Deformable Virtual Structures for Swarm Coordination

• Deformable Virtual Structures (DVS) are virtual coordination objects that continuously adjust their centroid, scale, and affine shape to guide swarm formations in dynamic, cluttered environments.
• They enable adaptive swarm flight by integrating centralized path planning with distributed local collision avoidance to maintain formation integrity amid varying agent numbers.
• DVS implementations leverage reduced-order models and low-dimensional deformation manifolds, balancing global formation objectives with local agent behaviors and practical safety constraints.

Deformable Virtual Structures (DVS) are virtual coordination objects whose geometry is allowed to deform while remaining a global guide for physical or simulated behavior. In the most explicit usage in the provided literature, DVS denotes a virtual formation geometry whose centroid motion, global scale, and affine shape deformation are planned continuously over time, so that a swarm can preserve formation integrity while adapting to narrow environments and changes in swarm cardinality (Zhou et al., 23 Sep 2025). In a broader research sense, closely related work treats DVS as a low-dimensional deformation manifold, a virtual elastic twin, or a physically constrained latent state that coordinates many points of a deformable body without requiring full online simulation at every degree of freedom (Tebyani et al., 2021).

1. Definition and conceptual scope

In the formation-planning formulation, DVS replaces a fixed rigid virtual body with a virtual formation geometry whose centroid motion, global scale, and affine shape deformation are all planned continuously over time. The resulting object is not merely a reference trajectory; it is a spatiotemporal template into which desired agent positions are embedded, after which each drone tracks its assigned moving target while enforcing local collision avoidance and dynamic feasibility. This is presented as a response to two failure modes of conventional formation methods: rigid formations become infeasible in narrow or cluttered environments, and fixed assignment schemes become brittle when the number of drones changes because agents join or leave (Zhou et al., 23 Sep 2025).

This usage is distinct from classical rigid virtual structures, which prescribe an undeformable body that may translate or rotate but does not scale or anisotropically deform. It is also distinct from purely distributed local methods that optimize only agent trajectories under local formation penalties. In the DVS formulation, deformation is a first-class planning variable, not a by-product of local reactivity. The virtual structure therefore mediates between global formation intent and local agent motion.

Related work broadens the interpretation of DVS beyond multi-agent flight. A geometric kinematic model for flexible voxel robots develops a reduced “Planes of Motion” representation in which structured deformation of a compliant lattice is expressed as a linear superposition of a small number of actuator-induced planar modes. That work does not use DVS terminology, but explicitly characterizes the model as a reduced deformation manifold and as a DVS-like shape-level representation that coordinates many physical nodes through a few internal variables (Tebyani et al., 2021). This broader interpretation is important because it places DVS at the intersection of formation control, reduced-order soft-body modeling, and physically informed virtual twins.

2. Mathematical structure of a DVS

The formal state of the formation-level DVS is built from the centroid position $p_W$, a scale factor $r$, and an affine deformation matrix restricted to the horizontal plane: $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$ By imposing $\alpha_x \cdot \alpha_y = 1$ and letting $\alpha = \alpha_x$, the affine deformation is reduced to a single scalar parameter. The intended effect is that the horizontal circular section becomes an equal-area ellipse, allowing the structure to narrow in one horizontal direction while expanding in the other without arbitrary collapse of the formation cross-section (Zhou et al., 23 Sep 2025).

If $p_{i,c}$ denotes the expected relative position of agent $i$ in the centroid frame before deformation, then the expected embedded position is

$$p_i = p_W + p_{i,c} \cdot \boldsymbol{A} \cdot \frac{\boldsymbol{R}}{\boldsymbol{R}_0}, \qquad \boldsymbol{R}_0^T = [r_0, r_0, 1], \quad \boldsymbol{R}^T = [r, r, 1].$$

The DVS trajectory itself is parameterized as

$\bm{\sigma}(t) = [p_W^T, r, \alpha]^T(t).$

The paper explicitly notes that the additional deformable degrees of freedom are scaling and planar affine anisotropy; no separate time-varying rotation matrix is introduced in the DVS trajectory equations (Zhou et al., 23 Sep 2025).

