Transition Control for a fixed-wing Vertical Take-Off and Landing Aircraft*

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Transition Control for a fixed-wing Vertical Take-Off and Landing Aircraft ⋆ P. Casau D. Cabecinhas C. Silvestre Instituto de Sistemas e Rob´ otica, Instituto Superior T´ecnico, Lisbon, Portugal (e-mail: [pcasau, dcabecinhas, cjs]@isr.ist.utl.pt). Abstract: This paper addresses the problem of autonomous transition between hover and level flight for a model-scale fixed-wing Unmanned Air Vehicle (UAV). The UAV flight envelope is divided into hover, transition and level flight. Each flight region employs different controllers and the overall system is robust provided that there exists appropriate switching logic. Classic linear optimal control techniques provide stabilization in hover and level flight as well as reference tracking during the transition between these two distinct regions of the flight envelope. The reference trajectories are generated with some physical insight about the system, using very few parameters. Simulation results demonstrate the performance and robustness of the proposed solution, namely under the effect of sensor noise. 1. INTRODUCTION In recent years, there has been an increasing interest in developing and using Unmanned Air Vehicles (UAVs) as tools for ocean surface data acquisition. However, the use of UAVs for ocean applications is still limited to a few scientific institutions scattered worldwide, and most vehicles have been designed to conduct simple survey missions that in general do not require close interaction between the operator and the environment. It is by now felt that the effective use of UAVs in demanding marine science applications must be clearly demonstrated, namely by evaluating the system in terms of adaptability to different missions scenarios, maritime launch and recovery, survivability, autonomy, endurance, payload performance and usability, and system integration with the existent marine science instrumentation. Meeting these stringent requirements poses considerable challenges namely on the design of fixed wing robotic air vehicles that can be easily used by marine scientists in the confined environment of an opportunity ship. Recent developments described in Green and Oh (2005), Adrian et al. (2007) and Desbiens et al. (2010) have shown that fixed-wing Vertical Take-Off and Landing (VTOL) aircrafts can perform both long endurance missions and precise maneuvering within exiguous environments. The versatility of such aircrafts combines helicopter precise trajectory tracking with conventional fixed-wing airplanes ability to cover large distances, delivering a final solution which largely exceeds the capabilities of its predecessors. However, the problem of achieving robust transitions between hover and level flight is difficult for its exquisite dynamics and relies on open-loop maneuvering once too often. The very different aircraft dynamics between hover and leveled flights suggest that supervisory control is a plausible solution for the given problem. Similar methodologies, like the ones described in Goebel et al. (2009) and Sanfelice and Teel (2007), have been successfully employed in a variety of applications. Controller switching during ⋆ This work was partially supported by Funda¸c˜ ao para a Ciˆ encia e a Tecnologia (ISR/IST plurianual funding) through the PIDDAC Program funds and by Project PTDC/MAR/64546/2006 OBSERVFLY. The work of D. Cabecinhas and P. Casau was supported with grants SFRH/BD/31439/2006 and SFRH/BD/70656/2010, from Funda¸ca ˜o para a Ciˆ encia e a Tecnologia.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

operating mode transitions adds discrete behavior to the continuous UAV model, creating a new layer of complexity that must be dealt with appropriately. Systems displaying both continuous and discrete behavior have been under an intense research effort over the last decade. This study has given rise to several concepts such as hybrid automata, see Marconi et al. (2009), and switched systems, see Liberzon and Morse (1999), which fall within the broader category of Hybrid Dynamical Systems, described in Goebel et al. (2009). The discrete behavior built into these systems may appear naturally for certain applications, such as UAV landing and take-off (see e.g. Gentili et al. (2008) and Cabecinhas et al. (2010)), but may also be the consequence of digital control or supervisory control, see Koutsoukos et al. (2000). The solution proposed in this paper employs supervisory control by dividing the aircraft flight envelope into the hover, transition and level flight regions. Linear optimal control techniques are employed for system stabilization in hover and level flight as well as reference tracking during transition. The linear controller provides robust transition between the two disjoint sets which characterize the hover and level flight regions of the flight envelope. This paper is organized as follows. Section 2 presents the 3-dimensional UAV nonlinear model, explaining thoroughly the gravity, propeller and aerodynamic interactions with the aircraft body. Section 3 justifies the overall system stability and robustness. Section 4 presents the controller structures which are employed for local stabilization and reference tracking. Finally, Section 5 presents the simulation results which are obtained by considering the existence of additive Gaussian white noise in the state measurements. 2. UAV NONLINEAR MODEL The UAV under analysis is the model-scale fixed wing aircraft, depicted in Figure 1a, which has a total of six actuators: two propellers, rudder, elevator, ailerons and flaps. The UAV nonlinear model described in this section is represented by the nonlinear differential equations ξ˙ = f (ξ, µ) system state ξ ∈ R14 and actuator input µ ∈ R6 given by

