aircraft routing under the risk of detection - department of industrial

Aircraft Routing under the Risk of Detection

Michael Zabarankin,1 Stan Uryasev,2 Robert Murphey3

1 Department of Mathematical Sciences, Stevens Institute of Technology,Castle Point on Hudson, Hoboken, New Jersey 07030

2 ISE Department, University of Florida, P.O. Box 116595, 303 Weil Hall,Gainesville, Florida 32611-6595

3 Air Force Research Lab, Munitions Directorate, Eglin Air Force Base

Received 10 April 2005; revised 14 April 2006; accepted 21 April 2006DOI 10.1002/nav.20165

Published online 8 June 2006 in Wiley InterScience (www.interscience.wiley.com).

Abstract: The deterministic problem for finding an aircraft’s optimal risk trajectory in a threat environment has been formulated.The threat is associated with the risk of aircraft detection by radars or similar sensors. The model considers an aircraft’s trajectoryin three-dimensional (3-D) space and represents the aircraft by a symmetrical ellipsoid with the axis of symmetry directing thetrajectory. Analytical and discrete optimization approaches for routing an aircraft with variable radar cross-section (RCS) subjectto a constraint on the trajectory length have been developed. Through techniques of Calculus of Variations, the analytical approachreduces the original risk optimization problem to a vectorial nonlinear differential equation. In the case of a single detectinginstallation, a solution to this equation is expressed by a quadrature. A network optimization approach reduces the original problemto the Constrained Shortest Path Problem (CSPP) for a 3-D network. The CSPP has been solved for various ellipsoid shapes anddifferent length constraints in cases with several radars. The impact of ellipsoid shape on the geometry of an optimal trajectory aswell as the impact of variable RCS on the performance of a network optimization algorithm have been analyzed and illustrated byseveral numerical examples. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 53: 728–747, 2006

Keywords: trajectory optimization; optimal path planning; risk minimization; risk of detection; analytical solution; networkoptimization; network optimization algorithm; Calculus of Variations; aircraft; radar cross-section

1. INTRODUCTION

The class of military and civil engineering applicationsdealing with optimal trajectory generation for space, air,naval, and land vehicles embraces several types of prob-lems with a variety of objectives, resource constraints, andcontrol limitations. Examples include minimizing risk ofaircraft detection by radars, sensors or Surface-to-Air mis-siles [4, 5, 15–17, 24, 26]; minimizing risk of submarinedetection by sensors [25]; minimizing cumulative radia-tion damage while passing through a contaminated area;finding optimal trajectories for multiple aircraft avoidingcollisions [17, 20]; minimizing propellant consumption bya spacecraft in interplanetary and orbit transfers [3]; andminimizing a weighted sum of fuel cost and time cost fora commercial plane.

This work was supported by Air Force Grant F49620-01-1-0338.Correspondence to: M. Zabarankin ([email protected])

A model for optimally routing an aircraft in a threat envi-ronment is developed based on specified mission objectives,available resources (fuel capacity), and limitations on air-craft control while minimizing the risk of exposure [24].Despite numerous studies in this area, only a few consideredrisk optimization problems with technological constraints.Zabarankin et al. [26] suggested analytical and discreteoptimization approaches for optimal risk path generationin two-dimensional (2-D) space with constant radar cross-section (RCS) of the aircraft, arbitrary number of sensors,and a constraint on path length.

This paper develops a 3-D model for minimizing the riskof aircraft detection by radars or sensors with variable RCSof the aircraft. A sensor is an antenna capable of receiving anisotropically radiated signal from the aircraft. In contrast, aradar is an antenna capable of transmitting a signal and receiv-ing the signal reflected off of the aircraft. Since a radar signalmust suffer a two-way transmission loss whereas a sensorsuffers only a one-way transmission loss, the received signal

© 2006 Wiley Periodicals, Inc.

Zabarankin, Uryasev, and Murphey: Aircraft Routing 729

strengths are strongly different functions of range in the twocases. The model is deterministic and static, since it assumesno uncertainty in aircraft detection and radar locations andconsiders neither aircraft kinematics equations nor parame-ters for aircraft control during a flight. The risk of detectionis assumed to be independent of aircraft speed. This model,being an extension of the 2-D problem for minimizing therisk of detection of the aircraft by sensors [26], considers

• aircraft trajectories in 3-D space;• variable RCS: an aircraft is modeled by an axisym-

metrical ellipsoid with the axis of ellipsoid symmetrydetermining direction of aircraft trajectory;

• the risk of detection is proportional to the aircraft’sRCS and reciprocal to the nth-power of the distancebetween the aircraft and a particular detecting instal-lation: n = 2 corresponds to a sensor, and n = 4corresponds to a radar;

• an arbitrary number of detecting installations;• a constraint on trajectory length.

The purpose of this model is to analyze the impact of vari-able RCS on the 3-D geometry of optimal trajectories subjectto a constraint on trajectory length and evaluate performanceof discrete optimization approaches with respect to runningtime and accuracy.

Analytic and discrete optimization approaches for solv-ing a trajectory optimization problem are developed withinthe framework of the suggested model. Through techniquesof Calculus of Variations, the risk minimization problemreduces to a nonlinear vectorial differential equation. In thecase of a single radar, we obtain an analytical solution to thisequation expressed by a quadrature. We should note that theRCS of a real air vehicle is not an ellipsoid, and, as a result,this model cannot be used for navigation of real aircrafts.However, to our knowledge, this is the first model for avoid-ing radar detection that takes into account variable RCS andstudies its impact on the geometry of optimal trajectories.Coupled with the analytical solutions approach, this modelserves as a platform for testing and analyzing the efficiencyof the proposed discrete optimization approaches, which thenwill be applied to the problem with actual RCS.

The efficiency of discrete optimization approaches in opti-mal risk path planning essentially depends on the type ofrisk functionals, technological constraints, and the scheme fortrajectory approximation [24]. Discrete approaches may ten-tatively be divided into three major categories: gradient-basedalgorithms, dynamic programming, and network optimiza-tion. Gradient-based algorithms are very efficient when therisk of detection is determined by smooth analytical function-als. However, gradient-based algorithms most often find onlylocally optimal solutions when risk functionals are noncon-vex. Many of the previous studies on trajectory generation

for military aircraft are concentrated on feasible directionalgorithms, and dynamic programming [4]. These methodstend to be computationally intensive and, therefore, are notwell suited for onboard applications. To improve computa-tion time, John and Moore [24] used simple analytical riskfunctions. Based on such an approach, they developed lat-eral and vertical algorithms to optimize flight trajectory withrespect to time, fuel, aircraft final position, and risk expo-sure. Recently, Tsitsiklis [22] and Polymenakos et al. [19]suggested Dijkstra-like and correcting-like methods for effi-ciently solving a continuous-space shortest path problemin 2-D plane. To find a globally optimal trajectory, theydiscretize the Hamilton–Jacobi equation, effectively fusinganalytical and discrete optimization techniques. Kim andHespanha considered a continuous shortest path problemin application to minimum-risk path planning for groupsof unmanned air vehicles (UAVs) [17]. However, theirmethods are not intended for solving optimization prob-lems with a constraint on the trajectory length. Zabarankinet al. [26] demonstrated the efficiency of a network optimiza-tion approach in solving risk minimization with a constrainton trajectory length and arbitrary number of sensors in 2-Dspace. The main advantages of the network optimizationapproach are that among all feasible trajectories in a net-work, it finds a globally optimal one; it can be appliedto the case with several detecting installations; it can beapplied to the case with actual nonsmooth RCS. Networkoptimization is closely related to mixed integer programming.Chaudhry et al. applied the latest approach to low observ-ability path planning for a UAV [5]. Also worth mentioningis the work by Inanc et al. [16], which considered nonlin-ear trajectory generation methods for UAVs with multipleradars.

The network optimization approach approximates anadmissible domain for the aircraft trajectory by a 3-D net-work and represents an aircraft’s trajectory by a path in thisnetwork. The optimal risk path generation problem subject toa constraint on trajectory length reduces to the ConstrainedShortest Path Problem (CSPP). To solve the CSPP, we sug-gest the Label Setting Algorithm (LSA) with a preprocessingprocedure [6, 7] and path smoothing. The efficiency of thenetwork optimization approach is demonstrated by severalnumerical examples for various ellipsoid shapes, constraintson trajectory length in the cases of one, two, and three radars.For the case with a single radar, we compare analytical andnumerical solutions and show that solutions coincide withhigh precision. However, it is also known that the CSPP is anNP-complete problem, and consequently, no exact polyno-mial algorithms should be expected. Numerical experimentsin a 3-D case show that LSA running time strongly dependson the shape of the ellipsoid. This phenomenon is analyzedfrom an optimization perspective, and an improvement forthe preprocessing procedure is suggested.

Naval Research Logistics DOI 10.1002/nav

730 Naval Research Logistics, Vol. 53 (2006)

Figure 1. Ellipsoid shape is defined by parameter κ = b/a.

The paper is organized as follows. Section 2 develops the3-D model for trajectory optimization with variable RCS sub-ject to a constraint on trajectory length. Section 3 derives thevectorial differential equation for finding the optimal trajec-tory in a general case and obtains an analytical solution tothis equation in the case of a single radar. Section 4 reducesthe optimal path planning to the CSPP and presents theLSA with the preprocessing procedure and path smoothing.Section 5 conducts numerical experiments with various ellip-soid shapes and constraints on trajectory length for one, two,and three radars. Section 6 analyzes the results of the numer-ical experiments from an optimization perspective. Section 7discusses numerical results and conclusions. The Appendixderives the necessary optimal conditions for the calculusof variations problem with a nonholonomic constraint andmovable end point.

