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Longitudinal Flight Dynamics of a Single Tilt-wing Unmanned Aerial Vehicle
19th IFAC Symposium on Automatic Control in Aerospace September 2-6, 2013. Würzburg, Germany
Longitudinal Flight Dynamics of a Single Tilt-wing Unmanned Aerial Vehicle Sungtae Hong ∗ Junho Jeong ∗ Seungkeun Kim ∗ Jinyoung Suk ∗ Ji In Jung ∗∗ ∗
Department of Aerospace Engineering, Chungnam National University, Daejeon 305-764, Republic of Korea (e-mail: [email protected], junho, skim78, [email protected]) ∗∗ Agency for Defense Development, Daejeon, 305-152, Republic of Korea (e-mail: [email protected]) Abstract: A single tilt-wing UAV has a new-concept hybrid configuration for adopting the advantages of both fixed-wing and rotary-wing aircraft. This paper presents longitudinal flight characteristics of the single tilt-wing UAV whose design is based on a tapered wing type aircraft with an actuating motor and a propeller mounted at each wing. Momentum theory is used to calculate thrust, and nonlinear modeling is performed considering lift and drag generated by slip stream effect of the propellers. Force and moment variation is considered for tilting angle change. Also, static trim on longitudinal axis is analyzed for operation airspeeds from 0 to 30m/s, and numerical simulation shows the flight dynamic characteristics of the UAV. It is verified that the single tilt-wing UAV has stable dynamic characteristics at fixed-wing mode. Keywords: Single tilt wing, Modeling, Static trim, Dynamic characteristics. 1. INTRODUCTION
2. DYNAMIC MODELLING 2.1 Dynamic modelling of the Single Tilt-wing UAV
Aircrafts can be classified as fixed-wing and rotary-wing. Fixed-wing aircrafts take the advantage of high lift-to-drag ratio, fuel-efficient flying, and high-speed flying. However, long runway and airport facilities are required in general for takeoff and landing. Rotary-wing aircrafts can take-off and landing vertically so that they does not need runway. They can also hover in the air performing the mission in one fixed point for a long duration. However, the rotary wing aircrafts cannot enjoy the advantage that fixed-wing can: high-speed and fuelefficiency as well as reliability. Therefore, it is strongly required to develop an aircraft that take advantage of both fixed-wing and rotary-wing aircrafts. A variety of V/STOL(Vertical/Short Take Off and Landing) aircrafts are developed in the mid1960s. Tilt-rotor, Tilt-wing, Tilt-jet and Tilt-duct aircrafts are most popular example of complex aircrafts. Hiller X-18 aircraft was developed as part of the US effort for army/navy/air-force tri-service plan, to provide vertical/short take-off and landing aircrafts. There has been vary few studies in this complicated type UAVs. JAXA and Chiba University in Japan have studied the aerodynamics of QTW(Quad Tilt Wing) UAVs[Muraoka et al. (2009), Kubo et al. (2010), Suzuki et al. (2010) and Nonami et al. (2010)]. Especially, JAXA has conducted the wind tunnel test to varify aerodynamic charatristics of its own UAV. In addition, The RWTH Aachen university in Germany has conducted a study on the aerodynamics of the tilt wing UAV[J. Holsten and Moormann (2012) and Ostermann et al. (2012)]. Chungnam national university in South Korea also had an extensive research on the dynamics of QTW UAV[Jung et al. (2013)]. This paper presents the modeling and flight dynamics of such a complex single tilt-wing UAV that has both vertical take-off and landing capability and flying just like fixed-wing aircraft. 978-3-902823-46-5/2013 © IFAC
Figure 1 shows the operational flight envelope for CTOL (Conventional Take Off and Landing), VTOL(Vertical Take Off and Landing) and tilt-wing aircrafts. The tilt-wing aircrafts have wide flight envelope from hover to high-speed and high altitude range.
Fig. 1. Flight envelope Assuming constant mass and mass distribution during flight, forces and moments due to aerodynamic/thrust/gravity are considered to derive the equation of motion for the developed single tilt-wing UAV in the body-fixed coordinate frame[Nelson (1989)]. 60
10.3182/20130902-5-DE-2040.00081
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
Thrust and induced velocity can be derived using vertical velocity and aerodynamic coefficient of the propeller[Johnson and Turbe (2006)]. vb = vz + 23 wr r 34 Ktwist τ = 14 (vb − vi ) wr r2 ρ∞ a0 bcr .
(3)
Where vz , wr , r, Ktwist , vi , ρ∞ , a0 , b and cr denote vertical velocity to the propeller disk, angular speed of the propeller, radius and twist of the propeller, induced velocity, lift coefficient, blade number of the propeller, and the chord of the propeller, respectively. Induced velocity can be derived as vi =
τ 2ρ∞ πr 2 v 0 .
