Things We Care About

• As the number of samples goes to infinity...
  • Probabilistic Completeness (PC): If a path exists, the probability it hasn't been found goes to zero.
  • Asymptotic Optimality (AO): The difference in cost of the current path and the optimal path goes to zero.
• Sample complexity bounds
  • What is the probability we have found a path after a certain number of iterations?
  • What is the expected number of iterations we need to find a path?

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

The RRT Algorithm

Clearance

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!

RRT is PC

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

The PRM Algorithm

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

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.

Expansiveness vs Clearance

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

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.