# Mathematical Modeling and Modal Switching Control of a Novel Tiltrotor UAV.

1. IntroductionTiltrotor aircraft attracted much researchers' attention in the field of aeronautics and astronautics since 1980s [1, 2]. The reason that tiltrotors have received such great attention is mainly because they combine the advantages of helicopters and fixed-wing aircraft [3, 4]. Helicopters can take off and land vertically, but they cannot fly horizontally at a high speed. Fixed-wing aircrafts can fly horizontally at a high speed, but they need a runway for takeoff and landing [4, 5]. Tiltrotor aircraft has the advantages of both helicopter and fixed-wing aircraft, which makes it have a wide application scene.

Bell Helicopter company developed the V22 Osprey tiltrotor aircraft in the 1990s, which became the world's first tiltrotor aircraft. Osprey immediately received widespread attention because of its excellent performance. At present, tiltrotor aircrafts have achieved great success in military and civil fields [6]. In military aspects, tiltrotor aircrafts can be used as carrier-based aircrafts and can complete the transportation task in harsh environment, such as deserts and islands. In civil aspects, they are often used in the tasks of pesticide spraying, power patrol inspection, terrain exploration, and disaster relief.

In previous studies on multimodal aircraft, many researchers have focused on quad tilt-wing (QTW) unmanned aerial vehicle (UAV) [3, 7-11]. This kind of aircraft transfers from quadrotor helicopter to airplane by turning wings. Oner [2] developed the QTW vehicle and the dynamic model is derived by using Newton-Euler formulation. And a linear quadratic regulator (LQR) controller was proposed to stabilize vertical flight. Benkhoud [3] utilized Model Predictive Control (MPC) approach to stabilize vehicle attitude and track its trajectory. And the simulation results showed the effectiveness of the approach. In engineering practice, the most widely used regulator control law is proportional, integral, and derivative control, referred to as PID control. Hancer [8] estimated the disturbance with a disturbance observer and utilized PID controller for robust hovering control. And the effectiveness of the controller was verified through experiments and simulations. Another kind of aircraft with similar performance called quad tiltrotor (QTR) UAV also attracted widespread attention [5, 6, 12-15]. This kind of aircrafts switches flight modal by turning rotors. In [12], a convertible QTR was designed and a nonlinear controller based on dynamic inversion was given. Moreover, the trajectory of the aircraft was tracked. Flores [4,16] have done some work on this subject. In [4], a QTR model with fixed wings was presented and a nonlineared controller was also given which is based on saturations and Lyapunov. And from the numerical simulations the control strategy obtained satisfactory results.

Some studies on the vertical takeoff and landing of quadrotor helicopter have achieved good results [1,17-21]. Sliding model controller was used in quadrotor helicopter altitude and attitude control [1, 22]. Xu [22] divided quadrotor helicopter model into actuated subsystem and underactuated subsystem. Furthermore, sliding mode control laws were designed for the two subsystems and the vehicle's altitude and attitude were stabilized. In [18], a feedback linearizing controller and backstepping-like control law was proposed to control vehicle's motion which is restricted to yaw and vertical motions.

In recent years, there have been a large number of studies on QTW and QTR. Most of them studied the horizontal and vertical flight of this type of aircraft. However, there are few studies on the transition process. Therefore, solving the problem of switching the flight modal stably for QTR or QTW has great practical significance.

In this paper, a novel tiltrotor aircraft model with dual tiltable rotors was proposed as you can see in Figures 1, 3, and 4. It can take off vertically with four rotors and after transition the vehicle can fly horizontally with the front dual rotors. And the back two rotors will stop when the transition mode is completed. The modal switching control strategy is showed in Figure 2. When the DTR is in the phase of VTOL, DTR's kinetics model is the same as the quadrotor helicopter and the attitude controller is the quadrotor helicopter attitude controller. After the DTR reaches the reference value, the decision-making module will switch the kinetics model and controller into transition modal at the same time. The remainder of this paper is arranged as follows. In Section 2, the DTR kinetics model will be presented. In Section 3, PID method was used to control the position and attitude of the vertical flight phase. In Section 4, a nonlinear controller will be designed to implement the transition. In Section 5, the takeoff experiment was carried out on DTR aircraft. In Section 6, we will give a conclusion about this part of work and point out deficiencies and future improvements.

