Abstract
In this paper, a control strategy based on the optimal control and subspace stabilization approach is developed to solve the two-point boundary value problem of a highly under-actuated quadrotor. To facilitate the development, the dynamic model of the quadrotor is firstly presented. Then the boundary value problem is mathematically formulated based on the optimal control theory. According to the problem formulation and utilizing the subspace stabilization approach, the control strategy is proposed to suppress the state-trajectory tracking errors and manipulate the quadrotor from a known initial state to the desired final state in a finite time horizon. As there exist input delays in real-time flights, the Smith predictor is designed to enhance the performance of the developed control strategy. Finally, an indoor experimental platform of the quadrotor is built and real-time experiments of the ball-batting is conducted with a coefficient of restitution of approximate 0.7 and a racket with diameter of 0.13 m. The experimental results show that the quadrotor can well establish the desired final state and bat the ball towards its target location (the deviation of position is less than 0.15 m), which verify the feasibility of the proposed control strategy.
Similar content being viewed by others
References
Kumar, V., Michael, N.: Opportunities and challenges with autonomous micro aerial vehicles. Int. J. Robot. Res. 31(11), 1279–1291 (2012)
Abdolhosseini, M., Zhang, Y., Rabbath, C.A.: An efficient model predictive control scheme for an unmanned quadrotor helicopter. J. Intel. Robotic Syst. 70(1-4), 27–38 (2013)
Lim, H., Park, J., Lee, D., Kim, H.: Build your own quadrotor: Open-source projects on unmanned aerial vehicles. IEEE Robotics Autom. Mag. 19(3), 33–45 (2012)
Bouabdallah, S.: Design and control of quadrotors with application to autonomous flying. Ph.D. thesis, Lausanne Polytechnic University (2007)
Pounds, P., Mahony, R.: Design principles of large quadrotors for practical applications. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 3265–3270 (2009)
Bouabdallah, S., Noth, A., Siegwart, R.: Pid vs lq control techniques applied to an indoor micro quadrotor. In: Proceedings of IEEE international conference on intelligent robots and systems, vol. 3, pp. 2451–2456 (2004)
Dong, W., Gu, G.Y., Zhu, X., Ding, H.: High-performance trajectory tracking control of a quadrotor with disturbance observer. Sensors and Actuators A Phys. 211, 67–77 (2014). doi:10.1016/j.sna.2014.03.011
Kendoul, F.: Nonlinear hierarchical flight controller for unmanned rotorcraft: design, stability, and experiments. J. Guid. Control. Dyn. 32(6), 1954–1958 (2009)
Tayebi, A., McGilvray, S.: Attitude stabilization of a vtol quadrotor aircraft. IEEE Trans. Control Syst. Technol. 14(3), 562–571 (2006)
Chamseddine, A., Zhang, Y., Rabbath, C.A., Theilliol, D.: Trajectory planning and replanning strategies applied to a quadrotor unmanned aerial vehicle. J. Guid. Control Dyn 35(5), 1667–1671 (2012)
He, R., Bachrach, A., Achtelik, M., Geramifard, A., Gurdan, D., Prentice, S., Stumpf, J., Roy, N.: On the design and use of a micro air vehicle to track and avoid adversaries. Int. J. Robot. Res. 29(5), 529–546 (2010)
Mahony, R., Kumar, V., Corke, P.: Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor. IEEE Robot Autom. Mag. 19(3), 20–32 (2012)
Orsag, M., Korpela, C., Oh, P.: Modeling and control of mm-uav: Mobile manipulating unmanned aerial vehicle. J. Int. Robot. Syst. 69(1-4), 227–240 (2013)
Metni, N., Hamel, T.: A uav for bridge inspection: Visual servoing control law with orientation limits. Automation in construction 17(1), 3–10 (2007)
