The Pennsylvania State University The Graduate School College of Engineering ACTIVE STABILIZATION OF SLUNG LOADS IN HIGH SPEED

Transcription

1 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

2 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

3 Abstract Helicopters performing external load missions are subject to instabilities that arise in high speed flight that limit their operational flight envelope. This thesis addresses the problem of active stabilization of slung loads in high speed flight. To demonstrate the method, simulations of a utility helicopter with a dynamic inversion controller (as its automatic flight control system) and a CONEX cargo container 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 evaluated 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 sustained limit cycle oscillations are avoided by the use of the controller. For severe turbulence levels, the controller did not provide any significant improvement. iii

4 Table of Contents List of Figures List of Tables List of Symbols Acknowledgments vii xi xii xv Chapter 1 Introduction Motivation Background Goal and Organization of the Thesis Chapter 2 Model Description Introduction External Load Model Isolated Load Dynamics Sling Cables Model Helicopter Model Coupled Helicopter-External Load System Load Stability in the Coupled Model Relative Cable Angles Dryden Wind Turbulence Model Low-Altitude Model Medium/High Altitudes Model Chapter 3 Controller Design 25 iv

5 3.1 Introduction Controller Design Process Description Design Example for Low Airspeed Design Example for High Airspeed Controller Design Summarized Chapter 4 Simulation Results Introduction Trimmed Cruise Flight Simulation at 25 kt Simulation at 97 kt Complex Maneuver Delayed Controller Activation Trimmed Cruise Flight with Time-Triggered Controller Simulation at 25 kt Simulation at 97 kt Turbulent Air Simulations Trimmed Cruise Flight Light Level of Turbulence Moderate Level of Turbulence Severe Level of Turbulence Complex Maneuver Light Level of Turbulence Moderate Level of Turbulence Severe Level of Turbulence Chapter 5 Conclusions and Future Works Conclusions Future Work Appendix A First Principles Physical Model 69 A.1 Introduction A.2 Helicopter and Load Reduced Lateral Models A.2.1 Model for Slow State Variables A.3 Helicopter Stability Augmentation System A.4 Control System for Helicopter-Load System v

6 Appendix B Airspeed Scheduled Controller Implementation 77 B.1 Introduction Appendix C Root Locus Analysis Code 83 C.1 Matlab Code Bibliography 98 vi

7 List of Figures 1.1 External load mission examples Cargo container with fins inclined in 33 degrees relative to the box (picture from [21]) Helicopter (H), load (L), and Earth (E) fixed coordinate system Isolated load equilibria points Limit cycle oscillations for 100 kt (168.8 ft/s) Equilibria points around 170 ft/s (101 kt) for the isolated load Helicopter flight control system model Helicopter inner loop dynamic inversion Effects observed in the model pole diagrams for the coupled system at 25 kt Effects observed in the model pole diagrams for the coupled system at 97 kt Relative cable angles sequence description Dryden medium/high altitude turbulence intensities and probability of exceedance [32] vii

8 3.1 Relative cable angles trimmed for the couple system at 100 kt New proposed controller design Lateral controller root locus design for 25 kt Longitudinal controller root locus design for 25 kt Lateral controller root locus design for 97 kt Longitudinal controller root locus design for 97 kt Relative cable angles simulation result for 25 kt Helicopter Euler angles simulation result for 25 kt Helicopter controls commands simulation result for 25 kt Relative cable angles simulation result for 97 kt Helicopter controls simulation result for 97 kt Helicopter Euler angles simulation result for 97 kt Relative cable angles simulation result for 97 kt, asymmetric LCO Helicopter Euler angles simulation result for 97 kt, asymmetric LCO Helicopter controls simulation result for 97 kt, asymmetric LCO Helicopter Euler angles for a complex maneuver simulation Relative cable angles for a complex maneuver simulation Relative cable angles results for 25 kt with controllers turned on at t = sec Example of excellent result for 97 kt viii

