Localization sensors: GNSS, RTK, UWB and odometry

A robot that works outdoors, or across a large building, needs to know where it is in a global frame: on a farm field, along a road, in a warehouse aisle. Some sensors measure that directly. Satellite navigation gives position anywhere with a clear view of the sky; real-time kinematic corrections sharpen it to centimetres; ultra-wideband radio beacons do a similar job indoors; and wheel odometry, visual odometry and the IMU fill the gaps between absolute fixes. This lesson covers the sensors and their error behaviour. How their measurements are combined into a single best estimate is the subject of ROB-310 and ROB-340.

How GNSS works

Global navigation satellite systems (GNSS) include the American GPS, the European Galileo, Russia's GLONASS and China's BeiDou, with more than 100 satellites in orbit between them. Each satellite broadcasts its precise orbit and a timing signal derived from atomic clocks. A receiver measures the apparent travel time of each signal, multiplies by the speed of light to get a pseudorange, and solves for its own position and its clock error. Four unknowns (x, y, z and receiver clock offset) need at least four satellites; more satellites improve accuracy and robustness, which is why multi-constellation receivers that track all four systems have become standard.

Here \( \rho_i \) is the pseudorange to satellite i, \( \mathbf{s}_i \) its position, \( \mathbf{x} \) the receiver position, \( \delta t \) the receiver clock error and \( \varepsilon_i \) everything else. That everything else is substantial: the ionosphere and troposphere delay the signals, satellite clock and orbit errors remain, signals reflect off buildings before arriving (multipath), and receiver noise adds a little. A standalone consumer receiver with a clear sky typically achieves a few metres horizontally, and much worse among tall buildings, under trees or near walls.

Dilution of precision and the error budget

The geometry of the visible satellites multiplies the ranging error. If all the satellites are clustered in one part of the sky, small range errors produce large position errors; if they are spread around the sky, the position is well constrained. This amplification is expressed as dilution of precision (DOP). Horizontal DOP (HDOP) of about 1 is excellent, 2 is good, and above 5 is poor. A useful rule of thumb is that the horizontal position error is roughly HDOP times the user range error.

Augmentation: SBAS, PPP and RTK

Several techniques remove most of the common errors. Satellite-based augmentation systems (SBAS) broadcast corrections from geostationary satellites; the North American system, WAAS, covers both Canada and the United States and brings accuracy to around 1 to 2 m, mainly for aviation. Precise point positioning (PPP) uses precise orbit and clock products to reach decimetre or centimetre accuracy with a single receiver, but traditionally needs many minutes to converge.

Real-time kinematic (RTK) positioning is the workhorse for robots that need centimetres. A base station at a precisely known position, either the robot owner's own or one in a network, tracks the same satellites as the robot's receiver (the rover) and sends corrections, usually over a radio link or the internet using the NTRIP protocol. Because base and rover see nearly the same atmospheric and satellite errors when they are within a few tens of kilometres, the differences cancel. RTK also uses the carrier phase of the signals, whose wavelength is about 19 cm for GPS L1, rather than just the code timing. Once the receiver has resolved the integer number of carrier wavelengths to each satellite, a fixed solution, it typically achieves about 1 to 2 cm horizontally. Before that, or when too few satellites are tracked, it reports a float solution at the decimetre level.

GNSS position alone does not give heading for a slow or stationary robot. Dual-antenna receivers solve this by measuring the vector between two antennas a metre or so apart with RTK precision, giving heading accurate to a fraction of a degree, which is why agricultural autosteer systems often carry two antennas.

Ultra-wideband and other indoor beacons

Satellite signals do not penetrate buildings usefully, so indoor robots need other references. Ultra-wideband (UWB) radio sends very short pulses spread over a bandwidth of 500 MHz or more, which makes it possible to time the arrival of the direct path precisely and to tell it apart from reflections. Two UWB devices measure the distance between them by two-way ranging: one sends a message, the other replies after a known delay, and the round-trip time gives the range to about 10 cm. A robot that measures ranges to three or more fixed anchors at known positions can find its position by trilateration, much like GNSS. UWB is used for warehouse robot positioning, tracking people and tools in factories, and keeping lawn-mowing robots inside their boundaries without buried wire.

Other indoor options include fiducial markers (printed tags such as AprilTags that a camera can locate precisely), reflectors used by laser-guided vehicles, magnetic tape or floor codes for automated guided vehicles, and Wi-Fi or Bluetooth signal strength, which is cheap but only accurate to metres. UWB is a radio transmitter and therefore needs certification; the same regulators named in lesson 5 set its power limits and permitted bands.

Wheel odometry and dead reckoning

Between absolute fixes, robots estimate motion from their own sensors, a process called dead reckoning. Wheel odometry integrates encoder readings (lesson 2) through the robot's kinematic model (ROB-210) to update position and heading. It is precise over short distances and available at high rates, but its errors accumulate without bound. Systematic errors include an incorrect wheel diameter, unequal wheel diameters and an incorrect track width; random errors include wheel slip, uneven floors and bumps. Heading errors are the most damaging, because each degree of heading error turns every subsequent metre of travel into sideways error.

Visual odometry and lidar odometry estimate motion by tracking features or matching scans between successive frames; they do not slip on wet floors but drift slowly too and fail in featureless or dynamic scenes. Combining wheel odometry, an IMU and visual or lidar odometry, then correcting the result with absolute fixes from GNSS, UWB or map matching, is the standard architecture of modern robot localization. The sensors in this lesson are the inputs to that architecture; lesson 11 introduces the simplest way to combine them.

import numpy as np
# Dead reckoning with a small left-wheel scale error on a "straight" 20 m drive
b, ds, steps = 0.40, 0.01, 2000          # track width (m), step (m), 2000 steps = 20 m
scale_left = 0.995                        # left wheel 0.5 % small
x = y = th = 0.0
for _ in range(steps):
    dl, dr = ds * scale_left, ds          # true wheel travel when encoders say ds each
    d, dth = (dl + dr) / 2, (dr - dl) / b
    x += d * np.cos(th + dth / 2); y += d * np.sin(th + dth / 2); th += dth
print(f"robot believes: x = 20.00 m, y = 0.00 m, heading 0.0 deg")
print(f"actually at:    x = {x:5.2f} m, y = {y:5.2f} m, heading {np.degrees(th):4.1f} deg")

robot believes: x = 20.00 m, y = 0.00 m, heading 0.0 deg
actually at:    x = 19.74 m, y =  2.48 m, heading 14.3 deg