Design of a simple controller for the flight mode ...

Design of a simple controller for the flight mode transition of a hybrid drone A. Flores, J. Sanchez, A. Montes de Oca, L. Arreola and G. Flores Abstract— This work presents the modeling and control of the transition maneuver of a class of hybrid Unmanned Aerial Vehicle (UAV): the tail-sitter. The modeling considers aerodynamic terms whereas the proposed controller is designed with the aim of achieve a transition flight from hover to cruise mode and vice-versa. The controller is designed to be applied in a real tailsitter platform, hence all the assumptions and considerations take into account a real scenario in which certain states are available. The key idea behind the controller design is the timescale separation between UAV’s attitude and position dynamics, in order to obtain a desired trajectory for the pitch angle, which is the responsible to achieve the transition maneuver. The controller design is based on Lyapunov approach and linear saturation functions. Simulations experiments demonstrate the effectiveness of the derived theoretical results. Index Terms— Convertible UAV; transition maneuver; VTOL; Lyapunov stability; saturation control; tail-sitter.

I. I NTRODUCTION UAV’s (aka drones) designs can be divided into two principal types: fixed-wing and the rotatory aircraft, both have their respectively advantages and disadvantages. For instance, fixed wing UAV can achieve long distance displacements in each flight session but it needs a runaway for taking off and landing, also they need a considerable area for performing positioning maneuvers. On the other hand, rotor aircraft can perform hovering flights, achieving smooth movements, and having the capability of performing rotational and lateral displacements; nonetheless this kind of aerial vehicles have several limiting aspects, like slow movement speed and greater energy consumption. As one could expect, these two types of drones have very different applications, however, one could obtain certain capabilities of these two by using the so-called hybrid convertible drone. This kind of drone will be the studied in this paper. In the past years several hybrid flight mode aircraft have been designed like the Unmanned Quad-TiltRotors, this model presented in [1] consists of a quad-rotor structure with wings aligned to the motors thrust direction, and its flight mode transition relies on the inclination of the four motors to obtain an horizontal thrust. An other configuration is the Hybrid UAV ULion [2], [3] which has a reconfigurable wing with a structure more similar to that of a fixed wing aircraft. It consists of two propulsion motors for aligned in the same axis, such motors rotate at different directions. During the hover flight mode the A. Montes de Oca, L. Arreola, A. Flores, J. Sanchez and G. Flores are with the Perception and Robotics Laboratory, Centro de Investigaciones ´ en Optica, Le´on, Guanajuato, Mexico, 37150. (email:[email protected], [email protected], [email protected], [email protected], [email protected]). This work was supported by the FORDECYT-CONACYT under grant 000000000292399.

Cruise

Cruise-Hover

Hover-Cruise

Hover Hover

Fig. 1: Flight mode change description hover-cruise and cruise-hover.

wings are folded and aligned to the fuselage and the motor thrust is directed vertically and while the cruise mode wings are placed perpendicularly to the fuselage obtaining the form of an airplane. Compared with the aforementioned hybrid UAV configurations, the Quadshot drone presented in [4], [5], [6] is the simplest model w.r.t mechanical design. Its also called tail-sitter configuration. The mechanical simplicity can be observed from its composition: it is only of a fixed-wing with four motors parallel to the plane formed by the wing. Their structure is completely fixed since it has not moving or rotating parts that modify its shape, a figure of this can be seen at Fig. 2. Notwithstanding all these advantages, their transition maneuver is considerable more complex than the rest of configurations, since it is required that the complete UAV body pivot around one of the axes in order to achieve the transition maneuver. Tail-sitter model is the case of study of this paper. This paper is focused to develop and implement a control system dedicated especially to the transition maneuver from hover to cruise flight mode and vice-versa. This approach not only considers aerodynamics effects, but also a real scenario in winch usual available states in practice are only needed. An other positive point is the simplicity of the controller avoiding switching dynamics between modes, having a sweet and bounded controllers, very desired properties in practice implementations. Most of the papers related with this topic tackle the problem from a simplified model without taking into account aerodynamics, bounded limits of actuators, or even smoothness of the controller. In [1], [7], [8] the control strategy consists in increasing the motor angles until they are horizontally aligned the, a switching strategy between flight modes is performed. In [9] authors model and develop a robust control system to control the two flight modes that includes aerodynamics and most of the external facts that could affect the aircraft dynamics. The negative point of this work is that they not get focus in the transition control,

