Probabilistic Robotics - Odometry - Velocity based Model

Autonomous navigation heavily relies on probabilities. The hard part isn't getting to build ROS based robots. But actually understand what are the underlying parameters and algorithms. One such book that is literally a bible to the approach of these algorithms is the book by S. Thrun, W. Burgard, and D. Fox - Probabilistic Robotics.

I first thought of directly publishing the odometry of Clerkbot using the ROS nodes. Not knowing the underlying models and states isn't very good for anyone interested in robotics. Here's my take on the Velocity based model based on the book suggested which is also the model which the base_local_planner works. This is because inherently navigation planning uses velocity commands to plan for obstacle avoidance. The odometer readings are only helpful after the command of control has been given.

There are two models into the probabilistic kinematics:

1. Odometric Model
2. Velocity Model

I would like to overemphasise one thing. That is in probabilistic robotics, as the name suggests, nothing is certain. You are constantly estimating states based on given inputs and due to noises present,  the uncertainty is shown with the help of probability distributions. So nothing is certain when you are dealing with noises present in almost all the states. The odomteric model can be realised with just one probability distribution,

This is when you want to estimate the state the pose of the robot at current time t=t, given the the control commands given at time t=t from the previous pose state at time t=t-1.

You can cross-verify here what I wanted to tell you in the previous para. That given a pose state of the robot and a control command given, you cannot be certain of the position of robot at time t=t+1 because of the noise in motors or odometers or whatever actuator you choose. You cannot be certain about your position, but you can devise a normal distribution of the position of the robot given you have the noise information.

Note: Odometry model is more accurate since you are getting values of the revolutions from encoders present. But for motion planning, obstacle avoidance velocity model wins the race.

Velocity model can be broken down to the following points:

1. The translational and rotational velocities given at an instant is v,w (read as mu and omega). Now, the radius r covered is, v/w.
2. Given the initial point of (x,y, theta), the final pose can be estimated by using the equation assuming error less state system.(Figures 2 and 3).
3. Final equations given the initial pose and after assuming constant angular and translational                                   velocity and delta time t. (Figure 4)

fig 2 : Rotational and translational motion

fig 3: Center of circle equations

fig 4: The final pose estimate(error less system)

Now this is where the concept of probabilistic kinematics and noise kicks in. Consider there are noises present in the rotational and translational velocity with a noise which has zero mean and a variance b.  Hence now the final velocities are, containing real world noises.

fig 5: Noise into the states