Safe Ergodic Control for Multi-Robot Systems via Quadratic Programming

Yo Toyomoto; Mohamed Elobaid; Bryce L. Ferguson; Alberto Quattrini Li; Shinkyu Park

cs.ROOct 4, 2026

Safe Ergodic Control for Multi-Robot Systems via Quadratic Programming

Authors: Yo Toyomoto, Mohamed Elobaid, Bryce L. Ferguson, Alberto Quattrini Li, Shinkyu Park

Organizations: Electrical and Computer Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia · Thayer School of Engineering at Dartmouth College, Hanover, NH, USA · Department of Computer Science, Dartmouth College, Hanover, NH, USA

Abstract

Ergodic control drives robots to spend time in each region in proportion to a spatial distribution of interest, making it well suited for dense spatiotemporal environmental monitoring. Existing safe ergodic controllers rely on offline trajectory optimization or hierarchical architectures, which limit real-time applicability and decouple the ergodicity objective from the safety constraint. This paper presents a quadratic programming (QP)-based controller that treats ergodicity and safety jointly. We first introduce a Gaussian-kernel ergodic metric that, unlike the classical indicator-based metric, is time differentiable. This allows the exponential decay of the metric to be imposed as a time-varying control barrier function (CBF) constraint, relaxed by a slack variable, alongside hard CBF constraints for region containment and inter-robot collision avoidance. We establish that the resulting QP remains feasible from any safe initial configuration and that, up to the slack term, the ergodic metric decays exponentially. Simulations show improved performance over baseline methods, and experiments with three aerial vehicles validate the work on real hardware.

Figures & tables

Fig. 1 : Safe ergodic control of multiple robots. Three drones move within a region $\mathcal{Q}$ so that the time spent along their trajectories (white curves) is distributed according to a desired distribution $\phi$ (color scale), while remaining inside $\mathcal{Q}$ and keeping a mutual distance of at least $r_{\min}$ (green circles of radius $r_{\min}/2$ ).

Parameter	Value	Parameter	Value
$\sigma_{\min}$	$0.1$	$\sigma_{\max}$	$2.0$
$\alpha(\theta)$	$\theta$	$\gamma$	$0.1$
$w$	$10^{12}u_{\max}$	$\beta_{\mathrm{b}}(\theta)$	$\theta$
$\beta_{\mathrm{c}}(\theta)$	$0.25\theta$	$r_{\min}$	$0.3$

TABLE I : Parameters used in the simulations.

Fig. 2 : Time evolution of the ergodic metric $\mathcal{E}$ in ( 7 ) for a single robot under a uniform target distribution scenario with the proposed controller (blue) and SMC baseline (red). Thin lines show ten trials with random initial positions; bold lines show their averages.

Fig. 3 : Time evolution of the ergodic metric $\mathcal{E}$ in ( 7 ) for three robots under a two-peak target distribution scenario with the proposed controller (blue) and hierarchical SMC–CBF baseline (red). Thin lines show ten trials with random initial positions; bold lines show their averages.

Fig. 4 : Robot position along $x$ , $y$ , and $z$ in the two-peak simulations with $\ell_{\phi}=1.0$ , $2.0$ , and $3.0\text{\,}\mathrm{m}$ (top to bottom). Gray dash-dotted lines mark the coordinates of the two peaks.

Fig. 5 : Time evolution of the ergodic metric $\mathcal{E}$ in ( 7 ) for a single robot under two-peak target distribution with separations $\ell_{\phi}\in\{1.0,2.0,3.0\}\,$\mathrm{m}$$ (solid) and under the uniform target distribution of the first scenario (dashed). — Fig. 5 : Time evolution of the ergodic metric $\mathcal{E}$ in ( 7 ) for a single robot under two-peak target distribution with separations $\ell_{\phi}\in\{1.0,2.0,3.0\}\,$ \mathrm{m}$$ (solid) and under the uniform target distribution of the first scenario (dashed).

Panel of Figure 7 — Fig. 6 : Snapshots of the three experiments with $n_{\phi}=2$ , $3$ , and $4$ peaks (left to right) at $t=$ 30\text{,}\mathrm{s} $(top) and $t=$150\text{\,}\mathrm{s}$ (bottom). Target distribution is overlaid in color (red high, blue low) and drone trajectories are shown in white.

Fig. 7 : Pairwise distances between drones $\|\bm{p}_{j}-\bm{p}_{k}\|_{2}$ for $n_{\phi}=2$ , $3$ , and $4$ (top to bottom). Legend entries indicate the pair $(j,k)$ . The dashed line marks $r_{\min}=$0.3\text{\,}\mathrm{m}$$ . — Fig. 7 : Pairwise distances between drones $\|\bm{p}_{j}-\bm{p}_{k}\|_{2}$ for $n_{\phi}=2$ , $3$ , and $4$ (top to bottom). Legend entries indicate the pair $(j,k)$ . The dashed line marks $r_{\min}=$ 0.3\text{,}\mathrm{m}$$ .

Fig. 8 : Time evolution of the ergodic metric $\mathcal{E}$ in ( 7 ) in the experiments with $n_{\phi}=2$ , $3$ , and $4$ peaks.

Fig. 9 : Drone trajectories in the $xy$ -plane in the two-peak experiment for $\sigma_{\max}=0.3$ , $1.0$ , and $2.0\text{\,}\mathrm{m}$ (left to right). Crosses mark the peak locations.

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Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada

Multi-Robot Systems Lyapunov Function

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This paper presents a novel density control framework for multi-robot systems with spatial safety and energy sustainability guarantees. Stochastic robot motion is encoded through the Fokker-Planck Partial Differential Equation (PDE) at the density level. Control Lyapunov and control barrier functions are integrated with PDEs to enforce target density tracking, obstacle region avoidance, and energy sufficiency over multiple charging cycles. The resulting quadratic program enables fast in-the-loop implementation that adjusts commands in real-time. Multi-robot experiment and extensive simulations were conducted to demonstrate the effectiveness of the controller under localization and motion uncertainties.

Abstract

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