free-forced-double-pendulum

Derivation, simulation, and control of a rigid-body double pendulum with a moving base, control torques, and an external wind force.

Model overview

The model consists of a 2 DOF double pendulum, where the DOF are defined as the segment angles $\theta_1$ and $\theta_2$, respectively, in the global reference frame. The base of the first pendulum segment is defined by a time-varying position $\mathbf{r}_a = [x_a, y_a]$ that is modeled as an input (not a DOF).

Each segment is modeled as a rigid-body with a length $L_\bullet$, mass $m_\bullet$, mass $m_\bullet$ and inertia $J_\bullet$. Additionally, each segment junction has a torsional stiffness $k_\bullet$ and damping $c_\bullet$. Torques acting on each segment $\tau_\bullet$ are also included to allow the module to be controlled, if desired. An external wind vector $\mathbf{w} = w \angle \zeta$ is modeled to create a force vector acting on the COM of each segment via damping $c_w$.

Model derivation

The dynamics equations of motion are derived symbolically via Lagrangian mechanics in a MATLAB Live Script: dynamics/free_forced_rigid_double_pendulum_with_wind_dynamics.mlx

The derivation code is designed to be easily modifiable. If one wishes to change certain model properties (for instance make damping proportion to velocity squared, etc.) the code can be modified and re-run.

Model simulation & control

The final output of the derivation code is the nonlinear state-space equations required to simulate the model, which can be copy-pasted for use with an ODE solver. The current dynamics are defined in the DoublePendulumModel class in double_pendulum.py in the f method

This repository makes use of the pybounds Python package (https://github.com/vanbreugel-lab/pybounds) to simulate and control the derived dynamics. It is recommended to install pybounds from source:

pip install pandas
pip install do-mpc
pip install git+https://github.com/vanbreugel-lab/pybounds

A nice feature of the simulation code is that it is easy to exclude certain parts of the dynamics by adjusting the model parameters. For instance, one can effectively turn off gravity, wind, given springs and dampers, etc., and even turn the double-pendulum into a single-pendulum by setting parameters to 0 accordingly.

Example simulations

Example simulations are provided as Jupyter notebooks

Simple open-loop

Include gravity, but no damping or springs

example_open_loop.ipynb

With model predictive control (MPC)

Precisely control the movement of the base, and the 1st segment angle, but leave the 2nd segment under open-loop control. Damping and springs are on. See pybounds for MPC guidance.

example_model_predictive_control.ipynb

With model predictive control + wind

Keep the base and 1st segment angle constant and let the wind move the 2nd segment. Damping and springs are on.

example_model_predictive_control_wind.ipynb