Development of the mathematical model for the tilt-rotor aircraft

In this paper, main challenges of the urban air mobility concept are discussed and comparison of the different aircraft types, capable of performing vertical take-off and landing for compliance to the urban aerial mobility concept is provided. The mathematical model for the tilt-rotor aircraft is derived by the Newton-Euler method. This model takes into account all the forces and moments applied to the aircraft. Aerodynamic characteristics for the developed aircraft were obtained using virtual aerodynamic blowing method. Simulation results of the tilt-rotor aircraft motion on the various flight stages, including vertical take-off and landing, horizontal flight and transition stages are presented.


Introduction
In recent years urban aerial vehicle applications have become trending worldwide. In this light, National Aeronautics and Space Administration (NASA, USA) has developed the Urban Air Mobility (UAM) concept -the system for air passenger and cargo transportation within an urban area, inclusive of small package delivery and other urban unmanned aircraft system services [1]. This concept has been approved by many countries and a huge number of research and developments have been already made to support it. In order to bring UAM to our everyday lives, a number of challenges must be solved [2,3].
The biggest challenge is safety. The new aerial mobility systems should contain very high safety threshold to ensure that the air transportations is the safest method of transportation there is.
The second challenge is collaboration. To bring these ideas and technologies to life would require the whole aviation industry working together to make this a success.
Administration challenges include airspace management and certification. Airspace diversity is increasing every year and maintaining it while keeping all air traffic moving safely and efficiently will be challenging on its own, but to do so while certifying an entirely new type of aircraft vehicles using a regulatory framework developed for traditional fixed-wing aircraft will be the significant challenge.
Other challenges related directly to aircraft development are autonomy and accessibility. Autonomy is the ability of the aircraft control system to control a vehicle and to respond to various unexpected events and hazards. Today there are two different approaches to achieve a fully autonomous aircraft. First one, fall-back pilot is a safety pilot system that is always online and is ready to take control of the vehicle at any time for any reason. The second one -full autonomy from the beginning of the development process. Using the fall-back pilot approach enables the aircraft to become airborne more quickly (decreases the development time), but it requires significant investment in systems that will become obsolete in the near future. Development of the fully autonomous aircraft eliminates the need for human-machine interfaces, but obtaining public acceptance and certification for this kind of vehicle can be a challenge. Accessibility challenge refers to the ability of the users (individual passengers and transportation companies) to access aerial vehicle framework. In most cases, this requires the vehicle to be able to perform the vertical take-off and landing (VTOL) within the city area.
In the department 305 (automated complexes of orientation and navigation systems) of the Moscow Aviation Institute (MAI) in collaboration with Experiment Automatization Department, the research is being carried to develop the unmanned aerial vehicle (UAV), capable of transporting passengers within the city area. During the research, we analyzed different types of vehicles capable of performing VTOL.
The first candidates were wingless multirotor vehicles [4]. The key advantages were simple vehicle structure and low efforts required to develop a control system. However, the disadvantages were severe: limited flight speeds, limited energy efficiency in cruise flight and no gliding capabilities (that also results in lesser energy efficiency and safety penalty).
The second candidates were fixed-wing lift & cruise vehicles [5]. These vehicles have medium energy efficiency since there is no need for all the rotors to be working in the same time, the control system is relatively easy to develop and gliding possibility allows to meet additional UAM requirements.
On the other hand, extra weight from carrying temporarily unused engines results in ineffective use of propulsion. And additional rotors lead to increased drag in horizontal flight (if not retractable).
The third candidate was tilt-rotor aircraft [6]. Tilt-rotor structure brings in good energy efficiency with effective use of propulsion systems and gliding possibility, good cargo lift parameters due to no extra weight for carrying temporarily unused engines. However the hardware structure provides a certain challenge to develop the rotating engine's mechanism, and flight control system is relatively hard to develop. Summarizing these results, tilt-rotor aircraft structure with four rotating engines (figure 1) was chosen since it is the most suited for the UAM applications with VTOL capability, huge cargo capacity and high energy efficiency. The purpose of this research is to develop the mathematical model for the described tilt-rotor aircraft. The development of this model is an essential requirement for conducting further research on the autonomous control systems for tilt-rotor aircraft.

