Modeling and Control of a Convertible VTOL Aircraft
Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006
WeA02.6
Modelling and Control of a Convertible VTOL Aircraft J. Escareño, S. Salazar and R. Lozano.
Abstract— The aim of this paper is to present the complete model of an unmanned convertible aerial vehicle in hover mode. This vehicle is capable of performing either in hover or in forward flight. A nonlinear control strategy is presented to stabilize the aircraft in hovering mode. An embedded low-cost pilot is described as well the experimental results of a hover flight using a prototype built in the laboratory. Index Terms— Aircraft control algorithm, VTOL, Nonlinear control, amplitude bounded input, Embbeded architecture.
Hover flight
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Forward flight
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Vertical take off
Vertical landing
I. INTRODUCTION Unmanned aerial vehicles (UAVs) are used in a wide variety of missions, like surveillance and reconnaissance, performed in different scenarios or environments. Therefore the task nature requires an appropriate flight profile vehicle in order to deal satisfactorily with the scenario’s drawbacks. For example, bridge inspection looking for structure’s cracks requires a vehicle with vertical take-off, hovering and lowspeed flight capabilities, then rotary-wing aircraft fits perfectly to that type of task. On the other hand, missions that require long-distance flights and relatively high forward velocity like forest fire detection or desert reconnaissance, usually use fixed-wing vehicles. However missions which involve the two scenarios described above require a vehicle which combines rotorcraft and fixed wing aircraft capabilities. We developed a convertible plane prototype capable of performing vertical take-off, hover and forward flight [see figure 1]. This type of convertible planes require specific control algorithms for handling take-off, hover, forward flight as well as the transition between those modes of operation. The aerodynamic configuration of the prototype we built is similar to the Boeing commercial aircraft Heliwing. There exist very few publications concerning the control of convertible planes. One of the few contributions in this area is given in [7] where H. Stone presented the control of the T-wing aircraft, which is a twin-engine tailsitter UAV that uses an LQR algorithm applied to the linearized hover dynamics. The main contribution of this paper is to provide a complete dynamical model of the convertible plane in hover mode. The nonlinear dynamical model is derived via the NewtonEuler formulation including the modelling of the gyroscopic effect and the adverse drag torque. In view of the physical limitations of the actuators, we propose a nonlinear control
Fig. 1.
algorithm satisfying the constraints on the inputs amplitude. We developed a specific embedded microcontroller architecture platform to be able to implement the control algorithm on-board. We present real-time experiments of the convertible plane in autonomous hover. The paper is organized as follows: Section II presents the complete dynamical model of the convertible VTOL. A control algorithm convergence analysis is presented in section III. Stabilization of the attitude and position of the aircraft is shown in simulation in section IV. The embedded architecture composed of the microcontroller, the sensors and filters is described in section V. The real-time application results of the stabilization of the VTOL in hover is presented in section VI. Finally some concluding remarks are given in section VII. II. DYNAMIC MODEL In this section we present the complete model of the convertible VTOL using a Newton-Euler formulation. The pitch, roll and yaw torques required for controlling the flying vehicle in hover are obtained from the speed difference between the two rotors and the control surfaces (aileron, elevon). The altitude of the vehicle is regulated by increasing or decreasing the propeller thrust. The roll torque is obtained from the difference of the rotors’ angular velocities. Since the control surfaces are submerged in the propeller slipstream (prop-wash), the aerodynamic forces are generated with the elevon and ailerons deflection to provide the pitch and yaw motion respectively [see figure 2]. Let I={iIx , jyI , kzI } denote the right hand inertial frame. B B Let B={iB x , jy , kz } denote the rigid-body frame with origin at the gravity center. Let the vector q = (ξ, η)T denote the generalized coordinates where ξ = (x, y, z)T ∈ 3 denotes the translation coordinates relative to the frame I, and η = (ψ, θ, φ)T ∈ 3 describes the vehicle orientation
Heudiasyc-UTC UMR 6599 Centre de Recherches de Royallieu B.P. 20529 60205 Compiegne France Tel.: + 33 (0)3 44 23 44 23 ; fax: +33 (0)3 44 23 44 77
Corresponding author [email protected] [email protected] [email protected]
1-4244-0171-2/06/$20.00 ©2006 IEEE.
UAV flightpath
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Aileron1 yB
L a1
δa xB la
Aileron2 L a2 zI
yI
xI Inertial frame
Fig. 2.
2) Moment of inertia: Matrix I represents the body moment of inertia whose value basically depends on the body mass distribution (for detail see [2]) and is decomposed as ⎞ ⎛ Ixx Ixy Ixz (5) I = ⎝Iyx Iyy Iyz ⎠ Izx Izy Izz
Elevator δe
Convertible VTOL
where
expressed in the classical yaw, pitch and roll angles (Euler angles) commonly used in aerodynamic applications [1]. The orientation of the convertible UAV is given by the rotation matrix RB→I . ⎞ ⎛ cθ cψ sφ sθ cψ − cφ sψ cφ sθ cψ + sφ sψ RB→I = ⎝ cθ sψ sφ sθ sψ + cφ cψ cφ sθ sψ − sφ cψ ⎠ −sθ sφ cθ cφ cθ
Ixx = m (y 2 + z 2 )dm, Ixy = Iyx = m −(xy)dm Iyy = m (x2 + z 2 )dm, Ixz = Izx = m −(xz)dm Izz = m (x2 + y 2 )dm, Iyz = Izy = m −(yz)dm