7250

10.3182/20110828-6-IT-1002.02711

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

7251

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

where subscript i ∈ {1, 2} in the formulæ above distinguishes each propeller , with the sign of the propeller’s moments depending on the direction of rotation. The total thrust and torque produced by the propellers is given by # " Tp1 + Tp2 0 (18) , mp =mp1 + mp2 . fp = 0 The aerodynamic forces are generated from the propeller slipstream flow and the free-stream flow. The two contributions are calculated separately and combined together in the end using superposition (this modeling strategy was also employed by Adrian et al. (2007)). Under this assumption, the propeller slipstream velocity up is given by p up = (8T )/(ρπd2 ), considering a steady, incompressible and inviscid flow. The lifting surfaces’ lift Li and drag Di is given by the following equations for any vB . iB ≥ 0 1 1 Li = ρAi kvB k2 CLi , Di = ρAi kvB k2 CDi , (19) 2 2 where Ai is the surface’s planform area. These aerodynamic forces greatly depend on the surface’s Coefficient of Lift (CL ) and the Coefficient of Drag (CD ), which are described by

CL2 , (20) πAe where α = arctan(w/u) is the surface’s angle of attack 2 , δj is the actuator deflection, CD0 is the parasitic Coefficient of Drag, A is the aspect ratio, and e is the Oswald’s efficiency. The Coefficient of Lift is given by  CLα α + CLδj δj , if − CLmax ≤ CL ≤ CLmax CL = . 0, otherwise (21) The coefficient CL is lower bounded at −CLmax and upper bounded at CLmax . These limits induce loss of lift (stall) at angles of attack such that α ∈ / [α, α] where CLmax − CLδj δj CLmax + CLδj δj , α(δj ) = − . α(δj ) = CLα CLα (22) CL = CL (α, δj ),

mw = rw × [0 0 Lw ]⊺ , (25) ⊺ mhs = rhs × [0 0 Lhs ] , (26) mvs = rvs × [0 Lvs 0]⊺ , (27) 1 Ma = − ρu2 Aa ra . jB CLδaw δa 2
1 (28) − ρ u2p1 + u2p2 Ap,a rp,a . jB CLδap,w δa , 2 1 Mdampq = − ρAp,hs CLαhs q(rp . iB − rhs . iB ) (up1 + up2 ) 2 1 − ρAhs CLαhs (rhs . iB )2 qu, (29) 2 1 Mdampr = − ρAp,vs CLαvs r(rp . iB − rvs . iB )(up1 + up2 ) 2 1 (30) − ρAvs CLβvs (rvs . iB )2 ru, 2 define the aileron moment, the elevator moment, the rudder moment, the wing moment and the damping moments due to pitch and yaw rotation, respectively. Substituting (7), (18), (19), and (24) into (4) and (5) one gets the aircraft dynamics and kinematics f (ξ, µ) thus concluding the UAV nonlinear model description. The system dynamics developed throughout this section enable the controller’s dimensioning and simulation provided in the following sections.

CD = CD0 +

One computes the wing lift Lw , the wing drag Dw , horizontal stabilizer’s lift Lhs and drag Dhs , the vertical stabilizer’s side force Lvs and drag Dvs , using the relations (20) and (19). The aerodynamic forces acting on the aircraft are given by " # Dw + Dhs + Dvs Lvs fa = (23) Lw + Lhs under the small angle approximation. The aerodynamic moment ma calculation requires the estimation of the aileron’s mean pressure center location ra , its slipstream mean pressure center rp,a location, the horizontal stabilizer’s aerodynamic center location rhs , the wing’s aerodynamic center location rw and the vertical stabilizer’s aerodynamic center location rvs . Given these parameters it is then computed by # " Ma (24) ma = mw + mhs + mvs + Mdampq , Mdampr where 2 For the vertical stabilizer the sideslip angle β = arcsin(v/kv k) is B used instead of the angle of attack

