plane isxt = [xt, yt]

T. The UAV located at x = [x, y]T performsstandoff target tracking with velocity vector x = [x, y]T andcourse χ. Without loss of generality, the position vector ofthe aircraft in the local coordinate system of the target can bepresented as xr = [xr, yr]

T, or radial distance r and bearingangle θ (in the local polar coordinate system).

A first-order course-hold loop control is used, and the

FIGURE 1. Scenario of standoff target tracking.

constant-speed aircraft kinematic model can be presented as x = s cos(χ)y = s sin(χ)χ = k(χd − χ)

(1)

where s is UAV speed, χd is the desired course, and k is timeconstants of the first-order kinematics of the course.

B. LYAPUNOV VECTOR FIELD GUIDANCE FRAMEWORKVarious vector field methods based on the Lyapunov functionwere utilized to guide the fixed-wing aircraft to maintaina steady distance to the target for continuous tracking inthe target local coordinate system. A series of Lyapunovvector fields were configured by different nondimensionalparameters through modification of the above research [23],whose framework is[

r

rθ

]=s

α

[−(r2 − r2d)c · rd · r

](2)

where c is a constant parameter defined as a guid-ance parameter here, α is normalization term of speedand satisfies constrain r2+(rθ)

2= s2, which leads to

α=√r4 + (c2 − 2)r2dr

2 + r2d.A= |r| denotes the radial convergence speed to the stand-

off circle, which can be called as a contraction component.Circulation component B =

∣∣∣rθ∣∣∣ presents the tangent speedof the aircraft with respect to the standoff circle, which is thenormal of the contraction component. Fig. 1 shows the abovecomponents for standoff target tracking in the describedsituation.

As can be seen from (2), the contraction component de-creases and the circulation component increases as guidanceparameter c becomes larger for the constant-speed kinematicmodel, where changes are nonlinear under the influenceof normalization term α. A positive or negative guidanceparameter decides between counterclockwise or clockwisestandoff target tracking, so we only discuss the case where

2 VOLUME 4, 2016



parameter c is positive. In other words, the UAV only tracksthe target counterclockwise in the following discussion.

To improve the convergence performance of the aircraftto the standoff circle and ensure the UAV maintains standoffdistance rd from the target, the contraction component shouldbe as large as possible when the aircraft is far away from thestandoff circle, and be zero while the aerial vehicle settlesonto the loiter circle. However, it is impossible to design theabove contraction component only by adjusting the value ofthe guidance parameter on the basis of [23]. Meanwhile, theguidance parameter was transfigured to a guidance functionof distance r to decrease convergence time [26], but it wasnot fast enough.

In the light of the above analysis and existing research,guidance term c should fulfill the following criteria:

1) the guidance parameter should be a continuous func-tion of radial distance to overcome the deficiency ofthe nondimensional parameter;

2) achieve maximum at r = rd for steady standoff track-ing;

3) and be zero when r → 0 or r → ∞ to guarantee fastconvergence.

A strictly mathematical notation is presented asc = c(r) > 0, r > 0rd = arg max

r(c(r))

limr→0,∞

c(r)→ 0(3)

Transforming (2) into the Cartesian coordinate system, thedesired velocity for the UAV relative to the target is

xd =

[xdyd

]=

s

αr

[ (r2 − r2d

)xr + crrdyr(

r2 − r2d)yr − crrdxr

](4)

For a stationary target, the course of the UAV can berepresented as

χ= arctan(y/x) (5)

whose derivative with respect to time is course rate χ

χ=xy − yxx2 + y2

(6)

Then, the curvature can be calculated as the quotient of thecourse rate and the UAV speed

κ=χ

s=

xy − yxs(x2 + y2)

(7)

Substituting (4) and its time derivative into (7) leads to acurvature [26] as

κ=scr3d((c

2 − 2)r2 + 2r2d)− αcrdr(r2 − r2d)sα3

(8)

The traditional vector field is a special case where theguidance parameter is constant at 2, and its curvature is κ2

κ2=4r3d

(r2 + r2d)2 (9)

It is worth noting that κ2 is a monotone decreasing func-tion of radial distance r. When the UAV is near the centerof the loiter circle, or r is close to 0, curvature κ2 obtains amaximum, which results in much redundant turning and timewasted on the path toward the loiter circle. Otherwise, whenthe aircraft is out of the loiter circle, or r > rd, curvature κ2is always positive, which causes the left turning of the aerialvehicle to achieve counterclockwise standoff tracking. Thischaracteristic further demands the UAV turn early even if itis far away from the standoff circle, so a massive amount oftime is consumed as well. When r=rd, curvature is a constant1rd

independent of the guidance parameter, which ensures theUAV flies on the circle with radius rd.

C. THE PROPOSED GUIDANCE FUNCTIONSAccording to the analysis in the previous section, two guid-ance functions satisfying the criteria of fast convergence inthe Lyapunov vector field framework are structured as

cn1 (r) =

{ (rrd

)n, r < rd(

rdr

)n, r ≥ rd

(10)

cn2 (r) =

{exp

[1−

(rdr

)n], r < rd

exp[1−

(rrd

)n], r ≥ rd

(11)

where n ≥ 0 is an undetermined parameter, and its effect onthe performance of the guidance vector field will be discussedin detail later. It is worth noting that cn=0

1 =cn=02 = 1 for

n = 0. The proposed guidance functions can be regardedas the approximation of a traditional vector field [28]. TheLyapunov vector field modified in [26] is a special case ofn = 1 for cn1 (r), and is only compared with traditional vectorfields considering different guidance parameters but does notfurther enhance convergence performance to the limit.