Notice that the aircraft is considered as a flat sheet aligned with the y − z plane [see figure 2]. Hence the application of the previous expressions to the convertible VTOL leads to a diagonal matrix ⎞ ⎛ 0 Ixx 0 (6) I = ⎝ 0 Iyy 0 ⎠ 0 0 Izz
where sβ := sin(β), cβ := cos(β). The rigid body motion equations (relative to B) subject to external force F B ∈ 3 and torque Γ ∈ 3 applied to the aircraft’s center of mass are given by the following expressions = =
F B + mGB Γc + Γg + Γq
A. Translational motion (1) (2)
1) Body frame: The equation (1) describes the translational motion and may be rewritten as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 mu˙ p mu fx ⎝ mv˙ ⎠+⎝q ⎠×⎝ mv ⎠ = ⎝fy ⎠+RI→B ⎝ 0 ⎠ (7) fz −mg mw˙ r mw
where m ¯ = diag(m) ∈ 3×3 , m ∈ denotes the vehicle mass, Ω is the angular velocity of the body frame, GB ∈ 3 is the gravity vector in B frame, v B = (u, v, w)T is the translational velocity of the aircraft’s center of mass, I ∈ 3×3 represents the tensor matrix, F B ∈ 3 is the total force acting on the aircraft including the weight, the rotors thrust, and the aerodynamic forces provided by the control surfaces, Γc ∈ 3 denotes the torque provided by the aircraft actuators, Γg denotes the gyroscopic torque and Γq is the drag torque generated by the propeller. The dynamic analysis requires the knowledge of the angular velocity Ω = (p, q, r)T and the inertia moments matrix I, which are discussed next. 1) Angular velocity: Let us denote by η = (ψ, θ, φ)T the Euler rotation angles relating the inertial frame and the ˙ θ, ˙ φ) ˙ T into the body-frame body frame. Projecting η˙ = (ψ, we obtain the expression for the angular velocity vector (Ω = Wn η) ˙ [5]. ⎛ ⎛ ⎞ p Ω = ⎝q ⎠ = Wn ⎝ r
⎞ ⎛ ˙ θ ⎞ ψ˙ φ˙ − ψs ˙ φ + ψs ˙ φ cθ ⎠ θ˙ ⎠ = ⎝ θc ˙ ˙ φ cθ ˙ −θsφ + ψc φ
⎞ 1 0 ⎠ 0
where p, q and r are the roll, pitch and yaw rates relative to ˙ =W ˙ n η+ the body axis. The time derivative Ω ˙ Wn η¨ is ⎞ ⎛ −cθ θ˙ψ˙ − sθ ψ¨ + φ¨ ˙ = ⎝−sφ sθ θ˙ψ˙ + cφ cθ φ˙ ψ˙ + cθ sφ ψ¨ − sφ φ˙ θ˙ + cφ θ¨⎠ (4) Ω −cφ sθ θ˙ψ˙ − sφ cθ φ˙ ψ˙ + cφ cθ ψ¨ − cφ φ˙ θ˙ − sφ θ¨
Center of mass
Le
m ¯ v˙ B + Ω × mv ¯ B ˙ + Ω × IΩ IΩ
0 cφ −sφ
−sθ Wn = ⎝ cθ sφ cθ cφ
zB
T r2
le
⎛
where
lr
where the RI→B transformation is such that RI→B = (RB→I )T . After some simple computations the equation (7) is rewritten as m(u˙ + qw − rv) m(v˙ − pw + ru) m(w˙ + pv − qu)
= = =
fx + sθ mg fy − cθ sφ mg fz − cθ cφ mg
or equivalently u˙ v˙ w˙
= = =
−(qw − rv) + fx /m − sθ g −(−pw + ru) + fy /m + cθ sφ g −(pv − qu) + fz /m + cθ cφ g
fx fy fz
= =
La1 − La2 − Le 0
=
Tr1 + Tr2 − (De + Da1 + Da2 )
with
(3)
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where Tri and Dri (for i = 1, 2) represent the thrust and the drag forces of the propeller respectively, Le and Lai denote the elevator and aileron lift forces respectively, De and Dai denote the elevon and aileron drag forces respectively. 2) Inertial frame: In this section we obtain the translational dynamics expressed in the inertial frame. In this case (1) becomes
1) Aerodynamic terms: The propeller lift and drag may be modelled by
(9)
Lw Dw
De Laj
(20) (21) (22)
Daj
=
ka υ 2 CDa δa (−kzB ) with j = 1, 2
(23)
(24)
where kυ > 0 is a constant. Remark: The convertible aircraft is composed of flat surfaces and therefore the associated derivatives of lift and drag coefficients are CL = 5/rad and CD ≈ 0. For flat surfaces the aerodynamic center is located at chord 4 .
which can also be written as r (Tr1 − Tr2 ) + q(−ωr1 + ωr2 ) e Le + p(ωr1 − ωr2 ) (13) a (La1 − La2 ) + r (Dr1 − Dr2 )
III. CONTROL STRATEGY
In this section we present a control algorithm using saturation functions which is applicable to a chain of integrators in cascade [8]. This control strategy will be used in the next section to control the longitudinal and lateral is equivalent to dynamics of the aircraft. In these dynamics appears, with ¨ (15)certains assumptions, a chain of integrators that involves (cθ θ˙ψ˙ + sθ ψ) 1 + [−qr(Iz − Iy ) + r (Tr1 − Tr2 ) + q(−ωr1 + ωr2 )] the control of the position through the control of the angles Ix (underactuated dynamics). ˙ (sφ sθ θ˙ψ˙ − cφ cθ φ˙ ψ˙ − cθ sφ ψ¨ + sφ φ˙ θ) It will be shown that after some change of variables and 1 after the introduction of a control input for the altitude, the [−pr(Ix − Iz ) + e Le + p(ωr1 − ωr2 )] + longitudinal as well as the lateral dynamic equations reduce cφ Iy to a chain of four integrators in cascade. We therefore focus ¨ (cφ sθ θ˙ψ˙ + sφ cθ φ˙ ψ˙ − cφ cθ ψ¨ + cφ φ˙ θ˙ + sφ θ) in this section in the control of the following system 1 [pq(Ix + Iy )a (La1 − La2 ) + r (Dr1 − Dr2 )] + cθ cφ Iz (25) x(4) (t) = u(t)
Solving (13) for η¨ leads to ˙ n η˙ − Ω × IΩ + Γc + Γg + Γq ) (14) η¨ = (IWn )−1 (−IW
ψ¨ =
(18) (19)
ke υ 2 CLe δe iB x ke υ 2 CDe δe kzB ka υ 2 CLa δa ± iB x with j = 1, 2
υ = kυ ωr
Substituting (3) into the LHS of (2) we get ˙ n η˙ + IWn η¨ + Ω × IΩ = Γc + Γg + Γq IW
θ¨ =
kw υ 2 CLw αw iB x =0 2 ke υ CDw αw (−kzB ) = 0
where δe and δa are the corresponding deflections. kr , kw , ke , ka which are positive constants that depend on the air density, the number of blades (in the case of kr ), the surface of the wing or blade, (CLr , CLw , CLe , CLa ) and (CDr , CDw , CDe , CDa ) represent the derivatives of the aerodynamic coefficients (depending on the shape of the wing or blade). Notice that in hover the air speed (prop-wash) υ depends of the rotor angular velocity ωr as follows
where Ir is the inertia moment of the propeller. • Finally, the drag torque, due to the propeller drag force Dri in free air may be written as ⎞ ⎛ 0 ⎠ 0 (12) Γq = ⎝ r (Dr1 − Dr2 )
which φ¨ =
= =
= = =
Le
The gyroscopic torque generated by the rotating blades turning with the body frame is modeled as ⎞ ⎛ 2 q(−ωr1 + ωr2 ) Γg = − Ir (Ω × ωri ) = ⎝ p(ωr1 − ωr2 ) ⎠ (11) i=1 0
