Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 3117–3144 | Cite as

Review of dynamic soaring: technical aspects, nonlinear modeling perspectives and future directions

  • Imran Mir
  • Sameh A. Eisa
  • Adnan Maqsood
Review
  • 113 Downloads

Abstract

In this paper, we present a comprehensive and detailed review of dynamic soaring process, and in particular, its application to unmanned aerial vehicles (UAVs). We start by explaining the biological inspiration that comes from soaring birds and how researchers have tried to utilize the dynamic soaring phenomenon/maneuver and apply it to UAVs. We present and discuss the fundamentals of wind shear models in both the linear and nonlinear cases. Moreover, a comprehensive parametric characterization of the key performance parameters for the dynamic soaring maneuver is given. Numerical methods for nonlinear trajectory optimization are summarized and methodologies capable of generating rapid solutions suitable for real-time implementation, are presented. Additionally, the paper introduces mathematical modeling and procedure to generate the optimized dynamic soaring trajectory. Through this paper, a consolidated platform is built, which not only covers technical aspects of advancements made over the passage of time, but also identifies and discusses the existing challenges. These challenges which are encountered by UAVs curtail the potential utility of dynamic soaring. Integrating dynamic soaring with morphology and inclusion of nonlinear control theory in the flight control system are introduced as a possible future research directions that may overcome the existing limitations.

Keywords

Dynamic soaring Nonlinear flight dynamics Nonlinear modeling Flight control system Morphing Optimal control Linear control Geometric nonlinear control 

Nomenclature

UAV

Unmanned aerial vehicle

sUAV

Small unmanned aerial vehicle

V

True air speed

\(\gamma \)

Flight path angle

\(\psi \)

Azimuth measured clockwise from the y-axis

x

Position vector along east direction

y

Position vector along north direction

h

Altitude

\(\alpha _{L\,=\,0}\)

Angle of attack at zero lift

b

Wing span

\(\Lambda \)

Sweep angle

\(C_\mathrm{L}\)

Lift coefficient

\(\phi \)

Bank angle

\(C_\mathrm{D}\)

Drag coefficient

E

Energy

M

Mass of the vehicle

g

Acceleration due to gravity

\(V_\mathrm{w}\)

Wind velocity

n

Load factor

\(n_{\max }\)

Maximum load factor

\((.)_{\max }\)

Maximum value of the variable

\((.)_{\min }\)

Minimum value of the variable

\(t_\mathrm{o}\)

Initial time

\(t_\mathrm{f}\)

Final time

\((.)_{\mathrm{t}_{o}}\)

Variable value at initial time

\((.)_{\mathrm{t}_{f}}\)

Variable value at final time

\(\dot{(\cdot )}\)

First-order time derivative

\(V_\mathrm{ref}\)

Reference wind speed

\(h_\mathrm{ref}\)

Reference altitude

\(h_{o}\)

Surface correctness factor

\(c_{\mathrm{l}_{\alpha }}\)

Lift curve slope of airfoil

\(C_{\mathrm{D}_{o}}\)

Zero lift drag coefficient

\(\rho \)

Density of the air

n

Load factor

S

Wing area

K

Induced drag factor

AR

Aspect ratio of the wing

L / D

Lift-to-drag ratio

m

meter

m/s

meter per sec

s

second

kg

kilogram

\({}^{\circ }\)

degree

NLP

Non linear programming

IPOPT

Interior point optimization

GPOPS

General purpose optimal control software

\(f(\varvec{x})\)

Drift vector

g

Control input field

V1,V2

Lie bracket between vector V1 and V2

LARC

Lie algebraic rank condition

Notes

Acknowledgements

Special thanks to Dr Haithem. E. Taha, Mechanical and Aerospace Engineering Department, UC Irvine for his continuous and sincere support throughout the phase of the research. The guidance and support extended by him contributed largely in making this research a success.

