Real-Time Configuration Space Transforms for Obstacle Avoidance

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 Cobs of two obstacles, is the union of the Cobs of each • Allows the authors to build up complicated Cobs 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 Cobs of two obstacles, is the union of the Cobs of each • Allows the authors to build up complicated Cobs 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 θ1=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 θ1=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

12. 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)