Ix p˙ + qr(Iz − Iy ) = Iy q˙ + pr(Ix − Iz ) = Iz r˙ − pq(Ix + Iy ) =
(17)
where υ is the air speed vector (prop-wash) which coincides with the wing zero-lift line, i.e. zero angle of attack (αw ). The lift and drag of the ailerons and elevon control surfaces are given by
B. Rotational motion The RHS terms of (2) correspond to the torques applied on the rigid-body which are described next: • The main torque provided by the actuators is given by ⎞ ⎛ r (Tr1 − Tr2 ) ⎠ e Le (10) Γc = ⎝ a (La1 − La2 ) •
with i = 1, 2
(16)
where ωri is the rotor’s angular velocity. The lift force of the wing is perpendicular to the air slipstream while the drag force is parallel to the air slipstream. In our case, given that the wings are flat, they are expressed as
which can be written as cψ cθ fx /m + (sψ sφ + cψ sθ cφ )fz /m sψ cθ fx /m + (−cψ sφ + sψ sθ cφ )fz /m sθ fx /m + cθ cφ fz /m − g
kr ωr2i CDr αr
=
Dr i
m ¯ v˙ I = RB→I F I + mGI x ¨ = y¨ = z¨ =
kr ωr2i CLr αr kzB with i = 1, 2
=
Tr i
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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006
WeA02.6 Define the positive function V4 = 12 z42 . Thus
We will define the following change of variables z1 z2 z3 z4
= = = =
x(3) z1 + x ¨ z2 + x ¨ + x˙ z3 + x ¨ + 2x˙ + x
V˙ 4 = −z4 (σd (z4 ))
it follows that after a finite time T4 we have |z4 | < d. Convergence analysis We will now prove that all the state x will converge to zero. Indeed, from (32) we conclude that z4 → 0. Similarly from (31) it follows that z3 → 0. From (30) we conclude that z2 → 0 and finally from (29) it follows that z1 → 0.
(26)
and the following control law u = −σa (z1 ) − σb (z2 ) − σc (z3 ) − σd (z4 ) where ση is a saturation function defined as ⎧ s>η ⎨ η s −η ≤s≤η ση (s) = ⎩ −η s < −η
(27)
A. Controller design In this section we will present a control strategy to stabilize the convertible plane in hover. For simplicity we will study separately the lateral, longitudinal and axial dynamics. The controller proposed in this section will be tested in simulation using the full model of the aircraft. Let us first consider the rotational dynamics (14-15) which has strong nonlinearities and couplings. Let us introduce the following change of input
(28)
We will show that the control law (5) will globally asymptotically stabilize the system (25) around the origin. State boundedness We will first prove that the state of the system is bounded. Let us consider the following positive function V1 = 12 z12 where z1 = x(3) (see (26)). Thus, z˙1 = x(4) = u(t) and so (see (5)) V˙ 1 = z1 (−σa (z1 ) − σb (z2 ) − σc (z3 ) − σd (z4 ))
˜ ˙ Γc = Γ(IW n ) + IWn η˙ + Ω × IΩ − Γg − Γq
(33)
˜ = (˜ where Γ τφ , τ˜θ , τ˜ψ )T . Then (15) leads to
(29)
φ¨ = θ¨ = ψ¨ =
note that if |z1 | > a and a > b + c + d then V˙ 1 < 0. Thus, there exists a finite time T1 such that for t > T1 then |z1 | < a. For t > T1 the control law (5) becomes
τ˜φ τ˜θ τ˜ψ
For simplicity we will study separately three different dynamics at hover. We will specifically study the lateral, the longitudinal and the axial dynamics. These simplified dynamical models are an acceptable approximation for the roll, pitch and yaw dynamical models specially when the aircraft is at hover. We will develop a control algorithm for each one of these simplified models which will lead us to the global control strategy. The following lateral, longitudinal and axial dynamics are obtained from (9) with m = 1. 1) Lateral dynamics (θ = 0, ψ = 0). In this case (9) reduces to
u = −z1 − σb (z2 ) − σc (z3 ) − σd (z4 ) ¨ (see (26)), it then follows that Since z2 = z1 + x z˙2 = x(4) + z1 = −σb (z2 ) − σc (z3 ) − σd (z4 ) Define the positive function V2 = 12 z22 then V˙ 2 = −z2 (σb (z2 ) + σc (z3 ) + σd (z4 ))
(32)
(30)
Note that if |z2 | > b and b > c + d , then V˙ 2 < 0. We conclude that there exists a finite time T2 such that for t > T2 then |z2 | < b. For t > T2 the control law becomes
y¨ = z¨ = φ¨ =
u = −z1 − z2 − σc (z3 ) − σd (z4 ) Given that z3 = z2 + x ¨ + x˙ (see (26)), we then have z˙3 = z˙2 + z2 = −σc (z3 ) − σd (z4 )
−fz sin φ fz cos φ − g τ˜φ
(34) (35) (36)
(31)
We propose the following control input to stabilize the altitude z rz + g fz = (37) cos φ
Similarly, if |z3 | > c and c > d then V˙ 3 < 0. Thus there exists a finite time T3 such that for t > T3 we have |z3 | < c. For t > T3 the control law becomes
where rz = −az1 dotz − az2 (z − zd ) with az1 , az2 > 0 and zd is the desired altitude. Note that z → zd and therefore rz → 0. Substituting (37) into (34) we get
u = −z1 − z2 − z3 − σd (z4 )
y¨ = − tan φ(rz + g)
Since z4 = z3 + x ¨ + 2x˙ + x (see (26)), we have for t > T
The control input τ˜φ will be chosen such that φ is close to the origin. Therefore the lateral dynamics reduces to four integrators in cascade as follows
Let us consider the positive function V3 = 12 z32 then V˙ 3 = −z3 (σc (z3 ) + σd (z4 ))
z˙4
= z˙3 + x(3) + 2¨ x + x˙ ¨ + x˙ = z˙3 + z2 + x = z˙3 + z3 = −σd (z4 )
y¨ = φ¨ =
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−gφ τ˜φ
(38)
(39)
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The saturation control algorithm presented in the previous section can be used to control the the y-position and the roll angle φ of above system. 2) Longitudinal dynamics (φ = 0, ψ = 0). In this case (9) reduces to x ¨ =
−fx cos θ + fz sin θ
(40)
z¨ = θ¨ =
−fx sin θ + fz cos θ − g
(41)
τ˜θ
(42)
0.1 phi theta psi
0.08 0.06
Orientation [rad]
0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1
0
5
10
15
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25
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Notice that when using the controller (37), we have in the limit fz ≈ g. Therefore we can use the following control input
Fig. 3.
