Affine Formation Control (AFC) Overview

• Affine formation control (AFC) is a framework that enables agents to maintain deformable geometric formations via arbitrary affine transformations such as translation, rotation, scaling, and shearing.
• It employs stress matrices, rigidity criteria, and equilibrium-unit methods to design distributed control laws ensuring convergence and stability in both leader–follower and leaderless architectures.
• Recent advances integrate event-triggered strategies, nonlinear control, and safety mechanisms to achieve robust, scalable, and obstacle-aware multi-agent formation maneuvers.

Affine formation control (AFC) is a formation–maneuvering framework in which a group of agents is required to maintain a geometric pattern that is allowed to move and deform according to arbitrary affine transformations of a given nominal formation. In its standard form, if the nominal configuration is $r=[r_1^T,\dots,r_n^T]^T\in\mathbb{R}^{dn}$, then a desired affine formation is

$$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$

so translations, rotations, scalings, shearings, reflections, and, in some formulations, coplanar or colinear degeneracies are admissible. Across the current literature, AFC appears as a distributed, graph-based control paradigm built around affine-invariant formation descriptions, stress matrices, leader–follower containment, and, increasingly, robust, safe, and scalable implementations for discrete-time, continuous-time, and control-affine multi-agent systems (Onuoha et al., 2024, Liu et al., 20 Jun 2025).

1. Mathematical formulation of affine formations

The basic AFC object is the affine orbit of a nominal configuration. For $r\in\mathbb{R}^{dn}$, the affine image set is

$$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$

An AFC problem asks for distributed control laws such that, for any desired time-varying affine image $x^*(k)\in\mathcal{A}(r)$, the follower states satisfy

$\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$

This definition makes the objective fundamentally different from rigid formation control: the target is not a single congruence class, but a family of configurations parameterized by affine maps (Onuoha et al., 2024).

A common alternative notation, used for general linear multi-agent systems, writes the target formation as

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$

where $\Upsilon(t)$ is the time-varying linear part and $\upsilon(t)$ is the translation. In that formulation, affine transformations include translation, rotation, scaling, shear, and coplanar or colinear degeneracies. The emphasis is that only leaders need to know the desired transformed positions explicitly; follower objectives are induced algebraically from the formation structure (Liu et al., 20 Jun 2025).

For planar formations, a complex-plane representation provides a compact affine parameterization. If $p=[p_1,\dots,p_n]^T\in\mathbb{C}^n$ and $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$0 is a reference shape, then an affine transformation can be written as

$$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$1

with complex coefficients $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$2. The corresponding affine shape set is the complex subspace

$$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$3

This formulation is particularly useful for analytic characterization of planar collective motions such as translation, scaling, shear, and certain rotation-induced motions in leaderless AFC (Bautista et al., 7 Apr 2026).

2. Stress matrices, rigidity, and affine localizability

Stress-matrix methods form the dominant algebraic core of AFC. Given a framework $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$4 in $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$5, an equilibrium stress is a set of edge weights $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$6 satisfying

$$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$7

The associated stress matrix $$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$8 is defined by

$$x_i^*(t)=\Theta(t)r_i+b(t),\qquad x^*(t)=[I_n\otimes \Theta(t)]\,r+1_n\otimes b(t),$$9

Unlike a standard Laplacian, the off-diagonal entries $r\in\mathbb{R}^{dn}$0 may be positive or negative, allowing tension/compression structure and enabling AFC in arbitrary dimensions (Onuoha et al., 2024).

The critical rigidity condition is usually universal rigidity. For generic configurations, universal rigidity is guaranteed if the graph is $r\in\mathbb{R}^{dn}$1-connected and there exists a positive semidefinite stress matrix $r\in\mathbb{R}^{dn}$2 with

$r\in\mathbb{R}^{dn}$3

Under these conditions, the nullspace of $r\in\mathbb{R}^{dn}$4 corresponds precisely to the affine image of the nominal configuration, so the formation geometry is encoded directly in the stress matrix. In large-scale AFC, the same condition appears as

$r\in\mathbb{R}^{dn}$5

with $r\in\mathbb{R}^{dn}$6 the augmented configuration matrix; this characterizes the affine equilibrium subspace and links AFC to universally rigid frameworks (Onuoha et al., 2024, Li et al., 27 Mar 2026).

