What Does Navigation Actually Mean?

Most people use the word navigation to mean “finding my position on a map”. That is only part of the problem. A navigation system usually tries to estimate a state: where something is, how fast it is moving, which way it is pointing, what time it is, and how uncertain all of those estimates are. Modern navigation is therefore closer to continuous state estimation than to simple map reading. Groves frames navigation as a broad field covering satellite navigation, inertial navigation, dead reckoning, feature matching and complete multisensor systems.¹

A useful first mental model is this:

$$\mathbf{x} = \begin{bmatrix} \mathbf{p} \\ \mathbf{v} \\ \boldsymbol{\theta} \\ \delta t \end{bmatrix}$$

where $\mathbf{p}$ is position, $\mathbf{v}$ is velocity, $\boldsymbol{\theta}$ is attitude or orientation, and $\delta t$ is clock error. Later in the series we will expand this state with sensor biases, scale-factor errors, covariance matrices and sometimes even map-matching hypotheses.

The problem

Imagine a car in a tunnel. A map app may still show a dot moving, but the phone may not be receiving usable GNSS signals. The dot is now being predicted from old position, estimated speed, heading, map constraints and inertial sensors. The system has not stopped navigating; it has switched from direct measurement to prediction.

A complete navigation answer should include:

• Position: Where am I?
• Velocity: How am I moving?
• Attitude: Which way am I facing?
• Time: What clock offset am I carrying?
• Uncertainty: How much should I trust this answer?

The last part matters because a wrong-but-confident navigation system is worse than one that admits it is uncertain.

The model

For a minimal 2D demo, use:

$$\mathbf{x} = [x, y, v, \psi]^T$$

where $x$ and $y$ are map coordinates, $v$ is speed and $\psi$ is heading. A simple constant-speed model is:

$$\dot{x} = v\cos\psi, \qquad \dot{y} = v\sin\psi$$

For a discrete time step $\Delta t$:

$$x_{k+1}=x_k+v_k\cos(\psi_k)\Delta t$$ $$y_{k+1}=y_k+v_k\sin(\psi_k)\Delta t$$

This is not yet a complete navigation system, but it introduces a state vector, a process model and a reason to care about errors.

Interactive demo: navigation state explorer

Use the explorer to see how position, heading, speed and uncertainty evolve together. Drag the state, adjust the process settings, and inject a noisy measurement to watch the estimate tighten.

What to watch

• Prediction moves the state smoothly according to the chosen speed, heading, and time step.
• The uncertainty ellipse grows during pure prediction.
• Adding a noisy position measurement pulls the state estimate and shrinks the covariance.
• Switching between standard mathematical angles and navigation heading conventions changes how the same angle value is interpreted.

Try this

Set the heading to $90^\circ$, speed to $10\,m/s$ and run for 10 seconds. In standard mathematical mode, the simulated vehicle should move roughly 100 metres in the positive $y$ direction. Then switch the convention to navigation heading, where $0^\circ$ is north and $90^\circ$ is east, and compare the motion.

Where this breaks

This first model ignores Earth curvature, acceleration, sensor errors, clock errors and coordinate frames. That is deliberate. The goal is to establish the core idea that navigation is state plus uncertainty evolving over time. Later posts add the missing pieces one by one.

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. ↩