A Review of Gradient Algorithms for Numerical Computation of Optimal Trajectories

Golfetto, Wander Almodovar; Fernandes, Sandro da Silva

doi:10.5028/jatm.2012.04020512

Abstract:

In this paper, two classic direct methods for numerical computation of optimal trajectories were revisited: the steepest descent method and the direct one based upon the second variation theory. The steepest descent method was developed for a Mayer problem of optimal control, with free final state and fixed terminal times. Terminal constraints on the state variables were considered through the penalty function method. The second method was based upon the theory of second variation and it involves the closed-loop solutions of a linear quadratic optimal control problem. The algorithm was developed for a Bolza problem of optimal control, with fixed terminal times and constrained initial and final states. Problems with free final time are also considered by using a transformation approach. An algorithm that combines the main characteristics of these methods was also presented. The methods were applied for solving two classic optimization problems - Brachistochrone and Zermelo - and their main advantages and disadvantages were discussed. Finally, the optimal space trajectories transference between coplanar circular orbits for different times of flight was calculated, using the proposed algorithm.

Keywords:
Optimization of trajectories; Numerical methods; Steepest descent method; Second-order gradient method

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REFERENCES

Betts, J.T., 1994, "Optimal Interplanetary Orbit Transfers by Direct Transcription", The Journal of the Astronautical Sciences, Vol. 42, No. 3, pp. 247-268.
Betts, J.T., 1993, "Using Sparse Nonlinear Programming to Compute Low Thrust Orbit Transfers", The Journal of the Astronautical Sciences, Vol. 41, No. 3, pp. 349-371.
Bliss, G.A., 1946, "Lectures on the Calculus of Variations". University of Chicago Press, Chicago, 296 p.
Bryson, A.E., Ho, Y.C., 1975, "Applied Optimal Control", John Wiley & Sons, New York, 481 p.
Bryson, A.E., Jr. , Denham, W.F., 1962, "A Steepest Ascent Method for Solving Optimum Programming Problems", Journal of Applied Mechanics, Vol. 84, No. 3, pp. 247-257.
Bullock, T.E., Franklin, G.F., 1967, "A Second Order Feedback Method for Optimal Control Computations", IEEE Transactions on Automatic Control, Vol. AC-12, No. 6, pp. 666-673.
Coverstone-Carroll, V., Williams, S.N., 1994, "Optimal Low Thrust Trajectories Using Differential Inclusion Concepts", The Journal of the Astronautical Sciences, Vol. 42, No. 4, pp. 379-393.
Elsgolts, L., 1977, "Differential Equations and the Calculus of Variations", Mir Publishers, Moscow, 440 p.
Gelfand, I.M., Fomin, S.V., 1963, "Calculus of Variations", Prentice-Hall, New Jersey, 232 p.
Hestenes, M. R., 1969, "Multiplier and gradient methods", Journal of Optimization Theory and Applications, Vol. 4, No. 5, pp. 303-320.
Hestenes, M. R., 1966, "Calculus of Variations and Optimal Control Theory", John Wiley & Sons, New York, 405 p.
Hussu, A., 1972, "The Conjugate-gradient Method for Optimal Control Problems with Undetermined Final Time", International Journal of Control, pp. 79-82.
Imae, J., 1998, "A Riccati-equation-based Algorithm for Continuous Time Optimal Control Problems", Optimal Control Applications & Methods, Vol. 19, No. 5, pp. 299-310.
Kelley, H.J., 1960, "Gradient Theory of Optimal Flight Paths", ARS Journal, Vol. 30, pp. 947-953.
Lasdon, L.S. et al, 1967, "The Conjugate Gradient Method for Optimal Control Problems", IEEE Transactions on Automatic Control, Vol. AC-12, No. 2, pp. 132-138.
Leitmann, G., 1981, "The Calculus of Variations and Optimal Control - An Introduction", Plenum Press, New York, 311 p.
Longmuir, A.G. and Bohn, E.V., 1969, "Second-Variation Methods in Dynamic Optimization", Journal of Optimization Theory and Applications, Vol. 3, No. 3, pp. 164-173.
Marec, J. P., 1976, Optimal space trajectories. New York, Elsevier, 329 p.
McDermott, Jr., M., Fowler, W.T., 1977, "Steepest-Descent Failure Analysis", Journal of Optimization Theory and Applications, Vol. 23, No. 2, pp. 229-243.
McGill, R., Kenneth, P., 1964, "Solution of Variation Problems by Means of a Generalized Newton-Raphson Operator", AIAA Journal, Vol. 2, No. 3, pp. 1761-1766.
McIntyre, J.E., 1968, "Guidance, Flight Mechanics and Trajectory Optimization. Vol. XIII - Numerical Optimization Methods", NASA CR - 1012, 105 p.
O'Doherty, R.J., Pierson, B.L., 1974, "A Numerical Study of Augmented Penalty Function Algorithms for Terminally Constrained Optimal Control Problems", Journal of Optimization Theory and Applications, Vol. 14, No. 4, pp. 393-403.
Pontryagin, L.S. et al, 1962, "The Mathematical Theory of Optimal Control Processes", John Wiley & Sons, New York, 357 p.
Quintana V. H., Davison, E. J., 1973, "A numerical method for solving control problems with unspecified terminal time". International Journal of Control, Vol. 17, No. 1, pp. 97-115.
Sage, A.P., White, C.C., 1977, "Optimum Systems Control", 2nd edition, Prentice - Hall Inc., Englewood Cliffs, New Jersey, 413 p.
Seywald, H., 1994, "Trajectory Optimization Based on Differential Inclusion", Journal of Guidance, Control and Dynamics. Vol. 17, No. 3, pp. 480-487
Silva Fernandes, S., Golfetto, W. A., 2007, "Numerical computation of optimal low-thrust limited-power trajectories-transfer between coplanar circular orbits", Mathematical Problems in Engineering.
Williamson, W.E., Tapley, B.D., 1972, "Riccati Transformations for Control Optimization Using the Second Variation", IEEE Transactions on Automatic Control, Vol. AC-17, No. 3, pp. 319-327.

