CBF_Obstacle_Avoidance

Discription

This collection of MATLAB scripts intends to study the performance of state-constrained controllers utilizing control barrier functions in the context of obstacle avoidance.

Concept of Control Barrier Functions

We consider a linear plant with parametric uncertainties of the form:

$$\begin{equation} \dot{x}(t) = f(x(t)) + g(x) u(t) \end{equation}$$

where $x(t) \in \mathbf{R}^{n}$ is a measurable state vector and $u(t) \in \mathbf{R}^{m}$ is a control input vector. The control input is assumed to be magnitude limited by $\vert u_0 \vert$, which leads to the following closed set for the control input space

$$\begin{equation} \mathcal{U} = \begin{Bmatrix} u \in \mathbf{R}^{m} : - u_0 \geq u(t) \geq u_0 \end{Bmatrix} \end{equation}$$

The objective is to determine a $u(t)$ for \eqref{LinearPlantModel} such that the plant state $x(t)$ tracks a desired reference $x_d(t)$ and that for any initial condition $x_0 := x(t_0) \in S$, it is ensured that the plant state vector $x(t)$ stays within the safe set $S \in \mathbf{R}^n$ i.e. the control input ensures that there is a CBF with $h(x,u) \geq 0$ for $\forall t \geq 0$.

A superlevel set $\mathcal{C} \in \mathbf{R}^n$, which we refer to as a safe set, is defined in the following form:

$$\begin{equation} \mathcal{C} = \begin{Bmatrix} x \in \mathbf{R}^n : h(x) \geq 0 \end{Bmatrix} \end{equation}$$

with $h: \mathbf{R}^n \times \mathbf{R}^p \to \mathbf{R}$ being a continuously differentiable function, called control barrier function (CBF).

The CBF can ensure for the presented control affine system that for any initial condition $x_0 := x(t_0) \in \mathcal{C}$, that $x(t)$ stays within $\mathcal{C}$ for any $t$, if there exists an extended class $\mathcal{K}$ functions $\alpha$ such that for all $x \in Int(\mathcal{C})$

$$\begin{equation} \sup_{u \in U} [L_f B(x) + L_g B(x) u + \alpha(h(x))] \geq 0 \end{equation}$$

Where $\alpha(h(x)$ often chosen to be

$$\begin{equation} \alpha(h(x)) = \gamma h(x) \end{equation}$$

with $\gamma$ being $\gamma > 0$.

Pointwise Controller and CBF-QP Safety Filter

A linear controller of the following form is defined:

$$\begin{align} u_d = -K \tilde{x} \end{align}$$

with $K$ being Hurwitz and $\tilde{x} = x - x_d$. The linear controller is tuned regarding the desired control performance but cannot generate safe commands by itself. Therefore the following CLF-QP safety filter is used to adapt $u_d$ such that $x(t)$ stays within $\mathcal{C}$ for any $t$.

$$\begin{align} &\min_{u \in \mathcal{U}} \begin{aligned}[t] &|u - u_d|_2 \end{aligned} \\ &\text{s.t.} \notag \\ & L_f h(x) + L_g h(x)u - \gamma h(x) \leq 0, \notag\\ & -u_0 \geq u \geq u_0, \notag \end{align}$$

After each simulation run, a plot with results is given out. An example of such a plot is given here:

Dependencies

The scripts use external libraries, which need to be installed.

• Export_fig

Further, the following MATLAB toolboxes are needed:

• Optimization Toolbox

The software was tested with MATLAB 2020b under Windows 11 Home.

License

The contents of this repository are covered under the MIT License.

License

The contents of this repository are covered under the MIT License.

References

We kindly acknowledge the following papers, which have been the foundation of the here presented scripts:

• [1] J. Zeng, B. Zhang, and K. Sreenath, “Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function,” in 2021 American Control Conference (ACC), May 2021, pp. 3882–3889.
• [2] A. D. Ames, X. Xu, J. W. Grizzle and P. Tabuada, "Control Barrier Function Based Quadratic Programs for Safety Critical Systems," in IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861-3876.