The curves of the two guidance functions as r changes areshown in Fig. 2, where the proposed guidance functions aredenoted with red and blue lines, the solid line is a case ofn = 2, and the dash-dot line is n = 15; the same as below. Asshown in Fig. 2, both the proposed guidance functions satisfythe fast convergence criteria analyzed in (4). Specifically, thelarger n is, the longer the distance between the two functionsis attached to the x-axis and steeper near rd. The secondguidance function is less than the first for the same r andn, which indicates the second guidance function cn2 (r) havea faster convergent performance.

Curves of contraction and circulation components withparameters n = 2, 15 are compared with the case of c = 2(denoted with a black line, the same as below) as shown inFig. 3. The black line is lower than the other curves for thecontraction component in Fig. 3(a) and the opposite situationis shown in Fig. 3(b), except r = 0 and r = rd, whichindicates that the proposed guidance functions increase thespeed of the aircraft toward the loiter circle as much aspossible, thus greatly improving the convergence speed ofstandoff tracking. With the increase of parameter n, there aremore ones in the contraction component and more zeros in

VOLUME 4, 2016 3



FIGURE 2. Proposed guidance functions with n = 2, 15.

the circulation component, which implies better convergenceperformance of the vector field.

Note: the contraction and circulation components are notstrictly symmetric about r = rd, even if the larger n is, thestronger the symmetry is. Therefore, different n should bedetermined in the case whether the initial aircraft position isin or out of the standoff circle.

Fig. 4(a) presents the traditional Lyapunov vector field in[28], and new Lyapunov vector fields with proposed guidancefunctions are shown in Fig. 4(b) and Fig. 4(c) where n = 2.Through comparison with the traditional vector field, we cansee that, in the proposed vector fields, the vectors far awayfrom the standoff circle point to the circle more radially,regardless of whether they are inside or outside the circle,which further confirms the contribution of the proposed guid-ance functions to tracking efficiency, as per previous analysis.

D. CURVATURE ANALYSISTo further analyze the influence of the proposed guidancefunctions on the curvature, the time derivative of the guidancefunctions can be obtained as

cn1 (r) =

{−n r

n−1

αrnd(r2 − r2d), r < rd

nrnd

αrn+1 (r2 − r2d), r ≥ rd(12)

and

cn2 (r) =

{−c2n rnd

αrn+1 (r2 − r2d), r < rd

c2nrn−1

αrnd(r2 − r2d), r ≥ rd

. (13)

Substituting (10)–(13) into (8), two curvature functionsbased on the proposed guidance functions can be presented asκcn1 and κcn2 . The results are too complicated to quantitativelyanalyze, so curvature curves were plotted in Fig. 5 when nwas equal to 2 and 15, as well as the curvature of the tradi-tional vector field, where the horizontal and vertical dashedlines denote κ = 1

rdand r = rd, respectively. Considering

the operational and physical performance of the aircraft, its

(a)

(b)

FIGURE 3. Curves of contraction and circulation components. (a) Contractioncomponent, (b) circulation component.

maximum turning rate is limited due to the maximum bankangle, so the curvature is subject to a saturation constraint as

|κ| ≤ χmax

s. (14)

where the constraint on the curvature was set as κmax = 0.01in Fig. 5.

In Fig. 5, when the aircraft was located in the centerof the standoff circle or radial distance r was far greaterthan rd, curvature curves based on the proposed guidancefunctions were all close to zero, which is significantly betterthan the curvature of the traditional method whose guidanceparameter is a constant at 2. The more points on both κcn1 andκcn2 closing to the x-axis as n increases, the more directlythe UAV flies onto the standoff circle, thereby speeding upconvergence. When r > rd, there is a negative part ofthe curvature except for the black solid line. It is exactlythe negative values that allow the aircraft to turn right ata position closer to the loiter circle and then turn left to

4 VOLUME 4, 2016



(a) (b) (c)

FIGURE 4. Lyapunov field vector with standoff distance of rd = 300m. (a) c = 2, (b) cn=21 , (c) cn=2

2 .

FIGURE 5. Curvature curves under saturation constraint.

achieve counterclockwise standoff tracking. Compared withthe traditional method whose curvature is always positive,the proposed vector field can greatly shorten convergencetime. When n becomes larger, the range of the curvature nearzero also becomes bigger, and the extremes of the curvatureare closer to the circle, leading to faster convergence speed.However, high requirements for the turning rate for UAVare put forward due to a large n. On the one hand, due tothe saturation constraint on the bank angle, the turning ofthe aircraft cannot complete the requirements of the controlcommand, which leads to course lag or even exceptionalcircumstances; on the other hand, when r is disturbed near rd,the curvature fluctuates violently around 1

rd, which causes the

aircraft to sway left and right, and unstable standoff trackingquickly and directly. In brief, the choice of parameter n needsto be related with specific scenarios, which is the focus of thenext section.

