Aerospace Research Station

Inertial Navigation System Fundamentals

Core principles of INS: IMU modelling, mechanization equations, and error propagation in strapdown systems.

December 1, 2025

#INS
#IMU
#strapdown
#navigation

What is an INS?

An Inertial Navigation System (INS) computes position, velocity, and attitude purely from onboard sensor measurements — no external signals required. The workhorse is the Inertial Measurement Unit (IMU), which bundles three accelerometers and three gyroscopes mounted along orthogonal axes.

Because the system is self-contained, INS is immune to jamming and works in GPS-denied environments: underground, underwater, in deep space, or in contested airspace. The cost is a fundamental flaw — errors integrate over time, so position drift grows without bound without external correction.

IMU Sensor Models

Accelerometers

A triad of accelerometers measures specific force $f^b$ in the body frame:

\tilde{f}^b = f^b + b_a + n_a

where $b_a$ is a slowly-varying bias (modelled as a random walk) and $n_a$ is white noise (vibration-induced). The accelerometer does not measure gravitational acceleration directly — gravity must be removed via the navigation equations.

Gyroscopes

MEMS gyros measure angular rate $\omega^b_{ib}$ of the body with respect to the inertial frame, expressed in body coordinates:

\tilde{\omega}^b_{ib} = \omega^b_{ib} + b_g + n_g

The bias $b_g$ is the dominant error source in low-cost MEMS gyros. Thermal variations cause it to drift by tens of deg/hr.

Strapdown Mechanization

Modern aircraft use strapdown (strapped-down) INS: the IMU is rigidly bolted to the airframe instead of sitting on a gimbal-stabilized platform. The attitude must be maintained in software.

The mechanization equations propagate three quantities simultaneously:

1. Attitude Update

Attitude is tracked as a quaternion $\mathbf{q}^n_b$ (navigation-to-body). The differential equation is:

\dot{\mathbf{q}}^n_b = \frac{1}{2}\mathbf{q}^n_b \otimes \boldsymbol{\Omega}^b_{nb}

where $\boldsymbol{\Omega}^b_{nb}$ is the angular velocity of the body relative to the navigation frame. In discrete time, the gyro measurement is integrated using the rotation vector approach to avoid first-order drift from naive Euler integration.

2. Velocity Update

Specific force is rotated from body to navigation frame and integrated:

\dot{\mathbf{v}}^n = C^n_b \, f^b + \mathbf{g}^n - (2\boldsymbol{\omega}^n_{ie} + \boldsymbol{\omega}^n_{en}) \times \mathbf{v}^n

The Coriolis and transport-rate terms $(2\boldsymbol{\omega}_{ie} + \boldsymbol{\omega}_{en}) \times \mathbf{v}^n$ are small for short flights but accumulate over time in precision applications.

3. Position Update

Position in curvilinear coordinates $(L, \lambda, h)$ — latitude, longitude, altitude — follows from velocity:

\dot{L} = \frac{v_N}{R_M + h}, \quad \dot{\lambda} = \frac{v_E}{(R_N + h)\cos L}, \quad \dot{h} = v_D

$R_M$ and $R_N$ are the meridian and normal radii of curvature of the WGS-84 ellipsoid.

Error Propagation

INS errors obey a linear state-space model. The 15-state error model tracks:

State	Symbol	Dimension
Attitude error	$\delta\boldsymbol{\psi}$	3
Velocity error	$\delta\mathbf{v}$	3
Position error	$\delta\mathbf{p}$	3
Accelerometer bias	$\delta b_a$	3
Gyro bias	$\delta b_g$	3

The error dynamics are:

\dot{\mathbf{x}} = F\mathbf{x} + G\mathbf{w}

where $F$ is the system matrix (contains rotation matrices, Earth-rate terms, gravity gradient), $G$ is the noise-coupling matrix, and $\mathbf{w}$ is zero-mean white noise.

For unaided INS, position error grows roughly as:

\sigma_p(t) \approx \frac{1}{2} \sigma_{b_a} t^2 + \frac{1}{\sqrt{3}} \sigma_{b_g} \|\mathbf{v}\| t^2

A gyro bias of 1 deg/hr produces roughly 1 nautical mile per hour of position drift — the standard figure of merit for tactical-grade INS.

GNSS/INS Integration

A loosely-coupled GNSS/INS Kalman filter fuses the 15-state INS error model with GNSS position and velocity measurements:

\mathbf{z}_k = H\mathbf{x}_k + \boldsymbol{\nu}_k

H = \begin{bmatrix} \mathbf{0}_{3\times6} & I_3 & \mathbf{0}_{3\times6} \end{bmatrix}

The filter estimates bias states continuously, feeding corrections back to the mechanization. During GNSS outages the INS coasts, accumulating error at the rate determined by residual bias after calibration.

Next Steps

The natural extensions from this baseline are:

Tightly-coupled integration — use raw pseudoranges instead of GNSS position fixes, maintaining lock through partial satellite visibility
Visual-inertial odometry (VIO) — fuse camera feature tracks with IMU to bound drift without GNSS
Magnetic aiding — use magnetometer + world magnetic model for heading during prolonged GPS denial
Barometric altitude aiding — bound vertical channel divergence cheaply