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. >

/pdf/a-motion-planner-for-nonholonomic-mobile-robots-1x3xuymw7z.pdf

A motion planner for nonholonomic mobile robots

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart & Leblond, 1998]. In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem.

Constrained optimization in classes of analytic functions with prescribed pointwise values

We study a Loewner type interpolation problem in the framework of matrix-valued H 2 functions. We establish a necessary and sufficient condition for a matrix-valued function given on an arc of circle to be the trace of such an H 2 function subject to some constraint.

https://www.ems-ph.org/journals/show_pdf.php?issn=0232-2064&vol=14&iss=2&rank=2

Loewner Interpolation in Matrix Hardy Classes

Let $\Omega_1,\Omega_2\subset {\mathbb C}$ be bounded domains. Let $\phi:\Omega_1\rightarrow \Omega_2$ holomorphic in $\Omega_1$ and belonging to $W^{1,\infty}_{\Omega_2}(\Omega_1)$. We study the composition operators $f\mapsto f\circ\phi$ on generalized Hardy spaces on $\Omega_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $\phi$, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact. Some of our results are new even for Hardy spaces of analytic functions.

Composition operators on generalized Hardy spaces

We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply
connected domain with Lipschitz boundary ∂Ω. Assuming Dirichlet and Neumann data available on Γ ⊂ ∂Ω
to be real-valued functions in W 1/2,2 (Γ) and L 2 (Γ) classes, respectively, we develop a non-iterative method for
solving this ill-posed Cauchy problem choosing L 2 bound of the solution on ∂Ω \ Γ as a regularizing parame-
ter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints
on the solution which, on practical side, can help incorporating outlying boundary measurements without
changing the boundary into a less regular one.

Recovery of harmonic functions in planar domains from partial boundary data respecting internal values

We investigate consistency properties of rational approximation of prescribed type in the weighted Hardy space 
$H^2_-(\mu)$
 for the exterior of the unit disk, where 
$\mu$
 is a positive symmetric measure on the unit circle 
${\mathbb T}$
 . The question of consistency, which is especially significant for gradient algorithms that compute local minima, concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational. In addition to describing some basic properties of the approximation problem, we prove for measures 
$\mu$
 having a rational function distribution (weight) with respect to arclength on 
${\mathbb T}$
 , that consistency holds only under rather restricted conditions.

Juliette Leblond

Papers

Constrained optimization in classes of analytic functions with prescribed pointwise values

Loewner Interpolation in Matrix Hardy Classes

Composition operators on generalized Hardy spaces

Recovery of harmonic functions in planar domains from partial boundary data respecting internal values

Weighted H 2 rational approximation and consistency