3. ROBUST TRANSITIONS The aircraft under consideration has vertical take-off and landing capabilities thus it is possible to divide the flight envelope into three distinct regions: hover, transition and level flight. Each region features its own distinct behavior thus requiring different controllers. The controllers used in hover, transition and level flight are identified by µH (ξ, ξ ⋆ (t)) ∈ U , µX (ξ, ξ ⋆ (t)) ∈ U , and µL (ξ, ξ ⋆ (t)) ∈ U , respectively, where U ⊂ R6 is expressed as U =[nmin , nmax ]2 × [δamin , δamax ] × [δemin , δemax ]× (31) [δrmin , δrmax ] × [δfmin , δfmax ]. and defines the actuator limits. The controllers are dependent on the current aircraft state ξ and some reference state trajectory ξ ⋆ (t). Moreover, each of these controllers is characterized by its domain of attraction which is used to define the different regions of the flight envelope. In this section we show that the overall system is robust provided that the maneuvers, in each operating mode, are locally tracked and that appropriate controller switching exists. For now, we assume that controllers exist for each operating mode. Their design is described in detail in the next section. The hover region of the flight envelope DH × U ⊂ R14 × R6 is defined around an equilibrium point with the following properties: • The orientation is described in the UEN reference frame by the Euler angles (φ, θ, ψ) = (0, 0, 0) [deg], i.e. the aircraft is facing the zenith; • The linear velocity is null, i.e. vB = [0 0 0]⊺ m/s. This equilibrium point specifications imply that the lifting surfaces do not produce lift at the equilibrium point and that the propellers’ thrust counteracts the gravity pull. The aircraft is locally stabilized around the hover equilibrium point (ξ H0 , µH0 ) ∈ R14 × R6 using the hover controller µH ∈ R6 such that DH is the controller’s domain of attraction.

7252

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

The level flight region of the flight envelope DL × U ⊂ R14 × R6 is defined around an equilibrium point with the following properties: • The angular velocity is null, i.e. ω B = [0 0 0]⊺ • The aircraft’s attitude is described in the NED reference frame by the Euler angles (φ, θ, ψ) = (0, 10, 0) [deg]; • Flap deflection is null, i.e. δf = 0◦ ; • The aircraft’s height is constant which in turn implies that α = θ; • The lateral velocity is null, i.e. v = 0. The remaining states defining the equilibrium point are computed a posteriori given the previous restrictions. It is very important to note that the pitch angle was not arbitrarily fixed, it corresponds to a worst-case scenario where the aircraft cruises at low speed and only 5 degrees away from stall 3 . In this flight region the aircraft behavior is very different from that of hover since it is the mainly the wing lift which is counteracting the gravity pull. Similarly to hover, the aircraft is locally stabilized near the level flight equilibrium point (ξ L0 , µL0 ) ∈ R14 × R6 using the level flight controller µL ∈ R6 such that DL is the controller’s basin of attraction. For any conventional fixed-wing VTOL aircraft, the two regions DL and DH are, most likely, disjoint. The transition region of the flight envelope connects these by means of a reference maneuver ξ ⋆X→L (t), defined for t ∈ [0, T ], such that for some ǫ and ǫX→L BǫX→L (ξ ⋆X→L (0)) ⊂ DH ,

Bǫ (ξ ⋆X→L (T )) ⊂ DL ,

where DL ⊆ DL 4 . Considering that switching from hover to transition occurs when ξ ∈ BǫX→L (ξ ⋆X→L (0)), that switching to level flight occurs when ξ ∈ DL and that the transition controller enables the aircraft state starting ǫX→L -close to the reference trajectory to be always ǫ-close to the reference trajectory (with ǫX→L ≤ ǫ) then controller switching is robust and does not compromise the stability of the overall system. Furthermore, the gap between the two sets DL and DL provides additional robustness to the system. The transition maneuvers generation relies on some intuitive insight about the aircraft system. First of all, it is desirable that the transition occurs on the vertical plane, therefore the references for the state variables which characterize the lateral motion are null, i.e. (v ⋆ (t), φ⋆ (t), ψ ⋆ (t)) = (0, 0, 0) for all t ≥ 0. Assuming that the aircraft motion occurs indeed in the vertical plane then its reference trajectory is defined by the triplet (u⋆ (t), w⋆ (t), θ⋆ (t)) for all t ∈ [0, T ]. During the transition from hover to level flight, the aircraft climbs upward before pitching down because the increased airflow (both from the propellers and the aircraft velocity) provides extra maneuverability and prevents the aircraft from ever entering stall. During the transition from level flight to hover, the aircraft velocity is kept high enough and is reduced to zero only when it is close to the hover equilibrium point, also preventing stall.

