DOCSLIB.ORG

Constrained State Estimation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constrained State Estimation 1 Constrained State Estimation - A Review Nesrine Amor, Ghulam Rasool and Nidhal C. Bouaynaya . Abstract—Increasingly for many real-world applications in The parametric techniques are based on the extended Kalman signal processing, nonlinearity, non-Gaussianity, and additional filter, unscented Kalman filter, ensemble Kalman filter and constraints are considered while handling dynamic state esti- moving horizon estimation. The non-parametric techniques are mation problems. This paper provides a critical review of the state of the art in constrained Bayesian state estimation for based on Sequential Monte Carlo as known particle filtering. linear and nonlinear state-space systems. Specifically, we provide This paper provides a critical review of constrained a review of unconstrained estimation using Kalman filters for Bayesian state estimation methods, i.e, Kalman filter, extended the linear system, and their extensions for nonlinear state-space Kalman filter, unscented Kalman filter, ensemble Kalman filter, system including extended Kalman filters, unscented Kalman moving horizon estimation and particles filters. filters and ensemble Kalman filters. In addition, we present the particle filters for non linear state space systems and discuss The paper is organized as follows: Section II presents recent advances. Next, we review constrained state estimation the problem statement. Section III reviews the unconstrained using all these filters where we highlighted the advantages and Bayesian state estimation framework. Section IV presents disadvantages of the different recent approaches. the literature available in constrained state estimation using Kalman filter, extended Kalman filter, unscented Kalman fil- I. INTRODUCTION ter, ensemble Kalman filter and moving horizon estimation. Due to physical laws, technological limitations, kinematic Section V introduces a large critical review of the constrained constraints, geometric considerations of many systems, such as particle filtering. Finally, Section VI summarizes the paper. material balance, bounds on actuators and plants, target speed constraints and road networks, the states of many dynamical II. PROBLEM STATEMENT systems are confined within constrained regions. Mathemati- cally, the constraints are given by a set of linear or nonlinear We consider a general state-space model defined by a state equalities or inequalities. In general, these constraints cannot transition and measurement models in a discrete form given be incorporated in the state-space model without a major by increase in the complexity of the model. Taking constraints x = f (x )+ w , (1) into account leads to physically meaningful and more accurate t+1 t t t h estimates. For instance, the exploitation of a known road yt = t(xt)+ vt, (2) network has been proven effective in tracking of ground where x Rnx and y Rny are, respectively, the hidden vehicles [1]. Similarly, in a maritime scenario, the knowledge t t state vector∈ with transition∈ probability density function (pdf) on shipping lanes and sea/land distinction can improve the p(x x − ), and the observation vector with conditional pdf tracking and detection performance [2]. t t 1 y |x at time instant . and are possibly nonlinear The general state space hidden Markov models provide p( t t) t ft ht state| transition and observation functions, respectively. an extremely flexible framework for modeling discrete-time nx and are state and output dimension. u and v are zero- dynamical systems. Dynamic systems are modeled using state ny t t mean state and observation white noise sequences with known arXiv:1807.03463v2 [eess.SP] 24 Dec 2018 evaluation and observation relations. The former captures the probability density functions (pdfs), respectively, u and evolution of the state with time and later provides noisy p( ) v . Both noise sequences are supposed to be uncorrelated measurement of a probably nonlinear function of the state. The p( ) with each other and the initial condition of the state x given great descriptive power of these models comes at the expense 0 by x . of intractability: it is impossible to obtain analytic solutions to p( 0) the inference problems with the exception of a small number of In the Bayesian context, an optimal state estimation of particulary simple cases. In the Bayesian framework, inference the state vector sequence xt given the available observations of the hidden state given all available observations at that history y1:t = [y1, ..., yt] up to time t, relies upon the posterior density p(x y ). Using Bayes rule and Chapman- time relies upon the posterior density function (pdf). For the t| 1:t linear and Gaussian estimation problems, optimal solution Kolmogorov equation, the posterior distribution can be com- can be obtained by the Kalman filter. For nonlinear and puted recursively using the following two-step