Kalman Filter

In statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. (Source-Wikipedia)

Rudolf E. Kálmán

The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

Algorithm

Gaussian

At the basis of the Kalman Filter is the Gaussian distribution, sometimes referred to as a bell curve or normal distribution. After a movement or a measurement update, it outputs a unimodal Gaussian distribution. This is its best guess at the true value of a parameter. A Gaussian distribution is a probability distribution, which is a continuous function. The probability that a random variable, x, will take a value between x₁ and x₂ is given by the integral of the function from x₁ to x₂.

Mean and Variance

A Gaussian is characterized by two parameters - its mean (μ) and its variance (σ²). The mean is the most probable occurrence and lies at the centre of the function, and the variance relates to the width of the curve. The term unimodal implies a single peak present in the distribution.

The formula for the Gaussian distribution,

Measurment Update

Calculate Posterior Mean

Calculate Posterior Vairance

The two Gaussians provide us with more information together than either Gaussian offered alone. As a result, our new state estimate is more confident than our prior belief and our measurement. This means that it has a higher peak and is narrower. You can see this in the graph above.

State Prediction

Posterior Mean, μ` = μ₁ + μ₂
Posterior Variance, σ² = σ²₁ + σ²₂

1D Kalman Filter

• 1 Dimensional Kalman Filter in C++