Towards faster roadmap algorithmsfor smooth and unbounded real algebraic sets
Remi Prebet, PolSys, LIP6, Sorbonne Universite JNCF 2021, March 2nd 2021
joint work with Mohab Safey El Din and Eric Schost
1
Motion planning problem
Semi-algebraic sets
S =M⋃i=1
{x ∈ Rn | fi(x) = 0 ∧ gi(x) > 0
}with (fi, gi) ⊂ R[x1, . . . , xn] finite
Motion planning problem
Data:
1. S ⊂ Rn semi-algebraic set
2. x,y ∈ S
Question:
Are x and y connected in S?
[LaValle, 2006]
2
First approach
Cylindrical Algebraic Decomposition (CAD) [Collins, 1973]
V ⊂ Rn alg. set of dimension d and defined by s polynomials of degree 6 D
Authors Complexities Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
A CAD adapted to a cylinder
[Basu & Pollack & Roy, 2009]
Goal: only describe the connectedness of S
Lower bound of complexity[Thom & Milnor, 1964]∣∣∣Connected components of S
∣∣∣ ≤ (sD)O(n)
3
First approach
Cylindrical Algebraic Decomposition (CAD) [Collins, 1973]
V ⊂ Rn alg. set of dimension d and defined by s polynomials of degree 6 D
Authors Complexities Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
A CAD adapted to a cylinder
[Basu & Pollack & Roy, 2009]
Goal: only describe the connectedness of S
Lower bound of complexity[Thom & Milnor, 1964]∣∣∣Connected components of S
∣∣∣ ≤ (sD)O(n)
3
Roadmaps
[Canny, 1988] Compute R ⊂ S one-dimensional, sharing its connectivity
Roadmap of a semi-algebraic set S
It is a semi-algebraic curve R ⊂ S, containing P finite and such that
for all connected components C of S: C ∩R is non-empty and connected
Proposition
x,y ∈P are connected in S ⇐⇒ they are in R
Interest?
Arbitrary dimension =⇒ Dimension 1
4
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Canny, 1988]
If V is bounded, W (π2, V ) ∪ F has dimension d− 1
and satisfies the Roadmap property.
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√
n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Canny, 1988]
If V is bounded, W (π2, V ) ∪ F has dimension d− 1
and satisfies the Roadmap property.
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√
n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Canny, 1988]
If V is bounded, W (π2, V ) ∪ F has dimension d− 1
and satisfies the Roadmap property.
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√
n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
Results based on a theorem in the bounded case
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
Results based on a theorem in the bounded case
Remove the boundedness
assumption is a costly step
5
Complexities
S ⊂ Rn semi alg. set of dimension d and defined by s polynomials of degree 6 D
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Author·s Complexity Assumptions
[Schwartz & Sharir, 1983] (sD)O(1)n
[Canny, 1993] (sD)O(n2)
[Basu & Pollack & Roy, 1999] sd+1DO(n2)
[Safey El Din & Schost, 2011] (nD)O(n√n) Smooth, bounded algebraic sets
[Basu & Roy & Safey El Din
& Schost, 2014](nD)O(n
√n) Algebraic sets
[Basu & Roy, 2014] (nD)O(n log2 n) Algebraic sets
[Safey El Din & Schost, 2017] O((nD)6n log2 d
)Smooth, bounded algebraic sets
Results based on a theorem in the bounded case
Remove the boundedness
assumption is a costly step
Necessity of a new theorem
in the unbounded case!
5
A challenging robotic application
Matrix M associated to a puma-type robot with a non-zero offset in the wrist
(v3 + v2)(1− v2v3) 0 A(v) d3A(v) a2(v23 + 1)(v22 − 1)− a3A(v) 2d3(v3 + v2)(v2v3 − 1)
0 v23 + 1 0 2a2v3 0 (a3 − a2)v23 + a2 + 2a30 1 0 0 0 2a30 0 1 0 0 0
v4 1− v24 0 d4(1− v24) −2d4v4 0
(v24 − 1)v5 4v4v5 (1− v25)(v24 + 1) (1− v25)(v24 − 1)d5 + 4d4v4v5 2d5v4(1− v25) + 2d4v5(1− v24) −2d5v5(v24 + 1)
where A(v) = (v23 − 1)(v22 − 1)− 4v2v3
A puma 560 [Unimation, 1984]
Fix arbitrary (a2, a3, d3, d4, d5) ∈ (Q∗+)5
a2, a3: distances d3, d4, d5: offsets
v2, v3, v4, v5: half-angle tangents of rotations
Cuspidality?
