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arxiv: 2508.02289 · v2 · pith:N2ZM4R7Wnew · submitted 2025-08-04 · 📡 eess.SY · cs.MA· cs.SY· math.OC

Distributed Non-Uniform Scaling Control of Multi-Agent Formation via Matrix-Valued Constraints

Tao He , Gangshan Jing This is my paper

Pith reviewed 2026-05-21 23:25 UTC · model grok-4.3

classification 📡 eess.SY cs.MAcs.SYmath.OC
keywords multi-agent formation controlnon-uniform scalingmatrix-valued constraintsdistributed control2-rooted graphposition-attitude formation
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The pith

New local matrix-valued constraints enable non-uniform scaling of multi-agent position formations by adjusting only two leaders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops distributed controllers for multi-agent formations that perform non-uniform scaling, meaning different stretch factors along orthogonal axes, without requiring uniform scaling or dense communication. These controllers rely on local matrix-valued constraints that each agent enforces using neighbor measurements, allowing the entire formation to reshape when two designated leaders move their positions. The same framework extends to joint position-attitude formations by incorporating separate scaling and translation rules for orientations. Global convergence to the desired scaled formation is proven when the underlying sensing graph satisfies the 2-rooted bidirectional condition, and the method needs fewer leaders and sparser graphs than prior affine formation techniques.

Core claim

Local matrix-valued constraints permit distributed non-uniform scaling and translation control of both position and attitude formations, with global asymptotic convergence guaranteed on any 2-rooted bidirectional sensing graph and with only two leaders required.

What carries the argument

Local matrix-valued constraints that encode independent scaling ratios along each coordinate axis and are enforced through neighbor-based feedback.

If this is right

  • Position formations can be stretched by different factors along x and y axes simply by relocating two leaders.
  • Attitude formations admit independent scaling and translation maneuvers under the same distributed law.
  • The required communication graph is sparser than the one needed for affine formation maneuver control.
  • Only two agents need to receive external commands, reducing the number of leaders compared with earlier methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The matrix constraint idea could extend to three-dimensional formations by replacing 2-by-2 matrices with 3-by-3 versions.
  • The approach might be combined with obstacle-avoidance terms while preserving the same convergence proof structure.
  • Time-varying leader trajectories could be tracked if the graph remains 2-rooted at every instant.

Load-bearing premise

The sensing graph among agents must be a 2-rooted bidirectional graph for the global convergence guarantee to hold.

What would settle it

A counter-example simulation on a connected but non-2-rooted graph in which the formation fails to converge to the commanded non-uniform scale would falsify the global convergence claim.

Figures

Figures reproduced from arXiv: 2508.02289 by Gangshan Jing, Tao He.

Figure 1
Figure 1. Figure 1: Non-uniform scaling transformation along arbitrary [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A DEP-induced graph with three DEPs. Lemma II.1. A bidirectional graph G is 2-rooted if and only if it contains a DEP-induced graph as its spanning subgraph. Proof: See Appendix VII-A. C. Affine Span and Diagonal Stability This section establishes the geometric and algebraic foun￾dations for formation stability. Definition II.5 (Affine Span [18]). The affine span of a set {xi} n i=1 is defined by S({xi} n … view at source ↗
Figure 3
Figure 3. Figure 3: Non-uniform scaling transformation in z = [cos θ,sin θ] ⊤ direction. E. Non-Uniform Scaling Transformation of Position Forma￾tion Before defining the non-uniform scaling transformation for position formations, we first introduce the concept of non￾uniform scaling transformation for a vector in R 2 . As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Attitude formation transformation. (a) Original forma [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Singular configuration. (a) Original positions of the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An example of the matrix-valued constraint. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Formation maneuver trajectories in 2-D space. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tracking errors. definition, any agent in GPh has two disjoint bidirectional paths in Lh to agents 1 and 2, respectively, h = 1, ..., κ. Since Lκ spans G, every agent in G is 2-reachable from {1, 2}, i.e., G is 2-rooted. (Necessity) Assume G is 2-rooted with roots {1, 2}. Ini￾tialize L0 = (V0, E0), where V0 = {1, 2} and E0 = ∅. For any agent k /∈ V0, since k is 2-reachable from {1, 2}, there must exist two… view at source ↗
read the original abstract

Distributed formation maneuver control refers to the problem of maneuvering a group of agents to change their formation shape by adjusting the motions of partial agents, where the controller of each agent only requires local information measured from its neighbors. Although this problem has been extensively investigated, existing approaches are mostly limited to uniform scaling transformations. This article proposes a new type of local matrix-valued constraints, via which non-uniform scaling control of position formation can be achieved by tuning the positions of only two agents (i.e., leaders). Here, the non-uniform scaling transformation refers to global scaling the position formation with different ratios along different orthogonal coordinate directions. Moreover, by defining scaling and translation of attitudes, we propose a distributed control scheme for scaling and translation maneuver control of joint position-attitude formations. It is proven that the proposed controller achieves global convergence, provided that the sensing graph among agents is a 2-rooted bidirectional graph. Compared with the affine formation maneuver control approach, the proposed approach leverages a sparser sensing graph, requires fewer leaders, and additionally enables scaling transformations of the attitude formation. A simulation example demonstrates our theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces local matrix-valued constraints to enable distributed non-uniform scaling control of multi-agent position formations, where only two leader agents are adjusted to achieve global scaling with different ratios along orthogonal directions. It extends the approach to scaling and translation maneuvers of joint position-attitude formations and proves global convergence of the closed-loop system when the sensing graph is 2-rooted and bidirectional. The method is positioned as requiring a sparser graph and fewer leaders than affine formation control while adding attitude scaling capability.