Attitude performance
1.2
− sin θg + θ fx = (43) cos θ which substituted into (40) gives us again four integrators in cascade
x−position y−position z−position
1 0.8
position [m]
0.6 0.4 0.2
x ¨ = θ¨ =
θ τ˜θ
0 −0.2 −0.4
5
10
Fig. 4.
15
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25
Position performance
0.1 phi control theta control psi control
0.05
0
(44)
Input control
τ˜ψ = −aψ1 ψ˙ − aψ2 ψ
0
Time [s]
We can use in this case also the saturation control algorithm presented in the previous section to control the x-position and the roll angle θ of the above system. 3) Axial dynamics (φ = 0, θ = 0). In this case the remaining dynamics corresponds to the yaw angle ψ. We finally propose the following control input
where aψ1 and aψ2 > 0 , which drives ψ to the origin.
−0.05
−0.1
−0.15
IV. SIMULATION RESULTS
−0.2
In this section we study the performance of the control laws (37), (43), (44) and (5) presented in the previous section when applied to the full nonlinear system (9). In this study we considered the following physical parameters [see table I] for the convertible VTOL prototype The following Parameter m g wide height Ix Iy Iz az1 az2 aψ1 aψ2
Value 0.5kg 9.8 sm2 0.6m 0.46m 0.0288kg · m2 0.020kg · m2 0.0088kg · m2 2 5 5 5
TABLE I P ROTOTYPE PHYSICAL PARAMETERS
initial conditions were taken for the simulation study: φ0 = 0.1, θ0 = 0.1, ψ0 = 1, x = 0.1, y = 0.1, z = 0.1. The desired altitude is set as zd = 1m. The control inputs, the attitude and the position’s behavior are depicted in the figures 3, 4 and 5 respectively. The simulations results show
−0.25
0
5
10
15
20
25
Time [s]
Fig. 5.
Control input performance
that the proposed controller satisfactorily stabilizes the full nonlinear system at hover, provided that the initial conditions are relatively close to the origin. V. EXPERIMENTAL RESULTS In this section is detailed the experimental prototype built in Heudiasyc laboratory, as well is shown the preliminary results of performing an autonomous hover flight. A. Experimental setup The convertible VTOL is driven by two DC motors. The propellers are connected to the motors through a 1:6 ratio gear. The aircraft structure is made of carbon fiber and the elevon and aileron are made of flat foam sheets. The convertible VTOL is equipped with a low-cost embedded microcontroller and a homemade inertial measurement unit (IMU). The IMU includes three surface-micromachining angular rate sensors (gyroscope ADXRS150) arranged ortogonally and a dual-axis micromachined silicon accelerometer 73
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-10 perturbation -15
-20
160
140
120
100
80
60 -25 40 -30 20
10
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Fig. 6.
Experimental platform
Fig. 8.
30
40
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Time(s)
Roll Angle and control input performance
pitch angle [ ]
˙ θ, ˙ ψ) ˙ (ADXL203). The IMU provides three angular rates (φ, and three angular positions (φ, θ, ψ). The control algorithm is implemented in dynamic C environment, the program code is downloaded to the rabbit RCM3400 microcontroller (on-board computer) which runs at 29.4 MHz, with 512K of flash memory. It has a 12 bit analog-digital converter which receives the IMU signals. The microcontroller sends the control inputs through the PWM outputs [see figure 7].
perturbation 10
5
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-5
0 10
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30
40
50
60
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Time(s)
Pitch Angle and control input performance yaw control (PWM)
30
4 PWM
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Time(s)
Fig. 9.
Input Capture
100
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yaw angle [ ]
1Dual-axis accelerometer
A D C
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Rabbit
3 gyrometers
200
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Radio receiver IMU
pitch control (PWM)
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perturbation 20
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-10
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power interface
-20
rotors
20 -30
servo:left aileron
10
20
30
servo: right aileron
40
50
60
Time(s)
10
20
30
40
Time(s)
servo: elevator Actuators
Fig. 10. Fig. 7.
Yaw Angle and control input performance
Embedded system
B. Attitude performance The experimental test were performed on the prototype previously described and it was focused to stabilize the aircraft at φ = 0, θ = 0 and ψ = 0. Preliminary experiments of the VTOL hovering autonomously run successfully and the results are shown in figures 8, 9 and 10 where it is shown each angle (roll, pitch and yaw) and its corresponding control input.
used to stabilize the aircraft at hover. The results were successfully tested in a convertible VTOL prototype platform with an embedded control algorithm. R EFERENCES [1] H. Goldstein, C.P. Poole and J.L Safko, "Classical Mechanics", Addison-Wesley Publishing Company, Inc., Massachusetts, 1983. [2] A. Bedford, and W. Fowler, "Dynamics", Addison-Wesley Publishing Company, 1989. [3] P. Castillo, R. Lozano A. Dzul, "Modelling and control of mini flying machines, Springer-Verlag, July 2005. [4] I. Fantoni and R. Lozano, "Nonlinear Control for Underactuated Mechanical Systems", Springer-Verlang, 2002. [5] B. Etkin and L. Reid, "Dynamics of Flight", J. Wiley & Sons, Inc., 1991. [6] B. L. Stevens and F.L. Lewis, "Aircraft Control and Simulation 2ed.", J. Wiley & Sons, Inc., 2003. [7] H. Stone, "Aerodynamic Modelling of a Wing-in-Slipstream TailSitter UAV", 2002 Biennial International Powered Lift Conference and Exhibit, Williamsburg, Virginia, Nov. 5-7, [8] Sussmann H, Sontag E and Yang Y. "A General Result on the Stabilization of linear System Using Bounded Control" IEEE TAC 1994 vol (12) 39 pp 2411 – 2425.
VI. CONCLUDING REMARKS We have presented in detail the dynamic equations of a convertible VTOL in hover mode. The equations were obtained using the Newton-Euler approach. After the introduction of a control input for the altitude, the dynamical model was transformed to obtain a simplified model for the longitudinal, lateral and axial dynamics. The resulting dynamical model is reduced to a chain of four integrators in cascade. A control scheme based on saturations was then
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Modelling and Control of a Convertible VTOL Aircraft J. Escareño, S. Salazar and R. Lozano.
