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Journal of Intelligent & Robotic Systems

, Volume 88, Issue 1, pp 147–162 | Cite as

Gaussian Process Model Predictive Control of an Unmanned Quadrotor

  • Gang Cao
  • Edmund M.-K. Lai
  • Fakhrul Alam
Article

Abstract

The Model Predictive Control (MPC) trajectory tracking problem of an unmanned quadrotor with input and output constraints is addressed. In this article, the dynamic models of the quadrotor are obtained purely from operational data in the form of probabilistic Gaussian Process (GP) models. This is different from conventional models obtained through Newtonian analysis. A hierarchical control scheme is used to handle the trajectory tracking problem with the translational subsystem in the outer loop and the rotational subsystem in the inner loop. Constrained GP based MPC are formulated separately for both subsystems. The resulting MPC problems are typically nonlinear and non-convex. We derived a GP based local dynamical model that allows these optimization problems to be relaxed to convex ones which can be efficiently solved with a simple active-set algorithm. The performance of the proposed approach is compared with an existing unconstrained Nonlinear Model Predictive Control (NMPC) algorithm and an existing constrained nonlinear GP based MPC algorithm. In the first comparison, simulation results show that the two approaches exhibit similar trajectory tracking performance. However, our approach has the advantage of incorporating constraints on the control inputs. In the second comparison, simulation results demonstrate that our approach only requires 20% of the computational time for the existing nonlinear GP based MPC.

