Multi-objective Compositions for Collision-Free Connectivity Maintenance in Teams of Mobile Robots

I. INTRODUCTION

• As robot teams and tasks grow more complex, designing one controller to meet several objectives becomes increasingly difficult, motivating formal objective composition.Examples include shape formation, collision avoidance, and connectivity maintenance.Source: §I · p. 1
• Existing approaches either do not extend easily beyond a specific task, lack provable guarantees, or provide task-specific solutions.The recentered barrier function was constructed for go-to-goal tasks, cascaded filters lack guarantees, and connectivity-preserving flocking is task-specific.Source: §I · p. 1
• Earlier multi-robot safety certificates assembled multiple barrier functions, but a common solution becomes unclear as the number of objectives increases.The paper motivates composing those barriers into a single barrier function so solutions ensuring multiple objectives always exist.Source: §I · p. 1
• Secondary controllers for collision avoidance and connectivity can dominate behavior when robots are too concentrated or scattered, preventing higher-level objectives from being achieved.The proposed secondary controller is minimally invasive, intervening when collisions or connectivity losses are imminent.Source: §I · p. 1
• Compositional barrier functions combine non-negotiable objectives using AND and OR operators, with conditions for composability and provable guarantees.The paper uses them to synthesize safety and connectivity certificates for teams performing general coordination tasks.Source: §I · p. 1

II. PIECEWISE SMOOTH CONTROL BARRIER FUNCTIONS

• Piecewise barrier functions provide the groundwork for composing barriers because AND and OR composition can make initially smooth functions piecewise smooth.The paper describes control barrier functions as guaranteeing forward invariance without explicitly computing forward reachable sets.Source: §II · p. 2
• A PCr-function is a continuous selection of finitely many Cr functions locally, and operations including pointwise maximum and minimum preserve this function class.The paper notes that PCr-functions are locally Lipschitz continuous and B-differentiable.Source: §II · p. 2
• The B-derivative of a piecewise function is selected directionally from the derivatives of its selection functions.The admissible controls are constrained using this derivative, and the system is assumed forward complete with locally Lipschitz dynamics.Source: §II · p. 2
• A piecewise barrier function can encode a desired set so that a suitable controller renders the set forward invariant.For the stated admissible control set, Theorem 2.1 gives this result for Lipschitz continuous controllers.Source: §II · p. 2
• Theorem 2.1 establishes forward invariance when the controller stays in the admissible set defined by the piecewise barrier condition.The proof uses the comparison lemma to show the barrier remains positive over time.Source: §II · p. 2

III. COMPOSITION OF MULTIPLE OBJECTIVES

• The method treats each objective as a desired set in which system states must remain, then composes multiple non-negotiable objectives with logical operators.The section presents this as a provably correct composition method with conditions for checking composability.Source: §III · p. 3
• The sum and product barrier constructions encode the union and intersection of objective sets, respectively, allowing two objectives to be composed with OR or AND.The set correspondence is established by Lemma 3.1.Source: §III · p. 3
• For a valid OR-composite barrier function, any Lipschitz continuous controller in its admissible control set makes the union of the two objective sets forward invariant.The composite barrier must be checked for validity before applying the theorem.Source: §III · p. 3
• The OR composition requires a validity check; its stated condition ensures a nonempty admissible control space, while an easier, more restrictive condition guarantees a common control satisfying both constraints.Source: §III · p. 3
• For a valid AND-composite barrier function, any Lipschitz continuous controller in its admissible control set makes the intersection of the two objective sets forward invariant.Source: §III · p. 3

• Synthesizing a single controller for multiple goals, including formation, area coverage, collision avoidance, and connectivity, is challenging.Source: §IV · p. 3
• The section applies compositional barrier functions to provide provable safety, defined as collision avoidance, and connectivity for mobile-robot teams.Source: §IV · p. 3
• The safety and connectivity requirements are considered alongside higher-level collective behaviors.Source: §IV · p. 3

A. Composite Safety and Connectivity Barrier Certificates

• The robot model uses position, velocity, and acceleration inputs, with velocity and acceleration bounded by β and α, respectively.Source: A. Composite Safety and Connectivity… · p. 3
• Pairwise safety constraints use the worst case of maximum braking to avoid collision, while connectivity constraints use maximum acceleration to avoid exceeding the distance bound.Source: A. Composite Safety and Connectivity… · p. 4
• The required graph specifies robot pairs that must remain within connectivity distance Dc, and the team is connected when its state stays in the resulting connectivity set for all time.Source: A. Composite Safety and Connectivity… · p. 4
• The composite target set intersects the team’s pairwise collision-safe sets with the sets for all required connectivity edges.The safety and connectivity sets are forward invariant under the theorem’s admissible controller.Source: A. Composite Safety and Connectivity… · pp. 4–5
• For any required connectivity graph, a valid composite barrier function and a Lipschitz continuous controller in its admissible set guarantee that the robot team remains safe and connected.The composite function must first be shown valid for the target set.Source: A. Composite Safety and Connectivity… · pp. 4–5

