Deadlock-free, Safe, and Decentralized Multi-Robot Navigation in Social Mini-Games via Discrete-Time Control Barrier Functions

TL;DR

Compared to both fully decentralized and centralized approaches with and without deadlock resolution capabilities, this approach results in safer, more efficient, and smoother navigation, based on a comprehensive set of metrics including success rate, collision rate, stop time, change in velocity, path deviation, time-to-goal, and flow rate.

Abstract

We present an approach to ensure safe and deadlock-free navigation for decentralized multi-robot systems operating in constrained environments, including doorways and intersections. Although many solutions have been proposed that ensure safety and resolve deadlocks, optimally preventing deadlocks in a minimally invasive and decentralized fashion remains an open problem. We first formalize the objective as a non-cooperative, non-communicative, partially observable multi-robot navigation problem in constrained spaces with multiple conflicting agents, which we term as social mini-games. Formally, we solve a discrete-time optimal receding horizon control problem leveraging control barrier functions for safe long-horizon planning. Our approach to ensuring liveness rests on the insight that \textit{there exists barrier certificates that allow each robot to preemptively perturb their state in a minimally-invasive fashion onto liveness sets i.e. states where robots are deadlock-free}. We evaluate our approach in simulation as well on physical robots using F$1/10$ robots, a Clearpath Jackal, as well as a Boston Dynamics Spot in a doorway, hallway, and corridor intersection scenario. Compared to both fully decentralized and centralized approaches with and without deadlock resolution capabilities, we demonstrate that our approach results in safer, more efficient, and smoother navigation, based on a comprehensive set of metrics including success rate, collision rate, stop time, change in velocity, path deviation, time-to-goal, and flow rate.

Figures (15)

• Figure 1: Examples/Counter-examples of social mini-games: Arrows indicate the direction of motion for two agents $1$ and $2$ toward their goals marked by the corresponding cross. The first scenario is a social mini-game since both the preferred trajectories of agents $1$ and $2$ are in conflict from some $t=a$ to $t=b$ where $b-a \geq \delta$. The second and third scenarios are not social mini-games as there are no conflicts. In the second scenario, there is no duration where agents intersect one another. In the third scenario, agent $2$ has an alternate conflict-free preferred trajectory to fall back on.
• Figure 2: Deriving the threshold, $\ell_\textnormal{thresh}$, in a symmetrical social mini-game when $\ell_j(p_t^i,v_t^i) = 0$. Robots are blue car-like systems of length $l$. Green and red circles represent starting and goal positions, respectively. (left) Step $1$: construct the worst case scenario when $\ell_j(p_t^i,v_t^i) = 0$. The second robot on the right must reach $\mathcal{Q}$ when the robot on the left passes fully through $\mathcal{Q}$. (middle) Step $2$: Assuming the second robot slows down by $\frac{v}{1 + \frac{l}{d}}$, $\ell_\textnormal{thresh}$ is obtained via Equation \ref{['eq: l_thresh']}. (right) Symmetric SMGs with arbitrary angles with the relative position vector configurations can be reduced to the SMG in Figure \ref{['fig: geom2']}.
• Figure 3: Velocity projection for two symmetrical SMGs with arbitrary angles $\theta$ with respect to the relative position vector. In these two examples, we examine when $\theta = \frac{\pi}{6}$ and $\theta = \frac{\pi}{3}$ (although any arbitrary $\left \lvert \theta \right\rvert < \frac{\pi}{2}$ may be chosen).
• Figure 4: Flowchart of the deadlock detect-and-prevent logic. This diagram illustrates at a high level the process for detecting and preventing deadlocks in our approach. It includes the foundational definition of social mini-games, followed by the prevention strategies enforced by liveness conditions. The interaction between these elements is visualized through connections between theoretical results in Section \ref{['sec: GT-resolution']}.
• Figure 5: Liveness set $\mathscr{C}_\ell(t)$ for the $2$ robot scenario: Velocities for two deadlocked agents are shown by the red point $v_{t,I}$. To resolve the deadlock, Equation \ref{['eq: perturbation']} is used to project $v_{t,I}$ onto $v^1_t = 2 v^2$. Robot $1$ increases its velocity component $v^1$, while robot $2$ decreases its velocity component $v^2$ to align with the barrier. If robot $1$ deviates from $v_{t,I}$ and decreases its speed to the new yellow point $v_{t,II}$, the new optimal perturbation will be onto $v^2_t = 2 v^1$. Thus, robot $2$ adjusts its strategy by increasing its speed to align with a new perturbation. Assuming no speed deviations, there will be a unique projection to one of the safety barriers.

Theorems & Definitions (25)

• Theorem 1: Laumond et al.laumond2005guidelines
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 2
• proof
• Definition 5
• Theorem 3
• proof