Online Control Barrier Functions
for Decentralized Multi-Agent Navigation

Assume agents and obstacles are disk-shaped with radii $\{R_{i}\}_{i=1}^{N}$ and $\{R_{\ell}\}_{\ell=1}^{M}$. Let $\{{\mathbf{p}}_{i}\}_{i=1}^{N}$, $\{{\mathbf{v}}_{i}\}_{i=1}^{N}$ and $\{{\mathbf{d}}_{i}\}_{i=1}^{N}$ be the positions, velocities and destinations of agents ${\mathcal{A}}$, which are determined by the internal states $\{{\mathbf{x}}_{i}\}_{i=1}^{N}$, and $\{{\mathbf{p}}_{\ell,o}\}_{\ell=1}^{M}$ be the obstacle positions. The goal is to move agents towards destinations while avoiding collision in a decentralized manner. The destination convergence is equivalent to the state convergence as $\lim_{t\to T}{\mathbf{x}}^{(t)}_{i}={\mathbf{x}}^{d}_{i}$ with $T$ the maximal time step for $i=1,...,N$. The collision avoidance is equivalent to the safety constraints on the agent states, i.e.,

		$\displaystyle{\mathcal{C}}^{(t)}_{i,\rm a}\!=\!\!\{{\mathbf{x}}^{(t)}_{i}\!\in\!\mathbb{R}^{n}\!\!\leavevmode\nobreak\ |\leavevmode\nobreak\ \!\|{\mathbf{p}}^{(t)}_{i}\!-\!{\mathbf{p}}^{(t)}_{j}\|\!\geq\!\frac{R_{i}\!\!+\!\!R_{j}}{2},\leavevmode\nobreak\ j\!\neq\!i\},$		(6)
		$\displaystyle{\mathcal{C}}^{(t)}_{i,\rm o}\!=\!\!\{{\mathbf{x}}^{(t)}_{i}\!\in\!\mathbb{R}^{n}\!\!\leavevmode\nobreak\ |\leavevmode\nobreak\ \!\|{\mathbf{p}}^{(t)}_{i}\!-\!{\mathbf{p}}^{(t)}_{\ell,\rm o}\|\!\!\geq\!\!\frac{R_{i}\!\!+\!\!R_{\ell}}{2}\!,\leavevmode\nobreak\ \!\ell\!=\!1,...,M\}.$		(7)

where $\|\cdot\|$ is the vector norm. This allows us to formulate the problem of multi-agent navigation as follows.

The condition (8) guarantees state convergence, i.e., navigation, and the condition (9) guarantees state safety, i.e., collision avoidance. Problem 1 is challenging because (i) decentralized policies generate control inputs with only local neighborhood information; and (ii), safety sets can be non-convex and time-varying, depending on moving agents.

We propose to solve Problem 1 with a CBF-CLF based quadratic programming (QP) controller. Specifically, at each time step $t$, we can formulate a QP problem as

		$\displaystyle\leavevmode\nobreak\ \underset{{\mathbf{u}}^{(t)}_{i}\in\mathbb{U}_{i},\delta_{i}^{(t)}\in\mathbb{R}}{\text{min}}$	$\displaystyle\|{\mathbf{u}}_{i}^{(t)}\|_{2}+\xi(\delta_{i}^{(t)})^{2}$		(10)
		s.t.	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\pounds_{f}h_{i,j}({\mathbf{x}}_{i}^{(t)})\!+\!\pounds_{g}h_{i,j}({\mathbf{x}}_{i}^{(t)}){\mathbf{u}}_{i}^{(t)}\!\!+\!\alpha_{i}(h_{i,j}({\mathbf{x}}_{i}^{(t)}))\!\geq\!0,$
	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{for all}\leavevmode\nobreak\ j=0,\dots,C_{i},$
	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\pounds_{f}V_{i}({\mathbf{x}}_{i}^{(t)})\!+\!\pounds_{g}V_{i}({\mathbf{x}}_{i}^{(t)}){\mathbf{u}}_{i}^{(t)}\!+\!\epsilon V_{i}({\mathbf{x}}_{i}^{(t)})\!+\!\delta_{i}^{(t)}\leq 0,$
	$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mathbf{x}}_{i}^{(t)}\in{\mathcal{C}}_{i}^{(t)},\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ i=1,\dots,N,$

