Distance-coupling as an Approach to Position and Formation Control

TL;DR

The paper addresses localization and formation control using only distance measurements to fixed anchors by introducing distance-coupling, which uses squared-distance differences to derive a linear position update $x = K\,(h(x) - b)$ and a linear controller $u = C\big(K\,(h(x) - b) - x^{eq}\big)$. It proves global exponential stability for the homing problem under ideal anchors ($C = \alpha I$, $\alpha<0$), and derives a region-of-attraction under orthogonal anchor transformations in 2D, along with a shifted equilibrium when anchors are rotated and translated. The approach is extended to multi-agent formation control, showing that equilibria correspond to formations congruent to the desired one, and establishing transformation invariance via Procrustes/Kabsch analysis; empirical results in 2D corroborate convergence under perturbations and rotations. Overall, the method offers a bearing- and relative-position-free framework with broad applicability to localization and distributed formation, scalable to higher dimensions.

Abstract

In this letter, we study the case of autonomous agents which are required to move to some new position based solely on the distance measured from predetermined reference points, or anchors. A novel approach, referred to as distance-coupling, is proposed for calculating the agent's position exclusively from differences between squared distance measurements. The key insight in our approach is that, in doing so, we cancel out the measurement's quadratic term and obtain a function of position which is linear. We apply this method to the homing problem and prove Lyapunov stability with and without anchor placement error; identifying bounds on the region of attraction when the anchors are linearly transformed from their desired positions. As an application of the method, we show how the policy can be implemented for distributed formation control on a set of autonomous agents, proving the existence of the set of equilibria.

Figures (7)

• Figure 1: Anchor set with $p=3$ such that $\mathcal{A}=\{ a_1,a_2,a_3 \}$ and $\mathcal{D}(x,A)=\{ d(x,a_1), d(x,a_2), d(x,a_3) \}$.
• Figure 2: Response of the distance-coupled controller \ref{['eq:distance-coupled_control']} with $x^{(\text{eq})}$ at the origin and for varying magnitudes of randomly generated measurement noise, $\omega_\varepsilon$. The agents are given arbitrary initial positions around the origin.
• Figure 3: Homing controller response with $x^{(\text{eq})} = 0$ and with variations in $R$ and $r$ simulated separately with (left) initial conditions around the origin, and (right) $x^{(0)} = x^{(\text{eq})}$.
• Figure 4: Distance-coupled controller response with the rotation, $R:\theta = \tfrac{3\pi}{4}$, of the anchor set (controller diverges).
• Figure 5: Distance-coupled formation control response for an arbitrary formation and with varying levels of randomly generated offsets from the equilibrium positions: $X^{(0)} = X^{(\text{eq})} + \omega_{\varepsilon = 25}$.

Theorems & Definitions (17)

• Proposition 1
• proof
• Remark
• Lemma 1
• proof
• Proposition 2
• proof
• Definition 1
• Definition 2
• Lemma 2