paolofrance/differential_gt

The solution of the two-players Linear-Quadratic Differential Game Theoretic problem, for the Cooperative and the Non-Cooperative cases

The differential GT package

This package implements the solution for the two-players Linear-Quadratic differential game-theoretic problem.

This implementation has been used in the works proposed in "Franceschi, Paolo, Nicola Pedrocchi, and Manuel Beschi. Human–Robot Role Arbitration via Differential Game Theory. IEEE Transactions on Automation Science and Engineering (2023).", and in Modeling and analysis of pHRI with Differential Game Theory.

This repository addressed the solution of the Non-Cooperative case, the so-called Nash Equilibrium, and the solution of the Cooperative case, the Pareto frontier.

Namely, it addresses the NC problem described as

$$\begin{split} & \min_{u_1}J_{1} (z,u_1,u_2) \\ & \min_{u_2}J_{2} (z,u_1,u_2) \\ s.t. & ; \dot{z} = Az+B_1 , u_1 + B_2, u_2 \\ & z(t_0) =z_0 \end{split}$$

and the Cooperative

$$\begin{split} & \min_{u}J_{c} = \int_{0}^{\infty}( z^T,Q_{c},z + u^T,R_{c},u ),dt \\ & s.t. ; \dot{z} = A_{c},z_{c}+B_{c}u_{c} \\ & z_c(t_0) =z_0 \end{split}$$

You can find more details in the related publications here and here, and in Engwerda, Jacob. LQ dynamic optimization and differential games. John Wiley & Sons, 2005..

Citation

If you are using this repository, please consider citing it as

@ARTICLE{10275780,
author={Franceschi, Paolo and Pedrocchi, Nicola and Beschi, Manuel},
journal={IEEE Transactions on Automation Science and Engineering}, 
title={Human–Robot Role Arbitration via Differential Game Theory}, 
year={2023},
volume={},
number={},
pages={1-16},
doi={10.1109/TASE.2023.3320708}}

TODO

1. Better documentation
2. purge from ros

Contacts

if you have any questions about this repository, are interested in applications, want to contribute or anything else, please contact me at [email protected] .

Acnowledgements

This repository was developed within the EU Horizon 2020 DrapeBot project context.