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Optimal Risk Path Algorithms

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Analytical and discrete optimization approaches for routing an aircraft in a threat environment have been developed. Using these approaches, an aircraft’s optimal risk trajectory with a constraint on the path length can be efficiently calculated. The analytical approach based on calculus of variations reduces the original risk optimization problem to the system of nonlinear differential equations. In the case of a single radarinstallation, the solution of such a system is expressed by the elliptic sine. The discrete optimization approach reformulates the problem as the Weight Constrained Shortest Path Problem (WCSPP) for a grid undirected graph. The WCSPP is efficiently solved by the Modified Label Setting Algorithm (MLSA). Both approaches have been tested with several numerical examples. Discrete nonsmooth solutions with high precision coincide with exact continuous solutions. For the same graph, time in which the discrete optimization algorithm computes the optimal trajectory is independent of the number of radars. The discrete approach is also efficient for solving the problem using different risk functions.
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1
OPTIMAL RISK PATH ALGORITHMS1
RESEARCH REPORT # 2001
4
Michael Zabarankin2
Risk Management and Financial Engineering Lab
Center for Applied Optimization
Department of Industrial and Systems Engineering
University of Florida, Gainesville, FL 32611
Date: March 29, 2001
Abstract
Analytical and discrete optimization approaches for routing an aircraft in a threat environment have been
developed. Using these approaches, an aircraft’s optimal risk trajectory with a constraint on the path
length can be efficiently calculated. The analytical approach based on calculus of variations reduces the
original risk optimization problem to the system of nonlinear differential equations. The solution of such a
system in the case of a single radar-installation is expressed by the elliptic sine. The discrete optimization
approach reformulates the problem as the Weight Constrained Shortest Path Problem (WCSPP) for a grid
undirected graph. The WCSPP is efficiently solved by the Modified Label Setting Algorithm (MLSA).
Both approaches have been tested with several numerical examples. Discrete nonsmooth solutions with
high precision coincide with exact continuous solutions. For the same graph, time in which the discrete
optimization algorithm computes the optimal trajectory is independent of the number of radars. The
discrete approach is also efficient for solving the problem using different risk functions.
Introduction
Optimal trajectory generation is a fundamental requirement for military aircraft flight
management systems. These systems are required to take advantage of all available information
in order to perform integrated task processing and reduce pilot workloads. The systems should
provide updates at regular time intervals sufficient for threat avoidance. To optimize flight
trajectory in a threat environment, a model for the risk of aircraft detection is developed based on
idealizing assumptions with respect to geometrical and physical aircraft properties. An ideal
flight trajectory for military operations meets the mission requirements within the constraints of
aircraft limitations while minimizing risk exposure. Several levels of information are used in the
selection of such a flight path and velocity profile. The trajectory can be a function of mission
requirements and the threat environment. The most challenging and general problem is finding a
minimal risk path, which depends on the locations of radar-installations or Surface Air Missiles
1 Research is supported by the Air Force grant F-08630-00-1-0001.
2 University of Florida, ISE, P.O. Box 116595, 303 Weil Hall Gainesville, FL 32611-6595; e-mail:
zabarank@ufl.edu.
2
(SAM), and is subject to technological constraints such as limits on flying time, fuel capacity and
trajectory length.
Optimal risk path generation is closely related to optimal search path planning for calculating
a route for a searcher to maximize the probability of target detecting. Basic results of the optimal
search theory can be found in [16,17,19]. The Special Issue on Search Theory [5] contains an
introductory paper by Stone and Washburn describing state-of-the-art research in this area. This
issue includes a survey of the literature on the search theory containing 239 references. The most
frequently used setup is to partition a search space into rectangular cells and to allocate search
efforts to these cells [16,19,22] (see, also, various approaches with this setup in [1,10,20]).
In spite of numerous studies in this area, only a few considered risk optimization problems
with technological constraints. The main purpose of this paper is to develop fast algorithms for
optimal risk path generation with a path length constraint. These algorithms are intended for
solving the optimization problem in online applications. We have considered two approaches for
solving the risk optimization problem: 1) analytical solutions based on calculus of variations, and
2) discrete optimization.
Using calculus of variations apparatus, we have derived the system of nonlinear differential
equations for finding the optimal risk trajectory with a path length constraint in a general case.
We have obtained the analytical solution of this system of differential equations in the case of a
single radar-installation. The solution is expressed by the elliptic sine. Using standard
mathematical software, it is illustrated with several numerical examples. Although we have made
significant progress in the development of the analytical approach, construction of an analytical
solution in the case with an arbitrary number of radars is still an open issue.
We have developed the discrete optimization approach for optimal path generation with a
path length constraint in the case with an arbitrary number of radars. Efficiency of discrete
optimization approaches for calculating optimal risk path essentially depends upon risk
definition, type of technological constraints, and the aircraft’s trajectory approximation (see, for
instance, [21] for discussions of these issues). Many of the previous studies on trajectory
generation for military aircraft are concentrated on feasible direction algorithms and dynamic
programming [6]. These methods tend to be computationally intense and, therefore, are not well
suited for onboard applications. To improve computation time, John and Moore [21] used simple
analytical risk functions. Based on such an approach, they developed lateral and vertical
algorithms to optimize flight trajectory with respect to time, fuel, aircraft final position, and risk
exposure. Nevertheless, these algorithms are not intended for solving optimization problems with
technological constraints, such as a constraint on the path length. In this paper, we use simple
analytical functions for defining the risk and reduce optimal risk path generation with a constraint
on the length to a network flow problem. An admissible domain for a flight route is approximated
by a grid undirected graph and an aircraft’s trajectory is presented by a path in the graph. The
optimal risk path problem is reformulated as the Weight Constrained Shortest Path Problem
(WCSPP) for the grid undirected graph. Several network flow optimization algorithms are
available for solving the WCSPP [7]. For our purposes, we use the Modified Label Setting
Algorithm (MLSA) with a preprocessing procedure developed and implemented in C++ code by
Boland and Dumitrescu [8]. A reader interested in the WCSPP and MLSA can find relevant
information in [7,8]. The original code solves problems with integer costs and weights. We have
adopted and applied this code for solving risk optimization problems with real costs and weights.
The efficiency of the discrete optimization approach is demonstrated with several numerical
examples (various numbers of radars in different locations). Computation time in these examples
is only several seconds, indicating that the MLSA is fast enough for use in online applications.
3
However, the performance of the algorithm is evaluated only numerically. Theoretical evaluation
of the algorithm’s complexity is beyond the scope of this study. For the case with a single radar,
we have compared analytical and numerical solutions and found that solutions coincide with high
precision. This validates both analytical and discrete optimization approaches developed in this
paper.