In leader–follower AFC, $r\in\mathbb{R}^{dn}$7 is partitioned as

$r\in\mathbb{R}^{dn}$8

If the leaders affinely span $r\in\mathbb{R}^{dn}$9 and $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$0 is nonsingular, then follower targets are uniquely determined by leader targets: $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$1 or, in stacked form,

$$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$2

This affine localizability result is the standard mechanism by which followers reconstruct their target positions without explicit access to $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$3 or $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$4 (Onuoha et al., 2024).

One of the main revisions of this picture is the claim that genericity is sufficient but not necessary. Earlier AFC works typically assumed generic nominal configurations and global construction of equilibrium weights. The equilibrium-unit framework replaces this with a local sufficient condition: if the directed graph is layerable, the leaders affinely span $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$5, and each follower together with its in-neighbors forms an equilibrium unit, then affine localizability holds. This removes the generic assumption and enables local equilibrium-unit construction, localized communication criteria, and self-reconstruction under node flow-in and flow-out (Zhu et al., 23 Feb 2025). A plausible implication is that affine localizability is better understood as a structural property of local geometric motifs and graph hierarchy than as a by-product of generic coordinates alone.

3. Dynamics and controller families

AFC has been developed for several dynamic classes. In discrete time, a basic model is the sampled single-integrator

$$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$6

while a more general model is

$$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$7

with $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$8 full column rank and $$\mathcal{A}(r)=\{x\in\mathbb{R}^{dn}:x=(I_n\otimes \Theta)r+1_n\otimes b,\ \Theta\in\mathbb{R}^{d\times d},\ b\in\mathbb{R}^d\}.$$9 stabilizable. These dynamics cover single, double, and triple integrators as special cases and also admit more complex linear agent models such as UAV lateral dynamics and vehicle platoons (Onuoha et al., 2024).

For discrete-time single-integrator followers with stationary leaders, the local stress-matrix controller is

$x^*(k)\in\mathcal{A}(r)$0

which yields error dynamics

$x^*(k)\in\mathcal{A}(r)$1

Under the assumptions of universal rigidity, $x^*(k)\in\mathcal{A}(r)$2, and positive real parts of the eigenvalues of $x^*(k)\in\mathcal{A}(r)$3, exponential stability is ensured when the sampling period satisfies

$x^*(k)\in\mathcal{A}(r)$4

where $x^*(k)\in\mathcal{A}(r)$5 is the smallest eigenvalue of $x^*(k)\in\mathcal{A}(r)$6. For dynamic leaders with time-varying velocities, a modified protocol compensates for leader motion between samples, and the sufficient discrete-time condition reduces to

$x^*(k)\in\mathcal{A}(r)$7

For general linear agents, a Riccati-based gain

$x^*(k)\in\mathcal{A}(r)$8

is used together with stress-based coupling to solve the AFC problem for followers tracking leader-induced affine maneuvers (Onuoha et al., 2024).

Communication constraints have motivated adaptive event-triggered AFC. For general linear systems

$x^*(k)\in\mathcal{A}(r)$9

distributed event-triggered leader and follower controllers use sampled states or observer states, adaptive thresholds, and compensation terms $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$0 satisfying

$\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$1

The resulting protocols handle both full-state and output-based settings, avoid Zeno behavior, and do not rely on predefined global information. In simulations, trigger counts remain below about $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$2 of time steps for state-based affine transformations and about $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$3 for output-based cases, while maintaining convergence to nominal and transformed formations (Liu et al., 20 Jun 2025).