The previous considerations justify the system’s stability in hover, transition and level flight employing the controllers µH , µX and µL , respectively. Although only the transition from hover to level flight is discussed in depth, the converse maneuver from level flight to hover is dealt similarly, considering a ξ ⋆X→H maneuver and appropriate controller switching. 4. LINEAR QUADRATIC REGULATOR In this section we describe and specify the controllers µL , µX and µH used for each of the considered flight regions. To stabilize the aircraft around the equilibrium points proposed in Section 3 we use classic LQR techniques. The controllers obtained using this technique are inherently robust to measurement noise and disturbances and preset high gain and phase margins, Franklin et al. (1994). The local controllers provide the full state feedback control law ˜ where ξ˜ = ξ − ξ and µ ˜ = −Kξ, ˜ = µ − µ0 . These µ 0 sublevel sets are identified as DH and DL for the hover and level flight regions as described in Section 3. The chosen control structure is that of D-methodology, described in Kaminer et al. (1995). Following this methodology, all integrators are moved to the plant input, and differentiators are added where they are needed to preserve the transfer functions and the stability characteristics of the closed-loop system. The D-methodology implementation has several important features, which include the following: i) auto-trimming property - the controller automatically generates adequate trimming values for the actuation signals and for the state variables that are not required to track reference inputs; ii) the implementation of anti-windup schemes is straightforward, due to the placement of the integrators at the plant input; iii) the controller gain K can be scheduled discontinuously without producing discontinuous control inputs. Since there are 6 actuators available we can select 6 states to be driven to zero at steady-state, by means of integral action, see Kaminer et al. (1995). We denote these integral states by ξ. Each operating mode has a different operating point and specifications which require distinct weightings (Q, R). Therefore controller dimensioning requires: i) Linearization around the chosen operating point (ξ q0 , µq0 ); ii) Selection of integrator states (ξ) choice according to the operating mode requirements; iii) Controllability evaluation; iv) (Q, R) weighting using Bryson’s trial-and-error method, Franklin et al. (1994).

Figure 1 depicts the different regions of the flight envelope as well as a concept on the transition logic previously explained. 3 Setting the pitch angle to values higher than 10◦ would be even more demanding but the system would lack robustness 4 B (p) represents a ball of radius ǫ around the point p, i.e. the set ǫ of points x such that kx − pk < ǫ.

Fig. 1. Hover, transition and level flight regions of the flight envelope depicting the transition logic concept.

7253

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

The next section presents the numerical analysis on the proposed control method as well as simulation results. 5. SIMULATION RESULTS This section is divided into three different subsections which present the simulation results and numerical analysis performed in each individual operating mode. The nonlinear model presented in Section 2 was used given the parameter estimation which was performed on the real world model-scale UAV depicted in Figure 1a. In the simulations, the state measurements used for feedback are corrupted with additive zero-mean white noise. The standard deviation of the propeller rotation speed and aircraft velocity measurement errors are 10 rps and 0.1 m/s, respectively. The attitude of the vehicle, in roll, pitch, and yaw Euler angles, is corrupted by noises with standard deviation of 0.1◦ for the roll, pitch and yaw. Finally the angular velocity measurements are corrupted with a 0.05◦ /s standard deviation noise.

The controller structure introduced in Section 4 is employed with error states ξ˜L =[n1 , n2 , u − u⋆ (t), v, w, p, q − q ⋆ (t), r, φ, θ − θ⋆ (t), ψ, z]⊺ , (35) ⋆ ⋆ ⊺ ξ L =[u − u (t), v, φ, θ − θ (t), ψ] . The integrator states in the u ˜ and θ˜ stabilize the forward velocity and the pitch angle at some desired values with zero steady state error. The state variable z stabilizes the aircraft at a given height. Level flight control stabilizes the aircraft at the equilibrium point in (34) as long as its initial state lies within the level flight domain, ξ(0) ∈ DL . The simulations results depicted in Figure 3 verify the condition [nmin , nmax ]2 × R6 × Bφ (0) × BθL (0)× L

× Bψ (0) × R2 × R≤0 × U ⊆ DL L

for ψ(0) = ψ L and θ(0) = θL with ψ L = θL = 15◦ and φL = 180◦ .

