Real-Time Configuration Space Transforms for Obstacle Avoidance

Wyatt S. Newman and Michael S. Branicky

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

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

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

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

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

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

Line Segments

• Just as a point splits a line in workspace, the curve formed in c-space by that point splits the shape

Line Segments

• A line segment has 2 such points• The resulting c-space obstacle is the set of curves in

between

Line Segments

• A line segment has 2 such points• The resulting c-space obstacle is the set of curves in

between

Real Robot Example

Real Robot Example

Circles

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)

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.”

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

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)