The paper is organized as follows: section 1 describes assumptions and the problem
statement; section 2 presents the analytical approach; section 3 considers the discrete
optimization approach; section 4 gives concluding remarks; the Appendix contains the derivation
of the system of differential equations for finding an optimal risk path with a constraint on the
path length in the case with an arbitrary number of radars and the derivation of the analytical
solution for this system in the case with a single radar.
1. Model description and setup of the optimization problem.
This paper develops a general methodology for optimal risk path planning with a constraint on
the path length. However, to be more specific, we focus primarily on risks related to radar
detection. For instance, for the case with two radars, Figures 1 and 2 illustrate an unconstrained
optimal trajectory and an optimal trajectory with a length constraint, respectively. Curves in these
figures correspond to level sets of the risk with the maximal values in areas close to the radars.
Risk declines when an aircraft is going away from radars and increases when the aircraft
approaches them. The threat models used in this study do not refer to any specific military
scenario. They are simple analytical models that characterize the general nature of threats.
-1 -0.5 00.5 1
-1
0
1
A
B
Radar 1 Radar 2
-1 -0.5 00.5 1
-1
0
1
A
B
Radar 1 Radar 2
Figure 1. Aircraft flies from point A to point B
trying to avoid two radars: optimal risk path
without length constraint.
Figure 2. Aircraft flies from point A to point B
trying to avoid two radars: optimal risk path with
length constraint.
To formulate a risk path optimization problem the following assumptions are used:
1. Horizontal plane model. Aircraft position is considered to be in a horizontal plane only.
2. Radar detection of the aircraft does not depend on the aircraft’s heading and climb angles.
3. Rotation angle does not depend on trajectory position.
4
4. An admissible domain for the aircraft’s trajectory is assumed to be a detection area for all
radar-installations. In other words, distance from the aircraft to each th
i radar-installation is
not greater than the th
i radar maximum detection range, i
Rmax, .
5. Risk is quantified in terms of the risk index per unit length for any particular aircraft location.
The simplified threat model assumes that risk index r is proportional to risk factor
σ
and
reciprocal to the squared distance from the aircraft position to the radar location (see, for
instance, [21]). The risk factor,
σ
, depends on the radar’s technical characteristics such as the
maximum detection range, the minimum detectable signal, the transmitting power of the
antenna, the antenna gain, and the wavelength of radar energy. It is considered that all of the
radar’s technical characteristics remain constant, hence, under such an assumption the risk
factor, i
σ
, is the constant for the th
i radar. If 22 )()( iii byaxd += is the distance from
the aircraft position, ),( yx , to the th
i radar location, ),( ii ba , then the risk index, i
r, at
trajectory point, ),( yx , is given by the formula 2
),(
=iii dyxr
σ
. Although, in this paper we
considered that risk is reciprocal to squared distance, this assumption is not critical for the
application of the developed methodology. For instance, the risk index can be expressed as
4
),(
=iii dyxr
σ
, which corresponds to the radar detection model with a signal reflected from
the aircraft. In this case, the risk factor i
σ
defines the aircraft radar cross section (RCS) for
the th
i radar (see, for instance, [18]).
6. At every point of the admissible deviation domain, the cumulative risk from N radar-
installations is evaluated as the sum of risks from each th
i radar, i. e. =
=== N
i
ii
N
i
idrr
1
2
1
σ
.
7. Aircraft velocity is assumed to be a constant, hence, time increment dt and unit length ds
are linearly dependent: dtVds 0
=. For convenience, we use unit length, ds , instead of time
increment, dt .
8. For any particular aircraft location, ),( yx , the risk per unit length, ds , is calculated as the
product of the risk index and unit length, ds , i. e.
=
=N
i
ii dsddsr
1
2
σ
.
9. The risk R accumulated along a path Ρ is presented by the expression
Ρ
=Ρ dsrR )(.
Based on model assumptions 1 9, the optimization problem with a constraint on the path length
is formulated in the following way. Let N be the number of radars and
()
ii ba , the location of the
th
i radar, where Ni ,1
=. The aircraft’s departure and destination points are ),A( 11 yx and
),B( 22 yx , respectively. The path Ρ from A to B is associated with the integrated risk, )(Ρ
R, and
the total length, )(Ρ
l. The optimal path *
Ρ is such a path Ρ, which minimizes )(Ρ
R subject to
length constraint *
)( ll Ρ . The optimization problem is presented in the form
,)(..
)(min
*
llts
R
Ρ
Ρ
Ρ(1)
where )(Ρ
R and )(Ρ
l are defined by the expressions
5
=
=Ρ B
A1
2
)( N
i
ii dsdR
σ
,(2)
=Ρ B
A
)( dsl .(3)
To solve problem (1)-(3), analytical and discrete optimization approaches are considered.
2. Analytical solutions approach for the risk path optimization problem.
We use calculus of variations apparatus to obtain an analytical solution for the risk optimization
problem with a constraint on the path length. This approach reduces the original problem to the
system of differential equations with respect to coordinates of the optimal trajectory. The
complexity of this system depends upon the type of constraints and the manner in which the
trajectory is defined.
The constraint on the trajectory length given in integral form (3) leads to an isoperimetric
problem of calculus of variations (general theory is given in [11,12]). We prefer not to use
integral constraints in the formulation of the problem. To reduce constraint (3) to an algebraic
form, coordinates of an unknown curve are presented as functions of the current length, s, of the
curve, i. e. )(sxx =, )(syy =. Such a parameterization is also known as the natural definition of
a curve. The relation between differentials of the curve arc length, ds , and curve coordinates, dx
and dy , such that 222 dydxds += , is reduced to an additional nonholonomic constraint for
derivatives of )(sx and )(sy . In other words, if we denote )()( sxsx ds
d
=
! and )()( sysy ds
d
=
!, then
)(sx
! and )(sy
! should satisfy the equality
()()
1)()( 22 =+ sysx !! . The distance between the th
i
radar location, ),( ii ba , and the aircraft position,
()
)(),( sysx , is defined by the expression
()()
22 )()( iii bsyasxd += . With calculus of variations, problem (1)-(3) is reduced to
minimization of functional )(lR with respect to unknown curve
()
)(),( sysx , where ls
0,
with boundary conditions ),( 11 yx , ),( 22 yx and constraint on the trajectory length, *
ll .
Problem (1)-(3) is reformulated in the form
dsd
lN
i
ii
yx
=
01
2
),(
min
σ
,(4)
subject to boundary conditions
,)(
,)(
,)0(
,)0(
2
2
1
1
yly
xlx
yy
xx
=
=
=
=(5)
and constraints
()()
1)()( 22 =+ sysx !! ,(6)
6
*
ll .(7)
Constraint *
ll is different from *
ll =, which is considered in classical calculus of variations
(see, for instance, [11,12]). In the case when *
ll , it is a problem of calculus of variations with a
moveable end point. The total variation of functional (4) must include the variation of the path
length, l, and, therefore, value l is determined from an additional condition.
Let us denote the risk function in integral (4) by function L, which depends on variables x
and y,
()
=
=N
i
ii dsysxL
1
2
)(),(
σ
.(8)
Calculus of variations problem (4)-(7) for finding the optimal trajectory,
()
)(),( sysx , is reduced
to the system of differential equations
()
()
=
=
yLyL
xLxL
Ly
Lx
ds
d
ds
d
!!!
!!!
λ
λ
(9)
with boundary conditions (5) and constraint (7), where L
λ
is a nonnegative constant appearing
due to length constraint (7). A detailed derivation of system (9) and the analytical solution for this
system in the case of a single radar are included in the Appendix.
Since equation (6) is the first integral of system (9), constraint (6) and system (9) are not
independent. While construction of an analytical solution of system (9) for a general form of
function L is an open issue, the numerical solution for system (9) with boundary conditions (5),
satisfying *
ll =, can be efficiently calculated by the method of weighted residuals [2,15].
However, a numerical solution of system (9) is beyond the scope of this paper.
We have derived the analytical solution of system (9) in the case of a single radar. Without
loss of generality, it is assumed that the radar is located at the origin of the system of coordinates,
i. e. 0
1=
a, 0
1=
b, and that the risk factor,
σ
, equals 1. To simplify calculations, the polar
system of coordinates is used. The polar radius,
ρ
, and polar angle, Ψ, are related to Cartesian
coordinates x and y in the following way
.
,
)(sin)()(
)(cos)()(
sΨssy
sΨssx
ρ
ρ
=
=
Function L is rewritten as
()
2
)(),(
=
ρρ
sΨsL . (10)
The optimal path has bounded length even in the case when length constraint (7) is relaxed. This
is caused by the fact that function L does not converge rapidly enough to zero when
ρ
tends to
infinity. The optimal solution for optimization problem (4)-(6) without constraint on the path
length is presented by an arc of the circle
7
()()
2
2
1
2
1)2(cos)2(sin)2( =+aCayCax , (11)
where constants a and C are determined from boundary conditions (5). In the case when the
constraint on the length is relaxed, the radar location, ),( 11 ba , and the aircraft’s departure and
destination points, ),( 11 yx and ),( 22 yx , lie on the same circle. Denoting 2
1
2
11 yx +=
ρ
,
2
2
2
22 yx +=
ρ
and the angle between vectors ),( 11 yx , ),( 22 yx by
ϑ
, such that
=+
21
2121
arccos
ρρ
ϑ
yyxx , (12)
the optimal risk, *
R, and unconstrained path length, l, are expressed in terms of 1
ρ
, 2
ρ
and
ϑ
2
12
2
12* )()(
21
1yyxxR +=
ρρ
, (13)
2
12
2
12 )()(
sin yyxxl +=
ϑ
ϑ
. (14)
If *
ll then length constraint (7) is inactive and l coincides with l. The optimal solution is
determined by (11)-(14). In the case when ll
*, constraint (7) becomes active and l coincides
with *
l. Using definition of the elliptic sine (see [3])
[]
()
===
φ
κ
φκκ
022 sin1
,sin),am(sin,sn t
dt
uuu ,
the optimal solution for the case ll
*, i. e. *
ll =, is presented in the form
[]
κρ
κ
κ
,sn)( 111 CΨΨ a+= +
+,
or
[]
[]
,sin,sn)(
,cos,sn)(
111 111
ΨCΨΨy
ΨCΨΨx
a
a
κ
κ
κ
κκ
κ
+= +=
+
++
+(15)
with boundary conditions
,arccos,
,arccos,
2
2
1
1
2
2
2
22
1
2
1
22
ρ
ρ
ρ
ρ
x
x
ΨΨyx
ΨΨyx
===+ ===+ (16)
where values 1
ρ
, 2
ρ
are defined above and
π
Ψ
0.
8
According to (15), the optimal risk, R, and optimal path length, l, are given
[]
()
dΨCΨR
Ψ
Ψ
a++= +
+
2
1
,sn 11
)1( 2
2
κκ
κ
κ
, (17)
[]
()
dΨCΨal
Ψ
Ψ
++= +
2
1
,sn1 11
21
κκ
κ
. (18)
Constants a,
κ
, and C are determined from boundary conditions (16) and equality (18) under
condition *
ll =.
First, the value of l should be calculated; if *
ll , the optimal solution is given by (11)-(14),
and if ll
* the optimal path is determined by (15)-(18).
To illustrate the analytical solution for the case of a single radar, we consider the following
numerical example. Suppose that the aircraft’s departure and destination points are A and B with
coordinates 0.25,0.25)(
=),( 11 yx , )(1.75,0.25
=),( 22 yx , respectively, and suppose that the
maximum value for the trajectory length is 3.2
=
*
l. The radar is located at (0,0) and the risk
factor 1
=
σ
. Values
ϑ
and l computed according to (12), (14) are 7arctg
4+=
π
ϑ
and
5.536
=
l. The length constraint is active, since *
ll . Parameters a,
κ
and, C can be
calculated using standard software; we have use package MATHEMATICA 4. For this case,
0.7378
=
κ
, 0.9414
=
a, 3.1419
=
C. In Figure 1, curve 1 (circle arc) represents the optimal
path without constraint on the length. The risk for the path and the path length are equal to 3.2
and 5.536, respectively. Curve 2 is the optimal path with constraint on the length, 3.2
=
*
l. The
risk for this path is 3.326. We compare the optimal risks and trajectory lengths not only for
these two curves, but also for the straight line between points A and B. The length of line AB
(curve 3), AB
l, equals 2.0, while the risk accumulated along this line is equal to 8.857.
An obvious conclusion from this particular example is that the increase of the curve length
considerably affects the risk only in an area close to the radar location.
9
-0.5 0.5 11.5 2x
0.5
1
1.5
2
y
AB
Radar
1
2
3
Figure 3. Optimal trajectories obtained by the analytical approach:
curve 1 optimal path without length constraint, R= 3.2, l= 5.536;
curve 2 optimal path with length constraint, R= 3.326, l= 3.2;
curve 3 shortest path, R= 8.857, l= 2.0.
Function )(*min lRR =, lll *AB , is concave and bounded, see Figure 4.
2.5 33.5 44.5 55.5 *
l
2
4
6
8
10 min
R
Figure 4. Dependence of the optimal risk on the trajectory length:
the increase of the curve length greatly decreases the risk only in
an area close to the radar location.
10
3. Discrete optimization approach for optimal risk path generation with a constraint on the
length.