Abstract— The aim of this paper is to present the complete model of an unmanned convertible aerial vehicle in hover mode. This vehicle is capable of performing either in hover or in forward flight. A nonlinear control strategy is presented to stabilize the aircraft in hovering mode. An embedded low-cost pilot is described as well the experimental results of a hover flight using a prototype built in the laboratory. Index Terms— Aircraft control algorithm, VTOL, Nonlinear control, amplitude bounded input, Embbeded architecture.
Hover flight
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Forward flight
Tr
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fixed-wing dynamics
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Hover flight
rotay-wing dynamics
Hover flight
Vertical take off
Vertical landing
I. INTRODUCTION Unmanned aerial vehicles (UAVs) are used in a wide variety of missions, like surveillance and reconnaissance, performed in different scenarios or environments. Therefore the task nature requires an appropriate flight profile vehicle in order to deal satisfactorily with the scenario’s drawbacks. For example, bridge inspection looking for structure’s cracks requires a vehicle with vertical take-off, hovering and lowspeed flight capabilities, then rotary-wing aircraft fits perfectly to that type of task. On the other hand, missions that require long-distance flights and relatively high forward velocity like forest fire detection or desert reconnaissance, usually use fixed-wing vehicles. However missions which involve the two scenarios described above require a vehicle which combines rotorcraft and fixed wing aircraft capabilities. We developed a convertible plane prototype capable of performing vertical take-off, hover and forward flight [see figure 1]. This type of convertible planes require specific control algorithms for handling take-off, hover, forward flight as well as the transition between those modes of operation. The aerodynamic configuration of the prototype we built is similar to the Boeing commercial aircraft Heliwing. There exist very few publications concerning the control of convertible planes. One of the few contributions in this area is given in [7] where H. Stone presented the control of the T-wing aircraft, which is a twin-engine tailsitter UAV that uses an LQR algorithm applied to the linearized hover dynamics. The main contribution of this paper is to provide a complete dynamical model of the convertible plane in hover mode. The nonlinear dynamical model is derived via the NewtonEuler formulation including the modelling of the gyroscopic effect and the adverse drag torque. In view of the physical limitations of the actuators, we propose a nonlinear control
Fig. 1.
algorithm satisfying the constraints on the inputs amplitude. We developed a specific embedded microcontroller architecture platform to be able to implement the control algorithm on-board. We present real-time experiments of the convertible plane in autonomous hover. The paper is organized as follows: Section II presents the complete dynamical model of the convertible VTOL. A control algorithm convergence analysis is presented in section III. Stabilization of the attitude and position of the aircraft is shown in simulation in section IV. The embedded architecture composed of the microcontroller, the sensors and filters is described in section V. The real-time application results of the stabilization of the VTOL in hover is presented in section VI. Finally some concluding remarks are given in section VII. II. DYNAMIC MODEL In this section we present the complete model of the convertible VTOL using a Newton-Euler formulation. The pitch, roll and yaw torques required for controlling the flying vehicle in hover are obtained from the speed difference between the two rotors and the control surfaces (aileron, elevon). The altitude of the vehicle is regulated by increasing or decreasing the propeller thrust. The roll torque is obtained from the difference of the rotors’ angular velocities. Since the control surfaces are submerged in the propeller slipstream (prop-wash), the aerodynamic forces are generated with the elevon and ailerons deflection to provide the pitch and yaw motion respectively [see figure 2]. Let I={iIx , jyI , kzI } denote the right hand inertial frame. B B Let B={iB x , jy , kz } denote the rigid-body frame with origin at the gravity center. Let the vector q = (ξ, η)T denote the generalized coordinates where ξ = (x, y, z)T ∈ 3 denotes the translation coordinates relative to the frame I, and η = (ψ, θ, φ)T ∈ 3 describes the vehicle orientation
Heudiasyc-UTC UMR 6599 Centre de Recherches de Royallieu B.P. 20529 60205 Compiegne France Tel.: + 33 (0)3 44 23 44 23 ; fax: +33 (0)3 44 23 44 77
Corresponding author [email protected] [email protected] [email protected]
1-4244-0171-2/06/$20.00 ©2006 IEEE.
UAV flightpath
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Aileron1 yB
L a1
δa xB la
Aileron2 L a2 zI
yI
xI Inertial frame
Fig. 2.
2) Moment of inertia: Matrix I represents the body moment of inertia whose value basically depends on the body mass distribution (for detail see [2]) and is decomposed as ⎞ ⎛ Ixx Ixy Ixz (5) I = ⎝Iyx Iyy Iyz ⎠ Izx Izy Izz
Elevator δe
Convertible VTOL
where
expressed in the classical yaw, pitch and roll angles (Euler angles) commonly used in aerodynamic applications [1]. The orientation of the convertible UAV is given by the rotation matrix RB→I . ⎞ ⎛ cθ cψ sφ sθ cψ − cφ sψ cφ sθ cψ + sφ sψ RB→I = ⎝ cθ sψ sφ sθ sψ + cφ cψ cφ sθ sψ − sφ cψ ⎠ −sθ sφ cθ cφ cθ
Ixx = m (y 2 + z 2 )dm, Ixy = Iyx = m −(xy)dm Iyy = m (x2 + z 2 )dm, Ixz = Izx = m −(xz)dm Izz = m (x2 + y 2 )dm, Iyz = Izy = m −(yz)dm