(4)
Fig. 2. Coordinate frame of the single tilt-wing UAV Where v 0 denotes the far-field velocity and can be given by U˙ = V R − W Q + (Fx /m)
v0 =
V˙ = W P − U R + (Fy /m) ˙ = U Q − V P + (Fz /m) W P˙ = (Iyy − Izz ) QR/Ixx + L/Ixx
(1)
f (vi ) = vi −
Forces and moments acting on the aircraft can be expressed as the total sum of individual terms as
P
Mi = MAwet + MAunwet + MAf reestream + MTL + MTR + MTT + Mother .
(5)
where
R˙ = (Ixx − Iyy ) P Q/Izz + N/Izz .
Fi = FAwet + FAunwet + FAf reestream + FTL + FTR + FTT + FG + Fother
2
vx 2 + vy 2 + (vz − vi ) .
Newton-Raphson method is used to obtain induced velocity for (4) and (5). f ( vi ) vij+1 = vij − df vj (6) dvi ( ij )
Q˙ = (Izz − Ixx ) P R/Iyy + M/Iyy
P
q
0.25(vb −vi )wr r 2 ρ∞ a0 bcr 2ρ∞ πr 2
√
vx 2 +vy 2 (vz −vi )
.
(7)
And the thrust on the rotors can be obtained by T = 2ρ∞ πr2 vi v 0 .
(8)
(2) 2.2 Slipstream effect
where Awet , Aunwet and Af reestream denote aerodynamic terms for wetted area due to the slipstream, unwetted area and free stream area. TL , TR and TT mean thrust force from left, right and tail propellers. G and other are gravity and other terms, respectively. Thrust generated by each prop can be calculated using momentum theory. Figure 3 shows the airflow through the propeller disk.
When the wing tilts, the propeller and the wing move together. Aerodynamic forces are produced by the wetted area in the slipstream as shown in Fig.4.
Fig. 4. Slipstream effect Lift coefficient in Fig. 4 can be expressed as (9), where the slope of the lift coefficient are set to be the conventional
Fig. 3. Flow around a rotor 61
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
value(0.1/deg). denotes the incidence angle of the propulsion system, which is set to be 2 degree in this configuration. Lift and drag of the wetted area can be given by (10). CLwet = CLα (αmotor,i )
(9)
Lwet = CLwet q S, Dwet = CDwet q S
(10)
Figure 5 shows the area of the wing wetted and unwetted by the slipstream.
Fig. 6. Unwetted area
Fig. 5. Wetted area, unwetted area and free stream area Lift and drag wetted in downwash can be obtained from lift coefficient using (10). Although part of wing wetted in downwash is not smooth, area of wing wetted in downwash from tilting point, Sw is calculated with Fig. 5. The downwash of propeller is wetted to trailing edge because of tilt wing shape. Moreover, the effect of induced drag is low because of low downwash due to vortex, which is characteristic of finite wing. Thus, only parasite drag is considered except low induced drag. q indicates dynamic pressure written as (11) where vi is the downwash velocity of propeller. q = 12 ρvi 2
Fig. 7. High angle of attack flight wind tunnel test (JAXA)[Kubo et al. (2010)]
(11)
Aerodynamic force based on slipstream is affected by induced velocity and does not vary with tilt angle. 2.3 Unwetted area There are parts of a wing which are not affected by the wake of the propeller, but by the freestream. Aerodynamics of these unwetted area shown in Fig. 6 is considered in this paper. Unlike the normal aerodynamics of the aircraft with the angle of attack below 20 degree, high angle of attack aerodynamics should be considered dealing with both vertical mode and transition mode for this UAV. A single tilt-wing UAV is inherently a stall-less aircraft, and it has still much higher lift with the corresponding drag at the very high angle of attack region. Figure 7 shows the wind-tunnel test result of the single tilt-wing UAV. From this figure, we can see that the lift does not vanish at the high angle of attack. Based on this test result and X-foil(an aerodynamic analysis tool), lift and drag coefficients are estimated in this paper, which is shown in Fig. 8.
Fig. 8. Assumption of CL and CD Final lift and drag can be obtained from unwetted area and aerodynamic coefficient with Fig. 8. Lift and drag are expressed as in (12). Luw = (Cl,uw qS)uw , Duw = (Cd,uw qS)uw .
(12)
Wetted area and unwetted area mainly act as drag in VTOL mode, and lift in CTOL mode.