where $\xi\in\mathbb{R}^{+}$ is a penalty weight for slack variable $\delta_{i}\in\mathbb{R}$ that is selected based on how strictly the CLF needs to be enforced, and $C_{i}$ is the number of CBF constraints based on agent $A_{i}$’s perceived agents and obstacles. For example, there are two CBF constraints w.r.t. the other agents and one w.r.t. the obstacle in Fig. 1. The control input is bounded by ${\mathbf{u}}_{i}^{(t)}\in\mathbb{U}_{i}$ given physical constraints of agent $A_{i}$. The QP is solved per time step to generate ${\mathbf{u}}_{i}^{(t)}$ until completion. The class $\mathcal{K}$ function $\alpha_{i}(\cdot)$ in the CBF constraint determines how strictly we want to enforce safety, and therefore will change the agent behavior to be either conservative or aggressive.

The CBF-CLF-QP controller solves problem (10) to generate a sequence of control inputs $\{{\mathbf{u}}_{i}^{(t)}\}_{t=0}^{T}$ for each agent $A_{i}$, providing goal-reaching convergence and safety guarantees. However, this may not be the case when problem (10) is unsolvable at some time step $t$ during navigation, i.e., when there is no feasible solution for problem (10) given CBF and CLF constraints at time step $t$. In this circumstance, an agent will stay safe at its current state but stop progressing towards its destination due to the lack of feasible control inputs, resulting in the failure of multi-agent navigation.

Specifically, given system dynamics (1), CBF constraints (5) and CLF constraints (3), the super-level set of agent $A_{i}$ that satisfies the constraints in problem (10) at time step $t$ is

$$\mathbb{U}^{(t)}_{i,\mathrm{CBF},\mathrm{CLF}}\!\!:=\!\!\left\{\!\!{\mathbf{u}}_{i}^{(t)}\!\!\!\;\middle|\;\!\!\begin{aligned} &\!\pounds_{g}h_{i,j}({\mathbf{x}}_{i}^{(t)}){\mathbf{u}}_{i}^{(t)}\!\!\geq\!\!-\pounds_{f}h_{i,j}({\mathbf{x}}_{i}^{(t)}\!)\\ &\!-\!\alpha_{i}\big{(}h_{i,j}({\mathbf{x}}_{i}^{(t)})\big{)},\text{for all}\leavevmode\nobreak\ j,\\ &-\pounds_{g}V_{i}({\mathbf{x}}_{i}^{(t)}){\mathbf{u}}_{i}^{(t)}\!\!\geq\!\pounds_{f}V_{i}({\mathbf{x}}_{i}^{(t)})\\ &+\epsilon V_{i}({\mathbf{x}}_{i}^{(t)})+\delta_{i}^{(t)}\end{aligned}\!\!\right\}\!\!.$$

(11)

By combining (11) with the physical constraints $\mathbb{U}_{i}$, the space of feasible solutions of agent $A_{i}$ is given by

$$\mathbb{U}_{i}\cap\mathbb{U}^{(t)}_{i,\mathrm{CBF},\mathrm{CLF}},\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ i=1,\ldots,N.$$

(12)

For conservative CBFs, the resulting constraints are strict and there may be no feasible solution in $\mathbb{U}^{(t)}_{i,\mathrm{CBF},\mathrm{CLF}}$ s.t. $\mathbb{U}_{i}\cap\mathbb{U}^{(t)}_{i,\mathrm{CBF},\mathrm{CLF}}=\varnothing$. For aggressive CBFs, the resulting constraints are relaxed and the agents may be too close to the obstacles or each other. In these cases, the QP controller may generate control inputs that require sudden changes beyond the agent’s physical capability $\mathbb{U}_{i}$ s.t. $\mathbb{U}_{i}\cap\mathbb{U}^{(t)}_{i,\mathrm{CBF},\mathrm{CLF}}=\varnothing$. Both scenarios lead to the infeasibility of problem (10) and, thus, navigation failure, indicating an inherent trade-off that is defined by CBF constraints – see Figs. 2(d)-2(e) for examples.