2. MODEL DEVELOPMENT

In this section, we develop a three-dimensional model thatminimizes the risk of aircraft detection with variable radarcross-section by radars or sensors subject to a constraint onthe length of the aircraft trajectory.

Suppose an aircraft must fly from point A(xA, yA, zA) topoint B(xB , yB , zB) in 3-D space minimizing the cost ofdetection by N radars located in the area of interest. Wemodel the aircraft by an axisymmetrical ellipsoid with theaxes’ lengths a, b, and b. The axis with length a is the axisof ellipsoid symmetry, which is aligned with the aircraft tra-jectory. Ellipsoid shape is defined by parameter κ = b/a.Cases of κ = 1, κ < 1, and κ > 1 correspond to a sphereand elongated and compressed ellipsoids, respectively (seeFigure 1).

Let vectors r = (x, y, z) and qi = (ai , bi , ci), i = 1, N ,determine the position of the ellipsoid geometrical centerand position of the ith radar, respectively. A trajectory ofthe ellipsoid’s center determines a path of the aircraft. Wedefine a trajectory as a function of its current length s, i.e.,

r(s) = (x(s), y(s), z(s)). Such a parameterization is alsoknown as the natural definition of a curve. Vector r(s) =dds

r(s) = (x(s), y(s), z(s)) determines a direction of theaircraft trajectory that coincides with the axis of ellipsoidsymmetry. Since (ds)2 = (dx)2 + (dy)2 + (dz)2, vectorr(s) must satisfy condition r2 = x2 + y2 + z2 = 1. Thelength of vector ri (s) = r(s) − qi = (x − ai , y − bi , z −ci), denoted by ‖ri (s)‖, defines the distance from the air-craft to the ith installation (see Figure 2), i.e., ‖ri (s)‖ =√

(x − ai)2 + (y − bi)2 + (z − ci)2.The RCS of the aircraft exposed to the ith radar at point

(x, y, z) is proportional to the area Si of the ellipsoid’sprojection onto the plane orthogonal to vector ri

RCSi = σi Si ,

where the constant coefficient σi depends on the radar’s tech-nical characteristics such as the maximum detection range,the minimum detectable signal, the transmitting power of theantenna, the antenna gain, and the wavelength of radar energy.

The magnitude of ellipsoid projection area is given by theformula Si = π b

√a2 sin2 θi + b2 cos2 θi , where θi is the

angle between vectors ri and r. Based on relation cos θi =

Figure 2. 3-D model for optimal path planning in a threat envi-ronment. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]



ri ·r‖ri‖ and using notation κ = b/a, we rewrite the formula forRCSi as

RCSi = σi π

(a2 + b2

2

)2κ

1 + κ2

√1 + (κ2 − 1)

(ri · r‖ri‖

)2

,

κ ∈ [0, +∞). (1)

The purpose of representing RCSi in the form of (1) is thefollowing. Since the aircraft has a limited size, we assume thevalue

√a2 + b2 (“diameter” of cross-section) to be constant

for all a and b. Hence, the form of the ellipsoid is definedby ratio b/a, i.e., parameter κ , only. For instance, the caseof κ = b/a = 0 corresponds to a “thickless” needle withlength a, while the case of κ = b/a → ∞ corresponds toa thickless disk with radius b. Note that the cross-section ofthe thickless needle always equals zero, whereas the cross-section of a thickless disk reduces to σi π a2 | cos θi |, whichis zero only when θi = π

2 .The risk function (also referred to as the cost function)

for detection of the aircraft by the ith radar in free space isproportional to RCSi and reciprocal to the nth power of thedistance between the aircraft and the ith radar, ‖ri‖n, see, forexample, [21]. The case of n = 2 corresponds to a sensor, andthe case of n = 4 corresponds to a radar. In general, we have

RCSi/‖ri‖n = σiπ

(a2 + b2

2

)2κ

1 + κ2

×

√1 + (κ2 − 1)

(ri ·r‖ri‖

)2

‖ri‖n.

Since the value of√

a2 + b2 is assumed to be constant, termπ(a2+b2

2 ) can be omitted, and the risk of detection by the ithinstallation takes the form

C(ri , r) = 2κσi

1 + κ2

√‖ri‖2 + (κ2 − 1)(ri · r)2

‖ri‖n+1. (2)

We assume that the risk of detection by N radars at pointr = (x, y, z) is the sum of risk functions (2) for all i = 1, N

L(r, r) =N∑

i=1

C(ri , r)

= 2κ

1 + κ2

N∑i=1

σi

√‖ri‖2 + (κ2 − 1) (ri · r)2

‖ri‖n+1. (3)

It should be noted that expression (3) is not the probabil-ity of detection; it rather defines a so-called cost function ofdetection at a particular point, which coincides with the totalenergy received by all the detecting installations. Integrating

(3) along aircraft trajectory with length l, we obtain the totalrisk of detection

F(r, r) =∫ l

0L (r(s), r(s)) ds. (4)

The risk minimization problem is finding a trajectory P =r(s) = (x(s), y(s), z(s)), 0 ≤ s ≤ l, from point rA =(xA, yA, zA) to point rB = (xB , yB , zB) with the minimal riskof detection subject to a constraint on trajectory length

minP

F(r, r)

s.t. r2 = 1,r(0) = rA, r(l) = rB , l ≤ l∗.

(5)

To solve problem (5), we develop approaches based oncalculus of variations and network optimization.

3. CALCULUS OF VARIATIONS APPROACH

In this section, we consider the case of a single radarand reduce the risk minimization problem (5) to a vecto-rial differential equation to obtain an analytical solution. Thisequation represents necessary conditions for a curve that min-imizes functional (4) with a movable end point subject tothe constraint r2 = 1. We refer the reader to the Appendixthat derives this equation through techniques of Calculus ofVariations in a general case.

Let function gi be defined by

gi(ri , r, λL) = 1

‖ri‖n−1√

‖ri‖2 + (κ2 − 1) (ri · r)2+ λL,

(6)

where λL is unknown constant, and let gi = dds

gi .

THEOREM 1 (necessary conditions for an optimal trajec-tory). An optimal solution to optimization problem (5) shouldnecessarily satisfy the vectorial differential equation

N∑i=1

σi

((ri

ri · r− r

)gi − r gi

)= 0, (7)

with boundary conditions

r(0) = rA, r(l) = rB , l ≤ l∗, (8)

and the constraint

φ(r) = r2 − 1 = 0. (9)

PROOF: The problem (5) is a particular case of the prob-lem (36)–(40) considered in the Appendix. For constraint (9),



we have ∂φ

∂ r = 2r, and, consequently, ∂φ

∂ r

/(r· ∂φ

∂ r ) ≡ r. Substi-tuting the last equality into the general vectorial differentialequation (45), derived in the Appendix, we obtain the vec-torial differential equation for finding an optimal trajectoryr(s), 0 ≤ s ≤ l,

∂L

∂r− d

ds

(∂L

∂ r+ r

(L − r · ∂L

∂ r+ cL

))= 0. (10)

Introducing a new constant λL by relation cL = 2κ1+κ2 λL and

using notations gi = g(ri , r, λL), gi = dds

g (ri , r, λL), wherefunction g(ri , r, λL) is defined by (6), we obtain the relations

L − r · ∂L

∂ r+ cL = 2κ

1 + κ2

N∑i=1

σi gi ,

∂L

∂r− d

ds

∂L

∂ r= 2κ

1 + κ2

N∑i=1

σi gi

(ri

ri · r

).

With these relations, equation (10) reduces to (7):

∂L

∂r− d

ds

(∂L

∂ r+ r

(L − r · ∂L

∂ r+ cL

))

= ∂L

∂r− d

ds

∂L

∂ r− d

ds

(r(

L − r · ∂L

∂ r+ cL

))

= 2κ

1 + κ2

N∑i=1

σi gi

(ri

ri · r

)− d

ds

(r

(2κ

1 + κ2

N∑i=1

σi gi

))

= 2κ

1 + κ2

(N∑

i=1

σi

((ri

ri · r− r

)gi − r gi

))= 0.

Note that equation (7) and constraint (9) are dependentin the sense that the scalar product of (7) and r reduces to∑N

i=1 σi((1 − r2) gi − r · r gi) = 0, which becomes identityif (9) is satisfied. �

Differential equation (7) may be solved numerically bya gradient-based algorithm; however, in this case we are notguaranteed a globally optimal solution. Deriving an analyticalsolution to (7)–(9) with an arbitrary number of radars reducesto finding the second integral for equation (7) (the first oneis the constraint (9)), which still remains an open issue. Thenext theorem formulates necessary conditions for an optimaltrajectory in the case of a single radar.

THEOREM 2 (necessary conditions: single radar). In thecase of a single radar, located at the origin of the system ofcoordinates (0, 0, 0), the following statements hold:

(1) An optimal trajectory is a planar curve in 3-D space.The trajectory’s plane is determined by [rA × rB] ·

r = 0, i.e., the plane passes through the origin of thesystem of coordinates and the starting and finishingtrajectory’s points.

