Abstract
Influence diagrams are decision-theoretic extensions of Bayesian networks. In this paper we show how influence diagrams can be used to solve trajectory optimization problems. These problems are traditionally solved by methods of optimal control theory but influence diagrams offer an alternative that brings benefits over the traditional approaches. We describe how a trajectory optimization problem can be represented as an influence diagram. We illustrate our approach on two well-known trajectory optimization problems – the Brachistochrone Problem and the Goddard Problem. We present results of numerical experiments on these two problems, compare influence diagrams with optimal control methods, and discuss the benefits of influence diagrams.
This work was supported by the Czech Science Foundation through projects 16-12010S (V. Kratochvíl) and 17-08182S (J. Vomlel).
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Notes
- 1.
This is an approximation only, but the smaller distance discretization step the smaller the approximation error.
- 2.
Bocop package implements a local optimization method. The optimal control problem is approximated by a nonlinear programming (NLP) problem using a time discretization. The NLP problem is solved by Ipopt, using sparse exact derivatives computed by ADOL-C.
- 3.
The initial mass of the rocket is the same for all methods but in the third plot of Fig. 4 we can see that the terminal mass slightly differs. The lowest fuel consumption is observed in case of RK method but we should conclude from that the RK method is optimal but rather that it has the largest approximation error.
- 4.
A bang-bang strategy is a strategy that consists of extreme values only, e.g. it consists of the full thrust and the zero thrust phases only. Bang-bang strategies are optimal solutions of a wide class of optimal control problems.
References
Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)
Bertsekas, D.P.: Dynamic Programming and Optimal Control, 2nd edn. Athena Scientific, Belmont (2000)
Goddard, R.H.: A method for reaching extreme altitudes, volume 71(2). Smithsonian Miscellaneous Collections (1919)
Howard, R.A., Matheson, J.E.: Influence diagrams. In: Howard, R.A., Matheson, J.E. (eds.) Readings on the Principles and Applications of Decision Analysis, vol. II, pp. 721–762. Strategic Decisions Group (1981)
Jensen, F.: Bayesian Networks and Decision Graphs. Springer, New York (2001)
Kratochvíl, V., Vomlel, J.: Influence diagrams for speed profile optimization. Int. J. Approximate Reasoning. (2016, in press). http://dx.doi.org/10.1016/j.ijar.2016.11.018
Miele, A.: A survey of the problem of optimizing flight paths of aircraft and missiles. In: Bellman, R. (ed.) Mathematical Optimization Techniques, pp. 3–32. University of California Press (1963)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann series in representation and reasoning. Morgan Kaufmann, Burlington (1988)
Seywald, H., Cliff, E.M.: Goddard problem in presence of a dynamic pressure limit. J. Guidance Control Dyn. 16(4), 776–781 (1992)
Team Commands, Inria Saclay: BOCOP: an open source toolbox for optimal control (2016). http://bocop.org
Tsiotras, P., Kelley, H.J.: Drag-law effects in the goddard problem. Automatica 27(3), 481–490 (1991)
Vomlel, J.: Solving the Brachistochrone Problem by an influence diagram. Technical report 1702.02032 (2017). http://arxiv.org/abs/1702.02032
Vomlel, J., Kratochvíl, V.: Solving the Goddard Problem by an influence diagram. Technical report 1703.06321 (2017). http://arxiv.org/abs/1703.06321
Zlatskiy, V.T., Kiforenko, B.N.: Computation of optimal trajectories with singular-control sections. Vychislitel’naia i Prikladnaia Matematika 49, 101–108 (1983)
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Vomlel, J., Kratochvíl, V. (2017). Solving Trajectory Optimization Problems by Influence Diagrams. In: Antonucci, A., Cholvy, L., Papini, O. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2017. Lecture Notes in Computer Science(), vol 10369. Springer, Cham. https://doi.org/10.1007/978-3-319-61581-3_14
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