2. The Mathematical Model of the Tiltrotor UAV

2.1. Coordinate System and Force Analysis. In order to establish the mathematical model of the aircraft and analyze the force, the inertial coordinate and body coordinate are introduced. As you can see in Figure 4, [C.sub.w]([x.sub.w], [y.sub.w], [z.sub.w]) is the inertial coordinate that the origin of the coordinate is any point in space and [x.sub.w] axis points north, [y.sub.w] axis points east, and the [z.sub.w] axis points vertically downwards to the center of the earth. [C.sub.b]([x.sub.b], [y.sub.b], [z.sub.b]) is the body coordinate that the origin of the coordinate is the center of the aircraft mass. [x.sub.b] points to the aircraft head, the [z.sub.b] axis is perpendicular to the [x.sub.b] axis and points to the earth, and the [y.sub.b] axis is determined by the right hand rule. The body coordinate is fixed relative to the fuselage and rotates as the aircraft rotates. The attitude angles [psi], [theta], [phi] represent [[GAMMA].sub.bw], pitch, and roll, respectively. The transformation matrix from body coordinate to inertial coordinate is represented as

[mathematical expression not reproducible] (1)

where [c.sub.[theta]] and [s.sub.[theta]] represent cos [theta] and sin [theta], respectively.

The aircraft is mainly subjected to the pull of four rotors [T.sub.i](i = 1,2,3,4), gravity mg, wings lift [T.sub.L], and air drag [T.sub.D]. The rotor tilting angle is expressed by [alpha], and then the pull along the [x.sub.b] axis can be expressed as

[T.sub.px] = ([T.sub.1] + [T.sub.2]) sin ([alpha]) (2)

where [T.sub.i] = k[[omega].sup.2.sub.i] and [[omega].sub.i] is the rotors' angular velocity. And the pull along the [z.sub.b] axis can be expressed as

[T.sub.pz] = ([T.sub.1] + [T.sub.2]) cos ([alpha]) + [T.sub.3] + [T.sub.4] (3)

The speed along the [x.sub.b] axis will keep increasing when the rotors are tilting. At this time, the lift of the wings and the air drag to the fuselage cannot be ignored. And the wings lift [T.sub.L] and the air drag [T.sub.D] can be calculated as follows [23]:

[T.sub.L] = [1/2] [epsilon] [rho] A[V.sup.2.sub.xz] (4)

[T.sub.D] = [p.sub.1] [V.sup.2.sub.xz] + [p.sub.2][V.sub.xz] (5)

where [epsilon] is the lift coefficient, [rho] is air density, A represents the area of the wing, and [p.sub.1] and [p.sub.2] are drag coefficient. [V.sup.2.sub.xz] represents the vector sum of [x.sub.b] axis and [z.sub.b] axis velocity, i.e.,

[V.sub.xz] = [square root of ([v.sup.2.sub.bx] + [v.sup.2.sub.bz])] (6)

Suppose L([+ or -][L.sub.x], [+ or -] [L.sub.y], [+ or -] [L.sub.z]) represents the coordinates of the rotor in the body frame; then the resultant force moment of the rotors [M.sub.t] = [[[M.sub.x], [M.sub.y], [M.sub.z]].sup.T] can be expressed as

[mathematical expression not reproducible]. (7)

Assuming that air drag and lift act on the geometric center of the two wings, then the moment of air drag and lift can be expressed as

[mathematical expression not reproducible] (8)

Due to the rotation of the rotors, the gyro effect and counteractive moment are produced. But when the tilting rotors have the same angular velocity as well as the fixed rotors, the gyro effect and counteractive moment will be eliminated. And in the following control strategy, the input [T.sub.1] = [T.sub.2] and [T.sub.3] = [T.sub.4] will be given. Therefore, the gyros effect and counteractive moment are not considered here. And gravity does not produce a moment due to it acts on the center of mass.

2.2. Kinetics Model. If v(p, q, r) represents the aircraft roll, pitch, and yaw rates in body coordinates, [gamma]([I.sub.x], [I.sub.y], [I.sub.z]) represents the rotary inertia of body coordinates axis. From the Newton-Euler equation, the relationship of angular acceleration and moment can be expressed as

[mathematical expression not reproducible] (9)

substituting (7) into (9),

[mathematical expression not reproducible] (10)

When the Euler angle is relatively small, it can be considered approximately as [mathematical expression not reproducible].

According to Newton's second law [mathematical expression not reproducible] and transforming the resultant force from body coordinate to inertial coordinate, the dynamic model of the aircraft can be expressed as

[mathematical expression not reproducible] (11)

where [mathematical expression not reproducible]

3. Vertical Takeoff Control Approach

In the phase of vertical takeoff, the classic quadrotor PID control method was used. The tilting angle [alpha] = 0 when the DTR takes off at a low speed. Therefore, air drag [T.sub.D] and wings lift [T.sub.L] will be ignored. The aircraft makes a small angle movement in the hovering state, and then the dynamic model can be expressed as

[mathematical expression not reproducible] (12)

The PID controller is a common feedback loop component in industrial control applications and consists of a proportional element P, an integral element I, and a derivative element D. The basis of PID control is proportional control, integral control can eliminate steady-state error but may increase overshoot, and derivative control can speed up large inertial system response speed and weaken overshoot trend.

