UNIVERSITY OF CALGARY. Control of an Unconventional VTOL UAV for Complex Maneuvers. Nasibeh Amiri A THESIS

Transcription

1 UNIVERSITY OF CALGARY Control of an Unconventional VTOL UAV for Complex Maneuvers by Nasibeh Amiri A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ELECTRICAL AND COMPUTER ENGINEERING CALGARY, ALBERTA January, 213 c Nasibeh Amiri 213

2 Abstract The increasing potential applications of Unmanned Aerial Vehicles (UAV) provides the motivations for numerous research to focus on developing fully autonomous and self guided UAVs with the purpose of controlling UAVs in confined environments. Current UAVs control systems are not able to offer the precise trajectory regulation required in autonomous flight technology. These systems fail to control aerial vehicles performing complex maneuvers through confined environments because current UAV designs do not have suitable control mechanisms providing agility and stability for the required maneuvers. New advances in control theory are required to overcome these limitations in order to enable aggressive autonomous vehicle maneuvering while adapting in real time to changes in the operational environment. This thesis addresses a control problem of an unconventional highly maneuverable Vertical Takeoff and Landing (VTOL) UAV, using tilted ducted fans as flight control mechanism. The main purpose of this research is to design a nonlinear control methodology that enables the vehicle to use the full potential of its flying characteristics for independent control of its six degree-of-freedom, including orientation and position of the UAV. This thesis investigates maneuvering inside obstructed environments in the presence of external disturbances such as wind, ground and wall effects. Achieving this goal is possible due to a revolution in aviation control by introducing Oblique Active Tilting (OAT) mechanism. Capabilities of OAT system will be fully used in controlling the UAV to enhance its maneuverability. ii

3 Acknowledgments I would like to gratefully and sincerely thank my supervisor, Dr. Robert Davies, and my co-supervisor, Dr. Alejandro Ramirez-Serrano, for their continuous support, generosity in sharing their knowledge and guidance during period of this research. Besides, I take this opportunity to express my gratitude to all staffs of the Department of Electrical and Computer Engineering in University of Calgary for providing an excellent working environment. I would like to acknowledge and thank all my friends for their love and efforts and for being the surrogate family which made me feel more at home. Particularly, I would like to express my gratitude to my very kind friend, Arya Janjani, for his time and help. Finally, yet importantly, I would like to thank my lovely parents, sisters, and brother-in-law whose unwavering love and support kept me motivated throughout the hardship of this experience. I offer my heartfelt thanks to my beloved Sepehr for his endless patience, kindness and encouragement when it was most required and his continued moral support that helped me a lot in completion of this project. iii

4 Table of Contents Abstract Acknowledgments Table of Contents List of Figures Nomenclature ii iii iv vii xi 1 Introduction Background of Unmanned Aerial Vehicles Examples of Unmanned Aerial Vehicles Introducing the evader Unmanned Aerial Vehicle Motivation General Problem Statement Thesis Outline Literature Review Overview of Previous Research on UAV Control Linear Control Techniques of UAV Flight Control Nonlinear Control Techniques of UAV Flight Control Control of evader Vehicle in the Literature Objectives and Goals Definitions Summary Modeling Introduction Lift-fan OAT Mechanism Special Characteristics of OAT Overview of soat and doat Lateral and Longitudinal Rotor Tilting VTOL Modeling Translational Dynamics Ground and wall effects iv

5 3.3.3 Rotational Dynamics Complete Dynamic Model of The evader Approximation of Equations of Motion Summary Feedback Linearization Control of evader Overview and Background of Feedback Linearization Controllability Modeling for Control evader s Feedback Linearization Design Nonlinear Adaptive Control Adaptive Control Design for the evader s Orientation Adaptive Control Design for the evader s Position Stability Analysis Robust Adaptive Feedback Linearization Simulation Results Results and discussion Integral Backstepping Control of evader Overview and Background of Backstepping Control Technique State-Space Model for Control Control System Objective Attitude Control Design Altitude and Position Controls Design Gradient Descent Optimization for Coefficient Tuning Adaptive Integral Backstepping Control Simulation Results Summary Sliding Mode Control for the evader Overview of Sliding Mode Control Sliding Surfaces Chattering Modeling for Control Sliding Mode Control based on Backstepping Controller Design Simulation Results Summary v

6 7 Neural Network Nonlinear Function Approximation Benefits of Neural Networks Problem Statement Function Approximation Feedforward networks Overview of Multi-Layer Perceptron Back-Propagation Training the MLP Neural Network for Actual Control Signal Approximation Generating data for neural network training Training One MLP network Training Six Parallel MLP Networks Summary Comprehensive Simulation Scenarios, Results and Discussion Scenario # Wind Buffeting (Scenario #3) Ground Effect (Scenario #12) Aggressive Maneuver Result Discussion Conclusion and future work Contribution Future Work Bibliography Appendices Appendix A Scope of Scenarios vi

7 List of Figures 1.1 Ducted fans of the unconventional highly maneuverable VTOL UAV Fans rotating around longitudinal (y-axis) and lateral (x-aixs) axes Dual-fan VTOL air vehicle having lateral and longitudinal tilting rotors prototype (evader) Fans tilted longitudinally 9 degrees for high speed forward flight [1] a) Oppositely spinning disks tilted equally towards one another generating gyroscopic moment τ gyro, b) The whole System rotated about y axis to a new attitude orientation [1] Schematic of VTOL aerial vehicle with dual-axis OAT mechanism [1] Schematic of the evader VTOL with a body fixed frame B and the inertial frame E. The circular arrows indicate the direction of rotation of each propeller [1] Control signals of FL control method in presence of white gaussian noise with mean = and variance =.1 [2] Regulation of orientation angles of the evader by FL controller with additive white noise (φ = 22.5,θ = 15,ψ = 18) Regulation of position of the evader by FL controller with additive white noise (x d = 3,y d = 4,z d = 2) Control signals of adaptive FL control method in presence of aerodynamic coefficient uncertainties and unknown mass Regulation of orientation angles of the evader by AFL controller (φ = 22.5,θ = 15,ψ = 18) Regulation of position of the evader by AFL controller (x d = 3,y d = 4,z d = 2) Parameter estimation of AFL control method (a 1 = J y J z, a 2 = J x, a 3 = J z J x, a 4 = J y, a 5 = J x J y, a 6 = J z, a 7 = a 8 = a 9 = m) The altitude output of FL and AFL controllers when the mass of the systems is changed. The FL controller failed to reach the desired altitude z d = 2 with almost.4 m steady state error Attitude control of evader s orientation and the corresponding control input signals of IB control method (φ = 1, θ = 35, ψ = 5 ) vii

8 5.2 Position(x, y) stabilization of the evader and the corresponding control input signals of IB control method Autonomous take-off, altitude control in hover and landing of the evader and the effect of tuning the IB controller gains by Gradient Descent algorithm Stabilization of roll, pitch and yaw angles by IB control method (left figure) and pitched stability of the evader at 25 in hover (right figure) Chattering due to delay in control switching Control input signals of SMC technique for performing pitched hover scenario (Scenario # 7) Orientation angles regulation in pitched hover stationary scenario(scenario # 7) Position regulation while stationary at pitched hover scenario(scenario # 7) Control input signals of SMC technique in presence of a strong sudden wind disturbance (unstructured uncertainty, Scenario # 8) Orientation angles and position regulation in hover pitched scenario with a strong sudden wind disturbance with magnitude 5 (Scenario # 8) Control input signals of SMC technique when performing Scenario # 2 and system parameters (mass and inertia matrix) are varying Orientation angles and position regulation with model parameter variations show robustness of SMC technique in presence of structured uncertainties Control input signals of SMC technique while picking up a heavy load (Scenario # 11) Orientation angles regulation error when the evader picks up a heavy load (Scenario # 11) Position regulation error when the evader picks up a heavy load scenario (Scenario # 11) Supervised learning block diagram Block diagram of an inverse function approximation system Structure of single-layer feedforward networks Structure of multi-layer feedforward neural networks neuron (1,i), (i = 1,2,...,p) in the hidden layer neuron (2,j), (j = 1,2,...,m) in the output layer Flowchart of training process in a two-layer perceptron network. This flowchart does not include the stopping criteria of the training process. 145 viii

9 7.8 Neural Network Training Performance, Best Validation Performance is at epoch Neural Network Training Error for α Neural Network Training Error for α Neural Network Training Error for β Neural Network Training Error for β Neural Network Training Error for ω Neural Network Training Error for ω Neural Network Training Performance to approximate α 1, Best Validation Performance is 5.285e-6 at epoch Neural Network Training Error for α Neural Network Testing Error for α Neural Network Training Performance to approximate α 2, Best Validation Performance is e-5 at epoch Neural Network Training Error for α Neural Network Testing Error for α Neural Network Training Performance to approximate β 1, Best Validation Performance is at epoch Neural Network Training Error for β Neural Network Testing Error for β Neural Network Training Performance to approximate β 2, Best Validation Performance is at epoch Neural Network Training Error for β Neural Network Testing Error for β Neural Network Training Performance to approximate ω 1, Best Validation Performance is at epoch Neural Network Training Error for ω Neural Network Testing Error for ω Neural Network Training Performance to approximate ω 2, Best Validation Performance is at epoch Neural Network Training Error for ω Neural Network Testing Error for ω Control input signals of FL, AFL and SMC controllers for orientation and position regulation in Scenario # Attitude outputs of the evader obtained by applying FL, AFL and SMC controllers in Scenario # Position outputs of the evader obtained by applying FL, AFL and SMC controllers in Scenario # ix

10 8.4 Control signals of AFL controller without robust modification in presence of wind disturbance. The control signals u 1, u 2 and u 3 go to infinity and make the evader unstable evader Orientation goes to infinity with AFL controller without robust modification in presence of wind disturbance Control signals of RAFL controller, with e-modification, in presence of wind disturbance Parameter estimation of RAFL controller with e-modification in presence of wind disturbance evader orientation with RAFL controller with e-modification in presence of wind disturbance evader position with RAFL controller with e-modification in presence of wind disturbance Schematic diagram of the evader which shows z, z, z cg Control signals of AFL with robust modification and SMC control in presence of ground effect disturbance Output orientation angles of evader obtained by applying AFL with robust modification and SMC control in presence of ground effect disturbance The Cartesian position output of evader obtained by applying AFL with robust modification and SMC control in presence of ground effect disturbance Three dimensional position output result obtained by applying RAFL control performing aggressive maneuver Three dimensional position output result obtained by applying SMC controller performing aggressive maneuver Control signals of RAFL controller in aggressive maneuver scenario Control signals of SMC controller in aggressive maneuver scenario Orientation of the evader performing aggressive maneuver obtained by applying RAFL controller Orientation of the evader performing aggressive maneuver obtained by applying SMC controller Position of the evader performing aggressive maneuver obtained by applying RAFL controller Position of the evader performing aggressive maneuver obtained by applying SMC controller Orientation tracking error of SMC controller in aggressive maneuver scenario Position tracking error of SMC controller in aggressive maneuver scenario.188 x

11 Nomenclature Abbreviations: AFL Adaptive Feedback Linearization BP Back Propagation CFD Computational Fluid Dynamic CG Center of Gravity CMG Control Moment Gyroscope DOF Degree of Freedom doat double-axis Oblique Active Tilting FL Feedback Linearization GE Ground Effect GWE Ground and Wall Effects IB Integral Backstepping IMU Inertial Measurement Unit IOL Input Output Linearization ISL Input State Linearization LQ Linear Quadratic LQR Linear Quadratic Regulator LP Linear Parameterizable MIMO Multiple Input Multiple Output MLP Multi Layer Perceptron OAT Oblique Active Tilting OLT Opposed Lateral Tilting PD Proportional Derivative PID Proportional Integral Derivative RAFL Robust Adaptive Feedback Linearization SAR Search and Rescue SISO Single Input Single Output SMC Sliding Mode Control soat Single-axis Oblique Active Tilting UAV Unmanned Aerial Vehicle UUB Uniformly Ultimately Bounded VTOL Vertical Take-off and Landing xi

12 1 List of Terms Variables: α 1, α 2 β 1, β 2 ω i, ω 1, ω 2 φ θ ψ r 1, r 2 E = {x E,y E,z E } B = {x B,y B,z B } : Longitudinal tilting angles, rotation of evader rotor about the vehicle s y-axis for right (#1) and left (#2) rotor, respectively : Lateral tilting angles, rotation of evader rotor about the vehicle s x-axis for right (#1) and left (#2) rotor, respectively : Propellers speeds, Rotational velocity of right (#1) and left (#2) rotor, respectively : Orientation roll angle : Orientation pitch angle : Orientation yaw angle : Rotor/disc #1, and #2, respectively : Right hand inertia frame (earth s frame) : Body fixed frame ζ(t) = : Position vector of UAV relative to the inertia frame of reference [x(t),y(t),z(t)] T η(t) = : Euler angle vector of UAV relative to the inertia frame of reference E [φ(t),θ(t),ψ(t)] T ζ : Translation velocity vector η : Rotation velocity vector T tot = [τ x,τ y,τ z ] T : Total torque in Newton-Euler equations applied to the body of vehicle relative to the body frame B F tot F grav F aero = [F ax,f ay,f az ] T F cg : Total force in Newton-Euler equations applied to the body of vehicle relative to the body fixed frame B : Gravity force : Aerodynamic forces : All forces applied to the center of gravity of vehicle s body relative to body fixed reference frame B

13 2 F E cg F E tot F E aero T aero = [T ax,t ay,t az ] T T gyro g r (z) d w (t) τ gyro τ prop τ thrust τ react τ x τ y τ z : All forces applied to the center of gravity of vehicle s body relative to inertia reference frame E : Total force in Newton-Euler equations applied to the body of vehicle relative to the inertia frame of reference E : Aerodynamic forces relative to inertia reference frame E : Aerodynamic torques relative to inertia reference frame E : Gyroscopic effects of vehicle s body and propellers : Ground effect function of altitude z : Wind gusts disturbance : Gyroscopic pitch moments : Fan-torque pitch moments : Thrust-vectoring pitch moments : Reactionary moments : Total torque along x-axis : Total torque along y-axis : Total torque along z-axis ν = [u v,v v,w v ] T : UAV body linear velocity vector Ω = [p v,q v,r v ] T : UAV body angular velocity vector v Ω m J = diag[j x,j y,j z ] J r T 1, T 2 D 1, D 2 Q 1, Q 2 C T C Q ρ : Derivative of UAV body linear velocity vector, Accelerator vector : Derivative of UAV body angular velocity vector : Mass of evader : Vehicle s body Inertia matrix : Propeller s inertia : Thrust force of right (#1) and left (#2) rotor, respectively : Drag force of right (#1) and left (#2) rotor, respectively : Net torque applied to right (#1) and left (#2) rotor shaft, respectively : Aerodynamic coefficient in thrust force : Aerodynamic coefficient in drag force : Density of air

14 3 A r r r R x (β i ) R y (α i ) : Rotor blade area : Radius of rotor blade : A counterclockwise rotation of a vector through angle β i about the x axis : A counterclockwise rotation of a vector through angle α i about the y axis R xy (β,α) i : Rotation matrix of vectors in coordinate frame attached to each rotor about lateral and longitudinal tilting angles R x (φ) : A counterclockwise rotation of a vector through angle φ about the x axis R y (θ) : A counterclockwise rotation of a vector through angle θ about the y axis R z (ψ) : A counterclockwise rotation of a vector through angle ψ about the z axis R yxz cg O l O h O A Gi Q i d i P i S Ω K fax K fay K faz k tax k tay k taz A B C n x x d f g : Rotation matrix whose Euler angles are φ, θ, ψ with x y z convention : Vehicle s centre of gravity (centre of mass) : Aerodynamic venter : Distance from the centre of propeller to the centre of the vehicle (O) : Distance from centre of the vehicle (O) to the centre of gravity (cg) : Gyroscopic moments : Propeller torques : Translational displacement of the ducts and the vehicles cg : Reactionary torques : Skew-symmetric matrix : Friction aerodynamic coefficient along x-axis affects total force : Friction aerodynamic coefficient along y-axis affects total force : Friction aerodynamic coefficient along z-axis affects total force : Friction aerodynamic coefficient along x-axis affects total torque : Friction aerodynamic coefficient along y-axis affects total torque : Friction aerodynamic coefficient along z-axis affects total torque : State matrix in linear state-space model : Control matrix in linear state-space model : Output matrix in linear state-space model : System order : State vector of initial conditions : State vector of desired values : Nonlinear function of : Nonlinear function of

15 4 u y 2 i r eq Γ a1 e φ1 e φ2 e θ1 e θ2 e ψ1 e ψ2 : Input vector : Second derivative of ith output y i of a system : Relative degree : Positive update gain of the parameter estimation update law in adaptive control method : Regulation error for φ(t) : Filtered regulation error for φ(t) : Regulation error for θ(t) : Filtered regulation error for θ(t) : Regulation error for ψ(t) : Filtered regulation error for ψ(t) d 1,...,d 6 : Additive external disturbances λ 1,...,λ 6 : Positive constant gain in integral backstepping control method χ 1,...,χ 6 : Integral of tracking errors e 1,...,e 11 : Roll tracking error e 2 : Angular velocity tracking error corresponding to roll angle S φ,s θ,s ψ : Sliding surfaces of roll, pitch and yaw orientation angles S x,s y,s z : Sliding surfaces of Euclidean position e φ,e θ,e ψ e x,e y,e z V W (1) ik W (2) jq d ref e k ˆ f u f 1 y k y j z i : Regulation errors of roll, pitch and yaw angles, respectively : Regulation errors of position x, y and z, respectively : Lyapunov function : Connection weight from the kth input to ith neurone in the first layer : Connection weight from the qth neurone in the first layer to the jth neurone in the output layer : Neural network desired (reference) output : Error vector of : Estimate of function f u : Inverse of function f : Neural network output for kth input point : jth output signal of the second layer of neural network : ith output signal of the first layer of neural network

16 Chapter 1 Introduction The use of Unmanned Aerial Vehicles (UAVs) has recently gained extensive interest due to their diverse potential applications. Different types of UAVs have been utilized in various civil, industrial and military applications such as search and rescue, weather research and environmental monitoring (e.g., Aerosonde), natural disaster risk management, pipeline inspection and high altitude military surveillance (e.g., MQ-1 Predator). In fact UAVs are becoming more attractive lately as the result of recent advancements in aerodynamics, propulsion, computers and sensor technology. However, current UAVs cannot be controlled to navigate autonomously in confined spaces. Therefore continuous effective improvement is essential in control mechanisms to support and secure multiple tasks being performed with a single airframe for complex missions in confined spaces. Current UAVs have different levels of autonomy for operation and control. Some UAV systems are controlled by an operator through a wireless connection from a ground control station (remote control). Some systems combine remote control and computerized automation. Some other systems are capable of semi-autonomous flight following pre-specified destinations. More sophisticated versions have built-in control and/or guidance systems to perform low-level human pilot duties such as speed and flight-path stabilization, and simple scripted navigation functions such as waypoint following. However only a small group of advanced UAV systems have the ability to 5

17 6 execute high-level operations in such a way that they can perform only by having the initial states and desired destinations known. Indeed, from this perspective, early UAVsarenotautonomousatall. Infact, thefieldofair-vehicleautonomyisarecently emerging field. Compared to the manufacturing of UAV flight hardware, the market for autonomous flight technology is fairly immature and undeveloped. Because of this, autonomy has been and may continue to be the bottleneck for future UAV developments, and the overall value and rate of expansion of the future UAV market could be largely driven by advances to be made in the field of autonomy. Technology development of a fully autonomous UAV, which refers to the technology that enables aircrafts to fly with reduced or no human intervention, comprises the following seven main categories: 1. Task allocation and scheduling: Determining the optimal distribution of tasks amongst a group of agents, considering different constraints such as time and equipment. 2. Communications: Communication management and coordination between multiple agents in the presence of imperfect information and missing data. 3. Path planning: Determining an optimal path for vehicle to move while meeting certain objectives and dealing with constraints, such as obstacles or fuel requirements. 4. Sensor fusion: Combining data from different sensor sources to be used in vehicle. 5. Trajectory generation (also named motion planning): Determining an optimal control movement to follow a given path or to go from one position to another. 6. Cooperative tactics: Formulating an optimal algorithm and spatial distribution of activities between agents in order to maximize chance of success in all given challenges.