III. DESIGN OF FAST VECTOR FIELD GUIDANCEAs per the above analysis, the proposed vector field con-structed with a small n has relatively long convergence timebut can guarantee directly stable tracking. On the contrary,the extremely large n makes the vector field more radial

toward the standoff circle; although this leads to a muchfaster arrival at the circle, this situation may need more timeon stabilization by means of the aircraft repeatedly traversingthe loiter circle. This makes UAV bear the maximum load fora long time and limits applications of standoff target tracking.

The ideal track should take into account both fast ap-proaching target and stable tracking target. Therefore, theselection of n should handle the trade-off between fast con-vergence and stability, which requires parameter n to be largeenough, and allows no traversing the standoff circle.

A. PARAMETER CHOICE FOR STATIONARY TARGETSAccording to the operational and physical performance of theaircraft, the maximum bank angle is restricted for UAV; thus,the maximum curvature of the track is limited. When thecurvature function is always within the saturation constraint,the UAV can follow the desired course determined by theproposed LVFG. In this case, optimal parameter nopt shouldbe selected to be as large as possible.

nopt= argn

(maxr

∣∣κcni ∣∣ ≤ κmax

)(15)

where κc generally refers to the curvature generated byproposed guidance function cni , i = 1, 2.

When the curvature function exceeds its threshold of acurvature constraint, the UAV turns at the minimum radiuswithin a voyage under the influence of the maximum bankangle. The schematic diagram of the trajectory is under theabove condition, shown as Fig. 6, where rmin is the minimumturning radius whose corresponding circle is expressed as asolid line and its center is at point o. Points pin and poutrepresent the tangent points when the aircraft enters and exitsthe circle, respectively.

It can be seen from Fig. 6 that, when the minimum turningradius circle and the standoff circle are separated, it takes ad-ditional time for the aircraft to move from point pout to steadystandoff tracking. Otherwise, when the n exceeds a value,the voyage formed by the UAV turning at the maximum bankangle intersects with the standoff circle, which leads to steadytracking after crossing over the standoff circle several times,increasing convergence time, and even threatens the safety of

VOLUME 4, 2016 5



FIGURE 6. Track diagram of unmanned aerial vehicle under performancesaturation constraints.

the UAV in some cases. Therefore, the optimal solution is thatthe aircraft should directly perform steady standoff trackingafter turning at the maximum bank angle, that is, optimalparameter n should make the minimum turning radius circleof UAV tangent to the standoff circle.

Maximum curvature κmax can be obtained from the perfor-mance constraint of the aircraft. While the guidance functionsand parameter n are given, the distance between point pinand the target can be obtained by combining the curvaturefunction. Angle β between aircraft speed and radial speedcan be obtained by the guidance law.

cosβ =

∣∣∣∣∣rθs∣∣∣∣∣ (16)

According to the triangle cosine theorem, the distancebetween the center of the minimum turning radius circle andthe target is

r2ou = r2min + r2in − 2rminrin cos < rmin, rin > (17)

where < rmin, rin > denotes the angle between rmin andrin, rin is distance vector from point pin to the target.Substituting (2) and (16), the above equation can be rewrittenas follows, based on the trigonometric function formula andthe geometric relationship.

r2ou =

r2min + r2in + 2cirminrdr

2in√

r4in+(c2i−2)r2dr2in+r

4d

, r > rd

r2min + r2in − 2cirminrdr

2in√

r4in+(c2i−2)r2dr2in+r

4d

, else

(18)It is easy to judge the positional relationship between the

minimum turning circle and the standoff circle according torou. If rou < δout (rou > δin), the two circles intersect,and the UAV trajectory fluctuates near the loiter circle, whereδout = rd + rmin and δin = rd − rmin are discriminantthresholds for the initial position of the aircraft inside andoutside the circle, respectively. If rou > δout (rou < δin), theminimum turning circle and the standoff circle are separated(inclusion), and the convergence is slow. For rou = δout(rou = δin), the two circles are tangential internally (exter-nally), and the aircraft can rapidly converge to the standoff

circle and directly form stable tracking where correspondingparameter is optimal for the given guidance function.

Substituting rou = δout or rou = δin into (18), since bothrin and ci are nonlinear functions related to parameter n, itis difficult to directly obtain a closed-form solution about n.It can be seen from previous analysis that the larger n is, thecloser the saturated points of curvature function are to rd,that is, the nearer the minimum turning circle is to the loitercircle. Therefore, rou is a monotone subtraction function forn, and the maximum of parameter n when rou is no less thanδout or no greater than δin can be selected as the optimalparameter, which is obtained based on the search method.The optimal parameter search algorithm developed for thestationary target is summarized in Algorithm 1.

Algorithm 1 Optimal parameter search for stationary targets.Input: Standoff radius rd; Maximum turning rate χmax;

UAV cruising speed s; Grid points of parameter n,n1, n2, · · · , nm; Given guidance function cni (r);

Output: Optimal parameter for given guidance functionniopt;

1: Calculate the minimum turning radius rmin

2: Calculate threshold δout or δin according to the initialposition of UAV.