The aforementioned issue is exacerbated when the environment becomes cluttered with increasing numbers of agents and obstacles, which makes it challenging to hand-tune CBFs. Furthermore, fixing CBFs during navigation may not effectively handle the dynamic nature of the environment with moving agents, and even well-tuned CBFs could suffer from performance degradation with environment changes. These observations motivate the use of time-varying CBFs based on instantaneously sensed states, to tune agents’ conservative and aggressive behavior. We refer to the latter as online CBFs. Define decentralized CBF-tuning policies as

$$\pi_{i,\mathrm{CBF}}\Big{(}\!\alpha_{i}\Big{|}{\mathbf{x}}_{i},\!\{{\mathbf{x}}_{j}\}_{j\in{\mathcal{N}}_{i}},\!\{{\mathbf{p}}_{\ell,\rm o}\}_{\ell\in{\mathcal{N}}_{i}}\!\Big{)},\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ i\!=\!1,...,N,$$

(13)

which generate the class ${\mathcal{K}}$ function $\alpha_{i}(\cdot)$, i.e., the CBFs, based on local neighborhood information. At each time step $t$, a new class ${\mathcal{K}}$ function $\alpha^{(t)}_{i}(\cdot)$ is generated for agent $A_{i}$ and passed into CBFs for solving (10) to compute the control input ${\mathbf{u}}_{i}^{(t)}$. Given any objective function $F(\{\pi_{i,\mathrm{CBF}}\}_{i=1}^{N},{\mathbf{X}}^{(0)},{\mathbf{X}}^{d})$ that represents the navigation performance, the initial states ${\mathbf{X}}^{(0)}$ and the target states ${\mathbf{X}}^{d}$, we can formulate the problem of online CBF optimization.

The CBF-tuning policy conducts online CBF adjustments based on the local state of a dynamic environment, which provides control feasibility where fixed CBFs would not be able to, while maintaining safety guarantees. The generated time-varying CBFs strike a balance between conservative and aggressive behaviors among different agents. For scenarios where agent trajectories are in conflict (e.g., several agents need to navigate through narrow space), this yields an inherent prioritization among agents and provides deconfliction for agent trajectories – see Fig. 3(a) for demonstration.

Given the destination ${\mathbf{d}}_{i}=[d_{i,1},d_{i,2}]^{\top}$ of agent $A_{i}$, define a Lyapunov function candidate as $V_{i}({\mathbf{x}})=(p_{i,1}-d_{i,1})^{2}+(p_{i,2}-d_{i,2})^{2}$ and the CLF constraint as

$$2({\mathbf{p}}_{i}-{\mathbf{d}}_{i})^{\top}{\mathbf{u}}_{i}+\epsilon V_{i}({\mathbf{x}}_{i})+\delta_{i}\leq 0$$