A related reduced-structure formalization appears in flexible voxel robots, where selected end-effector displacements are written as

$$\vec x_j(t) = \sum_{i=0}^{M-1} (\vec a_{d_i,n_j}) (c_{ij} q_i(t)).$$

Here the displacement of each end effector is a signed sum of basis vectors induced by control-node displacements. In that work, the reduced representation is interpreted as a “virtual linkage / virtual backbone surrogate” and as a “reduced deformation manifold induced by geometric planes.” Although it is not a swarm DVS, it exemplifies the same structural principle: a small number of virtual coordinates organizes coordinated motion of many physical points (Tebyani et al., 2021).

3. Repartitioning, planning, and distributed execution

A central feature of DVS in formation flight is number adaptation. The paper introduces PAAS, standing for uniform Partitioning and Assignment, with “PA” denoting Lloyd-based partitioning and “AS” denoting Hungarian assignment. The desired 3D shape is first sliced uniformly along the $r$0-axis into $r$1 layers with spacing equal to the required downwash safety distance. If the $r$2-th slice is $r$3 with area $r$4, then the layer populations $r$5 are chosen to be close to $r$6, subject to $r$7. Within each layer, expected positions are produced by minimizing

$r$8

with the Lloyd algorithm initialized by rectangular-grid seeds. Actual agents are then matched to the new slots by the Hungarian algorithm: $r$9 After repartitioning, the minimum pairwise slot spacing

$$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$0

is compared to the safety threshold

$$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$1

and the DVS scale is updated by

$$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$2

This mechanism allows the DVS to expand when more drones join and contract when drones leave, while preserving safety spacing and shape filling integrity (Zhou et al., 23 Sep 2025).

Global motion planning is performed in the higher-dimensional DVS state space. Primitive-based path search scores candidate paths using deformation cost, occupancy cost, and endpoint-alignment cost: $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$3 with

$$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$4

The occupancy term is defined through an obstacle indicator $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$5 over the deformed DVS volume, while the continuous DVS trajectory is refined by MINCO as a piecewise 5th-order polynomial trajectory in $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$6. Dynamic feasibility is enforced not only on the DVS center but on sampled boundary points of the deformed structure, so that if the outer contour satisfies velocity and acceleration limits, the interior embedded points do as well (Zhou et al., 23 Sep 2025).

Execution is hybrid centralized–distributed. A designated Deformation Guidance agent constructs the occupancy map, runs PAAS, performs DVS path search and DVS trajectory optimization, and broadcasts the spatiotemporal structure trajectory together with the assigned slot $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$7 for each drone. Each agent then solves its own local optimization with smoothness, travel-time, obstacle-clearance, reciprocal-avoidance, and formation-tracking terms. The formation term is a deviation penalty from the DVS-guided reference sequence: $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$8 Empirically, the method is reported to enable up to $$\boldsymbol{A} = \begin{bmatrix} \alpha_x & 0 & 0 \ 0 & \alpha_y & 0 \ 0 & 0 & 1 \end{bmatrix}, \qquad \alpha_x \cdot \alpha_y = 1.$$9 of swarm numbers to join or leave in cluttered environments while rapidly restoring the desired formation shape. In hardware, a 3-agent triangular formation joined by a fourth drone converged below $\alpha_x \cdot \alpha_y = 1$0 within $\alpha_x \cdot \alpha_y = 1$1 s, and narrow-gap traversal reduced formation width from $\alpha_x \cdot \alpha_y = 1$2 m to $\alpha_x \cdot \alpha_y = 1$3 m through spatiotemporal affine deformation rather than uniform scaling (Zhou et al., 23 Sep 2025).

