Abstract

In this paper, a momentum-preserving integration scheme is implemented for the simulation of single and cooperative multirotors with a flexible-cable suspended payload by employing a Lie group–based variational integrator (VI), which provides the preservation of the configuration manifold and geometrical constraints. Due to the desired properties of the implemented VI method, e.g., symplecticity, momentum preservation, and the exact fulfillment of the constraints, exponentially long-term numerical stability, and good energy behavior are obtained for more accurate simulations of aforementioned systems. The effectiveness of Lie group VI method with the corresponding discrete systems are demonstrated by comparing the simulation results of two example scenarios for the single and cooperative systems in terms of the preserved quantities and constraints, where a conventional fixed-step Runge–Kutta 4 (RK4) and variable-step integrator are utilized for the simulation of continuous-time models. It is shown that the implemented VI method successfully performs the simulations with a long-time stable behavior by preserving invariants of the system and the geometrical constraints, whereas the simulation of continuous-time models by RK4 and variable step are incapable of satisfying these desired properties, which inherently results in divergent and unstable behavior in simulations.

References

1.
Sreenath
,
K.
,
Lee
,
T.
, and
Kumar
,
V.
,
2013
, “
Geometric Control and Differential Flatness of a Quadrotor Uav With a Cable-Suspended Load
,”
52nd IEEE Conference on Decision and Control
, Florence, Italy, Dec. 10–13, pp.
2269
2274
.10.1109/CDC.2013.6760219
2.
Sreenath
,
K.
,
Michael
,
N.
, and
Kumar
,
V.
,
2013
, “
Trajectory Generation and Control of a Quadrotor With a Cable-Suspended Load—A Differentially-Flat Hybrid System
,”
IEEE International Conference on Robotics and Automation
, Karlsruhe, Germany, May 6-10, pp.
4888
4895
.10.1109/ICRA.2013.6631275
3.
Tang
,
S.
, and
Kumar
,
V.
,
2015
, “
Mixed Integer Quadratic Program Trajectory Generation for a Quadrotor With a Cable-Suspended Payload
,”
IEEE International Conference on Robotics and Automation
, Seattle, WA, May 26–30, pp.
2216
2222
.10.1109/ICRA.2015.7139492
4.
Godbole
,
A. R.
, and
Subbarao
,
K.
,
2019
, “
Nonlinear Control of Unmanned Aerial Vehicles With Cable Suspended Payloads
,”
Aerosp. Sci. Technol.
,
93
, p.
105299
.10.1016/j.ast.2019.07.032
5.
Guo
,
M.
,
Gu
,
D.
,
Zha
,
W.
,
Zhu
,
X.
, and
Su
,
Y.
,
2020
, “
Controlling a Quadrotor Carrying a Cable-Suspended Load to Pass Through a Window
,”
J. Intell. Rob. Syst.
,
98
(
2
), pp.
387
401
.10.1007/s10846-019-01038-6
6.
Goodarzi
,
F. A.
,
Lee
,
D.
, and
Lee
,
T.
,
2014
, “
Geometric Stabilization of a Quadrotor UAV With a Payload Connected by Flexible Cable
,”
American Control Conference
, Portland, OR, June 4–6, pp.
4925
4930
.10.1109/ACC.2014.6859419
7.
Wu
,
G.
, and
Sreenath
,
K.
,
2014
, “
Geometric Control of Multiple Quadrotors Transporting a Rigid-Body Load
,”
53rd IEEE Conference on Decision and Control
, Los Angeles, CA, Dec. 15–17, pp.
6141
6148
.10.1109/CDC.2014.7040351
8.
Sreenath
,
K.