3: for k = 1 to m do4: Obtain curvature function κcnk

i(r) in (8) by cnk

i (r)and its time derivative;

5: if maxr

∣∣∣κcnki

(r)∣∣∣ < χmax

s then6: niopt = nk and Continue;7: end if8: Calculate rin and rou according to (18);9: if r > rd & rou ≥ δout or r < rd & rou ≤ δin then

10: niopt = nk;11: end if12: end for

It is noted that, when the maneuverability of the UAV isweak or the motion space is limited under a certain initialposition and course conditions, such as position closing to thestandoff circle with the course pointing to the standoff circle,the proposed vector field guidance law is hard to design andsatisfy both fast convergence and avoidance of fluctuationonly by adjustment of parameter n. Due to the limited motionrange, when the aircraft initializes in the standoff circle,the above problems are particularly prominent, and can beresolved by reducing the speed of the airplane, reducing theturning radius, and improving maneuverability.

In practical applications, for the situations with the initialposition of the aircraft near the center of the standoff circle, itis usually applicable in non-hostile scenarios such as regionalsurveys and peripheral situation surveillance, therefore theconvergence speed and stability of the vector field are notrequired strictly. On the contrary, when the initial positionof the aircraft is farther away from the circle, especially fora dangerous region or hostile target, the standoff distanceis usually a safe distance. Hence, it is necessary to ensure

6 VOLUME 4, 2016



the aircraft safely completes the mission, which requires theaircraft to directly form stable standoff tracking.

Generally speaking, it is time-consuming to determineoptimal parameter nopt based on the grid-search method. Atthe same time, both the consumed time and the accuracy ofthe result rely heavily on the gird interval. Before the aircraftcarries out a real-time online mission, for a given UAVtype, its mobility (cruising speed, maximum turning rate) hasbeen decided, as well as standoff distance determined by themission. Therefore, the required parameter can be calculatedand selected offline after the mission is confirmed and beforethe UAV performs the mission.

B. PARAMETER CHOICE FOR MOVING TARGETSThe correction factor was introduced to modify the desiredcourse of the UAV for the case of a moving target, andwell-behaved guidance were performed [28]. Both the abovemethod and the proposed method are based on the samevector field guidance framework, therefore the correctionfactor can be used in the method mentioned in this paper.Consequently, the choice of optimal parameter nopt shouldbe modified for moving target standoff tracking as well.

If the UAV moves in the designed course of the vectorfield guidance in the local coordinate system of the target,the relationship between target velocity and UAV modifiedvelocity is as follows:

vg − vt = λvd (19)

where λ is the correction factor (denoted as α in the originalreference), vt is the target velocity vector, vg is the correctedUAV velocity vector in the global coordinate system, and s =|vg|, st = |vt|, sr = |λvd|.

Taking the modulus of the above equation to obtain thequadratic equation with respect to λ, the impossible negativeroot is discarded, and the larger positive root is the desiredcorrection factor.

The work of the correction factor on UAV velocity isshown in Fig. 7. Through the above modification, the UAVcan follow a desired course in the local coordinate system ofthe target according to the guidance vector field and realizestandoff target tracking. As we can see from Fig. 7, the rangeof the correction factor is λ ∈

[1− st

s , 1 + sts

], where the

extremum of λ is obtained when vg and vt are collinear. Thedifference of the UAV course before and after modificationis denoted as angle correction term ∆χ, whose range is∆χ ∈

[0, arcsin

(sts

)]regardless of its directionality.

In order to ensure sufficient time and space to veer whenthe UAV approaches the standoff circle, the minimum turningradius should be calculated according to the influence ofrelative velocity vr = vg − vt = λvd on course and speed.Therefore, < rmin, rin > in (17) can be corrected as

< rmin, rin>c =< rmin, rin>d ±∆χ (20)

where < rmin, rin>c denotes the corrected angle betweenrmin and rin, < rmin, rin>d denotes the included anglebased on guidance law before correction.

FIGURE 7. Work of correction factor on unmanned aerial vehicle (UAV)velocity.

1) Solution 1 with rough constraintsOn the one hand, due to the relative speed between the air-craft and the target, turning time is greatly compressed; on theother hand, the large angle correction term may limit turningspace for the aircraft. Accordingly, the rigorous case wheremaximum relative speed and maximum angle correction termare roughly selected is the simplest and most direct correctionsolution with the following rough constraints.

max (sr) = s+ st (21)

max(∆χ) = arcsin(sts

)(22)

2) Solution 2 with fine constraintsAs shown in Fig. 7, when the angle correction term equalsits minimum value of zero, the relative speed reaches itsmaximum value, and vg and vt are collinear and reverse;when the angle correction term is the maximum, vr and vtare vertical. For these reasons, it is virtually impossible tomake both the relative speed and the angle correction termapproach the maximum at the same time, and the relationshipbetween them is as follows.

s2 + s2r − 2ssr cos ∆χ = s2t (23)

Based on the above fine constraints, optimal parameternopt can be searched offline according to the grid points ofthe relative speed and the corresponding angle correctionterm calculated by (23). Algorithm 2 presents the specificimplementation flow of Solution 2.

IV. SIMULATION AND ANALYSISTo illustrate the performance of the proposed Lyapunovvector field, a simulation environment was designed as fol-lows. The control gain was set as k=30, the speed of theaircraft was s=20m/s, and standoff distance was rd=300m.