9 4.14 Example of good result for 97 kt Example of adequate result for 97 kt Time-triggered controller results summary for an airspeed of 97 kt Cruise flight at low altitude with light turbulence intensity and 97 kt airspeed Cruise flight at medium/high altitude with light turbulence intensity and 97kt airspeed Cruise flight at low altitude with moderate turbulence intensity and 97 kt airspeed Cruise flight at medium/high altitude with moderate turbulence intensity and 97 kt airspeed Cruise flight at medium/high altitude with moderate turbulence intensity and 97 kt airspeed (rotor span) Cruise flight at medium/high altitude with moderate turbulence intensity and 97 kt airspeed (40 minutes simulation) Relative roll Euler angle and load airspeed for the first LCO observed in 40 minutes simulation Relative roll Euler angle and load airspeed for the second LCO observed in 40 minutes simulation Relative roll Euler angle and load airspeed for the third LCO observed in 40 minutes simulation Cruise flight at low altitude with severe turbulence intensity and 97 kt airspeed Cruise flight at medium/high altitude with severe turbulence intensity and 97 kt airspeed Complex maneuver at low altitude with light turbulence intensity. 59 ix

10 4.29 Complex maneuver at medium/high altitude and light turbulence intensity Complex maneuver at low altitude and moderate turbulence intensity Load airspeed at low altitude for moderate turbulence intensity Complex maneuver with moderate turbulence intensity at medium/high altitude Load airspeed at medium/high altitude for moderate turbulence intensity Complex maneuver at low altitude with severe turbulence intensity Complex maneuver at medium/ high altitude with severe turbulence intensity A.1 Force and moments for the reduced lateral dynamic model A.2 SAS design: Root locus for yaw rate feedback A.3 SAS design: Root locus for roll angle feedback A.4 Helicopter block diagram with SAS A.5 Root locus diagram for the helicopter and load system A.6 Helicopter and load system with relative roll cable angle feedback. 76 B.1 Airspeed scheduled controller implementation in Simulink B.2 Reference relative cable angles for the airspeed scheduled controller 80 B.3 Lateral controller implementation in Simulink B.4 Lateral controller timer in Simulink x

11 List of Tables 2.1 Dryden wind turbulence model parameters Scheduled controller parameters Complex maneuver description xi

12 List of Symbols A, B, C, D State matrix, input matrix, output matrix, feedthrough matrix a, b Zero/pole coefficient in compensator b S C S e F H, F L, F C F S h H K K S l L Wingspan Cable damping coefficient Error state vector State vector functions for helicopter, load and DI controller Cable tension force Altitude for the Dryden wind turbulence model Forming filter transfer function Compensator gain Cable stiffness Cable position Turbulence scale length p, q, r Roll, pitch, and yaw rates p L r CH T EH Load relative position vector Cargo hook position vector Earth to helicopter body transformation matrix xii

13 T EL T HL Earth to load body transformation matrix Helicopter to load body transformation matrix u, v, w Inertial velocity components u V Control command vector Aircraft airspeed norm x, y, z Longitudinal, lateral, and vertical position x X A, X B, X C, X P y β 0, β 1S, β 1C l δ λ 0, λ 1S, λ 1C ν(t) ν φ, ν θ, ν VD, ν r σ State vector Lateral and longitudinal stick, collective, pedals Helicopter states output vector used for outer loop DI controller Main rotor flapping angles Cable stretch Vector of pilot control commands Dynamic inflow components DI controller pseudo-commands vector DI controller pseudo-commands for roll, pitch, aircraft vertical speed, and roll rate. 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

14 ( ) 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

15 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 Center 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. W911W The U.S. Government is authorized to reproduce and disxv

16 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 Government. xvi

17 Dedication To my family... xvii

18 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

19 Figure 1.1. External load mission examples 1 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 various 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 stabilization 1 Left: right: 2

20 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 performance 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 research is performed collaboratively by researchers from Penn State University and the Tel Aviv University and includes the development of active stabilization methods 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

21 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 characteristics 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 performance 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

22 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

23 Figure 2.1. Cargo container with fins inclined in 33 degrees relative to the box (picture from [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 theoretical 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

24 state vector being comprised of the load s inertial velocities, attitude angles, angular rates, and center of gravity position: x L = {u L, v L, w L, ψ L, θ L, φ L, p L, q L, r L, x N, y E, z D } (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 (ψ L ), then pitch (θ L ), and finally roll (φ L ). Figure 2.2. Helicopter (H), load (L), and Earth (E) fixed coordinate system 7

25 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

26 uation software package AUTO [27] into MATLAB. Figure 2.3 shows a bifurcation curve for the load roll angle, φ L, that was obtained 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 bifurcation 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 symmetric solution branch (with φ L = 0 ) and the second one includes two asymmetric branches (φ L 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 airspeed (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 ft/s (100 kt), three unstable equilibria ex- Figure 2.4. Limit cycle oscillations for 100 kt (168.8 ft/s) 9

27 ist: a symmetric equilibrium with φ L = 0 and two asymmetric equilibria with φ L = ±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 simulated 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 ft/s and a φ L = 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

28 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 φ L = 0. If another perturbation in the load decreases its airspeed, the load will remain stable and with φ L = 0 until its velocity reach a value close to 170 ft/s. From that moment on, the load roll angle will suddenly change to φ L = ±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, l i, and its rate of change, and is directed along the cable unit length vector, l i : F S,i = max ( K S,i l i + C S,i ( li ), 0 ) l i (2.2) where: l i = l i l i (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

29 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 K S,i and C S,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, x H, contains 12 rigid body states, x F, and 9 main rotor, x R, as follow: x F = {u H, v H, w H, p H, q H, r H, ψ H, θ H, φ H, x N, y E, z D } (2.4) x R = { β 0, β 1S, β 1C, β 0, β 1S, β } 1C, λ 0, λ 1S, λ 1C (2.5) x H = {x F, x R } (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 helicopter 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

30 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 (u N, v E, w D ) in the earth fixed coordinate system, and the helicopter heading (ψ): u = {u N, v E, w D, 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 = {X A, X B, X C, X p } (2.8) Figure 2.6. Helicopter flight control system model 13

31 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 y cmd is defined as: y cmd = [φ cmd θ cmd V Dcmd r cmd ] T (2.9) where φ, θ, V D, 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 = [ν φ ν θ ν VD ν r ] T (2.10) This vector is calculated by using the following equations: e φ = φ f φ (2.11) e θ = θ f θ (2.12) e r = r f r (2.13) 14

32 ν φ = φ f + K P e φ + K D ė φ + K I e φ dt (2.14) ν θ = θ f + K P e θ + K D ė θ + K I e θ dt (2.15) = V Df + K P e VD + K I e VD dt (2.16) ν r = ṙ f + K P e r + K I e r dt (2.17) ν VD 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): ẋ H F H (x H, x L, x C ) ẋ L = F L (x H, x L ) ẋ C F C (x H, x C, u) In the equations above, the functions F H, F L, and F C are the corresponding state vector functions that describe the helicopter and load dynamics and the DI controller, respectively; the vectors x H, x L, and x C are the helicopter, load, and DI controller state vector, respectively 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

33 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 mentioned 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, corresponding to the instability of the isolated load for this airspeed. These unstable modes moved from ± j1.166 to ± 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 25 kt 16

34 Figure 2.9. Effects observed in the model pole diagrams for the coupled system at 97 kt 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 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 p L ) using the Earth to helicopter 17

35 coordinate transform matrix T EH as can be seen in equation (2.18): p L = T EH (r L r H ) r CH (2.18) Where T EH is calculated following the order yaw (ψ), pitch (θ), and roll (φ), or the sequence 3-2-1: T EH = T φ T θ T ψ = Cθ H Cψ H Cθ H Sψ H Sθ H = Sφ H Sθ H Cψ H Cφ H Sψ H Cφ H Cψ H + Sφ H Sθ H Sψ H Sφ H Cθ H Sφ H Sψ H + Cφ H Sθ H Cψ H Sφ H Cψ H + Cφ H Sθ H Sψ H Cφ H Cθ H (2.19) The values of the relative cable angles were obtained assuming the transformation from (H) to (L), T HL, follows the order pitch (θ C ), roll (φ C ), and yaw (ψ C ) (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: T HL = T EL (T EH ) T (2.20) The relative cable angles φ C, θ C are calculated using the cable components of p L and ψ C is calculated by comparing the entries in T HL : ( φ C = asin y ) rp p L ) θ C = atan ψ C = atan2 ( xrp z rp ( Num Den ) (2.21) (2.22) (2.23) 18

36 Where p L = [x rp, y rp, z rp ] T and: Num = cos(θ H ) sin(θ L ) sin(φ H ) + cos(θ L ) {cos(ψ H )[cos(ψ L ) sin(θ H ) sin(φ H ) + + cos(φ H ) sin(ψ L )] + sin(ψ H )[ cos(φ H ) cos(ψ L ) + + sin(θ H ) sin(φ H ) sin(ψ L )]} (2.24) Den = cos(θ H ) cos(θ L ) sin(φ H ) sin(φ L ) + [cos(ψ H ) sin(θ H ) sin(φ H ) cos(φ H ) sin(ψ H )][cos(ψ L ) sin(θ L ) sin(φ L ) cos(φ L ) sin(ψ L )] + + [cos(φ H ) cos(ψ H ) + sin(θ H ) sin(φ H ) sin(ψ H )][cos(φ L ) cos(ψl) + + sin(θ L ) sin(φ L ) sin(ψ L )] (2.25) (a) (b) (c) Figure Relative cable angles sequence description 19

