Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability

This repository contains the code, data and plots of the paper Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability. The paper has been submitted at the ICRA 2024 Conference.

Abstract

Optimal behaviours of a system to perform a specific task can be achieved by leveraging the coupling between trajectory optimization, stabilization and design optimization. This approach proves particularly advantageous for underactuated systems, which are systems that have fewer actuators than degrees of freedom and thus require for more elaborate control systems. This paper proposes a novel co-design algorithm, namely Robust Trajectory Control with Design optimization (RTC-D). An inner optimization layer (RTC) simultaneously performs direct transcription (DIRTRAN) to find a nominal trajectory while computing optimal hyperparameters for a stabilizing time-varying linear quadratic regulator (TVLQR). RTC-D augments RTC with a design optimization layer, maximizing the system’s robustness through a time-varying Lyapunov-based region of attraction (ROA) analysis. This analysis provides a formal guarantee of stability for a set of off-nominal states.

The proposed algorithm has been tested on two different underactuated systems: the torque-limited simple pendulum and the cart-pole. Extensive simulations of off-nominal initial conditions demonstrate improved robustness, while real-system experiments show increased insensitivity to torque disturbances.

Content

In the examples folder the user can find all the code necessary to generate the data and plots that have been included in the paper. Running

python examples/simple_pendulum/resultsPlots.py
python examples/cart_pole/resultsPlots.py 

will show the final results of our approach. The main algorithm RTC-D and it's reduced version RTC are implemented for each system in the rtc_CMAES.py and rtcd_CMAES.py files respectively.

A set of example code is provided both for simple pendulum and cart-pole. Both the simulated and the experimental verification of the obtained results can be visualized by running the verificationPlots.py script. ROA estimation examples are implementedted in the lqr_roa.py and tvlqr_roa.py files. Also, a direct transcription trajectory optimization is included in a specific file named dirtran.py.

Installation

The use of a virtual environment for the installation is suggested as a common good programming choice. For example, the use of pipenv requires the following commands

pipenv shell
pipenv install software/python
pipenv install 'drake==1.5.0'    

A LaTeX distribution installation might be necessary for the correct visualization of the results. In case of error due to the version of a library please refer to versions available from March 2022.

Results

Torque-limited simple pendulum:

Cart-pole:

Aknowledgements and Citation (Preprint)

Code contained in software/python has been inspired by pre-existing software of DFKI GmbH. It's usage has been agreed with the owner.

Copyright on the material in this webpage has been transferred to IEEE for ICRA 2024: F. Girlanda, L. Maywald, S. Kumar and F. Kirchner, "Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability," 2024 IEEE International Conference on Robotics and Automation, Yokohama, Japan, 2024
A pre-print is now available in arXiv, you can cite it with the following BibTeX citation:

@misc{girlanda2024robust,   
  title={Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability},   
  author={Federico Girlanda and Lasse Shala and Shivesh Kumar and Frank Kirchner},  
  year={2024},  
  eprint={2403.10966},  
  archivePrefix={arXiv},  
  primaryClass={cs.RO}  
  }