Notice that the aircraft is considered as a flat sheet aligned with the y − z plane [see figure 2]. Hence the application of the previous expressions to the convertible VTOL leads to a diagonal matrix ⎞ ⎛ 0 Ixx 0 (6) I = ⎝ 0 Iyy 0 ⎠ 0 0 Izz
where sβ := sin(β), cβ := cos(β). The rigid body motion equations (relative to B) subject to external force F B ∈ 3 and torque Γ ∈ 3 applied to the aircraft’s center of mass are given by the following expressions = =
F B + mGB Γc + Γg + Γq
A. Translational motion (1) (2)
1) Body frame: The equation (1) describes the translational motion and may be rewritten as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 mu˙ p mu fx ⎝ mv˙ ⎠+⎝q ⎠×⎝ mv ⎠ = ⎝fy ⎠+RI→B ⎝ 0 ⎠ (7) fz −mg mw˙ r mw
where m ¯ = diag(m) ∈ 3×3 , m ∈ denotes the vehicle mass, Ω is the angular velocity of the body frame, GB ∈ 3 is the gravity vector in B frame, v B = (u, v, w)T is the translational velocity of the aircraft’s center of mass, I ∈ 3×3 represents the tensor matrix, F B ∈ 3 is the total force acting on the aircraft including the weight, the rotors thrust, and the aerodynamic forces provided by the control surfaces, Γc ∈ 3 denotes the torque provided by the aircraft actuators, Γg denotes the gyroscopic torque and Γq is the drag torque generated by the propeller. The dynamic analysis requires the knowledge of the angular velocity Ω = (p, q, r)T and the inertia moments matrix I, which are discussed next. 1) Angular velocity: Let us denote by η = (ψ, θ, φ)T the Euler rotation angles relating the inertial frame and the ˙ θ, ˙ φ) ˙ T into the body-frame body frame. Projecting η˙ = (ψ, we obtain the expression for the angular velocity vector (Ω = Wn η) ˙ [5]. ⎛ ⎛ ⎞ p Ω = ⎝q ⎠ = Wn ⎝ r
⎞ ⎛ ˙ θ ⎞ ψ˙ φ˙ − ψs ˙ φ + ψs ˙ φ cθ ⎠ θ˙ ⎠ = ⎝ θc ˙ ˙ φ cθ ˙ −θsφ + ψc φ
⎞ 1 0 ⎠ 0
where p, q and r are the roll, pitch and yaw rates relative to ˙ =W ˙ n η+ the body axis. The time derivative Ω ˙ Wn η¨ is ⎞ ⎛ −cθ θ˙ψ˙ − sθ ψ¨ + φ¨ ˙ = ⎝−sφ sθ θ˙ψ˙ + cφ cθ φ˙ ψ˙ + cθ sφ ψ¨ − sφ φ˙ θ˙ + cφ θ¨⎠ (4) Ω −cφ sθ θ˙ψ˙ − sφ cθ φ˙ ψ˙ + cφ cθ ψ¨ − cφ φ˙ θ˙ − sφ θ¨
Center of mass
Le
m ¯ v˙ B + Ω × mv ¯ B ˙ + Ω × IΩ IΩ
0 cφ −sφ
−sθ Wn = ⎝ cθ sφ cθ cφ
zB
T r2
le
⎛
where
lr
where the RI→B transformation is such that RI→B = (RB→I )T . After some simple computations the equation (7) is rewritten as m(u˙ + qw − rv) m(v˙ − pw + ru) m(w˙ + pv − qu)
= = =
fx + sθ mg fy − cθ sφ mg fz − cθ cφ mg
or equivalently u˙ v˙ w˙
= = =
−(qw − rv) + fx /m − sθ g −(−pw + ru) + fy /m + cθ sφ g −(pv − qu) + fz /m + cθ cφ g
fx fy fz
= =
La1 − La2 − Le 0
=
Tr1 + Tr2 − (De + Da1 + Da2 )
with
(3)
70
(8)
45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006
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where Tri and Dri (for i = 1, 2) represent the thrust and the drag forces of the propeller respectively, Le and Lai denote the elevator and aileron lift forces respectively, De and Dai denote the elevon and aileron drag forces respectively. 2) Inertial frame: In this section we obtain the translational dynamics expressed in the inertial frame. In this case (1) becomes
1) Aerodynamic terms: The propeller lift and drag may be modelled by
(9)
Lw Dw
De Laj
(20) (21) (22)
Daj
=
ka υ 2 CDa δa (−kzB ) with j = 1, 2
(23)
(24)
where kυ > 0 is a constant. Remark: The convertible aircraft is composed of flat surfaces and therefore the associated derivatives of lift and drag coefficients are CL = 5/rad and CD ≈ 0. For flat surfaces the aerodynamic center is located at chord 4 .
which can also be written as r (Tr1 − Tr2 ) + q(−ωr1 + ωr2 ) e Le + p(ωr1 − ωr2 ) (13) a (La1 − La2 ) + r (Dr1 − Dr2 )
III. CONTROL STRATEGY
In this section we present a control algorithm using saturation functions which is applicable to a chain of integrators in cascade [8]. This control strategy will be used in the next section to control the longitudinal and lateral is equivalent to dynamics of the aircraft. In these dynamics appears, with ¨ (15)certains assumptions, a chain of integrators that involves (cθ θ˙ψ˙ + sθ ψ) 1 + [−qr(Iz − Iy ) + r (Tr1 − Tr2 ) + q(−ωr1 + ωr2 )] the control of the position through the control of the angles Ix (underactuated dynamics). ˙ (sφ sθ θ˙ψ˙ − cφ cθ φ˙ ψ˙ − cθ sφ ψ¨ + sφ φ˙ θ) It will be shown that after some change of variables and 1 after the introduction of a control input for the altitude, the [−pr(Ix − Iz ) + e Le + p(ωr1 − ωr2 )] + longitudinal as well as the lateral dynamic equations reduce cφ Iy to a chain of four integrators in cascade. We therefore focus ¨ (cφ sθ θ˙ψ˙ + sφ cθ φ˙ ψ˙ − cφ cθ ψ¨ + cφ φ˙ θ˙ + sφ θ) in this section in the control of the following system 1 [pq(Ix + Iy )a (La1 − La2 ) + r (Dr1 − Dr2 )] + cθ cφ Iz (25) x(4) (t) = u(t)
Solving (13) for η¨ leads to ˙ n η˙ − Ω × IΩ + Γc + Γg + Γq ) (14) η¨ = (IWn )−1 (−IW
ψ¨ =
(18) (19)
ke υ 2 CLe δe iB x ke υ 2 CDe δe kzB ka υ 2 CLa δa ± iB x with j = 1, 2
υ = kυ ωr
Substituting (3) into the LHS of (2) we get ˙ n η˙ + IWn η¨ + Ω × IΩ = Γc + Γg + Γq IW
θ¨ =
kw υ 2 CLw αw iB x =0 2 ke υ CDw αw (−kzB ) = 0
where δe and δa are the corresponding deflections. kr , kw , ke , ka which are positive constants that depend on the air density, the number of blades (in the case of kr ), the surface of the wing or blade, (CLr , CLw , CLe , CLa ) and (CDr , CDw , CDe , CDa ) represent the derivatives of the aerodynamic coefficients (depending on the shape of the wing or blade). Notice that in hover the air speed (prop-wash) υ depends of the rotor angular velocity ωr as follows
where Ir is the inertia moment of the propeller. • Finally, the drag torque, due to the propeller drag force Dri in free air may be written as ⎞ ⎛ 0 ⎠ 0 (12) Γq = ⎝ r (Dr1 − Dr2 )
which φ¨ =
= =
= = =
Le