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2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
2.4 Total force and moment
FX = −mg sin θ + FAXwet + FAXunwet + FAXf reestream + FT X FY = mg sin φ cos θ + FAYwet + FAYunwet + FAYf reestream + FY,dynamic FZ = mg cos θ cos φ + FAZwet + FAZunwet + FAZf reestream + FT Z + FZ,dynamic L = LAwet + LAunwet + LAf reestream + LTM R + LDynamic M = MAwet + MAunwet + MAf reestream + MTM R + MTT R + MDynamic N = NAwet + NAunwet + NAf reestream + NTM R + NDynamic .
Thrust of motor and aerodynamic force of downwash are changed by direction of each force due to tilting angle shown in Fig. 9.
(16)
Fig. 9. Schematic of total force and moment configuration Aerodynamic and thrust forces acting on the UAV can be expressed as FAXwet = −(LL + LR )wet sin ξ − (DL + DR )wet cos ξ FAXunwet = −(DL + DR )unwet FAXf reestream = −(DL + DR )f reestream FT X = (TL + TR ) cos ξ FAYwet = 0, FAYunwet = 0 FAYf reestream = Ff use,side , FT Y = 0 FAZwet = −(LL + LR )wet cos ξ + (DL + DR )wet sin ξ FAZunwet = −(LL + LR )unwet FAZf reestream = −(LL + LR )f reestream FT Z = − (TL + TR ) sin ξ − TT + DT .
3. TRIM ANALYSIS AND DYNAMIC SIMULATION Static trim conditions are obtained from the nonlinear equation of motion to verify a variety of longitudinal flight characteristics. RPM, tilting angle, elevator angle, and longitudinal state variables are used to derive trim condition. Jacobian-based gradient method is used to obtain trim values. Trim conditions are calculated about flight velocity from static to 30m/s, and trim parameter control inputs are selected in RPM, tilting angle, and elevator displacement.
(13) T
R = [ R1 R2 R3 ] = [ FX − FX0 FZ − FZ0 M − M0 ] .
(18)
The Jacobian for state vector X is represented by Jij =
∂Ri ∂xj , J
=
∂R1 ∂R1 ∂R1 ∂x1 ∂x2 ∂x3 ∂R2 ∂R2 ∂R2 ∂x1 ∂x2 ∂x3 ∂R3 ∂R3 ∂R3 ∂x1 ∂x2 ∂x3
.
(19)
Using the Jacobian matrix, an increment of the state variable can be obtained. (14) −1
∆X = −k[J]
R −1
Xnew = X + −k[J]
R
(20)
Finally, the new state vector is updated by addition of increment to the current state vector. The state vector consisting of final trim parameters is calculated through iteration of this procedure until the residual reaches the preset threshold value.
Yawing moment is generated by the difference between the left and right thrusts and aerodynamic forces. NAwet = (LR − LL )wet lywet sin ξ + (DR − DL )wet lywet cos ξ NAunwet = (DR − DL )unwet lyunwet NAf reestream = (DR − DL )f reestream lyf reestream NT = (TL − TR ) lyL cos ξ.
(17)
3-residual vectors related to state vector X are set as
Rolling moment is calculated by using aerodynamic forces and difference of thrust occurred in left side and right side of main wing. MAwet = (LL + LR )wet lxwet cos ξ − (DL + DR )wet lxwet sin ξ + (LL + LR )wet lzwet sin ξ + (DL + DR )wet lzwet cos ξ MAunwet = (LL + LR )unwet lxunwet + (DL + DR )unwet lzunwet MAf reestream = (LL + LR )f reestream lxf reestream + (DL + DR )f reestream lzf reestream MT = (TL + TR ) lxL sin ξ − (TL + TR ) lzL cos ξ − (TT − DT ) lxT .
T
X = [ x1 x2 x3 ] = [ RP M ξ δE ] .
Figure 10 shows the trim result for the airspeed ranging from hover to 30m/s. As the airspeed increases, the aircraft shows smooth transition from hover to fixed-wing flying mode. RPM also has smooth change from 6000 to below 5000. The pitching moment can be vanished by applying equivalent elevator angle within 10 degree.
(15)
A nonlinear simulation for 50 seconds is performed to verify the flight dynamics and stability of the single tilt-wing UAV.
