MODELING OF TILTWING AIRCRAFT DYNAMICS AS LINEAR SYSTEM JOHANNA HOLSTEN* PHILIPP HARTMANN* DIETER MOORMANN* *Institute of Flight System Dynamics, RWTH Aachen University, Wüllnerstr. 7, 52064 Aachen, Germany [email protected] Abstract In this paper the nonlinear system dynamics of a tiltwing aircraft is linearized at a variation of operating points. Effect of linearization on dynamic behavior is analyzed in comparison to the original nonlinear model. Aircraft in tiltwing configuration combine the advantages of helicopters, such as hovering and vertical take-off and landing capabilities, with the advantages of conventional fixed- wing aircraft. During transition of tiltwing aircraft between hovering and aerodynamic horizontal forward flight, tilting the wing from vertical to horizontal position (and vice versa) poses a significant change in configuration. In combination with the given large velocity range this influences the control device effectiveness significantly. At the same time, tilting the wing provides an additional control variable. These aspects make designing a tiltwing flight controller complex. For controller design and analysis, linearization of the system dynamics as LTI model allows direct analysis of its stability margins. In this paper we show that the linear model of the tiltwing aircraft represents the nonlinear system sufficiently for controller analysis and design. Additionally it provides insight into changing dynamics and allows further analysis in the frequency domain. Keywords Tiltwing Aircraft; Flight System Dynamics. 1. Introduction Understanding and accurate modeling of systems behavior is a necessity when aiming at robust control performance. A linear model in contrast to a nonlinear model allows direct analysis and simplifies the control design process. Furthermore, state-space representations are more suitable for numerical calculations. Conclusions regarding state controllability and overall stability can be easily drawn (Skogestad and Postlethwaite, 2005). For tiltwing aircraft a linearization not only simplifies controller design but also provides additional insight into the changing flight dynamics over the complete flight envelope. The main challenge in controlling tiltwing aircraft is the flight state transition from thrust borne hovering to wing borne forward flight and vice versa. Developing the controller requires a system dynamics model. In the past different control concepts for tiltwing aircraft have been presented (Ostermann et al. , 2012; Hartmann et al. , 2016). Both control concepts were developed using nonlinear simulation based on wind tunnel and flight test data (Holsten et al. , 2011; Schütt et al. , 2014) and allow steady flight within the complete flight envelope. Research besides RWTH Aachen activities concerning tiltwing controller design and modeling has been done mainly with the primary aim to control transition from hover to cruise, but not allowing steady flight. Controller development for one specific flight state transition was done by Dickeson et al. (2007). In their contribution the aircraft dynamics was linearized for different tilt angles and based on the linearization, suitable transition trajectories were found. For these trajectories a H∞ transition controller was designed aiming at time optimal transition. Choosing the tilt angle as parameter for the linearization has some disadvantages, which will be discussed in this paper. Research about transition control for a tiltrotor has been presented by Ta et al. (2012). The aircraft is modeled mathematically and a cascaded PID structure in combination with a neural network is used as controller. Altitude control is excluded. The dynamics of tiltrotor aircraft differ significantly from those of a tiltwing aircraft, since the interaction between propulsion system and wing is not comparable. A in parts similar research to the presented approach comes from Sato et al. (2014). They present a control concept that divides the flight envelope into seven discrete points. At these design points a nominal model consisting of a linear aircraft motion model, an actuator model, a motor model and the primary flight control system is investigated. The linear aircraft motion model has a conventional structure with the velocities, angular velocities, pitch and roll angle as states. Eigenvalue plots with and without controller are presented. Since their contribution focusses on the controller development and performance the linear model and its validity over the complete flight envelope is not discussed in detail. In this study the dynamics of a tiltwing aircraft is linearized at different operating points and analyzed. The set of operating points is at a much closer grid than can be found in literature. Furthermore, the forward and vertical velocity are chosen as operating points, since the tilt angle alone is not sufficient when including vertical velocities. Additionally, the tilt angle is used as control input. The zero-input dynamic over the complete flight envelope as well as the corresponding transfer functions with and without controller are derived and analyzed. The linear models are validated against a 6-degree of freedom nonlinear aircraft simulation. In the following section tiltwing aircraft flight mechanics is described in general. In Section 3 the system dynamics is linearized and analyzed. In Section 4 the controller and controller linearization methods are presented. In Section 5 the open and closed loop system model is analyzed and discussed. Afterwards a conclusion is given. 2. Tiltwing Aircraft and General Assumptions In this section the general flight mechanics of tiltwing aircraft are described. Furthermore the challenges of the transition between aerodynamic flight and thrust-borne flight are illustrated in more detail. 2.1. Tiltwing Aircraft Flight Dynamics Aircraft in tiltwing configuration combine the advantages of helicopters, such as hovering and vertical take-off and landing capabilities, with the advantages of conventional fixed-wing aircraft by rotating the wing including the propulsion system around the lateral aircraft axis. In the past the institute of flight system dynamics of the RWTH Aachen University has developed different sized tiltwing aircraft in multiple projects (Ostermann et al. , 2012; Hartmann et al. 2016). The design of the tiltwing analyzed in this contribution, depicted in Fig. 1, is described in further detail in previous publications (Holsten et al. , 2011). Figure 1. The tiltwing demonstrator UAV close to hover flight. The tiltwing aircraft is able to perform trimmed flight over the entire flight envelope, including all velocities between aerodynamic and thrust born flight. The tilt angle σ decreases with increasing forward velocity. At low forward velocities the fuselage pitch is controlled to zero. In horizontal flight the control devices of the tiltwing aircraft resemble those of a conventional aircraft, including ailerons and elevator. Differential thrust is used for yaw control. In vertical flight the elevators are ineffective due to the missing dynamic pressure. Here pitch control is given through an auxiliary impeller, yaw control through the ailerons and roll control through differential thrust. For all tilt angles σ between zero and 90° the ailerons and differential thrust create a significant roll and yaw moment. Thus roll and yaw control within transition has to be balanced between differential thrust and aileron use. An overview on the control devices of a tiltwing is given in Fig. 2. Figure 2. Control devices of the tiltwing aircraft (Ostermann et al. , 2012). 2.2. Flight State Transition and Flight Envelope Trimmed flight at all forward velocities between aerodynamic and thrust-born flight requires continuous balancing of all forces and moments. Tilting the wing represents a significant change in configuration. It enables a large velocity range and is an additional control variable, but also influences the other’s control surface efficiencies. At large σ, flow separation occurs, although it is reduced by the propeller slipstream. During a slow continuous transition from hovering to aerodynamic horizontal flight σ is continuously reduced, while the aerodynamic lift increases with increasing forward velocity. A change in σ rotates the thrust vector and directly influences the angle of attack at the wing and with it the magnitude of lift and drag of the wing. The aerodynamic forces at the wing and the thrust do not act in the center of gravity and thus result in additional pitching moments. Regarding longitudinal motion at one forward and vertical velocity multiple trimmed operating points exist. This underdetermination is resolved by setting the pitch angle to zero and only using σ to maintain trimmed flight. Further insight into the unterdetermination and possible control solution is given by Hartmann et al. (2016). Releasing the pitch angle might bring benefits in controllability and robustness, since it allows to reduce the auxiliary thrust, but also makes the control system more complex. To further analyze the tiltwing aircraft dynamics at different operating points the nonlinear system is linearized. The linearization and analysis is described in the following sections. 3. Analysis of Tiltwing System Dynamics The analysis of the system dynamics of a linear model in state space representation is much simpler than the analysis of nonlinear systems. Keeping in mind the simplifications made, conclusions from the system dynamics of the linear model to