Efficient autonomous exploration in unknown obstacle cluttered environments with interior obstacles remains a challenging task for mobile robots. In this work, we present a novel exploration process for a non-holonomic agent exploring 2D spaces using onboard LiDAR sensing. The proposed method generates velocity commands based on the calculation of the solution of an elliptic Partial Differential Equation with Dirichlet boundary conditions. While solving Laplace's equation yields collision-free motion towards the free space boundary, the agent may become trapped in regions distant from free frontiers, where the potential field becomes almost flat, and consequently the agent's velocity nullifies as the gradient vanishes. To address this, we solve a Poisson equation, introducing a source point on the free explored boundary which is located at the closest point from the agent and attracts it towards unexplored regions. The source values are determined by an exponential function based on the shortest path of a Hybrid Visibility Graph, a graph that models the explored space and connects obstacle regions via minimum-length edges. The computational process we apply is based on the Walking on Sphere algorithm, a method that employs Brownian motion and Monte Carlo Integration and ensures efficient calculation. We validate the approach using a real-world platform; an AmigoBot equipped with a LiDAR sensor, controlled via a ROS-MATLAB interface. Experimental results demonstrate that the proposed method provides smooth and deadlock-free navigation in complex, cluttered environments, highlighting its potential for robust autonomous exploration in unknown indoor spaces.
Keywords: Monte Carlo integration (MCI); Poisson equation; autonomous exploration; hybrid visibility graph; walking on sphere (WoS).