The calculus of variations approach reduces the optimization problem to solving the system of
nonlinear differential equations. Construction of an analytical solution of this system for the case
of an arbitrary number of radars is still an open issue. There are various numerical methods
approximating a solution of the system, but that is not the focus of this paper. We propose the
discrete optimization approach to directly solve the original problem. This approach reduces
optimal risk path generation with a constraint on the length to the Weight Constrained Shortest
Path Problem for a grid undirected graph. In the case of an arbitrary number of radars, the
WCSPP can be efficiently solved by network flow optimization algorithms. However, the
computation time of these algorithms exponentially depends upon the precision prespecified for
the optimal trajectory.
We assume the admissible deviation domain for the aircraft’s trajectory to be an undirected
graph
()
AVG ,=, where },,1{ nV "= is the set consisting of n nodes and A is the set of
undirected arcs. A trajectory
()
(.)(.), yx is approximated by a path Ρ in the graph G, where path
Ρ is defined as a sequence of nodes p
jjj ,,, 10 " such that A
0=
j, B=
p
j and Ajj kk ,
1
for all k from 1 to p. To formulate (1)-(3) as a network optimization problem we use discrete
approximation for formulas (2) and (3) determining the risk and trajectory length, respectively,
∑∑===
=
B
A11 ,
1
2
11
p
k
jj
N
i
jjii
N
i
ii kkkk srdsd
σσ
, (19)
=
= p
k
jj kk
sds
1
B
A1, (20)
where kk jj
s1
is the length of arc kk jj ,
1 and kk jji
r1
, denotes the risk index for the arc
kk jj ,
1. If )(k
jx and )(k
jy are x and y coordinates for node k
j then arc length kk jj
s1
is
defined by the expression
()()
2
1
2
1)()()()(
1+=kkkkjj jyjyjxjxs kk . (21)
To derive the formula for the risk index kk jji
r1
, we compute the risk accumulated along the arc
kk jj ,
1 from the radar located at ),( ii ba (the risk factor i
σ
is omitted for convenience)
(
)
(
)
(
)
()()()()
ikikikik
jj
jx
jx ii
x
j
j
ibjyajxbjyajx
s
dx
byax
y
dsd kk
k
k
k
k
i
k
i
k
i
k
i
k
ajx
bjy
ajx
bjy
=
+
+
=
)()()()(
arctanarctan
)()(
)(1
11
)(
)( 22
2
21
11
)( )(
)( )(
1
1
.
The last formula has a simple interpretation. If we denote vector
()
ikik bjyajx )(,)( by k
ji,
r,
its length by |||| ,k
ji
r and the angle between two vectors 1
,
k
ji
r and k
ji,
r by kk jji 1
,
ϑ
, where kk jji 1
,
ϑ
satisfies condition
πϑ
kk jji 1
,
0 , then the last formula turns to the following form
11
||||||||sin ,,,
,
2
1
1
1
1
1kk
kk
kk
kk
k
kjiji
jj
jji
jji
j
j
i
s
dsd rr
=
ϑ
ϑ
,
and the risk index kk jji
r1
, is determined as
1
,
1
,
,
,
,||||||||
sin 1
1
1
1
=
kk
kk
kk
kk jiji
jji
jji
jji
rrr
ϑ
ϑ
. (22)
When kk jji 1
,
ϑ
tends to zero, the risk index has limit value 1
,
1
,|||||||| 1
kk jiji rr . In the limit case
0
1kk jj
s, i. e. kk jj
1, we have 0
1
,
kk jji
ϑ
, |||||||| ,, 1kk jiji rr
and expression (22)
coincides with the risk index definition for point k
j.
Figure 5 illustrates a network flow example for solving the risk minimization problem. The
thick broken line is a path in the area with two radars and kk jj ,
1 is an arc of this path. The
distance between nodes 1
k
j and k
j is arc length kk jj
s1
. Values |||| 1
,1
k
j
r, |||| ,1 k
j
r define
distances from radar 1 to nodes 1
k
j, k
j respectively. Magnitude kk jj 1
,1
ϑ
corresponds to the angle
between vectors 1
,1
k
j
r and k
j
,1
r.
kk jj 1
,1
ϑ
kk jj 1
,1
ϑ
kk jj 1
,1
ϑ
00.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
A
B
1
,1
k
j
r
Radar1 Radar 2
kk jj 1
,1
ϑ
k
j
,1
rk
j
1
k
j
kk jj
s1
Figure 5. A network flow example for solving the risk minimization problem:
the thick broken line AB is a path of the aircraft.
Using substitution
kkkkkk jj
N
i
jjiijj src 111 1,=
=
σ
, (23)
12
values )(Ρ
R and )(Ρ
l are rearranged in the form
=
=Ρ p
k
jj kk
cR
11
)( , (24)
=
=Ρ p
k
jj kk
sl
11
)( . (25)
Thus, each arc Ajj kk ,
1 is associated with its length kk jj
s1
and nonnegative cost kk jj
c1
,
defined by (21) and (23), respectively. Considering value kk jj
s1
as the arc’s weight we will use
)(Ρ
R to denote the cost of the path P and )(Ρ
l to denote the total weight accumulated along that
path. The path P is weight feasible if the total weight )(Ρ
lis at most *
l, i. e. *
)( ll Ρ . The Weight
Constrained Shortest Path Problem (WCSPP) is formulated in the following way. It is required to
find such a feasible path Ρ from point A to point B that minimizes cost )(Ρ
R
...
min
*
1
1
1
1
lsts
c
p
k
jj
p
k
jj
kk
kk
=
=
Ρ
(26)
The WCSPP (26) is closely related to the Shortest Path Problem with Time Windows (SPPTW)
and also to the Resource Constrained Shortest Path Problem (RCSPP), which uses a vector of
weights, or resources, rather than a scalar. These problems are solved in column generation
approaches for Vehicle Routing Problems with Time Windows (VRPTW) and in long-haul
aircraft routing problems. Algorithms for solving the WCSPP are divided into three major
categories: label-setting algorithms based on dynamic programming methods, scaling algorithms,
and algorithms based on the Lagrangean relaxation approach. The label setting algorithm is the
most efficient in the case when the weights are positive [9]. The subgradient optimization [4] and
cutting plane [13] methods are the core of the Lagrangean relaxation algorithm, which is efficient
for solving the Lagrangean dual problem of the WCSPP in the case of one resource. Scaling
algorithms use two fully polynomial approximation schemes for the WCSPP based on cost
scaling and rounding [14]. The first scheme is a geometric bisection search whereas the second
one iteratively extends paths. To solve the WCSPP, defined by (26), we use the Modified Label
Setting Algorithm (MLSA) with a preprocessing procedure [8].
