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.

References

References
1.
Ijspeert
,
A. J.
,
2014
, “
Biorobotics: Using Robots to Emulate and Investigate Agile Locomotion
,”
Science
,
346
(
6206
), pp.
196
203
.
2.
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
3.
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
.
4.
Kumar
,
V.
, and
Michael
,
N.
,
2012
, “
Opportunities and Challenges With Autonomous Micro Aerial Vehicles
,”
Int. J. Rob. Res.
,
31
(
11
), pp.
1279
1291
.
5.
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
.
6.
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
.
7.
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
.
8.
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
.
9.
Mellinger
,
D.
,
Shomin
,
M.
,
Michael
,
N.
, and
Kumar
,
V.
,
2013
,
Cooperative Grasping and Transport Using Multiple Quadrotors
,
Springer
,
Berlin
.
10.
U.K. Civil Aviation Authority Safety Regulation Group,
2006
, “
Helicopter External Load Operations
,” The Stationery Office, Norwich, Norfolk, UK, Document No. CAP 426.
11.
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
.
12.
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
.
13.
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
.
14.
John
,
H.
,
Khalid
,
S.
, and
William
,
S.
,
2008
, “
Useful Applications of Closed-Loop Signal Shaping Controllers
,”
Control Eng. Pract.
,
16
(7), p.
836846
.
15.
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
.
16.
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
.
17.
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
.
18.
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
.
19.
Malek-Zavarei
,
M.
, and
Jamshidi
,
M.
,
1987
,
Time-Delay Systems: Analysis, Optimization and Applications
,
Elsevier Science
, New York.
20.
Gu
,
K.
,
Chen
,
J.
, and
Kharitonov
,
V. L.
,
2003
,
Stability of Time-Delay Systems
,
Springer Science and Business Media
, New York.
21.
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
.
22.
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
.
23.
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
.
24.
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
.
25.
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.
26.
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
.
27.
Shu
,
C.
,
2000
,
Differential Quadrature and Its Application in Engineering
,
Springer
,
Berlin, Germany
.
28.
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
.
29.
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
.
30.
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
.
31.
Bert
,
C. W.
, and
Malik
,
M.
,
1996
, “
Differential Quadrature Method in Computational Mechanics: A Review
,”
ASME Appl. Mech. Rev.
,
49
(
1
), pp.
1
28
.
32.
Meyer
,
C. D.
,
2000
,
Matrix Analysis and Applied Linear Algebra
,
Siam
,
Philadelphia, PA
.
33.
Mann
,
B.
, and
Patel
,
B.
,
2010
, “
Stability of Delay Equations Written as State Space Models
,”
J. Vib. Control
,
16
(
7–8
), pp.
1067
1085
.
34.
Farkas
,
M.
,
1994
,
Periodic Motions
,
Springer-Verlag
,
New York
.
35.
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
.
36.
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
.
37.
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
.
38.
Lax
,
P. D.
,
2007
,
Linear Algebra and Its Applications
,
Wiley-Interscience
,
New York
.
39.
Yang
,
W. Y.
,
Cao
,
W.
,
Chung
,
T.-S.
, and
Morris
,
J.
,
2005
,
Applied Numerical Methods Using MATLAB
,
Wiley
,
Hoboken, NJ
.
40.
Poli
,
R.
,
Kennedy
,
J.
, and
Blackwell
,
T.
,
2007
, “
Particle Swarm Optimization
,”
Swarm Intell.
,
1
(
1
), pp.
33
57
.
41.
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
.
42.
Xue
,
D.
, and
Chen
,
Y.
, et al. .,
2013
,
System Simulation Techniques With MATLAB and Simulink
,
Wiley
,
Chichester, UK
.
43.
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
.
You do not currently have access to this content.