An overview of high-dimensional collision detection for the purpose of motion planning is in <A HREF="https://arxiv.org/abs/1806.07457">Petrović - Motion planning in high-dimensional spaces</A>:

> Grid-based approaches are resolution complete and often offer optimal
> solutions. However, the number of grid points grows exponentially in
> the configuration space dimension, which makes even the
> state-of-the-art methods inappropriate for very high-dimensional
> problems. Sampling-based approaches are efficient in most practical
> problems but offer weaker guarantees. They are probabilistically
> complete, however, they often require post-processing and can still be
> inefficient in very complex configuration spaces. Trajectory
> optimization approaches can solve high-dimensional motion planning
> problems quickly, but solutions are only locally optimal.