RLG Short Talk - November 22, 2024
Michigan Robotics 320, Lecture 15
Pathfinding with Rapidly-Exploring Random Tree, Knox
Probabilistic completeness of RRT for geometric and kinodynamic planning with forward propagation, Kleinbort et. al.
A path \(\gamma:[0,1]\to\mathcal C\) is \(\delta\)-clear if you can push a ball of radius \(\delta\) along the path without hitting an obstacle.
Note: defined for a specific path!
Klampt Documentation
Then the probability RRT fails to reach the goal in k iterations is at most \[\frac{1}{(m-1)!}k^mme^{-pk}\]
What happens as clearance increases?
Measure Theoretic Analysis of Probabilistic Path Planning, Ladd and Kavraki
Then the expected number of samples needed for the PRM to find a path is at most \[\frac{H(\frac{2L}{\epsilon})\mu(\mathcal C_\mathrm{free})}{\mu(B_{\epsilon/2}(\cdot))}\]
What happens as clearance increases?
The space on the right has low clearance by high expansiveness
Even though there are narrow passageways, it's still easy to cross the middle region
Randomized Single-Query Motion Planning in Expansive Spaces, Hsu
Theorem (Hsu): Suppose \(\mathcal C_\mathrm{free}\) is connected. Fix \(\gamma\in(0,1]\). If we randomly sample \(2n+2\) points, with \[n\ge 8\ln(8/\epsilon\alpha\gamma)/\epsilon\alpha+3/\beta,\] then with probability at least \(1-\gamma\), the resulting visibility graph is connected.
A note: the result is actually stated for possibly-disconnected spaces -- in that case, the subgraph corresponding to each connected component of \(\mathcal C_\mathrm{free}\) is connected.