Safe Formation Control of Open Multi-Robot Systems with Connectivity-Preserving Reconfiguration

I. INTRODUCTION

• Changing team membership and interaction links make collision avoidance and connectivity maintenance additional requirements for formation control.The motivation includes battery-limited aerial inspection, where depleted quadrotors return to charge and fresh vehicles take their place.Source: §I · p. 1
• Earlier open-network work addressed consensus and related problems,,,,,, while most approaches assumed unconstrained states or prescribed topology.Fixed-team formation control and first-order open-system BLFs do not directly resolve the second-order case, where the barrier gradient acts through velocity-error dynamics,,,,.Source: §I · p. 1
• The formation manager coordinates robot additions and removals, assigning prospective edges for joining robots or connectivity-critical departures.It establishes bridging edges before a departure that would disconnect the graph and temporarily relaxes prospective-edge upper-distance constraints while recovering the nominal range.Source: §I · pp. 1–2
• The switched open-team system is shown uniformly practically stable under a transition-dependent average dwell-time condition, and is validated in Gazebo with repeated UAV joining and departure.The stability result also establishes preservation of inter-robot constraints.Source: §I · p. 2

II. MODEL AND PROBLEM FORMULATION

• Robot-team composition and interaction topology remain constant between switching instants, which can represent admissions, bridge activation, or departures.The paper uses double-integrator robot dynamics and assumes commanded accelerations can be realized by a faster low-level controller away from aggressive maneuvers.Source: §II · p. 2

A. Interaction topology and inter-robot constraints

• Active undirected interaction edges must remain within admissible inter-robot distances, while the initial graph is assumed connected and its edges initially admissible.The common distance bounds encode collision separation and communication or sensing range.Source: A. Interaction topology and… · p. 2

B. Problem formulation

• Each formation edge is assigned a desired relative displacement between its two robots.Because the objective uses relative positions, translating all desired robot positions by the same amount leaves the formation unchanged.Source: B. Problem formulation · p. 2
• For each active mode, the control objective is asymptotic achievement of the prescribed formation while inter-robot constraints remain satisfied for all time.The formation requirement applies to all active edges and the safety/connectivity constraints to all robots.Source: B. Problem formulation · p. 3

A. Constrained formation control

• The controller uses backstepping and the gradient of a barrier-Lyapunov function to address formation control under collision-avoidance and connectivity constraints.The BLF remains finite inside its constraint set and diverges, along with its gradient, as the boundary is approached.Source: A. Constrained formation control · p. 3
• Each edge's BLF encodes the admissible distance constraints and diverges as the relative distance approaches either bound.The desired relative displacement is a global minimum of the underlying barrier term.Source: A. Constrained formation control · p. 3
• The assumed edge BLF has two isolated critical points: the desired equilibrium and a nondegenerate saddle induced by collision avoidance.The figure depicts the BLF surface and a one-dimensional cut showing the admissible regions and divergence at the distance constraints.Source: A. Constrained formation control · p. 3
• Backstepping first chooses a virtual velocity to drive formation errors, then uses commanded acceleration with damping and passivating terms to stabilize formation- and velocity-error dynamics.The virtual-velocity terms depend on relative states associated with the robot's incident edges.Source: A. Constrained formation control · p. 3

B. Open-Team Reconfiguration

• The formation manager schedules robot additions and removals and assigns the edges required to preserve connectivity during each reconfiguration.Source: B. Open-Team Reconfiguration · p. 4

1) Prospective edges:

• Prospective edges can be assigned before a topology switch, including when robots exceed the nominal upper-distance constraint, and use the forthcoming mode’s desired relative displacements.Source: 1) Prospective edges · p. 4
• The initialization margin ensures the relaxed-distance slack is at least ερ, and positive slack is equivalent to satisfying the relaxed upper-distance constraint.Source: 1) Prospective edges · p. 4
• The feasibility correction limits boundary contraction to what robot motion allows; when inactive, ρk decays linearly to zero in finite time, allowing edge promotion with the nominal BLF well defined.Source: 1) Prospective edges · p. 4

2) Robot addition:

