There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper presents the first randomized approach to kinodynamic planning (also known as trajectory planning or trajectory design). The task is to determine control inputs to drive a robot from an ...

Randomized kinodynamic planning

Real and Complex Analysis

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R/sup 2//spl times/S/sup 1/ of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning. >

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A motion planner for nonholonomic mobile robots

Given two oriented points in the plane, we determine and compute the shortest paths of bounded curvature joining them. This problem has been solved recently by Dubins in the no-cusp case, and by Reeds and Shepp otherwise. We propose a new solution based on the minimum principle of Pontryagin. Our approach simplifies the proofs and makes clear the global or local nature of the results.

Shortest paths of bounded curvature in the plane

Given two oriented points in the plane, the authors determine and compute the shortest paths of bounded curvature joining them. This problem has been solved by L.E. Dubins (1957) in the no-cusp case, and by J.A. Reeds and L.A. Shepp (1990) with cusps. A solution based on the minimum principle of Pontryagin is proposed. The approach simplifies the proofs and makes clear the global or local nature of the results. The no-cusp case and the more difficult case with cusps are discussed. >

https://hal.inria.fr/inria-00075059/document

We consider the class of C 2 , piecewise C 3 , planar paths joining two given conngurations (position, orientation, and curvature) X 0 and X f , and along which the derivative of the curvature (with respect to the arc length) remains bounded. We admit an innnite (countable) number of pieces, as long as the switching points do not accumulate more than a nite number of times. For generic X 0 and X f , we prove that the path of minimal length satisfying the constraint is such that : either it contains no line segment, or it contains innnitely many arcs of clothoid. As a consequence, the number of C 3 arcs involved in a shortest path may not be uniformly bounded with respect to X 0 and X f. the plane. Note sur les plus courts chemins dans le plan soumis a une contrainte sur la d eriv ee de la courbure R esum e : On consid ere la classe des chemins C 2 dans le plan, C 3 par mor-ceaux, joignant deux conngurations donn ees (position, tangente et courbure) X 0 et X f , le long desquels la d eriv ee de la courbure (par rapport a l'abs-cisse curviligne) reste born ee. On admet un nombre innni (d enombrable) de morceaux, mais seulement un nombre ni de points d'accumulations pour les points de commutations. Pour des X 0 et X f g en eriques, on prouve que le plus court chemin satisfaisant la contrainte est tel que : soit il ne contient pas de segment de droite, soit il contient aussi un nombre innni d'arcs de clothoide. En cons equence, le nombre de morceaux de classe C 3 constituant un plus court chemin n'est pas uniform ement born e par rapport a X 0 et X f. Mots-cl e : planiication de trajectoires, commande optimale, plus courts chemins contraints dans le plan.

/pdf/a-note-on-shortest-paths-in-the-plane-subject-to-a-25f08aqz8y.pdf

A note on shortest paths in the plane subject to a constraint on the derivative of the curvature

We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

https://iopscience.iop.org/article/10.1088/0266-5611/20/1/003/pdf

Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

Juliette Leblond

Papers

Shortest paths of bounded curvature in the plane

Shortest paths of bounded curvature in the plane

A note on shortest paths in the plane subject to a constraint on the derivative of the curvature

Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems

Recovery of pointwise sources or small inclusions in 2D domains and rational approximation