Mini Project Webpage

Project Details

Topic: Multi-Agent Safety-Critical Control for Complex Safety Requirements via Non-Smoooth Barrier Certificates.

Student: Clinton Enwerem (enwerem at umd dot edu)

Motivation & Summary

Intelligent cyber-physical systems, such as teams of aerial robots or autonomous vehicle platoons, face a fundamental challenge: they must safely navigate complex environments while completing their objectives efficiently. This challenge is non-trivial, as the sub-tasks that constitute the ensemble's objective may often be conflicting, so that it becomes imperative to conduct appropriate trade-offs between tasks. The inherent uncertainty in each agent's model and operational vicinity also imposes additional constraints on the multiagent system, making safe multiagent control an acute problem.

To further motivate the importance of safety, consider a platoon of connected autonomous vehicles on a busy highway, dispatched during trial runs to collect real-world driving data and validate new software. Besides achieving this tasking, the vehicles must remain within communication range, avoid accidents, and adhere to traffic regulations while also maintaining rider comfort. Despite sporting an array of exotic sensors and computing hardware, the vehicle ensemble may fail to achieve this mission under unfamiliar and challenging settings, leading to catastrophic [1] and sometimes fatal [2] consequences. Endowing intelligent collectives with the ability to safely complete missions with competing requirements is, thus, an important consideration.

In this project, we consider the problem of synthesizing control policies for collectives subject to complex safety constraints using Boolean logical composition of control barrier functions (CBFs) and model-predictive control. Due to the general non-smooth nature of the resulting Boolean-composed CBF, we also explore smoothing strategies to enable optimization-based control synthesis. Finally, as a proof of concept, we hope to validate the above ideas on a single-integrator collective.

Mathematical Preliminaries & Problem Definition

We consider nonlinear affine-in-control systems represented by dynamics of the form (subscript \(i\) denotes the \(i^{th}\) agent): \[\dot{x}_i = f(x_i) + g(x_i)u_i, \ i = 1, 2, \dots, N,\tag{1}\] with \(x_i \in \mathfrak{X} \subset \mathbb{R}^d\) and \(u_i \in \mathfrak{U} \subseteq \mathbb{R}^m\) denoting the state and control inputs, respectively. Admissible control signals take values in the set, \(\mathfrak{U}\), of piecewise continuous and absolutely-integrable functions. \(f\) and \(g\) are the drift and control vector fields of appropriate dimension, respectively, taken to be locally Lipschitz continuous. Let \(\alpha(\cdot)\) be a \(\kappa\)-class function, that is, \(\alpha(\cdot)\) is strictly increasing with \(\alpha(0) = 0\). Also let \(\mathfrak{C}\) denote the safe set (with boundary \(\mathfrak{C}\) and interior \(\text{Int}(\mathfrak{C})\)) for the ensemble, that is, the set of all desirable values of \(\mathrm{x} = {[x_i^T]^T}, i=1, 2, \dots, N\) (which we will call the ensemble state), for which the safety conditions are met. Letting \(h\) denote a sufficiently-smooth function, we can characterize \(\mathfrak{C}\) using the zero superlevel set of \(h\) as \[ \mathfrak{C} = \{\mathrm{x} \in \mathbb{R}^{Nd} : h(\mathrm{x}) \ge 0\}.\tag{2}\] Denoting the Lie derivatives of \(h(\cdot)\) along \(f\) and \(g\) respectively as \(L_fh(\cdot)\) and \(L_gh(\cdot)\), we say that \(h\) is a control barrier function (CBF) [3] defined on \(\mathfrak{C}\) if there exists a \(\kappa\)-class function, \(\alpha\) such that \[L_fh(\mathrm{x}) + L_gh(\mathrm{x})\mathrm{u} - \alpha(h(\mathrm{x})) \le 0.\tag{3}\] Given the foregoing background, we are interested in synthesizing controls \(\mathrm{u} \in \mathbb{R}^{Nm}\) for which the ensemble respects \(|\mathcal{I}|\) safety constraints, each encoded by a safety set, \(\mathfrak{C}_\star\), with corresponding barrier function, \(h_\star\), that is \[ \mathfrak{C}_\star = \{\mathrm{x} \in \mathbb{R}^{Nd} : h_\star(\mathrm{x}) \ge 0\}, \star = 1, 2, \dots, |\mathcal{I}|. \tag{4}\]

