Distributed Bearing-Only Protocol

• Distributed bearing-only protocol is defined as a class of algorithms enabling agents to achieve coordinated tasks using exclusively bearing measurements.
• The control law leverages local angular errors computed via cosine differences, with a tunable parameter a that switches between exponential and finite-time convergence.
• Its decentralized design ensures scalability, invariance under transformations, and practical implementation on systems with limited sensing capabilities.

A distributed bearing-only protocol refers to a class of algorithms and control laws that enable agents in a network (e.g., robots, sensors, UAVs) to achieve collective tasks—such as localization, formation control, consensus, or target circumnavigation—using exclusively inter-agent bearing (direction) measurements, without access to explicit relative distances or global positioning. This constraint, motivated by practical sensing limitations, demands the development of theory, algorithms, and analysis tools uniquely adapted to the geometry and algebraic structure induced by directional, often partial, inter-agent information.

1. Geometric Preliminaries and Sensed Information

Distributed bearing-only protocols are fundamentally built upon agents equipped with local sensors that can determine the bearings of neighbors in a shared or body-fixed frame. Consider $n$ agents in the plane, each with positions $p_i \in \mathbb{R}^2$, arranged in a circular or polygonal topology. Each agent $i$ measures the unit vectors $g_{i-1}$ and $g_i$, corresponding to the bearings to its two neighbors (using modulo-$n$ indexing). The measurable geometric primitive is the angle at agent $i$ subtended by its two neighbors, typically encoded by the cosine:

$$\epsilon_i = \cos\theta_i - \cos\theta_i^* = -g_i^\top g_{i-1} - \cos\theta_i^*,$$

where $\theta_i^*$ is the desired subtended angle at agent $i$.

The key property is that the inner product $-g_i^\top g_{i-1}$ maps directly onto the cosine of the angle between the two bearings—this algebraic structure enables agents to compute angular errors using only bearing measurements. The system's invariance under translation, rotation, and scaling reflects the projective nature of bearing-based shape determination: the target formation remains unchanged under these group actions, as bearings are unaffected.

2. Distributed Control Law Synthesis

Agents are assumed to evolve according to single-integrator dynamics:

$\dot{x}_i = u_i,$

where $u_i$ is the control input for agent $i$. The distributed bearing-only protocol is expressed as

$$\dot{x}_i = \operatorname{sgn}(\epsilon_i)\,|\epsilon_i|^a\,(g_i - g_{i-1}),$$

with $a \in (0,1]$ a tunable parameter, and $\operatorname{sgn}(\cdot)$ the signum function. For the special case $a=1$, the law reduces to a linear feedback in the error:

$\dot{x}_i = \epsilon_i\,(g_i - g_{i-1}),$

for $a\in(0,1)$, the term becomes homogeneous of sublinear degree, resulting in nonsmooth feedback—this induces finite-time convergence, exploiting the well-studied non-Lipschitz characteristics known from finite-time control theory.

This law is fully distributed: each agent updates based solely on its direct bearing measurements. It does not require inter-agent distance, position, or global orientation information, making it particularly suited for implementation on platforms with vision-based or restricted-view sensors.

3. Lyapunov-Based Stability and Convergence Analysis

The analysis of stability leverages a parameterized Lyapunov candidate:

$V = \frac{1}{a+1} \sum_{i=1}^n |\epsilon_i|^{a+1}.$

When $a=1$, this is quadratic and yields classical exponential convergence proofs. For $a\in(0,1)$, the function is non-quadratic, and the differential inequality for $V$ becomes

$\dot{V} \le -\frac{K}{\rho} V^{2a/(a+1)},$

where $K > 0$ and $\rho$ is the total length of the polygon. The exponent $2a/(a+1)$ dictates the stability type:

• For $a=1$, $2a/(a+1) = 1$, yielding exponential decay.
• For $a \in (0,1)$, $2a/(a+1) \in (0,1)$, and standard finite-time stability results guarantee convergence in finite time.

The Lyapunov argument is based on the fact that the mapping from bearing errors to the control input can be expressed via the incidence matrix $E$ of the circular graph and a diagonal matrix $D$ summarizing geometric relationships, ultimately leading to dissipativity of the control law.

4. Parameter Tuning and Stability Switching

The parameter $a$ acts as a smooth switch between exponential and finite-time convergence:

• $a=1$: linear feedback, local exponential stability.
• $a\in(0,1)$: nonlinear, stronger (homogeneous) feedback near equilibrium, finite-time convergence.

This tuning flexibility gives system designers the freedom to trade off between robustness to model uncertainties (exponential regime) or fast finite-time settling (homogeneous regime), which can be desirable for rapid deployment scenarios or systems with strict reconfiguration requirements.

5. Scalability, Invariance, and Protocol Properties

The protocol's core properties—distributedness, scalability, and invariance—stem from its reliance on minimal, local information and the inherent algebraic structure of the circular sensing topology:

• Distributed Implementation: Each agent responds only to its two neighboring bearings without global state knowledge.
• Scalability: The formation size $n$ is unrestricted due to the use of graph-theoretic incidence matrices.
• Flexibility: The formation is uniquely determined by specified angles but remains invariant under similarity transformations; collision avoidance is handled by showing boundedness of the total formation length for sufficiently small initialization errors.

Key practical implications:

• Collision avoidance is not explicitly enforced, but under small angle errors, agents' mutual distances remain bounded, preventing collisions.
• The angle-only constraint leads to broad invariance properties: formations are robust to uniform group movements and global scaling.
• The protocol is implementable with minimal hardware: bearing measurement (e.g., vision or compass), local computation of error, and actuation.

6. Implications for Real-World Distributed Multi-Agent Systems

This protocol is particularly relevant in scenarios where distance sensing is prohibitive or where communication overhead must be minimized, such as underwater robotics, formation flight of UAV swarms, or distributed ground sensor networks. Its robustness to limited information, ease of scalability, and analytic guarantees on convergence (both exponential and finite-time under appropriate settings) provide a compelling tool for coordinated control tasks.

In vision-based systems, where only the direction of a neighbor can be measured (not range), this protocol naturally fits and can be embedded in vision processing pipelines. No explicit synchronization or global orientation agreement is needed, as all operations are performed in the local coordinate frame, modulo the assumption that the agent can extract consistent bearing vectors to two neighbors.

7. Summary

The distributed bearing-only protocol described in the foundational work (Zhao et al., 2012) defines a control law of the form

$$\dot{x}_i = \operatorname{sgn}(\epsilon_i)\,|\epsilon_i|^a\,(g_i - g_{i-1}),$$

with the angle error $\epsilon_i = -g_i^\top g_{i-1} - \cos\theta_i^*$.

By tuning $a$, the protocol guarantees local exponential stability ($a=1$) or local finite-time stability ($a\in(0,1)$). The approach is scalable, requires only local bearing measurements, is robust under translation, rotation, and scaling, and is suited for deployment in systems constrained to partial measurements. The Lyapunov-based analysis rigorously characterizes stability properties, and practical considerations such as bounded agent displacements and implicit collision avoidance are addressed through geometric estimates. This protocol forms a foundational construct in the broader theory of distributed bearing-only control for coordinated multi-agent systems.