Every robot that slips beneath the ocean surface trades certainty for endurance. GPS signals die within moments of a dive, and the underwater gliders prized for staying at sea for weeks must steer by dead reckoning, their sense of position drifting quietly with every meter traveled. That drift creates a paradox at the heart of autonomous exploration: a vehicle can measure the temperature, chemical pollution, or light in the water around it exquisitely well, yet have only a fuzzy idea of where each measurement belongs on the map. Engineers at Michigan State University, working with a colleague now at the Johns Hopkins University Applied Physics Laboratory, have turned that fuzziness from a liability into an organizing principle. In a study published in the journal Autonomous Robots, Demetris Coleman and colleagues taught a miniature gliding robotic fish to reconstruct a three-dimensional light field inside a large indoor tank while deliberately starved of accurate positioning, using a multi-fidelity statistical model that ranks the robot’s own data by how much it trusts itself. The payoff was dramatic: mapping errors fell to less than a quarter of those produced by conventional methods that ignore the problem.

The target of the critique is one of robotics’ favorite statistical tools. When a mobile robot must characterize a spatial field it cannot exhaustively sample, for missions ranging from search and rescue to multi-target search and environmental monitoring, it builds a surrogate model from sparse measurements, and the workhorse of that task is Gaussian process regression. A Gaussian process is fully specified by a mean function and a covariance kernel whose shape is governed by tunable hyperparameters; fed noisy measurements at known coordinates, Bayesian updating returns both a predicted value and a confidence level at every unvisited point. Those confidences are exactly what an adaptive planner needs to decide where to swim next, which is why Gaussian processes underpin so much robotic information gathering. But the mathematics carries a quiet assumption: inputs must be precise. When samples are tagged with estimated rather than true positions, predictions warp. Underwater vehicles suffer this in a particularly cruel form, because their localization uncertainty is not constant. It grows and shrinks depending on how long the robot has been submerged since its last surfacing fix, while feature-based alternatives such as simultaneous localization and mapping demand rich environments and computing power that small, energy-frugal vehicles simply do not carry.

Earlier remedies for uncertain inputs, including Monte Carlo approximations of the posterior, kernel-function expectations, and training schemes that treat input noise like output noise, typically assumed localization error stays constant, or were designed for stationary sensor networks rather than a lone robot whose uncertainty rises and falls through every dive. The team’s alternative is disarmingly simple. After each maneuver, the robot’s state estimator attaches a covariance to its position estimate, and the trace of that covariance, a single number summarizing total uncertainty, is compared against user-defined thresholds. Each measurement is then filed into one of several fidelity bins: the best-localized samples go to the highest-fidelity dataset, the most badly misplaced to the lowest. Each bin trains its own Gaussian process, and the models are coupled in a nested, auto-regressive cascade. The coarsest level builds an admittedly warped picture of the field; each higher level learns a correction to the level beneath it, anchored to its better-localized measurements; and the top level’s prediction is the final answer. Because every bin observes the same physical sensor, the authors set the scaling coefficients between levels to one. The splitting carries a bonus, too: evaluating several small models scales roughly with the sum of the cubes of their dataset sizes, which is cheaper in the worst case than one giant model trained on everything at once.

A better map is only half the problem; the robot still has to decide where to swim. The researchers paired their model with a sampling-based trajectory planner descended from rapidly-exploring information gathering algorithms, which grow a graph of candidate paths and optimize an information objective under resource constraints. Their variant adds two twists tailored to localization uncertainty. First, nodes are placed only in regions where position can be pinned down with minimal error, essentially at the surface for an underwater vehicle, while the edges between them are full three-dimensional excursions into the uncertain depths, built from feasible motion primitives such as steady glides at chosen path angles, helical spirals, flat dives, and horizontal swimming. Second, every candidate trajectory must keep its worst predicted localization uncertainty below a hard limit, on top of an energy or time budget, with an extended Kalman filter propagating position covariance along each edge. The planner can then maximize one of two objectives: the expected information gain over a grid of test points, or an ergodic metric that scores how well a trajectory’s time-averaged coverage matches a target distribution. That target is a softmax over a blend of the model’s predicted mean and its predicted uncertainty, a dial that tunes the mission between exploiting known hotspots and exploring the unknown; in these experiments the dial was set firmly toward exploration.