(2) Let (ρ, ψ)be polar coordinates introduced in the tra-jectory’s plane. Then vectorial differential equation(7) along with (8) and (9) reduces to a nonlinear first-order differential equation with respect to functionρ = ρ(ψ)

1

ρn−2√

κ2(ρ ′ψ)2 + ρ2

+ λLρ2√(ρ ′

ψ)2 + ρ2= C,

(11)

with boundary conditions

ρ(ψA) = ρA, ρ(ψB) = ρB , (12)

and the constraint on trajectory length

∫ ψB

ψA

√(ρ ′

ψ)2 + ρ2 dψ = l∗, (13)

where λL is an unknown non-negative constant, and(ρA, ψA) and (ρB , ψB) determine points A and B inpolar coordinates, respectively.

PROOF: Since an analytical solution to (7) is derived in thecase of a single radar, without loss of generality, we assumethat the radar is located at the origin of the system of coordi-nates, that is, (a1, b1, c1) = (0, 0, 0), r1 = r, and σ1 = 1. Inthis case, functions L(r, r), g(r, r, λL) and equation (7) takethe form

L(r, r) = 2κ

1 + κ2

√‖r‖2 + (κ2 − 1)(r · r)2

‖r‖n+1,

g (r, r, λL) = 1

‖r‖n−1√

‖r‖2 + (κ2 − 1) (r · r)2+ λL,

( rr · r

− r)

g − r g = 0.

Producing the vector product of the last equation with thevector r, we obtain

d

ds([r × r] g) = 0,

which is equivalent to having the first integral

[r × r] g = C, (14)

where C = (C1, C2, C3) is a constant vector. Since (r · [r ×r]) = 0 and g(r, r, λL) ≡ 0, the scalar product of (14) and requals zero

C · r = 0,



which is the equation of a plane passing through the originof the system of coordinates. This means that an optimaltrajectory is a planar curve in 3-D space, i.e., all its points liewithin the same plane. Since points A and B also belong tothis plane, vectors rA and rB must satisfy equation C · r = 0.Consequently, vector C is parallel to [rA×rB], and the explicitexpression for the trajectory’s plane is given by

[rA × rB] · r = 0, (15)

orhxx + hyy + hzz = 0,

where (hx , hy , hz) are the components of vector [rA × rB](hx , hy , hz) = (yAzB −yBzA, zAxB − zBxA, xAyB −xByA).

In the trajectory’s plane, we introduce a system of polarcoordinates (ρ, ψ), whose origin coincides with (0, 0, 0).Let ψ be a counterclockwise angle producing left-handedscrew with the vector [rA × rB] and counted from theupper side of the plane xy. We parameterize the trajec-tory by ρ and ψ . This means that each trajectory’s point(x(ρ, ψ), y(ρ, ψ), z(ρ, ψ)) should satisfy (15) identically.Let angles α and β be defined by

cos α = hx√h2

x + h2y

, cos β = hz√h2

x + h2y + h2

z

,

sin α = hy√h2

x + h2y

, sin β =√

h2x + h2

y√h2

x + h2y + h2

z

.

The coordinates (x, y, z) of a trajectory’s points are deter-mined by

x(ρ, ψ) = ρ (sin α cos ψ − cos α cos β sin ψ),

y(ρ, ψ) = −ρ (cos α cos ψ + sin α cos β sin ψ),

z(ρ, ψ) = ρ sin β sin ψ .

Based on these relations, we have

[r × r] = −ρ2ψ[rA × rB]

‖[rA × rB]‖ .

Consequently, (14) reduces to a scalar equation

ρ2ψ g = C, (16)

where C is an unknown constant scalar. Since ‖r‖2 = ρ2,function g(r, r, λL) is expressed by

g(ρ, ρ, λL) = 1

ρn√

1 + (κ2 − 1)ρ2+ λL. (17)

In coordinates (ρ, ψ), constraint (9) takes the form

ρ2 + ρ2ψ2 = 1. (18)

Thus, equations (16), (17), and (18) form a system of dif-ferential equations for finding functions ρ(s) and ψ(s) withboundary conditions ρ(0) = ρA, ψ(0) = ψA, ρ(l∗) = ρB ,ψ(l∗) = ψB , where (ρA, ψA) and (ρB , ψB) are given by

ρA = ‖rA‖, ψA = arccos

(xA sin α − yA cos α

‖rA‖)

,

ρB = ‖rB‖, ψB = arccos

(xB sin α − yB cos α

‖rB‖)

.

Let ρ ′ψ = dρ

dψ. Solving equations ρ = ρ ′

ψ ψ and (18) with

respect to ρ and ψ , we have

ρ = ± ρ ′ψ√

(ρ ′ψ)2 + ρ2

, ψ = ± 1√(ρ ′

ψ)2 + ρ2.

Substituting these formulas into (16) and (17) and combin-ing the last two, we obtain a nonlinear first-order differentialequation (11). Since variable s was excluded from (17), theconstraint on trajectory length is expressed by integral (13).

Note that it does not matter what sign we choose for ψ inψ = ± 1√

(ρ ′ψ )2+ρ2

, since we can always change the sign of the

constant C in the right-hand side of equation (14) and denoteit by a new constant. �

Although an analytical verification of whether a particularextremal trajectory minimizes the functional in the case ofa single detecting installation is cumbersome, the graph ofthis trajectory immediately reveals what kind of an extremalit is. Indeed, if the trajectory moves away from a detectinginstallation, then it minimizes the risk, while if the trajectorymoves toward the detecting installation, then it maximizesthe risk.

THEOREM 3 (analytical solution: single radar). An ana-lytical solution to first-order differential equation (11) withconditions (12), (13) is given by the quadrature

ψ(ρ) = ψA ±∫ ρ

ρA

dτ√(υ∗(τ , λL, C))2 − τ 2

, (19)

where υ∗(ρ, λL, C) is a positive root of the algebraic equation(quartic equation)

f (υ) = ρn−2(Cυ − λLρ2

)√κ2υ2 + (1 − κ2)ρ2 − υ = 0,

(20)



and unknown constants λL and C are found from theconditions

(1)∫ ψB

ψAυ∗(ρ, λL, C) dψ = l∗ and ψ(ρB) = ψB if the

length constraint is active,(2) λL = 0 and ψ(ρB) = ψB if the length constraint is

inactive.

PROOF: The main technique for solving a first-order dif-ferential equation analytically is to explicitly express thederivative of an unknown function. Introducing an auxiliaryfunction

υ =√

(ρ ′ψ)2 + ρ2, (21)

we reduce (11) to the algebraic equation (20) with respect toυ, which is a particular case of the quartic equation

ρ2(n−2)(Cυ − λLρ2)2(κ2υ2 + (1 − κ2)ρ2) − υ2 = 0.

Explicit analytical expressions for four roots of any quarticequation may be represented by Cardan–Ferrari’s formulas.Suppose that υ∗(ρ, λL, C) is a root for (20), then based on(21), derivative ρ ′

ψ is expressed by

ρ ′ψ = ±

√(υ∗(ρ, λL, C))2 − ρ2,

which reduces to a quadrature for ψ = ψ(ρ):

ψ(ρ) = ±∫

dρ√(υ∗(ρ, λL, C))2 − ρ2

+ D.

Boundary conditions ψ(ρA) = ψA and ψ(ρB) = ψB areused to exclude constant D, and the quadrature takes theform (19). �

Quadrature (19) is considered an analytical solution, sincethe roots of the quartic equation (20) may be expressed byCardan–Ferrari’s formulas analytically. The issue of whetherin the case of a single radar, the functional in (5) achievesa global minimum at an extremal, as determined by (19), isbeyond the scope of this paper. There are two special caseswhen quadrature (19) is simplified.

EXAMPLE 1 (the optimization problem with no con-straint on trajectory length). If the optimization problem doesnot have a constraint on trajectory length, then an optimaltrajectory is the Rhodenea (rose function)

ρ(ψ) = C− 1n−1 sin

1n−1

((n − 1)

κ(ψ + D)

), (22)

where D = κn−1 arcsin(Cρn−1

A ) − ψA.

DETAIL: If there is no constraint on trajectory length,then λL = 0. Consequently, the only feasible root for (20),satisfying υ∗ > 0, is given by

υ∗ =√

1 − (1 − κ2)C2ρ2(n−1)

Cκρn−2.

Substituting the last expression into (19), we obtain a function

ψ(ρ) = ψA ± κ

n − 1arcsin

(Cτn−1

)∣∣ρρA

,

whose inverse is (22).

In the case of n = 2, function (22) graphs a circle passingthrough the origin of the system of coordinates and points A

and B. Figure 3 illustrates function ρ(ψ) = sin1

n−1 ( (n−1)

κψ)

for parameters n = 4 and κ = 0.5, 1.0, 2.0. Note that ifn > 2, constant C in (22) can be determined only when|ψB −ψA| < min{π , πκ

n−1 }, otherwise, a solution to (11)–(12)without constraint (13) will be unbounded.

EXAMPLE 2 (the case of sphere). If the aircraft is modeledby a sphere, i.e., κ = 1, then an optimal trajectory is presentedby the explicit quadrature

ψ(ρ) = ψA ± C

∫ ρ

ρA

τ n−2dτ√(λLτn + 1)2 − C2τ 2(n−1)

. (23)

In the case of n = 2, quadrature (23) reduces to the ellipticsine [26].