Compliance with ethical standards

Conflict of Interest

The authors of this paper have no conflict of interest to declare.

References

  1. 1.
    Wilson, J.: Sweeping flight and soaring by albatrosses. Nature 257(5524), 307–308 (1975)Google Scholar
  2. 2.
    Richardson, P.L.: How do albatrosses fly around the world without flapping their wings? Prog. Oceanogr. 88(1), 46–58 (2011)Google Scholar
  3. 3.
    Denny, M.: Dynamic soaring: aerodynamics for albatrosses. Eur. J. Phys. 30(1), 75 (2008)Google Scholar
  4. 4.
    Sachs, G., Traugott, J., Nesterova, A., Bonadonna, F.: Experimental verification of dynamic soaring in albatrosses. J. Exp. Biol. 216(22), 4222–4232 (2013)Google Scholar
  5. 5.
    Pennycuick, C.: The flight of petrels and albatrosses (Procellariiformes), observed in South Georgia and its vicinity. Philos. Trans. R. Soc. Lond. B Biol. Sci. 300(1098), 75–106 (1982)Google Scholar
  6. 6.
    Sachs, G., Traugott, J., Nesterova, A.P., Dell’Omo, G., Kümmeth, F., Heidrich, W., Vyssotski, A.L., Bonadonna, F.: Flying at no mechanical energy cost: disclosing the secret of wandering albatrosses. PLoS ONE 7(9), e41449 (2012)Google Scholar
  7. 7.
    Croxall, J.P., Silk, J.R., Phillips, R.A., Afanasyev, V., Briggs, D.R.: Global circumnavigations: tracking year-round ranges of nonbreeding albatrosses. Science 307(5707), 249–250 (2005)Google Scholar
  8. 8.
    Austin, R.: Unmanned Aircraft Systems: UAVS Design, Development and Deployment, vol. 54. Wiley, Hoboken (2011)Google Scholar
  9. 9.
    Langelaan, J.W., Roy, N.: Enabling new missions for robotic aircraft. Science 326(5960), 1642–1644 (2009)Google Scholar
  10. 10.
    Akhtar, N., Whidborne, J.F., Cooke, A.K.: Wind Shear Energy Extraction Using Dynamic Soaring Techniques. American Institute of Aeronautics and Astronautics AIAA, Reston (2009)Google Scholar
  11. 11.
    Grenestedt, J.L., Spletzer, J.R.: Optimal Energy Extraction During Dynamic Jet Stream Soaring. In: AIAA Guidance, Navigation, and Control Conference (2010)Google Scholar
  12. 12.
    Patel, C., Lee, H.-T., Kroo, I.: Extracting energy from atmospheric turbulence with flight tests. Tech. Soar. 33(4), 100–108 (2009)Google Scholar
  13. 13.
    Grenestedt, J.L., Spletzer, J.R.: Towards perpetual flight of a gliding unmanned aerial vehicle in the jet stream. 2010 49th IEEE Conference on Decision and Control (CDC), IEEE, pp. 6343–6349 (2010)Google Scholar
  14. 14.
    Cone, C.D.: Thermal soaring of birds. Am. Sci. 50(1), 180–209 (1962)Google Scholar
  15. 15.
    Raspet, A.: Biophysics of bird flight. Science 132(3421), 191–200 (1960)Google Scholar
  16. 16.
    Boslough, M.B.: Autonomous dynamic soaring platform for distributed mobile sensor arrays, Sandia National Laboratories, Sandia National Laboratories, Tech. Rep. SAND2002-1896 (2002)Google Scholar
  17. 17.
    Wood, C.: The flight of albatrosses (a computer simulation). Ibis 115(2), 244–256 (1973)Google Scholar
  18. 18.
    Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guidance Control Dyn. 21(2), 193–207 (1998)zbMATHGoogle Scholar
  19. 19.
    Gao, X.-Z., Hou, Z.-X., Guo, Z., Chen, X.-Q.: Energy extraction from wind shear: reviews of dynamic soaring. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 229(12), 2336–2348 (2015)Google Scholar
