The Robot That Finds the Middle of Everything — Using Only a Camera

There is a geometry problem so old and elegant that Pierre de Fermat posed a version of it in the 17th century: given a set of points, find the one spot whose total distance to all of them is minimized. It sounds academic. But this problem — now called the Fermat-Weber Location Problem, or FWLP — describes exactly what a surveillance drone should do to monitor several targets at once, or where a relay node in a sensor network should sit, or how a rescue robot should position itself to stay close to multiple survivors simultaneously.

The mathematics has been understood for decades. The trouble is that the clean theory assumes idealized robots — ones that can slide in any direction at will, like a hockey puck on frictionless ice. Real wheeled robots do not work that way. A unicycle-style robot, the kind that powers most ground vehicles and mobile platforms, must point its nose in the direction it wants to go before it can move forward. It cannot sidestep. This constraint — a "nonholonomic constraint," in the language of mechanics — turns a solved theoretical problem into a genuinely hard engineering one.

Now, Hong Liang Cheah, Mohammad Deghat, and Jose Guivant at the University of New South Wales have cracked it. In a new paper from UNSW's School of Mechanical and Manufacturing Engineering, they present the first bearing-only control laws that navigate a unicycle robot to the Fermat-Weber point without requiring distance measurements, GPS, or any information beyond the angles at which the robot can see its targets (Cheah et al., 2026). The solution handles not just stationary targets, but also the real-world reality of motor speed limits and even targets that are themselves moving.

The Science

The word "bearing-only" is the crux of why this is hard. A bearing is simply the angle at which you see something — the direction to a target, measured from your current heading. It is what a camera gives you naturally and cheaply. What it does not give you is distance. You can see that a target is "over there, at about 30 degrees to my right," but not whether it is 2 meters away or 200.

This matters because the Fermat-Weber point is implicitly defined by a weighted balance of directions. The robot is at the optimal spot when the weighted sum of all its bearing vectors to the beacons — each beacon's direction, scaled by its importance weight ${\gamma }_i$ — adds up to zero: ${\sum }_{i=1}^n{\gamma }_i{\mathbf{g}}_i=0_2$. When this vector sum is zero, the "pulls" in every direction cancel out, and you are at the optimum. Crucially, the robot can evaluate this condition using only bearing angles, even without knowing how far away each beacon is.

The challenge for the unicycle is that its motion is governed by two separate controls: a linear speed $\nu$ (how fast it moves forward or backward) and an angular rate $\omega$ (how fast it turns). The robot cannot move sideways; it is constrained to travel in the direction its heading vector $\mathbf{h}$ points. This couples position and orientation in a way that single-integrator models — the idealized "gliding" robots used in earlier FWLP work — simply do not have to contend with.

The research team at UNSW developed their control laws using Lyapunov stability theory, a mathematical framework for proving that a dynamical system will converge to a desired state. Think of a Lyapunov function as a carefully constructed energy measure: if you can show it always decreases over time, the system must settle where you want it. The team devised three progressively capable control laws — for stationary beacons, for actuator-limited robots, and for moving beacons — and proved convergence for each.

What They Found

The first and most fundamental result is a clean pair of control equations for the unconstrained case. The linear speed $\nu$ and angular rate $\omega$ are set proportional to how much the weighted bearing sum aligns with, or is perpendicular to, the robot's current heading:

$$\nu =k_p{\mathbf{h}}^{\top }\sum _{i=1}^n{\gamma }_i{\mathbf{g}}_i,\omega =k_h({\mathbf{h}}^{\perp }{)}^{\top }\sum _{i=1}^n{\gamma }_i{\mathbf{g}}_i$$

The intuition is elegant. If the weighted sum of bearings points mostly in the direction the robot is already facing ($\bm{h}^\top \sum \gamma_i \bm{g}_i$ is large and positive), the robot should drive forward. If the sum points mostly sideways relative to the heading (large $(\bm{h}^\perp)^\top \sum \gamma_i \bm{g}_i$), the robot should turn. The robot is simultaneously steering toward alignment and driving toward the point where bearing vectors balance.

In simulation, four unicycle agents placed at four different starting positions — all converging on four beacons arranged in a square, each beacon at a corner $(\pm 2,\pm 2)$ meters — all reached the Fermat-Weber point at the origin regardless of initial position or heading

. The tracking errors dropped smoothly to zero

, confirming asymptotic convergence.

The second result handles the uncomfortable reality that real motors have limits. Wheels spin only so fast; servo motors cap out. The paper introduces "saturated" control inputs — versions of $\nu$ and $\omega$ that are clipped at maximum forward speed ${\nu }_f$, maximum reverse speed ${\nu }_b$, and maximum turn rates ${\omega }_l$ and ${\omega }_r$. Rather than a proportional law, the robot now uses a saturation function: it applies full throttle or full turn when the bearing signal is large, and scales back proportionally when it is already close. Theorem 2 proves this saturated version still converges — a non-trivial result, because clipping introduces discontinuities that can destabilize naive control systems

.

