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.
Similar content being viewed by others
References
Artelys Knitro - Nonlinear optimization solver. https://www.artelys.com/knitro. Accessed 26 Oct 2017
Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286–1309 (2007)
Agwu, N.N., Martin, C.F.: Optimal control of dynamical systems: application to spline approximations. Appl. Math. Comput. 97, 99–138 (1998)
Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.A.: Quadratic order conditions for bang–singular extremals. Num. Alg. Contr. Optim. 2, 511–546 (2012)
Aronsson, G.: Perfect splines and nonlinear control theory. J. Approx. Theory 25, 142–152 (1979)
Augustin, D., Maurer, H.: Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems. Control Cybern. 29, 11–31 (2000)
Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM Publications, Philadelphia (2014)
Boissonnat, J.-D., Cérézo, A., Leblond, J.: Shortest paths of bounded curvature in the plane. Plus courts chemins de courbure borée dans le plan, INRIA internal report (1991)
Brunnett, G., Kiefer, J., Wendt, L.: Fair curves for motion planning. Int. J. Veh. Des. 21, 266–277 (1999)
Clarke, F.H., Vinter, R.B.: Applications of multiprocesses. SIAM J. Control Optim. 27, 1048–1071 (1989)
Dmitruk, A.V., Kaganovich, A.M.: Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete Contin. Dyn. Syst. 29, 523–545 (2011)
Dontchev, A.L.: Best interpolation in a strip. J. Approx. Theory 73, 334–342 (1993)
Dontchev, A.L., Kolmanovski, I.: Best interpolation in a strip II: reduction to unconstrained convex optimization. Comput. Optim. Appl. 5, 233–251 (1996)
Dontchev, A.L., Qi, H.-D., Qi, L., Yin, H.: A Newton method for shape-preserving spline interpolation. SIAM J. Optim. 13, 588–602 (2002)
Dubins, L.E.: On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. Am. J. Math. 79, 497–516 (1957)
Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, 2nd edn. Brooks/Cole Publishing Company / Cengage Learning, Boston (2003)
Fredenhagen, S., Oberle, H.J., Opfer, G.: On the construction of optimal monotone cubic spline interpolations. J. Approx. Theory 96, 182–201 (1999)
Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47, 99–131 (2005)
Goaoc, X., Kim, H.-S., Lazard, S.: Bounded-curvature shortest paths through a sequence of points using convex optimization. SIAM J. Comput. 42, 662–684 (2013)
Isaev, V.K.: To the theory of splines. Appl. Math. Comput. 217, 1095–1109 (2010)
Isaiah, P., Shima, T.: Motion planning algorithms for the Dubins tavelling salesperson problem. Automatica 53, 247–255 (2015)
Kaya, C.Y.: Markov–Dubins path via optimal control theory. Comput. Optim. Appl. 68(3), 719–747 (2017)
Kaya, C.Y., Lucas, S.K., Simakov, S.T.: Computations for bang–bang constrained optimal control using a mathematical programming formulation. Optim. Control Appl. Meth. 25(6), 295–308 (2004)
Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57(3), 685–702 (2014)
Kaya, C.Y., Noakes, J.L.: A global control law with implications in time-optimal control. In: Proceedings of 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, USA, 3823–3824 (1994)
Kaya, C.Y., Noakes, J.L.: Computations and time-optimal controls. Optim. Control Appl. Methods 17, 171–185 (1996)
Kaya, C.Y., Noakes, J.L.: Computational algorithm for time-optimal switching control. J. Optim. Theory App. 117, 69–92 (2003)
Kaya, C.Y., Noakes, J.L.: Finding interpolating curves minimizing \(L^\infty \) acceleration in the Euclidean space via optimal control theory. SIAM J. Control Optim. 51, 442–464 (2013)
Kreĭn, M.G., Nudel’man, A.A.: The Markov Moment Problem and Extremal Problems. American Mathematical Society (1977)
Looker, J.R.: Constant speed interpolating paths, Defence Science and Technology Organization technical report DSTO-TN-0989 (2011)
Markov, A.A.: Some examples of the solution of a special kind of problem on greatest and least quantities. Soobscenija Charkovskogo Matematiceskogo Obscestva 2–1(5,6), 250–276 (1889). (in Russian)
Maurer, H., Büskens, C., Kim, J.-H.R., Kaya, C.Y.: Optimization methods for the verification of second order sufficient conditions for bang–bang controls. Optim. Control Appl. Methods 26, 129–156 (2005)
McClure, D.E.: Perfect spline solutions of \(L_\infty \) extremal problems by control methods. J. Approx. Theory 15, 226–242 (1975)
Micchelli, C.A., Smith, P.W., Swetits, J., Ward, J.D.: Constrained \(L_p\) approximation. Constr. Approx. 1, 93–102 (1985)
Opfer, G., Oberle, H.J.: The derivation of cubic splines with obstacles by methods of optimization and optimal control. Numer. Math. 52, 17–31 (1988)
Osmolovskii, N.P., Maurer, H.: Applications to Regular and Bang–Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM Publications, Philadelphia (2012)
Savla, K., Frazzoli, E., Bullo, F.: Traveling salesperson problems for the Dubins vehicle. IEEE Trans. Auto. Control 53, 1378–1391 (2008)
Sussmann, H.J., Tang, G.: Shortest paths for the Reeds–Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control, Rutgers Center for Systems and Control (Sycon) Report 91–10 (1991)
Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Progr. 106, 25–57 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kaya, C.Y. Markov–Dubins interpolating curves. Comput Optim Appl 73, 647–677 (2019). https://doi.org/10.1007/s10589-019-00076-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-019-00076-y
Keywords
- Markov–Dubins path
- Interpolation
- Constrained curvature
- Optimal control
- Singular control
- Bang–bang control
- Abnormal optimal control problem