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GPS and RTK Positioning

GNSS Fundamentals

What Is GNSS

GNSS (Global Navigation Satellite System) is the umbrella term for satellite-based positioning technologies. GPS is a subset of GNSS.

Major Satellite Constellations

Constellation Country/Region Satellites Bands Accuracy (Civilian)
GPS USA 31 L1/L2/L5 ~2-5m
GLONASS Russia 24 G1/G2/G3 ~2-5m
Galileo EU 30 E1/E5a/E5b/E6 ~1-3m
BeiDou BDS China 45+ B1/B2/B3 ~2-5m
QZSS Japan 4 L1/L2/L5/L6 Regional augmentation
IRNSS/NavIC India 7 L5/S Regional coverage

Positioning Principle: Trilateration

Position is determined by measuring pseudoranges from the receiver to at least 4 satellites:

\[ \rho_i = \sqrt{(x - x_i^s)^2 + (y - y_i^s)^2 + (z - z_i^s)^2} + c \cdot \delta t \]

Where:

  • \(\rho_i\) is the pseudorange to the \(i\)-th satellite
  • \((x, y, z)\) is the receiver position
  • \((x_i^s, y_i^s, z_i^s)\) is the satellite position
  • \(c \cdot \delta t\) is the receiver clock bias

At least 4 satellites are needed: 3 position unknowns + 1 clock bias.

Error Sources

Error Source Magnitude Description
Ionospheric delay 1-50m Signal delay through the ionosphere
Tropospheric delay 0.5-5m Effect of water vapor in the atmosphere
Satellite orbit error 1-5m Satellite position prediction error
Satellite clock error 1-3m Satellite clock deviation from system time
Multipath effect 0.5-several m Signal reflected by buildings
Receiver noise 0.3-1m Hardware noise
Geometric Dilution of Precision (DOP) Multiplier Effect of satellite geometric distribution

RTK-GPS

RTK Principle

RTK (Real-Time Kinematic) eliminates most common errors through base-rover differential and achieves centimeter-level positioning using carrier phase observations.

Carrier phase observation:

\[ \Phi = \frac{1}{\lambda}(\rho + N\lambda + c \cdot \delta t - I + T + \epsilon) \]

Where:

  • \(\Phi\) is the carrier phase observation (cycles)
  • \(\lambda\) is the carrier wavelength (L1 ~ 19cm)
  • \(N\) is the integer ambiguity
  • \(I\) is the ionospheric delay
  • \(T\) is the tropospheric delay

Differential elimination: Base station and rover observe the same satellite simultaneously; common errors cancel in the difference:

\[ \Delta\Phi = \frac{1}{\lambda}(\Delta\rho + \Delta N \lambda + \Delta\epsilon) \]

The key is resolving the integer ambiguity \(\Delta N\); once correctly fixed, positioning accuracy reaches 1-2cm.

RTK System Components

graph LR
    A[GNSS Satellites] --> B[Base Station<br>Known Coordinates]
    A --> C[Rover<br>To Be Positioned]

    B --> |"Correction Data<br>RTCM"| D[Data Link<br>Radio/4G/NTRIP]
    D --> C

    C --> E[RTK Solution]
    E --> F[Centimeter-Level Position]

Positioning States

State Accuracy Description
Fix 1-2 cm Integer ambiguity correctly resolved
Float 10-50 cm Integer ambiguity not resolved
DGPS 0.5-1 m Pseudorange differential only
Single 2-5 m No differential correction

u-blox ZED-F9P

ZED-F9P is currently the most popular consumer/low-cost RTK GNSS receiver.

