Kalman filter

The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate. The estimate is updated using a state transition model and measurements. denotes the estimate of the system's state at time step k before the k-th measurement yk has been taken into account; is the corresponding uncertainty.

In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics.[1] The filter is named after Rudolf E. Kálmán.

Kalman filtering[2] has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically.[3] Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. Kalman filtering is also important for robotic motion planning and control,[4][5] and can be used for trajectory optimization.[6] Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters[7] provides a realistic model for making estimates of the current state of a motor system and issuing updated commands.[8]

The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current state variables, including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average, with more weight given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required.

Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán: "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear."[9] Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense,[10] although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian.[11]

Extensions and generalizations of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion,[12] and distributed sensor networks to develop distributed or consensus Kalman filtering.[13]

  1. ^ Lacey, Tony. "Chapter 11 Tutorial: The Kalman Filter" (PDF).
  2. ^ Fauzi, Hilman; Batool, Uzma (15 July 2019). "A Three-bar Truss Design using Single-solution Simulated Kalman Filter Optimizer". Mekatronika. 1 (2): 98–102. doi:10.15282/mekatronika.v1i2.4991. S2CID 222355496.
  3. ^ Paul Zarchan; Howard Musoff (2000). Fundamentals of Kalman Filtering: A Practical Approach. American Institute of Aeronautics and Astronautics, Incorporated. ISBN 978-1-56347-455-2.
  4. ^ Lora-Millan, Julio S.; Hidalgo, Andres F.; Rocon, Eduardo (2021). "An IMUs-Based Extended Kalman Filter to Estimate Gait Lower Limb Sagittal Kinematics for the Control of Wearable Robotic Devices". IEEE Access. 9: 144540–144554. Bibcode:2021IEEEA...9n4540L. doi:10.1109/ACCESS.2021.3122160. hdl:10261/254265. ISSN 2169-3536. S2CID 239938971.
  5. ^ Kalita, Diana; Lyakhov, Pavel (December 2022). "Moving Object Detection Based on a Combination of Kalman Filter and Median Filtering". Big Data and Cognitive Computing. 6 (4): 142. doi:10.3390/bdcc6040142. ISSN 2504-2289.
  6. ^ Ghysels, Eric; Marcellino, Massimiliano (2018). Applied Economic Forecasting using Time Series Methods. New York, NY: Oxford University Press. p. 419. ISBN 978-0-19-062201-5. OCLC 1010658777.
  7. ^ Azzam, M. Abdullah; Batool, Uzma; Fauzi, Hilman (15 July 2019). "Design of an Helical Spring using Single-solution Simulated Kalman Filter Optimizer". Mekatronika. 1 (2): 93–97. doi:10.15282/mekatronika.v1i2.4990. S2CID 221855079.
  8. ^ Wolpert, Daniel; Ghahramani, Zoubin (2000). "Computational principles of movement neuroscience". Nature Neuroscience. 3: 1212–7. doi:10.1038/81497. PMID 11127840. S2CID 736756.
  9. ^ Kalman, R. E. (1960). "A New Approach to Linear Filtering and Prediction Problems". Journal of Basic Engineering. 82: 35–45. doi:10.1115/1.3662552. S2CID 1242324.
  10. ^ Humpherys, Jeffrey (2012). "A Fresh Look at the Kalman Filter". SIAM Review. 54 (4): 801–823. doi:10.1137/100799666.
  11. ^ Uhlmann, Jeffrey; Julier, Simon (2022). "Gaussianity and the Kalman Filter: A Simple Yet Complicated Relationship" (PDF). Journal de Ciencia e Ingeniería. 14 (1): 21–26. doi:10.46571/JCI.2022.1.2. S2CID 251143915. See Uhlmann and Julier for roughly a dozen instances of this misconception in the literature.
  12. ^ Li, Wangyan; Wang, Zidong; Wei, Guoliang; Ma, Lifeng; Hu, Jun; Ding, Derui (2015). "A Survey on Multisensor Fusion and Consensus Filtering for Sensor Networks". Discrete Dynamics in Nature and Society. 2015: 1–12. doi:10.1155/2015/683701. ISSN 1026-0226.
  13. ^ Li, Wangyan; Wang, Zidong; Ho, Daniel W. C.; Wei, Guoliang (2019). "On Boundedness of Error Covariances for Kalman Consensus Filtering Problems". IEEE Transactions on Automatic Control. 65 (6): 2654–2661. doi:10.1109/TAC.2019.2942826. ISSN 0018-9286. S2CID 204196474.