Figure 3. Function ρ(ψ) = sin13( 3ψ

κ

). [Color figure can be

viewed in the online issue, which is available at www.interscience.wiley.com.]



Figure 4. Comparison of optimal trajectories in trajectories’ planefor the cases with a single sensor (n = 2) and single radar (n = 4)with the same constraint on the length, l∗ = 3.2. [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com.]

DETAIL: In the case of κ = 1, the root for (20) is given by

υ∗ = λLρn + 1

Cρn−2.

Substituting the last expression into (19), we obtain (23).

Figure 4 compares optimal trajectories with the same con-straint on trajectory length l∗ = 3.2 for a “spherical” aircraftin the cases of a single sensor (n = 2) and single radar(n = 4). As expected, an optimal trajectory is more sensitiveto a radar than to a passive sensor when in close proximity.

Figure 5a illustrates optimal planar trajectories for a spher-ical aircraft (κ = 1) for n = 4 with different constraintson trajectory length, l∗. In this example, the radar is locatedat (0, 0), and all trajectories have the same starting andending points: (xA, yA) = (−0.25, 0.25) and (xB , yB) =(1.75, 0.25), respectively. Figure 5c shows the same opti-mal trajectories for a spherical aircraft (κ = 1) for n = 4with the same constraints on trajectory length, l∗, in 3-Dspace with (xA, yA, zA) = (−0.25, 0.15, 0.2), (xB , yB , zB) =(1.75, 0.15, 0.2). The radar is located at (0, 0, 0). In the case ofa single sensor, n = 2, the analytical solution for an optimaltrajectory with a constraint on trajectory length is expressedby the elliptic sine, and similar figures can be found in [26].

The next theorem provides bounds for a real root υ∗ andconstant C for calculating (19) and (20).

THEOREM 4 (feasible root for the quartic equation). Let

λ−L = min

{n − 1

ρnA

,n − 1

ρnB

}

and λ+L = max

{n − 1

ρnA

,n − 1

ρnB

}.

If λL > 0, then

C ∈[0, n

(λL

n−1

)n−1n

], if λL ∈ [

λ−L , λ+

L

],[

0, min{λLρA + ρ

−(n−1)A , λLρB + ρ

−(n−1)B

}],

if λL /∈ [λ−

L , λ+L

], (24)

and equation (20) has a unique root υ∗ within interval[υmin, υmax], where

υmin = max

{ρ, C−1

(λLρ2 + ρ−(n−2) min

{1, κ−1

}),

ρ

κ

√max

{C−2ρ−2(n−1) + κ2 − 1, 0

}}, (25)

υmax = C−1 κ−1(κλLρ2 + ρ−(n−2)

+ Cρ√

max{1, κ2} − 1)

. (26)

PROOF: Bounds for C and a real root υ∗ are obtained byanalyzing equation (20) with respect to feasibility of υ∗ and

positiveness of λL. Using inequality υ =√

(ρ ′ψ)2 + ρ2 ≥ ρ,

or simply υ ≥ ρ, we reduce equation (20) to ρn−1(Cυ −λLρ2) − υ ≤ 0 or, equivalently, λLρn+1 ≥ (Cρn−1 − 1)υ.With υ ≥ ρ, the last inequality takes the form

C ≤ λLρ + ρ−(n−1). (27)

Since (27) holds for all feasible ρ, we have C ≤ minρ(λLρ +ρ−(n−1)). Expression λLρ+ρ−(n−1) is a convex function withrespect to ρ, achieving its global minimum at ρ0 = ( λL

n−1 )−1n .

Consequently, if min{ρA, ρB} ≤ ρ0 ≤ max{ρA, ρB}, thenρ0 is feasible and C ≤ λLρ0 + ρ

−(n−1)0 = n( λL

n−1 )n−1n . If

ρ0 /∈ [min{ρA, ρB}, max{ρA, ρB}], then we are not guaran-teed that in a particular example, function λLρ + ρ−(n−1)

will achieve its global minimum at ρ0, since ρ0 may not befeasible. However, in this case, at least the following weakestimate holds: C ≤ min{λLρA + ρ

−(n−1)A , λLρB + ρ

−(n−1)B }.

Based on inequalities υ ≥ 0 and λL ≥ 0, we show thatthe constant C is nonnegative. Indeed, rewriting (20) asρn−2(Cυ − λLρ2)

√κ2υ2 + (1 − κ2)ρ2 = υ ≥ 0, we obtain

Cυ ≥ λLρ2, and consequently, C ≥ 0. This finalizes theproof of formula (24).

Now we find interval [υmin, υmax] containing a single rootυ∗. Expressing λL from (20) and using condition λL ≥ 0, wederive a lower bound for υ∗. Namely, from λL = υ

ρ2

(C −

1

ρn−2√

κ2υ2+(1−κ2)ρ2

) ≥ 0, we obtain

υ ≥ ρ

κ

√max{C−2ρ−2(n−1) + κ2 − 1, 0}. (28)



Figure 5. Analytical trajectories for sphere (κ = 1.0) and elongated (κ = 0.1) and compressed (κ = 2.0) ellipsoids in the case of a singleradar. (a) 2-D, sphere (κ = 1.0), different length constraints; (b) 2-D, sphere (κ = 1.0), elongated (κ = 0.1), and compressed (κ = 2.0)ellipsoids, same length constraint, l∗ = 3.2; (c) 3-D, sphere (κ = 1.0), different length constraints; (d) 3-D, sphere (κ = 1.0), elongated(κ = 0.1), and compressed (κ = 2.0) ellipsoids, same length constraint, l∗ = 3.2. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

Then we use upper and lower estimates for√κ2υ2 + (1 − κ2)ρ2 depending on whether κ ≤ 1 or κ > 1.In the case of κ ≤ 1, we have κυ ≤√

κ2υ2 + (1 − κ2)ρ2 ≤υ. With this relation, equation (20) reduces to ρn−2κ(Cυ −λLρ2) ≤ 1 ≤ ρn−2(Cυ − λLρ2). Rearranging the lastinequalities, we obtain upper and lower bounds for υ whenκ ≤ 1

C−1(λLρ2 + ρ−(n−2)

) ≤ υ ≤ C−1(λLρ2 + κ−1ρ−(n−2)

).

(29)

Note that for κ = 1, (29) is an exact value for the root, i.e.,υ∗ = C−1(λLρ2 + ρ−(n−2)).

Similarly, in the case of κ > 1, we use κυ − ρ√

κ2 − 1 ≤√κ2υ2 + (1 − κ2)ρ2 ≤ κυ to reduce equation (20) to

ρn−2(Cυ − λLρ2)(κυ − ρ√

κ2 − 1) ≤ υ ≤ ρn−2κυ(Cυ −λLρ2), where the left inequality is simplified to υ ≥ρn−2υ(Cκυ − λLκρ2 − Cρ

√κ2 − 1). Consequently, we

obtain

C−1(λLρ2 + κ−1ρ−(n−2)

) ≤ υ

≤ C−1(λLρ2 + κ−1ρ−(n−2) + Cρ

√1 − κ−2

). (30)

Combining inequality υ ≥ ρ with (28), (29), and (30) forboth cases k ≤ 1 and κ > 1, we obtain (25) and (26).

Finally, to prove that equation (20) has a single root inthe interval [υmin, υmax], we show that the function f (υ) =ρn−2(Cυ − λLρ2)

√κ2υ2 + (1 − κ2)ρ2 − υ monotonically

increases on [υmin, υmax] and f (υmin) ≤ 0, f (υmax) ≥ 0.



Consider

d

dυf (υ) =

(Cρn−2

√κ2υ2 + (1 − κ2)ρ2 − 1

)

+ ρn−2(Cυ − λLρ2

)κ2υ√

κ2υ2 + (1 − κ2)ρ2.

For υ ≥ υmin, the first term in the right-hand side of thelast equality is always nonnegative because of (28), andthe numerator of the second term is always nonnegativebecause of (29) and (30). Consequently, d

dυf (υ) is positive on

[υmin, υmax]. This means that f (υ) is a monotonically increas-ing function. Since υmin is the maximum of three expressionsin (25), we check the sign of f (υ) for each expression.

The relation f (ρ) = ρn(C − λLρ − ρ−(n−1)) ≤ 0 holdsby virtue of (27).

For υmin = ρ

κ

√max{C−2ρ−2(n−1) + κ2 − 1, 0} we con-

sider two cases. If C−2ρ−2(n−1) + κ2 − 1 ≤ 0(when κ < 1) then υmin = 0 and f (υmin) =−λLρn+1

√1 − κ2 < 0. If C−2ρ−2(n−1) + κ2 − 1 > 0

then υmin = ρ

κ

√C−2ρ−2(n−1) + κ2 − 1 and f (υmin) =

C−1(Cυmin − λLρ2) − υmin = −C−1λLρ2 < 0.For υmin = C−1(λLρ2 + ρ−(n−2) min{1, κ−1}), based

on√

κ2υ2 + (1 − κ2)ρ2 ≤ υ min{1, κ}, f (υmin) ≤υmin(ρ

n−2(Cυmin − λLρ2) min{1, κ}−1) = υmin(min{1, κ−1}× min{1, κ} − 1) = 0.

Thus, we have shown that f (υmin) ≤ 0 for υmin =max{ρ, υmin, υmin}.

