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Finite time max-consensus for simultaneous target interception in switching graph topologies

Kushal P. Singh, Aditya K. Rao, Twinkle Tripathy

TL;DR

The paper addresses simultaneous interception of a stationary target by a group of heterogeneous pursuers modeled as unicycles under switching communication topologies. It introduces a max-consensus–based distributed guidance law that aligns each pursuer's estimated time-to-interception $\tilde{t}_i$ to a common value, with the leader being the maximum $\tilde{t}$ and capable of switching during the mission; pursuers follow trajectories formed by straight lines and circular arcs. Theoretical results establish finite-time consensus and interception for both static graphs and switching graphs, including dwell-time bounds when the leader loses global reachability, and allow for dynamic changes in the leader. Numerical simulations validate the finite-time convergence and demonstrate robust performance under various topology switching scenarios and node changes.

Abstract

In this paper, we propose a distributed guidance law for the simultaneous interception of a stationary target. For a group of `n' heterogeneous pursuers, the proposed guidance law establishes the necessary conditions on static graphs that ensure simultaneous target interception, regardless of the initial conditions of the pursuers. Building on these results, we also establish the necessary conditions for achieving simultaneous interception in switching graph topologies as well. The major highlight of the work is that the target interception occurs in finite time for both static and switching graph topologies. We demonstrate all of these results through numerical simulations.

Finite time max-consensus for simultaneous target interception in switching graph topologies

TL;DR

The paper addresses simultaneous interception of a stationary target by a group of heterogeneous pursuers modeled as unicycles under switching communication topologies. It introduces a max-consensus–based distributed guidance law that aligns each pursuer's estimated time-to-interception to a common value, with the leader being the maximum and capable of switching during the mission; pursuers follow trajectories formed by straight lines and circular arcs. Theoretical results establish finite-time consensus and interception for both static graphs and switching graphs, including dwell-time bounds when the leader loses global reachability, and allow for dynamic changes in the leader. Numerical simulations validate the finite-time convergence and demonstrate robust performance under various topology switching scenarios and node changes.

Abstract

In this paper, we propose a distributed guidance law for the simultaneous interception of a stationary target. For a group of `n' heterogeneous pursuers, the proposed guidance law establishes the necessary conditions on static graphs that ensure simultaneous target interception, regardless of the initial conditions of the pursuers. Building on these results, we also establish the necessary conditions for achieving simultaneous interception in switching graph topologies as well. The major highlight of the work is that the target interception occurs in finite time for both static and switching graph topologies. We demonstrate all of these results through numerical simulations.

Paper Structure

This paper contains 13 sections, 16 theorems, 18 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement
  4. Cooperative guidance law
  5. Attributes of circular trajectory
  6. Guidance law
  7. Consensus in static graphs
  8. Consensus in switching graphs
  9. Switching graphs with fixed nodes
  10. Switching graphs with changing nodes
  11. Feasible desired times of impact
  12. Simulation results
  13. Conclusion

Key Result

Lemma 1

Consider a set of $n$ pursuers with kinematics given by eqn. eq:kinematics. If a pursuer $i$, where $i\in \{1,2,...,n\}$, keeps on moving in a straight line, then $\tilde{t}_i(t)>0~\forall t\geqslant 0$ and it eventually increases monotonically.

Figures (10)

  • Figure 1: Basic engagement geometry
  • Figure 2: Motion of the pursuer
  • Figure 3: The figures show the plots of $t\sim\tilde{t}$ for different motions of pursuers $q$ and $m$.
  • Figure 4: Bounds on evolution of $\tilde{t}$
  • Figure 5: The figure illustrates the graphs for the time interval $[t_1,t_2+\delta_2)$ for a system of $5$ pursuers. The yellow node denotes the leader and the cyan graph represents the first graph where the switch occurs from $\mathcal{G}^c_{i-1}(t)$ to $\mathcal{G}^d_i(t)$. Dotted edges indicate edges removed from the previous graph. For $t\in[t_1,t_1+\delta_1)$, all graphs are of the form $\mathcal{G}^d_1(t)$, where the leader is not globally reachable. For $t\in[t_1+\delta_1)$, the graphs are of the form $\mathcal{G}^1_c(t)$ where the leader remains globally reachable at all times.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 2
  • ...and 23 more