Robust Attitude Control of Vectored Thrust Aerial Vehicles

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

Robust Attitude Control of Vectored Thrust Aerial Vehicles Makoto Kumon ∗ Jayantha Katupitiya ∗∗ Ikuro Mizumoto ∗ Graduate School of Science and Technology, Kumamoto University, 2-39-1, Kurokami, Kumamoto, 860-8555, Japan (e-mail: [email protected]). ∗∗ School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney NSW 2052, Australia (e-mail: [email protected]) ∗

Abstract: This paper proposes a controller to stabilize the attitude of Vectored Thrust Aerial Vehicle (VTAV). VTAV has high sensitivity to realize quick maneuver, which also means the system is easily effected by uncertainties or disturbances. In order to overcome these undesired effects, the controller is required to be robust. One of the major uncertainty factors of VTAV is uncertain aerodynamic forces, which leads a disturbance to the input to the dynamics. In order to compensate this, high gain output feedback approach with a filter inserted in the loop is proposed. This approach is able to ease Parallel Feedforward Compensator (PFC) selection that is necessary for the controller, considering both stability and disturbance attenuation simultaneously. Numerical simulations comparing with conventional methods showed validity of the proposed method. Keywords: Vectored Thrust Aerial Vehicle, Robust control, Dynamic output feedback, Parallel Feedforward compensator, Simple adaptive control 1. INTRODUCTION This paper proposes a control method for Vectored Thrust Aerial Vehicle (VTAV) attitude stabilization based on a high gain output feedback approach. VTAV, shown in Fig.1 is one of Unmanned Aerial Vehicle (UAV) that has three ducted engines and two of those engines can change their direction with respect to the body. Ducted fans are known to have high drag coefficients and the distribution of the required thrust into three smaller ducted fans partially help improve the drag coefficient. A schematic diagram of the control system is also shown in Fig.1. Gumstix

PC (User)

Tilt System Power System

Wi-Fi

On-Board Computer

RS-232

AHRS

I2C Bus

MicroController PWM (Speed)

Secondary Battery (7.4v)

PWM Signal

Electronic Speed Controller

Main Battery (22.2v)

Digital (Direction)

DC motor controller

DC Tilt Motor

DC motor controller

Ducted Fan

Tilt Gearbox

(a) VTAV

(b) Schematic of VTAV

Fig. 1. VTAV Ducted fan based systems have been attracting the attention of researchers in the recent past. The majority of the work has been directed to the modeling and control of single ducted fan which form some sort of an aircraft(Johnson 978-3-902661-93-7/11/$20.00 © 2011 IFAC

and Turbe (2006); Spaulding et al. (2005)). Control of a ducted fan against wind gusts in the presence of modeling uncertainties is presented in Pflimlin et al. (2007) with a robust controller based on backstepping method. A multi-input, multi-output sliding mode based controller is presented in Hess and Ussery (2004). The work in Naldi et al. (2008) presents a dynamic model that controls the ducted fan vertically up for hovering and then transiting to a near horizontal travel to achieve fast forward flight assisted by control surfaces. The work by Peddle et al. (2009) presents an alternative approach to achieve near hover through decoupling (the UAV has counter rotating fans) instead of using fully coupled high order dynamics. Applications of ducted fan based systems are presented in Pflimlin et al. (2006); Golightly and Jones (2005). This paper proposes a robust attitude controller for the VTAV with three ducted fans that is shown in Kumon et al. (2010). The controller is designed based on high gain feedback control (Mizumoto and Iwai (1996)) that is robust to the uncertainties of the system, and the gain is tuned adaptively (Mizumoto et al. (2003)). This type of controller is named Simple Adaptive Control because of its simple structure, but the method is only applicable if the target system satisfies Almost Strictly Positive Real (ASPR) condition (Kaufman et al. (1997)) or Output Feedback Exponentially Passive (OFEP) condition (Mizumoto et al. (2008); Fradkov and Hill (1998)). If the system does not satisfy those properties, additional filter is necessary to modify the system in the loop, that is named Parallel Feedforward Compensator (PFC, Barkana (1987); Iwai and Mizumoto (1994) ) since it is attached parallel

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to the original system in the control loop. Although PFC guarantees that the augmented system to control satisfies the conditions, it is not easy to tune PFC appropriately and PFC may lead undesired performance in disturbance attenuation in some cases if it is not well tuned. This paper proposes a method to overcome this difficulty by inserting the second filter in the loop. This allows a simple filter to work as the PFC, which is easy to tune. The paper is organized as follow: Section 2 briefly summarize the VTAV considered in this paper. Section 3 is the main part of the paper that proposes the controller and its performance was evaluated through numerical simulations (Section 4). Finally, conclusions will follow in Section 5. 2. VECTORED THRUST AERIAL VEHICLE 2.1 Dynamical Model The dynamics of VTAV can be written in the following form: d2 X 1 = f 1 (X 1 , X 2 , ω) − mg (1) dt2 d2 (2) J 2 X 2 = f 2 (X 2 , ω), dt where X 1 , X 2 , ω, m, J and g represent position, attitude of VTAV, rotor speed, weight and inertia of VTAV and gravitational acceleration. Details of (1) and (2) are shown in Appendix. m