Publication Dates

Publication in this collection
Apr-Jun 2012

History

Received
26 Jan 2012
Accepted
18 Mar 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] Betts, J.T., 1994, "Optimal Interplanetary Orbit Transfers by Direct Transcription", The Journal of the Astronautical Sciences, Vol. 42, No. 3, pp. 247-268.

[2] Betts, J.T., 1993, "Using Sparse Nonlinear Programming to Compute Low Thrust Orbit Transfers", The Journal of the Astronautical Sciences, Vol. 41, No. 3, pp. 349-371.

[3] Bliss, G.A., 1946, "Lectures on the Calculus of Variations". University of Chicago Press, Chicago, 296 p.

[4] Bryson, A.E., Ho, Y.C., 1975, "Applied Optimal Control", John Wiley & Sons, New York, 481 p.

[5] Bryson, A.E., Jr. , Denham, W.F., 1962, "A Steepest Ascent Method for Solving Optimum Programming Problems", Journal of Applied Mechanics, Vol. 84, No. 3, pp. 247-257.

[6] Bullock, T.E., Franklin, G.F., 1967, "A Second Order Feedback Method for Optimal Control Computations", IEEE Transactions on Automatic Control, Vol. AC-12, No. 6, pp. 666-673.

[7] Coverstone-Carroll, V., Williams, S.N., 1994, "Optimal Low Thrust Trajectories Using Differential Inclusion Concepts", The Journal of the Astronautical Sciences, Vol. 42, No. 4, pp. 379-393.

[8] Elsgolts, L., 1977, "Differential Equations and the Calculus of Variations", Mir Publishers, Moscow, 440 p.

[9] Gelfand, I.M., Fomin, S.V., 1963, "Calculus of Variations", Prentice-Hall, New Jersey, 232 p.

[10] Hestenes, M. R., 1969, "Multiplier and gradient methods", Journal of Optimization Theory and Applications, Vol. 4, No. 5, pp. 303-320.

[11] Hestenes, M. R., 1966, "Calculus of Variations and Optimal Control Theory", John Wiley & Sons, New York, 405 p.

[12] Hussu, A., 1972, "The Conjugate-gradient Method for Optimal Control Problems with Undetermined Final Time", International Journal of Control, pp. 79-82.

[13] Imae, J., 1998, "A Riccati-equation-based Algorithm for Continuous Time Optimal Control Problems", Optimal Control Applications & Methods, Vol. 19, No. 5, pp. 299-310.

[14] Kelley, H.J., 1960, "Gradient Theory of Optimal Flight Paths", ARS Journal, Vol. 30, pp. 947-953.

[15] Lasdon, L.S. et al, 1967, "The Conjugate Gradient Method for Optimal Control Problems", IEEE Transactions on Automatic Control, Vol. AC-12, No. 2, pp. 132-138.

[16] Leitmann, G., 1981, "The Calculus of Variations and Optimal Control - An Introduction", Plenum Press, New York, 311 p.

[17] Longmuir, A.G. and Bohn, E.V., 1969, "Second-Variation Methods in Dynamic Optimization", Journal of Optimization Theory and Applications, Vol. 3, No. 3, pp. 164-173.

[18] Marec, J. P., 1976, Optimal space trajectories. New York, Elsevier, 329 p.

[19] McDermott, Jr., M., Fowler, W.T., 1977, "Steepest-Descent Failure Analysis", Journal of Optimization Theory and Applications, Vol. 23, No. 2, pp. 229-243.

[20] McGill, R., Kenneth, P., 1964, "Solution of Variation Problems by Means of a Generalized Newton-Raphson Operator", AIAA Journal, Vol. 2, No. 3, pp. 1761-1766.

[21] McIntyre, J.E., 1968, "Guidance, Flight Mechanics and Trajectory Optimization. Vol. XIII - Numerical Optimization Methods", NASA CR - 1012, 105 p.

[22] O'Doherty, R.J., Pierson, B.L., 1974, "A Numerical Study of Augmented Penalty Function Algorithms for Terminally Constrained Optimal Control Problems", Journal of Optimization Theory and Applications, Vol. 14, No. 4, pp. 393-403.

[23] Pontryagin, L.S. et al, 1962, "The Mathematical Theory of Optimal Control Processes", John Wiley & Sons, New York, 357 p.

[24] Quintana V. H., Davison, E. J., 1973, "A numerical method for solving control problems with unspecified terminal time". International Journal of Control, Vol. 17, No. 1, pp. 97-115.

[25] Sage, A.P., White, C.C., 1977, "Optimum Systems Control", 2nd edition, Prentice - Hall Inc., Englewood Cliffs, New Jersey, 413 p.

[26] Seywald, H., 1994, "Trajectory Optimization Based on Differential Inclusion", Journal of Guidance, Control and Dynamics. Vol. 17, No. 3, pp. 480-487

[27] Silva Fernandes, S., Golfetto, W. A., 2007, "Numerical computation of optimal low-thrust limited-power trajectories-transfer between coplanar circular orbits", Mathematical Problems in Engineering.

[28] Williamson, W.E., Tapley, B.D., 1972, "Riccati Transformations for Control Optimization Using the Second Variation", IEEE Transactions on Automatic Control, Vol. AC-17, No. 3, pp. 319-327.

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