Decentralized Affine Transformation in Multi-Agent Systems

• Decentralized Affine Transformation is a leader–follower framework that enables multi-agent systems to perform time-varying affine maps for formation reshaping through local communication.
• It decomposes affine maps into rotation and strain components to characterize deformation while ensuring safety via principal-strain bounds and barycentric containment.
• Experimental validation with mini-quadcopters shows asymptotic convergence, tracking errors ≤1 cm, and safe navigation through obstacle-rich environments.

Searching arXiv for the cited work and closely related papers on decentralized affine transformation in multi-agent systems. Decentralized Affine Transformation (AT) is a leader–follower coordination framework for multi-agent systems in which a team realizes time-varying affine maps of the form $T(x)=Ax+b$ through local communication, allowing formation reshaping, obstacle-rich navigation, and formally characterized safety under contraction, expansion, rotation, and shear. In the formulation experimentally validated with mini-quadcopters, three leaders prescribe the affine motion of a planar team, while followers do not know the global affine parameters and instead infer their trajectories from fixed communication weights computed from the initial geometry; under stated graph, containment, and tracking assumptions, the team asymptotically converges to the leader-prescribed affine transformation while maintaining collision-free coordination (Mazmanyan et al., 13 Sep 2025).

1. Formal definition and geometric model

In the two-dimensional setting, affine transformation is written as

$T(x)=Ax+b,$

with $A\in\mathbb{R}^{2\times 2}$ and $b\in\mathbb{R}^2$. In the reported formulation, the desired transformed position of agent $i$ is expressed in full three-dimensional coordinates as

$p_i(t)=Q(t)a_i+d(t),$

where $Q\in\mathbb{R}^{3\times 3}$ is the Jacobian, $d\in\mathbb{R}^3$ is the translation, and $a_i$ is the agent’s initial position; motion is planar at a fixed altitude $Z$, so the essential deformation is two-dimensional (Mazmanyan et al., 13 Sep 2025).

The framework is explicitly designed to realize time-varying affine maps that combine translation, rotation, shear, and anisotropic scaling. A central feature is that AT permits aggressive changes in inter-agent distances. Those changes are not treated as incidental geometric effects but are formally characterized through the decomposition of the affine map. This differentiates AT from rigid formation methods, which preserve distances and therefore lack the same geometric flexibility in narrow or cluttered environments.

A closely related large-scale formulation appears in the multi-quadcopter literature, where the affine map is written as

$T(x)=Ax+b,$0

or equivalently $T(x)=Ax+b,$1 with $T(x)=Ax+b,$2. In that formulation, $T(x)=Ax+b,$3 leaders occupy the vertices of an $T(x)=Ax+b,$4-D simplex, and the remaining agents acquire the desired affine transformation through local communication (Rastgoftar et al., 2021). This suggests that the experimentally validated planar architecture is a specialized instance of a broader simplex-based AT class.

2. Leader–follower structure and containment

The experimentally validated two-dimensional construction uses three leaders at the vertices of a triangle and followers that start inside that triangle and remain inside under AT. Leaders know the prescribed transformation $T(x)=Ax+b,$5; followers do not. Instead, followers compute desired positions using local neighbor information and fixed convex weights determined from the initial spatial configuration (Mazmanyan et al., 13 Sep 2025).

Containment is expressed through barycentric coordinates. If a follower position $T(x)=Ax+b,$6 lies inside the leader triangle with vertices $T(x)=Ax+b,$7, then

$T(x)=Ax+b,$8

The area-based weights are

$T(x)=Ax+b,$9

with $A\in\mathbb{R}^{2\times 2}$0,

$A\in\mathbb{R}^{2\times 2}$1

An equivalent matrix form solves $A\in\mathbb{R}^{2\times 2}$2 with $A\in\mathbb{R}^{2\times 2}$3 and sets $A\in\mathbb{R}^{2\times 2}$4, $A\in\mathbb{R}^{2\times 2}$5, $A\in\mathbb{R}^{2\times 2}$6.

Implementation uses fixed communication weights from the initial geometry. For follower $A\in\mathbb{R}^{2\times 2}$7 with in-neighbors $A\in\mathbb{R}^{2\times 2}$8,

$A\in\mathbb{R}^{2\times 2}$9

These weights satisfy $b\in\mathbb{R}^2$0 and $b\in\mathbb{R}^2$1 when a valid local simplex is chosen. Because affine maps preserve barycentric combinations, followers remain in the leaders’ convex hull under affine deformation (Mazmanyan et al., 13 Sep 2025).

The larger-scale formulation generalizes this principle to $b\in\mathbb{R}^2$2 dimensions: $b\in\mathbb{R}^2$3 leaders are placed at the vertices of a $b\in\mathbb{R}^2$4-simplex, and followers use generalized barycentric weights to remain inside the convex hull. That formulation also describes a proximity-based communication topology in which each follower selects a minimal-radius containing simplex of neighbors from the initial geometry (Rastgoftar et al., 2021).