37 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: 2Lu H u (s) = σ u πv Lu s (2.26) V ( ) 1/6 π 0.8 H v (s) = σ v V. 4b ( S ( ) ) (2L w ) 1/ bS (2.27) πv s 2Lw H w (s) = σ w πv Lw s V ( 1 + 2L w V s) 2 (2.28) Where b S represents the aircraft wingspan, L u, L v, L w represent the turbulence scale lengths, and σ u, σ v, σ w represent the turbulence intensity components in the body frame 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: 2L w = h (2.29) L u = 2L v = h ( h) 1.2 (2.30) 20

38 From where, for an altitude of h = 1000 ft, the turbulence scale lengths are: L u = 1000 ft (2.31) L v = 500 ft (2.32) L w = 500 ft (2.33) On the other hand, the relationship between the turbulence intensity and the altitude is: σ w = 0.1W 20 (2.34) σ u σ w = σ v σ w = 1 ( h) 0.4 (2.35) Where W 20 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 σ u, σ v, and σ w are: Light turbulence level σ u = σ v = σ w = 2.5 ft/s (2.36) Moderate turbulence level σ u = σ v = σ w = 5 ft/s (2.37) Severe turbulence level σ u = σ v = σ w = 7.6 ft/s (2.38) Finally, in the current effort, the adopted wingspan (b S ) 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

39 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: L u = 2L v = 2L w = 1750 ft (2.39) Figure Dryden medium/high altitude turbulence intensities and probability of exceedance [32] 22

40 From where L u = 1750 ft and L v = L w = 875 ft. The turbulence intensity is defined as: σ = σ u = σ v = σ w (2.40) Which means that: σ RMS = 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 (b S ) 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

41 Parameters Low Altitude (1000 ft) Med/High Altitude (4000 ft) Light Moderate Severe Light Moderate Severe σ u (ft/s) σ v (ft/s) σ w (ft/s) L u (ft) L v (ft) L w (ft) b S (ft) 6.11 Table 2.1. Dryden wind turbulence model parameters 24

42 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

43 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

44 point by using the Simulink Linearization tool. The nonlinear system was defined by using the relative cable angles φ C (eq. (2.21)) and θ C (eq. (2.22)) as the system outputs and the pseudo-commands of the DI controller, ν φ and ν θ (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): ẋ = F (x, u) (3.1) y = G(x, u) (3.2) Where u = [ν φ, ν θ ] is the control vector, x is the state vectors that include the load, helicopter, DI controller, and relative cable angles states, and y = [φ C, θ C ] 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: ẋ = F x + F u (3.3) x x0,u 0 u x0,u 0 y = G x + G u (3.4) x x0,u 0 u x0,u 0 Then, the linear time invariant (LTI) system matrices are defined as: A = F (3.5) x x0,u 0 B = F (3.6) x x0,u 0 C = G (3.7) u x0,u 0 D = G (3.8) u x0,u 0 27

45 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: ẋ = 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) = K as + 1 bs + 1 (3.11) Figure 3.2. New proposed controller design 28

46 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 Design Example for Low Airspeed For this case the trim point was obtained by a load roll angle of φ L = 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

47 ratio of the lateral pendulum modes is ζ = for ω n = 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: K lat25kt = 0.8 a lat25kt = b lat25kt = 1 (3.12) By using the designed controller, the minimum damping ratio was increased to ζ = while keeping the natural frequency at ω n = 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 a long25kt, b long25kt, and K long25kt of 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 = (ω n = 1.36 rad/sec) to be increased to ζ = (ω n = 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

48 Figure 3.4. Longitudinal controller root locus design for 25 kt the values of K long25kt, a long25kt, and b long25kt used are: K long25kt = 6.84 a long25kt = 0.97 b long25kt = 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 (φ L > 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): K lat97kt = 1.2 a lat97kt = 0.33 b lat97kt = 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

