A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

Bekkers, EJ Erik; Duits, Remco; Mashtakov, Alexey; Sanguinetti, GR Gonzalo

doi:10.1137/15m1018460

Cited by 55 publications

(116 citation statements)

References 37 publications

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“…Then we show that all six integrals are functionally independent on an open dense domain in T * (SE (3)) and are in involution.…”

Section: Liouville Integrabilitymentioning

confidence: 85%

“…In problem P MEC on SE(3), we aim for a Lipschitzian curve γ : [0, T ] → SE (3), that satisfies the boundary conditions γ (0) = e := (0, I ) and γ (T ) = (x 1 , R 1 ) ∈ SE(3), and minimizes the integral of sub-Riemannian length…”

Section: Definitionmentioning

confidence: 99%

“…. , λ 6 linear on the fibers of cotangent bundle T * (SE (3)) and apply the Pontryagin maximum principle (PMP) to the problem P MEC , where the Hamiltonian H is expressed as…”

Section: Definitionmentioning

confidence: 99%

“…Next, in the application of PMP to the problem P MEC , we rely on the sub-Riemannian arclength parameter t, which is well-defined for all SR-geodesics in (SE (3), , G ξ ). Before a cusp occurs, recall …”

Section: Remarkmentioning

confidence: 99%

“…the variables λ i ). By analyticity, functional independence of the integrals on an open dense domain in T * (SE(3)) follows from linear independence of gradients of the integrals at a single point (q, λ) ∈ T * (SE (3)). Since we have Explicit integration of Eq.…”

Section: Independence Of Integralsmentioning

confidence: 99%

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On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

et al. 2016

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We consider the problem P curve of minimizing L 0 ξ 2 + κ 2 (s) ds for a curve x in R 3 with fixed boundary points and directions. Here, the total length L ≥ 0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ > 0 is constant. We lift problem P curve on R 3 to a sub-Riemannian problem P mec on SE(3)/({0} × SO (2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P curve . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.

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“…Then we show that all six integrals are functionally independent on an open dense domain in T * (SE (3)) and are in involution.…”

Section: Liouville Integrabilitymentioning

confidence: 85%

Section: Definitionmentioning

confidence: 99%

“…. , λ 6 linear on the fibers of cotangent bundle T * (SE (3)) and apply the Pontryagin maximum principle (PMP) to the problem P MEC , where the Hamiltonian H is expressed as…”

Section: Definitionmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Independence Of Integralsmentioning

confidence: 99%

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A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

Cited by 55 publications

References 37 publications

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions

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