In the case of υmax, determined by (25), we use√κ2υ2 + (1 − κ2)ρ2 ≥ κυ − ρ

√max{1, κ2} − 1 to show

that

f (υmax) ≥ ρn−2(Cυmax − λLρ2)

× (κυmax − ρ

√max{1, κ2} − 1

) − υmax

≥ ρn−2υmax(C κ υmax − Cρ

√max{1, κ2} − 1

− κλL ρ2) − υmax = 0.

Consequently, we have proved that f (υmin) ≤ 0 andf (υmax) ≥ 0, which, along with the condition of f (υ)’smonotonicity on [υmin, υmax], guarantee existence of only asingle root for f (υ) on [υmin, υmax]. �

EXAMPLE 3 (elongated and compressed ellipsoids inthe case of n = 4). Let points A and B have coordi-nates (xA, yA) = (−0.25, 0.25), (xB , yB) = (1.75, 0.25) inthe plane of the trajectory; and coordinates (xA, yA, zA) =(−0.25, 0.15, 0.2), (xB , yB , zB) = (1.75, 0.15, 0.2) in 3-Dspace. The radar is located at (0, 0, 0). Figures 5b and 5d com-pare optimal trajectories for a sphere, elongated (κ = 0.1) andcompressed (κ = 2.0) ellipsoids graphed in both the plane ofthe trajectory and 3-D space, respectively. Table 1 presents

Table 1. Minimal risk, λL, C, and C’s estimate (24) in Examples2 and 3.

κ l∗ Risk λL C 4(λL/3)3/4

1.0 2.6 9.7921 4.7636 5.3693 5.65811.0 3.2 8.4217 1.0401 1.7597 1.80731.0 4.0 7.9662 0.3007 0.7121 0.7126

0.1 3.2 0.4684 1.4517 2.2828 2.32070.5 3.2 3.9807 1.9934 2.9085 2.94392.0 3.2 12.464 0.6104 1.1431 1.2117

10.0 3.2 14.845 0.1091 0.2515 0.3331

values for the optimal risk, λL, C, and C’s estimate (24) asfunctions of κ and l∗ for Examples 2 and 3.

Analyzing optimal trajectories in Figures 5a–d and numer-ical results in Table 1, we conclude that

• the optimal risk is more sensitive to variation ofthe shape of the ellipsoid (parameter κ), than to thevariation of a trajectory’s total length, l∗;

• optimal trajectories for different κ (especially for κ >

1) are quite similar, which indicates that variation ofellipsoid shape has no strong effect on the geometryof an optimal trajectory;

• when close to an installation, an optimal trajectory ismore sensitive to a radar than to a sensor.

4. NETWORK OPTIMIZATION APPROACH

The calculus of variations approach reduces optimizationproblem (5) to a vectorial nonlinear differential equation (7)subject to (8) and (9). Obtaining an analytical solution tothis equation in the case of more than one detecting installa-tion is still an open issue. Various gradient-based techniquesare available for solving the equation numerically. However,regardless of the efficiency of those techniques, they provideonly locally optimal solutions.

As a result, we turn to a network optimization approach forfinding a globally optimal solution. This approach reduces thetrajectory length constrained problem to the CSPP for a 3-Dnetwork. The CSPP can be efficiently solved by network opti-mization algorithms. There are several advantages of usingnetwork optimization: among all feasible approximated tra-jectories in a considered network, it finds a globally optimalone; its complexity (running time in the worst case) dependsneither on the number of installations in a network nor onpower n in the risk functional (2); it can readily be appliedfor the case with an experimentally derived RCS (i.e., whenRCS is a nonsmooth function). However, because the CSPPis NP-complete, no polynomial algorithm solves the CSPPexactly. This means that in the worst case, the running timeof an algorithm solving the CSPP will exponentially dependon the number of arcs in a network.



We assume an admissible deviation domain for the aircrafttrajectory to be an undirected network G = (N , A), whereN = {1, . . . , n} is the set of n nodes, and A is the set ofundirected arcs. A trajectory (x(·), y(·), z(·)) is approximatedby a path P in the network G, where path P is defined as asequence of nodes 〈j0, j1, . . . , jp〉 such that j0 = A, jp = B

and 〈jk−1, jk〉 ∈ A for all k from 1 to p.Several schemes for approximation of optimization prob-

lem (5) are available. We consider one of them. Let vectorrjk

with components x(jk), y(jk) and z(jk) determine posi-tion of node jk . Then a path P = {rj1 , rj2 , . . . , rjp

} is apiece-wise linear curve with vertices at points rjk

, k = 1, p.Any point on the arc 〈jk−1, jk〉 can be determined by vectorrk(t) = (1 − t) rjk−1 + t rjk

with t ∈ [0, 1]. Thus, lengthdifferential ds and derivative r for each arc are

ds = ‖rjk− rjk−1‖ dt , rk = rjk

− rjk−1

‖rjk− rjk−1‖

, k = 1, p.

Approximating r, R and ds, we represent functional (4)and trajectory length by

F(r, r) ≈p∑

k=1

‖rjk− rjk−1‖

∫ 1

0L(rk(t), rk)dt

=p∑

k=1

C(rjk−1 , rjk), (31)

l ≈p∑

k=1

‖rjk− rjk−1‖, (32)

where ‖rjk− rjk−1‖ and C(rjk−1 , rjk

) are the length and riskindex of the arc 〈jk−1, jk〉, respectively. To derive the formulafor C(rjk−1 , rjk

) we compute the risk accumulated along thearc 〈jk−1, jk〉 from the ith radar located at qi = (ai , bi , ci).Substituting rik(t) = rk(t) − qi into (2), we have

C(rjk−1 , rjk) = 2κ

1 + κ2

N∑i=1

σi

∫ 1

0

√‖rik(t)‖2 ‖rjk

− rjk−1‖2 + (κ2 − 1)(rik(t) · (rjk

− rjk−1))2

‖rik(t)‖n+1dt

= 2κ

1 + κ2

N∑i=1

σi

∫ 1

0

√l2i,jk−1

sin2 ϕi,jk−1jk+ κ2

(li,jk−1 cos ϕi,jk−1jk

+ t �sjk−1jk

)2

(l2i,jk−1

sin2 ϕi,jk−1jk+ (

li,jk−1 cos ϕi,jk−1jk+ t �sjk−1jk

)2)n+1

2

dt , (33)

where

li,jk= ‖rjk

− qi‖, �sjk−1 jk= ‖rjk

− rjk−1‖,

and ϕi, jk−1jk∈ [0, π ] is the angle between vectors rjk−1 − qi

and rjk− rjk−1 (see Figure 6), i.e.,

ϕi, jk−1jk= arccos

((rjk−1 − qi ) · (rjk

− rjk−1)

‖rjk−1 − qi‖ ‖rjk− rjk−1‖

).

Figure 6 illustrates a 3-D network for solving the risk mini-mization problem. The seven-segment trajectory AB is a pathin the vicinity of the ith radar, while �sjk−1jk

is the length ofarc 〈jk−1, jk〉 between nodes jk−1 and jk in this path. Mag-nitude ϕi,jk−1jk

is the angle between vector rjk−1 − qi and arc〈jk−1, jk〉 directed from node jk−1 to node jk .

Integral (33) can efficiently be approximated by a Gaussianquadrature. If f (t) is a bounded smooth function on [0, 1]

then ∫ 1

0f (t) dt ≈

J∑j=1

hj f (tj ),

where hj and tj ∈ [0, 1] are known for any given J . We usedthe Gaussian quadrature with J = 16.

Figure 6. 3-D network for solving the risk minimization prob-lem. The piecewise linear trajectory AB is a path of the aircraft.[Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com.]



Consequently, through techniques of Calculus of Varia-tions, problem (5) is approximated by

minP

p∑k=1

C(rjk−1 , rjk)

s.t.p∑

k=1

‖rjk− rjk−1‖ ≤ l∗,

rj0 = rA, rjp= rB . (34)

To formulate (34) as a network optimization problem, let

cjk−1jk= C(rjk−1 , rjk

), �sjk−1jk= ‖rjk

− rjk−1‖,

and let R(P) and l(P) define the total cost (risk) and weight(length) accumulated along the path P , respectively,

R(P) =p∑

k=1

cjk−1jk, l(P) =

p∑k=1

�sjk−1jk.

Then each arc 〈jk−1, jk〉 ∈ A is assigned length �sjk−1jkand

nonnegative cost cjk−1jk. The path P is weight feasible if it

satisfies l(P) ≤ l∗. Consequently, the CSPP is to find such afeasible path P from point A to point B that minimizes costR(P), i.e.,

minP

p∑k=1

cjk−1jk

s.t.p∑

k=1

�sjk−1jk≤ l∗. (35)

The CSPP in the form of (35) is also referred to as theweight constrained shortest path problem (WCSPP) [6, 7],which under the assumption of cost and weight integrality, isknown to be NP-complete. The WCSPP is a special case ofthe Resource Constrained Shortest Path Problem (RCSPP),which uses a vector of weights, or resources, rather than scalarweight. For the relation of the WCSPP to other similar net-work optimization problems, see [7]. Algorithms for solvingthe CSPP are divided into three major categories: LSAs basedon dynamic programming methods; scaling algorithms; andalgorithms based on the Lagrangean relaxation. A LSA is themost efficient algorithm when the weights are positive [8].A Lagrangean relaxation algorithm is based on subgradientoptimization [2] and cutting plane [13] methods and is effi-cient for solving the Lagrangean dual for the CSPP in the caseof a single resource. Scaling algorithms use two fully poly-nomial approximation schemes for the CSPP based on costscaling and rounding [14]. The first scheme is a geometricbisection search whereas the second one iteratively extendspaths. We solve the CSPP (35) by the LSA with a preprocess-ing procedure [7] and path smoothing. The pseudo-codes for

Figure 7. Path smoothing.

the LSA and the preprocessing procedure for the RCSPP, i.e.,for the case of several resources, can be found in [7].

