Element-based Formation Control: a Unified Perspective from Continuum Mechanics

Kun Cao; Lihua Xie

eess.SYApr 5, 2026

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

Abstract

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 ( $\mathcal{V}_{e}$ ) 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.

Panel of Figure 2 — 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 $V(\mathbf{p})$ on a logarithmic scale, showing exponential convergence.

Panel of Figure 3 — 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 $\mathrm{CoV}(V)$ for rigidity-based (distance-, bearing-, and RoD-based) and element-based (with energy functions $\Psi_{\mathrm{TR}}$ , $\Psi_{\mathrm{TS}}$ , and $\Psi_{\mathrm{TRS}}$ ) controllers with different perturbations to the top-right node of $\mathbf{q}$ .

Multi-Robot Formation Control

Multi-robot formation control coordinates robots to maintain, change, or navigate in groups while meeting task and safety constraints.

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ArtGarage, Indian Institute of Science, India · Department of Aerospace Engineering, Indian Institute of Science, India
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Jul 18, 2026Tao He, Gangshan JingMulti-Robot Formation Control Multi-Agent Control
School of Automation, Chongqing University, Chongqing, 400044, PRC

Jun 22, 2026J/K move · Enter open · S save

Topological Online Learning for Displacement-based Formation Control

Saksham Sharma, Shubhankar Gupta, Sumant A Gunagi, Suresh Sundaram

ArtGarage, Indian Institute of Science, India · Department of Aerospace Engineering, Indian Institute of Science, India

Multi-Robot Formation Control Robust Control Multi-Robot Coordination Adaptive Control

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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.

Abstract

Multi-Robot Formation Control

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