5.1 Hover 5.3 Transition The chosen operating point for the Hover controller dimensioning is given by ξ H0 ≃[154.7 154.7 0 0 0 0 0 0 0 π/2 0 0 0 − 1]⊺ , (32) µH0 ≃[0.12 0.12 0 0 0 0]⊺ . The controller structure introduced in Section 4 is employed with ξ˜ =[n1 , n2 , u − u⋆ (t), v, w − w⋆ (t), p, H

q − q ⋆ (t), r, φ, θ − θ⋆ (t), ψ]⊺ (33) ξ H =[u − u⋆ (t), v, w − w⋆ (t), φ, θ − θ⋆ , ψ]⊺ . This particular choice of integrator states provides reference tracking (u⋆ , w⋆ , θ⋆ ) with zero steady state error as well as position stabilization. The hover control simulation depicted in Figure 2 demonstrates that the condition [nmin , nmax ]2 × R6 × Bφ (0) × BθH (0)× H

× Bψ (0) × R2 × R≤0 × U ⊆ DH H

is verified for ψ(0) = ψ H and θ(0) = θH with ψ H = θH = 15◦ and ψ H = 180◦ . 5.2 Level The chosen operating point for the Level controller dimensioning is given by ξ L0 ≃[112.4 112.4 10.8 0 1.9 0 0 0 0 π/18 0 − 1]⊺ , µH0 ≃[0.052 0.052 0 − 0.065 0 0]⊺ . (34)

X

q − q ⋆ (t), r, φ, θ − θ⋆ (t), ψ]⊺ (37) ⋆ ⋆ ⋆ ⊺ ξ X = [u − u (t), v, w − w (t), φ, θ − θ , ψ] , which identical to the design choice made for the hover controller. However, the transition flight controller spans a large set of operating points, which are defined by the reference maneuvers, thus requiring different LQR weightings. Simulation results show that the controller is able to stabilize the aircraft for the whole flight envelope. However, this feature comes at the cost of a more loosen reference tracking than that which is provided during hover and level flight. Transition: Hover to Level As stated in Section 3, the aircraft transition is to occur on the vertical plane thus 14

20

φ φ

12

15

θ θ

10

15

1.2

v v

1

w w

0

0

10 0

0

0.8 0.6 0.4 0.2 0

10

0

ψ ψ

5

0

0

Euler Angles [deg]

u u

Velocity [m/s]

1.4

Euler Angles [deg]

Velocity [m/s]

1.6

The transition flight does not have an equilibrium point in the same sense as level flight and hover but linearization around some equilibrium point is required in order to obtain the controller. The chosen operating point given by ξ X0 ≃[157.3 157.3 1 0 0 0 0 0 0 π/2 0 0 0 − 1]⊺ (36) µX0 ≃[0.12 0.12 0 0 0 0]⊺ is very similar to the hover operating point. However, its forward velocity of 1 m/s provides the linear model with characteristics which are present both in hover and level flight. Again, the controller structure described in Section 4 is employed with ξ˜ = [n1 , n2 , u − u⋆ (t), v, w − w⋆ (t), p,

u u

8

0

v v

6

0

w w

4

0

5 0 φ φ0

−5

2

−10

θ θ

0

−15

ψ ψ0

−5

0

−0.2 −0.4 0

2

4

6

Time [s]

8

10

−10 0

2

4

6

8

−2 0

10

2

4

6

Time [s]

Time [s]

Fig. 2. Hover simulation - Velocity and Euler Angles time evolution.

8

10

−20 0

2

4

6

8

10

Time [s]

Fig. 3. Level simulation - Velocity and Euler Angles time evolution.

7254

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

H

respectively. The lateral state variables are not presented because the deviations from the vertical plane xOz are negligible.

100

L

X

0

50 u(t) * u (t) w(t) * w (t) θ(t)

6. CONCLUSION

Pitch angle [deg]

Forward/Downward velocity [m/s]

20

*

θ (t) −20 0

2

4

6

8

0 10

Time [s]

Fig. 4. Transition from Hover to Level simulation - reference tracking.

H

X

0.3

L

REFERENCES

0 0.2 −5

−10

0.1

δe(t)

Input torque [N.m]

Elevator/Flap deflection [deg]

5

δ (t) f

τ (t) 1

τ (t) 2

0

2

4

6

The work presented in this paper tackled the problem of autonomous flight transition for a fixed-wing modelscale Vertical Take-Off and Landing (VTOL) aircraft. The Linear Quadratic Regulator (LQR) control solution was employed with different weightings in each of the three different regions composing the flight envelope: hover, transition and level flight. It was shown that appropriate switching between controllers did not compromise the overall system’s robustness. The proposed solution capabilities were proven within a computer simulation environment using corrupted state measures resulting from sensor noise.