  20. 20.
    Idrac, M.: Contributions à l’étude duvol des albatros. CR Acad. Sci. Paris 179, 28–30 (1924)Google Scholar
  21. 21.
    Idrac, P.: Experimental study of the soaring of albatrosses. Nature 115(2893), 532–532 (1925)Google Scholar
  22. 22.
    Walkden, S.: Experimental study of the soaring of albatrosses. Nature 116(2908), 25 (1925)Google Scholar
  23. 23.
    Pennycuick, C.: Gliding flight of the fulmar petrel. J. Exp. Biol. 37(2), 330–338 (1960)Google Scholar
  24. 24.
    Pennycuick, C.: Mechanics of flight. Avian Biol. 5, 1–75 (1975)Google Scholar
  25. 25.
    Pennycuick, C.: Power requirements for horizontal flight in the pigeon Columba livia. J. Exp. Biol. 49(3), 527–555 (1968)Google Scholar
  26. 26.
    Tucker, V.A., Parrott, G.C.: Aerodynamics of gliding flight in a falcon and other birds. J. Exp. Biol. 52(2), 345–367 (1970)Google Scholar
  27. 27.
    Hargrave, L.: Sailing Birds are Dependent on Wave-power (1899)Google Scholar
  28. 28.
    Jouventin, P., Weimerskirch, H.: Satellite tracking of wandering albatrosses. Nature 343(6260), 746–748 (1990)Google Scholar
  29. 29.
    Prince, P., Wood, A., Barton, T., Croxall, J.: Satellite tracking of wandering albatrosses (Diomedea exulans) in the South Atlantic. Antarct. Sci. 4(1), 31–36 (1992)Google Scholar
  30. 30.
    Alerstam, T., Gudmundsson, G.A., Larsson, B.: Flight tracks and speeds of Antarctic and Atlantic seabirds: radar and optical measurements. Philos. Trans. R. Soc. Lond. B Biol. Sci. 340(1291), 55–67 (1993)Google Scholar
  31. 31.
    Tuck, G.N., Polacheck, T., Croxall, J., Weimerskirch, H., Prince, P., Wotherspoon, S.: The potential of archival tags to provide long-term movement and behaviour data for seabirds: first results from Wandering Albatross Diomedea exulans of South Georgia and the Crozet Islands. Emu 99(1), 60–68 (1999)Google Scholar
  32. 32.
    Weimerskirch, H., Wilson, R.P.: Oceanic respite for wandering albatrosses. Nature 406(6799), 955–956 (2000)Google Scholar
  33. 33.
    Nel, D., Ryan, P.G., Nel, J.L., Klages, N.T., Wilson, R.P., Robertson, G., Tuck, G.N., et al.: Foraging interactions between Wandering Albatrosses Diomedea exulans breeding on Marion Island and long-line fisheries in the southern Indian Ocean. Ibis 144(3), 141–154 (2002)Google Scholar
  34. 34.
    Weimerskirch, H., Bonadonna, F., Bailleul, F., Mabille, G., Dell’Omo, G., Lipp, H.-P.: GPS tracking of foraging albatrosses. Science 295(5558), 1259–1259 (2002)Google Scholar
  35. 35.
    Wakefield, E.D., Phillips, R.A., Matthiopoulos, J., Fukuda, A., Higuchi, H., Marshall, G.J., Trathan, P.N.: Wind field and sex constrain the flight speeds of central-place foraging albatrosses. Ecol. Monogr. 79(4), 663–679 (2009)Google Scholar
  36. 36.
    Weimerskirch, H., Guionnet, T., Martin, J., Shaffer, S.A., Costa, D.: Fast and fuel efficient? Optimal use of wind by flying albatrosses. Proc. R. Soc. Lond. B Biol. Sci. 267(1455), 1869–1874 (2000)Google Scholar
  37. 37.
    Bevan, R., Woakes, A., Butler, P., Boyd, I.: The use of heart rate to estimate oxygen consumption of free-ranging black-browed albatrosses Diomedea melanophrys. J. Exp. Biol. 193(1), 119–137 (1994)Google Scholar