49 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, increasing the damping ratio from ζ = (ω n = 1.44 rad/sec) to ζ = (ω n = 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 diagram. 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 = and a natural frequency of ω n = 1.44 rad/sec can also be observed in the diagram. 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 = (and ω n = 1.65 rad/sec) by using the parameters in equation (3.14) for the longitudinal controller: K long97kt = 0.78 a long97kt = 1.5 b long97kt = 0.5 (3.14) 32

50 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

51 Velocity Lateral Controller Longitudinal Controller K lat a lat b lat K long a long b long Table 3.1. Scheduled controller parameters 34

52 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 Chapter 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

53 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 presented as severe symmetric LCO or milder asymmetric LCO. Simulations to verify the performance of the controller for these two cases were executed 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

54 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 (ψ C, red curve in Figure 4.1) presents a time to half of seconds (where the damping ratio obtained from the simulation of the nonlinear model was ζ = , 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 = , 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

55 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 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 doublet 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

56 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 φ L ) 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 impact 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

57 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

58 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 controller was off. These oscillations lead to significant lateral acceleration levels in the cockpit that would likely degrade flying qualities and increase the pilot s workload. 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 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/mission 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

59 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

60 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 amplitude 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 accelerated 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 oscillations 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 maneuver. 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 acceleration 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

61 Time Period [sec] Segment Description 0-20 Acceleration from hover to 97 kt Straight and level flight at 97 kt degrees right level turn at 97 kt 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 helicopter 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 observed. 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 Helicopter Euler angles for a complex maneuver simulation 44

62 Figure 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 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 (φ C ). These self-induced oscillations 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 longitudinal 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

63 potentially occur in practice and therefore needs to be analyzed 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 Simulation at 25 kt For this airspeed the cycle analyzed started at t = 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 t = sec Relative cable angles results for 25 kt with controllers turned on at 46

64 presented in section 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 = sec, where the controller effectiveness in this scenario is verified 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 = 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 Example of excellent result for 97 kt 47

65 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 (φ C and ψ C ) 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 (φ C and ψ C ) abruptly changed two or more times before finally stabilizing. Figure Example of good result for 97 kt 48

66 Figure Example of adequate result for 97 kt Figure 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

67 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, simulations 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 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

68 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 airspeed Cruise flight at low altitude with light turbulence intensity and 97 kt 51

69 Figure Cruise flight at medium/high altitude with light turbulence intensity and 97kt 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 Moderate Level of Turbulence Figures 4.19 and 4.20 presents the relative cable angles obtained when the simulation 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 observed 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

70 Figure Cruise flight at low altitude with moderate turbulence intensity and 97 kt airspeed Figure Cruise flight at medium/high altitude with moderate turbulence intensity and 97 kt airspeed 53

71 case, the higher turbulence intensity produced small oscillations that can be observed 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 relative 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 Cruise flight at medium/high altitude with moderate turbulence intensity and 97 kt airspeed (rotor span) 54

72 Figure Cruise flight at medium/high altitude with moderate turbulence intensity and 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 continuous 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 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

73 Figure Relative roll Euler angle and load airspeed for the first LCO observed in 40 minutes simulation Figure Relative roll Euler angle and load airspeed for the second LCO observed in 40 minutes simulation 56

74 Figure Relative roll Euler angle and load airspeed for the third LCO observed in 40 minutes simulation 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 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

75 Figure Cruise flight at low altitude with severe turbulence intensity and 97 kt airspeed Figure Cruise flight at medium/high altitude with severe turbulence intensity and 97 kt airspeed 58

76 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 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 Complex maneuver at low altitude with light turbulence intensity 59

77 Figure 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 Moderate Level of Turbulence For the case of low altitude with a moderate level of turbulence, the results are presented in Figure Here, it can be observed that the intensity level of turbulence was such that no severe symmetric LCO were observed when the controller 60

78 Figure Complex maneuver at low altitude and moderate turbulence intensity Figure 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

79 LCO during the 180 level turn. In order to understand the oscillations observed, Figure 4.31 shows the load airspeed 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 For the case of medium/high altitude, the relative cable angle results can be observed in Figure 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

80 Figure Complex maneuver with moderate turbulence intensity at medium/high altitude Figure Load airspeed at medium/high altitude for moderate turbulence intensity 63

81 Severe Level of Turbulence Figures 4.34 and 4.35 presents the relative cables angles for low altitude and medium/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 Complex maneuver at low altitude with severe turbulence intensity 64

82 Figure intensity Complex maneuver at medium/ high altitude with severe turbulence 65