Ch 12: INS/GNSS Integration — Groves GNSS/INS

Chapter 0: Why Integrate?

INS and GNSS have complementary strengths and weaknesses. Integration combines the best of both:

Property	INS	GNSS
Short-term accuracy	Excellent	Moderate (noise)
Long-term accuracy	Poor (drift)	Excellent (no drift)
Update rate	High (100–1000 Hz)	Low (1–20 Hz)
Environment	Self-contained	Needs sky view
Jamming/spoofing	Immune	Vulnerable
Outputs	Position, velocity, attitude	Position, velocity

The INS provides a smooth, high-rate, self-contained solution that drifts over time. GNSS provides an absolute position/velocity reference that corrects the drift. The Kalman filter fuses them optimally.

The integration also benefits GNSS: the INS-predicted velocity can aid the GNSS receiver by narrowing the code and carrier tracking loops, improving tracking sensitivity and robustness in poor signal environments (Chapter 8).

The integration corrects the INS. The Kalman filter uses GNSS measurements to estimate and correct INS errors: position error, velocity error, attitude error, accelerometer biases, and gyro biases. Once calibrated, the INS provides excellent performance even during GNSS outages.

Check: What is the fundamental reason INS and GNSS are complementary?

INS has excellent short-term accuracy but drifts; GNSS has bounded long-term accuracy but noisy short-term. Integration gets the best of both. INS and GNSS use the same sensors GNSS provides attitude measurements that the INS lacks

Chapter 1: Loosely Coupled Integration

In loosely coupled integration, the GNSS receiver computes its own position and velocity solution, which is then compared with the INS solution in an integration Kalman filter.

Architecture:

• The GNSS receiver runs its own navigation processor, producing position and velocity outputs.

• The INS runs its own navigation equations, producing position, velocity, and attitude.

• The integration filter computes the difference between the two solutions (the measurement innovation) and estimates corrections to the INS.

The measurement innovation is simply:

δz = (r̂_INS − r̂_GNSS, v̂_INS − v̂_GNSS)

Advantages:

• Simple — GNSS and INS are separate subsystems with a clean interface

• GNSS receiver can operate independently if the INS fails

• Easy to implement with commercial-off-the-shelf (COTS) equipment

Disadvantages:

• GNSS navigation processor requires at least 4 satellites for a solution; integration fails if fewer are visible

• Cannot use the INS to aid the GNSS tracking loops (no deep integration)

• GNSS measurement correlations are hidden inside the GNSS navigation solution

Closed-loop correction: The Kalman filter estimates are used to correct the INS navigation solution continuously. This keeps the INS errors small, ensuring the linearized error model remains valid. Without closed-loop correction, large errors can accumulate and the filter may diverge.

Check: What is the main limitation of loosely coupled integration?

It requires at least 4 satellites because the GNSS receiver must compute a full navigation solution before integration It cannot use gyroscope measurements It requires a high-quality INS

Chapter 2: Tightly Coupled Integration

In tightly coupled integration, the GNSS receiver's raw measurements (pseudo-ranges and pseudo-range rates from each satellite) are input directly to the integration Kalman filter, bypassing the GNSS navigation processor.

The measurement innovation compares each measured pseudo-range with the predicted pseudo-range from the corrected INS position:

δz_ρ,k = (˜ρ_C1 − ρ̂_C1, ˜ρ_C2 − ρ̂_C2, ... ˜ρ_Cn − ρ̂_Cn)

where ˜ρ_Cj is the corrected measured pseudo-range to satellite j, and ρ̂_Cj is predicted from the INS position, estimated clock offset, and satellite ephemeris.

The state vector adds GNSS receiver clock offset and drift to the INS error states:

x = (δψ, δv, δr, b_a, b_g, δρ_rc, δρ̇_rc)

Key advantage: Tight coupling can use measurements from any number of satellites, even fewer than four. With only one or two visible satellites, the INS provides the missing geometry. This is critical in urban canyons and during signal blockage.

Additional benefits:

• Better handling of measurement correlations and outliers (individual satellite measurements can be rejected)

• Per-satellite integrity monitoring

• More accurate modeling of measurement geometry

The key difference: Loosely coupled uses the GNSS solution (position/velocity). Tightly coupled uses the GNSS measurements (pseudo-ranges). This allows operation with fewer than 4 satellites and enables per-satellite quality control.

Check: Why can tightly coupled integration work with fewer than 4 visible satellites?

The INS provides the position/velocity geometry that the missing satellites would have provided; each satellite adds an independent measurement Tight coupling amplifies the satellite signals The receiver clock does not need to be estimated

Chapter 3: Deep Integration

Deep integration (also called ultra-tight coupling) extends the INS aiding all the way into the GNSS receiver's tracking loops. The INS-predicted velocity and acceleration are used to control the code and carrier NCOs (numerically controlled oscillators), replacing or supplementing the conventional tracking-loop discriminators.

