Position fixing, dead reckoning, inertial navigation, GNSS, feature matching, and how they all fit together.
You are driving through an unfamiliar city. Your phone shows a blue dot on a map, tracking you in real time. How does it know where you are? That question — how does a system determine its own position and velocity? — is the central problem of navigation.
There is no universally agreed definition, but for this book, navigation means: the automatic determination of the position and velocity of a moving body with respect to a known reference. Some systems also provide attitude (which way you're pointing), angular rate, and acceleration.
A navigation system is the device that does this automatically. A navigation sensor is a component that measures a physical property (like acceleration, radio signal timing, or magnetic field) from which the system computes its answer. The output is called the navigation solution.
| Concept | Meaning |
|---|---|
| Navigation solution | Position, velocity, and optionally attitude |
| Navigation sensor | Device measuring a physical quantity (accelerometer, GPS receiver, etc.) |
| Self-contained system | All hardware aboard the vehicle (e.g., INS) |
| Infrastructure-dependent | Requires external signals (e.g., GPS satellites) |
Positioning is a special case: determining position only, without velocity or attitude. Many systems (like GPS) are technically positioning systems, but operate fast enough that velocity can be derived from the rate of change of position.
Tracking differs from navigation: a third party determines your position without equipment on your vehicle. But a tracker can become a navigator simply by transmitting the results back to you.
Position fixing determines where you are right now by measuring your relationship to known reference objects. There are two main methods: bearing and ranging.
Bearing-based fixing measures the angle between the line of sight to a known object and a reference direction (usually north). Two bearing measurements from two known objects give you two lines — you're at the intersection. For 3D, add an elevation angle measurement.
Range-based fixing measures the distance to known objects. One range measurement places you on a sphere centered on that object. Two ranges give a circle (the intersection of two spheres). Three ranges narrow it to two points — usually one is obviously wrong (it might be inside the Earth or in space). A fourth measurement resolves any remaining ambiguity.
If you can measure both range and bearing to a single object, one reference is enough for a position fix. This is the principle behind radar.
Bearing measurements can be made with simple technology: a compass and a line of sight. The Sun, Moon, and stars have served as reference objects for centuries. Longitude at sea became practical only in the 1760s, when John Harrison built clocks accurate enough to keep time on transoceanic voyages.
Range measurements require more technology: radio signals, lasers, or radar. In passive ranging, you receive signals from transmitters (like GPS). In active ranging, you transmit a signal and receive the echo or response.
Dead reckoning (possibly from "deduced reckoning") takes a different approach: instead of asking "where am I?", it asks "how have I moved since I last knew where I was?"
The idea is simple. Start from a known position. Measure how far you've traveled and in what direction. Add that to your last position. Repeat.
Speed or distance is measured in the body frame (the vehicle's own axes), so a separate heading measurement is needed to convert that into movement in the reference frame (north/east/down). For 3D navigation, full three-axis attitude is required.
The critical weakness of dead reckoning: errors accumulate. Each small measurement error adds to the total. Over time, the position solution drifts further and further from truth. Position fixing doesn't have this problem — each fix is independent — but it requires external infrastructure that may not always be available.
| Property | Position Fixing | Dead Reckoning |
|---|---|---|
| Error behavior | Bounded (each fix is independent) | Grows with time |
| Availability | Requires external references | Self-contained, continuous |
| Starting info | None (self-referencing) | Needs known initial position |
| Example | GPS, Loran, celestial nav | INS, odometer, pedometer |
An inertial navigation system (INS) is the most complete dead reckoning system. It provides full 3D position, velocity, and attitude — all self-contained, with no external signals.
The hardware is an inertial measurement unit (IMU): three mutually orthogonal accelerometers and three gyroscopes, all rigidly mounted to the vehicle body.
Accelerometers measure specific force, not acceleration. Specific force is the acceleration due to all forces except gravity. A stationary accelerometer on a table reads 1g upward — the reaction to gravity — not zero. A gravity model is used to recover the true acceleration.
INS quality spans several orders of magnitude. Military aircraft INS: <1.5 km drift in the first hour, ~$100K. MEMS sensors in a phone: not suitable for standalone inertial navigation, but useful for short-term dead reckoning when integrated with GPS.
