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Prescribed Performance Distance-Based Formation Control of Multi-Agent Systems (Extended Version)
Farhad Mehdifar, Charalampos P. Bechlioulis, Farzad Hashemzadeh, Mahdi Baradarannia
TL;DR
The paper addresses robust distance-based formation control under external disturbances while maintaining connectivity and avoiding collisions. It designs prescribed-performance controllers and establishes convergence and maneuvering results under stated rigidity conditions.
Problem
External disturbances, collision avoidance, and connectivity maintenance remain insufficiently addressed in distance-based formation control, while LMI-based disturbance attenuation can increase controller complexity.
Method
The proposed protocol uses prescribed performance bounds and transformed distance errors to constrain formation behavior and maintain connectivity and collision avoidance.
Results
The analysis guarantees connectivity maintenance, collision avoidance, and infinitesimal rigidity, with exact zero error convergence when disturbances are absent or vanish.
Takeaways & Limitations
The approach extends to centroid maneuvering for nominal agents and supports minimally and infinitesimally rigid formations in two- or three-dimensional space.
Takeaways & Limitations
Future work must extend the approach to directed interactions, higher-order or nonlinear dynamics, and neighboring agents’ distance mismatch.
Abstract
from arXiv · showhide
This paper presents a novel control protocol for robust distance-based formation control with prescribed performance in which agents are subjected to unknown external disturbances. Connectivity maintenance and collision avoidance among neighboring agents are also handled by the appropriate design of certain performance bounds that constrain the inter-agent distance errors. As an extension to the proposed scheme, distance-based formation centroid maneuvering is also studied for disturbance-free agents, in which the formation centroid tracks a desired time-varying velocity. The proposed control laws are decentralized, in the sense that each agent employs local relative information regarding its neighbors to calculate its control signal. Therefore, the control scheme is implementable on the agents' local coordinate frames. Using rigid graph theory, input-to-state stability, and Lyapunov based analysis, the results are established for minimally and infinitesimally rigid formations in 2-D or 3-D space. Furthermore, it is argued that the proposed approach increases formation robustness against shape distortions and can prevent formation convergence to incorrect shapes, which is likely to happen in conventional distance-based formation control methods. Finally, extensive simulation studies clarify and verify the proposed approach.
1 Introduction
Distance-based formation control uses local relative measurements to establish desired shapes, but prior work leaves guaranteed performance under disturbances, connectivity, collision avoidance, and maneuvering insufficiently addressed. This paper proposes a decentralized prescribed-performance framework addressing these requirements for rigid formations.
- Distance-based control regulates inter-agent distances using relative neighbor positions, reducing implementation issues compared with position- or displacement-based schemes.
- Prior distance-based studies address stationary formation acquisition and moving formations, including constant-speed tracking, collision avoidance, finite-time convergence, and centroid tracking.
- External disturbances, collision avoidance, connectivity maintenance, and transient-response guarantees remain incompletely addressed together in prior distance-based formation control.
- The paper proposes robust distance-based formation acquisition with guaranteed transient performance, connectivity maintenance, and collision avoidance for minimally and infinitesimally rigid formations.
- The contributions include distance-based maneuvering with guaranteed performance and time-varying reference velocity for arbitrarily oriented local coordinate frames without relative-orientation measurements.
- The analysis combines rigid graph theory, input-to-state stability, and Lyapunov-based arguments, followed by extensive simulations.
2 Preliminaries
The preliminaries define graph representations, relative positions, distance-based rigidity, ambiguities, and stability tools used to analyze formation control. Minimal and infinitesimal rigidity provide the structural conditions underlying the paper’s formation results.
- Graphs: An undirected graph is represented by vertices, edges, neighbor sets, and an arbitrarily oriented incidence matrix.
- Graphs: The incidence matrix produces stacked relative position vectors, whose edge norms define the graph’s distance-based rigidity function.
- Rigidity: A rigid framework preserves the graph’s shape locally when edge lengths are maintained, and its rigidity matrix depends only on relative positions.
- Rigidity: Minimal rigidity means every edge constraint is necessary; in R2 and R3 it corresponds respectively to 2n −3 and 3n −6 edges.
- Rigidity: Infinitesimal rigidity concerns the rigidity matrix, while equivalent but non-congruent frameworks form the ambiguity set Amb(F).
- Rigidity: A sufficiently small shape perturbation preserves infinitesimal rigidity, whereas degenerate collinear or coplanar configurations can violate the required rigidity-matrix rank.
- Stability tools: The analysis also invokes input-to-state stability, Lyapunov criteria for ISS, and Young’s inequality.
3 Problem Statement
The problem concerns controlling disturbed agents toward a rigid target formation while preserving neighbor connectivity and avoiding collisions. The desired distance-error bounds also prescribe transient and steady-state performance.
- Agent model: Each agent follows single-integrator dynamics with position qi, control input ui, and an unknown bounded piece-wise continuous disturbance δi(t).
- Formation model: The desired formation is a minimally and infinitesimally rigid framework in two- or three-dimensional space, with actual positions sharing its graph.
- Distance errors: For each neighboring edge, the relative position is eqij = qi −qj and the distance error compares the actual inter-agent distance with its desired value.
- Safety constraints: Collision avoidance requires neighboring-agent distances to exceed the summed safety radii, while connectivity requires them to remain within their common sensing area.
- Control objective: The control objective is convergence to the isometric target formation while maintaining time-varying upper and lower distance-error performance bounds.
- Performance bounds: Choosing the initial performance bounds relative to sensing and safety distances ensures connectivity maintenance and collision avoidance for all t ≥0.
