# Robot Localization and Kalman Filters - Rudy Negenborn

The Kalman filter is the main algorithm to estimate dynamic systems specified with the state-space model. Actually, the state-space models and the Kalman filter models are often used as synonymous. The estimation and control of the problems of this methodology are based on stochastic models, assuming errors in the measures. The performance of the state-space model for a linear system captures a yn vector with nx1 order associated to an unknown xn vector with mx1 order, known as state vector. In speech processing, I assume the case with a signal received by a single microphone and additive noise. Let the signal measured by the microphone be given by:

Where yn is the observed signal, xn is the desired input and vn is the additive background noise (zero-mean noise). Furthermore, like xn is modeled as autoregressive, assume the standard LPC modeling for the speech signal over an analysis frame:

On the other hand, the last equation can be reformulated in a state space presentation with the state transition matrix or companion matrix:

Then, write the state-space form:

Where C is a matrix of the system, Wn+1 is the noise indoor and vn is the noise outdoor.

The first of these equations is known as process equation and the second one as measurement equation. The first equation shows the relation among previous states and futures states, while the second one gives us the correspondence between the internal state of the system and how it can be observed [10]. These equations are useful for most of the linear estimation methods, like the Kalman filter described above.

Figure 2: The kalman filter cycle

## Robot Localization and Kalman Filters ..

### In this thesis I look at two large research elds.

I know I am very late to this post, and I am aware that this comment could very well go unseen by any other human eyes, but I also figure that there is no hurt in asking. This article was very helpful to me in my research of kalman filters and understanding how they work. I would absolutely love if you were to do a similar article about the Extended Kalman filter and the Unscented Kalman Filter (or Sigma Point filter, as it is sometimes called). If you never see this, or never write a follow up, I still leave my thank you here, for this is quite a fantastic article.

### Airborne attitude estimation using a Kalman filter

Comparison between the unscented Kalman filter and the extended Kalman filter for the position estimation module of an integrated navigation information system Mathieu St-Pierre Electrical engineering

In this work we provide a thorough discussion of the robot localization problem and Kalman Filter techniques. First, we look at current methods to obtain location information, pointing out advantages and disadvantages. We formalize how to combine this information in a probabilistic framework and discuss several currently used methods that implement it. Second, we look at the basic concepts involved in Kalman Filters and derive the equations of the basic filter and commonly used extensions. We create understanding of the workings, while discussing the differences between the extensions. Third, we discuss and experimentally show how Kalman Filters can be applied to the localization problem. We look at system and measurement models that are needed by the filter; that is, we model a driving system, a GPS-like sensor, and a landmark-based sensor. We perform simulations using these models in our own general Kalman Filter simulator showing different behaviors when applying the Kalman Filter to the localization problem. In order to use the landmark-based sensor when it can not uniquely identify landmarks, we extend the Kalman Filter to allow for multiple beliefs.While localization is most commonly associated with GPS, many use cases remain where satellite-based navigation is too inaccurate or fails completely. In this seminar, we will present techniques usable for indoor localization of pedestrians. We will introduce several approaches using Inertial Measurement Units attached to the subject. Due to the strong drifting behavior of those units, several steps are necessary to provide feasible accuracy: the use of filter techniques and the use of Zero Velocity Updates. We will explain the required state-space

model and its application in recursive Bayesian filters like the Extended Kalman Filter or the Particle Filter. The use of aiding techniques is discussed and a map-aided, WiFi-initialized Particle Filter is presented.