(15)

for $i=1,...,N$. Define barrier function candidates as

	$\displaystyle h_{i,j,\mathrm{a}}({\mathbf{x}}_{i})$	$\displaystyle\!=\!(p_{i,1}\!-\!p_{j,1})^{2}\!+\!(p_{i,2}\!-\!p_{j,2})^{2}\!-\!(R_{i}\!+\!R_{j})^{2},$		(16)
	$\displaystyle h_{i,\ell,\mathrm{o}}({\mathbf{x}}_{i})$	$\displaystyle\!=\!(p_{i,1}\!-\!p_{\ell,1,\mathrm{o}})^{2}\!+\!(p_{i,2}\!-\!p_{\ell,2,\mathrm{o}})^{2}\!-\!(R_{i}\!+\!R_{\ell})^{2},$

where $h_{i,j,\mathrm{a}}$ is w.r.t. collision avoidance between the agents $A_{i}$ and $A_{j}$, and $h_{i,\ell,\mathrm{o}}$ is w.r.t. collision avoidance between agent $A_{i}$ and obstacle $O_{\ell}$ with ${\mathbf{p}}_{\ell,\mathrm{o}}=[p_{\ell,\mathrm{1,o}},p_{\ell,\mathrm{2,o}}]^{\top}$ the obstacle position. The resulting CBF constraints are

	$\displaystyle 2({\mathbf{p}}_{i}-$	$\displaystyle{\mathbf{p}}_{j})^{\top}{\mathbf{u}}_{i}-2({\mathbf{p}}_{i}-{\mathbf{p}}_{j})^{\top}\dot{{\mathbf{p}}}_{j}+\zeta_{i,\mathrm{a}}\big{(}h_{i,j,\mathrm{a}}({\mathbf{x}}_{i})\big{)}^{\eta_{i,\mathrm{a}}}\geq 0,$
		$\displaystyle 2({\mathbf{p}}_{i}-{\mathbf{p}}_{\ell})^{\top}{\mathbf{u}}_{i}+\zeta_{i,\mathrm{o}}\big{(}h_{i,\ell,\mathrm{o}}({\mathbf{x}}_{i})\big{)}^{\eta_{i,\mathrm{o}}}\geq 0,$		(17)

where $\zeta_{i,\mathrm{a}},\eta_{i,\mathrm{a}}$ are CBF parameters of agent $A_{i}$ w.r.t. the other agents and $\zeta_{i,\mathrm{o}},\eta_{i,\mathrm{o}}$ are w.r.t. the obstacles. In this context, each agent has two sets of CBF parameters for the other agents and for the obstacles, respectively. We can then specify the CBF-CLF-QP controller by substituting the CLF constraint (15) and CBF constraints (IV-A) into problem (10).

The CBFs are determined by the class ${\mathcal{K}}$ function $\alpha_{i}(\cdot)$ with parameters $\zeta_{i,\mathrm{a}},\eta_{i,\mathrm{a}}$ for the other agents and $\zeta_{i,\mathrm{o}},\eta_{i,\mathrm{o}}$ for the obstacles. This indicates that we can learn CBFs by learning CBF parameters $\boldsymbol{\zeta}_{i}=[\zeta_{i,\mathrm{a}},\zeta_{i,\mathrm{o}}]^{\top},\boldsymbol{\eta}_{i}=[\eta_{i,\mathrm{a}},\eta_{i,\mathrm{o}}]^{\top}$ at each agent $A_{i}$. Since it is challenging to explicitly model the relationship between CBF parameters and navigation performance, we formulate Problem 2 in the RL domain and learn CBF-tuning policies in a model-free manner.