Significance. If the central convergence claims hold, the work offers a meaningful advance in formation maneuver control by introducing matrix-valued constraints that support non-uniform (anisotropic) scaling with minimal leaders. This could reduce communication requirements relative to existing affine approaches and extends naturally to attitude formations. The explicit tie between the 2-rooted bidirectional graph condition and equilibrium uniqueness is a positive structural feature.

major comments (2)
  1. [Main convergence theorem / Lyapunov analysis] The global convergence proof for the position case (likely in the main theorem) relies on the 2-rooted bidirectional graph ensuring uniqueness of the equilibrium formation under the new matrix-valued constraints; however, the manuscript should explicitly derive or cite how the matrix constraints interact with the graph Laplacian to rule out non-scaling equilibria, as this is load-bearing for the 'global' claim.
  2. [Joint position-attitude control section] For the joint position-attitude extension, the definition of attitude scaling and translation needs to be shown to preserve the same graph condition without introducing additional equilibria; the current sketch leaves open whether the attitude dynamics couple back into the position error in a way that could violate global attractivity.
minor comments (3)
  1. [Introduction] Clarify the precise definition of 'non-uniform scaling transformation' early in the introduction, including how it differs from general affine transformations.
  2. [Control law definition] Add a brief remark on computational complexity or implementation of the matrix-valued constraints at each agent, as this affects practicality.
  3. [Simulation] The simulation example should include a quantitative comparison (e.g., convergence time or error norms) against an affine baseline under identical graph conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation of minor revision. We address each major comment below, clarifying the analysis and indicating where we will strengthen the manuscript.

read point-by-point responses

  1. Referee: The global convergence proof for the position case (likely in the main theorem) relies on the 2-rooted bidirectional graph ensuring uniqueness of the equilibrium formation under the new matrix-valued constraints; however, the manuscript should explicitly derive or cite how the matrix constraints interact with the graph Laplacian to rule out non-scaling equilibria, as this is load-bearing for the 'global' claim.

    Authors: We agree that an explicit derivation of the interaction would improve clarity. The equilibrium condition in Theorem 1 is obtained by setting the control input to zero, which yields that the stacked position vector lies in the kernel of a matrix constructed from the local matrix-valued constraints and the graph Laplacian. Because the graph is 2-rooted and bidirectional, the only solutions satisfying all local constraints simultaneously are the desired non-uniform scalings (plus translation). We will insert a short lemma immediately preceding the main theorem that derives this kernel characterization step by step and cites the relevant property of 2-rooted graphs. revision: yes

  2. Referee: For the joint position-attitude extension, the definition of attitude scaling and translation needs to be shown to preserve the same graph condition without introducing additional equilibria; the current sketch leaves open whether the attitude dynamics couple back into the position error in a way that could violate global attractivity.

    Authors: The attitude scaling and translation are defined via analogous matrix-valued constraints on the attitude variables, and the control laws are designed so that the position and attitude error systems remain decoupled. The composite Lyapunov function is the sum of two independent quadratic terms, each of which is strictly decreasing outside the desired set; cross terms do not appear because the sensing graph is bidirectional and the constraints act separately on position and attitude. We will expand the current sketch into a complete proof outline that explicitly verifies the absence of additional equilibria arising from coupling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proposes novel local matrix-valued constraints to enable non-uniform scaling of position formations using only two leaders and extends this to joint position-attitude formations. Global convergence is established via standard Lyapunov stability arguments tied explicitly to the 2-rooted bidirectional sensing graph condition for equilibrium uniqueness. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the proof structure follows conventional formation control techniques without importing uniqueness theorems or ansatzes from the authors' prior work as external facts. The central claims rest on the newly defined constraints and graph assumptions, which are independently verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The approach depends on the newly introduced matrix-valued constraints and the domain assumption of a 2-rooted bidirectional sensing graph; no free parameters or additional invented physical entities are described in the abstract.

axioms (1)
  • domain assumption The sensing graph among agents is a 2-rooted bidirectional graph.
    Explicitly required for the global convergence of the proposed controller.
invented entities (1)
  • Matrix-valued constraints no independent evidence
    purpose: To enable local enforcement of non-uniform scaling transformations in position and attitude formations.
    New mechanism proposed in the paper to achieve the non-uniform scaling capability.

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