The Preprocessing Procedure and Label Setting Algorithm (LSA) are two consecutive stages
of the MLSA. The LSA is the core of the MLSA, which integrates information obtained in
preprocessing. The objective of the preprocessing procedure is to reduce the original graph by
eliminating all arcs and nodes such that any path containing them is infeasible or does not
improve current cost upper bound. To discuss the algorithm in detail, let us denote the arc’s
nodes 1
k
j and k
j by i and j, respectively. For each node i, we consider the path obtained by
appending the least cost path from the source node s to i to the least cost path from i to the sink
node t. If the total cost accumulated along the new path is at least the current cost upper bound,
then the use of node i cannot improve a known feasible solution. Hence, node i and all arcs
incident to it can be deleted from the graph. If the total cost is less than the upper bound and the
13
path is feasible, then the upper bound can be updated and the process continues with the
improved upper bound. Similar, for each arc ji,, we consider the path obtained by appending
the least cost path from s to i to the least cost path from j to t, via arc ji,. If the total cost
accumulated along the new path is at least equal to the current cost upper bound, then we can
delete arc ji, from the graph. If the total cost is less than the upper bound and the path is
feasible then the upper bound can be updated. The preprocessing procedure is presented in the
pseudocode form below.
Preprocessing Algorithm for the WCSPP
Step 0: Let )1( = nCU where ji
Aji cC
=,
max .
Step 1: Find the shortest paths from source node A
=
s with arc costs given by ji
c.
Let c
js
Q be the least cost path from s to j and c
j
α
be the cost of the path: )(c
js
c
jQR
=
α
.
If there is no path from s to the sink node B
=
t then stop; the problem is infeasible.
If ( *
)( lQl c
ts ) then c
ts
Q is the optimal path.
Step 2: Find the shortest paths from all nodes to t with arc costs given by ji
c.
Let c
tj
Q be the least cost path from j to t and c
j
β
be the cost of the path: )(c
tj
c
jQR
=
β
.
Step 3: Find the shortest paths from s to all nodes with arc lengths given by ji
s
.
l
js
Q is the shortest path from s to j and l
j
α
is the length of this path: )(l
js
l
jQl
=
α
.
If ( *
)( lQl l
ts >) then stop; the problem is infeasible.
If ( *
)( lQl l
ts ) and ( UQR l
ts <)()
then set )(l
ts
QRU =.
Step 4: Find the shortest paths from all nodes to t with the arc lengths given by ji
s
.
l
tj
Q is the least length path from j to t and l
j
β
is the length of this path: )(l
tj
l
jQl
=
β
.
Step 5: For all },{\ tsVj do
if ( *
l
l
j
l
j>+
βα
) then delete node j and all arcs incident to it;
if ( U
c
j
c
j+
βα
) then delete node j and all arcs incident to it;
end
Step 6: For all Aji ,
do
if ( *
ls l
jji
l
i>++
βα
) then delete ji,
else if ( Uc c
jji
c
i++
βα
) then delete ji,
else if ( *
)()( lQlsQl c
tjji
c
is ++ ) then c
jji
c
icU
βα
++= ;
end
Step 7: If during steps 5 and 6 the graph changed then goto Step 1, else set c
t
L
α
= and stop.
Preprocessing has been completed.
14
The next stage after the preprocessing procedure is the Label Setting Algorithm. The idea of
the algorithm is to use a set of labels for each node and compare the labels to one another. Each
label on a node represents a different path from node s to that node and consists of a pair of
numbers representing the cost and weight of the corresponding path. No labels having the same
cost are stored and for each label on a node, any other label on that node with a lower cost must
have a greater weight. Let i
I be the index set of labels on node i and for each i
Ik let k
i
P
denote a path from s to i with weight k
i
W and cost k
i
C. Pair ),( k
i
k
iCW is the label of node i and
k
i
P is the path corresponding to it. For two labels ),( k
i
k
iCW and ),( q
i
q
iCW , corresponding to two
different paths k
i
P and q
i
P, respectively, ),( k
i
k
iCW dominates ),( q
i
q
iCW if q
i
k
iWW , q
i
k
iCC ,
and the labels are not equal. Label ),( k
i
k
iCW is efficient if it is not dominated by any other label
at node i, i. e. if (P))(P),( Rl does not dominate ),( k
i
k
iCW for all paths P from s to i. A path is
efficient if the label it corresponds to is efficient. The LSA finds all efficient labels in every node.
Starting without any labels on any node, except for label )0,0( on node s, the algorithm extends
the set of all labels by treating an existing label on a node, that is, by extending the corresponding
path along all outgoing arcs. Let i
L be the set of labels on node i and let ii IT index the labels
on node i, which have been treated. The algorithm proceeds until all labels have been treated,
i. e. until =
ii TI \ for all }{\ tVi .
The Modified Label Setting Algorithm (MLSA)
Step 0: Initialization
run Preprocessing Algorithm for the WCSPP to find U, c
j
β
, l
j
β
and c
tj
Q }{\ tVj .
Set )}0,0{(=
s
L and =
i
L for all }{\ sVi .
Initialize i
I accordingly for each Vi .
Set =
i
T for each Vi .
Step 1: Selection of the label to be treated
If =
)\( ii
Vi TI then stop; all efficient labels have been generated.
Else choose Vi and ii TIk \ so that k
i
W is minimal.
Step 2: Treatment of label ),( k
i
k
iCW
For all Aji , do
If ( *
lsW l
jji
k
i++
β
)
If ( UcC c
jji
k
i<++
β
)
If ),( ji
k
iji
k
icCsW ++ is not dominated by ),( q
j
q
jCW j
Iq
then set )},{( ji
k
iji
k
ijj cCsWLL ++= and update j
I
If ))(( *
lQlsW c
tjji
k
i++ then c
jji
k
icCU
β
++= .
end
Step 3: Set }{kTT ii = , goto to Step 1.
End.
15
The MLSA was implemented in C++ code by Boland and Dumitrescu [8]. The code was
originally designed for solving the WCSPP with integer costs and integer weights. Under the
assumption of cost and weight integrality, it was shown that the WCSPP is a NP-hard problem. In
the case of real costs and weights, no specific evaluation for the complexity of the MLSA is
provided. It is considered that in a worse case, the WCSPP can be solved in the time
exponentially depending on the number of arcs. The analysis of the MLSA’s complexity is
beyond the scope of this paper. The original code is adopted for solving the WCSPP with real
costs and real weights assigned by (22) and (23), respectively.
For numerical examples, we use an undirected graph with nodes located in the cross-points of
a square grid with the side length of T relative units. A set of arcs assigned for each node in the
graph is shown in figure 6. This set consists of 32 arcs. Each arc is counted twice, since the graph
is undirected. In figure 6, “1” represents four horizontal and four vertical arcs with the same
length of s
, “2” denotes eight diagonal arcs with the same length of s
2, and “3” denotes
sixteen long diagonal arcs with the same length of s
5. If m is the number of arcs on the grid
side, i. e. sTm = , then 2
)1( +
m is the number of nodes and )14(4
mm is the number of arcs
in the graph.
