Research Papers

An Efficient Approach for Stability Analysis and Parameter Tuning in Delayed Feedback Control of a Flying Robot Carrying a Suspended Load

[+] Author and Article Information
Wei Dong

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: dr.dongwei@sjtu.edu.cn

Ye Ding

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

Luo Yang

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: yangluo@sjtu.edu.cn

Xinjun Sheng

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: xjsheng@sjtu.edu.cn

Xiangyang Zhu

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mexyzhu@sjtu.edu.cn

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received January 30, 2018; final manuscript received March 7, 2019; published online April 9, 2019. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 141(8), 081015 (Apr 09, 2019) (10 pages) Paper No: DS-18-1049; doi: 10.1115/1.4043223 History: Received January 30, 2018; Revised March 07, 2019

This paper presents an accurate and computationally efficient time-domain design method for the stability region determination and optimal parameter tuning of delayed feedback control of a flying robot carrying a suspended load. This work first utilizes a first-order time-delay (FOTD) equation to describe the performance of the flying robot, and the suspended load is treated as a flying pendulum. Thereafter, a typical delayed feedback controller is implemented, and the state-space equation of the whole system is derived and described as a periodic time-delay system. On this basis, the differential quadrature method is adopted to estimate the time-derivative of the state vector at concerned sampling grid point. In such a case, the transition matrix between adjacent time-delay duration can be obtained explicitly. The stability region of the feedback system is thereby within the unit circle of spectral radius of this transition matrix, and the minimum spectral radius within the unit circle guarantees fast tracking error decay. The proposed approach is also further illustrated to be able to be applied to some more sophisticated delayed feedback system, such as the input shaping with feedback control. To enhance the efficiency and robustness of parameter optimization, the derivatives of the spectral radius regarding the parameters are also presented explicitly. Finally, extensive numeric simulations and experiments are conducted to verify the effectiveness of the proposed method, and the results show that the proposed strategy efficiently estimates the optimal control parameters as well as the stability region. On this basis, the suspended load can effectively track the pre-assigned trajectory without large oscillations.

