Displacement-Based Formation Control

Updated 22 November 2025

Displacement-based formation control is a method for coordinating multi-agent systems using prescribed inter-agent displacement vectors to achieve target geometric patterns.
It employs local control laws based solely on relative position measurements, ensuring distributed scalability across agents with diverse dynamics and sensing capabilities.
Robust design tools such as set invariance, port-Hamiltonian modeling, and Lyapunov analysis guarantee convergence despite sensor misalignments and environmental disturbances.

Displacement-based formation control is a foundational paradigm in the coordination of multi-agent systems, characterized by local control policies where each agent acts to achieve prescribed relative displacements with respect to its neighbors. The resulting methodology is distributed, scalable, and compatible with agents possessing varying dynamics, information structures, and sensing capabilities. Originating in both control theory and robotics, this framework has become central to geometric pattern formation, target tracking, and cooperative deployment across robotic, vehicular, and networked cyber-physical systems.

1. Mathematical Foundations and Distributed Architectures

At the core of displacement-based formation control is the encoding of formation objectives as prescribed inter-agent displacement vectors, facilitating control laws that only require each agent to access local relative position measurements (and possibly relative velocities). Consider $N$ agents in $\mathbb{R}^d$ , each with state $p_i\in\mathbb{R}^d$ and a dynamics model of the general form

$\dot{p}_i = v_i, \quad \dot{v}_i = u_i,$

or, in the simplest (single-integrator) case, $\dot{p}_i = u_i$ . The agents are connected via a communication or sensing graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ . The formation is specified by desired relative positions $z_{ij}^* = p_i^* - p_j^*$ for $(i,j)\in\mathcal{E}$ .

A prototypical control law for agent $i$ is

$u_i = -k_p \sum_{j\in\mathcal{N}_i} ( (p_i - p_j) - z_{ij}^* ) - k_v \sum_{j\in\mathcal{N}_i} ( (v_i - v_j) ),$

where $k_p, k_v > 0$ are gains and $\mathcal{N}_i$ represents the neighbors of $i$ . The global closed-loop dynamics can then be expressed using the incidence matrix $H$ associated with $\mathcal{G}$ , and the Laplacian $L = H^\top H$ , yielding system-wide convergence to the set $\{p: (H\otimes I_d)p = z^*\}$ up to rigid-body translations (Angheluţă et al., 15 Nov 2025, Marina, 2020).

A significant development in this context is the ability to perform distributed synthesis of inter-agent weights, particularly in directed graphs and under leader-follower architectures. If the communication topology contains a directed spanning tree rooted at a leader, it is possible to decentralize the computation of Laplacian weights so as to assign prescribed closed-loop poles and enforce $L_w F = 0$ , where $F$ is a vector of target positions (Mulla et al., 2013).

2. Robustness, Ultimate Bounds, and Set-Theoretic Guarantees

Measurement imperfection, environmental uncertainty, and actuator disturbances can substantially degrade performance. Several approaches address these issues:

Set-theoretic analysis: Utilizing robust positive invariance and ultimate boundedness, it is possible to analyze performance in the face of bounded measurement noise or persistent disturbances. The closed-loop error $e$ of a double-integrator system with bounded disturbances $\delta(t)\in\Delta$ admits an explicit robust positively invariant (RPI) set $\Omega_{UB}$ , computable via the Jordan decomposition of the closed-loop matrix and the structure of $\Delta$ . The control gains can be optimized, e.g., by minimizing the volume of the ultimate bound, to guarantee the tightest possible steady-state formation error in constrained environments (Angheluţă et al., 15 Nov 2025).
Robustness to sensor misalignment and scale variations: In undirected consensus-based displacement control, non-uniform scaling ( $\alpha_i$ ) or misalignment ( $R_i$ ) of agents’ local reference frames can result in global translation, steady-state drift, or geometric distortion of the formation. Calibration protocols and enforcing uniformity of $\alpha_i, R_i$ are critical to preserve correct convergence; otherwise, residual errors and nonzero velocities may occur (Marina, 2020).
Internal model-based disturbance rejection: For systems under persistent disturbances, such as sinusoidal or step signals, embedding internal-model dynamic compensators enables exact rejection of the disturbance, leading to asymptotic convergence to the displacement-based pattern. A typical realization involves virtual-spring energy storage between end-effectors and Lyapunov-based synthesis to prove invariance and exponential decay of errors (Wu et al., 2021).

3. Extensions to General Dynamics and Sensing Constraints

Displacement-based control frameworks have been extended to agents with diverse dynamics and sensing modalities:

Higher-order and nonholonomic agents: The barycentric-coordinate-based (BCB) architecture supports single integrator, double integrator, unicycle, and car-like models. By projecting the nominal single-integrator control into the agent’s available actuation directions (e.g., splitting into forward and angular components for unicycles), the BCB control law maintains global convergence and robustness to input saturation, model mismatch, and arbitrary local scale and rotation (within $\pm 90^\circ$ ) of the control input (Fathian et al., 2018).
Unknown agent orientations and negative couplings: When agents lack a global frame or possess local sensing subject to unknown rotations, geometric arguments—particularly those based on convex polytopes—can provide contractive set invariance properties, guaranteeing convergence to the target displacement pattern for admissible step sizes, even in the presence of negative (antagonistic) interaction weights (Li et al., 2023).
Port-Hamiltonian and passivity-based design: Port-Hamiltonian modeling facilitates physical interpretability, modularity, and energy-based stability proofs. The formation stabilization is cast as potential shaping in the edge (displacement) space, with velocity alignment handled by internal feedback, yielding global convergence on both tree and cycle graphs (Li et al., 2023).