Forces and moments of 6-DOF are summarized as 63
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
7000 0.1 u(m/s)
6000
0.05
RPM
5000 0 4000 0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25 time(sec)
30
35
40
45
50
3000 0.1 2000 0
10
15
20
25
30
0 w(m/s)
5
80
−0.1 −0.2
tilt angle(deg)
−0.3 60 −0.4 0 40 4 20
5
10
15
20
25
q(deg/s)
2 0 0
30
0
2
−2
0
−4 0
δele(deg)
−2 1
−4 −6
0.5 θ(deg)
−8 −10
0 −0.5
−12 0
5
10
15 time(sec)
20
25
30 −1 0
Fig. 10. Longitudinal trim condition
Fig. 12. Control response at CTOL mode
Figures 11 and 12 shows the elevator control input and corresponding longitudinal response for fixed-wing mode at the trimmed airspeed: V∞ = 30m/s. A 1 degree step input is applied at 1 second, and the resulting pitch response shows both well-damped short period motion and smooth phugoid motion. It can be seen that the longitudinal motion of the fixed-wing mode for the single tilt-wing UAV is similar to conventional aircraft, and the stability can be guaranteed.
Where, x = [u, w, q, θ]T . The flight characteristic is analyzed by obtain of damping ratio and frequency from system matrix, which is summarized in Table 1 for fixed-wing mode. Figure 13 shows poles corresponding to the system dynamics. Table 1. Longitudinal dynamics for fixed-wing mode Mode Short period Phugoid
δele(deg)
1
Eigen value −2.6766 ± 10.7711i −0.0626 ± 0.4481i
0.5
Frequency 11.1 0.452
Pole−Zero Map 15
0 5
10
15
20
25 time(sec)
30
35
40
45
10
50
0.115
0.085
0.056
0.036
0.26
14 0.016 12 10 6
Fig. 11. Control input at CTOL mode Linearized equation is more efficient for flight characteristic analysis and controller design than the nonlinear equation. In this paper, a numerical linearization is used for fixed-wing mode. A longitudinal motion of linearized equation is analysed, and the state variable are defined as F (x) = ∂F ∂x ∆x + x˙ = Ax + Bu.
0.17
8
Imaginary Axis
0
Damping 2.41 0.138
∂F ∂δE ∆δE
5 0.5
4 2
0 2 4
−5 0.5
6 8 −10
10
0.26 0.17
−15 −3
(21)
−2.5
0.115 −2
−1.5 Real Axis
0.056
0.085 −1
Fig. 13. Pole locations for fixed-wing mode 64
0.036 −0.5
12 0.016 14 0
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
4. CONCLUSIONS This paper presents the dynamic modelling of the single tiltwing UAV. Dynamic equation of motion is derived using momentum theory. Thrust is calculated using induced velocity by iterative Newton-Rapson method. The wing is divided by wetted, unwetted and free stream area to calculate the effect of aerodynamic force. A Jacobian matrix is used to calculate the trim condition at each airspeed. The trim plot shows smooth transition from hover to airspeed of 30m/s. A linearized model is obtained by numerical linearization, and flight characteristic is analyzed by eigenvalue analysis. Both numerical simulation and eigenvalue analysis show that the designed single tilt-wing UAV has stable dynamic characteristics at the fixed-wing mode. A more refined dynamic model will be obtained and the dynamic analysis will be conducted for the whole flight region including hover and transition modes. REFERENCES J. Holsten, T.O. and Moormann, D. (2012). Model validation of a tiltwing uav in transition phase applying windtunnel investigations and flight tests. Proceedings of ICAS 2012. Johnson, E.N. and Turbe, M.A. (2006). Modeling, control, and flight testing of a small-ducted fan aircraft. Journal of guidance, control, and dynamics, 29(4), 769–779. Jung, J., Hong, S., Kim, S., and Suk, J. (2013). A study on longitudinal flight dynamics of a qtw uav. Journal of the Korean Society for Aeronautical and Space Sciences, 41(1), 31–39. Kubo, D., Muraoka, K., and Okada, N. (2010). High angle of attack flight characteristics of a wing-in-propeller-slipstream aircraft. Proceedings of ICAS2010, ICAS2010-6.8, 3. Muraoka, K., Okada, N., and Kubo, D. (2009). Quad tilt wing vtol uav: Aerodynamic characteristics and prototype flight test. Proc., AIAA Unmanned Unlimited Conf., American Institute of Aeronautics and Astronautics, Seattle. Nelson, R.C. (1989). Flight stability and automatic control. McGraw-Hill New York. Nonami, K., Kendoul, F., Suzuki, S., Wang, W., and Nakazawa, D. (2010). Autonomous Flying Robots: Unmanned Aerial Vehicles and Micro Aerial Vehicles. Springer Publishing Company, Incorporated. Ostermann, T., Holsten, J., Dobrev, Y., and Moormann, D. (2012). Control concept of a tiltwing uav during low speed manoeuvring. Proceedings of ICAS 2012. Suzuki, S., Zhijia, R., Horita, Y., Nonami, K., Kimura, G., Bando, T., Hirabayashi, D., Furuya, M., and Yasuda, K. (2010). Attitude control of quad rotors qtw-uav with tilt wing mechanism. Journal of System Design and Dynamics, 4(3), 416–428.
