Inertial Navigation Systems

Autonomous Underwater Vehicles (AUVs) navigating without GPS typically fuse inertial measurements with acoustic Doppler Velocity Log (DVL) velocities and pressure-derived depth. Posing the navigation state on a Lie group improves accuracy and consistency. However, state-of-the-art filters based on the Invariant Extended Kalman Filter (IEKF) append the Inertial Measurement Unit (IMU) biases as a Euclidean extension, which breaks the group-affine structure required for exact log-linear error dynamics, causing the reported covariance to degrade alongside the estimate. We apply the Tangent-Group (TG) symmetry, which carries the biases within the geometry of the state space, to derive an Equivariant Filter (EqF) for this system, leaving zero linearization error in the navigation states and second-order error only in the biases. We develop an equivariant output model for the DVL, whose update incurs only third-order linearization error, together with a direct pressure output. Monte Carlo simulations benchmark the TG-EqF against a Two-Frame-Group IEKF and a Multiplicative EKF. The TG-EqF reduces error by 18--25% against both alternatives in each of attitude, velocity, and position. The main benefit is in the covariance it estimates: its Average Normalized Estimation Error Squared (ANEES) stays closer to its nominal value of one than that of the others. Offline analysis on AUV field data corroborates the findings of the simulations, demonstrating reduced position drift.

Recent developments in planetary exploration have shown the potential of Unmanned Aerial Vehicles (UAVs), such as the Ingenuity helicopter that provided valuable mapping data. However, limited payload capabilities constrain the flight times and compute available for localization, which restrict their applicability. By providing a tethered connection, issues such as battery and computational constraints are offloaded to the base rover. At the same time, the cable can be exploited for non-drifting localization. This work presents a novel Tether-Inertial Localization approach that uses tether length and angle measurements to estimate the UAV position relative to its base. The method combines a computationally efficient analytical catenary model with a Gaussian Process (GP) residual error compensation. This accounts for systematic sensor inaccuracies and model limitations. Experimental validation across circular, triangular, and figure-eight trajectories with tether lengths up to 4.5 m and a total flight time of 37 minutes demonstrates the effectiveness of the proposed approach. Using only tether-based position estimates for feedback, the analytical catenary model achieves an average RMSE of 7.4 cm, which is further reduced to 5.2 cm through GP-based residual compensation, one order of magnitude better than the state-of-the-art. These results establish Tether-Inertial Localization as a practical alternative to vision- and GNSS-based localization for Tethered Unmanned Aerial Vehicles (TUAVs).

The training of learned inertial odometry depends on dense, high-precision position ground truth from motion capture, visual-inertial odometry or SLAM, which is costly and hard to acquire at scale. We propose DINS-IO, which learns inertial odometry directly from raw IMU streams without position labels. Our key insight is that the strapdown INS velocity recursion is a strong, fully differentiable consistency prior: the predicted velocity, rotated into the navigation frame, must agree with the integrated specific force up to an unknown initial velocity and a constant accelerometer bias. We cast this constraint as a sliding-window least-squares problem with a globally shared bias, solve it in closed form, and use the solver residual as a self-supervised loss whose gradient flows back to the network through the analytic solution. To supply this per-sample constraint, we design a high-frequency network that emits dense body-frame velocity at the IMU rate. Since the self-supervised network learns consistent motion but its velocity is not yet metrically calibrated, we calibrate it to true metric velocity from a few labeled trajectories by directly supervising the predicted body-frame velocity and adapting only low-rank (LoRA) patches. On standard benchmarks, DINS-IO pretrained self-supervised and fine-tuned with a small fraction of labels matches or surpasses fully supervised baselines.

