A Total Energy Control System Design for the Transition Phase of a Tiltrotor Aerial Vehicle

2nd IFAC Workshop on Research, Education and Development of Unmanned Aerial Systems November 20-22, 2013. Compiegne, France

A Total Energy Control System Design for the Transition Phase of a Tiltrotor Aerial Vehicle ? Rogelio G. Hern´ andez-Garc´ıa ∗ Hugo Rodr´ıguez-Cort´ es ∗ ∗

Department of Electrical Engineering, Mechatronics Section, CINVESTAV-IPN, Mexico, D.F., 07360; (e-mail: [email protected], [email protected]).

Abstract: This paper presents a total energy control system design to take-off and transition flight for a tiltrotor aerial vehicle. The aerodynamic effect produced by the wake of the rotors on the wing and the horizontal stabilizer is taken into consideration. The lift, drag and pitch moment coefficients are modeled for angle of attack variations between one hundred eigthy degrees and minus one hunded eighty degrees based on preliminary wing tunnel tests. Numerical simulations are provided to verify the effectiveness of the proposed control strategy. 1. INTRODUCTION

gravity, the axis of rotation of the aerodynamic actuators can be aligned parallel to the longitudinal axis of the aircraft, becoming an airplane. Aerodynamic actuators with the special feature of functioning as rotors of helicopter and propellers of fixed-wing aircraft, are called proprotors Johnson [1974].

Aerial robotics represents one of the most dynamic markets in the aerospace industry. According to an estimate of Teal Group Co. 1 the total investment worldwide for development, research, testing and evaluation of these devices will be around $89 billion in this decade. In Mexico, there have been concrete manifestations of interest on this kind of technology, in both private companies and government, for aerial surveillance, aerial photography, coastal safeguard and forest monitoring.

Although the tiltrotor aerial vehicle combines the features of the helicopter and those of the fixed-wing aircraft, the flight controls are not combined. Each set of controls enter in function according to the operation mode. During helicopter mode, the flight controls are the cyclic and collective drivers that allow longitudinal and lateral as well as vertical motion Giulianetti and Dugan [2000]. This is possible thanks to the swashplate of each proprotor. Finally, yaw motion can produced by modifying, through the cyclic actuator, the plane of rotation of each proprotor. During airplane mode, there is a rudder that controls the yaw motion of the aircraft, an elevator that makes the aircraft to climb or descend (pitch) and a set of ailerons that control the roll motion of the aircraft.

According to Mettler [2003], the successful development of an autonomous aerial vehicle requires the solution of many different complex problems in engineering, such a modeling, control design, computer design and mechanical design. In this paper we address engineering problems related to modeling and control design for the development of a tiltrotor aircraft. The tiltrotor aircraft combines the advantages of the helicopter, such as hovering and vertical take-off and landing with those of an fixed wing aircraft, such as high speed and large autonomy. However, the tiltrotor is classified under the category of vertical takeoff and landing (VTOL) vehicles.

The critical flight phase for the tiltrotor aerial vehicle is the transition form helicopter mode to fixed-wing aircraft mode and vice verza. Principal problem of that aircraft is the phase transition. This flight phase is critical due to many reasons. One of them is that the flow induced by the proprotors, which is highly turbulent and therefore difficult to model, is interfered by the wing; in helicopter mode, the wing behaves like a quasi-flat plate that impedes the free flow of the wake, causing a resistance force in the direction of the gravity vector. The wake direction is modified as the aircraft performs the transition maneuver, attenuating gradually this wake effect on the wing. In airplane mode, the axes of rotation of the proprotors are almost or completely parallel to the longitudinal axis of the aircraft. In this configuration, the induced flow increases the air flow through the wing unfortunaletly with high vorticity. These aerodynamic phenomenon, as well as the pivoting of the nacelles of the proprotors which cause a shift of gravity center (CG) during the transition phase. Another reason for instability during the transition phase

This convertible aircraft is equipped with a conventional fixed-wing and powered by a pair of aerodynamic actuators, usually located on the tips of the wing. These aerodynamic actuators are mounted on pivoted engine nacelles that allow adjusting the direction of their thrust vector. For vertical take-off, the pilot aligns the axis of rotation of the aerodynamic actuators parallel to the gravity vector direction, in this configuration the aircraft can take-off as a helicopter. Once in the air, the thrust of the aerodynamic actuators can be tilted around the lateral axis of the aircraft, making appear forces that drive the aircraft forward. When the forward speed is enough for the wing to generate the lift necessary to counteract the force of ? This work was partially funded by CONACyT through the PNPC2013 program. 1 http://tealgroup.co/index.php/teal-group-products/interactivecatalog

