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

Authors: Zhiwei Kong; Qiang Lu
Journal: Journal of Robotics
 (2018)

---

## Abstract

This paper concentrates on the flight control of a novel tiltrotor aircraft with fixed wings. This kind of aircraft has two flight modes and a transition mode. In the phase of vertical take off and landing (VTOL), the aircraft can operate as a quadrotor helicopter. And in the phase of horizontal flight, the aircraft is in the normal airplane mode. The transition mode is between these two flight modes. In this part of work, a novel tiltrotor aircraft was presented since transition mode is achieved by tilting the front dual tiltrotor (DTR) and the mathematical model was established. The classical PID method was used during the phase of VTOL and the numerical results were given and the simulation shows good control effect. A nonlinear control law based on backstepping was proposed to achieve a stable transition from vertical flight to horizontal flight. And the numerical results show that the flight mode could transit stably which shows the effectiveness of the control approach. Finally, the vertical flight experiment has been carried out on DTR aircraft and the attitude was stable.


---

## Body

## 1. Introduction

Tiltrotor 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 Figures1, 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.Figure 1
Tilting angle.Figure 2
Modal switching control strategy.Figure 3
The top view of DTR.Figure 4
The inertial coordinate and body coordinate.


## 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 Figure4, Cw(xw,yw,zw) is the inertial coordinate that the origin of the coordinate is any point in space and xw axis points north, yw axis points east, and the zw axis points vertically downwards to the center of the earth. Cb(xb,yb,zb) is the body coordinate that the origin of the coordinate is the center of the aircraft mass. xb points to the aircraft head, the zb axis is perpendicular to the xb axis and points to the earth, and the yb 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 ψ,θ,ϕ represent yaw, pitch, and roll, respectively. The transformation matrix Γbw from body coordinate to inertial coordinate is represented as(1)Γbw=cψcθsϕsθcψ-cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ-sϕcψ-sθsϕcθcϕcθwhere cθ and sθ represent cos⁡θ and sin⁡θ, respectively.The aircraft is mainly subjected to the pull of four rotorsTi(i=1,2,3,4), gravity mg, wings lift TL, and air drag TD. The rotor tilting angle is expressed by α, and then the pull along the xb axis can be expressed as(2)Tpx=T1+T2sin⁡αwhere Ti=kωi2 and ωi is the rotors’ angular velocity. And the pull along the zb axis can be expressed as(3)Tpz=T1+T2cos⁡α+T3+T4The speed along thexb 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 TL and the air drag TD can be calculated as follows [23]:(4)TL=12ερAVxz2(5)TD=p1Vxz2+p2Vxzwhere ε is the lift coefficient, ρ is air density, A represents the area of the wing, and p1 and p2 are drag coefficient. Vxz2 represents the vector sum of xb axis and zb axis velocity, i.e.,(6)Vxz=vbx2+vbz2SupposeL(±Lx,±Ly,±Lz) represents the coordinates of the rotor in the body frame; then the resultant force moment of the rotors Mt=[Mx,My,Mz]T can be expressed as(7)Mt=T1sin⁡α0-T1cos⁡α×Lx-Ly-Lz+T2sin⁡α0-T2cos⁡α×LxLy-Lz+00-T3×-Lx-Ly-Lz+00-T4×-LxLy-Lz=-T1cos⁡α+T2cos⁡α-T3+T4Ly-T1+T2Lxcos⁡α-Lzsin⁡α+T3+T4Lx-T1+T2Lysin⁡αAssuming 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(8)Mld=-TD0-TL×0-Ly0+-TD0-TL×0Ly0=0Due 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 T1=T2 and T3=T4 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

