Non-smooth Control Barrier Functions for Stochastic Dynamical Systems

TL;DR

This paper provides formal guarantees on the safety of the system by leveraging the theoretical foundations of stochastic CBFs and non-smooth safe sets and enables the design of safe control strategies in uncertain and complex systems.

Abstract

Uncertainties arising in various control systems, such as robots that are subject to unknown disturbances or environmental variations, pose significant challenges for ensuring system safety, such as collision avoidance. At the same time, safety specifications are getting more and more complex, e.g., by composing multiple safety objectives through Boolean operators resulting in non-smooth descriptions of safe sets. Control Barrier Functions (CBFs) have emerged as a control technique to provably guarantee system safety. In most settings, they rely on an assumption of having deterministic dynamics and smooth safe sets. This paper relaxes these two assumptions by extending CBFs to encompass control systems with stochastic dynamics and safe sets defined by non-smooth functions. By explicitly considering the stochastic nature of system dynamics and accommodating complex safety specifications, our method enables the design of safe control strategies in uncertain and complex systems. We provide formal guarantees on the safety of the system by leveraging the theoretical foundations of stochastic CBFs and non-smooth safe sets. Numerical simulations demonstrate the effectiveness of the approach in various scenarios.

Figures (5)

• Figure 1: Illustration of a drone operating in a cluttered environment represented as a 2D occupancy grid map. The safe set is defined by an ESDF which is positive everywhere in the collision-free space. A stochastic wind disturbance acting on the drone is colored in red.
• Figure 2: Illustration of the partition-based CBF in a two dimensional state space $\mathcal{X}$. The dashed lines indicate the edges between partitions $\Phi_i$ and $\Phi_j$. An exemplary trajectory satisfying the safety constraint is shown in blue while the states at exit times $\tau_k$ are colored in red.
• Figure 3: Simulation results for a Boolean safety specification. The network coverage areas are shown by green circles located at $\mathcal{N}_1 = [0, -0.2]^T$ and $\mathcal{N}_2 = [0.5, 1.8]^T$ with radii $r_1=1.1$ and $r_2=1.4$, respectively. The obstacle is located at $\mathcal{O}=[1.2, 0.4]^T$ with $r_o=0.6$ and illustrated as a red circle. The left plot shows the stochastic evolution of the system if only the reference proportional controller is used while the right plot depicts the behavior when using the proposed NSCBFs.
• Figure 4: Simulation results for the multi agent swapping scenario. The agents initial positions are shown by the colored circles.
• Figure 5: Exemplary control input $u_x$ of agent one over time for two different settings. On the left, only the barrier condition in the current partition is satisfied while the right figure shows the control signal for an almost-active set.

Theorems & Definitions (10)

• Theorem 1: Thm. 3.21, leobacher2017strong
• Definition 1: Itô's Lemma
• Definition 2
• Lemma 1
• proof
• Definition 3
• Theorem 2
• proof
• Theorem 3
• proof