Article

Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization

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Abstract

In this paper, we propose a learning technique to solve parameter adjustment problems of Unscented Kalman Filter (UKF) for accurate localization. The parameter adjustment process of Kalman filters is very cumbersome for mobile robot developers and needs significant engineering cost and time. The parameters of UKF consist of three kinds of parameters as follows: I) the covariance matrix of input noise, II) the covariance matrix of measurement noise, III) Hyper-parameter of UKF. From a simulation result, it is shown that UKF localization performance critically depends on these Hyper-parameters. One of discriminative training methods is adopted to obtain the optimal parameters. The learning technique needs a highly accurate instrument for evaluating the filtering performance. A coordinate ascent algorithm is used as a training algorithm to optimize these parameters included in complicated functions. The technique promises to relieve mobile robot developers of the tedious task of adjusting several parameters by hand. Simulation and experimental results are presented. We demonstrate the effectiveness of the proposed learning method for localization, and the localization result using the proposed method outperforms the result of hand-tuning and the result of covariance learning.

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... For nonlinear systems an Unscented Kalman Filter (UKF) is proposed where unscented transformation (UT) is used to estimate uncertainties (Hartikainen et al. 2011). In Sakai and Kuroda (2010) a learning technique to help developers to adjust the UKF parameters (input-and measurement-noise covariance matrices and unscented transformation parameters) for an accurate localisation is used. In this work a statistical analysis of a lineralisation approach and UT is made to estimate covariances of the line segment nonlinear transformation. ...
... The appropriate operation of EKF depends on the correct assumption of uncertainties that influence state estimation process (Borges and Aldon 2003;Sakai and Kuroda 2010). In the following some directions and the validation of the proposed method for the estimation of observation uncertainty are given. ...
... where √ Z is obtained using Cholesky factorisation and c = α 2 (n + κ) is scaling parameter. The positive constants α, β and κ in the comparison are α = 0.01, β = 2 and κ = 1 (for more details see Sakai and Kuroda 2010;Hartikainen et al. 2011). ...
Article
In this paper an extended Kalman filter (EKF) is used in the simultaneous localisation and mapping (SLAM) of a four-wheeled mobile robot in an indoor environment. The robot’s pose and environment map are estimated from incremental encoders and from laser-range-finder (LRF) sensor readings. The map of the environment consists of line segments, which are estimated from the LRF’s scans. A good state convergence of the EKF is obtained using the proposed methods for the input- and output-noise covariance matrices’ estimation. The output-noise covariance matrix, consisting of the observed-line-features’ covariances, is estimated from the LRF’s measurements using the least-squares method. The experimental results from the localisation and SLAM experiments in the indoor environment show the applicability of the proposed approach. The main paper contribution is the improvement of the SLAM algorithm convergence due to the noise covariance matrices’ estimation.
... 24 Deploying a UKF or related nonlinear Kalman filter (KF) in an embedded system 25 provides numerous advantages over traditional computational platforms in the context 26 of biological systems. The use of embedded hardware dedicated to the specific 27 application allows for tight control over task scheduling, minimizes feedback latency in 28 closed-loop applications, and avoids interruptions in communication that can occur with 29 more complex operating systems. Embedded systems can often be converted into a 30 wearable form. ...
... The sigma point weights W i can be adjusted not only to 386 include or excludex from among the sigma points but also to incorporate prior 387 knowledge of the higher-order moments of the state distribution [17,19]. Methods for 388 tuning the weights are described in [28][29][30][31][32]. We weighted the sigma points by the 389 reciprocal of their total number. ...
... We weighted the sigma points by the 389 reciprocal of their total number. Finally, the noise covariance matrices Q and R may 390 also undergo tuning, as addressed in [28,31,33]. We held Q and R constant and inflated 391 the element in Q corresponding to the uncertainty of the unknown input δ. ...
Preprint
The unscented Kalman filter (UKF) is finding increased application in biological fields. While realizing a complex UKF system in a low-power embedded platform offers many potential benefits including wearability, it also poses significant design challenges. Here we present a method for optimizing a UKF system for realization in an embedded platform. The method seeks to minimize both computation time and error in UKF state reconstruction and forecasting. As a case study, we applied the method to a model for the rat sleep-wake regulatory system in which 432 variants of the UKF over six different variables are considered. The optimization method is divided into three stages that assess computation time, state forecast error, and state reconstruction error. We apply a cost function to variants that pass all three stages to identify a variant that computes 27 times faster than the reference variant and maintains required levels of state estimation and forecasting accuracy. We draw the following insights: 1) process noise provides leeway for simplifying the model and its integration in ways that speed computation time while maintaining state forecasting accuracy, 2) the assimilation of observed data during the UKF correction step provides leeway for simplifying the UKF structure in ways that speed computation time while maintaining state reconstruction accuracy, and 3) the optimization process can be accelerated by decoupling variables that directly impact the underlying model from variables that impact the UKF structure. Author summary Powerful statistical tools can be used to estimate unmeasured information in biological systems and predict the system’s future state. Translating these tools from a desktop computer to small, wearable or implantable platforms can unlock many benefits, but presents many design challenges, because tradeoffs between computational simplicity and system accuracy must be balanced. We present an approach to translating one such widely used statistical tool, the unscented Kalman filter, from a desktop computer to a miniature device. We demonstrate the approach with a case study centering on the neural circuits governing the mammalian sleep-wake cycle. By exploiting uncertainty in the biology being estimated and the corrective influence of biological measurements, we arrive at a design that is easily computable on the miniature device and maintains a high level of accuracy. We anticipate that our approach will aid others in designing such miniaturized systems for a broad range of applications.