Fang, Z., Gao, W.: Adaptive backstepping control of an indoor micro-quadrotor. Res. J. Appl. Scince Eng. Technol. 4(21), 4216–4226 (2012)
Oriolo, G., Nakamura, Y.: Control of mechanical systems with second-order nonholonomic constraints: Underactuated manipulators. In: Proceedings of the 30th IEEE Conference on Decision and Control, pp. 2398–2403 (1991)
Rui, C., Reyhanoglu, M., Kolmanovsky, I., Cho, S., McClamroch, N.: Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system. In: Proceedings of the 36th IEEE conference on decision and control, vol. 4, pp. 3998–4003. IEEE (1997)
Brockett, R.W., et al.: Asymptotic stability and feedback stabilization. Differential geometric control theory, pp. 181–191 (1983)
Mellinger, D., Michael, N., Kumar, V.: Trajectory generation and control for precise aggressive maneuvers with quadrotors. Int. J. Robot. Res. 31(5), 664–674 (2012)
Muller, M., Lupashin, S., D’Andrea, R.: Quadrocopter ball juggling. In: Proceedings of IEEE International Conference on Intelligent Robots and Systems, pp. 5113–5120 (2011)
Mueller, M.W., Hehn, M., D’Andrea, R.: A computationally efficient algorithm for state-to-state quadrocopter trajectory generation and feasibility verification. In: Proceedings of IEEE International Conference on Intelligent Robots and Systems, pp. 3480–3486 (2013)
Wikipedia: Coefficient of restitution http://en.wikiped-ia.org/wiki/Coefficient_of_restitution. Accessed 21 (2014)
Bouabdallah, S., Murrieri, P., Siegwart, R.: Design and control of an indoor micro quadrotor. In: Proceedings of IEEE International Conference on Robotics and Automation, vol. 5, pp. 4393–4398 (2004)
Dydek, Z.T., Annaswamy, A.M., Lavretsky, E.: Adaptive control of quadrotor uavs: A design trade study with flight evaluations. IEEE Trans. Control Syst. Technol. 21(4), 1400–1406 (2013)
Wikipedia: Configuration space. http://en.wikipedia.org/wiki/Configu-ration_space. Accessed 22 (2014)
Márton, L., Hodel, A.S., Lantos, B., Hung, J.Y.: Underactuated robot control: comparing lqr, subspace stabilization, and combined error metric approaches 55(10), 3724–3730 (2008)
Simmons, A.T., Hung, J.Y., Hodel, A.S.: A hybrid improvement to traditional nonlinear control. In: Proceedings of IEEE international symposium on industrial electronics, vol. 1, pp. 49–56. IEEE (2005)
Athans, M.: The role and use of the stochastic linear-quadratic-gaussian problem in control system design. IEEE Trans. Autom. Control 16(6), 529–552 (1971)
Ingimundarson, A., Hägglund, T.: Robust tuning procedures of dead-time compensating controllers. Control. Eng. Pract. 9(11), 1195–1208 (2001)
MathWorks: Control of processes with long dead time: The smith predictor. http://www.mathworks.cn/cn/help/control/examples/control-of-processes-with-long-dead-time-the-smith-predictor.html. Accessed 26 February (2014)
Nonomura, J., Nakashima, A., Hayakawa, Y.: Analysis of effects of rebounds and aerodynamics for trajectory of table tennis ball. In: Proceedings of SICE Annual Conference, pp. 1567–1572 (2010)
Yang, Z.J., Hara, S., Kanae, S., Wada, K., Su, C.Y.: An adaptive robust nonlinear motion controller combined with disturbance observer. IEEE Trans. Control Syst. Technol. 18(2), 454–462 (2010)
Vicon: Vicon mx systems. http://www.vicon.com/products/viconmx.html. Accessed 10 October (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dong, W., Gu, GY., Zhu, X. et al. Solving the Boundary Value Problem of an Under-Actuated Quadrotor with Subspace Stabilization Approach. J Intell Robot Syst 80, 299–311 (2015). https://doi.org/10.1007/s10846-014-0161-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-014-0161-3