Is there γ : [0, 1]→ R4 such that
γ(0) = v and γ(1) = v′
∀t ∈ [0, 1], detM(γ(t)) 6= 0?
Goal
Compute the number of connected components
of S ={v ∈ R4 | det(M(v)) 6= 0
}
6
Contributions
Computations - first contribution
• Computed a roadmap for the analysis of the kinematic singularities of a
puma-type robot
• This problem was considered as difficult using the current techniques
Roadmap algorithms can be used for solving robotic problems
New theoretical result - second contribution
• Stated and proved a new connectivity result for constructing roadmaps
• Holds in the case of unbounded real algebraic sets
• Assumptions hold generically
• An algorithm can be easily designed from it
We can obtain efficient algorithms for unbounded real algebraic sets
7
Cuspidality of a PUMA robot
Reduction
Consider S ={x ∈ Rn | f(x) 6= 0
}Assumption 1: S is bounded. [Canny, 1988]
For r > 0 large enough,
RoadMap(S ∩B(0, r)
)= RoadMap(S)
Assumption 2: S is an algebraic set [Canny, 1993]
For ε > 0 small enough,
Roadmap({f ≥ ε} ∩B(0, r)
)Roadmap
({f 6= 0} ∩B(0, r)
) ⋃Roadmap
({f ≤ −ε} ∩B(0, r)
)
Boundaries
Sufficient to compute the intersection of S ∩B(0, r) with the roadmaps of
S+ε = V(f − ε), S+
ε,r = V(f − ε, ||x||2 − r), S+r = V(||x||2 − r)
and S−ε = V(f + ε), S−ε,r = V(f + ε, ||x||2 − r), S−r = V(||x||2 − r).
8
Canny’s strategy
Projection through:
π2 : (x1, . . . , xn) 7→ (x1, x2)
W (π2, V ) critical locus of π2.
Intersects all the
connected components of V
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Projection through:
π2 : (x1, . . . , xn) 7→ (x1, x2)
W (π2, V ) critical locus of π2.
Intersects all the
connected components of V
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Projection through:
π2 : (x1, . . . , xn) 7→ (x1, x2)
W (π2, V ) critical locus of π2.
Intersects all the
connected components of V
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Projection through:
π2 : (x1, . . . , xn) 7→ (x1, x2)
W (π2, V ) critical locus of π2.
Intersects all the
connected components of V
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
Morse theory
“Scan” W (π2, V ) at the critical values
of π1
• We repair the connectivity failures
with critical fibers
• We repeat the process at
every critical value
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
W (π2, V ) polar variety
F critical fibers
Assumptions:
1. W (π2, V ) has dimension 1
2. F has dimension dim(V )− 1
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
W (π2, V ) polar variety
F critical fibers
Assumptions:
1. W (π2, V ) has dimension 1
2. F has dimension dim(V )− 1
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
W (π2, V ) polar variety
F critical fibers
Assumptions:
1. W (π2, V ) has dimension 1
2. F has dimension dim(V )− 1
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Canny’s strategy
W (π2, V ) polar variety
F critical fibers
Assumptions:
1. W (π2, V ) has dimension 1
2. F has dimension dim(V )− 1Genericity
Theorem [Canny, 1988]
W (π2, V )⋃F has dimension dim(V )− 1
and satisfies the Roadmap property
Roadmap property
∀C connected component,
C ∩R is non-empty and connected
9
Computation of critical loci
Critical points
x critical point of πi on V ⇐⇒{x ∈ reg(V ) | πi(TxV ) 6= Ci
}= W ◦(πi, V )
An effective characterisation
x critical point of πi on V Ji = Jac(h, [xi+1, . . . , xn]) where h ∈ I(V ) ⊂ R[x1, . . . , xn]