formulas: non-Gaussian state-space models, there are two fundamental techniques have been emerging: parametric and nonparametric. N. Amor is with the National Superior School of Engineers of Tunis (EN- Prediction step SIT), University of Tunis, Tunis, Tunisia, e-mail: ([email protected] G. Rasool and N. C. Bouaynaya are with Department of Electrical and Computer Engineering, Rowan University, New Jersey, USA, e-mail: (ra- p(xt y1:t−1)= p(xt−1 y1:t−1) p(xt xt−1) dxt−1, (3) [email protected], [email protected]) | Z | | 2 Update step B. Extended Kalman Filter (EKF) p(yt xt) p(xt y1:t−1) p(xt y1:t)= | | , (4) For the nonlinear model, the extended Kalman filter (EKF) | p(y x ) p(x y − ) dx t t t 1:t 1 t was formulated by a linearization procedure of nonlinear R | | In the nonlinear case, unfortunately, these equations are functions ft(xt) and ht(xt), using the Taylor series expansion. only a conceptual solution because the defined integrals are The error covariance is propagated in time using the linearized generally intractable. However, closed-form solutions in some functions, whereas the state estimate are propagated using special cases may exist, e.g., the Kalman filter for linear nonlinear functions. Specifically, eqs. (5a) and (6b) will use dynamics system, linear observation models and Gaussian nonlinear functions f(x) and hx, while all other relations densities for the noise sequences. In other cases, we resort utilize linearized forms of these functions (first-order Taylor + to several approximations [3]. series expansion) evaluated at the estimated state xˆt [5], [4]. However, computational complexity for calculation of Hessian matrices may prohibit its use [4]. In addition, the linearization III. UNCONSTRAINED STATE ESTIMATION of nonlinear system and measurement models may induce Generally, for tracking problems with linear and Gaussian errors in the estimation of the state, and in the worst-case, models, an optimal solution can be obtained using the Kalman the filter may diverge especially for highly nonlinear function Filter. In fact, many real-world applications such as target [6]. tracking, electric power systems, navigation and biomedical engineering, these linear and Gaussian assumptions do not hold. Furthermore, approximations are required for these nonlinear and non-Gaussian tracking problems. Thus, many approaches were introduced to solve this problem such as C. Unscented Kalman Filter (UKF) Extended Kalman Filter (EKF), Moving Horizon estimation (MHE), Ensemble Kalman Filter (EnKF), Unscented Kalman Filter (UKF) and Particle Filters (PF). In this section, we The Unscented Kalman Filter (UKF) was proposed as a introduce these filters for linear and nonlinear systems when method based on a mathematical approach called the ‘Un- there are no constraints on the system. scented Transform’ (UT). The UKF approximates the prob- ability distribution based on UT which uses a deterministic set of samples, called sigma points, to propagate the mean A. Kalman Filter and covariance. The calculated sigma points are propagated When the state transition and observation models are linear through the nonlinear function. The statistics of transformed with additive Gaussian noise, the Kalman Filter (KF) is the points can be calculated to form an estimate of the nonlinearly minimum mean square error (MMSE) estimator for linear and transformed mean and covariance [7], [6], [8]. The sigma points X Rnx , j = 0, 1,..., 2n are chosen deterministi- Gaussian dynamic systems. The KF estimates the unknown j,t ∈ state by propagating the mean and the covariance at each time cally as opposite to the particle filters. Consider xt with mean step [4], [5]. The Kalman filter defined by the two steps: the xˆt and covariance Pt. Let the matrix of all the sigma points be := [ ,..., ]. prediction step and the update step. Xt X1,t X2n,t + 1/2 + 1/2 =x ˆ 1 × + √n + λ 0 × (P ) (P ) , Prediction step Xt t 1 (2n+1) n 1 t − t h (7)i 1/2 − + where (.) is the Cholesky square root, n is the dimension xˆ = F − xˆ (5a) t t 1 t−1 of the system and λ > n. Further, we define weights for − + T − Pt = Ft−1Pt−1Ft−1 + Qt (5b) all sigma points γ = [γ0,γ1,...,γ2n] with the condition − T 2n using the relation Pxy = Pt Ht (5c) j=0 γj =1 − T P Py = HtPt Ht + Rt (5d) − yˆ = H xˆ (5e) t t t λ 1 γ = , γ = , j =1, 2,..., 2n. (8) 0 n + λ j 2(n + λ) Update step where Eqs. (7) and (8) represent the unscented transforma- −1 tion ( ). In the following, we summarize the UKF algorithm. Kt = PxyPy (6a) UT + − xˆt =x ˆt + Kt(yt yˆt) (6b) − − P + = P P P 1P T (6c) t t − xy y xy 3 Update step where the last term is the arrival cost and for t = h, Πt−h = , i.e., the initial covariance of the state estimate at time − + + Π0 t = γ, t = (ˆxt−1, Pt−1,n,λ), (9a) 0. The purpose for development of the MHE was to formulate X UT − + = f( ), j =0, 1,..., 2n, (9b) a mathematical optimization problem where constraints can be Xj,t Xj,t 2n naturally incorporated into the estimation framework [14].
