Netted Multi-Function Radars Positioning and Modes Selection by Non-Holonomic Fast Marching Computation of Highest Threatening Trajectories & by CMA-ES Optimization

Massively parallel computation of globally optimal shortest paths with curvature penalization

“…We describe the implementation of our massively parallel solver of generalized eikonal PDEs, assumed to be discretized in the form (8). The bulk of the method is split in three procedures, Algorithm 1, 2, and 3, discussed in detail in the corresponding sections.…”

“…The intruder has full knowledge of the radar configuration, and does its best to avoid detection, but is subject to maneuverability constraints as does a fast plane. Following References 7,8 the intruder is modeled as a Dubins vehicle, traveling at unit speed with a turning radius of 0.2, whose trajectory starts and ends at a given point $$$ {x}_{\ast}\in \Omega $$$ and which must visit a target keypoint $$$ {x}^{\ast}\in \Omega $$$ in between § . The problem takes the generic form $$$$ \underset{\xi \in \Xi}{\sup}\underset{\gamma \in \Gamma}{\operatorname{inf}}\mathcal{E}\left(\xi; \gamma \right), $$$$ where $$$ \Xi $$$ is the set of radar configurations, and $$$ \Gamma $$$ is the set of admissible trajectories.…”

“…Following (), a trajectory is represented as a pair , and its cost is defined as where denotes the Dubins cost (, right), and $$$ \rho \left(x,\theta; \xi \right) $$$ is an instantaneous probability of detection depending on the radar configuration $$$ \xi $$$, and on the intruder position $$$ x $$$ and orientation $$$ \theta $$$. We refer to Reference 8 for a discussion of the detection probability model, and settle for a synthetic and simplified yet already nontrivial construction. The detection probability is the sum of three terms $$$ \rho \left(x,\theta; \xi \right)={\sum}_{i=1}^3\tilde{\rho}\left(x,\theta; {y}_i,{r}_i,{v}_i\right) $$$, corresponding to as many radars, each of the following form illustrated on Figure 8, top left.…”

“…The variants of the eikonal PDE corresponding to these models are non‐holonomic (a degenerate form of anisotropy) and are posed on the three dimensional state space , which makes their numerical solution challenging. A dedicated variant of the fast marching method is presented in References 2,3, and together with earlier prototypes it has found applications in medical image segmentation 4‐6 as well as the configuration of surveillance systems 7,8 . However, a weakness of the fast marching algorithm is its sequential nature: the points of the discretized domain are accepted one by one in a specific order, namely by ascending values of the front arrival times, which imposes the use of a single CPU thread managing a priority queue.…”

Partial differential equations and variational methods for geometric processing of images

“…The goal is to maximize the probability of detection of the most dangerous trajectory between a given source and target, which will take advantage of any hideout in the terrain, blind spot or physical limitation in the radar network. The trajectory is only subject to a lower bound in the turning radius, due to the vehicle high speed (see the companion paper [29]).…”