{x ∈ V | rank Ji(x) < c
}All c-minors of Ji(x) vanish at x
(Lemma) c=n−dim(V )
Determinantal ideal
Torus of revolution axis directed by the vector −→x +−→z
Splitting in two sets =⇒ Degree reduction
10
Computation of critical loci
Critical points
x critical point of πi on V ⇐⇒{x ∈ reg(V ) | πi(TxV ) 6= Ci
}= W ◦(πi, V )
An effective characterisation
x critical point of πi on V Ji = Jac(h, [xi+1, . . . , xn]) where h ∈ I(V ) ⊂ R[x1, . . . , xn]
{x ∈ V | rank Ji(x) < c
}All c-minors of Ji(x) vanish at x
(Lemma) c=n−dim(V ) Determinantal ideal
Torus of revolution axis directed by the vector −→x +−→z
Splitting in two sets =⇒ Degree reduction
10
Computation of critical loci
Critical points
x critical point of πi on V ⇐⇒{x ∈ reg(V ) | πi(TxV ) 6= Ci
}= W ◦(πi, V )
An effective characterisation
x critical point of πi on V Ji = Jac(h, [xi+1, . . . , xn]) where h ∈ I(V ) ⊂ R[x1, . . . , xn]
{x ∈ V | rank Ji(x) < c
}All c-minors of Ji(x) vanish at x
(Lemma) c=n−dim(V ) Determinantal ideal
Torus of revolution axis directed by the vector −→x +−→z
Splitting in two sets =⇒ Degree reduction
10
Computation of critical loci
Critical points
x critical point of πi on V ⇐⇒{x ∈ reg(V ) | πi(TxV ) 6= Ci
}= W ◦(πi, V )
An effective characterisation
x critical point of πi on V Ji = Jac(h, [xi+1, . . . , xn]) where h ∈ I(V ) ⊂ R[x1, . . . , xn]
{x ∈ V | rank Ji(x) < c
}All c-minors of Ji(x) vanish at x
(Lemma) c=n−dim(V ) Determinantal ideal
Two kinds of critical points
x critical point of πi on V x ∈W2 (polar variety)
TxW2 ⊂ TxV is normal to Im(π1)
TxW2 is normal to Im(π1) or
TxW2 is normal to Im(π2) ⊃ Im(π1)
Splitting in two sets =⇒ Degree reduction
10
First contribution: application to a puma-type robot
Matrix M associated to a puma-type robot with a non-zero offset in the wrist
(v3 + v2)(1− v2v3) 0 A(v) d3A(v) a2(v23 + 1)(v22 − 1)− a3A(v) 2d3(v3 + v2)(v2v3 − 1)
0 v23 + 1 0 2a2v3 0 (a3 − a2)v23 + a2 + 2a30 1 0 0 0 2a30 0 1 0 0 0
v4 1− v24 0 d4(1− v24) −2d4v4 0
(v24 − 1)v5 4v4v5 (1− v25)(v24 + 1) (1− v25)(v24 − 1)d5 + 4d4v4v5 2d5v4(1− v25) + 2d4v5(1− v24) −2d5v5(v24 + 1)
S ={v ∈ R4 | det(M(v)) 6= 0
}
A puma 560 [Unimation, 1984]
First step
Max. degree without splitting: 1858 Max. time
Locus Max. degree msolve Maple
Critical points 934 3.4 min 63 min
Critical curves 220 77 min 280 min
Recursive step over 96 fibers
Data are for one fiber Max. time
Locus Max. degree msolve Maple
Critical points 38 2 s 3 s
Critical curves 21 3 s 40 s
Roadmap
Degree: 8168
Time: 3h22 (msolve)/ 20h (Maple)
11
A new connectivity result for
unbounded real algebraic sets
Second contribution: new connectivity result
Projection on 2 coordinates
π2 : Cn → C2
(x1, . . . ,xn) 7→ (x1,x2)
• W (π2, V ) polar variety
• F2 = π−11 (π1(K)) critical fibers
• K = critical points of π1 on W (π2, V )
Connectivity result [Canny, 1988]
If V is bounded, W (π2, V ) ∪ F2 has dimension d− 1
and satisfies the Roadmap property
12
Second contribution: new connectivity result
Projection on i coordinates
πi : Cn → Ci
(x1, . . . ,xn) 7→ (x1, . . . ,xi)
• W (πi, V ) polar variety
• Fi = π−1i−1(πi−1(K)) critical fibers
• K = critical points of π1 on W (πi, V )
Connectivity result [Safey El Din & Schost, 2011]
If V is bounded, W (πi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
No critical points!
