Satellite & Inertial Navigation (GPS/GNSS & INS/IRS)

2. Basic Concept & Physical Mechanism

At the continuum electromagnetic wave propagation level, space vehicles (SVs) transmit Right-Hand Circularly Polarized (RHCP) radio frequency carriers (e.g., GPS L1 at $1575.42\text{ MHz}$) modulated with Pseudo-Random Noise (PRN) ranging codes. The physical mechanism relies on measuring the exact transit time $\Delta t$ of the wavefront from satellite antenna phase center to receiver antenna. Because receiver clocks (typically crystal oscillators) are not synchronized with atomic GPS system time ($t_{GPS}$), clock offset $\delta t$ introduces an unknown error in every pseudorange measurement. Thus, four simultaneous measurements are mathematically mandatory.

3. Mathematical Derivation & Equations

Let the known Cartesian coordinates of satellite $i$ be $(X_i, Y_i, Z_i)$ and the unknown receiver position be $(x, y, z)$. The true geometric range $R_i$ is:

Due to receiver clock bias $\delta t$ multiplied by the speed of light $c$, the measured pseudorange $\rho_i$ differs from true range:

Where $\epsilon_i$ represents cumulative propagation errors (ionospheric, tropospheric, multipath). Linearizing via Taylor series expansion about an estimated receiver position $(x_0, y_0, z_0, \delta t_0)$ yields the matrix observation equation $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{v}$, solved via Ordinary Least Squares (OLS):

Where $\mathbf{H}$ is the $n \times 4$ geometry matrix containing Line-Of-Sight (LOS) unit vectors, and $\hat{\mathbf{x}}$ is the state vector $[\Delta x, \Delta y, \Delta z, c \delta t]^T$.

4. Dimensions & Units

• Fundamental Dimensions: Range $[L]$, Time $[T]$, Speed $[L \cdot T^{-1}]$
• SI Units: Meters ($m$), Seconds ($s$), Meters per second ($m/s$)
• CGS Units: Centimeters ($cm$), Seconds ($s$)
• Conversion Factors: 1 nanosecond of light travel time $\approx 0.29979$ meters distance.

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

Modern commercial flight decks utilize multi-constellation GNSS receivers (GPS, GLONASS, Galileo, BeiDou) integrated into Flight Management Systems (FMS) for RNAV (GPS) and RNP approaches down to RNP 0.1. DGCA CAR Section 2 Series 'I' Part VII mandates that aircraft operating under IFR with GPS primary means must possess predictive receiver autonomy and fault detection. Air Safety Officers investigating GNSS loss-of-navigation incidents examine RF interference, jamming, and spoofing events.

2. Basic Concept & Physical Mechanism

The accuracy of a GNSS position fix depends on two factors: User Equivalent Range Error (UERE) and Geometric Dilution of Precision (GDOP). Physically, if satellites are clustered closely together in one sector of the sky, small pseudorange measurement errors translate into massive intersecting volume errors (poor geometry). Conversely, widely dispersed satellites (one overhead, others near the horizon) create orthogonal intersection lines, minimizing position error. Atmospheric errors arise because signals pass through the ionosphere (free electrons causing group delay and phase advance proportional to Total Electron Content, TEC) and troposphere (water vapor and dry gases causing non-dispersive bending).

3. Mathematical Derivation & Equations

The covariance matrix of the position solution $\mathbf{P}$ is derived from the observation design matrix $\mathbf{H}$ and measurement variance $\sigma^2$:

Let $\mathbf{Q} = (\mathbf{H}^T \mathbf{H})^{-1}$ be partitioned into diagonal elements representing GDOP components in ENU (East, North, Up) coordinates:

Atmospheric ionospheric delay $I$ is modeled via the Klobuchar algorithm or dual-frequency ionosphere-free combinations:

4. Dimensions & Units

• DOP Values: Dimensionless scalar multipliers.
• Ionospheric Delay: Length $[L]$ in meters.
• TEC (Total Electron Content): $1\text{ TECU} = 10^{16}\text{ electrons/m}^2$.

