Safe and Stable Formation Control with Distributed Multi-Agents Using Adaptive Control and Control Barrier Functions

Multi-agent systems (MAS) have received much attention because of their potential in completing tasks that a single agent could not complete efficiently on their own. Examples include exploration, surveillance, reconnaissance, rescue, and failure-tolerance, which occur in various problems related to motion planning and robotics. Typical problems in the context of MAS include consensus, formation control, coordination, and synchronization. This paper pertains to formation control.

The concept of formation control can be classified into the formation tracking and formation producing. In the former, the agents maintain a desired trajectory while the configuration itself moves through space; this is of interest for problems related to rendezvous in air and space. In the latter, the objective is to converge to a static formation from some initial configuration of the agents, which is useful in tasks of surveillance and more generally sensor deployment. Our paper focuses on the latter. We will focus on a class of dynamic MAS which are subjected to parametric uncertainties, state-space constraints, and limited communication among the agents, and the goal is to achieve a real-time control solution that accomplishes a static formation task.

Several approaches have been reported in the literature to achieve formation control, but with a subset of the above features. The approaches in [1], [2], and [3] have addressed the formation control problem in the presence of parametric uncertainty using adaptive control. No constraints on the state or in the communication among agents are however considered. The solutions in [4], [5], and [6] focus on formation control with limited communication and parametric uncertainties, but do not address obstacles or other state-space constraints. The authors of [7, 8, 9, 10, 11, 12, 13] consider obstacles and formation control with limited communication, but assume full knowledge of the agent dynamics. The authors of [14] and [15] consider obstacles and parametric uncertainty but assume full communication between all agents.

In contrast to the above papers, the authors of [16, 17, 18] have addressed, as in this paper, formation control for distributed MAS amidst obstacles and parametric uncertainties. However the following distinctions can be made: The results in [16] limit the uncertainty to a constant additive disturbance that is unknown. The solutions in [17] and [18] consider general nonlinear dynamics that is unknown, and employ a neural-network based solution. In this paper, we propose an adaptive control solution for static formation control in the presence of parametric uncertainties, obstacles which introduce safety considerations, and limited communication among agents. The unique features of our proposed solution are (i) the use of a Graph Control Barrier Function that shapes the reference input so as to ensure safety, (ii) the use of a reference model that leverages the communication structure among the distributed MAS, and (iii) an analytically rigorous stability proof that guarantees boundedness of all signals and convergence to the desired formation.

Preliminaries are presented in Section II and the control problem in Section III. The main contribution of this manuscript, the development of an adaptive controller with stability and safety properties, is presented in Section IV. Section V includes numerical simulations. Proofs of the main contributions of the manuscript (Theorems 2 and 3) are presented in the appendix.

The problem we consider is this paper is the static formation control of distributed MAS consisting of $M$ agents indexed by $i\in\mathcal{V}=\{1,\ \cdots,M\}$, in the presence of parametric uncertainties and obstacles, with the goal of ensuring stability and safety. The dynamics of agent $i$ are given by

where $x_{i}(t)\in\mathbb{R}^{n}$ is the state of the agent, $u_{i}(t)\in\mathbb{R}^{m}$ is the control input of the agent. $A\in\mathbb{R}^{n\times n}$ is unknown, $\Lambda\in\mathbb{R}^{m\times m}$ is an unknown diagonal matrix with known sign, and $B\in\mathbb{R}^{n\times m}$ is known and it has full column rank. The term $\Xi\omega$ corresponds to nonlinearities present in the system where $\omega(x_{i},t)\in\mathbb{R}^{p}$ is known, but $\Xi\in\mathbb{R}^{m\times p}$ is unknown. For ease of exposition, we assume that $A,\ \Lambda$ and $\Xi$ are independent of $i$; extensions to the case when they depend on $i$ are straightforward. We introduce the following assumptions regarding the unknown parameters and nonlinearities in the dynamics:

Assumption 1: The matrix $\text{sign}(\Lambda)\Lambda$ is diagonal and positive definite, $\text{sign}(\Lambda)\Lambda\in\mathbf{S}_{++}^{m}$.

Assumption 2: The nonlinearity $\omega(x_{i},t)$ is a bounded signal for all $t\geq t_{0}$.

Assumption 3: A constant matrix $\theta^{\star}_{1}\in\mathbb{R}^{m\times n}$ exist such that

where $A_{m}\in\mathbb{R}^{n\times n}$ is a known Hurwitz matrix (Section IV-A1).

In addition to the parametric uncertainties, we assume that the problem has safety considerations in the form of state constraints. We now introduce the following definition that pertains to the safety of MAS.

Definition 5 [12]: A continuously differentiable function $h:\mathcal{X}^{M}\rightarrow\mathbb{R}$ is referred as a Graph Control Barrier Function (GCBF) if there exists $\alpha(h(\mathbf{x}_{\mathcal{N}_{i}}(t)))\in\mathcal{K}_{\infty}$ and a control law $k_{i}:\mathcal{X}^{M}\rightarrow\mathcal{U}\subseteq\mathbb{R}^{m}$ for each agent $i\in\mathcal{V}$ of the MAS, such that

where

for $u_{j}(t)=k_{j}(\mathbf{x}_{\mathcal{N}_{i}}(t))$ where $\mathbf{x}_{\mathcal{N}_{i}}$ is the joint state of the agents in neighbor set $\mathcal{N}_{i}$.