... In [15], a new version of the UKF has also been proposed that involves tuning of three parameters (α, β, and k) instead of one; while, there are few theoretical guidelines available as how to tune up such parameters. Although numerous heuristics including model-based optimisation, gradient-free optimisation, and learning techniques have been proposed to find near optimal values for such scaling parameters [16][17][18][19]. ...
... Due to symmetry of the sigma points, i.e. δx (i) = − δx (i + n) , all oddordered terms in (17) sum to zero (see (18)) Then ...
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This work, based on the standard unscented Kalman filter (UKF), proposes a modified UKF for highly non-linear stochastic systems, assuming that the associated probability distributions are normal. In the standard UKF with 2n + 1 sample points, the estimated mean and covariance match the true mean and covariance up to the third order, besides, there exists a scaling parameter that plays a crucial role in minimising the fourth-order errors. The proposed method consists of a computationally efficient formulation of the unscented transform that incorporates N - 1, N ≥ 2, constant parameters to scale 2n(N - 1) + 1 sample points such that the 2Nth order errors are minimised. The scaling parameters are obtained by solving a set of algebraic equations. Through rigorous analytical processes and numerical simulations, it is demonstrated that the new filter provides consistent estimates and the estimation error of the modified UKF is smaller than that of the standard UKF. With the help of a well-studied case, univariate non-stationary growth model, the authors evaluate the estimation performance of the new technique using 4n + 1 sample points over 100 independent runs.
... Several studies have investigated the selection of the proper scaling parameter of the UKF, taking both an online approach [25][26][27][28] and an offline approach [23,24]. However, the offline approach for the selection of the scaling parameter causes performance degradation when it is applied to the dynamic system. ...
Article
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In this paper, an adaptation method for adjusting the scaling parameters of an unscented Kalman filter (UKF) is proposed to improve the estimation performance of the filter in dynamic conditions. The proposed adaptation method is based on a sequential algorithm that selects the scaling parameter using the user-defined distribution of discrete sets to more effectively deal with the changing measurement distribution over time and avoid the additional process for training a filter model. The adaptation method employs regularized optimal transport (ROT), which compensates for the error of the predicted measurement with the current measurement values to select the proper scaling parameter. In addition, the Sinkhorn–Knopp algorithm is used to minimize the cost function of ROT due to its fast convergence rate, and the convergence of the proposed ROT-based adaptive adjustment method is also analyzed. According to the analysis results of Monte Carlo simulations, it is confirmed that the proposed algorithm shows better performance than the conventional algorithms in terms of the scaling parameter selection in the UKF.
... Here, we use the likelihood function as the criterion, which can be approximated by Gaussian distribution, expressed asp τ (z k |z k−1 ) N {ẑ k|k−1 , P z,k|k−1 } (36)where τ is designed to achieve the maximum likelihood [17]. ...
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To perform surveillance of a hypersonic cruise vehicle (HCV), space-based infrared system is a reliable and feasible means, which has been putting on the schedule for providing positioning and tracking information of the high-altitude unmanned vehicles. In this paper, a space-based HCV tracking method based on infrared satellite constellation is proposed. The method contains three main parts: constellation coverage analysis, a bearing-only positioning algorithm, and a tracking algorithm. For target tracking, an adaptive scaling unscented Kalman filter (ASUKF) is applied for high estimation performance. Simulation results are presented to show the effectiveness of the method.
... Next, parameter α i is adjusted in the PUKF in order to ensure fulfilment of Eq. (14) for the same data set. Such an off-line approach has also been proposed in [26] in a different context. Algorithm 1 states the proposed PUKF. ...
... Next, parameter α is adjusted in the PUKF in order to ensure fulfilment of Eq. (14) for the same data set. Such an offline approach has been also proposed in [33] in a different context. ...
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Accurate state estimation of large-scale lithium-ion battery packs is necessary for the advanced control of batteries, which could potentially increase their lifetime through e.g. reconfiguration. To tackle this problem, an enhanced reduced-order electrochemical model is used here. This model allows considering a wider operating range and thermal coupling between cells, the latter turning out to be significant. The resulting nonlinear model is exploited for state estimation through unscented Kalman filters (UKF). A sensor network composed of one sensor node per battery cell is deployed. Each sensor node is equipped with a local UKF, which uses available local measurements together with additional information coming from neighboring sensor nodes. Such state estimation scheme gives rise to a partition-based unscented Kalman filter (PUKF). The method is validated on data from a detailed simulator for a battery pack comprised of six cells, with reconfiguration capabilities. The results show that the distributed approach outperforms the centralized one in terms of computation time at the expense of a very low increase of mean-square estimation error.