Load more

Recommended publications
  • A Fusion Localization Method Based on a Robust Extended Kalman Filter and Track-Quality for Wireless Sensor Networks
    sensors Article A Fusion Localization Method based on a Robust Extended Kalman Filter and Track-Quality for Wireless Sensor Networks Yan Wang *, Huihui Jie and Long Cheng * Department of Computer and Communication Engineering, Northeastern University, Qinhuangdao 066004, Hebei Province, China * Correspondence: [email protected] (Y.W.); [email protected] (L.C.); Tel.: +86-188-9491-9683 (Y.W.) Received: 11 July 2019; Accepted: 19 August 2019; Published: 21 August 2019 Abstract: As one of the most essential technologies, wireless sensor networks (WSNs) integrate sensor technology, embedded computing technology, and modern network and communication technology, which have become research hotspots in recent years. The localization technique, one of the key techniques for WSN research, determines the application prospects of WSNs to a great extent. The positioning errors of wireless sensor networks are mainly caused by the non-line of sight (NLOS) propagation, occurring in complicated channel environments such as the indoor conditions. Traditional techniques such as the extended Kalman filter (EKF) perform unsatisfactorily in the case of NLOS. In contrast, the robust extended Kalman filter (REKF) acquires accurate position estimates by applying the robust techniques to the EKF in NLOS environments while losing efficiency in LOS. Therefore it is very hard to achieve high performance with a single filter in both LOS and NLOS environments. In this paper, a localization method using a robust extended Kalman filter and track-quality-based (REKF-TQ) fusion algorithm is proposed to mitigate the effect of NLOS errors. Firstly, the EKF and REKF are used in parallel to obtain the location estimates of mobile nodes.
    [Show full text]
  • Announcements Hidden Markov Models Filtering in HMM Discrete
    Announcements Final project: 45% of the grade, 10% presentation, 35% write-up CS 287: Advanced Robotics Presentations: in lecture Dec 1 and 3 Fall 2009 If you have constraints, inform me by email by Wednesday night, we will assign the others at random on Thursday Lecture 23: HMMs: Kalman filters, particle filters PS2: due Friday 23:59pm. Tuesday 4-5pm 540 Cory: Hadas Kress-Gazit (Cornell) Pieter Abbeel High-level tasks to correct low-level robot control UC Berkeley EECS In this talk I will present a formal approach to creating robot controllers that ensure the robot satisfies a given high level task. I will describe a framework in which a user specifies a complex and reactive task in Structured English. This task is then automatically translated, using temporal logic and tools from the formal methods world, into a hybrid controller. This controller is guaranteed to control the robot such that its motion and actions satisfy the intended task, in a variety of different environments. Hidden Markov Models Filtering in HMM Joint distribution is assumed to be of the form: Init P(x 1) [e.g., uniformly] P(X 1 = x 1) P(Z 1 = z 1 | X 1 = x 1 ) P(X 2 = x 2 | X 1 = x 1) P(Z 2 = z 2 | X 2 = x 2 ) … Observation update for time 0: P(X T = xT | X T-1 = x T-1) P(Z T = zT | X T = xT) For t = 1, 2, … Time update X1 X2 X3 X4 XT Observation update Z1 Z2 Z3 Z4 ZT For discrete state / observation spaces: simply replace integral by summation Discrete-time Kalman Filter Kalman Filter Algorithm Estimates the state x of a discrete-time controlled process
    [Show full text]
  • An Ensemble Kalman Filter for Feature-Based SLAM with Unknown Associations
