2-D Directed Formation Control Based on Bipolar Coordinates

TL;DR

This work addresses robust, coordinate-free 2-D formation control for directed, minimally persistent graphs by marrying bipolar coordinates with Prescribed Performance Control (PPC). The follower geometry is encoded by local bipolar coordinates $(r_k, \alpha_{kij})$, enabling two orthogonal error channels $(e_{r_k}, e_{\alpha_k})$ that are independently corrected along orthogonal directions, while a PPC framework imposes time-varying bounds $\rho_h(t)$ on errors to guarantee predefined transient and steady-state performance. A decentralized control law uses only bearing and distance-ratio information (obtainable from onboard vision) and is implementable in arbitrary local frames; an orientation-adjustment extension via a bearing error $e_{\beta}$ enables simultaneous scaling and rotation control. Theoretical results show (almost) global convergence to the target shape with bounded signals, and simulations confirm robustness to disturbances and effective maneuvering through constrained environments. The approach offers practically attractive, low-cost sensing requirements and establishes a foundation for extending to 3-D formations and more complex dynamics.

Abstract

This work proposes a novel 2-D formation control scheme for acyclic triangulated directed graphs (a class of minimally acyclic persistent graphs) based on bipolar coordinates with (almost) global convergence to the desired shape. Prescribed performance control is employed to devise a decentralized control law that avoids singularities and introduces robustness against external disturbances while ensuring predefined transient and steady-state performance for the closed-loop system. Furthermore, it is shown that the proposed formation control scheme can handle formation maneuvering, scaling, and orientation specifications simultaneously. Additionally, the proposed control law is implementable in agents' arbitrarily oriented local coordinate frames using only low-cost onboard vision sensors, which are favorable for practical applications. Finally, a formation maneuvering simulation study verifies the proposed approach.

Figures (10)

• Figure 1: (a) edge-angle in a triangular subgraph. (b) example of a desired formation (note that $d_{31}^{\ast} = d_{52}^{\ast}$ and $d_{42}^{\ast} = d_{43}^{\ast}$).
• Figure 2: (a) The virtual local Cartesian coordinate frame $\left\lbrace C_k \right\rbrace$ uniquely characterizes the position of agent $k\geq3$ with respect to its neighbors (agents $i$ and $j$). Instead of using the Cartesian coordinates in $\left\lbrace C_k \right\rbrace$ one can adopt bipolar coordinates \ref{['eq:edge_angle_bear']} and \ref{['eq:log_ratio']} in $\left\lbrace C_k \right\rbrace$ to determine agent $k$'s position. (b) Orthogonal bipolar coordinate basis $\widehat{r}_k$, $\widehat{\alpha}_k$ associated with agent $k\geq3$ and some of their isoquant curves.
• Figure 3: Given a desired sensing graph $\mathcal{G}$ as in Fig.\ref{['fig:angle_example']}, in each case the desired formation is characterized by different desired relative positions between agents 2 and 1, whereas the sets of desired edge-angles and ratio of the distances for followers ($i\geq3$) are the same. The dashed arrows show the local coordinate frame of agent 2 in which the formation orientation can be characterized by the desired bearing angle $\beta^{\ast}$. $p_{21,a}^{\ast}$ and $p_{21,b}^{\ast}$ have the same orientation but different length while $p_{21,a}^{\ast}$ and $p_{21,c}^{\ast}$ have different orientations with the same length.
• Figure 4: Starting from arbitrary initial positions, agents converge to the desired shape while following the leader's (agent 1's) motion. The scale and orientation of the formation is adjusted by agent 2 along the way. In particular, roughly around $t = 14$ agent 2 starts following a time-varying desired bearing and distance w.r.t. agent 1 that leads the formation to pass through a narrow passage (black curves).
• Figure 5: Agent 2's desired (time-varying) distance $d_{21}^{\ast}(t)$ and bearing angle $\beta^{\ast}(t)$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• Remark 4: PPC Design Philosophy
• Lemma 2
• Remark 5
• Remark 6
• Theorem 1
• Theorem 2