A separate leaderless line of work dispenses with explicit leaders entirely. Starting from a Laplacian that stabilizes a static affine shape, the edge weights are modified so that the resulting collective motion is characterized as a time-varying affine transformation of a reference configuration. In the complex-plane formulation, the modified Laplacian yields analytic eigenstructure inside the affine subspace, and the leaderless scheme allows agents to maintain distinct and possibly time-varying velocities while generating all linear combinations of translations, rotations, scaling, and shearing of a reference shape (Bautista et al., 7 Apr 2026). This suggests that AFC is not intrinsically tied to leader–follower containment, even though that architecture remains the most developed one.

4. Beyond canonical leader–follower AFC

Beyond stress-matrix containment, AFC has been generalized along at least three additional directions. The first is local graph construction. The equilibrium-unit construction method builds affine-localizable directed formations incrementally: leaders are initialized to affinely span $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$4, each follower selects $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$5 to $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$6 in-neighbors forming an equilibrium unit, and localized communication criteria update in-neighbor and out-neighbor sets under node additions or removals. In that setting, localized sensing-based affine formation maneuver control applies to $\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$7-th order integrator dynamics and provides self-reconstruction capability when nodes are added to or removed from the swarm (Zhu et al., 23 Feb 2025).

The second direction concerns control-affine nonlinear agents. For heterogeneous systems

$\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$8

interconnected through diffusive relative-output measurements, the edge variable

$\lim_{k\to\infty}\|x_f(k)-x_f^*(k)\|=0.$9

becomes the formation or synchronization coordinate. The design is decomposed into an edge-space step, which specifies feasible edge dynamics on the manifold $$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$0, and a lift step solving

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$1

to realize those edge dynamics through the allowable input directions. The central condition is admissibility,

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$2

and the paper derives generic matching-based certificates for this property on an associated bipartite graph. Although the primary focus there is synchronization with $$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$3, the same edge-space machinery is described as extending to arbitrary feasible edge targets and therefore to AFC-type shape objectives (Zelazo et al., 17 Mar 2026).

The third direction is output-feedback stabilization of control-affine systems by Lie-bracket approximations. For driftless control-affine dynamics

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$4

suitably chosen oscillatory inputs approximate Lie brackets and thereby recover ascent and descent directions of the output function $$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$5. The framework is applied to purely distance-based formation control for nonholonomic multi-agent systems, including unicycles on $$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$6, where distance-only output feedback yields practical and, under suitable rigidity assumptions, exponential stability of the desired formation set (Suttner, 2018). A plausible implication is that AFC for nonholonomic or strongly underactuated platforms can be analyzed within a broader control-affine and Lie-bracket synthesis framework rather than only through linear consensus analogies.

5. Robustness, topology changes, and safety

A recurring misconception is that rigidity alone guarantees robust AFC. The recent robustness literature states the opposite: the unavailability of local relative-position measurements, due to node failure or missing links, changes the underlying graph topology and subsequently causes instability and sub-optimal performance. In discrete-time single-integrator AFC, one proposed remedy is an estimation framework that adaptively fuses temporal information from agent dynamics with spatial information derived from affine-formation geometry. The framework includes Relative Kalman Filtering (RKF), Relative Affine Localization (RAL), consensus RAL, and Geometry-Aware RKF (GA-RKF). Its local convergence indicator

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$7

is shown, in the noiseless case, to be upper bounded by the global tracking error, and in the noisy case to satisfy an upper bound with an additive bias term. In simulation, GA-RKF preserves stability and convergence under random observation losses, switching topologies, and node departures, whereas no-estimator baselines and pure RKF can diverge when topology changes destroy the stabilizing assumptions of the nominal AFC graph (Li et al., 2024).

Safety introduces a separate layer of difficulty. For second-order follower dynamics

$$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$8

a barrier-modulated architecture combines nominal AFC with safety control: $$p^*(t)=[I_n\otimes \Upsilon(t)]\,a+\mathbf{1}_n\otimes \upsilon(t),$$9 where the modulation factor $\Upsilon(t)$0 smoothly attenuates formation tracking near safety boundaries. The safe set is defined through pairwise constraints

$\Upsilon(t)$1

and a higher-order control barrier function

$\Upsilon(t)$2

Two safety layers are developed. The first is an analytical barrier-gradient repulsive controller with explicit gain conditions guaranteeing forward invariance of the safe set by Nagumo’s theorem. The second is a data-driven optimal safety controller based on adaptive dynamic programming, in which an actor–critic neural network solves the HJB equation online using a danger-weighted state and a barrier penalty. For both controllers, the paper proves absolute collision avoidance together with uniformly ultimately bounded formation tracking errors (Naeem et al., 6 Jun 2026).