    An Ensemble Kalman Filter for Feature-Based SLAM with Unknown Associations Fabian Sigges, Christoph Rauterberg and Marcus Baum Uwe D. Hanebeck Institute of Computer Science Intelligent Sensor-Actuator-Systems Laboratory (ISAS) University of Goettingen, Germany Institute for Anthropomatics and Robotics Email: ffabian.sigges, [email protected] Karlsruhe Institute of Technology (KIT), Germany [email protected] Email: [email protected] Abstract—In this paper, we present a new approach for solving step of the filter, the posterior is then obtained using the the SLAM problem using the Ensemble Kalman Filter (EnKF). ensemble mean and an estimate of the sample covariance from In contrast to other Kalman filter based approaches, the EnKF the ensemble members. uses a small set of ensemble members to represent the state, thus circumventing the computation of the large covariance matrix Although being established in the geoscientific community, traditionally used with Kalman filters, making this approach a the EnKF has only recently received attention from the track- viable application in high-dimensional state spaces. Our approach ing community [3], [4]. In the case of SLAM, the EnKF is adapts techniques from the geoscientific community such as mostly just applied to specific parts of the whole problem. localization to the SLAM problem domain as well as using For example, in [5] the EnKF is used to generate a suitable the Optimal Subpattern Assignment (OSPA) metric for data association. We then compare the results of our algorithm with an proposal distribution for FastSLAM or in [6] the EnKF realizes extended Kalman filter (EKF) and FastSLAM, showing that our the merging operation on two approximate transformations.
    [Show full text]
  • MASTER's THESIS Performance Comparison of Extended and Unscented Kalman Filter Implementation in INS-GPS Integration
    2009:095 MASTER'S THESIS Performance comparison of Extended and Unscented Kalman Filter implementation in INS-GPS integration Joshy Madathiparambil Jose Luleå University of Technology Master Thesis, Continuation Courses Space Science and Technology Department of Space Science, Kiruna 2009:095 - ISSN: 1653-0187 - ISRN: LTU-PB-EX--09/095--SE Lulea University of Technology Performance comparison of Extended and Unscented Kalman Filter implementation in INS-GPS integration Erasmus Mundus Programme SpaceMaster Czech Technical University in Prague Prague,2009 Author: Joshy Madathiparambil Jose Acknowledgements I would like to thank my LTU supervisor Andreas Johanson for his helps and comments. I would like also to thank my supervisors Martin Hromcik at CTU and Martin Orejas, Honeywell Brno for their advice and timely help for completion of this thesis. i Abstract The objective of this thesis is to implement an unscented kalman filter for integrating INS with GPS and to analyze and compare the results with the extended kalman filter approach. In a loosely coupled integrated INS/GPS system, inertial measurements from an IMU (angular velocities and acceler- ations in body frame) are integrated by the INS to obtain a complete navi- gation solution and the GPS measurements are used to correct for the errors and avoid the inherent drift of the pure INS system. The standard approach is to use an extended kalman filter in complementary form to model the er- rors of the INS states and use the GPS measurements to estimate corrections for these errors which are then feedback to the INS. Although the unscented kalman filter is more computational intensive, it is supposed to outperform the extended kalman filter and be more robust to initial errors.
