Finding Interpolating Curves Minimizing $L^\infty$ Acceleration in the Euclidean Space via Optimal Control Theory
05 Feb 2013-Siam Journal on Control and Optimization (Society for Industrial and Applied Mathematics)-Vol. 51, Iss: 1, pp 442-464
TL;DR: The problem of finding an interpolating curve passing through prescribed points in the Euclidean space is studied as an optimal control problem and simple but effective tools of optimal control theory are employed.
Abstract: We study the problem of finding an interpolating curve passing through prescribed points in the Euclidean space. The interpolating curve minimizes the pointwise maximum length, i.e., $L^\infty$-norm, of its acceleration. We reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory. We characterize solutions associated with singular and nonsingular controls. Some of the results we obtain are new even for the scalar interpolating function case. We reduce the infinite-dimensional interpolation problem to an ensuing finite-dimensional one and derive closed form expressions for interpolating curves. Consequently we devise efficient numerical techniques and illustrate them with examples.
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TL;DR: In this paper, the structure and stability concepts for irreducible chains are treated, and they have to this point considered only the dichotomy between transient and recurrent chains, and not the notion of stable chains.
Abstract: In our treatment of the structure and stability concepts for irreducible chains we have to this point considered only the dichotomy between transient and recurrent chains.
66 citations
TL;DR: In this article, the authors study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins' result again, and characterize these solutions as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem.
Abstract: Markov–Dubins path is the shortest planar curve joining two points with prescribed tangents, with a specified bound on its curvature. Its structure, as proved by Dubins in 1957, nearly 70 years after Markov posed the problem of finding it, is elegantly simple: a selection of at most three arcs are concatenated, each of which is either a circular arc of maximum (prescribed) curvature or a straight line. The Markov–Dubins problem and its variants have since been extensively studied in practical and theoretical settings. A reformulation of the Markov–Dubins problem as an optimal control problem was subsequently studied by various researchers using the Pontryagin maximum principle and additional techniques, to reproduce Dubins’ result. In the present paper, we study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins’ result again. We prove that abnormal control solutions do exist. We characterize these solutions, which were not studied adequately in the literature previously, as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem. Moreover, we prove that any feasible path of the types mentioned in Dubins’ result is a stationary solution, i.e., that it satisfies the Pontryagin maximum principle. We propose a numerical method for computing Markov–Dubins path. We illustrate the theory and the numerical approach by three qualitatively different examples.
20 citations
TL;DR: Third and fourth order of accuracy stable difference schemes for the approximate solutions of hyperbolic multipoint nonlocal boundary value problem in a Hilbert space H with self-adjoint positive definite operator A are considered.
12 citations
TL;DR: A realistic generalization of the Markov–Dubins problem, which is concerned with finding the shortest planar curve of constrained curvature joining two points with prescribed tangents, is the requirement that the curve passes through a number of prescribed intermediate points/nodes.
Abstract: A realistic generalization of the Markov–Dubins problem, which is concerned with finding the shortest planar curve of constrained curvature joining two points with prescribed tangents, is the requirement that the curve passes through a number of prescribed intermediate points/nodes. We refer to this generalization as the Markov–Dubins interpolation problem. We formulate this interpolation problem as an optimal control problem and obtain results about the structure of its solution using optimal control theory. The Markov–Dubins interpolants consist of a concatenation of circular (C) and straight-line (S) segments. Abnormal interpolating curves are shown to exist and characterized; however, if the interpolating curve contains a straight-line segment then it cannot be abnormal. We derive results about the stationarity, or criticality, of the feasible solutions of certain structure. In particular, any feasible interpolant with arc types of CSC in each stage is proved to be stationary, i.e., critical. We propose a numerical method for computing Markov–Dubins interpolating paths. We illustrate the theory and the numerical approach by four qualitatively different examples.
10 citations
TL;DR: In this paper, the optimal relative backward error for the numerical solution of the Dahlquist test problem is computed by one-step methods using results from optimal control theory, but elementary methods can also be used here because the problem is so simple.
Abstract: We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces some new insights into the numerical solution of stiff problems.
6 citations
References
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Book•
01 Jan 1978
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
10,258 citations
TL;DR: A comprehensive description of the primal-dual interior-point algorithm with a filter line-search method for nonlinear programming is provided, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix.
Abstract: We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.
7,966 citations
"Finding Interpolating Curves Minimi..." refers methods in this paper
...4s, a popular optimization software based on an interior point method; see [37]....
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Book•
01 Jan 1993TL;DR: An efficient translator is implemented that takes as input a linear AMPL model and associated data, and produces output suitable for standard linear programming optimizers.
Abstract: Practical large-scale mathematical programming involves more than just the application of an algorithm to minimize or maximize an objective function. Before any optimizing routine can be invoked, considerable effort must be expended to formulate the underlying model and to generate the requisite computational data structures. AMPL is a new language designed to make these steps easier and less error-prone. AMPL closely resembles the symbolic algebraic notation that many modelers use to describe mathematical programs, yet it is regular and formal enough to be processed by a computer system; it is particularly notable for the generality of its syntax and for the variety of its indexing operations. We have implemented an efficient translator that takes as input a linear AMPL model and associated data, and produces output suitable for standard linear programming optimizers. Both the language and the translator admit straightforward extensions to more general mathematical programs that incorporate nonlinear expressions or discrete variables.
3,176 citations
"Finding Interpolating Curves Minimi..." refers methods in this paper
...As in the case of Problem (Pfd), we employ the AMPL and Ipopt suite for the discretized problem....
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...We first solve problem (Pfd) using AMPL and Ipopt to find α = 20.762605861013, λ11 = (−0.132870370719, −0.327962931793), λ21 = (1.000000000000, 0.027268487871), λ31 = (−0.867129629281, 0.300694443922), −1 0 1 2 3 4 −0.5 0 0.5 1 1.5 2 2.5 3 z1 z2 −1 0 1 2 3 4 −0.5 0 0.5 1 1.5 2 2.5 3 z1 z2 Fig....
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...We solve problem (Pfd) using AMPL and Ipopt to find α = 241.3193402320, λ11 = (0.018177821, 0.062302851, −0.051273586), λ21 = (−0.019455171, −0.473702614, 0.168980653), λ31 = (−0.409886118, 1.000000000, 0.039190136), λ41 = (0.779721907, −0.683683770, −0.153457943), λ51 = (−0.368558439, 0.095083533, −0.003439259)....
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...We use AMPL [17] as an optimization modeling language that employs Ipopt as a solver....
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...We solve problem (Pfd) using AMPL and Ipopt to find α = 55.935918218882, λ11 = (−0.019181930951, −0.017508694823), λ21 = (0.396919631617, −0.070361956619), λ31 = (−1.000000000000, 0.142414830069), λ41 = (0.622262299334, −0.054544178627)....
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TL;DR: The analysis utilizes a connection between the Kuhn-Tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle to determine the convergence rate for Runge-Kutta discretizations of nonlinear control problems.
Abstract: The convergence rate is determined for Runge-Kutta discretizations of nonlinear control problems. The analysis utilizes a connection between the Kuhn-Tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle. This connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function.
444 citations
"Finding Interpolating Curves Minimi..." refers background in this paper
...One should recall that the trapezoidal rule, being a second-order approximation, requires the state variables to be at least C(2) [18, 23]....
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271 citations