defense-dissertation

Parameterizations in the Configuration
Space and Approximations of Related
Surfaces
Dror Atariah
May 9th, 2014

Introduction
Motion
Planning
PhD. Defense (D. Atariah) 2

Introduction
Motion Planning Problem

Introduction
Boundary between Cfree and Cforb
O
O
O
O
A(q)

Parameterization
Motion
Planning
Parameterization

Parameterization
Vertex-Edge

Parameterization
Edge-Vertex

Parameterization
Video

Approximation
Motion
Planning
Parameterization
Approximat
ion

Approximation
Motivation

Approximation
Approximation of Saddles
Goal
Obtain a local and optimal (w.r.t. the vertical distance) approximation of
saddle surfaces.

Approximation
Results
Interpolating

Approximation
Results
Theorem (Non-Interpolating)
Let > 0 be ﬁxed and set T ⊂ R2 to be the triangle with the following
vertices:
p0 = (0, 0)
p1 = 2
√
1 +
1
√
3
, 1 −
1
√
3
, p2 = 2
√
1 −
1
√
3
, 1 +
1
√
3
If ˆT is the lifting of the triangle T to the oﬀset saddle S3
, then the
following hold
• Vertical distance: distV (S, ˆT ) =
• Area of projection to the plane:
area(T ) =
8
√
3
≈ 4.6188 > 4.4721 ≈ 2
√
5 = area(TI
)

CGAL
Motion
Planning
Parameterization
Approximat
ion Ar
rangements

CGAL
Arrangements of Polylines
Motivation
• Arrangements of rational curves
• Extend the support of 2D Arrangements to unbounded polylines
Contribution
• Improvement of the implementation
• 5% speedup of the runtime
• Code is more generic

Summary
Motion
Planning
Parameterization
Approximat
ion Ar
rangements

Parameterization
Outline of Appendix
1 Parameterization
2 Approximation

Parameterization
Rotating Robot
R0(0) a = P
raTheorem
Let Pa = { q ∈ C : a(q) = P }, then q ∈ Pa iﬀ
q =
P − Rφa
φ

Parameterization
Vertex-Edge Contact
bj
bj+1
P(t)
−nO
j
ρi
nA
i−1
nA
iai
ai−1
ai+1
−nO
j
nA
i−1
nA
i
S(t, φ) =
P(t) − Rφai
φ

Parameterization
Edge-Vertex Contact
ai
ai+1
ai,t
nA
i
(1 − t) EA
i
ωj
−nO
j−1
−nO
j
bj
bj−1
bj+1
S(t, φ) =
bj − Rφai,t
φ

Parameterization
Differential Geometry of E-V Case
• Surface Normal: Ns(t, φ) =
RφnA
i
ai,t, EA
i
• First Fundamental Forms: E = ai − ai+1
2, F = det(ai , ai+1) and
G = 1 + ai,t
2
• Gaussian Curvature: K(t) = −E2
ν4 , where ν =
√
EG − F2.
• Mean Curvature: H(t) = EF
2ν3
• supt∈R |K(t)| = |K(t )| = 1 where t = ai ,ai −ai+1
E and ai,t is the
closest point on the support line of EA
i to the origin
• Normal Curvature, Principal Curvatures and Principal Curvature
Directions

Parameterization
Rational Parameterization
Rθ
=
1
1 + τ2
1 − τ2 −2τ
2τ 1 − τ2 = Mτ
for τ = tan θ
2 ∈ (−∞, +∞).

Approximation
Outline of Appendix
1 Parameterization
2 Approximation

Approximation
The Vertical Distance
• For p, q ∈ S = { (x, y, z) : z = F(x, y) } we have
distV ( pq, S) =
1
4
|a11∆2
x + 2a12∆x ∆y + a22∆2
y |,
where ∆x = px − qx , ∆y = py − qy and F(x, y) is a quadratic form.
• distH(A, B) ≤ distV (A, B)
h
vx
y

Approximation
Simple Saddle (Interpolating)
For
S = (x, y, z) ∈ R3
: z = xy
we have:
distV (S, pq) =
1
4
|∆x ∆y |
|xy|=
4
p0

Approximation
For
S = (x, y, z) ∈ R3
: z = xy
we have:
distV (S, pq) =
1
4
|∆x ∆y |
|xy|=
4
p0
p1
e0

Approximation
For
S = (x, y, z) ∈ R3
: z = xy
we have:
distV (S, pq) =
1
4
|∆x ∆y |
|xy|=
4
|(x
−
ξ)y|=
4
p0
p1
e0

Approximation
For
S = (x, y, z) ∈ R3
: z = xy
we have:
distV (S, pq) =
1
4
|∆x ∆y |
|xy|=
4
|(x
−
ξ)y|=
4
p0
p1
e0p2,1
p2,2
p2,3
p2,4
p2,5
p2,6

Approximation
For
S = (x, y, z) ∈ R3
: z = xy
we have:
distV (S, pq) =
1
4
|∆x ∆y |
|xy|=
4
|(x
−
ξ)y|=
4
p0
p1
e0
T4(ξ)
e1e2
p2,1
p2,2
p2,3
p2,4
p2,5
p2,6

Approximation
Going Global

Approximation
Construction of non-interpolating case
p0
p1
p11
p22
p23
p24
p25
p26

Approximation
Interpolating vs. non-interpolating

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