    [Show full text]
  • A More Robust Unscented Transform
    A more robust unscented transform James R. Van Zandta aMITRE Corporation, MS-M210, 202 Burlington Road, Bedford MA 01730, USA ABSTRACT The unscented transformation is extended to use extra test points beyond the minimum necessary to determine the second moments of a multivariate normal distribution. The additional test points can improve the estimated mean and variance of the transformed distribution when the transforming function or its derivatives have discontinuities. Keywords: unscented transform, extended Kalman Filter, unscented filter, discontinuous functions, uncertainty distribution 1. INTRODUCTION The representation and estimation of uncertainty is an important part of any tracking application. For linear systems, the Kalman filter1 (KF) maintains a consistent estimate of the first two moments of the state distribution:the mean and the variance. Twice each update, it must estimate the statistics of a random variable that has undergone a transformation—once to predict the future state of the system from the current state (using the process model) and once to predict the observation from the predicted state (using the observation model). The Extended Kalman Filter (EKF)2 allows the Kalman filter machinery to be applied to nonlinear systems. In the EKF, the state distribution is approximated by a Gaussian random variable, and is propagated analytically through the first-order linearization (a Taylor series expansion truncated at the first order) of the nonlinear function. This can introduce substantial errors in the estimates of the mean and covariance of the transformed distribution, which can lead to sub-optimal performance and even divergence of the filter. Julier and Uhlmann have described the unscented transformation (UT) which approximates a probability distri- bution using a small number of carefully chosen test points.3 These test points are propagated through the true nonlinear system, and allow estimation of the posterior mean and covariance accurate to the third order for any nonlinearity.
    [Show full text]
  • Monte Carlo Methods and Skewed Kalman Filters for Orbit Estimation
    Monte Carlo Methods and Skewed Kalman Filters for Orbit Estimation Louis Tonc,∗ David Geller,y Geordie Richardsz April 20, 2018 Abstract in state space, far from the Gaussian class, and results in a wide discrepancy between the uncertainty of the filter Under reasonable orbital conditions, the Extended algorithm and the measurement noise. Kalman Filter (EKF) and Unscented Kalman Filter (UKF) both diverge after several updates from optical measure- POSITION (LVLH) 400 ments. We identify the source of divergence by studying Downrange (filter) a simplified model, and correct this problem by imple- 300 Downrange (monte carlo) menting a Monte Carlo based particle filter. However, the 200 computational cost of the particle filter is high, and future 100 work will implement a Skewed Unscented Kalman Filter 0 as a substitute for the particle filter. We present evidence that this skewed unscented Kalman filter will avoid diver- Position (km) -100 gence at a reduced cost compared to the particle filter, and -200 discuss Monte Carlo methods for validating this claim. -300 -400 0 1 2 3 4 5 6 7 8 9 1 Introduction Time (hours) The increasing abundance of orbital debris in Geostation- Figure 1: EKF Divergence in Downrange Position ary Orbit (GEO) represents a significant challenge for [LEO Orbit & Position Measurement at t = 3.75 hr] blue curve = filter variance, red curve = true variance the future of space travel and satellite operation. Avoid- ing collision events with high probability in real time will require orbit estimation algorithms that can utilize In particular, when tracking a debris object in GEO, sparse observations while maintaining computational ex- if optical measurements from a ground or space based pediency [2].
    [Show full text]
  • Extended Particle-Aided Unscented Kalman Filter Based on Self-Driving Car Localization
    applied sciences Review Extended Particle-Aided Unscented Kalman Filter Based on Self-Driving Car Localization Ming Lin and Byeongwoo Kim * Department of Electrical Engineering, University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan 44610, Korea; fl[email protected] * Correspondence: [email protected] Received: 8 June 2020; Accepted: 20 July 2020; Published: 22 July 2020 Abstract: The location of the vehicle is a basic parameter for self-driving cars. The key problem of localization is the noise of the sensors. In previous research, we proposed a particle-aided unscented Kalman filter (PAUKF) to handle the localization problem in non-Gaussian noise environments. However, the previous basic PAUKF only considers the infrastructures in two dimensions (2D). This previous PAUKF 2D limitation rendered it inoperable in the real world, which is full of three-dimensional (3D) features. In this paper, we have extended the previous basic PAUKF’s particle weighting process based on the multivariable normal distribution for handling 3D features. The extended PAUKF also raises the feasibility of fusing multisource perception data into the PAUKF framework. The simulation results show that the extended PAUKF has better real-world applicability than the previous basic PAUKF. Keywords: particle-aided unscented Kalman filter; non-Gaussian; sensor fusion; localization; unscented Kalman filter 1. Introduction In recent years, the self-driving car has become the hottest topic in the automotive industry [1–4]. To make a self-driving car drive conveniently and safely, accurate identification of the vehicle’s position is one of the most important parameters. Only if the self-driving car knows where it is can it execute the next decision.