00.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
1
1
1
22
22
3
33
3
3
33
3
Figure 6. A set of arcs assigned for each node in the graph:
1 horizontal and vertical arcs, 2 diagonal arcs, 3 long
diagonal arcs.
To compare the analytical and discrete optimization approaches, we consider the previous
example with a single radar. The radar location, the boundary value for the length constraint and
the departure and destination points for the aircraft are exactly the same. The graph has the
following characteristics: 2.1
=
T, 0.05
=s, 42
=
m. There are 1849 nodes and 28056 arcs
in this graph. All calculations have been performed using a PC 800 MHz with RAM of 256 Mb.
The computation time is approximately sec6. Figure 7 compares the analytical and discrete
optimization solutions. The smooth curve is the optimal trajectory obtained by the analytical
approach. The value of the risk accumulated along this trajectory equals 3.326. The nonsmooth
16
curve is the optimal trajectory obtained by the discrete optimization approach. In this case, the
trajectory has length 3.196 and risk value 3.360. For the discrete optimization solution, the
relative error of the optimal risk value is about 1%. This validates both approaches.
To estimate time required for calculating the optimal trajectory with higher precision, we
again consider the case with a single radar using grid undirected graph with parameters: 2.1
=
T,
0.025
=s, and 84
=
m. There are 7225 nodes and 112560 arcs in this graph. The solution
time is about sec200 . Figure 8 compares the analytical solution and discrete optimization
solutions with different precision (the discrete optimization solution with higher precision is
depicted by a dashed curve). The dashed trajectory has length 3.199 and accumulated risk
3.357. The relative risk error for the dashed curve in comparison with the risk value for the
analytical solution is about 0.9%. This indicates that the higher precision solution does not
essentially improve the lower precision solution and that practically both trajectories are
identical. In this case, the calculation time for the higher precision solution was about thirty times
greater than for the lower precision solution. However, risk was improved by only 0.1%. This
conclusion relates only to the considered example. In a general case, such an inference should be
made based on the evaluation of the MLSA’s complexity.
Computation time of the MLSA does not depend on the number of radars; it depends only on
the number of arcs. To demonstrate this advantage of the discrete optimization approach, we use
the same graph for calculating the optimal risk trajectories with a different number of radars. The
optimal risk trajectories in figures 9-12 are computed in the same time interval. In all considered
examples, grid undirected graph with parameters 2.0
=
T, 0.02
=s, 100
=
m is used, and the
risk factors i
σ
are assumed to be equal to 1.
17
0.5 11.5 x
0.2
0.4
0.6
0.8
1
1.2
1.4
y
AB
Radar
Figure 7. Comparison of the analytical and discrete optimization solutions:
the smooth curve is the optimal path obtained by the analytical approach,
R= 3.326, l= 3.2; the nonsmooth curve is the optimal path obtained by the
discrete optimization approach, R= 3.360, l= 3.196
0.5 11.5 x
0.2
0.4
0.6
0.8
1
1.2
1.4
y
AB
Radar
Figure 8. Comparison of the discrete optimization solutions with different precision:
the dashed curve is the optimal discrete solution with higher precision, R= 3.357,
l= 3.199; the solid nonsmooth curve is the optimal discrete solution with lower
precision, R= 3.360, l= 3.196; the smooth curve is the analytical solution, R= 3.326,
l= 3.2
18
-1 -0.5 0.5 1x
0.5
1
1.5
2
y
AB
Radar
Figure 9. Optimal path in the case with a single radar; 2.0
=
T, 0.02
=s, 100
=
m.
The nonsmooth curve is the optimal discrete solution; the smooth curve is the analytical
solution obtained by the calculus of variations approach.
-1 -0.5 0.5 1x
0.25
0.5
0.75
1
1.25
1.5
1.75
2
y
AB
Radar 1
Radar 2
Figure 10. Optimal path obtained by discrete optimization in the case with two radars;
2.0
=
T, 0.02
=s, 100
=
m.
19
-1 -0.5 0.5 1x
0.25
0.5
0.75
1
1.25
1.5
1.75
2
y
AB
Radar 1 Radar 3
Radar 2
Figure 11. Optimal path, symmetric location of three radars; 2.0
=
T, 0.02
=s, 100
=
m.
-1 -0.5 0.5 1x
0.25
0.5
0.75
1
1.25
1.5
1.75
2
y
AB
Radar 1
Radar 3
Radar 2
Figure 12. Optimal path, asymmetric location of three radars; 2.0
=
T, 0.02
=s, 100
=
m.
20
4. Concluding remarks and future directions of research.
We have developed analytical and discrete optimization approaches for calculating an optimal
risk path with a constraint on the path length. We have studied optimization problems with the
risk index in the form 2
d
σ
(
σ
is the risk factor, d is the distance to the radar). However, the
developed methodology is quite general and it can be used to generate optimal trajectories with
other risk functions, for instance, with the risk function defined as 4
d
σ
.
The analytical approach, based on calculus of variations, reduces the original problem to
solving the system of nonlinear ordinary differential equations. We have derived this system
using a general form of the risk function for the case with an arbitrary number of radars. For the
case of a single radar and the risk function in the form 2
d
σ
, we have obtained the analytical
solution of the system expressed by the elliptic sine. Using the analytical solution, it is shown that
the increase of the trajectory length greatly affects the risk only within an area close to the radar.
Although an analytical solution for the system of differential equations can be obtained for the
case with one radar and the risk function in the form 4
d
σ
, an analytical solution of this system
in the case with an arbitrary number of radars is still an open issue.
The discrete optimization approach reformulates the original problem as a network flow
optimization problem. Using approximation for the admissible domain by a grid undirected graph
and representation of a trajectory by a path in this graph, optimal risk path generation with a
constraint on the length is reduced to the Weight Constrained Shortest Path Problem (WCSPP).
The WCSPP is efficiently solved by the Modified Labeling Setting Algorithm (MLSA). The
optimization problem with about 30000 arcs requires approximately sec6 to be solved. This
time is suitable for online applications. However, the precision for the MLSA should be
reasonably specified, since the computation time exponentially depends on the number of arcs in
the graph. We have considered several examples with a different number of radars to investigate
the performance of the algorithm. The main advantages of the discrete optimization approach are:
1) it can account for an arbitrary number of radars; 2) the computation time does not depend upon
the number of radars; 3) the approach can be easily implemented for various risk functions.
We have compared the analytical and discrete optimization solutions for the case with a
single radar. These solutions coincide with high precision that verifies both approaches.
