Abstract
Range reduction techniques often considerably enhance the performance of algorithmic approaches for the solution of nonconvex problems. In this paper we propose a range reduction technique for a class of optimization problems with some special structured constraints. The procedure explores and updates the values associated to the nodes of a suitably defined graph. Convergence of the procedure and some efficiency issues, in particular related to the order into which the nodes of the graph are explored. The proposed technique is applied to solve problems arising from a relevant practical application, namely velocity planning along a given trajectory. The computational experiments show the efficiency of the procedure and its ability of returning solutions within times much lower than those of nonlinear solvers and compatible with real-time applications.
Similar content being viewed by others
References
Ausiello, G., Franciosa, P., Frigioni, D.: Directed hypergraphs: Problems, algorithmic results, and a novel decremental approach. In: Theoretical Computer Science. ICTCS 2001. Lecture Notes in Computer Science vol. 2202, pp. 312–328 (2001)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Basel (1997)
Belotti, P., Lee, J., Liberti, L., Margot, F., Wächther, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)
Benson, H.P.: Simplicial branch-and-reduce algorithm for convex programs with a multiplicative constraint. J. Optim. Theory Appl. 145, 213–233 (2010)
Bobrow, J., Dubowsky, S., Gibson, J.: Time-optimal control of robotic manipulators along specified paths. Int. J. Robot. Res. 4(3), 3–17 (1985)
Cabassi, F., Consolini, L., Laurini, M., Locatelli, M.: Convergence analysis of spatial-sampling based algorithms for time-optimal smooth velocity planning (2017) (submitted)
Caprara, A., Monaci, M.: Bidimensional packing by bilinear programming. Math. Program. 118, 75–108 (2009)
Caprara, A., Locatelli, M.: Global optimization problems and domain reduction strategies. Math. Program. 125, 123–137 (2010)
Caprara, A., Locatelli, M., Monaci, M.: Theoretical and computational results about optimality-based domain reductions. Comput. Optim. Appl. 64(2), 513–533 (2016)
Casado, L.G., Martinez, J.A., Garcia, I., Sergeyev, Y.D.: New interval analysis support functions using gradient information in a global minimization algorithm. J. Glob. Optim. 25, 345–362 (2003)
Chen, C., He, Y., Bu, C., Han, J., Zhang, X.: Quartic Bézier curve based trajectory generation for autonomous vehicles with curvature and velocity constraints. In: IEEE International Conference on Robotics and Automation (ICRA), pp. 6108–6113 (2014)
Consolini, L., Locatelli, M., Minari, A., Piazzi, A.: A linear-time algorithm for minimum-time velocity planning of autonomous vehicles. In: IEEE Proceedings of the 24th Mediterranean Conference on Control and Automation (MED) (2016)
Consolini, L., Locatelli, M., Minari, A., Piazzi, A.: An optimal complexity algorithm for minimum-time velocity planning. Syst. Control Lett. 103, 50–57 (2017)
Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
Faria, D.C., Bagajewicz, M.J.: Novel bound contraction procedure for global optimization of bilinear MINLP problems with applications to water management problems. Comput. Chem. Eng. 35, 446–455 (2011)
Frego, M., Bertolazzi, E., Biral, F., Fontanelli, D., Palopoli, L.: Semi-analytical minimum time solutions for a vehicle following clothoid-based trajectory subject to velocity constraints. In: 2016 European Control Conference (ECC), pp. 2221–2227 (2016)
Hamed, A.S.E., McCormick, G.P.: Calculation of bounds on variables satisfying nonlinear inequality constraints. J. Glob. Optim. 3, 25–47 (1993)
Hansen, P., Jaumard, B., Lu, S.H.: An analytical approach to global optimization. Math. Program. 52, 227–254 (1991)
Kant, K., Zucker., S.W.: Toward efficient trajectory planning: the path-velocity decomposition. Int. J. Robot. Res. 5(3), 72–89 (1986)
Li, X., Sun, Z., Kurt, A., Zhu, Q.: A sampling-based local trajectory planner for autonomous driving along a reference path. In: IEEE Intelligent Vehicles Symposium Proceedings, pp. 376–381 (2014)
Locatelli, M., Raber, U.: Packing equal circles in a square: a deterministic global optimization approach. Discrete Appl. Math. 122, 139–166 (2002)
Maranas, C.D., Floudas, C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–370 (1997)
Muñoz, V., Ollero, A., Prado, M., Simón, A.: Mobile robot trajectory planning with dynamic and kinematic constraints. In: Proceedings of the 1994 IEEE International Conference on Robotics and Automation, vol. 4, pp. 2802–2807, San Diego, CA (1994)
Nagy, A., Vajk, I.: LP-based velocity profile generation for robotic manipulators. Int. J. Control. (2017). https://doi.org/10.1080/00207179.2017.1286535
Ryoo, H., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–139 (1996)
Shen, P., Ma, Y., Chen, Y.: Global optimization for the generalized polynomial sum of ratios problem. J. Glob. Optim. 50, 439–455 (2011)
Slotine, J.J.E., Yang, H.S.: Improving the efficiency of time-optimal path-following algorithms. IEEE Trans. Robot. Autom. 5(1), 118–124 (1989)
Solea, R., Nunes, U.: Trajectory planning with velocity planner for fully-automated passenger vehicles. In: IEEE Intelligent Transportation Systems Conference, ITSC ’06, pp. 474–480 (2006)
Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming theory, algorithms, software, and applications. In: Nonconvex Optimization and Its Applications, vol. 65. Springer, Berlin (2003)
Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)
Tóth, B., Casado, L.: Multi-dimensional pruning from the Baumann point in an interval global optimization algorithm. J. Glob. Optim. 38, 215–236 (2007)
Velenis, E., Tsiotras, P.: Minimum-time travel for a vehicle with acceleration limits: Theoretical analysis and receding-horizon implementation. J. Optim. Theory Appl. 138(2), 275–296 (2008)
Verscheure, D., Demeulenaere, B., Swevers, J., Schutter, J.D., Diehl, M.: Time-optimal path tracking for robots: a convex optimization approach. IEEE Trans. Autom. Control 54(10), 2318 (2009)
Villagra, J., Milanés, V., Pérez, J., Godoy, J.: Smooth path and speed planning for an automated public transport vehicle. Robot. Auton. Syst. 60, 252–265 (2012)
Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. J. Glob. Optim. 14, 217–249 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cabassi, F., Consolini, L. & Locatelli, M. Time-optimal velocity planning by a bound-tightening technique. Comput Optim Appl 70, 61–90 (2018). https://doi.org/10.1007/s10589-017-9978-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-017-9978-6