Element-based Formation Control: a Unified Perspective from Continuum Mechanics
Authors: Kun Cao, Lihua Xie
Organizations: Department of Control Science and Engineering, College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China · Shanghai Institute of Intelligent Science and Technology, National Key Laboratory of Autonomous Intelligent Unmanned Systems, and Frontiers Science Center for Intelligent Autonomous Systems, Ministry of Education, Beijing 100816, China · School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798
This paper establishes a unified element-based framework for formation control by introducing the concept of the deformation gradient from continuum mechanics. Unlike traditional methods that rely on geometric constraints defined on graph edges, we model the formation as a discrete elastic body composed of simplicial elements. By defining a generalized distortion energy based on the local deformation gradient tensor, we derive a family of distributed control laws that can enforce various geometric invariances, including translation, rotation, scaling, and affine transformations. The convergence properties and the features of the proposed controllers are analyzed in detail. Theoretically, we show that the proposed framework serves as a bridge between existing rigidity-based and Laplacian-based approaches. Specifically, we show that rigidity-based controllers are mathematically equivalent to minimizing specific projections of the deformation energy tensor. Furthermore, we establish a rigorous link between the proposed energy minimization and Laplacian-based formation control. Numerical simulations in 2D and 3D validate the effectiveness and the unified nature of the proposed framework.
Figures & tables
Fig. 1 : (a) Simplicial complex representation of the multi-agent formation. The numbered circles and solid black lines denote the physical agents and the 1D communication topology, respectively. The macroscopic formation is tiled by five 2D simplicial elements (colored regions). The red triangular nodes ( Ve ) and dashed lines illustrate the corresponding dual graph, which models the adjacency and interaction pathways among neighboring elements. (b) Illustration of the deformation gradient mapping a reference element to its current configuration in 2D space.
Fig. 2 : Simulation results for 2D formation control ( N=6 ). (a) The trajectories of agents under different control laws. The colored arrows at the final positions of the Translation-and-Rotation-invariant and Similarity-invariant modes indicate the local coordinate frames of the agents, demonstrating that the final formation orientation is achieved without a global reference frame. (b) The evolution of the potential energy V(p) on a logarithmic scale, showing exponential convergence.
Fig. 3 : Simulation results for 3D formation control ( N=7 ). (a) 3D trajectories of the agents. The final configurations show the formation of a 3D heart-shaped structure (a pyramid with a heart base). (b) The energy evolution confirms the stability and fast convergence of the proposed methods in 3D space.
Fig. 4 : Heatmap of CoV(V) for rigidity-based (distance-, bearing-, and RoD-based) and element-based (with energy functions ΨTR , ΨTS , and ΨTRS ) controllers with different perturbations to the top-right node of q .
This paper addresses the problem of robust formation control by introducing Topological Online Learning for Displacement-based (TOLD) formation control, a real-time edge-level adaptation framework. Unlike conventional node-level robust controllers that regulate individual robot inputs without modifying the interaction topology, TOLD updates the interaction topology weights online to directly minimize formation distortion. Two strategies are proposed under the TOLD formation control framework: Online Gradient Flow (OGF) with unconstrained weights and Online Exponential Gradient Flow (OExpGF) with non-negative convex weights. Theoretical analysis establishes that, for single-integrator agents over directed graphs, OExpGF guarantees asymptotic consensus, while OGF ensures bounded formation distortion. Simulations with twelve robots under intermittent disturbances show 1.2%-33.14% median cumulative Root Mean Distortion Error reduction when augmenting TOLD with node-level controllers. Hardware experiments with Crazyflie 2.0 quadrotors demonstrate over 62% (OGF) and 31.4% (OExpGF) reduction in median formation distortion compared to fixed-weight consensus.
Saksham Sharma, Shubhankar Gupta, Sumant A Gunagi +1
ArtGarage, Indian Institute of Science, India · Department of Aerospace Engineering, Indian Institute of Science, India
This paper presents an energy-based controller for a multiagent robotic system designed to achieve and maintain a specific formation while moving on a plane and avoiding collisions between agents. The controller emulates a network of elementary spring-damper modules connecting pairs of agents. This network, with its de-energized states representing the desired formation, determines the system's dynamics, which is fully encapsulated by a bond graph model. The modeling is further enhanced through the introduction of a formation matrix, using a graph-theoretic approach, that describes both the distances and relative velocities among the agents of the arrangement. This matrix mathematically represents the interconnection and energy-exchange structure of the bond graph, allowing us to put it in correspondence with the control-by-interconnection CbI-scheme of the IDA-PBC theory, facilitating the solution of the formation control problem within the port-Hamiltonian system framework. Furthermore, the paper presents leader-following and position-based formation control systems based on the CbI scheme, including a stability analysis of the corresponding closed-loop systems. The theoretical findings are validated through numerical simulations across various scenarios.
Martín Crespo, Sergio Junco, Matías Nacusse
Automation and Control Systems Laboratory - FCEIA, National University of Rosario, Argentine
Non-uniform scaling control enables a multi-agent formation to adjust its shape by compressing or stretching independently along different coordinate axes through inter-agent interactions, offering high flexibility in complex environments. The fundamental idea is encoding the desired formation shape as the kernel of a matrix-valued Laplacian. In open multi-agent systems, however, changes in number of agents, number of edges, and leader selection dynamically alter this Laplacian, destroying the required spectral properties: positive semidefiniteness, correct kernel, and positive definiteness of the follower block (we summarize these properties as the formation spectrum). In this paper, we develop distributed protocols to strategically adjust partial weights of the Laplacian matrix for formation control in arbitrary dimensional space. By implementing the protocols, the desired formation spectrum can be preserved under dynamic topology changes including agent joining, edge addition, agent leaving, and edge removal, while any pair of agents can serve as leaders. Unlike existing Laplacian design methods for affine formation control under topology changes, the proposed approach requires a sparser sensing graph, avoids a predefined parent-child hierarchical structure, and supports leader reassignment. The effectiveness of the proposed protocols is validated through both theoretical analysis and numerical simulations.
Tao He, Gangshan Jing
School of Automation, Chongqing University, Chongqing, 400044, PRC