The platform anchoring all of this is the Miniglider, a miniature gliding robotic fish whose motion borrows more from buoyancy than from propellers. By adjusting its buoyancy and sliding an internal mass forward and backward, it glides through the water column in sawtooth paths at a fraction of the energy a swimming robot would burn, using its servo-driven tail only for low-speed maneuvers. The concept descends from pioneering autonomous gliders such as the Slocum concept of 1989 and the Spray glider, vehicles whose efficiency makes long-duration ocean missions possible precisely because they spend most of their time underwater, away from position fixes. The Miniglider’s dynamics are captured in a fifteen-state model spanning position, attitude, body-fixed velocities, the buoyancy actuator, the movable mass, and the tail angle, extended here with actuator dynamics, tail forces, and a corrected lift model, alongside an energy budget that charges the planner for every pump stroke, mass shift, and tail beat. That accounting is not decorative: it forces the planner to weigh the information value of a long glide against the cost of swimming, echoing the economics that make real ocean gliders attractive for months-long surveys.

The proving ground was a tank 4.6 meters long, 3.1 meters wide, and 1.2 meters deep, its floor and walls papered with fifteen-centimeter AprilTag fiducials. Three overhead cameras covering roughly 2.8 by 1.7 meters of the surface, together with a radio link, served as the stand-in GPS that the robot could use only when it came up for air; two tags mounted on its wings made those fixes possible. Two LED lights in the tank generated the artificial field, and the robot logged the blue-channel light intensity through its onboard color sensor at half a hertz, each reading paired with its position estimate and the growing uncertainty attached to it. A ground-truth model of the light field was assembled beforehand from about 1,100 teleoperated measurements, down-sampled by spatial density and smoothed with a Matérn kernel. The team then ran six trials of each of four strategies, single- or multi-fidelity models each driving either the information-gain or the ergodic objective, granting the planner about forty-five seconds per round and an energy budget of eighty units, doled out twenty at a time.

The scoreboard was lopsided. The single-fidelity planner that ignored localization error, treating every measurement as equally trustworthy, finished with an average weighted mean squared error of 3.803 when steered by information gain; its multi-fidelity counterpart scored 0.425. With the ergodic objective the pattern held: 2.862 for the oblivious model against 0.367 for the multi-fidelity one. In both pairings, sorting data by trust pushed the error below twenty-five percent of the baseline’s and shrank the variability across trials dramatically. The oblivious model’s failure mode is instructive. When two samples taken close together disagree because the robot misjudged where it was, a standard Gaussian process develops the geostatistical pathology known as the nugget effect, erupting into spuriously large, or even negative, predictions despite a sensor that only ever returns positive values. The multi-fidelity model makes some of the same early mistakes but can heal them: the moment a better-localized sample arrives nearby, the nested correction pulls the map back toward reality, and where no such sample exists the model simply withholds confidence. Curiously, the ergodic objective beat information gain under both models, suggesting that covering a field in proportion to its promise may guide exploration better than chasing individual measurements one at a time.

Simulations extended the result beyond the tank’s geometry. The team generated ninety synthetic datasets by replaying a reference trajectory through a ten-by-twenty-by-ten-meter volume while corrupting velocity measurements at three noise levels, then trained their model, a naive single-fidelity model, and a third baseline built to handle Gaussian input noise directly. Both uncertainty-aware approaches clearly beat the naive model, and the input-noise baseline edged out the multi-fidelity method on raw accuracy. The catch was time. On an ordinary laptop, training the input-noise baseline took roughly five to ten minutes per dataset of about two thousand points, while the multi-fidelity hyperparameters trained in seconds to a few minutes, a decisive margin on the modest computers that ride inside small, inexpensive robots. The authors also caution that the simulated positional errors stayed modest, and larger drifts of the kind real ocean gliders accumulate between surfacings may widen the gap in favor of the binned approach, which was designed precisely for those long, blind stretches.

