DEVELOPMENT OF A REAL-TIME FLIGHT SIMULATOR FOR …...experimental model helicopter. A mathematical model of the helicopter is developed to represent the dynamics of the real system

DEVELOPMENT OF A REAL-TIME FLIGHT SIMULATOR FOR AN EXPERIMENTAL MODEL HELICOPTER

Diploma Thesis

Cand. aer. Christian Munzinger

Atlanta, December 1998

Georgia Institute of Technology School of Aerospace Engineering

advised by: Dr. Anthony J. Calise Dr. J. V. R. Prasad

IFR

University of Stuttgart

SUMMARY

This work first describes the development of a real-time flight simulator for an R-50 experimental model helicopter. A mathematical model of the helicopter is developed to represent the dynamics of the real system. This simulation model is used to investigate and analyze the helicopter dynamics in a hovering flight condition. The importance of a control rotor used for stability augmentation of the helicopter is emphasized and investigated in more detail. Combining the model with further flight software and hardware, a flight simulator is obtained that is capable of real-time flight simulation. This simulator will be used in the future for detailed studies on new modern control algorithms used for helicopter flight control. Experimental flight tests with the real helicopter are performed and analyzed and allow the identification of a simplified linear model valid close to the hover flight condition. Results are shown and compared to the linearized model obtained from the simulation. The system identification employs a frequency response and a step response method that result in an approximate model for the helicopter dynamics. Linear models from simulation and flight tests are then used in a recently developed Neural Network Adaptive Nonlinear Flight Control System and applied to the real-time simulator and the real helicopter. The results of both applications are then briefly presented.

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ACKNOWLEDGEMENTS

This work was conducted by the School of Aerospace Engineering at the Georgia Institute of Technology in Atlanta. I want to thank Dr. Anthony J. Calise of the School of Aerospace Engineering in Atlanta and Dr. Klaus H. Well of the Institute of Flight Mechanics and Control in Stuttgart for their support. They both made it possible for me to study at Georgia Tech and enabled me to realize this work. I also want to thank Dr. Anthony J. Calise and Dr. J. V. R. Prasad who guided and assisted me throughout my work and studies and gave me all the support I needed during this time. Special thanks to Dr. Eric J. Corban of Guided Systems Technologies, Inc., who went with me through numerous hardware and software problems related to the flight test program. My respect and thanks to the pilot, Mr. Jeong Hur, and the whole flight team, who suffered only minor heart-attacks during some critical flight maneuvers that would have brought my work to a sudden end. Finally I want to thank all my colleagues and friends who supported me during this one year at Georgia Tech. Christian Munzinger Atlanta, Georgia December 1998

Contents

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CONTENTS

Summary .............................................................................................................................ii

Acknowledgements.............................................................................................................iii

Contents ............................................................................................................................iv

Nomenclature ......................................................................................................................vi

List of Figures......................................................................................................................x

List of Tables......................................................................................................................xii

Chapter 1 Introduction.....................................................................................................13

1.1 Simulation of Flight.................................................................................................. 13 1.2 Experimental Flight .................................................................................................. 15

Chapter 2 Helicopter Flight Dynamics............................................................................17

2.1 General Equations of Unsteady Motion ...................................................................... 17 2.2 The Small-Disturbance Theory .................................................................................. 21

Chapter 3 Helicopter Theory ..........................................................................................23

3.1 Main Rotor Reference Frames and Notations.............................................................. 23 3.2 Hover and Vertical Flight.......................................................................................... 26 3.3 Forward Flight ......................................................................................................... 28

3.3.1 Rotor Theory in Forward Flight....................................................................................................28 3.3.2 Influences of Rotor Effects and Rotor–Helicopter Interference ..................................................29

Chapter 4 Helicopter Stability and Control ...................................................................35

4.1 Helicopter Control.................................................................................................... 35 4.2 Helicopter Stability................................................................................................... 37

4.2.1 Hover...............................................................................................................................................38 4.2.2 Forward Flight................................................................................................................................42 4.2.3 Stability Augmentation with a Control Rotor ...............................................................................46

Chapter 5 Mathematical Modeling .................................................................................48

5.1 General Helicopter Model......................................................................................... 48 5.2 Rigid Body Model.................................................................................................... 49 5.3 Main Rotor Model.................................................................................................... 49 5.4 Control Rotor Model ................................................................................................ 58 5.5 Model of Fuselage, Wing and Tail ............................................................................. 63 5.7 Simulation Results for the Linarized Model................................................................ 65

Chapter 6 Real-Time Simulation: Hardware and Software .........................................73

6.1 Simulation Elements ................................................................................................. 73 6.2 Flight System Elements ............................................................................................ 76 6.3 Hardware-In-The-Loop-Simulation ........................................................................... 77

Contents

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Chapter 7 Validation of the Helicopter Simulation Model ...........................................79

7.1 Flight Test Data and Requirements ............................................................................ 79 7.2 System Identification Procedures ............................................................................... 81

7.2.1 Static Trim Values...........................................................................................................................81 7.2.2 Frequency Response Analysis........................................................................................................84 7.2.3 Step Response Analysis...................................................................................................................95

Chapter 8 Modern Adaptive Nonlinear Flight Control in Simulation and Real Flight ..........................................................................101

8.1 Flight Control System............................................................................................. 101 8.2 Simulation and Experimental Results ....................................................................... 105

References ........................................................................................................................108

Appendix A – R-50 Helicopter Data ..............................................................................110

Appendix B – Equations of Unsteady Motion of the Rigid Body................................112

Appendix C – System and Control Matrices ................................................................113

Appendix D – Results of Linear System Analysis for Hover.......................................115

Appendix E – R-50 Helicopter System Components....................................................118

Appendix F – Simulated and Experimental Bode Plots...............................................120

Nomenclature

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NOMENCLATURE

( L, M, N ) Components of moment about the CG, in body frame ft lb ( p, q, r ) Angular helicopter body rates, in body frame rad/sec ( u, v, w ) Velocity components relative to air expressed in body frame ft/sec ( u, v, w )E Velocity components relative to the Earth fixed frame ft/sec ( X, Y, Z ) Components of force acting along the (x, y, z) B axes lb ( x, y, z )a Helicopter aerodynamic coordinate frame ( x, y, z )B Helicopter body coordinate frame ( x, y, z )E Earth fixed coordinate frame ( φ, θ, ψ ) Euler angles rad a Two-dimensional constant lift curve slope 1/rad a0 Coning angle rad a1s First harmonic coefficient of longitudinal blade flapping rad with respect to shaft (positive for tilt back) ap, aq, ar Uncoupled stability derivatives with respect to 1/sec uncoupled body angular rates

ja Estimated system parameters A Rotor disk area ft2 A1, A2 First and second harmonic coefficient of lateral rad blade feathering A1,SP Lateral swashplate tilt relative to HP rad (positive for tilt right) AR Blade aspect ratio b1 First harmonic coefficients of lateral blade flapping rad with respect to feathering plane B Number of blades B1, B2 First and second harmonic coefficient of longitudinal rad blade feathering B1,SP Longitudinal swashplate tilt relative to HP rad (positive for tilt forward) b1s First harmonic coefficient of lateral blade flapping rad with respect to shaft (positive for tilt right) bp, bq, br Uncoupled control derivatives with respect to inputs resulting in uncoupled body angular rates c Mean blade chord length ft cD0 Mean profile drag coefficient cm Mean profile moment coefficient of control rotor dhub Horizontal hub distance from helicopter CG ft eMR Flap hinge offset of main rotor blade ft

Nomenclature

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E Vector of output errors E0 Mean harmonic coefficient of blade lag motion rad E1, E2 First and second harmonic cos-coefficients of rad blade lag motion F Matrix of state derivatives F1, F2 First and second harmonic sin-coefficients of rad blade lag motion

fwake Wake-function for low/high speed effects on dv

db s1 , du

da s1

G Matrix of control derivatives hhub Vertical hub distance from helicopter CG ft is initial shaft tilt (positive back) rad Ib Moment of inertia of blade about flapping hinge slug ft2 Ixy Product of helicopter inertia ∫ dmxy slug ft2

Ixz Product of helicopter inertia ∫ dmxz slug ft2

Iyz Product of helicopter inertia ∫ dmyz slug ft2

kMR Coefficient defining main rotor blade pitch due to swashplate tilt kβ Coefficient defining main rotor blade pitch due control rotor tilt K1 Cross-coupling coefficient due to delta-three-angle K2 Cross-coupling coefficient due to hinge offset Kc Total cross-coupling coefficient lb Length of aerodynamic blade section of control rotor ft lCR Length of control rotor bar ft

β&M Non-dimensional aerodynamic moment due to blade flapping velocity Mgust Control rotor moment due to wind velocity ft lb MT Torque ft lb Mµ Non-dimensional aerodynamic moment derivative with respect to rotor advance ratio R Blade radius ft T Thrust lb Td Time delay sec ν Total airspeed ft/sec V Velocity vector relative to the atmosphere ft/sec wblade Average velocity of main rotor blade relative to air ft/sec wr Velocity of rotor disk relative to air ft/sec x State vector of helicopter rigid body motion

Nomenclature

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Important Derivatives

duda s1 Derivative of longitudinal TPP tilt with respect to u-velocity rad/(ft/sec)

dvdb s1 Derivative of lateral TPP tilt with respect to v-velocity rad/(ft/sec)

1dAdL

Roll moment due to lateral cyclic pitch change ft lb/rad

sdbdL

1

Roll moment due to lateral TPP tilt ft lb/rad

sdadM

1

Pitch moment due to longitudinal TPP tilt ft lb/rad

1dBdM

Pitch moment due to longitudinal cyclic pitch change ft lb/rad

Greek Symbols

α Parameter vector in system identification β Blade flapping angle rad βc First harmonic coefficient of longitudinal blade flapping rad of control rotor with respect to shaft βc,CR Control rotor longitudinal TPP tilt rad βs First harmonic coefficient of lateral blade flapping rad of control rotor with respect to shaft βs,CR Control rotor lateral TPP tilt rad δ3 Delta-three-angle rad δcoll,MR Collective main rotor input rad δcoll,TR Collective tail rotor input rad δlat Lateral cyclic input rad δlong Longitudinal cyclic input rad δu Input vector to helicopter rigid body motion rad γ Lock number ϕd Phase shift due to time delay rad λ Directional parameter (-1=clockwise, 1=counterclockwise) νi induced velocity ft/sec θ Blade pitch angle rad θcoll Blade pitch due to pilot collective input rad θtwist Blade twist rad θ0 Collective main rotor blade pitch rad

Nomenclature

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θ0,CR Constant initial control rotor blade pitch rad ρ Density of air slug/ft3 τ Time constant ω Frequency rad/sec, Hz ω in Flap rate coefficient for in-axis-motion ωoff Flap rate coefficient for off-axis-motion Ω Rotor rotational speed rad/sec Ωf Coefficient defining change in natural main rotor frequency due to hinge offset ξ Limited extension parameter of control rotor ψ b Rotor blade azimuth rad

Abbreviations

CG Center of gravity coll Collective pitch DOF Degree of freedom HP Hub plane lat Lateral long Longitudinal MR Main rotor NN Neural network rpm Rotor rotational speed TR Tail rotor TPP Tip path plane

List of Figures

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LIST OF FIGURES

Figure 2.1: Body axes of the helicopter and notations............................................................................. 18

Figure 2.2: Wind axes of helicopter in forward flight .............................................................................. 19

Figure 2.3: Block diagram, vehicle with plane of symmetry, body axes, flat-earth approximation,

no wind [4].................................................................................................................................. 20

Figure 3.1: Rotor disk and notations........................................................................................................... 24

Figure 3.2: Hub plane, tip-path plane, body axes and notations ........................................................... 24

Figure 3.3: Fundamental blade motion....................................................................................................... 25

Figure 3.4: Rotor blade velocity in forward flight .................................................................................... 28

Figure 3.5: Offset of rotor blade flap hinge ............................................................................................... 31

Figure 3.6: Cross-coupling due to the delta-three-angle ........................................................................ 31

Figure 3.7: Rotor-Fuselage interference in (a) hover and (b) forward flight ....................................... 33

Figure 3.8: Mechanical linkages of the control rotor for the R-50 experimental helicopter ............. 34

Figure 4.1: Longitudinal hover poles dependent on normalized flap frequency ν =ωn /Ω ................ 40

Figure 4.2: Typical hover poles for decoupled longitudinal and lateral motion ................................. 41

Figure 4.3: Typical hover poles for coupled longitudinal and lateral motion ..................................... 41

Figure 4.4: Influence of forward speed and horizontal tail on longitudinal poles.............................. 44

Figure 4.5: Influence of forward speed on lateral poles .......................................................................... 45

Figure 4.6: R-50 control rotor providing rate feedback [29] ................................................................. 47

Figure 5.1: Control Rotor of the R-50 Helicopter (view from top).......................................................... 60

Figure 5.2: Poles of coupled longitudinal and lateral motion, no control rotor.................................. 68

Figure 5.3: Poles of coupled longitudinal and lateral motion, with control rotor............................... 70

Figure 5.4: Hover poles of longitudinal motion, with and without control rotor................................. 71

Figure 5.5: Hover poles of lateral motion, with and without control rotor........................................... 72

Figure 6.1: Elements of Simulation Software............................................................................................. 74

Figure 6.2: Display of the R-50 real-time simulator on PC-screen ........................................................ 75

Figure 6.3: Joint GST/Ga Tech real-time hardware-in-the-loop simulation facility [28] .................. 78

Figure 7.1: Trim table for R-50 (simulation), rearward to forward flight ............................................. 83

Figure 7.2: Block diagram of approximated linear helicopter dynamics for hover............................. 85

Figure 7.3: Frequency response in the pitch channel, ω = 0.75Hz, quδ =0.035 rad........................... 87

Figure 7.4: Experimental Bode plot for the pitch channel ....................................................................... 88

Figure 7.5: Experimental Bode plot for the roll channel ......................................................................... 89

Figure 7.6: Experimental Bode plot for the yaw channel......................................................................... 90

Figure 7.7: Experimental and simulated frequency response, 0.31sec time delay, pitch dynamics .. 94

List of Figures

xi

Figure 7.8: Response of identified model (7.17) compared with measured data.................................. 98

Figure 7.9: Response of identified model (7.18) compared with measured data.................................. 100

Figure 8.1: Neural Network Augmented Model Inversion Architecture................................................. 102

Figure 8.2: Multilayered network with one hidden layer......................................................................... 104

Figure 8.3: Simulated system response of R-50, doublet inputs in long. cyclic.................................... 105

Figure 8.4: Pitch response of R-50 in flight test with NN controller in pitch channel ......................... 106

Figure E.1: R-50 fully equipped during flight test .................................................................................... 118

Figure E.2: R-50 fully equipped on transport cart; GST ground control station in background ...... 118

Figure E.3: R-50 avionics box with on-board PC and sensor packet..................................................... 119

Figure E.4: R-50 horizontal tail for improved handling characteristics in forward flight................. 119

Figure F.1: Experimental and simulated frequency response, 0.31sec time delay, roll dynamics .... 120

Figure F.2: Experimental and simulated frequency response, 0.31sec time delay, yaw dynamics .... 121

List of Tables

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LIST OF TABLES

Table 4.1: Single rotor helicopter coupling sources [15]........................................................................ 36

Table 5.1: Trim values in simulated hover for R-50, with and without control rotor .......................... 66

Table 5.2: Analytically obtained system matrix in hover, no control rotor........................................... 67

Table 5.3: Analytically obtained control matrix in hover, no control rotor.......................................... 67

Table 5.4: Eigenvalues, damping and frequencies of hover modes, no control rotor ......................... 68

Table 5.5: Analytically obtained system matrix in hover, with control rotor........................................ 69

Table 5.6: Analytically obtained control matrix in hover, with control rotor....................................... 69

Table 5.7: Eigenvalues, damping and frequencies of hover modes, with control rotor ..................... 70

Table 7.1: Measured and simulated trim values for R-50 in hover and forward flight....................... 82

Table 7.2: Identified parameters and system delay in hover using experimental Bode plots ............ 92

Table 7.3: Identified parameters for decoupled lateral motion, step response .................................... 97

Table 7.4: Identified coupled model parameters for system including only angular rates................. 99

Chapter 1 Introduction

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Chapter 1

INTRODUCTION

During the early phase of design and development of an aircraft, it needs to be tested and performance limits need to be validated. For manned flight vehicles it is very common to use flight simulators before real flight tests, not only to reduce time and costs during the testing phase, but also to aid in the avoidance of possible loss of pilot and aircraft in case of a failure. The use of real-flight simulators is therefore mostly related to manned flight. In this work a simulation for a remotely controlled aircraft of much smaller size needs to be developed to test a modern controller. The approach and the mathematical model that will be used are similar to that of a manned helicopter; in fact the main part of the simulation is taken from a simulation of a full size helicopter. Several aspects arise from the very different size and therefore result in different dynamic behavior which need to be taken into account. A brief description of desired capability and capacity of real flight simulators in general will follow, and the still necessary results of experimental flight for validation purpose and to build up confidence in handling the real flight system will be pointed out.

1.1 Simulation of Flight

Rotorcraft and its unique capabilities of vertical take-off and landing, hover, vertical and forward flight are playing an important roll in commercial and military applications. Superior hover and low speed performance and agility are coupled with good flight characteristics even in fast forward flight. The rotor of a helicopter generates the predominant aerodynamic forces in all flight conditions and is source of forces and moments on the aircraft that control position, attitude and velocity. An increase of complexity with respect to rotor and blade dynamics and the still not sufficiently explained aerodynamic effects of rotor aerodynamic or rotor-body interference complicate the development of detailed mathematical models of rotors and helicopters in general. The use of mathematical models for simulation is therefore limited to some degree of accuracy and depends on the final objective of the simulation. In general, examining structural dynamics of single blades or blade sections needs a more accurate model than a simulation of flight mechanics and aircraft performance. The effort and expense put into


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modeling and simulation are justified by the helicopter’s unique capabilities, and the objective of simulation influences the level of math model fidelity directly related to the effort and expense required for a given task. High performance computers made an increase of accuracy of simulation possible, but the complexity of simulation still rules the capability of operating in real-time. The development of high-fidelity real-time simulators for research and design, concept validation and training is strongly dependent on available time and money. Once developed and validated, the simulator can be very efficiently used to support flight tests to evaluate handling qualities during the development and design phase of new or modified helicopter configurations or helicopter components. More advanced applications, as in unconventional configurations like tilt wings and tilt rotors, unmanned flight of full size aircraft, simulating emergency situations, validating and testing new control systems or simple pilot training, underline the usefulness of such a high-fidelity real-time simulator. Nevertheless, a highly accurate model of one component of a complex system does not necessarily mean that the simulation of the whole system behaves like the real physical system. This is especially true for a helicopter, since components like rotors, fuselage, wings, horizontal or vertical tail, engine and actuators interact with each other and influence the system response to external and internal disturbances. For application in helicopter controls, where the main objective is to control the dynamic behavior of the helicopter over some desired flight envelope, it is necessary to find a representative model that shows the same dynamic characteristics as the real aircraft. On the one hand a detailed model of the main rotor is desired since the dynamics are governed mainly by the main rotor, but on the other hand a too detailed description increases the complexity of the simulation and limits the capability of real-time simulation. Furthermore, most of the existing simulations make a very detailed knowledge of the simulated system necessary. This knowledge covers exact physical data of the aircraft geometry, airfoils of blades and wings and aerodynamic data that is gained in wind tunnel tests. For competitive reasons this data is handled by companies with care and is therefore generally not available. If a basic math model can be developed that depends only on basic data sources, then the inflexibility of very sophisticated models that are now in common use can be overcome. Additional individual vehicle components can be added and the existing model can be easily extended or refined. Such a so-called "Minimum-Complexity Helicopter Simulation Math Model" has been developed by NASA [1]. A modification of this simulation model will be used and applied to the remotely controlled experimental


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model helicopter R-50 (see Appendix E), built by YAMAHA [29]. Aerodynamic data from wind tunnel testing does not exist and the physical data of system components is known only approximately. The capability of the minimum-complexity math model to be flexible and easily extendible to other helicopter configurations is not only desired, it is of great importance for this application. A small-scale model helicopter shows a different aerodynamic behavior, since it is designed for a very different flight envelope than full-scale helicopters. Further undesirable effects of excessive model complexity are computational system delays, a great number of system parameters that need to be determined for each aircraft, inability to easily observe relationships between modeling parameters and model response (very important for handling quality simulation) and the inflexibility in temporarily removing undesired dynamics for debugging. The most important benefit of a minimum-complexity math model is the potential for a more clear understanding of the cause and the resulting effect. This aspect can become very important if the dynamic system itself is additionally complicated by a modern control system that does not allow a fast and easy engineering understanding of the control and response features. In summary, the main attribute of a simulator as an effective tool for controller design is the ability to produce desired results for a specific application and to operate over the full flight envelope (forward, rearward and sideward flight, hover, transition from hover to forward flight, vertical climb) with representative handling qualities. Through a man-in-the-loop simulation it also becomes a very powerful tool to identify critical man-machine or controller-machine interface issues and allow pilot training within a reasonable amount of time, costs and risk until confidence in flying with a new system or flight controller is gained. Even if satisfying results can be achieved with a high-fidelity real-time simulator, the results will not be sufficient unless they are confirmed in real flight. In the following, the objective and the importance of flight tests to validate results will be briefly described.

1.2 Experimental Flight

To increase the fidelity of the mathematical model and to decrease effort required to create and validate the model, a systematic system identification approach is used to identify parameters and data that tunes the simulator to fit the highly complex and nonlinear characteristics of helicopter flight. This approach will be described in Chapters 5 and 7. The data used for the identification process is produced during flight tests. The aircraft motion is measured, and a model that reflects the physical behavior of the system


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needs to be found and investigated. The complexity of helicopters, and for the application described in this work, the reduced size and low payload capability further reduce the possibility of adding numerous sensors and hardware to measure velocities, rates or angles in the rotating as well as in the non-rotating frame. The on-board system of the experimental helicopter consists of a 200 MHz Pentium based flight control processor and an integrated avionics system. Available on-board sensors include a 3-axis gyro and accelerometer package, differential GPS with 2 cm accuracy, 3-axis magnetometer and an 8 channel ultrasonic ranging system. A wireless digital data link provides a communication link with a mission control ground station. Representative flight tests are mainly basic maneuvers about hovering, level forward flight and step inputs to the control stick from carefully trimmed flight conditions. Once this basic validation is done, more complex maneuvers could be flown, measured and compared with the simulation results. It is also important to understand operating rules and human interaction with the flight system. The pilot information given in reports about handling qualities are not measurable with sensors and are also dependent on personal experience and pilot skills. However, to evaluate the overall accuracy of the simulation and its real-time capability, this aspect needs to be investigated. It is obvious that the visual channel strongly influences the pilot’s response to a directly observed change in aircraft motion. This is only one weak point of some real-time simulations, since the hard and software might not provide the pilot with the 3-D picture he is used to during flight. Flight tests will give us further information how these facts have to be evaluated and how they influence the simulation results. Engineering and piloted validation is therefore necessary, and the quantitative and qualitative handling quality flight tests are described in more detail in Chapter 7. In the following work flight tests will mainly be used to create the basis on that a fine-tuned model can be built on. This is a first important step for the procedure that will follow to evaluate and investigate advanced control algorithms on model helicopters. The Uninhabited Flight Research Facility (UFRF) located in the School of Aerospace Engineering at Georgia Tech was initiated in June 1997 and is dedicated to flight testing for this purpose. It presently contains two Yamaha model helicopters of the Type R-50. Testing capability and performance of the simulator is the main objective of this work. After validation, this simulator will be used to test a neural network (NN) based, adaptive flight controller, recently developed at Georgia Tech. More details on theory and design of the developed control algorithm are given in [25, 26, 27]; Chapter 8 only gives a brief overview of important aspects and should provide the reader with the basic understanding of Feedback Linearization and Adaptive Neural Networks in flight controls.

