Physics-Informed Neural Networks for Optimal Path Planning of Unicycle Robots with Regular Obstacles

Sammanfattning

Path planning is becoming increasingly relevant due to the advancement in self-driving vehicles and mobile robots. However, many of these systems are subject to non-holonomic constraints that restrict their motion. Generating feasible trajectories for such systems remain challenging, since the planned paths must simultaneously satisfy kinematic constraints and avoid obstacles in the environment. Classical methods often produce non-smooth or kinematically infeasible trajectories, while purely data-driven learning lacks formal guarantees on physical consistency. To address these limitations, this thesis investigates a Physics-Informed Neural Network (PINN) framework that embeds physics principles directly into the training process through loss functions. These functions are trained to promote consistency with the unicycle kinematics, ensure obstacle avoidance and regularize both translational and angular velocities. This is done in environments containing convex obstacles of regular geometry, specifically rectangular and circular shapes, which are represented using signed distance functions (SDFs). Four simulation experiments are conducted to evaluate the framework, which reveal that the obstacle geometry significantly influences the resulting trajectory. Rectangular obstacles consistently produce higher maximum curvature and a qualitatively different curvature profile compared to circular obstacles, with the curvature peaks corresponding to the leading and trailing corners of the rectangle. Beyond these empirical observations, a theoretical analysis based on the Euler--Lagrange framework is conducted on the loss functional to characterize the structural properties of the PINN framework. The analysis reveals that the relative magnitudes of the loss weights govern trade-offs between kinematic consistency, obstacle avoidance, and trajectory smoothness. The learned trajectory cannot satisfy all of them simultaneously and instead reflects a balance shaped by the chosen weights, formalizing a fundamental limitation of the framework.