Constrained Extended Kalman Filter Research Articles

UAV-UGV Collaborative Localisation with Minimum Sensing.

This paper presents a novel methodology to localise Unmanned Ground Vehicles (UGVs) using Unmanned Aerial Vehicles (UAVs). The UGVs are assumed to be operating in a Global Navigation Satellite System (GNSS)-denied environment. The localisation of the ground vehicles is achieved using UAVs that have full access to the GNSS. The UAVs use range sensors to localise the UGV. One of the major requirements is to use the minimum number of UAVs, which is two UAVs in this paper. Using only two UAVs leads to a significant complication that results an estimation unobservability under certain circumstances. As a solution to the unobservability problem, the main contribution of this paper is to present a methodology to treat the unobservability problem. A Constrained Extended Kalman Filter (CEKF)-based solution, which uses novel kinematics and heuristics-based constraints, is presented. The proposed methodology has been assessed based on the stochastic observability using the Posterior Cramér-Rao Bound (PCRB), and the results demonstrate the successful operation of the proposed localisation method.

Influence of Estimators and Numerical Approaches on the Implementation of NMPCs

Advanced control strategies, together with state-estimation methods, are frequently applied to nonlinear and complex systems. It is crucial to understand which of these are the most efficient methods for the best use of these approaches in a chemical process. In the current work, nonlinear model predictive control (NMPC) approaches were developed that considered three numerical methods: single shooting (SS), multiple shooting (MS), and orthogonal collocation (OC). Their performance was compared against the Van de Vusse reactor benchmark while considering set-point changes, unreachable set-point, disturbances, and mismatches. The results showed that the NMPC based on OC presented less computational cost than the other approaches. The extended Kalman filter (EKF), constrained extended Kalman filter (CEKF), and the moving horizon estimator (MHE) were also developed. The estimators’ performance was compared for the same benchmark by considering the computational cost and the mean squared error (MSE) for the estimated variables, thereby verifying the CEKF as the best option. Finally, the performance of the nine combinations of estimators and control approaches was compared to consider the Van de Vusse reactor and the same scenarios, thereby verifying the best performance of the CEKF with the OC. The present work can help with choosing the numerical method and the estimator for controlling chemical processes.

Recursive state and parameter estimation of COVID-19 circulating variants dynamics

COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.

Estimating Lower Body Kinematics Using a Lie Group Constrained Extended Kalman Filter and Reduced IMU Count

Goal: Gait monitoring is useful for diagnosing movement disorders or assessing surgical outcomes. This paper presents an algorithm for estimating pelvis, thigh, shank, and foot kinematics during walking using only two or three wearable inertial sensors. Methods: The algorithm makes novel use of a Lie-group-based extended Kalman filter. The algorithm iterates through the prediction (kinematic equation), measurement (pelvis position pseudo-measurements, zero-velocity update, and flat-floor assumption), and constraint update (hinged knee and ankle joints, constant leg lengths). Results: The inertial motion capture algorithm was extensively evaluated on two datasets showing its performance against two standard benchmark approaches in optical motion capture (i.e., plug-in gait (commonly used in gait analysis) and a kinematic fit (commonly used in animation, robotics, and musculoskeleton simulation)), giving insight into the similarity and differences between the said approaches used in different application areas. The overall mean body segment position (relative to mid-pelvis origin) and orientation error magnitude of our algorithm ( n=14 participants) for free walking was 5.93 ± 1.33 cm and 13.43 ±1.89 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> when using three IMUs placed on the feet and pelvis, and 6.35 ± 1.20 cm and 12.71 ±1.60 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> when using only two IMUs placed on the feet. Conclusion: The algorithm was able to track the joint angles in the sagittal plane for straight walking well, but requires improvement for unscripted movements (e.g., turning around, side steps), especially for dynamic movements or when considering clinical applications. Significance: This work has brought us closer to comprehensive remote gait monitoring using IMUs on the shoes. The low computational cost also suggests that it can be used in real-time with gait assistive devices.