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
Connectivity result [P. & Safey El Din & Schost, 2021]
If V is bounded, W (ϕi, V ) ∪ Fi has dimension max(i− 1, d− i+ 1)
and satisfies the Roadmap property
Critical point!
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
Morse theory
Two disjoint cases:
x ∈ ϕ−11 (K) or not
Sard’s lemma
ϕ−11 (K) is finite
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
Thom’s isotopy Lemma
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
Algebraic Puiseux Series
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
“Graded” roadmap property RM(x):
For all connected components C of V ∩ Rn ∩ϕ−11
((−∞, x]
)C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
12
Second contribution: new connectivity result
Non-negative proper polynomial map
ϕi : Cn −→ Ci
x 7→ (ψ1(x), . . . , ψi(x))
• W (ϕi, V ) generalized polar variety
• Fi = ϕi−1(ϕi−1(K)) critical fibers.
• K = critical points of ϕ1 on W (ϕi, V )
Roadmap property RM:
For all connected components C of V
C ∩ (Fi ∪W (ϕi, V )) is non-empty and connected
12
How to use it
Assumptions to satisfy in the new result
(P) ϕ1 is a proper map bounded from below
(D) dimWi = i− 1 and dimFi = n− d+ 1
A prototype
Take ϕ as:
ϕ =
(n∑
i=1
(xi − ai)2 , bT2−→x , . . . , bT
n−→x)
where ai ∈ R, bi ∈ Rn
Then choose a generic (a, b2, . . . , bn) ∈ Rn2
Semi-algebraic sets
A strategy to tackle unbounded semi-algebraic sets:
f ∈ R[x1, . . . , xn]
u new variable
f 6= 0 −→ uf = 1
f ≥ 0 −→ f = u2
f ≤ 0 −→ f = −u2
13
How to use it
Assumptions to satisfy in the new result
(P) ϕ1 is a proper map bounded from below
(D) dimWi = i− 1 and dimFi = n− d+ 1
A prototype
Take ϕ as:
ϕ =
(n∑
i=1
(xi − ai)2 , bT2−→x , . . . , bT
n−→x)
where ai ∈ R, bi ∈ Rn
Then choose a generic (a, b2, . . . , bn) ∈ Rn2
Semi-algebraic sets
A strategy to tackle unbounded semi-algebraic sets:
f ∈ R[x1, . . . , xn]
u new variable
f 6= 0 −→ uf = 1
f ≥ 0 −→ f = u2
f ≤ 0 −→ f = −u2
13
How to use it
Assumptions to satisfy in the new result
(P) ϕ1 is a proper map bounded from below
(D) dimWi = i− 1 and dimFi = n− d+ 1
A prototype
Take ϕ as:
ϕ =
(n∑
i=1
(xi − ai)2 , bT2−→x , . . . , bT
n−→x)
where ai ∈ R, bi ∈ Rn
Then choose a generic (a, b2, . . . , bn) ∈ Rn2
Semi-algebraic sets
A strategy to tackle unbounded semi-algebraic sets:
f ∈ R[x1, . . . , xn]
u new variable
f 6= 0 −→ uf = 1
f ≥ 0 −→ f = u2
f ≤ 0 −→ f = −u2
13
How to use it
Assumptions to satisfy in the new result
(P) ϕ1 is a proper map bounded from below
(D) dimWi = i− 1 and dimFi = n− d+ 1
A prototype
Take ϕ as:
ϕ =
(n∑
i=1
(xi − ai)2 , bT2−→x , . . . , bT
n−→x)
where ai ∈ R, bi ∈ Rn
Then choose a generic (a, b2, . . . , bn) ∈ Rn2
Semi-algebraic sets
A strategy to tackle unbounded semi-algebraic sets:
f ∈ R[x1, . . . , xn]
u new variable
f 6= 0 −→ uf = 1
f ≥ 0 −→ f = u2
f ≤ 0 −→ f = −u2
13
How to use it
Assumptions to satisfy in the new result
(P) ϕ1 is a proper map bounded from below
(D) dimWi = i− 1 and dimFi = n− d+ 1
A prototype
Take ϕ as:
ϕ =
(n∑
i=1
(xi − ai)2 , bT2−→x , . . . , bT
n−→x)
where ai ∈ R, bi ∈ Rn
Then choose a generic (a, b2, . . . , bn) ∈ Rn2
Semi-algebraic sets
A strategy to tackle unbounded semi-algebraic sets:
f ∈ R[x1, . . . , xn]
u new variable
f 6= 0 −→ uf = 1
f ≥ 0 −→ f = u2
f ≤ 0 −→ f = −u2
13
Algorithm
Considering an algebraic set V ⊂ Cn with dimension d
(V, d) d-roadmap of V
i = 2 Bounded!
