Adding noise to odometry data published by the Gazebo simulator

I've had funny time playing around with the Gazebo simulator for autonomous robot exploration. One thing I've encountered is that the odometry data provided by Gazebo is so perfect that, sometime, makes the simulation less realistic. I used a Turtlebot model as robot model in my simulations. Googling around, i didn't find any solution of adding noise to the odometry data of this robot (using the URDF file). I then decided to develop a dedicated ROS node allowing me to add some random noise to the Gazebo's odometry data.

The robot motion model

First thing fist, we need to understand the robot motion model. There are many motion models, but in the scope of this article, we focus only on the odometry motion model. Often, odometry is obtained by integrating sensor reading from wheel encoders, it measures the relative motion of the robot between time \(t\) and \(t-1\) or \((t-1,t]\). In 2D environment, a robot pose is represented by a point \((x,y)\) and an orientation (rotation angle) \(\theta\), so the robot pose at the time \(t-1\) and \(t\) are denoted by:

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