Mobile Robotics, 2nd Edition

7Kalman Filter

In Chapters 2 and 3 we looked at tools to control robots in a nonlinear manner. For this purpose, we have assumed that the state vector was completely known. However, this is not the case in practice. This vector must be estimated from sensor measurements. In the case where the only unknown variables are associated with the position of the robot, Chapter 5 gives guidelines to find them. In the more general case, filtering or state observation seeks to reconstruct this state vector as well as possible from all the data measured on the robot throughout time by taking into account the state equations. The aim of this chapter is to show how such reconstruction is performed, within a stochastic context in which the system to observe is assumed to be linear. This is the purpose of the Kalman filter [KAL 60] which will be developed in this chapter. The Kalman filter is used in numerous mobile robotics applications, even though the robots in question are strongly nonlinear. For such applications, the initial conditions are assumed to be relatively well-known in order to allow a reliable linearization.

7.1. Covariance matrices

The Kalman filter is mainly based on the concept of covariance matrix, which is important to grasp in order to understand the design and the utilization of the observer. This section recalls the fundamental concepts surrounding covariance matrices.

7.1.1. Definitions and interpretations

Let us consider two random vectors x ∈ ℝⁿ and y ∈ ℝ^m. The ...