How it works:

• The INS provides predicted range and range-rate to each satellite

• These predictions drive the code and carrier NCOs, keeping the tracking loops centered even when the signal-to-noise ratio is too low for conventional tracking

• The correlator I and Q outputs are fed directly to the integration filter instead of through a conventional discriminator

Benefits:

• Tracking sensitivity: Effective tracking-loop bandwidth can be reduced to <0.1 Hz (vs ~5 Hz conventional), gaining 15–20 dB of noise rejection

• Jamming resistance: Maintains tracking through much higher levels of interference

• Fast reacquisition: After signal blockage, the INS-predicted code phase and carrier Doppler allow near-instantaneous reacquisition

• Dynamics: High dynamics are handled by the INS, not the tracking loops

The cost: Deep integration requires access to the GNSS receiver's internal correlator outputs. This means it cannot be done with a COTS receiver — it requires either a custom receiver or a software-defined radio (SDR). The computational load is also higher.

Deep integration is the ultimate anti-jam technique. By using the INS to predict the GNSS signal dynamics, the receiver can track signals 20+ dB below the threshold of a standalone receiver. This is why military INS/GPS systems use deep integration in contested environments.

Check: How does deep integration improve GNSS tracking sensitivity?

The INS predicts signal dynamics, allowing the tracking loops to operate with very narrow bandwidth, which rejects more noise It amplifies the GNSS signals It uses multiple antennas

Chapter 4: System Model

The integration Kalman filter needs a system model that predicts how the INS errors evolve over time. The error state vector typically includes:

State	Dimension	What It Represents
Attitude error	3	Roll, pitch, heading errors of the INS
Velocity error	3	North, East, Down velocity errors
Position error	3	Latitude, longitude, height errors
Accel bias	3	X, Y, Z accelerometer biases
Gyro bias	3	X, Y, Z gyroscope biases
Clock offset	1	Receiver clock error (tight coupling)
Clock drift	1	Receiver clock drift (tight coupling)

This gives a typical state vector of 15 states (loosely coupled) or 17 states (tightly coupled). The system model describes how each error state propagates:

• Attitude error is driven by gyro bias and noise

• Velocity error is driven by attitude error (misrotated specific force), accelerometer bias, and noise

• Position error is the integral of velocity error

• Sensor biases are modeled as first-order Markov processes (slowly varying with a known correlation time)

• Clock offset is the integral of clock drift; clock drift is modeled as a random walk

Why sensor biases are estimated: An accelerometer bias of 1 mg (0.01 m/s²) causes a position error of 5 m after 1 minute of free-inertial navigation. A gyro bias of 1°/hr causes a heading error of 0.017° per minute, leading to growing cross-track position error. By estimating and correcting these biases, the INS accuracy during GNSS outages is dramatically improved.

Check: Why does the integration filter estimate accelerometer and gyro biases?

These biases are the dominant source of INS drift; estimating and correcting them dramatically improves INS accuracy during GNSS outages The biases are needed to compute the GNSS solution The biases change too rapidly to pre-calibrate

Chapter 5: Measurement Model

The measurement model relates the Kalman filter states to the measurements. It differs for loosely and tightly coupled integration.

Loosely coupled: The measurement innovation is the difference between INS and GNSS position and velocity solutions. The measurement matrix is approximately:

H ≈ [0₃ 0₃ −I₃ 0₃ 0₃; 0₃ −I₃ 0₃ 0₃ 0₃]

This says: the position measurement directly observes the position error states (−I₃), and the velocity measurement directly observes the velocity error states.

Tightly coupled: Each pseudo-range measurement depends on position, clock offset, and (through the lever arm) attitude. Each pseudo-range-rate measurement depends on velocity, clock drift, and attitude. The measurement matrix includes the line-of-sight vectors to each satellite.

Lever arm correction: The IMU and GNSS antenna are at different locations on the vehicle. The position and velocity of the antenna must be computed from the IMU-centered INS solution using the known lever arm and the vehicle attitude and angular velocity.

Estimation of attitude and IMU errors: How are attitude errors and gyro biases observable from position and velocity measurements? The attitude error causes a misrotation of the specific force. Under acceleration or during turns, this misrotated specific force produces a distinctive velocity error signature. The filter observes this signature and infers the attitude error. Vehicle dynamics (maneuvers) are essential for observability.

Check: How does the integration filter observe attitude errors from position/velocity measurements?

Attitude errors cause the specific force to be misrotated, producing a distinctive velocity error pattern during maneuvers that the filter can detect The GNSS measurements directly contain attitude information Attitude errors do not affect position or velocity

Chapter 6: State Observability

Observability determines which states the Kalman filter can estimate. A state is observable if the measurements, over time, contain enough information to determine it.

