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Ph.D. Thesis: Machine Learning-based Inertial Navigation.​
M.Sc. Thesis: Integrated Guidance / Estimation in Linear Quadratic Differential Games.

Selected Publications


8. “Learning Vehicle Trajectory Uncertainty” Barak Or and Itzik Klein. Under review, 2022. The linear Kalman filter is commonly used for vehicle tracking. This filter requires knowledge of the vehicle trajectory and the statistics of the system and measurement models. In real-life scenarios, prior assumptions made while determining those models do not hold. As a consequence, the overall filter performance degrades, and in some situations, the estimated states diverge. To overcome the uncertainty in the vehicle kinematic trajectory modeling, additional artificial process noise may be added to the model or different types of adaptive filters may be employed. This paper proposes a hybrid adaptive Kalman filter based on model and machine learning algorithms. First, recurrent neural networks are employed to learn the vehicle's geometrical and kinematic features. In turn, those features are plugged into a supervised learning model, thereby providing the actual process noise covariance to be used in the Kalman framework. The proposed approach is evaluated and compared to six other adaptive filters using the Oxford RobotCar dataset. The proposed framework can be implemented in other estimation problems to accurately determine the process noise covariance in real-time scenarios. Read Now
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7. “Adaptive Step Size Learning With Applications to Velocity Aided Inertial Navigation System” Barak Or and Itzik Klein. IEEE Access, 2022. Autonomous underwater vehicles (AUVs) are commonly used in many underwater applications. Recently, the usage of multi-rotor unmanned autonomous vehicles (UAV) for marine applications is receiving more attention in the literature. Usually, both platforms employ an inertial navigation system (INS) and aiding sensors for an accurate navigation solution. In AUV navigation, the Doppler velocity log (DVL) is mainly used to aid the INS, while for UAVs, it is common to use global navigation satellite systems (GNSS) receivers. The fusion between the aiding sensor and the INS requires a definition of step size parameter in the estimation process. It is responsible for the solution frequency update and, eventually, its accuracy. The choice of step size poses a tradeoff between computational load and navigation performance. Generally, the aiding sensors update frequency is considered much slower compared to the INS operating frequency (hundreds Hertz). Such a high rate is unnecessary for most platforms, specifically for low dynamics AUVs. In this work, a supervised learning-based adaptive tuning scheme to select the proper INS step size is proposed. To that end, a velocity error bound is defined, allowing the INS/DVL or the INS/GNSS fusion filter to act in sub-optimal working conditions, and yet minimize the computational load. Results from simulations and field experiment show the benefits of using the proposed approach. In addition, the proposed framework can be applied to any other fusion scenarios between any type of sensors or platforms.
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6. “A Hybrid Model and Learning-Based Adaptive Navigation Filter” Barak Or and Itzik Klein. IEEE Transactions on Instrumentation and Measurement, 2022. 
The fusion between an inertial navigation system and global navigation satellite systems is regularly used in many platforms such as drones, land vehicles, and marine vessels. The fusion is commonly carried out in a model-based extended Kalman filter framework. One of the critical parameters of the filter is the process noise covariance. It is responsible for the real-time solution accuracy, as it considers both vehicle dynamics uncertainty and the inertial sensors quality. In most situations, the process noise is covariance assumed to be constant. Yet, due to vehicle dynamics and sensor measurement variations throughout the trajectory, the process noise covariance is subject to change. To cope with such situations, several adaptive model-based Kalman filters were suggested in the literature. In this paper, we propose a hybrid model and learning-based adaptive navigation filter. We rely on the model-based Kalman filter and design a deep neural network model to tune the momentary system noise covariance matrix, based only on the inertial sensor readings. Once the process noise covariance is learned, it is plugged into the well-established, model-based Kalman filter. After deriving the proposed hybrid framework, field experiment results using a quadrotor are presented and a comparison to model-based adaptive approaches is given. We show that the proposed method obtained an improvement of 25% in position error. Furthermore, the proposed hybrid learning method can be used in any navigation filter and also in any relevant estimation problem. 
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5. “Learning Car Speed Using Inertial Sensors” Maxim Freydin and Barak Or. IEEE Sensors Letters, 2022. A deep neural network (DNN) is trained to estimate the speed of a car driving in an urban area using as input a stream of measurements from a low-cost six-axis inertial measurement unit (IMU). Three hours of data were collected by driving through the city of Ashdod, Israel in a car equipped with a global navigation satellite system (GNSS) real-time kinematic (RTK) positioning device and a synchronized IMU. Ground truth labels for the car speed were calculated using
the position measurements obtained at the high rate of 50 [Hz]. A DNN architecture with long short-term memory layers is
proposed to enable high-frequency speed estimation that accounts for previous inputs history and the nonlinear relation
between speed, acceleration, and angular velocity. A simplified aided dead reckoning localization scheme is formulated to
assess the trained model which provides the speed pseudo-measurement. The trained model is shown to substantially
improve the position accuracy during a 4 minutes drive without the use of GNSS position updates.  Read Now.