65
Longitudinal Flight Dynamics of a Single Tilt-wing Unmanned Aerial Vehicle Sungtae Hong ∗ Junho Jeong ∗ Seungkeun Kim ∗ Jinyoung Suk ∗ Ji In Jung ∗∗ ∗
Department of Aerospace Engineering, Chungnam National University, Daejeon 305-764, Republic of Korea (e-mail: [email protected], junho, skim78, [email protected]) ∗∗ Agency for Defense Development, Daejeon, 305-152, Republic of Korea (e-mail: [email protected]) Abstract: A single tilt-wing UAV has a new-concept hybrid configuration for adopting the advantages of both fixed-wing and rotary-wing aircraft. This paper presents longitudinal flight characteristics of the single tilt-wing UAV whose design is based on a tapered wing type aircraft with an actuating motor and a propeller mounted at each wing. Momentum theory is used to calculate thrust, and nonlinear modeling is performed considering lift and drag generated by slip stream effect of the propellers. Force and moment variation is considered for tilting angle change. Also, static trim on longitudinal axis is analyzed for operation airspeeds from 0 to 30m/s, and numerical simulation shows the flight dynamic characteristics of the UAV. It is verified that the single tilt-wing UAV has stable dynamic characteristics at fixed-wing mode. Keywords: Single tilt wing, Modeling, Static trim, Dynamic characteristics. 1. INTRODUCTION
2. DYNAMIC MODELLING 2.1 Dynamic modelling of the Single Tilt-wing UAV
Aircrafts can be classified as fixed-wing and rotary-wing. Fixed-wing aircrafts take the advantage of high lift-to-drag ratio, fuel-efficient flying, and high-speed flying. However, long runway and airport facilities are required in general for takeoff and landing. Rotary-wing aircrafts can take-off and landing vertically so that they does not need runway. They can also hover in the air performing the mission in one fixed point for a long duration. However, the rotary wing aircrafts cannot enjoy the advantage that fixed-wing can: high-speed and fuelefficiency as well as reliability. Therefore, it is strongly required to develop an aircraft that take advantage of both fixed-wing and rotary-wing aircrafts. A variety of V/STOL(Vertical/Short Take Off and Landing) aircrafts are developed in the mid1960s. Tilt-rotor, Tilt-wing, Tilt-jet and Tilt-duct aircrafts are most popular example of complex aircrafts. Hiller X-18 aircraft was developed as part of the US effort for army/navy/air-force tri-service plan, to provide vertical/short take-off and landing aircrafts. There has been vary few studies in this complicated type UAVs. JAXA and Chiba University in Japan have studied the aerodynamics of QTW(Quad Tilt Wing) UAVs[Muraoka et al. (2009), Kubo et al. (2010), Suzuki et al. (2010) and Nonami et al. (2010)]. Especially, JAXA has conducted the wind tunnel test to varify aerodynamic charatristics of its own UAV. In addition, The RWTH Aachen university in Germany has conducted a study on the aerodynamics of the tilt wing UAV[J. Holsten and Moormann (2012) and Ostermann et al. (2012)]. Chungnam national university in South Korea also had an extensive research on the dynamics of QTW UAV[Jung et al. (2013)]. This paper presents the modeling and flight dynamics of such a complex single tilt-wing UAV that has both vertical take-off and landing capability and flying just like fixed-wing aircraft. 978-3-902823-46-5/2013 © IFAC
Figure 1 shows the operational flight envelope for CTOL (Conventional Take Off and Landing), VTOL(Vertical Take Off and Landing) and tilt-wing aircrafts. The tilt-wing aircrafts have wide flight envelope from hover to high-speed and high altitude range.
Fig. 1. Flight envelope Assuming constant mass and mass distribution during flight, forces and moments due to aerodynamic/thrust/gravity are considered to derive the equation of motion for the developed single tilt-wing UAV in the body-fixed coordinate frame[Nelson (1989)]. 60
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2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
Thrust and induced velocity can be derived using vertical velocity and aerodynamic coefficient of the propeller[Johnson and Turbe (2006)]. vb = vz + 23 wr r 34 Ktwist τ = 14 (vb − vi ) wr r2 ρ∞ a0 bcr .
(3)
Where vz , wr , r, Ktwist , vi , ρ∞ , a0 , b and cr denote vertical velocity to the propeller disk, angular speed of the propeller, radius and twist of the propeller, induced velocity, lift coefficient, blade number of the propeller, and the chord of the propeller, respectively. Induced velocity can be derived as vi =
τ 2ρ∞ πr 2 v 0 .