(W2, 1)⋃
(F2, d− 1) (d− 1)-roadmap of V
Theorem
F2 = ϕ−11 (ϕ1(K))
with K finite
Complexity?
RoadmapBounded(F2) = O((nD)6n log2 d
)degree(F2)≤ D
+ Computation of W2 and F2 << O((nD)6n log2 d
)Complexity conjecture = O
((nD)6n log2 d
)
14
Algorithm
Considering an algebraic set V ⊂ Cn with dimension d
(V, d) d-roadmap of V
i = 2
(W2, 1)⋃
(F2, d− 1) (d− 1)-roadmap of V
[Safey El Din and Schost, 2017]
(W2, 1)⋃
(RF2 , 1) 1-roadmap of V
Theorem
RoadmapBounded
Complexity?
RoadmapBounded(F2) = O((nD)6n log2 d
)degree(F2)≤ D
+ Computation of W2 and F2 << O((nD)6n log2 d
)Complexity conjecture = O
((nD)6n log2 d
)
14
Algorithm
Considering an algebraic set V ⊂ Cn with dimension d
(V, d) d-roadmap of V
i = 2
(W2, 1)⋃
(F2, d− 1) (d− 1)-roadmap of V
[Safey El Din and Schost, 2017]
(W2, 1)⋃
(RF2 , 1) 1-roadmap of V
Theorem
RoadmapBounded
Complexity?
RoadmapBounded(F2) = O((nD)6n log2 d
)degree(F2)≤ D
+ Computation of W2 and F2 << O((nD)6n log2 d
)Complexity conjecture = O
((nD)6n log2 d
)14
Conclusion
Summary
Compute a roadmap for the analysis of the kinematic
singularities of a puma-type robot
New result in the unbounded case
Dimension assumptions hold generically
An algorithm can be developed
Future work
Applications:
◦ Implement an algorithm for describing the geometry of roadmaps
◦ Describe the geometry of the singularity set of a puma-type robot
(joint work with P.Wenger)
◦ Analysis of the kinematic singularities of a puma-type robot
Theoretical aspects:
◦ Develop an algorithm for constructing roadmaps of smooth
unbounded algebraic sets
◦ Generalize the result to smooth unbounded semi-algebraic sets
Credits: figures have been made using TEXgraph and Maple
Thank you for your attention!
15
Conclusion
Summary
Compute a roadmap for the analysis of the kinematic
singularities of a puma-type robot
New result in the unbounded case
Dimension assumptions hold generically
An algorithm can be developed
Future work
Applications:
◦ Implement an algorithm for describing the geometry of roadmaps
◦ Describe the geometry of the singularity set of a puma-type robot
(joint work with P.Wenger)
◦ Analysis of the kinematic singularities of a puma-type robot
Theoretical aspects:
◦ Develop an algorithm for constructing roadmaps of smooth
unbounded algebraic sets
◦ Generalize the result to smooth unbounded semi-algebraic sets
Credits: figures have been made using TEXgraph and Maple
Thank you for your attention! 15
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