Distributed 3D Source Seeking via SO(3) Geometric Control of Robot Swarms

TL;DR

The paper tackles 3D source seeking with swarms by formulating a geometric control law on the Lie group $SO(3)$, enabling reliable attitude alignment for drones with constant forward speed while avoiding Euler and quaternion pitfalls. A proportional-feed-forward controller is derived to align each agent’s body axis with a time-varying ascending direction $m_d$, with rigorous exponential convergence results under bounded unknown variations in the desired heading. A dispersion/maintenance analysis ensures the swarm deployment remains non-degenerate during transients, and the framework supports distributed covariance-based deployment considerations. Numerical simulations, along with open-source code, demonstrate practical effectiveness and robustness, highlighting the method’s potential for robust 3D swarm source seeking in realistic environments.

Abstract

This paper presents a geometric control framework on the Lie group SO(3) for 3D source-seeking by robots with first-order attitude dynamics and constant translational speed. By working directly on SO(3), the approach avoids Euler-angle singularities and quaternion ambiguities, providing a unique, intrinsic representation of orientation. We design a proportional feed-forward controller that ensures exponential alignment of each agent to an estimated ascending direction toward a 3D scalar field source. The controller adapts to bounded unknown variations and preserves well-posed swarm formations. Numerical simulations demonstrate the effectiveness of the method, with all code provided open source for reproducibility.

Figures (3)

• Figure 1: Illustration of a 3D unicycle robot. The robot moves with linear velocity $v = s00^\top$ and rotates with an angular velocity $\omega$. These velocities are expressed in the body-fixed frame $\mathcal{F}_B$, which is fixed at the body of the robot, but observed from the inertial frame $\mathcal{F}_E$. Vectors $\{e_x, e_y, e_z\}$ and $\{x_B, y_B, z_B\}$ denote the bases of $\mathcal{F}_E$ and $\mathcal{F}_B$, respectively.
• Figure 2: A swarm of $N=4$ 3D unicycle robots moving at a constant speed $s=0.6$space unit/time unit along $x_B$, tracking a time-varying desired attitude $R_d(t)$ with $x_{B_d}(t) = m_d(t)$. The time variation of $R_d(t)$ is given by the earth-fixed angular velocity vector $w_d(t) = R_d(t)^\top w^k + w^u$, where $\omega^k = [\pi/2,0,0]$ and $\omega^u = [0,0,-\pi/20]$, both in radians/time unit. The known component is $w^k$, while the unknown one satisfies $\|\omega^u\| \leq \omega_m^{\text{max}} = \pi/20$. Alignment control is given by \ref{['eq: omega_control_know']} with gain $k_\omega = \sqrt{2}\omega_m^{\text{max}}/\mu_{R_e}^*$ and $\mu_{R_e}^* = \delta^* = 0.4$radians for all robots. Left: alignment errors $\delta_i(t)$ and variation $\Delta \lambda_\text{min}(t)$. Right: robot trajectories, and body axes ($x_{B_i}$, red, $y_{B_i}$, green, $z_{B_i}$, blue).
• Figure 3: A swarm of $N=10$ unicycle robots with constant speed $s = 15$space unit/time unit align their body axis $x_B$ with the desired direction $x_{B_d}(t)=m_d(t)$ from bautista2025resilient to seek the source of a scalar field $\sigma(p)$. The desired attitude evolves with angular velocity $w_d(t)=R_d(t)^\top w^k + w^u$, where $\omega^k = [\pi,0,0]$radians/time unit is known and $w^u \in \mathbb{R}^3$ depends on the field and positions, as it is deeply explained in bautista2025resilient. Robots assume $\|w^u\| \leq \omega_m^{\text{max}} = \pi/4$radians/time unit, which holds when $p_c(t)$ is in a certain set where $\|\nabla\sigma(p_c(t))\| > \epsilon$ for a given $\epsilon > 0$bautista2025resilient. Alignment uses \ref{['eq: omega_control_know']} with $k_\omega = \sqrt{2} \omega_m^{\text{max}} / \mu_{R_e}^*$ and $\mu_{R_e}^* = 0.4$radians for all robots. Left: attitude error $\mu_{R_e}(t)$ and distance $\|p_i(t) - p_\sigma\|$ to the source. Right: scalar field projection, robot trajectories, and body axes ($x_{B_i}$, red, $y_{B_i}$, green, $z_{B_i}$, blue).