Robust extended Kalman filter estimation with moving window through a quadratic programming formulation

In this work, two new formulations for the extended (MW-EKF) and robust extended Kalman filter with moving window estimation (MW-REKF) are proposed. The MW-EKF and MW-REKF are formulated using an elegant quadratic programming problem that facilitates its implementation and decreases its computational cost. Besides that, the constrained extended Kalman filter (CEKF), constrained extended Kalman filter and smoother (CEKFS) and the moving horizon estimation (MHE) are compared in terms of computational cost and fit to the real data. The comparison is performed over a spherical-quadruple-tank model with different settings aiming to raise each approach's advantages and disadvantages. For both state and parameter estimation the MW-REKF has shown the smoothest and most robust behavior among all methodologies. This technique minimized the effect of the outliers, physical limitations, structural discrepancies, among others. The computational cost of the proposed techniques is only four times higher than CEKF and nine times smaller than the MHE.

Constrained extended Kalman filter design and application for on-line state estimation of high-order polymer electrolyte membrane fuel cell systems

In this paper an alternative approach to extended Kalman filtering (EKF) for polymer electrolyte membrane fuel cell (FC) systems is proposed. The goal is to obtain robust real-time capable state estimations of a high-order FC model for observer applications mixed with control or fault detection. The introduced formulation resolves dependencies on operating conditions by successive linearization and constraints, allowing to run the nonlinear FC model at significantly lower sampling rates than with standard approaches. The proposed method provides state estimates for challenging operating conditions such as shut-down and start-up of the fuel cell for which the unconstrained EKF fails. A detailed comparison with the unscented Kalman filter shows that the proposed EKF reconstructs the outputs equally accurate but nine times faster. An application to measured data from an FC powered passenger car is presented, yielding state estimates of a real FC system, which are validated based on the applied model.

Estimating Lower Limb Kinematics Using a Lie Group Constrained Extended Kalman Filter with a Reduced Wearable IMU Count and Distance Measurements.

Tracking the kinematics of human movement usually requires the use of equipment that constrains the user within a room (e.g., optical motion capture systems), or requires the use of a conspicuous body-worn measurement system (e.g., inertial measurement units (IMUs) attached to each body segment). This paper presents a novel Lie group constrained extended Kalman filter to estimate lower limb kinematics using IMU and inter-IMU distance measurements in a reduced sensor count configuration. The algorithm iterates through the prediction (kinematic equations), measurement (pelvis height assumption/inter-IMU distance measurements, zero velocity update for feet/ankles, flat-floor assumption for feet/ankles, and covariance limiter), and constraint update (formulation of hinged knee joints and ball-and-socket hip joints). The knee and hip joint angle root-mean-square errors in the sagittal plane for straight walking were and , respectively, while the correlation coefficients were and , respectively. Furthermore, experiments using simulated inter-IMU distance measurements show that performance improved substantially for dynamic movements, even at large noise levels ( m). However, further validation is recommended with actual distance measurement sensors, such as ultra-wideband ranging sensors.

Asphalt pavement modulus backcalculation using surface deflections under moving loads

AbstractThe measurement of pavement surface deflections under moving loads can be used for evaluation of pavement structural condition. The objective of this paper is to develop an innovative backcalculation method to calculate pavement layer moduli based on traffic speed deflectometer (TSD) measurements and analyze factors influencing backcalculation results. The backcalculation method is based on 2.5D finite element method (FEM) and constrained extended Kalman filter (CEKF). Field measurements at different pavement sections were used to verify the accuracy of backcalculation results. Parametric analysis was conducted to evaluate the influences of traffic loading and pavement structure variations in the field. It was found that the backcalculation results were considerably affected by the variation of asphalt layer thickness and dynamic loading due to pavement surface roughness. A heavy truck passing the adjacent lane during TSD testing could also cause variation of backcalculation results.

A constrained extended Kalman filter based on LS-VCE formulated by condition equations with prediction of cross-covariances

A constrained extended Kalman filter (CEKF) based on least-squares variance component estimation (LS-VCE) is generally developed by condition equations since the proper prediction of dispersion matrices is one of the main bottlenecks in the KF algorithms. Here we investigate four problems which have not been simultaneously considered yet. These problems are examination of non-linearty of dynamic model, VCE, general non-linear state constraints and fairly general stochastic model. Although a few contributions proposed some adaptive KF in particular based on Helmert’s VCE method, they developed their filters for special problems with some restrictive conditions such as independence of all variables and/or linearity of the dynamic model. Also some of these filters did not apply VCE methods to all parts of the dynamic model. In this contribution, we try to overcome all of these restrictions. Moreover, LS-VCE method gives some added advantages over other VCE methods. First the new formulation of CEKF is developed by condition equations with prediction of all possible cross-covariances as algorithm 1. Then the LS-VCE method is applied to it after some modifications which results in an adaptive constrained extended Kalman filter (ACEKF) as the second algorithm.