3. Communication model, decentralized update law, and convergence

The communication structure is a directed graph $b\in\mathbb{R}^2$5 with leader set $b\in\mathbb{R}^2$6 and follower set $b\in\mathbb{R}^2$7. The assumptions used in the convergence theorem are specific: leaders have no in-neighbors; each follower has exactly three in-neighbors; and there is at least one directed path from each leader to each follower (Mazmanyan et al., 13 Sep 2025).

Desired positions are defined differently for leaders and followers. For leaders,

$b\in\mathbb{R}^2$8

For followers,

$b\in\mathbb{R}^2$9

where the weights are constant and determined from the initial geometry. The key operational point is that followers use neighbors’ actual positions, not the global affine parameters.

At the aggregate level, the analysis introduces a weighted Laplacian-like matrix $i$0 with

$i$1

and a leader selector $i$2. The matrix $i$3 contains leaders’ identity rows and followers’ barycentric weights with respect to the leaders. Under the stated assumptions, the paper proves that $i$4 is Hurwitz and

$i$5

Using stacked coordinate vectors $i$6 and their desired counterparts $i$7, the relations

$i$8

imply

$i$9

and analogously for $p_i(t)=Q(t)a_i+d(t),$0. When low-level tracking ensures $p_i(t)=Q(t)a_i+d(t),$1 and leaders’ desired positions become constant after $p_i(t)=Q(t)a_i+d(t),$2, the team satisfies

$p_i(t)=Q(t)a_i+d(t),$3

so all agents asymptotically track the leader-prescribed AT (Mazmanyan et al., 13 Sep 2025).

The individual agent model is given abstractly by

$p_i(t)=Q(t)a_i+d(t),$4

with $p_i(t)=Q(t)a_i+d(t),$5 and gains chosen so that the invariant set

$p_i(t)=Q(t)a_i+d(t),$6

is asymptotically stable when $p_i(t)=Q(t)a_i+d(t),$7 is constant. In the large-scale MQS formulation, an analogous convergence result is obtained after feedback linearization, with stability linked to a Hurwitz matrix $p_i(t)=Q(t)a_i+d(t),$8 and bounded tracking error achieved for sufficiently large maneuver duration $p_i(t)=Q(t)a_i+d(t),$9 (Rastgoftar et al., 2021).

4. Decomposition of the affine map and safety conditions

A defining technical component of AT is the decomposition of the affine Jacobian to characterize how distances change under deformation. For planar AT, the paper uses the polar decomposition

$Q\in\mathbb{R}^{3\times 3}$0

where $Q\in\mathbb{R}^{3\times 3}$1 is a yaw rotation and $Q\in\mathbb{R}^{3\times 3}$2 is symmetric strain. The strain is then decomposed as

$Q\in\mathbb{R}^{3\times 3}$3

where $Q\in\mathbb{R}^{3\times 3}$4 defines the shear axes and

$Q\in\mathbb{R}^{3\times 3}$5

contains the principal strains in the deformation plane (Mazmanyan et al., 13 Sep 2025).

An equivalent characterization through singular values is also given. For $Q\in\mathbb{R}^{3\times 3}$6 with $Q\in\mathbb{R}^{3\times 3}$7 and $Q\in\mathbb{R}^{3\times 3}$8,

$Q\in\mathbb{R}^{3\times 3}$9

In the symmetric-strain case, the singular values coincide with the principal strains $d\in\mathbb{R}^3$0. These bounds quantify the extent to which pairwise separations contract or expand under AT.

Safety is enforced through a lower bound on the principal strains:

$d\in\mathbb{R}^3$1

where $d\in\mathbb{R}^3$2 is the minimum inter-agent separation in the reference configuration, $d\in\mathbb{R}^3$3 is a tracking error bound, and $d\in\mathbb{R}^3$4 is the agent radius. Maintaining this inequality guarantees safe inter-agent spacing despite aggressive contraction, expansion, and shear (Mazmanyan et al., 13 Sep 2025).

The larger-scale MQS framework extends the same idea by introducing both lower and upper eigenvalue bounds. There, safety decomposes into minimum inter-agent separation, maximum spread within a rigid-size containment ball, and bounded tracking error. The conditions are

$d\in\mathbb{R}^3$5

with

$d\in\mathbb{R}^3$6

In that formulation, obstacle avoidance is guaranteed by planning the center of the containment ball using A-star search and ensuring that the swept ball does not intersect obstacles (Rastgoftar et al., 2021).

A common misconception is that AT safety is achieved by online pairwise collision checking among all agents. The reported AT literature instead frames safety through containment and deformation constraints: leaders plan safe affine deformation, followers remain in the convex hull or containment set, and principal-strain or eigenvalue bounds guarantee spacing. This suggests a scalability advantage relative to approaches whose safety conditions grow with the number of pairwise interactions.