  38. 38.
    Rosén, M., Hedenstrom, A.: Gliding flight in a jackdaw: a wind tunnel study. J. Exp. Biol. 204(6), 1153–1166 (2001)Google Scholar
  39. 39.
    MacCready, P.B.: Optimum airspeed selector. Soaring (January–February), vol. 10(11) (1958)Google Scholar
  40. 40.
    Gordon, R.J.: Optimal dynamic soaring for full size sailplanes, Tech. rep., Air Force Inst of Tech Wright-Patterson AFB oh Dept of Aeronautics and Astronautics (2006)Google Scholar
  41. 41.
    Ariff, O., Go, T.: Waypoint navigation of small-scale UAV incorporating dynamic soaring. J. Navig. 64(1), 29–44 (2011)Google Scholar
  42. 42.
    Rao, A.V., Benson, D.A., Darby, C., Patterson, M.A., Francolin, C., Sanders, I., Huntington, G.T.: Algorithm 902: Gpops, a matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Trans. Math. Softw. (TOMS) 37(2), 22 (2010)zbMATHGoogle Scholar
  43. 43.
    Stull, R.B.: An Introduction to Boundary Layer Meteorology, vol. 13. Springer, Berlin (2012)zbMATHGoogle Scholar
  44. 44.
    Kaimal, J.C., Finnigan, J.J.: Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press, Oxford (1994)Google Scholar
  45. 45.
    Wharington, J.M.: Heuristic control of dynamic soaring. In: Control Conference, 2004. 5th Asian, Vol. 2, IEEE, pp. 714–722 (2004)Google Scholar
  46. 46.
    Lawrance, N.R., Sukkarieh, S.: A guidance and control strategy for dynamic soaring with a gliding UAV. In: IEEE International Conference on Robotics and Automation, 2009. ICRA’09. IEEE, pp. 3632–3637 (2009)Google Scholar
  47. 47.
    Idrac, P., Georgii, W.: Experimentelle Untersuchungen über den Segelflug, mitten im Fluggebiet grosser Segelnder Vögel (Geier, Albatros usw). Ihre Anwendung auf den Segelflug des Menschen...[Einleitung von Walter Georgii.]. R. Oldenbourg (1932)Google Scholar
  48. 48.
    Pennycuick, C.J.: Gust soaring as a basis for the flight of petrels and albatrosses (Procellariiformes). Avian Sci. 2(1), 1–12 (2002)Google Scholar
  49. 49.
    Sachs, G.: Minimum shear wind strength required for dynamic soaring of albatrosses. Ibis 147(1), 1–10 (2005)Google Scholar
  50. 50.
    Sachs, G., da Costa, O.: Optimization of dynamic soaring at ridges. In: AIAA Atmospheric Flight Mechanics Conference and Exhibit pp. 11–14 (2003)Google Scholar
  51. 51.
    Ariff, O., Go, T.: Dynamic soaring of small-scale UAVs using differential geometry. In: Proceedings of International Bhurban Conference on Applied Sciences and Technology (2010)Google Scholar
  52. 52.
    Zhao, Y.J.: Optimal patterns of glider dynamic soaring. Optim. Control Appl. Methods 25(2), 67–89 (2004)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Barnes, J.P.: How flies the albatross–the flight mechanics of dynamic soaring. Tech. rep., SAE Technical Paper (2004)Google Scholar
  54. 54.
    Richardson, P.L.: Upwind dynamic soaring of albatrosses and UAVs. Prog. Oceanogr. 130, 146–156 (2015)Google Scholar
  55. 55.
    Abdulrahim, M.: Flight dynamics and control of an aircraft with segmented control surfaces. In: 42nd AIAA Aerospace Sciences Meeting and Exhibit, 2004, pp. 2004–0128Google Scholar
  56. 56.
