An Introduction to Kalman Filter

In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

[1]  Frank Biocca,et al.  A Survey of Position Trackers , 1992, Presence: Teleoperators & Virtual Environments.

[2]  Ronald Azuma,et al.  Improving static and dynamic registration in an optical see-through HMD , 1994, SIGGRAPH.

[3]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[4]  Terrence P. McGarty.  Stochastic systems and state estimation , 1974 .

[5]  Guanrong Chen,et al.  Introduction to random signals and applied kalman filtering (second edition), Robert Grover Brown and Patrick Y. C. Hwang, John Wiley, New York, 1992, 512 p.p., ISBN 0–47152–573–1, $62.95 , 1992 .

[6]  Henry Fuchs,et al.  A real-time optical 3D tracker for head-mounted display systems , 1990, I3D '90.

[7]  Joost van Lawick van Pabst,et al.  Multisensor data fusion of points, line segments, and surface segments in 3D space , 1993, Other Conferences.

[8]  Alex Pentland,et al.  Synchronization in Virtual Realities , 1992, Presence: Teleoperators & Virtual Environments.

[9]  Woltring Hj,et al.  New possibilities for human motion studies by real-time light spot position measurement. , 1974 .

[10]  P PentlandAlex,et al.  Recursive Estimation of Motion, Structure, and Focal Length , 1995 .

[11]  Ronald Azuma,et al.  Space Resection by Collinearity: Mathematics Behind the Optical Ceiling Head-Tracker , 1991 .

[12]  Robert K. Rebo,et al.  A Helmet-Mounted Virtual Environment Display System , 1989, Defense, Security, and Sensing.

[13]  Douglas G. Kelly,et al.  Introduction to Probability , 1993 .

[14]  Pierrick Grandjean,et al.  3-D modeling of indoor scenes by fusion of noisy range and stereo data , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[15]  Rama Chellappa,et al.  Estimation of Object Motion Parameters from Noisy Images , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Jack B. Kuipers.  SPASYN-an electromagnetic relative position and orientation tracking system , 1980, IEEE Transactions on Instrumentation and Measurement.

[17]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[18]  Vernon L. Chi,et al.  Noise Model and Performance Analysis Of Outward-looking Optical Trackers Using Lateral Effect Photo Diodes , 1995 .

[19]  Ronald Azuma,et al.  Tracking a head-mounted display in a room-sized environment with head-mounted cameras , 1990, Defense, Security, and Sensing.

[20]  Eric Michael Fuchs,et al.  Inertial head-tracking , 1993 .

[21]  K. Åström.  Introduction to Stochastic Control Theory , 1970 .

[22]  Tomasz Mazuryk,et al.  Two‐step Prediction and Image Deflection for Exact Head Tracking in Virtual Environments , 1995, Comput. Graph. Forum.

[23]  J.C.K. Chou,et al.  Quaternion kinematic and dynamic differential equations , 1992, IEEE Trans. Robotics Autom..

[24]  J. A. Bather,et al.  Optimization of Stochastic Systems: Topics in Discrete-Time Dynamics , 1989 .

[25]  Ralph. Deutsch,et al.  Estimation Theory , 1966 .

[26]  Ron Daniel,et al.  Specification and design of input devices for teleoperation , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[27]  Greg Welch,et al.  SCAAT: incremental tracking with incomplete information , 1997, SIGGRAPH.

[28]  WangJih-fang,et al.  A real-time optical 3D tracker for head-mounted display systems , 1990 .

[29]  Mark A. Livingston,et al.  Superior augmented reality registration by integrating landmark tracking and magnetic tracking , 1996, SIGGRAPH.

[30]  Richard M. Stanley.  Optimal Estimation With an Introduction to Stochastic Control , 1988 .

[31]  F. Raab,et al.  Magnetic Position and Orientation Tracking System , 1979, IEEE Transactions on Aerospace and Electronic Systems.

[32]  O. Jacobs,et al.  Introduction to Control Theory , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[33]  F. Lewis.  Optimal Estimation: With an Introduction to Stochastic Control Theory , 1986 .

