Introduction to the Bayes Filter and Related Models

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Introduction to the Bayes Filter and Related Models

As I mentioned above, the estimate the state x of a system given observations z and robot controls u. The Bayes filter is a framework for recursive state estimation.

There are two steps in Bayes filter, prediction and correction step.

In the prediction step, thanks to the recursive form, the state(t) can be estimated from the state(t-1). In the correction model, sensor - observation - will check that prediction is reasonable. We called the probabilistic models in the prediction model as a motion model and the one in the correction model as a sensor or an observation model. The Bayes Filter is a framework for recursive state estimation. So it is good to use the specialized Bayes Filter in each problem. Think about is it linear or nonlinear? Gaussian distribution only? Parametric or non-parametric? like those.

Motion Models

Robot motion is inherently uncertain. How can we model this uncertainty? We will specify a posterior probability that action u carries the robot from x to x'.

In practice, one often finds two types of motion models, odometry-based and velocity-based. 

Odometry Model

Odometry model is for systems that are equipped with wheel encoders. It is also called to the rotation-transformation-rotation model because the position of the robot is explained in terms of coordinates in cartesian coordinate and the angle. The robot position is (x_bar, y_bar, theta_bar) and it moves to (x_bar_prime, y_bar_prime, theta_bar_prime). The odometry information u is (delta_rot1, delta_trans, delta_rot2). Because it is on the cartesian coordinate, the odometry information can be explained like formula below.

Do not forget we are using probability distribution! So the noise in odometry is also shown as the probability distribution, for example, Gaussian noise.

Velocity-Based Model

Velocity-Based model is usually used when no wheel encoders are available. The robot position is (x_bar, y_bar, theta_bar) and it moves to (x_bar_prime, y_bar_prime, theta_bar_prime) and the velocity information u is (v, w).

But there is a problem in Velocity-based model. The robot has the final orientation because it moves on a circle. I introduce an additional noise term on the final orientation to fix this problem.

Sensor Model

The sensor model is used in the correction step to check the predicted belief is reasonable with the observation that the sensor detected.

• Model for Laser Scanners
• Beam-Endpoint Model
• Ray-Cast Model
• Model for Perceiving Landmarks with Range-Bearing Sensors

Summary

• Bayes filter is a framework for state estimation.
• Motion model and sensor model are the central models in the Bayes filter.
• Standard models for robot motion and laser-based range sensing.