(4)
Fig. 2. Coordinate frame of the single tilt-wing UAV Where v 0 denotes the far-field velocity and can be given by U˙ = V R − W Q + (Fx /m)
v0 =
V˙ = W P − U R + (Fy /m) ˙ = U Q − V P + (Fz /m) W P˙ = (Iyy − Izz ) QR/Ixx + L/Ixx
(1)
f (vi ) = vi −
Forces and moments acting on the aircraft can be expressed as the total sum of individual terms as
P
Mi = MAwet + MAunwet + MAf reestream + MTL + MTR + MTT + Mother .
(5)
where
R˙ = (Ixx − Iyy ) P Q/Izz + N/Izz .
Fi = FAwet + FAunwet + FAf reestream + FTL + FTR + FTT + FG + Fother
2
vx 2 + vy 2 + (vz − vi ) .
Newton-Raphson method is used to obtain induced velocity for (4) and (5). f ( vi ) vij+1 = vij − df vj (6) dvi ( ij )
Q˙ = (Izz − Ixx ) P R/Iyy + M/Iyy
P
q
0.25(vb −vi )wr r 2 ρ∞ a0 bcr 2ρ∞ πr 2
√
vx 2 +vy 2 (vz −vi )
.
(7)
And the thrust on the rotors can be obtained by T = 2ρ∞ πr2 vi v 0 .
(8)
(2) 2.2 Slipstream effect
where Awet , Aunwet and Af reestream denote aerodynamic terms for wetted area due to the slipstream, unwetted area and free stream area. TL , TR and TT mean thrust force from left, right and tail propellers. G and other are gravity and other terms, respectively. Thrust generated by each prop can be calculated using momentum theory. Figure 3 shows the airflow through the propeller disk.
When the wing tilts, the propeller and the wing move together. Aerodynamic forces are produced by the wetted area in the slipstream as shown in Fig.4.
Fig. 4. Slipstream effect Lift coefficient in Fig. 4 can be expressed as (9), where the slope of the lift coefficient are set to be the conventional
Fig. 3. Flow around a rotor 61
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
value(0.1/deg). denotes the incidence angle of the propulsion system, which is set to be 2 degree in this configuration. Lift and drag of the wetted area can be given by (10). CLwet = CLα (αmotor,i )
(9)
Lwet = CLwet q S, Dwet = CDwet q S
(10)
Figure 5 shows the area of the wing wetted and unwetted by the slipstream.
Fig. 6. Unwetted area
Fig. 5. Wetted area, unwetted area and free stream area Lift and drag wetted in downwash can be obtained from lift coefficient using (10). Although part of wing wetted in downwash is not smooth, area of wing wetted in downwash from tilting point, Sw is calculated with Fig. 5. The downwash of propeller is wetted to trailing edge because of tilt wing shape. Moreover, the effect of induced drag is low because of low downwash due to vortex, which is characteristic of finite wing. Thus, only parasite drag is considered except low induced drag. q indicates dynamic pressure written as (11) where vi is the downwash velocity of propeller. q = 12 ρvi 2
Fig. 7. High angle of attack flight wind tunnel test (JAXA)[Kubo et al. (2010)]
(11)
Aerodynamic force based on slipstream is affected by induced velocity and does not vary with tilt angle. 2.3 Unwetted area There are parts of a wing which are not affected by the wake of the propeller, but by the freestream. Aerodynamics of these unwetted area shown in Fig. 6 is considered in this paper. Unlike the normal aerodynamics of the aircraft with the angle of attack below 20 degree, high angle of attack aerodynamics should be considered dealing with both vertical mode and transition mode for this UAV. A single tilt-wing UAV is inherently a stall-less aircraft, and it has still much higher lift with the corresponding drag at the very high angle of attack region. Figure 7 shows the wind-tunnel test result of the single tilt-wing UAV. From this figure, we can see that the lift does not vanish at the high angle of attack. Based on this test result and X-foil(an aerodynamic analysis tool), lift and drag coefficients are estimated in this paper, which is shown in Fig. 8.
Fig. 8. Assumption of CL and CD Final lift and drag can be obtained from unwetted area and aerodynamic coefficient with Fig. 8. Lift and drag are expressed as in (12). Luw = (Cl,uw qS)uw , Duw = (Cd,uw qS)uw .
(12)
Wetted area and unwetted area mainly act as drag in VTOL mode, and lift in CTOL mode.
