Rule-Based Lloyd Algorithm for Multi-Robot Motion Planning and Control with Safety and Convergence Guarantees

TL;DR

This work addresses distributed multi-robot motion planning without inter-robot communication by introducing a Rule-Based Lloyd (RBL) algorithm that reshapes safety-aware cells and weighting to guarantee collision-free convergence toward goal regions. It combines Lloyd-based coverage control with learning-enabled extensions (LLB) and a model predictive control (MPC) layer to handle dynamic constraints, enabling applicability to holonomic and nonholonomic platforms. The authors prove safety and convergence properties, validate them through extensive simulations across diverse scenarios and robot types, and corroborate the results with real-world experiments. The proposed framework offers asynchronous operation, robustness to heterogeneity, and competitive performance relative to state-of-the-art methods, with the LLB variant providing a practical safety-backed learning paradigm for faster goal achievement.

Abstract

This paper presents a distributed rule-based Lloyd algorithm (RBL) for multi-robot motion planning and control. The main limitations of the basic Loyd-based algorithm (LB) concern deadlock issues and the failure to address dynamic constraints effectively. Our contribution is twofold. First, we show how RBL is able to provide safety and convergence to the goal region without relying on communication between robots, nor synchronization between the robots. We considered different dynamic constraints with control inputs saturation. Second, we show that the Lloyd-based algorithm (without rules) can be successfully used as a safety layer for learning-based approaches, leading to non-negligible benefits. We further prove the soundness, reliability, and scalability of RBL through extensive simulations, comparisons with the state of the art, and experimental validations on small-scale car-like robots, unicycle-like robots, omnidirectional robots, and aerial robots on the field.

Figures (17)

• Figure 1: We depicted the main parameters and the variables associated with robot $i$. The robots are indicated with blue circles. Notice that the centroids' position strongly depend on the spreading factor $\beta_i$ in \ref{['eq:rho']}. In particular, $\beta_i \rightarrow \infty$ cancels any attraction towards the goal $\bar{p}_i$ (e.g., $c_{\mathcal{S}_i} \equiv p_i)$. On the other hand, $\beta_i \rightarrow 0$ pushes the centroids towards the goal position $\bar{p}_i$, keeping them inside their cells.
• Figure 4: Overview of the learned policy network. The observed robots are encoded through a self-attention block (SAB) and decoded with a multi-headed attention block (MAB) and three fully connected layers (FC) that take $s_i$ as query. The resulting vector is mapped to the parameters of the diagonal Gaussian distribution, $\mathcal{N}(\mu_{a'_t},\sigma^2_{a'_t})$, and value function estimate, $V^{\pi}(s^t_i,\mathcal{O}^t_i)$.
• Figure 5: Circle crossing: simulation results with $N= [5,10,25,50,300]$ holonomic robots. The value $\beta^D=0.5$ is the same for all the robots.
• Figure 6: Circle crossing: simulation results with $N=20$ holonomic robots with random encumbrance $r$. On the left hand side $\beta^D_i = 0.5$ for all the robots, On the right hand side $\beta^D_i$ is random and different from robot to robot.
• Figure 7: Dependence of the travel time with respect to $\beta^D_i$. Each dot represents the performance of one robot. In red the left hand side simulation in Figure \ref{['fig:simul1']}, in blue the right hand side simulation in Figure \ref{['fig:simul1']}.

Theorems & Definitions (6)

• Theorem 1: Safety
• Definition 1: Deadlock
• Definition 2: Live-lock
• Lemma 1: Deadlock in Symmetries
• Theorem 2: Convergence
• Remark 1: Numerical Simulations and Asynchrony