Ifν(p,q,r) represents the aircraft roll, pitch, and yaw rates in body coordinates, γ(Ix,Iy,Iz) represents the rotary inertia of body coordinates axis. From the Newton-Euler equation, the relationship of angular acceleration and moment can be expressed as(9)Ixp˙+Iz-Iyqr=ΣMxIyq˙+Ix-Izrp=ΣMyIzr˙+Iy-Ixpq=ΣMzsubstituting (7) into (9),(10)p˙=Iy-IzIxqr-Lycos⁡αIx-T1+T2+LyIx-T3+T4q˙=Iz-IxIyrp-Lxcos⁡α-Lzsin⁡αIyT1+T2+LxIyT3+T4r˙=Ix-IyIzpq+Lysin⁡αIz-T1+T2When the Euler angle is relatively small, it can be considered approximately as p=ϕ˙,  q=θ˙, and  r=ψ˙.According to Newton’s second lawF→=mv˙→ and transforming the resultant force from body coordinate to inertial coordinate, the dynamic model of the aircraft can be expressed as(11)mx¨=cψcθTpx-sϕsψ+cψsθcϕTpz+2TL-cψcθTDmy¨=cθsψTpx-cϕsθsψ-sϕcψTpz+2TL-cθsψTDmz¨=-sθTpx-cϕcθTpz+2TL+mg+sθTDϕ¨=c1θ˙ψ˙-c2cos⁡α-T1+T2+c3-T3+T4θ¨=c4ψ˙ϕ˙-c5cos⁡α-c6sin⁡αT1+T2+c7T3+T4ψ¨=c8ϕ˙θ˙+c9sin⁡α-T1+T2where c1=Iy-Iz/Ix,c2=c3=Ly/Ix,c4=Iz-Ix/Iy,c5=c7=Lx/Iy,c6=Lz/Iy,c8=Ix-Iy/Iz,c9=Ly/Iz.


## 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 Figure4, Cw(xw,yw,zw) is the inertial coordinate that the origin of the coordinate is any point in space and xw axis points north, yw axis points east, and the zw axis points vertically downwards to the center of the earth. Cb(xb,yb,zb) is the body coordinate that the origin of the coordinate is the center of the aircraft mass. xb points to the aircraft head, the zb axis is perpendicular to the xb axis and points to the earth, and the yb 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 ψ,θ,ϕ represent yaw, pitch, and roll, respectively. The transformation matrix Γbw from body coordinate to inertial coordinate is represented as(1)Γbw=cψcθsϕsθcψ-cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ-sϕcψ-sθsϕcθcϕcθwhere cθ and sθ represent cos⁡θ and sin⁡θ, respectively.The aircraft is mainly subjected to the pull of four rotorsTi(i=1,2,3,4), gravity mg, wings lift TL, and air drag TD. The rotor tilting angle is expressed by α, and then the pull along the xb axis can be expressed as(2)Tpx=T1+T2sin⁡αwhere Ti=kωi2 and ωi is the rotors’ angular velocity. And the pull along the zb axis can be expressed as(3)Tpz=T1+T2cos⁡α+T3+T4The speed along thexb 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 TL and the air drag TD can be calculated as follows [23]:(4)TL=12ερAVxz2(5)TD=p1Vxz2+p2Vxzwhere ε is the lift coefficient, ρ is air density, A represents the area of the wing, and p1 and p2 are drag coefficient. Vxz2 represents the vector sum of xb axis and zb axis velocity, i.e.,(6)Vxz=vbx2+vbz2SupposeL(±Lx,±Ly,±Lz) represents the coordinates of the rotor in the body frame; then the resultant force moment of the rotors Mt=[Mx,My,Mz]T can be expressed as(7)Mt=T1sin⁡α0-T1cos⁡α×Lx-Ly-Lz+T2sin⁡α0-T2cos⁡α×LxLy-Lz+00-T3×-Lx-Ly-Lz+00-T4×-LxLy-Lz=-T1cos⁡α+T2cos⁡α-T3+T4Ly-T1+T2Lxcos⁡α-Lzsin⁡α+T3+T4Lx-T1+T2Lysin⁡αAssuming 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(8)Mld=-TD0-TL×0-Ly0+-TD0-TL×0Ly0=0Due 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 T1=T2 and T3=T4 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