We start by defining a partially observable Markov decision process. At each time $t$, agents are defined by states ${\mathbf{X}}^{(t)}\!=\!\{{\mathbf{x}}_{i}^{(t)}\}_{i=1}^{N}$. Each agent $A_{i}$ observes its local state ${\mathbf{x}}_{i}^{(t)}$, communicates with its neighboring agents, and senses its neighboring obstacles to collect the neighborhood information $\{{\mathbf{x}}_{j}^{(t)}\}_{j\in{\mathcal{N}}_{i}}$ and $\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell\in{\mathcal{N}}_{i}}$. The CBF-tuning policy $\pi_{i,\rm CBF}$ generates CBF parameters $\boldsymbol{\zeta}_{i}^{(t)}$ and $\boldsymbol{\eta}_{i}^{(t)}$, which is a distribution over $\boldsymbol{\zeta}_{i}^{(t)}$, $\boldsymbol{\eta}_{i}^{(t)}$ conditioned on ${\mathbf{x}}_{i}^{(t)}$, $\{{\mathbf{x}}_{j}^{(t)}\}_{j\in{\mathcal{N}}_{i}}$, $\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell\in{\mathcal{N}}_{i}}$. The CBF parameters $\boldsymbol{\zeta}_{i}^{(t)}$, $\boldsymbol{\eta}_{i}^{(t)}$ are fed into the QP controller (10), which generates the control action ${\mathbf{u}}_{i}^{(t)}$ that drives the local state ${\mathbf{x}}_{i}^{(t)}$ to ${\mathbf{x}}_{i}^{(t+1)}$ based on the agent’s dynamics (14). The reward function $r_{i}({\mathbf{X}}^{(t)})$ represents the instantaneous navigation performance of agent $A_{i}$ at time $t$, which consists of two components: (i) the navigation reward $r_{i,\textrm{nav}}$ and (ii) the QP’s feasibility reward $r_{i,\textrm{infs}}$, i.e.,

$\displaystyle r_{i}^{(t)}({\mathbf{X}}^{(t)})=r_{i,\text{nav}}^{(t)}({\mathbf{X}}^{(t)})+\beta_{i}r_{i,\text{infs}}^{(t)}({\mathbf{X}}^{(t)}),$

(18)

where $\beta_{i}$ is the regularization parameter. The first term represents the task-relevant performance of agent $A_{i}$, while the second term corresponds to the feasibility of the QP controller (10) with the generated CBF parameters, e.g., it penalizes the scenario where the QP controller has no feasible solution with overly conservative or aggressive CBFs. The total reward of agents is $r^{(t)}=\sum_{i=1}^{N}r_{i}^{(t)}$. With the discount factor $\gamma$ that accounts for the future rewards, the expected discounted reward can be represented as

$\displaystyle R\big{(}{\mathbf{X}}^{(0)}\!,{\mathbf{X}}^{d}\!,\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell=1}^{M}|\{\pi_{i,\rm CBF}\}_{i=1}^{N}\big{)}\!=\!\mathbb{E}\Big{[}\!\sum_{t=0}^{\infty}\gamma^{t}r^{(t)}\!\Big{]}\!,$

(19)

where $\mathbb{E}[\cdot]$ is w.r.t. CBF-tuning policies. The expected discounted reward in (19) corresponds to the objective function in Problem 2, which transforms the problem into the RL domain. By parameterizing the policies $\{\pi_{i,\rm CBF}\}_{i=1}^{N}$ with information processing architectures $\{\boldsymbol{\Phi}_{i}({\mathbf{x}}_{i}^{(t)},\{{\mathbf{x}}_{j}^{(t)}\}_{j\in{\mathcal{N}}_{i}},\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell\in{\mathcal{N}}_{i}},\boldsymbol{\theta}_{i})\}_{i=1}^{N}$ of parameters $\{\boldsymbol{\theta}_{i}\}_{i=1}^{N}$, the goal is to learn optimal parameters $\{\boldsymbol{\theta}_{i}^{*}\}_{i=1}^{N}$ that maximize $R({\mathbf{X}}^{(0)},{\mathbf{X}}^{d},\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell=1}^{M}|\{\boldsymbol{\theta}_{i}\}_{i=1}^{N})$. We solve the latter by updating $\{\boldsymbol{\theta}_{i}\}_{i=1}^{N}$ through policy gradient ascent.

We parameterize CBF-tuning policies with GNNs, which allow for decentralized execution. They are inherently permutation equivariant (independent of agent ordering), and hence, generalize to unseen agent constellations [36, 37, 38].

