Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dependent Reachable Sets for the Constant Bearing Pursuit Strategy

Venkata Ramana Makkapati, Tulasi Ram Vechalapu, Vinodhini Comandur, Seth Hutchinson

TL;DR

This work analyzes the dependent reachable set (DRS) for a follower that uses constant bearing pursuit to track a leader in the plane. It derives geometric bounds and an exact region for early times ($0\le t\le t_2$) and provides a tight bound plus a conjecture for later times when the follower's reach encases the leader's. An optimization problem linked to the constant bearing strategy is formulated and explored, with results supported by simulations that reveal ellipse-based switch loci. The findings elucidate the interplay between pursuit geometry and reachability, with implications for pursuit-evasion and cooperative guidance under constant bearing laws.

Abstract

This paper introduces a novel reachability problem for the scenario where one agent follows another agent using the constant bearing pursuit strategy, and analyzes the geometry of the reachable set of the follower. Key theoretical results are derived, providing bounds for the associated dependent reachable set. Simulation results are presented to empirically establish the shape of the dependent reachable set. In the process, an original optimization problem for the constant bearing strategy is formulated and analyzed.

Dependent Reachable Sets for the Constant Bearing Pursuit Strategy

TL;DR

This work analyzes the dependent reachable set (DRS) for a follower that uses constant bearing pursuit to track a leader in the plane. It derives geometric bounds and an exact region for early times () and provides a tight bound plus a conjecture for later times when the follower's reach encases the leader's. An optimization problem linked to the constant bearing strategy is formulated and explored, with results supported by simulations that reveal ellipse-based switch loci. The findings elucidate the interplay between pursuit geometry and reachability, with implications for pursuit-evasion and cooperative guidance under constant bearing laws.

Abstract

This paper introduces a novel reachability problem for the scenario where one agent follows another agent using the constant bearing pursuit strategy, and analyzes the geometry of the reachable set of the follower. Key theoretical results are derived, providing bounds for the associated dependent reachable set. Simulation results are presented to empirically establish the shape of the dependent reachable set. In the process, an original optimization problem for the constant bearing strategy is formulated and analyzed.

Paper Structure

This paper contains 8 sections, 6 theorems, 33 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Reachable Sets
  4. Dependent Reachable Sets
  5. Theoretical Results
  6. Simulation Results
  7. The Optimization Problem
  8. Conclusion

Key Result

Lemma 3.2

The reachable set of an agent in a one-dimensional environment with the dynamics at time $t=T \in [0, \infty)$ is $[-T, T]$, where $x(t) \in \mathbb{R}$, the control function $w \in \mathcal{W}$, and $\mathcal{W}$ is the set of all piecewise continuous functions in time $t\in [0,\infty)$ with range $\{-1, 1\}$.

Figures (7)

  • Figure 1: Schematics of the two distinct scenarios that are considered to characterize the dependent reachable set for the constant bearing strategy
  • Figure 2: Characterization of the DRS for $0\leq t \leq t_2$
  • Figure 3: Independent agent's path for the control function in (\ref{['eq:switch_func_2D_2']}), and the resulting points of both agents for a given vertical coordinate $y$.
  • Figure 4: Characterization of the DRS for $t_2 < t \leq t_c$
  • Figure 5: DRS for the limiting case $v_D = v_I$
  • ...and 2 more figures

Theorems & Definitions (14)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 4 more