978-3-902823-57-1/2013 © IFAC

52

10.3182/20131120-3-FR-4045.00043

IFAC RED-UAS 2013 November 20-22, 2013. Compiegne, France

is caused by the gyroscopic moments due to the proprotors, whose inertia is important with respect to the total inertia of the aircraft Haixu et al. [2010]. There are reported some dynamic models for the tiltrotor aerial vehicle. Miller and Narkiewicz [2006] presents a dynamic model which is not tested for the transition phase. For numerical simulations their model uses design parameters from the V-22 aircraft, although many of them had to be guessed due to lack of information. The model is not validated experimentally. However, the authors concluded that their model works correctly based on the numerical simulations. In Peng et al. [2010] a longitudinal dynamic model for the tiltrotor aerial vehicle in the transition phase is reported. The authors also propose an automatic control design based on path-tracking gain scheduling by Zero Phase Error Tracking Control. The proposed controller is tested by means of numerical simulations.

Fig. 1. External moments and forces applied to tiltrotor aircraft. Yong; and Jihong [2008] aircraft, Iyy is the inertia around the lateral axis of the aircraft and the notation cos(x) = cx and sin(x) = sx has been introduced. Moreover, X, Z are the forces along the longitudinal and verticals axes and M is the moment around the lateral axis of the aircraft, this is, X = FTxb − mgsθ − Lsα − Dcα Z = −FTzb + mgcθ + Lcα − Dsα M = −zH sη FTxb + (xR − zH cη )FTzb + Mac
with α = arctan w u the angle of attack. From Figure 1 is easy to verify that FTxb = T c(η−a1s ) FTzb = −T s(η−a1s ) with T the thrust generated by the proprotor, η the tilt angle of the nacelle of the proprotor and a1s the longitudinal cyclic angle of the tip path plane of the proprotor.

In this work, we propose a total energy control system design for the transition phase of a tiltrotor aircraft. The total energy control system strategy was proposed in Lambregts [1983b,c,a] for fixed-wing aircrafts and is based on careful selected outputs related to the change rates of the Hamiltonian and Lagrangian functions of the aircraft. We propose a local stability analysis of the closed-loop system. Numerical simulations show the successful transition from helicopter mode to fixed-wing aircraft mode. Control strategies based on TECS have been successfully used in industrial applications for fixed-wing aircraft Bruce [1987]. In the case of helicopters, control strategies based on TECS have been proposed but only tested in simulation Chen et al. [2007]. The implementation of this control technique for a hybrid aircraft, like the tiltrotor, to the best knowledge of the authors has not been proposed in the literature.

In order to design the total energy based control strategy, we write the longitudinal tiltrotor aircraft dynamics in wind axes coordinates, this is, mV˙ = T c(α+η−a1s ) − D − mgsγ mV γ˙ = −T s(α+η−a1s ) − L − mgcγ (2) θ˙ = q Iyy q˙ = M √ where V = u2 + w2 is the aircraft velocity and γ = θ − α is the flight path angle. It is assumed that the aerodynamic forces and moments appearing in equation (2) have the following structure

The remainder of this paper is organized as follows. Section 2 presents the longitudinal dynamic model of the tiltrotor aircraft. Section 3 introduces the control strategy for automatic take-off and transition maneuver from helicopter mode to fixed-wing aircraft mode. Section 4 verify the stability properties of the closed–loop dynamics at different trim points. Section 5 validates the derived TECS controller by means of numerical simulations, and finally, Section 6 gives some concluding remarks and future perspectives. 2. LONGITUDINAL DYNAMIC MODEL OF THE TILTROTOR AIRCRAFT

1 ρ(V + vi )2 SCL (α, q, δe ) 2 1 D = ρ(V + vi )2 SCD (α, q, δe ) 2 1 Mac = ρ(V + vi )2 S¯ cCM (α, q, δe ) 2 where ρ is the air density, c¯ is the wing aerodynamic mean chord, S is the wing area, δe is the elevator control input, vi is the induced velocity by the proprotor. CL , CD , CM are the lift, drag and pitch moment coefficients respectively. L=

Under the assumption that the tiltrotor aircraft behaves as a rigid body, the differential equations that describe its longitudinal dynamics, in the body reference system, are the following u˙ = X/m − gsθ − qw w˙ = Z/m + gcθ + qu θ˙ = q