VOLUME 4, 2016 7



Algorithm 2 Optimal parameter search for moving targetInput: Standoff radius rd; Maximum turning rate χmax;

UAV cruising speed s; Target speed st; Given guidancefunction cni (r); Grid points of relative speed, s1r , s2r , · · · ,spr ;

Output: Optimal parameter for given guidance functionniopt;

1: Initialize niopt to zero;2: for j = 1 to p do3: Calculate minimum turning radius rmin;4: Calculate angle correction term ∆χ in (23);5: Correct < rmin, rin > using ∆χ as (20);6: Calculate optimal parameter nj by Algorithm 1;7: if niopt < nj then8: niopt = nj ;9: else

10: Break;11: end if12: end for

Assuming the maximum course rate was χmax=0.2rad/s, andthe stationary target was located at (0 m, 0 m).

A. INITIAL UAV POSITION INSIDE STANDOFF CIRCLEThe initial position of the aircraft was set as (5 m, 0 m),close to the target, and its initial course was 0 deg. Fig. 8gives the standoff circle of the target and UAV trajectoriesbased on different LVFGs for 50 s. It can be seen from thefigure that the main component of aircraft speed based on thetraditional Lyapunov vector field (solid black line) is used forveering instead of flying radially to the standoff circle, whichforms stable standoff tracking after the aircraft turns about360 deg. Using the proposed vector fields, the aircraft canquickly approach the standoff circle from the beginning on,and then turn left to directly perform stable tracking, whichconverges faster.

To more intuitively show convergence performance underdifferent parameters, convergence error is defined as the ab-solute of the distance difference from the target and standoffdistance.

er = |r − rd| (24)

Maximum distance r to the target in standoff tracking, andtime taken for the traditional and the proposed vector fieldsto converge to 0.5%rd were investigated, respectively, andthe results are shown in Table 1. At the same time, integratedtime absolute error (ITAE) was used to measure the trackingperformance of a guidance law with different guidance func-tions and parameters. ITAE is defined as follows, and ITAEcurves as time changes are shown in Fig. 9.

JITAE =

∫ t

0

τer(τ)dτ (25)

It can be seen from Table 1 and Fig. 9, the traditionalguidance method is of the slowest convergence speed, and

FIGURE 8. UAV trajectories when initial position is inside standoff circle.

FIGURE 9. Integrated time absolute error (ITAE) results when initial UAVposition is inside standoff circle.

even simulation time needs to be extended to 100 s beforea time result can be obtained. Meanwhile, its ITAE is alsoalways the largest and escalates, which indicates the aircraftnever arrives to the loiter circle within 50 s. For the proposedguidance vector fields, the increase rate of ITAE significantlydecreased after 8 s, which presents that the UAV can quickly

TABLE 1. Convergence performance when UAV initial position is (5 m, 0 m).

Guidance term configuration 0.5%rd/s max(r)/mn = 2 71.01 300.01

c1

n = 2.0 35.63 299.92n = 15.0 14.87 353.90nopt = 3.7 17.97 300.16

c2

n = 2.0 28.53 300.07n = 15.0 14.77 382.05nopt = 2.2 19.47 300.14

8 VOLUME 4, 2016



FIGURE 10. UAV trajectories when UAV initial position is outside standoffcircle.

converge to the circle. The larger parameter n is, the shorterthe time to converge to 0.5%rd. However, when parametern is too large, such as n = 15.0, time to first reach 0.5%rddramatically decreases, and the corresponding ITAE is alsothe smallest, temporarily near 15 s. The advantage of alarge n in ITAE was not maintained to the end, and theaircraft traversed the loiter circle twice (the maximum ofradial distance is larger than the standoff distance) beforeconverging to steady tracking. The vector field adopted withthe optimal parameter searched offline (highlighted in bold)appears to have no oscillation and could directly converge tothe standoff circle. The corresponding ITAE became horizon-tal after 20 s because of continuous standoff target trackingon the standoff circle.

B. INITIAL UAV POSITION OUTSIDE THE STANDOFFCIRCLEAssuming that the initial position of the aircraft is (–900m, 0 m), which is located outside the standoff circle, anexperiment was simulated for 100 s, and the other simulationconditions remained the same. Fig. 10 and Fig. 11 showthe UAV trajectories and ITAE variation curves, and thecorresponding convergence performance is given in Table 2.When a traditional vector field is adopted, the aircraft slowlyconverges (simulation time is extended to 200 s) because itonly turns left to settle onto the loiter circle due to always-positive curvature, as per the previous analysis in Section II-B. However, using the guidance vector field designed in thepaper, especially in the case of cn=12.0

1 or cn=6.02 (red or blue

dashed line) where most of the curvature tends to zero, theUAV can directly approach the standoff circle. Meanwhile,the negative part of the curvature makes the aircraft turnright as late as possible at a position that guarantees notcrossing the standoff circle, and then turn left to directly formstable tracking. In the above cases, ITAE always remains ata low level for almost the entire time compared with theother cases. Table 2 further illustrates the superiority andeffectiveness of the proposed optimal parameter based offlinesearch, avoiding the slow convergence speed of a small n andthe vibration of a large n.

Compared with the second guidance function, the first

FIGURE 11. ITAE results when UAV initial position is outside standoff circle.

TABLE 2. Convergence performance when UAV initial position is (–900 m, 0m).

