Optimal navigation in a noisy environment

TL;DR

This work addresses efficient navigation to a fixed target in noisy environments by proposing intermittent directional resetting as a minimal control strategy. It combines experiments with a differential-drive robot, an Active Brownian Particle (ABP) toy model, and scaling analysis to reveal a universal trade-off between noise-induced deviation and reorientation costs, yielding an optimal reset frequency $\alpha_{\mathrm{opt}}=\sqrt{D/\tau_{0}}$ and a non-monotonic dependence of mean target-hit time $\langle \tau \rangle$ on $\alpha$ via $\frac{\Delta\langle \tau\rangle}{D}=\frac{1}{\alpha}+\tau_{0}\frac{\alpha}{D}$. The study finds Gaussian first-passage-time distributions for rescaled times and non-Gaussian angular dispersion, with angular statistics collapsing onto a universal form under resets. These results are demonstrated to hold across experiments, ABP simulations, and scaling theory, suggesting that intermittent, low-overhead course corrections provide a robust navigation principle for noisy environments. The findings have implications for designing smart active matter and resilient robotic systems, including micro-robotic delivery and rescue missions, where continuous feedback is costly or impractical.

Abstract

Navigating toward a known target in a noisy environment is a fundamental problem shared across biological, physical, and engineered systems. Although optimal strategies are often framed in terms of continuous, fine-grained feedback, we show that efficient navigation emerges from a far simpler principle: natural wandering punctuated by intermittent course corrections. Using a controlled robotic platform, active Brownian particle simulations, and scaling theory, we identify a universal trade-off between noise-induced deviation and the finite cost of reorientation, yielding an optimal course correction frequency governed by only a few system parameters. Despite their differing levels of complexity, our experiment and theory collapse onto common quantitative signatures, including first-passage time distribution and non-Gaussian angular dispersion. Our results establish intermittent course-correction as a minimal and robust alternative to continuous feedback, offering a unifying guiding principle for point-to-point navigation in complex environments.

Figures (4)

• Figure 1: (a) Representative trajectories for different values of course-correction frequency $\alpha$ for a robot traveling from the $\epsilon$-neighborhood ($\epsilon=17~\text{cm}$) of region $A$ (grey circle) to that of region $B$ (red circle), separated by $AB\equiv L=150~\text{cm}$. At time $t$, the robot is at position $\mathbf{r}(t)$ with orientation $\theta(t)$ relative to the $x$-axis and heading vector $\hat{n}(t)=(\cos\theta(t),\,\sin\theta(t))$. The line of sight defines the angle $\psi(t)$, and at reset events at fixed intervals $1/\alpha$ the heading is realigned towards the target, i.e. $\theta(t)\to -\psi(t)$. The robot is considered to have reached the goal once it enters the $\epsilon$-neighborhood of $B$. The color gradient (blue to red) of the trajectories indicate the passage of time, including time delays from angular reset. (b) A sample evolution of the orientation $\theta$ (in degrees) with time $t$ (in seconds) for $\alpha=0.1$Hz. The red points mark the initiation of the resetting events. (c) Picture of differential-drive robot used in the experiment.
• Figure 2: Normalized mean target-hitting time $\Delta\langle \tau \rangle/D$ versus $\alpha/D$ for the experiment and the ABP-simulation. The experimental results are for the single-robot and the double-robot problems. The solid lines show non-monotonic dependence of the form in \ref{['eq:mfpt_rescaled']}. The inset shows the mean power consumption with the solid lines representing an empirical fit $P_0 + P_1/\alpha + P_2 \alpha$, with ($P_0=526,\; P_1=43,\; P_2=2.6$) for the single-robot case and ($P_0=509,\; P_1=344,\; P_2=4.7$) the two-robot problem.
• Figure 3: (a) Logarithm of normalized histogram for rescaled target-hitting time $P(\tau/\sigma_\tau)$ versus $-(\tau-\langle\tau\rangle)^2/\sigma_\tau^2$ for different $\alpha/D$, where $\sigma_\tau$ is the standard deviation of $\tau$. Data collapse to a straight line confirms a Gaussian distribution. Inset: the corresponding histogram in linear scale, from experiment ($\alpha/D=22.5$) and simulation ($\alpha/D=20.0$), with a gaussian fit indicated by the solid curves. (b) Normalized standard deviation as a function of $\alpha/D$ on a log-log plot. The dashed line indicates a power-law $(\alpha/D)^{-3/2}$.
• Figure 4: Distribution of angle $\theta$, rescaled by its standard deviation $\sigma_\theta$, for different values of rescaled reset frequencies $\alpha/D$ indicated in the legends. The solid line is the non-Gaussian distribution in \ref{['eq:theta_distribution 1']}, while the dashed line shows a Gaussian fit. The scaling collapse of all data, experimental and numerical, demonstrates a universal angular statistics that evades central limit regime.