making a simple switch between the two flight modes while the aircraft is increasing or decreasing its orientation angle. In [10] the transition is made manually an the control relies in modifying the flight modes control weights according to the percentage of the transition progress. In [11] it is implemented a controller in SO(3) for the transition maneuver of a tail-sitter UAV. The controller is designed in the 6DOF, however the research lack of a formal proof which demonstrate the effectiveness of their result. The remainder of this paper is organized as follows. In Section II it is defined the problem to be solved and system model is presented. Section III presents the control system proposed and the followed strategy to ensure a successful flight mode transition with its respective stability proof. In section IV it is presented simulations experiments carried out to demonstrate the effectiveness of the theoretical results. Finally, some conclusions and further work are described in Section V. II. P ROBLEM F ORMULATION

Switch mode signal Hover mode Control

Transition completed

Transition completed Flight mode transition control

Cruise mode Control

Switch mode signal

Error or cancelation

>50% completed

N

Flight mode?

Cruise

Hover

Y Hover Flight mode? Cruise

Fig. 3: Flow digram of the aircraft full control system.

for the transition maneuver appears [12]-[13]. The forces that affect the behavior of the aircraft are shown in Fig. 4. According with the frames shown at Fig. 4, the system can

Figs. 1 and 2 depict the tail-sitter drone used in this study. This UAV have three modes of operation: a) hover; b) cruise; and c) transition. The last mode consists in the transition phase between cruise (hover) to hover (cruise); this flying mode will be investigated in this paper. From hover to cruise mode the attitude changes from 90, in the x − u axis, to approximately 0 to 12 degrees, which is the range for hover mode. It is the same for cruise to hover mode but with the angles reversed. The actuators are six: four rotors and

T1

u L V

T2

N

x,u D w

y,v

z,w

Fig. 4: Main system forces. N

E D

Fig. 2: System frames and aircraft used. two servos which manipulate the pitch and roll angle. It is important to mention that the presence of these six actuators results in an over-actuated roll and pitch subsystems. In fig. 3 is represented the full control system flow. A. System Model The analysis and control design is implemented on the (x − z) plane, i.e. the inertial frame and (u, w) from the body frame. This is the plane where the most interesting dynamics

be described as follows ( u˙ = −D cos α + L sin α + T − g sin θ − qw Σ1 w˙ = −D sin α − L cos α + g cos θ + qu ( θ˙ = q Σ2 q˙ = τ,

(1) (2)

since the orientation subsystem is full independent of the translational dynamics, but the translational dynamics does not, the system can be divided into two separated subsystems where ∑ 2 is the orientation and ∑ 1 the translation dynamics. The u and w states are the horizontal and the vertical velocities respectively according to the body frame, in this case, the position of the aircraft is not really important since it is preferable to know the direction of displacement of it.

w , (3) α = tan u finally T and τ are the two control inputs that will modify the accelerations and rotations of the whole system. Because of ∑ 2 is a faster dynamic subsystem than ∑ 1 , the angle position should not be controlled as fast as it can so it could cause instability in the translational state as a result of the orientation dependence in ∑ 1 . This way, it is crucial to do a time scale separation [14], [15] that help determine how the subsystem ∑ 2 should be controlled allowing the evolution of the states of each subsystem be similar and converging in a finite time. Since the modeling of the aircraft is expressed in body frame, the rotation matrix that describes the translation from the NED (North, East, Down) coordinate frame to the body frame is defined as      x cos θ 0 sin θ N y  =  0 1 0  E  , (4) z − sin θ 0 cos θ D

1.5 1 0.5 0

β0

Cl

θ and q are the y-axis angle rotation and angular velocity between the inertial and the body frame, this angle could be taken as the most important state of the whole system because it intertwines the two subsystems (orientation and translation). D, L and g are the drag, lift and gravity forces respectively, α is the angle of attack (AoA) that depends of the air vector and the body velocities where −1