What the study ultimately offers is a template for autonomy wherever the world’s positioning infrastructure runs out, whether under ice, in deep water, in disaster zones where GPS is jammed or absent, and in search, rescue, or pollution-tracking missions where the map must be built by a machine that cannot always know where it is. The framework asks for no new sensors; it requires only that a robot grade its own data by trust and let its planner respect that grading. The authors point to clear next steps: learning the model’s hyperparameters online during a mission, refining trajectory estimates with techniques such as motion tomography before feeding data to the map, finding principled ways to choose the number of fidelity levels, and scheduling the swing between exploration and exploitation over a mission’s lifetime. With the code released openly and the experimental evidence in hand, a robotic fish that answers “where am I?” and “what is out there?” together, while flagging which of its answers are shaky, marks a quietly radical step for machines that work where satellites cannot follow.

Phytoplankton are microscopic organisms that form the foundation of marine food webs and contribute more than half of the oxygen produced by Earth’s biosphere. Their distribution, however, is far from smooth. Ocean currents, eddies, internal waves, sunlight, nutrients, temperature and grazing by zooplankton can gather them into transient patches that shift through space and time. These structures may extend horizontally across kilometers but vary sharply with depth, sometimes forming narrow layers below the surface. Chlorophyll A is useful because its concentration generally tracks the amount of phytoplankton present, although the relationship depends on species and environmental conditions. Satellite ocean-color measurements can reveal broad surface patterns, but clouds, suspended particles and dissolved organic matter can obscure the signal. More importantly, satellites cannot reliably see blooms that begin deep underwater. An AUV carrying a fluorometer can instead measure chlorophyll directly while moving through the water column.

The new system treats the changing chlorophyll field as a four-dimensional problem: north-south position, east-west position, depth and time. Its statistical engine is a Gaussian random field, a mathematical model that represents how measurements at nearby locations are related. The researchers modeled the logarithm of chlorophyll A rather than the raw concentration, a transformation that helps accommodate strongly skewed biological data and allows the modeled quantity to vary across the full real-number line. Before the mission begins, the model is given a depth-dependent mean and correlation scales describing how quickly chlorophyll patterns change laterally, vertically and over time. The correlations in the horizontal plane and depth follow Matérn functions, which can represent moderately smooth environmental variation, while the time correlation follows an exponential form suited to less predictable fluctuations. Every new fluorometer reading updates the model, changing both the predicted chlorophyll level and the uncertainty at nearby unvisited locations.

The vehicle then evaluates possible future trajectories using a decision rule called expected improvement. At each candidate point, the algorithm estimates the probability that the vehicle will find a chlorophyll value higher than the best one observed so far, as well as the size of the potential gain. Mathematically, if the predicted log-chlorophyll value has mean (m), uncertainty (v), and the current maximum measurement is (x_{text{max}}), expected improvement combines the term ((m-x_{text{max}})Phi((m-x_{text{max}})/v)) with an uncertainty term involving the normal probability density. The result rewards both exploitation—returning to areas likely to contain intense chlorophyll—and exploration, where uncertainty is large enough that a previously unknown hotspot might be discovered. This balance is crucial. A strategy based only on predicted intensity can become trapped around a local maximum, while a strategy based only on variance may spend too little time sampling the biologically important regions.

Path selection is divided into two linked stages designed to match the limitations of an underwater robot. First, while near the surface, the AUV chooses among seven possible lateral directions arranged like the spokes of a spider web. It selects the direction whose prospective transect offers the greatest expected improvement. The second stage chooses depths along that route. The vehicle’s diving angle limits how rapidly it can move vertically; in the Norwegian trials, a 10-degree limit allowed roughly 17 meters of vertical movement for every 100 meters traveled laterally. Nine possible depth profiles were evaluated during each transect. The vehicle also returned to the surface after 800 meters or 15 minutes, whichever came first, so that it could obtain a GPS position and correct accumulated navigation error. This surface reset sacrifices some sampling time, but it prevents uncertainty in dead-reckoned position from growing too large, particularly in strong currents.