Parameter identification of a differentiable Bouc-Wen model using constrained extended Kalman filter

Hysteresis is of critical importance to structural safety under severe dynamic loading conditions. One of the widely used hysteretic models for civil structures is the Bouc-Wen model, the effectiveness of which depends on suitable model parameters. The locally non-differentiable governing equation of the conventional Bouc-Wen model poses difficulty on existing identification algorithms, especially the extended Kalman filter, which relies on linearized system equations to propagate state estimates and covariance. In addition, the standard extended Kalman filter usually does not incorporate parameter constraints, and therefore may result in unreasonable estimates. In this article, a modified and differentiable Bouc-Wen model, together with a constrained extended Kalman filter (CEKF), is proposed to identify the hysteretic model parameters in a reliable way. The partial derivatives of the differentiable Bouc-Wen model with respect to hysteretic parameters can be easily calculated for implementing the identification algorithm. Constrained extended Kalman filter restricts the Kalman gain to ensure that the estimates of parameters satisfy constraints from physical laws. Parameter identification using simulated and experimental data collected from a four-story structure demonstrates that constrained extended Kalman filter can achieve more reliable identification results than the standard extended Kalman filter.

On general constrained extended Kalman filter formulated by condition equations: three algorithms

Three constrained extended Kalman filters (CEKF) are developed by making use of condition equations which equations allows one to predict directly the residuals of all variables. The first one is a general CEKF algorithm in which it is supposed that all of the observation equations, system equations and constraints of a dynamic problem are non-linear functions. Although the constrained Kalman filter was already investigated by a few contributions, they assumed some restrictive conditions such as linearity of constraints and/or equations. Moreover, this generalization helps one to deal with problems which encounter with raw GPS data. In some problems, constrains can be expressed by a quadratic form. Hence, the second algorithm proposes a CEKF solution with quadratic constraints. In this algorithm, the constraints are not linearized. Eventually, in case of using refined GPS data in which quadratic constraints must be imposed to the state vector, the third algorithm is developed. In this algorithm one does not require to linearize any part of the dynamic model. Rigorous prediction of posterior dispersion (variance-covariance) matrix of the unknown parameters is the other attainment of this contribution. A numerical example demonstrates the efficiency of the proposed algorithm.

Parametric study of sensor placement for vision-based relative navigation system of multiple spacecraft

In order to overcome the limited range of GPS-based techniques, vision-based relative navigation methods have recently emerged as alternative approaches for a high Earth orbit (HEO) or deep space missions. Therefore, various vision-based relative navigation systems use for proximity operations between two spacecraft. For the implementation of these systems, a sensor placement problem can occur on the exterior of spacecraft due to its limited space. To deal with the sensor placement, this paper proposes a novel methodology for a vision-based relative navigation based on multiple position sensitive diode (PSD) sensors and multiple infrared beacon modules. For the proposed method, an iterated parametric study is used based on the farthest point optimization (FPO) and a constrained extended Kalman filter (CEKF). Each algorithm is applied to set the location of the sensors and to estimate relative positions and attitudes according to each combination by the PSDs and beacons. After that, scores for the sensor placement are calculated with respect to parameters: the number of the PSDs, number of the beacons, and accuracy of relative estimates. Then, the best scoring candidate is determined for the sensor placement. Moreover, the results of the iterated estimation show that the accuracy improves dramatically, as the number of the PSDs increases from one to three.

Improved Calibration Method for Dynamic Traffic Assignment Models: Constrained Extended Kalman Filter