At a more abstract dynamical-systems level, strong chain control sets for affine control systems on non-compact manifolds provide a language for robust reachable formation sets and asymptotic behavior. By embedding affine systems on $\Upsilon(t)$3 into bilinear systems on the Poincaré sphere, strong chain control sets are preserved under topological conjugacy, giving a compactified perspective on controllability regions, transitions between formation regimes, and behavior at infinity (Colonius et al., 2024). This suggests a rigorous bridge between local AFC controller design and global qualitative analysis of reachable formation domains.

6. Scalability, topology design, and applications

As AFC moved from tens to hundreds of agents, stress-matrix design became a computational bottleneck. One convex optimization line starts from the complete graph and designs the stress vector directly, rather than fixing a rigid graph in advance. For sparse and fast-converging AFC, the objective trades $\Upsilon(t)$4-sparsity of the stress vector against a trace surrogate for the nonzero spectrum of the stress matrix, subject to the affine-nullspace condition and spectral constraints. In simulation, this approach yields more sparse graphs while admitting faster convergence compared to state-of-the-art URF-based methods (Li et al., 25 Aug 2025).

Large-scale AFC goes further by combining optimized stress design with multicluster decomposition. For a generic configuration, a convex SDP produces a sparse, fast-converging stress matrix; for rotationally symmetric configurations, the unique-stress-identifier reduction groups symmetry-equivalent edges, reducing the variable count from $\Upsilon(t)$5 to $\Upsilon(t)$6. The multicluster control framework then decomposes a large swarm into overlapping clusters linked by bridging nodes. In the mean, the resulting dynamics remain an AFC system, while rank conditions on the bridge configuration determine whether the swarm exhibits a single collective affine motion or partially independent inter-cluster motions. The overall design is reported as compatible with a swarm size of several hundred agents with fast formation convergence, as compared to up to only a few dozen agents by existing methods, and the USI method gives about a $\Upsilon(t)$7 speedup on a $\Upsilon(t)$8-node truncated icosahedron (Li et al., 27 Mar 2026).

Application work has increasingly coupled AFC with planning and obstacle avoidance. In multi-UAV obstacle fields, a BI-APF-RRT planner produces a smooth centroid path, while an affine transformation matrix with rotation and non-uniform scaling deforms the formation online according to path direction and obstacle distance. The scaling rule

$\Upsilon(t)$9

causes the formation to shrink near obstacles and recover afterward. In a U-shaped trap environment, APF-only path planning plus affine control yields $\upsilon(t)$0 obstacle avoidance success, whereas BI-APF-RRT plus affine control yields $\upsilon(t)$1. In a Monte Carlo study with $\upsilon(t)$2 obstacles randomly placed in a $\upsilon(t)$3 area over $\upsilon(t)$4 runs, BI-APF-RRT compared with BI-RRT reduces average iterations by about $\upsilon(t)$5, average path length by about $\upsilon(t)$6, and average planning time by about $\upsilon(t)$7, while keeping formation deformation error bounded around $\upsilon(t)$8 with small fluctuations (Wu et al., 29 Jun 2026).

Taken together, these developments indicate that AFC has evolved from a stress-matrix theory for affine-invariant shape maintenance into a broad systems framework encompassing discrete-time and continuous-time designs, leader–follower and leaderless motion generation, event-triggered and output-based control, topology-aware estimation, HOCBF-based safety, ADP-based optimal avoidance, and large-scale multicluster deployment. This suggests that the central invariant across the literature is not a single controller architecture, but the affine-geometric principle that the controlled formation should remain on, or converge to, an affine orbit of a nominal configuration while local interactions determine how that orbit is tracked, deformed, or safeguarded.