Path smoothing means that in a path, we choose only thosearcs which produce the angle with a preceding arc not greaterthan α (see Figure 7). This constraint can be imposed formodeling turning rate limits of a real aircraft. Moreover,path smoothing significantly reduces the number of treatedlabels in the LSA. Given two arcs 〈jk−1, jk〉 and 〈jk , jk+1〉in a network, the path smoothing is expressed by inequalityejk−1jk

· ejkjk+1 ≥ cos α, where eij is the unit vector alongarc 〈i, j〉.

5. NUMERICAL EXPERIMENTS

Let network G be a 3-D grid of nodes of size nx ×ny ×nz

with edges oriented along coordinate axes x, y, z and havingnx , ny , and nz segments in each edge, respectively. To avoid“naive discretization,” sometimes referred to as the digitiza-tion bias [22], we assign not only axis and diagonal but alsoso-called “long-diagonal” arcs connecting opposite vertexesof any two neighbor cubes. The structure of arcs assigned inthe 3-D network is shown in Figure 8. The total number ofnodes and arcs in the undirected G with this structure are

|N | = (nx + 1)(ny + 1)(nz + 1),

|A| = 2(49 nxnynz − 8 nxny − 8 nxnz

− 8 nynz + nx + ny + nz) ∼ O(|N |3),for nx ≥ 1, ny ≥ 1, and nz ≥ 1. Similarly, in the 2-Dcase, the network G is a 2-D grid of nodes of size nx × ny

with edges oriented along coordinate axes x, y and havingnx and ny segments in each edge, respectively. The structureof arcs assigned in 2-D network is shown in Figure 9. The2-D network is a special case of 3-D one with nz = 0 andcontains (nx + 1)(ny + 1) nodes and 2(8 nxny − nx − ny)

arcs, nx ≥ 1 and ny ≥ 1.Nodes of the 3-D network are associated with integer vec-

tors (i, j , k) forming a 3-D integer grid. The set of arc weights(lengths) in the grid with the structure of arcs as shownin Figure 8 is {1,

√2,

√3,

√5,

√6, 3}. In numerical exper-

iments, arc weights were approximated by integers {1000,1414, 1732, 2236, 2449, 3000}. In assigning real arc length,



Figure 8. Structure of arcs in every node in a 3-D network. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com.]

all these values are scaled by an appropriate coefficientdepending on actual size of a network. Finding a constrainedshortest path in a network with the integer lengths of arcs isapproximately 1.5 times faster than that with real lengths ofarcs. However, because of the integer approximation of arcweights, the actual weight of a constrained shortest path inthe network may be slightly greater than an assigned weightconstraint, while the corresponding optimal risk value may belesser than that obtained by the analytical solutions approach.Numerical experiments were conducted on four sets of data.Set 1 is a 2-D network with a single radar; Set 2 is a 3-Dnetwork with a single radar; Set 3 is a 3-D network with tworadars; and Set 4 is a 3-D network with three radars (seeTable 2). In all data sets, the risk of detection (33) is calcu-lated only for radars, i.e., for n = 4. In the 3-D case, the CSPPis solved by the LSA with and without path smoothing. In allnumerical experiments with the path smoothing, α = π

6 . APC with Xeon 3.08 GHz and 3.37 Gb of RAM was used.

We calculate constrained shortest paths for different ellip-soid shapes (parameter κ) and length constraints, l∗, andcompare optimal values of the risk with those obtained ana-lytically. The following parameters are analyzed: number ofnodes left after preprocessing, Nprep; cost upper bound in pre-processing, U ; preprocessing time, Tprep, in seconds; numberof labels treated in the LSA, Nlabels; and running time of theLSA, TLSA, in seconds. In the 3-D case, we analyze the impact

of path smoothing on LSA running time. Table 3 presentsnumerical results of 2-D and 3-D network optimization fordifferent values of parameter κ , length constraints l∗, in thecases of one, two, and three radars.

Figures 10a–d compare analytical and discrete optimiza-tion trajectories with the following parameters:

(a) 2-D space; n = 4; κ = 1.0; l∗ = 2.6, 3.2, 4.0.(b) 2-D space; n = 4; κ = 0.1, 1.0, 2.0; l∗ = 3.2.

Figure 9. Structure of arcs in every node in a 2-D network:“1,” axis arcs, “2,” diagonal arcs, “3,” long diagonal arcs.



Table 2. Data sets for numerical experiments.

Graph parameters Set 1, 2-D Set 2, 3-D Set 3, 3-D Set 4, 3-D

Size of the network 2.3 × 2.3 2.3 × 1.0 × 1.25 2.3 × 1.0 × 1.25 2.3 × 1.0 × 1.25nx × ny 46 × 46 46 × 20 × 25 46 × 20 × 25 46 × 20 × 25Left-bottom corner (−0.55, 0) (−0.55, 0, 0) (−0.3, 0, 0) (−0.1, 0, 0)Length of axis arcs 0.05 0.05 0.05 0.05Number of nodes 2,209 25,662 25,662 25,662Number of arcs 33,672 2,213,062 2,213,062 2,213,062

Point A (−0.25, 0.25) (−0.25, 0.15, 0.2) (0, 0.5, 0) (0, 0.75, 0)Point B (1.75, 0.25) (1.75, 0.15, 0.2) (2, 0.5, 0) (2, 0.75, 0)Radar 1 (0, 0) (0, 0, 0) (1, 0, 0) (1, 0, 0)Radar 2 — — (0.5, 1, 0) (0.5, 1.25, 0)Radar 3 — — — (1.5, 1, 0)

(c) 3-D space; n = 4; κ = 1.0; l∗ = 2.6, 3.2, 4.0.(d) 3-D space; n = 4; κ = 0.1, 1.0, 2.0; l∗ = 3.2.

The smooth curves are the optimal trajectories obtained bythe analytical approach and the piece-wise linear curves arethose obtained by solving the CSPP. The analytical and cor-responding discretely optimized trajectories are very close toone anther, clearly validing both approaches.

Figures 11a–d illustrate discrete optimization trajectoriesin 3-D space with two and three radars for the followingparameters: n = 4; l∗ = 3.2; κ = 0.1, 1.0, 2.0. Of course,there is no analytical result for these cases to compare with,but the trajectories are similar in appearance to those obtainedfor a single radar and clearly exhibit risk avoidance.

6. ANALYSIS OF NUMERICAL RESULTS

Analyzing the results presented in Table 3 and Figures10a–d and 11a–d, we make the following conclusions:

• Optimal trajectories obtained by analytical and dis-crete optimization approaches are very close whichvalidates both approaches.

• For all tested values κ and length constraints l∗, pathsmoothing substantially reduces running time, seeFigure 12a. The difference in risk values for optimalpaths obtained with and without smoothing is negli-gibly small. Moreover, in most cases correspondingoptimal paths coincide.

• LSA running time, TLSA, is very sensitive to the shapeof ellipsoid, for instance, in the case of a single radarand same length constraint, l∗ = 3.2, running timefor κ = 10 is about 1000 times greater than that forκ = 0.1.

• In the case of a single radar, the optimal risk increaseswith parameter κ; however, in examples with two andthree radars, the optimal risk for a spherical RCSis greater than that for elongated and compressed

ellipsoids. Thus, in the case of several radars, asphere, being uniformly exposed to all radars, mayaccumulate the greatest total risk along a trajectory.

• TLSA strongly depends on a path length constraint, l∗.• There is no strong correlation between TLSA and the

number of radars. Depending on ellipsoid shape itmay decrease (κ = 1.0), increase (κ = 2.0), or vary(κ = 1.0).

• Running time of the preprocessing procedure is about2% of the total computational time and indicates nopredictive power for TLSA.

• In examples with several radars, optimal trajectorieswith the same constraint on the length but differentellipsoid shapes are relatively close to each other,which suggests that in general, ellipsoid shape has nostrong effect on the geometry of an optimal trajectory.

Figure 12a shows TLSA as function of κ for the 3-D networkwith a single radar with and without the network smooth-ing. In the case of very elongated ellipsoids (ellipsoid withκ = 0.1 is almost a needle), the excessive TLSA can beexplained by lowest risk accumulations in directions radialto a radar, which are the ones producing greatest total lengthsfrom point A to point B. This complicates comparison oflabels in risk minimization, while balancing length constraint.This idea is supported by the fact that in generating an optimaltrajectory for a compressed ellipsoid with κ = 10.0, TLSA ismerely several seconds. In this case, because of compressedgeometry (for instance, a disk flying along its axis of sym-metry), the risk of detection accumulates slower in directionstransverse to a radar and those direction are the ones produc-ing lowest total lengths from point A to point B. It is worthmentioning that for small values of κ , the path smoothingreduces TLSA more efficiently.