(1)

q˙ = M/Iyy

2.1 Rotor inflow dynamics

where u, w are the longitudinal and vertical velocities, θ is the pitch angle, q is the pitch angular velocity, g is the gravity acceleration constant, m is the mass of the

The inflow dynamics result from the highly turbulent velocity vector field produced by the airflow passing 53

IFAC RED-UAS 2013 November 20-22, 2013. Compiegne, France

where V˜˙ = V˙ d − V˙ and γ˜ = γ − γd . The TECS strategy described in Lambregts [1983b,c,a] defines the control signals as follows

through the proprotor. This velocity vector field is threedimensional and can be decomposed into normal and parallel, to the plane of rotation, components. Due to the high turbulence, the airflow is not uniformily distributed along the tip path plane described by the proprotors, moreover this airflow is time varying Leishman [2000].

Z θ0 = KEp He + KEi

The inflow dynamics can be characterized by the kinetic energy that the proprotor transfers to air mass under its influence. This kinetic energy is translated into a velocity known as induced velocity Prouty [1995]. It can be calculated by means of ! r ΩRab σ 32θ0 vi = −1 + 1 + (3) 16 ab σ

He (τ )dτ

Z

0 t

a1s = KLp Le + KLi

(8) Le (τ )dτ + kpa θ + kda q

0

and Z δe = KLp Le + KLi

t

Le (τ )dτ + kpδ θ + kdδ q

(9)

0

with KEp , KEi , KLp , KLi , kpa , kda , kdδ and kpδ are positive gains. Due to the fact that He and Le depend on the control inputs, θ0 and a1s , the TECS strategy for the tiltrotor aircraft in (8) produces an implicit definition of the control inputs θ0 and a1s . In order to construct an explicit definition for these control inputs, we exploit the fact that the tiltrotor aircraft translational dynamics can be rewriten in the following form V˙ D + sγ = Ep θ0 c(α+η−a1s ) − g mg (10) L V γ˙ + cγ = −Ep θ0 s(α+η−a1s ) − g mg then, the energy rate and energy distribution errors can be synthesized as follows D V˙ d + sγ + sγ˜ − Ep θ0 c(α+η−ˆa1s −kpa θ−kda q) + He = g mg V˙ d D Le = + sγ − sγ˜ − Ep θ0 c(α+η−ˆa1s −kpa θ−kda q) + g mg (11) where we have defined a1s = a ˆ1s + kpa θ + kda q (12) Hence, from the first two equations of (8) and equation (11) it is possible to define " ! 1 V˜ D a ˆ1s = KLp kV + sγ − sγ˜ + 1 + KLp g mg +KLi η2 ] " ! 1 V˜ D θ0 = KEp kV + sγ + sγ˜ + 1 + Ep KEp g mg +KEi η1 ] (13) ˜ where it has been considered that V˙ d = kV Vg with kV a positive gain and η˙ 1 = He η˙ 2 = Le The proportional derivative controllers on the pitch angle (12) and (9) are designed to stabilize the tiltrotor aircraft rotational dynamics. Notice that the controller (12) has authority during the helicopter mode and it is disabled when η < 60o ; while the controller (9) has authority during the fixed-wing aircraft mode.

where Ω is the angular velocity of the proprotor, R is the proprotor radius, ab is the profile blade slope, σ is the proprotor solidity and θ0 is the pitch angle of the blade. We assume that the thrust can be modeled by T = Ep mgθ0

t

(4)

where Ep > 1 is the power excess of the power plant with respect to the aircraft total weight. 3. CONTROL DESIGN FOR AUTOMATIC TRANSITION OF THE TILTROTOR AIRCRAFT The design of the controller is based on the total energy control system (TECS). TECS was introduced in Lambregts [1983b,c,a]. It is based on the properties of the time derivatives of the Hamiltonian and Lagrangian functions of the aircraft translational dynamics. The Hamiltonian and Lagrangian functions are defined as Z t 1 H = mV 2 + mg V (τ )sγ (τ )dτ 2 Z0 t (5) 1 L = mV 2 − mg V (τ )sγ (τ )dτ 2 0 The Hamiltonian and Lagrangian time derivatives are ! ˙ V H˙ = mgV + sγ g ! (6) ˙ V L˙ = mgV − sγ g According to Lambregts, the time derivative of the Hamiltonian function is proportional to the rate of total energy; while the time derivative of the Lagrangian function is proportional to the energy distribution. From (6) we can deduce that the rate of total energy and the energy distribution are proportional to the time derivative of the Hamiltonian and Lagrangian time derivatives respectively, with mgV the proportionality term. Since TECS is based on driving the energy rate and the energy distribution to desired values, the energy rate and energy distribution errors are defined as follows V˜˙ + sγ˜ He = g (7) V˜˙ Le = − sγ˜ g