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2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
2.4 Total force and moment
FX = −mg sin θ + FAXwet + FAXunwet + FAXf reestream + FT X FY = mg sin φ cos θ + FAYwet + FAYunwet + FAYf reestream + FY,dynamic FZ = mg cos θ cos φ + FAZwet + FAZunwet + FAZf reestream + FT Z + FZ,dynamic L = LAwet + LAunwet + LAf reestream + LTM R + LDynamic M = MAwet + MAunwet + MAf reestream + MTM R + MTT R + MDynamic N = NAwet + NAunwet + NAf reestream + NTM R + NDynamic .
Thrust of motor and aerodynamic force of downwash are changed by direction of each force due to tilting angle shown in Fig. 9.
(16)
Fig. 9. Schematic of total force and moment configuration Aerodynamic and thrust forces acting on the UAV can be expressed as FAXwet = −(LL + LR )wet sin ξ − (DL + DR )wet cos ξ FAXunwet = −(DL + DR )unwet FAXf reestream = −(DL + DR )f reestream FT X = (TL + TR ) cos ξ FAYwet = 0, FAYunwet = 0 FAYf reestream = Ff use,side , FT Y = 0 FAZwet = −(LL + LR )wet cos ξ + (DL + DR )wet sin ξ FAZunwet = −(LL + LR )unwet FAZf reestream = −(LL + LR )f reestream FT Z = − (TL + TR ) sin ξ − TT + DT .
3. TRIM ANALYSIS AND DYNAMIC SIMULATION Static trim conditions are obtained from the nonlinear equation of motion to verify a variety of longitudinal flight characteristics. RPM, tilting angle, elevator angle, and longitudinal state variables are used to derive trim condition. Jacobian-based gradient method is used to obtain trim values. Trim conditions are calculated about flight velocity from static to 30m/s, and trim parameter control inputs are selected in RPM, tilting angle, and elevator displacement.
(13) T
R = [ R1 R2 R3 ] = [ FX − FX0 FZ − FZ0 M − M0 ] .
(18)
The Jacobian for state vector X is represented by Jij =
∂Ri ∂xj , J
=
∂R1 ∂R1 ∂R1 ∂x1 ∂x2 ∂x3 ∂R2 ∂R2 ∂R2 ∂x1 ∂x2 ∂x3 ∂R3 ∂R3 ∂R3 ∂x1 ∂x2 ∂x3
.
(19)
Using the Jacobian matrix, an increment of the state variable can be obtained. (14) −1
∆X = −k[J]
R −1
Xnew = X + −k[J]
R
(20)
Finally, the new state vector is updated by addition of increment to the current state vector. The state vector consisting of final trim parameters is calculated through iteration of this procedure until the residual reaches the preset threshold value.
Yawing moment is generated by the difference between the left and right thrusts and aerodynamic forces. NAwet = (LR − LL )wet lywet sin ξ + (DR − DL )wet lywet cos ξ NAunwet = (DR − DL )unwet lyunwet NAf reestream = (DR − DL )f reestream lyf reestream NT = (TL − TR ) lyL cos ξ.
(17)
3-residual vectors related to state vector X are set as
Rolling moment is calculated by using aerodynamic forces and difference of thrust occurred in left side and right side of main wing. MAwet = (LL + LR )wet lxwet cos ξ − (DL + DR )wet lxwet sin ξ + (LL + LR )wet lzwet sin ξ + (DL + DR )wet lzwet cos ξ MAunwet = (LL + LR )unwet lxunwet + (DL + DR )unwet lzunwet MAf reestream = (LL + LR )f reestream lxf reestream + (DL + DR )f reestream lzf reestream MT = (TL + TR ) lxL sin ξ − (TL + TR ) lzL cos ξ − (TT − DT ) lxT .
T
X = [ x1 x2 x3 ] = [ RP M ξ δE ] .
Figure 10 shows the trim result for the airspeed ranging from hover to 30m/s. As the airspeed increases, the aircraft shows smooth transition from hover to fixed-wing flying mode. RPM also has smooth change from 6000 to below 5000. The pitching moment can be vanished by applying equivalent elevator angle within 10 degree.
(15)
A nonlinear simulation for 50 seconds is performed to verify the flight dynamics and stability of the single tilt-wing UAV.
Forces and moments of 6-DOF are summarized as 63
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
7000 0.1 u(m/s)
6000
0.05
RPM
5000 0 4000 0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25 time(sec)
30
35
40
45
50
3000 0.1 2000 0
10
15
20
25
30
0 w(m/s)
5
80
−0.1 −0.2
tilt angle(deg)
−0.3 60 −0.4 0 40 4 20
5
10
15
20
25
q(deg/s)
2 0 0
30
0
2
−2
0
−4 0
δele(deg)
−2 1
−4 −6
0.5 θ(deg)
−8 −10
0 −0.5
−12 0
5
10
15 time(sec)
20
25
30 −1 0
Fig. 10. Longitudinal trim condition
Fig. 12. Control response at CTOL mode
Figures 11 and 12 shows the elevator control input and corresponding longitudinal response for fixed-wing mode at the trimmed airspeed: V∞ = 30m/s. A 1 degree step input is applied at 1 second, and the resulting pitch response shows both well-damped short period motion and smooth phugoid motion. It can be seen that the longitudinal motion of the fixed-wing mode for the single tilt-wing UAV is similar to conventional aircraft, and the stability can be guaranteed.
