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MPC Tutorial

This repository provides a set of Python scripts and Jupyter notebooks solving toy optimal control problems (simple pendulum, cartpole, etc.). The goal is to familiarize students with the core concepts and methods from Optimal Control theory (continuous and discrete time).

In particular, the following methods are implemented:

  • Dynamic Programming (Value Iteration)
  • HJB PDE numerical resolution
  • Pontryagin Maximum Principle (a.k.a. indirect optimal control)
  • Direct Optimal Control (using SQP or DDP, based on Crocoddyl API)

🚀 Interactive Cloud Execution

You can run the notebooks directly in your web browser without installing anything locally:

1. Google Colab

Open the notebooks directly in Google Colab using the badges below.

Important

Google Colab uses condacolab to install compiled robotics dependencies. Run the setup cell at the top of the notebook. The first time, Colab may automatically restart the runtime after installing Conda. This is expected. After the restart, run the same setup cell again, then continue with the notebook.

  • Intro: Open in Colab
  • Part 1 (Optimal Control): Open in Colab
  • Part 2 (Constraints & SQP): Open in Colab
  • Part 3 (MPC & DDP): Open in Colab

2. Binder

Click the badge below to start a full JupyterLab environment in Binder with all dependencies pre-installed:

Binder

💻 Local Installation

We support two ways to install dependencies locally: Pixi (recommended, modern and fast) and Conda.

Option A: Pixi (Recommended)

Pixi is a modern package creator and manager. It installs all dependencies (including Python and system libraries) locally inside a .pixi/ folder without touching your global system environment.

  1. Install Pixi (if you don't have it):

    curl -fsSL https://pixi.sh/install.sh | bash

    (Restart your terminal after installation)

  2. Run a script directly: Pixi will automatically install everything needed on the first run:

    pixi run pendulum-ocp

    Other tasks: pendulum-bellman, cartpole-ocp, kuka-ocp, kuka-mpc.

  3. Start Jupyter Lab:

    pixi run lab

Option B: Conda

If you prefer traditional Conda/Mamba:

  1. Create the environment:

    conda env create -f environment.yaml

  2. Activate the environment:

    conda activate mpc_tutorial

  3. Install the package and mim_robots:

    pip install git+https://github.com/machines-in-motion/mim_robots.git --no-deps

  4. Run a script:

    python pendulum/pendulum_ocp.py

Create the conda environment manually

The conda environment was created using the following command:

conda create -n mpc_tutorial -c conda-forge --override-channels \
  python=3.12 pip matplotlib \
  mim-solvers=0.2.0 \
  crocoddyl=3.2.0 \
  pinocchio=3.8.0 \
  pinocchio-python=3.8.0 \
  libpinocchio=3.8.0 \
  coal=3.0.1 \
  coal-python=3.0.1 \
  libcoal=3.0.1 \
  ipython \
  matplotlib \
  numpy<2.0.0

🛠️ Dependencies

The main dependencies installed automatically by the setup tools are:

  • Crocoddyl (Contact-Consistent Robot Co-dynamics)
  • Pinocchio (Rigid Body Dynamics)
  • mim_solvers (Fast solvers for optimal control)
  • mim_robots (Robot descriptions and wrappers)
  • pybullet, osqp, numpy, scipy, matplotlib, jupyterlab

Acknowledgements and References

Some KUKA examples in this repository are freely adapted from my companion repository minimal_examples_crocoddyl, which provides compact examples for using Crocoddyl.

This tutorial is also inspired by standard references in numerical optimization and numerical optimal control, in particular: