active stabilization of slung loads in high speed …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
College of Engineering
ACTIVE STABILIZATION OF SLUNG LOADS IN HIGH SPEED
FLIGHT USING CABLE ANGLE FEEDBACK
A Thesis in
Aerospace Engineering
by
Mariano D. Scaramal
© 2018 Mariano D. Scaramal
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2018
The thesis of Mariano D. Scaramal was reviewed and approved∗ by the following:
Joseph F. Horn
Professor of Aerospace Engineering
Thesis Co-Adviser
Jacob Enciu
Assistant Research Professor of Aerospace Engineering
Thesis Co-Adviser
Amy R. Pritchett
Professor of Aerospace Engineering
Head of the Department of Aerospace Engineering
∗Signatures are on file in the Graduate School.
ii
AbstractHelicopters performing external load missions are subject to instabilities that arise
in high speed flight that limit their operational flight envelope. This thesis ad-
dresses the problem of active stabilization of slung loads in high speed flight. To
demonstrate the method, simulations of a utility helicopter with a dynamic inver-
sion controller (as its automatic flight control system) and a CONEX cargo con-
tainer were used. An airspeed scheduled controller utilizing cable angle feedback
to the primary dynamic inversion controller was designed for the nonlinear coupled
system by the classic root locus technique. Nonlinear simulations of straight and
level flight at different airspeeds were used to validate the controller performance
in stabilizing the load pendulum motions. Controller performance was also evalu-
ated in a complex maneuver and in more demanding scenarios by adding different
levels of atmospheric turbulence to the previous cases. The results show that the
use of cable angle feedback provides or improves system stability when turbulence
is not included in the simulation. When light/moderate turbulence is present sus-
tained limit cycle oscillations are avoided by the use of the controller. For severe
turbulence levels, the controller did not provide any significant improvement.
iii
Table of Contents
List of Figures vii
List of Tables xi
List of Symbols xii
Acknowledgments xv
Chapter 1Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Goal and Organization of the Thesis . . . . . . . . . . . . . . . . . 3
Chapter 2Model Description 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 External Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Isolated Load Dynamics . . . . . . . . . . . . . . . . . . . . 82.3 Sling Cables Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Helicopter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Coupled Helicopter-External Load System . . . . . . . . . . . . . . 15
2.5.1 Load Stability in the Coupled Model . . . . . . . . . . . . . 152.5.2 Relative Cable Angles . . . . . . . . . . . . . . . . . . . . . 17
2.6 Dryden Wind Turbulence Model . . . . . . . . . . . . . . . . . . . . 202.6.1 Low-Altitude Model . . . . . . . . . . . . . . . . . . . . . . 202.6.2 Medium/High Altitudes Model . . . . . . . . . . . . . . . . 22
Chapter 3Controller Design 25
iv
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Controller Design Process Description . . . . . . . . . . . . . . . . . 25
3.2.1 Design Example for Low Airspeed . . . . . . . . . . . . . . . 293.2.2 Design Example for High Airspeed . . . . . . . . . . . . . . 31
3.3 Controller Design Summarized . . . . . . . . . . . . . . . . . . . . . 33
Chapter 4Simulation Results 354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Trimmed Cruise Flight . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Simulation at 25 kt . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Simulation at 97 kt . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Complex Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Delayed Controller Activation . . . . . . . . . . . . . . . . . . . . . 45
4.4.1 Trimmed Cruise Flight with Time-Triggered Controller . . . 464.4.1.1 Simulation at 25 kt . . . . . . . . . . . . . . . . . . 464.4.1.2 Simulation at 97 kt . . . . . . . . . . . . . . . . . . 47
4.5 Turbulent Air Simulations . . . . . . . . . . . . . . . . . . . . . . . 504.5.1 Trimmed Cruise Flight . . . . . . . . . . . . . . . . . . . . . 50
4.5.1.1 Light Level of Turbulence . . . . . . . . . . . . . . 514.5.1.2 Moderate Level of Turbulence . . . . . . . . . . . . 524.5.1.3 Severe Level of Turbulence . . . . . . . . . . . . . . 57
4.5.2 Complex Maneuver . . . . . . . . . . . . . . . . . . . . . . . 574.5.2.1 Light Level of Turbulence . . . . . . . . . . . . . . 594.5.2.2 Moderate Level of Turbulence . . . . . . . . . . . . 604.5.2.3 Severe Level of Turbulence . . . . . . . . . . . . . . 64
Chapter 5Conclusions and Future Works 665.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Appendix AFirst Principles Physical Model 69A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2 Helicopter and Load Reduced Lateral Models . . . . . . . . . . . . 69
A.2.1 Model for Slow State Variables . . . . . . . . . . . . . . . . 71A.3 Helicopter Stability Augmentation System . . . . . . . . . . . . . . 73A.4 Control System for Helicopter-Load System . . . . . . . . . . . . . 75
v
Appendix BAirspeed Scheduled Controller Implementation 77B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Appendix CRoot Locus Analysis Code 83C.1 Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography 98
vi
List of Figures
1.1 External load mission examples1 . . . . . . . . . . . . . . . . . . . . 2
2.1 Cargo container with fins inclined in 33 degrees relative to the box(picture from [21]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Helicopter (H), load (L), and Earth (E) fixed coordinate system . . 7
2.3 Isolated load equilibria points . . . . . . . . . . . . . . . . . . . . . 8
2.4 Limit cycle oscillations for 100 kt (168.8 ft/s) . . . . . . . . . . . . 9
2.5 Equilibria points around 170 ft/s (101 kt) for the isolated load . . . 10
2.6 Helicopter flight control system model . . . . . . . . . . . . . . . . 13
2.7 Helicopter inner loop dynamic inversion . . . . . . . . . . . . . . . . 14
2.8 Effects observed in the model pole diagrams for the coupled systemat 25 kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.9 Effects observed in the model pole diagrams for the coupled systemat 97 kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.10 Relative cable angles 2-1-3 sequence description . . . . . . . . . . . 19
2.11 Dryden medium/high altitude turbulence intensities and probabil-ity of exceedance [32] . . . . . . . . . . . . . . . . . . . . . . . . . . 22
vii
3.1 Relative cable angles trimmed for the couple system at 100 kt . . . 26
3.2 New proposed controller design . . . . . . . . . . . . . . . . . . . . 28
3.3 Lateral controller root locus design for 25 kt . . . . . . . . . . . . . 29
3.4 Longitudinal controller root locus design for 25 kt . . . . . . . . . . 31
3.5 Lateral controller root locus design for 97 kt . . . . . . . . . . . . . 32
3.6 Longitudinal controller root locus design for 97 kt . . . . . . . . . . 33
4.1 Relative cable angles simulation result for 25 kt . . . . . . . . . . . 36
4.2 Helicopter Euler angles simulation result for 25 kt . . . . . . . . . . 37
4.3 Helicopter controls commands simulation result for 25 kt . . . . . . 38
4.4 Relative cable angles simulation result for 97 kt . . . . . . . . . . . 39
4.5 Helicopter controls simulation result for 97 kt . . . . . . . . . . . . 40
4.6 Helicopter Euler angles simulation result for 97 kt . . . . . . . . . . 40
4.7 Relative cable angles simulation result for 97 kt, asymmetric LCO . 41
4.8 Helicopter Euler angles simulation result for 97 kt, asymmetric LCO 42
4.9 Helicopter controls simulation result for 97 kt, asymmetric LCO . . 42
4.10 Helicopter Euler angles for a complex maneuver simulation . . . . . 44
4.11 Relative cable angles for a complex maneuver simulation . . . . . . 45
4.12 Relative cable angles results for 25 kt with controllers turned on att = 29.05 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.13 Example of excellent result for 97 kt . . . . . . . . . . . . . . . . . 47
viii
4.14 Example of good result for 97 kt . . . . . . . . . . . . . . . . . . . . 48
4.15 Example of adequate result for 97 kt . . . . . . . . . . . . . . . . . 49
4.16 Time-triggered controller results summary for an airspeed of 97 kt . 49
4.17 Cruise flight at low altitude with light turbulence intensity and 97kt airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.18 Cruise flight at medium/high altitude with light turbulence inten-sity and 97kt airspeed . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.19 Cruise flight at low altitude with moderate turbulence intensity and97 kt airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.20 Cruise flight at medium/high altitude with moderate turbulenceintensity and 97 kt airspeed . . . . . . . . . . . . . . . . . . . . . . 53
4.21 Cruise flight at medium/high altitude with moderate turbulenceintensity and 97 kt airspeed (rotor span) . . . . . . . . . . . . . . . 54
4.22 Cruise flight at medium/high altitude with moderate turbulenceintensity and 97 kt airspeed (40 minutes simulation) . . . . . . . . . 55
4.23 Relative roll Euler angle and load airspeed for the first LCO ob-served in 40 minutes simulation . . . . . . . . . . . . . . . . . . . . 56
4.24 Relative roll Euler angle and load airspeed for the second LCOobserved in 40 minutes simulation . . . . . . . . . . . . . . . . . . . 56
4.25 Relative roll Euler angle and load airspeed for the third LCO ob-served in 40 minutes simulation . . . . . . . . . . . . . . . . . . . . 57
4.26 Cruise flight at low altitude with severe turbulence intensity and 97kt airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.27 Cruise flight at medium/high altitude with severe turbulence inten-sity and 97 kt airspeed . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.28 Complex maneuver at low altitude with light turbulence intensity . 59
ix
4.29 Complex maneuver at medium/high altitude and light turbulenceintensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.30 Complex maneuver at low altitude and moderate turbulence intensity 61
4.31 Load airspeed at low altitude for moderate turbulence intensity . . 61
4.32 Complex maneuver with moderate turbulence intensity at medi-um/high altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.33 Load airspeed at medium/high altitude for moderate turbulenceintensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.34 Complex maneuver at low altitude with severe turbulence intensity 64
4.35 Complex maneuver at medium/ high altitude with severe turbu-lence intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.1 Force and moments for the reduced lateral dynamic model . . . . . 70
A.2 SAS design: Root locus for yaw rate feedback . . . . . . . . . . . . 73
A.3 SAS design: Root locus for roll angle feedback . . . . . . . . . . . . 74
A.4 Helicopter block diagram with SAS . . . . . . . . . . . . . . . . . . 75
A.5 Root locus diagram for the helicopter and load system . . . . . . . 75
A.6 Helicopter and load system with relative roll cable angle feedback . 76
B.1 Airspeed scheduled controller implementation in Simulink . . . . . . 79
B.2 Reference relative cable angles for the airspeed scheduled controller 80
B.3 Lateral controller implementation in Simulink . . . . . . . . . . . . 81
B.4 Lateral controller timer in Simulink . . . . . . . . . . . . . . . . . . 82
x
List of Tables
2.1 Dryden wind turbulence model parameters . . . . . . . . . . . . . . 24
3.1 Scheduled controller parameters . . . . . . . . . . . . . . . . . . . . 34
4.1 Complex maneuver description . . . . . . . . . . . . . . . . . . . . . 44
xi
List of Symbols
A, B, C, D State matrix, input matrix, output matrix, feedthrough matrix
a, b Zero/pole coefficient in compensator
bS Wingspan
CS Cable damping coefficient
e Error state vector
FH, FL, FC State vector functions for helicopter, load and DI controller
FS Cable tension force
h Altitude for the Dryden wind turbulence model
H Forming filter transfer function
K Compensator gain
KS Cable stiffness
l Cable position
L Turbulence scale length
p, q, r Roll, pitch, and yaw rates
pL Load relative position vector
rCH Cargo hook position vector
TEH Earth to helicopter body transformation matrix
xii
TEL Earth to load body transformation matrix
THL Helicopter to load body transformation matrix
u, v, w Inertial velocity components
u Control command vector
V Aircraft airspeed norm
x, y, z Longitudinal, lateral, and vertical position
x State vector
XA, XB, XC , XP Lateral and longitudinal stick, collective, pedals
y Helicopter states output vector used for outer loop DI con-troller
β0, β1S, β1C Main rotor flapping angles
∆l Cable stretch
δ Vector of pilot control commands
λ0, λ1S, λ1C Dynamic inflow components
ν(t) DI controller pseudo-commands vector
νφ, νθ, νVD, νr DI controller pseudo-commands for roll, pitch, aircraft vertical
speed, and roll rate.
σ Turbulence intensity
φ, θ, ψ Roll, pitch, and yaw Euler angles˙( ) Time rate of change
( ) Unit vector
||.|| Vector norm
( )C Relative cable angles
( )cmd Commands
( )f Filtered commands
xiii
( )F Fuselage
( )H Helicopter
( )L Load
( )N,E,D North, east, down
( )rp Relative position
( )R Rotor
( )RMS Root mean square
( )u,v,w Airspeed components
xiv
Acknowledgments
Firstly, I would like to start by thanking Dr. Jacob Enciu and Dr. Joseph Horn for
the great opportunity to conduct this research in the Vertical Lift Research Cen-
ter of Excellence (VLRCOE) as well as for all the recommendations, observations,
guidance, and everything I learnt from them.
I would also like to acknowledge my colleagues from the University of Buenos Aires,
especially to Lic. Susana Gabbanelli, Dr. Leonardo Rey Vega, Dr. Juan Giribet,
and Dr. Daniel Vigo, who were all very patient and helpful with my time at Penn
State University.
This work was also possible thanks to the people working in the BecAr Programme
and the Argentinean Fulbright Foundation. I would like to thank them not only
for the financial support but also for all the help and orientation they gave me.
Of course, nobody has been more important to me in the pursuit of this project
than the members of my family. I would like to thank my mother, whose love
and guidance through my studies made it possible for me to be here, and to my
brother, who makes me realize about the important things.
Finally but not least, I would like to make a special mention to my wife, whose
love has become the main reason for improving myself in everything I do.
This research was partially funded by the Government under Agreement No.
W911W6-17-2-0003. The U.S. Government is authorized to reproduce and dis-
xv
tribute reprints for Government purposes notwithstanding any copyright notation
thereon. The views and conclusions contained in this document are those of the
authors and should not be interpreted as representing the official policies, either
expressed or implied, of the Aviation Development Directorate or the U.S Govern-
ment.
xvi
Dedication
To my family...
xvii
Chapter 1 |Introduction
1.1 Motivation
External load missions are among the most significant tasks that a helicopter can
perform. Carriage of external loads for either civil or military objectives is used
in rescue missions, transport of consumable products to flood zones, fire-fighting,
transport of military equipment to bases close to enemy territories, and other
situations (Figure 1.1). In all of these cases the helicopter flight speed during the
mission has a high impact on the mission safety and efficiency. Nonetheless, the
dynamics of the external load is usually not a part of the helicopter design process.
Therefore, external load carriage can lead to a degradation in the stability and
control of the coupled helicopter and external load system during forward flight.
The factors that generate these instabilities include the load pendulum dynamics,
the load aerodynamics, the rotorcraft dynamics, and the pilot’s compensation [1]-
[3].
1.2 Background
In the past, several techniques for passive and active stabilization of slung loads
were analyzed in various studies [4]-[8]. One approach for the avoidance of slung
1
Figure 1.1. External load mission examples1
load instabilities during flight involved the use of a flight director that provided
pilots with guidance cues for damping the load pendulum modes [9]-[12]. For var-
ious reasons, neither of these technical solutions culminated into an operational
system.
In recent years, studies have been conducted for the use of load state feedback
to the primary control system of the rotorcraft for increasing the load damping or
improving the handling qualities of the coupled systems. In [13], Krishnamurthi
and Horn demonstrated stability in hover and low speed flight by the use of a
primary flight control law based on relative cable angle measurements and lagged
relative cable angle feedback (LCAF). In [14], Ottander et al. simulated and flight
validated slung load station keeping above a moving vehicle using a combination of
input shaping and delayed swing feedback. In [15], Ivler et al. designed a primary
flight control based on rate and angle feedback tested on a UH-60 RASCAL. In the
study presented in [16], a control system based on the classical root locus technique
was used to design a load damping architecture for hover and low speed flight by
using LCAF. In [17], Patterson et al. developed and flight demonstrated a hybrid
solution consisting of an active cargo hook and a flight control load stabilization1Left: http://www.vortexxmag.com, right: http://fightersweep.com/3041/milestone-
monday-ch-47-chinook-54
2
mode in the primary control system using LCAF. Recently, the stabilization of
external loads in forward flight has been demonstrated by a collaborative research
between the Technion University, and the US Army. The stabilization methods
used in this research included passive stabilization using rear mounted fixed fins
[18] and active rotational stabilization using controlled anemometric cups [19].
Both methods were demonstrated in flight and produced an extended carriage
envelope of approximately 120 kt for box-like loads that are currently limited to
60 kt. Although these methods provide stability to the system, their operational
implementation implies a drag penalty, as well as some logistic problems and per-
formance degradations like preparing the loads for flight or reducing the amount
of cargo load due to the hardware weight used to achieve stability.
A research program for the development of stabilization methods of external loads
during high speed flight was more recently initiated by the US ARMY. The re-
search is performed collaboratively by researchers from Penn State University and
the Tel Aviv University and includes the development of active stabilization meth-
ods for external load carriage and their validation by real-time piloted simulations
and hardware in the loop wind tunnel tests.
1.3 Goal and Organization of the Thesis
In the work presented in this thesis, we extend the concepts shown in [13] and
[16] from hover to forward flight. The root locus technique is used to design an
airspeed scheduled controller to stabilize the slung load at airspeed ranging from
hover to high speed flight. The studies conducted here are focused on the design
and simulation for a coupled controlled system of a UH-60 Black Hawk utility
helicopter and an external load. The UH-60 helicopter uses a dynamic inversion
(DI) controller to provide stability and trajectory control to the clean aircraft. The
external load model is that of a 2500 lb empty CONEX cargo container fixed with
3
33 degree rear mounted fins that prevent rotation but do not guarantee stability
throughout the flight envelope. This particular load was chosen for this study due
to the availability of a high fidelity dynamic model that was validated in both
dynamic wind tunnel tests and flight tests. The controller achieves its objective by
providing additive control signal to the existing baseline controller in the UH-60
helicopter. Although the control method is applied to a load that includes fins,
the designed method is intended to be applied to any external load that does not
rotate about the cable axis.
The outline of the thesis is as follows: In Chapter 2 the basic dynamic character-
istics and model of the isolated external load, the sling cables, the helicopter, and
the coupled helicopter-external load system are described. Chapter 3 presents the
details of the controller design. In Chapter 4 the results of the simulations are
then presented showing the controller performance for cruise level flight for two
airspeeds in which the system without the relative cable angle feedback controller
is unstable or marginally stable. For this maneuver, the controller is also analyzed
when it is turned on during oscillatory responses. In addition, the controller per-
formance during a complex maneuver is examined. These simulations are then
followed by adding different levels of atmospheric turbulence to the helicopter and
the load models, and rechecking the controller performance. Finally, Chapter 5
presents the conclusions and future works.
4
Chapter 2 |Model Description
2.1 Introduction
A first principles physical model was developed to study the feasibility of using
relative cable angle feedback to stabilize the load and helicopter in high airspeed
(see Appendix A). The positive results from this analysis supported the next phase
of the research. More precise simulation results, need validated models. For the
aircraft, an UH-60 GenHel Black Hawk model was used, and for slung load, an
empty CONEX cargo container model which was validated in test flight and wind
tunnel tests [20] was used. These models, along with the sling cables, integrate to
create a coupled system. This system presents instabilities for airspeeds close to
100 kt and very low damping for the airspeed range between 15 kt and 35 kt.
2.2 External Load Model
The external load model used is that of an 8ft x 6ft x 6ft CONEX cargo container
with two rear mounted stabilization fins. The fins prevent load rotation but do not
guarantee load stability throughout the helicopter flight envelope. This model was
selected due to its extensive use in the studies mentioned previously. The two fins
are inclined in 33 degrees relative to the box side faces, trailing edge out (Figure
5
Figure 2.1. Cargo container with fins inclined in 33 degrees relative to the box (picturefrom [21])
2.1). The model also assumes a total weight of 2489 lb, which represents an empty
container with the four sling cables. For this study, the load center of gravity was
set to be 0.3 ft aft of the CONEX geometric center. This makes the load unstable
at an airspeed of 100 kt, selected as the target airspeed for load stabilization in
the current research.
The dynamic model described above has been thoroughly validated using dedicated
wind tunnel tests and flight tests. The aerodynamic model of the fin stabilized load
uses static aerodynamic forces and moment coefficients measured in a wind tunnel
for the complete load (fins included). These coefficients are augmented by a theo-
retical calculation to include the fins quasi-steady damping effect (due to the arm
between the fins and the load center of gravity). This approach was validated by
dedicated dynamic wind tunnel tests (see [22] for details).
The load’s equations of motion are implemented as a state space model with the
6
state vector being comprised of the load’s inertial velocities, attitude angles, an-
gular rates, and center of gravity position:
xL = {uL, vL, wL, ψL, θL, φL, pL, qL, rL, xN , yE, zD} (2.1)
The angular rates and inertial velocities are given in a load-fixed coordinate system
(L) located at the center of mass, with the x axis pointing forward, y axis pointing
right and z axis pointing down (Figure 2.2). The position vector is given in an
Earth fixed NED inertial system (E), with the x axis pointing to the north, y axis
pointing to the east, and z axis pointing down. The load attitude angles defining
the transformation from (E) to (L) follow the conventional Euler angle order yaw
(ψL), then pitch (θL), and finally roll (φL).
Figure 2.2. Helicopter (H), load (L), and Earth (E) fixed coordinate system
7
2.2.1 Isolated Load Dynamics
It is convenient to present a more detailed analysis of the isolated load dynamics
to explain in Chapter 3 the results obtained in the simulations.
The dynamic analysis of the system was performed using the continuation and
bifurcation tools of Dynamical Systems Theory (DST). For this analysis, wind
tunnel speed was used as the continuation parameter. The use of DST provides
a comprehensive approach for the description of the slung load dynamics so that
load stability can be efficiently evaluated for the entire relevant airspeed range
of interest. The dynamic characteristics of the system are determined through
the study of equilibria, solution trajectories, solutions periodicity and transition
to chaos [23]-[25]. This approach had been applied before for the analysis of the
fins stabilized CONEX and showed good agreement with wind tunnel tests results
[20]. In the current study, the continuation and bifurcation analysis was performed
using the Dynamical Systems Toolbox [26], which is an integration of the contin-
Figure 2.3. Isolated load equilibria points
8
uation software package AUTO [27] into MATLAB.
Figure 2.3 shows a bifurcation curve for the load roll angle, φL, that was ob-
tained for the model of the isolated external load in a wind tunnel. In this figure,
the blue curve segments indicate stable equilibria while the red dashed segments
present unstable equilibria. The purple pentagrams denote pairs of Hopf bifurca-
tion points, in between which limit cycle oscillations (LCO) exist. The bifurcation
curve demonstrates the nonlinear nature of the system, as multiple equilibria exist
for a single airspeed in large parts of the airspeed domain. The described equilibria
points can be observed in two types of solution branches. The first one is a sym-
metric solution branch (with φL = 0◦) and the second one includes two asymmetric
branches (φL 6= 0◦) between 68 ft/s (40 kt) and 170 ft/s (100 kt).
Regarding the stability, the symmetric branch stability can be found for low air-
speed (except for the range between 31 ft/s and 54 ft/s) and high airspeed. For
the case of the asymmetric branches, the stability varies with the load’s airspeed.
At the design point airspeed of 168.8 ft/s (100 kt), three unstable equilibria ex-
Figure 2.4. Limit cycle oscillations for 100 kt (168.8 ft/s)
9
ist: a symmetric equilibrium with φL = 0◦ and two asymmetric equilibria with
φL = ±10.9◦. Solution trajectories for these points are characterized by sustained
LCO. Figure 2.4 shows the trajectory time histories (load Euler angles) of the sim-
ulated system at 100 kt. An initial excitation was applied to the load 2 seconds
into the simulation through a doublet in the lateral cargo hook position, otherwise
kept fixed. The load yaw, pitch and roll angles time plots show two distinct LCO
patterns: a symmetric LCO about the center solution branch (dash-dotted red
lines) and an asymmetric LCO about the asymmetric branches (blue solid line).
The intensity of the excitation doublet determines which of the two trajectories is
taken. Note that in the actual physical system, the trajectories may shift between
the two solutions due to external disturbances such as atmospheric turbulence.
The equilibria points around 170 ft/s (101 kt) present an interesting behavior that
can be seen in Figure 2.5. Here, it can be observed that a hysteresis effect exists,
which can be explained as follow: assuming that the load is at an initial airspeed
of 168.8 ft/s and a φL = 10.9◦, instabilities will be present in the form of LCO.
If a perturbation in the load increases its velocity, the load roll angle will follow
Figure 2.5. Equilibria points around 170 ft/s (101 kt) for the isolated load
10
the dashed arrow pointing to the right. If the velocity is high enough the load
will become stable and after certain velocity value the load roll angle will “jump”
from the asymmetric branch to the symmetric branch where φL = 0◦. If another
perturbation in the load decreases its airspeed, the load will remain stable and
with φL = 0◦ until its velocity reach a value close to 170 ft/s. From that moment
on, the load roll angle will suddenly change to φL = ±10◦ and the load will become
unstable, presenting LCO in its Euler angles.
From this, it can be assumed that the load perturbations induce cyclic airspeed
changes. The multiple solutions and the hysteresis characteristic around 170 ft/s
(101kt) complicate the controller design for the entire airspeed range
2.3 Sling Cables Model
The external load is carried by the helicopter using four identical sling cables of 18.7
ft length. The cables connect the four upper corners of the load to the helicopter
cargo hook. Each one of these cables is modeled as a combination of a linear
spring and a linear damper, and assumed to carry only tension forces (excluding
compression forces and bending or torsion moments). The tension force in the
ith cable is calculated from the cable stretch, ∆li, and its rate of change, and is
directed along the cable unit length vector, li:
FS,i = max(KS,i∆li + CS,i
(∆li
), 0)· li (2.2)
where:
li = li
||li||(2.3)
The cable vectors are calculated from the positions of the helicopter cargo hook
and the four attachment points on the load upper surface. These, in turn, depend
11
on the helicopter and load position, attitude, and geometric properties. Cable
directions are defined positive for vectors originating from the cargo hook and
pointing into the load attachment point. Stiffness and damping values of 9645
lb/ft and 30.3 lb.sec/ft were used for KS,i and CS,i, respectively. These values
were obtained previously by the US ARMY using a dynamic shaker test.
2.4 Helicopter Model
As mentioned before, a model of a UH-60 Black Hawk helicopter was used for
this research. The helicopter nonlinear model is largely based on the GENHEL
engineering simulation of the UH-60 helicopter [28]. The model utilizes a simplified
version of the rotor model compared to the one used in the original simulation.
Blade lag dynamics are neglected, a linear lift aerodynamic model is used for the
blade sections and approximate closed form expressions are utilized for the main
rotor total hub aerodynamic loads. The model follows [29] but uses a hinge offset
representation rather than a center spring model. The dynamic inflow model used
is that of [31]. Like for the load, the helicopter model is formulated using state
variables. The 21 element state vector, xH , contains 12 rigid body states, xF , and
9 main rotor, xR, as follow:
xF = {uH , vH , wH , pH , qH , rH , ψH , θH , φH , xN , yE, zD} (2.4)
xR ={β0, β1S, β1C , β0, β1S, β1C , λ0, λ1S, λ1C
}(2.5)
xH = {xF ,xR} (2.6)
Similar to the external load, the inertial velocities and the angular rates of the
helicopter are given in a fuselage fixed coordinate system (H) located at the heli-
copter center of mass. The helicopter position is given in the earth fixed coordinate
system (E). The transformation from (E) to (H) follows the conventional Euler an-
12
gle order presented earlier. The state vector of the main rotor includes the first
harmonic flapping angles of the tip path plane and their rates of change, and the
main rotor dynamic inflow components. The tail rotor is modeled using simplified
closed form expressions for the force and moment coefficients.
The helicopter model includes a dynamic inversion (DI) controller acting as an
automatic flight control systems (AFCS). The controller includes an outer loop
trajectory following model and an inner loop pitch and roll attitude, yaw rate,
and vertical speed controller (Figure 2.6). The outer loop is designed to follow a
desired reference trajectory, u, defined by the combination of the vector of inertial
velocity components (uN , vE, wD) in the earth fixed coordinate system, and the
helicopter heading (ψ):
u = {uN , vE, wD, ψ} (2.7)
The inner loop then uses a dynamic inversion of a piecewise reduced order linear
model of the helicopter to produce the vector of control commands, δ [13]. This
vector includes the set of cyclic pitch, collective pitch and tail rotor pitch commands
required to follow the desired trajectory:
δ = {XA, XB, XC , Xp} (2.8)
Figure 2.6. Helicopter flight control system model
13
Figure 2.7. Helicopter inner loop dynamic inversion
As the DI controller provides the desired stability and control characteristics for the
baseline helicopter (without the external load), the stability augmentation system
(SAS) of the UH-60 Black Hawk was not included in the model.
A more detailed description of the DI controller can be observed in Figure 2.7. This
controller was designed by linearizing the helicopter model about a trim point for
different airspeeds (scheduled controller) in order to stabilize the nonlinear system.
In it, the command vector ycmd is defined as:
ycmd = [φcmd θcmd VDcmdrcmd]T (2.9)
where φ, θ, VD, and r are the roll attitude, pitch attitude, vertical speed and yaw
rate, respectively. The vector ν(t) is the pseudo-command vector, which will be
part of the controller design that will be explained in Chapter 3, and is defined as:
ν = [νφ νθ νVDνr]T (2.10)
This vector is calculated by using the following equations:
eφ = φf − φ (2.11)
eθ = θf − θ (2.12)
er = rf − r (2.13)
14
νφ = φf +KP eφ +KDeφ +KI
∫eφdt (2.14)
νθ = θf +KP eθ +KDeθ +KI
∫eθdt (2.15)
νVD= VDf
+KP eVD+KI
∫eVD
dt (2.16)
νr = rf +KP er +KI
∫erdt (2.17)
where the subscript f denotes filtered values by using the command filters. An
analysis of the stability of the error dynamics can be found in [30].
2.5 Coupled Helicopter-External Load System
For the studied configuration, it was assumed that the load was connected to the
helicopter cargo hook by a swivel, which enabled free yaw rotations of the load
with a negligible resisting friction moment. The new coupled system created in this
way was studied by using its state-space representation, obtained by combining the
load and the helicopter model (including the flight control system):
xH
xL
xC
=
FH(xH,xL,xC)
FL(xH,xL)
FC(xH,xC,u)
In the equations above, the functions FH, FL, and FC are the corresponding
state vector functions that describe the helicopter and load dynamics and the DI
controller, respectively; the vectors xH, xL, and xC are the helicopter, load, and
DI controller state vector, respectively.
2.5.1 Load Stability in the Coupled Model
To analyze the stability, the linear models of the isolated load, the helicopter, and
the coupled system where the load is connected to the helicopter, were obtained
15
via the Simulink Linearization tool. Via an eigenvalue analysis of these models,
it could be seen that when the load was connected to the helicopter, the unstable
load modes were moved towards the left half plane and, in some cases, the system
become stable. This can be verified by observing the poles of the previously men-
tioned models as they are presented in Figure 2.8 for the low airspeed of 25 kt.
In this figure, the two poles located in the right half plane (red squares markers)
indicate that the isolated load model was unstable for this airspeed (condition that
can be verified in Figure 2.3). However, when the isolated load was connected to
the helicopter, the load poles were moved to the left half plane (blue crosses near
the imaginary axis), making the system stable.
On the other hand, Figure 2.9 presents an example for a higher airspeed of 97 kt.
As expected, the red square markers were located in the right half plane, corre-
sponding to the instability of the isolated load for this airspeed. These unstable
modes moved from 0.043± j1.166 to 0.002± j1.442 (closer to the imaginary axes)
when the load was connected to the helicopter. However, even with the new loca-
Figure 2.8. Effects observed in the model pole diagrams for the coupled system at 25kt
16
Figure 2.9. Effects observed in the model pole diagrams for the coupled system at 97kt
tion of the poles, the load was still unstable, as the final location of the poles was
still in the right half plane.
This analysis was done for the airspeed range between hover and 130kt. The results
for the linear system approximation showed that the coupled system was unstable
only for airspeeds between 96kt and 105kt.
2.5.2 Relative Cable Angles
The relative cable angle is based on the relative angle of a hypothetical line from the
cargo hook to the load center of mass. These angles are defined by the orientation
of the load relative to the rotorcraft as described below.
Figure 2.2 presented the helicopter, load, and Earth coordinate systems. The load
relative cable angles were calculated by obtaining the distance from the cargo hook
to the load center of mass (norm of the vector pL) using the Earth to helicopter
17
coordinate transform matrix TEH as can be seen in equation (2.18):
pL = TEH (rL − rH)− rCH (2.18)
Where TEH is calculated following the order yaw (ψ), pitch (θ), and roll (φ), or
the sequence 3-2-1:
TEH = Tφ Tθ Tψ =
=
CθHCψH CθHSψH −SθH
SφHSθHCψH − CφHSψH CφHCψH + SφHSθHSψH SφHCθH
SφHSψH + CφHSθHCψH SφHCψH + CφHSθHSψH CφHCθH
(2.19)
The values of the relative cable angles were obtained assuming the transformation
from (H) to (L), THL, follows the order pitch (θC), roll (φC), and yaw (ψC) (Figure
2.10). Using this order, the transformation matrix is derived. It can also be
calculated from the transformation matrices from (E) to (H) and (L), respectively:
THL = TEL (TEH)T (2.20)
The relative cable angles φC , θC are calculated using the cable components of pLand ψC is calculated by comparing the entries in THL:
φC = asin(− yrp||pL||
)(2.21)
θC = atan(xrpzrp
)(2.22)
ψC = atan2(NumDen
)(2.23)
18
Where pL = [xrp, yrp, zrp]T and:
Num = − cos(θH) sin(θL) sin(φH) + cos(θL) {cos(ψH)[cos(ψL) sin(θH) sin(φH) +
+ cos(φH) sin(ψL)] + sin(ψH)[− cos(φH) cos(ψL) +
+ sin(θH) sin(φH) sin(ψL)]} (2.24)
Den = cos(θH) cos(θL) sin(φH) sin(φL) + [cos(ψH) sin(θH) sin(φH) −
− cos(φH) sin(ψH)][cos(ψL) sin(θL) sin(φL)− cos(φL) sin(ψL)] +
+ [cos(φH) cos(ψH) + sin(θH) sin(φH) sin(ψH)][cos(φL) cos(ψL) +
+ sin(θL) sin(φL) sin(ψL)] (2.25)
(a)(b)
(c)
Figure 2.10. Relative cable angles 2-1-3 sequence description
19
2.6 Dryden Wind Turbulence Model
In order to further verify the performance of the designed controller, simulations
with various levels of atmospheric turbulence were performed. For this objective,
the Dryden Wind Turbulence model for continuous gusts was used. From this
model, wind turbulence was injected to the load and helicopter systems separately.
The wind turbulence was created by using white noise in forming filters, which were
derived from the spectral square roots of the spectrum equations presented in [32].
The filters used for this research are related to [33] and their transfer function in
the Laplace transform domain are:
Hu(s) = σu
√2LuπV
.1
1 + Lu
Vs
(2.26)
Hv(s) = σv
√0.8V.
(π
4bS
)1/6
(2Lw)1/3(1 +
(4bS
πV
)s) (2.27)
Hw(s) = σw
√2LwπV
.1 + 2
√3Lw
Vs(
1 + 2Lw
Vs)2 (2.28)
Where bS represents the aircraft wingspan, Lu, Lv, Lw represent the turbulence
scale lengths, and σu, σv, σw represent the turbulence intensity components in the
body frame.
2.6.1 Low-Altitude Model
The Dryden Wind Turbulence model is an altitude dependent model. For an
altitude below 1000 ft, [33], the model assumes the following relationship between
the altitude h and the turbulence scale lengths:
2Lw = h (2.29)
Lu = 2Lv = h
(0.177 + 0.000823h)1.2 (2.30)
20
From where, for an altitude of h = 1000 ft, the turbulence scale lengths are:
Lu = 1000 ft (2.31)
Lv = 500 ft (2.32)
Lw = 500 ft (2.33)
On the other hand, the relationship between the turbulence intensity and the
altitude is:
σw = 0.1W20 (2.34)σuσw
= σvσw
= 1(0.177 + 0.000823h)0.4 (2.35)
Where W20 is the wind speed at 20 feet. This speed depends of the level of the
turbulence, for light turbulence it is 15 knots (25.3 ft/s), for moderate turbulence
it is 30 knots (50.6 ft/s), and for severe turbulence it is 45 knots (76 ft/s). For
these cases, the corresponding values of σu, σv, and σw are:
Light turbulence level → σu = σv = σw = 2.5 ft/s (2.36)
Moderate turbulence level → σu = σv = σw = 5 ft/s (2.37)
Severe turbulence level → σu = σv = σw = 7.6 ft/s (2.38)
Finally, in the current effort, the adopted wingspan (bS) for the helicopter and the
load was the load span width (6.11 ft). A sensitivity analysis was run to validate
the use of this wingspan value aginst the rotor diameter, which is discussed in
Section 4.5.
21
2.6.2 Medium/High Altitudes Model
For this altitude range, due to the objective of using wind turbulence model for
testing the performance of the designed controller, the altitude for the worst case
scenario was selected. Figure 2.11 presents the altitude (in thousands of feet) as a
function of the root mean square value of the turbulence amplitude (in ft/s). It can
be observed that the highest turbulence intensity for the light and the moderate
intensities cases is given for an altitude around 4000 ft, which corresponds to the
medium/high altitude section of the model (used for altitudes above 2000 ft). For
this altitude, the turbulence scale lengths are:
Lu = 2Lv = 2Lw = 1750 ft (2.39)
Figure 2.11. Dryden medium/high altitude turbulence intensities and probability ofexceedance [32]
22
From where Lu = 1750 ft and Lv = Lw = 875 ft. The turbulence intensity is
defined as:
σ = σu = σv = σw (2.40)
Which means that:
σRMS =√
3σ (2.41)
As for the low altitude case, three different scenarios were selected according the
turbulence intensity level: light, moderate and severe. From Figure 2.11, equation
(2.41), and with an altitude of 4000 ft, the turbulence intensity values obtained
for the three scenarios are:
Light turbulence level → σ = 4 ft/s (2.42)
Moderate turbulence level → σ = 6 ft/s (2.43)
Severe turbulence level → σ = 13 ft/s (2.44)
As in the case of low-altitude, the adopted wingspan (bS) for the helicopter and
the load was the load span (see Section 4.5 for a sensitivity analysis).
Table 2.1 summarize the parameters for the low and medium/high altitude for the
light, moderate, and severe turbulence levels.
23
Parameters Low Altitude (1000 ft) Med/High Altitude (4000 ft)Light Moderate Severe Light Moderate Severe
σu(ft/s) 2.5 5 7.6 4 6 13σv(ft/s) 2.5 5 7.6 4 6 13σw(ft/s) 2.5 5 7.6 4 6 13Lu(ft) 1000 1000 1000 1750 1750 1750Lv(ft) 500 500 500 875 875 875Lw(ft) 500 500 500 875 875 875bS (ft) 6.11
Table 2.1. Dryden wind turbulence model parameters
24
Chapter 3 |Controller Design
3.1 Introduction
In previous studies [13] and [15], relative cable angle feedback (RCAF) has been
used effectively to stabilize the helicopter-load coupled system in hover and low
airspeeds. In the current effort, it was intended to expand this range of operation
by stabilizing the coupled system from hover to 130 kt. By taking into account that
in high airspeeds the aerodynamic forces are more important than in low airspeed,
trim points of the nonlinear system were found for different airspeeds. Then, high
order linearized models around these trim points were obtained. Finally, these
models allowed the design of an airspeed scheduled controller by using the root
locus technique ([34], [35]).
3.2 Controller Design Process Description
As mentioned in previous chapters, the proposed controller is a scheduled controller
for the airspeed range between hover and 130 kt. The concept of this controller
is simple: for different airspeeds the coupled system is linearized about a trim
point and then a controller is design to stabilize the linear system. With enough
trim points, the nonlinear system will be stabilized by using the gain scheduling,
25
basically interpolating the controller parameters when the system is between trim
points.
The design process started by creating a trim script in Matlab where initially the
isolated load was trimmed. For this objective, the initial guess used corresponded
to the load roll angle described in Figure 2.3, where the upper asymmetric branch
was used for airspeeds between 40 kt and 100 kt. Then, the isolated helicopter was
trimmed using the sling cable forces calculated from the trimmed load as external
forces. Finally, the coupled system was trimmed by using the previous two stages
as the initial guess for its trim point. Something to remark is that, even with this
procedure, for certain airspeeds the trim algorithm was not able to converge. The
solution to this problem was obtained by trimming the system at a close airspeed
velocity and then use this trim point as the initial guess for the desired airspeed.
Using this method, the coupled system was verified to be trimmed for airspeeds
from hover to 130 kt by using steps of 1 kt. Figure 3.1 presents an example of the
trimmed relative cable angles for an airspeed of 100 kt (see Appendix C for the
trim algorithm).
The second step involved the linearization of the nonlinear system around the trim
Figure 3.1. Relative cable angles trimmed for the couple system at 100 kt
26
point by using the Simulink Linearization tool. The nonlinear system was defined
by using the relative cable angles φC (eq. (2.21)) and θC (eq. (2.22)) as the system
outputs and the pseudo-commands of the DI controller, νφ and νθ (eq. (2.10)), as
the input signals (see Figure 3.2). In this way, the nonlinear system is expressed
as a state space model by using equations (3.1) and (3.2):
x = F (x,u) (3.1)
y = G(x,u) (3.2)
Where u = [νφ, νθ] is the control vector, x is the state vectors that include the
load, helicopter, DI controller, and relative cable angles states, and y = [φC , θC ]
is the output of the system.
By obtaining the linear approximation using Taylor series, the small variations
∆x, ∆u, and ∆y of equations (3.1) and (3.2) can be obtained as:
∆x = ∂F
∂x
∣∣∣∣∣x0,u0
∆x + ∂F
∂u
∣∣∣∣∣x0,u0
∆u (3.3)
∆y = ∂G
∂x
∣∣∣∣∣x0,u0
∆x + ∂G
∂u
∣∣∣∣∣x0,u0
∆u (3.4)
Then, the linear time invariant (LTI) system matrices are defined as:
A = ∂F
∂x
∣∣∣∣∣x0,u0
(3.5)
B = ∂F
∂x
∣∣∣∣∣x0,u0
(3.6)
C = ∂G
∂u
∣∣∣∣∣x0,u0
(3.7)
D = ∂G
∂u
∣∣∣∣∣x0,u0
(3.8)
27
Using equations (3.5)-(3.8) in (3.3) and (3.4), and dropping the ∆ symbol, the
linear system can be expressed with equations (3.9) and (3.10) as:
x = Ax +Bu (3.9)
y = Cx +Bu (3.10)
Following the linearization, the system was reduced using a minimal realization
algorithm, where the poles and zeros separated by a distance less than 10−6 were
removed. In this way, the zero-pole diagram is clearer by removing the zero and
poles that cancel each other.
The final step involved using the linearized model previously obtained for the design
of the two lead/lag compensators (for the relative pitch and roll cable angles) with
the objective of stabilizing the coupled system and maximize the damping ratio of
the load pendulum modes. The compensators transfer function was integrated by
the parameters K, a, and b, were determined by the root locus technique:
T (s) = Kas+ 1bs+ 1 (3.11)
Figure 3.2. New proposed controller design
28
Here the values of a and b define the controller as a lead (a>b) or lag (a<b)
compensator, and K is its gain. This results, in 6 parameters (3 for the lateral
controller and 3 for the longitudinal controller) to be defined for each airspeed in
the scheduled controller. Different controller designs were tested and, in most of
the cases, the lag controllers presented better performance, matching the results
obtained in [13] and [16] for hover and low airspeed.
Through this process the original helicopter DI controller was modified to include
the compensation for external load carriage by simply adding the relative cable
angle feedback block that can be observed in Figure 3.2.
3.2.1 Design Example for Low Airspeed
For this case the trim point was obtained by a load roll angle of φL = 0◦ and an
airspeed of 25 kt. This particular airspeed was selected because it is an example
in which the system is marginally stable and presents lightly damped oscillations
(LDO) in its outputs.
Figure 3.3 presents the root locus for the lateral controller obtained with the Matlab
tool controlSystemDesigner. This figure also shows that the minimum damping
Figure 3.3. Lateral controller root locus design for 25 kt
29
ratio of the lateral pendulum modes is ζ = 4.1 10−3 for ωn = 1.07 rad/sec, this
low damping ration is the indication of LDO. The longitudinal pendulum modes
can also be found in the middle of the green circles. These poles were close to
zeros, making them impossible to be significantly moved by using a controller.
This characteristic is due to the decoupling between the lateral and longitudinal
dynamics at low airspeed. The decoupling eased the design process, as the lateral
and longitudinal controllers could be design independently. As will be shown in
the following section, when the airspeed increases, the longitudinal and lateral
pendulum modes become coupled, making the design task more complex.
Figure 3.3 also shows the compensator’s zero and pole with a black circle and a
black cross, respectively. For this configuration, the controller’s parameters are:
Klat25kt= 0.8 alat25kt
= 1.111 blat25kt= 1 (3.12)
By using the designed controller, the minimum damping ratio was increased to ζ =
0.028 while keeping the natural frequency at ωn = 1.07 rad/sec. The trade-off for
this increase in the damping ratio is the reduction in the damping ratio of the other
lateral pendulum mode, however, the results obtained with this design presented
significant improvements that will be shown in Chapter 4. Figure 3.4, shows the
root locus diagram for the longitudinal controller (without the lateral controller
applied). As with the lateral controller diagram, the lateral load pendulum modes
(green circles) were not able to be moved with the values of along25kt, blong25kt
, and
Klong25ktof the longitudinal controller due to the decoupled dynamics characteristic
previously mentioned. However, these parameters allowed the minimum damping
ratio of the longitudinal pendulum modes ζ = 0.0884 (ωn = 1.36 rad/sec) to be
increased to ζ = 0.186 (ωn = 1.32 rad/sec). As in the previous case, a trade-off
with the phugoid helicopter pole damping ratio had to be made to achieve this.
This was indicated in Figure 3.4 by a black arrow. For this zero-pole constellation
30
Figure 3.4. Longitudinal controller root locus design for 25 kt
the values of Klong25kt, along25kt
, and blong25ktused are:
Klong25kt= −6.84 along25kt
= 0.97 blong25kt= 10
3.2.2 Design Example for High Airspeed
For the high airspeed example, a velocity in the unstable range between 96kt and
105kt was selected. In this case the trim point was chosen for an airspeed of 97 kt
and a positive load roll angle (φL > 0◦), corresponding to the equilibrium point in
the upper asymmetric branch, see Figure 2.3.
Figure 3.5 presents the root locus diagram for the lateral controller. In this figure
the position of the compensator’s pole and zero can be observed, which correspond
to the design parameters in equation (3.13):
Klat97kt= −1.2 alat97kt
= 0.33 blat97kt= 1.3 (3.13)
Unlike the previous case, for this airspeed the lateral and longitudinal dynamics
were coupled. This can be concluded because a change in any of the values of
31
Figure 3.5. Lateral controller root locus design for 97 kt
a, b, or K, modified all the six pendulum load modes. Among these poles there
were two poles on the right half plane highlighted in Figure 3.5 as unstable modes.
With the values in (3.13) these poles were moved to the left half plane, increas-
ing the damping ratio from ζ = −1.22 10−3 (ωn = 1.44 rad/sec) to ζ = 0.0138
(ωn = 1.45 rad/sec) and, in this way, stabilizing the coupled system.
On the other hand, Figure 3.6 presents the longitudinal dynamics root locus di-
agram. As in the previous case, a variation in the values of a, b, and K were
able to modified the position of all the load modes in the complex plane. In this
case, the unstable load pendulum modes with a damping ratio of ζ = −1.2 10−3
and a natural frequency of ωn = 1.44 rad/sec can also be observed in the dia-
gram. By following the design premise of maximizing the damping ratio of the
load pendulum modes, the damping ratio of the unstable poles were increased to
ζ = 8.4 10−2 (and ωn = 1.65 rad/sec) by using the parameters in equation (3.14)
for the longitudinal controller:
Klong97kt= 0.78 along97kt
= 1.5 blong97kt= 0.5 (3.14)
32
Figure 3.6. Longitudinal controller root locus design for 97 kt
3.3 Controller Design Summarized
The previous section described the design method for two airspeeds. By repeating
this method for different airspeeds between hover and 130kt the airspeed scheduled
controller was defined. Table 3.1 presents all the parameters for the scheduled
controller, were the parameters that stabilize the coupled system for the airspeed
of 100kt were not possible to find due to the nonlinearities and the hysteresis effect
that occurred at that airspeed.
An implementation print of the airspeed scheduled controller in Simulink can be
found in Appendix B.
33
Velocity Lateral Controller Longitudinal ControllerKlat alat blat Klong along blong
1 -1 0.1 2.5 -6 1 105 -10.1 0.58 10 -1 0.5 1010 -5 0.1 6.7 -0.615 0.97 215 10 0.46 20 -1 0.1 220 1.1 0.1 1 -3 0.97 1025 0.8 1.111 1 -6.84 0.97 1030 1 0.25 0.59 -4 0.83 1035 5 0.1 5 -4 0.67 1040 -5 0.33 3.3 -1 0.1 1.345 -1.5 0.2 8.9 -4.74 0.56 1050 -0.67 0.067 1.3 -0.14 0.1 155 -0.4 0.76 0.47 -2.81 0.4 1060 -2.7 0.33 2.5 -0.17 0.1 0.865 -2.1 0.17 2.7 -1.4 0.5 2.570 -4 0.33 5 -2 0.56 575 -5 0.5 5 -2 0.5 580 -2 0.57 3.3 -1 0.5 3.385 -2.8 0.25 3.3 -0.45 1 1.190 -5 0.5 17 -2 1 595 -0.92 1 1.2 0.72 0.88 0.3397 -1.2 0.33 1.3 0.78 1.5 0.5100 - - - - - -105 10 0.29 2 -1.14 0.74 1.4110 -5.22 0.1 5 -2 1.3 2.5115 -5.6 0.16 5 -2.5 0.67 1.4120 -10.1 0.25 10 -0.5 0.77 1125 -10 0.32 10 -1 0.77 1130 -4.68 0.25 5 -1.2 0.77 1
Table 3.1. Scheduled controller parameters
34
Chapter 4 |Simulation Results
4.1 Introduction
This chapter shows the simulation results obtained with the designed controller.
Different scenarios were designed to verify the effectiveness of the controller. The
first set of simulations is for a trimmed cruise flight, which can be considered a
baseline test. These tests were executed for the low and high airspeed presented
in Chapter 3. The second set of simulations is for a more demanding scenario in
which a complex maneuver combining four segments was used. The third set of
tests was designed to verify the controller performance when it was turned on when
instabilities were developed. Finally, the previously mentioned scenarios, trimmed
cruise flight and complex maneuver, were modified to include wind turbulence with
light, moderate, and severe turbulence levels.
4.2 Trimmed Cruise Flight
For the two airspeeds used as examples for the controller design procedure in Chap-
ter 3, 25 kt for low airspeed and 97 for high airspeed, a simulation for a trimmed
cruise flight was executed.
Once the coupled system was trimmed at the corresponding airspeed, the simula-
35
tion started and a perturbation at t = 3 sec was applied. Such a perturbation was
a combination of a roll doublet and an increase in the load velocity (a “push”).
As previously mentioned in Chapter 2, for an airspeed in the range of 96kt to
105kt and depending on the level of the perturbation, instabilities can be pre-
sented as severe symmetric LCO or milder asymmetric LCO. Simulations to verify
the performance of the controller for these two cases were executed.
4.2.1 Simulation at 25 kt
For this airspeed the simulation showed the presence of LDO, a lightly damped
oscillatory response to the push applied 3 seconds after the simulation started.
The undesired characteristic of these oscillations are related to two factors: its long
duration, which could easily be more than 300 seconds, and its large initial value
(which actually depends on the excitation level) that induces lateral accelerations
in the cockpit, which for a long period of time reduces the pilots ride qualities.
Figure 4.1 presents the relative cable angles for 25 kt constant airspeed cruise
Figure 4.1. Relative cable angles simulation result for 25 kt
36
maneuver where LDO can be observed. The improvements achieved with the
controller are noticeable. The yaw angle time history when the controller is not
active (ψC , red curve in Figure 4.1) presents a time to half of 115.5 seconds (where
the damping ratio obtained from the simulation of the nonlinear model was ζ =
5.4 10−3, which is close to the one obtained from the linear model). With the
controller on, the time to half amplitude is reduced to 14.2 seconds (ζ = 4.7 10−2,
increased by a factor of 10), which is approximately 12.5% of the previous value.
As mentioned in the previous chapter, this improvement has an impact in the
helicopter dynamics. Figures 4.2 and 4.3 show that the helicopter Euler angles
and the helicopter control commands (simulation time shown was reduced to 60
seconds). In both of these figures, it can be observed that when the controller
was on, the responses presented higher levels of oscillations at the beginning of the
simulation as compared to the case in which the controller was off. This difference
can mostly be observed in the helicopter roll Euler angle (Figure 4.2) and in the
collective and lateral commands (Figure 4.3). However, the oscillations in the
Figure 4.2. Helicopter Euler angles simulation result for 25 kt
37
Figure 4.3. Helicopter controls commands simulation result for 25 kt
helicopter Euler angles were damped in less than 60 seconds, and for the case of
the helicopter commands, the small differences were far from making the helicopter
control commands reach their mechanical limits (which could bring saturation
problems) and they were also quickly damped. It is important to note that the
LDO were mostly impacting the load Euler angles, which present similar results
to those presented for the relative cable angles (Figure 4.1). Nevertheless, these
oscillations (as well as the previously shown) were quickly damped by the stability
improvement granted by the designed controller.
4.2.2 Simulation at 97 kt
For this airspeed, the instabilities were presented as LCO rather than LDO. Similar
to the case of 25 kt, the coupled model was perturbed with a lateral stick dou-
blet and load push applied 3 seconds after the simulation started. As previously
mentioned, the intensity of the perturbation will produce severe symmetric LCO
or milder asymmetric LCO responses. The characteristics of the LCO presented
38
Figure 4.4. Relative cable angles simulation result for 97 kt
at this airspeed were similar to that previously explained for the isolated load at
an airspeed of 100 kt. The instabilities at this airspeed make this case a more
demanding scenario than the 25 kt airspeed. However, the controller allowed the
system to quickly achieve stability no matter the type of LCO.
Figure 4.4 presents the relative cable angle results for an airspeed of 97 kt. In this
figure it can be observed that, for this case, the instabilities were severe symmetric
LCO. By knowing that these results were similar to the load Euler angles, it can
be seen that the severe symmetric oscillation (at least 40 degrees peak-to-peak for
φL) could lead to the load striking the helicopter’s tail boom and, in this way,
endanger the crew and the mission. However, when the controller was used, these
oscillations were quickly damped. Figure 4.5 presents the helicopter controls for
this simulation, in this figure it can be observed that the controller’s higher im-
pact was in the initial 10 seconds. In that interval, the helicopter’s longitudinal,
collective, and pedals controls were slightly increased and then all oscillations were
damped. On the other hand, when the controller was turned off the helicopter com-
39
Figure 4.5. Helicopter controls simulation result for 97 kt
Figure 4.6. Helicopter Euler angles simulation result for 97 kt
mands present oscillations that impacted in the helicopter Euler angles. Figure 4.6
presents the helicopter Euler angles, for the same simulation, showing the level of
40
Figure 4.7. Relative cable angles simulation result for 97 kt, asymmetric LCO
oscillations to which the helicopter and the crew would be subjected if the con-
troller was off. These oscillations lead to significant lateral acceleration levels in
the cockpit that would likely degrade flying qualities and increase the pilot’s work-
load.
The relative cable angles can be observed in Figure 4.7, where asymmetric LCO
were present. Comparing this figure with Figure 4.4, it is easy to observe that the
sustained oscillations in the roll angle present a lower peak-to-peak amplitude of
7.49◦ (compared to the 29.45◦ from the symmetric LCO) and a higher frequency of
1.36 rad/sec (compared to the 0.83 rad/sec for the symmetric LCO). In the case of
the helicopter (Figure 4.8) and load (relative cable angles presents similar results,
Figure 4.7) roll angle, when the controller is off, the oscillation amplitudes were
also less severe and their impact on the flight qualities and safety of the crew/mis-
sion would likely be smaller than in the case of the severe symmetric LCO. It can
also be observed in figures 4.7 and 4.8 that the damped oscillations (controller on)
that started at t = 10 seconds presented a higher initial amplitude. This can be
41
Figure 4.8. Helicopter Euler angles simulation result for 97 kt, asymmetric LCO
Figure 4.9. Helicopter controls simulation result for 97 kt, asymmetric LCO
42
explained by observing the helicopter controls in Figure 4.9. This figure shows
that the initial 20 seconds of the results obtained with the controller on presented
higher amplitude oscillations in the helicopter controls, which increased the ampli-
tude of the oscillations in the helicopter. However, this increase in the amplitude
of the controls was far from making them reach their mechanical limits and it is a
small price to pay in order to subside the LCO in 70 seconds.
4.3 Complex Maneuver
As explained earlier, the design process was repeated for airspeeds from hover to
130 kt in 5 kt steps (or smaller steps where needed) in order to secure stability.
In this way, an airspeed scheduled controller assembled from 56 separate lag and
lead controllers was obtained (Table 3.1). To verify the correct operation of the
scheduled control system in a more demanding scenario, a complex maneuver was
simulated. The maneuver started with the helicopter in hover from where it accel-
erated to 97 kt in 20 seconds and stayed trimmed at that airspeed for 70 seconds
(which, as seen in the previous section, is the necessary time to damp the oscilla-
tions that last longer, the asymmetric LCO). After that time, the helicopter made
a 180 degrees right level turn at 97 kt which took 40 seconds to complete, and
finally, resumed straight and level flight at that airspeed for 40 additional seconds.
It is important to mention that no perturbations were used during the maneuver.
Table 4.1 describes this maneuver.
Figure 4.10 presents the helicopter Euler angles throughout the complex maneu-
ver. In it, the initial variation in the pitch angle is related to the acceleration that
the helicopter is performing at the beginning of the maneuver. When the accel-
eration is terminated, the pitch angle remains at the negative trim value required
for flight at the constant airspeed of 97 kt. After 90 seconds of simulation, the
180◦ right level turn began and the roll and yaw angles changed (lateral dynamic);
43
Time Period[sec] Segment Description
0 - 20 Acceleration from hover to 97 kt20 - 90 Straight and level flight at 97 kt90 - 130 180 degrees right level turn at 97 kt130 - 200 Straight and level flight at 97 kt
Table 4.1. Complex maneuver description
the turn was completed when the yaw angle reached 180 degrees. Then, the he-
licopter continued in a straight and level flight at 97 kt and the helicopter Euler
angles presented the same response than in the previous similar segment (from 20
to 90 seconds). Besides the description of the maneuver that this figure provides,
it is important to note the oscillations that were self-induced (no perturbation was
added to the simulation) during the straight and level flight segments.
In Figure 4.11 the relative cable angles obtained from the simulation can be ob-
served. This figure shows that when the controller is off, severe symmetric LCO
can be observed in the straight and level flight segments and milder asymmetric
Figure 4.10. Helicopter Euler angles for a complex maneuver simulation
44
Figure 4.11. Relative cable angles for a complex maneuver simulation
LCO in the right level turn. When the controller was on, the oscillations in these
segments were damped, providing stability in short time.
The perturbations in the load were similar to those in Figure 4.11. When the
controller was off, severe self-induced symmetric LCO (with more than 30 degree
peak-to-peak value) can be observed in the roll angle (φC). These self-induced os-
cillations were observed in the straight and level flight segments along with milder
asymmetric LCO for the 180 degrees right level turn. However, as for the case of
the relative cable angles, these LCO where damped when the lateral and longitu-
dinal controllers were on.
4.4 Delayed Controller Activation
A preliminary study showed sensitivity to the controller activation time due to
the nonlinear nature of the system. For this reason a delayed controller activation
analysis was performed. In addition to being a more challenging scenario, it can
45
potentially occur in practice and therefore needs to be analyzed.
4.4.1 Trimmed Cruise Flight with Time-Triggered Controller
Trimmed cruise flight maneuvers were used for analysis of the system behavior for
delayed controller activation following appearance of oscillations. These simpler
maneuvers allowed easier comparisons of the different results obtained. For both
airspeeds tested (25 kt and 97 kt), the controller was turned on at 20 different
consecutive time points during a single cycle of the oscillatory response.
4.4.1.1 Simulation at 25 kt
For this airspeed the cycle analyzed started at t = 26.25 sec and for a time cycle
of T = 6.21 sec the N = 20 time points where the controller was activated were
separated by ∆t = T/N = 0.31 sec.
As expected for this airspeed, the controller performance was similar to the one
Figure 4.12. Relative cable angles results for 25 kt with controllers turned on att = 29.05 sec
46
presented in section 5.2.1 for the 20 different test cases. As an example of the
results obtained, Figure 4.12 shows the response when the controller was turned
on at the peak of the cycle, at t = 29.05 sec, where the controller effectiveness in
this scenario is verified.
4.4.1.2 Simulation at 97 kt
Due to the proximity of 97 kt to the hysteresis effect zone (101kt) and the fact
that turning on the controller during the LCO introduces a perturbation in the
system, it is expected that the results differ depending on the time frame in which
the controller was turned on. The current analysis allowed to observe if in this
scenario the controller stabilized the system. The analyzed cycle used started at
t = 24.43 sec, where the time period was T = 7.54 sec. With N = 20, the time
interval between the points where the controller was turned on was ∆t = T/N =
0.38 sec.
For all the 20 simulations, the controller was able to achieve stability when it
Figure 4.13. Example of excellent result for 97 kt
47
was switched on in the middle of the oscillation. However, as mentioned before,
different results were obtained. To categorize the results, they were divided in three
different sets according to the oscillatory response obtained when the controller was
activated. For simplicity, the sets were named: excellent results, good results, and
adequate results. For the first case, Figure 4.13 shows an example of excellent
results, it can be observed that after the controller was turned on (25.5 seconds)
the system was stabilized quickly. In Figure 4.14, an example of a good result is
shown. For this case, in the roll and yaw angles (φC and ψC) at t = 40 seconds,
it can be seen that the system stabilized after making an abrupt change in the
relative cable angles. This abrupt change was due to hysteresis effect explained in
Chapter 2 and this set of results is characterized by having one abrupt change.
Finally, in Figure 4.15, an example of an adequate result is presented. This set of
results contain the cases in which the perturbation energy was such that the roll
and yaw angles (φC and ψC) abruptly changed two or more times before finally
stabilizing.
Figure 4.14. Example of good result for 97 kt
48
Figure 4.15. Example of adequate result for 97 kt
Figure 4.16. Time-triggered controller results summary for an airspeed of 97 kt
In Figure 4.16 the results obtained for the 20 different time points in which the
controller was turned on are summarized. In this figure it can be observed that the
effect of the controller activation time presents a lower impact when the controller
49
is turned on in the green round dots. Those are the recommended moment for
which the controller should be turned on in order to obtain the best performance
and avoid large oscillations. The blue square dots present the points in which
the controller was turned on and good results were obtained. Finally, the red
pentagram points present the results where more than one large oscillation was
presented when the controller was turned on.
It should be noted that despite the differences in the times required for oscillations
decay, the controller was able to achieve stability in all the cases tested.
4.5 Turbulent Air Simulations
To further test the controller performance in more demanding conditions, simula-
tions in turbulent air were executed. With this objective in mind, wind turbulence
was generated with the Dryden Wind Turbulence Model and added to the load
and helicopter airspeed during the simulation. As mentioned in Chapter 3, the
wind turbulence was generated for two different altitudes (1000 ft for low altitude
and 4000 ft for medium/high altitude) and three different intensities of turbulence
(light, moderate, and severe intensity). The turbulence model parameters used in
each simulation can be observed in Table 3.1.
4.5.1 Trimmed Cruise Flight
For the trimmed cruise flight, simulations for 97 kt airspeed are presented here
because of the LCO present when the controller was turned off. The simulation
scenario was the same as that presented in the previous sections, with a doublet
and an initial “push” perturbation that was used for exciting the symmetric LCO.
The maneuver duration for the constant airspeeds was increased to 200 seconds in
order to verify that the continued perturbations provided by the turbulence did
not destabilize the coupled system when the controller was on.
50
4.5.1.1 Light Level of Turbulence
In Figure 4.17, the relative cable angle results for a constant airspeed of 97 kt and
a low altitude (σ = 2.5 ft/s) are shown. In this figure, the severe symmetric LCO
can be observed after the initial perturbation when the controller was off. Close
to 140 seconds after the simulation started, the LCO fade due to a lower value
in the load airspeed as a consequence of the continuous perturbation provided by
the turbulence. However, with the controller on, the results did not present LCO
during the entire simulation.
On the other hand, Figure 4.18 presents the relative cable angles results for a
medium/high altitude. In this case, when the controller was off the LCO were
present for around 50 seconds before they were damped. Once again with the
controller on, no LCO were observed during the entire simulation.
From these results it can be observed that although the LCO is finally disappearing
when the controller is off, having large oscillations even for low periods of time
Figure 4.17. Cruise flight at low altitude with light turbulence intensity and 97 ktairspeed
51
Figure 4.18. Cruise flight at medium/high altitude with light turbulence intensity and97kt airspeed
creates a ride quality problem and significantly increases pilot’s workload, which is
a safety of flight issue. It can be also be concluded that the controllers stabilized
the system when light intensity turbulence was present. However, the continuous
perturbation introduced by the turbulence produced small oscillations when the
controller was on, which were not seen when the controller was off and the LCO
were damped.
4.5.1.2 Moderate Level of Turbulence
Figures 4.19 and 4.20 presents the relative cable angles obtained when the simula-
tion was executed with moderate turbulence and for low altitude and medium/high
altitude, respectively.
For a low altitude (Figure 4.19) and when the controller was off, LCO were ob-
served from the initial perturbation to 130 seconds, when they faded due to the
effects of the continuous perturbation in the load airspeed. Unlike in the previous
52
Figure 4.19. Cruise flight at low altitude with moderate turbulence intensity and 97kt airspeed
Figure 4.20. Cruise flight at medium/high altitude with moderate turbulence intensityand 97 kt airspeed
53
case, the higher turbulence intensity produced small oscillations that can be ob-
served after the LCO faded. When the controller was on, no LCO were observed.
However, jumps between the asymmetric branches of the isolated load (section 3.2)
where observed during the simulation.
For the case of medium/high altitude (Figure 4.20) and the controller off, the rela-
tive cable angles present LCO during the first 25 seconds of the simulation. For the
rest of the simulation some jumps were seen along some oscillations that took place
in the range between 100 seconds and 150 seconds. However, with the controller
on, the system did not present severe symmetric LCO but the small oscillations
observed with light turbulence become milder asymmetric LCO.
As mentioned in Section 2.6, the cargo load width span was used for the wind
turbulence model. With this, the worst case scenario in which all the turbulence
energy was concentrated in a smaller span was used. By using the rotor span, the
results obtained were slightly different, as can be observed in Figure 4.21 for the
case of medium/high altitude with moderate turbulence intensity at 97 kt airspeed.
Figure 4.21. Cruise flight at medium/high altitude with moderate turbulence intensityand 97 kt airspeed (rotor span)
54
Figure 4.22. Cruise flight at medium/high altitude with moderate turbulence intensityand 97 kt airspeed (40 minutes simulation)
Longer duration simulations were executed in order to verify the behavior of the
controller when LCO appeared during the simulation as a product of the continu-
ous perturbation provided by the turbulence. Figure 4.22 present the relative cable
angles for a simulation of 2400 seconds (40 minutes). In this figure, short duration
LCO in three different time frames can be observed. The first from 0 seconds to
30 seconds, the second from 340 seconds to 380 second, and the last from 1320
seconds to 1390 seconds. Figures 4.23-4.25 present the relative roll cable angle
and the load airspeed for each one of these time frames. In these figures, it can
be observed that the LCO were originated by an abrupt increase and reduction
(doublet) of the load airspeed when it was higher than 99 kt (hysteresis zone). On
the other hand, when the airspeed falls below 96 kt (where the coupled system is
stable) the LCO faded. These figures also showed no presence of LCO when the
controller was on, just a large transitory oscillations as for the case in Figure 4.24.
55
Figure 4.23. Relative roll Euler angle and load airspeed for the first LCO observed in40 minutes simulation
Figure 4.24. Relative roll Euler angle and load airspeed for the second LCO observedin 40 minutes simulation
56
Figure 4.25. Relative roll Euler angle and load airspeed for the third LCO observed in40 minutes simulation
4.5.1.3 Severe Level of Turbulence
In Figures 4.26 and 4.27 the relative cable angles for the severe level of turbulence
at low altitude and medium/high altitude, respectively, can be observed. In both
cases, when the controller was off, no LCO were observed due to the high level
of turbulence, only transitory oscillations were detected. When the controller was
on no LCO were detected, however the transitory oscillations were also seen in
this case. It is important to note that the controller was not designed to provide
suppression of transient oscillations. All in all, for this particular case of turbulence,
the impact of the controller is small.
4.5.2 Complex Maneuver
Of all the test scenarios presented previously, the complex maneuver was the most
demanding scenario and was therefore even more challenging when wind turbulence
57
Figure 4.26. Cruise flight at low altitude with severe turbulence intensity and 97 ktairspeed
Figure 4.27. Cruise flight at medium/high altitude with severe turbulence intensityand 97 kt airspeed
58
was included. For the following simulation results the scenario used is the one
described in section 4.3, where no perturbation other than the wind turbulence
was applied.
4.5.2.1 Light Level of Turbulence
For this level of turbulence, Figure 4.28 presents the relative cable angles for the
low altitude case. Like in the case in which no turbulence was present, with the
controller off, self-induced LCO were present in the first segment of cruise level
flight (between 21 seconds and 90 seconds). However, the turbulence level at the
end of the simulation damped the LCO for the second segment of cruise level flight
(from 130 seconds to 200 seconds). For the 180◦ level turn segment (90 seconds
to 130 seconds) milder asymmetric oscillations were observed. On the other hand,
when the controller was on, the LCO in the cruise level flight and in the 180◦
level turn were damped. Nevertheless, when the second cruise level flight segment
started, large symmetric oscillation were observed (at 130 seconds) but they were
Figure 4.28. Complex maneuver at low altitude with light turbulence intensity
59
Figure 4.29. Complex maneuver at medium/high altitude and light turbulence intensity
damped by the controller, achieving stability at the end of the simulation.
In Figure 4.29, the results for the medium/high altitude can be observed. In this
case, when the controller was off, the severe symmetric LCO were only observed
in the first cruise level flight segment but they were damped by the intensity of
the turbulence before this segment concluded. As in the previous case, milder
asymmetric LCO were observed in the level turn segment, however, for this case
the oscillations amplitude were higher. Contrarily, when the controller was on, no
symmetric or asymmetric LCO were observed. However, as in the previous case,
when the 180◦ level turn segment was finished a large transient oscillation was
observed, but it was damped faster than in the low altitude case.
4.5.2.2 Moderate Level of Turbulence
For the case of low altitude with a moderate level of turbulence, the results are
presented in Figure 4.30. Here, it can be observed that the intensity level of turbu-
lence was such that no severe symmetric LCO were observed when the controller
60
Figure 4.30. Complex maneuver at low altitude and moderate turbulence intensity
Figure 4.31. Load airspeed at low altitude for moderate turbulence intensity
was off; only the milder asymmetric oscillation during the level turn segment were
present. However, when the controller was on it presented transient oscillations
during the cruise level flight segments, but it was able to subside the asymmetric
61
LCO during the 180◦ level turn.
In order to understand the oscillations observed, Figure 4.31 shows the load air-
speed as a function of time for this simulation. It can be seen that at t =
140 seconds the load airspeed was less than 95 kt and in less than ten seconds
the load was moving at 101 kt, entering the hysteresis zone. This change in the
airspeed was the cause of the large oscillation that started at t = 140 seconds.
It can also be observed that the significant variations in airspeed kept the load
within the hysteresis zone for an important part of the segment, generating the
large oscillations presented in Figure 4.30.
For the case of medium/high altitude, the relative cable angle results can be ob-
served in Figure 4.32. The results when the controller was off present no severe
symmetric LCO, only transient oscillations in the level turn segment. When the
controller was on, no LCO was observed in the cruise level flight and the transient
oscillations in the level turn segment were subsided. However, small oscillations
were observed in the cruise level flight along with large transient oscillations at
the beginning and the end of the turn level flight segment. Figure 4.33 presents
the load airspeed for the case of medium/high altitude. In this figure, the large
oscillations that can be observed in Figure 4.32 around 100 seconds correspond to
portion in which the load airspeed is equal or greater than 100 kt, which is the
hysteresis zone. The same conclusions can be arrived for the oscillations observed
around 140 seconds. However, besides all these oscillations presented for this level
of turbulence, the controller managed to avoid the presence of severe symmetric
LCO for low and medium/high altitude.
62
Figure 4.32. Complex maneuver with moderate turbulence intensity at medium/highaltitude
Figure 4.33. Load airspeed at medium/high altitude for moderate turbulence intensity
63
4.5.2.3 Severe Level of Turbulence
Figures 4.34 and 4.35 presents the relative cables angles for low altitude and medi-
um/high altitude, respectively. From these figures, it can be observed that no
improvement was provided by the controllers in any of these cases. However, as in
the previous cases, no severe symmetric LCO were observed when the controller
was on.
Figure 4.34. Complex maneuver at low altitude with severe turbulence intensity
64
Figure 4.35. Complex maneuver at medium/ high altitude with severe turbulenceintensity
65
Chapter 5 |Conclusions and Future Works
5.1 Conclusions
Stabilization of a test slung load at high airspeeds was achieved by using a relative
cable angle feedback to the primary flight control system. The airspeed scheduled
control systems was evaluated under different maneuvers and turbulence level from
which the following conclusions can be drawn:
1. The lead/lag relative cable angle feedback strategy can be used for load
stabilization in forward flight.
2. The scheduled controller approach by linearization of the nonlinear model
proved to be a feasible way to improve stability for low speeds, and provide
stability for high airspeeds.
3. For the ideal case where no turbulence was present, the controller was able
to stabilize the system for the conditions in which LCO were persistent. For
lower airspeeds, it was able to improve the stability of the coupled system
by increasing the damping ratio of the load pendulum modes.
4. The effectiveness of the controller usually depends on the time in which it
was turned on. However, the use of relative cable angles feedback always pro-
66
vided stabilization when the controller was turned on during the oscillatory
response.
5. The controller was able to respond well for light and moderate turbulence
levels by quickly damp the LCO; however, transient oscillations were ob-
served.
6. In the vicinity of the hysteresis zone, persistent excitation provided by at-
mospheric turbulence is able to generate/fade LCO by changing the load
airspeed.
7. In general, the controller is effective in providing load stability a complex
maneuver. In ideal conditions (no turbulence) the airspeed scheduled con-
troller presented significant improvements by quickly damping LCO. For the
case of light/moderate turbulence, LCO at the beginning of the simulation
were damped by the controller but after that the improvements due to the
controller were not significant.
8. For the severe turbulence case, the controller did not present significant im-
provements for straight and level flight and for the complex maneuver. Fur-
ther, as the turbulence omits the appearance of sustained instabilities, the
stability of the system is generally sufficient without the controller. Still,
these are considered extreme test conditions that can possibly be avoided by
changing the flight level.
9. In any of the simulations with wind turbulence the controller prevented the
development of sustained LCO.
5.2 Future Work
Future work can be conducted to improve the stability of the system for airspeeds
in the hysteresis zone (between 99kt and 102 kt). For this objective, combining
67
relative cable angle feedback with a different active stabilization technique (such as
an active cargo hook) can be studied. It is also advisable, using the advantage of
having validated models, use nonlinear controllers methods like model predictive
control (MPC).
On the other hand, future research can also be aligned with the next generation
of helicopters which are able to flight at velocities exceeding 200 kt. By using
models for compound helicopters as the Piasecki X-49 or for tilt rotor aircrafts as
the V-22 Osprey, new control systems can be designed for larger loads moved at
higher speeds.
Load stabilization feasibility can also be check for using relative cable angles feed-
back in a dual point carriage configuration.
68
Appendix A|First Principles Physical Model
A.1 Introduction
A first principles physical model was designed to analyze the feasibility of the
research objectives. The model is a reduced lateral dynamic model of a UH-60
Black Hawk GenHel model and a CONEX cargo container with stabilizing fins
at 33 degrees with respect to the side box faces (both described in Chapter 2),
which Dr. Enciu provided for this research. This appendix explains the helicopter
model reduction, its stability augmentation system, and the roll relative cable angle
feedback designed to increase the stabilization of the helicopter and load system.
A.2 Helicopter and Load Reduced Lateral Models
The reduced lateral model was obtained by taking into account the forces and
moments applied in the system that can be observed in Figure A.1. The model
was obtained by using lagrangian mechanics, linearize the equation of motion for
an airspeed of u0 = 100 kt (168.8 ft/s), and then express the model in a state-space
form as can be observed in equation (A.1):
x = Ax +Bu (A.1)
69
Figure A.1. Force and moments for the reduced lateral dynamic model
where:
A =
Yv Yp 0 g −µg Fr − u0
Lv Lp 0 0 0 LrYv−FLv
l− Lv Yv − Lp FLv 0 − g
l(1+µ) 0
0 1 0 0 0 0
0 0 1 0 0 0
Nv Np 0 0 0 Nr
(A.2)
B =
Yδa Yδr
Lδa Lδr
−Lδa −Lδr
0 0
0 0
Nδa Nδr
(A.3)
70
and µ = mL/mH . The velocity of the system was defined as u0 = 168.8 ft/s, l is
the length of the cable, g = 32.174 ft/s2 is the gravity acceleration, the parameter
Y is the aerodynamic force applied to the helicopter in the yh direction, L and N
are the aerodynamic moments about the xh and the zh axes (Figure A.1, where
xh is defined by the right hand rule), respectively, and finally, the parameter FL is
the aerodynamic force acting on the load in the yh direction. For the parameters
Y, L, N, and FL, the subindex indicates the variable to which the parameter was
differentiated (Yv = ∂Y/∂v). Finally, the state vector was:
x = [vy, pH , pL, φH , φL, rH ]T (A.4)
and the control variables:
u =
δlatδdir
(A.5)
A.2.1 Model for Slow State Variables
To obtain the parameters for the model in equation (A.1) an UH-60 Black Hawk
GenHel non-linear model was used. The state variables from this model are pre-
sented in equation A.6:
X = [u, v, w, p, q, r, φ, θ, ψ, XN , YE, ZD,
β0, βls, βlc, β0, βls, βlc, λ0, λls, λlc]T
(A.6)
The state vector was divided to state variables related to fast and slow dynamics.
The fast dynamics are associated with the rotor state variables (from β0 to λlc)
and the slow dynamics are related to the first 12 state variables (from u to ZD).
71
In this way, equation (A.1) can be expressed as:
xsxf
=
Ass Asf
Afs Aff
xsxf
+
Bs
Bf
u (A.7)
By taking into account that the variables with fast dynamics will be stabilized
faster than the state variables from slow dynamics then, for steady-state, it is
reasonable to assumed that xf = 0, and from equation (A.7):
xf = 0 = Afsxs + Affxf +Bfu (A.8)
xf = −(A−1ff )Afsxs − Asf (A−1
ff )Bfu (A.9)
With this result, the slow state variables will be:
xs = Assxs + Asfxf +Bsu (A.10)
xs = Assxs − Asf (A−1ff )Afsxs − Asf (A−1
ff )Bfu+Bsu (A.11)
xs =[Ass − Asf (A−1
ff )Afs]
xs +[Bs − Asf (A−1
ff )Bf
]u = Asxs +Bsu (A.12)
Finally, equation (A.12) was used as the linearized state variable model for this
analysis, where the slow state variables expressed as xs are:
xs = [v, p, r, φ]T (A.13)
And the control variables:
u =
δlatδdir
(A.14)
By proceeding in the same way with the wind tunnel validated model of a load
(which is explained in Chapter 2), the parameters for the equations (A.2) and
72
(A.3) were obtained.
A.3 Helicopter Stability Augmentation System
In order to obtain a stable helicopter system a stability augmentation system
(SAS) was needed. By using the isolated helicopter model for the slow dynamics
from equation (A.12), where its parameters were obtained from a UH-60 Black
Hawk GenHel model, the matrices As and Bs of the model were obtained for the
lateral dynamics with the state variables in equation (A.13). The obtained model
corresponded to a MIMO model with two inputs given by δlat and δdir (see equation
(A.14)). By using the root locus technique two proportional feedback loops were
designed for the SAS. The first one between the input δdir and the yaw rate (r)
and the second between the input δlat and roll Euler angle (φ).
Figure A.2 presents the root locus when the loop between δdir and the yaw rate
was closed. In it, it can be observed that the stabilization design was oriented to
improving the stability of the Dutch roll modes by increasing the damping ratio
Figure A.2. SAS design: Root locus for yaw rate feedback
73
from ζ = 0.157 to ζ = 0.954 by using a gain of Kr = −39.2. However, the increase
of the damping ratio of these poles moved the unstable spiral mode even more
to the right, making this mode more unstable. In order to fix this, the next loop
closed was between δlat and the roll Euler angle, designed to move this pole as much
as possible into the left half plane. Figure A.3, present the root locus diagram for
this case, where the new pole constellation was obtained with a gain of Kφ = 5.9,
and making the system stable. With this SAS, the inputs to the helicopter were
redefined as can be seen in Figure A.4, where:
δp = δdir −Krr δa = δlat −Kφφ (A.15)
In Figure A.5, the displacement of the poles with the different controllers that
integrate the SAS can be observed with thicker blue crosses. These crosses mark
the original location of the helicopter poles, before the relative cable angle feedback
was applied.
Figure A.3. SAS design: Root locus for roll angle feedback
74
Figure A.4. Helicopter block diagram with SAS
A.4 Control System for Helicopter-Load System
The current model configuration obtained in the previous section is one of a reduced
lateral dynamics for a helicopter UH-60 Black Hawk with a designed SAS. In this
section, a control system based on the relative roll cable angles (RCA) for the
helicopter and load system is designed to ensure the stability of this system. For
Figure A.5. Root locus diagram for the helicopter and load system
75
this model the relative roll cable angles were defined as follow:
φC = φL − φH (A.16)
Figure A.5 shows the root locus diagram for the helicopter and load system, where
the transfer function in equation (A.17) was used to displace the poles and improve
the stability:
TRCA(s) = −7.813 [0.81 s+ 1] (A.17)
It can be noticed in Figure A.5 that for the original configuration the system was
stable with a damping ratio of ζ = 0.218. With the compensator added, this
damping ratio was increased to ζ = 0.837, an increase of 384% in damping ratio.
Finally, Figure A.6 presents the final block diagram for the helicopter and load
system with the relative roll cable angle feedback.
Figure A.6. Helicopter and load system with relative roll cable angle feedback
76
Appendix B|Airspeed Scheduled ControllerImplementation
B.1 Introduction
Figure B.1 presents the Simulink implementation for the airspeed scheduled cable
angle feedback controller. In this figure two manual switches used for the activation
of the lateral and longitudinal controllers can be observed. Due to the linear nature
of the proposed controller, the relative cable angles variations, ∆φC = φC − φC0
and ∆θC = θC − θC0 , were obtained by using the reference relative cable angles
(φC0 and θC0) from the Simulink block that can be observed in Figure B.2. These
reference values were calculated by using a low pass filter to average the variations
that may occur in the relative cable angle (φC and θC). In this way, the instabilities
or perturbations in the system are removed in order to provide a reference level.
Figure B.3 shows the implementation of the lateral controller (the longitudinal
controller only differs in the values of K, a, and b) where three tables contains the
values of K, A, and B as a function of the norm of the velocity vector. Then, with
these values the relative cable angles were filtered by implementing the lead/lag
77
controller with the following mathematical relationship:
Kas+ 1bs+ 1 = K
a
b+K
1− a/bbs+ 1
Finally, Figure B.4 presents the lateral/longitudinal controller timer used to verify
the controller performance when it was activated during the LCO.
78
Fig
ure
B.1
.Airs
peed
sche
duledcontrolle
rim
plem
entatio
nin
Simulink
79
Fig
ure
B.2
.Referen
cerelativ
ecablean
gles
fortheairspe
edsche
duledcontrolle
r
80
Fig
ure
B.3
.La
teralc
ontrollerim
plem
entatio
nin
Simulink
81
Fig
ure
B.4
.La
teralc
ontrollertim
erin
Simulink
82
Appendix C|Root Locus Analysis Code
C.1 Matlab Code
clear all; clc; close all
warning(’off’)
analysis = 0; % analysis = 1, return root locus diagrams
velc = 97; % [knots]
% Find first guess for the initial condition
[trimPoints, idx] = FindInitCond(velc);
%Try to initiate the simulation with the initial conditions
UH60andSL_Init
% Linearize the model to design the controller
Trim_Point_Generator
%% Find the initial conditions to find the trim point
% [vel, idx, minDist] = FindInitCond(VelIn, opt)
83
% VelIn: the velocity in knots to find the initial conditions.
% opt: ’positive’ (default), ’negative’ or ’zero’ branch.
% IC: the initial conditions.
% idx: the index in the which it was found the closest velocity.
% minDist: the distance abs(val-vel).
function [IC, idx, minDist] = FindInitCond(val, opt)
load(’InitCondGuess.mat’);
%% SELECTION OF THE BRACH
% IT ONLY USE THE POSITIVE BRANCH
if not(exist(’opt’))
opt = ’positive’;
end;
opt = lower(opt);
if strcmp(opt, ’positive’)
shft1 = 3668; % Positive branch begin
shft2 = 6026; % Positive branch end
else
if strcmp(opt, ’negative’)
shft1 = 1310; % Negative branch begin
shft2 = 3668; % Negative branch end
else
if strcmp(opt, ’zero’)
shft1 = 0; % Zero branch begin
84
shft2 = 1310; % Zero branch end
else
error(’Value of opt not recognized’);
end;
end;
end;
v = Par(shft1+1:shft2);
%% FIND INITIAL CONDITIONS
val = val*1.688; %This should be the value passed to the function
tmp = abs(v-val);
[minDist, idx] = min(tmp);
% vel = v(idx); % In case it is needed it can be return
idx = shft1 + idx;
IC = X(idx,:);
%% UH60_SL_INIT
% This code is used for initialization and trim of the UH60
% helicopter and slung LOAD model, prior to starting the
% "UH60_SLoad" Simulink simulation. First, the LOAD is
% trimmed, then the helicopter is trimmed using the sling
% cable loads calculated from the load trim.
ICL = trimPoints;
[vels, cant] = convergTrimPoint(velc);
velc = vels(1);
85
addpath(’C:\Users\mds68\Box Sync\Spring 2017\AERSP 596...
\AERSP 596\H60Sim’)
addpath(’C:\Users\mds68\Box Sync\Spring 2017\AERSP 596...
\AERSP 596\LoadModel’)
%% Read LOAD Data and Simulation Control Files and
%Initialize LOAD States
% Read LOAD data files and simulation control file
[HELICOPTER,LOAD,SYSTEM]=SL_Read_SYSTEM_Data_Files();
% Initialize States
SL_Initialize_SYSTEM_States_and_Parameters;
% set the value of gravitational accel. as set in LAOD input file
% This value will also be used by the helicopters controllers
g=LOAD.CONST.g;
%% Set Target Trim Conditions
VnTrim=velc.*1.688; VeTrim=0.; VdTrim=0.; PsiDotTrim=0;
PsiTrim=atan2(VeTrim,VnTrim); % It is assumed that
% initially the helicopters yaw angle is equal to the trajectory
%azimuth.
%% Trim Payload Block
% Thefollowing two line are needed because ’SL_PayloadBlock’ uses
%an isolated LOAD model (no ’SYSTEM’ fields)
LOAD.ATMOS=SYSTEM.ATMOS;
LOAD.SIM=SYSTEM.SIM;
load_system(’SL_PayloadBlock’); %%%
CHarm=[0.20, 0, 4.3917]; % This is the cargo hook arm assumed
%during isolated payload trim, when the actual CHarms were
%not yet read from files.
86
% Set initial conditions for u0 and initial guesses for x0 & y0
LOAD.SIM.Wind(1,:)=-[VnTrim,VeTrim,VdTrim];
xyz0 = HELICOPTER.Position+CHarm+ICL(10:12);
pqr0 = ICL(7:9);
psi0 = ICL(4); theta0 = ICL(5); phi0 = ICL(6);
att0 = [psi0,theta0,phi0]; % Attitude vector (roll angle is first)
uvw0 = ICL(1:3);
airspeed0 = [VnTrim,VeTrim,VdTrim];
groundSpeed0 = [0,0,0];
% Set LOAD initial state vector - Symmetric branch, 4KLB LOAD
x0 = [uvw0’;att0’;pqr0’;xyz0’];
u0 = CHarm’; % Set initial conditions for trim
% Set LOAD initial output vector.
y0 = [airspeed0’;pqr0’;att0’;groundSpeed0’;...
xyz0’;att0’;LOAD.Weight/4*[0,0,1,0,0,1,0,0,1,0,0,1]’];
[x,u,y,dx] = trim(’SL_PayloadBlock’,x0,u0,y0,[],[1:3],[],[],...
[1:9],[0,1e-5,1e-5,1e-6,zeros(1,9),10000]);
close_system(’SL_PayloadBlock’,0); % Close system without saving
disp(’LOAD Trim completed.’);
disp([’psi_L:’,num2str(x(4)*180/pi),’ theta_L:’,...
num2str(x(5)*180/pi),’ phi_L:’,num2str(x(6)*180/pi)]);
% Cable forces on cargo hook
Fcables = y(19:30);
% Sum cable forces
Fcables_sum=sum(reshape(Fcables,[3,4]),2);
% Store trim results in LOAD sturcture for used as initial
87
%conditions for
% the UH60_SL simulation
LOAD.STATES.Euler = x(4:6)’;
LOAD.STATES.Position = x(10:12)’;
LOAD.STATES.Rate = x(7:9)’;
LOAD.STATES.Velocity = [VnTrim,VeTrim,VdTrim]*...
(angle2dcm(x(4),x(5),x(6)))’;
% Set LOAD ground speed to equal target trim ground speed
%(this is done artificallt because the isolated load is trimmed
%assuming it is fixed, with an incomming airspeed)
Y0L = y; xL = x; x0L = x0;
%% Read Helicopter Data and Trim Helicopter
% Data file names for helicopter
H_fname=’C:\Users\mds68\Box Sync\Spring 2017\AERSP 596\AERSP...
596\UH60_SLUNG\UH60_data’;
% Read data files and Trim Helicopters
[simprop]=SL_HELICOPTER_Read_and_Trim(H_fname,...
HELICOPTER.Position,Fcables_sum,SYSTEM.ATMOS.Density,...
[VnTrim,VeTrim,VdTrim,PsiDotTrim]);
% Set same controller gains for all helicopters
%% Set Controller Gains and Initial Positions and Control Commands
load controlgains_noTurnCoord;
% Trim control commands
U0H=simprop.trimprop.CONTROL0;
% Trim state vector
X0H=simprop.trimprop.X0IC;
% Set helicopter position (x,y,z)
X0H(10:12)=HELICOPTER.Position;
88
% Set cargo hook arms (dx,dy,dz)
chArm=simprop.acprop.chprop.chArm;
disp(’HELICOPTER Trim completed.’);
disp([’psi_H:’,num2str(X0H(9)*180/pi),’ theta_H:’,...
num2str(X0H(8)*180/pi),’ phi_H:’,num2str(X0H(7)*180/pi)]);
%% Trim Slung Load System
% Load slung load system
load_system(’UH60_SLoad_Block_w_OL’);
% The first VnTrim in X0 should be sqrt(VnTrim^2 + VeTrim^2)
X0 = [X0H;VnTrim;zeros(15,1);X0H(1:3);LOAD.STATES.Euler’;...
LOAD.STATES.Position’;LOAD.STATES.Rate’;...
LOAD.STATES.Velocity’];
Y0 = [X0H(7:9);VnTrim;VeTrim;VdTrim;X0H(10:12);U0H;Y0L(7:9);...
Y0L(4:6);LOAD.STATES.Velocity’;Y0L(13:15);Y0L(16:18)];
dx0 = zeros(size(X0));
dx0(10:12) = [VnTrim,VeTrim,VdTrim]’;
% Set xdot for [x,y,z] of HELICOPTER and LOAD to equal trim
%ground velocity
dx0(44:46) = [VnTrim,VeTrim,VdTrim]’;
U0 = [VnTrim; VeTrim; VdTrim; PsiDotTrim];
[x,u,y,dx,options] = trim(’UH60_SLoad_Block_w_OL’, X0, U0,...
Y0, [4:6, 9], [], [], dx0, [1:22, 23:52],...
[1,1e-5,1e-6,1e-6,zeros(1,9),50000]);
89
if cant > 1
for vel = vels(2:end)
VnTrim = vel*1.688;
% Try calculating this as the previous one. The values
%of U0H might be the wrong ones.
Y0 = y;
dx0 = zeros(size(x));
dx0(10:12) = [VnTrim,VeTrim,VdTrim]’;
% Set xdot for [x,y,z] of HELICOPTER and LOAD to equal
%trim ground velocity
dx0(44:46) = [VnTrim,VeTrim,VdTrim]’;
U0 = [VnTrim; VeTrim; VdTrim; PsiDotTrim];
[x,u,y,dx,options] = trim(’UH60_SLoad_Block_w_OL’, x, ...
U0, Y0, [4:6, 9], [], [], dx0, [1:22, 23:52], ...
[1,1e-5,1e-6,1e-6,zeros(1,9), 10000]);
end;
end;
X0H = x(1:21); U0H = y(10:13);
LOAD.STATES.Euler = x(41:43)’;
LOAD.STATES.Position = x(44:46)’;
LOAD.STATES.Velocity = x(50:52)’;
LOAD.STATES.Rate = x(47:49)’;
% Close system without saving
close_system(’UH60_SLoad_Block_w_OL’, 0);
90
function [vels c] = convergTrimPoint(vel)
vels = [];
M = zeros(45,2);
%Vel. Need; Vel. Dependency
M(1,1) = 0; M(1,2) = 1; M(2,1) = 1; M(2,2) = 2;
M(3,1) = 2; M(3,2) = 5; M(4,1) = 3; M(4,2) = 5;
M(5,1) = 4; M(5,2) = 5; M(6,1) = 7; M(6,2) = 6;
M(7,1) = 8; M(7,2) = 6; M(8,1) = 9; M(8,2) = 6;
M(9,1) = 13; M(9,2) = 1; M(10,1) = 15; M(10,2) = 16;
M(11,1) = 43; M(11,2) = 42; M(12,1) = 44; M(12,2) = 43;
M(13,1) = 45; M(13,2) = 48; M(14,1) = 46; M(14,2) = 48;
M(15,1) = 47; M(15,2) = 48; M(16,1) = 48; M(16,2) = 51;
M(17,1) = 49; M(17,2) = 51; M(18,1) = 50; M(18,2) = 51;
M(19,1) = 51; M(19,2) = 52; M(20,1) = 52; M(20,2) = 54;
M(21,1) = 53; M(21,2) = 54; M(22,1) = 81; M(22,2) = 80;
M(23,1) = 87; M(23,2) = 86; M(24,1) = 90; M(24,2) = 89;
M(25,1) = 93; M(25,2) = 92; M(26,1) = 97; M(26,2) = 96;
M(27,1) = 101; M(27,2) = 102; M(28,1) = 104; M(28,2) = 105;
M(29,1) = 110; M(29,2) = 109; M(30,1) = 114; M(30,2) = 113;
M(31,1) = 115; M(31,2) = 113; M(32,1) = 116; M(32,2) = 115;
M(33,1) = 117; M(33,2) = 115; M(34,1) = 118; M(34,2) = 117;
M(35,1) = 119; M(35,2) = 117; M(36,1) = 120; M(36,2) = 117;
M(37,1) = 121; M(37,2) = 120; M(38,1) = 122; M(38,2) = 120;
M(39,1) = 123; M(39,2) = 120; M(40,1) = 124; M(40,2) = 123;
M(41,1) = 125; M(41,2) = 123; M(42,1) = 126; M(42,2) = 125;
M(43,1) = 127; M(43,2) = 125; M(44,1) = 128; M(44,2) = 127;
91
M(45,1) = 129; M(45,2) = 127; M(46,1) = 130; M(46,2) = 129;
M(47,1) = 131; M(47,2) = 130; M(48,1) = 132; M(48,2) = 130;
if (vel <= 132)&&(vel >= 0)
c = 1;
vels(c) = vel;
else
error(’The velocity is out of the range [0, 130] knots’);
end
idx = find(M(:,1) == vel);
if ~isempty(idx)
c = c + 1; % Only one previous velocity needed.
vels(c) = M(idx,2);
idx = find(M(:,1)==vels(c)); %previous has a dependency?
while ~isempty(idx)
c = c + 1;
vels(c) = M(idx,2);
idx = find(M(:,1)==vels(c));
end % One or more previous velocity are needed.
end
% Flip the vector because first we need to calculate the
% dependency velocity.
vels = fliplr(vels);
return
92
%% Exact linearization of the Simulink model
% UH60_SLoad_Block_w_OL
% This MATLAB script is the command line equivalent of
% the exact linearization tab in linear analysis tool
% with current settings. It produces the exact same
% linearization results as hitting the Linearize button.
% MATLAB(R) file generated by MATLAB(R) 8.6 and Simulink
% Control Design (TM) 4.2.1.
% Generated on: 27-Jul-2017 14:22:28
%% Specify the model name
model = ’UH60_SLoad_Block_w_OL’;
%% Specify the analysis I/Os
% Create the analysis I/O variable IOs1
io(1) = linio(’UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/...
Dynamic Inversion Inner Loop CLAW/From’,1,’output’);
io(2) = linio(’UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/...
Dynamic Inversion Inner Loop CLAW/Pseudo cmds ...
transfer block/nu’,1,’input’);
%% Specify the operating point
% Create the operating point variable op_trim1 using model
%initial condition as a starting point
op = operpoint(’UH60_SLoad_Block_w_OL’);
% Set the states in the model with different values than
%model initial condition
93
% State (1) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Pitch CF/Integrator
op.States(1).x = x(30);
% State (2) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Pitch CF/Integrator1
op.States(2).x = x(31);
% State (3) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Roll CF/Integrator
op.States(3).x = x(28);
% State (4) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Roll CF/Integrator1
op.States(4).x = x(29);
% State (5) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Vz CF/Integrator
op.States(5).x = x(32);
% State (6) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Command Filters/Yaw CF/Integrator
op.States(6).x = x(33);
% State (7) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Integrators/Integrator
op.States(7).x = x(34:35);
% State (8) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Integrators/Integrator1
op.States(8).x = x(36);
% State (9) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/Integrators/Integrator2
op.States(9).x = x(37);
% State (10) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
94
%Inversion Inner Loop CLAW/u,v,w washout/Integrator
op.States(10).x = x(38);
% State (11) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/u,v,w washout/Integrator1
op.States(11).x = x(39);
% State (12) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Dynamic
%Inversion Inner Loop CLAW/u,v,w washout/Integrator2
op.States(12).x = x(40);
% State (13) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/
%Integrator
op.States(13).x = x(1:21);
opspec.States(13).Known(2) = true;
% State (14) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
%Loop DI/Integrator
op.States(14).x = x(26);
% State (15) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
%Loop DI/Integrator4
op.States(15).x = x(27);
% State (16) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
%Loop DI/XY Velocity Control/Integrator1
op.States(16).x = x(22);
% State (17) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
%Loop DI/XY Velocity Control/Integrator2
op.States(17).x = x(23);
% State (18) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
%Loop DI/XY Velocity Control/Integrator3
op.States(18).x = x(24);
% State (19) - UH60_SLoad_Block_w_OL/Subsystem/HELICOPTER/Outer
95
%Loop DI/XY Velocity Control/Integrator4
op.States(19).x = x(25);
% State (20) - UH60_SLoad_Block_w_OL/Subsystem/LOAD Dynamics/
%Payload EOM/Euler Angles Integrator
op.States(20).x = x(41:43);
% State (21) - UH60_SLoad_Block_w_OL/Subsystem/LOAD Dynamics/
%Payload EOM/pqr Integrator
op.States(21).x = x(47:49);
% State (22) - UH60_SLoad_Block_w_OL/Subsystem/LOAD Dynamics/
%Payload EOM/uvwBody Integrator
op.States(22).x = x(50:52);
% State (23) - UH60_SLoad_Block_w_OL/Subsystem/LOAD Dynamics/
%Payload EOM/xyzNED Integrator
op.States(23).x = x(44:46);
% Set the inputs in the model with different values than
%model initial condition
% Input (1) - UH60_SLoad_Block_w_OL/Trajectory_Cmd
op.Inputs(1).u = U0;
%% Linearize the model
msys = linearize(model,io,op);
%% Analysis
if analysis ~= 1
% Phi output, Lateral input
mysys = tf(msys(2, 1));
verifySystem(mysys, 1e-6);
% Theta output, Collective input
96
mysysLong = tf(msys(1, 2));
verifySystem(mysysLong, 1e-6);
end
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Bibliography
[1] Ronen, T., Bryson, A. E., and Hindson, W. S., “Dynamics of a Helicopter witha Sling Load,” AIAA Atmospheric Flight Mechanics Conference, Williams-burg, VA, August 18-20, 1986.
[2] Lusardi, J. A., Blanken, C. L., Braddom, S. R., Cicolani, L. S., and Tobias, E.L., “Development of External Load Handling Qualities Criteria,” AmericanHelicopter Society 66th Annual Forum Proceedings, Phoenix, AZ, May 11-132010.
[3] Gera, J., and Farmer, S.W., “A Method of Automatically Stabilizing Heli-copter Sling Loads,” NASA TN D-7593, 1974
[4] Watkins, T. C., Sinacori, J. B., and Kesler, D. F., “Stabilization of Exter-nally Slung Helicopter Loads [Final Report, 1 Jul. 1972 - 31 Oct. 1973],”USAAMRDL-TR-74-42, August 1974.
[5] Liu, D. T., “In-Flight Stabilization of Externally Slung Helicopter Loads [FinalReport, 25 Jun. 1970 - 17 Jun. 1972],” USAAMRDL Tech. Rept. 73-5, May1973.
[6] Asseo, S. J., and Whitbeck, R. F., “Control Requirements for Sling-Load Sta-bilization in Heavy Lift Helicopters,” Journal of the American Helicopter So-ciety, Vol. 18, (3), July 1973, pp. 23-31.
[7] Micale, E. C., and Poli, C., “Dynamics of Slung Bodies Utilizing a RotatingWheel for Stability,” Journal of Aircraft, Vol. 10, (12), 1973, pp. 760-763.
[8] Feaster, L. L., “Dynamics of a Slung Load,” PhD Thesis, Department of Me-chanical and Aerospace Engineering, Massachusetts University, Amherst, June1975.
[9] Hamers, M., and Bouwer, G., “Flight Director For Helicopter with SlungLoad,” 30th European Rotorcraft Forum, Marseilles, France, September 14-16,2004.
98
[10] Hamers, M., and Bouwer, G., “Helicopter Slung Load Stabilization Using aFlight Director,” American Helicopter Society International 60th Annual Fo-rum, Grapevine, TX, June 1-3, 2005.
[11] Hamers, M., “Flight Director for Handling of Helicopter Sling Loads,” 31stEuropean Rotorcraft Forum, Florence, Italy, September 13-15, 2005.
[12] Hamers, M., Von Hinuber, E., and Richter, A., “Flight Director for Slung LoadHandling - First Experiences on CH53,” 33rd European Rotorcraft Forum,Kazan, Russia, September 11-13, 2007.
[13] Krishnamurthi, J. and Horn, J.F., “Helicopter Slung Load Control UsingLagged Cable Angle Feedback,” Journal of the American Helicopter Society,vol. 60, no. 2, 2015.
[14] Ottander, J. A., and Johnson, E. N., “Precision Slung Cargo Delivery onto aMoving Platform,” AIAA Modeling and Simulation Technologies Conference,AIAA Paper 2010-8090, 2010. doi:10.2514/6.2010-8090
[15] Ivler, C.M., Powell, J.D., Tischler, M.B., Fletcher, J.W., Ott, C, “Design andFlight Test of a Cable Angle Feedback Flight Control System for the RASCALJUH-60 Helicopter,” Journal of the American Helicopter Society, vol. 59, no.4, 2014.
[16] Patterson, B., Ivler, C.M., Hayes, P., “External Load Stabilization ControlLaws for an H-6 Helicopter Testbed,” Proceedings of the American HelicopterSociety 70th Annual Forum, Montreal, Canada, May 2014.
[17] Patterson, B., Enns, R., King, C., Kashawlic, B., Mohammed, S., Lukes, G.,and The Boeing Company, “Design and Flight Test of a Hybrid ExternalLoad Stabilization System for an H-6 Helicopter Testbed,” Presented at theAmerican Helicopter Society, 71st Annual Forum, Virginia Beach, Virginia,May 5-7, 2015.
[18] Raz, R., Rosen, A., Carmeli, A., Lusardi, J., Cicolani, L. S., and Robinson,D., “Wind Tunnel and Flight Evaluation of Passive Stabilization of a CargoContainer Slung Load,” Journal of the American Helicopter Society, Vol. 55,(3), July 2010, pp. 0320011- 03200118.
[19] Cicolani, L., Ivler, C., Ott, C., Raz, R., and Rosen, A., “Rotational Stabiliza-tion of Cargo Container Slung Loads,” American Helicopter Society Interna-tional 69th Annual Forum, Alexandria, VA, May 21-23, 2013.
[20] Enciu, K., and Rosen, A., “Nonlinear Dynamical Characteristics of Fin-Stabilized Underslung Loads,” AIAA Journal, Vol. 53, (3), March 2015, pp.723-738.
99
[21] Enciu, J., Singh, A, and Horn, J. F., âĂIJStabilization of External Loads inHigh Speed Flight Using an Active Cargo Hook,âĂİ 43rd European RotorcraftForum, Milan, Italy, September 12-15, 2017
[22] Enciu, K., and Rosen, A., “Aerodynamic Modeling of Fin Stabilized Under-slung Loads,” Aeronautical Journal, Vol. 119, (1219), September 2015, pp.1073-1103.
[23] Nayfeh, A. H., and Balachandran, B., “Applied Nonlinear Dynamics: Analyt-ical, Computational, and Experimental Methods,” Wiley-VCH Verlag GmbHand Co. KGaA, Weinheim, 1995.
[24] Strogatz, S. H., “Nonlinear Dynamics and Chaos”, Perseus Books, Reading,1994.
[25] Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems andChaos, Springer, New York.
[26] “Dynamical Systems Toolbox,” Coetzee, E., 2011,https://www.mathworks.com/matlabcentral/fileexchange/32210-dynamical-systems-toolbox [retrieved 1 March, 2012].
[27] “AUTO-07P, Software Package,” Computational Mathematicsand Visualization Laboratory, C. U., Montreal, Canada, 2007,http://indy.cs.concordia.ca/auto/ [retrieved March 1, 2012].
[28] Howlett, J. J., “UH-60A Black Hawk Engineering Simulation Program: Vol-ume I - Mathematical Model,” SER 70452 (NASA Contractor Report 166309),December 1981.
[29] Padfield, G. D. (1966) Helicopter Flight Dynamics: The Theory and Applica-tions of Flying Qualities and Simulation Modelling, AIAA Inc., WashingtonDC.
[30] Stevens, B. L., Lewis, F. L., Johnson, E. N., “Aircraft Control and Simulation:Dynamics, Controls Design, and Autonomous Systems,” third edition, Wiley-Blackwell, 2015.
[31] Pitt, D. M., and Peters, D. A., “Theoretical Prediction of Dynamic InflowDerivatives,” Vertica, Vol. 5, (1), pp. 21-34.
[32] “Generate continuous wind turbulence with Dryden velocity spectra -Simulink,” https://www.mathworks.com/help/aeroblks/drydenwindturbulencemodelcontinuous.html, Accessed: 2018-05-22.
[33] U.S. Department of Defense, “Flying Qualities of Piloted Aircraft.” Depart-ment of Defense Handbook. MIL-HDBK-1797B. Washington, DC, 2012.
100
[34] Ogata, K. (2010) Modern Control Engineering, fifth edition, Prentice Hall.
[35] Nise, N. S. (2008) Control Systems Engineering, fifth edition, John Wiley &Sons, Inc.
101