Chapter2 Helicopter Flight Dynamics

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Chapter 2

HELICOPTER FLIGHT DYNAMICS

Like most other flight vehicles, the helicopter body is connected to several elastic bodies such as rotor, engine and control surfaces. The physical nature of this system is very complex in shape and motion, and simple mathematical modeling seems not to be very precise. Nonlinear aerodynamic forces and gravity act on the vehicle, and flexible structures increase complexity and make a realistic analysis difficult. Several assumptions can be made to reduce this complexity to formulate and solve relevant problems. This chapter describes assumptions necessary for a satisfactory modeling of the helicopter motion and introduces the fundamental motion of the flight vehicle in general. Some features for the helicopter case are emphasized and explained with respect to stability analysis and system identification as needed.

2.1 General Equations of Unsteady Motion

Derived from first principles, equations can be found that describe the aircraft as a rigid body with six degrees of freedom (DOF’s), free to move in the atmosphere. Aerodynamic forces and moments and gravity are incorporated directly in those equations. In [5] the derivation of the general motion of a mass particle is given, then the dynamic and kinematic equations for an arbitrary deformable vehicle in flight are derived. Treating the earth as flat and stationary in inertial space simplifies the model significantly. For most problems of airplane flight this is acceptable, and for an experimental helicopter flying at low speed at very low altitude this approximation is valid. The derived equations contain only a few further assumptions:

• The aircraft can be treated as a rigid body with any number of rigid spinning rotors. • There is a plane of symmetry, so that Ixy = Iyz = 0. • The axes of spinning rotors are fixed in the direction relative to the body axes and

have constant angular speed relative to the body axes. The third assumption seems to be rather crude for the helicopter case since helicopter motion is mainly controlled by tilting the main rotor relative to the body axes and therefore creates additional moments and forces. This assumption is justified by assuming


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only changes in rotor tilt of a few degrees relative to the body axes. This is in general true and no modification needs to be made at this point. For deriving the equations of motion, it is convenient to choose body axes since all of the inertias remain constant in the body frame. The x-axis is fixed to a longitudinal reference line in the aircraft; the y-axis is oriented to the right and the z-axis downward. A common assumption is the exact symmetry of the aircraft with respect to the xz-plane. The used body reference frame and its notations for forces, moments and angular rates about its axis are shown in Figure 2.1.

xB, X, L, p

yB ,Y, M, q

zB, Z, N, r

Figure 2.1: Body axes of the helicopter and notations

Another reference frame that is very useful in formulating the equations of motion is the earth-fixed frame. With the assumption of a flat and stationary earth, this frame becomes an inertial system in which Newton's laws are valid. The origin is arbitrary, the x-axis is horizontally pointing in any convenient direction, the z-axis is pointing vertically downward. The y-axis is perpendicular to both. For all frames of reference a right-handed coordinate frame is assumed. The center of gravity (CG) of the aircraft is equal to the mass center, and its location is given by its Cartesian coordinates relative to the earth-fixed frame. To describe the aerodynamic forces the wind frame becomes important, since all of these forces depend on the velocity relative to the surrounding air mass. This wind frame is as well fixed to the aircraft, but the x-axis is now oriented along the velocity vector V of the vehicle relative to the atmosphere. The z-axis lies again in the plane of symmetry;


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the y-axis is perpendicular to both. The origin is again located at the CG of the aircraft. The wind frame and notations for a helicopter in forward flight are shown in Figure 2.2.

xB

yB ,ya

zB

xa

za

V

Figure 2.2: Wind axes of helicopter in forward flight

In the case of no velocity relative to the atmosphere, which occurs for a helicopter in hover with no additional wind, the wind frame is not defined. That makes it necessary to change back to the body axis if the helicopter motion from or to hover condition needs to be computed. This again increases the complexity of the computations, but high performance computers used in flight simulation allow an easy transformation to any desired frame without remarkable effort. Important transformation matrices can be found in [4, 5]. The resulting general equations of unsteady motion are also taken from [4] and listed in Appendix B. This set of equations includes kinematic and dynamic equations and is presented with respect to body axes. For the case of no wind the equations can be viewed in a block diagram shown in Figure 2.3. The mathematical system consists of 12 independent equations and the same number of dependent variables. These variables are:

Position of the Center of Gracity (CG): xE, yE, zE Attitude angles (Euler angles): φ, θ, ψ Velocities relative to the earth fixed frame: uE, vE, wE Angular velocities: p, q, r


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Force Equationsp , q , ru , v , w

Control forces

θ , φ u

w

v

Moment Equations

u , v , w

p , q , r

Control Moments

p

q

r

Kinematics 1p , q , r

φ

θ ψ

θ , φ

Kinematics 2u , v , w

xE

θ, φ ,ψ yE

zE

Figure 2.3: Block diagram, vehicle with plane of symmetry, body axes, flat-earth approximation,

no wind [4]

Each block represents a set of equations with inputs and outputs. The generated outputs on the right-hand side are the inputs to the left-hand side. Control forces and moments are dependent on the control inputs. For a helicopter, these inputs are collective and two cyclic (lateral and longitudinal) stick inputs to the main rotor, pedal inputs controlling the tail rotor and the throttle controlling the power. In Chapter 5 the effects on the main rotor tilt and body reactions due to control inputs are examined in more detail. For stability and control analysis these equations of motion are frequently linearized about a specific flight condition. From some steady flight condition it is assumed that the aircraft motion consists of only small deviations from this reference condition. Since the system of linear equations can also be used for identification purposes of the developed simulation math model for several flight conditions, relevant aspects of the small-disturbance theory will be formulated in the following. With respect to the control aspects it should be pointed out that various flight control structures require at least an approximate linear model for a vehicle, valid for one or even various flight conditions. This allows vehicle dynamics to be easily inverted and used, for example, in methods based on an inverting control scheme [20].


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2.2 The Small-Disturbance Theory

The nonlinear equations of motion listed in Appendix B are still accurate and therefore very useful for engineering purposes when linearized about some unaccelerated steady-state flight condition. Aerodynamic effects can be assumed to be linear functions of disturbances, and the values of linear and angular velocity perturbations are usually small for many cases [4]. Limitations for this method are disturbances or flight maneuvers that result in large changes of angles and rates and cause large nonlinearities, as for example in flight at high angle-of-attack. The detailed derivation of the basic equations is given in [4]. The linearization process assumes small disturbances, so only first-order terms are kept, and squares and products are assumed to be negligible. For a steady-state flight condition all disturbances are set equal to zero. Linear relations to eliminate reference forces and moments acting on the vehicle in this trimmed flight condition are obtained. Then the classic assumption of linear aerodynamic theory allows us to express aerodynamic forces in terms of stability or, more generally, in aerodynamic derivatives. In determining these derivatives more effort is necessary, and engineers use analytical and experimental means to find reasonable and accurate results. At this point the equations of motion for airplanes and helicopters need to be treated separately. An assumption of decoupled equations of longitudinal and lateral airplane motion is in general not valid for the helicopter. Derivatives of lateral forces and moments (Y, L, N) with respect to longitudinal motion variables (u, w, q) are no longer zero, and some derivatives with respect to rate changes of variables, often negligible in the airplane case, need to be considered for a helicopter. The system and control matrices F and G for hover listed in Appendix A show all of the important gravitational terms that can be obtained analytically and partial derivatives arising from aerodynamic forces and moments necessary to describe the linear set of equations for a helicopter. The linear, first-order set of differential equations is then of the form uGxFx δ⋅+⋅=& , (2.1)

where x represents the perturbation of state variables uB, wB, θB, qB, vB, pB, φB and rB from a steady-state reference flight condition, the trim state. The control vector δu contains deviations from the trim control inputs δlong, δcoll,MR, δlat and δcoll,TR. The trim states as part of the elements inside the matrix are noted with the subscript 0. It is


22

convenient to use only the variable names in this matrix form instead of adding another subscript to denote perturbations. This linear representation is only valid for the initial angular velocities (pB, qB and rB) equal to zero. The system matrix F includes derivatives due to small perturbations of system states; the control matrix G represents the derivatives due to small perturbations of control inputs. It can be seen that the throttle is not considered to be a control input. For a wide range of flight conditions, the rotational speed of the rotor does not change, and a variation of throttle is made only to adjust power keeping some desired rotational rotor speed constant. One way to obtain the force and moment derivatives is to sequentially perturb the states and control inputs, positively and negatively from trim values by some small amount ∆. Then the forces and moments due to both perturbed conditions are computed, and the derivatives can be obtained by the following equation.

( ) ( )

XXu

X u u X u uuu = ≅

+ − −⋅

∂∂

0 0

2∆ ∆

∆ (2.2)

The force X and the state u in this equation represent all the forces, moments, states and control inputs in the equations of motion. This approach is used in the later described simulation routine to compute the linear system matrices for any desired trimmed flight condition. Linear system analysis is very useful and convenient to examine eigenvalues or eigenvectors, system responses to step inputs, frequency response and other stability characteristics of a dynamic system. The system matrices for hover are analyzed in more detail in Chapter 5. As this chapter has summarized the dynamics of the helicopter treated as a rigid body, the following chapter introduces the main features of helicopter theory. The main rotor is essentially responsible for thrust, control forces and moments and is therefore the main subject of the following investigation. Rotor dynamics and aerodynamics influence the previously mentioned rigid body dynamics, result in cross-coupling of longitudinal and lateral motion, and affect the stability of the dynamic system. Chapters 3 and 4 summarize the most important aspects of helicopter theory with respect to the main rotor and its dynamics.

Chapter 3 Helicopter Theory

23

Chapter 3

HELICOPTER THEORY

As previously mentioned, helicopter dynamics and aerodynamics are mainly affected by the main rotor. Therefore this chapter will first introduce new main rotor reference frames and notations that are useful to describe the main rotor in detail. Then it will cover important aspects of hover and vertical flight and discuss forward flight. A section on helicopter stability and control follows. Finally the influence of an additional control or servo rotor to improve stability and handling qualities is described, and the importance of this dynamic subsystem is pointed out. In general simplifying assumptions are introduced to allow a faster analysis of the complex main rotor system, and some of the introduced aspects will be neglected in some applications.

3.1 Main Rotor Reference Frames and Notations

Basically there are three different rotor reference frames used in this main rotor analysis. The first reference frame is the rotor disk as shown in Figure 3.1. Basic variables necessary to define and derive the basic equations of main rotor and blade motion are described. The orientation of rotor rotation in the equations derived is counterclockwise. Since the main rotor for the experimental helicopter, as for most remotely controlled helicopters, rotates clockwise, this major difference can be compensated with a simple sign change in some equations. This additional parameter describing the direction of rotation will be pointed out when necessary. Further sign changes in force and moment components due to torque and tail rotor will also be necessary, as explained in the mathematical modeling of main and tail rotor in Chapter 5. It is important to mention that the azimuth angle ψ of the blade is defined as zero in the downstream direction. Azimuth and rotational speed Ω are for now defined positive counterclockwise. The two additional rotor reference frames are shown in Figure 3.2. The hub plane (HP) axes are defined with respect to the main rotor hub that remains fixed relative to the rigid body of the helicopter. The tip-path plane (TPP) axes are defined with respect to the motion described by the main rotor blade tips. A first simplifying assumption is that the thrust vector is always perpendicular to the TPP, which is true in hover and vertical flight and still very accurate in forward flight. The tilt of the TPP with respect to the HP can be


24

defined by the two angles, a1s and b1s, referring to longitudinal and lateral tilt of the TPP. For some helicopters the shaft and therefore the HP is designed with a forward tilt by a small angle, is, with respect to the helicopter body axes frame. This shaft tilt is not shown in Figure 3.2, but it will be included in the final force and moment equations for the helicopter math model.

ψ = 0°

ψ = 270°

ψ = 90°

ForwardVelocity

V

ψ = 180°ψ

RotorDisk

Blade

Figure 3.1: Rotor disk and notations

xB

yB

zB

xHP

zHP yHP, yTPP

xTPP

a1s

T

zTPPzTPP

xHP, xTPP

b1s

zHP yHP

yTPP

T

TTR

b body axes

TPP tip path plane

HP hub plane

Figure 3.2: Hub plane, tip-path plane, body axes and notations


25

The motion of the single main rotor blades governs the final tilt of reference planes. The basic blade motion as treated in this analysis is essentially rigid body rotation about the root attached to the hub. The degrees of freedom are the angles β, ζ, θ shown in Figure 3.3. The angle of rotation β about an axis in the disk plane, perpendicular to the blade axis is called flap angle. The lag angle ζ is defined by the rotation about an axis normal to the disk plane, parallel to the rotor shaft, and the pitch angle θ is the angle of rotation about an axis in the disk plane parallel to the blade spar. For a detailed main rotor analysis more complex motion than this fundamental blade motion needs to be considered. In this work it is assumed that only the basic flap and pitch motion contribute to major force and moment calculations and describe the most important rotor characteristic influencing stability and control of the rigid helicopter.

ζ

RotorShaft

Ω

β

θ

Blade

Figure 3.3: Fundamental blade motion

The steady-state blade motion is periodic around the azimuth. Using a Fourier series expansion, the flap, lag and pitch motion can be written as

...2sin2cossincos

...2sin2cossincos...2sin2cossincos

22110

22110

22110

+⋅−⋅−⋅−⋅−=

+⋅+⋅+⋅+⋅+=−⋅−⋅−⋅−⋅−=

ψψψψθθ

ψψψψζψψψψβ

BABA

FEFEEbabaa

(3.1)

Since the mean and first harmonics (subscript 0 and 1) are most important to rotor performance and control, all higher order terms will be neglected. The accuracy of the


26

model remains high whereas the analysis is simplified. The most important angles used in Equations 3.1 are the coning angle a0, pitch and roll angles of the TPP, a1 and b1, collective pitch, θ0, and cyclic pitch, A1 and B1, commanded by the pilot. The notation is mainly taken from [1, 6]. It is convenient to compute main rotor forces and moments in the TPP and then transform them into the HP or directly into the body axes frame. The components can then be easily added to aerodynamic and gravitational forces and moments acting on the rigid body. The nature of main rotor forces and moments is briefly described in the following for different flight conditions, hover, vertical flight and forward flight.

3.2 Hover and Vertical Flight

Hover and vertical flight implies axial symmetry of the rotor and can therefore be treated as a special case. Analysis is greatly simplified compared to forward flight described later, and the necessary equations can be written in a nearly closed form. Momentum theory and lift line theory will be used to determine inflow velocities, thrust and power for main and tail rotor. Momentum theory treats the rotor as an actuator disk with zero thickness and circular surface, able to support a pressure difference and thus accelerate air through the disk. This resulting airflow is called induced inflow. An approximation of uniform inflow over the rotor disk is valid in hover and vertical flight and also represents a rough estimation for forward flight. A more accurate modeling, as described by vortex wake theory and dynamic wake theory [6, 8], will increase computational cost and is therefore not always useful for most real-time simulations. For modeling forward flight, a triangular induced velocity field can be used to increase accuracy. To compute thrust and induced velocity in general, momentum theory is applied to a specific inflow model. In any case the iteration of thrust and inflow velocity converges quickly and gives a good estimate of induced velocity for most flight conditions. More detailed information on rotor wake theory can be found in [6, 8, 13, 14]. Uniform induced velocity yields minimum induced power loss of an ideal rotor for given thrust. Other power losses due to non-uniform inflow, non-optimal rotor design, blade drag and swirl in the wake need to be considered if accuracy is to be increased. For a finite number of blades additional tip losses reduce effective thrust, affect inflow characteristics and increase power losses of the rotor. Engine and transmission losses also affect power calculations, but for most applications a detailed engine model is not necessary. The constant rotor speed assumption through the entire flight envelope


27

simplifies this influence on power calculations. Especially in hover, interference between main rotor, fuselage and tail rotor cause additional power losses and need to be considered. For forward flight this effect becomes less important due to increasing other aerodynamic forces. In the case of vertical flight an additional climb or descent velocity of the helicopter needs to be added, and total power must be further changed due to necessary climb power. To compute forces and moments acting on the single blades due to velocity relative to the surrounding air, blade element theory is used. Linear lifting-line theory is applied to a rotating wing, the rotor blade, and it is assumed that the aerodynamic forces are produced by the two-dimensional airfoil of each blade section. The induced angle of attack at a blade section influences again induced velocity, and therefore again wake-body interference and further related aerodynamic aspects. Blade element theory is capable of dealing with the detailed flow and blade loading and allows a very accurate description of rotor aerodynamics, rotor performance and flight characteristics. To compute the blade angle of attack a first estimation of the induced velocity at the rotor disk is needed and can be provided by the momentum theory. The mathematical modeling of the previously mentioned aspects of rotor aerodynamics is described in Chapter 5. Furthermore, parameters might be used to adjust constraints or aerodynamic limits like stall speed or maximum side velocity. If the mathematical modeling of some phenomena can not be easily done, the existing model needs to be fine-tuned for this very specific experimental helicopter. Helicopter stability and control is based on the equations of motion for force and moment equilibrium on the entire aircraft. In hover, all of the forces and moments are due to gravity, main rotor, tail rotor or rotor-body interference. Thus the main rotor primarily governs the stability characteristics of the helicopter for this flight condition. More detailed remarks about stability and control for hover and forward flight are made in Chapter 4 . The following section introduces some very important aerodynamic features of the main rotor in forward flight. The resulting forces and moments strongly influence flight characteristics of the helicopter and are therefore very important for a desired accurate modeling and the overall simulation of flight.


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3.3 Forward Flight

Rotational motion of the rotor and translational motion of the helicopter are combined and become a source of additional complexity of rotor theory in forward flight. Axisymmetry as assumed in hover or vertical flight is no longer valid, because the aerodynamic environment varies periodically with rotation of rotor blades. The velocity of the blade relative to the air now governs blade motion. The resulting blade motion and its influence on forces and moments are the subject of this chapter.

3.3.1 Rotor Theory in Forward Flight

Necessary background of rotor theory in forward flight is provided in this section and important aspects are explained. For more detailed derivations see [6, 8, 9]. In Figure 3.4 the rotor blade velocity is shown dependent on the azimuth. The velocity of the advancing blade relative to the air is higher than that of the retreating blade due to forward velocity and rotation of the blade. Because of this periodic motion, with the fundamental frequency equal to the rotor speed Ω, the blade aerodynamics as well as the blade dynamics are first considered.

V

Ω

AdvancingSide

RetreatingSide

Ω r + V

x

yReverseFlow

Figure 3.4: Rotor blade velocity in forward flight


29

Momentum theory can again be used to obtain the power due to induced velocity. At high forward speed this power loss is small compared to other components due to high velocity, and the assumption of uniform inflow over the entire disk is again valid. But for transition from hover to fast forward flight the inflow model must be treated more carefully. This approach is given in [8]. In [1] a mathematical formulation is presented for the entire range from hover to fast forward flight assuming uniform inflow over the rotor disk and a triangular velocity field for the rotor wake acting on the fuselage (see also Chapter 5). Lifting-line theory for single blade sections integrated over the blade length is used to derive the force and moment equations for forward flight. Several assumptions can be made to simplify the analysis and to obtain the equations of blade motion. The following chapter introduces and explains the influences of blade motion on rigid body dynamics qualitatively. This provides also the background for the equations used to describe those influences if desired and necessary for the analysis in Chapter 5.

3.3.2 Influences of Rotor Effects and Rotor–Helicopter Interference

Influences on power losses, thrust and blade motion depend on blade and rotor geometry, aerodynamic phenomena and dynamic characteristics of the rotating system and the rigid body motion. The most important effects are briefly mentioned in the following sections.

Nonuniform Inflow

Using a linear variation over the disk instead of a uniform inflow can extend the computation of induced velocity. Additional coefficients can be found dependent on the forward speed of the helicopter that define a linear distribution of inflow at the rotor. Typical coefficients result in an inflow model with a small induced velocity at the leading edge of the rotor disk and about twice the mean value at the trailing edge. The mean value can still be obtained with the assumption of uniform inflow. This kind of analysis can be implemented in existing models, and improvements are expected in computation of mean and first harmonic quantities influencing rotor performance and blade flapping. If higher harmonics are subject of the analysis, the much more complicated and nonlinear inflow models become very important. Inflow variation mainly affects the rotor cyclic flapping and cyclic pitch trim; longitudinal flapping due to inflow variation is small.


30

Tip Loss and Root Cutout

Finite number of blades instead of a solid rotor disk result in an additional performance loss. At blade tips the lift decreases to zero, thrust will therefore be reduced and induced power loss will increase. This tip loss can be accounted for by introducing a tip loss factor such that there is drag but no lift starting at the radial position defined by this factor. Major influences can be found on the thrust magnitude. Near the blade root the airfoil of the blade is different compared to the rest of the blade. Due to reverse flow at the blade root in forward flight (Figure 3.4), the blade profile is cut out near the blade root in order to decrease drag. In general, the influences on thrust and flapping moment are small and can be neglected as long as performance and control aspects are concerned.

Effects of Natural Frequency on Flap Motion

Assuming that the flap hinge of a blade is located at the center of rotation without a spring producing any additional moment, the natural frequency of the flapping motion is equal to the frequency of the rotational speed. The resulting moment at the hinge is zero since the blade is free to move about this hinge. As soon as a spring, some hinge offset or both (Figure 3.5) is introduced, the resulting moment on the rotor hub is no longer equal to zero. The additional force of the spring and the offset of the hinge must be considered in the derivation of the flapping equation of motion. The offset results in a moment arm for centrifugal, inertial and aerodynamic forces and the additional hub moment must be added to the moments acting on the rigid body. The natural frequency of the flap motion for a blade with hinge offset and spring becomes larger than the rotational natural frequency. The primary effect of the hinge offset and the spring on the flap response is a coupling of longitudinal and lateral control due to this change of natural frequency. Some rotor systems instead are designed without a flap hinge and are called hingeless rotors. They can be treated similarly as described in this section, and [6, 8] deal with further details of these kind of rotors. At this point it should be pointed out that an additional control or servo rotor described later, strongly influences the helicopter stability and performance. This kind of rotor system can approximately be treated like a teetering rotor. In most applications, for a teetering rotor the coning angle, a0, of the blades in Equation 3.1 can be disregarded.


