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

We address the formation control problem for open multi-robot systems (OMRS), i.e., systems in which robots may join or leave the team during operation and new interaction links are established over time, subject to inter-robot collision-avoidance and connectivity-maintenance constraints. The robots are modeled by double integrators, and interact over a dynamic undirected graph. We design a distributed controller based on the gradient of a barrier-Lyapunov function. To enable team reconfiguration, we introduce a formation manager that coordinates robot additions and removals and establishes prospective edges whenever needed, either to connect a joining robot to the team or to preserve connectivity before a robot departs. The upper-distance constraints of prospective edges are temporarily relaxed through auxiliary dynamics that preserve feasibility while progressively recovering the nominal interaction range. The resulting open-team dynamics are modeled as a switched system, for which we establish uniform practical stability for almost all initial conditions under a transitiondependent average dwell-time condition. Finally, the proposed approach is validated in realistic Gazebo simulations with dynamically simulated quadrotors undergoing repeated joining and departure maneuvers.

We consider an aerial multi-robot team whose composition may change during a long-duration mission as robots join or leave, e.g., to recharge or due to temporary unavailability. The resulting time-varying team is modeled as a switched system [6]. Specifically, let σ: R≥0 → P denote a switching signal describing the current composition of the team, where ϕ = σ(t) ∈ P denotes the active mode. For any interval t ∈ [tl, tl+1), the composition of the team is constant and consists of Nϕ robots, where tl and tl+1 are consecutive switching instants. A switching instant corresponds to a change in the team composition and/or established interaction topology, such as robot admission, bridge activation, or robot departure.

At each mode ϕ ∈ P, the translational dynamics of robot i ∈ [1, Nϕ] are modeled as:

ẋi = vi, (1a)

v̇i = ui, (1b)

The robots exchange information over an undirected interaction graph Gϕ = (Vϕ, Eϕ), where Vϕ is the set of robots participating in the mission and Eϕ ⊆ Vϕ2 is the set of Mϕ active interaction links. We refer to the interaction links as edges, denoted εk with k ≤ Mϕ, and to a pair of robots sharing an edge as neighbors. Accordingly, Ni,ϕ:= [j ≤ Nϕ: εk = (i, j) ∈ Eϕ] denotes the neighbor set of robot i.

For every edge εk = (i, j) ∈ Eϕ, define the relative displacement δk and relative distance dk between two robots as:

δk:= xi − xj

dk:= δk = xi − xj. (2)

We assume, for simplicity, common lower and upper admissible inter-robot distances dmin > 0 and dmax > dmin. The admissible set associated with mode ϕ is then:

Ik,ϕ:= δk ∈ R3: dmin < dk < dmax, εk ∈ Eϕ. (3)

The objective is to design a distributed control law ui such that, for each active mode ϕ ∈ P of the open multi-robot system:

1. the prescribed formation is asymptotically achieved, exk,σ(t) (t) → 0, vi,σ(t) (t) → 0. Moreover, both critical points are isolated.

2. the inter-robot constraints remain satisfied, dmin < dk < dmax, ∀ εk = (i, j) ∈ Eσ(t), for all t ≥ 0.

To address the formation-control problem under interrobot collision avoidance and connectivity-maintenance constraints, we design a distributed controller based on backstepping and the gradient of a barrier Lyapunov function (BLF). We first recall the definition of a BLF:

Definition 1: Consider the system ẋ = f (x) and let I be an open set containing the origin. A BLF is a positive definite function W: I → R≥0, x 7→ W (x), that is C 1, satisfies Ẇ (x) ≤ 0, and has the property that W (x) → ∞, ∇W (x) → ∞ as x → ∂I.

Using (5), the constraint set (3) associated with edge εk ∈ Eϕ can be expressed in formation-error coordinates as:

Ik,ϕ = [exk,ϕ ∈ R3: dmin < exk,ϕ + δdk,ϕ < dmax, k ≤ Mϕ].