8

0 10

Time [s]

Fig. 5. Transition from Hover to Level simulation - actuator inputs. its reference trajectory is fully defined by the references (u⋆ (t), w⋆ (t), θ⋆ (t)). The chosen reference trajectories are ramps that take the aircraft from some starting point ξ ⋆X→L (0) ∈ DH to the equilibrium point (or any other point within DL ). These ramps are given by  if t ∈ [0, t0 ) r0 r(t) = r0 + mr (t − t0 ) if t ∈ [t0 , t1 ] , (38)  r1 if t ∈ (t1 , T ]

0 where mr = rt11 −r −t0 is the ramp slope, t0 is the ramp’s starting time, r0 is the trajectory’s starting point, t1 is the ramp’s ending time and r1 is the trajectory’s ending point. The values used for each of the references (u⋆ (t), w⋆ (t), θ⋆ (t)) for the hover to level transition can be checked in Figure 4. In this particular simulation, the aircraft starts its transition to level flight at the hover equilibrium point and, since the reference transition trajectory starts at u = 1 m/s, a step input of the same magnitude. Figure 4 depicts the reference trajectory tracking and Figure 5 depicts the actuator inputs. The aircraft starts in hover, switches to the transition region of the flight envelope at time t = 1.8 s when the aircraft state is ǫX→L close to the reference trajectory ξ ⋆X→L , and switches to level flight at t = 3.6 s when ξ ∈ DL with n DL = (ξ, µ) ∈ DL : |φ| ≤ φX→L ∧ o |θ| ≤ θX→L ∧ |ψ| ≤ ψ X→L ,

where φX→L = 13◦ , θX→L = 13◦ and ψ X→L = 180◦ , guaranteeing that DL ⊂ DL . This behavior is due to the particular reference maneuver choice which has slopes mu = 4.9 m/s, mw = 0.96 m/s and mθ = −45 deg/s, for the forward velocity, downward velocity and pitch angle,

Adrian, F., McGrew, J.S., Valenti, M., Levine, D., and How, J.P. (2007). Hover, transition, and level flight control design for a single-propeller indoor airplane. In AIAA Guidance, Navigation and Control Conference. Anderson, J.D. (1991). Fundamentals of Aerodynamics. McGraw-Hill. Cabecinhas, D., Naldi, R., Marconi, L., Silvestre, C., and Cunha, R. (2010). Robust take-off and landing for a quadrotor vehicle. In IEEE Conference on Robotics and Automation. Desbiens, A.L., Asbeck, A., and Cutkosky, M. (2010). Hybrid aerial and scansorial robotics. In IEEE Conference on Robotics and Automation. Etkin, B. and Reid, L.D. (1996). Dynamics of Flight Stability and Control. John Wiley and Sons, Inc. Franklin, G.F., Powell, J.D., and Emami-Naeini, A. (1994). Feedback Control of Dynamic Systems. AddisonWesley. Gentili, L., Naldi, R., and Marconi, L. (2008). Modeling and control of vtol uavs interacting with the environment. In Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, 1231 –1236. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2009). Hybrid dynamical systems. IEEE Control Systems Magazine. Green, W.E. and Oh, P.Y. (2005). A mav that flies like an airplane and hovers like a helicopter. In Proceeding of the 2005 IEEE/ASME. Kaminer, I., Pascoal, A.M., Khargonekarg, P.P., and Coleman, E.E. (1995). A velocity algorithm for the implementation of gain-scheduled controllers. Automatica, 31, 1185–1191. Koutsoukos, X.D., Antsaklis, P.J., Stiver, J.A., and Lemmon, M.D. (2000). Supervisory control of hybrid systems. In Proceedings of the IEEE. Liberzon, D. and Morse, S. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine. Marconi, L., Naldi, R., and Gentili, L. (2009). A control framework for robust practical tracking of hybrid automata. In Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Mises, R.V. (1959). Theory of Flight. General Publishing Company. Sanfelice, R.G. and Teel, A.R. (2007). A throw-and-catch hybrid control strategy for robust global stabilization of nonlinear system. In American Control Conference.

7255