Ifν(p,q,r) represents the aircraft roll, pitch, and yaw rates in body coordinates, γ(Ix,Iy,Iz) represents the rotary inertia of body coordinates axis. From the Newton-Euler equation, the relationship of angular acceleration and moment can be expressed as(9)Ixp˙+Iz-Iyqr=ΣMxIyq˙+Ix-Izrp=ΣMyIzr˙+Iy-Ixpq=ΣMzsubstituting (7) into (9),(10)p˙=Iy-IzIxqr-Lycos⁡αIx-T1+T2+LyIx-T3+T4q˙=Iz-IxIyrp-Lxcos⁡α-Lzsin⁡αIyT1+T2+LxIyT3+T4r˙=Ix-IyIzpq+Lysin⁡αIz-T1+T2When the Euler angle is relatively small, it can be considered approximately as p=ϕ˙,  q=θ˙, and  r=ψ˙.According to Newton’s second lawF→=mv˙→ and transforming the resultant force from body coordinate to inertial coordinate, the dynamic model of the aircraft can be expressed as(11)mx¨=cψcθTpx-sϕsψ+cψsθcϕTpz+2TL-cψcθTDmy¨=cθsψTpx-cϕsθsψ-sϕcψTpz+2TL-cθsψTDmz¨=-sθTpx-cϕcθTpz+2TL+mg+sθTDϕ¨=c1θ˙ψ˙-c2cos⁡α-T1+T2+c3-T3+T4θ¨=c4ψ˙ϕ˙-c5cos⁡α-c6sin⁡αT1+T2+c7T3+T4ψ¨=c8ϕ˙θ˙+c9sin⁡α-T1+T2where c1=Iy-Iz/Ix,c2=c3=Ly/Ix,c4=Iz-Ix/Iy,c5=c7=Lx/Iy,c6=Lz/Iy,c8=Ix-Iy/Iz,c9=Ly/Iz.


## 3. Vertical Takeoff Control Approach

In the phase of vertical takeoff, the classic quadrotor PID control method was used. The tilting angleα=0 when the DTR takes off at a low speed. Therefore, air drag TD and wings lift TL will be ignored. The aircraft makes a small angle movement in the hovering state, and then the dynamic model can be expressed as(12)mx¨=-sϕsψ+cψsθcϕT1+T2+T3+T4my¨=-cϕsθsψ-sϕcψT1+T2+T3+T4mz¨=-cϕcθT1+T2+T3+T4+mgϕ¨=c1θ˙ψ˙+c2T1-T2-T3+T4θ¨=c4ψ˙ϕ˙+c5-T1-T2+T3+T4ψ¨=c8ϕ˙θ˙+c9-T1+T2-T3+T4The 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(13)x¨=sϕsψ+cψsθcϕu1y¨=cϕsθsψ-sϕcψu1z¨=cϕcθu1+gϕ¨=c1θ˙ψ˙+u2θ¨=c4ψ˙ϕ˙+u3ψ¨=c8ϕ˙θ˙+u4The trajectory of the aircraft (x,y,z) and the yaw ϕ were tracked only since this is an underactuated system. Furthermore, to track the desire attitude θd,ϕd, replace the virtual inputs u1 in (13) as follows:(14)ux1=sϕsψ+cψsθcϕu1uy1=cϕsθsψ-sϕcψu1uz1=cϕcθu1And the control law is designed as(15)ux1=-rxpx-rxi∫xdt-rxdx˙uy1=-rypy-ryi∫ydt-rydy˙uz1=-rzpze-rzi∫zedt-rzdz˙e-g+z¨dwhere ze=z-zd and  zd is the desire altitude. To satisfy the control law (15) the desired attitude angle θd,ϕd should be tracked. The attitude angle error is defined as(16)θe=θ-θd,θ˙e=θ˙-θ˙dϕe=ϕ-ϕd,ϕ˙e=ϕ˙-ϕ˙dwhere θd,ϕd can be calculated as(17)θd=arcsin⁡cψ2ux1+cψsψuy1uz1ϕd=arctan⁡cψsψux1-cψ2uy1uz1The attitude system control law is designed as(18)u2=-rp2ϕe-ri2∫ϕedt-rd2ϕ˙e+ϕ¨du3=-rp3θe-ri3∫θedt-rd3θ˙e+θ¨du4=-rp4ψe-ri4∫ψedt-rd4ψ˙e+ψ¨dThe numerical results are showed in Figures5, 6, 7, and 8. Figure 5 shows the aircraft’s takeoff trajectory from (3,3) to (0,0) and the target altitude zd=15. Figure 6 shows the control inputs changes over time. In Figure 7, θ and ϕ converge to 0 and ψ reaches the target value π/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.Figure 5
DTR takeoff trajectory.Figure 6
The control inputs of takeoff.Figure 7
DTR attitude tracking effect.Figure 8
DTR position tracking effect.