Accurate and robust attitude estimation is a key challenge for autonomous vehicles, particularly in GNSS-denied conditions and during highly accelerated flight. In such conditions, Inertial Measurement Units (IMUs) alone are insufficient for reliable tilt estimation due to the ambiguity between gravitational and inertial accelerations. Although auxiliary velocity sensors such as GNSS, Pitot tubes, Doppler radar, or Visual Inertial Odometry are commonly used, they may be unavailable, intermittent, or costly. This paper introduces a barometer-aided attitude estimation architecture that exploits barometric altitude measurements to provide complementary information on the vehicle's vertical motion, thereby enhancing attitude estimation within nonlinear observers on SO(3). The contributions are twofold. First, we design a deterministic Riccati observer cascaded with a complementary filter, ensuring almost-global asymptotic stability (AGAS) under a uniform observability (UO) condition while preserving the geometric structure of the attitude dynamics. Second, we propose a nonlinear observer evolving on SO(3)xR2, which integrates IMU measurements as inputs and barometer and magnetometer measurements as outputs within a unified framework, guaranteeing local exponential stability (LES) under relaxed uniform observability conditions. The proposed approaches are validated using both simulated and real flight data. The results demonstrate that barometer-aided estimation provides a lightweight, reliable, and effective complementary sensing modality for attitude estimation in minimal-sensing configurations, offering a practical alternative when conventional velocity measurements are unavailable or degraded.

Reliable localization in GNSS-denied environments remains a fundamental challenge for intelligent vehicles, as inertial navigation systems accumulate unbounded drift without external correction. Existing approaches provide drift correction through dedicated infrastructure, expensive external sensors, or complex multi-sensor fusion, each introducing practical deployment barriers. We propose Evidential Velocity Correction using Mamba (EVC-Mamba), a learning-based architecture that transforms onboard vehicle sensor data into a virtual velocity sensor for IMU drift correction without additional hardware. A Mamba-based selective state space model captures the temporal dynamics of vehicle motion, while evidential deep learning with a Normal-Inverse-Gamma distribution provides principled uncertainty quantification. The resulting uncertainty-aware velocity estimate is incorporated as a virtual correction measurement into an Error-State Extended Kalman Filter to reduce position drift. Evaluation on real-world vehicle data demonstrates that inertial navigation using the proposed velocity correction achieves localization accuracy within 10% of a dedicated external velocity sensor across different outage durations. The proposed architecture supports real-time onboard deployment at 40 Hz on edge hardware, enabling reliable localization during prolonged GNSS outages.

In many multisensor systems, measurements from different sensors are subject to unknown relative time delays. Accurate state estimation requires that delays be accounted for and, when possible, calibrated online. We consider the case of aided navigation, where measurements from a single aiding sensor are subject to an unknown but constant delay relative to the inertial measurement stream, and study the identifiability of the resulting system. Critically, identifiability depends not only on the temporal structure of the measurements, but also on the shape of the vehicle trajectory: some trajectories are sufficiently informative to support unique recovery of the delay and the navigation state, while others are not. Using the special Galilean group, we characterize these uninformative (or degenerate) trajectories and relate them to a continuous symmetry of the delayed measurement model, providing geometric insight into identifiability failures. We show that the class of trajectories for which identifiability fails is larger than previously reported, and connect our characterization to the familiar linearized, Jacobian-based analysis. Although our development is motivated by aided navigation, the underlying ideas apply more broadly to estimation problems on Lie groups with delayed measurements.

The accuracy of consumer-grade inertial navigation is bottlenecked by the stochastic noise of Micro-Electro-Mechanical Systems (MEMS). Traditional deterministic neural architectures often succumb to ``estimation jittering,'' sacrificing high-frequency kinematic fidelity for numerical stability. We propose PedestrianDiffusion, a multimodal spectral-domain generative framework reformulating dense 6D state estimation as a continuous conditional denoising process. By operating in the frequency domain, our formulation bounds the spectral covariance, acting as a mathematical preconditioner to stabilize the reverse diffusion trajectory. Furthermore, we introduce a zero-shot semantic conditioning mechanism leveraging vision-language embeddings as categorical priors to generalize across heterogeneous sensor noise profiles. To address the computational intractability of generative tracking, we deploy a single-step deterministic probability flow ODE solver ($T=1$). This yields high-capacity asynchronous batch trajectory refinement, establishing the viability of generative architectures for asynchronous batch trajectory refinement on edge hardware. Extensive evaluations on the OxIOD, RIDI, RoNIN, and TLIO benchmarks demonstrate that PedestrianDiffusion achieves state-of-the-art performance, exhibiting unprecedented robustness to impulse perturbations and coupled 6D kinematic drift. This work provides a rigorous algorithmic blueprint for next-generation Neural Inertial Measurement Units (N-IMUs).

