wheelbot-v2.5

This repository contains all files required to build a Wheelbot v2.5.

Project page: https://sites.google.com/view/wheelbot/start

/CAD files

Contains the CAD files of the Wheelbot v2.5 as stl-files.

3D printer used: Markforged Onyx One
Material: Onyx
Layer height: 0.2 mm
Use default settings for all else.
See the following links for tips on printing:
https://markforged.com/resources/learn/design-for-additive-manufacturing-plastics-composites/3d-printing-strategies-for-composites/composites-3d-printing-design-tips
https://static.markforged.com/downloads/CompositesDesignGuide.pdf

The folder also contains the technical drawing for the cut copper rings forming the reaction wheel and the pdf "wheelbot v2.5 assembly view.pdf" that lets you interact with the complete Wheelbot's assembly (requires Adobe Acrobate reader).

/motherboard circuitry

Contains the circuit layouts of the motherboard that connects to the Maevarm M2 and which also supplies the uDriver-v2 with power.

/simulation/matlab-symbolic-derivation-EOM

Contains the matlab files used to symbolically derive the EOM of the Wheelbot.

FlywUni_symbolic_derive.m: Derives the Euler-Lagrange equations of a unicycle reaction wheel robot using the parameters set in "config_FlyUni" and the "Langrange.m" defined in external libraries.

FlywUni_symbolic_linearize.m: Reshapes the symbolic ODE obtained from "FlywUni_symbolic_derive", brings it into the form for the standard matlab ODE solvers and saves it as

save('EQS_matrices_nonlin.mat', 'M_nonlin', 'RHS_nonlin')

Also linearizes the set of equations and saves the linearization as

save('EQS_matrices.mat', 'M', 'RHS')

FlywUni_symbolic_analyze.m: Loads the pre-computed dynamic equations, computes a trajectory using a standard ODE solver (RK45), and plots trajectory and total energy of the system.

/simulation/simulink-model

Contains the simulink model used to tune the estimator and LQR controller. Recommended matlab version is R2020a.

The file "s00_config" contains the simulation settings including the exact estimates for the mass and inertia of the Wheelbot v2.5 which we obtained from CAD.

Before running "s01_unicycle.slx", you need to run "s00_designLQR".

/firmware

Contains the firmware required to run Wheelbot v2.5

/firmware/M2-on-wheelbot

Contains the firmware that runs on a Maevarm M2 that is attached to the motherboard of the Wheelbot.

/firmware/M2-wifi-dongle

Contains the firmware that runs on a Maevarm M2 being connected to a PC and that handles the wifi communication with the Wheelbot.

/firmware/computer-python-interface

Python program running on Ubuntu 18.04 LTS collecting incoming data/outgoing commands from/to the wifi dongle.

Erratum

In the initial publication, in Equation (3), the transform from averaged body rates ${}^{\text{B}}\omega_i$ to Euler rates was incorrectly denoted as

$$\begin{bmatrix} \dot{q}_{1, \mathrm{G}} \\\ \dot{q}_{2, \mathrm{G}} \\\ \dot{q}_{3, \mathrm{G}} \end{bmatrix} = \begin{bmatrix} e_1^{\mathrm{T}} R_2 \\\ e_2^{\mathrm{T}} \\\ e_3^{\mathrm{T}} R_1 R_2 \end{bmatrix} \sum_{k=1}^4 \frac{{ }^B \omega_i(k)}{4},$$

corresponding to

$$\left[\begin{array}{c} \dot{q}_{1, \mathrm{G}} \\\ \dot{q}_{2, \mathrm{G}} \\\ \dot{q}_{3, \mathrm{G}} \end{array}\right]=\left[\begin{array}{c} R_2^T e_1 & e_2 & R_2^T R_1^T e_3 \end{array}\right]^{\top} \sum_{i=1}^4 \frac{{ }^B \omega_i(k)}{4}.$$

However, the correct transform from averaged body rates ${}^{\text{B}}\omega_i$ to Euler rates is

$$\left[\begin{array}{c} \dot{q}_{1, \mathrm{G}} \\\ \dot{q}_{2, \mathrm{G}} \\\ \dot{q}_{3, \mathrm{G}} \end{array}\right]=\left[\begin{array}{c} R_2^T e_1 & e_2 & R_2^T R_1^T e_3 \end{array}\right]^{-1} \sum_{i=1}^4 \frac{{ }^B \omega_i(k)}{4}.$$

While we implemented the faulty transform in the Wheelbot's state estimation routine (see main.c, Line 340), the error had not been noticed during experimentation as the robot remained close to its upright equilibrium. In turn, $q_1$ and $q_2$ remained near zero such that both transforms became almost identical, as

$$\left[\begin{array}{c} R_2^T e_1 & e_2 & R_2^T R_1^T e_3 \end{array}\right]^{\top} = \left[\begin{array}{c} \cos(q_2) & 0 & \sin(q_2) \\\ 0 & 1 & 0 \\\ -\cos(q_1) \sin(q_2) & \sin(q_1) & \cos(q_1) \cos(q_2) \end{array}\right],$$ $$\left[\begin{array}{c} R_2^T e_1 & e_2 & R_2^T R_1^T e_3 \end{array}\right]^{-1} = \left[\begin{array}{c} \cos(q_2) & 0 & \sin(q_2) \\\ \tan(q_1) \sin(q_2) & 1 & -\tan(q_1) \cos(q_2) \\ -\sin(q_2) / \cos(q_1) & 0 & \cos(q_2) / \cos(q_1) \end{array}\right].$$

Importantly, our results on tilt estimation using accelerometers as depicted in Figure 10 are not affected by this error. In the first experiment (Figure 10, left) the robot's tilt angles were kept at zero. In the second experiment (Figure 10, right), $q_1 \approx 0$ such that $\tan(q_1) \approx q_1$. In turn, both transforms deviated only marginally from each other in both experiments.

We added a Jupyter notebook to the projects Github repository detailing the calculation of the transform from body rates to Euler rates.