## 4. Transition Flight Control Approach

In the transition mode, rotor1 and rotor2 are always given the same inputT1=T2 as well as rotor3 and rotor4 T3=T4. This input allocation method has a greater influence on θ than ϕ and ψ. Moreover, define the input τ1=T1=T2 and τ2=T3=T4. Therefore, the dynamic model during transition can be expressed as(19)x¨=2τ1sαcψcθm-2sϕsψ+cψsθcϕmτ1cα+τ2+TL-cψcθmTDy¨=2τ1sαcθsψm-2cϕsθsψ-sϕcψmτ1cα+τ2+TL-cθsψmTDz¨=-2τ1sαsθm-2cϕcθmτ1cα+τ2+TL+g+sθmTDϕ¨=c1θ˙ψ˙θ¨=c4ψ˙ϕ˙-2τ1c5cα-c6sα+2c7τ2ψ¨=c8ϕ˙θ˙When the rotors tilting angle change from 0 toπ/2, the key point to design the control law is to keep the height of the aircraft from falling and the pitch angle θ should be close to 0. That means the resultant force on the zw axis should be in the negative direction, i.e.,(20)sθTpx+cϕcθTpz+2TL+sθTD≧mgTherefore, to realize the tilting process stably the main task under this input strategy is to control the following subsystem:(21)z¨=-2τ1sαsθm-2cϕcθmτ1cα+τ2+TL+g+sθmTDθ¨=c4ψ˙ϕ˙-2τ1c5cα-c6sα+2c7τ2In addition, at the equilibrium point of the systemsin⁡θ≈θ,cos⁡θ≈1. Then, the nonlinear system is partially linearized as follows:(22)z¨=-2τ1sαθm-2cϕmT+g+θmTDθ¨=c4ψ˙ϕ˙-2τ1K+2c7τ2where K=c5cα-c6sα,  T=τ1cα+τ2+TL.To stabilize the transition mode system (22), the backstepping approach is used to design the control laws τ1 and τ2. 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 θ¨ and z¨ could be separated to design the control law, respectively. If a=m(-z3-k3z˙3+g-k4z4)+TDθ-2cϕTL-cϕK/c7,  b=2sαθ+2cϕcα+2cϕK/c7. The control law can be given as τ1=a/b and  τ2=k1z˙1+z1+k2z2+2aK/b/2c7.Proof.
The process of proof will be divided into three parts. In the first part, the Lyapunov function will be constructed and the stability ofθ¨ subsystem will be proved. In the second part, the stability of the z¨ 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 ofθ¨ Subsystem.The pitch subsystem is designed as follows:(23)ω˙1=ω2ω˙2=c4ψ˙ϕ˙-2τ1K+2c7τ2where ω1=θ. The virtual input uθ is introduced which is equal to the pitch rate ω˙1. The error of pitch is defined as(24)z1=θd-ω1Construct the Lyapunov function,(25)Vz1=12z12The derivative of the Lyapunov function is(26)V˙z1=z1z˙1=-z1uθTo guarantee V˙(z1)<0, suppose that(27)uθ=k1z1where k1>0. The error of the pitch rate is defined as(28)z2=uθ-ω2The derivative of z2 is(29)z˙2=k1z˙1-ω˙2And z˙1 can be obtained from (24) and (28),(30)z˙1=-k1z1+z2Construct the Lyapunov function,(31)Vz1,z2=12z12+12z22And the derivative of V(z1,z2) is(32)V˙z1,z2=z1z˙1+z2z˙2=z1-k1z1+z2+z2k1z˙1-ω˙2To make V˙(z1,z2)<0, suppose that(33)k1z˙1-ω˙2=-k2z2-z1where k2>0. Thus,(34)V˙z1,z2=-k1z12-k2z22And V˙(z1,z2)<0 satisfies the condition for system stabilization.
