State Estimation and Kalman Filtering Zeeshan Ali Sayyed

What is State Estimation? We need to estimate the state of not just the robot itself, but also of objects which are moving in the robot’s environment. For instance, other cars, people, deers, etc. Localization Tracking

Why do we need it? The world is stochastic and not deterministic There are errors in the motors or transition mechanism of the robot. There are errors in the sensors on the robot. Sometimes, we also need to predict future states so as to plan accordingly. For instance, apply brakes if we are about to collide with another car.

What is Localization? Imagine a robot in a simple world. The robot doesn’t know where it is in the world frame of reference. Estimating the position and state of the robot in this world making use of the limited information available to the robot is called Localization. Localization is a form of State Estimation where we estimate the state of the robot in the given world.

Example of Localization

Belief of a Robot What is belief? How do we represent it? How do we start when we have absolutely no information? How do we update belief?

How do we start? Uniform Distribution – This shows we have absolutely no information about the location of the robot

Quiz There are 4 possible places where the robot can be. What is the probability that the robot is in the 3 rd place, given absolutely no other information?

Incorporating Sensor Measurements The belief after we incorporate the sensor measurements is called Posterior Belief.

How do we do that in practice? There are a variety of techniques for incorporating sensor input into our belief. The simplest one is a simple product. For instance, Consider the following world Let’s say the robot observes Yellow. What do we do? 0.2 ? ? ? ? ?

Incorporating Transition of Robot This is technically called Convolution.

How do we do that in practice? Assume a cyclic world. What happens, say, if the robot moves 2 steps forward?

Final Localization This technique is referred to as Monte Carlo Localization

Modelling Noisy Sensors 0.2

Modelling Noisy Transition

Representation of things we have learned

Introducing Kalman Filters Kalman Filters used for both Localization as well as Tracking. It is very similar to Monte Carlo Localization It one of the most popular state estimation technique is use, not only in robotics but in many other fields. It deals in Continuous State Spaces (What do they mean?).

Gaussian

Comparison of Means and Variances

Representing Belief and Measurement The belief and sensor measurement, both are represented by Gaussians. Gaussian with high variance implies uncertainty and low variance implies certainty. Example on board

Kalman Filter Algorithm Incorporate Sensor Measurements Bayes Rule Incorporate Transition Update Total Probability Transition Update Measurement Update

Incorporating Sensor Measurements Can you say anything about the posterior?

Multiplication of two Gaussians Addition of two Gaussians

Incorporating Transition Update When we move, we tend to lose information. Therefore, the variance of the belief increases. Simple add the two Gaussians using the previous formula. That’s the Kalman Filter for a simple one dimensional case!

Generalized Kalman Filter We assume we have a linear transition and observation (sensing) models.

Kalman Filter Algorithm 1. Algorithm Kalman_filter(  t-1,  t-1, u t, z t ): 2. Prediction: Correction: Return  t,  t 27

28 Kalman Filter Summary Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k n 2 ) Optimal for linear Gaussian systems! Most robotics systems are nonlinear!