Where, x = [u, w, q, θ]T . The flight characteristic is analyzed by obtain of damping ratio and frequency from system matrix, which is summarized in Table 1 for fixed-wing mode. Figure 13 shows poles corresponding to the system dynamics. Table 1. Longitudinal dynamics for fixed-wing mode Mode Short period Phugoid
δele(deg)
1
Eigen value −2.6766 ± 10.7711i −0.0626 ± 0.4481i
0.5
Frequency 11.1 0.452
Pole−Zero Map 15
0 5
10
15
20
25 time(sec)
30
35
40
45
10
50
0.115
0.085
0.056
0.036
0.26
14 0.016 12 10 6
Fig. 11. Control input at CTOL mode Linearized equation is more efficient for flight characteristic analysis and controller design than the nonlinear equation. In this paper, a numerical linearization is used for fixed-wing mode. A longitudinal motion of linearized equation is analysed, and the state variable are defined as F (x) = ∂F ∂x ∆x + x˙ = Ax + Bu.
0.17
8
Imaginary Axis
0
Damping 2.41 0.138
∂F ∂δE ∆δE
5 0.5
4 2
0 2 4
−5 0.5
6 8 −10
10
0.26 0.17
−15 −3
(21)
−2.5
0.115 −2
−1.5 Real Axis
0.056
0.085 −1
Fig. 13. Pole locations for fixed-wing mode 64
0.036 −0.5
12 0.016 14 0
2013 IFAC ACA September 2-6, 2013. Würzburg, Germany
4. CONCLUSIONS This paper presents the dynamic modelling of the single tiltwing UAV. Dynamic equation of motion is derived using momentum theory. Thrust is calculated using induced velocity by iterative Newton-Rapson method. The wing is divided by wetted, unwetted and free stream area to calculate the effect of aerodynamic force. A Jacobian matrix is used to calculate the trim condition at each airspeed. The trim plot shows smooth transition from hover to airspeed of 30m/s. A linearized model is obtained by numerical linearization, and flight characteristic is analyzed by eigenvalue analysis. Both numerical simulation and eigenvalue analysis show that the designed single tilt-wing UAV has stable dynamic characteristics at the fixed-wing mode. A more refined dynamic model will be obtained and the dynamic analysis will be conducted for the whole flight region including hover and transition modes. REFERENCES J. Holsten, T.O. and Moormann, D. (2012). Model validation of a tiltwing uav in transition phase applying windtunnel investigations and flight tests. Proceedings of ICAS 2012. Johnson, E.N. and Turbe, M.A. (2006). Modeling, control, and flight testing of a small-ducted fan aircraft. Journal of guidance, control, and dynamics, 29(4), 769–779. Jung, J., Hong, S., Kim, S., and Suk, J. (2013). A study on longitudinal flight dynamics of a qtw uav. Journal of the Korean Society for Aeronautical and Space Sciences, 41(1), 31–39. Kubo, D., Muraoka, K., and Okada, N. (2010). High angle of attack flight characteristics of a wing-in-propeller-slipstream aircraft. Proceedings of ICAS2010, ICAS2010-6.8, 3. Muraoka, K., Okada, N., and Kubo, D. (2009). Quad tilt wing vtol uav: Aerodynamic characteristics and prototype flight test. Proc., AIAA Unmanned Unlimited Conf., American Institute of Aeronautics and Astronautics, Seattle. Nelson, R.C. (1989). Flight stability and automatic control. McGraw-Hill New York. Nonami, K., Kendoul, F., Suzuki, S., Wang, W., and Nakazawa, D. (2010). Autonomous Flying Robots: Unmanned Aerial Vehicles and Micro Aerial Vehicles. Springer Publishing Company, Incorporated. Ostermann, T., Holsten, J., Dobrev, Y., and Moormann, D. (2012). Control concept of a tiltwing uav during low speed manoeuvring. Proceedings of ICAS 2012. Suzuki, S., Zhijia, R., Horita, Y., Nonami, K., Kimura, G., Bando, T., Hirabayashi, D., Furuya, M., and Yasuda, K. (2010). Attitude control of quad rotors qtw-uav with tilt wing mechanism. Journal of System Design and Dynamics, 4(3), 416–428.
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