Part II: We Will Prove the Stability ofz¨ Subsystem.The altitude subsystem is designed as(35)ξ˙1=ξ2ξ˙2=-2τ1sαθm-2cϕmT+g+θTDmwhere ξ1=z. The virtual input uz is introduced which is equal to the altitude rate z˙. The altitude error is defined as(36)z3=zd-ξ1Construct the Lyapunov function,(37)Vz3=12z32The derivative of the V(z3) is(38)V˙z3=z3z˙3=-z3uzSuppose that uz=k3z3. Thus,(39)V˙z3=-k3z32where k3>0. The altitude rate error is defined as(40)z4=uz-ξ2From (37) and (41) z3 can be obtained as(41)z3=-k3z3+z4Construct the Lyapunov function V(z3,z4) as(42)Vz3,z4=12z32+12z42The derivative of the V(z3,z4) is(43)V˙z3,z4=z3z˙3+z4z˙4=z3-k3z3+z4+z4k3z˙3-ξ˙2To make V˙(z3,z4)<0, suppose that(44)k3z˙3-ξ˙2=-k4z4-z3where k4>0. Thus,(45)V˙z3,z4=-k3z32-k4z42And V˙(z3,z4)<0 satisfies the condition for system stabilization.
Part III: The Control Inputs Will Be Given.The first equation of τ1 and τ2 can be obtained from (23) and (33),(46)k1z˙1-c4ψ˙ϕ˙+2τ1K-2c7τ2=-k2z2-z1
The second equation ofτ1 and τ2 can be obtained from (35) and (44),(47)k3z˙3+2sαθτ1m+2cϕTm-g-TDθm=-k4z4-z3
Thus, the control lawτ1 and τ2 can be calculated by (46) and (47),(48)τ1=abτ2=k1z˙1+z1+k2z2+2aK/b2c7This proof is completed.The numerical results for the DTR UAV have been showed in Figures10, 11, and 12. The Figure 9 represents the tilting angle signal source which changes from 0 to π/2 in 3 to 6 seconds. The system parameters are listed in Table 1.Table 1
Simulation parameters of the DTR vehicle.
Parameters Unit Value m kg 4.50 I x k g . m 2 0.039 I y k g . m 2 0.042 I z k g . m 2 0.076 L x m 0.32 L y m 0.30 L z m 0.11 ρ k g / m 3 1.29 A m 2 0.20 k 1 1 1.20 k 2 1 1.20 k 3 1 2.20 k 4 1 2.20Figure 9
Tilting angle signal source.Figure 10
Altitude control during transition mode.Figure 11
Attitude control during transition mode.Figure 12
DTR trajectory from takeoff to horizontal flight.In Figure10, 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, θ changes within 10 degrees and eventually converges to 0 from the beginning of the transition to the end. ϕ shows a slight changes and ψ 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 Figures13–16 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.Figure 13
The pitch and desired pitch in experiment.Figure 14
The roll and desired roll in experiment.Figure 15
The yaw and desired yaw in experiment.Figure 16
The flight height of the DTR aircraft.Figure 17
Overview of the developed DTR aircraft.Figure 18
The DTR aircraft vertical flight experiment.


## 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.



---

*Source: 8641731-2018-08-07.xml*