Mathematical model
The definitions of the main aircraft wing-body reference framed and corresponding conversion matrices describing rotation from one frame to another are well known [7]. In this paper, there are two coordinate O x y z is attached to the centre of gravity of the aircraft. The origin point g O of the inertial earth-fixed frame g g g g O x y z is located at aircraft's starting point. Axis g g O x aligns with the north direction and axis g g O z -with the east. Described reference coordinate frames are shown in figure 2. According to the principle of coordinate transfer [8], the rotation matrix describing the rotation from the earth-fixed frame to the body-fixed frame is given by cos cos sin sin cos cos sin cos sin sin cos cos sin sin cos cos sin cos sin sin cos cos sin sin sin sin cos cos son where , ,    are the yaw, pitch and roll angles of the aircraft. The nonlinear dynamic model of the tilt-rotor aircraft motion can be derived in the form of a Newton-Euler formulation [9]. For the body-fixed frame it can be expressed as follows: ; , where m is the mass of the aircraft, represent the total force and moment vectors in the body-fixed frame, is the vector of angular rotations in the body-fixed frame, and b I is the inertia tensor of the rigid body and can be expressed as The forces and moments in (2) are contributed by gravity, aerodynamics and the propulsion system. The can be expressed as where g is the gravity acceleration constant.
Aerodynamic forces in (4) are presented as follows [10]: where , , sin cos 0 P P P P P P P P where 1 2 3 4 , , , P P P P are the thrust force generated by each of the aircraft's engines,  is the tilting angle of the engines.
Aerodynamic moments in (4) are presented as follows [10]: where , , x y z m m m are aerodynamic parameters of the aircraft described in section 3 of this paper, z l is the wingspan of the aircraft, x l is the aircraft's length. The moment vector generated by the propulsion systems can be expressed as l P P P P l P P P P k P P P P l P P l P P l P P l P P where xe l is the distance from b b O x axis to the point of application of the thrust force shown in figure 3a, r k is the reactive torque coefficient, 1  Taking into consideration all forces and moments described in (4)- (9), the full mathematical model for the tilt-rotor aircraft motion in the body-fixed frame can be derived: To transform these equations to the earth-fixed frame aircraft's orientation angles must be found from translation and rotation kinematic equations derived according to the transformation relationship between coordinate systems.
Given (11), the velocity of the aircraft in the earth-fixed frame can be expressed as The coordinates and orientation angles of the tilt-rotor aircraft can be found by integrating expressions (11)- (12). In conclusion, the six degrees of freedom nonlinear dynamic model of the tiltrotor aircraft can be described by the equations (10)-(12).

Aerodynamics research
To determine aerodynamic coefficients of the tilt-rotor aircraft we used flow 5 software pack [11], as it provides various in-depth methods for aerodynamic analysis, such as Lifting-Line Theory, Vortex Lattice Method, and Uniform Density Triangular Panel method which we used in this research. Advantage of using this software is the ability to quickly and conveniently obtain results for the analysis of the developed preliminary designs of wings, fuselages and other aerodynamic elements of aircraft.
In order to analyze aerodynamic effects, rear wing profile design, shown in figure 4 was developed [12].   When designing the surface of the UAV fuselage we used the ability of flow5 software to import a surface from a CAD file as shown in figure 6.
The results of simulation results for designed profiles allowed us to obtain aerodynamic coefficients as functions of the current attack angle and aileron deflection angle. In this research, we assumed that the slippage angle equals zero, and the ailerons of both wings (front and rear) are deflected symmetrically. We also didn't take into account the aerodynamic interference from UAV's motors.
For the simulation purposes attack angle  and slippage angle  can be defined as follows:  figure 7. These functions will be used in the simulation to determine aerodynamic forces and moment affecting the aircraft. The real aerodynamic parameters may differ, however. This will require further research in the aerodynamic tube once the prototype of the tilt-rotor aircraft is developed.  Figure 8 shows that velocity of the aircraft in the earth-fixed frame changes according to the motion stage. During vertical take-off yg V increases and slowly falls down as UAV is approaching target safe attitude. During the transition from helicopter mode to plane mode, horizontal velocity xg V increases to provide sufficient lift force for the UAV to stay in the air. During horizontal flight stage, vertical velocity is essentially zero and the velocity vector only has projections on axis g g O x and axis g g O z , once a coordinated turn is made. During the transition to the helicopter mode, all velocities come to a halt as the aircraft performs aerodynamic deceleration. During vertical landing yg V has negative values and slowly decreases as UAV is approaching the landing pad.
In figure 9 the horizontal trajectory of the aircraft is presented. Transition stages are marked with black circles. Simulation time took 420 seconds and UAV translocation during this time equals to 9450 meters with an average velocity of 22.5 m/s.
The simulation results proved that the developed mathematical model of the tilt-rotor aircraft is adequate on all flight stages and in both flight modes. The analysis shows that it is possible to maintain stable flight during the transition process from one flight mode to another and the development of the corresponding control system is subject to subsequent research. Derived mathematical model correlates with the results of similar research for slightly different UAV designs [6,7], but the proposed method for accounting for aerodynamic forces appears to be more accurate as it is based on full-scale aerodynamic analysis for the given tilt-rotor aircraft design.

Conclusion
This paper described the main challenges of the UAM concept and provides a comparison of the different aircraft types to be used as an autonomous transportation vehicle, capable of performing VTOL. The development process of the mathematical model of the tilt-rotor aircraft is presented along with the final results -differential motion equations. Aerodynamic research for the developed aircraft model was carried out with flow5 software and approximate functions for aerodynamic coefficients were obtained. Simulation results of the tilt-rotor aircraft motion on every flight stage proved the adequacy of the developed mathematical model.