The state constraints are captured by the safe set $S_{i}$ for each agent $i\in\mathcal{V}$ which corresponds to the obstacle-free region of the state space. Associated with this safe set is a GCBF $h(x_{i})$, which implies the existence of a $u_{i}$ in (6) that guarantees (8). We assume that the GCBF $h$ satisfies conditions in [23] such that the resulting controller is Lipschitz.

The overall problem statement is therefore the following: given that the initial condition $x_{i}(t_{0})\in\mathcal{S}_{i}$ for all $i\in\mathcal{V}$ and a desired position in the static formation $x_{i}^{\star}\in\mathcal{S}_{i}$ for all $i\in\mathcal{V}$, the problem is to find a control policy of the form

that guarantees that $x_{i}(t)\in\mathcal{S}_{i}$ for all $t\geq t_{0}$, and $x_{i}(t)\rightarrow x_{i}^{\star}$ in the presence of unknown parameters. In addition, we require that each control policy $k_{i}:[\mathbb{R}^{n}]^{|\mathcal{N}_{i}|+1}\rightarrow\mathbb{R}^{m}$ of agent $i$ must depend only on its own state $x_{i}$ and the state of neighboring agents $x_{j}$, for $j\in\mathcal{N}_{i}$.

The controller that we propose includes an adaptive component and a safety filter. In order to guide the adaptive control design, a reference model is suitably chosen. In Section IV-A, we design such a reference model, the corresponding adaptive controller, and show that the adaptive controller can enable the MAS to reach a desired formation. In Section IV-B, we integrate a safety filter into the whole adaptive control design and show that the desired formation can be reached even while ensuring the safety constraints. The corresponding results are stated in Theorem 2, Corollary 1, and Theorem 3.

The proposed controller is applied to a two-dimensional obstacle avoidance problem. The dynamics of each agent are given by

such that $x_{i}$ and $y_{i}$ are respectively the horizontal and vertical positions of the agents, actuation has been compromised ($\Lambda=0.7I$) and the nonlinearities are characterized by

The reference model is defined by

and the communication graph shown in Fig. 2. To guarantee the safety of each agent, the following CBF constraint is chosen for each obstacle

where $(\bar{x}_{l},\bar{y}_{l})$ and $r_{l}$ are the position and radius of the l-th circular obstacle.

The parameters $\hat{\theta}_{k}^{(i)}$, for all $i\in\mathcal{V}$ and $k=\{1,2,3\}$, are initialized assuming the agents dynamics are given by

where $\bar{A}=0.16I$, $\bar{\Lambda}=I$, and

Fig. 3 shows the trajectories of the agents using only the adaptive control described in Section IV-A2 without any safety filter. Fig. 4 shows the trajectories of the agents with the modified QP-GCBF filter in (25) but without adaptation. Finally, Fig. 5 shows our proposed integrative adaptive controller with the modified GCBF and the graph-based reference model as in (11). For the last figure, a $E_{h}=0.6$, $\alpha_{0}=15$ and $\upsilon=0.05$ were chosen. The superior performance of our proposed controller compared with adaptation but no safety as well as safety but with no adaptation is clear from these figures. When the adaptive controller is employed but without any CBF, agent 2 collides with an obstacle along the trajectory (Fig. 3). Only using the modified GCBF but without adaptation allows the MAS to remain safe, but the agents do not reach the desired formation (Fig. 4). In contrast, with our proposed approach in this paper, agents reach the desired formation without collision (Fig. 5). We refer the reader to aaclab.mit.edu for animations of the results. It was observed that a larger $E_{h}$, that can be determined using known upper bounds on the parametric uncertainties, led to a more conservative performance with the trajectories staying far from the obstacle during the transient phase of the adaptation.

It’s crucial to emphasize that the system successfully accomplishes the task without collisions, despite operating under unstable dynamics and partially known nonlinearities. Furthermore, it manages to do so even when the actuation is compromised in an unknown manner.

In this paper, we consider the problem of static formation control with distributed MAS in the presence of parametric uncertainties and limited communication. The class of problems considered is nonlinear systems that are feedback linearizable, with states accessible for measurement. The goal is to ensure that the MAS stay inside a safe set with the overall closed-loop system remaining stable while meeting the formation goals. Our approach is a combination of adaptive control and control barrier functions, with the former providing a means for accommodating to parametric uncertainties and the latter providing a safety filter that ensures that the states stay within a safe region and remain forward-invariant. The innovations are the design of a GCBF suitably modified to account for parametric uncertainties and the design of a graph-based reference model which serves as a desired dynamics that ensures a safe formation. Theoretical results are provided that guarantee global boundedness and safety against obstacles in the overall state space, and convergence to the desired formation. Numerical results show the effectiveness of the proposed method.