Time-optimal velocity planning by a bound-tightening technique

  • Federico Cabassi
  • Luca Consolini
  • Marco Locatelli


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.


Range reduction Velocity planning Minimum-time problems Local search 


  1. 1.
    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)Google Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Basel (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benson, H.P.: Simplicial branch-and-reduce algorithm for convex programs with a multiplicative constraint. J. Optim. Theory Appl. 145, 213–233 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bobrow, J., Dubowsky, S., Gibson, J.: Time-optimal control of robotic manipulators along specified paths. Int. J. Robot. Res. 4(3), 3–17 (1985)CrossRefGoogle Scholar
  6. 6.
    Cabassi, F., Consolini, L., Laurini, M., Locatelli, M.: Convergence analysis of spatial-sampling based algorithms for time-optimal smooth velocity planning (2017) (submitted)Google Scholar
  7. 7.
    Caprara, A., Monaci, M.: Bidimensional packing by bilinear programming. Math. Program. 118, 75–108 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Caprara, A., Locatelli, M.: Global optimization problems and domain reduction strategies. Math. Program. 125, 123–137 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caprara, A., Locatelli, M., Monaci, M.: Theoretical and computational results about optimality-based domain reductions. Comput. Optim. Appl. 64(2), 513–533 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Consolini, L., Locatelli, M., Minari, A., Piazzi, A.: An optimal complexity algorithm for minimum-time velocity planning. Syst. Control Lett. 103, 50–57 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    Hamed, A.S.E., McCormick, G.P.: Calculation of bounds on variables satisfying nonlinear inequality constraints. J. Glob. Optim. 3, 25–47 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hansen, P., Jaumard, B., Lu, S.H.: An analytical approach to global optimization. Math. Program. 52, 227–254 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kant, K., Zucker., S.W.: Toward efficient trajectory planning: the path-velocity decomposition. Int. J. Robot. Res. 5(3), 72–89 (1986)CrossRefGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    Locatelli, M., Raber, U.: Packing equal circles in a square: a deterministic global optimization approach. Discrete Appl. Math. 122, 139–166 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maranas, C.D., Floudas, C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–370 (1997)CrossRefGoogle Scholar
  23. 23.
    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)Google Scholar
  24. 24.
    Nagy, A., Vajk, I.: LP-based velocity profile generation for robotic manipulators. Int. J. Control. (2017).
  25. 25.
    Ryoo, H., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–139 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Shen, P., Ma, Y., Chen, Y.: Global optimization for the generalized polynomial sum of ratios problem. J. Glob. Optim. 50, 439–455 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    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)Google Scholar
  30. 30.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tóth, B., Casado, L.: Multi-dimensional pruning from the Baumann point in an interval global optimization algorithm. J. Glob. Optim. 38, 215–236 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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)CrossRefGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Federico Cabassi
    • 1
  • Luca Consolini
    • 1
  • Marco Locatelli
    • 1
  1. 1.Dipartimento di Ingegneria e ArchitetturaUniversità di ParmaParmaItaly

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