Parameter Value
Supported Constellations GPS + GLONASS + Galileo + BeiDou
Bands L1 + L2 (dual-frequency)
RTK Accuracy Horizontal 1cm + 1ppm, Vertical 1.5cm + 1ppm
Convergence Time <10s (RTK Fix)
Update Rate Up to 20Hz (RTK 8Hz)
Interfaces UART x3, I2C, SPI, USB
Power ~150mW
Price ~$200 (module) 
# Using ublox driver in ROS2
sudo apt install ros-humble-ublox-gps

ros2 launch ublox_gps ublox_gps_node-launch.py \
    device:=/dev/ttyACM0

NTRIP Protocol

NTRIP (Networked Transport of RTCM via Internet Protocol) transmits RTK correction data over the internet, eliminating the need to build your own base station.

# Connect to NTRIP service
str2str -in ntrip://username:password@caster_host:2101/mountpoint \
        -out serial://ttyUSB0:115200

Common CORS (Continuously Operating Reference Station) services in China: Qianxun (qxwz.com), China Mobile OnePoint, etc.

Indoor Positioning Alternatives

GPS/GNSS is unavailable indoors, requiring alternative positioning solutions.

UWB (Ultra-Wideband)

Parameter Description
Principle Ultra-wideband pulse ToF ranging
Accuracy 10-30 cm
Range 10-100m
Representative Chips Decawave DW1000 / DW3000
Advantages High accuracy, anti-multipath
Disadvantages Requires deploying anchors
Price Anchor ~\(30/each, Tag ~\)15/each

Positioning methods:

  • TWR (Two-Way Ranging): Bidirectional ranging
  • TDoA (Time Difference of Arrival)
\[ d_{ij} = c \cdot \frac{t_{\text{round}} - t_{\text{reply}}}{2} \]

AprilTag Positioning

Parameter Description
Principle Visual marker detection + PnP pose estimation
Accuracy 1-5 cm (depends on distance and marker size)
Range 0.5-5m (depends on marker size and camera resolution)
Cost Only printed markers + camera needed
Disadvantages Markers must be visible, affected by lighting

Other Indoor Positioning Technologies

Technology Accuracy Cost Description
WiFi fingerprinting 1-5m Low Uses existing APs
BLE Beacon 1-3m Low Bluetooth beacons
LiDAR SLAM 5-10cm Medium No infrastructure needed
Visual SLAM 5-20cm Low Camera only
Motion capture <1mm High Vicon/OptiTrack

GPS + IMU Fusion

Fusion Motivation

GPS Characteristics IMU Characteristics After Fusion
Low frequency (1-20Hz) High frequency (200-1000Hz) High-frequency output
No drift Long-term drift Good short and long term
May be interrupted Continuously available Continuously available
Variable accuracy Short-term precise Consistently precise

EKF Fusion

Using Extended Kalman Filter (EKF) to fuse GPS and IMU:

Prediction step (IMU-driven, high frequency):

\[ \hat{\mathbf{x}}_{k|k-1} = f(\hat{\mathbf{x}}_{k-1}, \mathbf{u}_k^{\text{IMU}}) \]
\[ \mathbf{P}_{k|k-1} = \mathbf{F}_k \mathbf{P}_{k-1} \mathbf{F}_k^T + \mathbf{Q}_k \]

Update step (GPS observation, low frequency):

\[ \mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{H}_k^T (\mathbf{H}_k \mathbf{P}_{k|k-1} \mathbf{H}_k^T + \mathbf{R}_k)^{-1} \]
\[ \hat{\mathbf{x}}_k = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k (\mathbf{z}_k^{\text{GPS}} - h(\hat{\mathbf{x}}_{k|k-1})) \]

ROS2 robot_localization

sudo apt install ros-humble-robot-localization

# Configure EKF fusion of GPS + IMU
ros2 launch robot_localization dual_ekf_navsat.launch.py

Configuration files need to specify:

  • ekf_local: Fuses IMU + wheel odometry (local frame)
  • ekf_global: Fuses GPS + IMU (global frame)
  • navsat_transform: GPS coordinate conversion

References

  • u-blox ZED-F9P Integration Manual
  • GPS: Theory, Algorithms and Applications - Xu & Xu
  • Decawave DW1000 Datasheet
  • ROS2 robot_localization Documentation
  • Qianxun Developer Documentation

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