Dead reckoning

What is dead reckoning?

Dead reckoning is the oldest and most universal method of navigation. The idea is beautifully simple: if you know where you started, and you keep track of every movement you make — each step’s direction and distance — then you can compute where you are at any moment by summing them up. No map, no landmarks, no satellites. Just integration of your own motion. The name may come from “deduced reckoning” (ded. reckoning in old logbooks), or possibly from navigating “dead” — without wind or current references. Either way, the principle is the same: position = origin + ∑ displacements. It is the method used by every inertial navigation system ever built, by every animal that returns to its nest after foraging, and by every sailor who has ever crossed open water without sight of land.

Why does it drift?

The trouble with dead reckoning is that it is an integrating system, and integration accumulates errors. If your compass is off by half a degree on a single step, that half-degree error gets baked into every subsequent position estimate. There are two types of error, and they behave very differently. Random noise — small unpredictable fluctuations in each measurement — causes the error to grow as σ√N, where σ is the noise per step and N is the number of steps. This is the same square-root law that governs random walks and diffusion. Systematic bias — a consistent offset like a miscalibrated compass — is far worse: the error grows linearly as βN. A submarine with a gyroscope that drifts 0.01° per hour will be miles off course after a day. Random errors are forgiving; bias is relentless. The error chart in this simulation lets you see both regimes directly: the concave √N curve of pure noise versus the straight line of bias.

Desert ants and path integration

The Saharan desert ant Cataglyphis fortis is perhaps the most celebrated dead reckoner in nature. It forages across featureless salt pans at temperatures that would kill most insects, following a tortuous search path that can extend hundreds of meters from the nest. When it finds food, it turns and walks in a nearly straight line directly back to its nest entrance — a hole a few millimeters wide in a vast, flat expanse. The classic experiments of Rüdiger Wehner and Martin Müller in the 1980s and 1990s demonstrated that Cataglyphis performs path integration using two inputs: a step counter (it literally counts its strides) and a sky compass (it reads the polarization pattern of sunlight to determine heading). The ant continuously integrates these signals into a “home vector” that points from its current position back to the nest. In a famous displacement experiment, Wehner and Müller captured ants that had just found food and moved them to a distant, featureless test field. The ants walked exactly the distance and direction that would have taken them home — then searched, bewildered, at the spot where the nest should have been. They had no landmark memory, no path retracing. Pure path integration. Try the “Desert ant” preset and hit “Return to nest” to see this in action.

Polynesian wayfinding and the etak system

The navigators of the Pacific Islands crossed thousands of miles of open ocean in double-hulled canoes, centuries before European contact, without instruments. Their method was a form of dead reckoning so sophisticated that Western researchers struggled to understand it for decades. The key concept is etak, a system in which the navigator holds the canoe mentally stationary and imagines the islands, stars, and ocean moving past. A reference island off to one side “moves” under successive star positions on the horizon, and the navigator tracks progress by which star the reference island currently lies beneath. This is dead reckoning, but with the coordinate frame inverted — the vessel is the fixed origin, and the world moves. Edwin Hutchins, in his landmark work on distributed cognition, argued that Polynesian navigation is not an inferior version of Western chart-based navigation but a different — and in some ways more elegant — cognitive technology: a system in which knowledge is distributed across the navigator’s mental model, the star compass, the wave patterns, and the social structure of the crew. The navigators also used wave piloting: reading the interference patterns of ocean swells refracted around distant islands to detect land far beyond visual range. Their periodic “star fixes” — checking the position of known stars at the horizon — served the same function as the correction intervals in this simulation: resetting accumulated drift. Try the “Polynesian canoe” preset to see how periodic corrections tame the growth of error.

Modern inertial navigation

Every submarine, aircraft, and spacecraft carries an inertial navigation system (INS) — a dead reckoning computer built from gyroscopes and accelerometers. The gyroscopes measure rotation (to track heading), and the accelerometers measure linear acceleration (which is integrated twice to get displacement). Early INS used mechanical spinning gyroscopes; modern systems use ring laser gyros or fiber-optic gyros, which measure the Sagnac effect — the phase shift between two laser beams traveling in opposite directions around a closed loop. A nuclear submarine under the Arctic ice cap may navigate for weeks using only its INS, accumulating drift on the order of 1–2 nautical miles per day — remarkable precision, but still drift. The drift is dominated by bias in the gyroscopes: a ring laser gyro with a bias stability of 0.003°/hour will still accumulate heading error over time. This is why submarines surface for GPS fixes whenever possible, and why ballistic missile submarines carry the most expensive gyroscopes ever manufactured. The “Submarine” preset in this simulation models this scenario: long straight segments, very low noise, slight bias, and infrequent corrections.

The correction problem — from landmark fixes to Kalman filters

The fundamental limitation of dead reckoning — unbounded error growth — means that every practical navigation system needs a way to correct. The simplest correction is a landmark fix: at some point you look up, see something whose position you know, and reset your estimated position to match. Polynesian navigators took star fixes. Sailors used celestial navigation. Pilots checked ground features. In this simulation, a “correction” resets the dead reckoned position to the true position at regular intervals. The optimal approach to combining continuous dead reckoning with intermittent corrections is the Kalman filter, developed by Rudolf Kalman in 1960. The Kalman filter maintains a probability distribution over the agent’s position: between corrections, the distribution spreads (reflecting growing uncertainty from integration); at each correction, it narrows (incorporating the new information). GPS itself is the ultimate correction — a global network of atomic clocks broadcasting time signals that let any receiver compute its position to within meters. GPS does not navigate; it corrects. The INS navigates. Modern systems fuse both: the INS provides smooth, high-rate position updates between GPS fixes, and GPS resets the drift. The result is better than either system alone.

The universal lesson

Dead reckoning reveals a deep principle that extends far beyond navigation: any system that estimates its state by integrating incremental updates will drift without external ground truth. This is true of inertial navigation, but it is also true of beliefs updated by successive pieces of evidence (where confirmation bias acts like compass bias), of scientific knowledge that accumulates through citation chains without periodic replication (a kind of epistemic dead reckoning), and of organizations that track their goals through internal metrics without external validation. The square-root law is forgiving — random errors grow slowly — but bias is not. A small systematic distortion, invisible on any single step, becomes dominant over time. The only remedy is the same one that navigators discovered millennia ago: look up. Check your position against something you did not compute. Take a fix. The correction does not have to be frequent; even rare corrections dramatically reduce drift. But without any correction at all, every integrating system eventually loses its way.