The gyroscopic torque generated by the rotating blades turning with the body frame is modeled as ⎞ ⎛ 2 q(−ωr1 + ωr2 ) Γg = − Ir (Ω × ωri ) = ⎝ p(ωr1 − ωr2 ) ⎠ (11) i=1 0
Ix p˙ + qr(Iz − Iy ) = Iy q˙ + pr(Ix − Iz ) = Iz r˙ − pq(Ix + Iy ) =
(17)
where υ is the air speed vector (prop-wash) which coincides with the wing zero-lift line, i.e. zero angle of attack (αw ). The lift and drag of the ailerons and elevon control surfaces are given by
B. Rotational motion The RHS terms of (2) correspond to the torques applied on the rigid-body which are described next: • The main torque provided by the actuators is given by ⎞ ⎛ r (Tr1 − Tr2 ) ⎠ e Le (10) Γc = ⎝ a (La1 − La2 ) •
with i = 1, 2
(16)
where ωri is the rotor’s angular velocity. The lift force of the wing is perpendicular to the air slipstream while the drag force is parallel to the air slipstream. In our case, given that the wings are flat, they are expressed as
which can be written as cψ cθ fx /m + (sψ sφ + cψ sθ cφ )fz /m sψ cθ fx /m + (−cψ sφ + sψ sθ cφ )fz /m sθ fx /m + cθ cφ fz /m − g
kr ωr2i CDr αr
=
Dr i
m ¯ v˙ I = RB→I F I + mGI x ¨ = y¨ = z¨ =
kr ωr2i CLr αr kzB with i = 1, 2
=
Tr i
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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006
WeA02.6 Define the positive function V4 = 12 z42 . Thus
We will define the following change of variables z1 z2 z3 z4
= = = =
x(3) z1 + x ¨ z2 + x ¨ + x˙ z3 + x ¨ + 2x˙ + x
V˙ 4 = −z4 (σd (z4 ))
it follows that after a finite time T4 we have |z4 | < d. Convergence analysis We will now prove that all the state x will converge to zero. Indeed, from (32) we conclude that z4 → 0. Similarly from (31) it follows that z3 → 0. From (30) we conclude that z2 → 0 and finally from (29) it follows that z1 → 0.
(26)
and the following control law u = −σa (z1 ) − σb (z2 ) − σc (z3 ) − σd (z4 ) where ση is a saturation function defined as ⎧ s>η ⎨ η s −η ≤s≤η ση (s) = ⎩ −η s < −η
(27)
A. Controller design In this section we will present a control strategy to stabilize the convertible plane in hover. For simplicity we will study separately the lateral, longitudinal and axial dynamics. The controller proposed in this section will be tested in simulation using the full model of the aircraft. Let us first consider the rotational dynamics (14-15) which has strong nonlinearities and couplings. Let us introduce the following change of input
(28)
We will show that the control law (5) will globally asymptotically stabilize the system (25) around the origin. State boundedness We will first prove that the state of the system is bounded. Let us consider the following positive function V1 = 12 z12 where z1 = x(3) (see (26)). Thus, z˙1 = x(4) = u(t) and so (see (5)) V˙ 1 = z1 (−σa (z1 ) − σb (z2 ) − σc (z3 ) − σd (z4 ))
˜ ˙ Γc = Γ(IW n ) + IWn η˙ + Ω × IΩ − Γg − Γq
(33)
˜ = (˜ where Γ τφ , τ˜θ , τ˜ψ )T . Then (15) leads to
(29)
φ¨ = θ¨ = ψ¨ =
note that if |z1 | > a and a > b + c + d then V˙ 1 < 0. Thus, there exists a finite time T1 such that for t > T1 then |z1 | < a. For t > T1 the control law (5) becomes
τ˜φ τ˜θ τ˜ψ
For simplicity we will study separately three different dynamics at hover. We will specifically study the lateral, the longitudinal and the axial dynamics. These simplified dynamical models are an acceptable approximation for the roll, pitch and yaw dynamical models specially when the aircraft is at hover. We will develop a control algorithm for each one of these simplified models which will lead us to the global control strategy. The following lateral, longitudinal and axial dynamics are obtained from (9) with m = 1. 1) Lateral dynamics (θ = 0, ψ = 0). In this case (9) reduces to
u = −z1 − σb (z2 ) − σc (z3 ) − σd (z4 ) ¨ (see (26)), it then follows that Since z2 = z1 + x z˙2 = x(4) + z1 = −σb (z2 ) − σc (z3 ) − σd (z4 ) Define the positive function V2 = 12 z22 then V˙ 2 = −z2 (σb (z2 ) + σc (z3 ) + σd (z4 ))
(32)
(30)
Note that if |z2 | > b and b > c + d , then V˙ 2 < 0. We conclude that there exists a finite time T2 such that for t > T2 then |z2 | < b. For t > T2 the control law becomes
y¨ = z¨ = φ¨ =
u = −z1 − z2 − σc (z3 ) − σd (z4 ) Given that z3 = z2 + x ¨ + x˙ (see (26)), we then have z˙3 = z˙2 + z2 = −σc (z3 ) − σd (z4 )
−fz sin φ fz cos φ − g τ˜φ
(34) (35) (36)
(31)
We propose the following control input to stabilize the altitude z rz + g fz = (37) cos φ
Similarly, if |z3 | > c and c > d then V˙ 3 < 0. Thus there exists a finite time T3 such that for t > T3 we have |z3 | < c. For t > T3 the control law becomes
where rz = −az1 dotz − az2 (z − zd ) with az1 , az2 > 0 and zd is the desired altitude. Note that z → zd and therefore rz → 0. Substituting (37) into (34) we get
u = −z1 − z2 − z3 − σd (z4 )
y¨ = − tan φ(rz + g)
Since z4 = z3 + x ¨ + 2x˙ + x (see (26)), we have for t > T
The control input τ˜φ will be chosen such that φ is close to the origin. Therefore the lateral dynamics reduces to four integrators in cascade as follows
Let us consider the positive function V3 = 12 z32 then V˙ 3 = −z3 (σc (z3 ) + σd (z4 ))
z˙4
= z˙3 + x(3) + 2¨ x + x˙ ¨ + x˙ = z˙3 + z2 + x = z˙3 + z3 = −σd (z4 )
y¨ = φ¨ =
72
−gφ τ˜φ
(38)
(39)
45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006
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The saturation control algorithm presented in the previous section can be used to control the the y-position and the roll angle φ of above system. 2) Longitudinal dynamics (φ = 0, ψ = 0). In this case (9) reduces to x ¨ =
−fx cos θ + fz sin θ
(40)
z¨ = θ¨ =
−fx sin θ + fz cos θ − g
(41)
τ˜θ
(42)
0.1 phi theta psi
0.08 0.06
Orientation [rad]
0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1
0
5
10
15
20
25
Time [s]
Notice that when using the controller (37), we have in the limit fz ≈ g. Therefore we can use the following control input
Fig. 3.