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

In Indian airspace, the GPS Aided GEO Augmented Navigation (GAGAN) SBAS system broadcasts differential corrections and integrity messages, reducing vertical error to < $1.5\text{ meters}$, enabling LPV (Localizer Performance with Vertical guidance) approaches at Indian airports. Air Safety Officers must evaluate terrain masking and satellite geometry alerts (e.g., sudden HDOP spikes above 4.0 during banking maneuvers where wing blanking occurs).

2. Basic Concept & Physical Mechanism

To perform position navigation, 4 satellites are required. If an aircraft tracks 5, 6, or more satellites, the system is overdetermined. RAIM exploits this redundancy by computing multiple position solutions using subsets of available satellites (e.g., leaving one satellite out at a time). If a single satellite's clock or ephemeris corrupts, the residuals (errors between measured and expected pseudoranges) of the subset containing that satellite will diverge statistically from the rest. The algorithm performs a Chi-Square ($\chi^2$) hypothesis test on the sum of squared residuals to isolate and exclude the faulty satellite (FDE - Fault Detection and Exclusion).

3. Mathematical Derivation & Equations

Let $n$ be the number of visible satellites ($n \ge 5$ for fault detection, $n \ge 6$ for fault exclusion). The residual vector $\mathbf{r}$ is computed from the measurement residual projection:

The test statistic $SSE$ (Sum of Squared Errors) is:

Under nominal conditions, $SSE / \sigma^2$ follows a Central Chi-Square distribution with $(n - 4)$ degrees of freedom. The decision threshold $T$ for a given false alarm rate $\alpha$ is:

If $SSE > T$, the null hypothesis (all satellites healthy) is rejected, and a RAIM warning is triggered.

4. Dimensions & Units

• Pseudorange Residuals: Meters $[L]$ squared for SSE ($m^2$).
• False Alarm Rate ($\alpha$): Dimensionless probability (typically $10^{-5}$ per flight hour).
• Protection Level (PL): Meters $[L]$.

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

Before commencing an RNAV (GPS) approach in IMC, flight crews are required by DGCA operational circulars to check "RAIM Availability Prediction" via FMS or dispatch sheets. If satellite geometry is poor (e.g., due to an in-orbit satellite outage or masking), the FMS flags "RAIM UNAVAILABLE", forcing the crew to revert to conventional VOR/ILS ground-based approaches. Air Safety Officers review RAIM alert logs during incident investigations involving runway deviations.

2. Basic Concept & Physical Mechanism

Newton's Second Law of Motion forms the core physical mechanism. Accelerometers measure specific force ($\mathbf{f}$), which is the vector difference between acceleration in inertial space and gravitational acceleration ($\mathbf{f} = \mathbf{a} - \mathbf{g}$). Gyroscopes measure angular rotation rates relative to inertial space using either mechanical angular momentum conservation, Ring Laser Gyros (RLG) exploiting the Sagnac effect, or Fiber Optic Gyros (FOG). By continuously integrating specific force accounting for aircraft attitude (provided by gyros) and subtracting Earth's gravity, the computer extracts velocity and double-integrates to yield position.

3. Mathematical Derivation & Equations

Let $\mathbf{f}^b$ be specific force measured in body frame, transformed to navigation frame (North, East, Down) via direction cosine matrix $\mathbf{C}_b^n$:

Where $\boldsymbol{\omega}_{ie}^n$ is Earth's rotation rate vector, $\boldsymbol{\omega}_{en}^n$ is transport rate due to aircraft movement over curved Earth, and $\mathbf{g}^n$ is local gravity. Velocity $\mathbf{v}$ and position $\mathbf{p}$ are obtained by integration:

4. Dimensions & Units

• Specific Force: Acceleration $[L \cdot T^{-2}]$, units $\text{m/s}^2$ or $g$-units ($9.80665\text{ m/s}^2$).
• Angular Rate: $[\theta \cdot T^{-1}]$, units degrees per hour ($^\circ/\text{hr}$) or radians per second ($\text{rad/s}$).
• Position: Angular degrees ($^\circ$) or meters ($m$).