... In this case, the scaling parameter is a function of the state-space dimension only. In [Sakai and Kuroda, 2010], a technique for the off-line scaling parameter selection using a training procedure was proposed. Contrary to the previous procedure, adaptive techniques for the parameter selection has been of interest in past few years [Duník et al., 2010, Turner and Rasmussen, 2012. ...
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... Regarding the choice of the tuning parameter, a couple of results have already been proposed in the literature. The authors in [64] provide an off-line way of computing it by maximizing the likelihood function with a training set of data. In [22], an on-line method of computing the tuning parameter by means of maximizing a Gaussian approximation of the likelihood function is proposed. ...
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In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and cross-covariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-of-the-art methods.
... Regarding the choice of the tuning parameter, a couple of results have already been proposed in the literature. The authors in [60] provide an off-line way of computing it by maximizing the likelihood function with a training set of data. In [19], an on-line method of computing the tuning parameter by means of maximizing a Gaussian approximation of the likelihood function is proposed. ...
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Abstract-This paper introduces a kinematic model of a deep-sea mining vehicle in presence of sliding parameters. The model describes both the noises features of sliding parameters and the deep-sea condition features. To handle sliding parameters noises, a recursive algorithm to minimize difference between the filter-computed and the actual innovation covariance is adopted, which is a novel integrated navigation method based on unscented Kalman filters (UKF). Taking into account the influence of measurement data delay, UKF fuses the localization information of long base line (LBL) sonar localization system and dead-reckoning (DR) to perform the state estimation. Simulation results show that the adaptive UKF has better localization estimation than a normal UKF for a deep-sea tracked vehicle (DTV).
Conference Paper
In this paper, we propose an efficient solution to 6 degrees of freedom (6DOF) localization using unscented Kalman filter for planetary rovers. The solution is a technique augmented the unscented Kalman filter for accurate 6DOF localization, named augmented unscented Kalman filter (AUKF). The AUKF is designed to deal with problems which occur on other planets: wheel slip, visual odometry error, and gyro drift. To solve the problems, the AUKF estimates the slippage ratio in an augmented state vector, the accuracy of the visual odometry with the number of inliers among feature points, and sensor usefulness with gyrodometry model. Experimental results of rover runs over rough terrain are presented, the effectiveness of the AUKF and its each component is shown.
Conference Paper
In this paper, we propose stereo vision based visual odometry with an effective feature sampling technique for untextured outdoor environment. In order to extract feature points in untextured condition, we divide an image into some sections and affect suitable processes for each section. This approach can also prevent concentration of feature points, and the influence with a moving object can be reduced. Robust motion estimation is attained using the framework of 3-point algorithm and RANdom SAmple Consensus (RANSAC). Moreover, the accumulation error is reduced by keyframe adjustment. We present and evaluate experimental results for our system in outdoor environment. Proposed visual odometry system can localize the robot's position within 4% error in untextured outdoor environment.
Conference Paper
Kalman filters are a workhorse of robotics and are routinely used in state-estimation problems. However, their performance critically depends on a large number of modeling parameters which can be very difficult to obtain, and are often set via significant manual tweaking and at a great cost of engineering time. In this paper, we propose a method for automatically learning the noise parameters of a Kalman filter. We also demonstrate on a commercial wheeled rover that our Kalman filter's learned noise covariance parameters—obtained quickly and fully automatically—significantly outperform an earlier, carefully and laboriously hand-designed one.
Article
In this paper, we propose an efficient solution to 6-d.o.f. localization using an unscented Kalman filter (UKF) for planetary rovers. The solution is a technique using a modified UKF for accurate planetary rover localization. The technique is able to cope with problems that occur frequently on other planetary surfaces: wheel slip, pulsive noise of visual odometry (VO) and gyro drift. To solve these problems, the technique estimates a slippage ratio with the UKF, the accuracy of the VO with the number of inliers of feature points and sensor usefulness with a gyrodometry model. Experimental results of rover runs over rough terrain are presented and discussed; the effectiveness of proposed method and its components is shown.
Article
In this paper we propose a novel method for nonlinear, non-Gaussian, on-line estimation. The algorithm consists of a particle filter that uses an unscented Kalman filter (UKF) to generate the importance proposal distribution. The UKF allows the particle filter to incorporate the latest observations into a prior updating routine. In addition, the UKF generates proposal distributions that match the true posterior more closely and also has the capability of generating heavier tailed distributions than the well known extended Kalman filter. As a result, the convergence results predict that the new filter should outperform standard particle filters, extended Kalman filters and unscented Kalman filters. A few experiments confirm this prediction.
Advances in derivative-free state estimation for nonlinear systems
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