4. Observability and state representation

The DVS literature in the provided corpus repeatedly links deformable virtual structure quality to observability. DOFS proposes a real-world 3D deformable-object dataset with “full spatial information” obtained through a transparent operating plane, using 6 Intel RealSense D435i RGB-D cameras, of which 4 view the object from above and the sides and 2 from below. The current dataset contains 150 pinches, or 4500 frames total, with 30 steps per pinch. Each frame includes a 3D deformed mesh, 3D positions of two fingers, multi-view RGB-D images, registered 3D point clouds, and 3D occupancy with semantics. The occupancy grid uses voxel size $\alpha_x \cdot \alpha_y = 1$4 m, and semantic labels are assigned for “the operating plane, two fingers, and deformable objects” (Zhang et al., 2024).

The conceptual importance of DOFS for DVS is its insistence that hidden support-side geometry can significantly affect dynamics. The paper illustrates this with solid-versus-hollow plasticine examples that look identical from visible viewpoints but respond differently under the same action. This suggests that a DVS state space should likely be full-spatial and interaction-aware rather than restricted to partial top-view observations. The available representations—RGB-D, registered point clouds, framewise meshes, and semantic occupancy—also define several candidate parameterizations for a DVS state, from observation models to mesh- or occupancy-based dynamics (Zhang et al., 2024).

A complementary line of work estimates deformable state from sparse sensing through reduced-order mechanics. MORPH-DSLAM uses a CAD-based volumetric model, a hyperelastic or possibly visco-hyperelastic constitutive law, and model order reduction to estimate the full 3D displacement field

$\alpha_x \cdot \alpha_y = 1$5

from monocular video. The method also recovers internal strain and stress fields, including in occluded regions, and reports 30 fps on CPU with online minimization time smaller than SLAM computation time. The price of this physical completeness is strong prior knowledge: known geometry, material model, and parameterized boundary conditions are assumed (Badias et al., 2020).

VIRDO addresses the same representational problem from a visio-tactile implicit perspective. It defines a nominal implicit SDF $\alpha_x \cdot \alpha_y = 1$6, a force/contact code $\alpha_x \cdot \alpha_y = 1$7 from contact points $\alpha_x \cdot \alpha_y = 1$8 and reaction force $\alpha_x \cdot \alpha_y = 1$9, and a deformation field $\alpha = \alpha_x$0 so that deformed geometry is represented by

$\alpha = \alpha_x$1

Because the deformation field maps deformed-space points back into nominal space, it induces dense unsupervised correspondences to a canonical reference configuration. This makes VIRDO particularly relevant to DVS observers and canonical-coordinate objectives under partial visio-tactile sensing (Wi et al., 2022).

5. Physics, interaction, and rendering

A mature DVS generally requires more than geometry and slot assignment: it needs a deformation model, interaction grounding, and a rendering or feedback layer. “Virtual Elastic Objects” presents an integrated capture-to-physics-to-rendering loop. It uses a 19-camera static rig, transparent fishing lines, and a tracked compressed air stream to collect force-annotated videos; reconstructs 4D geometry with Non-Rigid Neural Radiance Fields; infers spatially varying per-point Lamé parameters $\alpha = \alpha_x$2 under a quasistatic nonlinear Neo-Hookean model; simulates novel interactions with a mesh-free point-based solver; and renders new deformations by constructing a backward warp into a canonical radiance field. The PLUSH dataset contains 12 soft everyday objects, and the paper reports an average test-set point-cloud error of $\alpha = \alpha_x$3 mm average point distance, $\alpha = \alpha_x$4 mm at the 95th percentile, and $\alpha = \alpha_x$5 mm max (Chen et al., 2022).

DeformMaster pushes this integration toward online interactive world models. It represents deformable state with MPM particles

$\alpha = \alpha_x$6

combines differentiable MPM with a bounded neural residual,

$\alpha = \alpha_x$7

grounds sparse hand motion through Distributed Compliant Actuators, and models heterogeneous materials with a Mixture of Constitutive Experts over Neo-Hookean, fixed Corotated, and St. Venant–Kirchhoff stresses. Appearance is driven by Gaussian splatting coupled to the particle state, and the reported online playground supports interaction-point selection, keyboard-based manipulation, material parameter adjustment, synchronized novel-view rendering, and speeds over 15 fps (Li et al., 10 May 2026).