Guidance term configuration 0.5%rd/s min(r)/mn = 2 112.57 300.34

c1

n = 2.0 56.20 300.17n = 15.0 30.63 289.71

nopt = 12.0 31.53 300.10

c2

n = 2.0 55.17 300.17n = 15.0 30.03 254.84nopt = 6.0 31.60 300.17

guidance function always has a weak advantage. The mainreason may be that the first guidance function has a milderinfluence on the guidance law than the second for the samechange on parameter n. While the same grid interval wasapplied to search parameter, the final result for the firstguidance function was perhaps closer to the real optimalparameter, so the performance of the proposed vector fieldcould be improved. However, the influence of this differ-ence on tracking performance is limited. The choice of thesecond guidance function can reduce the search scope andimprove search speed. Therefore, the guidance function canbe flexibly selected according to the above characteristics ofdifferent guidance functions in practice.

C. STANDOFF TRACKING FOR MOVING TARGETThe target vehicle is assumed to move counterclockwisearound a circle with a radius of 1000 m, and the target speedis 10 m/s. The relationship curves between relative speed andoptimal parameter nopt based on different solutions are givenin Fig. 12, where Solutions 1 and 2 are denoted by the dottedline and solid line, respectively. The minimum of each curvewas selected to design the proposed vector field guidance law.For the same guidance function, and the parameter searchedwith Solution 2 was larger than Solution 1 to meet the fastconvergence criteria.

Fig. 13 shows UAV trajectories according to differentparameters determined by Solutions 1 and 2, where Fig. 13(a)

VOLUME 4, 2016 9



FIGURE 12. Relationship curves between relative speed and the optimalparameter based on different solutions.

TABLE 3. Convergence performance when UAV initial position is (–900 m, 0m).

Guidance term configuration 0.5%rd/s min(r)/mn = 2 120.13 300.19

c1ns1 = 4.0 49.43 300.08ns2 = 4.7 48.53 300.08

c2ns1 = 2.3 51.33 300.08ns2 = 2.8 50.20 300.08

gives the UAV trajectories in target coordinates, and thetracking results in the global coordinate system are given inFig. 13(b), as well as the standoff circle at last moment. Itshows that the proposed guidance vector field designed by theoffline searched parameter can effectively track the movingtarget and maintain stable standoff distance.

It can be seen from the performance of the track shown inTable 3 that the convergence time of the proposed methodswas significantly shorter than that of the traditional method(120.13 s) for a moving target. For the same guidance func-tion, the optimal parameter searched according to Solution 2presented faster convergence speed and better guidance per-formance, but the search process of Solution 2 has the highercomputational complexity of O(mp), where m and p are thenumber of grid points of parameter n and relative speed,respectively. The maximum relative speed was only selectedusing rough constraints without searching, so the complexityof Solution 1 is O(m). Combined with the characteristics ofthe guidance functions in the previous subsection, for themoving target, the first guidance function and Solution 2can be combined to improve standoff tracking performance,or the combination of the second guidance function andSolution 1 could be used to enhance search efficiency at thecost of small convergence time.

In addition, the UAV kept a distance of 900 m from thetarget, and all phases relative to the target were simulated tocheck the reliability of the search method for nopt. Severalrough situations, where the initial UAV position was 900

(a)

(b)

FIGURE 13. Standoff tracking for moving target. (a) UAV trajectories relativeto the target. (b) UAV and target trajectories, and the standoff circle at lastmoment.

m away from the target and all phases relative to the targetwere traversed, were simulated to demonstrate that there isno case of the UAV crossing the standoff circle, and prove thereliability of the search method for nopt. In fact, until n = 5.0for the first guidance function or n = 2.9 for the second,UAV briefly enters the inside of the standoff circle whenthe aircraft is in some particular orientation to the target.This phenomenon shows that the optimal parameter obtainedby the proposed Solution 2 with fine constraints basicallyapproaches the practical optimal parameter.

V. CONCLUSIONThis paper presented fast Lyapunov vector field guidancefor standoff target tracking with short convergence time andno traversing the circle orbit. Two guidance functions wereproposed to fulfill the criteria of fast convergence based onanalysis of the contraction and circulation components in theLyapunov vector field guidance framework. The parameter inthe guidance function was optimized under the physical satu-ration constraints of the aircraft based on an offline search.For a moving target, fast Lyapunov vector field guidancewas implemented by the introduction of a correction factor

10 VOLUME 4, 2016



and angle correction term, and the corresponding optimalparameter search method was presented with a rough or fineconstraint. A series of simulations were performed to demon-strate the lower time consumption and the stability of theguidance method designed with the proposed guidance func-tions and the searched optimal parameter estimates. Differentguidance functions and solutions for the moving target can beused in combination according to mission requirements fortracking performance and search time.

There are many potential modifications for applicationconsiderations. For uncertain target speed and unknownbackground airspeed, they can be regarded as the disturbancespeed with determined interval, and analysis of a movingtarget can be referenced to provide the optimal parameterwith a loose constraint. Therefore, the proposed fast guidancemethod can be extended to wide application scenarios.