Attitude performance
1.2
− sin θg + θ fx = (43) cos θ which substituted into (40) gives us again four integrators in cascade
x−position y−position z−position
1 0.8
position [m]
0.6 0.4 0.2
x ¨ = θ¨ =
θ τ˜θ
0 −0.2 −0.4
5
10
Fig. 4.
15
20
25
Position performance
0.1 phi control theta control psi control
0.05
0
(44)
Input control
τ˜ψ = −aψ1 ψ˙ − aψ2 ψ
0
Time [s]
We can use in this case also the saturation control algorithm presented in the previous section to control the x-position and the roll angle θ of the above system. 3) Axial dynamics (φ = 0, θ = 0). In this case the remaining dynamics corresponds to the yaw angle ψ. We finally propose the following control input
where aψ1 and aψ2 > 0 , which drives ψ to the origin.
−0.05
−0.1
−0.15
IV. SIMULATION RESULTS
−0.2
In this section we study the performance of the control laws (37), (43), (44) and (5) presented in the previous section when applied to the full nonlinear system (9). In this study we considered the following physical parameters [see table I] for the convertible VTOL prototype The following Parameter m g wide height Ix Iy Iz az1 az2 aψ1 aψ2
Value 0.5kg 9.8 sm2 0.6m 0.46m 0.0288kg · m2 0.020kg · m2 0.0088kg · m2 2 5 5 5
TABLE I P ROTOTYPE PHYSICAL PARAMETERS
initial conditions were taken for the simulation study: φ0 = 0.1, θ0 = 0.1, ψ0 = 1, x = 0.1, y = 0.1, z = 0.1. The desired altitude is set as zd = 1m. The control inputs, the attitude and the position’s behavior are depicted in the figures 3, 4 and 5 respectively. The simulations results show
−0.25
0
5
10
15
20
25
Time [s]
Fig. 5.
Control input performance
that the proposed controller satisfactorily stabilizes the full nonlinear system at hover, provided that the initial conditions are relatively close to the origin. V. EXPERIMENTAL RESULTS In this section is detailed the experimental prototype built in Heudiasyc laboratory, as well is shown the preliminary results of performing an autonomous hover flight. A. Experimental setup The convertible VTOL is driven by two DC motors. The propellers are connected to the motors through a 1:6 ratio gear. The aircraft structure is made of carbon fiber and the elevon and aileron are made of flat foam sheets. The convertible VTOL is equipped with a low-cost embedded microcontroller and a homemade inertial measurement unit (IMU). The IMU includes three surface-micromachining angular rate sensors (gyroscope ADXRS150) arranged ortogonally and a dual-axis micromachined silicon accelerometer 73
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roll control (PWM)
45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006
roll angle [ ]
0
-5
-10 perturbation -15
-20
160
140
120
100
80
60 -25 40 -30 20
10
20
30
40
50
60
10
20
Time(s)
Fig. 6.
Experimental platform
Fig. 8.
30
40
50
60
50
60
50
60
Time(s)
Roll Angle and control input performance
pitch angle [ ]
˙ θ, ˙ ψ) ˙ (ADXL203). The IMU provides three angular rates (φ, and three angular positions (φ, θ, ψ). The control algorithm is implemented in dynamic C environment, the program code is downloaded to the rabbit RCM3400 microcontroller (on-board computer) which runs at 29.4 MHz, with 512K of flash memory. It has a 12 bit analog-digital converter which receives the IMU signals. The microcontroller sends the control inputs through the PWM outputs [see figure 7].
perturbation 10
5
0
-5
0 10
20
30
40
50
60
10
30
40
Time(s)
Pitch Angle and control input performance yaw control (PWM)
30
4 PWM
20
Time(s)
Fig. 9.
Input Capture
100
50
yaw angle [ ]
1Dual-axis accelerometer
A D C
150
-15
Rabbit
3 gyrometers
200
-10
Radio receiver IMU
pitch control (PWM)
250
15
perturbation 20
10
0
-10
120
100
80
60
40
power interface
-20
rotors
20 -30
servo:left aileron
10
20
30
servo: right aileron
40
50
60
Time(s)
10
20
30
40
Time(s)
servo: elevator Actuators
Fig. 10. Fig. 7.
Yaw Angle and control input performance
Embedded system
B. Attitude performance The experimental test were performed on the prototype previously described and it was focused to stabilize the aircraft at φ = 0, θ = 0 and ψ = 0. Preliminary experiments of the VTOL hovering autonomously run successfully and the results are shown in figures 8, 9 and 10 where it is shown each angle (roll, pitch and yaw) and its corresponding control input.
used to stabilize the aircraft at hover. The results were successfully tested in a convertible VTOL prototype platform with an embedded control algorithm. R EFERENCES [1] H. Goldstein, C.P. Poole and J.L Safko, "Classical Mechanics", Addison-Wesley Publishing Company, Inc., Massachusetts, 1983. [2] A. Bedford, and W. Fowler, "Dynamics", Addison-Wesley Publishing Company, 1989. [3] P. Castillo, R. Lozano A. Dzul, "Modelling and control of mini flying machines, Springer-Verlag, July 2005. [4] I. Fantoni and R. Lozano, "Nonlinear Control for Underactuated Mechanical Systems", Springer-Verlang, 2002. [5] B. Etkin and L. Reid, "Dynamics of Flight", J. Wiley & Sons, Inc., 1991. [6] B. L. Stevens and F.L. Lewis, "Aircraft Control and Simulation 2ed.", J. Wiley & Sons, Inc., 2003. [7] H. Stone, "Aerodynamic Modelling of a Wing-in-Slipstream TailSitter UAV", 2002 Biennial International Powered Lift Conference and Exhibit, Williamsburg, Virginia, Nov. 5-7, [8] Sussmann H, Sontag E and Yang Y. "A General Result on the Stabilization of linear System Using Bounded Control" IEEE TAC 1994 vol (12) 39 pp 2411 – 2425.
VI. CONCLUDING REMARKS We have presented in detail the dynamic equations of a convertible VTOL in hover mode. The equations were obtained using the Newton-Euler approach. After the introduction of a control input for the altitude, the dynamical model was transformed to obtain a simplified model for the longitudinal, lateral and axial dynamics. The resulting dynamical model is reduced to a chain of four integrators in cascade. A control scheme based on saturations was then
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