Analysis of Sampling-Based Planners

Thomas Cohn

RLG Short Talk - November 22, 2024

Properties of Space

Configuration space $\mathcal C$ and collision-free space $\mathcal C_\mathrm{free}\subseteq\mathcal C$
Reachable set $\mathcal R(x)$ , for $x\in\mathcal C_\mathrm{free}$
- Set of all configurations that can be reached from $x$
$\ell$ -reachable set $\mathcal R_\ell(x)$ , for $x\in\mathcal C_\mathrm{free}$
- Set of all configurations that can be reached in $\ell$ steps by a local planner
Visible set $\mathcal V(x)$ , for $x\in\mathcal C_\mathrm{free}$
- Equivalent to 1-reachable set for a straight-line local planner.
The volume of a set $X$ is denoted $\mu(X)$

Setup for the RRT Algorithm

Start configuration $x_\mathrm{init}\in\mathcal C_\mathrm{free}$
Goal region $X_\mathrm{goal}\subseteq\mathcal C_\mathrm{free}$
- In practice, we can choose a small ball around a goal configuration $x_\mathrm{goal}\in\mathcal C_\mathrm{free}$
$\eta$ step size
$k$ iteration limit

RRT Sample Complexity

Let $p$ be the probability a sample is drawn from a ball
Let $m$ be the number of balls needed to cover the path
Let $k$ be the number of iterations

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?

$p$ increases
$m$ decreases

PRM Sample Complexity

Let $L$ be the length of the path
Let $\epsilon$ be the clearance of the path
Let $H(n)$ be the $n$ th Harmonic number

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?

$H(\tfrac{2L}{\epsilon})$ decreases
$\mu(B_{\epsilon/2}(\cdot))$ increases

A Stronger Bound: Expansiveness

Fix $\alpha,\beta\in(0,1]$
Fix $S\subseteq\mathcal C_\mathrm{free}$
The $\beta$ -lookout of $S$ is $\{p\in S:\mu(\mathcal R_\ell(p)\setminus S)\ge\beta\mu(\mathcal R(S)\setminus S)\}$
A set $S$ is $(\alpha,\beta)$ -expansive if for any connected $S'\subseteq S$ , $\mu(\beta\mathrm{-lookout}(S'))\ge\alpha\mu(S')$
$\mathcal C_\mathrm{free}$ is $(\alpha,\beta)$ -expansive if $\forall p\in\mathcal C_\mathrm{free}$ , $\mathcal{R}(p)$ is $(\alpha,\beta)$ -expansive
- If $\alpha'\le\alpha$ and $\beta'\le\beta$ , then $\mathcal C_\mathrm{free}$ is also $\alpha',\beta'$ -expansive.

Further Definitions

$\mathcal C_\mathrm{free}$ is $\epsilon$ -good, for $\epsilon\in(0,1]$ , if $\forall p\in\mathcal C_\mathrm{free}$ , $\mu(\mathcal V(p))\ge\epsilon\mu(\mathcal C_\mathrm{free})$
- Intuition: the visibility set of each point is at least an $\epsilon$ fraction of the free space
A set of points $p_i$ in $\mathcal C_\mathrm{free}$ provides adequate coverage if $\mu\left(\mathcal C_\mathrm{free}\setminus\bigcup_{i}\mathcal V(p_i)\right)\le\frac{\epsilon}{2}\mu(\mathcal C_\mathrm{free})$
- Intuition: the points not visible from any $p_i$ are at most an $\epsilon/2$ fraction of the free space

Sample Complexity of PRM

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.

Analysis of Sampling-Based Planners Thomas Cohn RLG Short Talk - November 22, 2024