Achieving Asymptotic Near-Optimality Without elta-Similarity

arXiv:2609.04464v1 Announce Type: new Abstract: Sampling-based motion planning algorithms are a popular class of trajectory planning algorithm due to their speed in complex, high-dimensional environments and ability to handle kinodynamic constraints, specifically through the use of forward dynamics propagation. Many such planners claim to achieve asymptotic near-optimality by proving the almost sure sampling of trajectories that are close to an optimal trajectory in the state space, known as elta-similar trajectories. This paper shows that the proof behind asymptotic elta-similarity relies on an unstated assumption that elta-similar trajectory segments will always be kept once sampled. This assumption does not hold in general. A problematic case, referred to as ``crowding out,'' is described, where locally low-cost paths prevent trajectories that are elta-similar to the optimal trajectory from being added to the tree. It is shown, however, that asymptotic near-optimality guarantees can still be achieved without guarantees of elta-similar solution trajectories when crowding out is properly accounted for. An example environment and system are provided where crowding out is shown to occur, demonstrating a scenario where inductively sampling a elta-similar solution trajectory is impossible.

Source: arXiv cs.RO | 2026-09-07