Introduction
Kalman filters are a versatile tool for tracking, estimating, and predicting a wide range of dynamic models and systems. They have been used extensively in many fields such as aviation, robotics, medicalImage processing and computer vision. Kalman filters are also widely used in engineering and control theory applications.
Definition
Kalman filter is an optimal recursive state estimation algorithm and is widely used to estimate the state of dynamic systems from incomplete, uncertain and noisy measurements. Kalman filter is a two-stage iterative estimator. In the first stage, the Kalman filter estimates the current state based on the previous measurements and the system dynamics. In the second stage, the Kalman filter improves the estimation by taking into account the measurement noises.
Theory
The dynamics of the system are characterized by a linear dynamic equation which is given by:
zᵢ⁺¹= Azᵢ + Buᵢ
where zᵢ is the state vector at time instant ‘i’, uᵢ is the control vector and A and B are known constant matrices.
The measurement equation is given by
yᵢ= Czᵢ + wᵢ
where C is an observation matrix and wᵢ is the measurement noise.
The Kalman filter combines the system equations and the measurement equation to obtain a single equation for estimating the current state zᵢ. This equation is given by:
zᵢ⁺ˢ=Azᵢ + Buᵢ + K(yᵢ-Czᵢ)
where K is the Kalman gain. The Kalman gain is determined iteratively based on the current estimate of the system state and measurement noise.
Principles
Kalman filter is an algorithm used to estimate the current state from a stream of noisy and incomplete measurements. The two main principles used in Kalman filter are the prediction principle and the measurement principle.
The prediction principle states that the current state can be estimated based on the previous state and the system dynamics. The prediction principle is given by:
zᵢ⁺¹= Azᵢ + Buᵢ
The Measurement Principle states that the current state can be improved by taking into account the measurement noise. The measurement principle is given by
zᵢ⁺¹=zᵢ + K(yᵢ-Czᵢ)
where K is the Kalman gain.
Applications
Kalman filters have been used in various fields such as control systems, robotics, medicalImage processing, and computer vision.
Robotics: In robotics, Kalman filters are used to control and estimate the state of mechanical manipulators. The filters can be used to track the motion of robots with accuracy, which is essential for tasks such as material handling and assembly.
Medical Image Processing: In medical applications, Kalman filters are used to track the movement of medical images by predicting the changes in the image with time. The filters can process series of medical images to estimate the movement of organs and tissues.
Computer Vision: Kalman filters are also used in computer vision applications. They are used to estimate the motion of objects from a series of frames of an image. The filters can be used to track and monitor the position of objects in an image with accuracy and precision.
Conclusion
Kalman filters are a powerful and versatile tool for tracking and estimating the state of dynamic systems. They have been used in various fields such as robotics, medicalImage processing, and computer vision. The Kalman filter combines the system equations and the measurement equation to obtain a single equation for estimating the current state. The two main principles used in Kalman filter are the prediction principle and the measurement principle. Kalman filters have enabled more reliable and accurate state tracking and estimation in dynamic systems.
参考资料：
https://www.intechopen.com/books/neural-networks-in-control-systems/kalman-filtering-concepts-and-applications
https://academic.oup.com/bioscience/article/66/11/901/320815
https://www.sciencedirect.com/topics/engineering/kalman-filter