Closed-loop (or feedback) error-state Kalman filters with their relatives and offspring are the state-of-the-art in modern aided inertial navigation research. Estimated inertial navigation system (INS) errors are continually fed back to the INS to correct the nominal system state before subsequent predictions. Conversely, in safety-critical aeronautical applications, open-loop (or feedforward) systems are an undisputed standard, where the inertial mechanization is strictly decoupled to allow for operational independence and fault isolation of computing units. We assess the performance impacts of this architectural choice beyond qualitative system-safety justifications using a standard inertial mechanization in geodetic coordinates and direct position aiding. Simulations using a variety of inertial sensor error characteristics, ranging from consumer to navigation grade systems, showcase the trade-off between smooth information fusion for high-end IMUs using an open-loop filter and the inherent long-term stability of the closed-loop architecture.

Global navigation systems require state estimation algorithms that handle Earth's curvature, Earth's rotation, and gravitational variations. These factors can typically be neglected in local navigation algorithms for robots, drones, etc. In classical error-state Kalman Filtering (ESKF) the error state dynamics are trajectory-dependent. Invariant ESKFs utilize Lie Group symmetries to represent the error, which can render error propagation trajectory-independent for group-affine systems. Choosing between a standard filter (where position and velocity errors are defined additively in the navigation frame), a left-invariant filter (where errors are represented in the body frame) and a right-invariant filter (where errors are represented in the navigation/world frame) depends on system dynamics and sensor configuration. This note presents the mathematical formulas for four classical and invariant ESKFs for globally applicable aided inertial navigation systems. It is intended to serve as a systematic reference for comparison and implementation.

Inertial navigation systems aided by three-dimensional landmark measurements constitute a fundamental problem in robotic perception and state estimation. Classical SO(3)-based Extended Kalman Filter (SO(3)-EKF) approaches provide practical solutions, but suffer from the false observability problem, in which the filter becomes overconfident in unobservable directions, leading to degraded estimation performance. The Invariant EKF (IEKF) addresses this limitation by reformulating the system dynamics as a group-affine system on a Lie group, although its measurement update does not fully satisfy certain state compatibility properties. More recently, the Iterated Invariant EKF (IterIEKF) was proposed to further improve the IEKF by ensuring, in the low-noise regime, that the estimated state remains on the observed state manifold while the uncertainty is confined to its tangent space. In this work, we formulate and apply the IterIEKF to landmark-based inertial 3D localization for the first time. Through numerical simulations, we show that the proposed approach outperforms the classical SO(3)-EKF, the Iterated SO(3)-EKF, and the IEKF in terms of both estimation accuracy and consistency.

The Doppler velocity log (DVL) velocity measurements are critical to the accuracy of autonomous underwater vehicle (AUV) navigation solutions and, consequently, to mission success. To ensure accurate measurements, the DVL is commonly calibrated before mission start while the AUV sails on the water surface, receiving global navigation satellite system (GNSS) signals that provide accurate reference measurements. Conventionally, Kalman filter-based approaches are employed during calibration to estimate the scale factor and misalignment errors. However, in certain environments, GNSS signals may be unavailable, rendering conventional calibration impossible and forcing the use of uncalibrated DVL measurements, which degrades navigation performance. To address this limitation, this work proposes information-aided calibration (IAC) with two main contributions: first, improving the accuracy of conventional Kalman filter-based calibration in GNSS-enabled environments, and second, enabling GNSS-free DVL self-calibration. Using real-world AUV datasets, the proposed IAC models achieve up to a 20% average improvement in GNSS-enabled environments and up to a 35% improvement in velocity vector estimation during GNSS-free DVL self-calibration. Overall, the proposed approach improves navigation accuracy, reduces navigation drift, and consequently enhances mission reliability.

We propose a robust Extended Kalman Filter (EKF) architecture for land navigation using an array of hundreds of low-cost micro-electromechanical systems (MEMS) inertial sensors. The main challenges in this setting are bursty sensor-specific bias errors, bias drift, and the need to aggregate many inertial measurements without increasing the computational burden of the navigation filter. To address these challenges, we introduce Robust Inertial Sensor Array Fusion (RISAF), a pre-filtering framework that combines dynamic percentile gating with real-time bias tracking before the EKF prediction step. The proposed aggregation suppresses anomalous sensor readings and compensates for individual sensor drift while preserving the vehicle-level kinematic signal. Because the resulting fused inertial measurements are passed to a standard EKF, the navigation filter retains a minimal state vector and supports real-time execution. We evaluate RISAF through extensive simulations and real-world field tests in GNSS-denied environments, with the data provided as supplementary material. Compared with a baseline that averages the sensor readings, RISAF achieves substantially improved azimuth accuracy and reduced drift accumulation. These results demonstrate that robust fusion of large MEMS inertial arrays can bridge a substantial part of the gap between cost-effective hardware and tactical-grade inertial navigation performance.

