Multi-Robot Rigid Formation Navigation

• The paper presents a coordination strategy where robots maintain fixed inter-agent distances while traversing environments.
• It details distributed control and gradient-based methods that enable precise collective translation and rotation of robot formations.
• The research integrates motion planning and collision avoidance with graph rigidity theory to ensure system robustness and scalability.

Multi-robot rigid formation navigation is the coordinated movement of multiple robots maintaining a prescribed inter-agent geometry, typically defined by fixed inter-robot distances, while traversing an environment. Rigid formation navigation encompasses distributed control design, optimal and reactive planning, graph-theoretic rigidity, and robustness to sensing, communication, and actuation constraints. The field is technically characterized by the mathematical theory of rigidity, decentralized algorithms for control and estimation, and a variety of motion planning and collision avoidance frameworks—each motivated by practical applications ranging from cooperative transportation and surveillance to multi-robot mapping and resource-limited embedded deployment.

1. Mathematical Foundations of Rigid Formation

Rigid formation navigation relies fundamentally on the theory of graph rigidity. An agent network is represented by an undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, where each node $i$ (agent) has a position $x_i\in\mathbb{R}^m$ ($m=2$ or $3$), and each edge $k=(i,j)$ encodes a distance constraint $d_{ij}$. The formation's rigidity is characterized via the rigidity matrix $R(x) = D_z^T(B^T\otimes I_m)$, where $D_z = \text{diag}(z_1,\ldots,z_{|\mathcal{E}|})$, $B$ is the incidence matrix, and $z_k = x_i - x_j$. Infinitesimal rigidity holds if $\text{rank}\; R(x) = mn-\frac{m(m+1)}{2}$, and a minimally rigid graph has $|\mathcal{E}|=mn-\frac{m(m+1)}{2}$ edges. The rigid shape space is

$$\mathcal{D} = \{\,z\,|\,\|z_k\| = d_k,\,\,\forall k\,\}$$

ensuring the maintenance of all pairwise specified distances modulo global translation and rotation (Marina et al., 2016).

2. Distributed Control of Rigid Formations

A widely-adopted approach for rigid formation maintenance is gradient-based control, where a global potential

$$V(x) = \frac12\sum_{(i,j)\in\mathcal{E}}(\|x_i - x_j\| - d_{ij})^2$$

guides robot actuators: $$u_i = -\nabla_{x_i}V = -\sum_{j\in\mathcal{N}_i} (\|x_i-x_j\|-d_{ij})\frac{x_i-x_j}{\|x_i-x_j\|}.$$ This guarantees asymptotic convergence to the desired formation shape (up to a rigid body transformation), but does not generate net translation or rotation of the collective—i.e., the system asymptotically comes to rest in $\mathcal{D}$ (Marina et al., 2016).

To achieve active translation or rotation, mismatches in the inter-agent distance constraints are introduced as explicit motion parameters. By assigning each agent on edge $k$ slightly different target distances $(d_k \pm \mu_k/c)$ and incorporating these mismatches in the distributed control law,

$u = -cR(x)^T e + (B\otimes I_m)A(\mu)z,$

one induces steady-state collective motion of the formation at arbitrary velocities and angular rates. The choice of $\mu$ encodes the desired translation $v$ and (for $m=2$) rotation $\omega$, yielding at steady-state

$\dot x^*_i = v + \omega J(x^*_i - x_c).$

A rigorous Lyapunov analysis establishes local exponential convergence to the prescribed formation shape and collective motion (Marina et al., 2016).

3. Motion Planning and Optimality in Rigid Formations

Optimal control of rigid formations traditionally requires trajectory planning for all agents under coupled holonomic constraints. However, for identical holonomic agents, the formation's motion can be decomposed into the translation and rotation of its center of mass (CoM) plus internal constraints. Pontryagin's minimum principle yields that the optimal control policy for each agent is equivalent to that for the CoM and the orientation variable: $$J = \frac{1}{2}\int_0^{t_f} \sum_{i=1}^N \|u_i(t)\|^2 dt = \frac{1}{2}\int_0^{t_f} \left( N\|u_c(t)\|^2 + \omega^2 I \right) dt,$$ where $u_c$ is the CoM translational velocity, $\omega$ is the angular speed, and $I$ is the formation's moment of inertia. The control synthesis thus reduces to a 3-DOF problem, scaling efficiently for large $N$ (Narayanan et al., 2023). Simulation results confirm that the per-agent optimal control derived from this reduction precisely recreates the multi-agent optimal trajectory subject to rigid formation constraints.

Discrete-time frameworks that support synchronous execution under communication delays and packet loss, as in the hold-and-hit protocol, further extend the robustness and real-world deployability of optimal rigid formation navigation. Coupled with intra-cycle error-minimizing quadratic programming, these methods guarantee that experimental inter-distance and orientation errors remain within stringent bounds even under severely unreliable wireless networks (Yang et al., 3 Oct 2025).