Figure 12c illustrates TLSA as function of treated labels,Nlabels, for a 3-D network with a single radar and various κ andl∗ with and without smoothing. TLSA almost linearly dependson Nlabels. Figure 12b shows strong correlation between



Tabl

e3.

Res

ults

ofnu

mer

ical

expe

rim

ents

inso

lvin

gth

eC

SPP.

Para

met

ers

Prep

roce

ssin

gpr

oced

ure

Lab

el-s

ettin

gal

gori

thm

(LSA

)L

SAw

ithpa

thsm

ooth

ing

Set

κl ∗

Ris

kU

Npr

Tpr

Rl ∗

−l

NL

TL

SAR

l ∗−

lN

LT

LSA

11.

02.

69.

7921

91.3

483

70.

359

9.98

380.

0080

85,4

610.

611

1.0

3.2

8.42

1791

.34

1260

0.31

38.

5042

0.00

0522

4,52

42.

051

1.0

4.0

7.96

6235

.69

1703

0.50

08.

0045

0.00

6742

3,05

64.

75

10.

13.

20.

4684

15.9

412

810.

484

0.48

820.

0010

329,

517

3.64

10.

53.

23.

9807

66.8

412

650.

329

4.06

380.

0068

292,

594

2.92

12.

03.

212

.464

12.8

676

40.

547

12.5

190.

0042

60,6

190.

41

21.

02.

69.

7921

91.3

411

518

9.51

610

.251

0.00

021,

936,

613

766

10.2

510.

0002

1,08

0,09

129

92

1.0

3.2

8.42

1791

.34

2252

114

.234

8.52

520.

0020

6,64

4,60

865

988.

5263

0.00

364,

841,

651

4287

21.

04.

07.

9662

8.90

221

175

18.0

318.

0410

0.00

038,

066,

613

7253

8.04

150.

0019

6,84

9,79

959

37

20.

13.

20.

4684

15.9

422

598

16.0

940.

5547

0.00

139,

930,

869

1152

00.

5566

0.00

085,

262,

314

5066

20.

53.

23.

9807

66.8

422

553

14.8

754.

1109

0.00

378,

427,

875

9069

4.12

320.

0014

5,09

1,68

147

152

2.0

3.2

12.4

6412

.97

1554

322

.187

12.5

300.

0011

2,52

9,26

612

2012

.530

0.00

022,

015,

458

846

30.

13.

2—

3.08

620

342

16.5

470.

9219

0.00

129,

993,

019

9838

0.92

300.

0007

6,42

8,63

555

573

1.0

3.2

—15

.37

1926

712

.766

4.89

110.

0007

6,73

7,16

653

924.

8914

0.00

075,

095,

577

3708

32.

03.

2—

7.53

615

968

16.8

284.

3208

0.00

073,

617,

699

2011

4.33

000.

0022

2,81

8,01

213

35

40.

13.

2—

19.3

720

766

14.7

181.

6353

0.00

017,

898,

013

6733

1.63

980.

0001

5,29

6,12

340

564

1.0

3.2

—28

.74

1966

016

.641

9.07

310.

0010

5,62

3,46

039

999.

0943

0.00

304,

413,

319

2916

42.

03.

2—

22.2

618

377

15.6

258.

6230

0.00

103,

982,

280

2286

8.62

300.

0009

3,15

5,04

416

34

Not

e.“S

et”

isth

enu

mbe

rof

data

set;

κ=

b/a

isth

era

tioof

ellip

soid

axes

;l∗

isth

eco

nstr

aint

onpa

thle

ngth

;“R

isk”

isop

timal

valu

eof

the

risk

obta

ined

anal

ytic

ally

;Npr

isnu

mbe

rof

node

sle

ftaf

ter

the

prep

roce

ssin

g;T

pr

isru

nnin

gtim

eof

the

prep

roce

ssin

gin

seco

nds;

Ris

the

risk

ofan

optim

alpa

th;l

∗−l

isth

edi

ffer

ence

betw

een

the

cons

trai

nton

path

leng

than

dth

ele

ngth

ofan

optim

alpa

th;N

Lis

the

num

ber

ofla

bels

trea

ted

inth

eL

SA;T

LSA

isru

nnin

gtim

eof

the

LSA

inse

cond

s.



Figure 10. Comparison of analytical (smooth) and discrete optimization (piece-wise linear) trajectories for sphere (κ = 1.0) and elongated(κ = 0.1) and compressed (κ = 2.0) ellipsoids in the case of a single radar. (a) 2-D, sphere (κ = 1.0), different length constraints; (b) 2-D,elongated (κ = 0.1) and compressed (κ = 2.0) ellipsoids, same length constraints, l∗ = 3.2; (c) 3-D, sphere (κ = 1.0), different lengthconstraints; (d) 3-D, elongated ellipsoid (κ = 0.1), length constraint l∗ = 3.2. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

Nlabels and the number of labels left after the preprocessing,Npr , plotted for all κ and l∗ in the case of a single radar.Based on results presented in Table 3, cost upper boundsare not close enough to optimal values of the risk obtainedanalytically. Obviously, the closer the upper bound, U , is tothe optimal cost, the fewer number of labels, Nlabels, will betreated. This fact suggests developing preprocessing proce-dures for obtaining more accurate cost upper bounds. Suchprocedures may be based on Lagrange relaxation [7, 13, 14].

7. CONCLUSIONS

We have developed a three-dimensional deterministicmodel for routing an aircraft with a variable radar cross-section in a threat environment. The threat is associated withthe risk of detection by active radars or passive sensors. To

investigate dependence of the risk of detection on variableRCS, we have modeled the aircraft by a symmetrical ellipsoidwith the axis of symmetry aligned with the aircraft trajec-tory. The model considers that the risk of detection is thesum of risks from all installations in the area of interest,where the risk to be detected by a particular installation isproportional to the area of ellipsoid projection and recipro-cal to the nth-power of the distance between the aircraft andthis particular installation. We have developed analytical anddiscrete optimization approaches for solving the risk mini-mization problem with variable RCS and arbitrary numberof detecting installations subject to a constraint on trajectorylength.

The analytical approach, based on calculus of variations,has reduced the optimization problem to solving a vecto-rial nonlinear differential equation. In the case of a single



Figure 11. Optimal trajectories in the case of two and three radars for compressed ellipsoid (κ = 2.0), sphere (κ = 1.0) and elongatedellipsoid (κ = 0.1) with the same length constraint, l∗ = 3.2. (a) 2 radars; (b) 2 radars, front view; (c) 3 radars; (d) 3 radars, side view. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

installation and arbitrary ellipsoid shape, an analytical solu-tion to the differential equation has been expressed by aquadrature. From numerical experiments based on analyticalsolutions, we have concluded that

• the optimal risk is more sensitive to the variation ofellipsoid shape than to the variation of trajectory totallength;

• optimal trajectories for different κ (especially forκ > 1) are close to each other, which indicates thata variation of ellipsoid shape has no strong effect onthe geometry of an optimal trajectory;

• when close to an installation, an optimal trajectoryis more sensitive to a radar than to a sensor, andin regions remote from the installation the effect isopposite.

The network optimization approach has reduced the riskminimization problem to the CSPP. To solve the CSPP, wehave suggested the LSA with a preprocessing procedure and

path smoothing. The path smoothing has been imposed asa necessary constraint for modeling turning rate limits ofan aircraft. We have tested the algorithm for various ellip-soid shapes, various constraints on trajectory length andwith one, two and three radars. Based on results of discreteoptimization, we have made the following conclusions:

• In the case of a single radar, all optimal trajectoriesobtained by the discrete approach for various κ and l∗are sufficiently close to the corresponding analyticaltrajectories;

• The path smoothing significantly reduces LSA run-ning time, while preserving accuracy of optimaltrajectories;

• LSA running time is extremely sensitive to the shapeof the ellipsoid;

• In all examples, ellipsoid shape has no strong effecton the geometry of an optimal trajectory;

• Algorithm running time strongly depends on thetrajectory length constraint.



Figure 12. (a) Dependence of LSA running time on the shape of ellipsoid, κ (3-D network, single radar): curve “1,” no smoothing, curve“2,” smoothing is used; (b) number of labels treated versus number of nodes left after preprocessing (3-D network, single radar): curves“1” and “2” correspond to LSA with and without path smoothing, respectively; (c) LSA running time versus number of labels treated: 3-Dnetwork, single radar. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

APPENDIX

Minimization of a Functional with NonholonomicConstraint and Movable End Point:

Necessary Conditions

In this section, we derive necessary conditions for minimization of a func-tional with a nonholonomic constraint and a moveable end point. Theseconditions are expressed in the form of a vectorial nonlinear differentialequation, which plays a central role in solving optimization problem (5) inthe case of a single detecting installation. Consider the following problemof Calculus of Variations

minr

�(r, r, l), (36)

�(r, r, l) =∫ l

0L(r(s), r(s))ds, (37)

r(0) = r1, r(l) = r2, (38)

φ(r(s)) = 0, (39)

l ≤ l∗, (40)

where r(s) = (x(s), y(s), z(s)) and R(s) = (x(s), y(s), z(s)).