18 7 7. Trajectory regulation: The specific control mechanisms required to constrain a vehicle within some deviation from a trajectory. In the present study the ultimate focus is on developing a control mechanism to improve trajectory tracking and set point regulation. Thus secure performance is guaranteed in various possible extreme conditions such as complex agile maneuvers in confined spaces. 1.1 Background of Unmanned Aerial Vehicles As briefly discussed in the previous section, the term of UAV refers to aircrafts that are designed to operate with no human pilot on-board [3]. Consequently, UAVs have been considered for many applications with the purpose of reducing the human involvement, and in turn, minimizing mission limitations where human presence is dangerous, as in a case of searching for people trapped in a fire, or finding sources of dangerous chemicals at industrial accident sites. Conventional UAVs are typically classified in two main groups: fixed-wing and rotor crafts. Each of these two types has advantages and disadvantages depending on the aimed mission and the characteristics of the environment in which the desired task is to be executed. Conventional fixedwing aircrafts are capable of achieving long lasting flights, long distance ranges and high forward speeds that are not attainable in traditional rotor crafts. However maneuverability is limited for fixed-wing vehicles. Therefore, they are not suitable for operations in confined spaces. Conventional fixed-wing aircrafts require a constant forward speed to generate lift. On the other hand, rotary-wing aircrafts, such as helicopters, have the advantage of being able to hover and perform Vertical Take Off and Landing (VTOL) without a need for runways in a limited space. Additionally,

19 8 rotor crafts have some additional advantages including the ability to fly stationary in hover, omni-directionality and VTOL capability. However, traditional VTOL vehicles are usually highly affected by wind and ground effect disturbances. Moreover, their big rotors decrease maneuverability, causing a limitation in application of rotary-wing aircrafts in confined spaces. Having rotary-wing aircrafts advantages in mind, from stability perspective, although fixed-wing aircrafts are generally internally stable [4], the rotary-wing aircrafts dynamics are naturally unstable without closed-loop control [5]. This intuitive characteristic makes the control system design more challenging for rotary-wing UAVs. In what follows and throughout this thesis the term UAV refers to a rotary-wing UAV Examples of Unmanned Aerial Vehicles The analysis of control methods and the investigation of their performances are focused on civilian UAV missions in this thesis. Hence, a brief historical development of the civil UAV sector is presented here. The following UAVs are examples of some of the more prominent civilian UAV systems that are considered to be operational. A more complete list of civilian UAVs is presented in [6]. 1. AEROSONDE: The AEROSONDE UAV was developed by Aerosonde Pty, Ltd. of Australia. It was originally designed for meteorological reconnaissance and environmental monitoring although it has found additional missions. AEROSONDEs are currently being operated by NASA Goddard Space Flight Center for earth science missions. 2. ALTAIR: ALTAIR was built by General Atomics Aeronautical Systems Incorporated as a high altitude version of the Predator aircraft. It has been

20 9 designed for increased reliability. It comes with a fault-tolerant flight control system and triplex avionics. It is operated by General Atomics although NASA Dryden Flight Research Center maintains an arrangement to conduct Altair flights. 3. ALTUS I/ALTUS II: The ALTUS aircrafts were developed by General Atomics Aeronautical Systems Incorporated, San Diego, CA, as a civil variant of the U.S. Air Force Predator. Although ALTUS is similar in appearance with Predator, it has a slightly longer wingspan and is designed to carry atmospheric sampling and other instruments for civilian scientific research missions in place of the military reconnaissance equipment carried by the Predators. 4. CIPRAS: The Office of Naval Research established CIRPAS in the spring of CIRPAS provides measurements from an array of airborne and groundbased meteorological, aerosol and cloud particle sensors, and radiation and remote sensors to the scientific community. The data is reduced at the facility and provided to the user groups as coherent data sets. The measurements are supported by a ground based calibration facility. CIRPAS conducts payload integration, reviews flight safety, and provides logistical planning and support as part of its research and test projects around the world. 5. RMAX: The Yamaha RMAX helicopter has been around since about It has been used for both surveillance and crop dusting, and other agricultural purposes. 6. Quad-rotor: Although the first successful quad-rotors flew in the 192s [7], no practical quad-rotor helicopters have been built until recently, largely

21 1 due to the difficulty of controlling four motors simultaneously with sufficient bandwidth. Recently, quad-rotor design has become one of the most popular designs for small UAVs. The dynamic model of the quad-rotor helicopter has six outputs while it only has four independent inputs. Therefore the quad-rotor is an under-actuated system and it is not possible to control all its outputs at the same time. Due to the fact that new aerial vehicles have no conventional design basis, many research groups build their own tilt-rotor vehicles according to their desired technical properties and objectives. Some examples of these tilt-rotor vehicles are large scale commercial aircrafts like Boeing s V22 Osprey [8], Bell s Eagle Eye [9] and smaller scale vehicles like Arizona State University s HARVee [1] and Compigne University s BIROTAN [11] which consist of two rotors. Some other examples of tilt-rotor vehicles with quad-rotor configurations are Boeing s V44 [12] and Chiba University s QTW UAV [13]. In fact none of these UAVs can be deployed in confined spaces. The focus of this research is on developing a control system for an advanced unconventional VTOL UAV with high maneuverability and capability, named evader, to provide secure performance in confined spaces. The evader is introduced briefly in Section 1.2 and more comprehensively in Chapter Introducing the evader Unmanned Aerial Vehicle The capability of small UAVs which only need small ground spaces to fly has become a priority element in the design and development of unmanned vehicles in this modern era [14]. In this respect, UAVs that are small, autonomous, and have high maneuverability have been considered in recent years. Furthermore the ducted fan configuration

22 11 has gained more interest (Fig. 1.1). Enclosing the rotors within a frame, ducted fan rotors, would conclude better rotor protection from breaking during collisions, permit flights in obstacle-dense environments among other aspects beyond the scope of this thesis such as aerodynamic characteristics. It decreases the risk of damaging the vehicle, or its surroundings. Moreover, with the helicopters limitations in both flying in closed environments and forward speed, development of alternate VTOL air vehicles has been increasingly considered by many researchers [15], [16], [17], and [18]. The most popular small helicopter type UAV in the literature is the quad-rotor. Although quad-rotors are small and have diverse advantages over traditional helicopter designs in the case of small electrically actuated aircraft [8], they do not have high maneuverability in confined spaces due to their under-actuated property. The evader is a novel VTOL UAV which is targeted for operations in confined spaces. In order to achieve this goal, the vehicle exploits a new mechanism of dual ducted fans with a lateral and longitudinal rotor tilting mechanism to provide the agility characteristic needed for missions in confined spaces. The novelty of the design is the degrees of freedom of the fans which can rotate along both longitudinal and lateral axes as shown in Fig This mechanism utilizes the inherent gyroscopic properties of tilting rotors and driving torques of the fans for vehicle pitch control, and eliminates the need for external control elements or lift devices. The special characteristics of this new design, which will be discussed in more details in Chapter 3, offer unique capabilities such as inclined hovering, a task which is not theoretically possible by other type of VTOLs [19]. As a result of the unique characteristics of the evader, it is a potential alternative of VTOL for complex maneuvers in urban areas or inside confined spaces. Throughout present thesis, in order to successfully perform these missions, an accurate nonlinear model of the vehicle s dynamics is developed,

23 12 Figure 1.1: Ducted fans of the unconventional highly maneuverable VTOL UAV. Figure 1.2: Fans rotating around longitudinal (y-axis) and lateral (x-aixs) axes. and control methodologies are designed based on the UAV dynamic model to precisely track the trajectory (position and orientation) of complex maneuvers. This research is focused on this kind of VTOL UAV. The weight of the prototype vehicle, that the simulations of this thesis are based on its parameters, is approximately 6.5 kg and the fans are cm. The fans rotational speeds must be about 6 rpm to produce enough thrust (equal to the weight of the vehicle) for hover flight. This m long and m wide aerial vehicle is one of the first of its kinds among tilt-wing vehicles on that scale range. 1.3 Motivation As mentioned, the development of fully autonomous and self guided UAVs will result in minimizing the risk to and the cost of human life. UAVs have been used in various anti-terrorist and accident-related missions and emergencies, sometimes with success, but more often just confirming their potential. Although UAVs showed their potential, they were not completely reliable, accurate and capable of performing the tasks needed. They were not able to perform diverse tasks in an obstructed unknown urban environment (e.g., Search and Rescue (SAR) and patrol operations). The main

24 13 motivation of this work is to deploy UAVs in confined spaces such that they can be used in operations that may not be possible today such as search for victims in a collapsed building. Despite continued research has recently resulted in relative success and considerable enhancements in the UAV design, there still remain a number of major challenges in the mentioned seven fields in Section 1. The main challenges associated with the UAV controller design that require huge efforts can be listed as follow: open loop instability High degree of coupling among different state vectors and different variables Highly nonlinear behavior Diverse sources of noise and disturbances Very fast dynamics especially in the case of small model UAVs Thus, designing a nonlinear control that demonstrates the high performance and robust stability encounter with significant disturbances is a challenging control problem for UAVs. In fact, a great amount of innovative work must still be done across a number of disciplines before the full potential of UAVs would be achieved. A number of examples of such challenges in the scope of this research interest are: 1. Overcoming the nonlinearity characteristics of UAV flying vehicle such as open-loop instability and very fast dynamics to achieve a perfect control and tracking for complex maneuvers. 2. Applying one type of controller for position and orientation trajectory regulation.

25 14 3. Investigating the effect of changes in aerodynamic of the vehicle on the control system for UAV Flying in close proximity of solid boundaries (e.g., ground and walls). 4. Flying autonomously in presence of model inaccuracies (parametric uncertainties), unmodeled dynamics and external disturbances. 5. Flying in confined spaces, which is a challenge for both aerodynamic and control system design. 6. Performing aggressive maneuvers with agility and stability. 1.4 General Problem Statement UAVs have been considered for many applications with the purpose of reducing the human involvement where human presence is dangerous and in turn, reducing mission limitations. All these applications demand advanced robotics technologies, leading ultimately to fully autonomous, specialized, and reliable UAVs. In order to achieve the stated mission, without a need to have an expert pilot, certain levels of autonomy are needed for the vehicle to maintain its stability and follow a desired path, under embedded guidance and control algorithms. The level of autonomy of current UAV systems, in terms of their control systems for precise trajectory tracking, varies greatly. Recent advances in technology, including sensors and micro controllers, now allow small electrically actuated UAVs and Micro UAVs to be built relatively easily and cost effectively. These small UAVs, such as small quad-rotors, have completely new applications and would be able to fly either indoors or outdoors. Indoor flight offers

26 15 some challenging requirements in terms of size, weight and maneuverability of the vehicle. Combined indoor and outdoor flying also requires a more advanced on-board automation system. Inside a building, not much space for maneuvering is available, but many obstacles exist. Therefore, a very accurate stabilization of the platform, a highly precise trajectory tracking, and a highly maneuverable UAV are necessary in order to guarantee a higher degree of autonomy. Certain control systems enable certain UAVs to operate in cluttered environments, but not for indoor confined spaces or urban confined environments. A number of other challenges are associated with small UAV control systems in each of these environments. For instance, wind gusts in outdoor spaces, the wall effect when flying in close proximity to buildings in urban areas and the ground effects while flying close to the ground surface, are few examples. In order to extend the range of current applications of UAVs to areas such as search and rescue, as well as indoor surveillance, a reliable control methodology and an agile UAV configuration are required that could facilitate highly complex maneuvers in confined spaces. To effectively overcome this difficulty, the following problem statement is formulated in this thesis: Develop a control methodology for the new configuration of a small VTOL UAV, evader, to successfully maneuver through confined 3D spaces in the presence of external disturbances such as wind gusts, ground and wall effects and perform complex tasks such as inclined hover and aggressive maneuvers with agility and stability. Motivated by the goal to design a controller which is robust to external disturbances and capable of adapting to changes in model parameters as well as sensor noise, the controller is designed in this thesis for orientation and position regulation

27 16 and trajectory tracking with capability of independent control of all 6 degrees of freedom, including pitch, roll, and yaw angles, and altitude, lateral, and longitudinal translations. 1.5 Thesis Outline This thesis is organized as follows: Chapter 2 provides a literature review of the control techniques and methods that have been studied on UAVs. This chapter also includes objectives and goals to be addressed in this thesis. Chapter 3 provides the detailed discussion of the characteristic of the evader and how it produces the essential moments, as well as the nonlinear dynamic model of the evader vehicle. The following three chapters are focused on the proposed control methodologies and how these approaches are employed to achieve successful control in various flight scenarios. This includes a design of feedback linearization, adaptive feedback linearization and adaptive feedback linearization with robust modification techniques in Chapter 4, Integral backstepping and adaptive integral backstepping controllers in Chapter 5, and sliding mode control technique in Chapter 6. Chapter 7 addresses the problem of approximating the relation between virtual and actual control signals using a neural network as a nonlinear function approximator. A comprehensive set of simulations on adverse flight conditions such as windy situations and ground effects, and on performing aggressive maneuvers are presented in Chapter 8. Finally, Chapter 9 provides a list of contributions and future work of this research.

28 Chapter 2 Literature Review Successful implementation of a UAV depends on the level of controllability and flying capabilities. Throughout the years, different control methods have achieved different levels of success in controlling UAVs. These methods can be classified in two main categories: i) linear control and ii) nonlinear control methods. This chapter discusses these different methods and their limitations. This chapter starts with a comprehensive survey of works that have been done so far, related to UAV control, in Section 2.1, followed by the objectives and goals of the present research in Section 2.2. Definitions of terms used in this thesis to investigate the problem in hand are introduced in Sections 2.3. The main key points of this chapter are summarized in Section Overview of Previous Research on UAV Control As stated in Chapter 1, UAVs have recently attracted considerable interest for a wide variety of applications in the civilian world, including monitoring of traffic conditions, recognition and surveillance of vehicles, and search and rescue operations [2]. In order for a UAV to accomplish its tasks in each of these applications, a fully autonomous flight system is needed. In addition, fully autonomous flight system technology requires high-authority control systems, such as position and orientation control systems and trajectory tracking systems. So far, various control method- 17

29 18 ologies have been developed for controlling UAVs, ranging from classical linear and nonlinear techniques to fuzzy control intelligent approaches. However, the fact is that these techniques have been applied mostly on helicopter type air vehicles. The existing works can be subdivided into two main categories in term of the tasks of control systems: the first class addresses the stabilization (regulation) problem, whereas the second group deals with solving the trajectory tracking issues. In stabilization problems, a control system, called stabilizer (or a regulator), should be designed to make the state of the closed-loop system stable around an equilibrium (operating) point. Example of stabilization task is altitude control of UAVs. In tracking control problems, the design objective is to construct a controller, called tracker, so that the system output tracks a given time-varying trajectory. Making a UAV fly along a specified path is a tracking control task. In the following sections, some of these works in the literature on controlling rotary-wing VTOL aircrafts (e.g., helicopter and quad-rotor) by linear and nonlinear control approaches for the purpose of stabilization and trajectory tracking are presented Linear Control Techniques of UAV Flight Control Cranfield University s Linear Quadratic Regulator (LQR) controller [21], Swiss Federal Institute of Technology s Proportional-Integral-Derivative(PID), Linear Quadratic (LQ) controllers [16] and Lakehead University s PD 2 [22] controller are examples of the controllers developed on quad-rotors linearized dynamic models. The result of the work of Pounds et. al. shows that linear controls successfully stabilized the prototype X-4 Flyer in the presence of step disturbances [23]. The vehicle uses tuned plant dynamics with an on-board embedded attitude controller to stabilize flight. Later the same research group tested a newer Mark II prototype of quad-rotor with

30 19 a linear single-input single-output (SISO) controller to regulate its attitude without disturbances [24]. The controller designed by Pounds et. al. stabilized the dominant decoupled pitch and roll modes, and used a model of disturbance inputs to estimate the performance of the UAV. The disturbances experienced by the attitude dynamics were expected to take the form of aerodynamic effects propagated through variations in the rotor speed. Therefore, the sensitivity model was developed for the motor speed controller to predict the displacement in position due to a motor speed output disturbance. The desired position variations were in the order of.5 m and the success of the controller to regulate attitude was at low speeds. The second iteration testbed of the Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control (STARMAC-II) quad-rotor prototype achieved free-flight hovering using PID control [25], where it was noted that wind disturbances caused the control to fail. In [25], a number of issues were observed in quad-rotor aircrafts, operating at higher speeds and in the presence of wind disturbances. Hoffman et. al. in [25] have explored the resulting forces and moments applied to the quad-rotor related to three aerodynamic effects. The first group of the mentioned effects results from total thrust variation not only with the power input, but also with the free stream velocity and the angle of attack in respect to the free stream. This type of effect would impact altitude control. The second effect results from differing inflow velocities experienced by the advancing and retreating blades, and it may lead to blade flapping, which includes roll and pitch moments on the rotor hub as well as a deflection of the thrust vector. The third group of effects that have been researched by Hoffman et. al. deals with the interferences caused by the vehicle body in the slip stream of the rotor. They may result in unsteady attitude-tracking difficulties. The impact of these type of effects may be significantly reduced by airframe modifications.

31 2 In summary, according to the result of [25], existing models and control techniques are inadequate for accurate trajectory tracking at higher speeds and in uncontrolled (unknown or not-engineered) environments. Later, the same research group worked on outdoor trajectory tracking [26]. Hoffman et. al in [26] presented a trajectory tracking algorithm to follow a desired path. The proposed control law in [26] tracks line segments connected to sequences of waypoints at a desired velocity. The discussed trajectory tracking algorithm has been experimentally tested to track a path indoors with 1 cm accuracy and outdoors with 5 cm accuracy. Another prototype called OS4 achieved autonomous flight where a linear Proportional Derivative (PD) control maintained stable hover providing robustness to small disturbances [27]. Although, the results of linear control approaches including PD, PD 2, PID, LQ and LQR methods are sufficiently good, either for stabilizing specific operating points such as hover flight, or small displacements from hover, this stabilization can only be achieved at low velocities and with small additive aerodynamic disturbances. It is worth noting here that unstable situations may occur as rotor speed increases and also in presence of external disturbances such as wind gusts and ground effects. Unstable situations makes untethered flight almost impossible for flying vehicles [24]. Moreover, in real conditions the use of a classical linear control is limited to a small neighbourhood around the operating point. Due to the fact that tracking complex trajectories involves far away operation from neighbourhood of operating points, performing difficult maneuvers, which require complex trajectory following control, are not achievable with linear control theory. However, since the goal of this thesis is to design a controller for the evader UAV to perform difficult tasks and maneuvers such as tracking complex trajectories and regulation in presence of external disturbances, classical linear controllers are not applicable, and nonlinear control approaches are

32 21 required Nonlinear Control Techniques of UAV Flight Control Nonlinear controls can substantially expand the region of controllable flight angles compared to linear controls. For instance, Spectrolutions HMX-4 is a tethered quadrotor that uses state inputs from a camera fed into a feedback linearization control without disturbances [28]. In this study, ground and on-board cameras, were used to estimate the full six degrees of freedom of the helicopter. The pose estimation algorithm is compared through simulation to some other feature based pose estimation methods and is shown to be less sensitive to feature detection errors. Backstepping controllers have been used to stabilize and perform output tracking control [28]. Nonlinear controls also achieved robustness to impulse disturbances, both in simulation [29], [3] and using a test-stand experiment [31], [32]. In [33] and [34], a nested-saturations controller stabilized a Draganfly III in the presence of impulse disturbances, and results were compared to linear feedback controls. An algorithm was introduced by Hauser et al. to control the VTOL based on an approximate inputoutput linearization procedure that achieves bounded tracking [35]. A non-linear small gain theorem was proposed in [36], for stabilizing a VTOL, which proved the stability of a controller based on nested saturations. An extension of the algorithm proposed by Hauser was presented in [37], finding a flat output of the system that was used for tracking control of the VTOL in the presence of unmodeled dynamics. The forwarding technique developed in [38] was used in [39] to propose a control algorithm for the VTOL. This approach leads to a Lyapunov function which ensures asymptotic stability. Other techniques based on linearization were also proposed in [4]. Marconi proposed a control algorithm of the VTOL for landing on a ship whose deck oscillates

33 22 [41]. They designed an internal model-based error feedback dynamic regulator that is robust with respect to uncertainties. [42] presented an algorithm to stabilize a VTOL aircraft with a strong input coupling, using a smooth static state feedback. An approach based on Lyapunov analysis to control the VTOL which can lead to further developments in nonlinear systems is presented in [43]. The controller has been tested in numerical simulations, but also in a real-time application. It simplified the tuning of the controller parameters. [44] proposed control strategy which aims to be both adaptive to model uncertainty (payloads) as well as robust to disturbances. There is a summary of UAVs stabilization literature review available in [44]. Reference [45] provides a review of adaptive intelligent approaches for robust control of a helicopter. Uncertainties associated with dynamic models lead to a more challenging control design. Different strategies have been proposed to deal with uncertain quad-rotor model, such as adaptive control, neural network based control, sliding mode control, H control and so on. In [46], a direct adaptive control algorithm was designed for the tracking control of a quad-rotor UAVs roll, pitch, yaw angles, together with altitude while compensating for the model parameter uncertainties. A reference system corresponding to a virtual UAV, which contains a third order oscillator, was utilized to track the desired trajectory. In [47], a backstepping based approach was used for quad-rotor UAV control, while two neural networks were used to approximate the uncertain aerodynamic components. More literature review for quad-rotor UAV control can be found in [48]. There are also robust controllers designed for quad-rotor systems. A sliding mode disturbance observer was presented in [49] to design a robust flight controller for a quad-rotor vehicle. This controller allowed continuous control, robust to external disturbance, model uncertainties and actuator failure. Robust adaptive-fuzzy control

34 23 was applied in [5]. This controller showed a good performance against sinusoidal wind disturbance. Mokhtari presented robust feedback linearization with a linear generalized H controller, and the results showed that the overall system was robust to uncertainties in system parameters and disturbances, when weighting functions are chosen properly [51]. In [52], a robust dynamic feedback controller of Euler angles is proposed using estimates of wind parameters. This controller performed well under wind perturbation and uncertainties in inertia coefficients. In [53], a sliding mode controller was suggested. Due to the under-actuated property of a quad-rotor helicopter, they divided a quad-rotor system into two subsystems: a fully-actuated subsystem and an under-actuated subsystem. Two separate controllers were designed for these subsystems. A PID controller was applied to the fully actuated subsystem and a sliding mode controller was designed for the under-actuated subsystem. Because of the advantage of a sliding mode controller, namely insensitivity to uncertainties, it robustly stabilized the overall system under parametric uncertainties. A pre-trained neural network stabilized a Draganfly II quad-rotor in hover without disturbances[54]. Adaptive neural network controls successfully stabilized quad-rotors in simulation [55], [56]. Most of these methods in prior works have focused on simple trajectories, especially in case of having uncertainties associated with dynamic model of UAVs. The simple external disturbances such as impulse signals were studied in most of the previous works. However, even few studies on sinusoidal wind disturbances have not achieved the ability of performing complex tasks and executing in confined spaces. In fact, performing aggressive maneuvers providing agility and stability with rotary-wing aircrafts have not received much attention in the literature.

35 Control of evader Vehicle in the Literature A great deal of research has been done on controlling rotary-wing aerial vehicles such as helicopters and quad-rotors [57], [46], [58], but controlling VTOLs having a lateral and longitudinal rotor tilting ability, such as with evader, is new in the literature. Previous research on evader by Gary Gress mainly discussed the evader mechanism and the potential of better control responses and independent 6-axis control [59], [1], [19]. Gress linearized the equation of motion in pitch by assuming small values for a propeller tilt angle and used a simple linear proportional controller. He investigated the predicted pitch response of the MicroVader UAV to positive control input (positive angles) of tilt angle for various oblique angles [1]. The feedback proportional controller was applied on the evader for regulation of the vehicle pitch angle for different propeller speeds and different longitudinal angles [6]. Although the above-mentioned research by Gress, in evader control and operation, investigated the potential of an OAT mechanism to produce sufficient moments to change the orientation of the UAV, the orientation and position set point regulation and trajectory tracking of this vehicle to be used as part of an autonomous flight system, which are the main focus of the present research, were not studied before. In fact, all the methods presented in Sections and have been studied and applied on helicopter type of UAVs especially on quad-rotor vehicles, but evader is still a new subject with lots of room for research and experiment. Hence, this thesis provides a comprehensive study of different nonlinear control approaches on evader aerial vehicles in order to find the best choice of control methodology for this vehicle in order to implement an autonomous UAV.

36 Objectives and Goals In terms of UAV control, most prior works have focused on simple trajectories at low velocities. Performing complex and aggressive maneuvers providing agility and stability with rotary-wing aircrafts has not been studied much in literature. Additionally, previous treatments of rotary-wing vehicle dynamics have often ignored known aerodynamic effects of rotor craft vehicles. At slow velocities, such as during hovering period, ignoring aerodynamic effects is indeed a reasonable assumption. However, even at moderate velocities, the impact of the aerodynamic effects resulting from variation in air speed is significant (e.g., wind gusts). Preliminary results of the inclusion of aerodynamic phenomena in vehicle and rotor design show promise in flight tests, although an instability currently occurs as rotor speed increases, making untethered flight of the vehicle impossible [24]. Although many of the effects have been discussed in the helicopter literature (e.g, flow simulations of a helicopter in low speed forward flight in ground effect) [61], [7], [62], their influence on evader type of UAV has not been explored. Considering the shortcomings described above and motivated by the overall objective of developing an aerial vehicle, capable of performing complex and agile tasks in confined spaces, this thesis is focused on the following objectives: 1. The first step of every control design is modeling the system in order to be controlled. The performance of the UAV controller will be dependent on the availability of a sufficiently accurate vehicle model. Thus, the complete dynamic model of a VTOL aerial vehicle having a lateral and longitudinal rotor tilting mechanism (e.g., evader) is derived based on a first principles approach. Chapter 3 is devoted to development the complete dynamic model of the evader.