The calibration of dynamic traffic assignment (DTA) models involves the estimation of model parameters to best replicate real-world measurements. Good calibration is essential to estimate and predict accurately traffic states, which are crucial for traffic management applications to alleviate congestion. A widely used approach to calibrate simulation-based DTA models is the extended Kalman filter (EKF). The EKF assumes that the DTA model parameters are unconstrained, although they are in fact constrained; for instance, origin–destination (O-D) flows are nonnegative. This assumption is typically not problematic for small- and medium-scale networks in which the EKF has been successfully applied. However, in large-scale networks (which typically contain numbers of O-D pairs with small magnitudes of flow), the estimates may severely violate constraints. In consequence, simply truncating the infeasible estimates may result in the divergence of EKF, leading to extremely poor state estimations and predictions. To address this issue, a constrained EKF (CEKF) approach is presented; it imposes constraints on the posterior distribution of the state estimators to obtain the maximum a posteriori (MAP) estimates that are feasible. The MAP estimates are obtained with a heuristic followed by the coordinate descent method. The procedure determines the optimum and are computationally faster by 31.5% over coordinate descent and by 94.9% over the interior point method. Experiments on the Singapore expressway network indicated that the CEKF significantly improved model accuracy and outperformed the traditional EKF (up to 78.17%) and generalized least squares (up to 17.13%) approaches in state estimation and prediction.

A constrained extended Kalman filter for the optimal estimate of kinematics and kinetics of a sagittal symmetric exercise

This paper presents a method for real-time estimation of the kinematics and kinetics of a human body performing a sagittal symmetric motor task, which would minimize the impact of the stereophotogrammetric soft tissue artefacts (STA). The method is based on a bi-dimensional mechanical model of the locomotor apparatus the state variables of which (joint angles, velocities and accelerations, and the segments lengths and inertial parameters) are estimated by a constrained extended Kalman filter (CEKF) that fuses input information made of both stereophotogrammetric and dynamometric measurement data. Filter gains are made to saturate in order to obtain plausible state variables and the measurement covariance matrix of the filter accounts for the expected STA maximal amplitudes. We hypothesised that the ensemble of constraints and input redundant information would allow the method to attenuate the STA propagation to the end results. The method was evaluated in ten human subjects performing a squat exercise. The CEKF estimated and measured skin marker trajectories exhibited a RMS difference lower than 4mm, thus in the range of STAs. The RMS differences between the measured ground reaction force and moment and those estimated using the proposed method (9N and 10Nm) were much lower than obtained using a classical inverse dynamics approach (22N and 30Nm). From the latter results it may be inferred that the presented method allows for a significant improvement of the accuracy with which kinematic variables and relevant time derivatives, model parameters and, therefore, intersegmental moments are estimated.

Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator

In this paper, a secure chaotic scheme for communications in noisy public channel is proposed. This scheme is based on the concept of carrier encryption in addition to the typical data encryption techniques. At the transmitter end, the unified chaotic system with adaptive parameter and a hyperchaotic Rossler system with uncertain parameters are coupled, constrained and used as a new hyperchaotic system which generates waveforms that are different from those of any known chaotic oscillator. After modulating one of the outputs of the system with the encrypted data signal, the outputs of the system are encrypted using a set of pre-defined encryption rules and transmitted to the receiving end through a noisy public communications channel. At the receiving end, the received outputs are decrypted and the transmitted data are retrieved by reconstructing the constrained hyperchaotic signals using the novel discrete-time iterative decomposed uncertain constrained extended Kalman filter (IDUCEKF). The proposed state estimator, besides being used to handle the estimation problem of uncertain constrained nonlinear dynamical systems, reduces the required processing time and gives good numerical performance. Simulation results are firstly presented to illustrate the applicability of the IDUCEKF in synchronizing the states of the constrained hyperchaotic system. Then, the proposed secure communication scheme is applied to transmit images, and the quality of the transmission process is assessed. The obtained results show the effectiveness of the proposed approach.

Constrained Extended Kalman Filter for Target Tracking in Directional Sensor Networks

The target tracking problem in directional sensor networks (DSNs) is attracting increasing attention. Unlike the traditional omnidirectional sensor, a directional sensor has a special angle of view. It can offer direction information rather than just the sensing signal measurement with respect to the detected target. The existing tracking approaches in DSNs always separately consider the direction and measurement information; they hardly promise the tracking performance of minimum variance. In this paper, the field of view of directional sensor is approximated to a rectangle; as such the constrained area in which the target is bound to be is constructed. Then, the target tracking problem is formulated as a constrained estimation problem, and a constrained extended Kalman filter (CEKF) tracking algorithm integrating the direction and measurement information is presented; its structural and statistical properties are rigorously derived. It is proved that CEKF is the linear unbiased minimum variance estimator, and CEKF can yield a smaller error covariance than the unconstrained traditional extended Kalman filter using only sensor measurements. Simulation results show that the CEKF has superior tracking performance for directional wireless networks.