A functional has an extremum at extremal r∗, if its total variation at r∗equals zero. Note that variations δx, δy, and δz are dependant by virtue ofnonholonomic constraint (39). Moreover, minimization of functional (37)subject to constraint (40) is the problem with the movable end point, r(l).This means that variation of the curve length, δl, is not zero.

We apply the Lagrange multiplier method to problem (36)–(39). Con-straint (40) will be discussed later. Let �(L, φ, λ, l) = ∫ l

0 (L(r, r) +λ(s)φ(r))ds be a Lagrangian for (36)–(39). By definition, the variation ofthe Lagrangian is expressed by

δ� =∫ l

0(δL(r, r) + λ δφ(r) + φδλ)ds + (L + λ φ)|s=l δl

=∫ l

0

(∂L

∂r· δr + ∂L

∂ r· δr + λ

∂φ

∂ r· δr + φδλ

)ds + (L + λφ)|s=l δl

=∫ l

0

[(∂L

∂r− d

ds

∂L

∂ r− d

ds

(λ

∂φ

∂ r

))· δr + φδλ

]ds

+[(

∂L

∂ r+ λ

∂φ

∂ r

)· δr

]∣∣∣∣s=l

+ (L + λφ)|s=l δl,



where δr = (δx, δy, δz), δr = (δx, δy, δz), ∂L∂r = (

∂L∂x

, ∂L∂y

, ∂L∂z

)and ∂L

∂ r =(∂L∂x

, ∂L∂y

, ∂L∂z

). Note that δr|s=l = 0, since l is varied, and s = l is no longer

a boundary point. Based on boundary conditions (38), the variation δr at thestarting and finishing points s = 0 and s = l + δl, respectively, should bezero, i.e., δr(0) = 0 and δr(l+δl) = 0. The last condition is used to calculatethe variation δr at s = l. Namely, from δr(l + δl) ≡ δr(l) + rδl = 0, weobtain δr(l) = −rδl. With the last equality, the variation δ� takes the form

δ� =∫ l

0

[(∂L

∂r− d

ds

∂L

∂ r− d

ds

(λ

∂φ

∂ r

))· δr + φδλ

]ds

+[L + λ φ −

(∂L

∂ r+ λ

∂φ

∂ r

)· r]∣∣∣∣

s=l

δl.

Since constraint (39) is relaxed, all three variations δx, δy and δz are indepen-dent. Consequently, the necessary conditions for an extremum, i.e., δ� = 0,reduce to constraint (39) and equations

∂L

∂r− d

ds

∂L

∂ r− d

ds

(λ

∂φ

∂ r

)= 0, (41)

and [L −

(∂L

∂ r+ λ

∂φ

∂ r

)· r]∣∣∣∣

s=l

= 0. (42)

Vectorial equation (41) has the first integral. Indeed, consider the scalarproduct of (41) and r

(∂L

∂r− d

ds

∂L

∂ r− d

ds

(λ

∂φ

∂ r

))· r = 0.

Note that the left-hand side of this equality is a total differential. Thus,integrating the last expression, we obtain

L − r · ∂L

∂ r− λ

(r · ∂φ

∂ r

)= C, (43)

where C is an arbitrary constant.Based on (43), the Lagrange multiplier takes the form

λ(s) =(

L − r · ∂L

∂ r+ cL

)/(r · ∂φ

∂ r

), (44)

where cL = −C. Substituting (44) into (41), we obtain the followingvectorial differential equation

∂L

∂r− d

ds

(∂L

∂ r+

∂φ∂ r

r · ∂φ∂ r

(L − r · ∂L

∂ r+ cL

))= 0. (45)

Equation (45) along with the nonholonomic constraint (39) and boundaryconditions (38) is the necessary condition for an extremum. Note that equa-tions (45) and (39) are dependent in the sense that the scalar product of (45)and r reduces to r · ∂φ

∂ r = 0, which is the differential for (39).In the case when constraint l ≤ l∗ is active, i.e., l = l∗, the length of a

trajectory is fixed, and therefore, the variation δl should equal zero by defini-tion. Consequently, in this case, equation (42) is excluded from determiningan optimal trajectory. If constraint l ≤ l∗ is inactive, then from (42) and (43)we obtain cL = 0.

An extremal r∗ solving (45), (38), and (39) either minimizes or maxi-mizes functional (37). Sufficient conditions for r∗ to minimize the functionalrequire the second variation of �(L, φ, λ∗, l) at r∗ to be greater than zero.Consideration of this issue is beyond the scope of this paper.

ACKNOWLEDGMENTS

We are grateful to the anonymous referees for their valuablecomments and suggestions, which greatly helped to improvethe quality of the paper.

REFERENCES

[1] H. Bateman and A. Erdelyi, Higher transcendental functions.Vol. 3, McGraw–Hill, 1955.

[2] J.E. Beasley and N. Christofides, An algorithm for the resourceconstrained shortest path problem, Networks 19 (1989),379–394.

[3] R.H. Bishop, D.L. Mackison, R.D. Culp, and M.J. Evans,Spaceflight Mechanics 1999. Univelt, Inc., Collection:Advances in the Astronautical Sciences, Vol. 102, Part I,1999.

[4] Y.K. Chan and M. Foddy, Real Time Optimal Flight Path Gen-eration by Storage of Massive Data Bases. Proceedings of theIEEE NEACON 1985, Institute of Electrical and ElectronicsEngineers, New York, 1985, 516–521.

[5] A. Chaudhry, K. Misovec, and R. d’Andrea, Low ObservabilityPath Planning for an Unmanned Air Vehicle Using Mixed Inte-ger Linear Programming. 43rd IEEE Conference on Decisionand Control, Paradise Island, Bahamas, 2004.

[6] I. Dumitrescu and N. Boland, Algorithms for the weightconstrained shortest path problem, ITOR 8 (2001), 15–29.

[7] I. Dumitrescu and N. Boland, Improved preprocessing, label-ing and scaling algorithms for the weight-constrained shortestpath problem, Networks 42(3) (2003), 135–153.

[8] M. Desrochers and F. Soumis, A generalized permanent label-ing algorithm for the shortest path problem with time windows,INFOR 26 (1988), 191–212.

[9] J.N. Eagle and J.R. Yee, An optimal branch-and-bound proce-dure for the constrained path, moving target search problem,Oper Res 38(1) (1990), 110–114.

[10] Electronic Warfare and Radar Systems Engineering Hand-book, Naval Air Warfare Center Weapons Division (https://ewhdbks.mugu.navy.mil).

[11] L.E. Elsgolc, Calculus of Variations. Franklin Book Company,Inc., 1961.

[12] I.M. Gelfand, R.A. Silverman, and S.V. Fomin, Calculus ofVariations. Dover, 2000.

[13] G.Y. Handler and I. Zang, A dual algorithm for the constrainedshortest path problem, Networks 10 (1980), 293–309.

[14] R. Hassin, Approximated schemes for the restricted shortestpath problem, Math Oper Res 17 (1992), 36–42.

[15] J. Hebert, D. Jacques, M. Novy, and M. Pachter, CooperativeControl of UAVs. AIAA Guidance, Navigation, and ControlConference and Exhibit, Montreal, Canada, 2001.

[16] T. Inanc, K. Misovec, and R.M. Murray, Nonlinear TrajectoryGeneration for Unmanned Air Vehicles with Multiple Radars.43rd IEEE Conference on Decision and Control, ParadiseIsland, Bahamas, 2004.

[17] J. Kim and J.P. Hespanha, Discrete Approximations to Con-tinuous Shortest-Path: Application to Minimum-Risk PathPlanning for Groups of UAVs. 42nd IEEE Conference onDecision and Control, Hawaii, 2003.

[18] M.B. Milam, K. Mushambi, and R.M. Murray, A New Com-putational Approach to Real-Time Trajectory Generation forConstrained Mechanical Systems, 39th IEEE Conference onDecision and Control, Sydney, Australia, 2000.



[19] L.C. Polymenakos, D.P. Bertsekas, and J.N. Tsitsiklis,Implementation of effcient algorithms for globally optimaltrajectories, IEEE Trans Auto Contr 43(2) (1998), 278–283.

[20] A. Richards and J.P. How, Aircraft Trajectory PlanningWith Collision Avoidance Using Mixed Integer Linear Pro-gramming. IEEE American Control Conference, Anchorage,Alaska, 2002, 1936–1941.

[21] M.I. Skolnik, Radar Handbook, 2nd ed. New York: McGraw-Hill, 1990.

[22] J.N. Tsitsiklis, Efficient algorithms for globally opti-mal trajectories, IEEE Trans Auto Contr 40(9) (1995),1528–1538.

[23] L.C. Thomas and J.N. Eagle, Criteria and approximate meth-ods for path-constrained moving-target search problems, NavRes Logist 42 (1995), 27–38.

[24] J.L. Vian and J.R. More, Trajectory optimization with riskminimization for military aircraft, AIAA J Guid Contr Dynam12(3) (1989), 311–317.

[25] A.R. Washburn, Continuous Autorouters with an Applica-tion to Submarines. Research Report, NPSOR-91-05, NavalPostgraduate School, Monterey, CA, 1990.

[26] M. Zabarankin, S. Uryasev, and P. Pardalos, Optimal Risk PathAlgorithms. Cooperative Control and Optimization (R. Mur-phey and P. Pardalos ed.), Kluwer Academic, Dordrecht, 2002,vol. 66, 271–303.


aircraft routing under the risk of detection - department of industrial

Documents