- Extension: Distance-based formation centroid maneuvering is additionally studied for nominal single-integrator agents.
4 Controller Design and Stability Analysis
The controller uses prescribed distance-error bounds and transformed-error dynamics to preserve rigidity, connectivity, and collision avoidance while maintaining bounded closed-loop behavior under disturbances. Stability analysis establishes forward completeness, prescribed performance, and robustness properties, with correct-shape convergence requiring an appropriate initial formation condition.
- Bound Selection: Distance-error bounds are selected to preserve infinitesimal rigidity, connectivity, and collision avoidance throughout the formation evolution.The bounds are chosen from initial distance errors, sensing and geometric radii, and robustness constants, using a distributed algorithm.
- Transformed Error System: Keeping the transformed errors bounded enforces prescribed performance while the selected performance functions determine the distance-error transient and steady-state behavior.The transformed variables also support internal stability and bounded control inputs.
- Main Result: Theorem 1 establishes that the proposed control law guarantees prescribed performance, connectivity maintenance, and collision avoidance for minimally and infinitesimally rigid desired formations.The result applies to agents in two- or three-dimensional space under the stated initial-condition and bound-selection assumptions.
- Stability Analysis: The proof establishes bounded closed-loop signals and forward completeness by showing transformed errors remain in a compact subset of their admissible domain.Input-to-state stability yields an ultimate bound under bounded disturbance-related input, independent of the maximal-solution time.
- Disturbance Effects: Zero or vanishing disturbances yield exact error convergence, whereas non-vanishing disturbances produce an ultimate error set that can be narrowed by reducing the steady-state performance bounds.The convergence speed is affected by the constants aij governing lower bounds on error-convergence speed.
- Correct-Shape Convergence: For minimally rigid target formations, zero distance errors can still lead to undesired reflected or otherwise ambiguous shapes, so correct convergence requires an initial condition closer to the desired formation.The proposed scheme is reported to prevent disturbance-driven transient motion from moving the formation toward such undesired shapes, improving robustness against shape distortions.
5 Extension to Formation Centroid Maneuvering
The extension addresses centroid maneuvering for disturbance-free agents while preserving the desired formation, prescribed performance, connectivity, and collision-avoidance guarantees. A single leader receives the bounded time-varying centroid-velocity command, and the resulting decentralized controller achieves exact centroid-velocity tracking.
- Problem formulation: A single leader receives the desired time-varying centroid velocity, while the remaining agents maintain the formation shape as the centroid maneuvers.The desired velocity is known only to the leader.
- Problem formulation: The control objective is to ensure the prescribed formation constraints together with centroid velocity tracking, connectivity maintenance, and collision avoidance.The target condition is q̇_c(t) = v_d(t).
- Main result: Theorem 2 states that control law (63) guarantees q̇_c(t) = v_d(t) and the prescribed-performance formation conditions.These guarantees solve the centroid maneuvering problem for the stated agent dynamics.
- Main result: The centroid velocity follows the desired command because the single leader’s pinning contribution reduces the centroid dynamics to q̇_c(t) = v_d(t).The reduction uses the leader’s exclusive access to the reference velocity.
- Robustness: External disturbances can produce small centroid-velocity fluctuations, while vanishing disturbances preserve the desired formation behavior asymptotically.The paper notes that the desired centroid-velocity amplitude is typically larger than disturbance amplitudes in practice.
- Implementation and scope: The maneuvering controller is decentralized and implementable in arbitrarily oriented local coordinate frames, extending prior work to time-varying reference velocities.The leader additionally requires the total number of agents and access to the desired velocity.
6 Simulation Results
Simulations evaluate decentralized formation-control protocols on rigid tetrahedron and pentagon formations under increasing external disturbances. The prescribed-performance law maintains the desired shape more robustly than the conventional robust comparison law, while simulations also examine formation maneuvering.
- Three simulation examples evaluate the effectiveness of the proposed decentralized control protocols.
- Tetrahedron formation: A tetrahedron formation is tested in three-dimensional space using a minimally and infinitesimally rigid graph with six specified inter-agent edges.
- Robustness comparison: The proposed control law (39) is compared with a modified conventional law (66) using prescribed-performance bounds and three disturbance levels.
- Robustness comparison: Under disturbance δ(t), both controllers reach the desired shape, but law (66) does not keep errors within the prescribed-performance bounds and is less robust to formation distortions.
- Robustness comparison: With disturbance magnitude 2δ(t), law (66) fails to ensure convergence to the desired shape, whereas the prescribed-performance law still converges correctly.
- Robustness comparison: With disturbance 4δ(t), law (66) does not ensure the desired shape, while law (39) maintains convergence without modifying controller gains.
7 Conclusion
The paper concludes with a decentralized robust formation controller that guarantees prescribed performance, maintains connectivity, avoids collisions, and supports centroid maneuvering. Its results apply to rigid formations in 2-D or 3-D space, while directed interactions and higher-order or nonlinear agents remain future extensions.
- The proposed decentralized controller provides guaranteed performance for single-integrator agents affected by unknown external disturbances.
- Predefined performance bounds address connectivity maintenance and collision avoidance among neighboring agents.
- For nominal agents, the extension enables formation maneuvering with the centroid tracking a predefined time-varying velocity available only to the leader.
- The controllers are independent of a global coordinate system and target formations are defined by minimally and infinitesimally rigid graphs in 2-D or 3-D space.
- The prescribed transient response is reported to prevent convergence to undesired formation shapes despite external disturbances.
- Future work includes directed interactions, higher-order and nonlinear agent dynamics, and distance mismatch among neighboring agents.