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Ijspeert, A. J. , 2014, “ Biorobotics: Using Robots to Emulate and Investigate Agile Locomotion,” Science, 346(6206), pp. 196–203. [CrossRef] [PubMed]
Lungu, M. , 2012, “ Stabilization and Control of a Uav Flight Attitude Angles Using the Backstepping Method,” Int. J. Aerosp. Mech. Eng., 6(1), pp. 53–60. https://waset.org/publications/6207/stabilization-and-control-of-a-uav-flight-attitude-angles-using-the-backstepping-method
Lungu, M. , and Lungu, R. , 2013, “ Adaptive Backstepping Flight Control for a Mini-Uav,” Int. J. Adapt. Control Signal Process., 27(8), pp. 635–650. [CrossRef]
Kumar, V. , and Michael, N. , 2012, “ Opportunities and Challenges With Autonomous Micro Aerial Vehicles,” Int. J. Rob. Res., 31(11), pp. 1279–1291. [CrossRef]
Goodarzi, F. A. , Lee, D. , and Lee, T. , 2015, “ Geometric Adaptive Tracking Control of a Quadrotor Unmanned Aerial Vehicle on SE(3) for Agile Maneuvers,” ASME J. Dyn. Syst., Meas., Control, 137(9), p. 091007. [CrossRef]
Sanchezorta, A. , Parravega, V. , Izaguirreespinosa, C. , and Garcia, O. , 2015, “ Position–Yaw Tracking of Quadrotors,” ASME J. Dyn. Syst., Meas., Control, 137(6), p. 061011. [CrossRef]
Palunko, I. , Fierro, R. , and Cruz, P. , 2012, “ Trajectory Generation for Swing-Free Maneuvers of a Quadrotor With Suspended Payload: A Dynamic Programming Approach,” International Conference on Robotics and Automation (ICRA), Saint Paul, MN, May 14–18, pp. 2691–2697.
Faust, A. , Palunko, I. , Cruz, P. , Fierro, R. , and Tapia, L. , 2013, “ Learning Swing-Free Trajectories for UAVs With a Suspended Load,” International Conference on Robotics and Automation (ICRA), Karlsruhe, Germany, May 6–10, pp. 2691–2697.
Mellinger, D. , Shomin, M. , Michael, N. , and Kumar, V. , 2013, Cooperative Grasping and Transport Using Multiple Quadrotors, Springer, Berlin.
U.K. Civil Aviation Authority Safety Regulation Group, 2006, “ Helicopter External Load Operations,” The Stationery Office, Norwich, Norfolk, UK, Document No. CAP 426.
Adams, C. , Potter, J. , and Singhose, W. , 2015, “ Input-Shaping and Model-Following Control of a Helicopter Carrying a Suspended Load,” J. Guid. Control Dyn., 38(1), pp. 94–105. [CrossRef]
Potter, J. J. , Adams, C. , and Singhose, W. , 2015, “ A Planar Experimental Remote-Controlled Helicopter With a Suspended Load,” IEEE-ASME Trans. Mechatronics, 20(5), pp. 2496–2503. [CrossRef]
Orszulik, R. , and Shan, J. , 2011, “ Vibration Control Using Input Shaping and Adaptive Positive Position Feedback,” J. Guid. Control Dyn., 34(4), pp. 1031–1044. [CrossRef]
John, H. , Khalid, S. , and William, S. , 2008, “ Useful Applications of Closed-Loop Signal Shaping Controllers,” Control Eng. Pract., 16(7), p. 836846.
Ziyad, M. , Ali, N. , and Nayfeh, N. , 2005, “ Sway Reduction on Quay-Side Container Cranes Using Delayed Feedback Controller: Simulations and Experiments,” J. Vib. Control, 11(8), p. 11031122.
Pereira, E. , Trapero, J. R. , Díaz, I. M. , and Feliu, V. , 2012, “ Adaptive Input Shaping for Single-Link Flexible Manipulators Using an Algebraic Identification,” Control Eng. Pract., 20(2), pp. 138–147. [CrossRef]
Lee, T. , 2018, “ Geometric Control of Multiple Quadrotor UAVs Transporting a Cable-Suspended Rigid Body,” IEEE Trans. Control Syst. Technol., 26(1), pp. 255–264. [CrossRef]
Steed, E. , Quesada, E. S. E. , Rodolfo, L. , Carrillo, G. , Ramirez, A. , and Mondie, S. , 2016, “ Algebraic Dominant Pole Placement Methodology for Unmanned Aircraft Systems With Time Delay,” IEEE Trans. Aerosp. Electron. Syst., 52(3), pp. 1108–1119. [CrossRef]
Malek-Zavarei, M. , and Jamshidi, M. , 1987, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science, New York.
Gu, K. , Chen, J. , and Kharitonov, V. L. , 2003, Stability of Time-Delay Systems, Springer Science and Business Media, New York.
Dong, W. , Gu, G. , Zhu, X. , and Ding, H. , 2016, “ A High-Performance Flight Control Approach for Quadrotors Using a Modified Active Disturbance Rejection Technique,” Rob. Auton. Syst., 83(1), pp. 177–187. [CrossRef]
Armah, S. K. , Yi, S. , and Choi, W. , 2016, “ Design of Feedback Control for Quadrotors Considering Signal Transmission Delays,” Int. J. Control, Autom. Syst., 14(6), pp. 1395–1403. [CrossRef]
Amelin, K. , Tomashevich, S. , and Andrievsky, B. , 2015, “ Recursive Identification of Motion Model Parameters for Ultralight UAV,” Proc. Int. Fed. Autom. Control, 48(11), pp. 233–237.
Yi, S. , Nelson, P. W. , and Ulsoy, A. G. , 2013, “ Proportional-Integral Control of First-Order Time-Delay Systems Via Eigenvalue Assignment,” IEEE Trans. Control Syst. Technol., 21(5), pp. 1586–1594. [CrossRef]
Insperger, T. , and Stépán, G. , 2011, Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications, Vol. 178, Springer Science and Business Media, New York.
Dong, W. , Ding, Y. , Zhu, X. , and Ding, H. , 2015, “ Optimal Proportional–Integral–Derivative Control of Time-Delay Systems Using the Differential Quadrature Method,” ASME J. Dyn. Syst., Meas., Control, 137(10), p. 101005. [CrossRef]
Shu, C. , 2000, Differential Quadrature and Its Application in Engineering, Springer, Berlin, Germany.
Fung, T. , 2001, “ Solving Initial Value Problems by Differential Quadrature Method–Part 1: First-Order Equations,” Int. J. Numer. Methods Eng., 50(6), pp. 1411–1427. [CrossRef]
Quan, J. , and Chang, C. , 1989, “ New Insights in Solving Distributed System Equations by the Quadrature Method: I—Analysis,” Comput. Chem. Eng., 13(7), pp. 779–788. [CrossRef]
Ding, Y. , Zhu, L. , Zhang, X. , and Ding, H. , 2013, “ Stability Analysis of Milling Via the Differential Quadrature Method,” ASME J. Manuf. Sci. Eng., 135(4), p. 044502. [CrossRef]
Bert, C. W. , and Malik, M. , 1996, “ Differential Quadrature Method in Computational Mechanics: A Review,” ASME Appl. Mech. Rev., 49(1), pp. 1–28. [CrossRef]
Meyer, C. D. , 2000, Matrix Analysis and Applied Linear Algebra, Siam, Philadelphia, PA.
Mann, B. , and Patel, B. , 2010, “ Stability of Delay Equations Written as State Space Models,” J. Vib. Control, 16(7–8), pp. 1067–1085. [CrossRef]
Farkas, M. , 1994, Periodic Motions, Springer-Verlag, New York.
Sheng, J. , and Sun, J. , 2005, “ Feedback Controls and Optimal Gain Design of Delayed Periodic Linear Systems,” J. Vib. Control, 11(2), pp. 277–294.
Wu, D. , and Sinha, S. , 1994, “ A New Approach in the Analysis of Linear-Systems With Periodic Coefficients for Applications in Rotorcraft Dynamics,” Aeronaut. J., 98(971), pp. 9–16. [CrossRef]
Ding, Y. , Zhu, L. , Zhang, X. , and Ding, H. , 2012, “ Response Sensitivity Analysis of the Dynamic Milling Process Based on the Numerical Integration Method,” Chin. J. Mech. Eng., 25(5), pp. 940–946. [CrossRef]
Lax, P. D. , 2007, Linear Algebra and Its Applications, Wiley-Interscience, New York.
Yang, W. Y. , Cao, W. , Chung, T.-S. , and Morris, J. , 2005, Applied Numerical Methods Using MATLAB, Wiley, Hoboken, NJ.
Poli, R. , Kennedy, J. , and Blackwell, T. , 2007, “ Particle Swarm Optimization,” Swarm Intell., 1(1), pp. 33–57. [CrossRef]
Ding, Y. , Niu, J. , Zhu, L. , and Ding, H. , 2015, “ Differential Quadrature Method for Stability Analysis of Dynamic Systems With Multiple Delays: Application to Simultaneous Machining Operations,” ASME J. Vib. Acoust., 137(2), p. 024501. [CrossRef]
Xue, D. , and Chen, Y. , et al. ., 2013, System Simulation Techniques With MATLAB and Simulink, Wiley, Chichester, UK.
Dong, W. , Gu, G. Y. , Zhu, X. , and Ding, H. , 2014, “ High Performance Trajectory Tracking Control of a Quadrotor With Disturbance Observer,” Sens. Actuators, A, 211, pp. 67–77. [CrossRef]


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Fig. 1

The illustration of the suspended transportation with flying robot

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Fig. 2

The control structure of feedback control for the flying robot carrying a suspended load

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Fig. 3

Control structure of a typical input shaping with linear feedback

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Fig. 4

The contour plot of spectral radius regarding parameters h and Kp: (a) DQM and (b) LMI

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Fig. 5

The responses of different control strategies

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Fig. 6

The 3D plot of the ITAEC evaluation

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Fig. 7

The stability region investigation of system (27) with input shaper: (a) 3D stability region and (b) τs = 1.01

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Fig. 8

The gradient of the spectral radius

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Fig. 9

The response of delayed feedback control system with different parameters

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Fig. 10

The flying robot testbed utilized by this work

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Fig. 11

The real-time flight results of the open-loop transportation and delay feedback control: (a) normal and (b) delayed feedback control

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Fig. 12

The cross tracking error of the suspended load

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Fig. 13

The response of the suspended load with external disturbance



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