37 26 2. The developed 6 Degree of Freedom (DOF) nonlinear dynamic model of the vehicle in this thesis accounts for various parameters which affect the dynamics of a flying structure, such as gyroscopic effects and ground effects. The nonlinear state-space model of the VTOL vehicle under investigation is presented for the first time in this thesis. 3. There are more advantages associated with the OAT mechanism of evader than just stability and controllability in the conventional sense, which have not been explored yet. Examining these properties by applying a proper choice of controller to verify the OAT capabilities, is one of the goals of this project. To achieve this goal, various simulation experiments are tested to investigate the characteristics of this unique UAV. 4. This project is a comprehensive study on controlling the evader aerial vehicle with nonlinear control techniques, namely: 1) feedback linearization, 2) adaptive feedback linearization, 3) adaptive feedback linearizarion with robust modification (in Chapter 4), 4)integral backstopping, 5) adaptive integral backstepping (in Chapter 5), 6) sliding mode approach (in Chapter 6). 5. Along with the presented nonlinear methodologies, another objective of this thesis is to achieve full six degrees of freedom control including the position and orientation stabilization and regulation, which is a unique characteristics of evader due to the fact that the rotary wing UAVs are usually under-actuated vehicles, and controlling all six outputs of interest at the same time is impossible. 6. Using the above-mentioned nonlinear control techniques to investigate the capabilities of the evader such as performing aggressive and agile maneuvers,

38 27 maneuvering close to ground and wall surfaces for indoor and outdoor missions, taking off and landing from sloped surfaces, and tracking an object while pointing to it that requires the pitched hover capability. 7. Another focus of the present research is to investigate the application of different control approaches on the evader to obtain asymptotic stability. Trajectory tracking and set point regulation are desired with complete performance. For the purpose of this study, complete performance is defined as a performance that provides the asymptotic stability of the tracking method with structured (e.g., modelling errors, unmodeled dynamics, sensor noise) and unstructured uncertainties (e.g., external disturbances). 8. The designed control structure requires achieving both robustness and high performance in presence of disturbances. The robustness of the flight controller is defined as its ability to compensate for: 1) external disturbances such as wind gusts and ground and wall effects, 2) model parameter uncertainties in terms of changing payload and aerodynamic parameters, and 3) sensor noise for attitude control signals. 9. Real actuator signals can not be obtained directly as an outcome of a control algorithm. This problem makes the feasibility of control approaches a difficult task to investigate. A neural network mapping is utilized to verify the feasibility issue and to obtain the amount of system inputs including longitudinal and lateral angles of each duct, and the rotor speeds of each of them. In summary, the ultimate goal of this project is providing a powerful control system for a fully autonomous UAV, which would be capable of doing agile and

39 28 aggressive maneuvers for indoor and outdoor applications. For this purpose, the focus of this thesis is on the accurate stabilization and precise trajectory tracking in different flight scenarios with sensor noise, model inaccuracies and unmodeled dynamics. 2.3 Definitions A set of fundamental definitions are used for interpretation of a number of terms to address the problem at hand within this thesis. Some of these definitions are specifically defined for this thesis. 1. Autonomy: The ability to execute processes or missions using on-board decision capabilities. 2. Agility: Agility refers to being able to execute controllable maneuvers under high g forces on complex flight trajectories, very much like piloted fighter aircrafts do. 3. Aggressive maneuvers: Aggressive maneuverability in this thesis is in the sense of attitude control: 1) controlling in the whole range of the attitude angles of the UAV, and 2) tracking a given trajectory at the highest possible velocity. If aggressive maneuverability in these terms is achieved, the controller described here executes, in a stable and robust manner: 1) tracking of trajectories describing curvilinear translational (or horizontal) motion at relatively high speed and constant altitude, and 2) set-point regulation for fast translational acceleration/deceleration, hovering, and climb. 4. Asymptotic stability: An equilibrium point is asymptotically stable if it

40 29 is stable, and if in addition there exists some r > such that x() < r implies that x(t) as t [63]. Asymptotic stability means that the equilibrium is stable, and in addition, states started close to actually converge to as time goes to infinity. 5. Asymptotic tracking: Asymptotic tracking implies that perfect tracking is asymptotically achieved. 6. Confined environment: Environments with high environment density measure. In confined environments, the distance between the UAV and obstacles is usually smaller than in cluttered environments. 7. Exponential stability: An equilibrium point is exponentially stable if there exist two strictly positive numbers γ and λ such that t >, x(t) γ x() e λt in some ball B r around the origin [63]. In other words, it means that the state vector of an exponentially stable system converges to the origin faster than an exponential function. 8. Lie derivatives: Let h : R n R be a smooth scalar function, and f : R n R n be a smooth vector field on R n, then the Lie derivative of h with respect to f is a scalar function defined by L f = hf. 9. Lie Bracket: Let f and g be two vector fields on R n. The Lie bracket of f

41 3 and g is a third vector field defined by [f,g] = gf fg 1. Maneuvering: Maneuver is a tactical move, or series of moves, that improves or maintains a UAV s strategic situation in a competitive environment or avoids a worse situation. 11. Perfect tracking (perfect control): When the closed-loop system is such that proper initial states imply zero tracking error for all the times, y(t) = y d (t), t. 12. Relative degree: The number of times the output of a system needs to be differentiated to generate an explicit relationship between the output y and the input u. 13. Robustness: The robustness of the flight controller is defined as its ability to compensate for external disturbances. 14. Reliable control (reliability): The ability of a UAV flight system to adapt to system or hardware failures is called reliability, which is a key technology for flying UAVs. Considering that, the most critical system for the aircraft is the flight control system, having a reliable control is critical. One approach to improve system reliability is simply to increase the redundancy of flight systems. This comes with both an initial cost and an on-going weight penalty. Another approach would be adding on-board intelligence to recognize and remedy a failure.

42 Smooth function: A function that has derivatives of all orders is called a smooth function. 16. Stability: The equilibrium state x = is said to be stable if, for any R >, there exists r >, such that if x() < r, then x(t) < R for all t. Otherwise, the equilibrium point is unstable [63]. Qualitatively, a system is described as stable if starting the system somewhere near its desired operating point implies that it will stay around the point ever after. 19. Zero-dynamics (internal dynamics): The zero-dynamics for a nonlinear system is defined to be the internal dynamics of the system when the system output is kept at zero by the input. 2.4 Summary The present study is devoted to developing and discussing a set of perfect control laws to achieve high maneuverability and high reliability for autonomous flight in highly cluttered environments. The focus of this research is on a newly built configuration of small rotary-wing VTOL aerial vehicle with ducted fans, each of which has two rotors named evader. A control law needs to be designed to employ the unique flying capabilities of the evader UAV using the novel OAT mechanism. By applying the suggested nonlinear control to the nonlinear dynamics of the evader, this thesis pursues the goal of performing aggressive motions such as tracking of trajectories describing sharp (aggressive) curvilinear translational (or horizontal) motion at relatively high speed, take off and landing from severely sloped surfaces, and stationary pitched hover, which is not possible in any other manned or unmanned aerial vehicles. Diverse sources of noise and disturbance plus very fast dynamics, especially in the

43 32 case of small UAVs such as evader, adds up to the challenges associated with the open loop unstable systems (e.g., rotary-wing UAVs). Therefore, a stability problem should be carefully studied. It is desired to develop a control algorithm that would simultaneously stabilize the orientation and position of the evader UAV based on its nonlinear dynamic model, without making simplification by linearizing the model. That is to achieve the independent six degrees-of-freedom (DOF) of control which includes three orientation angles of pitch, roll and yaw and three Cartesian position variables in 3D space (x, y, z). The model includes frictions due to the aerodynamic torques, drag forces and gyroscopic effects as well as independent tilting of the right and left rotors(different tilting angles) and independent(not necessarily the same) rotor speeds. Moreover, motivated by the desire of maintaining stability in the presence of model uncertainty and external disturbances (such as wind and ground effects generated by the down wash produced by the vehicle itself), the control strategy is aimed to be both adaptive to model uncertainties as well as robust to external disturbances. According to the dynamic model of the vehicle, the resulting control signals from the output of control methodologies are virtual signals. It is important to note that actual control signals should be obtainable from virtual control signals, otherwise the resulting controller would not be a practical design. As a result, another challenge associated with controlling the evader is finding the actual control signals corresponding to actuator signals. Present study takes the advantage of neural network method in order to calculate actual control signals from virtual control signals by approximating the mapping relation between virtual and actual control signals of the system.

44 Chapter 3 Modeling The first step in designing a control system for a given physical plant (the evader in this thesis) is to derive a model that captures the key dynamics of the plant in the operational range of interest. The presented model in [] by G. Gress was the soat equation of motion for the evder tilting its propellers by the same angle in the same but opposite oblique directions. This model only represents the pitching moments results about the aircraft pitch axis. In this thesis, the highly maneuverable characteristics of the evader are modeled to represent the autonomous UAV with double Oblique Active Tilting (doat) mechanism for the first time in the literature. This mechanism allows the vehicle to navigate in confined environments [64], by enabling the vehicle to respond fast and with agility to obstacles. Following traditional modeling approaches, a complete dynamic model of this unconventional UAV is developed using a Newton-Euler formulation. The dynamic model of the evader developed in this research includes reactionary moments, ground effects and aerodynamic friction effects, and considers the capability of independent movements of each rotor about both the rotor s own x-axis and y-axis. When compared to previous models of this vehicle, the developed new model makes the model more realistic and more reliable for complex maneuvers. The lateral (β 1, β 2 ) and longitudinal (α 1, α 2 ) angles, and the propellers speeds (ω 1, ω 2 ) are input signals in the model. The vehicle s orientation angles (φ, θ, ψ), the x and y vehicle position, and UAV s altitude z are the six outputs 33

45 34 in the model manipulated via α 1,α 2,β 1,β 2,ω 1 and ω 2. This chapter focuses on a lift-fan doat or Opposed Lateral Tilting (OLT) mechanism which is used in the evader UAV as a control device. The capability of providing the three required moments for control including pitch, roll, and yaw moments, and the way in which the doat mechanism produces them, are discussed in this chapter. Moreover, a theoretical analysis of the doat vehicle control response is described 3.1 Introduction The ability of small or medium air vehicles to access and operate in confined and obstructed environments is required for air mobility to enable aerial transportation systems and UAV complex mission execution (e.g. search and rescue within collapsed buildings). Satisfying this condition necessitates VTOL aerial vehicles to be highly agile, highly maneuverable, and more compact for a given payload, while maintaining effective vehicle control, which becomes more difficult as the vehicle s size reduces. When the moment arms decrease in length, the control of the vehicle requires larger forces, which conventional control devices (e.g. ailerons) can not provide. To obtain an effective control for a compact UAV, the vehicle should be able to provide sufficient moments for control regardless of its dimensions. A control device that does not rely on moment arms is a gyroscope. It has been used before in satellites, missile guidance, and space stations [65]. A gyroscope generates the large moments required to change the attitudes of satellites and space stations within short time periods [66]. A traditional mechanical Control Moment Gyroscope (CMG) has been proposed in the literature for attitude stability and control [66], [67], but weight limitations make it impractical for small aerial vehicles. Along with

46 35 this, Gary Gress found that utilizing the vehicle s lift-fans as CMGs offers a powerful control system with minimal weight and independent of vehicle geometry or scale [1]. He discovered that the tilt rotor-based mechanism can provide hover stability of a small UAV by using the gyroscopic nature of two tilting rotors [68]. Furthermore, due to the helicopters limitations in both close environments and forward speed, development of alternate VTOL air vehicles has been considered by many researchers [15], [18], [69]. Therefore, a combination of VTOL capability with efficient, high-speed cruise flight plus high maneuverable characteristics in tilt-rotor aircrafts have the potential to revolutionize UAVs. A good example of this type of VTOL aerial vehicles is VTOL having lateral and longitudinal rotor tilting mechanisms that give them the unique ability to maneuver in confined spaces. This entirely new system, which uses only the dual lift-fans for control, has been developed recently [1]. It utilizes the inherent gyroscopic properties and driving torques of the fans for vehicle pitch control, and it eliminates the need for external control elements or lift devices. The system allows for agile and compact VTOL air vehicles by generating pure and extensive moments rather than just forces. Figure 3.1 shows the University of Calgary prototype of the evader UAV that utilizes the doat concepts proposed in [68]. From model simulations verified by several scenarios the doat control mechanism has shown to be very promising. 3.2 Lift-fan OAT Mechanism In the lift-fan OAT mechanism, unlike other tilt rotor UAVs, propellers can tilt independently in two directions (lateral and longitudinal with respect to the vehicle s

47 36 Figure 3.1: Dual-fan VTOL air vehicle having lateral and longitudinal tilting rotors prototype (evader). frame of reference) providing stability and control in hover mode. This mechanism provides the required lift and control moments without the need for any helicopter type cyclic controls. This is unique because most of the UAVs with tilt rotor design have helicopter type cyclic controls. Cyclic controls are not compact and require relatively slow turning with large diameter rotors which are not desirable for the purpose of performing aggressive maneuvers, especially when flying in confined spaces. In this thesis, aggressive maneuver refers to the maneuver with fast changing complex flight trajectories such as acrobatic maneuvers, like loops and barrel rolls, and cuban eight maneuvers. In OAT design, the roll movement is obtained by differential propeller speeds. The yaw angle can be controlled through differential longitudinal tilting. The gyroscopic moment issued from opposed lateral tilting, together with the torque generated by the collective longitudinal tilting, allow to obtain a significant pitching moment. In the following section, we provide a brief description of how this novel vehicle (Fig. 3.1) operates and how the essential moments are produced.

48 37 Figure 3.2: Fans tilted longitudinally 9 degrees for high speed forward flight [1] Special Characteristics of OAT The OAT design comprises a differential or opposed lateral tilting element for generating gyroscopic and fan-torque pitching moments, in addition to the collective longitudinal tilting component, which produces pitch moments from thrust vectors, as well as forces for controlling horizontal motion. This mechanism can contribute more than just stability and control in the conventional sense. Using the dual-axis version makes it possible to have an independent control of all six axes [59]. 1. High Speed Flight: Transition to high speed forward flight or airplane mode is achieved by tilting the fans longitudinally 9 degrees (Fig. 3.2), during which longitudinal stability is maintained by lateral tilting and by the horizontal stabilizer at the rear of the aircraft. Because VTOL air vehicles do not require runways their lifting surface areas need not be as large as those of a conventional airplane. As a result, there is no need for conventional control surfaces (except the horizontal stabilizer) and associated dual control system, thereby reducing weight, complexity, and cost. Furthermore, because the entire wing-halves (fan shrouds) tilt, and differential longitudinal tilting of the fans generates a gyroscopic rolling moment (whether in hover or airplane mode), roll rates of the vehicle are substantially higher than those using a conventional wing with ailerons.

49 38 2. Gyroscopic pitch moments: Tilting both spinning fans simultaneously towards or away from one another laterally produces gyroscopic moments perpendicular to their tilt axes in the right angles direction. This moment, τ gyro, changes the vehicle s attitude as illustrated in Fig. 3.3, and this is the moment used to initiate control and dynamically stabilize the pitch attitude of the developed evader UAV. Returning the spinning discs to their neutral orientation will stop rotation of the vehicle in the case of space vehicles, where it will rest at the new attitude. In aerial vehicles, however, there are aerodynamic and inertial forces which tend to terminate, limit, or enhance the vehicle s rotation without returning the fans to neutral. 3. Fan-torque pitch moments: When using lift-fans as CMGs for air vehicle pitch control, there is another pitching moment associated with the fans lateral tilting, which is a fan-torque pitching moment. Unlike the gyroscopic moment, a fan-torque will remain after the tilting has stopped. Without this moment the fans would have to be tilting continuously to generate gyroscopic moments in order to reach a desired pitch angle or to compensate for a pitch disturbance. With the fans net torques providing a static pitching moment, these aerial vehicles have the potential to remain level in hover despite any pitch imbalances. So they have the ability to pitch while stationary, a particularly advantageous feature allowing direct targetpointing and VTOL take off and land from sharply inclined surfaces. Till now, only tandem-rotor helicopters can achieve pitched hover stationary [7]. 4. Thrust-vectoring pitch moments: The fan net torque may be insufficient to provide the static restoration. Therefore, to improve the vehicle s static stability in all instances, an additional pitch control moment is obtained by collectively tilting the fans in the longitudinal direction while simultaneously tilting them laterally. These improvements all derive from the resulting characteristic of non-vertical thrust vector,

50 Figure 3.3: a) Oppositely spinning disks tilted equally towards one another generating gyroscopic moment τ gyro, b) The whole System rotated about y axis to a new attitude orientation [1]. 39

51 4 which also provides more direct horizontal motion control. Therefore, the fans tilting for full and proper pitch control of the UAV will be in oblique direction. Hence the name of this control method in either of its two executions is single-axis or dual-axis OAT Overview of soat and doat 1. Single-axis Oblique Active Tilting (soat): In the simplest method, called single-axis OAT or soat, the fans or propellers tilt about a fixed and oblique horizontal axis, and the corresponding tilt path lies along a vertical plane oriented at a fixed angle α from the longitudinal direction. As a result, the rotating disc changes its lateral and longitudinal tilting following a predefined (fixed) curve. Thus, independent lateral and longitudinal disc tilting is not provided. This is due to the fact that the lateral tilt (β) is coupled with the longitudinal tilt (α). The tilt angle β is measured along the tilt-path plane, and is zero when the propeller spin axis is vertical. The soat mechanism provides full, helicopter-like pitch control of the vehicle. Moreover, it also improves stability and control in yaw and roll, either by reducing their high degree of coupling intuitively associated with dual-fan rotor crafts, or by taking advantage of that coupling. This distinct superiority, together with its simplicity, makes soat an exceptional choice of control method for small UAVs (at the expense of reducing the maneuvering capabilities of the vehicle, e. g. very limited pitched hover). 2. Dual-axis Oblique Active Tilting (doat): In this mechanism the fans tilt independently about the x and y axis providing lateral (β i ) and longitudinal (α i ) tilting angles, respectively (Fig. 3.4). There is much more to be gained by taking full advantage of the dual-axis OAT capability, like the potential of better control

52 41 Figure 3.4: Schematic of VTOL aerial vehicle with dual-axis OAT mechanism [1]. response for independent 6-axis control, vertical take offs and landings from severely sloped terrain, remaining perfectly level in hover, remaining stationary while pitching and yawing to track a target, and extreme maneuvering in three dimensional space [59]. The capabilities of doat are still an open area of research and exploration. To investigate these capabilities and verify the characteristics of this control mechanism, in this thesis, a full model of the dual-fan VTOL aerial vehicle with lateral and longitudinal tilting rotors is derived, which represents a general dynamic model for this kind of vehicle and can be used to explore the features of both soat and doat. In this thesis, the focus is on doat as soat is a specific case of doat. 3.3 Lateral and Longitudinal Rotor Tilting VTOL Modeling In this section, the translational and the rotational dynamic equations of the tilt-rotor aerial vehicle are developed, and a state-space model is suggested for the developed

53 42 UAV prototype. In this modeling, a general form of the evader is considered, having a doat ability in which each of its ducts can have independent (different) lateral and longitudinal angles and different propeller s speeds that have not been considered in previous works; see [7] and [1]. Due to the complexity of the vehicle and its mathematical model, a set of four assumptions are used in this research: Assumptions 1. The UAV structure is rigid and symmetrical, 2. The centre of mass is fixed below the origin of the body fixed frame B, 3. The propellers are rigid and have a fixed pitch blade, and 4. Thrust and drag forces are proportional to the square of the propeller s speed. Let E = {x E,y E,z E } represent the right hand inertial frame and B = {x B,y B,z B } represent the body fixed frame, as can be seen in Fig The evader is studied as a vehicle with six Degree of Freedom (DOF). It changes its position along three coordinate axes, longitudinal x, lateral y and vertical z. Its attitude is described by three angles, roll φ, pitch θ, and yaw ψ as shown in Fig The Euclidean position of the UAV with respect to E is represented by ζ(t) = [x(t),y(t),z(t)] T R 3, and the Euler angles of the UAV with respect to E are represented by η(t) = [φ(t),θ(t),ψ(t)] T R 3 where φ(t) is the roll, θ(t) is the pitch and ψ(t) is the yaw angle. Rotating the blades around the vehicle s y axis defines the longitudinal tilting angle (α i ). Considering the right rotor as rotor number one (r 1 ) and the left one as rotor number two (r 2 ), α 1 and α 2 are used to refer to the longitudinal angles of r 1 (rotor/disc #1) and r 2 (rotor/disc #2), respectively. β i, (i = 1,2 for each rotor) is the lateral tilting angle, which denotes rotating the blades around the vehicle s x axis (Fig. 3.5). The equations of motion for a rigid body subject to body force F tot R 3 and

54 43 Figure 3.5: Schematic of the evader VTOL with a body fixed frame B and the inertial frame E. The circular arrows indicate the direction of rotation of each propeller [1]. torque T tot R 3 applied to the center of mass are given by Newton-Euler equations with respect to the coordinate frame B and can be written as: mi 3 3 J V Ω + Ω mv Ω JΩ = F tot T tot where V R 3 is the body linear velocity vector, Ω R 3 is the body angular velocity vector, m R specifies the mass, J R 3 3 is the body inertia matrix, and I 3 3 is an identity matrix. A short list of main parameters and effects acting on the evader is listed in Table 3.1.