    [Show full text]
  • Adaptive Filter Based on Monte Carlo Method to Improve the Target Tracking in Radar Systems
    Adaptive lter based on Monte Carlo method to improve the target tracking in radar systems Khaireddine Zarai ( [email protected] ) Universite de Tunis El Manar Cherif Adnane Universite de Tunis El Manar Research Keywords: Radar, Monte Carlo, Extended KALMAN Filter, Tracking, PF, Random target Posted Date: February 28th, 2020 DOI: https://doi.org/10.21203/rs.2.24844/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Adaptive filter based on Monte Carlo method to improve the target tracking in radar systems Zarai Khaireddine1, Cherif Adnane2 1,2Analysis and signal processing of electrical and energetic systems. Faculty of sciences of Tunis .Tunisia [email protected] , [email protected], [email protected] Abstract The state estimation and tracking of random target is an attractive research problem in radar system. The information received in the radar receiver was reflected by the target, that it is received with many white and Gaussian noise due to the characteristics of the transmission channel and the radar environment. After detection and location scenarios, the radar system must track the target in real time. We aim to improve the state estimation process for too random target at the given instant in order to converge to the true target state and smooth their true path for a long time, it simplifies the process of real-time tracking. In this framework, we propose a new approach based on the numerical methods presented by MONTE CARLO (MC) counterpart the method conventionally used named Extended KALMAN Filter (EKF), we showed that the first are more successful.
    [Show full text]
  • The Iterated Extended Kalman Particle Filter
    The Iterated Extended Kalman Particle Filter Li Liang-qun, Ji Hong-bing,Luo Jun-hui School of Electronic Engineering, Xidian University ,Xi’an 710071, China Email: [email protected] Abstract— Particle filtering shows great promise in addressing Simulation results are given in Section 5.Conclusions are a wide variety of non-linear and /or non-Gaussian problem. A presented in Section 6. crucial issue in particle filtering is the selection of the importance proposal distribution. In this paper, the iterated extended 2. PARTICLE FILTER kalman filter (IEKF) is used to generate the proposal distribution. The proposal distribution integrates the latest 2.1 Modeling Assumption observation into system state transition density, so it can match Consider the nonlinear discrete time dynamic system: the posteriori density well. The simulation results show that the new particle filter superiors to the standard particle filter and xkk= fx( k−1,vk−1) (1) the other filters such as the unscented particle filter (UPF), the extended kalman particle filter (PF -EKF), the EKF. zhkk= ( xk,ek) (2) where f :ℜnnxv×ℜ →ℜnxand h :ℜ×nnxeℜ→ℜnz represent the 1. INTRODUCTION k k system evolution function and the measurement model Nonlinear filtering problems arise in many fields n n function. x is the system state at time , z is including statistical signal processing, economics, statistics, xk ∈ℜ k zk ∈ℜ biostatistics, and engineering such as communications, radar n m the measurement vector at time k . vk −1 ∈ Rand eRk ∈ tracking, sonar ranging, target tracking, and satellite represent the process noise and the measurement noise navigation [1-7].