[34]  R. Bucy,et al.  Filtering for stochastic processes with applications to guidance , 1968 .

[35]  Yaakov Bar-Shalom,et al.  Estimation and Tracking: Principles, Techniques, and Software , 1993 .

[36]  J. Deyst,et al.  Conditions for asymptotic stability of the discrete minimum-variance linear estimator , 1968 .

[37]  Richard L. Holloway,et al.  Registration errors in augmented reality systems , 1996 .

[38]  Otmar Loffeld,et al.  Multisensor data fusion for automated guided vehicles , 1994, Other Conferences.

[39]  Michael J. Griffin,et al.  Compensating Lags in Head-Coupled Displays Using Head Position Prediction and Image Deflection , 1992 .

[40]  Roy Kalawsky,et al.  The science of virtual reality and virtual environments - a technical, scientific and engineering reference on virtual environments , 1993 .

[41]  G. J. Geier,et al.  GUIDANCE SIMULATION AND TEST SUPPORT FOR DIFFERENTIAL GPS (GLOBAL POSITIONING SYSTEM) FLIGHT EXPERIMENT , 1987 .

[42]  Robert Grover Brown,et al.  Introduction to random signals and applied Kalman filtering : with MATLAB exercises and solutions , 1996 .

[43]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[44]  Ronald Azuma,et al.  A demonstrated optical tracker with scalable work area for head-mounted display systems , 1992, I3D '92.

[45]  Ronald Azuma,et al.  Predictive tracking for augmented reality , 1995 .

[46]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[47]  Satoru Emura,et al.  Sensor fusion based measurement of human head motion , 1994, Proceedings of 1994 3rd IEEE International Workshop on Robot and Human Communication.

[48]  Nathaniel I. Durlach,et al.  Telepresence, time delay and adaptation , 1991 .

[49]  A.H. Haddad,et al.  Applied optimal estimation , 1976, Proceedings of the IEEE.

[50]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[51]  F. Ham,et al.  Observability, Eigenvalues, and Kalman Filtering , 1983, IEEE Transactions on Aerospace and Electronic Systems.

[52]  James S. Meditch,et al.  Stochastic Optimal Linear Estimation and Control , 1969 .

[53]  Nathaniel I. Durlach,et al.  Virtual Reality: Scientific and Technological Challenges , 1994 .

[54]  H. Sorenson.  Least-squares estimation: from Gauss to Kalman , 1970, IEEE Spectrum.

[55]  S. Julier,et al.  A General Method for Approximating Nonlinear Transformations of Probability Distributions , 1996 .

[56]  Susumu Tachi,et al.  Sensor Fusion based Measurement of Human Head Motion , 1993 .

[57]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice , 1993 .

[58]  Chris Shaw,et al.  On temporal-spatial realism in the virtual reality environment , 1991, UIST '91.

[59]  Richard A. Brown,et al.  Introduction to random signals and applied kalman filtering (3rd ed , 2012 .

[60]  H.F. Durrant-Whyte,et al.  A new approach for filtering nonlinear systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[61]  John F. Hughes,et al.  Autocalibration for virtual environments tracking hardware , 1993, SIGGRAPH.

[62]  Yves Demazeau,et al.  Principles and techniques for sensor data fusion , 1993, Signal Process..

[63]  C. Atkeson,et al.  Kinematic features of unrestrained vertical arm movements , 1985, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[64]  Susumu Tachi,et al.  Compensation of time lag between actual and virtual spaces by multi-sensor integration , 1994, Proceedings of 1994 IEEE International Conference on MFI '94. Multisensor Fusion and Integration for Intelligent Systems.

[65]  Mark R. Mine.  Characterization of End-to-End Delays in Head-Mounted Display Systems , 1993 .

[66]  H. Woltring,et al.  New possibilities for human motion studies by real-time light spot position measurement. , 1974, Biotelemetry.

[67]  P. B. Liebelt.  An Introduction To Optimal Estimation , 1967 .