README.md

Intro

The "joints" of the wheelbot are in order:

1. The yaw angle $\psi$; its derivative is the yaw rate~$\dot{\psi}$.
2. The roll angle $\phi$; its derivative is the roll rate~$\dot{\phi}$.
3. The pitch angle $\theta$; its derivative is the pitch rate~$\dot{\phi}$.
4. The driving wheel angle $q_w$ that is actually doesn't appear in the equations of motions; its derivative is the wheel rotational velocity $\dot{q}_W$; the input torque applied to this joint is $\tau_W$.
5. The reaction wheel angle $q_w$ that is actually doesn't appear in the equations of motions; its derivative is the wheel rotational velocity $\dot{q}_R$; the input torque applied to this joint is $\tau_R$.

The generalized coordinates are: $q=[\psi,\phi,\theta,q_W, q_R]^\top$

The inputs of the wheelbot are: $u=[\tau_W, \tau_R]$

Continuous time

In continuous time, the dynamics are of the form: $M(q)\ddot{q}+b(q,\dot{q})+g(q,u)=0$. The mass matrix $M$, coriolis and centripedal terms $b$, and gravitational terms and inputs $g$ are also parametric in the mass and inertia of the wheels W and R, and the body of the robot, and in the wheel radius and distance from wheel rotational axis to body center of mass (assumed to be in the center of the body).

In this repository, you will find functions fM,fb,fg for a generic Wheelbot type of robot alongside specific fM_WheelbotvXX, fb_WheelbotvXX,fg_WheelbotvXX functions, where the mass and inertia parameters of the specific hardware versions are already substituted in.

Python example

Inside the python folder, you find a python package with the dynamics and parameters for the new Mini Wheelbot alongside an acados simulation example.

Installation

Install acados following the official installation instructions here. You might need to append acados to your path, you can conveniently do this in your .bashrc or manually, whenever you need acados:

export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:"/path/to/acados/lib"
export ACADOS_SOURCE_DIR="/path/to/acados"

I recommend to proceed with a python virtual environment from here:

python -m venv wheelbot-venv
source wheelbot-venv/bin/activate

Now install the Python interface for acados by calling:

pip install -e /path/to/acados/interfaces/acados_template

This should automatically also install casadi. Now, you are ready to install the wheelbot-dynamics package as:

pip install -e /path/to/wheelbot-dynamics-functions/python

Running the example

To run the example, you can

cd /path/to/wheelbot-dynamics-functions/python/wheelbot-dynamics/examples
python wheelbot_example.py

You should see a plot with all states of the wheelbot.