Tutorial on Aided Inertial Navigation Systems: A Modern Treatment Using Lie-Group Theoretical Methods

Abstract

This tutorial presents a control-oriented introduction to aided inertial navigation systems using a Lie-group formulation centered on the extended Special Euclidean group SE_2(3). The focus is on developing a clear and implementation-oriented geometric framework for fusing inertial measurements with aiding information, while making the role of invariance and symmetry explicit. Recent extensions, including higher-order state representations, synchronous observer designs, and equivariant filtering methods, are discussed as natural continuations of the same underlying principles. The goal is to provide readers with a coherent system-theoretic perspective that supports both understanding and practical use of modern aided inertial navigation methods.

Figures (10)

• Figure 1: INS kinematics setting. The North--East--Down (NED) inertial frame $\{\mathcal{I}\}$ is defined by the basis $(e_1,e_2,e_3)$. The body-fixed IMU frame $\{\mathcal{B}\}$ has basis vectors $(e_{1b},e_{2b},e_{3b})$. The vehicle position expressed in $\{\mathcal{I}\}$ is $p$, and its orientation is represented by the rotation matrix $R \in \mathbb{SO}(3)$, which maps body-frame vectors into the inertial frame.
• Figure 2: MEMS gyroscope structure used in inertial sensing. (a) A two-dimensional schematic of a vibrating MEMS gyroscope in which a driven proof mass experiences a Coriolis-induced motion proportional to the angular rate. (b) A tri-axis MEMS gyroscope with three orthogonal vibrating structures that measure the body-frame angular-velocity components about the $x$, $y$, and $z$ axes.
• Figure 3: MEMS accelerometer structure used in inertial sensing. (a) A two-dimensional view of a capacitive one-axis MEMS accelerometer in which a suspended proof mass deflects under acceleration, producing a differential change in capacitance. (b) A tri-axis accelerometer consisting of three orthogonal sensing axes that measure the specific-force components along the $x$, $y$, and $z$ body-frame directions.
• Figure 4: Schematic overview of the recursive bias-free INS estimation architecture. The state is propagated forward in time using high-rate IMU measurements through the Lie-group process model \ref{['eq:strapdown_group']} (prediction step), whose exact discrete-time update is given in \ref{['eq:hatX_update']}, together with the associated covariance propagation in \ref{['eq:Sigma-next-4th-A']}. The state is then intermittently corrected using measurements from aiding sensors (update step) via the computation of the innovation \ref{['eq:innovation']}. The Kalman gain $K_k$ is computed according to \ref{['eq:gain1']}--\ref{['eq:gain2']}.
• Figure 5: Exact versus classical IMU discretization for circular motion. Comparison of discrete-time propagation under piecewise-constant IMU inputs. The left panel shows the exact Lie-group propagation on $\mathbb{SE}_2(3)$ for multiple sampling intervals $\Delta t$, where the circular trajectory is preserved independently of $\Delta t$. The right panel shows the classical INS discretization ($J_\ell \approx I$, $Q_\ell \approx \tfrac{1}{2}I$), for which discretization errors accumulate as $\Delta t$ increases, resulting in visible geometric distortion of the trajectory.

Theorems & Definitions (6)

• lemma 1: Exact per-sample IMU increment
• lemma 2: IMU increment noise perturbation
• lemma 3: Group preintegration factorization
• theorem 1: UGES of the translational subsystem
• theorem 2: Almost global convergence on $\mathbb{SE}_5(3)$
• remark 1