Motivated by the observation that CBFs need only relative information (e.g., relative positions between agents and obstacles) [cf. (IV-A)], we design a translation-invariant GNN that leverages message passing mechanisms to generate CBF parameters with relative information. For each agent $A_{i}$ with its local state ${\mathbf{x}}_{i}$, the states of neighboring agents $\{{\mathbf{x}}_{j}\}_{j\in{\mathcal{N}}_{i}}$ and the positions of neighboring obstacles $\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell\in{\mathcal{N}}_{i}}$, it generates CBF parameters with the message aggregation functions ${\mathcal{F}}_{\rm m,a},{\mathcal{F}}_{\rm m,o}$ and the feature update function ${\mathcal{F}}_{\rm u}$ as

	$\displaystyle(\boldsymbol{\zeta}_{i}^{(t)}\!,\boldsymbol{\eta}_{i}^{(t)})$	$\displaystyle\!=\!\boldsymbol{\Phi}_{i}({\mathbf{x}}_{i}^{(t)},\{{\mathbf{x}}_{j}^{(t)}\}_{j\in{\mathcal{N}}_{i}},\{{\mathbf{p}}_{\ell,\mathrm{o}}\}_{\ell\in{\mathcal{N}}_{i}},\boldsymbol{\theta}_{i})$		(20)
		$\displaystyle\!=\!{\mathcal{F}}_{\rm u}\Big{(}\!\sum_{j\in{\mathcal{N}}_{i}}\!{\mathcal{F}}_{\rm m,a}({\mathbf{x}}_{j}\!-\!{\mathbf{x}}_{i})\!+\!\!\sum_{\ell\in{\mathcal{N}}_{i}}\!{\mathcal{F}}_{\rm m,o}({\mathbf{p}}_{\ell,\mathrm{o}}\!-\!{\mathbf{p}}_{i})\!\Big{)},$

where $\boldsymbol{\theta}_{i}$ are the function parameters of ${\mathcal{F}}_{\rm m,a},{\mathcal{F}}_{\rm m,o}$ and ${\mathcal{F}}_{\rm u}$. By sharing ${\mathcal{F}}_{\rm m,a},{\mathcal{F}}_{\rm m,o}$ and ${\mathcal{F}}_{\rm u}$ over all agents, we have $\boldsymbol{\theta}_{1}=\cdots=\boldsymbol{\theta}_{N}$ and thus $\boldsymbol{\Phi}_{1}=\cdots=\boldsymbol{\Phi}_{N}$.

The GNN-based policy has the following properties:

1. 1.

   Decentralized execution: The functions ${\mathcal{F}}_{\rm m,a},{\mathcal{F}}_{\rm m,o},{\mathcal{F}}_{\rm u}$ require only neighborhood information and the policy can be executed in a decentralized manner.

2. 2.

   Translation invariance: The policy uses relative information and is invariant to translations in $\mathbb{R}^{2}$.

3. 3.

   Permutation equivariance: The functions ${\mathcal{F}}_{\rm m,a},{\mathcal{F}}_{\rm m,o},{\mathcal{F}}_{\rm u}$ are homogeneous and the policy is equivariant to permutations (i.e., agent reorderings).

We consider an environment shown in Fig. 2(a). The agents have radius $0.15$m, and are initialized randomly in the top region of the workspace and tasked towards goal positions in the bottom region. The obstacles are of radius $0.5$m, and are distributed between the initial and goal positions of agents.

To show the trade-off between conservative and aggressive behavior inherent in CBFs, we select the minimal and maximal values from time-varying CBF parameters in Figs. 2(b)-2(c) (dashed lines), corresponding to the most conservative and aggressive CBFs, and perform navigation for fixed CBFs [21, 25] with these selected parameters. For the minimal values in Fig. 2(d), $A_{2}$, $A_{3}$ keep still and $A_{1}$, $A_{4}$ move along the environment boundary, with overly conservative trajectories, and only $A_{1}$ reaches its destination. For the maximal values in Fig. 2(e), while all agents aggressively move towards destinations, $A_{3}$ and $A_{4}$ get stuck before the narrow passage between $O_{3}$ and $O_{4}$. This is because the agents are too close to each other and the obstacles, where their controller has no feasible solution. This highlights the importance of our approach, which provides online CBFs.