In aided inertial navigation, measurements from different sensors are often subject to unknown relative time delays. Consider a single aiding sensor whose measurements have an unknown but constant delay relative to the inertial-measurement data stream. We study the identifiability of the delay and the initial navigation state that parameterizes the trajectory. Identifiability depends on both the temporal structure of the aiding measurements and the form of the trajectory itself. Our geometric analysis shows that, for a larger class of uninformative (i.e., degenerate) trajectories than has previously been reported, the delayed measurement model admits a continuous symmetry that prevents unique delay-and-state recovery.

Video Diffusion Models (VDMs) is constrained by immense computational costs. While offline calibration-based acceleration suffers from calibration data dependency, prohibitive calibration duration, and susceptibility to distribution shifts, offline calibration-free methods eliminate these hurdles. However, since they rely on instantaneous zero-order approximations where the mapping between input and output differences varies in real-time, they are susceptible to observational noise and ignore the intrinsic momentum within the diffusion trajectory. In this paper, we propose NaviCache, a plug-and-play test-time self-calibration method re-conceptualizing feature evolution as an Inertial Navigation System (INS) problem. NaviCache bridges the fundamental domain gap and the non-stationary nature of diffusion by modeling the relative coupling between input and output variations. We introduce a dual-state estimation architecture that adaptively tracks the feature change ratio and its latent drift, initialized via a specialized Initial Alignment phase. By integrating a time-dependent noise schedule with an uncertainty-aware Measurement Update mechanism, NaviCache provides a theoretically grounded mechanism for error-bounded computation skipping. Extensive experiments on the HunyuanVideo, Wan, and Open-Sora series demonstrate that NaviCache exhibits more accurate error judgment for computation skipping and achieves outstanding comprehensive performance.

Autonomous Underwater Vehicles (AUVs) rely heavily on the fusion of inertial sensors and Doppler velocity logs (DVLs) for navigation. In standard autonomous navigation systems, the DVL measures four beam velocities, thereby enabling the estimation of the AUV velocity vector. However, during real-world missions, the DVL may receive noisy or incomplete beam measurements due to marine obstacles, seabed reflections, or environmental disturbances. Furthermore, some low-cost underwater platforms operate without inertial sensors to reduce system complexity and cost. In such cases, reliable estimation of the AUV velocity vector in real-world missing beam scenarios becomes challenging, leading to degraded navigation solutions. To circumvent these challenges and enable resilient underwater navigation, we propose DVL-DeepONet, a physics-guided deep neural operator framework along with three variants. The proposed models are designed to estimate DVL-based velocity information under multiple operational scenarios, including (i) noise-resilient estimation in coupled inertial/DVL measurements, (ii) DVL-only learning, and (iii) beam measurement recovery. By learning a nonlinear operator that maps temporal inertial/DVL observations directly to vehicle velocity while enforcing DVL measurement physics through a consistency constraint, the proposed approach enables robust velocity estimation even under degraded sensing conditions. The proposed framework is validated using real-world AUV experiments, comprising a cumulative path length of approximately 10,000 m. Experimental results demonstrate that the proposed DVL-DeepONet architectures outperform baseline model-based approaches and learning-based algorithms by 40%.

Electromagnetic Navigation Systems (eMNS) have gained considerable attention for minimally invasive surgery and targeted drug delivery. While most of the literature relies on quasi-static control of these systems, recent work has demonstrated the benefits of dynamic approaches. However, trajectory tracking far from equilibrium states remains largely unaddressed. We close this gap by demonstrating the first swing-up of a magnetically actuated inverted pendulum using the clinically-ready Navion eMNS. Although the inverted pendulum is not clinically relevant in itself, the proposed method utilizes torques and forces as control objectives, making it applicable to other magnetically actuated devices such as catheters and guidewires. Our approach combines trajectory optimization that accounts for internal eMNS dynamics with time-varying Linear Quadratic Regulator (LQR) state feedback and Iterative Learning Control (ILC), which leverages previous trial data and the system's dynamic model to progressively refine the feedforward command. While LQR alone fails due to the complex phenomena of magnetic actuation, ILC enables successful swing-up within six iterations. Furthermore, post-experimental analysis reveals that the learned ILC correction closely matches the torque discrepancy predicted by high-fidelity magnetic field model calibration, suggesting learning and adaptation as a promising tool to deal with uncertainties in electromagnetic actuation arising, e.g., from patient-specific physiological motion patterns and field model calibration inaccuracies.