Well-observable states:

• Position error — directly observed by GNSS measurements

• Velocity error — directly observed by GNSS measurements

• Clock offset and drift — directly observed

Conditionally observable states (require dynamics):

• Heading error: Only observable during turns or lateral accelerations. A vehicle flying straight and level has an unobservable heading error because the heading error produces the same velocity signature as a cross-track accelerometer bias.

• Roll/pitch errors: Become separable from horizontal accelerometer biases only during maneuvers with vertical acceleration changes.

• Gyro biases: Observable through the attitude error dynamics during maneuvers.

• Accelerometer biases: Observable when the vehicle accelerates in different directions.

Practical implication: For best integration filter performance, the vehicle should maneuver. Aircraft perform S-turns for heading observability. Ground vehicles benefit from turns at intersections. A vehicle moving in a straight line will have poor heading and bias calibration.

The heading observability problem: During straight-line motion, heading error and cross-track accelerometer bias produce identical velocity errors. The filter cannot tell them apart. Only a turn — which rotates the body-frame biases relative to the navigation frame — breaks this ambiguity. This is why INS alignment procedures always include turns.

Check: Why is heading error unobservable during straight-line flight?

During straight flight, heading error and cross-track accelerometer bias produce identical velocity error signatures; only a turn can distinguish them The GNSS receiver cannot measure heading The heading gyro is turned off during straight flight

Chapter 7: Advanced Topics

Differential GNSS integration: When DGNSS or RTK corrections are available, the GNSS measurement accuracy improves dramatically (meter to centimeter level). The integration filter's measurement noise covariance is reduced accordingly, giving the GNSS measurements more weight. Carrier-phase integer ambiguities may be estimated as additional filter states.

GNSS attitude integration: When multiple GNSS antennas provide an attitude measurement (Chapter 8), this can be integrated with the INS attitude, directly observing the attitude error states. This eliminates the need for maneuvers to observe heading error and greatly speeds up the convergence of the integration filter.

Large heading errors: The standard error-state Kalman filter assumes small attitude errors. When the heading error is large (e.g., during initial alignment of a low-cost IMU), the small-angle approximation fails. Solutions include using an unscented Kalman filter (UKF) or particle filter, or parameterizing the heading error differently.

Advanced IMU error modeling: The standard first-order Markov model for sensor biases is a rough approximation. Better performance can be achieved with second- or third-order autoregressive models tuned to each sensor type, or frequency-domain approaches that do not require a priori assumptions about the time variation.

Smoothing: For post-processing applications (surveying, geo-referencing), a Kalman smoother uses measurements from both before and after the time of interest to estimate the INS errors. This effectively halves the INS drift period during GNSS outages, reducing maximum position error by up to a factor of 4.

Check: Why does GNSS attitude (from multiple antennas) improve integration filter convergence?

It directly observes the attitude error states, eliminating the need for vehicle maneuvers to achieve heading observability It provides more satellite measurements It reduces the receiver clock error

Chapter 8: Integration Simulation

This simulation shows loosely coupled INS/GNSS integration. The INS (orange) drifts over time. GNSS measurements (teal dots) arrive periodically. The integration filter (white) corrects the INS drift. During a GNSS outage, the calibrated INS continues with reduced drift.

INS/GNSS Loosely Coupled Integration

Check: What happens to the integrated solution during a GNSS outage?

The solution relies on the INS alone, but the previously calibrated sensor biases mean it drifts much more slowly than an uncalibrated INS The solution immediately becomes invalid The GNSS solution continues from the last known position

Chapter 9: Summary

Key takeaways:
• INS and GNSS are complementary: INS = smooth, self-contained, drifts; GNSS = absolute, noisy, needs sky
• Loosely coupled: uses GNSS position/velocity solution; requires ≥4 satellites; simple, COTS-compatible
• Tightly coupled: uses raw pseudo-ranges; works with <4 satellites; per-satellite quality control
• Deep integration: INS aids GNSS tracking loops; 20+ dB jamming resistance; requires custom receiver
• State vector: attitude error (3), velocity error (3), position error (3), accel bias (3), gyro bias (3), clock (2)
• System model: attitude error driven by gyro bias; velocity error driven by attitude error and accel bias
• Observability requires maneuvers: heading error is unobservable during straight-line motion
• GNSS attitude directly observes heading error; smoothing reduces outage drift by up to 4×
• Closed-loop correction keeps errors small and the linearized model valid
• The integration calibrates INS biases, improving standalone INS performance during GNSS outages

Check: What is the key advantage of tightly coupled over loosely coupled integration?

It can use measurements from fewer than 4 satellites because the INS provides the missing geometry It is simpler to implement It uses less processing power