5. Experimental validation with mini-quadcopters

The experimental validation uses six Crazyflie 2.1 quadrotors indoors, with Vicon motion capture at 100 Hz, a central ground station running Crazyswarm2, and a 2.4 GHz Crazyradio PA. The leaders are $d\in\mathbb{R}^3$7 and the followers are $d\in\mathbb{R}^3$8. Motion is planar at fixed altitude $d\in\mathbb{R}^3$9, and initial positions are specified in YAML (Mazmanyan et al., 13 Sep 2025).

The experiment consists of three 10 s phases with smooth transitions generated by the quintic polynomial

$a_i$0

Phase	Mode	Parameters
AT1	Pure contraction	$a_i$1
AT2	Rigid body motion	translation + rotation, shape fixed
AT3	Precise deformation	$a_i$2

Across the maneuver, the team undergoes an overall translation of $a_i$3 in $a_i$4. Leaders execute trajectories generated from $a_i$5 and $a_i$6, while followers compute decentralized desired positions as weighted combinations of three neighbors’ actual positions using barycentric weights from the initial geometry. The ground station facilitates message passing, but the logic of trajectory acquisition remains decentralized (Mazmanyan et al., 13 Sep 2025).

The reported quantitative outcomes are specific. Tracking errors are $a_i$7 for both leaders and followers. Altitude control remains within $a_i$8 about the constant desired height of $a_i$9. Inter-agent distances remain at least $Z$0 throughout. In corridor navigation through a $Z$1 wide cardboard corridor, the minimum clearance is approximately $Z$2 and no collisions occur.

The safety-bound calculation is also instantiated numerically. The experiment uses $Z$3, $Z$4, and $Z$5, giving

$Z$6

The minimum principal strains maintained in the experiment are $Z$7. A related explicit statement in the experimental synthesis is that the minimum separation is enforced by $Z$8, resulting in inter-agent distances greater than $Z$9 at all times (Mazmanyan et al., 13 Sep 2025).

These experiments validate two claims that are central to the framework: asymptotic convergence of decentralized AT and safe navigation of a multi-agent team through obstacle-laden environments under affine deformation.

6. Generalizations, limitations, and adjacent literatures

The reported framework is explicitly generalizable. In $T(x)=Ax+b,$00 dimensions, AT requires $T(x)=Ax+b,$01 leaders at the vertices of a $T(x)=Ax+b,$02-simplex; followers remain inside the convex hull via generalized barycentric weights. The method is described as scalable because followers use only local communication with fixed weights and the computation cost is minimal and independent of team size. Robustness is stated in conditional form: as long as leader-to-follower reachability holds and weights remain nonnegative, containment and convergence persist; leader motion planning may use optimal control, A*, or PSO (Mazmanyan et al., 13 Sep 2025).

The limitations are equally explicit. The experimental validation is planar; full 3D requires four leaders and careful altitude shaping. The communication topology is fixed, with three neighbors per follower and weights determined from the initial geometry. Topology changes or link failures can violate nonnegativity or reachability and break containment. Message passing in the experiment uses a central ground station, so fully onboard decentralized implementation would require reliable onboard estimation and radio networking. Safety also depends on the leaders’ planned deformation satisfying the $T(x)=Ax+b,$03 condition; overly aggressive deformations or poor obstacle planning can violate the stated guarantees (Mazmanyan et al., 13 Sep 2025).

Comparisons to adjacent methods are sharply drawn in the source material. Rigid formations preserve distances but do not offer the same maneuvering flexibility in tight spaces. Consensus-only methods ensure agreement but not geometric containment or safety under deformation. Potential fields and control barrier functions can encode safety but are described as scaling poorly because of quadratic constraints and possible infeasibility, whereas AT’s safety condition is expressed through principal-strain bounds (Mazmanyan et al., 13 Sep 2025).

There is also a terminological distinction worth making. A separate literature studies decentralized or distributed synchronization of affine transformations between Euclidean coordinate systems, emphasizing transitive consistency of relative linear or affine maps on graphs. That problem concerns estimating globally consistent transformations such as $T(x)=Ax+b,$04 from pairwise measurements, with centralized methods for affine synchronization and distributed protocols provided only for orthogonal matrices in the cited work (Thunberg et al., 2015). This is conceptually different from AT in multi-agent guidance, where the objective is not synchronization of coordinate frames but decentralized acquisition of a leader-prescribed affine deformation by a moving formation.

Taken together, the literature portrays decentralized AT as a geometric coordination framework in which affine-map preservation, barycentric containment, graph-theoretic stability, and strain-based safety bounds are integrated into a single leader–follower architecture. A plausible implication is that its principal research frontier lies at the interface of richer deformation planning, stronger robustness to communication and estimation imperfections, and fully onboard decentralized implementation.