55 44 Table 3.1: Main physical parameters and effects acting on the evader VTOL UAV with respect to the inertial frame E. Effect Source Symbol used in the model Forces and torques induced The rotation of two propellers Ftot E = R yxz [F x,f y,f z ] on the vehicle T tot = [τ x,τ y,τ z ] T Aerodynamic friction UAV motions F E aero = [F ax,f ay,f az ] T T aero = [T ax,t ay,t az ] T Gyroscopic effects Change in the orientation of T gyro = [T gx,t gy,t gz ] T rigid body and propeller s plane Gravity effect Center of mass position F grav = [,, mg] T External disturbances Ground effect, wind gusts g r (z),d w (t) Translational Dynamics In this section, the translational dynamics of motion for evader are defined. Aerodynamic forces and moments are derived using a combination of momentum and blade element theory [7]. The VTOL has one left and one right motor with propellers. The direction of the thrust can be redirected by tilting the propellers laterally and longitudinally. A voltage applied to each motor results in a net torque being applied to the rotor shaft, Q i, which results in a propeller speed, ω i, which in turn results in a thrust, T i. In other words, a propeller produces thrust by pushing air in a direction perpendicular to its plane of rotation. Forward velocity causes a drag force, D i, on the rotor that acts opposite to the direction of travel. The thrust and drag forces

56 45 produced by duct/propeller i can be defined below as in [71]: T i = 1 2 C TρA r r 2 rω 2 i D i = 1 2 C DρA r r 2 rω 2 i (3.1) where A r is the blade area, ρ is the density of air, r r is the radius of the blade, ω i is the angular velocity of propeller i, and C T > and C D > are aerodynamic coefficients depending on the blade geometry and the fluid density of the medium (air in this case). T i (i = 1,2) represents the thrust force produced by the right and left propellers, respectively (see Fig. 3.5). During hover, it can be assumed that the thrust and drag are proportional to the square of the propellers rotational speed. Thus the thrust and drag forces are given by T i C T ω 2 i D i C D ω 2 i (3.2) The rotation matrices are comprised of the ducts pitch (α i ) and roll (β i ) manipulation which transform the thrust vectors to the force vectors applied to the vehicle s centre of gravity (cg), as it can be seen in (3.6) below using the rotational matrices R y (α i ) and R x (β i ). R y (α i ) = cos(α i ) sin(α i ) 1 sin(α i ) cos(α i ) (3.3)

57 46 R x (β i ) = 1 cos(β i ) sin(β i ) sin(β i ) cos(β i ) (3.4) R xy (β,α) i = R x (β i )R y (α i ) (3.5) R xy (β,α) i = cos(α i ) sin(α i ) sin(β i )sin(α i ) cos(β i ) sin(β i )cos(α i ) cos(β i )sin(α i ) sin(β i ) cos(β i )cos(α i ) (3.6) The forces applied to the vehicle s cg corresponding to the body fixed frame B are represented by (3.7), in which the thrust vectors (T 1,T 2 ) are multiplied by rotation matrices. F B cg = R xy (β,α) 1 T 1 e 3 +R xy (β,α) 2 T 2 e 3 R xy (β,α) 1 D 1 (e 1,e 2 ) R xy (β,α) 2 D 2 (e 1,e 2 ) (3.7) where e 1 = [ 1 ] T, e 2 = [ 1 ] T and e 3 = [ 1 ] T define the longitudinal, lateral and vertical axes respectively. The drag forces in (3.7) can be considered as a disturbance in the translational dynamic of the evader. Therefore, (3.7) can be written as: Fcg B = sinα 1 T 1 +sinα 2 T 2 sinβ 1 cosα 1 T 1 sinβ 2 cosα 2 T 2 = F x F y (3.8) cosβ 1 cosα 1 T 1 +cosβ 2 cosα 2 T 2 F z It is very important to consider the vehicle s orientation when calculating the Cartesian equations of motion. Similar to most aerial vehicles, this type of tilt-rotor UAV can control its Cartesian position with its attitude. The Cartesian equations of

58 47 motion can be derived by multiplying the force vector (F B cg) by the rotation matrix (R yxz ) to give the force vector applied to the inertial frame (E). The rotation matrix usedinourdevelopmentisintheformr yxz, withrespecttotheright-handconvention using the rotational matrices R y (θ), R x (φ), and R z (ψ). The total force F E tot acting on the vehicle s center of gravity is the sum of the lift and drag forces F E cg created by the rotors, the gravity F grav, the aerodynamic forces F E aero, and the ground effect g r (z), namely: F E tot = F E cg +F grav +F E aero +g r (z)e 3, F E tot = R yxz F B cg +(,, mg) T +R yxz F B aero +g r (z)e 3. (3.9) In forward or sideways flight (horizontal motion), the main rotor downwash is deflected by the fuselage, creating a drag force along the x and y axes additional to the velocity-induced drag force. The aerodynamic friction effect Faero B on the evader body during horizontal motion is modeled in body fixed frame based on the model in [72] as: Faero B = 1C 2 D x A x ρẋ ẋ 1C 2 D y A y ρẏ ẏ 1C 2 D z A z ρż ż K fax = K fay ζ 2 (t) (3.1) K faz where C Dx, C Dy, C Dz represent longitudinal drag coefficients in x, y and z directions respectively, A x, A y, A z are the cross-sectional areas of the evader to the body fixed axes. In the right hand side of (3.11), the terms 1C 2 D x A x ρ, 1C 2 D y A y ρ and 1C 2 D z A z ρ are considered as friction aerodynamic coefficients representing by K fax,

59 48 K fay, K faz respectively and ζ(t) is the vehicle s body linear velocity vector. F E aero in (3.9) is obtained by multiplying the rotation matrix and the aerodynamic friction force in body fixed frame as below: Faero E = R yxz K fax K fay ζ 2 (t) (3.11) K faz Itisshownin[73]thatfuselagedragisnegligibleinhoverandcanbeomittedfromthe hovermodel. ThedragfrictioncoefficientsC Dx, C Dy, C Dz oftheevaderarenotknown and F aero in translational dynamic of the evader is considered in simulations of this research. It can be added to simulations as a disturbance. The term g r (z) represents the ground effect experienced during landing and flying close to the ground surface. Computational Fluid Dynamic (CFD) simulation results show that the ground effect affects the UAV when it is below a certain altitude [74]. This certain level is assumed to be z. g r (z) = a z (z+z cg) 2 az (z +z cg) 2 < z z else (3.12) where a z is the ground effect constant and z cg is the z component of the vehicle s center of gravity. Due to the fact that it is very difficult to derive the exact equations for the ground effect, the term g r (z) is usually considered an unknown perturbation in designing a controller, which requires compensation or adaptation. In this work, we have used the result of the available CFD research on ducted fans in [75] to model g r (z). More explanation of ground and wall effects is given in the next section.

60 Ground and wall effects Ground effects are related to a reduction of the induced airflow velocity. The evader, similar to any other aerial vehicle, experiences ground/wall effects (GWE) during flights in confined spaces. Characterizing these effects on the evader or similar vehicles, which use more than one fan, is of great complexity as the interaction between the fans can also affect the aerodynamic performance. However, if the fans are sufficiently far from each other (i.e., > 3r r where r r is the rotor radius), such interactions can be negligible. The principal need is to find a model of this effect for the evader to improve the autonomous take off and landing controllers. In unconventional UAVs such as the evader, the rotor can roll and be positioned at different orientations, or it may operate in proximity to a lateral wall as well as the ground. In order to have successful autonomous flights in confined spaces, it is necessary to examine these effects. The effects of solid surfaces (i.e., ground and side wall) on the performance of one tilting fan of unconventional UAVs, such as evader, is computed with CFD analysis in [75]. The interactions between the two fans were assumed to be negligible as in the evader prototype the separation between the fans is larger than 3r r [75]. The result of this investigation is used in Chapter 8, Section 8.3, to model g r (z) in (3.9), based on (8.1) in this study, and added to the model of the vehicle. The influence of the ground effects in controlling the evader, while maneuvering close to the ground is investigated in this thesis in Chapter Rotational Dynamics Inthissection, allthemajortorquesactingonthevehicleinordertodrivetheangular acceleration equations of motion are presented. It has been identified that there are

61 5 four major torques acting on the vehicle that need to be considered when controlling a VTOL with doat: gyroscopic moments, propeller torques, thrust vectoring moments and reactionary torques. 1. Gyroscopic moments (τgyro,i i = 1,2): One of the primary torques acting on the vehicle are the gyroscopic torques created when tilting the ducts. Forcing propellers to perform laterally in opposite directions will create gyroscopic moments which are perpendicular to their respective spin and tilt axes. They are created about a perpendicular axis from the orthogonal axis of rotation of the propeller and the orthogonal tilt axis. It is worth noting that each duct creates its own torques about its principle axis, where ω i will be positive or negative depending on the direction of rotation, and each duct is capable of pitch (α i ) and roll (β i ) motions. For example, with the propeller spinning clockwise and positive tilt rotation velocity for α i, there is a reactionary torque created perpendicular to the orthogonal axis of the spinning propeller and the tilt axis of β i. These moments are defined by the cross product of thekineticmoments(j re ω i )ofthepropellers, wherej re R 3 3 = diag(j rx,j ry,j rz )is the inertia matrix of the spinning part of the fan around tilt axis and the tilt velocity vector, as below: τgyro i = J rx ω i J ry ω i J rz ω i β i α i = J rz ω i α i J rz ω i βi J ry ω i βi +J rx ω i α i (3.13) According to the measurements of inertia for the evader in [1], J rx,j ry,j rz can be considered the same J rx = Jry = Jrz = J r. Therefore, the gyroscopic moments τ i gyro

62 51 expressed in the rotor frames are: τgyro i = J r ω i α i J r ω i βi (3.14) J r ω i βi +J r ω i α i In order to express these gyroscopic moments in the body fixed frame with respect to the vehicle s cg, the above equations should be multiplied by the rotational matrices R xy (β,α) 1 and R xy (β,α) 2, as below: τ gyro = R xy (β,α) 1 τ 1 gyro +R xy (β,α) 2 τ 2 gyro (3.15) R xy (β,α) i τgyro i = cosα i J r ω i α i sinα i J r ω i βi +sinα i J r ω i α i sinβ i sinα i J r ω i α i +cosβ i J r ω i βi +sinβ i cosα i J r ω i βi sinβ i cosα i J r ω i α i cosβ i sinα i J r ω α i +sinβ i J r ω i βi cosβ i cosα i J r ω i βi +cosβ i sinα i J r ω i α i (3.16) 2. Propeller torques (τ i prop,i = 1,2): As the blades rotate, they are subject to drag forces which produce torques around the aerodynamic centre O. These moments act in opposite direction relative to ω i. Q 1 = (,, Q 1 ) T Q 2 = (,,Q 2 ) T (3.17) The positive quantities Q i can be approximated as Q i C Q ω 2 i [76]. Therefore, these torques can be written in B as:

63 52 τ prop = 2 R xy (β,α) i Q i = i=1 sin(α 1 )Q 1 +sin(α 2 )Q 2 sin(β 1 )cos(α 1 )Q 1 sin(β 2 )cos(α 2 )Q 2 (3.18) cos(β 1 )cos(α 1 )Q 1 +cos(β 2 )cos(α 2 )Q 2 3. Thrust vectoring moments (τthrust i,i = 1,2): These torques are derived based on the thrust vector T i and the translational displacement of the ducts and the vehicle s cg, represented as a vector d i. The torques are the cross-product of the thrust vector, with respect to the vehicle s cg, and the displacement vector d i which can be defined in B as d 1 = (, l o,h o ) T and d 2 = (,l o,h o ) T. = τ thrust = (R xy (β,α) 1 T 1 ) d 1 +(R xy (β,α) 2 T 2 ) d 2 cosβ 1 cosα 1 T 1 l o sinβ 1 cosα 1 T 1 h o cosβ 2 cosα 2 T 2 l o sinβ 2 cosα 2 T 2 h o sinα 1 T 1 h o sinα 2 T 2 h o sinα 1 T 1 l o +sinα 2 T 2 l o (3.19) 4. Reactionary torques (τ i react,i = 1,2): The reactionary torques are comprised of the counter torques experienced by the duct with respect to the vehicle s cg and the tilting rotations of the ducts. τ react = R xy (β,α) 1 τ 1 react +R xy (β,α) 2 τ 2 react (3.2)

64 53 τreact i = J px βi J py α i (3.21) wherej pe R 3 3 = diag(j px ),J py,j pz isthepropellergroup(includingthefanandall its associated rotating components) inertia matrix. According to [1] J px = Jpy = J p and the reactionary moments can be written as: τreact i = J p βi J p α i (3.22) τ react = cosα 1 J P β1 +cosα 2 J p β2 sinβ 1 sinα 1 J p β1 +cosβ 1 J p α 1 +sinβ 2 sinα 2 J p β2 +cosβ 2 J p α 2 cosβ 1 sinα 1 J p β1 +sinβ 1 J p α 1 cosβ 2 sinα 2 J p β2 +sinβ 2 J p α 2 (3.23) Finally, the complete expression of the torque vector, with respect to cg of the vehicle and expressed in B, is: T tot = τ gyro +τ thrust +τ prop τ react (3.24) T tot = [τ x,τ y,τ z ] T (3.25)

65 Complete Dynamic Model of The evader The evader UAV has six Degrees of Freedom (DOF) according to the reference frame B: three translation velocities v = [v 1,v 2,v 3 ] T and three rotation velocities Ω = [Ω 1,Ω 2,Ω 3 ] T. The relation existing between the velocity vectors (v,ω) and ( ζ, η) is: ζ = R t v Ω = R r η (3.26) where R t and R r are respectively the transformation velocity matrix and the rotation velocity matrix between E and B such as: R t = C φ C ψ S φ S θ C ψ C φ S ψ C φ S θ C ψ +S φ S ψ C θ S ψ S φ S θ S ψ +C φ C ψ C φ S θ S ψ S φ C ψ (3.27) S φ S φ C θ C φ C θ and R r = 1 S θ C φ C θ S φ (3.28) S φ C φ C θ where S (.) and C (.) are the respective abbreviations of sin(.) and cos(.). One can write R t = R t S(Ω) where S(Ω) denotes the skew symmetric matrix such that S(Ω)ν = Ω ν for the vector cross-product and any vector ν R 3. In other

66 55 words, for a given vector Ω, the skew-symmetric matrix S(Ω) is defined as follows: S(Ω) = Ω 3 Ω 2 Ω 3 Ω 1 Ω 2 Ω 1 (3.29) The derivation of (3.26) with respect to time gives ζ = R t v+ R t v = R t v+r t S(Ω)v = R t ( v+ω v) Ω = R r η +( Rr φ φ+ Rr θ θ) η (3.3) Using the Newton s laws in the reference frame E, about the evader UAV subjected to forces F tot and moments T tot applied to the epicenter, the dynamic equation motions are obtained as: F tot = m v+ω (mv) T tot = J Ω+Ω (JΩ) (3.31) where m is the mass and J = diag[j x,j y,j z ] is the total inertia matrix of the evader, F tot, T tot include the external forces and torques, developed in the epicenter of the vehicle according to the direction of the reference frame B, such as: F tot = F cg +F aero +F grav T tot = T +T aero (3.32) where the forces F cg,f aero,f grav and the torques T,T aero are explained in Sections and 3.3.3, and G = [,,g] T is the gravity vector (g = 9.81m.s 2 ). The equation of the dynamic of rotation of the evader, expressed in the reference

67 56 frame E, is: F cg = mrt 1 ζ +mrt 1 G+ f T = JR r η +J( Rr φ φ+ Rr θ θ) η +R r η JR r η + t (3.33) In (3.33), F aero and T aero are considered as disturbances f, t, respectively, and it is assumed that pitch angle satisfy the following inequalities: π 2 < θ(t) < π 2 (3.34) so that the inverse of matrix R r defined in (3.28) exists Approximation of Equations of Motion In this section the simplified dynamic model of the evader is presented in which the rate of change of the orientation angles ( η) and the body angular velocities (Ω) are assumed to be approximately equal. This assumption is valid if the perturbations from hover flight are small. Equation (3.35), outlines Euler s rigid body motion equations for the vehicle s principle angular acceleration roll ( φ), pitch( θ) and yaw ( ψ), considering the vehicle s principle axis of inertia (J x, J y and J z ) and the sum of torques (T tot = [τ x,τ y,τ z ] T ). J x φ+(jz J y ) ψ θ = τ x J y θ+(jx J z ) φ ψ = τ y J z ψ +(Jy J x ) θ φ = τ z (3.35) In order to derive the vehicle s equations of motion of rotational dynamics, all torque expressions in (3.16), (3.18), (3.19) and (3.23) are replaced in the right hand

68 57 side of (3.35) as follow: J x φ = ψ θ(j y J z ) cosα 1 J r ω 1 α 1 sinα 1 J r ω 1 β1 +sinα 1 J r ω 1 α 1 cosα 2 J r ω 2 α 2 sinα 2 J r ω 2 β2 +sinα 2 J r ω 2 α 2 h o (sinβ 1 cosα 1 T 1 +sinβ 2 cosα 2 T 2 )+l o (cosβ 1 cosα 1 T 1 cosβ 2 cosα 2 T 2 ) sinα 1 Q 1 +sinα 2 Q 2 +cosα 1 J p β1 +cosα 2 J p β2 J y θ = φ ψ(j z J x ) sinβ 1 sinα 1 J r ω 1 α 1 +cosβ 1 J r ω 1 β1 +sinβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) sinβ 2 sinα 2 J r ω 2 α 2 +cosβ 2 J r ω 2 β2 +sinβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) +sinβ 1 cosα 1 Q 1 sinβ 2 cosα 2 Q 2 h o (sinα 1 T 1 +sinα 2 T 2 ) +sinβ 1 sinα 1 J p β1 +cosβ 1 J p α 1 +sinβ 2 sinα 2 J p β2 +cosβ 2 J p α 2 J z ψ = θ φ(jx J y )+cosβ 1 sinα 1 J r ω 1 α 1 +sinβ 1 J r ω 1 β1 cosβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) +cosβ 2 sinα 2 J r ω 2 α 2 +sinβ 2 J r ω 2 β2 cosβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) cosβ 1 cosα 1 Q 1 +cosβ 2 cosα 2 Q 2 l o (sinα 1 T 1 sinα 2 T 2 ) cosβ 1 sinα 1 J p β1 +sinβ 1 J p α 1 cosβ 2 sinα 2 J p β2 +sinβ 2 J p α 2 (3.36) Thereby, the dynamic model of the evader including translational and rotational dynamics is presented in (3.37), where τ x, τ y, and τ z are specified in (3.36). For the sake of simplicity, this model is used for designing controllers for the evader in next chapters of this thesis. However, the procedure of designing controllers remains the same for the model presents in Section 3.4.

69 58 Table 3.2: Parameters of the vehicle. parameter value parameter value m 6.5 kg J r kg.m 2 g 9.81 m.s 1 J x.13 kg.m 2 l o.4 m J y kg.m 2 h o.8 m J z kg.m 2 C T.1 J p kg.m 2 J x φ = ψ θ(j y J z )+τ x J y θ = φ ψ(j z J x )+τ y J z ψ = θ φ(jx J y )+τ z mẍ = C φ C ψ F x +(S φ S θ C ψ C φ S ψ )F y +(C θ S θ C ψ +S φ S ψ )F z mÿ = C θ S ψ F x +(S φ S θ S ψ +C φ C ψ )F y +(C φ S θ S ψ S φ C ψ )F z m z = S φ F x +S φ C θ F y +C φ C θ F z mg +g r (z) (3.37) where F x = sinα 1 T 1 +sinα 2 T 2 F y = sinβ 1 cosα 1 T 1 sinβ 2 cosα 2 T 2 F z = cosβ 1 cosα 1 T 1 +cosβ 2 cosα 2 T 2 Table (3.2) shows the parameters of the evader that have been used in simulations of this study. These parameters are based on the measurements in [1] for the evader. 3.5 Summary A description of a lift-fan OAT mechanism is presented in this chapter which has the potential to provide VTOL capability with efficient, and highly maneuverable characteristics in tilt-rotor aircrafts. Moreover, the modeling of a small VTOL UAV with

70 59 lift-fan OAT system, which is able to perform maneuvers in hover, lateral and forward flight, is discussed. This UAV has the capability to pitched hover (something that no other aircraft can execute). It uses the inherent gyroscopic properties and driving torques of the fans for vehicle pitch control, and eliminates the need for external control elements or lift devices. Therefore, in order to control this unconventional UAV for complex missions, a complete dynamic model of it is developed in this chapter, in which aerodynamic friction effects, ground effects, and reactionary moments are included. In addition, the lateral and longitudinal tilting angles of evader vehicle are not forced to be the same in this model, and each fan rotates independently. In other words, a dynamic model of doat mechanism is developed in this thesis as opposed to soat model which was presented in previous works. There are more advantages associated with OAT systems, which have not been explored yet. Examining all these properties, by applying proper choice of nonlinear controller to verify the doat capability, based on the developed nonlinear dynamic model in this chapter, is what is intended in the following chapters.

71 Chapter 4 Feedback Linearization Control of evader It is well known that a conventional linear control (PID or PD) can stabilize a traditional VTOL aircraft(e.g., helicopter) in noncritical conditions(e.g., without external disturbances such as wind gusts) or around a specific operating point [3]. In real conditions the use of a classical linear control is limited to a small neighborhood around the operating point. However, the goal of this thesis is to design a controller for the evader UAV to perform difficult tasks and maneuvers, and follow complex trajectories in presence of external disturbances. Classical linear controllers are not applicable, and nonlinear control approaches are required. Nonlinear control approaches can be divided into two major directions. First, the ones that employ the use of Lyapunov functions and the second group dealing with Feedback Linearization (FL). Feedback linearization can be defined as methods of transforming original system models into equivalent models of a simpler form. This approach is completely different from conventional (Jacobian) linearization, due to the fact that FL is achieved by exact state transformation and feedback, rather than by linear approximations of the system dynamics. The central idea of FL is to algebraically transform nonlinear system dynamics into (fully or partly) linear ones, so that linear control techniques can be applied. This chapter is focused on FL, while Lyapunov-based techniques are investigated in the next chapters (Chapter 5 and 6) of this thesis. 6

72 61 Feedback linearization controllers can be directly applied to nonlinear dynamics without linear approximations of the given system. Although these types of controller are simple to implement, model uncertainty can cause performance degradation or instability of the closed-loop system, because it uses inverse system dynamics as part of the control input to cancel nonlinear terms. One solution to deal with the problem caused by model uncertainties is adaptive control methods. In this chapter, the review and background of FL methodologies is discussed in Section 4.1. In Section 4.2 the dynamic model presented in Chapter 3 for the UAV of interest (e.g., evader) is modified for controller design. For this, a state-space model is developed for the evader vehicle. Then, in Section 4.3 the feedback linearization controller is designed to regulate the Cartesian positions (x, y, z) and orientation angles (φ,θ,ψ) to desired values of the UAV. The design of the corresponding adaptive nonlinear control and robust adaptive control are described in Sections 4.4 and 4.6, respectively. Then, Section 4.7 presents a number of simulation results of the proposed nonlinear controllers based on FL and adaptive control methodologies. Finally, Section 4.8 discusses the results and provides a summary of this chapter. 4.1 Overview and Background of Feedback Linearization The basic idea of FL is first to transform a nonlinear system into a linear system, and then use the well-known and powerful linear design techniques to complete the control design. Such concepts are applicable to well-known important classes of nonlinear systems, namely input-state linearizable or minimum-phase systems, which typically requires full system state measurement. One of such mechanisms is the Input-Output Linearization (IOL) technique [77].