We follow a backstepping procedure by first treating the robot velocity vi as a virtual control input for the position subsystem (1a). Specifically, we design a virtual velocity vi,ϕ that drives the formation errors toward the desired formation:

X∗vi,ϕ(exk,ϕ) = −k1aij,ϕ ∇ij W̃ϕ (exk,ϕ), i ≤ Nϕ, j∈Ni,ϕ εk =(i,j) (8)

For each k ≤ Mϕ, we associate with εk a BLF 1 W̃k,ϕ:

W̃k,ϕ (exk,ϕ) = 2/exk,ϕ 2 + Bk,ϕ (exk,ϕ + δdk,ϕ), (9)

where Bk, ϕ (exk,phi + δdk,phi) is continuously differentiable on Ik,ϕ and encodes the constraints defined in (7). It is nonnegative, satisfies Bk, ϕ (δdk,phi) = 0 and Bk, ϕ (exk,phi +δdk,phi) → ∞ as either δk → dmin or δk → dmax.

The actual control input ui is then designed to stabilize the coupled formation- and velocity-error dynamics through damping and passivating terms arising from the backstepping construction:

Xui = −k2aij,ϕ ṽi,ϕ − ṽj,ϕ − k3 ṽi,ϕ j∈Ni,ϕ εk =(i,j) - X∗aij,ϕ ∇ij W̃phi (exk,phi) + v̇i, j∈Ni,εk =(i,j) (12)

We consider a formation manager with knowledge of the current team composition, interaction topology, and robot states. The formation manager schedules robot additions and removals and assigns the edges required to preserve connectivity during each reconfiguration.

1. Prospective edges: Prior to a topology switch, the formation manager may assign a set Eϕp of prospective edges, i.e., edges that are intended to be added to the interaction graph. In particular, a prospective edge εk = (i, j) may be assigned at time ta when the corresponding robots do not satisfy the nominal upper-distance constraint, i.e., dk (ta) = δk (ta) > dmax. A prospective edge is treated as an ordinary interaction, except that its upper-distance constraint is temporarily relaxed from dmax to dmax + ρk, where ρk ≥ 0 is an auxiliary relaxation state whose dynamics drive the relaxed bound back toward the nominal one. Until the prospective edge is physically established, the relative state required by its controller is provided either through multi-hop communication over the current interaction graph, when available, or by the formation manager. For each prospective edge εk = (i, j) ∈ Eϕp, define the relaxed admissible set:

ρ Ik,ϕ:= exk,ϕ ∈ R3: dmin < exk,ϕ + δdk,ϕ < dmax + ρk (13)

For each mode ϕ ∈ P, the corresponding BLF is:

X X ρ W̄ϕ (ex,phi, ρϕ) = W̃ϕ,k (exk,phi) + W̃ϕ,k (exk,phi, ρk), p k∈Eϕ k∈Eϕ (14)

The relaxation is progressively removed in order to recover the nominal interaction range. We consider the projected linear nominal contraction:

• ρ̇ = [−cρ]ρk, cρ > 0, where, given v ∈ R and a ∈ R≥ 0, [v]+a = v if a > 0 and [v]+a = max[0, v] if a = 0. However, the contraction rate of the relaxed boundary dmax + ρk must be compatible with the dynamics of the inter-robot distance dk so as to preserve sk > 0. To this end, we impose:

ṡk ≥ −λρ sk, λρ > 0 (18)

Condition (18) limits the rate at which the margin sk can decrease. Specifically, for sk (ta) > 0, it implies sk (t) ≥ sk (ta)e−λρ (t-ta) > 0, and therefore guarantees preservation of the relaxed upperdistance constraint. Since, for εk = (i, j), [δk,ϕ] = [vi - vj], d˙k = k dk condition (18) is equivalent to ṡk = ρ̇k − d˙k. Accordingly, we select the relaxation rate as the nominal linear decay (17) whenever it satisfies the above condition, and otherwise project it onto the admissible boundary from (21):

• ρ̇ = max [−cρ]ρk, d˙k − λρ sk, (22)

When Eϕp ≠ ∅, the derivative of (37) carries the additional P ∂W ρ term k∈E p ∂ρϕ,k ρ̇k. Enlarging ρk relaxes the upper bound k ϕ ρ in (13) and hence decreases the barrier, so ∂Wϕ,k/∂ρk ≤ 0. From (22), if (17) is active then ρ̇k 0. In the latter case it remains bounded from (18). ρ Consequently there exists s ≥ 0 such that P ∂Wϕ,k/∂ρk ≤ s, and (39) is replaced by V̇ϕ ≤ k∈Eϕ ∂ρk ρ̇k −γϕ Vϕ + s.

