Understanding the Kalman filter with a simple radar example

Kalman Filter

"If you can't explain it simply, you don't understand it well enough."

Albert Einstein

Introduction to Kalman Filter

The Kalman Filter is an algorithm for estimating and predicting the state of a system in the presence of uncertainty, such as measurement noise or influences of unknown external factors. The Kalman Filter is an essential tool in areas like object tracking, navigation, robotics, and control. For instance, it can be applied to estimate the trajectory of a computer mouse by reducing noise and compensating for hand jitter, resulting in a more stable motion path.

In addition to engineering, the Kalman Filter finds applications in financial market analysis, such as detecting stock price trends in noisy market data, and in meteorological applications for weather prediction.

Although the Kalman Filter is a simple concept, many educational resources present it through complex mathematical explanations and lack real-world examples or illustrations. This gives the impression that the topic is more complex than it actually is.

This guide presents an alternative approach that uses hands-on numerical examples and simple explanations to make the Kalman Filter easy to understand. It also includes examples with bad design scenarios where Kalman Filter fails to track the object correctly and discusses methods for correcting such issues.

By the end, you will not only understand the underlying concepts and mathematics but also be able to design and implement the Kalman Filter on your own.

Kalman Filter Learning Paths

This project explains the Kalman Filter at three levels of depth, allowing you to choose the path that best fits your background and learning goals:

• Single-page overview (this page): A concise introduction that presents the main ideas of the Kalman Filter and the essential equations, without derivations. This page explains the core concepts and overall structure of the algorithm using a simple example, and assumes basic knowledge of statistics and linear algebra.
• Free, example-based web tutorial: A step-by-step online tutorial that builds intuition through numerical examples. The tutorial introduces the necessary background material and walks through the derivation of the Kalman Filter equations. No prior knowledge is required.
• Kalman Filter from the Ground Up (book): A comprehensive guide that includes 14 fully solved numerical examples, with performance plots and tables. The book covers advanced topics such as nonlinear Kalman Filters (Extended and Unscented Kalman Filters), sensor fusion, and practical implementation guidelines.

The prediction requirement

We begin by formulating the problem to understand why we need an algorithm for state estimation and prediction. To illustrate this, consider the example of a tracking radar: Suppose we have a radar that tracks an aircraft. In this scenario, the aircraft is the system, and the quantity to be estimated is its position, which represents the system state.

The radar samples the target by steering a narrow pencil beam toward it and provides position measurements of the aircraft. Based on these measurements, we can estimate the system state (the aircraft's position). To track the aircraft, the radar must revisit the target at regular intervals by pointing the pencil beam in its direction. This means the radar must be able to predict the aircraft's future position for the next beam. If it fails to do so, the beam may be pointed in the wrong direction, resulting in a loss of track. To make this prediction, we need a model that describes the system's behavior over time, known as the dynamic model.

To simplify the example, let us consider a one-dimensional world in which the aircraft moves along a straight line. The system state is defined as the range of the airplane from the radar, denoted by ( r ). Let us assume that at time ( t_{0} ), the radar measures the aircraft's range and velocity. The measured range is 10,000 meters, and the velocity is 200 meters per second. Assuming a sampling interval of 5 seconds, the predicted position at time ( t_{1} ) is:

[ r_{t_{1}} = r_{t_{0}} + \Delta r = 10,000 + 200 \cdot 5 = 11,000m ]

In real life, things are more complex. Radar measurements are affected by measurement noise. Another issue is the accuracy of the dynamic model, as unpredictable influences like wind are referred to as process noise. The Kalman Filter is an optimal algorithm that provides both an estimate and a prediction along with a measure of their uncertainty.

Kalman Filter example

In this example, the system state is described by both the aircraft range ( r ) and velocity ( v ). We define the system state by the vector ( \boldsymbol{x} ):

[ \boldsymbol{x}=\left[\begin{matrix}r\v\\end{matrix}\right] ]

Because the system state includes more than one variable, we use linear algebra tools, such as vectors and matrices, to describe the mathematics of the Kalman Filter.