Dead Reckoning: Navigating with Speed and Direction

Dead reckoning is the idea that if you know where you started, which way you went and how fast you travelled, you can estimate where you are now. It is one of the oldest navigation ideas and it is still central to modern systems. Groves treats dead reckoning as part of the broader family of navigation technologies alongside GNSS, inertial navigation and environmental feature matching.¹

The problem

Dead reckoning feels simple:

Start here. Travel this far in this direction. Update position.

The catch is that every small heading or speed error is integrated into position. The system may look smooth, but smooth does not mean correct.

The model

In a local north-east frame:

$$\dot{n} = v\cos\psi$$ $$\dot{e} = v\sin\psi$$

where $n$ is north position, $e$ is east position, $v$ is speed and $\psi$ is heading.

Discrete propagation is:

$$n_{k+1}=n_k+v_k\cos\psi_k\Delta t$$ $$e_{k+1}=e_k+v_k\sin\psi_k\Delta t$$

If heading has a constant error $\delta\psi$, the cross-track error grows approximately with distance:

$$e_{cross} \approx s\sin(\delta\psi)$$

For small angles:

$$e_{cross} \approx s\delta\psi$$

where $s$ is distance travelled and $\delta\psi$ is in radians.

Interactive demo: dead-reckoning drift simulator

This simulator keeps a truth route and a dead-reckoned estimate separate. The truth vehicle follows the selected path, while the estimate only sees speed, heading, time, and the biases you inject.

What to look for

• With zero speed bias, zero heading bias, and zero noise, the estimate should sit on top of the truth path.
• A fixed heading bias creates growing cross-track error, especially on the straight route.
• A fixed speed bias creates along-track error even when heading is correct.
• Lower update rates and added heading noise make the live error history less stable.
• The error vector and summary panel show the north, east, and total error building in real time.

Try this

Set heading bias to just $1^\circ$ and drive 10 km. The approximate cross-track error is:

$$10000\sin(1^\circ) \approx 175\,m$$

That small heading error is enough to put the estimate on the wrong road.

Where this breaks

This demo assumes you already know heading and speed. Inertial navigation is harder because it must infer orientation and velocity from gyroscopes and accelerometers. This post is the conceptual bridge into INS: inertial navigation is high-rate dead reckoning with much stricter sensor and error requirements.²

Footnotes

1. Paul D. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition, Artech House, 2013. https://uk.artechhouse.com/Principles-of-GNSS-Inertial-and-Multisensor-Integrated-Navigation-Systems-Second-Edition-P1557.aspx - Core textbook reference for navigation systems, GNSS, INS, multisensor integration, coordinate frames, dead reckoning, feature matching and environmental aiding. Accessed 2026-06-11. ↩

2. Paul D. Groves, “Navigation Using Inertial Sensors”, IEEE Aerospace and Electronic Systems Magazine, 2015. https://ieee-aess.org/media/navigation-using-inertial-sensors - Tutorial covering inertial sensor technology, strapdown navigation, alignment, zero updates, motion constraints, pedestrian dead reckoning and fault detection. Accessed 2026-06-11. ↩