Autonomous Execution of Aircraft Supermaneuvers with Switching Nonlinear Backstepping Control Majid Moghadam∗ Controls and Avionics Research Group, Aerospace Research Center, Istanbul Technical University, Istanbul, 34469, Turkey N. Kemal Urey and Gokhan Inalhanz Department of Aeronautical Engineering Istanbul Technical University, Istanbul, 34469, Turkey Control system design for agile maneuvering aircraft poses several challenges, such as underactuated nonlinear dynamics, input saturation limits and loss of control effectiveness in high angle of attack regions of the flight envelope. Previous work in the field developed a variety of switched nonlinear control methods to enable precise tracking of agile maneuver profiles. However, demonstrating autonomous execution of challenging agile maneuvers, such as the ones that require simultaneous tracking of both translational and attitude state variables is still an open problem. In this study, we develop switched nonlinear backstepping control laws tailored towards the execution of such agile maneuvers. In particular, we demonstrate the applicability of our design on a high fidelity F-16 model and two supermaneuvers: Pugachev's Cobra and Rolling Circle. In addition, we present a numerical study for analysis of stability of these controllers by introducing the notion of Region of Recoverability (ROR). ROR plots outline the subsets of the state space where the switching of controllers is feasible, which helps the designer to assess the stability of switched control system. Nomenclature VT Airspeed, ft/s X;¯ Y;¯ Z¯ Body force components, lb α Angle of attack, rad L;¯ M;¯ N¯ Angular moments, lb.ft β Side slip angle, rad g Gravitational acceleration, ft/s2 q0; q1; q2; q3 Quaternion components ci Inertia parameters φ, θ; Euler roll, pitch, and yaw angles, rad q¯ Dynamic pressure, lb/ft2 p; q; r Roll, pitch, and yaw rates, rad/s S Wing area, ft2 γ Flight path angle, rad B Wing span, ft χ Course angle, rad c¯ Mean aerodynamic chord, ft Np Inertial north position, ft C? Aerodynamic coefficients Ep Inertial east position, ft δa; δe; δr Control surface deflections, deg h Height, ft δlef Leading-edge flap deflection, rad 2 D; L; Y Drag, lift, and side forces, lb hE Angular momentum of engine, slug.ft /s T Thrust force, lb Tw=b Body to wind frame rotation matrix 2 2 g1; g2; g3 Gravity components, m/s Ps Static pressure, lb/ft m Mass, slug ref Reference input ∗Graduate Research Assistant, Department of Aeronautical and Astronautical Engineering, [email protected] yAssistant Professor, Department of Aeronautical and Astronautical Engineering, [email protected]. zProfessor, Department of Aeronautical and Astronautical Engineering, [email protected]. 1 of 26 American Institute of Aeronautics and Astronautics I. Introduction Agile maneuvers are often executed by piloted fighter aircraft for gaining the advantage in combat and/or evading threats. Such maneuvers are also commonly used in aerobatics shows. These maneuvers usually feature i) high angle of attack, ii) high body angular rates and iii) simultaneous tracking of attitude and translational states, such as rolling the aircraft while keeping the altitude constant. In the last two decades, design of feedback control systems that can track such maneuvers in an autonomous manner attracted a lot of interest in control and aerospace communities. This trend is partially correlated with the increasing number of threats to unmanned combat aerial vehicles (UCAVs). In order to evade these threats, UCAVs need to perform agile maneuvers, in addition to the classical autonomous control modes such as waypoint tracking.1 Another potential source of motivation is to perform automated aerobatics shows.2 Design of such control systems is challenging due to a number of reasons, such as highly nonlinear dynamics of the aircraft and envelope saturation limits. The main objective of this work is to push the state of the art in nonlinear flight control systems design to enable execution of challenging agile maneuvers. A. Previous Work Similar to most nonlinear systems, stabilizing and controlling nonlinear six-degree-of-freedom (6DOF) air- craft dynamics to perform supermaneuvers initiated with design and implementation of linear control strate- gies.3 However, in most aggressive maneuvers, aircraft states deviate significantly from the equilibrium point around which the dynamics are linearized, which leads to loss of performance and even instability in some cases. Hence, nonlinear control strategies such as nonlinear dynamic inversion (NDI),4 sliding mode control (SMC),5 gain scheduling,6 and backstepping method (BS)7 have gained substantial interest in control system design for tracking agile maneuvers. The simplicity of the design methodology and implementation made