We perform our approach in three distinct navigation scenarios: Narrow Passage, Cross, and Singularity. For the fixed CBFs, we employ an exhaustive grid-search to find optimal parameters $(\zeta,\eta)$ from $[0.1,10]\times[1.0,2.0]$ for $100$ combinations.

Fig. 3 shows the performance of our approach and the optimal fixed CBFs in our three scenarios. Overall, our results show that, while we exhaustively traversed the parameter space, there still exist scenarios where there is no solution for fixed CBFs. In contrast, the online CBFs can solve these infeasible scenarios (i.e., all agents reach their destinations successfully). For the Narrow Passage scenario in Figs. 3(a)-3(b) and the Cross scenario in Figs. 3(c)-3(d), our approach deconflicts agents by prioritizing them with varying degrees of conservative / aggressive behaviors. For the Singularity scenario in Figs. 3(e)-3(f), the agent and its destination are aligned with an obstacle in the middle. The fixed CBF-based controller generates controls on the edge or vertex of the admissible control set, and hence, agents get stuck in local minima [40], while our approach helps agents escape such conditions due to online CBF tuning.

We show the generalization of our approach by testing the trained policy on previously unseen environments. We consider two baselines: (i) optimal fixed CBFs with exhaustive grid-search and (ii) time-varying CBFs with random parameters. The first searches the parameter space exhaustively and selects the optimal values for fixed CBFs. As existing works primarily concentrate on identifying the optimal fixed CBFs for specific tasks [41, 11, 42], we consider this baseline to approximate the optimal performance of controllers with fixed CBFs. Note that it is inefficient and included here solely for reference. The second selects random CBF parameters every $10$ time steps, which is as efficient as our approach.

We consider larger environments with $8$ obstacles, where the maximal time step is $750$. We train our approach for online CBFs and conduct grid-search for fixed CBFs in the environment shown in Fig. 4(a), and test them by randomly shifting initial, goal and obstacle positions. The performance is measured by two metrics: (i) Success weighted by Path Length (SPL) [43] and (ii) the percentage to the maximal speed (PCTSpeed). The former is a stringent measure combining the success rate and the path length, while the latter represents the ratio of the average speed to the maximal one.

Fig. 4(b) shows the results averaged over $20$ random initializations. Our approach outperforms the baselines in both metrics, with higher expectations and lower standard deviations. This corresponds to theoretical findings that (i) online CBFs coordinate agents’ conservative / aggressive behaviors based on instantaneous environment states, which allows smooth navigation without congestion for higher expected performance; (ii), online CBFs deconflict agents to solve infeasible scenarios, which improves robustness for lower standard deviations. Random CBFs exhibit a higher SPL than fixed CBFs because they deconflict some infeasible scenarios by randomly prioritizing agents and have a higher success rate. However, random CBFs show a lower PCTSpeed because randomly coordinating agents exhibits poor performance, even though navigation tasks are successful.

We conduct real-world experiments to validate our approach. We consider a narrow passage scenario that requires deconfliction for multi-robot navigation. We use four customized DJI Robomasters with Raspberry Pi. Each robot has a partially observable space with a sensing range of $2$m, and employs an external telemetry (OptiTrack) for localization. We use ROS2 as communication middle-ware. At each time, the robot receives its current state and the neighbors’ states, and deploys the decentralized controller for navigation.

Fig. 5 shows that our approach steers all robots to their destinations without collision, because online CBFs deconflict robots by adapting between conservative and aggressive constraint values. When performing the same experiment with (optimized) fixed CBFs, robots fail to navigate through the narrow passage because online deconfliction is not facilitated. This insight corroborates our theoretical analysis and numerical simulations [44, Video Link].