The third and most sophisticated result addresses moving beacons — the scenario where every target is drifting at the same constant velocity ${\mathbf{v}}^{\ast }$. This is practically significant: think of a robot trying to optimally supervise a convoy of vehicles, or a drone tracking a school of fish. The moving-beacon control law adds a third internal state, $\mathbf{\varphi }$, which acts as an adaptive "velocity estimator." The robot implicitly learns the beacon drift rate from the way its bearing measurements evolve, without ever being told ${\mathbf{v}}^{\ast }$ directly. In steady state, the robot ends up moving at the beacon velocity while sitting exactly at the moving Fermat-Weber point

. The tracking error converges to zero even as the entire constellation of beacons translates across the plane

.

Physical experiments on a real wheeled robot corroborated the simulation results

. The robot, equipped only with a camera and onboard processing, successfully navigated to and held the Fermat-Weber point among stationary beacons, with the saturated control law keeping the motor commands within hardware limits throughout.

Why This Changes Things

The significance of "bearing-only" deserves emphasis. The alternative — measuring distances to beacons using LiDAR, ultrasound, or radio time-of-flight — requires active sensors that consume power, add weight, cost money, and fail in dusty, wet, or electromagnetically noisy environments. A small ground robot or drone navigating using only a camera is a fundamentally more deployable system. It can be cheaper, lighter, and more rugged.

The FWLP itself is more versatile than it might first appear. The weights ${\gamma }_i$ attached to each beacon can encode priority: a surveillance robot might weight a high-risk site more heavily, pulling the Fermat-Weber point closer to it. A search-and-rescue robot might assign higher weights to survivors with critical injuries. The optimal point shifts accordingly, and the same control law converges to the new optimum automatically.

Prior work had solved bearing-only FWLP for single-integrator agents (Trinh et al., 2015) and double-integrator agents (Le-Phan et al., 2025), but both of those idealizations break down for wheeled robots. A single-integrator model pretends the robot can apply a velocity in any direction at any instant — physically impossible for a vehicle that must steer first. A double-integrator model (which accounts for acceleration) can be made to mimic a unicycle via a mathematical transformation, but only when the robot is already moving; it is undefined when the robot is stopped, a condition that arises constantly in practice. The UNSW team's contribution fills a genuine gap between theory and deployable hardware.

There is also a conceptual elegance in the Lyapunov proofs that deserves appreciation. For the stationary case, the energy function $V_1={\mathbf{e}}^{\top }W(\mathbf{g}-{\mathbf{g}}^{\ast })$ — roughly, how far the current bearing configuration is from the optimal one — decreases at a rate proportional to $-k_p\Vert {\mathbf{h}}^{\top }{\sum }_i{\gamma }_i{\mathbf{g}}_i{\Vert }^2$. This quantity is zero only when the robot is aligned with the Fermat-Weber condition or has already reached it. LaSalle's invariance principle then rules out the first possibility as a steady state, leaving the second — arrival at ${\mathbf{p}}^{\ast }$ — as the only attractor. For the moving-beacon case, the Lyapunov function $V_2$ adds terms tracking how far the robot's internal velocity estimate is from the true beacon velocity, and how far the robot's heading is from the desired travel direction. The time derivative neatly telescopes to ${\dot{V}}_2=-k_1\Vert {\mathbf{h}}^{\top }{\sum }_i{\gamma }_i{\mathbf{g}}_i{\Vert }^2-\Vert ({\mathbf{h}}^{\perp }{)}^{\top }\mathbf{\varphi }{\Vert }^2$, negative semidefinite, guaranteeing convergence.

The control gains $k_p$, $k_h$, $k_1$, $k_2$, and $k_3$ are tunable parameters that let practitioners trade off speed of convergence against smoothness of trajectory. Simulation used $k_p=0.5$, $k_h=1$ for the basic case, and $k_1=0.5$, $k_2=1$, $k_3=1$ for the moving-beacon case — modest, practical values that produced clean convergence in both simulation and hardware.

What's Next

Several important open questions remain. The paper assumes all beacons move with the same constant velocity; a more general scenario where beacons drift at different or time-varying velocities would require a more sophisticated estimator. The collision-avoidance assumption — that the robot never runs into a beacon — is treated as a simplifying condition rather than an actively enforced constraint. Real deployments would need integration with obstacle-avoidance modules.

The current analysis is also restricted to two-dimensional motion. The unicycle model in 2D is the paper's focus, though the authors note it can in principle be extended to 3D nonholonomic agents using the same mathematical framework (Zhao, 2018). Extending bearing-only FWLP to aerial vehicles or underwater robots operating in three dimensions would be a natural and valuable next step.

Multi-robot extensions are perhaps the most tantalizing direction. If a team of unicycle robots each independently runs a bearing-only FWLP controller, do they collectively form a useful distributed sensor network without any inter-robot communication? The mathematics of bearing-based formation control (Zhao and Zelazo, 2016b) suggests this is possible, and combining the FWLP optimality result with multi-agent coordination could yield cooperative surveillance systems that self-organize without infrastructure.

For now, the result stands on its own considerable merits. A wheeled robot, given nothing but the angles at which it sees its targets, can navigate provably and reliably to the point in space that minimizes the total weighted distance to all of them — even as those targets move. The gap between elegant 17th-century geometry and deployable 21st-century robotics just got meaningfully smaller.