    [Show full text]
  • Non-Linear State Error Based Extended Kalman Filters with Applications to Navigation Axel Barrau
    Non-linear state error based extended Kalman filters with applications to navigation Axel Barrau To cite this version: Axel Barrau. Non-linear state error based extended Kalman filters with applications to navigation. Automatic. Mines Paristech, 2015. English. tel-01247723 HAL Id: tel-01247723 https://hal.archives-ouvertes.fr/tel-01247723 Submitted on 22 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INSTITUT DES SCIENCES ET TECHNOLOGIES École doctorale nO84: Sciences et technologies de l’information et de la communication Doctorat européen ParisTech THÈSE pour obtenir le grade de docteur délivré par l’École nationale supérieure des mines de Paris Spécialité “Informatique temps-réel, robotique et automatique” présentée et soutenue publiquement par Axel BARRAU le 15 septembre 2015 Non-linear state error based extended Kalman filters with applications to navigation Directeur de thèse: Silvère BONNABEL Co-encadrement de la thèse: Xavier BISSUEL Jury Brigitte d’ANDREA-NOVEL, Professeur, MINES ParisTech Examinateur Xavier BISSUEL, Expert en navigation inertielle, SAGEM Examinateur Silvère BONNABEL, Professeur, MINES Paristech Examinateur Jay FARRELL, Professeur, UC Riverside Examinateur Pascal MORIN, Professeur, Université Pierre et Marie Curie Rapporteur Christophe PRIEUR, Directeur de recherche, Gipsa-lab, CNRS Rapporteur T Pierre ROUCHON, Professeur, MINES Paristech Président du jury H MINES ParisTech Centre de Robotique È 60 bd Saint-Michel, 75006 Paris, France S E 1 Contents 1 Introduction 5 1.1 Estimating the state of a dynamical system .
    [Show full text]
  • Kalman Filtering and Friends: Inference in Time Series Models
    Kalman filtering and friends: Inference in time series models Herke van Hoof slides mostly by Michael Rubinstein Problem overview • Goal – Estimate most probable state at time k using measurement up to time k’ k’<k: prediction k’=k: filtering k’>k: smoothing • Input – (Noisy) Sensor measurements – Known or learned system model (see last lecture) • Many problems require estimation of the state of systems that change over time using noisy measurements on the system Applications • Ballistics • Robotics – Robot localization • Tracking hands/cars/… • Econometrics – Stock prediction • Navigation • Many more… Example: noisy localization Position at t=0 https://img.clipartfest.com Measurement at t=1 t=2 t=3 t=4 t=5 t=6 Example: noisy localization Position at t=0 https://img.clipartfest.com Measurement at t=1 t=2 Smoothing: where was I in the past t=3 t=4 t=5 t=6 Example: noisy localization Position at t=0 https://img.clipartfest.com Measurement at t=1 t=2 Smoothing: where was I in the past t=3 t=4 t=5 Filtering: where am I now t=6 Example: noisy localization Position at t=0 https://img.clipartfest.com Measurement at t=1 t=2 Smoothing: where was I in the past t=3 t=4 t=5 Filtering: where am I now t=6 Prediction: where will I be in the future Today’s lecture • Fundamentals • Formalizing time series models • Recursive filtering • Two cases with optimal solutions • Linear Gaussian models • Discrete systems • Suboptimal solutions Stochastic Processes • Stochastic process – Collection of random variables indexed by some set – Ie.
    [Show full text]
  • 1D Kalman Filtering) to Solve This…
    Temporal inference and SLAM 16-385 Computer Vision http://www.cs.cmu.edu/~16385/ Spring 2018, Lecture 25 Course announcements • Homework 7 will be posted and will be due on Sunday 6th. - You can use all of your remaining late days for it. • RI Seminar this week: Vladlen Koltun, “Learning to drive”, Friday 3:30-4:30pm. - Very exciting speaker, you should all attend. - Make sure to go early as the room will be packed. - Do you want me to move my office hours so that you can make it to the talk? Overview of today’s lecture • Temporal state models. • Temporal inference. • Kalman filtering. • Extended Kalman filtering. • Mono SLAM. Slide credits Most of these slides were adapted from: • Kris Kitani (16-385, Spring 2017). Temporal state models Represent the ‘world’ as a set of random variables location on the ground plane position in the 3D world position and velocity position, velocity and location of landmarks Object tracking (localization) e.g., location on the ground plane Object location and world landmarks (localization and mapping) e.g., position and velocity of robot and location of landmarks The state of the world changes over time The state of the world changes over time So we use a sequence of random variables: The state of the world changes over time So we use a sequence of random variables: The state of the world is usually uncertain so we think in terms of a distribution How big is the space of this distribution? If the state space is the location on the ground plane is the probability over all possible trajectories through a
    [Show full text]