Title: Configuration Space of 2T 1R Robot
1Configuration Space of 2T1R Robot
2Last Lecture
- Minkowski sum
- Configuration space of 2T, 3T robots
3Polygonal robot translating in 2-D workspace
configuration space
workspace
4Polygonal robot translating rotating in 2-D
workspace
configuration space
workspace
5Polygonal robot translating rotating in 2-D
workspace
q
y
x
6Contact Surfaces (C-surfaces)
- A C-surface arises from a contact between
features of the robot and the obstacle
R
O
R
O
Type A contact
Type B contact
7Type A Contact Surface
APPLAi,j
vi R(q) . (bj-1 bj ) ? 0 ?
Contact is feasible when
vi R(q) . (bj1 bj ) ? 0
bj-1
ai1(q)
O
R
bj
bj1
vi R(q)
ai(q)
8Type A Contact Surface
C-surface is a ruled surface
Contact surface
fi,jA(q) 0
fi,jA(q) viR(q) . (bj ai(q) )
where
bj-1
ai1(q)
O
R
bj
bj1
vi R(q)
ai(q)
9Type A C-constraint
CONSTAi,j (q)
APPLAi,j(q) ? fi,jA(q) ? 0
Similarly, you can define a C-constraint for Type
B contact Type B C-surface is also a ruled surface
102D Translation and Rotation
Obstacles
Robot
11Contact Surfaces
3,929 contact surfaces
12Representation of C-obstacle
- How can we represent C-obstacle in terms of
C-surfaces? - Non-convex case
- Resort to convex decomposition
For the case of a convex robot and a convex
obstacle,
CONSTAi,j (q) ? CONSTBi,j(q) is true for all
contacts (edge-vertex pairs)
q ? CO
13Free Space and Contact Surfaces
- F is bounded by the C-surfaces
C-surfaces
F
C-obstacle
14Free Space Computation
- To obtain the free space requires computing
arrangement of the C-surfaces
15Arrangement
- Arrangement A(S) of a set S of geometric objects
Halperin 1997 Agarwal Sharir 2000
Decomposition of space into relatively open
connected cells of dimensions 0,...,d
Arrangement of lines (clipped within a window)
16Free Space Computation
- Compute an arrangement of the C-surfaces
- Compute intersections between the C-surfaces
- Retain the appropriate portions of the arrangement
F
C-obstacle
17Free Space Computation
- Arrangement computation is difficult
- Computing surface-surface intersection is prone
to robustness problems - Typically O(n2) number of contact surfaces
- Contact surfaces are non-linear
18Free Space Approximation
- We have developed an accurate and efficient
approximate algorithm Varadhan and Manocha 2004
- Provides certain geometric and topological
guarantees on the approximation - Approximation is close to the boundary of the
free space - It has the same number of connected components
and genus as the exact Minkowski sum
19Free Space Approximation
3,929 contact surfaces
202T1R Gears
Goal
Start
212T1R Gears
222T1R GearsPath in Configuration Space
Path
Goal
Start
23Next lecture
- Probabilistic roadmap methods