73 62 IOL is a control technique where the output y of the system is differentiated as many times as required so that the input signal u appears explicitly in the output equation. The Input-State Linearization (ISL) [77] is a special case of IOL where the output function leads to a relative degree n. Therefore, if a system is input-output linearizable with relative degree n, it must be input-state linearizable. If the relative degree r eq is less than the system order n, then there will be internal (unobservable) dynamics in the feedback linearized system. The IOL will be discussed in this thesis. Due to the under-actuated property of quad-rotors and other helicopter type UAVs, they are categorized under non-minimum phase nonlinear systems. A nonminimum phase system has (n r eq ) internal dynamics, or zero-dynamics, which play an important role in the stability of the whole dynamics of the system. These unobservable states may be stable or unstable. However, the stability or at least controllability of these unobservable states should be addressed to ensure these states do not cause problems in practice. For such systems with zero-dynamics, perfect or asymptotic convergent tracking error can not be achieved by FL, which can only partially linearize the non-minimum phase nonlinear system. Instead, only small tracking error for the desired trajectories of interest is achievable. There are some methods to cope with this restriction of FL method for under-actuated systems [78]. Some of them which have been applied to quad-rotor helicopters are as below: 1. Output-redefinition method: One of the most common approaches for controlling non-minimum phase systems is the output-redefinition method [79]. The principle of this method is to redefine the output function y 1 = h 1 (x) so that the resulting zero-dynamic is stable. For example, in the case of quad-rotors, choosing altitude, pitch, roll, and yaw angles as outputs of the system will introduce unstable zero-dynamics which will result in drifting the vehicle in the x y plane [8]. How-

74 63 ever, by selecting position (x,y,z) and yaw angle as outputs the system will not have unstable zero-dynamics. From this example, it is observed that FL is able to only track four out of six specific outputs of quad-rotors and stabilize the other two to zero. 2. Output-differentiation method: Another practical approximation is using successive differentiations of the output [81]. This method neglects the terms containing the input and keeps differentiating the selected output a number of times equal to the system order, so that there is no zero-dynamics. Because of the number of differentiations of the system dynamic in this method, it can only be applied to the system if the coefficient of u values at the intermediate steps are small. Otherwise, because of the high-order derivative terms arising from the differentiation of the dynamic equations, FL controllers are quite sensitive to external disturbance or sensor noise. However, this method provides the opportunity of arbitrarily selecting the desired four outputs of quad-rotor systems, but differentiation makes this approach noise sensitive. In this chapter, it is verified that FL is a proper choice of control for the evader UAV. This is due to the fact that the system does not have any of the above problems. In other words, the evader is a full state-linearizable vehicle and does not have zerodynamics Controllability In this section the controllability of the evader system based on the model of the vehicle in Section of Chapter 3 is investigated. The controllability of a linearized system is investigated based on the following controllability condition: A continuous time-invariant linear state-space model of the form ẋ = Ax + Bu is controllable if

75 64 and only if: [ rank B AB A 2 B... A n 1 B ] = n (4.1) where A R n n is sate matrix and B R n m is control matrix in the linear statespace model and n is the order of the system. The controllability condition for linear systems turns into an input-state linearizable condition for single-input nonlinear systems of the form ẋ = f(x)+g(x)u, with f and g being smooth vector fields. Therefore, based on a theorem 6.2 in [63], if the vector fields { g,ad f g,...,ad n 1 f g } are linearly independent, the system is input-state linearizable. In this expression ad f g represents the Lie bracket of the two vector fields f and g which is defined as: ad f g = gf fg (4.2) In the Lie bracket ad f g, ad stands for adjoint. The concepts of input-state linearization and input-output linearization can be extended for MIMO systems of the form ẋ = f(x)+g(x)u y = h(x) (4.3) where x is n 1 the state vector, u is the m 1 control input vector of components u i, y is the m 1 vector of system outputs of components y i, f and h are smooth vector fields, and G is a n m matrix whose columns are smooth vector fields g i. Inputoutput linearization of MIMO systems is obtained by differentiating the outputs y i until the inputs appear, similarly to the SISO systems. Assume that r i is the smallest

76 65 integer such that at least one of the inputs appears in y (r i) i, then y (r i) i = L r i f h i + m j=1 L gj L r i 1 f h i u j (4.4) with L gj L r i 1 f h i (x) for at least one j. If the partial relative degrees r i, i = 1,...,m are all well defined, the system (4.3) is then said to have relative degree (r 1,...,r m ) at x, and the scalar r = r r m is called the total relative degree of the system at point x [63]. Thevectorfieldsf andg i areformallydefinedinsection4.3throughoutthedesign development of FL controller for evader. The investigation of the controllability of the evader UAV verified that the system has relative degree (r 1 = r 2 = r 3 = r 4 = r 5 = r 6 = 2) and the total relative degree (r = r 1 +r 2 +r 3 +r 4 +r 5 +r 6 ) of its model is equal to the vehicle s order (n = 12). Hence, the system is input-state linearizable and both stabilization and tracking can be achieved for the evader without any worry about the stability of the internal dynamics as shown in Section Modeling for Control This section presents the state-space model of the evader UAV suitable for designing feedback linearization nonlinear control laws. The nonlinear dynamic model (3.37) developed in Chapter 3 is used to write the system in state-space form ẋ = f(x,u) with u input vector and x state vector chosen as follows: [ x = φ φ θ θ ψ ψ x ẋ y ẏ z ż ] T, (4.5)

77 66 [ u = u 1 u 2 u 3 u 4 u 5 u 6 ] T. (4.6) The expressions for u 1,..,u 6 are defined in Section of Chapter 3 as: u 1 = cosα 1 J r ω 1 α 1 sinα 1 J r ω 1 β1 +sinα 1 J r ω 1 α 1 cosα 2 J r ω 2 α 2 sinα 2 J r ω 2 β2 +sinα 2 J r ω 2 α 2 h o (sinβ 1 cosα 1 T 1 +sinβ 2 cosα 2 T 2 )+l o (cosβ 1 cosα 1 T 1 cosβ 2 cosα 2 T 2 ) sinα 1 Q 1 +sinα 2 Q 2 +cosα 1 J p β1 +cosα 2 J p β2 u 2 = sinβ 1 sinα 1 J r ω 1 α 1 +cosβ 1 J r ω 1 β1 +sinβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) sinβ 2 sinα 2 J r ω 2 α 2 +cosβ 2 J r ω 2 β2 +sinβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) +sinβ 1 cosα 1 Q 1 sinβ 2 cosα 2 Q 2 h o (sinα 1 T 1 +sinα 2 T 2 ) (4.7) +sinβ 1 sinα 1 J p β1 +cosβ 1 J p α 1 +sinβ 2 sinα 2 J p β2 +cosβ 2 J p α 2 u 3 = cosβ 1 sinα 1 J r ω 1 α 1 +sinβ 1 J r ω 1 β1 cosβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) +cosβ 2 sinα 2 J r ω 2 α 2 +sinβ 2 J r ω 2 β2 cosβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) cosβ 1 cosα 1 Q 1 +cosβ 2 cosα 2 Q 2 l o (sinα 1 T 1 sinα 2 T 2 ) cosβ 1 sinα 1 J p β1 +sinβ 1 J p α 1 cosβ 2 sinα 2 J p β2 +sinβ 2 J p α 2

78 67 u 4 = C φ C ψ F x +(S φ S θ C ψ C φ S ψ )F y +(C θ S θ C ψ +S φ S ψ )F z u 5 = C θ S ψ F x +(S φ S θ S ψ +C φ C ψ )F y +(C φ S θ S ψ S φ C ψ )F z u 6 = S φ F x +S φ C θ F y +C φ C θ F z F x = sinα 1 T 1 +sinα 2 T 2 F y = sinβ 1 cosα 1 T 1 sinβ 2 cosα 2 T 2 F z = cosβ 1 cosα 1 T 1 +cosβ 2 cosα 2 T 2 Therefore the vehicle s state-space model is given by f(x,u) = φ ψ θ( Jy Jz J x ) ktax J φ2 x Jr J x Ω θ + 1 J x u 1 θ φ Jz Jx ψ( J y ) ktay J θ2 y Jr J y Ω φ+ 1 J y u 2 ψ θ Jx Jy φ( J z ) ktaz J z ψ J z u 3 ẋ K fax m ẋ2 + 1 m u 4 ẏ K fay m ẏ2 + 1 m u 5 ż K faz m ż2 g + 1 m u 6 (4.8) 4.3 evader s Feedback Linearization Design The feedback linearization technique is based on inner and outer loops of the controller. The input-output linearization-based inner loop uses the full-state feedback to globally linearize the nonlinear dynamics of selected controlled outputs. Each of

79 68 the output channels is differentiated sufficiently many times until a control input component appears in the resulting equation. Using the Lie derivative, the inputoutput linearization technique transforms the nonlinear system into a linear and noninteracting system in the Brunovsky form [63]. The outer controller adopts a classical polynomial control law for the new input variable of the resulting linear system. This can be described for the evader vehicle by linear state-space equations via the states transformation z l = z l (x) and nonlinear state feedback u = u(x,v l ), where x = (x 1,x 2,...,x 12 ) T is the state vector of the nonlinear system, z l = (z l1,z l2,...,z l12 ) T is the state vector and v l = (v l1,v l2,v l3,v l4,v l5,v l6 ) T is the input of the linear system resulting from the transformation. The transformation of nonlinear form into a linear and controllable form is given by ż l = Az l +Bv l y = Cz l (4.9) where A R and B R 12 6 are control and input matrices and C R 6 12 is the output matrix of the linear system. The evader s model can be rewritten from (4.8) as: ẋ 1 = x 2, x 2 = (x 4 x 6 ) (Jy Jz) J x ktax J x x 2 2 Jr J x Ωx 4 + u 1 J x ẋ 3 = x 4, x 4 = (x 2 x 6 ) (Jz Jx) J y ktay J y x 2 4 Jr J y Ωx 2 + u 2 J y ẋ 5 = x 6, x 6 = (x 2 x 4 ) (Jx Jy) J z ktaz J z x u 3 J z ẋ 7 = x 8,ẋ 8 = K fax m x2 8 + u 4 m ẋ 9 = x 1,ẋ 1 = K fay m x2 1 + u 5 m ẋ 11 = x 12,x 12 = K faz m x2 12 g + u 6 m (4.1)

80 69 The system model in (4.1) is transformed into an affine nonlinear form given by (4.11). ẋ = f(x)+ 6 i=1 g iu i, y = h(x), (4.11) where u i (i = 1,...,6) are control variables, y is a 6 1 output function vector, f and h are smooth vector fields. The function vectors f(x) R 12, g i R 12 and output vector y from (4.1) are given by [ f(x) = x 2 x 4 x 6 a 1 a 2 x 2 2 a 3 Ωx 4 x 4 x 2 x 6 a 4 a 5 x 2 4 a 6 Ωx 2 x 6 x 2 x 4 a 7 a 8 x 2 6 ] T x 8 a 9 x 2 8 x 1 a 1 x 2 1 x 12 a 11 x 2 12 [ ] T g 1 (x) = 1 J x [ ] T g 2 (x) = 1 J y [ ] T g 3 (x) = 1 J z [ ] T g 4 (x) = 1 m [ ] T g 5 (x) = 1 m [ ] T g 6 (x) = 1 m [ ] T y(x) = x 1 x 3 x 5 x 7 x 9 x 11 (4.12) where a 1 = Jy Jz J x, a 2 = ktax J x, a 3 = Jr J x, a 4 = Jz Jx J y, a 5 = ktay J y, a 6 = Jr J y, a 7 = Jx Jy J z, a 8 = ktaz J z, a 9 = K fax, a m 1 = K fay and a m 11 = K faz. The controllability of the system is m

81 7 given by the controllability matrix as: [ g ad f g ], Next the relative degree of the orientation (rotation) subsystem is explored. By definition, if 1)TheLiederivativeofthefunctionL k fh(x)alongg equalszeroinaneighbourhood of x, i.e., L g L k f h(x) =,k < r eq i 1; 2) The Lie derivative of the function L k 1 f h(x) along the vector field g(x) is not equal to zero, i.e., L g L k 1 f h(x) ; then this system is said to have relative degree r. The rotation subsystem of the evader has a relative degree of six (r eq = 6), and the relative degree of the corresponding translation subsystem is six as well. Thus, the relative degree of the whole system is = 12. In this thesis, the whole system was considered as two subsystems: i) rotation and ii) translation subsystems used to simplify the design process. However, the design procedure for the whole system remains completely the same as what is described in this section. For the rotation subsystem, the output y i (i = 1,2,3) are given by y (r 1) 1 y (r 2) 2 y (r 3) 3 y (r 4) 4 y (r 5) 5 y (r 6) 6 = L r 1 f h 1(x) L r 2 f h 2(x) L r 3 f h 3(x) L r 4 f h 4(x) L r 5 f h 5(x) L r 6 f h 6(x) +E u 1 u 2 u 3 u 4 u 5 u 6, (4.13) where y (r i) i represents the r i th derivative of y i and E is a 6 6 invertible matrix, also

82 71 called the decoupling matrix, given by E(x) = 1 J x 1 J y 1 J z 1 m 1 m 1 m The linearizing control law u = (x,v l ), by which the nonlinear feedback exactly compensates the system nonlinearities, is defined as u = E 1 v l1 v l2 v l3 v l4 v l5 v l6 L 2 f h 1(x) L 2 f h 2(x) L 2 f h 3(x) L 2 f h 4(x) L 2 f h 5(x) L 2 f h 6(x). (4.14) Using the control law defined in (4.14), the following decoupled set of equations are obtained: which represents a linear system. y (r i) i = v li, i = 1,...,6. (4.15) As mentioned above, in this control approach we seek a transformation that transforms the nonlinear system into an equivalent linear system. The main concern in this type of control is coping with disturbances. Even though the control may have good

83 72 results around equilibrium points of the system such as in hover, if the system goes far from equilibrium points by a sudden unexpected disturbances such as wind gusts or adding a payload, this control would fail to offer perfect control due to the fact that FL is based on the exact cancellation of nonlinearities, which is, in practice, not often practically possible. Most likely only the approximations of the model parameters are available. The suggested approach in such situations is an adaptive nonlinear control methodology which is discussed and designed for the evader in the next section. The basic objective of adaptive control is to maintain consistent performance of a system in the presence of uncertainty or unknown variation in plant parameters. 4.4 Nonlinear Adaptive Control The controller designed in the preceding section guarantees that in the presence of uncertain bounded nonlinearities, the closed-loop state remains bounded. In this section, and in the remainder of the thesis, the uncertainties are more specific. They consist of unknown constant parameters, which appear linearly in the system equations. In the presence of such parametric uncertainties, we will be able to achieve both boundedness of the closed-loop states and convergence of the tracking error to zero. Many dynamic systems to be controlled have constant or slowly-varying uncertain parameters. An adaptive control system can thus be regarded as a control system with on-line parameter estimation. The dynamic behavior of a UAV depends on its speed, and maneuvering situation. Also if UAV s mission is to pick up some loads, it has to manipulate weights, and mass distribution is changing. In addition it is not reasonable to assume that the inertial parameters of the load and of the UAV

84 73 are well known and constant. If controllers with constant gains are used and the load parameters are not accurately known, UAV motion can be either inaccurate or unstable. An adaptive controller differs from an ordinary controller in that the controller parameters are variable, and there is a mechanism for adjusting these parameters online, based on signals in the system. Many formalisms in nonlinear control can be used to synthesize mechanism, such as Lyapunov theory, hyperstability theory, and passivity theory [82]. Lyapunov theory is used in this thesis to design an adaptation law which would guarantee that the control system remains stable and the tracking error converges to zero as the parameters are varied. The controller designed in the previous section employed static feedback, whereas the controller in this section will, in addition, employ a form of nonlinear integral feedback. The underlying idea in the design of this dynamic part of feedback is parameter estimation. The dynamic part of the controller is designed as a parameter update law with which the static part is continuously adapted to new parameter estimates, hence its name: Adaptive control law. Adaptive controllers are dynamic and therefore more complex than the static controllers designed in Section 4.3. On the other hand, as it is shown in simulations, an adaptive controller guarantees not only that the plant state x remains bounded, but also that it tends to a desired constant value (regulation) or asymptotically tracks a reference signal (tracking), This section describes an adaptive feedback linearization controller. We define a suitable feedback control and adaptation rules so that a trajectory of the system follows desired references under model parameter uncertainty.

85 Adaptive Control Design for the evader s Orientation Because of the presence of unknown parameters in the dynamics of roll angle in (4.8), an adaptive control input u 1 (t) needs to be developed to regulate φ(t) to its desired value φ d. Let us consider the regulation error for φ(t) to be defined as: e φ1 = φ φ d (4.16) A filtered regulation error system, denoted by e φ2 (t) R can thus be defined as: e φ2 = e φ1 +e φ1 (4.17) The time derivative of (4.17) gives the open-loop regulation error. J x ė φ2 = ψ θa 1 +a 2 ė φ1 +u 1 (4.18) where unknown constants a 1, a 2 R are defined as a 1 = J y J z and a 2 = J x. The control input u 1 (t) needs to be designed based on the Lyapunov approach, and the open-loop regulation error system in (4.18), and is given as below: u 1 = â 1 ψ θ â 2 e φ1 e φ2 (4.19) where â 1,â 2 R are dynamic estimates for unknown parameters a 1,a 2. These two parameters are generated according to the following update laws: â 1 = Γ a1 e φ2 ψ θ â2 = Γ a2 e φ2 ė φ1 (4.2)

86 75 where Γ a1,γ a2 R are some positive update gains. After substituting (4.19) into (4.8), the closed-loop dynamic is obtained as: J x ė φ2 = ã 1 ψ θ +ã 2 ė φ1 e φ2 (4.21) where ã 1 = a 1 â 1 and ã 2 = a 2 â 2 are parameter estimation errors. Following the same approach described above for regulating pitch and yaw angles, and considering the following four regulation errors and filtered regulation system error defined as: e θ1 = θ θ d e θ2 = ė θ1 +e θ1 (4.22) e ψ1 = ψ ψ d e ψ2 = ė ψ1 +e ψ1 (4.23) the control inputs u 2 (t) and u 3 (t) are obtained as u 2 = â 3 φ ψ â 4 ė θ1 e θ2 (4.24) u 3 = â 5 θ φ â6 ė ψ1 e ψ2 (4.25) where a 3 = J z J x, a 4 = J y, a 5 = J x J y, and a 6 = J z. The result is the update laws defined as: â 3 = Γ a3 e θ2 φ ψ â4 = Γ a4 e θ2 ė θ1 (4.26) â 5 = Γ a5 e ψ2 θ φ â6 = Γ a6 e ψ2 ė ψ1

87 Adaptive Control Design for the evader s Position The dynamic equation of the UAV for altitude is given as (Equation (3) in Section 3.4.1): m z = u 6 mg (4.27) It is assumed that m in unknown. An adaptive control input u 6 (t) can be developed in the same way as developing input signals for controlling orientation angles. After defining e z1 = z z d, e z2 = ė z1 +e z1 and getting the time derivative of the dynamic of the altitude error, the open-loop error system is given as mė z2 = a 7 g +a 7 ė z1 +u 6 (4.28) where a 7 = m is unknown. Having the error dynamic system provided in (4.28), the control input u 6 (t) is designed as below u 6 = â 7 g â 1 ė z1 e z2 (4.29) where â 7 is an estimate of unknown parameter a 7 which is generated through its dynamic update law â 7 = Γ a7 (e z2 ė z1 ge z2 ). (4.3) After substituting (4.29) into (4.27), the closed-loop dynamic is obtained as mė z2 = ã 7 g +ã 7 ė z1 e z2 (4.31)

88 77 where ã 7 is the parameter s estimation error. The stability of the closed-loop system (4.31) is proved based on the Lyapunov approach and discussed in the next section (Section 4.5). Following the same approach presented above for altitude, the UAV s longitudinal and lateral position x(t) and y(t), respectively, are regulated to their desired position x d and y d by applying u 4 and u 5. u 4 = â 8 ė x1 e x2, â 8 = Γ a8 ė x1 e x2 (4.32) u 5 = â 9 ė y1 e y2, â 9 = Γ a9 ė y1 e y2 (4.33) 4.5 Stability Analysis Theorem 1: The adaptive controller proposed in (4.19) with the updating laws in (4.2) ensures the boundedness of the closed loop system, and the UAV s roll angle φ(t) is regulated to its desired angle φ d asymptotically in the sense that lim t e φ1 (t) =. (4.34) Proof: To prove the above theorem, a Lyapunov function candidate V 1 (t) is chosen as V 1 = 1 2 e2 φ J xe 2 φ Γ 1 a 1 ã Γ 1 a 2 ã 2 2. (4.35) The time derivative of V 1 (t) along (4.17), (4.2) and (4.21) is V 1 = e 2 φ 1 e 2 φ 2 +e φ1 e φ2. (4.36)

89 78 By invoking the young s inequality, V1 (t) is restricted to the upper bound V e2 φ e2 φ 2. (4.37) Since V 1 (t) of (4.35) is a non-negative function, we can conclude that V 1 (t) L, hence e φ1 (t),e φ2 (t), ã 1 (t),ã 2 (t) L and e φ1, e φ2 L. We can show that e φ1 (t), φ(t), â 1 (t), â 2 (t) L by utilizing (4.16), (4.17) and (4.2) respectively. It is now easy to conclude that u 1 (t), â 1 (t), â 2 (t) L from (4.19) and (4.2). Then (4.18) can be used to show that ė φ2 (t) L, which implies that φ(t) L according to (4.16) and (4.17). With the above information, Lemma A.3 in [83] is invoked to conclude that lim t e φ1 (t) =. (4.38) Thus the UAV s roll angle φ is regulated to its desired value φ d with the control input of (4.19) and the error reaches zero asymptotically. 4.6 Robust Adaptive Feedback Linearization The traditional techniques for robust adaptive control include parameter projection [84], deadzone [84], and e-modification [85]. The e-modification method appears attractive because it requires neither a-priori training on the model (like projection), nor knowledge of the bounds on disturbances (like deadzone), in order to guarantee uniformly ultimately bounded (UUB) signals. To make the adaptive controller designed in Sections and robust to each bounded approximation error and wind disturbances, the e-modification term is added to parameter update laws for â 1,..., â 9. The control laws of robust adaptive feedback linearization approach

90 79 become as follows: â 1 = Γ a1 (e φ2 ψ θ ν 1 e φ1 â 1 ) â 2 = Γ a2 (e φ2 ė φ1 ν 2 e φ2 â 2 ) â 3 = Γ a3 (e θ2 φ ψ ν 3 e θ2 â 3 ) â 4 = Γ a4 (e θ2 θ 1 ν 4 e θ2 â 4 ) â 5 = Γ a5 (e ψ2 θ φ ν5 e ψ2 â 5 ) â 6 = Γ a6 (e ψ2 ψ 1 ν 6 e ψ2 â 6 ) (4.39) â 7 = Γ a7 (e z2 ė z1 ge z2 ν 7 e z2 â 7 ) â 8 = Γ a8 (ė x1 e x2 ν 8 e x2 â 8 ) â 9 = Γ a9 (ė y1 e y2 ν 9 e y2 â 9 ) In this e-modification term ν 1,...,ν 9 are robust modification gains which are positive constants. The adaptive control laws u 1,...,u 6 are the same as in Sections and In Chapter 8, the wind disturbance of the form d w (t) = Disturbance Magnitude(1 + 5sin(2πt)) is added to all states (roll, pitch, yaw, x,y,z) to model blowing wind in all directions corresponding to vehicle s frame of reference. The result of this simulation is presented and discussed in Chapter 8, where comprehensive simulation scenarios are applied to the evader UAV. 4.7 Simulation Results The simulation Scenario #1 is defined as follows: the initial condition is x = [] 12 12, and the desired value is set arbitrarily to x d = [22.5,,15,,18,,3,,4,,2,] T. Table 4.1 shows the results of the FL controller applied on the evader, designed in Section 4.3. The parameters of the vehicle that have been used in simulations of

91 8 this study are shown in Table (3.2) of Chapter 3. The FL controller gains for this simulation are set to k 1 = k 3 = k 5 = 25,k 2 = k 4 = k 6 = k 8 = k 1 = k 12 = 1,k 7 = k 9 = 2, and k 11 = 15. Figure 4.1 shows the six control signals, and Figs. 4.2 and 4.3 show the regulation of position and orientation in the presence of gaussian noise of mean= and variance=.1. According to the analysis of the errors in [2] which is done with the tools that are used for the IMU static data analysis (Allan Variance, PSD, etc. ), it is assumed that the IMU error follows the usual error models associated to inertial sensors (white noise, random walk and Gauss Markov). The short-term error component of an IMU, which is added to this simulation, is made up of white Gaussian noise [86] and the long-term error component is created with a 1st order Gauss-Markov process. Table 4.2 shows the results of applying adaptive feedback linearization (AFL) controller on evader for Scenario 1. The controller gains for parameter estimation updating laws (4.2), (4.26), (4.3), (4.32), and (4.33) are selected as: Γ a1 = 1, Γ a2 = 1, Γ a3 = 1, Γ a4 = 1, Γ a5 = 1, Γa 6 = 1, Γ a7 = 1, Γ a8 = 1, and Γ a9 = 1. Figures 4.5 and 4.6 show the φ(t), θ(t), ψ(t), x(t), y(t), and z(t) output signals which have been regulated to their desired values. It can be seen that in the 5 seconds window of the simulation, all the outputs reache their desired values. Figure 4.4showsthecontrolinputstoforcetheeVadertogofromzeroinitialconditiontothe reference point. The unknown parameters a 1,...,a 9 are estimated with the updating laws, as it can be seen in Fig In real applications the parameters are changing and we may not know the exact value of them. Thus, assuming to know the exact model, like in FL method, in not a valid assumption in real world. In other words, FL does not guarantee robustness in the face of parameter uncertainty or disturbances. To show the lack of robustness of

92 81 Table 4.1: The result of FL controller for position and orientation regulation simulation Without Noise With Noise time(sec) error time(sec) error Roll φ 2.66 (deg) (deg) Pitch θ 2.44 (deg) (deg) Yaw ψ 2.53 (deg) (deg) Position x (m) (m) Position y 2.27 (m) (m) Altitude z (m) 2.62 (m) Table 4.2: The result of AFL controller for position and orientation regulation simulation Without Noise With Noise Time(sec) error Time(sec) error Roll φ (deg) (deg) Pitch θ (deg) (deg) Yaw ψ (deg) (deg) Position x (m) (m) Position y (m) (m) Altitude z (m) (m) the FL controller in presence of parameter uncertainty and the effect of variation in the parameters of the dynamic model, in Scenario# 2, all variables initial conditions, desired values and controller gains are fixed as in Scenario # 1 except the mass of the evader. Figure 4.8 shows that the AFL controller adapts itself and reaches a desired value of altitude, but the FL controller can not make the altitude error zero. 4.8 Results and discussion Feedback linearization is based on the idea of transforming nonlinear dynamics into a linear form by using state feedback, with input-state linearization corresponding to complete linearization and input-output linearization to partial linearization. The method can be used for both stabilization and tracking control problems.