    Wickenheiser, A.M., Garcia, E.: Optimization of perching maneuvers through vehicle morphing. J. Guidance Control Dyn. 31(4), 815–823 (2008)Google Scholar
  57. 57.
    Akhtar, N.: Control system development for autonomous soaring (2010)Google Scholar
  58. 58.
    Sukumar, P.P., Selig, M.S.: Dynamic soaring of sailplanes over open fields. J. Aircr. 50(5), 1420–1430 (2013)Google Scholar
  59. 59.
    Lawrance, N., Acevedo, J., Chung, J., Nguyen, J., Wilson, D., Sukkarieh, S.: Long endurance autonomous flight for unmanned aerial vehicles. AerospaceLab 8, 1 (2014)Google Scholar
  60. 60.
    Mir, I., Maqsood, A., Eisa, S.A., Taha, H., Akhtar, S.: Optimal morphing-augmented dynamic soaring maneuvers for unmanned air vehicle capable of span and sweep morphologies. Aerosp. Sci. Technol. 79(1), 17–36 (2018)Google Scholar
  61. 61.
    Berger, M., Göhde, W.: Zur Theorie des Segelfluges von Vögeln über dem Meere. Zool. Jb. Physiol 71, 217–224 (1965)Google Scholar
  62. 62.
    Bonnin, V., Bénard, E., Moschetta, J.-M., Toomer, C.: Energy-harvesting mechanisms for UAV flight by dynamic soaring. Int. J. Micro Air Veh. 7(3), 213–229 (2015)Google Scholar
  63. 63.
    DSKinetic Kernel Description. http://www.dskinetic.com/, Accessed (2019)
  64. 64.
    Gao, X.-Z., Hou, Z.-X., Guo, Z., Fan, R.-F., Chen, X.-Q.: Analysis and design of guidance-strategy for dynamic soaring with UAVs. Control Eng. Pract. 32, 218–226 (2014)Google Scholar
  65. 65.
    Deittert, M., Richards, A., Toomer, C., Pipe, A.: Dynamic soaring flight in turbulence. In: AIAA Guidance, Navigation and Control Conference, Chicago, Illinois, pp. 2–5 (2009)Google Scholar
  66. 66.
    Mir, I., Maqsood, A., Akhtar, S.: Optimization of dynamic soaring maneuvers for a morphing capable UAV. In: AIAA Information Systems-AIAA Infotech@ Aerospace, p. 0678 (2017)Google Scholar
  67. 67.
    Sachs, G., Mayrhofer, M.: Shear wind strength required for dynamic soaring at ridges. Tech. Soar. 25(4), 209–215 (2001)Google Scholar
  68. 68.
    Zhao, Y.J., Qi, Y.C.: Minimum fuel powered dynamic soaring of unmanned aerial vehicles utilizing wind gradients. Optim. Control Appl. Methods 25(5), 211–233 (2004)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Fırtın, E., Güler, Ö., Akdağ, S.A.: Investigation of wind shear coefficients and their effect on electrical energy generation. Appl. Energy 88(11), 4097–4105 (2011)Google Scholar
  70. 70.
    Shen, X., Zhu, X., Du, Z.: Wind turbine aerodynamics and loads control in wind shear flow. Energy 36(3), 1424–1434 (2011)Google Scholar
  71. 71.
    Liu, D.-N., Hou, Z.-X., Guo, Z., Yang, X.-X., Gao, X.-Z.: Bio-inspired energy-harvesting mechanisms and patterns of dynamic soaring. Bioinspir. Biomim. 12(1), 016014 (2017)Google Scholar
  72. 72.
    Langelaan, J.W., Spletzer, J., Montella, C., Grenestedt, J.: Wind field estimation for autonomous dynamic soaring. In: 2012 IEEE International Conference on Robotics and Automation (ICRA), IEEE, pp. 16–22 (2012)Google Scholar
  73. 73.
    Bencatel, R., Girard, A., Abdelhafiz, M., Sousa, J.: Shear wind estimation (2011)Google Scholar
  74. 74.
    Lawrance, N.R.: Autonomous soaring flight for unmanned aerial vehicles. Ph.D. Thesis, University of Sydney (2011)Google Scholar
  75. 75.
