Continuous-Time Gradient-Proportional-Integral Flow for Provably Convergent Motion Planning with Obstacle Avoidance

This paper presents a novel continuous-time gradient-proportional-integral flow (GPIF) for motion planning with obstacle avoidance. We first frame the motion planning task as a constrained optimization problem, which is relaxed to be an unconstrained optimization problem that can be locally solved via a gradient flow approach using functional analysis. To enforce constraints, the proposed GPIF augments the gradient flow dynamics with proportional and integral feedback terms. Under reasonable assumptions formulated as linear matrix inequalities, we prove that the GPIF can generate optimal control trajectories with guaranteed exponential convergence. Numerical simulations validate the algorithm's efficacy, focusing on simple car navigation in cluttered environments. Simulations show that even after discretization for practical implementation, the GPIF method retains computational efficiency, enabling both offline planning and real-time online execution.