31

RotorShaft

Ω

Blade

β

Flap Hinge

Flap Hinge Offset

e

Figure 3.5: Offset of rotor blade flap hinge

A further coupling of blade pitch and flap motion arises from the geometry used to control blade pitch. For a pitch bearing outboard of the flap hinge, the blade experiences a pitch change due to flapping displacement of the blade if the pitch link is not in line with the flapping hinge (see Figure 3.6).

δ3

Blade

ΩHub

Flap Hinge

Pitch Horn

Figure 3.6: Cross-coupling due to the delta-three-angle

The δ3-angle between the virtual hinge axis and the real flap hinge axis defines the nature and magnitude of this coupling effect. It also introduces an aerodynamic spring, and the effective natural frequency of the flap motion is again increased and influences the flapping response.


32

Flap Motion due to Pitch and Roll Velocities

This effect becomes important to flying qualities since additional damping is added due to the effect of roll or pitch velocities on the rotor tilt. Gusts change the attitude of the helicopter and therefore the tilt of the rotor shaft. The change in main rotor tilt lags behind by an amount proportional to pitch or roll rate and rotor moment of inertia due to this additional damping term. The TPP wants to follow the shaft tilt due to a pitch or roll rate of the body as long as there is a hinge that provides a moment from the shaft to the rotor blades. Asymmetric flapping velocities over the azimuth with respect to the shaft results in different aerodynamic forces and moments over the azimuth, and the TPP follows the shaft. Furthermore, looking at pitch or roll rates it can be seen that the blade angle-of-attack also changes. To maintain equilibrium, the blade compensates for this with off-axis flapping. A pure pitch rate for example results in a change of lateral flapping. It is obvious that this aspect introduces an additional cross-coupling term and becomes important for evaluating flying and handling qualities.

Dihedral Effect

As in the airplane case, this effect is desirable since it helps the pilot to fly the aircraft. For wind producing sideslip angles, the aircraft tends to roll away from the approaching wind. For helicopters, the source for this positive dihedral effect is the blade flapping. Since for zero side slip angle the blades over tail and nose experience no velocity due to pure forward flight, a non-zero side slip angle results in additional velocities for those blade positions over the azimuth. As a result of wind coming directly from the right, for example, the blade over the nose will become the retreating blade, and the blade over the tail the advancing blade (for a rotor spinning counterclockwise). The rotor plane will flap down on the left. Therefore thrust is tilted to the left, and the helicopter rolls away from the wind. Even so the advancing and retreating blades are different for a rotor spinning clockwise, the dihedral effect results in a tilt of the rotor plane to the same side. This dihedral effect is mainly responsible for the phugoid-like response of the helicopter in forward flight.

Rotor-Body Interference

Due to induced velocity of the main rotor, the additional airflow over the surface of the helicopter body causes drag counteracting the thrust created by the main rotor. For some helicopter configurations with additional wings or other aerodynamic surfaces, this effect becomes even more important. As Figure 3.7 illustrates, this aerodynamic aspect is


33

different for hover and forward flight [7]. Assuming that the shape of the rotor wake can be described in such a simple manner, the influence of this wake in hover contributes mainly to the horizontal force component. In forward flight this wake is diverted due to the airspeed and also acts on the tail rotor, horizontal tail and vertical tail. The transition from hover to forward flight is an intermediate condition partially affecting all components and causing great difficulty. The exact shape of this rotor wake is furthermore not known and its influence on body, wings or horizontal tail can only be approximated if a simple model is desired.

(a) (b)

Figure 3.7: Rotor-Fuselage interference in (a) hover and (b) forward flight

Main Rotor Control

Among different types of rotors, different types of rotor control schemes were developed. The conventional way is through cyclic pitch changes of the individual blades. A pilot stick input is transformed by means of links and actuators into a tilt of the swashplate. The swashplate tilt occurs in the non-rotating and in the rotating frame and is then transferred to the individual blades through further mechanical links. This can be achieved directly over linkages and joints, or, as in the case of a so-called servo or control rotor, via an additional smaller rotor mounted on top of the main rotor. In some literature this smaller rotor is referred to as a fly bar, servo or Hiller rotor. The mechanical linkages of the R-50 hub from swashplate to control and main rotor are shown in Figure 3.8. The TPP of the control rotor then governs the lateral and longitudinal inputs of the main rotor.


34

Figure 3.8: Mechanical linkages of the control rotor for the R-50 experimental helicopter

One advantage is, that rotor systems of this kind prevent a feedback of rotor forces into the control system of the non-rotating frame. Forces and therefore the loads on swashplate actuators are minimized. Another even more important effect is the additional stability produced by the control rotor if arranged properly. Additional damping is added to the rotating system, which, as seen in Chapter 4, provides pitch and roll rate feedback as well as translational velocity feedback to the main rotor. The dynamic effects of the control rotor on stability are described in general. In Section 5.7 the influences of the control rotor on the rigid body dynamics of the helicopter are evaluated and investigated for the linearized equations of motion for the R-50 helicopter. This chapter referred to individual and coupled effects of aerodynamic, dynamic and kinematic features and their influence on blade motion or body dynamics. The overall stability of a helicopter is discussed in the next chapter. Since, for simulation, the response of the physical system is much more important than individual blade motion, the background for stability and control analysis important for helicopter dynamics are provided.

Chapter 4 Helicopter Stability and Control

35

Chapter 4

HELICOPTER STABILITY AND CONTROL

This work refers to a single rotor two bladed helicopter with a tail rotor. Therefore any further explanations and analysis are related to only this kind of helicopter. To keep the model more general, non-rotating surfaces generating additional lift through wings and a horizontal tail are not excluded from the analysis. For the investigation of the simulated model helicopter additional wings are not present. To provide better flight characteristics in forward flight a simple horizontal tail was added. In the mathematical model the general case of a single main rotor and tail rotor helicopter is treated. This chapter will first describe how control is accomplished for this kind of helicopter and summarizes how the control inputs can influence off-axis motion. Stability aspects of the decoupled and coupled rigid body dynamics are then presented for hover and forward flight. A final section on stability augmentation with a control rotor follows.

4.1 Helicopter Control

Direct control of the helicopter is obtained mainly by controlling moments. Except the vertical force component of the main rotor thrust, which can be controlled directly, all main rotor control inputs result in a tilt of the main rotor and produce a moment about the aircraft CG. Commanded are therefore changes in pitch and roll angles, resulting in lateral and longitudinal forces and finally in the desired translational helicopter motion. Controlling particular moments also makes some compensating control inputs in other axes necessary, since most inputs are coupled with off-axis motion as mentioned in the previous chapter. The pilot's controls consist of a collective stick to control the vertical force, a cyclic stick to control longitudinal and lateral moments, pedals to control the yaw moment via the tail rotor thrust and the throttle to adjust rotor speed. Collective stick is used to trim the thrust of the main rotor for some desired forward flight condition and for height control in hover. This input changes the collective pitch of all blades equally, so that only the magnitude of thrust, not the orientation of the thrust vector, is influenced; similarly for the pedal input, which provides torque balance and directional control from the tail rotor. Only the collective blade pitch of the tail rotor is changed to control tail rotor thrust and the resulting yaw moment due to the moment arm relative to the center of gravity. To


36

vary thrust and forward speed, and to maintain the desired constant rotor speed, the required rotor power changes and throttle must be adjusted. Since a speed governor on the engine can manage this, the assumption of constant rotor speed over the entire flight envelope in regards to helicopter performance, is valid. The engine dynamics are assumed to be fast compared to the rigid-body dynamics and are therefore neglected. An increase in throttle and engine power results in higher available power and does not directly influence the helicopter stability. Cyclic pilot stick displacements are connected to the blade pitch, such that the rotor tilts to the desired direction. In small manned helicopters this can be done directly by mechanical linkages, for bigger helicopters electro-hydraulic actuators can be used to convert rotor control inputs. In the case of the remotely controlled model helicopter, purely electric actuators are mounted on the aircraft. For full-sized helicopters it is important for the pilot to have a proper feedback of control forces due to pitch moments of the blades to the pilot's stick to improve handling qualities. A mechanical linkage automatically provides this feedback. If actuators are used, an artificial-feel-unit can simulate these forces in the pilot's stick.

Response Input

Pitch Roll Yaw Climb or Descent

Longitudinal Stick

Pure (Prime) 1. Lateral flapping due to longitudinal stick

2. Lateral flapping due to load factor

Negligible Desired for vertical flight path control in forward flight

Lateral Stick 1. Longitudinal flapping due to lateral stick

2. Longitudinal flapping due to roll rate

Pure (Prime) 1. Undesired in hover, caused by directional stability

2. Desired for turn coordination and heading control in forward flight

Descent with bank angle at fixed power

Pedals (Rudder)

Negligible 1. Roll due to tail rotor thrust

2. Roll due to side slip

Pure (Prime) Undesired due to power changes in hover

Collective 1. Transient longitudinal flapping with load factor

2. Steady longitudinal flapping due to climb and descent in forward flight caused by rotor flapping

3. Pitch due to change in horizontal tail lift

1. Transient lateral flapping with load factor

2. Steady lateral flapping due to climb and descent

3. Side slip induced by power change causes roll due to dihedral effect

Power change varies requirement for tail rotor thrust

Pure (Prime)

Table 4.1: Single rotor helicopter coupling sources [15]


37

To find the controls required in a trimmed flight condition, all forces and moments on the helicopter must be zero. An iterative routine is necessary that varies the pilot inputs until the six force and moment components are simultaneously zero. Predicting the control inputs for a trim condition is difficult due to the complexity of the rotor dynamics. For validation purposes it is therefore necessary to compare simulated results with flight experiments. Since for some identification studies linear models about steady-state flight conditions are computed, it is very important to verify that the system response about these steady-state trim conditions correspond with the real system. The basic behavior of rotor control from hover to forward flight is briefly described in the following. Summarizing sources of inertial and aerodynamic coupling of longitudinal and lateral helicopter motion, Table 4.1 describes various sources of cross-coupling. In forward flight, a longitudinal cyclic input creates a lateral moment on the rotor disk necessary to cancel the changes of blade-angle-of-attack due to flapping and to compensate for the higher velocity of the advancing blade. Due to hinge offset and hinge spring there is also a cross-coupling effect and the phase shift of input and TPP tilt is less than 90°. A lateral tilt of the TPP and a resulting lateral moment for longitudinal cyclic input proportional to the natural frequency of the rotating system develop and partially compensate for the effect of the higher velocity on the advancing blade. To maintain forward flight a longitudinal cyclic input (cyclic forward stick) is required to change the TPP tilt as speed increases. A lateral cyclic input (cyclic left/right stick, dependent on direction of rotor rotation) is required to compensate for the lateral TPP tilt due to lateral flapping.

4.2 Helicopter Stability

In terms of dynamic stability and response to control inputs, the rigid body degrees of freedom are mainly involved in the flight dynamic analysis. Separate longitudinal and lateral motion can usually be assumed to simplify analysis and to observe the most important stability characteristics. A further simplification is to use only low frequency dynamics of the main rotor. No additional degrees of freedom are therefore added to the system. In fact, the low frequency model for the main rotor response is a very good approximation even for a more complex analysis. Later it will be shown how the dynamic characteristics of the rigid body are influenced by an additional control rotor. In contrast to the main rotor dynamics, these rotor dynamics are then coupled with the rigid body dynamics. Also the coupling of longitudinal and lateral rigid body motion can be considerable and important for handling qualities. The use of a fully coupled simulation


38

model for the experimental helicopter characteristics will allow a more realistic representation of the dynamics. In general, a helicopter shows different characteristics in hover and forward flight. The most important stability characteristics will now be investigated for decoupled longitudinal and lateral dynamics to show the basic features of helicopter motion. If principal aircraft axes are assumed, inertial cross-coupling of yaw and roll can be neglected. Furthermore yaw motion is assumed to be fully decoupled from the other degrees of freedom. The following analysis is a summary of the basic helicopter motion for the flight conditions hover and forward flight and explains the typical helicopter behavior. More details can be found in [6, 8].

4.2.1 Hover

Vertical force equilibrium is given by the equation of motion for the helicopter vertical velocity Ez& . Collective pitch control is directly related to main rotor thrust, and for now it is assumed that there is no pitch-flap-coupling. The resulting first-order differential equation describing vertical dynamics has only a single pole. The time constant increases with rotor speed, blade loading and gross weight. This root is in general small, justifying the low frequency rotor response with respect to vertical motion. Rotor speed will always be assumed to remain constant to simplify analysis. A variable rotor speed would add another degree of freedom and modeling height control would become more difficult.

For the yaw motion, only moments due to main rotor torque and tail rotor thrust will be considered. Perturbations due to side velocity will be included. The low frequency response of the tail rotor thrust leads to a first-order differential equation for the yaw rate. The time constant is approximately the same as the time constant of vertical motion. Since sideward velocity changes with a change in tail rotor thrust, the lateral translation and yaw motion in general are coupled. This coupling is small compared to other coupling effects. A change in lateral cyclic causes a sideward velocity and requires small pedal input to maintain heading. For constant rotor speed, a change in thrust will also vary the main rotor torque, and therefore couple vertical and yaw control. To maintain the heading while thrust is changed, a coordinated pedal input is necessary. The tail rotor is operating in adverse aerodynamic environment due to the main rotor wake, fuselage and vertical tail. Modeling all these aspects is very complex and can often be simplified by an approximation only. However, these effects will become important in forward flight since


39

yaw damping and directional control are greatly influenced. Because of only low yaw damping in hover, a helicopter is very sensitive to tail rotor thrust changes. Most real helicopters therefore require at least yaw rate feedback to show handling qualities that allow a reasonable control over the helicopter. Longitudinal dynamics consist of the pitch motion, longitudinal velocity and vertical velocity. Corresponding longitudinal inputs are longitudinal cyclic stick, longitudinal gust velocity and collective pitch. The characteristic equation has three solutions representing the open loop poles of the longitudinal dynamics. One is a stable root on the real axis, the other two are a mildly unstable complex pair. This instability is a result of the coupling between pitch moment due to longitudinal velocity and the longitudinal component of the gravitational force due to pitch. The stable real root is mainly due to main rotor pitch damping. It can be found that the frequency of the roots is small compared to the rotor speed, justifying the assumption of only important low frequency response. The real root is about as fast as the vertical root mentioned previously. Since period and time to double amplitude of longitudinal dynamics are large, the motion in hover is still controllable by the pilot. However, it requires the pilot to know about the influence of cross-coupling and compensation in order to stabilize the unstable mode. As described in detail in [8] several feedback loops can be introduced to improve stability characteristics in hover. One possibility is a lagged feedback of the pitch rate, introducing damping to the system. This kind of feedback can be provided by a mechanical feedback system. Since this method is commonly used, it will be described and examined in detail in Section 5.7. The additional lagged pole of this feedback subsystem should be to the left of the pitch root to satisfy the intention of damping and improving stability. Considering short and long time response to cyclic control and longitudinal wind velocity, it can be found that the primary response is pitch acceleration. Hence the longitudinal response to a gust is small. Because of low damping, the response to cyclic control is usually large for roll and pitch motion, depending on inertia and position of CG relative to the rotor hub.


40

Re s

Im s

ν =1.0

ν =1.0

1.11.15 1.05

1.15

Figure 4.1: Longitudinal hover poles dependent on normalized flap frequency ν =ωn /Ω

The natural frequency of the main rotor also influences the rigid body dynamics, such that additional cross-coupling is introduced for any natural frequency different from the rotor rotational speed. An additional spring or damper in the flap hinge and hinge offset from the axis of rotor rotation changes this natural frequency and results in off-axis motion due to purely longitudinal inputs. The importance of the natural frequency is shown in Figure 4.1, where the pole position of the longitudinal motion is shown as a function of the flap frequency normalized with respect to the main rotor rotational speed. The lateral dynamics contain lateral velocity and roll angle, lateral cyclic control and lateral wind velocity. The basic physical system of lateral and longitudinal motion is similar, except that the roll moment of inertia is much smaller than the pitch moment of inertia. This increases the magnitude of roll stability derivatives relative to pitch derivatives. Poles can again be found as one real stable pole due to roll damping and an unstable complex conjugate pole pair due to the rotor dihedral effect and speed stability. Damping moments in roll and pitch are similar in hover. However, the roll inertia is smaller than the pitch inertia, and the lateral mode has therefore a higher frequency than the longitudinal mode. The smaller roll inertia results in a shorter period and less damping of lateral modes which makes it more difficult for the pilot to control the lateral motion in hover. As for the longitudinal dynamics, rate and attitude feedback would be required to stabilize the system. A control rotor can again provide the mechanical rate feedback of the roll rate and improve stability characteristics. Figure 4.2 shows a typical plot of decoupled hover poles for longitudinal and lateral motion of a single main rotor helicopter.


41

Re s

Im slong. poleslat. poles

-0.050.025

0.04

-0.1

-0.04

Figure 4.2: Typical hover poles for decoupled longitudinal and lateral motion

Even for a natural frequency equal to the rotor rotational speed, there is a cross-coupling of lateral and longitudinal dynamics due to the moments of inertia. For small enough roll inertia, one of the oscillatory modes might be stabilized, and the other might be destabilized as shown in Figure 4.3.

Re s

Im slong. poleslat. poles

-0.05 0.025

0.04

-0.1

-0.04

0.04

Figure 4.3: Typical hover poles for coupled longitudinal and lateral motion

Even if the roots are not significantly influenced, roll and pitch motions are coupled, and the dynamic behavior compared to the uncoupled dynamics changes. Hence the coupled system should be considered if handling qualities and dynamic response are the subject of the investigation. Later in Section 5.7 the poles are examined for the R-50 experimental helicopter. Most helicopters typically show similar characteristics to those just described.


42

However, the very different mass inertia of the model helicopter will result in differences compared to a full-size helicopter. With an increase of forward speed the dynamics of the helicopter change. Additional aerodynamic forces and moments are acting on the helicopter due to the higher body velocity. The most important aspects are discussed in the following section.

4.2.2 Forward Flight

Centrifugal forces due to angular body velocities, aerodynamic forces on fuselage and tail, and rotor forces proportional to forward speed arise in forward flight and significantly influence handling qualities. Rotor forces and body accelerations are now an additional source for the coupling of longitudinal and lateral dynamics. In this section, it will be assumed that lateral velocity, yaw rate and roll attitude can be analyzed separately from longitudinal, vertical velocity and pitch rate. For the final analysis of the simulation in Chapter 7 all six coupled DOF’s are examined. To investigate the dynamics of the helicopter in forward flight the rigid body equations are linearized about some trimmed flight condition. Zero angular rates and constant forward speed are assumed. The differential equations of motion perturbed from the given trim condition allow the computation of stability derivatives, which are the elements of the system and control matrices F and G in Appendix C. The forces acting on the helicopter are aerodynamic forces on the fuselage, horizontal and vertical tail, horizontal stabilizer and wings as present in some helicopter configurations. To demonstrate basic characteristics of helicopter handling qualities, it is sufficient to assume that the tail rotor thrust produces a yaw moment only. It can be found that due to a pitch rate a vertical acceleration evolves, and that a lateral acceleration is generated due to the yaw rate. These forces introduce coupling of longitudinal, lateral and vertical dynamics. However, inertial cross-coupling of roll and yaw will be neglected (Ixz = 0). From the low frequency response of the system, rotor forces and moments can be found. An important stability is the pitching moment due to angle-of-attack changes of the helicopter. A downward vertical velocity of the helicopter results in an increase of the rotor blade angle-of-attack and finally in a nose up pitching moment. Increasing angle-of-attack produces a pitch up moment, further increasing angle-of-attack due to rotor dynamics and aerodynamics. The main rotor is therefore a source of an angle-of-attack instability for the helicopter in forward flight. Aerodynamic effects of non-rotating aerodynamic surfaces like wings, vertical and horizontal tail and fuselage are similar to fixed wing aircraft. Therefore those stability


43

and control derivatives are similar. The fuselage drag produces a damping force, and all influences of the airframe are proportional to the square of forward speed. Damping derivatives with respect to translational velocities are thus increase with an increase in flight speed. Another contribution to instability arises from the helicopter fuselage angle-of-attack changes, resulting in an unstable pitch and yaw moment. The horizontal tail adds a pitch moment due to angle-of-attack changes and counteracts the destabilizing effect of the main rotor. It also adds pitch damping due to the helicopter pitch rate. Lift of the vertical tail resulting in a side force counteracts the destabilizing yaw moment of the fuselage due to side velocity. The major effect of the vertical tail is yaw damping and directional stability as for a fixed wing aircraft. Additional drag from a horizontal and vertical tail further increases speed stability. The complex interactions of main rotor, tail rotor, fuselage, vertical and horizontal tail often prevent a simple analysis. In forward flight, the aerodynamic characteristics of the tail and the tail rotor are strongly influenced by the shape of the fuselage and the rotor wake produced by the main rotor. Experimental data from wind tunnel testing or flight tests can be used to model these aerodynamic effects, and good results can be achieved with this approach. The uncoupled longitudinal dynamics consisting of longitudinal velocity, pitch attitude and vertical velocity are controlled by collective and longitudinal cyclic stick inputs. Vertical and longitudinal gusts are included as well. The characteristic equation shows two negative real poles for vertical and pitch motion and a slightly unstable long period oscillation, similar to those obtained for hover. With increasing speed the longitudinal modes vary as illustrated in the following Figure 4.4. Shown are the roots with and without horizontal tail to point out its importance for the angle-of-attack stability. Damping becomes less without horizontal tail for increasing speed. Including the horizontal tail usually moves the long period hover mode into the left half-plane, and pitch and vertical real roots are transformed into a short period highly damped oscillatory mode. Since the tail effectiveness is reduced at lower speed due to dominating rotor and fuselage wakes, this result may not always be true. But the basic tendency can always be found for most helicopter analysis, and handling qualities are improved significantly. The primary response of the helicopter to control inputs and gusts is again vertical and pitch accelerations with minimal longitudinal acceleration. Controlling the longitudinal motion is therefore, as in hover, indirect and can be observed only in the long-term response. For further simplified analysis, short and long term response could be treated separately as done in [8].


