Distributed Circumnavigation Using Bearing Based Control with Limited Target Information

In this paper, we address the problem of circumnavigation of a stationary target by a heterogeneous group comprising of $\textbf{n}$ autonomous agents, having unicycle kinematics. The agents are assumed to have constant linear speeds, we control only the angular speeds. Assuming limited sensing capabilities of the agents, in the proposed framework, only a subset of agents, termed as \textit{leaders}, know the target location. The rest, termed as \textit{followers}, do not. We propose a distributed guidance law which drives all the agents towards the desired objective; global asymptotic stability (GAS) is ensured by using Zubov’s theorem. The efficacy of the approach is demonstrated through both numerical simulations and hardware experiments on the TurtleBots utilising OptiTrack motion capture system.

💡 Research Summary

The paper tackles the problem of enabling a heterogeneous group of autonomous agents, modeled as unicycles with constant forward speeds, to perform coordinated circumnavigation of a stationary target while controlling only their angular velocities. Recognizing that in many practical scenarios only a subset of agents can directly sense the target location, the authors partition the team into leaders, which have access to the target’s position, and followers, which do not. The followers are allowed to sense only bearing (heading) information from neighboring agents, not range.

A directed sensing graph G_S captures which agents can measure the heading and line‑of‑sight (LOS) angle of others. From G_S a communication graph G_C is derived by letting each follower select its nearest sensed neighbor as its unique out‑neighbour; leaders are defined as sink nodes (no outgoing edges). The key structural assumption (Assumption 1) is that every follower has at least one directed path to a leader in G_C, ensuring that target information can propagate through the network.

The core contribution is a bearing‑based distributed guidance law for followers. Defining two error variables for a follower i with out‑neighbour j: e_i1 = γ_j – γ_i (heading difference) and e_i2 = λ_ij – γ_i (difference between LOS angle to j and i’s heading), the proposed angular‑velocity control law is

  ˙γ_i = C₁ (γ_j – γ_i) + C₂ sin(λ_ij – γ_i),

where C₁, C₂ > 0 are design gains. Lemma 3 establishes that if the out‑neighbour j already moves on a stable circular orbit of radius R_j and angular speed ω_j, then maintaining constant e_i1 and e_i2 guarantees that i will also move on a concentric circle with the same angular speed. The proof shows that under these conditions the relative geometry between i and j forms a rigid triangle, forcing i’s trajectory to be a scaled and rotated copy of j’s trajectory centered at the same point.

Stability is proved using Zubov’s theorem, which provides a constructive way to define a Lyapunov‑like function V(z) that satisfies a partial differential equation linking its gradient to a positive‑definite function h(z). By shifting the equilibrium to the origin (z_i = e_i – ē_i) and constructing V and h appropriately, the authors demonstrate that the error dynamics converge to zero globally, i.e., all followers asymptotically inherit the leader’s angular speed and radius. An inductive argument extends this result from a single follower‑leader pair to the entire directed communication graph, establishing global asymptotic stability (GAS) for the whole network.

Simulation studies explore various initial configurations and leader‑follower assignments, confirming that the proposed law drives all agents to the desired circular formation regardless of starting positions. Hardware validation is performed with five TurtleBot 3 platforms (two leaders, three followers) inside an OptiTrack motion‑capture arena. Only bearing measurements are fed to the controllers; no range sensors are used. The experiments show successful convergence to a common circular orbit around the target, with smooth angular motion and minimal steady‑state error, thereby corroborating the theoretical claims and highlighting the practical advantage of requiring only angular control.

The paper’s contributions can be summarized as follows: (1) a novel bearing‑only distributed guidance law that works under limited target information and heterogeneous agent speeds; (2) a rigorous global stability proof using Zubov’s theorem, providing explicit region‑of‑attraction characterization; (3) experimental demonstration on real robots, showing that the method is implementable with low‑cost sensors and simple angular‑velocity actuation. Potential extensions include handling moving targets, coping with communication dropouts, accommodating non‑identical forward speeds, and extending the framework to three‑dimensional aerial or underwater vehicles.