Accurate and robust localization remains a fundamental challenge for autonomous ground vehicles. In this work, we propose a hybrid neural inertial navigation framework that integrates a wheel-mounted inertial sensors, enforced periodic trajectories, and a simple, efficient neural network capable of regressing vehicle displacement with GNSS position updates in an error-state extended Kalman filter. The periodic trajectories increase the inertial signal-to-noise ratio, allowing the network to use only inertial readings to estimate displacement. The approach is validated through real-world experiments using multiple wheel-mounted inertial sensors. Experimental results demonstrate that the proposed method achieves a significant improvement in positioning accuracy, reducing the position root mean squared error by approximately 46 % compared to standard wheel-mounted inertial sensor fusion with GNSS updates.

Accurate and reliable navigation is essential for autonomous ground vehicle operations. Standard INS/GNSS fusion relies on GNSS position updates, which provide limited observability of orientation and inertial sensor error states, particularly during low-dynamic motion. In this work, we propose utilizing past GNSS measurements alongside a motion model to extract meaningful vehicle acceleration information. This acceleration measurement is then integrated into the INS/GNSS filter to improve its robustness and accuracy. The proposed approach is evaluated on two real-world unmanned ground vehicle datasets collected from different mobile platforms and inertial sensor grades. Results demonstrate consistent positioning accuracy improvements relative to the standard position-aided filter, with mean position root mean square error improvements of 11.40 % and 20.74 % on the two datasets, respectively.

Tightly coupled SLAM formulations under mixed-rate sensing often bind temporal processing, local geometric association, estimator formulation, and map-update policy into method-specific designs. Such binding makes it difficult to vary one design choice without re-engineering the rest of the state-estimation process. This paper presents FUSE, a framework for unified state estimation in vehicular and robotic SLAM systems. FUSE organizes the state-estimation interface around observation ingestion, propagation, update, and state query, and uses this interface to separate temporal processing, residual-ready local geometric association, estimator formulation, and map-update policy. A LiDAR--IMU instantiation is developed to examine the framework under mixed-rate sensing and directional degeneracy, where high-rate inertial propagation, LiDAR-triggered geometric update, residual screening, and degeneracy-aware correction operate through the same interface boundaries. On a 418~m loop-corridor sequence, the instantiation reports a 1.626 m end-to-end trajectory error, corresponding to a 7.9% relative error reduction compared with Faster-LIO, the lowest-error baseline on this sequence. The results support FUSE as a framework for organizing state-estimation design choices and show how the evaluated instantiation regularizes updates along weakly observable directions.

Many Inertial Navigation Systems (INS) use Global Navigation Satellite System (GNSS) position as the primary measurement to drive filter performance and bound error growth. However, commercial-grade GNSS receivers introduce unknown measurement delays ranging from 50 ms to 300 ms depending on sensor quality and operating mode. Such time delays can significantly degrade INS performance unless they are explicitly compensated for. Existing algorithms commonly estimate this delay offline, run the filter concurrently with GNSS measurements using buffered Inertial Measurement Unit (IMU) data, and predict the current state by forward-integrating buffered inertial measurements via IMU preintegration. The state-of-the-art online method is an Extended Kalman Filter (EKF) that explicitly models the time delay as a state parameter, which defines the preintegration duration. This paper introduces a novel geometric framework for modeling time-delayed INS, in which Galilean symmetry is leveraged to provide a joint representation of space and time for consistent state estimation. An Equivariant Filter (EqF) is derived for the coupled estimation of navigation states and time delay. Validation is performed on two fixed-wing Uncrewed Aerial Vehicles (UAV) with GNSS time lags of 90 ms and 120 ms. The test flights last two to three minutes. Simulations further investigate delays up to 500 ms and provide a statistical comparison against the state-of-the-art EKF. Results show that the EqF preserves accuracy and consistency, while the EKF lacks consistency and its performance degrades significantly with increasing measurement delays.