Key Results & Approach

It is widely known that affine-in-control nonlinear dynamical systems can be forced to desired states via controllers synthesized from CBF-constrained quadratic programs [3]. However, when several safety requirements must be met, one runs the risk of landing at an intractable quadratic program. Performing minimum or maximum operations over the set of associated CBFs also poses the challenge of being non-smooth so that one cannot apply a gradient-based synthesis approach. Moreover, composing CBFs via Boolean logic also leads one to a non-smooth CBF [4], motivating the need for gradient-free or appropriate smoothing techniques.

Thus, to synthesize controllers satisfying the foregoing requirements, we will adopt the following approach, similar in spirit to [4] and described (roughly) as follows:

  1. Elect barrier functions capturing each safety requirement and perform compositions via Boolean logic to obtain a single CBF for the ensemble.
  2. Consider the system as an ensemble and adopt robust model-predictive control to find controllers satisfying (4).

Connections to Robot Decision-Making & General Applications

Since robots are embodied autonomous agents fitted with components to interact with the material world, it is important to incorporate safety constraints into their planning and decision-making modules. The need for safety becomes all the more important in robot teams, where safe coordination at the agent level does not necessarily translate to the same for the ensemble. In the literature, a common approach for ensuring multi-agent safety is through the imposition of constraints for collision avoidance. However, as recent mishaps involving autonomous vehicles have shown [1, 2], notions of safety in practical settings are typically not singular and may often be conflicting. And while CBFs are a popular technique for safety-aware controller synthesis, little research is available in the literature on their utility in multi-agent systems tasked with complex safety requirements, thus motivating the focus for this project.

While we aim to consider two independent constraints in this project, we note that real-life safety constraints will mostly be more involved. However, the methods here can easily be scaled to address more complicated constraints, captured by appropriate CBFs. Nevertheless, progress in the aforementioned front can be applied to collective robot control under varied tasking, autonomous vehicle platooning, and time-optimal, resource-robust, and fault-tolerant control for networked cyber-physical systems, among other applications.

Open Problems

A number of open problems exist in CBF-enabled controller synthesis, mostly due to the technique's reliance on strong deterministic assumptions viz. affine-in-control dynamics, convex objectives, and well-defined tasks or constraints encoded by smooth functions. However, as already alluded to earlier, even under these convenient assumptions, extending CBF thinking to autonomous ensembles with complex tasking is not straightforward. And while one can construct singular CBFs that capture several requirements in one function, such functions are usually non-smooth. Learning-based methods circumvent this challenge by reconstructing the CBF directly from system trajectories via an appropriate neural network architecture and loss function, or defining safety constraints in terms of graphical objects, as in the recent studies on Graph CBFs.

  • [1] B. Wessling, “Cruise recalls 300 robotaxis in response to crash with bus,” The Robot Report. Accessed: Oct. 4, 2023. [Online]. Available:

  • [2] “17 fatalities, 736 crashes: The shocking toll of Tesla’s Autopilot,” Washington Post. Accessed: Oct. 04, 2023. [Online]. Available:

  • [3] Aaron D. Ames, Jessy W. Grizzle, and Paulo Tabuada. Control barrier function based quadratic programs with application to adaptive cruise control. In 53rd IEEE Conference on Decision and Control, pages 6271–6278, December 2014. ISSN: 0191-2216.

  • [4] Paul Glotfelter, Jorge Cortes, and Magnus Egerstedt. Nonsmooth Barrier Functions With Applications to Multi-Robot Systems. IEEE Control Systems Letters, 1(2):310–315, October 2017.