NDI and SMC attractive choices for researchers.8{12 Separating nonlinear dynamics into inner and outer loop dynamics and designing controller for each loop have been investigated by Snell.9 However, the inability of this method in stabilizing the aircraft while tracking translational variables that arises due to non-minimum phase (NMP) characteristic of the equations of motion4 motivated researchers to look for alternative solutions. Fiorentini and Serrani13 used a suitable redefinition of the internal dynamics and presented an adaptive flight path angle trajectory tracking control method. A number of early studies10, 14, 15 neglected NMP characteristics of the system on controller design and studied the effects of NMP for specific types of aircraft. On the other hand, SMC has the additional advantage of being robust to uncertainties in the system model. In addition, it is pos- sible to design a dynamic compensator to slightly overcome the NMP effect on controlling the aircraft.16 Chattering of the control inputs because of the switching nature of the design methodology in SMC is one of the disadvantages of this method, which can be addressed by using higher order sliding mode (HOSM) techniques.17, 18 However, because of the NMP dynamics, none of the aforementioned control techniques could demonstrate consistently superior tracking performance for challenging agile maneuvers that require simultaneous tracking of attitude and translational variables. The issues with NMP dynamics popularized alternative approaches such as BS. Backstepping is a Lyapunov based design method that gained a lot of attention in last decade. In this method, each state can be stabilized using either control input or another state as a virtual control that manipulates the state trajectory. A global stabilizing controller in each step is designed using Lyapunov's direct method.4 Compared to NDI and SMC, backstepping offers a more flex- ible way of dealing with nonlinearities and internal dynamics of highly coupled nonlinear dynamics. More importantly, the control design does not necessarily suffer from NMP dynamics. Aircraft flight path angle control using BS method is studied by.19{21 In these works, aerodynamic coefficients are usually modeled as data lookup tables with considerable uncertainties. Adaptive and incremental backstepping methods have been used widely in the literature to make the controller robust to model uncertainties.22{27 A comparison between NDI and BS control in stabilizing the flight path angle of the aircraft is also investigated.28 However, it should be noted that due to underactuated nature of the aircraft dynamics, even with the BS method, most of the supermaneuvers cannot be tracked with a single controller, hence a switching strategy is needed. It is seen that, theoretically or practically, different types of agile maneuvers like Herbst,3 rapid turning,27 hover flight,29 and many kinds of maneuvers have been performed/analyzed in the previous work. However, a large portion of these maneuvers does not challenge the coupled nonlinear dynamics of the aircraft. For example, assume an aircraft that flies with a roll angle of 90◦ and tends to hold the altitude and course 2 of 26 American Institute of Aeronautics and Astronautics angle at a constant value. In this orientation, most of the basic intuition regarding flight dynamics no longer hold true. For instance, lift force generated by the elevator deflection does not result in acceleration in longitudinal plane, instead, it produces an acceleration in lateral plane. Vice versa, side force pushes the aircraft in vertical plane. Designing nonlinear controllers for performing such maneuvers is not trivial, as the state-input relation cannot be decoupled into independent equations. The proposed controller should stabilize the aircraft while performing the desired aggressive maneuver. In addition, it is also a point of interest to execute more than one maneuver in a row to be able to perform a complete predefined aerobatic scenario. Frazzoli et al.30 used discrete maneuvers for motion planning of agile vehicles. Ure and Inalhan1 presented a multi-modal flight control framework and flight path planning to enable the vehicle to perform agile maneuvers. Attitude transition modes were used to recover the UAV's orientation while passing between maneuvers. B. Contributions In this work, we present a switched BS based approach for tracking agile maneuvers that demand simulta- neous tracking of translational and attitude variables. Our contributions can be listed as follows: • We show that by designing a suitable switching strategy it is possible to execute highly challenging agile maneuvers that require simultaneous tracking of both attitude (such as roll and pitch angle) and trans- lational