Decentralized Algorithms for 3D Symmetric Formations in Robotic Networks: a Contraction Theory Approach

Abstract: This paper presents decentralized algorithms for formation control of multiple robots in three dimensions. Specifically, we leverage the mathematical properties of cyclic pursuit along with results from contraction and partial contraction theory to design decentralized control algorithms that ensure global convergence to symmetric formations. We first consider regular polygon formations as a base case, and then extend the results to Johnson solid and other polygonal mesh formations. The algorithms are further augmented to allow control over formation size and avoid collisions with other robots in the formation. The robustness properties of the algorithms are assessed in the presence of bounded additive disturbances and their effect on the quality of the formation is quantified. Finally, we present a general methodology for embedding the control laws on complex dynamical systems, in this case, quadcopters, and validate this approach via simulations and experiments on a fleet of quadcopters.

• The paper's main contribution is the design of decentralized control algorithms that leverage contraction theory to guarantee convergence to desired 3D symmetric formations.
• It employs cyclic pursuit and null space representations to derive linear constraints that ensure formation stability and robustness against bounded disturbances.
• Simulations and quadcopter experiments validate the approach, demonstrating scalability, adaptability, and precise control in dynamic 3D environments.

Decentralized Algorithms for 3D Symmetric Formations in Robotic Networks: A Contraction Theory Approach

This paper addresses the formation control problem within decentralized multi-robot systems operating in three-dimensional (3D) spaces. The study contributes to the development of control algorithms designed to guide multiple robots into symmetric formations, utilizing the mathematical underpinnings of cyclic pursuit and contraction theory. The proposed methodologies are implemented in a decentralized framework, emphasizing robustness, scalability, and independence from centralized control.

The architecture of the proposed control system leverages contraction theory to ensure convergence to desired formations, with robots forming regular polygons or polyhedra in 3D. The foundational structure of these formations is defined through null space representations, where the constraints are articulated as linear equations governing the relative positions and orientations of robots. This encoding facilitates the precise mathematical analysis necessary for ensuring that the entire system converges to a specific geometric configuration.

A key advancement is the algorithm's ability to extend formation control from simple regular polygons to complex polyhedral shapes, including Johnson solids and arbitrary convex polygonal meshes. The paper capitalizes on lexicon from nonlinear system theory, using contraction and partial contraction frameworks to guarantee stability and convergence within these spaces. Through this approach, the control laws ensure that formations, once initiated, asymptotically move towards the desired configuration, negating divergent trajectories.

The algorithms are evaluated for their robustness in the presence of bounded disturbances. Performance bounds are derived, measuring the deviation from a nominal formation under disturbances, capturing how well the system's structure absorbs disruptions and maintains its integrity. The authors further refine the control strategies to allow manipulation of formation size and inter-robot distance, a feature crucial when operating in dynamic environments with shifting constraints.

Practical validation is realized through simulations and experimental setups, using these algorithms onboard quadcopters. A two-tiered control paradigm is employed where high-level symmetric cyclic control laws generate velocity references followed by lower-level velocity tracking controllers to enforce these references on quadcopter dynamics. Experiments confirm the efficacy of the proposed architecture in achieving precise formation control among multiple aerial agents.

Implications for these findings can extend into domains requiring distributed robotic networks, evident in applications like environmental sampling, orbiting satellite networks, and coordinated drone-based event coverage, illustrating the potential for expansive deployment of decentralized control paradigms. Future work could involve optimizing these formation algorithms against performance metrics like energy efficiency or minimizing convergence time, as well as accommodating unexpected internal or environmental dynamics, ensuring broader applicability of the system.

This paper provides a rigorous exploration of decentralized multi-robot formation control, setting a structured pathway for enhancements in coordination strategies of autonomous agents in three-dimensional environments.