93 82 2 Control Signal u1 4 Control Signal u4 1 2 u1 u4 u Controller u Controller u3 2 u Controller u Controller u6 3 u3 1 u Figure 4.1: Control signals of FL control method in presence of white gaussian noise with mean = and variance =.1 [2]. Roll[degree] 4 2 The roll angle The pitch angle 2 Pitch[degree] The yaw angle 2 Yaw[degree] Figure 4.2: Regulation of orientation angles of the evader by FL controller with additive white noise (φ = 22.5,θ = 15,ψ = 18). z[m] x[m] The Cartesian position x system output desired x(t) y[m] The Cartesian position y The Cartesian position z Figure 4.3: Regulation of position of the evader by FL controller with additive white noise (x d = 3,y d = 4,z d = 2).

94 83 6 Control signal u1 3 Control signal u4 u u Control signal u Control signal u5 u2 2 u Control signal u Control signal u u3 1.5 u Figure 4.4: Control signals of adaptive FL control method in presence of aerodynamic coefficient uncertainties and unknown mass. Roll[degree] The roll angle x [m] The Cartesian position x system output desired x(t) Pitch[degree] Yaw[degree] The pitch angle The yaw angle y [m] The Cartesian position y z [m] The altitude z Figure 4.5: Regulation of orientation anglesoftheevaderbyaflcontroller(φ = 22.5,θ = 15,ψ = 18). Figure 4.6: Regulation of position of the evader by AFL controller (x d = 3,y d = 4,z d = 2).

95 a a1 a2 2 1 a2 a a time time a4.2 a5 1.5 time a6 a4 5 a5.4 a time time time a7.4.2 a7 a8.4.2 a8 a9.4.2 a time time time Figure 4.7: Parameter estimation of AFL control method (a 1 = J y J z, a 2 = J x, a 3 = J z J x, a 4 = J y, a 5 = J x J y, a 6 = J z, a 7 = a 8 = a 9 = m). 2 FL controller 2 AFL controller Altitude z [m] Altitude z [m] Figure 4.8: The altitude output of FL and AFL controllers when the mass of the systems is changed. The FL controller failed to reach the desired altitude z d = 2 with almost.4 m steady state error.

96 85 The simulation results obtained show that the proposed controller is able to stabilize the evader even for relatively critical initial conditions. However, the feedback linearization method has some important limitations. In feedback linearization an important assumption is that the model dynamics are perfectly known and can be canceled entirely. Full state measurement is necessary in implementing the control law. Many efforts are being made to construct observers for nonlinear systems and to extend the separation principle to nonlinear systems. Finding convergent observers for nonlinear systems is difficult. Besides, the lack of a general separation principle, which would guarantee that the straightforward combination of a stable state feedback controller and a stable observer will guarantee the stability of the closed-loop system, is another difficulty associated to this problem. Moreover, no robustness is guaranteed in the presence of parameter uncertainty or unmodeled dynamics. This problemisduetothefactthattheexactmodelofthenonlinearsystemisnotavailable in performing feedback linearzation. Adaptive control based on feedback linearization method has been successfully developed for evader. However, adaptive control technique is only applicable for nonlinear control problems that satisfy the following conditions: The nonlinear plant dynamics can be linearly parameterized. The full state is measurable. Nonlinearities can be canceled stably (i.e., without unstable hidden modes or dynamics) by the control input if the parameters are known. However, in next chapters it will be revealed that there are other types of uncertainties in the evader dynamic model in addition to the linearly parameterizable nonlineari-

97 86 ties. For example, there are additive disturbances in control input signal, which make the adaptive control approach unable to provide asymptotic stability.

98 Chapter 5 Integral Backstepping Control of evader Backstepping is a technique providing a recursive method of designing stabilizing controls for a class of nonlinear systems that are transformable to a strict feedback system. Backstepping can force a nonlinear system to behave like a linear system in a new set of coordinates in the absence of uncertainties. However, backstepping and other forms of feedback linearization such as IOL and ISL, addressed in the previous chapter, require cancellations of nonlinearities, even those which are helpful for stabilization and tracking. A major advantage of backstepping over feedback linearization is that it has the flexibility to avoid cancellations of useful nonlinearities and pursue the objectives of stabilization and tracking, rather than that of linearization. In this chapter, an integral backstepping (IB) control technique is proposed to improve the pitch, yaw, and roll stability of the evader UAV. The controller is able to simultaneously stabilize the 6 outputs of the system (i.e., orientation angles and x, y, z). Position and orientation outputs are regulated to their desired values and all six of them (φ,θ,ψ,x,y,z) track the desired reference trajectories at the same time. Thus, unlike backstepping control of VTOL (vertical take off and landing) UAVs such as quad-rotors, there is no need for inner-loop and outer-loop control. Simulation results using backstepping verified the potential of the evader as a small UAV used in different scenarios such as autonomous take off and landing and tracking time-varying trajectories in order to maneuver inside obstructed environments. Simulation scenar- 87

99 88 ios presented in this chapter include attitude and position control, and stabilization and autonomous take off and landing, which show promising results. The remainder of this chapter is organized as follows: Section 5.1 discusses the background of backstepping control technique. The state-space model of the vehicle for backsteeping design is suggested in Section 5.2, followed by an explanation of the controller design procedure, first for the vehicle s attitude and then for its altitude and position, in Section 5.3. The designed backstepping controller s gains are tunned based on gradient descent optimization method in Section 5.4. Simulation results are also shown. Adaptive backstepping controller design for the evader dynamic model with parametric uncertainty is presented in Section 5.5. The conclusion and discussion are summarized in Section Overview and Background of Backstepping Control Technique Backstepping techniques provide an easy method to obtain a control algorithm for nonlinear systems. Several controllers based on backstepping technique have been developed for controlling rotary aerial vehicles such as quad-rotors and helicopters [87], [88]. Madani et. al. designed a full-state backstepping technique based on the Lyapunov stability theory and backstepping sliding mode control to perform hover and tracking of desired trajectories [3], [89]. Another controller proposed by Castillo et. al. used the backstepping technique and saturation functions. Using saturation function in the control law guaranteed attitude and control inputs boundedness in the presence of perturbations in the angular displacement [34]. However, this controller was designed for a linear system and can not work properly out of the hover operation point. Metni et. al. used backstepping techniques to derive an adaptive

100 89 nonlinear tracking control law for a quad-rotor system, using visual information [9]. This control law uses visual information and defines a desired trajectory by a series of prerecorded images. The above-mentioned papers, all studied quad-rotor helicopter control and showed that backstepping control method can effectively deal with the under-actuated property of quad-rotors. However, due to the under-actuated property of quad-rotor UAVs, only four outputs out of six outputs of the system are controllable independently. For example, the controller can set the quad-rotor tracks three Cartesian positions (x,y,z) and the yaw angle (ψ) to their desired values and stabilize the roll (φ) and pitch (θ) angles to zero. Improvements have been introduced by combining integral action within the control law, which consequently results in guaranteed asymptotic stability, as well as steady state errors cancellation due to integral action [88]. The idea of adding integral action in the backstepping design was first introduced by Kanellakopoulos in [91] to increase the robustness against external disturbances and model uncertainties. In this chapter of this thesis, the application of integral backstepping controller proposed in [91] is extended to the evader UAV for stabilization at hover and trajectory tracking maneuvers. The goal of using this control method on the evader is to allow this vehicle to use the full potential of its flying characteristics, enabled by the OAT mechanism, in order to maneuver in confined spaces. The backstepping controller presented in this thesis is a Multi-Input Multi-Output (MIMO) IB controller, which enables the evader to track all six outputs of the system including the three Cartesian positions (x, y, z) and the three orientation angles (φ, θ, ψ). The design methodology is based on the Lyapunov stability theory. Various simulations on the evader s dynamic model show that the control law stabilizes the whole system with zero steady state tracking error.

101 9 After designing the IB control, the controller gains are tunned by employing a gradient descent technique to avoid trial and error procedure. Optimization-based methods help to systematically accelerate the multiple-parameter tuning process. With optimization-based techniques, controller gains are tuned based on optimization of defined performance indices to improve transient stability, which would enhance the UAV s maneuvering performance. That is, the problem of setting nonlinear control gains is formulated as an optimization problem based on gradient descent optimization method including system and control constraints. 5.2 State-Space Model for Control The state-space form of the model of the targeted OAT mechanism can be written as in Chapter 3, with the following u input and x state vectors: [ x = φ φ θ θ ψ ψ x ẋ y ẏ z ż ] T (5.1) where: x 1 = φ x 7 = x x 2 = x 1 = φ x 8 = x 7 = ẋ x 3 = θ x 9 = y x 4 = x 3 = θ x 1 = x 9 = ẏ (5.2) x 5 = ψ x 11 = z x 6 = x 5 = ψ x 12 = x 11 = ż [ ] T u = u 1 u 2 u 3 u 4 u 5 u 6 (5.3)

102 91 where u 1 = τ x = cosα 1 J r ω 1 α 1 sinα 1 J r ω 1 β1 +sinα 1 J r ω 1 α 1 cosα 2 J r ω 2 α 2 sinα 2 J r ω 2 β2 +sinα 2 J r ω 2 α 2 h o (sinβ 1 cosα 1 T 1 +sinβ 2 cosα 2 T 2 )+l o (cosβ 1 cosα 1 T 1 cosβ 2 cosα 2 T 2 ) sinα 1 Q 1 +sinα 2 Q 2 +cosα 1 J p β1 +cosα 2 J p β2 u 2 = τ y = sinβ 1 sinα 1 J r ω 1 α 1 +cosβ 1 J r ω 1 β1 +sinβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) sinβ 2 sinα 2 J r ω 2 α 2 +cosβ 2 J r ω 2 β2 +sinβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) (5.4) +sinβ 1 cosα 1 Q 1 sinβ 2 cosα 2 Q 2 h o (sinα 1 T 1 +sinα 2 T 2 ) +sinβ 1 sinα 1 J p β1 +cosβ 1 J p α 1 +sinβ 2 sinα 2 J p β2 +cosβ 2 J p α 2 u 3 = τ z = cosβ 1 sinα 1 J r ω 1 α 1 +sinβ 1 J r ω 1 β1 cosβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) +cosβ 2 sinα 2 J r ω 2 α 2 +sinβ 2 J r ω 2 β2 cosβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) cosβ 1 cosα 1 Q 1 +cosβ 2 cosα 2 Q 2 l o (sinα 1 T 1 sinα 2 T 2 ) cosβ 1 sinα 1 J p β1 +sinβ 1 J p α 1 cosβ 2 sinα 2 J p β2 +sinβ 2 J p α 2 and u 4 = C φ C ψ F x +(S φ S θ C ψ C φ S ψ )F y +(C θ S θ C ψ +S φ S ψ )F z u 5 = C θ S ψ F x +(S φ S θ S ψ +C φ C ψ )F y +(C φ S θ S ψ S φ C ψ )F z u 6 = S φ F x +S φ C θ F y +C φ C θ F z F x = sinα 1 T 1 +sinα 2 T 2 F y = sinβ 1 cosα 1 T 1 sinβ 2 cosα 2 T 2 F z = cosβ 1 cosα 1 T 1 +cosβ 2 cosα 2 T 2 (5.5) The definition of all parameters and variables used here are the same as the ones defined in Section 3.3 of Chapter 3. The following state-space expression is obtained

103 92 for the dynamics of the evader: f(x,u) = φ ψ θa 1 b 1 φ2 b 2 Ω θ +c 1 u 1 θ φ ψa 2 b 3 θ2 b 4 Ω φ+c 2 u 2 ψ θ φa 3 b 5 ψ 2 +c 3 u 3 ẋ 1 m u 4 b 6 ẋ 2 ẏ 1 m u 5 b 7 ẏ 2 ż 1 m u 6 g b 8 ż 2 (5.6) with: a 1 = Jy Jz J x c 1 = 1 J x b 1 = ktax J x b 2 = Jr J x b 7 = K fay m a 2 = Jz Jx J y c 2 = 1 J y b 3 = ktay J y b 4 = Jr J y b 8 = K faz m (5.7) a 3 = Jx Jy J z c 3 = 1 J z b 5 = ktaz J z b 6 = K fax m It is advantageous to note in the latter system that the orientation angles and their time derivatives do not depend on the translation components. On the other hand, translations depend on the UAV s orientation angles. As a result, one can ideally imagine the overall system described by (5.6) as integration of two subsystems: i) the angular rotations, and ii) the linear translations. Thus, to avoid the difficulties of directly designing a MIMO integral backstepping controller for the entire dynamical system with six degrees of freedom, the dynamical model of the evader is divided

104 93 into two subsystems. Subsequently, the IB method is applied to design the controller for each of the two subsystems. 5.3 Control System Objective The main objective of this section is to describe a backstepping controller for the evader ensuring that its position x(t), y(t), z(t), and its orientation φ(t), θ(t), ψ(t) track the desired trajectory φ d (t), θ d (t), ψ d (t), x d (t), y d (t), z d (t) asymptotically. This is achieved as described in Sections and Attitude Control Design This section presents the roll control derivation of the UAV, based on IB. The same approach is applied for pitch and yaw control as well. Let us consider the first two-state subsystem in (5.6) for the roll control as below: x 1 = x 2 = φ x 2 = ψ θa (5.8) 1 b φ2 1 b 2 Ω θ +c 1 u 1 The first step in IB control design is to consider the tracking-error. The roll tracking error e 1 and its derivative with respect to time are considered first. e 1 = φ d φ de 1 dt = φ d Ω x (5.9) This definition specifies the control objective, where the recursive methodology will systematically drive the tracking error to zero. Herein, Lyapunov function (5.1),

105 94 which is positive definite around the desired position, is used for stabilizing the tracking error e 1 : V(e 1 ) = 1 2 e2 1, (5.1) V(e 1 ) = e 1 ( φ d Ω x ) (5.11) If the angular velocity Ω x of the vehicle is considered to be the control input in (5.3), it would be straightforward to choose Ω x so that exponential convergence for the system is guaranteed. One example of such selection is : Ω x = φ d +c 1 e 1 where c 1 is a positive number that determines the error convergence speed. Thus the derivative of the Lyapunov function is negative definite and consequently the error converges exponentially to zero ( V = c 1 e 2 1 ). Hence, e 1 = is ultimately asymptotically stable. However, the vehicle s roll angular speed Ω x is not the control input and it is only a system variable and has its own dynamics. So, a desired behavior is set for it and it is considered as a virtual control: Ω xd = c 1 e 1 + φ d +λ 1 χ 1 (5.12) The integral action in the backstepping design is to ensure the convergence of the tracking error to zero at the steady state, despite the presence of disturbances and model uncertainties, with c 1 and λ 1 being positive constants, and χ 1 = t e 1(τ)dτ being the integral of the roll tracking error. So, the integral term is now introduced in (5.12). Knowing that Ω x has its own error e 2, its dynamics are computed by using

106 95 (5.12) as follows: de 2 dt = c 1( φ d Ω x )+ φ d +λ 1 e 1 φ (5.13) where e 2, the vehicle s roll angular velocity tracking error, is defined by: e 2 = Ω xd Ω x (5.14) Therefore, this dynamic error can be compensated by defining the velocity tracking error and its derivative. Using (5.12) and (5.14) the roll tracking error dynamic is written as: de 1 dt = c 1e 1 λ 1 χ 1 +e 2 (5.15) By replacing φ in (5.13) by its corresponding expression from model (5.6),the control input u 1 appears in (5.16) de 2 dt = c 1( φ d ω x )+ φ d +λ 1 e 1 θ ψa 1 +a 2 φ2 +a 3 Ω θ b 1 u 1 (5.16) Now the augmented Lyapunov function is: V(e 2 ) = λ 1 2 χ e e2 2 The real control input now appeares in (5.16). So, using (5.9), (5.15) and (5.16) the tracking errors of the position e 1, of the angular speed e 2, and of the integral position tracking error χ 1 are combined to obtain: de 2 dt = c 1( c 1 e 1 λ 1 χ 1 +e 2 )+ φ d +λ 1 e 1 θ ψa 1 +a 2 φ2 +a 3 Ω θ τ x /J x (5.17)

107 96 where τ x is the overall rolling torque. The desirable dynamic for the angular speed tracking error is: de 2 dt = c 2e 2 e 1 (5.18) where c 2 is a positive constant which determines the convergence speed of the angular speed loop. This is achieved if one chooses the control input u 1 as: u 1 = + 1 b 1 [(1 c 2 1 +λ 1 )e 1 +(c 1 +c 2 )e 2 c 1 λ 1 χ 1 + φ d θ ψa ] (5.19) 1 +a 3 Ω θ Considering the above control signal, the derivative of the Lyapunov function is semi-negative definite and satisfies V = c 1 e 2 1 c 2 e 2 2. By the definition of Lyapunov function and its non-positive derivative, the position tracking error e 1, the velocity tracking error e 2, and the integral action χ 1, are bounded signals. Thus, the conclusion is the boundedness of all the internal signals in the closed-loop control system. Thus the derivatives of the error signals, ė 1 and ė 2, are bounded as well. The closed loop system consisting of the vehicle rolling model (5.8), the controller (5.19), and the integral action χ 1 has a global uniformly stable equilibrium at e = [e 1,e 2 ] T =. This guarantees the global boundedness of the states x 1,x 2, the integral action χ 1, and the control action u 1, and lim t e(t) =, i.e. subsequently, lim t [φ(t) φ d (t)] =. To find out the proof and read more on Lyapunov global stability theorem see [63]. Similarly, the corresponding pitch and yaw controls for the evader UAV are: u 2 = + 1 b 2 [(1 c 2 3 +λ 2 )e 3 +(c 3 +c 4 )e 4 c 3 λ 2 χ 2 + θ d φ ψa 2 ] (5.2)

108 97 u 3 = + 1 b 3 [(1 c 2 5 +λ 3 )e 5 +(c 5 +c 6 )e 6 c 5 λ 3 χ 3 + ψ d θ φa 3 ] (5.21) To identify and tune the values of the control law coefficients (c 1,c 2,c 3,c 4,c 5 and λ 1,λ 2,λ 3 ), the gradient descent optimization technique is used Altitude and Position Controls Design Within this thesis, and to find the control law to keep the distance of the UAV from the ground at a desired value, the altitude tracking error is defined as (ground and wall fluid flow effects are neglected): e 7 = z d z (5.22) The speed tracking error is: e 8 = c 7 e 7 + z d +λ 4 χ 4 ż (5.23) The control law is then: u 6 = m [ g + ( 1 c 2 7 +λ 4 ) e7 +(c 7 +c 8 )e 8 c 7 λ 4 χ 4 ] (5.24) where (c 7,c 8,λ 4 ) are positive constants. Position control keeps the vehicle over the desired point. In what follows, the vehicle s position is referred to as the (x,y) horizontal position with regard to a starting point. Horizontal motion is achieved by orienting the thrust vector towards the desired direction of motion. This is typically done by rotating the vehicle itself in

109 98 the case of a quad-rotor. For the evader, this is achieved by rotating the vehicle as well as the ducted fans using their lateral and longitudinal rotation characteristics. According to (5.6), the same IB approach is applicable for controlling (x, y). Position tracking errors for x and y are defined as: e 9 = x d x e 11 = y d y (5.25) Accordingly, the speed tracking errors are: e 1 = c 9 e 9 + x d +λ 5 χ 5 ẋ e 12 = c 11 e 11 + y d +λ 6 χ 6 ẏ (5.26) The control laws are then: u 4 = m[(1 c 2 9 +λ 5 )e 9 +(c 9 +c 1 )e 1 c 9 λ 5 χ 5 ] u 5 = m[(1 c λ 6 )e 11 +(c 11 +c 12 )e 12 c 11 λ 6 χ 6 ] (5.27) where (c 9,c 1,c 11,c 12,λ 5,λ 6 ) are positive constants. 5.4 Gradient Descent Optimization for Coefficient Tuning Gradient descent is a function optimization method which uses the derivative of a function and the idea of steepest descent. This technique works iteratively to find feasible results according to the constraints of the problem. To avoid a trial and error procedure for obtaining controller gains, the controller coefficients (c 1,...,c 12,λ 1,...,λ 6 ) are tuned by applying a gradient descent method, which enhances the UAV s maneuvering performance. With this technique, gains are

110 99 Figure 5.1: Attitude control of evader s orientation and the corresponding control input signals of IB control method (φ = 1, θ = 35, ψ = 5 ). tuned based on optimization of defined performance indices for improving transient stability. The problem of setting nonlinear control gains is formulated as an optimization problem including system and control constraints. Optimization-based methods help to systematically accelerate the multiple-parameter tuning process. 5.5 Adaptive Integral Backstepping Control The task of nonlinear design is much more challenging in the presence of uncertainty. Backstepping with a general form of bounded uncertainties with unknown bounds is a key tool used to achieve boundedness without adaptation. When the uncertainty is in the form of constant but unknown parameters, then a more suitable form of the backstepping is adaptive backstepping, developed in this section. For example, for evader UAV, the value of inertia matrix and the mass of the vehicle are uncertain

111 1 parameters. As the vehicle is capable of picking up different loads during a given mission, the total mass of the vehicle may change which is a priori unknown. Adaptive backstepping approach can help in such situations and thus enable UAVs in general, and the evader in particular, to perform more complex missions. In order to design an adaptive controller for the evader, the same assumptions presented in Chapter 4 are considered. That is, the parametric value of inertial matrix diag(j x,j y,j z ), and the mass of the vehicle, m, are unknown. For the sake of simplifying the controller design, the aerodynamic friction is not considered in the model for adaptive integral backstopping control design. The following control law is chosen for adaptive controller: u 1 = 1ˆb1 [ (1 c 2 1 +λ 1 ) e1 +(c 1 +c 2 )e 2 c 1 λ 1 χ 1 + φ d θ ψâ 1 ] (5.28) where â 1 is an estimate of a 1 = J y J z and ˆb 1 is an estimate of b 1 = 1 J x. Now, the update laws for parameter estimates should be derived to complete the adaptive design. For this purpose, the parameter estimation error signals are defined as: b1 = b 1 ˆb 1 ã 1 = a 1 â 1 (5.29) In order to obtain the update laws for parameter estimates, the Lyapunov design approach is utilized. The Lyapunov energy function for the closed-loop system is chosen as: V = λ 1 2 χ e e γ 1 ã γ 2 b2 2 (5.3) In the above Lyapunov function, γ 1, and γ 2 are adaptive gains which are positive constants, and they determine the convergence speed of the estimates.