A major engineering challenge was making the calculations fast enough for a relatively small onboard computer. Conventional spatio-temporal models often rely on dense grids covering an entire survey area. Updating a Gaussian model on such a grid can require matrix operations whose computational cost rises approximately as the cube of the number of conditioning measurements. The researchers therefore used a grid-free design. The AUV retained observed locations and values rather than maintaining a permanent high-resolution map, and it generated only the small sets of points needed to compare candidate paths immediately ahead. The system also thinned the stored data when the mission became computationally demanding, removing redundant nearby observations and measurements far from the vehicle. Because spatially distant data have limited influence on local predictions—a property related to the screening effect in kriging—this reduction was designed to preserve useful accuracy while keeping response times manageable. The onboard platform was a Light Autonomous Underwater Vehicle equipped with an NVIDIA Jetson TX2 and integrated with robotic software used to exchange sensor and navigation data.

Before going to sea, the team tested the strategy in 100 simulated chlorophyll landscapes, each covering a 4-by-4-kilometer area and extending to 75 meters depth. The virtual vehicle had four hours to survey, traveled at 1 meter per second and periodically surfaced. Expected improvement was compared with maximum variance, maximum expected intensity, probability of improvement and a systematic lawnmower pattern. The principal test classified a hotspot as a location above the 90th percentile of chlorophyll values across the simulated field and mission. Expected improvement increased the fraction of time spent in these top-concentration areas more rapidly than the other adaptive methods and eventually stabilized at the highest level. It also explored hotspot clusters more effectively than the strategy based on maximum expected intensity, which sometimes remained focused on one region after finding a promising signal. Maximum variance visited slightly more clusters overall, but did not examine them as thoroughly. The results indicate that the best strategy depends on the goal: broad uncertainty reduction across an entire field favored systematic or variance-driven paths, whereas locating and characterizing intense patches favored expected improvement.

The field demonstration took place in the Frohavet region near Mausund, roughly 100 kilometers from Trondheim, during two missions on June 6 and 7, 2024. The vehicle used a RBR Tuner Cyclops7 fluorometer to guide its decisions and carried additional instruments for offline comparison, including a conductivity-temperature-depth sensor and a SilCam imaging system for zooplankton. After an initial dive to 70 meters, the adaptive controller directed the AUV mainly toward depths between about 10 and 30 meters, where chlorophyll readings were highest. On the first day, the strongest layer occupied approximately 0 to 25 meters; on the second, it was centered slightly deeper, around 10 to 30 meters, and appeared narrower. The observations also revealed a sharp transition in temperature and salinity near 40 meters, consistent with a seasonal thermocline separating warmer, fresher surface water from colder, saltier water below. Chlorophyll declined rapidly beneath the well-mixed upper layer, suggesting that the robot was tracking a biologically distinct near-surface structure rather than simply responding to a gradual vertical trend.

The researchers also found evidence that phytoplankton-rich water was associated with concentrations of the copepod Calanus finmarchicus, a common zooplankton grazer and an important food source for larger marine animals. The SilCam photographed a small illuminated volume of water at one frame per second, and images were later segmented and classified with a convolutional neural network. The clearest relationship appeared at depths of roughly 10 to 25 meters and at chlorophyll readings around 2 to 4 in the study’s measurement scale, where images contained more suspected Calanus individuals. The result is consistent with copepods gathering where phytoplankton is abundant, although it does not yet provide a calibrated estimate of population size or biomass. Motion blur caused by the AUV’s operating speed made species identification difficult, and copepods may have avoided the vehicle’s hydrodynamic disturbance. The authors therefore describe the relationship as suggestive rather than definitive. Future versions could assimilate chlorophyll, temperature, salinity and image-derived plankton data simultaneously, allowing robots to seek regions that satisfy several biological objectives at once. For now, the work shows how an underwater robot can turn sparse observations into an adaptive biological survey, seeking not merely to pass through the ocean but to follow its most important living signals.