44

Re s

Im s

forward flight roots

without horizontal tail


with horizontal tailhover roots

Figure 4.4: Influence of forward speed and horizontal tail on longitudinal poles

In general, pitch control sensitivity is high, and a longitudinal command results in a reasonable normal acceleration. Nevertheless, there is a delay after applying cyclic control input until normal acceleration develops. It can cause difficulties in helicopter control if this lag is too large. To evaluate handling qualities with respect to this characteristic, the time until maximum acceleration is achieved can be used. Common specifications define the desired response due to step inputs and are given in [8]. The goal is to ensure satisfying acceleration and handling qualities during flight maneuvers. Important influence of pitch and roll damping can again be provided by rate feedback with a stabilizer bar or a control rotor as already mentioned earlier. Longitudinal flying qualities are much improved, since steady-state angular rates develop faster, and a faster response of normal acceleration due to control input is achieved. Lateral dynamics of forward flight consist of side velocity, yaw rate and roll attitude, including side gust velocity. Yaw velocity of the body axes results in a lateral acceleration. Main rotor side forces and moments due to a yaw rate are small. Likewise small are torque changes due to power changes compared to the yaw moment produced by the tail rotor thrust. Therefore, the coupling of yaw and lateral dynamics is mainly due to the yaw moment produced by lateral velocity and the lateral acceleration produced by the yaw rate. Directional stability is governed by the influence of side velocity on the tail


45

rotor thrust. As in hover, the lateral dynamics result in a real stable yaw root, a real roll root, and an often slightly unstable lateral oscillation. For increasing forward speed, directional stability is increased, and the inertial coupling transforms the oscillatory mode in hover into a stable, short period oscillation. A typical root locus plot for the lateral dynamics is shown in Figure 4.5 for increasing forward speed.

uncoupled roll roots


hover roots

Re s

Im s

Figure 4.5: Influence of forward speed on lateral poles

Both real roots move to the right; the yaw mode might even approach the origin for a further increase in forward speed. In contrast to hover the lateral velocity produces a force opposing the motion. Lateral cyclic input commands mainly a roll rate with a small lag due to inertial forces; lateral velocity builds up again slowly. As for the longitudinal motion the lateral velocity can only be influenced indirectly through a commanded roll rate. To include all aerodynamic effects due to fuselage, rotor and tail interference, the fully coupled dynamics should be considered. All the influences should be modeled and included again in the force and moment calculations. Depending on the application, some of these effects may be neglected.


46

4.2.3 Stability Augmentation with a Control Rotor

As mentioned in the previous chapters, handling qualities of a helicopter can be much improved by using an automatic control system like a control rotor. An additional gyro sensing the inertial forces is introduced which acts on the main rotor. In the case of a control or servo rotor this control system is fully mechanical. The control rotor is attached to the rotor shaft on top of the main rotor, with its blades parallel to the main rotor plane with a 90° phase shift. It rotates therefore with the same angular velocity about the shaft axis as the main rotor. As for the main rotor, the motion of the smaller control rotor can be described by a lateral and longitudinal tilt of the rotor plane relative to the non-rotating hub frame. When roll and pitch inertia damping is added, this system acts as a mechanical damper as long as no airfoil or an additional blade section is attached to the control rotor. In Section 5.4 the equations for the control rotor with aerodynamic blade sections are given, and the similarity with the main rotor is used to derive the equations of motion. A direct mechanical link connects control rotor and main rotor blades. The control rotor plane tilt is transferred to the main rotor blades as cyclic blade pitch. Blade-feathering moments are also transferred back from the main rotor blades to the control rotor. If cyclic control inputs are fed through the control rotor to the main rotor blades, the control system provides lagged rate feedback of helicopter pitch and roll motion. This greatly improves handling qualities in hover and forward flight. The damping for roll and pitch axes is equal for this mechanical system. The control rotor with two rods and small airfoils on each end as designed for the R-50 Helicopter is shown in Figure 4.6. A rotor of this kind is often called a Hiller control rotor. More details on the mechanical linkage can be found in Figure 3.8 and [29]. The pilot controls cyclic pitch of the control rotor airfoils, producing an aerodynamic moment on the gyro resulting in a tilt of the control rotor plane. Some control rotor designs might also provide a direct input from the swashplate to the main rotor blades. Both inputs of control rotor tilt and direct pilot input are then mechanically mixed inputs to the main rotor. An additional effect of the Hiller control rotor important for full size helicopters is that control forces are less sensitive to the main rotor motion itself. Feedback of control forces into the pilot sticks or into the actuators can be reduced. This might not be always desirable since the pilot likes to feel the necessary control forces to keep the aircraft in a specific flight condition. This force can of course be provided by an artificial-feel-unit, resulting in a higher complexity of the control system. The control rotor shown in Figure 4.7 is a typical control rotor as used for two-bladed, single-rotor helicopters. The


47

mechanical linkage can be fairly complex depending on the desired portion of input going through the control rotor or directly to the main rotor. The system used for the R-50 experimental helicopter is described in Chapter 5.4 based on measured geometry of the linkage and rotor data available.

Figure 4.6: R-50 control rotor providing rate feedback [29]

Some assumptions are made to simplify modeling and assure real-time capability. The mathematical models for the helicopter simulation will be described in the following chapter for the single components, and the basic dynamic features will then be examined for a linearized helicopter system.

Chapter 5 Mathematical Modeling

48

Chapter 5

MATHEMATICAL MODELING

This chapter describes the mathematical helicopter model that is used in the current version of the real-time flight simulator. It is based on a Minimum-Complexity Helicopter Simulation Math Model developed by NASA [1]. Additional components like control rotor, actuator models and sensor models are added to increase the fidelity of the simulation.

5.1 General Helicopter Model

When the Minimum-Complexity Helicopter Simulation Math Model was developed by NASA in 1988, aspects like computational delays, costs and inflexibility of commonly used complex simulation models made a real-time simulation not very efficient with respect to better engineering understanding of specific handling quality features. In addition, a detailed knowledge of a specific helicopter was and still is not always given. This fact further complicates the implementation of any helicopter of interest into a real-time simulation process. The goal was to develop a build-up of individual vehicle components described by equations addressing the features associated with those components. Only basic data sources such as flight manuals or system component specifications should be necessary to achieve satisfactory results. The model that was used throughout the entire flight envelope was then further investigated and validated for some specific helicopter systems. Refinements and extended modeling was necessary to obtain satisfactory results for specific helicopters. Important features of the desired math model are:

• First-order flapping dynamics for main rotor (coupled or uncoupled) • Main rotor induced velocity computation • Rigid body degrees of freedom • Realistic power requirements over the desired flight envelope • Rearward and sideward flight without computational singularities • Hover dynamic modes (longitudinal and lateral hover cubic, rotor-body coupling

with flapping) • Dihedral effect


49

• Correct transition from hover to forward flight • Potential for rotor RPM variation

More detailed aspects on some of the components used and modified are given in the following chapter. A model for engine RPM variation is about to be developed at Georgia Tech but not yet included in the simulation software.

5.2 Rigid Body Model

The six rigid body DOF model was already introduced in Chapter2.1. For manned or unmanned simulation of flight, this model is accurate in describing system response for flight mechanical and controlling purposes. Therefore the basis for the helicopter model are the equations of unsteady motion of the rigid body as already mentioned and is derived in [4]. The equations were rederived in the body axes system to avoid singularities due to undefined values for angle-of-attack and side-slip-angle in hover condition. These equations are listed in Appendix B. The equations are in general valid over the entire flight envelope for a typical aircraft. For rotorcraft gyroscopic effects due to main and tail rotor become important, and the force and moment components for a two-bladed main rotor, a tail rotor and a control rotor are added. The assumption of the xB-zB- plane as a plane of symmetry is also made. The equations will be given for a counterclockwise spinning main and tail rotor (viewed from above and left respectively). A change of directional rotation will affect some of the derived expressions. Introducing a directional variable, λ, the direction of rotation can be chosen arbitrarily, dependant on the system that needs to be simulated. The use of this parameter is based on a tail rotor in ‘pull configuration’. This implies that the vertical tail is always mounted in the downstream direction of the tail rotor. Since rotating systems add dynamics to the rigid body equations, the two most important subsystems, main and control rotor, are described in the next section in more detail.

5.3 Main Rotor Model

Responsible for the unique and complex characteristics of a helicopter, the main rotor model described in [1] is modified in order to fit the very different dynamics of the model helicopter it is applied to and still guarantee real-time capability. For the final implementation in the simulator the nonlinear dynamic equations for the main rotor are substituted by its steady-state representation. This is justified for several reasons, as described later in this chapter.


50

The basis for the model is taken from [13], extended in [14] and simplified in [2, 3] to represent the desired first-order flapping model of the main rotor dynamics. Higher-order flapping dynamics were not considered. This first-order model results in a TPP lag affecting the main rotor thrust vector orientation and therefore the immediate response to control inputs. Uniform inflow distribution is first assumed to compute thrust and induced velocity. Moment equations are rederived from [15, 16] in the body-axis system. Based on classical momentum theory the equations to compute thrust and induced velocity can be found to be

( )4

2aBcRvwT iblade

Ω⋅−=

ρ (5.1)

and

2ˆ

22ˆ 2222

2 vA

Tvvi −

⋅

+

=

ρ , (5.2)

where

⋅+⋅Ω⋅+= twistcollrblade Rww θθ

43

32

, (5.3)

( ) vbuiaww sssr ⋅−⋅++= 11 , (5.4)

( )ir vwwvuv ⋅−⋅++= 2ˆ 2222 , (5.5)

and 2RA π= . (5.6)

Since all angles are small, it is assumed that sin(α)≈α and cos(α)≈1. In Equation 5.3 the collective input θcoll to the main rotor blades occurs, influencing directly main rotor thrust. Most blade designs are based on a fixed change of pitch angle over the blade radius, the so-called blade twist θtwist. Assuming a linear twist from the root to the blade tip is mostly sufficient for this kind of analysis. The total velocity of the main rotor blade relative to the air mass is defined by wblade, wr is velocity of the main rotor disk relative to air due to body velocities ub, vb and wb. The velocities need to be transformed with the TPP tilt angles a1s, b1s and the shaft tilt angle is from the body frame to the TPP. Induced velocity and thrust can be computed for a given thrust with only a few iterations using Equations 5.1-5.5 until convergence is reached.


51

The dihedral effect of the main rotor describing the change of TPP tilt due to lateral and longitudinal velocity is dependent on thrust and induced velocity respectively. The equation for the lateral tilt derivative can be found in [6] to be

+

⋅⋅

Ω−=

2821 TTs CaC

Rdvdb

σ , (5.7)

where σ = Bc/πR is the rotor solidity. Assuming equal effects on lateral and longitudinal motion, the longitudinal derivative is

bb dv

dbduda 11 −= . (5.8)

The assumption of constant derivatives in hover is true as long as uniform inflow is assumed. For fast forward flight non-uniform inflow becomes necessary and this assumption is only an approximation. From experimental results on full size helicopters a more accurate estimation is given by introducing an empirical variable, fwake, considering low or high-speed effects on those derivatives, as described in [1]. This variable will be used in the dynamic rotor equations. With respect to cross-coupling of longitudinal and lateral TPP tilt two already mentioned components need to be considered. Mechanical linkages from swashplate and control rotor to the main rotor cause an additional flap angle due to a commanded blade pitch change. The geometry of this linkage and the rotor hub design result in an off-axis motion. Defined by the angle between flap hinge axis and an imaginary line drawn from the hinge to the pitch horn of the main rotor blade, as shown in Figure 3.6, the so called delta-3-angle, a cross-coupling coefficient is obtained in the form 31 tan δ=K . (5.9)

This coefficient also describes the mechanical feedback from flap angle to blade pitch, automatically decreasing pitch for an increasing flap angle. The second effect of cross-coupling can be derived from the fact that a flapping hinge with an offset from the axis of rotation also changes the natural frequency of the dynamic system as mentioned in Chapter 3.3.2. Since the R-50 helicopter is designed with such a


52

hinge offset, this additional effect might become important and therefore needs to be considered in the flapping equations. With the ratio computed by the hinge offset eMR relative to the main rotor radius RMR, this coefficient can be written as

f

MR

MR

MR

Re

KΩΩ

⋅⋅=43

2 , (5.10)

where ΩMR is the rotor rotational speed. The coefficient

⋅+⋅

Ω⋅=Ω

MR

MRMRf R

e38

116

γ (5.11)

describes the change of natural frequency due to the hinge offset. This coefficient is also a function of the so-called Lock number, expressing the ratio of aerodynamic and blade inertial forces. With the assumptions of a blade section two-dimensional lift curve slope (a≈6.0), approximately constant over the entire blade length, and with a constant chord length c, this non-dimensional Lock number for the main rotor can be written as

MRb

MR IRca

⋅⋅⋅=

4ργ . (5.12)

The blade chord length, usually also a function of the radius, was found to be constant except at the blade root, where a profile modification avoids an excessive aerodynamic loss due to reverse flow. For a given blade inertia Ib relative to the flap hinge, the total cross-coupling coefficient Kc can then simply be written as the sum of the previously computed coefficients 21 KKK c += . (5.13)

In the final analysis of the helicopter dynamic response, this coefficient might be adjusted to fit experimental data, to correct for unmodeled spring-damper dynamics in the hinge or to compensate for further unmodeled cross-coupling effects not considered in the given approximation.


53

Due to the interaction of the rotor wake with the fuselage especially at low speed, the dihedral effect will become more effective. This will directly influence the cyclic flapping due to changes in forward or sideward velocity. The empirical value for fwake dependants on flight speed and can be chosen based on experience or experimental data, if available and necessary. For the R-50 flying at low speed, this variable will always be equal to 1. The final equations for the TPP tilt angles relative to the HP can then be written in the form

)21(11,111 wake

sscMRs fu

duda

bKBaa ⋅+⋅⋅+⋅⋅+−= λ (5.14)

and )1(11,111 wake

sscMRs fv

dvdb

aKAbb +⋅⋅+⋅⋅−+= λ , (5.15)

with b1s and a1s as the lateral and longitudinal TPP tilt relative to the HP, the TPP tilt angles with respect to the swashplate are defined by b1 (lateral) and a1 (longitudinal). The influence of the cross-coupling is dependent on the direction of rotor rotation. Therefore the directional parameter needs to be considered in those terms. The inputs to the blade pitch angle are given by A1,MR and B1,MR. These are the resulting blade pitch changes commanded by the pilot stick inputs, transmitted by the swashplate tilt and the mechanical linkage from swashplate to the pitch horn. The cross-coupling in Equations 5.14 and 5.15 influences the TPP tilt and depends on the direction of rotor rotation. The lateral and longitudinal swashplate tilt A1,SP and B1,SP commanded by the pilot is distributed by mechanical linkages to the control rotor as well as directly to the main rotor. The largest portion is fed through the control rotor. The resulting tilt of the control rotor plane due to control rotor blade pitch changes, commands the main rotor blade pitch through mechanical linkages. The resulting main rotor input can then be defined as CRsSPMRMR kAkA ,,1,1 ββ ⋅+⋅= (5.16)

and CRcSPMRMR kBkB ,,1,1 ββ ⋅−⋅= , (5.17)

independent on the direction of rotor rotation. The coefficients kSP and kβ prescribe the amount of commanded swashplate tilt and resulting control rotor plane tilt finally acting on the main rotor blade pitch. The additional states βs,CR and βc,CR describe the lateral longitudinal TPP tilt of the control rotor with respect to the HP as shown in Chapter 5.4. The equations of motion for the control rotor are given in the next chapter, the


54

coefficients kMR and kβ are found by measuring linkage lengths and angles of pitch changes due to a swashplate tilt. The collective input commanded by the pilot is fed directly to the main rotor and therefore does not appear explicitly in the main rotor flapping equations. It is assumed that the collective pitch directly influences the induced velocity through the rotor disk and therefore thrust. For an equally over the rotor azimuth distributed change of blade pitch due to collective stick input and resulting swashplate displacement, it can also be assumed that effects on lateral and longitudinal TPP tilt are small and will be neglected. The induced velocity and thrust iteration uses the collective input while computing induced velocity due to collective pitch and linear blade twist in Equation 5.3. The final equations of main rotor flapping motion in the rotating rotor frame relative to the swashplate can be written in the simplified form qbaa offin −⋅⋅−⋅−= 111 ωλω& (5.18)

and pabb offin −⋅⋅+⋅−= 111 ωλω& . (5.19)

The simplified flap rate coefficients ω in and ωoff for in- and off-axis changes of the flap angle can be computed with no flap-cross-coupling, 0=offω and fin Ω=ω , (5.20)

or including the flap-cross-coupling,

2

1

ΩΩ

+

Ω=

f

MR

MRoffω and off

f

MRin ωω ⋅

ΩΩ

= . (5.21)

Equations 5.21 give an approximation for the effect of cross-coupling in the flap rates including cross-coupling in the flap angles. In [1] it is also mentioned, that a better fit to experimental data might be obtained for a neglected rate-cross-coupling in the simulation. With respect to the hinge offset one more aspect should be considered. A pilots cyclic input causes a change in flapping and finally TPP tilt angles, also causing an aerodynamic blade moment acting on the helicopter only for a nonzero hinge offset. For no hinge offset the blades are free to flap and no moments are transferred to the rotor hub.


55

The only rotor moments acting on a helicopter with no hinge offset (eMR/RMR=0) are generated by rotor forces and the moment arm given by the hub distance to the helicopter CG. An approximation for the in-axis blade moment due to cyclic inputs with a hinge offset is given in [6] by Equation 5.22. Equation 5.23 represents the aerodynamic and inertial flap-cross-coupling moment including a hinge offset. The computed moments can be written in the body-fixed frame as

MR

MRTipMR R

eVRcaB

dAdL

dBdM ⋅⋅⋅⋅⋅⋅⋅==

61

222

11

ρ , (5.22)

and

⋅

⋅⋅Ω⋅⋅⋅⋅⋅⋅==

MR

MRMRMR

ss ReaRRcB

dbdL

dadM

γρ

λ22

11

)(43 . (5.23)

In the previous chapter it has already been mentioned, that the main rotor dynamics are usually very fast compared to the rigid body dynamics of the helicopter. Therefore a so-called quasi-steady-state model of the main rotor can be used to describe the dynamics of the body-rotor dynamics. This approach is also very useful since additional dynamics result in additional states that need to be integrated. This should be avoided to maintain real-time capability of the code. The DOF’s to be added are longitudinal and lateral tilt angles of the main rotor TPP and of the control rotor TPP. This would result in a model extension from 6 to 10 DOF’s. Investigating the influence of control rotor in Chapter 5.7, it is found that the control rotor dynamic modes are about as fast as the helicopter rigid body modes. The control rotor dynamic modes become coupled with rigid body modes, and a steady-state model for the control rotor is not sufficient to represent the real system dynamics. To avoid the 10 DOF model the steady-state model for the main rotor will be used to represent the rotating subsystem dynamics. Characteristics with respect to flight dynamics and control aspects remain accurate for this approximation. Nevertheless the two control rotor states are added to the equations of rigid body motion. This model is described in Chapter 5.4 and in [10] in more detail. The control rotor is treated like a smaller rotor with only a limited blade profile. The following section gives the quasi-steady-state equations for the main rotor derived from the previous dynamic equations.


56

Quasi-Steady-State Equations for Main Rotor Dynamics

For the valid steady-state blade motion it is assumed that the desired orientation of the TPP is reached instantaneously. In fact, the transient flap motion is in general a highly damped oscillation with an approximate time constant τ ≈16/γ (see [6]). The resulting time-to-half amplitude, t1/2=0.693⋅τ, typically corresponds to 90° of the rotor azimuth, and the transient response dies out after less than one revolution of the main rotor. Steady-state equations are obtained by the necessary conditions for steady-state 0,0 11 == ba && . (5.24)

Solving Equations 5.14 and 5.15 for a1 and b1 and substituting in Equations 5.18 and 5.19, the steady-state equations for the main rotor TPP angles relative to the HP are

( ) ( )( )34243122

21

1

1TTpTTTqT

TTa offinoffins ⋅⋅−⋅−⋅+⋅⋅−⋅+⋅⋅

+= ωλωωλω (5.25)

and ( )21341

1

1TaTTp

Tb soffins ⋅−⋅⋅−⋅−⋅= ωλω . (5.26)

The terms T1, T2, T3 and T4 represent abbreviations given in Equations 5.27. ( )coffin KT ⋅−−= ωω1

( )offcin KT ωωλ +⋅−⋅=2 (5.27)

⋅+⋅⋅−= )21(1,13 wake

sMR fu

duda

BT

+⋅⋅+= )1(1,14 wake

sMR fv

dvdb

AT

The final Equations 5.25 and 5.26 are then used to compute the TPP tilt angles a1s and b1s relative to the main rotor HP. The assumption of a thrust vector perpendicular to the TPP is used to transform the thrust components into the body-fixed frame by


57

( )ssMRbMR iaTX +⋅−= 1, sin , (5.28)

( )sMRbMR bTY 1, sin⋅= (5.29)

and ( ) ( )sssMRbMR biaTZ 11, coscos ⋅+⋅−= . (5.30)

For a given flight condition, the main rotor power calculations need to include induced power Pi, profile power Ppr due to friction between air and blade surface, climb power Pc and parasite power Ppa to overcome fuselage drag. The final equations are given in Equations 5.31.

( ) ( )( )2220 6.442

vuRRRcbc

P MRMRMRD

pr +⋅+Ω⋅Ω⋅⋅⋅⋅

⋅= ρ (5.31)

iMRi vTP ⋅= (5.32)

Ec zgmP &⋅⋅= (5.33)

( ) ( ) ( )( )iFusFusFuspa vwZvYuXP −⋅−⋅−⋅−= (5.34)

Equation 5.31 represents a commonly used approximation for profile power, where the effective frontal area of the main rotor producing the aerodynamic drag is given in [6] by cD0⋅b⋅c⋅RMR. The representative profile drag coefficient of the blade is defined by cD0,

taken from airfoil data for a particular rotor blade profile. Force components XFus, YFus and ZFus are computed in the body-fixed frame. The main rotor torque can then be computed by

MR

MRMRT

PM

Ω=, , (5.35)

where cpaiprMR PPPPP +++= . (5.36)

Main rotor moments have to be transformed into the body-fixed frame, and the final equations are


58

( )sMRsshubMRbMR bkAadAdL

bdbdL

hYL 11,111

11

, ⋅−+⋅⋅+⋅+⋅= λ , (5.37)

( )ssMR

shubMRhubMRbMR akBbdB

dMa

dadM

hXdZM 1111,1

11

, ⋅−+−⋅⋅+⋅+⋅−⋅= λ (5.38)

and MRTbMR MN ,, ⋅= λ . (5.39)

The vertical and horizontal distance from hub to helicopter CG represent the moment arms hhub and dhub, measured in the body-axis frame along the zB- and xB-axis. It is assumed that there is a negligible offset of the hub in direction of the yB-axis. Since for dynamic characteristics the most important components are the main and control rotor, only those two subsystems are described in more detail. Further components like fuselage, tail rotor, horizontal tail and wings will be briefly mentioned, but are described in more detail in [1, 8]. Steady-state equations for the main rotor were assumed to be sufficient in the previous chapter. In the following chapter the dynamic control rotor equations of motion are given, which will strongly affect the helicopter dynamics. Some results comparing the dynamic characteristics of the helicopter with and without control rotor will be shown in Chapter 5.7.

