Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost function is numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditions of the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.

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Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

10.36227/techrxiv.15057798 ◽

2021 ◽

Author(s):

Wei Liao ◽

Taotao Liang ◽

Xiaohui Wei ◽

Jizhou Lai ◽

Qiaozhi Yin

Keyword(s):

Cost Function ◽

Level Sets ◽

Bellman Equation ◽

Reachable Sets ◽

Time Horizons ◽

Hamilton Jacobi Bellman Equation ◽

Novel Method ◽

Running Cost ◽

Hamilton Jacobi Bellman ◽

Grid Interpolation

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Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets

2019 IEEE 58th Conference on Decision and Control (CDC) ◽

10.1109/cdc40024.2019.9029193 ◽

2019 ◽

Cited By ~ 1

Author(s):

Morgan Jones ◽

Matthew M. Peet

Keyword(s):

Bellman Equation ◽

Reachable Sets ◽

Hamilton Jacobi Bellman Equation ◽

Hamilton Jacobi Bellman ◽

Outer Bounds

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Quadratization of Hamilton-Jacobi-Bellman Equation for Near-Optimal Control of Nonlinear Systems

2020 59th IEEE Conference on Decision and Control (CDC) ◽

10.1109/cdc42340.2020.9303913 ◽

2020 ◽

Author(s):

Arash Amini ◽

Qiyu Sun ◽

Nader Motee

Keyword(s):

Optimal Control ◽

Nonlinear Systems ◽

Bellman Equation ◽

Hamilton Jacobi Bellman Equation ◽

Hamilton Jacobi Bellman

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Interception of automated adversarial drone swarms in partially observed environments

Integrated Computer-Aided Engineering ◽

10.3233/ica-210653 ◽

2021 ◽

pp. 1-14

Author(s):

Daniel Saranovic ◽

Martin Pavlovski ◽

William Power ◽

Ivan Stojkovic ◽

Zoran Obradovic

Keyword(s):

Bellman Equation ◽

Pid Controllers ◽

Partially Observed ◽

Hamilton Jacobi Bellman Equation ◽

Partial Feedback ◽

Point Solution ◽

Physical Limitations ◽

Defensive Mechanisms ◽

Hamilton Jacobi Bellman ◽

New Framework

As the prevalence of drones increases, understanding and preparing for possible adversarial uses of drones and drone swarms is of paramount importance. Correspondingly, developing defensive mechanisms in which swarms can be used to protect against adversarial Unmanned Aerial Vehicles (UAVs) is a problem that requires further attention. Prior work on intercepting UAVs relies mostly on utilizing additional sensors or uses the Hamilton-Jacobi-Bellman equation, for which strong conditions need to be met to guarantee the existence of a saddle-point solution. To that end, this work proposes a novel interception method that utilizes the swarm’s onboard PID controllers for setting the drones’ states during interception. The drone’s states are constrained only by their physical limitations, and only partial feedback of the adversarial drone’s positions is assumed. The new framework is evaluated in a virtual environment under different environmental and model settings, using random simulations of more than 165,000 swarm flights. For certain environmental settings, our results indicate that the interception performance of larger swarms under partial observation is comparable to that of a one-drone swarm under full observation of the adversarial drone.

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