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A differential game approach to intrinsic encirclement control

Panpan Zhou, Yueyue Xu, Yibei Li, Bo Wahlberg, Xiaoming Hu

TL;DR

This work tackles intrinsic encirclement control for two competing agent groups by formulating a non-cooperative differential game whose Nash equilibrium induces a formation manifold of two concentric circles, invariant to rotation and translation. The authors design inter-agent topologies and edge weights so that the manifold \mathcal{M}_{\rm T}(x^*) is asymptotically stable, with Nash equilibrium controls given by u^a = -B^T ∂V^a/∂x and u^p = -B^T ∂V^p/∂x derived from coupled HJB equations. A key contribution is achieving encirclement and counter-encirclement purely from inter-agent interactions and network geometry, without prescribing fixed positions, and verifying effectiveness through numerical simulations. The approach advances robust, scalable multi-agent formation control by leveraging intrinsic symmetry, invariant manifolds, and non-quadratic cost structures, with potential extensions to higher-order dynamics and directed networks.

Abstract

This paper investigates the encirclement control problem involving two groups using a non-cooperative differential game approach. The active group seeks to chase and encircle the passive group, while the passive group responds by fleeing cooperatively and simultaneously encircling the active group. Instead of prescribing an expected radius or a predefined path for encirclement, we focus on the whole formation manifold of the desired relative configuration, two concentric circles, by allowing permutation, rotation, and translation of players. The desired relative configurations arise as the steady state resulting from Nash equilibrium strategies and are achieved in an intrinsic way by designing the interaction graphs and weight function of each edge. Furthermore, the asymptotic convergence to the desired manifold is guaranteed. Finally, numerical simulations demonstrate encirclement and counter-encirclement scenarios, verifying the effectiveness of our strategies.

A differential game approach to intrinsic encirclement control

TL;DR

This work tackles intrinsic encirclement control for two competing agent groups by formulating a non-cooperative differential game whose Nash equilibrium induces a formation manifold of two concentric circles, invariant to rotation and translation. The authors design inter-agent topologies and edge weights so that the manifold \mathcal{M}_{\rm T}(x^*) is asymptotically stable, with Nash equilibrium controls given by u^a = -B^T ∂V^a/∂x and u^p = -B^T ∂V^p/∂x derived from coupled HJB equations. A key contribution is achieving encirclement and counter-encirclement purely from inter-agent interactions and network geometry, without prescribing fixed positions, and verifying effectiveness through numerical simulations. The approach advances robust, scalable multi-agent formation control by leveraging intrinsic symmetry, invariant manifolds, and non-quadratic cost structures, with potential extensions to higher-order dynamics and directed networks.

Abstract

This paper investigates the encirclement control problem involving two groups using a non-cooperative differential game approach. The active group seeks to chase and encircle the passive group, while the passive group responds by fleeing cooperatively and simultaneously encircling the active group. Instead of prescribing an expected radius or a predefined path for encirclement, we focus on the whole formation manifold of the desired relative configuration, two concentric circles, by allowing permutation, rotation, and translation of players. The desired relative configurations arise as the steady state resulting from Nash equilibrium strategies and are achieved in an intrinsic way by designing the interaction graphs and weight function of each edge. Furthermore, the asymptotic convergence to the desired manifold is guaranteed. Finally, numerical simulations demonstrate encirclement and counter-encirclement scenarios, verifying the effectiveness of our strategies.

Paper Structure

This paper contains 8 sections, 9 theorems, 58 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Symmetry and invariance on manifolds
  4. Properties of rotation group and translation group actions
  5. Topology design
  6. Game analysis
  7. Numerical simulation
  8. Conclusion

Key Result

Proposition 1

Each manifold $\mathcal{M}_{kl}$ is closed. For any $k_1,k_2=1,\cdots,o^a$, $l_1,l_2=1,\cdots,o^p$, $\mathcal{M}_{k_1,l_1}$, $\mathcal{M}_{k_1,l_2}$, $\mathcal{M}_{k_2,l_1}$ and $\mathcal{M}_{k_2,l_2}$ are either equal or disjoint.

Figures (4)

  • Figure 1: Illustration of the relative positions of active players under the topology (d), when $m=7$.
  • Figure 2: Results of the case $m=6$, $n=6$. (a) The active group achieves encirclement control under the settings $\alpha_1=16$, $\alpha_2=50$, $\beta_1=8$, $\beta_2=7$. (b) The passive group achieves the counter-encirclement control under the settings $\alpha_1=16$, $\alpha_2=5$, $\beta_1=8$, $\beta_2=7$. (c) Topology of the active group. (d) Topology of the passive group.
  • Figure 3: Results of the case $m=7$, $n=5$. (a) The active group achieves encirclement control under the settings $\alpha_1=7$, $\alpha_2=40$, $\beta_1=3$, $\beta_2=2$. (b) The passive group achieves the counter-encirclement control under the settings $\alpha_1=7$, $\alpha_2=1$, $\beta_1=3$, $\beta_2=2$. (c) Topology of the active group. (d) Topology of the passive group.
  • Figure 4: Results of the case $m=12$, $n=4$. (a) The active group achieves encirclement control under the settings $\alpha_1=6.5$, $\alpha_2=40$, $\beta_1=8$, $\beta_2=2$. (b) The passive group achieves the counter-encirclement control under the settings $\alpha_1=7$, $\alpha_2=1$, $\beta_1=6$, $\beta_2=2$. (c) Topology of the active group. (d) Topology of the passive group.

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 4 more