REFERENCES[1] I. Mahmud and Y.-Z. Cho, “Detection Avoidance and Priority-Aware

Target Tracking for UAV Group Reconnaissance Operations,” Journalof Intelligent & Robotic Systems, vol. 92, no. 2, pp. 381–392, 2018.[Online]. Available: https://doi.org/10.1007/s10846-017-0745-9

[2] Y. Liu, Q. Wang, H. Hu, and Y. He, “A Novel Real-Time Moving TargetTracking and Path Planning System for a Quadrotor UAV in UnknownUnstructured Outdoor Scenes,” IEEE Transactions on Systems, Man, andCybernetics: Systems, pp. 1–11, 2018.

[3] D. Solazzo, J. B. Sankey, T. T. Sankey, and S. M. Munson, “Mapping andmeasuring aeolian sand dunes with photogrammetry and LiDAR fromunmanned aerial vehicles (UAV) and multispectral satellite imagery onthe Paria Plateau, AZ, USA,” Geomorphology, vol. 319, pp. 174–185,2018. [Online]. Available: http://pubs.er.usgs.gov/publication/70198467

[4] K. Harikumar, J. Senthilnath, and S. Sundaram, “Multi-UAV OxyrrhisMarina-Inspired Search and Dynamic Formation Control for Forest Fire-fighting,” IEEE Transactions on Automation Science and Engineering,vol. 16, no. 2, pp. 863–873, 2019.

[5] Z. Zhou, C. Zhang, C. Xu, F. Xiong, Y. Zhang, and T. Umer, “Energy-Efficient Industrial Internet of UAVs for Power Line Inspection in SmartGrid,” IEEE Transactions on Industrial Informatics, vol. 14, no. 6, pp.2705–2714, 2018.

[6] H. Oh, S. Kim, H. S. Shin, and A. Tsourdos, “Coordinated standoff track-ing of moving target groups using multiple UAVs,” IEEE Transactions onAerospace and Electronic Systems, vol. 51, no. 2, pp. 1501–1514, 2015.

[7] R. Rysdyk, “UAV Path Following for Constant Line-of-Sight,”2nd AIAA "Unmanned Unlimited" Conf. and Workshop &Exhibit, no. September, pp. 1–10, 2003. [Online]. Available:http://arc.aiaa.org/doi/10.2514/6.2003-6626

[8] R. Wise and R. Rysdyk, “UAV Coordination for AutonomousTarget Tracking,” in AIAA Guidance, Navigation, and ControlConference and Exhibit, 2006, pp. 1–22. [Online]. Available:http://arc.aiaa.org/doi/10.2514/6.2006-6453

[9] S. Park, “Circling over a target with relative side bearing,” Journal ofGuidance, Control, and Dynamics, vol. 39, no. 6, pp. 1450–1456, 2016.

[10] S. Park, J. Deyst, and J. How, “A New Nonlinear GuidanceLogic for Trajectory Tracking,” in AIAA Guidance, Navigation, andControl Conference and Exhibit, 2004, pp. 1–16. [Online]. Available:http://arc.aiaa.org/doi/10.2514/6.2004-4900

[11] S. Park, J. Deyst, and J. P. How, “Performance and Lyapunov Stabilityof a Nonlinear Path Following Guidance Method,” Journal of Guidance,Control, and Dynamics, vol. 30, no. 6, pp. 1718–1728, 2007. [Online].Available: http://arc.aiaa.org/doi/10.2514/1.28957

[12] M. Kim and Y. Kim, “Multiple UAVs nonlinear guidance laws forstationary target observation with waypoint incidence angle constraint,”International Journal of Aeronautical and Space Sciences, vol. 14, no. 1,pp. 67–74, 2013.

[13] E. Johnson, A. Calise, and J. Corban, “A Six Degree-of-Freedom AdaptiveFlight Control Architecture for Trajectory Following,” in AIAA Guidance,Navigation, and Control Conference and Exhibit, no. August, 2002, pp.1–11.

[14] C. Xia and A. Yudi, “Multi-UAV path planning based on Improved NeuralNetwork,” in CCDC 2018, 2018, pp. 354–359.

[15] D. A. Lawrence, “Lyapunov Vector Fields for UAV Flock Coordination,”in 2nd AIAA Unmanned Unlimited Systems, Technologies, and Opera-tions Conference, no. September, 2003.

[16] D. Nelson, D. Barber, T. McLain, and R. Beard, “Vector fieldpath following for small unmanned air vehicles,” in 2006 AmericanControl Conference, no. July, 2006, pp. 1–8. [Online]. Available:http://ieeexplore.ieee.org/document/1657648/

[17] D. R. Nelson, D. B. Barber, T. W. McLain, and R. W. Beard, “VectorField Path Following for Miniature Air Vehicles,” IEEE Transactions onRobotics, vol. 23, no. 3, pp. 519–529, 2007.

[18] S. R. Griffiths, “Vector Field Approach for Curved Path Following forMiniature Aerial Vehicles,” in AIAA Guidance, Navigation, and ControlConference, vol. 61, no. August, 2006, pp. 63–64.

[19] Y. Liang, Y. Jia, Z. Wang, and F. Matsuno, “Vector Field Guidance forThree-Dimensional Curved Path Following with Fixed-Wing UAVs,” in2015 American Control Conference, Chicago, IL, USA, 2015, pp. 5980–5985.