Keywords

Quadrotor trajectory tracking Model predictive control Gaussian process 

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References

  1. 1.
    Abdolhosseini, M., Zhang, Y., Rabbath, C. A.: An efficient model predictive control scheme for an unmanned quadrotor helicopter. J. Intell. Robot. Syst. 70(1-4), 27–38 (2013)CrossRefGoogle Scholar
  2. 2.
    Alexis, K., Nikolakopoulos, G., Tzes, A.: Switching model predictive attitude control for a quadrotor helicopter subject to atmospheric disturbances. Control. Eng. Pract. 19(10), 1195–1207 (2011)CrossRefGoogle Scholar
  3. 3.
    Berkenkamp, F., Schoellig, A. P.: Learning-based robust control: Guaranteeing stability while improving performance. In: IEEE/RSJ Proceedings of International Conference on Intelligent Robots and Systems (IROS) (2014)Google Scholar
  4. 4.
    Bouabdallah, S., Noth, A., Siegwart, R.: PID vs LQ control techniques applied to an indoor micro quadrotor. In: IEEE/RSJ Proceedings of International Conference on Intelligent Robots and Systems (IROS), vol. 3, pp. 2451–2456. IEEE (2004)Google Scholar
  5. 5.
    Candela, J. Q., Girard, A., Larsen, J., Rasmussen, C. E.: Propagation of uncertainty in bayesian kernel models-application to multiple-step ahead forecasting. In: IEEE Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol. 2, pp. II–701. IEEE (2003)Google Scholar
  6. 6.
    Cao, G., Lai, E. M. K., Alam, F.: Gaussian process based model predictive control for linear time varying systems. In: International Workshop on Advanced Motion Control (AMC Workshop). IEEE (2016)Google Scholar
  7. 7.
    Cao, G., Lai, E.M.K., Alam, F.: Gaussian process model predictive control of unknown nonlinear systems. IET Control Theory & Applications. arXiv:1612.01211. Accepted for publication (2016)
  8. 8.
    Cao, G., Lai, E. M. K., Alam, F.: Gaussian process model predictive control of unmanned quadrotors. In: International Conference on Control, Automation and Robotics (ICCAR). IEEE (2016)Google Scholar
  9. 9.
    Deisenroth, M. P.: Efficient reinforcement learning using Gaussian processes. Ph.D. thesis, Karlsruhe Institute of Technology (2010)Google Scholar
  10. 10.
    Diehl, M., Ferreau, H. J., Haverbeke, N.: Efficient numerical methods for nonlinear MPC and moving horizon estimation International Workshop on Assessment and Future Directions on Nonlinear Model Predictive Control, pp 391–417. Springer, Pavia (2008)Google Scholar
  11. 11.
    Dierks, T., Jagannathan, S.: Output feedback control of a quadrotor UAV using neural networks. IEEE Trans. Neural Netw. 21(1), 50–66 (2010)CrossRefGoogle Scholar
  12. 12.
    Doherty, P., Rudol, P.: A UAV search and rescue scenario with human body detection and geolocalization. In: Advances in Artificial Intelligence, pp. 1–13. Springer (2007)Google Scholar
  13. 13.
    Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley-Interscience Publication (1987)Google Scholar
  14. 14.
    Girard, A., Rasmussen, C. E., Candela, J. Q., Murray-Smith, R.: Gaussian process priors with uncertain input – Application to multiple-step ahead time series forecasting. In: Advances in Neural Information Processing Systems (NIPS), pp. 545–552. MIT (2003)Google Scholar
  15. 15.
    Grancharova, A., Johansen, T. A., Tøndel, P.: Computational aspects of approximate explicit nonlinear model predictive control. In: Proceedings of the International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, pp. 181–192. Springer (2007)Google Scholar
  16. 16.
    Grancharova, A., Kocijan, J., Johansen, T. A.: Explicit stochastic predictive control of combustion plants based on Gau- ssian process models. Automatica 44(6), 1621–1631 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control–Theory and Algorithms. Springer-Verlag, London (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Han, F., Feng, G., Wang, Y., Zhou, F.: Fuzzy modeling and control for a nonlinear quadrotor under network environment. In: IEEE 4th Annual International Conference on Cyber Technology in Automation Control, and Intelligent Systems (CYBER), pp. 395–400. IEEE (2014)Google Scholar
  19. 19.
    Hemakumara, P., Sukkarieh, S.: Non-parametric UAV system identification with dependent Gaussian processes. In: IEEE Proceedings of International Conference on Robotics and Automation (ICRA), pp. 4435–4441. IEEE (2011)Google Scholar
  20. 20.
    Hemakumara, P., Sukkarieh, S.: UAV parameter estimation with multi-output local and global Gaussian process approximations. In: IEEE Proceedings of International Conference on Robotics and Automation (ICRA), pp. 5402–5408. IEEE (2013)Google Scholar
  21. 21.
    Huang, M., Xian, B., Diao, C., Yang, K., Feng, Y.: Adaptive tracking control of underactuated quadrotor unmanned aerial vehicles via backstepping. In: American Control Conference, pp. 2076–2081. IEEE (2010)Google Scholar
  22. 22.
