Abstract
We present a framework wherein the trajectory optimization problem (or a problem involving calculus of variations) is formulated as a search problem in a discrete space. A distinctive feature of our work is the treatment of discretization of the optimization problem wherein we discretize not only independent variables (such as time) but also dependent variables. Our discretization scheme enables a reduction in computational cost through selection of coarse-grained states. It further facilitates the solution of the trajectory optimization problem via classical discrete search algorithms including deterministic and stochastic methods for obtaining a global optimum. This framework also allows us to efficiently use quantum computational algorithms for global trajectory optimization. We demonstrate that the discrete search problem can be solved by a variety of techniques including a deterministic exhaustive search in the physical space or the coefficient space, a randomized search algorithm, a quantum search algorithm or by employing a combination of randomized and quantum search algorithms depending on the nature of the problem. We illustrate our methods by solving some canonical problems in trajectory optimization. We also present a comparative study of the performances of different methods in solving our example problems. Finally, we make a case for using quantum search algorithms as they offer a quadratic speed-up in comparison to the traditional non-quantum algorithms.
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References
Bernoulli, J.: Problema Novum ad Cujus Solutionem Mathematici Invitantur. Acta Erud. 15, 264–269 (1696)
Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–207 (1998)
Bliss, G.A.: Calculus of Variations. Mathematical Association of America Chicago, Chicago (1925)
Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46(4–5), 493–506 (1998)
Brooks, S.H.: A discussion of random methods for seeking maxima. Oper. Res. 6(2), 244–251 (1958)
Bulger, D., Baritompa, W.P., Wood, G.R.: Implementing pure adaptive search with Grover’s quantum algorithm. J. Optim. Theory Appl. 116(3), 517–529 (2003)
Bulger, D.W.: Combining a local search and Grover’s algorithm in black-box global optimization. J. Optim. Theory Appl. 133(3), 289–301 (2007)
Cowling, I.D., Whidborne, J.F., Cooke, A.K.: Optimal trajectory planning and LQR control for a quadrotor UAV. In: UKACC International Conference on Control (2006)
Dürr, C., Høyer, P.: A Quantum Algorithm for Finding the Minimum. arXiv preprint arXiv:quant-ph/9607014 (1996)
Elsgolc, L.D.: Calculus of Variations. Courier Corporation, North Chelmsford (2012)
Fahroo, F., Ross, I.M.: Direct trajectory optimization by a Chebyshev pseudospectral method. J. Guid. Control Dyn. 25(1), 160–166 (2002)
Feynman, R.P.: The principle of least action in quantum mechanics. In: Brown, L.M. (ed.) Feynman’s Thesis—A New Approach To Quantum Theory, pp. 1–69. World Scientific (2005). https://doi.org/10.1142/5852
Gelfand, I.M., Silverman, R.A., et al.: Calculus of Variations. Courier Corporation, North Chelmsford (2000)
Goldstein, H.: Classical Mechanics. Pearson Education India, New Delhi (2011)
Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219. ACM (1996)
Lara, P.C.S., Portugal, R., Lavor, C.: A new hybrid classical-quantum algorithm for continuous global optimization problems. J. Glob. Optim. 60(2), 317–331 (2014)
Mureşan, M.: Soft landing on the moon with mathematica. Math. J 14, 14–16 (2012)
Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Patel, N.R., Smith, R.L., Zabinsky, Z.B.: Pure adaptive search in Monte Carlo optimization. Math. Program. 43(1–3), 317–328 (1989)
Rahman, Q.I., Schmeisser, G., et al.: Analytic Theory of Polynomials. Number 26. Oxford University Press, Oxford (2002)
Rieffel, E.G., Polak, W.H.: Quantum Computing: A Gentle Introduction. MIT Press, Cambridge (2011)
Vanderbilt, D., Louie, S.G.: A Monte Carlo simulated annealing approach to optimization over continuous variables. J. Comput. Phys. 56(2), 259–271 (1984)
Von Stryk, O., Bulirsch, R.: Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37(1), 357–373 (1992)
Woodhouse, R.: A Treatise of Isoperimetrical Problems, and the Calculus of Variations. J. Smith, Cambridge (1810)
Yanofsky, N.S., Mannucci, M.A., Mannucci, M.A.: Quantum Computing for Computer Scientists, vol. 20. Cambridge University Press, Cambridge (2008)
Zabinsky, Z.B., Smith, R.L.: Pure adaptive search in global optimization. Math. Program. 53(1–3), 323–338 (1992)
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Shukla, A., Vedula, P. Trajectory optimization using quantum computing. J Glob Optim 75, 199–225 (2019). https://doi.org/10.1007/s10898-019-00754-5
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DOI: https://doi.org/10.1007/s10898-019-00754-5
Keywords
- Trajectory optimization
- Calculus of variations
- Global optimization
- Quantum computation
- Randomized search algorithm
- Brachistochrone problem