[20] M. A. H. Abozied and S. Qin, “High performance path following for UAVbased on advanced vector field guidance law,” in 2016 IEEE AdvancedInformation Management, Communicates, Electronic and AutomationControl Conference, IMCEC 2016, 2017, pp. 555–564.

[21] H. Chen, “UAV Path Planning with Tangent-plus-Lyapunov Vector FieldGuidance and Obstacle Avoidance,” IEEE Transactions on Aerospaceand Electronic Systems, vol. 49, no. 2, pp. 840–856, 2013. [Online].Available: https://ieeexplore.ieee.org/document/6494384

[22] H. Chen, K. Chang, and C. S. Agate, “Tracking with UAV using Tangent-plus-Lyapunov Vector Field Guidance,” in 12th International Conferenceon Information Fusion, 2009, pp. 363–372.

[23] S. Lim, Y. Kim, D. Lee, and H. Bang, “Standoff target tracking using avector field for multiple unmanned aircrafts,” Journal of Intelligent andRobotic Systems: Theory and Applications, vol. 69, no. 1-4, pp. 347–360,2013.

[24] S. Lim, W. Jung, and H. Bang, “Vector field guidance for path followingand arrival angle control,” in 2014 International Conference on UnmannedAircraft Systems, ICUAS 2014 - Conference Proceedings, no. May, 2014,pp. 329–338.

[25] W. Jung, S. Lim, D. Lee, and H. Bang, “Unmanned Aircraft Vector FieldPath Following with Arrival Angle Control,” Journal of Intelligent andRobotic Systems: Theory and Applications, vol. 84, no. 1-4, pp. 311–325,2016.

[26] A. A. Pothen and A. Ratnoo, “Curvature-Constrained Lyapunov VectorField for Standoff Target Tracking,” Journal of Guidance, Control, andDynamics, vol. 40, no. 10, pp. 2729–2736, 2017. [Online]. Available:https://arc.aiaa.org/doi/10.2514/1.G002281

[27] S. Zhu, D. Wang, and C. B. Low, “Ground target tracking using UAV withinput constraints,” Journal of Intelligent and Robotic Systems: Theory andApplications, vol. 69, no. 1-4, pp. 417–429, 2013.

[28] E. W. Frew, D. A. Lawrence, and S. Morris, “Coordinated StandoffTracking of Moving Targets Using Lyapunov Guidance Vector Fields,”Journal of Guidance, Control, and Dynamics, vol. 31, no. 2, pp. 290–306,2008. [Online]. Available: http://arc.aiaa.org/doi/10.2514/1.30507

[29] T. H. Summers, M. R. Akella, and M. J. Mears, “Coordinated StandoffTracking of Moving Targets: Control Laws and Information Architec-tures,” Journal of Guidance, Control, and Dynamics, vol. 32, no. 1, pp.56–69, 2008.

[30] S. Yoon, S. Park, and Y. Kim, “Circular motion guidance law for co-ordinated standoff tracking of a moving target,” IEEE Transactions onAerospace and Electronic Systems, vol. 49, no. 4, pp. 2440–2462, 2013.

[31] N. M. T. Kokolakis and N. T. Koussoulas, “Coordinated Standoff Trackingof a Ground Moving Target and the Phase Separation Problem,” 2018International Conference on Unmanned Aircraft Systems, ICUAS 2018,no. June, pp. 473–482, 2018.

[32] M. Deghat, I. Shames, B. D. Anderson, and C. Yu, “Target localization andcircumnavigation using bearing measurements in 2D,” IEEE Conferenceon Decision and Control, pp. 334–339, 2010.

[33] I. Shames, S. Dasgupta, B. Fidan, and B. D. Anderson, “Circumnavigationusing distance measurements under slow drift,” IEEE Transactions onAutomatic Control, vol. 57, no. 4, pp. 889–903, 2012.

VOLUME 4, 2016 11



SHUN SUN received the B.S. degrees in com-munication engineering from China Universityof Petroleum, Qingdao, China, in 2014 and theM.S. degree in information and communicationsengineering from, Yantai, China, in 2016. He iscurrently pursuing Ph.D. degree at Naval AviationUniversity, Institute of Information Fusion, Yantai,China.

He received joint education by National Univer-sity of Defense Technology, College of Electronic

Science, in 2017. His research interest includes information fusion, passivelocalization, collaborative control.

HAIPENG WANG received doctor’s degree fromNaval Aviation University in 2012. He is currentlyan Associate Professor of Naval Aviation Uni-versity. His research interests cover the generalarea of intelligent perception and fusion, big datatechnology and application. He also serves as aReviewer of the several distinguished journals asIET RSN and IEEE AES.

JUN LIU received the M.S. degree in informationand communication engineering from Naval Avi-ation University in 2015. He is currently pursuingPh.D. degree in information and communicationengineering from Naval Aviation University. Hisresearch interests include information fusion, dis-tributed state estimation, and situation awareness.

YOU HE received doctor’s degree from TsinghuaUniversity in 1997. He was a visiting scholar atBrunswick University of Technology in Germany.He is currently a Full Professor and a DoctorTutor of naval aviation university, IET Fellow. Hisresearch interests cover the general area of infor-mation fusion theory and technology, equipmentsimulation, big data technology and application.

12 VOLUME 4, 2016

Documents

Fast Lyapunov Vector Field Guidance for Standoff Target