Collaborative Safety-Critical Formation Control with Obstacle Avoidance

TL;DR

This work forms a control barrier function (CBF) based safety filter control law for a generic distributed formation controller and extends the previously developed collaborative safety framework to an obstacle avoidance problem for agents with acceleration control inputs, and incorporates multi-obstacle collision avoidance into the collaborative safety framework.

Abstract

This work explores a collaborative method for ensuring safety in multi-agent formation control problems. We formulate a control barrier function (CBF) based safety filter control law for a generic distributed formation controller and extend our previously developed collaborative safety framework to an obstacle avoidance problem for agents with acceleration control inputs. We then incorporate multi-obstacle collision avoidance into the collaborative safety framework. This framework includes a method for computing the maximum capability of agents to satisfy their individual safety requirements. We analyze the convergence rate of our collaborative safety algorithm, and prove the linear-time convergence of cooperating agents to a jointly feasible safe action for all agents under the special case of a tree-structured communication network with a single obstacle for each agent. We illustrate the analytical results via simulation on a mass-spring kinematics-based formation controller and demonstrate the finite-time convergence of the collaborative safety algorithm in the simple proven case, the more general case of a fully-connected system with multiple static obstacles, and with dynamic obstacles.

Figures (6)

• Figure 1: An example of when agent $i \in [n]$ is constrained by two neighbors, where $u_i \in \mathbb{R}^2$. The constrained control space for agent $i$, $\mathcal{U}_i^s$, is shaded green, with the feasibly safe control actions for neighbors 1 and 2 shaded in blue and red, respectively. Both $\overline{\mathcal{U}}_{1i}$ and $\overline{\mathcal{U}}_{2i}$ are individually feasible, but jointly infeasible, with the set of feasibly safe control actions for neighbors 1 and 2 shown in the purple-shaded region. The compromise-seeking action $\overline{u}_i \in \mathbb{R}^2$ chosen by Algorithm \ref{['alg:getClosestPoint']} is marked on the boundary of $\mathcal{U}_i^s$, which is the closest action in $\mathcal{U}_i^s$ to the feasibly safe control actions for both neighbors $\overline{\mathcal{U}}_{1i} \cap \overline{\mathcal{U}}_{2i}$.
• Figure 2: An illustration of Assumptions \ref{['assume:tree_structure']} and \ref{['assume:only_1hop_requests']} for a given system, where, at the start of the first round of negotiation, requests are sent by the 1-hop neighborhood $\mathcal{N}_i$ (orange) of only one root agent $i \in [n]$ (blue), which are received by both $i$ and the 2-hop neighborhood (green) of $i$.
• Figure 3: The trajectories of a 7-agent tree-structure formation network avoiding a single obstacle, where each agent is given a constant control signal directing it in the positive $x$ direction. Each agent implements safety filtering according to Algorithm \ref{['alg:colab_safety']} and \ref{['eq:problem_statement']} to avoid the obstacle while maintaining a formation behavior, according to \ref{['eq:formation_dynamics_mass_spring']} and \ref{['eq:formation_controller_mass_spring']}.
• Figure 4: The safety-filtered control signals for each agent in the $\vec{x}$ component (top) and $\vec{y}$ component (middle) of $u_i^s$, which are computed using Algorithm \ref{['alg:colab_safety']} and \ref{['eq:problem_statement']}, during the traversal of the formation around the obstacle shown in Figure \ref{['fig:obs_trajectory_7a']}. We also show the number of collaborative rounds $\tau$ (bottom) at each time-step of the simulation, where the sampling rate is 100 Hz.
• Figure 5: The trajectories of an 8-agent fully-connected formation network avoiding an obstacle field, where each agent is given a constant control signal directing it in the positive $x$ direction. Each agent implements safety filtering according to Algorithm \ref{['alg:colab_safety']} and \ref{['eq:problem_statement']} to avoid obstacles while maintaining a formation behavior, according to \ref{['eq:formation_dynamics_mass_spring']} and \ref{['eq:formation_controller_mass_spring']}.

Theorems & Definitions (36)

• Definition 1
• Lemma 1
• Definition 2
• Lemma 2
• proof
• Definition 3
• Definition 4
• Lemma 3
• proof
• Lemma 4