5.4 Control Rotor Model

The importance of the control rotor due to damping added to the system and its function as a rate feedback system were already discussed. In order to model the helicopter more accurate it is necessary to investigate the control rotor in more detail. Therefore this rotating system is treated as a smaller rotor with similar DOF’s. The kind of control rotor used at the R-50 (Figure 4.7) is called a teetering rotor. More details on those rotor systems can be found in [6, 8, 15]. Perhinschi investigated in [10] the influence of a control rotor on the linear system dynamics of a model helicopter as part of a controller design. The basic equations of motion are taken from main rotor flapping equations with the similar assumptions made previously. Additionally, since a teetering rotor is modeled, the coning angle is negligible. With only small blades at the rod ends, the aerodynamic forces are small compared to the inertial forces. This will result in a very small Lock number for the control rotor.


59

Writing the equation of basic blade motion for a clockwise rotating system, CRbCRsCRbCRcCR ,,,, sincos ψβψββ ⋅−⋅−= , (5.40)

and computing moment equilibrium for a rotating and flapping blade, we obtain the moment equation in the form

CRb

CRaCRCRCR I

M

,

,2 =⋅Ω+ ββ&& . (5.41)

Equations 5.40 and 5.41 include only variables referring to the control rotor. Notice that the control rotor is mounted on top of the main rotor with a 90° phase shift. The moment due to aerodynamic forces on the control rotor blade is represented by Ma,CR. An additional moment due to gyroscopic effects of the rotating control rotor in the rotating rigid body system needs to be considered. CRbCRCRbCRgyro pqM ,, cos2sin2 ψψ ⋅⋅Ω⋅−⋅⋅Ω⋅−=∆ (5.42)

The angular velocities of the rigid body motion p and q as well as the rotational speed ΩCR of the control rotor are included. Since both rotating systems, control and main rotor, are linked on top of each other, their rotational speed is the same. The aerodynamic moment due to blade flapping and feathering, rigid body roll and pitch rates and additional moments due to changes in wind velocities (Mgust) can be found to be

( )

( )+⋅⋅

Ω⋅+⋅⋅

Ω⋅−

⋅−⋅−⋅+⋅

Ω⋅−⋅⋅Ω⋅=

CRbgustCRbCR

CRbCR

CRbCRCRbCRCRCR

CRba

Mqp

BAIM

,,,

,,1,,12

,

cos8

sin8

sincos88

ψψγ

ψγ

ψψγ

βγ

ξ &

(5.43)

In Equation 5.43 the additional gust moment is written in a general form as a function of the control rotor blade azimuth ψ b,MR. The equation of feathering blade motion is of the form CRbCRCRbCRCR BA ,,1,,1 sincos ψψθ ⋅−⋅−= , (5.44)


60

since there is no collective input into the control rotor. Collective input from the pilot is directly fed to the main rotor blades. The so-called limited extension, ξ, of the control rotor blade is introduced to describe the limited aerodynamic force due to the only small profile at the end of the bar. It is assumed that the aerodynamic moment of the blade section of the control rotor (Ma,CR) is equal to the difference of the aerodynamic moment (Ma,ent) of a blade with a profile over the entire control rotor radius RCR, and the moment (Ma,lim) of a blade whose profile would extend from the hub to the point where the true control rotor blade starts Rlim (see Figure 5.1).

R CRl b R lim

Ω CR

Figure 5.1: Control Rotor of the R-50 Helicopter (view from top)

The aerodynamic moment equation therefore becomes lim,,, aentaCRa MMM −= . (5.45)

With the linear blade theory it follows that mCRCRmCRCRCRCRa RcaRcaM cc 24

lim24

, ⋅Ω⋅⋅⋅⋅−⋅Ω⋅⋅⋅⋅= ρρ . (5.46)

The chord length cCR is measured, and the moment coefficient cm is estimated from available airfoil data and assumed to be constant. The limited radius can be expressed as

bCR lRR −=lim and substituting in 5.46 it produces

−−⋅⋅⋅Ω⋅⋅⋅=

4

42, 11c

CR

bCRmCRCRCRa R

lRcaM ρ . (5.47)


61

The last term in parenthesis is defined as the limited extension parameter

−−=

4

11CR

b

Rl

ξ . (5.48)

A linear lift curve slope aCR can be computed as a function of the blade aspect ratio AR

AR

aCR 21

2

+=

π , with

CRb

CR

cll

AR⋅

=2

. (5.49)

The gust terms in 5.43 due to change of translational velocity are estimated according to [8 and 10]. The flapping components can be found to be

⋅Ω

−⋅

=∆

CRCR

CRs Rv

M

M

β

µβ&

, (5.50)

and CRCR

CRc Ru

M

M

⋅Ω⋅

=∆

β

µβ&

, , (5.51)

where

CR

TT Ca

CM

⋅+

⋅⋅

=24

12σµ (5.52)

and

( )

CR

T RR

alRcl

C

⋅Ω⋅⋅

⋅⋅

−⋅Ω⋅⋅

= 22

0

22

2

π

θ

. (5.53)

The aerodynamic derivative due to the flapping velocity is approximately equal to

8ξγ

β

⋅=&M . (5.54)


62

This represents a very good approximation for hover and slow forward flight considering the limited extension coefficient and the small Lock number for the control rotor. For fast forward flight this approximation is only a rough estimate. Additionally, other aerodynamic effects due to increasing velocity become more important, and the constant moment derivative with respect to flapping velocity can still be used for most applications. Substituting these equations in Equations 5.50 and 5.51, the additional flapping angles due to wind velocity are obtained as

vR

CRCRs ⋅

⋅Ω⋅⋅−=∆

ξγθ

β ,0, (5.55)

and uR

CRCRc ⋅

⋅Ω⋅⋅=∆

ξγθ

β ,0, . (5.56)

The constant control rotor blade pitch θ0,CR can be measured. Computing the derivatives with respect to time in Equation 5.40 and substituting in 5.41, the harmonic balancing method can be applied. Two equations are obtained including in- and off-axis flap velocities. Neglecting the very small amount of cross-coupling for the control rotor, the equations can be written with respect to the hub plane in the form

pq

A CRsCRCR

CRsCR

CRs +

∆+

Ω+−⋅

⋅⋅Ω−⋅

⋅⋅Ω−= ,,1,, 1616

βξγ

βξγ

β& (5.57)

qp

B CRcCRCR

CRcCR

CRc −

∆+

Ω−−⋅

⋅⋅Ω+⋅

⋅⋅Ω−= ,,1,, 1616

βξγ

βξγ

β& . (5.58)

These equations are independent of rotor rotational direction since cross-coupling terms are totally neglected. With Equations 5.57 and 5.58 there are two additional DOF’s added to the six rigid body DOF’s used for the helicopter simulation model. The 8 DOF model will be used for further analysis. The control rotor inputs are given by the geometry of the mechanical linkage of control rotor and swashplate, represented by the coefficient kCR in Equations (5.59).


63

For the R−50 it is measured that kCR = 1.95, and it follows that SPCRCR AkA ,1,1 ⋅=

(5.59) SPCRCR BkB ,1,1 ⋅=

The tilt of the control rotor plane represented by the two additional DOF’s also represents the output of the control rotor and via further mechanical links finally the input to the main rotor as already described in Equations 5.16 and 5.17. The helicopter components not yet described in detail will be summarized in the following section.

5.5 Model of Fuselage, Wing and Tail

To complete the modeling, equations are needed expressing the aerodynamic effects arising from fixed aerodynamic surfaces like wings, fuselage, horizontal and vertical tail, but also the important influence of the tail rotor mainly responsible for yaw stability and control. This chapter intends to give a brief overview of the used models. For details References [1, 6, 8] are very helpful.

Tail Rotor

The tail rotor of the R-50 in a “pull-configuration” mounted on the left (viewed from aft) is modeled similar to the main rotor. Flapping degrees of freedom and any cross-coupling are neglected. Thrust, induced velocity and power calculations including drag and induced power are used similarly as already explained in the main rotor model. The only control input to the tail rotor is collective pitch, influencing directly induced velocity and therefore thrust. The rpm of the tail rotor is assumed to be constant since the main and tail rotor are directly coupled over a belt drive. For power calculations the additional power to run the tail rotor also needs to be considered.

Fuselage

The fuselage is modeled as a virtual flat plate creating no lift, but drag only. A quadratic aerodynamic form expresses drag in forward and backward flight limiting the maximum speed. Drag is also computed for sideward flight, and rotor related down wash results in additional drag forces on the fuselage. These effects are directly related to power losses for that the main rotor has to compensate. The quadratic form is taken from [1] representing a simplified application of Lambs theory [17]. Only approximated data is


64

necessary to compute the fuselage forces and moments. The estimated effective fuselage areas with respect to all three axes and the estimated position of the center of pressure of the fuselage are the main data for fuselage force calculations. Note, that the modeled experimental helicopter has a position of CG different from the original helicopter. Not yet considered changes of mass and moments of inertia due to system modifications for the flight tests need to be measured and computed (e.g. data link, sensor packages, power supply and on-board PC).

Wings, Horizontal and Vertical Tail

For the horizontal and vertical tail it is assumed that mainly lift is produced. The helicopter as purchase from YAMAHA does not include a horizontal tail. For better handling quality and for mounting GPS- and data link antenna a horizontal tail was designed (see Appendix E). Since drag due to these components is small compared to their vertical aerodynamic forces, additional drag will not be considered in this model. Nevertheless, the effects of aerodynamic stall are included in the equations. The basic model is again the quadratic form previously mentioned. For the vertical tail it must be considered that it experiences airflow due to the tail rotor induced velocity. Dependent on the flight condition, the horizontal tail in general might be located in the down wash field of the main rotor. To include this effect in the computations, a down wash field of triangular shape has been assumed in [1]. For additional wings the lift and drag components can also be computed similar to the horizontal tail approach. Additional drag terms are then becoming important for further power calculations. As the tail a wing experiences the additional airflow due to the main rotor downwash field and influences the aerodynamic wing forces and moments. The R-50 helicopter does not possess such an additional wing. The calculation is therefore not necessary but will be included in the simulation model for a use in future applications. After modeling the important helicopter components and combining them in the simulation, a linear model can be extracted from the equations of motion as mentioned in Chapter 2. Results and the analysis for model structures including and neglecting the effect of the control rotor are shown in the following section.


65

5.7 Simulation Results for the Linarized Model

After modeling the single components in the previous chapter, it is of interest to determine how similar the overall system dynamics are to the real helicopter. Therefore this section comments on a linearized model obtained by the small-disturbance theory described in Chapter 2.2, for which the dynamics can be written in the previously introduced form (Equation 2.1) uGxFx δ⋅+⋅=& .

Of further interest is the influence of the control rotor on the pole position of the rigid body modes. This chapter investigates the dynamic characteristics mainly for hover, since a comparison with flight test data for forward flight was not possible at this time. Limited range of the pilots transmitter to guarantee safe operation does not allow a steady-state forward flight for the time necessary to collect reasonable flight data. The maneuvers that needed to be flown for the purpose of system identification of the unmanned helicopter were already challenging enough without the risk of loosing control of the aircraft. More details on the system identification and the flight tests can be found in Chapter 7. The actual validation of the simulation model with flight tests was done only for the hover condition for the reasons given above. Since simulation is not restricted with respect to time and limited range, almost all maneuvers and flight conditions can be simulated. Using small perturbations in the states and control inputs from the trimmed flight condition, the system response of the nonlinear model is computed and force and moment derivatives can be obtained. These stability and control derivatives represent the elements of the system and control matrices F and G. The investigated system obtained from the simulation is the 8 DOF system consisting of the 6 rigid body DOF’s and the two additional control rotor states. This linearized system is given based on the following definition of state and input vectors [ ] T

CRcCRsrpvqwux ,, ββφθ= , (5.60)

=

=

PitchCollRotorTailTiltSwashplateLateral

TiltSwashplatealLongitudinPitchCollRotorMain

AB

u

TRcoll

SP

SP

MRcoll

.

.

,

,1

,1

,

θ

θ

δ . (5.61)


66

To compare this model including the control rotor with a simulation model neglecting the control rotor, the two state variables for the control rotor TPP tilt angles βs,CR and βc,CR are dropped. The mixing coefficients of the input to main rotor and control rotor are kMR=1 and kβ=0. Furthermore it should be mentioned that the inputs to the simulation without control rotor, A1 and B1, are directly given to the main rotor blades, such that a swashplate tilt results in a blade pitch change of the same magnitude. This results in a one-to-one transformation of swashplate tilt to main rotor blade pitch, for both collective and cyclic inputs. As already mentioned this one-to-one transformation of cyclic inputs is not true for the system including the control rotor. Therefore differences in the G-matrix are expected in the elements related to cyclic inputs. Appendix C shows all elements of the system and controls matrices for the extended model as functions of force and moment derivatives, trim states, basic helicopter data given by mass and inertia about principal axis and control rotor data. The assumptions made in Chapter 5.4 to simplify the control rotor modeling are already included. The trimmed flight conditions at which the system is linearized are computed by iteratively varying the helicopter states and inputs until force and moment equations result in the desired flight condition. For hover, the sum of all forces and moments on the helicopter CG have to be zero. The values of trim states for hover are dependent on helicopter forces and moments only. The control rotor does not generate any additional forces or moments transferred to the helicopter body. Therefore the only difference for both systems in trim, with or without control rotor, can be found in the required control inputs.

θ0

[rad]

φ0

[rad]

B1,SP (long. cycl.)

[rad]

A1,SP (lat. cycl.)

[rad]

collMR

[rad]

collT R

[rad]

Hover, no Control rotor -0.0464 0.0561 -0.0447 -0.0108 0.1085 0.0046

Hover,

with Control Rotor

-0.0464 0.0561 -0.007 -0.0291 0.1085 0.0046

Table 5.1: Trim values in simulated hover for R-50, with and without control rotor

The linearized matrices for this flight condition excluding the control rotor are given in Tables 5.2 and 5.3, with the longitudinal and lateral derivatives listed in the matrix form given in Appendix C. Columns and rows are marked with the states and inputs they are referring to.


67

u w q theta v p phi r

u -0.0469 -0.0296 1.4106 -32.1394 -0.0034 0.21 0 0

w -0.0311 -0.6892 -0.1084 1.4901 -0.0032 -0.0012 -1.8032 0

q 0.1559 -0.068 -5.8228 0 0.0465 0.7644 0 0

theta 0 0 0.9984 0 0 0 0 -0.0561

v 0.0047 -0.0039 0.2102 0.0837 -0.0998 -1.443 32.0888 0.4197

p 0.2176 -0.0148 -2.3849 0 -0.4276 -17.881 0 0.428

phi 0 0 -0.0026 0 0 1 0 -0.0464

r -0.0116 -0.2397 0 0 0.2881 0.1293 0 -1.7523

Table 5.2: Analytically obtained system matrix in hover, no control rotor

coll MR B1 A1 coll TR

u -18.1857 32.5193 3.4817 0

w -391.015 -1.4548 -0.3507 0

q -30.9937 -85.857 -50.0334 0

theta 0 0 0 0

beta_c_cr 0 -2.1633 0 0

v -2.3784 -3.4852 32.5522 -43.3123

p -9.0806 -155.97 497.447 -43.3504

phi 0 0 0 0

r -75.505 0 0 180.7083

beta_s_cr 0 0 2.1633 0

Table 5.3: Analytically obtained control matrix in hover, no control rotor

To investigate the helicopter dynamics without the control rotor, the eigenvalues for the system matrix are computed. Damping and frequencies of dynamic modes are listed in Table 5.4. Figure 5.2 illustrates the pole position of the hover modes in the complex plane for the coupled longitudinal and lateral motion. The two fastest real modes are the roll and pitch mode. Both are strongly dependent on the helicopter moment of inertia and the position of the CG, not exactly known for the investigated model helicopter. As shown in Figure 4.1 for the longitudinal motion, the pole position is also changing with the varying natural frequency of the flap motion. Mainly responsible for a change in this natural frequency is the flapping hinge offset. Decreasing the estimated equivalent hinge offset for the R-50 rotor system will further reduce the frequency of the roll and pitch mode. Adjusting the parameters can be used to obtain better agreement with the experimental results discussed in Chapter 7. Heave and yaw modes have a similar relationship as discussed in Chapter 4. Both oscillatory modes are slightly unstable at almost the same frequency.


68

Re Im Damping Freq. (rad/sec)

Mode

0.0976 1.0065 -0.0965 1.0113 Lateral Oscillation

0.0976 -1.0065 -0.0965 1.0113

0.0102 0.8473 -0.012 0.8474 Longitudinal Oscillation

0.0102 -0.8473 -0.012 0.8474

-0.6887 0 1 0.6887 Heave

-1.8383 0 1 1.8383 Yaw

-2.1633 0 1 2.1633 Long. Control Rotor

-2.1633 0 1 2.1633 Lat. Control Rotor

-6.1697 0 1 6.1697 Pitch mode

-17.811 0 1 17.8108 Roll mode

Table 5.4: Eigenvalues, damping and frequencies of hover modes, no control rotor

Poles of Lateral and Longitudinal Motion in Hover, no Control Rotor

-1.5

-1

-0.5

0

0.5

1

1.5

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

Re

Im

Figure 5.2: Poles of coupled longitudinal and lateral motion, no control rotor


69

Including the control rotor states in the linearization about hover, the results for the system and control matrices are obtained and shown in Table 5.5 and 5.6. The same analysis of eigenvalues leads to results listed in Table 5.7 and Figure 5.3.

u w q theta beta_c_cr v p phi r beta_s_cr

u -0.0469 -0.0296 1.4106 -32.1394 -19.8693 -0.0034 0.21 0 0 2.1273

w -0.0311 -0.6892 -0.1084 1.4901 0.8889 -0.0032 -0.0012 -1.8032 0 -0.2143

q 0.1559 -0.068 -5.8228 0 52.4587 0.0465 0.7644 0 0 -30.5704

theta 0 0 0.9984 0 0 0 0 0 -0.0561 0

beta_c_cr 0.0101 0 -1 0 -2.1633 0 -0.0237 0 0 0

v 0.0047 -0.0039 0.2102 0.0837 2.1295 -0.0998 -1.443 32.0888 0.4197 19.8894

p 0.2176 -0.0148 -2.3849 0 95.3006 -0.4276 -17.881 0 0.428 303.9402

phi 0 0 -0.0026 0 0 0 1 0 -0.0464 0

r -0.0116 -0.2397 0 0 0 0.2881 0.1293 0 -1.7523 0

beta_s_cr 0 0 0.0237 0 0 -0.0101 -1 0 0 -2.1633

Table 5.5: Analytically obtained system matrix in hover, with control rotor

coll MR B1 A1 coll TR

u -18.1857 11.2387 1.2033 0

w -391.015 -0.5028 -0.1212 0

q -30.9937 -29.6722 -17.2916 0

theta 0 0 0 0

beta_c_cr 0 -4.2184 0 0

v -2.3784 -1.2045 11.25 -43.312

p -9.0806 -53.9049 171.918 -43.35

phi 0 0 0 0

r -75.505 0 0 180.708

beta_s_cr 0 0 4.2184 0

Table 5.6: Analytically obtained control matrix in hover, with control rotor


70

Re Im Damping Freq. (rad/sec) Mode


0.0309 -0.7663 -0.0402 0.7669

-0.0043 0.6423 0.0067 0.6423 Longitudinal Oscillation

-0.0043 -0.6423 0.0067 0.6423

-0.6838 0 1 0.6838 Heave

-1.9241 0 1 1.9241 Yaw

-4.0211 7.7222 0.4619 8.7064 Pitch + longitudinal CR

-4.0211 -7.7222 0.4619 8.7064

-10.011 15.3329 0.5467 18.3116 Roll + lateral CR

-10.011 -15.333 0.5467 18.3116

Table 5.7: Eigenvalues, damping and frequencies of hover modes, with control rotor

Poles of Lateral and Longitudinal Motion in Hover, with Control Rotor

-20

-15

-10

-5

0

5

10

15

20

-12 -10 -8 -6 -4 -2 0 2

Re

Im

Figure 5.3: Poles of coupled longitudinal and lateral motion, with control rotor

Two new complex pole pairs are found at frequencies similar to those obtained for the pitch and roll mode without control rotor. Stability of both oscillatory modes is improved, and the longitudinal oscillatory mode even becomes stable. Heave and yaw modes show only small changes. To illustrate the strong influence of the control rotor, Figure 5.4 and 5.5 show the longitudinal and lateral modes for both cases with and without control rotor


71

respectively in one plot. Also shown are the control rotor poles for the case of no coupling between control rotor and rigid body.

Open Loop Poles of Longitudinal Motion, Hover

-10

-8

-6

-4

-2

0

2

4

6

8

10

-7 -6 -5 -4 -3 -2 -1 0 1

Re

Im

no Control Rotor

decoupled Control Rotor

with Control Rotor

Figure 5.4: Hover poles of longitudinal motion, with and without control rotor

Figure 5.4 shows that the fast pitch mode and the longitudinal control rotor mode become one complex pole pair from the previous picture. As discussed in Chapter 4.2.3 the control rotor modes are expected to be located to the left of the fastest rigid body modes to increase damping. This result can not be achieved with the data set used for the simulation model. Changing the mass inertia of the helicopter and again the cross-coupling effect due to the hinge offset, we can push the fast pitch mode to the right. Varying the control rotor data should be avoided. The blade inertia is already calculated accurately, and other parameters do not have a big influence on the control rotor mode. Unless the neglected cross-coupling terms for the control rotor states are considered, there are no changes in the control rotor modes expected. Those coupling terms are considered to be less critical than the more important unknown helicopter roll and pitch inertia.


72

Open Loop Poles of Lateral Motion, Hover

-20

-15

-10

-5

0

5

10

15

20

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

Re

Im

no Control Rotor

decoupled Control Rotor

with Control Rotor

Figure 5.5: Hover poles of lateral motion, with and without control rotor

The picture for the lateral motion shows similar changes. The new complex pair evolved from the fast roll mode that coupled with the lateral control rotor mode. The roll motion is much faster since roll inertia is much smaller than pitch inertia. Effects of inaccurate helicopter data become therefore even more important in the lateral motion than in the longitudinal dynamics. In both axis only small changes can be observed in the remaining modes. Nevertheless, it becomes clear that the control rotor adds damping to the system as expected from the previous analysis in Chapter 4.

Chapter 6 Real-Time Simulation: Hardware and Software

73

Chapter 6

REAL-TIME SIMULATION: HARDWARE AND SOFTWARE

For the purpose of testing modern control schemes for flight control in a simulator, a real-time simulation is necessary that provides the same immediate response of a helicopter model as the real physical system. The mathematical models from the previous chapters combined with the equations of unsteady motion provide this basis for the real-time simulation code. This chapter describes how the simulation code is implemented together with other necessary components like flight system hardware and software, pilot-system interface and real-time 3-D visualization of flight.

6.1 Simulation Elements

Various components of the simulation software are presented and summarized in Figure 6.1, showing also the order in which computations are done. In Chapter 6.3 these components are shown in interaction with other components of the real-time-hardware-in-the-loop simulation facility. This section gives a brief overview of how the software components work together to provide a real-time system response similar to the real aircraft. Some components listed in Figure 6.1 are optional, and their influence on the system or finally the controller performance can be studied separately. For a given pilot or controller input the simulated actuator response is transmitted to the control rotor, main rotor and tail rotor, resulting in the desired calculated blade pitch. The amount of input fed through the control rotor and directly to the main rotor depends on the linkage geometry. Additional wind components due to wind gusts can be added if desired. The terrain model is important for simulation of ground effect during take-off, low altitude flight and landing is about to be developed, and therefore not yet included in the simulation. The force and moment calculations include the interaction of several helicopter components like main rotor-fuselage, main rotor-horizontal tail, main rotor-tail rotor and tail rotor-vertical tail. Engine modeling is neglected in the current version. Therefore it was assumed that the engine is capable of providing enough torque to maintain a constant rpm for main and tail rotor. The non-linear equations for forces and moments are then used to compute changes in the aircraft motion.


74

Command Filter

Operator Inputs (Pilot)

Actuator Model

Controller

Tail Rotor Main Rotor

Gust Model

Terrain Model

System Response

Rigid BodyEquations of Motion

Summation of Forcesand Moments

SensorModel

3-D-Visualization

Aerodynamic Interaction between

Main Rotor, Tail Rotor, Fuselage

Linear PerturbationModel (optional)

Control Rotor

Figure 6.1: Elements of Simulation Software


75

Integration over the simulation time results in non-linear motion for the 6 DOF rigid body helicopter plus the additional control rotor states. The resulting vehicle response drives a PC−based real-time 3-D visualization, and it is input to a sensor model of the on-board motion pack, including models for sensor noise, bias and sensor latency. The controller-in-the-loop receives the simulated sensor data as inputs and can correct for any deviations from the desired control objective. The pilot obtains his feedback via the visual queue provided by the PC-display only. The screen displays a helicopter and a simple model of the surrounding terrain, including a grid on the earth surface for better orientation as shown in Figure 6.2.

Figure 6.2: Display of the R-50 real-time simulator on PC-screen

A cockpit view and different views from the outside of the aircraft are possible to create a picture as real as possible for the pilot. Improving displays and the visual pilot-system interface will allow an even better simulation of the real system. Further improvements in this direction would therefore be desirable. A further component allows the computation of trim for each desired flight condition in the entire flight envelope of the aircraft. Using a linear perturbation routine this part also provides the linear stability and control matrices for a trimmed flight condition. This linearized model is very useful for system identification and validation as mentioned in Chapter 5. It should be noticed that the Adaptive Neural Network Controller that needs to be tested with the simulator also needs an approximation of the linear system matrices for hover condition. In the simulation these matrices can be accurately calculated by this routine and the controller should show good results close to hover even without the network. The adaptive controller is based on an inverting control


76

scheme that is accurate if the inverse of the system and control matrix is known. Chapter 8 and [25] provide details on the controller. The trim routine can be either used for advance calculations of trim tables, or it can be activated during the simulation process via the keyboard. The command filter and controller in Figure 6.1 are components of the flight system including pilot-controller interface, hardware and software switches and a ground control station providing system information and communication with the on-board flight control computer. The elements of the flight system will be described in the next section.

6.2 Flight System Elements

While the elements of the simulation mostly contain mathematical models and software code, the flight system includes most of the hardware components like actuators and control surfaces, mechanical linkages and the ground control station. The part of most interest is the flight control system. Coded in the C programming language, it is able to provide the real time simulation in double precision with an update rate of 100 Hz on a 200 MHz Pentium-based Single Board Computer (SBC). A previously developed commercial-grade flight control system, known as the R1 [28], provides management of sensor data, hardware and software interface to actuators and pilot, and it manages the telemetry links to the ground control station. The flight control system can consist of any appropriately coded flight control scheme developed for helicopter control. The control system of recent interest is the Adaptive Nonlinear Flight Controller described in detail in [25] and summarized in Chapter 8. With any control system integrated in the unmanned R-50 helicopter, the result is a very capable testbed for evaluating or testing modern control schemes and investigating handling quality and system performance improvements for aircraft or controller. During the hardware-in-the-loop simulation the real helicopter is used with all of its on-board system components. All the commands and signals are generated and processed as in real flight. Only sensor data is generated by the simulation code. This allows a detailed examination of actuator and system response to changes in controller parameters, variables or controller configuration. The best controller performance can be obtained by fine tuning the controller in simulated flight while observing helicopter motion on the display and actuator displacements on the real system to assure proper behavior and actuator response in each flight phase. Pictures of the R-50 helicopter manufactured by YAMAHA and ubgraded by Georgia Tech and GST are given in Appendix E. Further details can be obtained from the flight manual [29]. The upgrade of the Flight Control System as used in flight tests, allows a high speed data transmission between the four Motorola processors, an eight channel ultrasonic ranging system, differential GPS of optional accuracy down to 2cm, the Attitude and Heading Reference System


77

(AHRS) based on a 3-axis magnetometer and a 3-axis state rate sensor. In planned future flight tests the AHRS will be substituted by a complete GPS-aided inertial navigation platform for further improvement of data accuracy and evaluation of the controller and system performance. The performance of navigation and sensor components, as well as PC performance also influence data transfer speed. It can cause time delays, noise or an undesirable large and varying bias in sensor data used by the controller, causing undesirable non-linearities, unknown dynamics or even instability. It was also observed during tests that the transfer rate of sensor data to the ground station via the data link is very limited, and that saving the data on board at too high frequency causes a delay in the data processing and controller computations. Therefore the frequency and size of the data set saved for evaluation purposes must be kept as low as possible.

6.3 Hardware-In-The-Loop-Simulation

The real-time-hardware-in-the-loop test facility is schematically illustrated in Figure 6.2 from [28]. The left half of the figure shows necessary model components to simulate the whole helicopter system as described in section 6.1. The elements of the flight system in the right half represent the equivalent hardware components and necessary software to combine both. An interesting opportunity of this facility would be the optional replacement of parts of the simulation as actuators, mechanical linkages and control surfaces by their mechanical equivalence. The real dynamic response of actuators could be measured or investigated, real time ability of the on-board flight control computer can be tested, and the data transfer to the ground control station can be optimized while the helicopter is still on ground. The simulated system response is visualized by the pilot on the PC-screen (Figure 6.2) while commanding the necessary inputs via the transmitter also used in real flight. The advantage of this simulation is that theoretically for a successful controller test in real flight, the mathematical helicopter model simply needs to be replaced by the real helicopter. The better the mathematical simulation of the whole system and all important aspects like disturbances, noise and time delays, the easier is the transition from simulation to real flight, and perhaps more important, the lower the risk of loosing the aircraft because of system failure.


78

True System Dynamics

True Sensor Data

Software Switch

Controller

Command Filters

Software Switch

True Actuator Response

Safety Switch

ActuatorInterface

Logic

PilotInterface

Logic

Simulated Actuator Dynamics

Software Switch

Simulated Mechanical Linkage

Sensed Linkage Position

Artificial Loads

Summation of Forces and Moments

Rigid Body Equations of Motion

Sensor Models

RotorModels

FuselageAero

EngineModels

Copy of Actuator Interface Logic

GustModel

TerrainModel

PilotInputs

SerialTx

PWMTx

OperatorInputs

Real-Time 3-D Flight Animation Tool Ground Control Station

SIMULATION ELEMENTS FLIGHT SYSTEM

Figure 6.3: Joint GST/Ga Tech real-time hardware-in-the-loop simulation facility [28]

The true system dynamics in flight are input to sensors and controller, and all simulation models are taken “out of the loop”. If further tuning of controller parameters should be necessary, changes can still be made in flight via the data link from the Ground Control Station. More advanced applications like high performance maneuvers or simulation of subsystem failures to test the controller performance can easily be introduced during flight by limiting actuator displacements, introducing high latency or sensor failure. These advanced procedures might follow a more general test. The purpose of a basic test program is to assure reliability and the full functionality of the controller in normal flight conditions. The advanced test procedures should be subject of intense real-time simulations before being applied to the real system. A very important option that should always be available during all flight tests, is a switch back to the open loop system to give the pilot full authority. It can also be switched to the already approved and fully tested control system, the Yamaha Attitude Control System (YACS) with the pilot in the loop. Before the simulation can be used for detailed flight tests, model validation is important to gain confidence and assure the correct response of the math model compared to the real helicopter. The validation procedures and some results are presented in the following chapter.

Chapter 7 Validation of the Helicopter Simulation

79

Chapter 7

VALIDATION OF THE HELICOPTER SIMULATION MODEL

In the previous chapters background information for understanding helicopter and rotor dynamics were presented, and a math model for simulating rigid body and rotor dynamics, including the control rotor, were described. The purpose of this chapter is to evaluate and validate this model by comparing simulated results with flight test data. The complexity of the system combined with limitations related to the operation of the remotely controlled helicopter make the flight test program a difficult task. Improvements and further investigations related to the system identification and model validation are recommended, but require changes in the system to improve the data quality and quantity. Improvements are necessary related to eventual time delay in the data processing and increasing the data transfer rate to the ground station.

7.1 Flight Test Data and Requirements

The test site located 40 miles south of Atlanta allows flight maneuvers in the entire range of the radio transmitter. A mobile ground station operated by Guided Systems Technologies, Inc., (GST) is available and includes devices for data transfer, flight recording and flight surveillance. Sensor data is processed on board and sent to the ground station via a data link. Information from the ground station to the on-board system to switch controller modes, change system configurations in flight, or to activate the on-board data collection is transferred through the same data link. For the system identification sensor data of high accuracy and low noise level is desirable. The configuration of the data link was originally optimized for system surveillance in flight, allowing a data transfer of numerous parameters and variables at only a relatively low frequency. Maximum performance of the data link without loosing the real-time capability of on-board computations is found at a transfer rate of 5Hz, processing 66 variables and parameters including sensor data, controller variables for later flight tests, GPS signals and pilot commands. This data can be stored on ground for the entire flight test. This low transfer rate will limit the accuracy of the system identification especially with respect to the high frequent system response. A second recording device is available on board the helicopter. This system runs on a 200MHz PC with an available flash disk memory of 20KB. Sensor data and controller


80

calculations are processed on board. To assure real-time performance the sampling rate is limited to a minimum of 100Hz. Saving data on board puts an additional load on the system, and the minimum sampling rate can not be maintained if frequency, number of saved variables or recording time is too large. For the actual system configuration it is possible to achieve a recording frequency of 10Hz for the duration of several flight maneuvers, with the number of recorded variables limited to 8. Thus the system identification process was hampered by a low data transfer rate. These limitations require a careful investigation of useful system identification procedures to support a validation of the simulation. Two appropriate identification procedures are presented in this chapter. Angular rate sensors measure the body rotational velocities in the body-fixed frame. Translational velocities are obtained by a GPS receiver of high accuracy, transmitting data in a vehicle carried north-east-down reference frame. Euler angles are computed by integrating angular rates and compensated for bias and measurement noise before the data is saved. For further data evaluation, the velocity components need to be transformed from the vehicle carried frame to the body-fixed frame, and the bias needs to be removed. The transformation matrix Lbv is given by the following equation

⋅⋅−

⋅⋅⋅+

⋅⋅

⋅⋅+

⋅⋅⋅−

⋅⋅

−⋅⋅

=

θφψφ

ψθφψφ

ψθφ

θφψφ

ψθφψφ

ψθφ

θψθψθ

coscos)cossinsinsin(cos

)sinsincossin(cos

cossin)coscos

sinsin(sin)sincos

cossin(sin

sinsincoscoscos

bvL , (7.1)

and the velocity components in the body-fixed frame are computed as

bzyx

Lwvu

bv

v

&&&

+

⋅=

. (7.2)

The vector b

v contains the bias for all three axis measured on ground or in hover, where

all angular rates and translational velocities are known to be zero.


81

7.2 System Identification Procedures

One approach chosen to validate the simulation model is to experimentally identify the parameters of an assumed linear system for a known flight condition, and to compare the results to the linear system obtained from the simulation by small perturbation theory. The aircraft motion is measured and a model has to be found reflecting the same physical behavior of the real system. A valuable tool for model structure and fidelity assessment is frequency response analysis. This approach will be described in Section 7.2.2 to identify the most important system parameters. Section 7.2.3 presents a time domain approach by looking at a static trim position like hover and then applying step inputs to investigate the dynamic system response in each axis. This will probably result in a less reliable estimate, because the least-square method for estimating the approximated state-space model is sensitive to noise, low recording rate of sensor data, measurement errors and initial starting values for the estimated parameters. Flight conditions that should be investigated are hover and forward flight, whereas for a remotely controlled helicopter a steady-state forward flight condition is hard to maintain due to the limiting transmitter range. Therefore the parameter identification is presented for the most important parameters in hover only. Further system identification tests to improve accuracy and usefulness of the real-time simulation is necessary and will be the subject of a more detailed study for the R-50 experimental helicopter in the near future.

7.2.1 Static Trim Values

First, the sensor data is recorded in a hover flight condition at about 30 feet altitude to determine trim values without any disturbances or influences due to ground effect. Measured trim values for hover are listed in Table 7.1 and can be directly compared to the trim values obtained from the simulation. Also listed are the measured and simulated control inputs for hover. Trim values for the simulated model partially do not agree with test data. However, the avionics box including navigation system, sensor pack and PC is not included in the simulation. Furthermore, changes in the CG position and helicopter inertia due to the additional component added to the helicopter are completely neglected. The simulated collective main rotor input, mainly dependant on the helicopter gross weight, is smaller compared to test results. Pitch and roll angles are strongly dependent on the position of the CG relative to the rotor hub. Adding estimates for the avionics box to the system data,


82

the trim values change towards the experimentally obtained values as it can be seen in Table 7.1 A more detailed model of the avionics box has to be computed and included in the helicopter data set for an accurate simulation. The analysis in this chapter is based on the helicopter model including the approximated changes due to the avionics box listed below.

θ0

[rad]

φ0

[rad]

B1,SP (long. cycl.)

[rad]

A1,SP (lat. cycl.)

[rad]

collMR

[rad]

collTR

[rad]

Flight Test, Hover 0.0525 0.0661 0.0204 -0.0089 0.1567 0.232

Simulation, Hover

(no avionics box) -0.0464 0.0561 -0.007 -0.0291 0.1085 0.0046

Simulation, Hover

(incl. avionics box:

25 lbm, ∆xCG=-3in, ∆zCG =7in)

0.0073 0.0427 0.0064 0.004 0.1291 0.0166

Simulation,

Forward Flight at 20fps

(incl. avionics box)

0.0068 0.0334 0.007 0.0688 0.1136 0.0198

Table 7.1: Measured and simulated trim values for R-50 in hover and forward flight

For steady-state forward flight at 20fps, the simulated trim values and the control inputs are also listed. No flight test data is yet available to compare with, but as described in Chapter 4, the influence of forward speed is observable in increasing cyclic control inputs to compensate for the cross-coupling and aerodynamic rotor effects due to the higher velocity at the advancing blade. Less input in collective main rotor pitch is required in forward flight, reflecting the decrease in desired thrust for an increasing forward velocity. In Figure 7.1 a trim table is computed, and the results are shown as a function of forward speed from –20fps to +20fps. Roll angle, pitch angle and the inputs show reasonable values during transition from hover to forward or backward flight. The trim values show consistent results throughout the investigated speed range, and compare qualitatively with results that can be found in literature [6, 7, 11]. The control rotor requires a relatively large input in the longitudinal cyclic control to maintain forward speed. This commanded swashplate tilt results in a blade pitch of the control rotor and then, through mechanical linkages, in a blade pitch of the main rotor. Responsible for


83

TPP and thrust tilt is the actual main rotor blade pitch, which is less than the computed pilot input. Cyclic inputs are strongly dependent on the approximated mechanical mixing of direct and indirect inputs from swashplate to control and main rotor. Ground tests visualizing the real resulting control and main rotor TPP tilt due to a swashplate tilt could provide more information to model this mixing more accurately. The present model is based on geometry and single blade pitch measurements over the blade azimuth. The position of the CG and mass inertia are not accurately known and therefore also contribute to the only slowly varying pitch and roll angle with changes in forward speed. With increasing velocity the estimated center of aerodynamic pressure of the fuselage becomes more important. This variable was estimated based on the computed center of fuselage projections in all three axes. Changing the flap hinge offset, and therefore the influence of cross-coupling and additional moments on the body, affects the required lateral cyclic input needed to compensate for this effect in forward flight. The previously mentioned parameters should be tuned to achieve good agreement between the simulation and the real helicopter.

R-50 Trim Table (incl. Control Rotor)

-0.1

-0.05

0

0.05

0.1

0.15

-20 -15 -10 -5 0 5 10 15 20

Forward Speed [fps]

[rad]

Coll. Main Rotor Pitch

Long. Cyclic Pitch

Lat. Cyclic Pitch

Coll. Tail Rotor Pitch

Body Pitch Angle

Body Roll Angle

Figure 7.1: Trim table for R-50 (simulation), rearward to forward flight


84

7.2.2 Frequency Response Analysis

From the determined hover condition, the goal is now to command computer generated sinusoidal inputs of specific amplitude and frequencies in each axis and record the system response. The aircraft should therefore always be in a flight condition near hover. The flight tests are performed mainly open loop, and the pilot is advised to command only low frequent inputs to keep the helicopter near hover. Frequency sweeps with varying frequencies in a single flight maneuver could not be flown due to the difficulty in maintaining full control over the helicopter throughout the entire maneuver. Those inputs used for flight test with full size helicopters allow in general a relatively easy and accurate system identification with reasonable results. In the yaw channel of the R-50 a yaw damper is present providing rate feedback to the yaw input. The yaw motion is therefore highly damped, and the response expected in yaw will be different compared to the simulation neglecting this rate feedback. For future tests this rate feedback should be removed by calculating the amount of feedback and removing it from the final input to the collective tail rotor pitch. This will be necessary for future tests where the NN Adaptive Controller is supposed to provide this feedback. In addition, mechanical coupling of collective tail and main rotor collective pitch should be modeled and analytically removed. Since the vertical motion due to collective inputs is not investigated, this mechanical coupling is also neglected in the simulation model. The collective main rotor input is kept constant for all flown test maneuvers.

Experimental Bode Plots

Frequency response methods as developed by Bode in the 1930s, are very powerful when modeling transfer functions from physical data. The concept is that in steady-state, sinusoidal inputs to a linear system result in a sinusoidal response of the same frequency as the input, but they differ in amplitude and phase angle. Measuring these differences and concluding how the desired system can be best approximated is the objective of generating the experimental Bode plots [18]. The ratio of the output sinusoid’s magnitude to the input sinusoid’s magnitude is defined as the magnitude response, the difference in phase between output and input as the phase response. Both responses are functions of frequency and represent only the steady-state sinusoidal response of the investigated system.


85

The following assumptions are made to simplify the analysis: • Only angular rates in all three axes are investigated and their dynamics are decoupled. • The steady-state response of the helicopter can be approximated at low frequencies by

a first-order system. • Delay due to play in mechanical linkages and signal processing in the electrical

systems result in pure phase shift that needs to be included in the analysis.

Thus the resulting linear system for all three axes can be presented in the following block diagram (Figure 7.2). The inputs to the uncoupled system for roll, pitch and yaw motion are pure lateral cyclic, longitudinal cyclic and collective tail rotor pitch respectively. Outputs are roll, pitch and yaw rate. Equations are presented for the pitch channel only, but the approach is identical for roll and yaw motion.

pdTse ,⋅−ppp ubpap δ⋅+⋅−=&

ppuδ

rdTse ,⋅− rrr ubrar δ⋅+⋅−=&r

ruδ

qdTse ,⋅−qqq ubqaq δ⋅+⋅−=&

qquδ

Figure 7.2: Block diagram of approximated linear helicopter dynamics for hover

Writing the transfer function for the pitch channel it follows that

)(

,

q

Tsq

q as

eb

uq qd

+

⋅=

⋅−

δ , (7.3)


86

with Td,q representing the time delay in the pitch channel and s the Laplace transform. The parameters aq and bq are to be identified from the frequency response data. Substituting s = jω in the Equation 7.3, we obtain the pitch transfer function

( ))(

,

q

Tjq

q aj

ebjG

qd

+

⋅=

⋅−

ωω

ω

. (7.4)

The magnitude of the frequency response is then

( ) ( )

1ˆˆ

2

+

⋅=

q

q

q

qq

a

a

b

uq

jGω

ωδ

ω . (7.5)

The phase angle due to pure time delay can be written as

( ) qdqd T ,, ⋅= ωωϕ , (7.6)

resulting in a total phase angle expressed as the phase response

( ) qdq

q Ta ,

1tan ⋅−

−= − ω

ωωϕ . (7.7)

For ω << 1 in Equation 7.5, an estimate independent of the low frequency response, written as

q

q

q a

b

uq

≈δ

ˆ . (7.8)

Figure 7.3 shows a typical test result for the sinusoidal response of the helicopter in the

pitch channel.


87

Pitch Channel Frequency ResponseSinusoidal Input: Frequency 0.75 Hz, Amplitude 0.035 rad

-0.120

-0.100

-0.080

-0.060

-0.040

-0.020

0.000

0.020

0.040

0.060

0.0 5.0 10.0 15.0 20.0 25.0

Time [sec]

Am

plit

ud

e

Long. Cyclic Input [rad]

Pitch Rate [rad/sec]

Figure 7.3: Frequency response in the pitch channel, ω = 0.75Hz, quδ =0.035 rad

Measuring time differences and given the frequency of the exciting sinusoidal input, the total phase shift ϕq(ω) is ( ) tq ∆⋅= ωωϕ , (7.9)

with ∆t is the measured time difference. The phase shift due to system dynamics and pure delay are included in Equation 7.9. Results for a typical test data are plotted in a Bode diagram (Figure 7.4). The mean value of at least three data points in the time history plot for one flight maneuver is used to compensate for measurement errors. The plots show results for more than one measurement at each frequency. Different sections in the time history plot are picked for each measurement. The data needs to be corrected for bias and trim value. In the pitch channel a positive input (forward tilt of swashplate) results in a negative body pitch rate (forward pitch of helicopter).


88

Experimental Bode Plot, Long. Cyclic Input to Pitch Rate

0

5

10

15

20

25

1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

-350

-300

-250

-200

-150

-100

-50

0

1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure 7.4: Experimental Bode plot for the pitch channel


89

Experimental Bode Plot, Lat. Cyclic Input to Roll Rate

0

5

10

15

20

25

1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

-250

-200

-150

-100

-50

0

1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure 7.5: Experimental Bode plot for the roll channel


90

Experimental Bode Plot, Coll. Tail Rotor Input to Yaw Rate

0

2

4

6

8

10

12

14

16

18

20

1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

-250

-200

-150

-100

-50

0

1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure 7.6: Experimental Bode plot for the yaw channel


91

It can be seen, that the assumption of a first-order system in the pitch, yaw and roll rate dynamics is only approximately true for low frequencies up to ω ≈ 5Hz. The following calculations are therefore only valid for the low frequency response. For higher frequencies the behavior is more like a second-order response with a resonance frequency for the pitch response at about 9 rad/sec. This mode can be identified as the longitudinal control rotor mode coupled with the longitudinal fast pitch mode of the body dynamics. The Bode plot in Figure 7.5 for the roll rate response due to lateral cyclic input shows a second-order response with a natural frequency at about 13 rad/sec identified as the fast roll mode. The yaw response due collective tail rotor input is shown in Figure 7.6. If we want to compare these results with simulation results, a time delay needs to be considered resulting in a pure phase shift in all three axes. An approximation for this time delay can be computed using the Bode plots and the previously made assumption of a first-order response at low frequencies. The tested frequency range is chosen based on estimated maximum limits causing uncontrollable instabilities, and lower limits mainly restricted by the helicopter response. For low frequencies, flight conditions develop which are too dangerous to be maintained for the time necessary to record the steady-state response. Furthermore, pilot inputs of about the same magnitude as the sinusoidal input are observed in the test data at very low frequencies. An accurate analysis for those cases is consequently not possible since the pure sinusoid can not be extracted from the test data to measure phase shift and magnitude. Controllable flight maneuvers down to 0.5Hz in the pitch axis, 0.3Hz in the roll axis and 0.2Hz in the yaw axis are possible. As a maximum limit ω = 2Hz is chosen in all three axes. To obtain approximations for the system parameters of the assumed first-order system described in Equation 7.3, in each Bode plot at least two data points in the low frequency range are used. Knowing the phase shift as expressed in Equation 7.7 for two different frequencies, we have two equations with two unknowns, the system parameter aq and the time delay Td. Those equations can be solved numerically for aq and Td. Equation 7.5 can then be solved for the unknown bq. The results for this approximate identification are shown in Table 7.2 for all three axes.


92

States

Parameters

p q r

-a -5.6 -2.772 -1.838

b 25.6 -14.716 10.34

Td 0.112 0.115 0.099

Table 7.2: Identified parameters and system delay in hover using experimental Bode plots

Several system components can be made responsible for the time delay, which could cause problems for the later implementation of the new control architecture. Processing sensor data, sampling time synchronization, a Bessel anti-aliasing filter for the rate sensor, play in mechanical linkages and finally actuators contribute to a time delay that can add up to a significant value important for stability and control issues. The next section compares the experimentally obtained results with simulation results, and the observed time delay not present in the linear simulated model will be investigated.

Simulated Bode Plots

Generating the corresponding Bode plots for the linearized helicopter model in hover is much simpler, since all stability derivatives have already been computed. The linear model is further reduced to a system with decoupled longitudinal and lateral motion. For the decoupled longitudinal motion the following state-space model taken from the fully coupled computation of the linear system matrices (see Appendix C) is assumed, including the longitudinal control rotor tilt and the approximated changes in mass and CG position for the additional avionics box (Equation 7.10). Inputs to the collective main rotor pitch are not considered. The longitudinal cyclic input δlong remains as the only longitudinal input. The lateral model is similar and can be extracted from the tables in Appendix D. Results for the longitudinal model are shown in Figure 7.7 for the transfer function from longitudinal cyclic input to pitch rate. Experimental data and simulated data are plotted in the same picture. Results are shown for a pure time delay of Td = 0.31sec acting on the system obtained from the simulation. Very good agreement of simulated and experimental data can be seen.


93

long

CRcCRc

qwu

qwu

δ

βθ

βθ

⋅

−

−+

⋅

−−

−−−−−−−−−

=

2184.406267.38

0698.02579.11

1633.20100101.0009991.0002896.6809424.6002.02373.01233.02358.00236.05727.00027.09033.191731.32413.10039.00553.0

,,&

&&&&

(7.10)

The simulated time delay of 0.31sec seems to be very large, even if all the mentioned components are considered. This leads to the assumption that there might be some delay in the data processing, creating a large phase shift related to not synchronized input and output data collection or other effects introducing some time delay. Further tests with the real system revealed that the primary contributor of time delay for very low frequency is the analog anti-aliasing filter at the input to the rate sensors. Another source was found in the message passing structure in the software. The remaining measured time delay from sensor input to actuator output varies from 0.03-0.1sec. This is due to the fact that the various components in the path have finite duration processes that are unsynchronized and of varying frequencies. The goal for further flight tests is to reduce the time delay from rate measurement to actuator response to less than 0.01sec. This amount of time delay would be acceptable to achieve high bandwidth control. Comparing the experimentally obtained (Table 7.2) and simulated (Appendix D) stability and control derivatives for the same case, it is seen that simulated results show a much higher sensitivity to perturbations of states and control inputs. Differences in the control matrix expressed by the elements bp, bq, br can be partially related to not exactly modeled changes of CG position and mass properties due to hardware components added to the original helicopter. The already mentioned additional yaw damper present in the real open loop system further contributes to the differences in sensitivity and stability. The used model for the control rotor also includes an approximation defining the mixing of direct inputs from the swashplate to the main rotor and inputs via the control rotor tilt as described in Chapter 5. A significant influence is expected from the coefficients


94

defining this mixing of inputs to the main rotor cyclic pitch, directly related to the rate response of the helicopter. The geometric flap hinge offset also contributes to the sensitivity of system response, since rotor moments are directly related to this input parameter to the simulation.

Bode Plot, Pitch Dynamics, 0.31sec Time Delay

0

5

10

15

20

25

1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

Flight Test

Simulation

-400

-350

-300

-250

-200

-150

-100

-50

0

1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure 7.7: Experimental and simulated frequency response, 0.31sec time delay, pitch dynamics


95

Comparing the natural frequencies of the dynamic modes from flight test data with those from simulation (Figure 7.8 and Appendix F), we find relatively good agreement in the roll, pitch and yaw response. In those results the same amount of time delay is added as that identified in the pitch channel. Due to the additional yaw damper in the real system, the results for the yaw channel show big differences in the magnitude response, whereas the frequency response is still similar (Figure F.2). The following system identification method will focus on the validation of the already obtained values and on the identification of other important system and control derivatives

7.2.3 Step Response Analysis

This more deterministic identification technique also assumes a linear relationship between state derivatives x& , states x and control inputs δu. The goal is to predict the measured states based on a linear combination of measured states and inputs defined by constant parameters ai. The estimated parameters will be compared to those obtained with the frequency analysis and those calculated from the simulation model. The fully coupled system as presented in Appendix C for the general case, and in Appendix D for results obtained from the simulation, contain too many parameters to be properly identified. Furthermore some of the parameters are not even practically identifiable with the input-output flight test data available. The flight tests, consisting of step and doublet inputs over a short recording time, do not provide optimal excitation of the state response. Some parameters can be assigned based on the analytically obtained perturbation matrices as given in Appendix C.

• Zero terms in the system and controls matrices F and G are not identified. • Cross-coupling terms will be neglected. • Parameters based on geometric functions of trim values are not identified. • Parameters relating p to φ, q to θ and r to ψ are set equal to 1. • Collective input is neglected and not changed during the flight maneuvers. • Immediate response of the helicopter is mainly resulting in a change of angular

rates.

The numerical problem is to minimize the error between experimental and simulated output data for an estimated set of identifiable parameters. A performance function is defined as the sum of the squared errors between simulated and experimental output data,


96

( ) 2

1

2

21

21

EieVt

i

⋅=⋅= ∑=

, (7.11)

where YYE ˆ−= (7.12) and ( )jaufY ˆ,ˆ δ= . (7.13)

Estimated parameters ja are the elements of the estimated model that gives the

estimated output Y if the known measured input data set δu is input to the estimated model. Minimizing the performance function to find the best solution for the parameters in the sense of least-square requires computational steps in the direction of a decreasing V, and finally finding the minimum value of the performance function for an estimated combination of parameters. A powerful method to find steps in this direction is given by a Levenberg-Marquardt method [19]. It evaluates the gradient of the performance function as the partial derivatives with respect to the parameters to be identified, known as the Jacobian matrix

ja

VJ

ˆ∂∂

= . (7.14)

An approximated Hessian matrix representing the second partial derivatives can be defined for a performance function of the form of a sum of squares as JJH T= . (7.15) The gradient of the error function V is then eJg T= , where e is the vector of the output

errors, and the update law for the parameters is

[ ] eJIJJaa TTkk ⋅⋅+−=

−+

1

1 ˆˆ µ , (7.16)

known as the Levenberg-Marquardt-Algorithm. For the scalar µ = 0 Equation 7.16 represents a pure Newton algorithm using the approximated Hessian matrix H. For large µ it becomes the gradient descent method with a small step size. The Newton method is in general better and converges faster for a minimum search near the error minimum.


97

Thus µ is decreased after each successful step towards a minimum and increased when a step would increase the value of the performance function. As a result, the performance function will always be reduced for all iterations of the algorithm. The problem of finding local minima instead of the desired absolute minimum is still

present and makes it necessary to start from a different initial guess for the parameter

vector [ ] Tjaaa ˆ...ˆˆˆ 21=α and compare the results of several solutions.

The previously described approach is applied to the decoupled lateral motion of the helicopter and to a system with model structure similar to that assumed for the frequency analysis in the previous chapter. Results are listed in the following figures and tables. For the longitudinal motion a successful identification was not possible. Poor velocity data, and noise in the angle and angular rate measurements, prevented a successful identification with this approach. Different and more powerful step response identification procedures might lead to reasonable results.

For a model of the following form

lat

a

r

pv

aa

aaaaa

r

pv

δφφ

⋅

+

⋅

−

−=

00

6.25

838.100524.0010

06.5070.32 8

76

54

321

&

&&&

, (7.17)

the results obtained from the Bode plot analysis are set as fixed. Furthermore, parameters that can be computed by geometric functions or that are zeros as given in the system matrix F in Appendix C are again not identified. Error calculations include velocity, roll rate and roll angle, the error in the yaw rate is neglected. The identified system with the parameters in Table 7.3 results in a step response shown in Figure 7.8 together with the measured output data. Control rotor states are not measured and no information about its dynamics can be obtained by this very limited analysis method. Values for the control rotor derivatives obtained from the simulation could be set as known and fixed in an extended model that includes control rotor states. This approach did not result in a better solution. For the system shown in Equation 7.17, the parameters listed in Table 7.3 are identified.

a1 a2 a3 a4 a5 a6 a7 a8

-0.0474 -11.1943 -0.9378 0.0247 -3.3562 0.0026 0.6824 2.499

Table 7.3: Identified parameters for decoupled lateral motion, step response


98

Step Response, Lateral Cyclic Input

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6 7 8 9 10

Time [sec]

Rol

l Rat

e [r

ad/s

ec]

Lateral Cycl. Input [rad]

Simulation

Flight Data

-5

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10

Time [sec]

Bod

y-v-

velo

city

[ft

/sec

]

Simulation

Flight Data

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Time [sec]

Ro

ll A

ng

le [

rad

]

Simulation

Flight Data

Figure 7.8: Response of identified model (7.17) compared with measured data


99

A model similar to the one identified with the frequency method is now assumed, except that cross-coupling terms are included.

lonaa

rqp

aaaa

aaa

rqp

δ⋅

+

⋅

=

000 9

8

76

54

321

&&&

, (7.18)

As an initial guess only the parameters known from the frequency analysis are given, and off-diagonal terms are set to zero. The following parameters are identified and the resulting step response is shown in Figure 7.9.

a1 a2 a3 a4 a5 a6 a7 a8 a9

-4.9153 0.5747 0.5641 -0.8205 -2.7600 -5.9641 -2.9033 2.5303 -11.4687

Table 7.4: Identified coupled model parameters for system including only angular rates.

The diagonal terms are relatively close to the identified parameters from the frequency response method. Note that this is the response due to longitudinal cyclic input only. Since most of the cross-coupling terms except rate cross-coupling are neglected, a good fit in the roll and yaw channels are not expected. Good agreement is also found for the pitch derivative ( bq ) in the control matrix. An identification of system models including velocity data is not presented, since no reasonable results were obtained. This might lead to the conclusion that the velocity data achieved from the GPS system is not very reliable. More emphasis on the system identification is needed in the future to make the simulator more useful for flight tests. Comparing the identified parameters with the linear matrices from the simulator, we can see differences in sign and magnitude. The identified parameters strongly depend on the helicopter inertia as can be seen in matrix F (Appendix C). Without accurate helicopter inertia data and a good estimation for the position of the CG, these significant differences in control and system matrix elements will remain. Some of these differences can also be a result of the neglected effects from inertial or aerodynamic cross-coupling. Further identification procedures should therefore emphasize those cross-coupling terms, also because they influence the natural flapping frequency and therefore might increase stability as shown in Figure 4.1. A method to identify the control rotor parameters by looking at the step response is not yet possible, since the control rotor states are not measurable with the currently used equipment. Improvements in this direction are planned but not yet fully developed.


100

Step Response, Longitudinal Cyclic Input

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Time [sec]

Pitc

h R

ate

[rad

/sec

]

Long. Cyclic Input [rad]SimulationFlight Data

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Time [sec]

Ro

ll R

ate

[rad

/sec

]

Simulation

Flight Data

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Time [sec]

Yaw

Rat

e [r

ad/s

ec]

Simulation

Flight Data

Figure 7.9: Response of identified model (7.18) compared with measured data

Chapter 8 Modern Adaptive Nonlinear Flight Control in Simulation and Real Flight

101

Chapter 8

MODERN ADAPTIVE NONLINEAR FLIGHT CONTROL IN

SIMULATION AND REAL FLIGHT

Artificial Neural Networks (NN’s) function as highly nonlinear adaptive control elements and have great advantages over conventional linear parameter adaptive controllers. To examine performance and limits of such a controller, the School of Aerospace Engineering at the Georgia Institute of Technology has implemented a recently developed NN Adaptive Flight Controller in the experimental helicopter R-50 previously described and modeled. The goal is to apply this modern controller and show its capability in normal and critical flight conditions. In [26] the theoretical development of a direct adaptive tracking control architecture for flight control using NN’s is presented. This work is extended and applied on helicopter flight control in [27]. In [25] the same controller is also applied to a tiltrotor aircraft, and some stability issues for the controller architecture are discussed. This chapter first describes the controller architecture and summarizes its main features. Then some results for a simulated flight maneuver are shown and compared to recently obtained experimental results. In flight tests, the NN Adaptive Controller was first implemented in the pitch channel. Finally, some comments on future work in this field concerning simulated and real flight conclude this chapter. Improvements necessary to facilitate an implementation of modern controllers in the real system are still required and are related to the simulator and hardware components like data link, on-board system, sensor performance and accuracy.

8.1 Flight Control System

The changing flight characteristics of a helicopter in different flight conditions provide a challenging opportunity for the application of adaptive flight control laws. Two basic types of control augmentation for aircraft are rate command and attitude command systems, each with advantages and disadvantages dependent on the flight condition and desired task. The system first implemented in the R-50 controller is a rate command system.


102

The basis for the control scheme is a NN Augmented Model Inversion Architecture shown in Figure 8.1 for the pitch channel. Roll and yaw channel controllers are constructed similarly.

Kps1

Kd

+q~

Pitch ChannelNeural Network

HelicopterDynamics

ApproximateInverse

Transformation

Normalization

X

Uq

BiasUAD

ControlSurface

Deflections

X

δ

Sensed State, q

UUc

1 OrderCommand

Filter

st

RCAH(Rate Command,Attitude Hold)

PitchRate

Command

Pitch Channel Example

qPilot qc

qc•

LON

qq

q

Figure 8.1: Neural Network Augmented Model Inversion Architecture

As a feedback linearization method dynamic model inversion [20] is used, where the model inversion is based on linearized dynamics about the hover condition only. The states included in the model inversion scheme are the angular rates p, q and r. The dynamic equation is thus given by δωω ⋅+⋅+⋅= BAxA 211& , (8.1)

where A1, A2 and B represent the aerodynamic stability and control derivatives at the nominal trim point, respectively. The rates about the body-axes are contained in the vector ω = [p q r ]T. The main rotor collective input is treated in this application as an additional slow translational state, such that x1 = [ u v w δcoll,MR]T. The remaining inputs are the controls resulting in a direct moment about the fixed body axes δ = [δlat δlon δcoll,TR]T. Slow and fast states are separated and any cross-coupling between them is neglected for the inverting controller. The NN is used to adapt to errors caused by this linearized inverted model for hover. A ‘pseudo control’ U of the form ADCC UUU −+= ω& (8.2)

is designed for the rotational degrees of freedom ( p, q, r ), including the output of the

linear controller operating on an error signal q~ (see Figure 8.1), the adaptive control term


103

Uad from the NN and the filtered commanded acceleration Cω& . The term Uc specifies the

tracking error transient for the linear controller

∫⋅+⋅=t

tIPC dqKqKU

0

~~ τ . (8.3)

With T

cccc rqp ][ &&&& =ω as the vector of filtered commanded angular accelerations the left-

hand side of Equation 8.1 is set equal to the desired accelerations, which are in this case identically the pseudo controls for each channel. Substituting cω& in Equation 8.1 using

Equation 8.2 and solving for the required control, we obtain

ωδ ⋅−⋅−⋅= −211

1 AxAUB . (8.4)

Since neither A1 and A2 , nor B are exactly known, it follows that ωδ ⋅−⋅−⋅= −

2111 ˆˆˆˆ AxAUB , (8.5)

Furthermore, since the elements in x1 are not presently sensed on board the helicopter, it is assumed that x1 = 0 in the implementation. The inversion error for each channel can be written in the form δωωε ˆˆˆ

2 ⋅+⋅−= BA& . (8.6)

This inversion error can be expressed as a function of states and pseudo controls. The tracking-error dynamics for the pitch channel can finally be written as

( )∫ −=⋅+⋅+t

tqADIP UdqKqKq

0

2~~~ ετ& , (8.7)

with the left-hand side representing the error dynamics and the right-hand side describing the network compensation error, which is a forcing function of the tracking error dynamics.


104

The adaptation law assuring boundedness of the tracking error for a linear-in-the-parameters network can be written in the form ( )UXWU T

AD ,β⋅= (8.8)

where W is the vector of network weights, and β is a vector of basis functions. In [25] it has been proven, that errors and network weights remain bounded, and that the errors asymptotically approach zero if the network is capable of perfectly mapping the function given in Equation 8.6. This will depend on the network architecture. The design in the R-50 flight controller employs a multilayered NN with sigmoidal activation functions σ(⋅) in the hidden layer, capable of approximately reconstructing the inversion error. NN’s of this structure are also called universal approximators. The inputs to the multilayered NN with one hidden layer shown in Figure 8.2 consist of the state variables, the pseudo control and bias terms. Variable network weights of the first-to-second layer interconnection are noted with V, whereas the second-to-third layer interconnection weights are represented by W.

(⋅)σ

(⋅)σ

(⋅)σ

(⋅)σ

x

V W

Input Layer Hidden LayerOutput Layer

Figure 8.2: Multilayered network with one hidden layer

A further problem related to adaptive systems and specifically unmodeled dynamics, can be solved by using NN’s. Nonlinear unmodeled dynamics might cause instabilities. A Nonlinear Dynamic Damper was recently employed at Georgia Tech, which can work in conjunction with a multilayered NN. This additional component is also added to the


105

tested flight control system as a part of the network learning algorithm, to examine its performance in the presence of unmodeled dynamics like actuators or unknown delays. Performance improvements are expected mainly related to command tracking and robustness during normal operation but also in the case of system failure.

8.2 Simulation and Experimental Results

The results in Figure 8.3 show the simulated system response to a commanded doublet input in the pitch channel. Also considered in this simulation result are sensor noise and bias, actuator dynamics, time delay and wind gusts acting on the system. The adaptive NN controller compensates for nearly all the motion in the other channels due to cross-coupling effects of rotor or body dynamics. It can be seen in the plot, that the response follows the filtered command very well.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

11 13 15 17 19 21 23 25 27 29

Time [sec]

Pit

ch C

han

nel

Co

mm

and

an

d R

esp

on

se

Pitch Rate Command [rad/sec]Pitch Rate [rad/sec]Pitch Attitude [rad]

Figure 8.3: Simulated system response of R-50, doublet inputs in long. cyclic

It should be also mentioned, that in this result the used linearized model for hover is also a model with only non-zero diagonal elements. All off-diagonal elements in the system matrix obtained from the linearized simulation model are set to zero. A further artificial


106

model error is introduced by multiplying all system matrix elements by 5. The controller performance can be evaluated for different magnitudes of the introduced error. In flight tests the NN adaptive controller was first tested in the real-time simulator to assure a reliable switching between an operation with and without the controller, and to verify the stable dynamics of the NN. The controller was first activated in the pitch channel only. The remaining channels were flown open-loop by the pilot. Starting with the inverting controller only, learning rates, robustifying components and parameters of the nonlinear dynamic damper were raised slowly to see the influence on the system response and tracking quality. The linearized system matrices are taken from the system identification results of the frequency response method in Chapter 7. It was assumed that only diagonal terms are of importance and all off-diagonal terms were set to zero. The results given in Figure 8.4 show a typical input in the pitch channel and the resulting systems response, including noise, bias, time delay and actuator dynamics. A similar behavior than in Figure 8.3 can be seen. Tuning the parameters of the NN controller may improve the tracking of the commanded input and minimize the oscillations observed in the pitch rate response.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

2020 2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042 2044 2046

Time Step Number (~0.25 sec each)

Pitc

h C

hann

el C

omm

and,

Res

pons

e an

d C

ontr

ol Pitch Rate Command [rad/sec]

Pitch Rate [rad/sec]

Pitch Attitude [rad]

Figure 8.4: Pitch response of R-50 in flight test with NN controller in pitch channel


107

The shown results give confidence in the use of the simulator to test modern NN control architectures. Including sensor noise and bias, actuator dynamics, time delays and wind gusts, the developed real-time simulator provides a very useful tool to find limits of the controller performance and to identify the influence of various controller parameters on the system performance. Influences of various components of the simulation and disturbances can be investigated separately, and the controller can be tested for its ability to compensate for unmodeled dynamics. Further flight tests are necessary to find the optimal setting of controller parameters for the experimental helicopter. Also of interest is the tracking performance of the controller during high performance maneuvers. These tests will probably be the subject of flight tests in the near future. To improve the capability of the simulator and to further validate the simulation model, more detailed studies in the field of system identification are necessary.

References

108

REFERENCES

[1] R.K. Heffley, M. A. Mnich: Minimum-Complexity Helicopter Simulation Math Model, NASA CR 177476, USAAVSCOM TR 87-A-7, California, April 1988

[2] R.K. Heffley, S.M. Bourne, H.C. Curtiss, JR., W.S. Hindson, R.A. Hess: Study of Helicopter Roll Control Effectiveness Criteria, NASA CR 117404, USAAVSCOM TR 85-A-5, California, April 1986

[3] R.K. Heffley, S.M. Bourne, M. A. Mnich: Helicopter Roll Control Effectiveness Criteria, Manudyne Report 83-2-2 (forthcoming NASA CR/AVSCOM TR), May 1987

[4] B. Etkin, L.D. Reid: Dynamics of Flight, Stability and Control, 3rd Edition, John Wiley and Sons, Inc., 1996

[5] B. Etkin: Dynamics of Atmospheric Flight, John Wiley and Sons, Inc., 1972

[6] R.W. Prouty: Helicopter Performance, Stability, and Control, Robert E. Krieger Publishing Company, Florida 1990

[7] R.W. Prouty: Helicopter Aerodynamics, Rotor and Wing International, Philips Publishing, Inc., 1985

[8] W. Johnson: Helicopter Theory, Dover Publications, Inc., New York 1980

[9] A. Gessow, G.C. Meyers, JR.: Aerodynamics of the Helicopter, Frederick Ungar Publishing Co., New York 1967

[10] M.G. Perhinschi, J.V.R. Prasad: A Simulation of An Autonomous Helicopter, Proceedings of the Remotely Piloted Vehicles 13th International Conference, Bristol, United Kingdom, 30 March-1 April 1998,

[11] G.D. Padfield: Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling, AIAA Education Series, 1996

[12] E.N. Johnson, P.A. DeBitetto: Modeling and Simulation for Small Autonomous Helicopter Development, AIAA Modeling and Simulation Technologies Conference, 1997

[13] M.A. Glauert: A General Theory of Autogyro, Royal Aeronautical Establishment R. and M. No. 1111, November 1926

[14] C.N.H. Lock: Further Development of Autogyro Theory, Royal Aeronautical Establishment R. and M. No. 1127, March 1927

References

109

[15] P.D. Talbot, B.E. Tinling, W.A. Decker, R.T.N. Chen: A Mathematical Model of a Single Main Rotor Helicopter for Piloted Simulation, NASA-TM-84281, California, September 1982

[16] R.T.N. Chen: Effects of Primary Rotor Parameters on Flapping Dynamics, NASA TP-1431, January 1980

[17] H. Lamb: Hydrodynamics, Dover, New York 1945

[18] N.S. Nise: Control Systems Engineering, The Benjamin/Cummings Publishing Company, Inc., 1992

[19] W.H. Press, S.A. Teukolsky, W.T Vetterling, B.P. Flannery: Numerical Recipes In Fortran, The Art of Scientific Computing, 2nd Edition, Cambridge University Press, 1992

[20] L.-J. E. Slotine, W. Li: Applied Nonlinear Control, Prentice Hall, Inc., 1991

[21] D.A. Teare: Modeling and System Identification for Rotorcraft, Thesis, Georgia Institute of Technology, Atlanta, March 1988

[22] W.D. Lewis: An Aerolastic Model Structure Investigation for a Manned Real-Time Rotorcraft Simulation, Thesis, Georgia Institute of Technology, Atlanta, August 1992

[23] S.H. Sturisky: A Linear System Identification and Validation of an AH-64 Apache Aerolastic Simulation Model, Thesis, Georgia Institute of Technology, Atlanta, July 1993

[24] M. Hossein, M.B. Tischler: An Empirical Correction Method for improving Off-Axes Response Prediction in Component Type Flight Mechanics Helicopter Models, AGARD-CP-592, April 1997

[25] A.J. Calise, R.T. Rysdyk: Nonlinear Adaptive Flight Control using Neural Networks, IEEE Control System Magazine, Vol. 18, No. 6, December 1998

[26] B.S. Kim, A.J. Calise: Nonlinear Flight Control Using Neural Networks, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 1, January-February 1997

[27] J. Leitner, A.J. Calise, J.V.R. Prasad: Analysis of Adaptive Neural Networks for Helicopter Flight Control, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 5, September-October 1997

[28] J.E. Corban, A.J. Calise, J.V.R. Prasad: Implementation of Adaptive Nonlinear Control for Flight Test on an Unmanned Helicopter, Proceedings of the 37th Annual IEEE Conference on Decision and Control, December, 1998

[29] YAMAHA: Unmanned Helicopter R-50 Instruction Manual, 1st Edition, YAMAHA Motor Co., Ltd., Aeronautics Operation, May 1995

Appendix A – R-50 Helicopter Data

110

APPENDIX A – R-50 HELICOPTER DATA

/*** R50 Parameters ***/

/*** Loading Parameters ***/

FS_CG = 24.99 ; /* Location of CG in inches from the zero fuselage station */

WL_CG = 20.154 ; /* Vertical CG location in inches above the zero waterline */

WT = 97.85 ; /* Vehicle Take-Off Gross Weight in lbs. (weight is assumed constant) */

IX = 0.1997*7.345; /* slug-ft2 */

IY = 0.6231*7.345; /* slug-ft2 */

IZ = 0.6*7.345; /* slug-ft2 */ IXZ = 0.0; /* slug-ft2 */

/*** Main Rotor Parameters ***/

FS_HUB = 27.50; /* station line of main rotor hub, measure aft of the zero fuselage station */

WL_HUB = 42.25; /* waterline location of the main rotor hub */

IS = 0.0; /* Main rotor shaft tilt backward of vertical (radians) */

E_MR = 0*2.25/12.0; /* geometric main rotor flapping hinge offset or effective hinge offset */

I_B = 0.86754; /* flapping inertia of a single blade about the flapping hinge */

R_MR = 5.05; /* main rotor radius in feet */

A_MR = 6.0; /* effective lift curve slope of main rotor (per radian) */

RPM_MR = 870.0; /* nominal main rotor anglular velocity (rev. per min) */

CDO = 0.010; /* effective profile drag for main rotor blade cross section */

B_MR = 2.0; /* number of main rotor blades */

C_MR = 0.354; /* blade chord in feet */

TWST_MR = 0.0; /* effective blade twist in radians */

K1 = 0.000; /* tangent of delta-3, (effective pitch-flap coupling) */

/* based on flaping hinge geometry */

DIR = -1.0; /* rotor spin direction -1 = clockwise */

/*** Fuselage Parameters***/

FS_FUS = 25.25; /* station line of center of pressure of fuselage (in.), including hub drag */

WL_FUS = 6.904; /* waterline of center of pressure of fuselage (in.), including hub drag */

XUU_FUS = -2.322; /* effective frontal area (ft2) corresponding to profile drag in the x axis */

YVV_FUS = -7.849; /* effective side area (ft2) for sideward flight */

ZWW_FUS = -6.960; /* effective planview area (ft2) for vertical flight and download in hover */

/*** Wing Parameters ***/ /*** Note that the R-50 does not have a wing */

FS_WN = 0.0; /* fuselage station of aerodynamic center of wing (in.) */

WL_WN = 0.0; /* waterline location of aerodynamic center of wing (in.) */

ZUU_WN = 0.0; /* (ft2) */

ZUW_WN = 0.0; /* (ft2) */

ZMAX_WN = 0.0; /* (ft2) */

B_WN = 1.0; /* Wing span (ft), Arbitrary non-zero number to avoid division by zero */

Appendix A – R-50 Helicopter Data

111

/*** Horizontal Tail Parameters ***/

FS_HT = 64.50; /* (in.) fuselage station of aerodynamic center of horizontal tail */

WL_HT = 27.30; /* (in.) waterline location of aerodynamic center of horizontal tail */

ZUU_HT = 0.0; /* (ft2) effective lift per unit dynamic pressure at zero angle of attack */

/* relative to fuselage reference system */

ZUW_HT = -1.9374; /* (ft2) effective variation in circulation lift */

/* (approx. the neg. product of lift curve slope and surface area) */

ZMAX_HT = -1.2536; /* (ft2) max force generated by the horizontal tail when stalled */

/*** Vertical Tail Parameters ***/

FS_VT = 99.0; /* (in.) station location for the effective aerodynamic center of the */

/* vertical fin */

WL_VT = 29.5; /* (in.) vertical position of the vertical fin aerodynamic center */

YUU_VT = 0.0; /* net y-force per unit dynamic press for zero sideslip */

/* Assume symmetric airfoil at 0 incidence */

YUV_VT = -0.6876; /* sideforce arising from a side-velocity component */

/* (approx. equal to lift curve slope times the net fin area) */

YMAX_VT = -0.2487; /* sets the maximum sideforce generated by the vertical tail at stall */

/*** Tail Rotor Parameters ***/

FS_TR = 97.5; /* (in.) station line of the tail rotor hub */

WL_TR = 25.85; /* (in.) waterline location of the tail rotor hub */

R_TR = 0.853; /* radius of the tail rotor in feet */

A_TR = 3.0; /* effective lift curve slope of tail rotor */

SOL_TR = 0.10807; /* solidity of the tail rotor (ratio of blade area to disk area) */

RPM_TR = 5400.0; /* angular velocity of the tail rotor (rev. per minute) */

TWST_TR = 0.0; /* effective twist of the tail rotor blade */

B_TR = 2.0; /* number of tail rotor blades */

C_TR = 0.1458; /* (ft.) tail rotor blade chord */

/*** Control Rotor Parameters ***/

R_CR = 1.8537; /* radius of control rotor in feet */

lb_CR = 0.4921; /* lenght of control rotor blade in feet */

A_CR = 2.657; /* 2*pi/(1+2/AR) and AR = lb_CR*lb_CR/(lb_CR*C_CR) */

C_CR = 0.328; /* chord length in feet */

I_B_CR = 0.04566; /* control rotor blade inertia in slug*ft^2 */

/*** Other Modeling Parameters ***/

VTRANS = 30.0; /* speed for transition from dihedral wake function in rotor tip */

/* path plane dynamics (set empirically) */

Appendix B – Equations of Unsteady Motion of the Rigid Body

112

APPENDIX B – EQUATIONS OF UNSTEADY MOTION OF THE RIGID BODY

Force and Moment Equations:

( )( )( )

( )( ) ( )

( ) qrIIIpqpIrIN

rpIIIrpqIM

pqIIIqrrIpILqupvwmmgZ

pwruvmmgY

rvqwummgX

zxxyzxz

zxzxy

zxyzzxx

EEE

EEE

EEE

⋅+−⋅+⋅−⋅=

−⋅+−⋅+⋅=

⋅−−⋅+⋅−⋅=−+⋅=⋅⋅+

−+⋅=⋅⋅+

−+⋅=⋅−

&&

&

&&&

&

&

22

coscos

sincos

sin

φθ

φθ

θ

Kinematic Equations:

( )

( ) θφφψφφθ

θφφφ

φθφθψ

φθψφθ

θψφ

seccossinsincos

tancossin

sincoscos

sincoscos

sin

⋅⋅+⋅=⋅−⋅=

⋅⋅+⋅+=

⋅−⋅=

⋅⋅+⋅=

⋅−=

rqrq

rqp

r

q

p

&

&

&

&&

&&&&

( )( )

( )( )

θφθφθ

ψφψθφ

ψφψθφψθ

ψφψθφ

ψφψθφψθ

coscoscossinsin

cossinsinsincos

coscossinsinsinsincos

sinsincossincos

sincoscossinsincoscos

⋅⋅+⋅⋅+⋅−=

⋅−⋅⋅⋅+

⋅+⋅⋅⋅+⋅⋅=

⋅+⋅⋅⋅+

⋅−⋅⋅⋅+⋅⋅=

EEEE

E

EEE

E

EEE

wvuz

w

vuy

w

vux

&

&

&

Appendix C – System and Control Matrices

113

APPENDIX C – SYSTEM AND CONTROL MATRICES

Ω⋅⋅−

⋅−−

⋅

Ω⋅⋅−

⋅⋅

−

⋅+

⋅+++++++

⋅+

⋅+++++++

⋅⋅−⋅+⋅−

−

⋅⋅

⋅⋅⋅−−⋅−+

⋅⋅+⋅−−

=

16000

160010

16

016

00116

016

00

00

00tancos0100tansin00

00

coscossinsin

00sin0000cos00

00

cossinsincos

0cos

,0

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0000

000000

00

000000

000

ξγξγθ

ξγθξγ

θθθφ

θφθφ

φφ

θφθφ

θ

βδδ

βδδ

βδδ

βδδ

βδ

βδ

βδ

βδ

βδ

βδ

βδ

βδ

CR

CR

CR

CR

c

axaxz

c

cxcxz

c

rxrxz

c

pxpxz

c

vxvxz

c

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c

wxwxz

c

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c

axzaz

c

cxzcz

c

rxzrz

c

pxzpz

c

vxzvz

c

qxzqz

c

wxzwz

c

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y

a

y

c

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r

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p

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v

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q

y

w

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u

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acrpvqwu

l

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mY

gmY

mY

mY

kI

Mk

IM

IM

IM

IM

IM

IM

IM

km

Zk

mZ

mrZ

gvmZ

mZ

gumZ

mZ

mZ

km

Xk

mX

vmX

mX

mX

gwmX

mX

mX

F

Appendix C – System and Control Matrices

114

⋅Ω⋅⋅

⋅Ω⋅⋅

−

+⋅

+⋅

++

+⋅

+⋅

++

⋅−⋅⋅

⋅⋅

⋅⋅

⋅⋅

=

016

00

0016

0

0000

sinsin

0000

00

CR

CR

c

pxzpzMR

c

axaxzMR

c

cxcxz

c

exexz

c

pxzpzMR

c

axzazMR

c

cxzcz

c

exzez

MRa

MRce

y

pMR

y

aMR

y

c

y

e

pMR

aMR

ce

pMR

aMR

ce

k

k

I

NILIk

INILI

kI

NILII

NILI

I

NILIk

INILI

kI

NILII

NILI

gkm

Yk

mY

mY

I

Mk

IM

kI

MI

Mm

Zk

mZ

km

Zm

Zm

Xk

mX

km

Xm

X

G

ξγ

ξγ

θφ

δδδδδδδδ

δδδδδδδδ

δδδ

δδδδ

δδδδ

δδδδ

=

CRs

CRc

r

pv

qwu

x

,

,

ββ

φ

θ

( )2xzzxc IIII −⋅=

=

=

TRcoll

lat

long

MRcoll

pace

u

,

,

δδδ

δ

δδδδ

δ

tiltswashplatelateral

tiltswashplateallongitudin

lat

long

δ

δ

Appendix D – Results of Linear System Analysis for Hover

115

APPENDIX D – RESULTS OF LINEAR SYSTEM ANALYSIS FOR HOVER

Control rotor decoupled from rigid body dynamics (rotor rotating clockwise): F-Matrix:


u -0.0469 -0.0296 1.4106 -32.1394 0 -0.0034 0.21 0 0 0

w -0.0311 -0.6892 -0.1084 1.4901 0 -0.0032 -0.0012 -1.8032 0 0

q 0.1559 -0.068 -5.8228 0 0 0.0465 0.7644 0 0 0

theta 0 0 0.9984 0 0 0 0 0 -0.0561 0

beta_c_cr 0.0101 0 -1 0 -2.1633 0 -0.0237 0 0 0

v 0.0047 -0.0039 0.2102 0.0837 0 -0.0998 -1.443 32.0888 0.4197 0

p 0.2176 -0.0148 -2.3849 0 0 -0.4276 -17.881 0 0.428 0

phi 0 0 -0.0026 0 0 0 1 0 -0.0464 0

r -0.0116 -0.2397 0 0 0 0.2881 0.1293 0 -1.7523 0

beta_s_cr 0 0 0.0237 0 0 -0.0101 -1 0 0 -2.1633

G-Matrix:

trim:

coll MR B1 A1 coll TR theta -0.0464

u -18.1857 32.5193 3.4817 0 phi 0.0561

w -391.015 -1.4548 -0.3507 0 coll MR 0.1085

q -30.9937 -85.857 -50.0334 0 B1 -0.0447

theta 0 0 0 0 A1 -0.0108

beta_c_cr 0 -2.1633 0 0 coll TR 0.0046

v -2.3784 -3.4852 32.5522 -43.3123 a1s 0.0453

p -9.0806 -155.97 497.447 -43.3504 b1s -0.0059

phi 0 0 0 0

r -75.505 0 0 180.7083

beta_s_cr 0 0 2.1633 0

Eigenvalues of F:



0.0976 -1.0065 -0.0965 1.0113

0.0102 0.8473 -0.012 0.8474 Longitudinal Oscillation

0.0102 -0.8473 -0.012 0.8474

-0.6887 0 1 0.6887 Heave

-1.8383 0 1 1.8383 Yaw

-2.1633 0 1 2.1633 Long. Control Rotor

-2.1633 0 1 2.1633 Lat. Control Rotor

-6.1697 0 1 6.1697 Pitch mode

-17.811 0 1 17.8108 Roll mode


116

Control rotor coupled with rigid body dynamics (rotor rotating clockwise): F-Matrix:


u -0.0469 -0.0296 1.4106 -32.1394 -19.8693 -0.0034 0.21 0 0 2.1273

w -0.0311 -0.6892 -0.1084 1.4901 0.8889 -0.0032 -0.0012 -1.8032 0 -0.2143

q 0.1559 -0.068 -5.8228 0 52.4587 0.0465 0.7644 0 0 -30.5704

theta 0 0 0.9984 0 0 0 0 0 -0.0561 0

beta_c_cr 0.0101 0 -1 0 -2.1633 0 -0.0237 0 0 0

v 0.0047 -0.0039 0.2102 0.0837 2.1295 -0.0998 -1.443 32.0888 0.4197 19.8894

p 0.2176 -0.0148 -2.3849 0 95.3006 -0.4276 -17.881 0 0.428 303.9402

phi 0 0 -0.0026 0 0 0 1 0 -0.0464 0

r -0.0116 -0.2397 0 0 0 0.2881 0.1293 0 -1.7523 0

beta_s_cr 0 0 0.0237 0 0 -0.0101 -1 0 0 -2.1633

G-Matrix:

trim:

coll MR B1 A1 coll TR theta -0.0464

u -18.1857 11.2387 1.2033 0 phi 0.0561

w -391.015 -0.5028 -0.1212 0 coll MR 0.1085

q -30.9937 -29.6722 -17.2916 0 B1 -0.0447

theta 0 0 0 0 A1 -0.0108

beta_c_cr 0 -4.2184 0 0 coll TR 0.0046

v -2.3784 -1.2045 11.25 -43.312 a1s 0.0453

p -9.0806 -53.9049 171.918 -43.35 b1s -0.0059

phi 0 0 0 0

r -75.505 0 0 180.708

beta_s_cr 0 0 4.2184 0

Eigenvalues of F:



0.0309 -0.7663 -0.0402 0.7669


-0.0043 -0.6423 0.0067 0.6423

-0.6838 0 1 0.6838 Heave

-1.9241 0 1 1.9241 Yaw


-4.0211 -7.7222 0.4619 8.7064

-10.011 15.3329 0.5467 18.3116 Roll + lateral CR

-10.011 -15.333 0.5467 18.3116


117

Control rotor and estimated avionics box included (rotor rotating clockwise): F-Matrix:


u -0.0553 0.0039 1.413 -32.1731 -19.9033 -0.004 0.2104 0 0 2.1309

w -0.0027 -0.5727 -0.0236 -0.2358 -0.1233 0.004 -0.0112 -1.3749 0 0.1946

q 0.2373 0.002 -6.9424 0 68.2896 0.0595 0.5964 0 0 -32.255

theta 0 0 0.9991 0 0 0 0 0 -0.0427 0

beta_c_cr 0.0101 0 -1 0 -2.1633 0 -0.0237 0 0 0

v 0.0059 0.0049 0.2104 -0.0101 2.1309 -0.0871 -1.4622 32.1437 0.2913 19.903

p 0.2791 0.0307 -1.8597 0 100.6205 -0.6653 -21.5131 0 0.8169 353.6293

phi 0 0 0.0003 0 0 0 1 0 0.0073 0

r 0.0022 -0.2894 0 0 0 0.2551 0.2469 0 -1.4638 0

beta_s_cr 0 0 0.0237 0 0 -0.0101 -1 0 0 -2.1633

G-Matrix:

trim:

coll MR B1 A1 coll TR theta 0.0073

u 2.3713 11.2579 1.2053 0 phi 0.0427

w -322.9734 0.0698 0.1101 0 coll MR 0.1291

q 7.3491 -38.6267 -18.2444 0 B1 0.004

theta 0 0 0 0 A1 0.0064

beta_c_cr 0 -4.2184 0 0 coll TR 0.0166

v 2.9838 -1.2053 11.2578 -31.0981 a1s -0.0072

p 18.8329 -56.914 200.0234 -86.3006 b1s 0.009

phi 0 0 0 0

r -87.6774 0 0 156.1621

beta_s_cr 0 0 4.2184 0

Eigenvalues of F:


-0.0203 0.702 0.0289 0.7023 Lateral Oscillation

-0.0203 -0.702 0.0289 0.7023


-0.0328 -0.5605 0.0584 0.5615

-0.5749 0 1 0.5749 Heave

-1.4704 0 1 1.4704 Yaw


-4.4905 -8.5089 0.4667 9.6211

-11.9143 15.9267 0.599 19.8899 Roll + lateral CR

-11.9143 -15.9267 0.599 19.8899

Appendix E – R-50 Helicopter System Components

118

APPENDIX E – R-50 HELICOPTER SYSTEM COMPONENTS

Figure E.1: R-50 fully equipped during flight test

Figure E.2: R-50 fully equipped on transport cart; GST ground control station in background

Appendix E – R-50 Helicopter System Components

119

Figure E.3: R-50 avionics box with on-board PC and sensor packet

Figure E.4: R-50 horizontal tail for improved handling characteristics in forward flight

Appendix F – Simulated and Experimental Bode Plots

120

APPENDIX F – SIMULATED AND EXPERIMENTAL BODE PLOTS

Bode Plot, Roll Dynamics, 0.31sec Time Delay

-5

0

5

10

15

20

25

30

1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

Flight TestSimulation

-700

-600

-500

-400

-300

-200

-100

0

100

1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure F.1: Experimental and simulated frequency response, 0.31sec time delay, roll dynamics

Appendix F – Simulated and Experimental Bode Plots

121

Bode Plot, Yaw Dynamics, 0.31sec Time Delay( yaw damper incl. in flight test )

0

5

10

15

20

25

30

35

40

45

50

0.1 1.0 10.0 100.0

Frequency [rad/sec]

Gai

n [d

B]

Flight Test

Simulation

-400

-350

-300

-250

-200

-150

-100

-50

0

0.1 1.0 10.0 100.0

Frequency [rad/sec]

Pha

se [°

]

Figure F.2: Experimental and simulated frequency response, 0.31sec time delay, yaw dynamics

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DEVELOPMENT OF A REAL-TIME FLIGHT SIMULATOR FOR …...experimental model helicopter. A mathematical model of the helicopter is developed to represent the dynamics of the real system