−1 −1.5 −10

−5

0

5

10

15

5

10

15

α

0.25 β0

Cd

0.15

0.1

0.05

0 −10

where

−5

0 α

x˙ = u y˙ = v z˙ = w For simplicity the controller is designed on the body frame, avoiding extra terms from the aforementioned rotation matrix. 1) Aerodynamic forces: The proposed model considers aerodynamic terms explained next. The lift and drag forces are functions of several variables, but the most important of these variables are: the air speed and the lift and drag coefficients. Lift and drag are defined as follows L

= K1CLV 2

D = K1CDV 2

√ where V = u2 + w2 is the air speed magnitude [16]. The drag and lift coefficients (CL ,CD ) for the airfoil used in this experiment are described at Fig. 5 according to the angle of attack (AoA) (α). Such coefficients correspond to a symmetrical airfoil NACA-0020 shown at Fig. 6, they were obtained by numerical simulations made in the software XFR5 with many different Reynold numbers. Also it was designed some different airfoils that include aileron angles (β ) to obtain different lift and drag coefficients which will help the aircraft to produce lift and also produce a part of the torque needed to rotate. Finally, K1 defined as K1 = ρS 2 describes the wing hitting area (S), air pressure (ρ) that affects the complete amount of lift or drag force.

Fig. 5: Lift and drag coefficients w.r.t α angle obtained from XFR5 software and [17], where α = 6 degrees corresponds to the best AoA according to the relation between the Cl and the Cd.

0.1 0 −0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 6: Airfoil normalized profile used in the aircraft (NACA0020).

The control inputs for the system (1-2) are the thrust T and the torque generated by the difference of thrust between rear and from rotors and from the aileron movement τ. These terms can be modeled as T

= T1 + T2

(5)

τ

= τβ + τL(u,w) + (T1 − T2 )

(6)

where τβ is the part of torque generated by the propellers wind stream in the aileron; τL(u,w) is the torque induced by wind stream shocking in the wing area and the rest of the torque is originated by the difference force between the front

and back pair of motors. The total thrust force is calculated by the sum of each motor thrust of the aircraft. B. Problem Statement Design a control system that let an hybrid flight mode aircraft to change securely from hover to cruise flight mode and vice versa through the difference of thrust of its motors and with the help of its ailerons. That control should be continuous, stable, with low computational hardware requirements and easy to implement. III. M AIN R ESULT In this section it is explained the control design technique that was carried out to develop the control system which will help the aircraft to achieve an stable and safe flight mode transition. After, the proposed control will be presented and its characteristics for ensure that the control will keep stable. Finally the demonstration for the stability of the proposed control system will be developed through the Lyapunov’s stability theory. A. Control design technique The key idea behind the proposed controller is explained next. Based on time-scale separation principle presented in the UAVs, as one of the co-authors and others have previously demonstrated in [14], [15], it is possible to use the variable θ as a virtual controller for subsystem (1). Once this virtual controller is designed, its value must be tracked by the subsystem (2). Such subsystem is enough faster than (1), then it is possible to control the complete system by this approach. Let’s start with the design of the virtual controller for (1). In this case the states u and w should converge to a predefined values according to the appropriate characteristics for the flight mode in question. Said that, subsystem (1) can be rewritten as p 2 (7) u˙ = f1 (u, w) + T − 1 − ε 2 w˙ =

f2 (u, w) + ε

T , by nature, can not be negative, and in fact T is limited by the motor thrust capacity. • Since (7) holds, therefore 0 ≤ ε ≤ 1 • θ must be bounded from −10 to 100 degrees to ensure a secure transition; these limits are chosen by the control designer. It is fundamental that ud and wd have to be defined according to the speed characteristics of each flight mode, as it is known, if the flight transition is from hover to cruise, u should increase to a value in which the lift force is sufficient to maintain the aircraft at a constant altitude. On the other hand, w must be designed w.r.t. u in such a way that the angle of attack of the aircraft is the most effective angle for flying, in this case, 6 degrees. In the same way, when the transition is from cruise to hover, both speeds must be reduced in such a way that the AoA decreases to the point of being equal to zero. Knowing this, a correlation could be made between these two variables (u, w) to obtain the graphs of the desired speeds, i.e. (ud , wd ). Once the virtual control input ε for replace θ in (1) is designed, it is time to design a control system for (7) and (8) according to the new control inputs, in this case T is maintained from the original system, and now ε that was created as a virtual control input, which will help to momentarily replace the orientation state. As can be seen in (7), the virtual control variable is limited to 0 ≤ ε ≤ 1, so the design for this control will be by saturated control mode, which consists of applying linear saturation functions in chain with the order to maintain said control within the range of desired values. Where the saturation function defined in [18] σ (s) is a continuous non decreasing function satisfying (a) sσ (s) > 0 ∀s 6= 0 (b) | σ (s) |≤ M ∀s ∈ R (c) σ (s) = s when | s |≤ L. •