4. Maneuvering, Pattern Generation, and Large-Scale Deployments

Displacement-based controllers naturally generate coordinated geometric patterns and support dynamic maneuvers:

Translating and rotating formations: By introducing asymmetry or time-variation in the Laplacian weights (motion parameters), the desired centroid velocity or coordinated rotation can be encoded explicitly, ensuring that the formation converges exponentially to the intended shape while translating at a specified velocity, subject to constraints on the perturbation amplitude to maintain stability (Marina, 2020).
Pattern formation in large-scale swarms: Adaptive, communication-free displacement-based laws leverage only range sensing and compass information to enable the emergence of regular lattices (triangular, square, etc.) in systems with hundreds of agents. Gains can be adapted online based on local angular-regularity error, and these scalable systems exhibit robustness to noise, agent failures, and pattern switching (2207.14567).
Coupled oscillator approaches and moving target tracking: The use of relative localization (e.g., via RLSE with forgetting factors) and phase oscillators ensures persistent excitation, which underpins the exponential convergence of localization and tracking errors in dynamic enclosing tasks, such as UAVs circling a moving target (Liu et al., 2023).
PDE approximations for continuum swarms: In the large-agent limit, displacement-based control protocols can be approximated by parabolic PDEs (e.g., heat equations with periodic boundary conditions), with leader-based absolute measurements injected via boundary feedback. Design guidelines for target decay rates and robustness to delay are obtained via Lyapunov-Krasovskii arguments and LMIs (Wei et al., 2019).

5. Hybrid and Heterogeneous Constraints

Modern frameworks integrate displacement-based control with additional geometric specifications to achieve richer objectives:

Bearing and angle constraints: Controllers based on the projection of displacement errors onto the orthogonal complement of desired bearings support translation and formation tracking without explicit communication or velocity information sharing. Infinitesimal rigidity (bearing, distance, angle) is pivotal for ensuring uniqueness of the formation up to symmetry classes and Lyapunov invariance (Tran et al., 2022, Li et al., 2023).
Mixed constraint formulations: Compound port-Hamiltonian models admit the simultaneous enforcement of displacement, distance, bearing, and angle constraints, resulting in almost-global convergence to complex formation manifolds under assumptions of suitable composite rigidity and constraint Jacobian full rank (Li et al., 2023).

6. Implementation Guidelines and Practical Considerations

The deployment of displacement-based formation controllers in physical multi-agent platforms hinges on:

Locality and decentralization: All control and estimation laws are constructed to rely strictly on local (possibly relative) sensing without global information or inter-agent communication, promoting scalability and real-world applicability (Fathian et al., 2018, 2207.14567).
Gain tuning and adaptive choices: For tightness of formation (small ultimate bounds), gains are optimized off-line via set-theoretic volume minimization or online via adaptation laws responsive to neighborhood regularity error. Care is taken to ensure that stability margins remain robust to switching topologies and varying neighborhood sizes (Angheluţă et al., 15 Nov 2025, 2207.14567).
Robustness to environmental and sensing uncertainties: Calibration, contractive invariance, and Lyapunov/passivity-based analysis provide resilience to noise, delays, and bounded mismodeling, with explicit expressions for residual errors and domains of validity (Marina, 2020, Angheluţă et al., 15 Nov 2025).
Collision avoidance and reconfiguration: Geometric projections or control direction rotations modulate local control actions to prevent collisions, allowing for online repatterning and dynamic adaptability (Fathian et al., 2018, 2207.14567).

7. Theoretical Guarantees and Limitations

Theoretical guarantees for displacement-based formation control are anchored by Lyapunov stability theory, passivity arguments, contractive-set geometry, and explicit eigenstructure assignment. Under generic assumptions of global rigidity, nondegenerate graph topology (typically requiring a directed or undirected spanning tree), and feasible sensing or actuation models, convergence is exponential to the prescribed displacement pattern modulo group symmetries (translation, rotation, scale in 2D/3D). Set-theoretic and port-Hamiltonian methods enable quantifiable performance bounds.

Limitations arise in the presence of nonlinearly coupled dynamics, switching graphs with severe disconnectivity, unbounded disturbance, or extreme measurement ambiguity. Extensions to nonlinear or nonholonomic agents often require auxiliary arguments or linearization analyses. Computational bottlenecks may emerge in mixed-integer or global optimization formulations when obstacle avoidance is combined with tight formation constraints (Angheluţă et al., 15 Nov 2025).

Displacement-based formation control thus provides a mathematically rigorous, scalable, and robust basis for multi-agent coordination, underpinned by a rich set of analytic and practical tools spanning graph theory, set invariance, passivity, and geometric control. It represents a mature and evolving field foundational to modern collective robotics and autonomous networked systems (Mulla et al., 2013, Marina, 2020, Li et al., 2023, Angheluţă et al., 15 Nov 2025, Fathian et al., 2018).