The Geometry of Hidden Modes in Distance-Based Formation Control

TL;DR

This work addresses how disturbances affect distance-based formation control when sensing and actuation are locally constrained. It develops a geometric input-output framework that identifies the uncontrollable subspace as the local rotational subspace $\mathcal{T}_i$, containing both the global rigid-body rotations about the input node $i$ ($\mathcal{R}_i(p^*)$) and locally hidden deformations, and links disturbance rejection to the alignment of the input with the local rotational component via the coefficient $c_r=\langle [v_r]_i, w_0\rangle$. The authors derive a complete decomposition of hidden modes, show that for a single actuator the uncontrollable subspace equals $\mathcal{T}_i$, and demonstrate a shape-recovery dichotomy: if the input is orthogonal to the local rotational direction, the formation undergoes a pure translation and achieves perfect shape recovery; otherwise, excitation of $\mathcal{R}_i(p^*)$ induces persistent shape distortion. A case study on a minimally rigid four-agent planar formation provides concrete validation. This framework clarifies how geometry constrains the dynamic response and guides disturbance mitigation in decentralized formation control.

Abstract

This paper presents a geometric input-output analysis of hidden modes in distance-based formation control. We study the linearized dynamics under a gradient control law to characterize the system's structural limitations and their dynamic consequences. Our main contribution is a unified geometric framework for uncontrollable modes. We first prove that uncontrollable rigid-body modes are pure rotations about the input node, defining a global rotational subspace $\mathcal{R}_i$. To generalize this, we introduce the local rotational subspace, $\mathcal{T}_i$, which contains all motions, including deformations, that are locally invisible to the controller at node $i$. These two geometric objects provide a complete decomposition of the uncontrollable subspace. Finally, we demonstrate the dynamic implications of this structure by proving that the system's ability to recover its shape is determined by an input's alignment with the local component of the standard rotational rigid-body mode, directly linking the geometry of hidden modes to disturbance rejection. We illustrate our results with a case study.

Figures (5)

• Figure 1: Geometric interpretation of the hidden RBM subspaces, illustrated for the planar case ($d=2$). The dashed gray lines show the framework's configuration for visual context. The red arrows depict a motion in the uncontrollable rotational subspace $\mathcal{R}_i(p^*)$, a pure rotation of the entire framework about the actuated node $i$ (red). The dashed red lines emphasize that this global rotation depends on the relative position of every node with respect to node $i$. Similarly, the blue arrows and dashed lines illustrate a motion in the unobservable rotational subspace $\mathcal{R}_j(p^*)$, a pure rotation about the measured node $j$ (blue).
• Figure 2: Geometric interpretation of the local rotational subspaces, illustrated for the planar case ($d=2$). The union of the thick black and dashed gray lines represents the edges of the sensing graph ${\mathcal{G}}$. The thick black lines emphasize the star of edges incident to the actuated node $i$ (red) and the measured node $j$ (blue). The local rotational subspace $\mathcal{T}_i$ is defined by these incident edges at $i$. The red arrows show elementary motions in $\mathcal{T}_i$, which are locally invisible to the controller at $i$. Similarly, the blue arrows show motions in $\mathcal{T}_j$, which are locally invisible to the measurement at $j$. Each arrow is orthogonal to its corresponding edge vector, representing a purely rotational motion of a neighbor around its central node.
• Figure 3: Shape recovery with an orthogonal input. (a) An impulse orthogonal to the local rotational RBM direction at the actuated node results in a pure translation. (b) The edge length errors decay to zero.
• Figure 4: Shape distortion with an aligned input. (a) An impulse with a component along the local rotational RBM direction excites a rotational motion. (b) The formation deforms, resulting in persistent edge length errors.
• Figure 5: Geometric illustration of the input-to-RBM map. The set of all reachable RBMs forms the blue "controllable plane." This plane is orthogonal to the "uncontrollable direction" $n_c$. The horizontal plane represents the subspace of pure translations ($c_r=0$). The intersection of these two planes is the 1D subspace of outcomes that result in perfect shape recovery.