Defined the new dynamics and the control characteristics, it is now possible to propose a preliminary control system for ε and T where

(8) ε

where

T ε

= cos θ

f1 (u, w) = −D cos α + L sin α f2 (u, w) = −D sin α − L cos α, Now, before proposing the control algorithm we should define some natural constrains and assumptions that will help to design in a simple manner the controller. Such assumptions and constrains are • In cruise flight mode it is natural to consider that u >> w holds, i.e. the horizontal velocity is much greater than vertical velocity. • In hover mode, u ≈ 0 and w ≈ 0, since there is no displacement in horizontal and vertical directions. • Because of (3), u must never be zero


= −σ2 f2 (u, w) + σ1 (w − wd ) − w˙ d (9) p 2 2 = −σ3 (u − ud ) + 1 − ε − f1 (u, w) + u˙d , (10)

and after simulate the subsystem (7)-(8) with the controls proposed below, the evolution of the control input ε obtained, effectively describes a secure trajectory for θ according to the translation dynamics. B. Proposed control system Once the virtual control ε was designed and several simulations were carried out, it was possible to obtain the evolution of the control ε while the velocities u and w converges to the desired values, as it is well known, the variable (ε) was used to replace the state θ of the complete system so that, in this way, a continuous function can be defined for the desired value for θ in such a way that it is now possible to design a control for the orientation dynamics

according to the translation dynamics time speed. Now, going back to the original system defined as =

u˙

f1 (u, w) + T − g sin θ − qw

w˙ = f2 (u, w) + g cos θ + qu θ˙ = q = τ

q˙

(11) (12) (13) (14)

and substituting the controls

x˙3

= −σ (x1 ) − (x4 + qd )(x2 + wd ) (21) f2 (x1 + ud , x2 + wd ) + g cos(x3 + θd ) = (22) + (x4 + qd )(x1 + ud ) − w˙ d = x4 (23)

x˙4

= −kθ (x3 ) − kq (x4 )

x˙1 x˙2

(24)

now it is possible to define the second control input τ which will modify the orientation dynamics in order to finally make the transition change of the aircraft, this control as can be seen, it is easier to define since the dynamics of this subsystem has been considered only with the control input, therefore, τ control is proposed as

since said before, in this case x3 and x4 states are independent of x1 and x2 states, and because it is well known that the structure of that subsystem is GAS then it will be analyzed the stability of the two first states, proposing the Lyapunov’s candidate function as

τ = −kθ (θ − θd ) − kq (q − qd )

(15)

1 1 V (x) = x12 + x22 , 2 2

T = −σ (u − ud ) + g sin θ − f1 (u, w) + u˙d .

(16)

and

(25)

where with this, the orientation subsystem can follow a safe trajectory such that it does not cause instability or undesirable behavior in the translation subsystem since it has the characteristic of being a slower system as mentioned above. As it can be seen, these two input controls are not controlling directly the state w, but because of defining a θd and a ud , it can be ensure an stable w state behavior. Theorem 1: Being the system dynamics (11)-(14) where u 6= 0. And applying the control inputs (15) and (16), where σ (·) is a linear saturation function for given L, M positive values such that L < M, with ud and θd continuous differentiable functions as the desired state values. Then, for a set of initial conditions u, w according to the actual system restrictions, the system will converge to the desired values in a finite time. Proof: Applying a coordinate change over the original system with

V˙ (x) = x1 x˙1 + x2 x˙2 after applying the substitution it obtains V˙ (x) = − x1 σ (x1 ) + x2 ( f2 (x1 + ud , x2 + wd ) + g cos(θd ) − w˙ d ), as the states x3 and x4 where demonstrated to be GAS, it is possible to suppose that they will be converge to zero in a finite time, so this states and qd (angular speed) values can be expected to be zero. Now, as the definition of σ , the argument x1 σ (x1 ) is always positive ∀x 6= 0. And as a property of f2 (·, ·), where the sign of the function is the counter of the input argument x2 and assuming that w˙ d ≈ 0, because of θd is known, it can be supposed that g cos θd is always positive, so, for ensure stability in x2 , | f2 (·, ·) |> g cos θd . Demonstrating local asymptotically stability.