    Akhtar, N., Whidborne, J., Cooke, A.: Real-time optimal techniques for unmanned air vehicles fuel saving. Proc. Instit. Mech. Eng. Part G J. Aerosp. Eng. 226(10), 1315–1328 (2012)Google Scholar
  76. 76.
    Akhtar, N., Cooke, A.K., Whidborne, J.F.: Positioning algorithm for autonomous thermal soaring. J. Aircr. 49(2), 472–482 (2012)Google Scholar
  77. 77.
    Sachs, G., Grüter, B.: Dynamic soaring- kinetic energy and inertial speed. In: AIAA Atmospheric Flight Mechanics Conference, p. 1862 (2017)Google Scholar
  78. 78.
    Sachs, G., da Costa, O.: Dynamic soaring in altitude region below jet streams. In: AIAA Guidance, Navigation and Control Conference, no. AIAA Paper vol. 6602, pp. 21–24 (2006)Google Scholar
  79. 79.
    Bousquet, G.D., Triantafyllou, M.S., Slotine, J.-J.E.: Optimal dynamic soaring consists of successive shallow arcs. J. R. Soc. Interface 14(135), 20170496 (2017)Google Scholar
  80. 80.
    Silva, W., Frew, E.W.: Experimental assessment of online dynamic soaring optimization for small unmanned aircraft. AIAA SciTech Forum, 2016, pp. 2016–0252Google Scholar
  81. 81.
    Bower, G.C.: Boundary layer dynamic soaring for autonomous aircraft: design and validation. Ph.D. thesis, Stanford University Stanford (2011)Google Scholar
  82. 82.
    Sachs, G.: Optimal wind energy extraction for dynamic soaring. In: Miele, A., Salvetti, A. (eds.) Applied Mathematics in Aerospace Science and Engineering. Mathematical Concepts and Methods in Science and Engineering, vol. 44, pp. 221–237. Springer, Boston, MA (1994)Google Scholar
  83. 83.
    Shaw-Cortez, W.E., Frew, E.: Efficient trajectory development for small unmanned aircraft dynamic soaring applications. J. Guidance Control Dyn 38, 519–523 (2015)Google Scholar
  84. 84.
    Kahveci, N.E., Ioannou, P.A.: Adaptive steering control for uncertain ship dynamics and stability analysis. Automatica 49(3), 685–697 (2013)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Zhang, L., Gao, H., Chen, Z., Sun, Q., Zhang, X.: Multi-objective global optimal parafoil homing trajectory optimization via Gauss pseudospectral method. Nonlinear Dyn. 72(1–2), 1–8 (2013)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Cheng, X., Li, H., Zhang, R.: Autonomous trajectory planning for space vehicles with a Newton-Kantorovich/convex programming approach. Nonlinear Dyn. 89(4), 2795–2814 (2017)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Ghasemi, S., Nazemi, A., Hosseinpour, S.: Nonlinear fractional optimal control problems with neural network and dynamic optimization schemes. Nonlinear Dyn. 89(4), 2669–2682 (2017)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Qiu, H., Duan, H.: Receding horizon control for multiple UAV formation flight based on modified brain storm optimization. Nonlinear Dyn. 78(3), 1973–1988 (2014)Google Scholar
  89. 89.
    Rao, A.V.: A survey of numerical methods for optimal control. Adv. Astronaut. Sci. 135(1), 497–528 (2009)Google Scholar
  90. 90.
    Bellman, R.: Dynamic Programming. Princeton Univ, Princeton (1957)zbMATHGoogle Scholar
  91. 91.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)MathSciNetzbMATHGoogle Scholar
  92. 92.
    Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The mathematical theory of optimal processes (Russian), English translation by KN Trirogoff, ed. by LW Neustadt (1962)Google Scholar
  93. 93.
    Gerdts, M.: Direct shooting method for the numerical solution of higher-index DAE optimal control problems. J. Optim. Theory Appl. 117(2), 267 (2003)MathSciNetzbMATHGoogle Scholar
  94. 94.
    Diedam, H., Sager, S.: Global optimal control with the direct multiple shooting method. Optim. Control Appl. Methods 39, 449–470 (2016)MathSciNetzbMATHGoogle Scholar
  95. 95.
    Cannataro, B.Ş., Rao, A.V., Davis, T.A.: State-defect constraint pairing graph coarsening method for Karush-Kuhn-Tucker matrices arising in orthogonal collocation methods for optimal control. Comput. Optim. Appl. 64(3), 793–819 (2016)MathSciNetzbMATHGoogle Scholar
  96. 96.
    Huntington, G.T., Rao, A.V.: Comparison of global and local collocation methods for optimal control. J. Guidance Control Dyn. 31(2), 432 (2008)Google Scholar
  97. 97.
    Schwartz, A.L.: Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems. Ph.D. thesis, University of California, Berkeley (1996)Google Scholar
  98. 98.
    Reddien, G.: Collocation at Gauss points as a discretization in optimal control. SIAM J. Control Optim. 17(2), 298–306 (1979)MathSciNetzbMATHGoogle Scholar
  99. 99.
    Herman, A.L., Conway, B.A.: Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules. J. Guidance Control Dyn. 19(3), 592–599 (1996)zbMATHGoogle Scholar
  100. 100.
    Darby, C.L., Hager, W.W., Rao, A.V.: An hp-adaptive pseudospectral method for solving optimal control problems. Optim Control Appl. Methods 32(4), 476–502 (2011)MathSciNetzbMATHGoogle Scholar
  101. 101.
    Wiegand, A., et al.: ASTOS User Manual, vol. 17. Astos Solutions GmbH, Unterkirnach (2010)Google Scholar
  102. 102.
    Härer, A., Matha, D., Kucher, D., Sandner, F.: Optimization of offshore wind turbine components in multi-body simulations for cost and load reduction. In: Proceedings of the EWEA Offshore, pp. 1–7 (2013)Google Scholar
  103. 103.
    Sachs, G., Knoll, A., Lesch, K.: Optimal utilization of wind energy for dynamic soaring. Tech. Soar. 15(2), 48–55 (1991)Google Scholar
  104. 104.
    Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H.: User’s guide for NPSOL (version 4.0): a Fortran package for nonlinear programming. Tech. rep., Stanford Univ CA Systems Optimization Lab (1986)Google Scholar
  105. 105.
    Liu, Y., Longo, S., Kerrigan, E.C.: Nonlinear predictive control of autonomous soaring UAVs using 3DOF models. Control Conference (ECC), 2013 European, IEEE, pp. 1365–1370 (2013)Google Scholar
  106. 106.
    Patterson, M.A., Rao, A.V.: GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming. ACM Trans. Math. Softw 41(1), 1 (2014)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Becerra, V.M.: PSOPT Optimal Control Solver User Manual. University of Reading, Reading (2010)Google Scholar
  108. 108.
    Rutquist, P., Edvall, M.: PROPT-MATLAB Optimal Control Software. Tomlab Optimization, Inc., Pullman, WA (2010)Google Scholar
  109. 109.
    Betts, J.T.: Practical methods for optimal control and estimation using nonlinear programming. SIAM, Philadelphia (2010)zbMATHGoogle Scholar
