Article

Kalman Filtering With Adaptive Step Size Using a Covariance-Based Criterion

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Abstract

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.

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... Hence, tracking a vehicle involves a discrete realization of continuous motion. Such realization requires a step size selection, usually made by the designer according to the scenario and computational constraints [12]- [14]. Moreover, to save power and extend the sensor/system life, the number of samples received from each source should be determined such as the information quality is maintained and the computational load is minimized [15]. ...
... It improves the energy efficiency during target tracking scenarios. In [12], a simple criterion was suggested to define the step size for sensor measurements to minimize computational load and still provide moderate navigation performance. This approach is based on the predictor and corrector of the linear discrete Kalman Filter (KF) [18], [19], where the main idea is to keep the discretized implementation of the continuous process with a lower numerical error. ...
... The inertial sensors operate in a much faster frequency (tens or hundreds of Hertz) than the aiding sensors (several Hertz). As a consequence, approaches like in [12], [15]- [17], are not suitable for such setups, as they assume constant step sizes. In order to avoid the need of constant step size, recent works explore the possibilities of using classical machine learning (ML) or deep learning (DL) based approaches [21]. ...
Preprint
Full-text available
Autonomous underwater vehicles (AUV) 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, 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 the 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 high rate is unnecessary for most platforms, specifically for low dynamics AUVs. In this work, a supervised machine 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 to act in a 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.
... The EKF is the most commonly used filtering algorithm for a nonlinear system with additive white Gaussian noise to determine the location and velocity of the target in the environment [30]. The EKF transforms a linear state-space continuous-time dynamic model into a discrete-time dynamic model with an Euler approximation for the system matrix [31], enabling determination of the expected continuous-time location and velocity of the moving target. Compared with the multilateration-based positioning algorithm, the EKF leverages the estimation accuracy by including prior knowledge of the target kinematics; therefore, this approach is more precise and reliable than others, especially in cases in which faster measurements are obtained. ...
... Eq. (28) shows that every step in the EKF is impacted by k t  , and the accuracy of the observations in Eq. (29) increases with decreasing k t  as the numerical error from integration and division decreases [31], [33]. However, an excessively small k t  will introduce time consumption for the EKF. ...
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... Hence, tracking a vehicle involves a discrete realization of continuous motion. Such realization requires a step size selection, usually made by the designer according to the scenario and computational constraints [12], [13], [14]. Moreover, to save power and extend the sensor/system life, the number of samples received from each source should be determined such as the information quality is maintained and the computational load is minimized [15]. ...
... However, the step size does not necessarily depend on the vehicle speed and/or the distance to target. In [12], a simple criterion was suggested to define the step size for sensor measurements to minimize computational load and still provide moderate navigation performance. This approach is based on the predictor and corrector of the linear discrete Kalman filter (KF) [18], [19], where the main idea is to keep the discretized implementation of the continuous process with a lower numerical error. ...
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Autonomous underwater vehicles (AUV) 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, 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 the 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 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 a 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.
... The Kalman filter (KF) is widely used for vehicle tracking tasks [1]- [8] and vehicle trajectory smoothing [4], [9]- [12]. One of the challenges to consider when applying a KF for tracking applications is the modeling of the vehicle trajectory, as expressed by the system matrix and associated process noise covariance. ...
... One of the challenges to consider when applying a KF for tracking applications is the modeling of the vehicle trajectory, as expressed by the system matrix and associated process noise covariance. Commonly, the constant velocity (CV) or a constant acceleration (CA) models are employed for a wide range of vehicle tracking problems [1], [12]- [15]. In the CV model, the underlying assumption is that the vehicle travels with constant velocity. ...
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A novel approach for vehicle tracking using a hybrid adaptive Kalman filter is proposed. The filter utilizes recurrent neural networks to learn the vehicle’s geometrical and kinematic features, which are then used in a supervised learning model to determine the actual process noise covariance in the Kalman framework. This approach addresses the limitations of traditional linear Kalman filters, which can suffer from degraded performance due to uncertainty in the vehicle kinematic trajectory modeling. Our method is evaluated and compared to other adaptive filters using the Oxford RobotCar dataset, and has shown to be effective in accurately determining the process noise covariance in real-time scenarios. Overall, this approach can be implemented in other estimation problems to improve performance.
... As a result, the problem of designing a robust Kalman filter in practical applications where the knowledge of noise distribution is missing or imprecise is a big challenge for both researchers and developers. The authors of [7][8][9][10][11][12] proposed methods, which include so-called adaptive Kalman filtering, to estimate signal states and noise simultaneously, which works well when there is a lot of data used to obtain certain accurate performances in the entire estimated period. Later on, minimax-based Kalman filters [13][14][15] and finite impulse response Kalman filters [16][17][18][19] were presented. ...
... According to the state-space model described in (9) and (10), if x k  is the least-square estimate of  x k 1 , then the stochastic process of the following: ...
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... The inference-based models can be further divided into the probabilistic interference models and deterministic optimization models [20], [21]. The probabilistic interference models use techniques such as KF [22], extended KF [23], and particle filter [24], [25] to estimate an object's state based on previous observation data [26], [27], [28]. The deterministic optimization models aim to find the maximum probability of tracking the object, often using a posteriori to solve the problem [29], [30]. ...
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... In this manner, similar to step length estimation in PDR, we estimated the peak-topeak change in distance of the quadrotor. 2) The figure-eight trajectory was used to show the benefits of a criterion that acts as a guideline for a reasonable choice for the step size in Kalman filtering (KF) as by [43]. In KF, a trade-off exists when selecting the filter step size. ...
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... These four techniques are Bayesian, correlation, Maximum Likelihood Estimation (MLE), and covariance matching. These techniques have been applied in various land, air, and space applications [13][14][15][16][17]. ...
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  • J M Mendel
J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control. London, U.K.: Pearson Education, 1995.