    Klenske, E. D., Zeilinger, M. N., Scholkopf, B., Hennig, P.: Gaussian process-based predictive control for periodic error correction. IEEE Trans. Control Syst. Technol. (2015)Google Scholar
  23. 23.
    Kocijan, J., Murray-Smith, R.: Murray-Smith nonlinear predictive control with a Gaussian process model. In: In, R., Shorten, R. (eds.) Switching and Learning in Feedback Systems, pp 185–200. Springer, Heidelberger, Berlin, Germany (2005)CrossRefGoogle Scholar
  24. 24.
    Kocijan, J., Murray-Smith, R., Rasmussen, C. E., Girard, A.: Gaussian process model based predictive control. In: American Control Conference, vol. 3, pp. 2214–2219. IEEE (2004)Google Scholar
  25. 25.
    Kocijan, J., Murray-Smith, R., Rasmussen, C. E., Likar, B.: Predictive control with Gaussian process models Proceedings of IEEE Region 8 EUROCON 2003: Computer as a Tool, vol. A, pp 352–356. IEEE, Ljubljana (2003)Google Scholar
  26. 26.
    Likar, B., Kocijan, J.: Predictive control of a gas–liquid separation plant based on a Gaussian process model. Comput. Chem. Eng. 31(3), 142–152 (2007)CrossRefGoogle Scholar
  27. 27.
    Liuzzi, G., Lucidi, S., Sciandrone, M.: Sequential penalty derivative-free methods for nonlinear constrained optimization. SIAM J. Optim. 20(5), 2614–2635 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lucidi, S., Sciandrone, M., Tseng, P.: Objective-derivative-free methods for constrained optimization. Math. Program. 92(1), 37–59 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Madani, T., Benallegue, A.: Sliding mode observer and backstepping control for a quadrotor unmanned aerial vehicles. In: American Control Conference, pp. 5887–5892. IEEE (2007)Google Scholar
  30. 30.
    Mesbah, A.: Stochastic model predictive control: An overview and perspectives for future research. IEEE Control Systems Magazine. Accepted (2016)Google Scholar
  31. 31.
    Metni, N., Hamel, T.: A UAV for bridge inspection: Visual servoing control law with orientation limits. Autom. Construct. 17(1), 3–10 (2007)CrossRefGoogle Scholar
  32. 32.
    Pan, Y., Theodorou, E.: Probabilistic differential dynamic programming. In: Advances in Neural Information Processing Systems (NIPS), pp. 1907–1915 (2014)Google Scholar
  33. 33.
    Quiñonero-Candela, J., Rasmussen, C. E.: A unifying view of sparse approximate Gaussian process regression. J. Mach. Learn. Res. 6, 1939–1959 (2005)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Raffo, G. V., Ortega, M. G., Rubio, F. R.: An integral predictive/nonlinear H\(\infty \) control structure for a quadrotor helicopter. Automatica 46(1), 29–39 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  36. 36.
    Valipour, M.: Application of new mass transfer formulae for computation of evapotranspiration. J. Appl. Water Eng. Res. 2(1), 33–46 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Valipour, M.: How much meteorological information is necessary to achieve reliable accuracy for rainfall estimations? Agriculture 6(4), 53 (2016)CrossRefGoogle Scholar
  38. 38.
    Valipour, M.: Optimization of neural networks for precipitation analysis in a humid region to detect drought and wet year alarms. Meteorol. Appl. 23(1), 91–100 (2016)CrossRefGoogle Scholar
  39. 39.
    Valipour, M.: Variations of land use and irrigation for next decades under different scenarios. IRRIGA: Braz. J. Irrig. Drain. 1(01), 262–288 (2016)CrossRefGoogle Scholar
  40. 40.
    Valipour, M., Sefidkouhi, M. A. G., Raeini, M., et al.: Selecting the best model to estimate potential evapotranspiration with respect to climate change and magnitudes of extreme events. Agric. Water Manag. 180, 50–60 (2017)CrossRefGoogle Scholar
  41. 41.
    Voos, H.: Nonlinear and neural network-based control of a small four-rotor aerial robot. In: 2007 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 1–6. IEEE (2007)Google Scholar
  42. 42.
    Wang, Y., Boyd, S.: Fast model predictive control using online optimization. IEEE Trans. Control Syst. Technol. 18(2), 267–278 (2010)CrossRefGoogle Scholar
  43. 43.
    Yannopoulos, S. I., Lyberatos, G., Theodossiou, N., Li, W., Valipour, M., Tamburrino, A., Angelakis, A. N.: Evolution of water lifting devices (pumps) over the centuries worldwide. Water 7(9), 5031–5060 (2015)CrossRefGoogle Scholar
  44. 44.
    Yiqing, L., Xigang, Y., Yongjian, L.: An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Comput. Chem. Eng. 31(3), 153–162 (2007)CrossRefGoogle Scholar
  45. 45.
    Zuo, Z.: Trajectory tracking control design with command-filtered compensation for a quadrotor. IET Control Theory Appl. 4(11), 2343–2355 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Engineering and Advanced TechnologyMassey UniversityAucklandNew Zealand
  2. 2.Department of Information Technology and Software EngineeringAuckland University of TechnologyAucklandNew Zealand

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