B. Minimally Invasive Optimization based Controller

• The nominal controller assumes safety and connectivity, and the QP lets it execute as long as it satisfies the composite barrier certificates.Source: B. Minimally Invasive Optimization… · p. 5
• A QP-based controller modifies a nominal command with minimal possible impact in the least-squares sense when safety or connectivity violations are imminent.Source: B. Minimally Invasive Optimization… · p. 5
• The QP-based controller unifies the nominal controller’s higher-level objectives with safety and connectivity requirements encoded by the barrier certificates.Source: B. Minimally Invasive Optimization… · p. 5

C. Maintaining Dynamical Connectivity Graphs

• Dynamic connectivity allows the team to remain connected while switching among allowable graphs as robot states and the environment change.The team must satisfy at least one of the allowable connectivity graphs at any time.Source: C. Maintaining Dynamical Connectivity… · p. 5
• The composite PBF encodes the requirement that the ensemble state remain in the set for safe, dynamically connected operation.The safety and dynamical connectivity guarantees are obtained using a composite PBF.Source: C. Maintaining Dynamical Connectivity… · p. 6
• The admissible control space defined by the composite PBF is always non-empty.This follows from the stated validity of the composite PBF on the safe, dynamically connected set.Source: C. Maintaining Dynamical Connectivity… · p. 6
• A Lipschitz continuous controller within the admissible control space guarantees that the team remains safe and dynamically connected.The guarantee applies to any set of allowable connectivity graphs and the composite PBF defined for it.Source: C. Maintaining Dynamical Connectivity… · p. 6

V. ROBOTIC IMPLEMENTATIONS

• The certificates were tested on a team of four Khepera robots.Realtime robot positions were tracked with the Optitrack Motion Capture System; communications and controls used ROS.Source: §V · p. 6
• Optitrack tracked the robots’ realtime positions during the implementation.Multi-robot communications and controls were executed on the Robot Operating System (ROS).Source: §V · p. 6
• The nominal waypoint controller directed each robot through three waypoints in sequence without accounting for safety or connectivity.The planned waypoints were intentionally arranged to create multiple potential collisions.Source: §V · p. 6

A. Composite Safety Barrier Certificates

• The composite safety barrier certificates wrapped the nominal waypoint controller through a QP-based strategy to avoid all possible pairwise collisions.Connectivity constraints were not included in this experiment.Source: A. Composite Safety Barrier Certificates · p. 6
• The nominal waypoint trajectories were planned for four robots, with each line representing a robot’s trajectory under the waypoint controller.The planned waypoints were designed to create potential collisions.Source: Figure 1 · p. 6
• All inter-robot distances remained above the 0.15 m safety distance, and the robots visited their waypoints without collisions.Without connectivity constraints, the team sometimes disconnected, including splitting into two parts.Source: Figure 2 · p. 6, A. Composite Safety Barrier Certificates · p. 6

B. Composite Safety and Connectivity Barrier Certificates

• The composite PBF encoded collision avoidance, a permanent robot 2–3 connection, and alternative links from robots 1 and 4 to robot 2 or 3.The certificates were wrapped around the waypoint controller using the QP-based strategy.Source: B. Composite Safety and Connectivity… · pp. 6–7
• All safety and connectivity requirements were satisfied: pairwise distances exceeded the safety threshold, while the specified connectivity links stayed within range.Robot 2 remained connected to robot 3; robot 1 and robot 4 each remained connected to robot 2 or 3.Source: B. Composite Safety and Connectivity… · p. 7, Figure 3 · p. 7
• The robots skipped the last waypoint because robot 1 could not reach it without breaking connectivity to robots 2 and 3.The experiment indicates that not all higher-level objectives are compatible with the safety and connectivity constraints.Source: B. Composite Safety and Connectivity… · p. 7

VI. CONCLUSION AND FUTURE WORK

• AND and OR operators compose multiple non-negotiable objectives with provable guarantees, subject to stated composibility conditions.The resulting safety and connectivity certificates are combined with general higher-level objectives using an optimization-based controller.Source: §VI · p. 7
• Robotic experiments validated the effectiveness of the synthesized composite safety and connectivity barrier certificates on a team of four mobile robots.The Fig. 5 experiment used Ds = 0.15m and Dc = 0.6m.Source: §VI · p. 7, Figure 5 · p. 8
• Fig. 4 depicts four robots executing waypoint controllers regulated by safety barrier certificates, with trajectories, positions, velocities, and waypoint markers shown.Fig. 5 depicts waypoint controllers regulated by safety and connectivity certificates, with projected inter-robot connectivity lines.Source: Figure 4 · p. 7, Figure 5 · p. 8