We consider the edge-based formulation where the constrained formation objective in (6)-(7) becomes the stabilization of the origin in formation error coordinates. The closed-loop system for t ∈ [tl, tl+1), in spanning-tree formation error coordinates, can be expressed as:

ėxtϕ (t) = −k1 [Let,ϕ ⊗ I3]∇W̄t,ϕ (extϕ (t)) + [Et,ϕ ⊗ I3]ṽϕ (t),

and ṽ˙ ϕ (t) = −k2 [Lϕ ⊗ I3]ṽϕ (t) − k3 ṽϕ(t) -[Et,phi ⊗ I3]∇W̄t,phi (extϕ (t)).

The resulting open-team dynamics are modeled as a switched system with varying topology and dimension, for which uniform practical stability and preservation of the inter-robot constraints are established under a transitiondependent average dwell-time condition.

Proposition 1: Consider the OMRS (1), under Assumptions 1-2, in closed loop with the switching control law (12), where W̄t,ϕ is defined in (14). Let ϕ, ϕ̂ ∈ P be any two consecutive modes, where ϕ̂ precedes ϕ. Define omegaϕ,ϕ̂ = 2/k1 λmin (Le) t,ϕ, 2k3), with κ2 > 0 a positive γϕ = min[κ2 constant. Let c̄ = minϕ∈P W̄t,ϕ (e∗t,phi) > 0, where e∗t,phi denotes the saddle point of W̄t,phi, and for c ∈ (0, c̄) define Sϕ:= [ext,ϕ ∈ Iϕ: W̄t,ϕ (extphi) ≤ c]. If the switching signal σ admits a transition-dependent average dwell time satisfying ln(omegaϕ,ϕ̂) τ ϕ,ϕ̂ ≥ γ ϕ (35), then the origin of the closed-loop system (33)-(34) is uniformly practically stable for all initial conditions such that (extϕ (0), ṽϕ (0)) ∈ Sϕ ×R 3Nphi. Moreover, for every mode ϕ ∈ P, the constraints set Iϕ defined in (8) is forward invariant.

The approach is validated in realistic Gazebo simulations with multiple UAVs repeatedly joining the team from, and returning to, a charging station. The proposed framework is validated in a ROS 2–Gazebo simulation with seven quadrotors. The results show that the proposed controller and formation manager preserve collision avoidance and connectivity while repeatedly adding and removing robots from a multi-UAV system. The complete evolution of the experiment, including robot admission, bridge formation, departures, and return-to-charge maneuvers, is shown in the accompanying video.

Overall, the results show that the proposed controller and formation manager preserve collision avoidance and connectivity while repeatedly adding and removing robots from a multi-UAV system. The complete evolution of the experiment is shown in Fig. 3. The center panels report the formation control performance. Starting from a large initial error, the desired formation is rapidly recovered. Reconfiguration events introduce temporary transients, as expected from the changes in the desired formation and interaction graph, after which the errors converge again. The composite Lyapunov function exhibits the same behavior, decreasing between consecutive reconfiguration events. The left panel of Fig. 3 shows the evolution of the inter-robot constraints. Throughout the simulation, the minimum distance between any two physical robots remains above the collision-avoidance bound, while every established interaction remains within the nominal connectivity range. Prospective interactions are allowed to start outside the nominal range; in the considered mission they are initialized at distances of up to approximately 2.35 m. The corresponding relaxed bounds dmax +ρk preserve feasibility while the prospective robots approach one another, with ρk converging to zero before the associated interactions are promoted to established edges. In particular, three bridge interactions are successfully created before the three planned robot departures. Finally, the right panels of Fig. 3 highlight the open nature of the system, showing the changes in the number of robots and interaction edges throughout the mission.

VI. CONCLUSIONS

We presented a BLF-based distributed control framework for formation control of OMRS with robots joining and leaving the team over time. The proposed formation manager coordinates these reconfigurations by defining new edges to be formed for connectivity-critical departures and temporarily relaxing their connectivity constraints through auxiliary dynamics that tend to recover the nominal interaction range in finite time, while the controller enforces collision avoidance and connectivity constraints. The resulting switched closed-loop system was shown to be uniformly practically stable under a transition-dependent average dwell-time condition, and the approach was validated in realistic multi-UAV Gazebo simulations.