The resultant force of the four rotors is used as the control input, and the dynamic model can be writen as

[mathematical expression not reproducible] (13)

The trajectory of the aircraft (x, y, z) and the yaw [phi] were tracked only since this is an underactuated system. Furthermore, to track the desire attitude [[theta].sub.d], [[phi].sub.d], replace the virtual inputs [u.sub.1] in (13) as follows:

[mathematical expression not reproducible] (14)

And the control law is designed as

[mathematical expression not reproducible] (15)

where [z.sub.e] = z - [z.sub.d] and [z.sub.d] is the desire altitude. To satisfy the control law (15) the desired attitude angle [[theta].sub.d], [[phi].sub.d] should be tracked. The attitude angle error is defined as

[mathematical expression not reproducible] (16)

where [[theta].sub.d], [[phi].sub.d] can be calculated as

[mathematical expression not reproducible] (17)

The attitude system control law is designed as

[mathematical expression not reproducible] (18)

The numerical results are showed in Figures 5, 6, 7, and 8. Figure 5 shows the aircraft's takeoff trajectory from (3,3) to (0,0) and the target altitude [z.sub.d] = 15. Figure 6 shows the control inputs changes over time. In Figure 7, [theta] and [phi] converge to 0 and [psi] reaches the target value [pi]/6. Figure 8 shows the position changes along the axis of the inertia coordinate. From the simulation results the PID control approach shows good control effect on attitude and position in vertical flight phase.

4. Transition Flight Control Approach

In the transition mode, rotor1 and rotor2 are always given the same input [T.sub.1] = [T.sub.2] as well as rotor3 and rotor4 [T.sub.3] = [T.sub.4]. This input allocation method has a greater influence on [theta] than [phi] and y. Moreover, define the input [[tau].sub.1] = [T.sub.1] = [T.sub.2] and [[tau].sub.2] = [T.sub.3] = [T.sub.4]. Therefore, the dynamic model during transition can be expressed as

[mathematical expression not reproducible] (19)

When the rotors tilting angle change from 0 to [pi]/2, the key point to design the control law is to keep the height of the aircraft from falling and the pitch angle [theta] should be close to 0. That means the resultant force on the [z.sub.w] axis should be in the negative direction, i.e.,

[mathematical expression not reproducible] (20)

Therefore, to realize the tilting process stably the main task under this input strategy is to control the following subsystem:

[mathematical expression not reproducible] (21)

In addition, at the equilibrium point of the system sin [theta] [approximately equal to] [theta], cos [theta] [approximately equal to] 1. Then, the nonlinear system is partially linearized as follows:

[mathematical expression not reproducible] (22)

where K = [c.sub.5][c.sub.[alpha]] - [c.sub.6] [s.sub.[alpha]] T = [[tau].sub.1][c.sub.[alpha]] + [[tau].sub.2] + [T.sub.L].

To stabilize the transition mode system (22), the backstepping approach is used to design the control laws [[tau].sub.1] and [[tau].sub.2]. The basic idea of the backstepping approach is to decompose a complex nonlinear system into several subsystems that do not exceed the system order. Then, design partial Lyapunov functions and intermediate virtual control for each subsystem until back to the entire system. Finally, they are integrated to complete the design of the entire control law.

Theorem 1. Consider that the transition mode system (22) is a multi-input and multioutput coupling system. The equations of [mathematical expression not reproducible] could be separated to design the control law, respectively. If [mathematical expression not reproducible]. The control law can be given as [mathematical expression not reproducible].

Proof. The process of proof will be divided into three parts. In the first part, the Lyapunov function will be constructed and the stability of [??] subsystem will be proved. In the second part, the stability of the [??] subsystem will be proved. In the third part, the control input of the stabilization system will be given.

Part I: We Will Prove the Stability of [??] Subsystem. The pitch subsystem is designed as follows:

[mathematical expression not reproducible] (23)

where [[omega].sub.1] = [theta]. The virtual input [u.sub.[theta]] is introduced which is equal to the pitch rate [[??].sub.1]. The error of pitch is defined as

[z.sub.1] = [[theta].sub.d] - [[omega].sub.1] (24)

Construct the Lyapunov function,

V([z.sub.1]) = [1/2] [z.sup.2.sub.1] (25)

The derivative of the Lyapunov function is

[mathematical expression not reproducible] (26)

To guarantee [??]([z.sub.1]) < 0, suppose that

[u.sub.[theta]] = [k.sub.1][z.sub.1] (27)

where [k.sub.1] > 0. The error of the pitch rate is defined as

[z.sub.2] = [u.sub.[theta]] - [[omega].sub.2] (28)

The derivative of [z.sub.2] is

[mathematical expression not reproducible] (29)

And [[??].sub.1] can be obtained from (24) and (28),

[[??].sub.1] = -[k.sub.1] [z.sub.1] + [z.sub.2] (30)

Construct the Lyapunov function,

V([z.sub.1], [z.sub.2]) = [1/2] [z.sup.2.sub.1] + [1/2] [z.sup.2.sub.2] (31)

And the derivative of V([z.sub.1], [z.sub.2]) is

[mathematical expression not reproducible] (32)

To make [??]([z.sub.1], [z.sub.2]) < 0, suppose that

[mathematical expression not reproducible] (33)

where [k.sub.2] > 0. Thus,

[??] ([z.sub.1], [z.sub.2]) = - [k.sub.1][z.sup.2.sub.1] - [k.sub.2] [z.sup.2.sub.2] (34)

And [??]([z.sub.1],[z.sub.2]) < 0 satisfies the condition for system stabilization.

Part II: We Will Prove the Stability of [??] Subsystem. The altitude subsystem is designed as

[mathematical expression not reproducible] (35)

where [[xi].sub.1] = z. The virtual input [u.sub.z] is introduced which is equal to the altitude rate [??]. The altitude error is defined as

[z.sub.3] = [z.sub.d] - [[xi].sub.1] (36)

Construct the Lyapunov function,

V ([z.sub.3]) = [1/2] [z.sup.2.sub.3] (37)

The derivative of the V([z.sub.3]) is

[mathematical expression not reproducible] (38)

Suppose that [u.sub.z] = [k.sub.3][z.sub.3]. Thus,

[??] ([z.sub.3]) = -[k.sub.3][z.sup.2.sub.3] (39)

where [k.sub.3] > 0. The altitude rate error is defined as

[z.sub.4] = [u.sub.z] - [[xi].sub.2] (40)

From (37) and (41) [z.sub.3] can be obtained as

[z.sub.3] = -[k.sub.3][z.sub.3] + [z.sub.4] (41)

Construct the Lyapunov function V([z.sub.3], [z.sub.4]) as

V([z.sub.3], [z.sub.4]) = [1/2] [z.sup.2.sub.3] + [1/2] [z.sup.2.sub.4] (42)

The derivative of the V([z.sub.3], [z.sub.4]) is

[mathematical expression not reproducible] (43)

To make [??]([z.sub.3],[z.sub.4]) < 0, suppose that

[mathematical expression not reproducible] (44)

where [k.sub.4] > 0. Thus,

[??]([z.sub.3],[z.sub.4]) = -[k.sub.3][z.sup.2.sub.3] -[k.sub.4][z.sup.2.sub.4] (45)

And [??]([z.sub.3], [z.sub.4]) < 0 satisfies the condition for system stabilization.

Part III: The Control Inputs Will Be Given. The first equation of [[tau].sub.1] and [[tau].sub.2] can be obtained from (23) and (33),

[mathematical expression not reproducible] (46)

The second equation of [[tau].sub.1] and [[tau].sub.2] can be obtained from (35) and (44),

[mathematical expression not reproducible] (47)

Thus, the control law [[tau].sub.1] and [[tau].sub.2] can be calculated by (46) and (47),

[[tau].sub.1] = a/b [[tau].sub.2] = [k.sub.1] [[??].sub.1] + [z.sub.1] + [k.sub.2] [z.sub.2] + 2aK/b/2[c.sub.7] (48)

This proof is completed.

The numerical results for the DTR UAV have been showed in Figures 10,11, and 12. The Figure 9 represents the tilting angle signal source which changes from 0 to [pi]/2 in 3 to 6 seconds. The system parameters are listed in Table 1.

In Figure 10, the DTR UAV takes off vertically until it reaches the target height d1 = 15m. The rotors start to tilt in the 3rd second and end in the 6th second and the height also reaches the reference height zd2 = 40m. After transition the vehicle flies horizontally at the reference height. In Figure 11, d changes within 10 degrees and eventually converges to 0 from the beginning of the transition to the end. 0 shows a slight changes and y slightly deviates from the reference value of the vertical takeoff. However, these tiny changes will not affect the balance of the aircraft. The Figure 12 shows the complete trajectory from takeoff to horizontal flight.

