Achieving Asymptotic Near-Optimality Without $δ$-Similarity

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 $δ$-similar trajectories. This paper shows that the proof behind asymptotic $δ$-similarity relies on an unstated assumption that $δ$-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 $δ$-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 $δ$-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 $δ$-similar solution trajectory is impossible.