On the choice of importance distributions for unconstrained and constrained state estimation using particle filter

Recursive state estimation of constrained nonlinear dynamical system has attracted the attention of many researchers in recent years. For nonlinear/non-Gaussian state estimation problems, particle filters have been widely used (Arulampalam et al. [1]). As pointed out by Daum [2], particle filters require a proposal distribution and the choice of proposal distribution is the key design issue. In this paper, a novel approach for generating the proposal distribution based on a constrained Extended Kalman filter (C-EKF), Constrained Unscented Kalman filter (C-UKF) and constrained Ensemble Kalman filter (C-EnkF) has been proposed. The efficacy of the proposed state estimation algorithms using a particle filter is illustrated via a successful implementation on a simulated gas-phase reactor, involving constraints on estimated state variables and another example problem, which involves constraints on the process noise (Rao et al. [10]). We also propose a state estimation scheme for estimating state variables in an autonomous hybrid system using particle filter with Unscented Kalman filter as a proposal and unconstrained Ensemble Kalman filter (EnKF) as a proposal. The efficacy of the proposed state estimation scheme for an autonomous hybrid system is demonstrated by conducting simulation studies on a three-tank hybrid system. The simulation studies underline the crucial role played by the choice of proposal distribution in formulation of particle filters.

Model based estimation and optimal control of fed-batch fermentation processes for the production of antibiotics

This contribution addresses the application and comparison of model based estimation, optimization, and control methods for fed-batch bioprocesses. For the application of model based control, appropriate knowledge of the system’s state is required. The estimation quality of two constrained optimization based state estimation algorithms, namely the Bayesian maximum a posteriori based Constrained Extended Kalman-Filter (CEKF) and the Moving-Horizon-State-Estimation (MHE) is compared to the classical unconstrained Extended Kalman-Filter (EKF). The comparison is based on Monte Carlo simulations of a small mechanical and a high order grey-box model of a biological system. Moreover, EKF, CEKF, and MHE usually introduced separately, are described in a coherent setting starting from the same maximum a posteriori estimation problem. Finally the well-known nonlinear model predictive control (NMPC) is compared with a control strategy that uses online optimization of the complete future trajectory. Extensive Monte Carlo simulation studies and a real world application are considered.

CONSTRAINED EXTENDED KALMAN FILTER FOR NONLINEAR STATE ESTIMATION

Constraints on state variables are commonly encountered in dynamic state estimation in the form of algebraic equality and/or inequality constraints. For weakly nonlinear systems, the extended Kalman filter (EKF) has found numerous uses as a suboptimal state estimator. Unfortunately the structure of the filter does not include constraints on the states. The failure of unconstrained EKF is frequently cited as motivation for moving horizon estimation (MHE) methods for constrained state estimation. However, work on actually imposing the constraints in the existing EKF framework is scarce. This paper presents analytical solutions to the state constrained EKF (CEKF) for a class of linear constraints. It is possible to implement the CEKF efficiently with little additional computation cost and avoid expensive online optimization in MHE. The MHE is a general suboptimal strategy to impose constraints on states, noise processes and inputs, but for a class of state constraints, the proposed CEKF is sufficient. The performance of CEKF is illustrated with a simulation study of a nonlinear batch reactor.

Online monitoring of semi-continuous emulsion copolymerization: Comparing constrained extended Kalman filtering to feed-forward calorimetry

Semi-continuous emulsion copolymerization (SCEP) is one of the major processes for the production of polymer latexes. For tailor-made copolymers, SCEP is employed because better control of polymer quality is possible compared to batch operation. To efficiently produce high quality polymers, recipe optimization and good feedback process control is required. For optimization or control purposes it is necessary to measure important product qualities such as the composition of the copolymer or the average chain length of the macromolecules. Measurements of these properties for this multi-phase process are expensive and usually not available online. State estimation and reaction calorimetry are possibilities to overcome this problem. Starting from a complex model for the considered process we develop a simplified model which enables us to estimate the concentrations of the monomers in the particle phase and thus the composition of the copolymer using a Constrained Extended Kalman Filter (CEKF). We compare this to earlier approaches based on simple feed-forward calorimetry and show that it is more robust to modelling errors and to the choice of initial value.