x1

= u − ud

IV. S IMULATION E XPERIMENTS

x˙1

= u˙ − u˙d

x2

= w − wd

x˙2

= w˙ − w˙ d

In this Section a MATLAB simulation is performed, where two experiments are conducted: transition from hover to cruise mode and vice-versa. The results are explained next.

x3

= θ − θd

x4

= x˙3 = q − qd

A. Hover to cruise flight mode

the new new system has the form f1 (x1 + ud , x2 +wd ) + T − g sin(x3 + θd )

x˙1

=

x˙2

=

x˙3

= x4

(19)

x˙4

= τ

(20)

− (x4 + qd )(x2 + wd ) − u˙d f2 (x1 + ud , x2 +wd ) + g cos(x3 + θd ) + (x4 + qd )(x1 + ud ) − w˙ d

(17) (18)

As said before, during hovering, the initial conditions before the transition considers that (u, w) are approximated to zero and θ = 90. Applying this to the simulation the results are shown in Fig. 7 where θ , α and τ control evolution over time from the hover mode to the cruise mode set at 12 degrees approximately. Fig. 8 depicts the thrust control as the u and w evolve over time. According to the conditions established for the hover mode, at he end of the transition it is achieved u >> w, this is due to the conditions of cruise flight mode.

B. Cruise to hover flight mode 1 0.8 Units/sec

Now the simulation results for the opposite flight transition are presented with u = 1, w ≈ 0 and θ ≈ 12 as the initial conditions. Fig. 9 shows the system evolution w.r.t. the desired angle position and its respective AoA; in Fig. 10 it can be see that the velocities decrease at ≈ 0 which corresponds to the hover condition.

u ud

0.6 0.4 0.2 0

0

5

10

15

0.1 100

Degrees

60

Units/sec

θ θd

80

0.08

0.04 0.02

40

0

20 0

w wd

0.06

0 0

5

10

5

10

15

10

15

15 10 8 Thrust

α [Degrees]

6

4

6 4 2

2

0 0

5 time [sec]

0

5

10

15

Fig. 8: u and w velocities during the transition from hover to cruise flight mode, also the thrust output control is presented.

0.4

100

0

80

−0.2 0

5

10

15

Degrees

τ

0.2

θ θd

60 40

time [sec] 20 0

Fig. 7: θ and α angle evolution according to the θd signal, also presents the control output τ.

0

5

10

15

0

5

10

15

0

5

10

15

8

In this paper a simple control strategy for the transition maneuver of the tail-sitter UAV is proposed. Such controller is based on the time-scale separation and on saturation functions. The design is based on Lyapunov approach. Simulations demonstrate the effectiveness of the controller for achieving transition from hover to cruise mode and viceversa. It is important to mention that the controller have the peculiarity that it does not present any switching, it is smooth and it take into account the saturation limits imposed by the actuators. Such characteristics are useful for implementation in a real UAV. In a further work, we can execute a real experiment with the physical prototype of the UAV and check the behavior of the real system, also it could be possible to design a more robust control that consider external disturbance as part of the modeling, such as wind gusts terms. Also the control can be completed by analyzing the 6-DOF and finally perform a full real simulation of the transition control.

4 2 0 −2

4 3 2 τ

V. C ONCLUSIONS

α [Degrees]

6

1 0 −1

time [sec]

Fig. 9: θ and α angle evolution according to the θd signal, also presents the control output τ. R EFERENCES [1] C. Papachristos, K. Alexis, and A. Tzes, “Hybrid model predictive flight mode conversion control of unmanned quad-tiltrotors,” in 2013

1

u ud

Units/sec

0.8 0.6 0.4

[12] [13]

0.2 0

[14] 0

5

10

15

[15]

Units/sec

0.15

w wd

0.1

0.05

[16] [17]

0 0

5

10

15

[18] 10

Thrust

8 6 4 2 0

5

10

15

time [sec]

Fig. 10: u and w velocities during the transition from cruise to hover flight mode, with its respective thrust output control.

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