Compatible CLF and CBFs

Verifying and synthesizing compatible Control Lyapunov Function (CLF) and Control Barrier Function (CBF) through Sum-of-Squares.

Example

A quadrotor stabilized by our compatible CLF-CBF-QP controller, avoiding the ground obstacle, flying to the target from different initial states.

quadrotor_compatible_clf_cbf.mp4

References

• Verification and Synthesis of Compatible Control Lyapunov and Control Barrier Functions
  Hongkai Dai*, Chuanrui Jiang*, Hongchao Zhang and Andrew Clark
  IEEE Conference on Decision and Control (CDC), 2024

Background

For a continuous-time control affine system $$\dot{x} = f(x) + g(x)u, u\in\mathcal{U},$$ we want to synthesize a controller that can stabilize the system, while also maintain safety. To this end, we will introduce Control Lyapunov Function (CLF) for stability, and Control Barrier Function (CBF) for safety.

The Control Lyapunov Function (CLF) $V(x)$ satisfies

$$ \exists u\in\mathcal{U}, L_f V(x) + L_g V(x)u \leq -\kappa_VV(x)$$

where $L_fV(x)$ is the Lie-derivative $\frac{\partial V}{\partial x}f(x)$, similarly $L_gV(x)$ is the Lie-derivative $\frac{\partial V}{\partial x}g(x)$. $\kappa_V>0$ is a given constant.

The Control Barrier Function (CBF) $h(x)$ satisfies

$$ \exists u \in\mathcal{U}, L_fh(x) + L_gh(x)u \geq -\kappa_hh(x),$$

$\kappa_h>0$ is a given constant.

Compatible CLF/CBF

We say a CLF $V(x)$ is compatible with a CBF $h(x)$ if there exists a common action $u$ that satisfies both the CLF condition and the CBF condition simultaneously

$$\exists u\in\mathcal{U}, \begin{cases}L_fV(x)+L_gV(x)u\leq -\kappa_VV(x)\\ L_fh(x)+L_gh(x)u\geq-\kappa_hh(x)\end{cases}.$$

We will certify and synthesize such compatible CLF and CBFs through Sum-of-Squares optimization.

Getting started

Installation

Create a virtual environment.

$ python -m venv venv
$ source venv/bin/activate

Run

$ pip install -e .

to install the dependencies and the package. After the installation, please make sure that you can use Mosek solver. Please set the environment variable that points to the Mosek license as

export MOSEKLM_LICENSE_FILE=/path/to/mosek.lic

Using Drake

We formulate and solve the optimization problem using Drake. To get started with Drake, you can checkout its tutorials (on the Drake webpage, navigate to "Resources"->"Tutorials"). Drake hosts several tutorials on deepnote, and there is a section of tutorials on Mathematical Programming. You might want to pay special attention to "Sum-of-squares optimization" as we will use it in this project.

Contributing

To maintain quality of the code, we suggest in each pull request, please do the followings

• Add unit tests. We use pytest framework.
• Format the code. You can run

  black ./
  

  to format each python code using black formatter.

Each pull request need to pass the CI before being merged. We also use reviewable to review the code.