• Because the existing graph is connected, any established edge between the joining robot and a current team member suffices to preserve connectivity after the addition.Source: 2) Robot addition · p. 4
• The joining event is performed once the prospective edges have been established with ρk = 0.Source: 2) Robot addition · p. 4

3) Robot removal:

• If removing robot r leaves the graph connected, it can depart directly; otherwise, the manager selects prospective bridging edges between the resulting connected components.Source: 3) Robot removal · p. 4
• The manager assigns a desired formation for the new mode, with bridging edges using desired relative displacements associated with that forthcoming mode.Source: 3) Robot removal · pp. 4–5
• Robot r remains active while bridging edges are established, and departure occurs simultaneously with edge activation once every required bridge has ρk = 0.Source: 3) Robot removal · p. 5
• The resulting graph is connected, and its desired relative displacements are consistent with the new interaction topology.Source: 3) Robot removal · p. 5

• In edge-based coordinates, the constrained formation objective becomes stabilization of the origin in formation-error coordinates.Source: A. Edge-based formulation · p. 5
• Partitioning graph edges into spanning-tree and remaining edges yields a reduced-order stability problem in the spanning-tree edge states.Source: A. Edge-based formulation · p. 5
• At switching instants, the edge-state transition accounts for changes in error-state dimension and bounded discontinuities associated with newly activated edges or changed neighbor sets.Source: A. Edge-based formulation · p. 5

B. Stability under topology switching

• A transition-dependent average dwell time bounds switching frequency relative to each mode’s active duration.It counts transitions between consecutive modes and compares them with time spent in the destination mode.Source: B. Stability under topology switching · p. 6
• Under the stated assumptions and dwell-time condition, the closed-loop system is uniformly practically stable and each mode’s constraint set is forward invariant.The guarantee applies to initial conditions in Sϕ × R3Nϕ.Source: B. Stability under topology switching · p. 6
• The stability bound accounts for energy added at switches, with the largest number of newly activated edges contributing to that bound.The proof also assumes finitely many possible switching modes.Source: B. Stability under topology switching · p. 6
• Relaxation dynamics can add a bounded term to the Lyapunov derivative, and that term becomes zero once every prospective edge is activated.The resulting derivative bound uses Θς in place of Θ.Source: B. Stability under topology switching · p. 6
• The invariant region excludes the constraint boundary, where the barrier function would diverge, and contains no critical point other than the origin.The proof uses this boundary behavior to establish forward invariance.Source: B. Stability under topology switching · p. 7

V. SIMULATION RESULTS

• Seven quadrotors are simulated with full six-degree-of-freedom rigid-body and rotor dynamics, while a low-level geometric controller realizes the desired translational acceleration.Five robots begin as a chain; two more request admission before three robots request departure.Source: §V · p. 7
• Formation error starts large, recovers rapidly, and converges again after temporary transients at reconfiguration events.The composite Lyapunov function also decreases between consecutive reconfiguration events.Source: §V · p. 7
• Prospective interactions begin outside the nominal range, at distances up to approximately 2.35 m, while their relaxed bounds preserve feasibility until ρk returns to zero.The prospective edges are promoted to established edges after the relaxation has converged.Source: §V · p. 7
• The mission preserves collision avoidance and connectivity through two robot admissions and three departures, using bridge formation before connectivity-critical removals.The left panel tracks established and prospective interaction distances against nominal and relaxed bounds.Source: §V · p. 7, Figure 3 · p. 8

VI. CONCLUSIONS

• The paper presents a distributed BLF-based framework for formation control when robots join or leave over time.The robots’ reconfiguration is coordinated by a formation manager.Source: §VI · p. 8
• For connectivity-critical departures, the manager forms new edges and temporarily relaxes their connectivity constraints while auxiliary dynamics recover the nominal interaction range.The controller enforces collision-avoidance and connectivity constraints.Source: §VI · p. 8
• The switched closed-loop system is shown uniformly practically stable under a transition-dependent average dwell-time condition, and the approach is validated in realistic multi-UAV Gazebo simulations.Source: §VI · p. 8