112 11 The rest of the design approach is the same as in Section in Chapter 4, which will give the adaptation laws for parameter estimates, â 1,ˆb 1 as below: â 1 = γ 1 e 2 θ ψ ˆb 1 = γ 2 e 2 { (1 c 2 1 +λ 1 )e 1 +(c 1 +c 2 )e 2 c 1 λ 1 χ 1 + φ d θ ψâ 1 } (5.31) In (5.31), all parameters, c 1,c 2,λ 1,χ 1,e 1,e 2,θ,ψ, and φ d are defined to be the same as in Section Simulation Results In this section, simulation results for several scenarios are presented in order to observe the effectiveness of the derived model and the performances of the proposed control law. We considered the case of stabilization and a trajectory tracking problem. The following simulations are based on the evader s dynamic model presented in Equations (5.6) and (5.7). In what follows the following UAV parameters from Table 3.2 in Chapter4areused: m = 6.5kg, g = 9.8m/s 2, J r = kg.m 2, J x =.13kg.m 2, J y = kg.m 2, J z = kg.m 2, l =.4m, and h =.8m. In the first simulation, the attitude control is considered. The initial condition of the orientation angles and their derivatives are at zero. The desired values of the vehicle orientation were placed at (φ d,θ d,ψ d ) = (1,35,5 ). The attitude control results of the Backstepping approach and the obtained control signals can be seen in Fig The results of stabilization of x and y position and the related control signals are shown in Fig For the purpose of autonomous take off and landing, the vehicle is forced to follow the squared signal of altitude. The results are depicted in Fig.5.3 for different coeffi-

113 12 Figure 5.2: Position (x, y) stabilization of the evader and the corresponding control input signals of IB control method. Table 5.1: The IB controller gains. c 1 =.3 c 4 = 2 c 7 = 1 c 1 =.5 λ 1 =.5 λ 4 =.7 c 2 = 1.55 c 5 =.1 c 8 =.5 c 11 = 1.2 λ 2 = 1 λ 5 =.1 c 3 =.1 c 6 = 1.5 c 9 = 1.4 c 12 =.5 λ 3 = 1 λ 6 =.1 Table 5.2: The IB controller gains for autonomous take off and landing scenario. In this table k 1 = 1 c λ 6, k 2 = c 11 +c 12, k 3 = c 11 λ 6. The blue curve The red curve The green curve k 1 = k 1 = k 1 = 9322 k 2 = k 2 = k 2 = k 3 = k 3 = k 3 =

114 Figure 5.3: Autonomous take-off, altitude control in hover and landing of the evader and the effect of tuning the IB controller gains by Gradient Descent algorithm. 13

115 14 Figure 5.4: Stabilization of roll, pitch and yaw angles by IB control method (left figure) and pitched stability of the evader at 25 in hover (right figure). cients in control law in (5.24). The gradient descent optimization method was used to apply the desired constraints to the problem and force the output of the IB controller to track desired trajectories with the error restrictions within an acceptable range [92]. Therefore, the coefficients of the controller {c 1,c 2,...,c 12,λ 1,...λ 6,} have been adjusted by the gradient descent optimization algorithm. Applying the optimization algorithm improves the controller design by estimating and tuning its parameters. Different objectives of the optimization technique help improve system performance and reduce control efforts. For the purpose of automatically tuning and optimizing controller gains of this thesis work, the rise-time and overshoot constraints were chosen as objectives. Whenever the optimization solver finds a solution that meets the design requirements within the parameter bounds, (e.g controller gains must be positive), the local minimum is found and the optimization process terminates. The effect of this optimization can be seen in Fig The position initial value

116 15 is (x(),y(),z()) = (1,1,) m and the desired value of altitude is fixed at 2 m. The effect of controller coefficient optimization is illustrated in this figure as well. The first and second curves in Fig. 5.3 on top are infeasible answers because they cause the vehicle crash on the ground. The third curve on top in this figure is a feasible answer obtained from optimization process. The coefficients of control law (5.24) for this answer are obtained as follows: c 7 = , c 8 = , and λ 4 = 1. On bottom of Fig. 5.3 all these three curves are shown in the same coordinate to provide a better sense of comparison. Figure 5.4 shows the stabilization of roll, pitch and yaw angles. In this simulation the initial values of Euler angles is considered at (φ = 1,θ = 35,ψ = 5 ) and the final values are (φ =,θ = 25,ψ = ). The vehicle becomes stable at 25 pitch angle while is at hover. 5.7 Summary In this Chapter, an Integral Backstepping control approach was proposed to control both position and orientation of an evader. The IB controller is presented based on the derived dynamic model of this special UAV, which enables us to design 6 control laws. The controller design is divided to six subsystems, each of which is designed through similar procedure. The main advantage of the above design method is that difficulties of designing a controller for the entire system are avoided. The recursive Lyapunov methodology in the backstepping technique ensures the system stability, and the integral action increases the system robustness against disturbances and model uncertainties. Furthermore, gradient descent optimization algorithm was applied on controller to find and adjust the coefficients, which makes the results to be even more promising. As a design tool, backstepping is less restric-

117 16 tive than feedback linearization of the pervious Chapter. In some situations, it can overcome singularities such as lack of controllability [82]. Simulation results show that the proposed algorithm is capable of controlling the nonlinear model of the dual-fan VTOL air vehicle with lift-fan oblique active tilting mechanism. It also fulfills the position and orientation stabilization task as well as the ability of pitched hover. It provides the required stabilization to perform aggressive maneuvering and reliable navigation. More importantly, the model presented in this thesis, along with the proposed controller, show the ability of performing 6DOF control without the need to cope with the under-actuated property of helicopter type aerial vehicles. This chapter validated the usefulness of this method and verified that the evader has a lot more to offer in autonomous flight control of UAVs. The adaptive IB controller provides asymptotic stability in presence of constant parametric uncertainties. The parametric uncertainties has to appear linearly in the dynamic model of the system. Now the question is what would happen to UAV if it flies outdoor and a strong wind blows, or if, in an indoor flight, the evader gets too close to a wall or ground. In these situations the system is under external disturbances and adaptive control methods can not guarantee the system stability anymore. This issue is discussed in the next chapter.

118 Chapter 6 Sliding Mode Control for the evader The controllers designed in the preceding chapter guarantee that in the presence of uncertain bounded nonlinearities the closed-loop state remains bounded. Adaptive control can deal with uncertainties by tuning of the parameters online, but generally is able to achieve only asymptotical convergence of the tracking error to zero. As it is mentioned in previous chapters, adaptive control method is based on the assumption that the structure of the system model is known with unknown slow-varying system parameters, where the parameters appear linear. But several issues, such as transient performance, unmodeled dynamics, disturbances such as wind and ground effects, and not linear parameterizable uncertainties often complicate the adaptive approach [93], [94]. In this chapter, the focus is on a useful and powerful robust control scheme to deal with the uncertainties, nonlinearities, and bounded external disturbances using the sliding mode control (SMC) scheme. These and other uncertainties may come from unmodeled dynamics, variations in system parameters, or approximations of complex plant behaviours. In robust control designs, a fixed control law based on a priori information of the uncertainties is typically designed to compensate for their effects, 17

119 18 and exponential convergence of the tracking error to a (small) ball centred at the origin is obtained (see the definition of exponential stability in Chapter 2). Robust control has some advantages over the adaptive control, such as its ability to deal with disturbances, quickly-varying parameters, and unmodeled dynamics [63]. Sliding controller design provides a systematic approach to the problem of maintaining stability and consistent performance despite modelling imprecisions. An overview of the sliding-mode nonlinear controller and main concepts of SMC design, such as sliding surface, are introduced in Section 6.1. Section 6.2 represents the state-space form of the evader model to facilitate the SMC design. The following section describes how to design a SMC for the evader UAV. Section 6.4 then presents studies of some simulations to investigate the robustness of SMC methodology for the evader maneuvers, such as pitched hover maneuvers, while a sudden wind may be affecting the vehicle s behavior. The final section of this chapter summarizes the main concepts of this chapter. 6.1 Overview of Sliding Mode Control From the control point of view, modelling inaccuracies can be classified into two major kinds: i) structured or parametric uncertainties, which correspond to inaccuracies in the terms included in the model and ii) unstructured uncertainties or unmodeled dynamics, which correspond to inaccuracies in the system order. Modelling inaccuracies has strong effects on nonlinear control systems. Therefore, practical designs must

120 19 address them explicitly. Two major approaches to dealing with model uncertainty are robust control and adaptive control. In Chapter 4 of this thesis, adaptive control for the evader was employed. In this chapter the focus is on robust control approaches. The typical structure of a robust controller is composed of a nominal part, similar to a feedback linearizing or inverse control law, and of additional terms to cope with model uncertainty. The structure of an adaptive controller is similar, but the difference is that the model is actually updated during operation, based on the measured performance. A simple approach to robust control is the so-called sliding control methodology. Perfect performance can be achieved in the presence of arbitrary parameter inaccuracies Sliding Surfaces Sliding mode control is a high-speed switched feedback control. It is also known as variable structure control (VSC) in the literature. The most important task in the SMC methodology is to design a switched control that drives the plant state to the switching surface and maintains it on the surface upon interception. A Lyapunov approach is used to characterize this task. A generalized Lyapunov function, that characterizes the motion of the state trajectory to the sliding surface, is defined in terms of the surface. For each chosen switched control structure, one chooses the gains so that the derivative of this Lyapunov function is negative definite, thus guaranteeing motion of the state trajectory to the surface. After proper design of the surface, a

121 11 switched controller is constructed so that the tangent vectors of the state trajectory point towards the surface such that the state is driven to and maintained on the sliding surface. Such controllers result in discontinuous closed-loop systems. To be more precise, the gains in each feedback path switch between two values according to a rule that depends on the value of the state at each instant. The purpose of the switching control law is to drive the nonlinear plant s state trajectory onto a prespecified (user-chosen) surface in the state-space and to maintain the plant s state trajectory on this surface for subsequent time. The surface is called a switching surface. When the plant state trajectory is above the surface, a feedback path would have a specific gain and then, a different gain if the trajectory drops below the surface. This surface defines the rule for proper switching. This surface is also called a sliding surface (sliding manifold). Ideally, once intercepted, the switched control maintains the plant s state trajectory on the surface for all subsequent time, and the plant s state trajectory slides along this surface. The motion of the system as it slides along boundaries of the control structures is called a sliding mode [95]. Intuitively, sliding mode control uses practically infinite gain to force the trajectories of a dynamic system to slide along the restricted sliding mode subspace. Trajectories from this reduced-order sliding mode have desirable properties (e.g., the system naturally slides along it until it comes to rest at a desired equilibrium). The main strength of sliding mode control is its robustness. Because the control can be as simple as a switching between two states (e.g., on/off ), it does not need to be precise

122 111 and will not be sensitive to parameter variations that enter into the control channel. Additionally, because the control law is not a continuous function, the sliding mode can be reached in finite time (i.e., better than asymptotic behavior). However, real implementations of sliding mode control approximate the theoretical behavior of sliding along the surface with a high-frequency and generally non-deterministic switching control signal that causes the system to chatter in a tight neighborhood of the sliding surface [96]. In summary, the motion consists of a reaching phase during which trajectories starting off the manifold S = move toward it and reach it in finite time, followed by a sliding phase, during which, the motion is confined to the manifold S = and the dynamics of the system are represented by a reduced-order model with exponentially stable error dynamics. The S = is called the sliding mode, and the control law manifold u = κ(x)sgn(s) is called sliding control mode Chattering It should be noted that the controller is discontinuous at S =. Thus, sliding mode control must be applied with more care than other forms of nonlinear control that have more moderate control action. In particular, due to the effects of sampling, switching and delays in the actuators used to implement the controller, and other imperfections, the hard SMC action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics [97]. Figure 6.1 shows how delays can cause

123 112 Figure 6.1: Chattering due to delay in control switching. chattering. It depicts a trajectory in the region S > heading toward the sliding manifold S =. The trajectory first hits the manifold at a point P a. In ideal sliding mode control, the trajectory should start sliding on the manifold from a point P a. In reality, there will be a delay between the time the sign of S changes and the time the control switches. During this delay period, the trajectory crosses the manifold into the region S <. Chattering results in low control accuracy, high heat losses in electrical power circuits, and high wear of moving mechanical parts. It may also excite unmodeled high frequency dynamics, which degrade the performance of the system and may potentially even lead to instability.

124 113 One commonly used method to eliminate the effects of chattering is to replace the switching control law by a saturating approximation within boundary layer around the sliding surface [63], [98], and [99]. Inside the boundary layer, the discontinuous switching function κsgn(s) is approximated by a continuous function to avoid discontinuity of the control signals. Even though the boundary layer design can alleviate the chattering phenomenon, this approach, however, provides no guarantee of convergence to the sliding mode, and involves a trade-off between chattering and robustness, and results in the existence of the steady state error. 6.2 Modeling for Control From Chapter 3, the vehicle s model can be written in the following form with additive external disturbances d 1,...,d 6. [ φ = 1 J x (J y J z ) θ ψ ] k φ2 tax J r Ω θ +u 1 +d 1 [ θ = 1 J y (J z J x ) ψ φ k ] θ2 tay J r Ω φ+u 2 +d 2 [ ψ = 1 J z (J x J y ) θ φ k ] taz ψ 2 +u 3 +d 3 ẍ = 1 m [ K faxẋ 2 +u 4 +d 4 ] (6.1) ÿ = 1 m [ K fayẏ 2 +u 5 +d 5 ] z = 1 m [ K fazż 2 mg +u 6 +d 6 ]

125 114 Considerthestate-spacemodelpresentedin(6.1),wherethevectory = [φ,θ,ψ,x,y,z] T is the output of interest, the vector u = [u 1,u 2,u 3,u 4,u 5,u 6 ] T is the control input, and x = [x 1,x 2,...,x 12 ] T is the vehicle s state vector such as x = [φ, φ,θ, θ,ψ, ψ,x,ẋ,y,ẏ,z,ż] T. The state-space model can be written in the following closed form: ẋ = f(x)+b(x)u+d(x) (6.2) with f(x), b(x), d(x) defined as follows:

126 115 where x 2 a 1 x 4 x 6 a 2 x 2 2 a 3 Ωx 4 x 4 a 4 x 2 x 6 a 5 x 2 4 a 6 Ωx 2 x 6 a f(x) = 7 x 2 x 4 a 8 x 2 6 x 8 a 9 x 2 8 x 1 a 1 x 2 1 x 12 a 11 x 2 12 g a 1 = Jy Jz J x a 5 = ktay J y a 9 = K fax m a 2 = ktax J x a 6 = Jr J y a 1 = K fay m a 3 = Jr J x a 7 = Jx Jy J z a 11 = K faz m a 4 = Jz Jx J y a 8 = ktaz J z (6.3)

127 116 b(x) = b 1 b 2 b 3 b 4 b 4 (6.4) b where b 1 = 1 J x, b 2 = 1 J y, b 3 = 1 J z, and b 4 = 1 m. And finally d(x) is defined as: [ d(x) = d 1 d 2 d 3 d 4 d 5 d 6 ] T (6.5) In(6.2)thenonlinearfunctionf(x)isnotexactlyknown,andthecontrolgainb(x) is of known sign but unknown exact value. f(x) and b(x) are both upper bounded by known, continuous function of x and external disturbance b(x). The inertia of

128 117 a mechanical system is only known to a certain accuracy, and friction models only describe part of the actual friction forces. The control problem is to get the state x to track a specific time varying state x d in the presence of model imprecision on f(x) and b(x). Remark: The disturbance d(x) can describe uncertainties such as inaccurate torques and lifts of the rotors, the ground effects, wind disturbance, and the bias between the geometric centre and its centre of gravity. 6.3 Sliding Mode Control based on Backstepping In this thesis a 2 step approach for the design of the controller is taken. The two steps are: 1. Defining the sliding mode. This is a surface that is invariant of the controlled dynamics, where the controlled dynamics are exponentially stable, and where the system tracks the desired set-point. 2. Defining the control that drives the state to the sliding mode in finite time Controller Design The first step in designing the sliding mode controller is similar to the one used for the backstepping approach, except S φ (Surface) as defined in (6.6) is used instead of e 2 : S φ = λ 1 e 1 +x 2 ẋ 1d (6.6)

129 118 For the second step the following augmented Lyapunov function is considered: V(e 1,S φ ) = 1 2 (e2 1 +S 2 φ) The chosen law for the attraction surface is the time derivative of (6.6) satisfying (SṠ < ): Ṡ φ = k 1 sgn(s φ ) λ 1 S φ = λ 1 ė 1 +ẋ 2 ẍ 1d (6.7) = λ 1 (x 2 ẋ 1d ) ẍ 1d +a 1 x 4 x 6 +b 1 u 1 +d 1 (t) As for the backsteppning approach the control u 1 is extracted: u 1 = 1 b 1 [ a 1 x 4 x 6 +ẍ 1d k 1 sgn(s φ ) λ 2 1e 1 2λė 1 ] (6.8) Using the backstepping approach as a recursive algorithm for the synthesis of control-law, all the stages of calculation concerning the tracking errors and Lyapunov functions can be simplified in the following way: x i x id i {1,3,5,7,9,11} e i = λ j e (i 1) +x i ẋ (i 1)d i {2,4,6,8,1,12} (6.9)

130 119 with λ j >, j [1,6], and V i = 1 2 e2 i i {1,3,5,7,9,11} 1 (V 2 i 1 +e 2 i) i {2,4,6,8,1,12} (6.1) The choice of the sliding surfaces is based upon the synthesized tracking errors which permitted us the synthesis of stabilizing control laws. Thus, from (6.9) the dynamic sliding surfaces, S φ,s θ,s ψ,s x,s y and S z, are defined as: S φ = λ 1 e 1 +ė 1 S θ = λ 2 e 3 +ė 3 S ψ = λ 3 e 5 +ė 5 S x = λ 4 e 7 +ė 7 S y = λ 5 e 9 +ė 9 S z = λ 6 e 11 +ė 11 (6.11) As e 1 = φ φ d, e 3 = θ θ d, e 5 = ψ ψ d, e 7 = x x d, e 9 = y y d, and e 11 = z z d, and for simplicity in notations, the switching surfaces can be rewritten as:

131 12 S φ = λ 1 e φ +ė φ S θ = λ 2 e θ +ė θ S ψ = λ 3 e ψ +ė ψ S x = λ 4 e x +ė x S y = λ 5 e y +ė y S z = λ 6 e z +ė z (6.12) To synthesize a stabilizing control law by sliding mode, the necessary sliding condition (SṠ < ) must be verified; so the synthesized stabilizing control laws are as follows: u 1 = 1 b 1 [ a 1 x 4 x 6 + φ ] d (t) 2λ 1 ė φ (t) λ 2 1e φ (t) k 1 sgn(s φ ) u 2 = 1 b 2 [ a 2 x 2 x 6 + θ ] d (t) 2λ 2 ė θ (t) λ 2 2e θ (t) k 2 sgn(s θ ) u 3 = 1 b 3 [ a 3 x 2 x 4 + ψ ] d (t) 2λ 3 ė ψ (t) λ 2 3e ψ (t) k 3 sgn(s ψ ) u 4 = 1 b 4 [ a 9 ẋ 2 +ẍ d (t) 2λ 4 ė x (t) λ 2 4e x (t) k 4 sgn(s x )] u 5 = 1 b 4 [ a 1 ẏ 2 +ÿ d (t) 2λ 5 ė y (t) λ 2 5e y (t) k 5 sgn(s y )] u 6 = 1 b 4 [ a 11 ż 2 g + z d (t) 2λ 6 ė z (t) λ 2 6e z (t) k 6 sgn(s z )] (6.13) 6.4 Simulation Results In order to test the developed SMC a number of simulations were tested. Herein, four of such simulations are presented, which show the performance of the developed

132 121 SMC. The following four scenarios are presented: 1) pitched hover, 2) pitched hover under wind disturbances, 3) Scenario # 2 with varying model parameters, and 4) picking up a load. 1. Pitched hover scenario (Scenario #7): The first simulation is the pitched hover maneuver, where the vehicle is commanded to hover in a maneuver that no aircraft known to have performed. In this case study, the system initial condition and the desired states are chosen as below: x = [22.5,,,,18,,5,,4,,3,] T x d = [,,22.5,,,,,,,,3,] T In other words, the controller is commanded to maintain the evader at the same altitude at 3 m while pitching its nose from to 22.5, and regulate all other states to zero. The results are illustrated in Figs. 6.2, 6.3, and 6.4. For this simulation, the switching functions and controller gains are selected according to Table 6.1. Figure 6.2 shows the six control signals u 1,...,u 6. The chattering effect is obviously seen in this figure. Figure 6.3 shows that the controller is able to regulate the evader s roll and yaw angles to zero, while keeping the vehicle s pitch angle at Pitched hover windy scenario (unstructured uncertainty, Scenario #8): The next scenario is the evader in a pitched hover condition facing a sudden strong sinusoidal wind disturbance. The initial condition and desired values of states