, and
Kumar
,
V.
,
2013
, “
Dynamics, Control and Planning for Cooperative Manipulation of Payloads Suspended by Cables From Multiple Quadrotor Robots
,”
rn
,
1
(
r2
), p.
r3
.
9.
Lee
,
T.
,
2014
, “
Geometric Control of Multiple Quadrotor UAVs Transporting a Cable-Suspended Rigid Body
,”
53rd IEEE Conference on Decision and Control
, Los Angeles, CA, Dec. 15–17, pp.
6155
6160
.10.1109/CDC.2014.7040353
10.
Goodarzi
,
F. A.
, and
Lee
,
T.
,
2015
, “
Dynamics and Control of Quadrotor UAVs Transporting a Rigid Body Connected Via Flexible Cables
,”
American Control Conference (ACC)
, Chicago, IL, July 1–3, pp.
4677
4682
.10.1109/ACC.2015.7172066
11.
Simo
,
J.
,
Tarnow
,
N.
, and
Wong
,
K.
,
1992
, “
Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
100
(
1
), pp.
63
116
.10.1016/0045-7825(92)90115-Z
12.
Marsden
,
J. E.
, and
West
,
M.
,
2001
, “
Discrete Mechanics and Variational Integrators
,”
Acta Numer.
,
10
(
1
), pp.
357
514
.10.1017/S096249290100006X
13.
Hairer
,
E.
,
Wanner
,
G.
, and
Lubich
,
C.
,
2006
,
Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
,
Springer
,
Berlin/Heidelberg
.
14.
Lee
,
T.
,
Leok
,
M.
, and
McClamroch
,
N. H.
,
2007
, “
Lie Group Variational Integrators for the Full Body Problem
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
29–30
), pp.
2907
2924
.10.1016/j.cma.2007.01.017
15.
Manchester
,
Z. R.
, and
Peck
,
M. A.
,
2016
, “
Quaternion Variational Integrators for Spacecraft Dynamics
,”
J. Guid. Control Dyn.
,
39
(
1
), pp.
69
76
.10.2514/1.G001176
16.
Lee
,
T.
,
McClamroch
,
N. H.
, and
Leok
,
M.
,
2005
, “
A Lie Group Variational Integrator for the Attitude Dynamics of a Rigid Body With Applications to the 3D Pendulum
,”
Proceedings of 2005 IEEE Conference on Control Applications
, Toronto, ON, Canada, Aug. 28–31, pp.
962
967
.10.1109/CCA.2005.1507254
17.
Lee
,
T.
,
Leok
,
M.
, and
McClamroch
,
N. H.
,
2007
,
Discrete Control Systems
, arXiv preprint arXiv:0705.3868.
18.
Nikhilraj
,
A.
,
Simha
,
H.
, and
Priyadarshan
,
H.
,
2019
, “
Optimal Energy Trajectory Generation for a Quadrotor UAV Using Geometrically Exact Computations on se(3)
,”
IEEE Control Syst. Lett.
,
3
(
1
), pp.
216
221
.10.1109/LCSYS.2018.2874103
19.
Nordkvist
,
N.
, and
Sanyal
,
A. K.
,
2010
, “
A Lie Group Variational Integrator for Rigid Body Motion in SE (3) With Applications to Underwater Vehicle Dynamics
,”
49th IEEE Conference on Decision and Control (CDC)
, Atlanta, GA, Dec. 15–17, pp.
5414
5419
.10.1109/CDC.2010.5717622
20.
Schultz
,
J.
,
Flaßkamp
,
K.
, and
Murphey
,
T. D.
,
2017
, “
Variational Integrators for Structure-Preserving Filtering
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
2
), p.
021005
.10.1115/1.4034728
21.
Leok
,
M.
, and
Shingel
,
T.
,
2012
, “
General Techniques for Constructing Variational Integrators
,”
Front. Math. China
,
7
(
2
), pp.
273
303
.10.1007/s11464-012-0190-9
22.
Davis
,
P. J.
, and
Rabinowitz
,
P.
,
1984
, “
Chapter 2—Approximate Integration Over a Finite Interval
,”
Methods of Numerical Integration
, 2nd ed.,
P. J.
Davis
and
P.
Rabinowitz
, eds.,
Academic Press
, Cambridge, MA, pp.
51
198
.
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