2. Fundamental Principle of Track Deception
The fundamental principle of track deception is to exploit the ranging principle of common pulse system radar. The UAVs, equipped with DRFM, retransmit intercepted radar signals after a delay, causing the radar to perceive these signals as phantom targets. Multiple continuous phantom waypoints can thereby form a phantom track.
The radar network system often employs the homology test rules to verify identified targets. When radars transmit the position information of a target to the fusion center, the fusion center considers it a valid track point if multiple radars report the same spatial positions simultaneously.
Figure 1 illustrates how multiple continuous valid track points can form a reasonable track, with dashed lines representing the lines of sight of radars and wavy arrows indicating the flight trajectories of UAV or phantom target. To deceive the radar network system, the UAV swarm must create spatially overlapping phantom targets for multiple radars at continuous intervals, passing the homology test and generating a phantom track.
To successfully conduct phantom track deception, some assumptions are made to simplify the considered problem as follows:
Assumption 1: In the radar network system, the position information and transmitting signal parameters of each radar are known, and all radars operate as single-input single-output systems;
Assumption 2: The UAV can only deceive the same radar to generate one or more phantom targets at the same time;
Assumption 3: The flight velocity, flight height, pitch angle, and course angle of the UAV are all controlled within a reasonable range. The UAV can fly to the specified position at the specified times according to the preset flight path to retransmit the specified parameter signals.
4. Solution Technique
The formulated optimization model in Equation (16) is a mixed-integer programming, multivariable, and non-linear optimization problem, which is difficult to solve directly. Due to the LOS criterion, the UAVs must be positioned along the line connecting the phantom targets and the radar. Although directly solving problem (16) may yield an optimal solution, finding a feasible solution that simultaneously ensures efficient path planning and maximizes the number of phantom targets satisfying the LOS criterion is challenging and time-consuming. Therefore, in this paper, we leverage a three-stage solution methodology by incorporating the mission planning based on platform reuse and PSO to tackle this problem. The specific solving stages are as follows:
Stage (1) Problem Partition: Since the radar, UAVs, and phantom targets must strictly adhere to the LOS criterion, obtaining a feasible solution is challenging if simultaneously considering the mission planning and trajectory optimization. Therefore, Equation (16) is partitioned as
and
Stage (2) The Mission Planning for UAV Swarm: As shown in
Figure 3, multiple phantom tracks generated by the same UAV also adhere to the LOS criterion. Hence, a phantom track generation method based on the proportional factor is first proposed.
In the space cartesian coordinate system with radar as the origin, as shown in
Figure 2, the proportional factor
of the
-th phantom target associating with the
-th phantom target at time
is defined as
where
and
denote the
-th phantom target and
-th phantom target range from radar, respectively;
denotes the constant proportional factor, which gives the basic range of motion for the
-th phantom target; and
denotes the fluctuation proportional factor at time
, which makes the trajectory of the
-th phantom target different from that of the
-th phantom target.
Therefore, the coordinates of the
-th phantom target at time
are
Building on this basis and considering the hardware constraints, it is assumed that each UAV can generate at most two phantom targets. Equation (17) is then solved by conducting the mission planning based on platform reuse.
Suppose that the phantom target needs to pass the homology test of
radars. As shown in
Table 1, for the first phantom track, the 1st to
-th UAVs generate the first phantom track and pass the homology test. For the
-th phantom track
, the
-th UAV is used as the first UAV in this group, and
additional UAVs are introduced to ensure that the generated
-th phantom track passes the homology test. Subsequently, the task assignment for the remaining UAVs is carried out sequentially.
This mission planning ensures that at least one UAV in each UAV group generates two phantom targets, and the number of UAVs required to generate
phantom tracks is as follows:
where
denotes the number of UAVs.
This is fewer than the UAVs required under the mission planning where UAVs are not reused.
Given
UAVs
, the maximum number of phantom tracks by applying the above mission planning can be computed as follows:
where
represents the maximum number of generated phantom tracks and
represents the round-down operator.
If
is not an integer, some UAVs in the swarm will not be utilized. Therefore, consider dividing the UAV swarm into
clusters, and for each cluster, assign the UAV-generated phantom targets according to the mission planning given in
Table 1, ensuring that each UAV is utilized efficiently.
Specifically, if the number of phantom tracks generated by the
-th UAV cluster is
, then the number of UAVs required by the above mission planning is
where
denotes the number of UAVs in the
-th UAV cluster.
Since the number of UAVs in the UAV swarm is
, then
By substituting Formula (23) into Equation (24), we obtain the following result:
where
represents the number of phantom targets generated by UAV swarm.
Therefore, to maximize the number of phantom tracks generated by the UAV swarm,
must be minimum and satisfy
. So
can be computed as follows:
where
represents the round-down operator. By substituting
into Formula (25), we can obtain
.
After obtaining
and
from Formula (26), the number of phantom tracks
generated by each UAV cluster can be allocated, and it needs to satisfy Equation (27):
After obtaining
, the number of UAVs and the mission planning of the
-th UAV cluster can be obtained from Formula (23) and
Table 1.
For a given phantom target, the UAV being first utilized should select a radar that has not yet been deceived by other UAVs in the group. Furthermore, a UAV that has generated a phantom target should continue to deceive the radar that was deceived when the UAV was first utilized.
Stage (3) The Trajectory Optimization for UAV Swarm with Given Mission Planning: After conducting mission planning, the trajectories of UAVs can be identified by solving the optimization model (18). This optimization model is a multivariable, non-linear optimization problem. Given the PSO algorithm’s fast convergence and strong global optimization capabilities in high-dimensional problems, we employ this algorithm to solve Equation (18).
The idea behind the PSO algorithm originates from the study of bird flocking behavior, where birds find the optimal destination by sharing information collectively as a group. Assuming that the group consists of
particles, the position vector and velocity vector of the
-th particle in the search space in the
-th generation are, respectively,
where every vector in
represents the vector composed of the initial UAV flight height and the velocity angles at every time;
represents the rotation angles vector of the phantom track; and
represents the particle velocity of the above parameters.
Each iteration cycle of PSO generates a new position state according to its position vector, velocity vector, individual history information, population information, and disturbance. In the algorithm, the
-th particle in the
-th generation on the
-th dimension is calculated as follows:
where
denotes inertia weight;
and
denotes learning factor;
and
are randoms on
;
and
denote the velocity of
-th particle in the
-th and (
t + 1)-th generation on the
-th dimension, respectively;
and
denote the position of
-th particle in the
-th and
-th generation on the
-th dimension, respectively; and
and
denote the best position of the
-th particle and the swarm in the
-th generation on the
-th dimension, respectively. The detailed steps of the PSO algorithm for trajectory optimization are outlined in Algorithm 1, which provides a framework for obtaining the optimal flight trajectories and phantom track rotation angles while adhering to the constraints imposed by UAV kinematic performance and phantom track rotation angle.
Algorithm 1: The Detailed Steps of the PSO Algorithm for Trajectory Optimization in UAV Swarm |
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