Navigating Around Convex Sets

Veerman, J. J. P.

doi:10.1080/00029890.2020.1743596

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…In [1] (see also [10]), a related, but more complicated, counter-example was constructed which carries over to cylinders in R 3 . It says that here is a convex C 1,1 cylinder such that that the projection Π onto this cylinder does not have a derivative.…”

Section: Remarkmentioning

confidence: 99%

“…In many applications it is important to track the point Π(γ(t)) on the surface of the body nearest to the moving object 1 . In [10], a method to compute and track the projection was considered. Instead, here we consider the question whether this projection can take geodesics to (reparametrized) geodesics.…”

Section: Introductionmentioning

confidence: 99%

“…Because of the smoothness and the convexity, we can smoothly coordinatize the space Ω surrounding the convex body by using these coordinate patches as follows [10]:…”

Section: Introductionmentioning

confidence: 99%

“…where n is the unit normal to K 0 . These 3-dimensional coordinate patches form a differentiable atlas of Ω. Denote by Π : Ω → K 0 the orthogonal 'projection' from Ω onto K 0 , defined as follows [10]: γ := Π(z) is the unique point on K 0 nearest to z ∈ Ω. Clearly, the inverse of Π at a point γ of K 0 consists of a ray normal to K 0 at γ.…”

Section: Introductionmentioning

confidence: 99%

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Geodesics on Regular Constant Distance Surfaces

Veerman

2023

Preprint

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Suppose that the surfaces K0 and Kr are the boundaries of two convex, complete, connectedC2 bodies in R3. Assume further that the (Euclidean) distance between any point x in Krand K0 is always r (r > 0). For x in Kr, let Pi(x) denote the nearest point to x in K0. Weshow that the projection Pi preserves geodesics in these surfaces if and only if both surfacesare concentric spheres or co-axial round cylinders. This is optimal in the sense that the mainstep to establish this result is false for C1;1 surfaces. Finally, we give a non-trivial example of a geodesic preserving projection of two C2 non-constant distance surfaces. The question whether for any C2 convex surface S0, there is a surface S whose projection to S0 preserves geodesics is open. 2020 Mathematics Subject Classification: 52A15, 53A05.

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Because of the smoothness and the convexity, we can smoothly coordinatize the space Ω surrounding the convex body by using these coordinate patches as follows [10]:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geodesics on Regular Constant Distance Surfaces

Veerman

2023

Preprint

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“…Further, P C is not generally directionally differentiable [17], [18] unless secondorder regularity assumptions on C are satisfied [19], [20]. An up-to-date review of this subject including detailed examples can also be found in [21]. We avoid these technicalities because we require directional differentiability only along a trajectory (c.f.…”

Section: Solution Approach and Related Workmentioning

confidence: 99%

On the Differentiability of Projected Trajectories and the Robust Convergence of Non-Convex Anti-Windup Gradient Flows

Hauswirth

Dörfler

Teel

2020

IEEE Control Syst. Lett.

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This paper concerns a new class of discontinuous dynamical systems for constrained optimization. These dynamics are particularly suited to solve nonlinear, non-convex problems in closed-loop with a physical system. Such approaches using feedback controllers that emulate optimization algorithms have recently been proposed for the autonomous optimization of power systems and other infrastructures. In this paper, we consider feedback gradient flows that exploit physical input saturation with the help of anti-windup control to enforce constraints. We prove semi-global convergence of "projected" trajectories to firstorder optimal points, i.e., of the trajectories obtained from a pointwise projection onto the feasible set. In the process, we establish properties of the directional derivative of the projection map for non-convex, prox-regular sets.

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Geodesics on regular constant distance surfaces

Veerman

2023

J. Geom.

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Navigating Around Convex Sets

Cited by 5 publications

References 13 publications

Geodesics on Regular Constant Distance Surfaces

Geodesics on Regular Constant Distance Surfaces

On the Differentiability of Projected Trajectories and the Robust Convergence of Non-Convex Anti-Windup Gradient Flows

Geodesics on regular constant distance surfaces

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