Interactive DVS also depends on contact handling. A haptic rendering framework for interacting deformable objects formulates contact with Signorini’s law, friction with Coulomb’s law, and multicontact coupling through the Delassus operator

$\alpha = \alpha_x$8

A Gauss–Seidel-type solver is used instead of a polyhedral friction-cone approximation, and a global corotational formulation dissociates the mass–stiffness ratio from the simulation time step. The resulting system demonstrates stable and realistic 6D haptic feedback in a clipping task, making it directly relevant to DVS applications involving insertion, snap-fit behavior, and frictional manipulation (0804.0561).

Within VR content creation, VR-Doh exemplifies a design-oriented DVS instantiation. It is described as an open-source VR modeling system for elastoplastic objects that customizes the Material Point Method for real-time hand-induced large deformations, enhances 3D Gaussian Splatting for rendering, supports both contact- and gesture-based hand–object interactions, and uses localized simulation, particle-level collision handling, and decoupled physical and appearance representations to maintain responsiveness. The system supports object creation and editing and produces simulation-ready assets (Luo et al., 2024).

6. Limits, neighboring concepts, and acronym ambiguity

The provided literature also shows that DVS remains a heterogeneous concept rather than a single settled formalism. In the formation-planning setting, the method is explicitly not fully distributed because a Deformation Guidance agent performs centralized map construction, repartitioning, and DVS optimization. The affine deformation is deliberately restricted to the horizontal plane, path search uses primitives with constant deformation parameters along each primitive, and the paper does not provide formal proofs of global convergence or collision freedom. It also states that agent entry detection and failure detection are not addressed (Zhou et al., 23 Sep 2025).

Related systems reveal recurring boundary conditions. MORPH-DSLAM assumes known CAD geometry, known or well-approximated material behavior, and deformations that remain within a precomputed reduced manifold (Badias et al., 2020). Virtual Elastic Objects require known external forces during capture and use a quasistatic solver with simple penalty contact rather than full frictional contact mechanics (Chen et al., 2022). DeformMaster is scene-specific, requires heavy preprocessing, finds planar objects more challenging than linear or volumetric ones, and performs per-scene fitting rather than zero-shot generalization (Li et al., 10 May 2026). DOFS, while highly relevant for state representation, does not itself define a virtual structure, virtual fixture, or control law (Zhang et al., 2024). Taken together, these works suggest that present DVS practice is strongest when geometry, forces, or constitutive structure are at least partially known.

A common source of confusion is the acronym itself. “Demonstrating DVS: Dynamic Virtual-Real Simulation Platform for Mobile Robotic Tasks” uses DVS to mean Dynamic Virtual-Real Simulation Platform, not Deformable Virtual Structures. That paper concerns virtual-real synchronization, dynamic pedestrians, scene editability, annotated synthetic data, and motion-capture-based benchmarking rather than deformable virtual-structure kinematics or control (Zheng et al., 26 Apr 2025). Another adjacent but non-equivalent usage appears in navigation among deformable obstacles, where compliance-aware obstacle envelopes modulate a learned dynamical system; this is DVS-adjacent but explicitly not a true deformable virtual structure formulation (Chen et al., 25 Jun 2025).

In summary, Deformable Virtual Structures designate a family of methods that use a deformable virtual object—explicitly parameterized, reduced-order, or implicitly represented—to coordinate motion, encode deformation constraints, or maintain a physically meaningful virtual twin. In their most explicit current form, DVS provide a continuously deformable formation scaffold for adaptive swarm flight. In adjacent formulations, they appear as reduced deformation manifolds, semantic occupancy states, visio-tactile canonical spaces, or physics-neural virtual assets. The unifying idea is not a specific solver, but the use of a structured deformable virtual entity as the intermediary between sensing, planning, interaction, and physical execution (Zhou et al., 23 Sep 2025).