The proposed framework is validated in a ROS 2–Gazebo simulation with seven quadrotors. Gazebo simulates the full six-degree-of-freedom rigid-body dynamics of each vehicle together with the individual rotor dynamics. The distributed controller developed in the previous sections provides the desired translational acceleration, which is realized by a lowlevel geometric controller [24] that computes the commanded total thrust and attitude, followed by rotor allocation. A charging station is included in the environment, where robots that are not currently part of the team remain parked and from which newly joining robots are deployed (Fig. 2). For the numerical validation, the edge potentials are instantiated using the weighted recentered barrier function of [27]. For each k ≤ Mϕ, let δk,ϕ = exk,ϕ + δdk,ϕ denote the corresponding relative displacement. Let κ1,k = d2min / (1/2 δd,phi 2 (δd,phi 2 −dmin 2)) and κ2,k = 2 dmax-δdk,ϕ squared. We use min k k [κ1,k] for the scaling factor.

The overall results show that the proposed controller and formation manager preserve collision avoidance and connectivity while repeatedly adding and removing robots from a multi-UAV system. The complete evolution of the experiment is shown in Fig. 3. The center panels report the formation control performance. Starting from a large initial error, the desired formation is rapidly recovered. Reconfiguration events introduce temporary transients, as expected from the changes in the desired formation and interaction graph, after which the errors converge again. The composite Lyapunov function exhibits the same behavior, decreasing between consecutive reconfiguration events. The left panel of Fig. 3 shows the evolution of the inter-robot constraints. Throughout the simulation, the minimum distance between any two physical robots remains above the collision-avoidance bound, while every established interaction remains within the nominal connectivity range. Prospective interactions are allowed to start outside the nominal range; in the considered mission they are initialized at distances of up to approximately 2.35 m. The corresponding relaxed bounds dmax +ρk preserve feasibility while the prospective robots approach one another, with ρk converging to zero before the associated interactions are promoted to established edges. In particular, three bridge interactions are successfully created before the three planned robot departures. Finally, the right panels of Fig. 3 highlight the open nature of the system, showing the changes in the number of robots and interaction edges throughout the mission.

---

As a fastidious and diligent researcher, I have analyzed the provided scientific paper, Safe Formation Control of Open Multi-Robot Systems with Connectivity-Preserving Reconfiguration. This work addresses the complex problem of maintaining desired formations for multi-robot systems (like UAVs) where robots can dynamically join or leave the team.

Based on this research, here are the specific improvements to AI systems and what those improved systems can achieve:

---

The core contribution of this paper is a robust, distributed control framework that handles dynamic topology changes while guaranteeing both collision avoidance and connectivity preservation. Improving AI systems using these principles leads to agents capable of operating in highly unstructured, mobile, and unpredictable environments.

Here are the specific improvements:

1. Improve the formation control layer from static/fixed-topology consensus to a fully dynamic, switched system control law based on Barrier Lyapunov Functions (BLFs).

2. Integrate a dedicated Formation Manager module capable of proactively managing team membership (joining/leaving) and dynamically establishing necessary communication links.

3. Implement an auxiliary dynamics mechanism that allows the system to temporarily relax strict distance constraints during reconfiguration maneuvers, ensuring feasibility is maintained while the system converges back to its nominal safe operating range.

4. Develop a stability analysis framework that utilizes transition-dependent average dwell-time conditions to rigorously prove uniform practical stability for the entire switched system under dynamic topology changes.

The resulting improved AI systems can perform the following specific tasks:

1. Enhanced Cooperative Aerial Missions (e.g., Search and Rescue, Infrastructure Inspection):

2. Robust Operation in Unpredictable Environments: The agents can maintain a precise geometric formation (e.g., a line, circle, or complex shape) even when teammates are lost or new members join mid-mission (like robots returning to a charging station).

3. Guaranteed Safety During High-Dynamic Reconfiguration: Unlike standard controllers that might violate collision avoidance during robot swaps, this system ensures that as robots join or leave, the necessary communication links are established and the minimum separation distance is always maintained, preventing collisions and loss of coordination.

4. Autonomous Dynamic Team Management: The system can autonomously decide when to request a robot to join (e.g., a returning drone) or when a robot can safely depart (e.g., due to battery depletion), automatically calculating the required bridge connections needed to keep the remaining team connected during the transition.

5. Resilient Communication and Sensing Management: The system is designed to be resilient against intermittent communication links; it can temporarily rely on relaxed constraints during link establishment, ensuring control authority is never lost during critical maneuvers.