Acknowledgments
We want to thank Capt. R. Pendleton, USAF for helping with the development of the model for
aircraft detection in a threat environment. We are also grateful to Prof. N. Boland and
I. Dumitrescu for the informative discussions and for providing C++ code for the Modified
Labeling Setting Algorithm that was used for conducting numerical experiments.
References
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Applied Probability, Vol. 31, # 2, 1994, pp. 438 457.
[2] Aziz, A. K., “Numerical Solution of Boundary Value Problems for Ordinary Differential
Equations,” Academic Press, New York, 1975.
21
[3] Bateman, H. and Erdelyi, A., “Higher Transcendental functions,” Vol. 3, Mc Graw-Hill Book
Company, Inc., 1955.
[4] Beasley, J. E. and Christofides, N., “An Algorithm for the Resource Constrained Shortest
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[17] Mangel, M., “Search Theory,” Lecture Notes, Springer-Verlag, Berlin, 1984.
[18] Skolnik, M. I., “Radar Handbook,” McGraw-Hill, New York, second edition, 1990.
[19] Stone, L. D., “Theory of Optimal Search,” Academic Press, New York, San Francisco,
London, 1975.
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Moving-Target Search Problems,” Naval Research Logistics, Vol. 42, 1995, pp. 27 38.
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Vol. 31, 1983, pp. 739-751.
22
Appendix
This appendix contains the derivation of the system of differential equations for determining an
optimal risk path with a constraint on the path length and the analytical solution of this system for
the case with a single radar.
We start with formulation of the calculus of variation problem with a nonholonomic
constraint and a moveable end point
),,,,(min
,lyxyx
yx
!!
Φ, (a.1)
()
dssysxsysxLlyxyx
l
=Φ 0
)(),(),(),(),,,,( !!!! , (a.2)
,)(
,)(
,)0(
,)0(
2
2
1
1
yly
xlx
yy
xx
=
=
=
=(a.3)
()
0)(),( =
sysx !!
ϕ
, (a.4)
*
ll . (a.5)
A necessary condition for the existence of a functional extremum requires the total variation of
the functional to be equal to zero. Constraint (a.5) implies that this is the problem with the
movable end point,
()
)(),( lylx , (variation of the total curve length, l, is not equal to zero). It
should be taken into account that variations x
δ
and y
δ
are dependent due to nonholonomic
constraint (a.4). Usually, to separate differential expressions in the functional variation, the
Lagrange multiplier method is used. However, compared to the traditional approach, here, the
multiplier )(s
λ
depends upon variable s, since two degrees of freedom, variables x and y, are
used to formulate the problem. Applying the Lagrange multiplier method to problem (a.1)-(a.5),
the total variation of functional (a.2) with constraints (a.4) and (a.5) is rearranged in the form
()
[]
()
[]
()
[]
()
.0)(
)()(
0
+
+
+
=Φ
=
lyxsLyLxL
dsysLLxsLL
ls
yxyx
l
y
ds
d
y
ds
d
yx
ds
d
x
ds
d
x
δϕϕλ
δϕλδϕλδ
!!!!
!!!!
!!!!
(a.6)
Assuming variation x
δ
to be independent, and choosing )(s
λ
to turn the expression
(
)
y
ds
d
y
ds
d
ysLL !!
ϕλ
)(
at the variation y
δ
to zero, equality (a.6) is reduced to the system of differential equations
()
()
.
,
0)(
0)( =
=
y
ds
d
y
ds
d
y
x
ds
d
x
ds
d
x
sLL
sLL
!!
!!
ϕλ ϕλ
(a.7)
The equation defining the moveable end point, l, is given by
23
[]
()
0)( =
+
=ls
yxyx yxsLyLxL !!!! !!!!
ϕϕλ
. (a.8)
It can be shown that equations of system (a.7) have the first integral. Summing the first equation
multiplied by x
! with the second one multiplied by y
!, we have
()
()
(
)
(
)
0)()( =
+
y
ds
d
y
ds
d
yx
ds
d
x
ds
d
xsLLysLLx !!!! !!
ϕλϕλ
.
The left-hand side of this equality is a total differential. After integration it turns to the expression
(
)
constyxsLyLxL yxyx =
+
!!!! !!!!
ϕϕλ
)( . (a.9)
Lagrange multiplier, )(s
λ
, is derived from (a.9)
0,)( =
+
+
=const
yx
LyLxL
sL
yx
yxL
λ
ϕϕ
λ
λ
!!
!!
!!
!! . (a.10)
Substituting (a.10) into (a.7), we obtain the system of differential equations for determining )(sx
and )(sy . In the case when constraint *
ll is active, i. e. *
ll =, equation (a.8) is excluded from
the system for determining an optimal solution, since the total curve length is fixed and,
therefore, the variation of l equals zero by definition. If constraint *
ll is inactive, then from
(a.8) and (a.9) we have 0=
L
λ
.
For the case of optimization problem (1)-(3), function L defines the risk index at the point
),( yx , and hence, it depends on variables x and y only,
()()
()()
=+
=N
iii
i
bsyasx
sysxLsysxsysxL
122 )()(
)(),()(),(),(),(
σ
!! . (a.11)
Function
ϕ
represents reformulated nonholonomic constraint (6)
()()()
01)()()(),( 22 =+= sysxsysx !!!!
ϕ
. (a.12)
Substituting value 2=
+
yx yx !! !!
ϕϕ
into formula (a.10), Lagrange multiplier, )(s
λ
, is rewritten as
()
L
sysxLs
λλ
+= )(),()(2 . (a.13)
Using (a.7), (a.8), (a.11) and (a.13), the original optimization problem (1)-(3) for defining the
optimal trajectory
()
)(),( sysx is reduced to the system of differential equations
()
()
=
=
yLyL
xLxL
L
ds
d
y
L
ds
d
x
!!!
!!!
λ
λ
(a.14)
24
with boundary conditions (a.3) and (a.5). Only two equations from (a.12), (a.14) are independent,
since equation (a.12) is the first integral of system (a.14). A pair of independent equations is
chosen from (a.12) and (a.14) to simplify the derivation of the solution. Certainly, the obtained
solution must satisfy (a.12) and (a.14). Obtaining an analytical solution for system (a.14) in the
case of an arbitrary number of radars is still an open issue. We present the analytical solution of
system (a.14) in the case with a single radar. Without loss of generality, let us assume that the
radar is located at the origin of the system of coordinates, i. e. 0
1=
a, 0
1=
b, and that the risk
factor 1
=
σ
. In such a case,
()
1
222
),(
+== yxdyxL . (a.15)
To simplify further transformations, we use the polar system of coordinates. The polar radius
ρ
and polar angle Ψ are related to Cartesian coordinates x and y in the following way
,
,
)(sin)()(
)(cos)()(
sΨssy
sΨssx
ρ
ρ
=
=(a.16)
.
,
cossin
sincos
ΨΨΨy
ΨΨΨx
!