The TECS strategy synthetized by equation (9) and (13) is able to drive the tiltrotor aircraft to desired references for Vd and γd and allows to keep the nacelle angle η as 54

IFAC RED-UAS 2013 November 20-22, 2013. Compiegne, France

an extra control input. In order to be able to perform the transition from helicopter mode to fixed-wing aircraft mode and viceverza it is necessary to correctly define the desired references for Vd , γd and η.

1.5

1

0.5

0

Cl

4. A PREELIMINARY CLOSED–LOOP STABILTY ANALYSIS

−0.5

−1

Note that the tiltrotor aircraft dynamics (2) in closed– loop with the controller defined by equations (9) and (13) is described by a differential equation of the form Angle of attack (α) [degrees] χ˙ = f (χ, t) (14) Fig. 2. CL vs. angle of attack (α). Number of Reynolds of where  > 148500. χ = V˜ γ˜ θ q η1 η2 and   D ˜ k V − E gθ c − − gs V p 0 (α+ηd −a1s ) (γd −˜ γ)   m      1 L  γ˙ d − −Ep gθ0 s(α+ηd −a1s ) − − gc(γd −˜γ )    m Vd − V˜     q        1 f =   (xR s(ηd −a1s ) − zH sa1s )Ep mgθ0 + Mac   Iyy     ˜ V D   Angle of attack (α) [degrees] kV + s(γd −˜γ ) + sγ˜ − Ep θ0 c(α+η−a1s ) +     g mg     V˜ D Fig. 3. CD vs. angle of attack (α). Number of Reynolds of kV + s(γd −˜γ ) − sγ˜ − Ep θ0 c(α+η−a1s ) + g mg 148500. Notice that in (14), the tilt angle of the nacelles is now considered as an additional control input. −1.5

−2 −200

−150

−100

−50

0

50

100

150

200

−150

−100

−50

0

50

100

150

200

−150

−100

−50

0

50

100

150

200

4

3.5

3

Cd

2.5

2

1.5

1

0.5

0 −200

0.8

0.6

Unfortunately, the closed–loop dynamics is quite nonlinear so that it will be difficult to find a Lyapunov function to show its stability properties. In this paper, we propose to establish some stability properties of the closed–loop dynamics relaying in stability tools for linear systems. For, we compute the equilibrium points of the closed–loop dynamics, which are given by the solution of the following set of nonlinear algebraic equations ¯ D −Ep g θ¯0 c(α+η − gsγd = 0 ¯ a1s ) − d −¯ m   ¯ L 1 −Ep g θ¯0 s(α+η − − gc = 0 γ˙ d − γd ¯ a1s ) d −¯ Vd m q¯ = 0

0.4

Cm

0.2

0

−0.2

−0.4

−0.6

−0.8 −200

Angle of attack (α) [degrees]

Fig. 4. Cm vs. angle of attack (α). Number of Reynolds of 148500. mode the tiltrotor will be flying at an angle of attack α = −π/2. This flight condition makes necessary to have models for the aerodynamic forces and moments at high angles of attack. In Crespo Pacheco [2011] the drag, lift and pitch moment coefficients for the wing profile NACA 8H12, assuming an infinite aspect ratio, for angles of attack from −π to π have been measured by means of a wind tunnel test. These aerodynamic coefficients are shown in Figures 2, 3 and 4, respectively. In this paper we assume that these coefficients are equal to the aerodynamic coefficients of the tiltrotor aircraft.

 1  ¯ ac = 0 (xR s(ηd −¯a1s ) − zH sa¯1s )Ep mg θ¯0 + M Iyy ¯ D sγd − Ep θ¯0 c(α+η = 0 ¯ a1s ) + d −¯ mg where   ¯  1 D a ¯1s = KLp sγd + 1 + KLp mg +KLi η¯2 ] (15)   ¯  1 D ¯ θ0 = KEp sγd + 1 + Ep KEp mg +KEi η¯1 ] The flight envelope of the tiltrotor aircraft requires that the aerodynamic forces and moments can be computed for high angles of attack. For instance, during the helicopter

