Bearing-based Circumnavigation with Collision Avoidance in Time-varying Graphs under Limited Target Information

The gist: The proposed guidance law works for both static and time-varying interaction graphs under limited target information, achieving asymptotic convergence to circumnavigation while enforcing collision avoidance through a logarithmic Barrier Lyapunov Function.

Problem Formulation

The problem investigates distributed circumnavigation of a stationary target by a heterogeneous team of agents, where access to the target is restricted to a subset called leaders and followers The objective is to design angular-speed control inputs that guarantee asymptotic circumnavigation for all agents while avoiding inter-agent collisions The admissible set Xadm is defined by the constraint that the inter-agent distance must satisfy Pi − Pk > 2Rs, ∀i ≠ k

Collision Avoidance Methodology

Inter-agent collision avoidance is enforced through a logarithmic Barrier Lyapunov Function (BLF), which guarantees forward invariance of the collision-free set Collision avoidance can be guaranteed by enforcing αij < π/3 at all times, where αij is the angular width subtended by another agent at its own centre The design admissible interval Di is defined as n e j i2 ∈ R: e j i2 < α¯ijo, which defines an open domain on which the BLF-based controller is constructed

Guidance Law for Followers

For follower agents, the guidance law is designed using only the heading angle of its designated out-neighbour and the corresponding line-of-sight (LOS) bearing information The distributed guidance law is given by ui = 1/Ai Ci + z1γ˙ j + κi(z 2 1 + ζ 2 2), where κi ∈ R+, and it renders Ai positively invariant while guaranteeing that z1(t) → 0 and ζ2(t) → 0 This ensures that e j i1(t) → e¯ j i1 and e j i2(t) → e¯ j i2, leading to asymptotic circumnavigation with the same angular speed as the leader at the terminus of its directed path

Handling Time-Varying Graphs

The framework is extended to piecewise-static time-varying graphs where switching times have no finite accumulation point, and there exists a finite time Ts ≥ 0 such that G(t) = G¯ for all t ≥ Ts By applying Theorem 3, the collision-free set remains forward invariant for all time and every agent asymptotically circumnavigates the target in the limiting graph G¯ Furthermore, Corollary 1 specifies conditions under which node-entry and node-exit events do not prevent asymptotic convergence

Results and Convergence

Numerical simulations confirm that all follower agents synchronize with the leader’s angular speed and converge to concentric circular paths around the target in static interaction topologies The results also demonstrate that even under switching interaction topologies with a fixed number of agents, all follower agents achieve synchronisation and converge to concentric circular trajectories The framework robustly guarantees coordinated circumnavigation and collision avoidance under both time-varying interactions and dynamically varying agent populations

Conclusion

The proposed framework presents a distributed solution to the circumnavigation problem for a heterogeneous group of unicycle agents around a stationary target, with explicit collision-avoidance guarantees The guidance design acts only on the angular speeds, while the linear speeds are assumed constant and heterogeneous A barrier Lyapunov function-based guidance law was developed to enforce forward invariance of the collision-free set while driving the agents towards the desired circular formation The framework was then extended to piecewise-constant time-varying interaction graphs and to node-entry/node-exit events Numerical simulations demonstrated the effectiveness of the proposed approach in all three cases considered Future work will focus on strengthening the switching graph analysis beyond piecewise static graphs, extending the framework to moving targets and environmental obstacles, and incorporating communication delays and sensing uncertainty into the design.

--- Page 1 ---

Bearing-based Circumnavigation with Collision Avoidance in Time-varying Graphs under Limited Target Information is a distributed control strategy for heterogeneous agents to circumnavigate a stationary target while ensuring collision avoidance, which is critical for applications like boundary surveillance and asset protection

How it works

The system distinguishes between leaders, who know the target location, and followers, who rely only on local neighbour information The guidance design acts only on the angular speeds, while the linear speeds are assumed constant and heterogeneous

Collision Avoidance Methodology

Inter-agent collision avoidance is enforced through a logarithmic Barrier Lyapunov Function (BLF), which guarantees forward invariance of the collision-free set Collision avoidance can be guaranteed by enforcing αij < π/3 at all times, where αij is the angular width subtended by another agent at its own centre

Handling Time-Varying Graphs

The framework is extended to piecewise-static time-varying graphs where switching times have no finite accumulation point, and there exists a finite time Ts ≥ 0 such that G(t) = G¯ for all t ≥ Ts By applying Theorem 3, the collision-free set remains forward invariant for all time and every agent asymptotically circumnavigates the target in the limiting graph G¯

Conclusion

The proposed framework presents a distributed solution to the circumnavigation problem for a heterogeneous group of unicycle agents around a stationary target, with explicit collision-avoidance guarantees The guidance design acts only on the angular speeds, while the linear speeds are assumed constant and heterogeneous A barrier Lyapunov function-based guidance law was developed to enforce forward invariance of the collision-free set while driving the agents towards the desired circular formation The framework was then extended to piecewise-constant time-varying interaction graphs and to node-entry/node-exit events Numerical simulations demonstrated the effectiveness of the proposed approach in all three cases considered.

1. This system can perform distributed circumnavigation of a stationary target by heterogeneous agents while guaranteeing inter-agent collision avoidance through a logarithmic Barrier Lyapunov Function (BLF). This is achieved by controlling only angular speeds to develop a distributed guidance law for followers, which works for both static and time-varying interaction graphs.

2. The AI system can operate robustly in dynamic network environments by handling piecewise-static time-varying graphs with nodeentry and node-exit events, as described in Theorem 3. This allows the agents to maintain collision avoidance and asymptotic convergence even when the number of agents or connectivity changes over time.

3. The system can leverage limited information efficiently by allowing followers to rely solely on local angular measurements and information from a single out-neighbour, which is suitable for practical implementation as it requires no range measurements and no state memory.

4. The AI can maintain formation shape control in dynamic graphs by utilizing the BLF-based guidance law, which ensures that for every follower, the error dynamics converge to zero, meaning e j i1(t) → e¯ j i1 and e j i2(t) → e¯ j i2.

5. In scenarios with multiple leaders, the system can adopt a distributed formation control strategy where leaders determine their desired radii based on local information, ensuring leader-leader collision avoidance is guaranteed by maintaining inter-agent distances above a threshold.

In this paper, we study distributed circumnavigation of a stationary target by a heterogeneous team of agents. Each agent is modelled as a disk rather than a point mass to account for its physical dimensions. The target location is assumed to be accessible only to a small subset of agents, called leaders. The rest, called followers, therefore use only local information available from their designated out-neighbour in the interaction graph characterised by the selection of nearest neighbours. By controlling only angular speeds, we develop a distributed guidance law to circumnavigate a stationary target. The proposed guidance law works for both static and time-varying interaction graphs. Inter-agent collision avoidance is enforced through a logarithmic Barrier Lyapunov (BLF) Function, which guarantees forward invariance of the collision-free set. We show that every follower converges to circumnavigation about the same target as the leader at the end of its directed path in the interaction graph, provided the initial conditions are admissible. Numerical simulations illustrate the effectiveness of the proposed method for both static and time-varying topologies.