Although Global Navigation Satellite Systems (GNSS) provide a general solution for bike tracking outdoors, there still exist complex riding environments where only inertial navigation systems work, such as urban canyons. Despite decades of research, localization using only low-cost inertial sensors still faces challenges such as cumulative drifts and poor robustness caused by filtering methods. Furthermore, sensors such as visual and LiDAR could provide reliable measurements, but they are not suitable for large-scale deployment. In this paper, we propose an inertial tracking framework that integrates bicycle mechanical constraints with a mixture-of-experts model. Specifically, we leverage multiple expert modules to capture shared representations and weight them through the gating mechanism, thus improving multi-task learning performance and enabling uncertainty-aware trajectory estimation. Furthermore, based on the mechanical transmission between the pedal and the rear wheel of a bike, we explore the intrinsic relationship between the rider's periodic pedalling behaviors and acceleration variations, and convert such patterns into bike's wheel speed for dynamic calibration. Experiments with real-world riding data from shared bikes of the DiDi ride-hailing platform demonstrate that our system improves the accuracy of baselines by at least 12%, with wheel speed errors below 0.5 m/s at 95-percentile.

Autonomous underwater vehicles (AUVs) have become indispensable for deep-sea exploration, spanning critical scientific research and commercial applications. The rapid attenuation of electromagnetic waves renders satellite radio signals unavailable, while the dynamic unpredictability of the marine environment presents formidable navigation challenges. This chapter explores recent advancements in AI-aided AUV positioning, specifically focusing on advanced sensor fusion architectures that integrate inertial navigation systems with Doppler velocity logs and cameras. Beyond traditional model-based filtering, we examine the transformative emergence of AI-driven learning approaches in enhancing inertial dead-reckoning tasks and adaptive fusion algorithms. By addressing these recent milestones, this chapter provides a comprehensive roadmap for achieving the high-precision navigation essential for autonomous underwater missions.

Frequency Modulated Continuous Wave (FMCW) radar is a promising sensor for aided inertial navigation, due to its robustness in environments that challenge traditional alternatives, such as LiDAR and vision. However, its widespread adoption is hindered by complex, noisy measurements, which make reliable estimation difficult. This manuscript addresses these challenges by analyzing the fundamental measurement relations of FMCW radar sensing and developing a reliable estimator. Noise models are derived by applying first principles to the underlying signal processing of a typical radar sensor. These models guide the design of a factor graph-based estimator, utilizing a first-order approximation for the measurement noise propagation. The approach is first examined through simulation, evaluating the significance of different noise sources, the validity of the first-order approximation, and the state-dependent nature of the covariance expressions. Extensive experiments demonstrate the superior robustness and accuracy of the proposed method across diverse field environments and flight profiles, including beyond the radar's standard operating range. Furthermore, the experiments confirm the insights from the simulation regarding the behavior and performance of different estimator configurations relative to their operating conditions. The evaluation data and estimator implementation are made available at https://github.com/ntnu-arl/rig.

An accurate initial heading angle is essential for efficient and safe navigation across diverse domains. Unlike magnetometers, gyroscopes can provide accurate heading reference independent of the magnetic disturbances in a process known as gyrocompassing. Yet, accurate and timely gyrocompassing, using low-cost gyroscopes, remains a significant challenge in scenarios where external navigation aids are unavailable. Such challenges are commonly addressed in real-world applications such as autonomous vehicles, where size, weight, and power limitations restrict sensor quality, and noisy measurements severely degrade gyrocompassing performance. To cope with this challenge, we propose a novel diffusion denoiser-aided gyrocompass approach. It integrates a diffusion-based denoising framework with an enhanced learning-based heading estimation model. The diffusion denoiser processes raw inertial sensor signals before input to the deep learning model, resulting in accurate gyrocompassing. Experiments using both simulated and real sensor data demonstrate that our proposed approach improves gyrocompassing accuracy by 26% compared to model-based gyrocompassing and by 15% compared to other learning-driven approaches. This advancement holds particular significance for ensuring accurate and robust navigation in autonomous platforms that incorporate low-cost gyroscopes within their navigation systems.