Having the aerodynamic coefficients, the equilibrium points for three conditions were computed. There flight conditions are vertical climb (helicopter mode), transition from helicopter mode to fixed-wing aircraft mode, with a flight angle path of 45o and straight and level flight (fixedwing mode). The equilibrium points are summarized in Tables 1, 2 and 3. 55

IFAC RED-UAS 2013 November 20-22, 2013. Compiegne, France

Table 1. Operation values at the trim points ηd π/2 π/4 0

γd -π/2 π/4 0

αd -π/2 −π/4 0

0.6

Pitch angle ( θ ) [rad]

Operation Climb Transition Level flight

Vd 2 0 15

0.4 0.2 0 −0.2 0

10

20

30

40

50

60

70

80

90

50

60

70

80

90

Trajectory angle (γ ) [rad]

time [s]

Table 2. Aerodynamic coefficients at the trim points Operation Climb Transition Level flight

¯L C 0.0512 1.3229 0.1792

¯D C 3.4994 1.6217 0.01707

¯M C 0.6145 −0.1707 0.0204

Level flight

0.60807

−0.66627

−0.71378

δ¯e

−0.5245

−0.2376

0.07832

a ¯∗1s

−0.002555

−0.0603

−0.00762

Horizontal velocity, u [m/s]

Transition

As a preeliminary stability analysis we have verified that the linearized model of the closed–loop dynamics has all the eigenvalues with negative real part. It is clear that this is not a complete stability analysis as the linearized dynamics is time variant.

1 2 9600 πh

+ 12 π

20

30

40

time [s]

30 20 10 0 −10 0

10

20

30

40

50

60

70

80

90

50

60

70

80

90

time [s] 2.5 2 1.5 1 0.5 0 −0.5 −1

0

10

20

30

40

5

He [Nm/s]

4

3

2

1

0 0

10

20

30

40

time [s]

50

60

70

80

90

0 −1 −2

Le [Nm/s]

−

10

Fig. 6. (top) Vertical velocity, (bottom) horizontal velocity.

In order to verify the performance of the proposed controller numerical simulations are presented. The desired trajectories for the tilt angle of the nacelles, the flight path angle and the titlrotor aircraft velocity were defined as follows 1 1 ηd = 1728000 πh3 − 9600 πh2 + 12 π 1 3 1728000 πh

0 0

time [s]

5. NUMERICAL SIMULATIONS

γd =

1

0.5

Vertical velocity, w [m/s]

Climb

1.5

Fig. 5. (top) Pitch angle θ, (bottom) flight path angle γ.

Table 3. Control inputs at the trim points Control inputs θ¯0

2

(16)

−3 −4 −5 −6 −7



Climb velocity = 2m/s, Vd = Forward velocity = 15m/s where h is the height. The parameters of the tiltrotor aircraft are m = 4Kg, Iyy = 6.076Kg − m2 , c¯ = 0.36m, S = 0.85m2 , ρ = 0.9667kg/m3 , xR = 0.01m, n = 5400RP M , ab = 5.7rad−1 , b = 6, c = 0.05m and zH = 0.08m.

−8 −9 0

10

20

30

40

50

60

70

80

90

time [s]

Fig. 7. (top) Energy rate error, (bottom) energy distribution error. from the helicopter mode to the fixed-wing mode, which requires high horizontal velocity. Figure 7 shows the time responses of the energy rate and energy distribution errors. Note that they have considerable changes during the trasition phase. Finally, Figures 8 and 9 present the translational behavior of the aircraft. As it can be observed, in Figure 8, the aircraft horizontal displacement is almost null during the first 25sec, because the aircraft takes-off vertically. During the transition maneuver, the aircraft is moving horizontally. On the other hand, in Figure 9, it can be observed that the aircraft climbs until 55 meters during take-off. Then, it starts its transition maneuver with a higher ascensional rate. Finally, the aircraft begins its steady and level flight.

Figure 5 shows the time response of the pitch and flight path angles. Note that the tiltrotor aircraft performs a positive pitch oscillation (nose-up) during the first five seconds of the take off procedure. Then, the pitch angle stabilizes at 0.2rad. During transition (25-30 sec), the pitch angle of the aircraft increases. Finally, in the fixedwing mode the pitch angle stabilizes around 0.15 rad. On the other hand, it is important to observe that during take off (helicopter mode) the aircraft flights vertically, γ ∈ [0.8, 1.5] rad, to reach the vertical velocity of reference. Finally, during the fixed-wing mode the flight path angle converges to zero.