!
!
!
!
!
ρρ ρρ
+= = (a.17)
Using (a.16), (a.17) and rearranged function L
()
2
)(),(
=
ρρ
sΨsL ,
system (a.14) and equation (a.12) are converted to equalities
()
()
,
,
0))(cossin(sin2
0))(sincos(cos2
23
23
=+++ =++
L
ds
d
L
ds
d
ΨΨΨΨ
ΨΨΨΨ
λρρρρ λρρρρ
!
!
!
!(a.18)
1)()( 222 =+ Ψ
!
!
ρρ
. (a.19)
Equations of system (a.18) have the first integral. Subtracting the second equation multiplied by
Ψ
cos from the first one multiplied by Ψ
sin , we obtain
(
)
(
)
0))(cossin(cos))(sincos(sin 22 =+++
L
ds
d
L
ds
dΨΨΨΨΨΨΨΨ
λρρρλρρρ
!
!
!
!.
After algebraic transformations
0
)( )(
22
2=
++
++
L
L
ds
d
Ψ
Ψ
λρ λρ
ρ
ρ
!
!
!! .
Being a total differential, the left-hand side of the equation above is integrated and the right-hand
side is converted to an unknown constant a. Using the equality obtained by integration, value Ψ
!
is presented as the function of
ρ
25
0,
12>
+
=a
a
Ψ
L
ρλ
!. (a.20)
By substituting (a.20) into (a.18) and (a.19), we obtain the differential system for defining )(s
ρ
and )(sΨ
0
1
1)()(
2
222
>
+
=
=+
a
a
Ψ
Ψ
L
ρλ
ρρ
!
!
!
. (a.21)
To solve system (a.21), the relation between differentials
ρ
and Ψ is used, i. e. if )(Ψ
ρρ
=
then
Ψ
dΨ
d!
!
ρ
ρ
=. (a.22)
Based on (a.21) and (a.22), function )(Ψ
ρ
must satisfy the nonlinear ordinary differential
equation
()
2
2
22
21
ρρλ
ρ
+=
L
a
dΨ
d. (a.23)
The solution of (a.23) is reduced to the integral representation for )(
ρ
ΨΨ =, i. e.
()
C
a
d
aΨ
L
+
+
=22
2
2
1
ρρλ
ρ
. (a.24)
Taking into account condition L
a
λ
2
2>, we make the following change of the variable in
integral (a.24)
()()()
222224
21112,
τκτρλρλατρ
=+= LL a,
where
α
and
κ
are new constants defined through a and L
λ
as
0,10,,
2
42 1
22 =
=
LL
L
LL aaa
λκλκα
λλλ
κ
. (a.25)
Function Ψ is expressed by the elliptic integral of the first kind
[]
CaC
d
aΨL
L
+=
+
=
κρ
τκτ τ
α
κ
λ
λ
κ
,arcsinF
11 222 . (a.26)
26
Inverting function (a.26) with respect to variable
ρ
, and using the expression for the elliptic sine
(see, for instance, [3]),
[]
()
===
φ
κ
φκκ
022 sin1
,sin),am(sin,sn t
dt
uuu ,
we obtain the solution for equation (a.23) in the form
CCCΨΨ aa
LL
L
=
+=
κ
λ
κ
λ
λ
κ
κρ
22 ,,sn)( . (a.27)
To simplify relations between constants L
λ
, a, and
κ
in (a.25), parameter L
λ
is expressed by a
and
κ
()
10,
12
2
+
=
κ
κ
κ
λ
a
L. (a.28)
Substituting the value of (a.28) into expression (a.27), the final representation for )(Ψ
ρ
is
[]
κρ
κ
κ
,sn)( 111 CΨΨ a+= +
+. (a.29)
Based on (a.16) and (a.29), we have the analytical solution for the optimal trajectory ),( yx
[]
[]
,sin,sn)(
,cos,sn)(
111 111
ΨCΨΨy
ΨCΨΨx
a
a
κ
κ
κ
κκ
κ
+= +=
+
++
+(a.30)
with boundary conditions
,arccos,
,arccos,
2
2
1
1
2
2
2
22
1
2
1
22
ρ
ρ
ρ
ρ
x
x
ΨΨyx
ΨΨyx
===+ ===+ (a.31)
where 2
1
2
11 yx +=
ρ
, 2
2
2
22 yx +=
ρ
and
π
Ψ
0 by definition.
Using the arc length differential in the form
(
)
dΨads L
21 1
ρλ
+= , derived from (a.20), the total
length and optimal risk for the trajectory
()
)(),( sysx are
[]
()
dΨCΨal
Ψ
Ψ
++= +
2
1
,sn1 11
21
κκ
κ
, (a.32)
[]
()
dΨCΨR
Ψ
Ψ
a++= +
+
2
1
,sn 11
)1( 2
2
κκ
κ
κ
. (a.33)
27
Values for a,
κ
, and C should be determined in terms of *
l and boundary conditions ),( 11 yx ,
),( 22 yx . In the case when the length constraint is relaxed, the optimal path has bounded length
denoted by l. If *
ll , then the length constraint, *
ll , is inactive and l coincides with l. To
determine the value of l, the optimal path without constraint on the length should be calculated.
For the case when the length constraint is relaxed we have 0=
L
λ
. This implies 0
=
κ
and the
elliptic becomes
[]
uu sin0,sn =. Expression (a.29) is simplified
()
CΨaΨ+= sin)( 1
ρ
,
and the optimal solution is presented
()()
()()
.2coscos)2()(
,2sinsin)2()(
1
1
CΨCaΨy
CΨCaΨx
+= ++=
(a.34)
System (a.34) represents the circle in the parametric form. Excluding parameter Ψ from the
system, we obtain the well-known representation for the circle
()()
2
2
1
2
1)2(cos)2(sin)2( =+aCayCax . (a.35)
Based on solution (a.35), unknown constants a and C are determined from the boundary
conditions ),( 11 yx and ),( 22 yx . If
ϑ
is the angle between vectors ),( 11 yx and ),( 22 yx
++ +
=2
2
2
2
2
1
2
1
2121
arccos
yxyx
yyxx
ϑ
, (a.36)
then
2
12
2
12 )()(
sin
yyxx
a+
=
ϑ
, (a.37)
and value l for the unconstrained path length is determined by aand
ϑ
al
ϑ
=. (a.38)
Notice that a is always nonnegative, since formula (a.36) defines the value of
ϑ
in interval
πϑ
0 and, therefore, 0sin
ϑ
for all
ϑ
.
Consequently, if *
ll then the optimal solution is given by (a.35), (a.37), and in the case
when ll
*, the optimal path is determined by system (a.30), where a,
κ
, and, C must satisfy
equation (a.32) under condition *
ll =.
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