4. “LengthNet: Length Learning for Planar Euclidean Curves” Barak Or and Ido Amos. The Eurographics Association, Smart Tools and Apps for Graphics, 2021. We used a deep learning (DL) model to solve a fundamental problem in differential geometry. One can find many closed-form expressions for calculating curvature, length, and other geometric properties in the literature. As we know these properties, we are highly motivated to reconstruct them by using DL models. In this framework, our goal is to learn geometric properties from many examples. The simplest geometric object is a curve, and one of the fundamental properties is the length. Therefore, this work focuses on learning the length of planar sampled curves created by a simulation. The fundamental length axioms were reconstructed using a supervised learning approach. Following these axioms, a DL-based model, which we named LengthNet, was established. For simplicity, we focus on the planar Euclidean curves. Read Now.


3. “Kalman Filtering with Adaptive Step Size Using a Covariance based Criterion” Barak Or, Ben-Zion Bobrovsky, and Itzik Klein.  IEEE Transactions on Instrumentation and Measurement, 2021.
In Kalman filtering, a trade-off exists when selecting the filter step size. Generally, a smaller step size improves the estimation accuracy, yet with the cost of a high computational load. To mitigate this trade-off influence on performance, a criterion that acts as a guideline for a reasonable choice of the step size is proposed. This criterion is based on the predictor-corrector error covariance matrices of the discrete Kalman filter. In addition, this criterion is elaborated to an adaptive algorithm, for the case of the time-varying measurement noise covariance. Two simulation examples and a field experiment using a quadcopter are presented and analyzed to show the benefits of the proposed approach. Read Now.
 

2. “Optimal Disturbance Attenuation Approach with Measurement Feedback to Missile Guidance” Barak Or, Joseph Z. Ben-Asher, and Isaac Yaesh. AIAA Journal of Guidance, Control, and Dynamics, 2021.
Pursuit-evasion differential games using the Disturbance Attenuation approach are revisited. Under this approach, the pursuer actions are considered to be control actions, whereas all external actions, such as target maneuvers, measurement errors, and initial position uncertainties, are considered to be disturbances. Two open issues have been addressed, namely the effect of noise on the control gains, and the effect of trajectory shaping on the solution. These issues are closely related to the question of the best choice for the disturbance attenuation ratio. Detailed analyses are performed for two simple pursuit-evasion cases: a Simple Boat Guidance Problem and Missile Guidance Engagement. Read Now.


1. “Imperfect Information Game for a Simple Pursuit-Evasion Problem” Barak Or, Joseph Z. Ben-Asher and Isaac Yaesh. EuroGNC (5th CEAS Conference on GNC) AIAA conference, 2019.
Differential Games for pursuit-evasion problems have been investigated for many years. Differential games, with linear state equations and quadratic cost functions, are called Linear Quadratic Differential Games (LQDGs). In these games, one defines two players a pursuer and an evader, where the former aims to minimize, and the latter aims to maximize the same cost function (zero-sum games). The main advantage of using the LQDG formulation is that one gets Proportional Navigation (PN) like solutions with continuous control functions. One approach which plays a main role in the LQDG literature is Disturbance Attenuation (DA), whereby target maneuvers and measurement errors are considered as external disturbances. In this approach, a general representation of the input-output relationship between disturbances and output performance measures is the DA function (or ratio). This function is bounded by the control. This work revisits and elaborates upon this approach. We introduce the equivalence between two main implementations of the DA control. We then study a representative case, a “Simple Pursuit Evasion Problem”, with perfect and imperfect information patterns. By the derivation of the analytical solution for this game, and by running some numerical simulations, we develop the optimal solution based on the critical values of the DA ratio. The qualitative and quantitative properties of the Simple Pursuit Evasion Problem, based on the critical DA ratio, are studied by extensive numerical simulations and are shown to be different from the fixed DA ratio solutions. Read Now.

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