  110. 110.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Mathematical Programming Language, Citeseer (1987)Google Scholar
  111. 111.
    Cizniar, M., Fikar, M., Latifi, M.A.: MATLAB Dynamic Optimisation Code DYNOPT. User’s Guide, Technical report. KIRP FCHPT STU, Bratislava (2006)Google Scholar
  112. 112.
    Lawrance, N.R., Sukkarieh, S.: Wind energy based path planning for a small gliding unmanned aerial vehicle. In: AIAA Guidance, Navigation, and Control Conference, pp. 10–13 (2009)Google Scholar
  113. 113.
    Bird, J.J., Langelaan, J.W., Montella, C., Spletzer, J., Grenestedt, J.: Closing the loop in dynamic soaring. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, National Harbor, MD, USA, pp. 13–17 (2014)Google Scholar
  114. 114.
    Hassan, A.M., Taha, H.E.: Geometric control formulation and nonlinear controllability of airplane flight dynamics. Nonlinear Dyn. 88, 1–19 (2017)MathSciNetGoogle Scholar
  115. 115.
    Mir, I., Taha, H., Eisa, S.A., Maqsood, A.: A controllability perspective of dynamic soaring. Nonlinear Dyn. (2018).  https://doi.org/10.1007/s11071-018-4493-6 CrossRefGoogle Scholar
  116. 116.
    Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, vol. 49. Springer, Berlin (2004)zbMATHGoogle Scholar
  117. 117.
    Kalman, R.E., Ho, Y.C., Narendra, K.S.: Controllability of linear Dynamical systems. Contrib. Diff. Equ. 1, 189–213 (1963)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Cariñena, J.F., Núñez, J.F.: Geometric approach to dynamics obtained by deformation of time-dependent Lagrangians. Nonlinear Dyn. 86(2), 1285–1291 (2016)MathSciNetzbMATHGoogle Scholar
  119. 119.
    Bianchini, R.M., Stefani, G.: Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28(4), 903–924 (1990)MathSciNetzbMATHGoogle Scholar
  120. 120.
    Brunovsky, P.: Local controllability of odd systems. Math. Control Theory 1, 39–45 (1974)Google Scholar
  121. 121.
    Crouch, P.E., Byrnes, C.I.: Local accessibility, local reachability, and representations of compact groups. Theory Comput. Syst. 19(1), 43–65 (1986)MathSciNetzbMATHGoogle Scholar
  122. 122.
    Hermes, H.: On local controllability. SIAM J. Control Optim. 20(2), 211–220 (1982)MathSciNetzbMATHGoogle Scholar
  123. 123.
    Jurdjevic, V., Kupka, I.: Polynomial control systems. Math. Ann. 272(3), 361–368 (1985)MathSciNetzbMATHGoogle Scholar
  124. 124.
    Sussmann, H.J.: A general theorem on local controllability. SIAM J. Control Optim. 25(1), 158–194 (1987)MathSciNetzbMATHGoogle Scholar
  125. 125.
    Aguilar, C.O., Lewis, A.D.: Small-time local controllability for a class of homogeneous systems. SIAM J. Control Optim. 50(3), 1502–1517 (2012)MathSciNetzbMATHGoogle Scholar
  126. 126.
    Birdsall, D.: Flight stability and automatic control—second edition, Nelson RC, The McGraw-Hill Companies, 1221 Avenue of the Americas, New York, NY 10020-1095, USA1998. 441pp. Illustrated. Aeronaut. J. 102(1015), 299–299 (1998)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Research Center for Modeling and SimulationNational University of Sciences and Technology (NUST)IslamabadPakistan
  2. 2.Mechanical and Aerospace Engineering DepartmentUniversity of CaliforniaIrvineUSA

Personalised recommendations