Distributed Safe Control Design and Probabilistic Safety Verification for Multi-Agent Systems

TL;DR

The proposed distributed algorithm addresses infeasibility issues of existing schemes via a cooperation mechanism between agents via a cooperation mechanism between agents, and quantifies safety for multi-agent systems probabilistically by means of CBFs.

Abstract

We propose distributed iterative algorithms for safe control design and safety verification for networked multi-agent systems. These algorithms rely on distributing a control barrier function (CBF) related quadratic programming (QP) problem assuming the existence of CBFs. The proposed distributed algorithm addresses infeasibility issues of existing schemes via a cooperation mechanism between agents. The resulting control input is guaranteed to be optimal, and satisfies CBF constraints of all agents. Furthermore, a truncated algorithm is proposed to facilitate computational implementation. The performance of the truncated algorithm is evaluated using a distributed safety verification algorithm. The algorithm quantifies safety for multi-agent systems probabilistically by means of CBFs. Both upper and lower bounds on the probability of safety are obtained using the so called scenario approach. Both the scenario sampling and safety verification procedures are fully distributed. The efficacy of our algorithms is demonstrated by an example on multi-robot collision avoidance.

Figures (9)

• Figure 1: Pictorial illustration of a connected network $\mathcal{G}$ with 9 agents, where agents $1, 2,$ and $3$ form the sub-network $\mathcal{G}_1$ with safe set $\mathcal{S}_1$, agents $1, 4, 5, 6$ and $7$ form the sub-network $\mathcal{G}_2$ with a safe set $\mathcal{S}_2$, and agents $7, 8,$ and $9$ form the sub-network $\mathcal{G}_3$ with a safe set $\mathcal{S}_3$. The set of agents in each sub-network is given by $\mathcal{V}_1=\{1,2,3\}$, $\mathcal{V}_2=\{1,4,5,6,7\}$, and $\mathcal{V}_3=\{7,8,9\}$. It can be observed that agent $1$ belongs to two sub-networks, $\mathcal{G}_1$ and $\mathcal{G}_2$, and thus $\mathcal{C}_1=\{1,2\}$. Similarly, agent $2$ belongs only to $\mathcal{G}_1$, and agent $7$ belongs to both $\mathcal{G}_2$ and $\mathcal{G}_3$, giving $\mathcal{C}_2=\{1\}$ and $\mathcal{C}_7=\{2,3\}$.
• Figure 2: Trajectory of ten robots swapping positions according to Algorithm \ref{['al:dcbf']}. Robots with the same color are swapping positions, and avoiding collision with the others.
• Figure 3: Trajectory of ten robots swapping positions while avoiding collision by means of Algorithm \ref{['al:truncated']}, with $\eta=30$.
• Figure 4: Evolution of the relaxation parameters $\rho_{\mathrm{sum}}^0(\boldsymbol{x})$ and $\rho_{\mathrm{sum}}^{30}(\boldsymbol{x})$ evaluated at the state trajectory, across algorithm iterations.
• Figure 5: Bar graph for the violation probability, and (with dashed lines) the theoretical bounds $[\underline{\epsilon},\overline{\epsilon}]$.