3D Directed Formation Control with Global Shape Convergence using Bispherical Coordinates

TL;DR

This work presents a 3D coordinate-free formation-control framework for directed leader–follower graphs using bispherical coordinates to uniquely characterize and steer the shape of a multiagent formation. By formulating three orthogonal formation variables per follower and designing decentralized control laws that act along the corresponding bispherical basis directions, the authors achieve (almost) global convergence to the desired 3D shape while relying only on vision-based bearing and distance-ratio measurements in local frames. The stability analysis uses a cascade of almost global asymptotic stability results, ensuring boundedness of errors and collision avoidance, with the added capability of scalable formation by adjusting the leader–follower distance. The approach is validated via simulations showing convergence to a unit octahedron and successful scaling, highlighting practical applicability with low-cost onboard vision sensors and acyclic triangulated directed sensing graphs.

Abstract

In this paper, we present a novel 3D formation control scheme for directed graphs in a leader-follower configuration, achieving (almost) global convergence to the desired shape. Specifically, we introduce three controlled variables representing bispherical coordinates that uniquely describe the formation in 3D. Acyclic triangulated directed graphs (a class of minimally acyclic persistent graphs) are used to model the inter-agent sensing topology, while the agents' dynamics are governed by single-integrator model. Our analysis demonstrates that the proposed decentralized formation controller ensures (almost) global asymptotic stability while avoiding potential shape ambiguities in the final formation. Furthermore, the control laws are implementable in arbitrarily oriented local coordinate frames of follower agents using only low-cost onboard vision sensors, making it suitable for practical applications. Finally, we validate our formation control approach by a simulation study.

Figures (5)

• Figure 1: (a) A graph constructed under Assumption 1 with 5 agents, which consists of two tetrahedral subgraphs. The blue tetrahedron has a positive volume $(V_{1234}>0)$ and the red one has a negative volume $(V_{1235}<0)$. (b) Position of vertex 4 makes a tetrahedron with positive volume $V_{1234}>0$ (blue) while its reflected position 4' leads to the same volume ($|V_{1234}|=|V_{1234'}|$) with a negative sign $V_{1234'}<0$ (red). (c) Tetrahedron $ABCD$ and three of its edge lengths, face angles, and dihedral angles.
• Figure 2: (a) Showing direction of $\hat{\varphi}$ for three cases of agent $l$ positions (indicated by $l$, $l'$ and $l"$). Note that agents $i$, $j$, $l$, and $l'$ are on the same plane but different half-planes leading to opposite $\hat{\varphi}$ directions. (b) 2D view of $\{C_l \}$ showing $\hat{\varphi}$ and $\varphi$ for some different positions of agent $l$. (indicated by $l$, $l'$ and $l"$)
• Figure 3: 2D view of $i-j-l$ plane in Fig. \ref{['frame3d']}. In the entire upper half plane (blue) $\hat{\varphi}$ is pointing outwards, while it points inwards in the lower half plane (red). $Y^\prime_l$ is rotated version of $Y_l$ around $X_l$ lying down in $i-j-l$ plane. This figure depicts the projection of the isoquant surfaces of $\xi$, $\eta$ on $i-j-l$ plane and their corresponding orthogonal basis vectors $\hat{\xi}$ and $\hat{\eta}$ for two different positions of $l$ and $l'$. Note that $\hat{\xi}$ and $\hat{\eta}$ always lie on the $i-j-l$ plane, which are perpendicular to $\hat{\varphi}$.
• Figure 4: (a) Trajectory of agents until $t=10$. (b) Scaling simulation: agents scaling up after $t=10$ because of a change in $d_{21}^*$.
• Figure 5: (a) Sensing graph among agents. (b) Formation errors vs time.

Theorems & Definitions (11)

• Remark 1
• Lemma 1
• proof
• Remark 2
• Remark 3
• Remark 4
• Lemma 2
• proof
• Remark 5
• Theorem 1