The world's first satellite navigation system was the U.S. Navy's Transit (1964). It used Doppler positioning from low-altitude satellites — one fix per satellite pass, with up to 100 minutes between passes. Useful for ships and geodesy, but impractical for aircraft or cars.
GPS (development started 1973, IOC 1993) changed everything. Instead of Doppler, it uses passive ranging from a constellation of 24+ satellites at ~20,200 km altitude. Signals from at least four satellites are available almost everywhere on Earth, all the time.
The principle: each satellite broadcasts a synchronized timing signal. The receiver measures the time of arrival, subtracts the time of transmission (encoded in the signal), and multiplies by the speed of light to get a pseudo-range. "Pseudo" because the receiver clock isn't perfectly synchronized — but that clock error is common to all measurements, so it becomes a fourth unknown solved alongside the three position coordinates.
Where ρj is the pseudo-range to satellite j, tsa is time of signal arrival, tst is time of signal transmission, and c is the speed of light.
Three global systems exist or are planned: GPS (USA), GLONASS (Russia), and Galileo (EU). Regional systems include Beidou/Compass (China), QZSS (Japan), and IRNSS (India). Collectively: GNSS.
Feature matching determines position by comparing observed features of the environment with a database — much as you would compare landmarks with a map.
Terrain-referenced navigation (TRN) is the best-known example. An aircraft's radar altimeter measures height above terrain. Subtract the aircraft's barometric altitude to get terrain height. Compare a sequence of terrain height measurements against a terrain database to determine position. Accuracy: ~50 m, best over hilly terrain.
Image matching captures images (camera, SAR, or laser scanner) and compares features with a database. It works better over flat terrain with distinctive features — complementary to TRN. Accuracy: ~20 m.
Map matching exploits constraints: cars drive on roads, trains run on tracks, pedestrians don't walk through walls. By following a dead reckoning solution on a map, corrections can be applied when the solution strays outside permitted areas.
No single navigation technology is perfect for all applications. Position fixing (GNSS) gives long-term accuracy but can be blocked or jammed. Dead reckoning (INS) gives continuous output but drifts. Feature matching works in some environments but not others. The solution: integration.
The dead reckoning system provides the continuous solution. Position fixes from GNSS (or other systems) are used by an estimation algorithm to correct the DR solution's drift.
The estimation algorithm at the heart of integration is usually based on the Kalman filter (Chapter 3). It optimally combines the dead reckoning predictions with position fix measurements, weighting each by its uncertainty.
Different applications need different sensor combinations:
| Application | Typical Sensors |
|---|---|
| Commercial airliner | INS + GPS + baro + ILS (landing) |
| Military aircraft | INS + GPS + TRN + Doppler radar |
| Submarine | INS + sonar + depth sensor + gravity gradiometry |
| Road vehicle | GPS + odometer + compass + map matching |
| Pedestrian | GPS + MEMS IMU + baro + WLAN + compass |
Integrity monitoring ensures the navigation solution is trustworthy. Fault detection tells you something is wrong. Fault isolation identifies which sensor failed. Fault exclusion removes the faulty data and verifies the result. For safety-critical applications (civil aviation), this must be formally certified.
This introductory chapter has laid out the landscape of navigation technology. Here are the key takeaways:
| Technology | Type | Strengths | Weaknesses |
|---|---|---|---|
| INS | Dead reckoning | Continuous, high bandwidth, self-contained | Drifts over time, expensive |
| GNSS | Position fixing | Accurate, global, cheap receivers | Signal blockage, jamming, low bandwidth |
| Terrestrial radio | Position fixing | GNSS backup, some indoor coverage | Limited coverage, 2D only |
| TRN / image matching | Feature matching | Works without radio signals | Database-dependent, terrain-dependent |
| Odometer / PDR | Dead reckoning | Low cost, simple | Lower accuracy than INS |
The next chapter dives into the mathematical foundations: coordinate frames, attitude representations, and Earth models. These aren't just formalities — they're the language every navigation equation is written in.