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

Modern transport aircraft (e.g., Boeing 777, Airbus A350) use Laser-gyro IRS units providing autonomous attitude, heading, and velocity data to PFDs, autopilot, and FMS. DGCA CAR Section 2 requires dual or triple redundant IRS installations for extended twin-engine operations (ETOPS). Air Safety Officers investigating total electrical or GPS jamming incidents rely on IRS standalone dead-reckoning capabilities to verify aircraft trajectories.

2. Basic Concept & Physical Mechanism

Because INS integrates sensor outputs, any constant gyro drift ($\epsilon$) creates an attitude error that grows linearly with time, which in turn tilts the accelerometer platform relative to local gravity. This tilt causes the accelerometer to sense a component of gravity as horizontal acceleration, causing position error to grow with a $t^2$ (quadratic) or $t^3$ (cubic) dependency. Maximilian Schuler discovered in 1923 that if a pendulous mechanical system is tuned to have a natural pendulum period equal to the orbital period of a satellite skimming Earth's surface ($T = 2\pi \sqrt{R/g} \approx 84.4\text{ minutes}$), the platform remains perfectly aligned with the local vertical despite vehicle accelerations.

3. Mathematical Derivation & Equations

Let $R$ be Earth's radius ($6,371,000\text{ m}$) and $g$ be gravitational acceleration ($9.80665\text{ m/s}^2$). The Schuler oscillation period $T_s$ is derived from pendulum mechanics:

The error differential equation for a Schuler-tuned horizontal channel exhibits undamped harmonic oscillation at frequency $\omega_s = 1/\sqrt{RG}$:

4. Dimensions & Units

• Schuler Period: Time $[T]$, units seconds ($s$) or minutes ($min$).
• Drift Rate: Angular rate $[\theta \cdot T^{-1}]$, $^\circ/\text{hr}$ or nautical miles per hour ($\text{NM/hr}$).
• Earth Radius $R$: Length $[L]$, meters ($m$).

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

In transoceanic flights where VHF and radar ground aids are unavailable, IRS drift determines dead-reckoning navigation accuracy. Civil aviation regulations mandate maximum allowable IRS drift rates (typically $< 2\text{ NM/hr}$ in non-updated free-inertial mode). Air Safety Officers analyzing oceanic waypoint excursions examine IRS alignment procedures and drift logs.

2. Basic Concept & Physical Mechanism

Individually, GPS and INS have complementary error characteristics: GPS suffers from cycle slips, low update rates ($1-10\text{ Hz}$), and susceptibility to RF jamming/multipath, but has zero long-term drift. INS has exceptional bandwidth ($>100\text{ Hz}$), smooth short-term outputs, and immunity to RF interference, but drifts unbounded over time. By fusing them using a Kalman Filter, the system continuously estimates IRS sensor errors (gyro/accelerometer biases) using GPS position and velocity updates as measurements, correcting the inertial wander while maintaining navigation continuity during momentary GPS dropouts.

3. Mathematical Derivation & Equations

The linear continuous-time state-space error model of the integrated system is:

Where state vector $\mathbf{x}$ includes position errors, velocity errors, platform tilt angles, gyro biases, and accelerometer biases. The discrete measurement model (difference between GPS and INS positions) is:

The Kalman Filter recursively executes Time Update (Prediction) and Measurement Update (Correction) using Kalman Gain $\mathbf{K}_k$:

4. Dimensions & Units

• State Vector $\mathbf{x}$: Mixed dimensions (meters, m/s, radians, $^\circ/\text{hr}$).
• Kalman Gain $\mathbf{K}_k$: Dimensionless weighting matrix blending measurements and model estimates.
• Update Rate: Frequency $[T^{-1}]$, typically $50\text{ Hz} - 400\text{ Hz}$.

5. Reference Values Table

6. Aircraft Application & Air Safety Officer Connection

Modern flight management systems (FMS) blend GPS and IRS sensors into a unified position estimate labeled "GPS/IRS" or "MIXED" on the ND (Navigation Display). If GPS experiences RF spoofing or jamming over conflict zones, tightly coupled Kalman filtering detects the measurement residual anomaly, rejects GPS data, and seamlessly continues dead reckoning via IRS without navigation jumps. Air Safety Officers investigate GPS jamming reports by analyzing FMS navigation mode transition logs.