The goal of this toolbox is to analyze empirically the optimization performance of several algorithms for nonlinear control from an optimization viewpoint on several synthetic benchmarks.

A companion report ilqc_algos is available and details the implementation of the algorithms in a common differentiable programming framework. A theoretical analysis of the algorithms under appropriate assumptions is presented in the manuscript ilqc_theory.

Consider controlling a system such as a car to make it perform a task such as racing a track in a finite time. The movement of the system is generally determined by a nonlinear differential equation driven by a control input. To determine an optimal control law for a given task determined by some costs, the system is usually discretized to define an optimization problem in a sequence of control variables. The resulting optimization problem is generally non-convex, yet, experiments demonstrate that nonlinear control algorithms tailored for such problems can exhibit fast convergence to reasonable or even optimal controllers. The objective of this toolbox is to study such phenomena.

Algorithms for discretized nonlinear control problems exploit the dynamical structure of the problem to compute candidate improved solutions at each step as detailed in ilqc_algos. These algorithms can be classified depending on the approximations they used on the problem at each iteration: linear/quadratic approximations of the dynamics, linear/quadratic approximations of the costs. The algorithms we consider in this toolbox are

• a gradient descent (linear approx. of the dynamics, linear approx. of the costs),
• a Gauss-Newton method and its Differentiable Dynamic Programming (DDP) variant (linear approx. of the dynamics, quadratic approx. of the costs),
• a Newton method and its DDP variant (quadratic approx. of the dynamics, quadratic approx. of the costs).

Furthermore, we consider several line-search strategies to update the current candidate solution (see ilqc_algos for more details). In addition to the implementation of the above algorithms for finite-horizon control, we implemented a model predictive controller for autonomous car racing on different tracks.

To study these algorithms we consider the task of swinging up a fixed pendulum or a pendulum on a cart and autonomous car racing with either a simplified model of a car or a bicycle model of a car. The latter is a model developed by Liniger et al, 2017 by carefully studying the real dynamics of a miniature car, see e.g. this video. We reimplemented these dynamics in Pytorch to further study the numerical performance of optimization algorithms in realistic settings.

To install the dependencies, create a conda environment with conda env create -f ilqc.yml Note that to visualize the environments, a specific version of pyglet needs to be installed, due to backward incompatible changes done by pyglet. Activate the environment, using conda activate ilqc and install pytorch (see https://pytorch.org/ to find the adequate command line for your OS); for example on a mac, do conda install pytorch -c pytorch.

To optimize for example the racing of a simple model of a car on a simpe track create a python file in the repository such as

import matplotlib.pyplot as plt
import seaborn as sns
import torch

from envs.car import Car
from algorithms.run_min_algo import run_min_algo

torch.set_default_tensor_type(torch.DoubleTensor)

# Create nonlinear control task
env = Car(model='simple', track='simple', cost='exact', reg_bar=0., horizon=50)

# Optimize the task with a DDP algorithm using linear quadratic approximations
cmd_opt, _, metrics = run_min_algo(env, algo='ddp_linquad_reg', max_iter=20)

# Visualize the movement
env.visualize(cmd_opt)

# Plot the costs along the iterations of the algorithm
sns.lineplot(x='iteration', y='cost', data=metrics)
plt.show()

A set of default experiments is present in finite_horizon_control/fhc_example.py. You can simply run the file from the root of the repository using python finite_horizon_control/fhc_example.py to observe optimal controllers for the tasks described above.

To observe a controller computed by a model predicitve control approach run python model_predictive_control/mpc_example.py.

A set of notebooks is avaialble to describe the models considered for modeling autonomous car racing with different nonlinear control algorithms in ilqc.ipynb. To run them, simply type jupyter notebook from the root of the folder after having activated the environment.

The experiments presented in ilqc_algos can be reproduced by running python finite_horizon_control/compa_algos.py. Output figures are saved in the folder finite_horizon_control. For ease of test, the results have been saved in advance. If you prefer to rerun all experiments simply erase the folder results.

You can report issues and ask questions in the repository's issues page. If you choose to send an email instead, please direct it to Vincent Roulet at [v][my_last_name]@google.com and include [ilqc] in the subject line.

If this software is useful for your research, please consider citing it as

@article{roulet2022iterative,
  title={Iterative Linear Quadratic Optimization for Nonlinear Control: Differentiable Programming Algorithmic Templates},
  author={Roulet, Vincent and Srinivasa, Siddhartha and Fazel, Maryam and Harchaoui, Zaid},
  journal={arXiv preprint arXiv:2207.06362},
  year={2022}
}

You may also consider taking a look at the following packages:

• Trajax: Nonlinear control algorithms in JAX
• Aligator: An efficient and versatile trajectory optimization library for robotics and beyond.

Vincent Roulet
Siddhartha Srinivasa
Maryam Fazel
Zaid Harchaoui

This code has a GPLv3 license.