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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 \[ \mathfrak{C}_\star = \{\mathrm{x} \in \mathbb{R}^{Nd} : h_\star(\mathrm{x}) \ge 0\}, \star = 1, 2, \dots, |\mathcal{I}|. \tag{4}\] |
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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:
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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. |
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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. |
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