133 122 Table 6.1: The SMC controller gains and sliding surfaces λ 1 = 2 S φ = 2e φ +ė φ k 1 = 1 λ 2 = 2 S θ = 2e θ +ė θ k 2 = 1 λ 3 = 2 S ψ = 2e ψ +ė ψ k 3 = 1 λ 4 = 1 S x = 1e x +ė x k 4 = 1.5 λ 5 = 1 S y = 1e y +ė y k 5 = 1.5 λ 6 = 1 S z = 1e z +ė z k 6 = 1.5 are the same as previous scenario, and it is assumed that after 1.5 s, the wind blows for.2 s. Wind as an external disturbance is formulated with d 1 (t) = d 2 (t) = d 3 (t) = d 4 (t) = d 5 (t) = d 6 (t) = 5(1+5sin(2πt)) in evader model in (6.5). It is also assumed that wind blows in all directions effects on all states. The results in Figs. 6.5 and 6.6 illustrate the robustness of sliding mode controller in presence of unstructured uncertainties. This property insures the stability of evader, while external disturbances such as wind, ground and wall effects affect the vehicle during its maneuvers. 3. Scenario # 2 with varying model parameter (structured uncertainty): The simulation scenario # 2 used in Chapter 4, Section 4.7, is simulated again here to verify the robustness of the designed SMC controller to parameter variations. The scenario is defined as follows: the initial condition is x = [,,,,,,,,,,,] T, and the desired value is x d = [22.5,,15,,18,,3,,4,,2,] T. In this simulation the mass of the evader is changed with the same amount as in Chapter 4. Moreover, the inertia matrix of the vehicle is also changed by 2%. Another simulation was made on the evader model with mass and inertia matrix uncertainty and the added

134 123 random noise. This noise can be considered as a measurement noise. The results were compared with the results of simulation without parameter uncertainty, and also with the scenario with only the parameter uncertainty but no added noise, in Table 6.2. As it can be seen easily from this table, after 1 second has passed, the error for all six outputs are almost zero for all mentioned scenarios. The results in Table 6.2 verify the robustness of the proposed designed controller in presence of structured (parametric) uncertainty. Figures 6.7 and 6.8 show the control input signals, the orientation output, and the position output of the evader, respectively, in this simulation scenario. 4. Picking-up-a-load scenario (Scenario #11): In this simulation, the evader is commanded to pick up a heavy load of 3.5 kg after 1 second into the simulation. It is assumed that this load adds a sudden disturbance to the pitch (θ) and the pitch angular velocity ( θ) of the vehicle. The results are depicted in Figs. 6.9, 6.1, and 6.11, which show the ability of the controller in handling this situation. The system pitch returned to a steady state in.25 s. The steady state error of altitude is only 4 cm and reached after.5 s. 6.5 Summary A simple approach to robust control is the so-called sliding control methodology. The aimofaslidingcontrolleristodesignacontrollawtoeffectivelyaccountforparameter uncertainty, including imprecision on the mass properties or loads, inaccuracies in the

135 124 Table 6.2: The results of SMC controller for position and orientation regulation simulation (Scenario # 2) in three different conditions: i) the parameters of the dynamic model are known and do not change, ii) the mass and inertia matrix of the vehicle have uncertainty and change during the flight, iii) the parameters of the dynamic model have uncertainty and change plus there is additive white noise due to the sensor measurement noise. i) Fixed parameters ii) Varying parameters iii) Additive sensor noise time (sec) error (deg, m) error (deg, m) error (deg, m) Roll φ Pitch θ Yaw ψ Position x Position y Altitude z Control Signal u1 4 Control Signal u4 5 2 u1 u4 u Control signal u2 1 u Control signal u5 4 2 u Control signal u u Control signal u Figure 6.2: Control input signals of SMC technique for performing pitched hover scenario (Scenario # 7).

136 125 Roll[degree] Yaw[degree] 2 1 The roll angle The pitch angle 2 Pitch[degree] The yaw angle x[m] 4 2 The Cartesian position x system output desired x(t) The Cartesian position y y[m] The Cartesian position z 3 z[m] Figure 6.3: Orientation angles regulation in pitched hover stationary scenario (Scenario # 7). Figure 6.4: Position regulation while stationary at pitched hover scenario (Scenario # 7). torque constants of the actuators, friction, and so on. Although perfect tracking can be achieved in principle in the presence of arbitrary parameter inaccuracies, such performance is obtained at the cost of extremely high control activity. The drawback is that this high control activity may excite high frequency dynamics neglected in the course of modeling. In practice, this corresponds to modification of control laws by replacing a switching, chattering control law by its smooth approximation. The switching control laws derived above can be smoothly interpolated in boundary layers, so as to eliminate chattering, thus leading to a tradeoff between parametric uncertainty and tracking performance. In this chapter, a robust controller based on SMC methodology was designed for

137 126 2 Control Signal u1 4 Control Signal u4 u1 u4 2 u2 u Control signal u Control signal u u5 u Control signal u Control signal u Figure 6.5: Control input signals of SMC technique in presence of a strong sudden wind disturbance (unstructured uncertainty, Scenario # 8). Roll[degree] 2 1 The roll angle Pitch[degree] Yaw[degree] 4 2 The pitch angle The yaw angle y[m] z[m] x[m] The Cartesian position x The Cartesian position y The Cartesian position z Figure 6.6: Orientation angles and position regulation in hover pitched scenario with a strong sudden wind disturbance with magnitude 5 (Scenario # 8).

138 127 u1 1 Control Signal u Control signal u2 1 u Control Signal u Control signal u5 u2 u Control signal u Control signal u6 2 u3 u Figure 6.7: Control input signals of SMC technique when performing Scenario # 2 and system parameters (mass and inertia matrix) are varying. Roll[degree] Pitch[degree] Yaw[degree] 4 2 The roll angle The pitch angle The yaw angle x[m] 4 2 The Cartesian position x The Cartesian position y 6 y[m] The Cartesian position z 2 z[m] Figure 6.8: Orientation angles and position regulation with model parameter variations show robustness of SMC technique in presence of structured uncertainties.

139 128 2 Control Signal u1 2 Control Signal u4 u1 2 u Control signal u Control signal u5 2 u2 5 u Control signal u Control signal u6 12 u3 u Figure 6.9: Control input signals of SMC technique while picking up a heavy load (Scenario # 11). evader, which achieves exponential tracking control of a desired trajectory where the plant dynamics contain uncertainty and bounded non-lp disturbances. Simulation results demonstrate the robustness of the controllers to sensor noise, exogenous perturbations, parametric uncertainty, and plant nonlinearities, while simultaneously exhibiting the capability to follow a reference trajectory. However, in order to compensate for uncertainties in the model and external disturbances, large input gains are required which makes a serious limitation in power-limited systems such as small UAVs like evader. Moreover, the chattering phenomena may cause the rotors to rotate and tilt in a different direction very fast. This may makes using the SMC impossible for evader even though it has a robustness property which is necessary for

140 129 1 The error of roll angle 5 The error of x Roll[deg] x[m] The error of pitch angle 5 Pitch[deg] The error of yaw angle The error of y 5 y[m] The error of z.4 Yaw[deg] z[m] Figure 6.1: Orientation angles regulation error when the evader picks up a heavy load (Scenario # 11). Figure 6.11: Position regulation error when the evader picks up a heavy load scenario (Scenario # 11). control of such a nonlinear dynamics system. It is now time to investigate the feasibility of the designed controllers including SMC and other controllers discussed in previous chapters. This issue will be discussed in the next chapter.

141 Chapter 7 Neural Network Nonlinear Function Approximation In previous chapters of this thesis we have derived the dynamic model of evader UAV and then designed several control laws to do specific tasks such as regulation to an arbitrary desired position and orientation, tracking a time varying trajectory in 3D space, and taking off and landing. All of the designed controllers in previous chapters are based on six control signals u 1,...,u 6, such as six control signals in Equations (5.3) and (4.8) in Chapters 5 and 4, respectively. However, control signals u 1,...,u 6 are not the real actuator signals. They are nonlinear functions of the actual control signals α 1, α 2, β 1, β 2, ω 1, ω 2, that would be used to actually control the actuators influencing them. The applicability of previously designed controllers, the evader and doat mechanism, lies in the existence of a feasible (in range) combination of longitudinalandlateralangles,androtorangularspeeds,namelyα 1,α 2,β 1,β 2,ω 1,ω 2, to produce the necessary control signals. Thus the question is how to solve equations of u 1,...,u 6 based on α 1, α 2, β 1, β 2, ω 1, ω 2 to obtain actual control signals. Since these equations do not have a mathematical systematic solution, it is not possible to obtain the exact values of α 1, α 2, β 1, β 2, ω 1, ω 2. Hence, we proposed to utilize nonlinear function approximator approaches to find an approximation of these functions in a 13

142 131 waythatbyenteringthesixcontrolsignalsu 1,...,u 6, thefunctionapproximatorwould give us α 1, α 2, β 1, β 2, ω 1, ω 2 as its outputs. To address this problem, a Multi Layer Perceptron(MLP) neural network is trained in this thesis as an inverse mapping of the u 1,...,u 6 functions, using supervised learning with back-propagation (BP) algorithm. By applying this approach, we find an approximation of the actual control signals of doat mechanism, which has not been addressed in the literature before. The rest of this chapter is organized as follows. The benefits of using neural networks for the specific problem of this research are first discussed in Section 7.2, followed by an explanation of feedforward networks in Section 7.3. The MLP networks are introduced in Section 7.4. Section 7.5 explains the BP learning algorithm. The application and the results of the MLP neural network with BP learning algorithm for our specific problem of approximation of the inverse mapping between α 1, α 2, β 1, β 2, ω 1, ω 2 and u 1,...,u 6 is investigated in Section 7.6. Finally, a summary of this chapter and the main observations and findings of it are discussed in Section Benefits of Neural Networks The neural network derives its computing power through, first, its massively parallel distributed structure and, second, its ability to learn and therefore generalize. Generalization refers to the neural network producing reasonable outputs for inputs not encountered during training (learning). These two information-processing capabilities make it possible for neural networks to solve complex (large-scale) problems

143 132 that are currently intractable such as the problem of solving the nonlinear functions of u 1,...,u 6 based on six unknown variables α 1, α 2, β 1, β 2, ω 1, ω 2, which does not have any conventional mathematical solution. The use of neural networks offers the following useful properties and capabilities: 1. Nonlinearity. A neuron is basically a nonlinear device. Consequently, a neural network, made up of an interconnection of neurons, is itself nonlinear. Moreover the nonlinearity is of a special kind in the sense that it is distributed throughout the network. Nonlinearity is a highly important property, particularly if the underlying physical mechanism responsible for the generation of an input signal is inherently nonlinear such as the evader s dynamic in the specific problem at hand in this thesis. 2. Input-Output Mapping. A popular concept of learning called supervised learning (Fig. 7.1) involves the modification of the weights of a neural network by applying a set of labeled training samples or task examples. Each example consists of a unique input signal and the corresponding desired response. The network is presented and an example picked at random from the set, and the weights (free parameters) of the network are modified so as to minimize the difference between the desired response and the actual response of the network produced by the input in accordance with an appropriate statistical criterion. The training of the network is repeated for many examples in the set until the network reaches a steady state, where there are no further significant changes in the weights. The previously applied training examples may be reapplied during the training session but in a different order.

144 133 Figure 7.1: Supervised learning block diagram. Thus the network learns from the examples by constructing an input-output mapping for the problem at hand. Lack of existing systematic mathematical approaches for the problem of finding actual control signals of UAV flight makes us to think of its solution as an input-output mapping problem. 3. Adaptivity. Neural networks have a built-in capability to adapt their weights to changes in the surrounding environment. In particular, a neural network trained to operate in a specific environment can be easily retrained to deal with minor changes in the operating environmental conditions. Moreover, when it is operating in a nonstationary environment (i.e. one whose statistics change with time), a neural network can be designed to change its weights in real time (adaptive neural network). As a general rule, it may be said that the more adaptive we make a system in a properly

145 134 designed fashion, assuming the adaptive system is stable, the more robust its performance will likely be when the system is required to operate in a non-stationary environment. It should be emphasized, however, that adaptivity does not always lead to robustness. Indeed, it may do the very opposite. For example, an adaptive system with short time constants may change rapidly and therefore tend to respond to spurious disturbances, causing a drastic degradation in system performance. 7.2 Problem Statement Recalling the equations of u 1,...,u 6 from Chapter 3, we have: u 1 = cosα 1 J r ω 1 α 1 sinα 1 J r ω 1 β1 +sinα 1 J r ω 1 α 1 cosα 2 J r ω 2 α 2 sinα 2 J r ω 2 β2 +sinα 2 J r ω 2 α 2 h o (sinβ 1 cosα 1 T 1 +sinβ 2 cosα 2 T 2 )+l o (cosβ 1 cosα 1 T 1 cosβ 2 cosα 2 T 2 ) sinα 1 Q 1 +sinα 2 Q 2 +cosα 1 J p β1 +cosα 2 J p β2 (7.1) u 2 = sinβ 1 sinα 1 J r ω 1 α 1 +cosβ 1 J r ω 1 β1 +sinβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) sinβ 2 sinα 2 J r ω 2 α 2 +cosβ 2 J r ω 2 β2 +sinβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) +sinβ 1 cosα 1 Q 1 sinβ 2 cosα 2 Q 2 h o (sinα 1 T 1 +sinα 2 T 2 ) +sinβ 1 sinα 1 J p β1 +cosβ 1 J p α 1 +sinβ 2 sinα 2 J p β2 +cosβ 2 J p α 2

146 135 u 3 = cosβ 1 sinα 1 J r ω 1 α 1 +sinβ 1 J r ω 1 β1 cosβ 1 cosα 1 (J r ω 1 β1 J r ω 1 α 1 ) +cosβ 2 sinα 2 J r ω 2 α 2 +sinβ 2 J r ω 2 β2 cosβ 2 cosα 2 (J r ω 2 β2 J r ω 2 α 2 ) cosβ 1 cosα 1 Q 1 +cosβ 2 cosα 2 Q 2 l o (sinα 1 T 1 sinα 2 T 2 ) cosβ 1 sinα 1 J p β1 +sinβ 1 J p α 1 cosβ 2 sinα 2 J p β2 +sinβ 2 J p α 2 u 4 = C φ C ψ F x +(S φ S θ C ψ C φ S ψ )F y +(C θ S θ C ψ +S φ S ψ )F z u 5 = C θ S ψ F x +(S φ S θ S ψ +C φ C ψ )F y +(C φ S θ S ψ S φ C ψ )F z u 6 = S φ F x +S φ C θ F y +C φ C θ F z F x = sinα 1 T 1 +sinα 2 T 2 F y = sinβ 1 cosα 1 T 1 sinβ 2 cosα 2 T 2 F z = cosβ 1 cosα 1 T 1 +cosβ 2 cosα 2 T 2 In order to be able to control the evader for autonomous flight, based on the proposed model with six control inputs, the actual control signals, α 1, α 2, β 1, β 2, ω 1, ω 2, areneededtobeobtained. Astherearenomathematicalsystematicapproachesto solve the system of nonlinear equations in (7.1), we are seeking for an approximation of the six functions in (7.1) so that by giving u 1,...,u 6 as an input, the approximation would give us α 1, α 2, β 1, β 2, ω 1, ω 2. Therefore, we have to find the inverse mapping between these variables. For this purpose, the neural network is trained as an inverse

147 136 model of the functions in (7.1), using supervised learning (Fig. 7.2). The network input is the outputs of each controller, and the network output is the functions input in (7.1). The question is whether neural network is able to map such a complex function with six inputs and six outputs or not. Before answering this question, first the problem of inverse mapping is formulized as a function approximation problem in the next section Function Approximation GivenasetofN differentpointsinapdimensionalinputspace,x k R p,k = 1,2,...,N and a corresponding set of N points in a m dimensional output space, d k R m,k = 1,2,...,N, it is desired to find a mapping function ˆf : R p R m that fulfills the relationship, such that ˆf(xk ) = d k, k = 1,2,...,N. (7.2) The actual nonlinear input-output mapping between x k and d k is denoted as f(x k ) = d k, (7.3) where f(.) is assumed to be unknown. The objective for this approximation task is ˆf(x k ) f(x k ) < ǫ, (7.4)

148 137 where ǫ is a small positive number. Provided that the size N of the training set is large enough and the network is equipped with an adequate number of free parameters (weights), then the approximation error ǫ can be made small enough for the task (universal approximation theorem [1], [11], [12]). In this thesis, the ability of a neural network to approximate an unknown inputoutput mapping function is used for inverse system identification as its structure is shown in Fig Notice that the place of d k and x k have been reversed in the structure of inverse system identification and d k and x k denote the input vector and desired output of the unknown inverse system, respectively. Equation (7.5) describes the input-output relation of an unknown inverse system. x k = f 1 (d k ) (7.5) In the case of Fig. 7.2, y k denotes the output vector of the neural network which is the vector of actual control signals produced in response to an input vector d k. The difference between the desired output vector x k (associated with d k ) and the neural network output y k provides the error vector e k. A neural network is utilized to approximate the inverse function f 1. The error vector e k is used to adjust the free parameters of the neural network. One of the possibles way to perform the parameter adjustment is to use an objective function whereby the goal is to adjust

149 138 Figure 7.2: Block diagram of an inverse function approximation system. the parameters of the network so as to minimize the objective function. A choice of objective function is the error function given by: J(W) = 1 2 m (d j y j ) T (d j y j ) (7.6) j=1 7.3 Feedforward networks Neural networks are adaptive nonlinear systems that adjust their parameters automatically in order to minimize a performance criterion. Identification of the neural models involves learning, which is covered extensively in [13]. There are two different configurations of feedforward networks: i) single-layer feedforward networks (Fig. 7.3) and ii) multi-layer feedforward networks (Fig. 7.4).

150 139 Figure 7.3: Structure of single-layer feedforward networks. Multi-layer feedforward neural networks, or more commonly known as Multi-Layer Perceptrons (MLP) have very quickly become the most widely encountered neural networks, particularly within the area of systems and control [14]. Usually, backpropagation rule or delta learning rule is used to train multi-layer feedforward neural network. Feedforward networks are widely used in classification [15], and approximation theory [16], [17].

151 Figure 7.4: Structure of multi-layer feedforward neural networks. 14

152 Overview of Multi-Layer Perceptron In this thesis, MLP network is used, which is very popular in the applied fields as well as in theoretical research. The reasons for this popularity might be as follows: Simplicity. Scalability. Property to be a general function approximator. Adaptivity. These features make MLP networks a suitable choice of network for inverse function approximation mapping. For multi-layer perceptrons, weight learning is most commonly carried out by the method of back-propagation [18]. In this approach the network outputs are first compared with a set of desired values for those outputs. The error function (7.6), on the output layer only, must first be minimized by a best selection of output layer weights. Once the output layer weights have been selected the weights in the hidden layer next to the output can be adjusted by employing a linear back-propagation of error term from the output layer. A full description of back-propagation for both static and dynamic MLPs can be found in Cichocki and Unbehauen [19]. On the down side, back-propagation is a nonlinear steepest descent type algorithm, and it may either converge on local minima or be extremely slow to converge. On the

153 142 positive side, however, a MLP with only one hidden layer is sufficient to approximate any continuous function [1]. 7.5 Back-Propagation Back-Propagation (BP) is a specific technique for implementing gradient descent in weight space for a multilayer feedforward network. The BP algorithm provides a way to calculate the gradient of the error function efficiently using the chain rule of differentiation. The error after initial computation in the forward pass is propagated backward from the output units, layer by layer. This algorithm involves minimization of an error function in the least mean square, trained by applying gradient descent method [11]. A general structure of a two-layered feedforward neural network, with p neurons in the first layer and m neurons in the second layer is depicted in Fig The detailed structure of the input and output neurons are shown in Figs. 7.5 and 7.6. In this thesis, a two-layer linear-output feedforward network (MLP network) is used. The flowchart of training the MLP network to learn the inverse mapping between actual control signals (α 1, α 2, β 1, β 2, ω 1, ω 2 ) and controller control signals (u 1,...,u 6 ) is shown in Fig This flowchart summarizes the training process in a two-layer perceptron network. The variables and scripts that are used as superscripts or subscripts in this flowchart are defined as follows: k = 1,2,...,n (dimension of input layer), j = 1,2,...,m (dimension of output layer),

154 143 Figure 7.5: neuron (1,i), (i = 1,2,...,p) in the hidden layer Figure 7.6: neuron (2,j), (j = 1,2,...,m) in the output layer x k,(k = 1,2,...,n) input signal of order n (input layer), x k = [x 1 x 2...x n ] T, N samples of input elements k = 1,2,...,N, z i,(i = 1,2,...,p) output signal of the first layer (hidden layer), z k = [z 1 z 2...z p ] T the outputs of all neurons in the hidden layer, y j,(j = 1,2,...,m)outputsignalofsecondlayer(outputlayer),y k = [y 1 y 2...y m ] T the output vector of neural network, W (1) ik,(i = 1,2,...,p),(k = 1,2,...,n) the weights corresponding to first layer neurons, the connection weight from the kth input to ith neuron in first layer, W (2) jq,(j = 1,2,...,m),(q = 1,2,...,p) the connection weight from the qth neuron in the first layer to the jth neuron in the output layer.

155 144 In this flowchart a set of N input data x k = [x 1,...,x n ] is first presented to the input layer. The output from this layer are then fed as inputs to the hidden layer and subsequently the outputs from the hidden layer are fed as weighted inputs (i.e., the outputs from the hidden layer are multiplied by the weights (W (1) ik x k) to the second layer which is a output layer in this flowchart. The hidden layer has activation function of sigmoid. Thus, the accumulative weighted inputs (S (1) i = n k= W(1) ik x k) ofallneuronsinthehiddenlayerfirstgointoasigmoidfunction(z i = σ(s (1) i ))andare then fed to the next layer as inputs. Despite the fact that the output layer performs based on a linear function, the same process as for the hidden layer happens in the output layer, and the output of this layer (y j = p q= W(2) jq z q) is the response of neural network to the input x k. The output of neural network is then compared with the desired ones and the error goes back to update the weights of output and hidden layers through BP algorithm as illustrated in Modifications of Weights sections in the flowchart in Fig Training the MLP Neural Network for Actual Control Signal Approximation As mentioned before, for approximation or fitting problem, a neural network has to map between a data set of numeric inputs and a set of numeric targets. A twolayer feed-forward with sigmoid hidden neurons and linear output neurons is trained, which based on universal approximation theorem, can fit multi-dimensional mapping

156 Figure 7.7: Flowchart of training process in a two-layer perceptron network. This flowchart does not include the stopping criteria of the training process. 145