5. Experimental Results

The vertical flight experiment was performed on the DTR aircraft platform which is developed at laboratory. The fuselage framework is built by elastomeric polyurethane (EPU) and carbon fiber. The tilting angle is controlled by two actuators. In addition, the DTR aircraft is equipped with GPS, airspeed meter, and antenna for communication with the ground station software. The 10000mha high-rate lithium polymer battery is chosen and connected with electronic governor to supply the rotors. Moreover, the control system integrates 3-axis accelerometer and gyroscope to provide aircraft's attitude. The aircraft is equipped with the high precision digital air pressure sensor MS-5611 used to obtain height and connected with the compass through the I2C interface to calibrate the flight direction. The flight control system is based on PIXHAWK Autopilot. PIXHAWK is a dual-processor flight controller consisting of a 32 bit STM32F427 Cortex M4 core processor which contains 256KB RAM and 2 MB flash and a 32 bit STM32F103 coprocessor which is used to ensure safety when the core processor crashes.

The experiment was conducted in an outdoor environment without wind. The experiment results can be obtained by establishing a connection between the ground station software Mission Planner and the aircraft. The experiment results are showed in Figures 13-16 and Figure 18. Figures 13-16 show the DTR aircraft attitude and flight height. And the DTR aircraft is showed in Figure 17. Figure 18 shows the DTR aircraft's flight in the real experiment and the aircraft flight attitude was stable.

6. Conclusion

A novel tiltrotor aircraft has been presented with two tiltable rotors and two fixed rotors. The kinetics model of the DTR aircraft has been established. And control laws for vertical flight and transition are designed, respectively. The PID controller was designed for the vertical flight and a nonlinear controller based on backstepping approach was designed for the transition. The numerical results have shown the effectiveness of the control approaches for vertical flight and transition. Moreover, the vertical flight experiment was performed on the DTR aircraft platform and the flight attitude was stable. In this part of work, wind disturbance is not considered. Therefore, the robustness of the proposed approach needs to be further verified. The elevator and rudder are not used during the transition and the DTR aircraft attitude is completely controlled by rotors. In the next work, the disturbance will be considered into the kinetics model. The elevator and rudder will also be used to control the attitude of the DTR aircraft.

https://doi.org/10.1155/2018/8641731

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Zhejiang Provincial Natural Science Foundation of China under Grant LY18F030008 and the National Natural Science Foundation of China under Grant 61375104.

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Zhiwei Kong and Qiang Lu (iD)

School of Automation, Hangzhou Dianzi University, Hangzhou, China

Correspondence should be addressed to Qiang Lu; lvqiang@hdu.edu.cn

Received 17 May 2018; Accepted 26 July 2018; Published 7 August 2018

Academic Editor: L. Fortuna

Caption: Figure 1: Tilting angle.

Caption: Figure 2: Modal switching control strategy.

Caption: Figure 3: The top view of DTR.

Caption: Figure 4: The inertial coordinate and body coordinate.

Caption: Figure 5: DTR takeoff trajectory.

Caption: Figure 6: The control inputs of takeoff.

Caption: Figure 7: DTR attitude tracking effect.

Caption: Figure 8: DTR position tracking effect.

Caption: Figure 9: Tilting angle signal source.

Caption: Figure 10: Altitude control during transition mode.

Caption: Figure 11: Attitude control during transition mode.

Caption: Figure 12: DTR trajectory from takeoff to horizontal flight.

Caption: Figure 13: The pitch and desired pitch in experiment.

Caption: Figure 14: The roll and desired roll in experiment.

Caption: Figure 15: The yaw and desired yaw in experiment.

Caption: Figure 16: The flight height of the DTR aircraft.

Caption: Figure 17: Overview of the developed DTR aircraft.

Caption: Figure 18: The DTR aircraft vertical flight experiment.

Table 1: Simulation parameters of the DTR vehicle. Parameters Unit Value m kg, 4.50 [I.sub.x] kg.[m.sup.2] 0.039 [I.sub.y] kg.[m.sup.2] 0.042 [I.sub.z] kg.[m.sup.2] 0.076 [L.sub.x] m 0.32 [L.sub.y] m 0.30 [L.sub.z] m 0.11 [rho] kg/[m.sup.3] 1.29 A [m.sup.2] 0.20 [k.sub.1] 1 1.20 [k.sub.2] 1 1.20 [k.sub.3] 1 2.20 [k.sub.4] 1 2.20

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Title Annotation: | Research Article |
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Author: | Kong, Zhiwei; Lu, Qiang |

Publication: | Journal of Robotics |

Date: | Jan 1, 2018 |

Words: | 4743 |

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