1. Real-Time Configuration Space Transforms for Obstacle Avoidance Wyatt S. Newman and Michael S. Branicky

2. Summary Explicit computation of configuration space – Useful for planning and control “Primitives” allow for generalization across environments – Points, lines, circles – 3D equivalents Techniques are not general across manipulators – Derived for 2 kinds in the paper

3. Key Properties Set Union Property – The two C obs of two obstacles, is the union of the C obs of each – Allows the authors to build up complicated C obs out of simple primitives Set Containment Property – If an obstacle is contained inside another, only the C-space of the outer one matters – The authors only have to consider the boundaries of obstacles

4. Key Properties Set Union Property – The two C obs of two obstacles, is the union of the C obs of each – Allows the authors to build up complicated C obs out of simple primitives Set Containment Property – If an obstacle is contained inside another, only the C-space of the outer one matters – The authors only have to consider the boundaries of obstacles

5. Points Two link planar manipulator Point obstacle at distance d from the origin on the x-axis Link 1 only collides at θ 1 =0 Link 2 collisions are computed using inverse kinematics for a series of points along the link

6. Points Translation property – If the point is not on the x-axis, it just shifts this c-space shape – e.g. If the point is at a 45 degree angle from the x-axis, then the shape will be centered around θ 1 =45 degrees

7. Lines A line is just a series of points (union property) The authors show what happens for a line normal to the x-axis and distance d from the origin These circles are actually filled in, but because of the containment property we only have to worry about borders

8. Line Segments Just as a point splits a line in workspace, the curve formed in c-space by that point splits the shape

9. Line Segments A line segment has 2 such points The resulting c-space obstacle is the set of curves in between

10. Line Segments A line segment has 2 such points The resulting c-space obstacle is the set of curves in between

11. Real Robot Example

13. Circles

14. Generalization to 3D Points, lines, and circles generalize to points, planes, and spheres Done for a R-R planar manipulator with a base joint that changes the “slice” (plane)

15. Nice Insight “For serial links numbers sequentially from the ground to the most distal link, link “i” obstacles require an i-dimensional configuration space representation.”

16. Limitations The translation properties in this paper are specific to the kinematics of the manipulator It only generalizes to 3D in certain cases Even the shown extension to 3D is a little forced if the links have non-negligible width – The “slices” are an oversimplification This appears to get intractable quickly – The authors only go up to 3DoF

17. How to update it? Computing high- dimensional c-space is expensive even today If explicit c-space is really needed, it can be approximated with a sampling method (like PRMs)