1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/2003 Probabilistic Roadmaps Sampling and Connection Strategies (2/2)

2. Multi-Query Strategies Narrow-passage sampling (medial-axis transform)  Paper 1

3. Single-Query Strategies mbmbmbmb mgmgmgmg Diffusion Adaptive step Lazy collision checking  Paper 2

4. Configuration of a Rigid Body reference point x y  robot reference direction workspace In 3D q = (x,y,z,a,b,g) q = (x,y,z) + r11 r12 r13 r21 r22 r23 r31 r32 r33 r i1 2 + r i2 2 + r i3 2 = 1 r i1 r j1 + r i2 r 2j + r i3 r j3 = 0 det(R) = +1 R The c-space is a 6-D space (manifold) embedded in a 12-D Cartesian space. It is denoted by R 3 xSO(3)

5. Other representation of SO(3) Unit quaternion: (cos  /2, n 1 sin  /2, n 2 sin  /2, n 3 sin  /2)

6. Metric (in Paper 1)  q = (x,y,z,R)  q’ = (x’,y’,z’,R’)  R  R’ rotation of  around n  d(q,q’) = [(x’-x) 2 +(y-y’) 2 +(z-z’) 2 ] 1/2 +  |  | where  > max distance from reference point to body’s boundary  closest point of a configuration to obstacle obtained by translation only

7. Hybrid Strategies Motivations: 1. Connections are much more expensive (ratio of 10 to 100) to check than sampled configurations 2. It takes many queries (~100) to amortize the cost of pre-computing a multi-query roadmap  Phase 1: Pick milestones using “smart” multi-query strategies (gaussian, bridge, MAT, etc)  Phase 2: Run a single-query strategy (diffusion, lazy-collision checking, etc) using both milestones generated in Phase 1 and new milestones

8. Biased Sampling