6. CONCLUSION

Figure 6 shows the time responses of the vertical and horizontal velocities. As it can be observed, the aircraft climbs with almost zero horizontal velocity and vertical velocity at 2 m/s, the desired reference. During the transition phase, the vertical velocity decreases while the horizontal velocity increases as it was expected. The aircraft is going

We have presented a control strategy based on the Total Energy Control System for a tiltrotor aircraft. Since, the resulting closed–loop dynamics is described by a highly nonlinear system we have proposed some preeliminary 56

IFAC RED-UAS 2013 November 20-22, 2013. Compiegne, France

Omar Crespo Pacheco. Propuesta de dise˜ no aerodin´ amico de un aerogenerador de eje vertical. Bachelor thesis. Ing. Aeron´autica, Escuela Superior de Ingenier´ıa Mec´ anica y El´ectrica. Instituto Polit´ecnico Nacional, February 2011. M. D. Maisel; D. J. Giulianetti and D.C. Dugan. The history of the xv-15 tilt rotor research aircraft: From concept to flight. In NASA, editor, Monography, volume 4 of 5, chapter 8, page 222. NASA, 2000. Li Haixu, Qu Xiangju, and Wang Weijun. Multi-body motion modeling and simulation for tilt rotor aircraft. Chinese Journal of Aeronautics, 23(4):415 – 422, 2010. ISSN 1000-9361. Wayne Johnson. Dynamics of tilting proprotor aircraft in cruise flight. Technical report, NASA D-7677, 1974. AA Lambregts. Functional integration of vertical flight path and speed control using energy principles. In Proc. 1st Annu. NASA Aircraft Controls Workshop, pages 389–409, 1983a. AA Lambregts. Vertical flight path and speed control autopilot design using total energy principles. AIAA paper, 83, 1983b. AA Lambregts. Integrated system design for flight and propulsion control using total energy principles. In American Institute of Aeronautics and Astronautics, Aircraft Design, Systems and Technology Meeting, Fort Worth, TX, volume 17, 1983c. J. Gordon Leishman. Principles of helicopter aerodynamics. Cambridge aerospace series. Cambridge University Press, Cambridge, New York, 2000. ISBN 0-521-660602. Bernard Mettler. Identification modeling and characteristics of miniature rotorcraft. Kluwer Academic Publishers, 2003. M. Miller and J. Narkiewicz. Tiltrotor modelling for simulation in various flight conditions. Journal of theoretical and applied mechanics, (2):881–906, 2006. Chih-Cheng Peng, Thong-Shing Hwang, Shiaw-Wu Chen, Ching-Yi Chang, Yi-Ciao Lin, Yao-Ting Wu, Yi-Jing Lin, and Wei-Ren Lai. Zpetc path-tracking gainscheduling design and real-time multi-task flight simulation for the automatic transition of tilt-rotor aircraft. In Robotics Automation and Mechatronics (RAM), 2010 IEEE Conference on, pages 118–123, 2010. doi: 10.1109/RAMECH.2010.5513203. Raymond W Prouty. Helicopter performance, stability, and control. 1995. Yang Xili; Fan Yong; and Zhu Jihong. Transition flight control of two vertical/short takeoff and landing aircraft. Journal of Guidance, Control, and Dynamics, 31(2), 2008.

1000 900 800

Displacement [m]

700 600 500 400 300 200 100 0 0

10

20

30

40

50

60

70

80

90

70

80

90

time [s]

Fig. 8. Translational aircraft movement. 180 160 140 120

Height [m]

100 80 60 40 20 0

−20 0

10

20

30

40

50

60

time [s]

Fig. 9. Aircraft vertical displacement.

Fig. 10. Prototype tiltrotor aircraft stability conclusions based on linear systems theory. The validity of the proposed control strategy has been tested by means of numerical simulations. Right now, we are working to establish more precise stability conclusions of the resulting closed–loop dynamics as well as on the design of a tiltrotor aircraft to test experimentally the proposed controller. The first version of our tiltrotor aircraft is shown in Figure 10. REFERENCES Kevin R. Bruce. NASA B737 flight test results of the total energy control system [microform] : final report / Kevin R. Bruce. National Aeronautics and Space Administration ; National Technical Information Service, distributor [Washington, DC : Springfield, Va, 1987. Sheng-Wen Chen, Pang-Chia Chen, Ciann-Dong Yang, and Yaug-Fea Jeng. Total energy control system for helicopter flight/propulsion integrated controller design. Journal of guidance, control, and dynamics, 30(4):1030– 1039, 2007. 57