4. Decentralized and Distributed Planning Frameworks

Consensus-based approaches for formation-level planning represent rigid formations by similarity transforms of a fixed base configuration: $p_i = R(\phi)S(s)p^0_i + t,$ where $R(\phi)$ is a planar rotation, $S(s)$ anisotropic scaling, and $t$ translation. Each robot maintains a local estimate $\eta_i=(\phi_i, s_i, t_i)$ and executes a loop consisting of desired-velocity tracking (using Jacobian pseudoinverse), consensus correction (Laplacian term), constraint satisfaction (soft/hard projection to scaling safe sets), and velocity recovery. This enables each agent to track feasible formation parameters and maintain rigidity, collision-avoidance, and constraint satisfaction in a computationally efficient, fully distributed fashion (Mikkelsen et al., 2023, Mikkelsen et al., 27 Aug 2024). For instance, in simulation, a 9-robot system achieves loop rates exceeding 7 kHz with per-cycle computational cost under 23 $\mu$s on general-purpose hardware (Mikkelsen et al., 2023).

Recent advances include probabilistic collision avoidance via chance constraints on inter-robot distances under state uncertainty. By expressing safety constraints as linearized inequalities on the scaling and orientation parameters and projecting decentralized parameter updates in each cycle, real-time and scalable rigid-formation navigation with provable probabilistic guarantees is realized experimentally (Mikkelsen et al., 27 Aug 2024).

5. Rigidity Preservation, Estimation, and Connectivity

Beyond fixed-shape maintenance, dynamic environments and performance guarantees require continuous quantification and maximization of formation rigidity. The rigidity index, defined as the fourth smallest eigenvalue $\lambda_4$ of the formation rigidity Laplacian $E=R^TWR$, acts as a scalar indicator of infinitesimal rigidity (with three zero eigenvalues representing the allowable rigid-body motions in the plane). A gradient-based distributed controller maximizes $\lambda_4$, keeping the formation rigid throughout navigation and implicitly ensuring collision avoidance and graph connectivity (Sun et al., 2013).

Fully distributed estimation of $\lambda_4$ and the associated eigenvector is achieved via a shifted inverse power method implemented as a local Jacobi-overrelaxation scheme with dynamic consensus averaging. This architecture runs in real time, requires only neighbor-to-neighbor state exchange, and demonstrably maintains rigidity and connectivity under agent mobility and leader-follower maneuvers (Sun et al., 2013).

6. Path Planning, Obstacle Avoidance, and Flexibility

In cluttered spaces, motion planners must compute feasible and optimal trajectories for the entire formation in configuration space, representing both pose $(x,y,\theta)$ and formation shape. Graph-based planners discretize this high-dimensional space, identify feasible configurations by explicit collision-checking for all robots, generate connectivity graphs, and apply breadth-first or Dijkstra search for pathfinding and optimality. Empirical results on four- and six-robot rigid teams in obstacle-dense arenas demonstrate the critical dependence of planner performance on discretization granularity and configuration-space densification (Liu et al., 2022).

Alternative quasi-centralized methods invoke a virtual-agent trajectory, with individual robots attracted to the minima of formation potential fields (FPF) centered on the virtual agent. This paradigm enables real-time, collision-free navigation that is flexible to nonholonomic motion and allows limited formation deformation near obstacles (G et al., 2021). Inter-robot and robot-obstacle repulsive potentials, combined with decentralized force-balance controllers, further ensure trajectory feasibility and collision avoidance.

7. Practical Implementation, Robustness, and Experimental Validation

The reviewed methods emphasize decentralization, scalability, and robustness to sensing errors, communication delays, and network unreliability. Key implementation practices include:

• Local gradient evaluation based exclusively on relative positions and neighbor communication, obviating the need for a global coordinate frame (Marina et al., 2016, Sun et al., 2013).
• Real-time computational efficiency enabling deployment on resource-limited microcontrollers, with minimal communication per cycle and computational load (Mikkelsen et al., 2023, Yang et al., 3 Oct 2025, Mikkelsen et al., 27 Aug 2024).
• Robustness to packet loss and delay via discrete-time communication-control synchronization protocols (Yang et al., 3 Oct 2025).
• Explicit probabilistic constraint handling for safety under uncertainty (Mikkelsen et al., 27 Aug 2024).
• Experimental validation across application scenarios, including cooperative board transportation by four robots over WiFi (inter-distance error $\pm0.069$ m, orientation error $\pm19.15^\circ$ at $0.1$ m/s) (Yang et al., 3 Oct 2025), real-time navigation among static and dynamic obstacles, and formation shape recovery post-obstruction (Mikkelsen et al., 2023, Marina et al., 2016).

Ongoing research focuses on robust multi-robot formation navigation in dynamic and partially observable environments, scalability to large teams, extension to non-Euclidean (e.g., manifold) formation spaces, time-varying formation shapes, and hybrid integration with vision and communication-aware protocols.

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Representative References (arXiv ids):

• Distributed controller for formation-motion control (Marina et al., 2016)
• Decentralized formation parameter consensus (Mikkelsen et al., 2023)
• Rigid-formation navigation with probabilistic collision avoidance (Mikkelsen et al., 27 Aug 2024)
• Rigidity-preserving distributed control (Sun et al., 2013)
• Synchronous discrete-time implementation over wireless (Yang et al., 3 Oct 2025)
• Optimal path planning via CoM equivalence (Narayanan et al., 2023)
• Graph-based sampling motion planning (Liu et al., 2022)
• Quasi-centralized virtual-agent navigation (G et al., 2021)