2.2 Attitude Model In this paper, an attitude subsystem of the dynamical system (1) and (2) is considered to control. Quantities to control are selected to roll, pitch and height that are denoted by φ, θ and z respectively, and three control inputs that are speed of three fans are utilized and denoted by ω = (ω0 , ω1 , ω2 )T . Although the main aim of this paper is to derive a controller for attitude (φ and θ), height z is included in order to make the model well-determined because the system has three inputs. Since the objective of the controller is to stabilize the VTAV attitude close to the origin, it is natural to assume that roll and pitch angles and their derivatives are close to 0. This implies that the control inputs are also close to constant values that are referred as “trim inputs” later, and denoted by ω t = (ω0t , ω1t , ω2t ). In order to derive the controller, a linearized system with respect to φ, θ and z around the origin is considered, and it can be written as d2 d X = A X + Bδω, dt2 dt

(3) T

where X = (φ, θ, z)T and δω = (δω0 , δω1 , δω2 ) is defined i by ωi = ωit + δω assuming δω ≪ 1. A is given as ωit ωit ! 0 −W 0 A = Jd W 0 0 , 0 0 0 where Jd and W show the inertia of a rotor which leads gyroscopic effects, and a positive constant respectively.

Since B is invertible, let introduce the decoupling matrix D such that BD = I. Define the new input vector v = (vφ , vθ , vz )T that satisfies Dv = δω. Assuming that the gyroscopic effect is negligibly small, or if the inertia of the rotor Jd is small, and also recalling that the state is considered to be close to the equilibrium, the above model can be simplified into three SISO double integrators as d2 X = v. dt2 2.3 Saturation of input The inputs ωi are limited within the range from 0 to Ωi where Ωi shows the upper bound of the input that is determined by the physical characteristics of the engine. This limitation is modeled by a saturation function sat() in computing the aerodynamic force fi of the dynamics (1) and (2) as 2

fi = αi {sat0,Ωi (ωi )} , where αi shows the aerodynamic coefficient of i-th engine. satxl ,xu (x) is defined as ( xl (x ≤ xl ) satxl ,xu (x) = x (xl < x < xu ) xu (xu ≤ x) 2.4 Disturbance It is usual that the plant has uncertainties. For VTAV, aerodynamic forces are hard to model precisely, and they may cause significant change in the dynamics. In this paper, the effect is modeled as a disturbance added to the input since the aerodynamic forces affecting to the rigid body are close to the effect of the change of inputs. 3. CONTROLLER 3.1 Proposed control system This paper proposes the control system using a filter, denoted by F (s) with adaptive gains and PFC as shown in the figure (Fig.2). Since z is not considered, the corresponding input vz is given as 0. d in Fig.2 shows the disturbance to the system that includes trim inputs mismatch and signals caused by parametric uncertainties. The reference signal r is set to 0 in this case. PFC shows the parallel feed-forward compensator and it is given as first order systems for φ and θ as   qθ qφ , ,0 . GP F C = diag s + pφ s + pθ Denote the output of PFC as y P F C = GP F C v and define T the augmented output y a as y a = F (s) (φ, θ, z) + y P F C . The input v is defined as v = −diag(Kφ , Kθ , 0)y a , where K⋆ shows an adaptive gain that is updated by the following law: d K⋆ = γ⋆ ya 2⋆ − σ⋆ K⋆ , dt

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wt r

+ −

K

D

y

d

sat

+ ++ +

VTAV

1 s2

F (s) + +

PFC ya Fig. 2. Control loop

y

wt d r

+ −

K

sat

+ + D + + ∆

VTAV

1 s2

CPID + +

PFC ya Fig. 3. Control loop with unmodeled dynamics where ya ⋆ shows the element of y a and γ⋆ and σ⋆ show appropriate constants. As shown in the previous section, the nominal model can be given as s12 . When the filter F (s) is given as a PID type filters for φ and θ,  KI,φ F (s) = CPID := diag KP,φ + + sKD,φ , s  KI,θ KP,θ + + sKD,θ , s then the dynamics (dashed box in the figure) can be written as   KD s 2 + KP s + KI 1 KI = × K + + sK (4) P D 2 s s s3 which is ASPR (Kaufman et al. (1997)). This implies that the high gain output feedback approach might be applicable, if the saturation does not harm ASPR-ness. In order to consider the saturation into account, PID parameters should be tuned appropriately. Roughly speaking, the smaller PID gains are, the less the saturation effects. Multiplicative unmodeled dynamics shown in Fig. 3 is also considered. Engine dynamics, uncertain aerodynamic effects or servo dynamics can be included in this dynamics. This dynamics can vary the relative degree of the plant, which might break ASPR conditions. PFC is introduced to compensate this uncertainty to keep the relative degree less than or equal to 1. 3.2 PFC design Under the existence of disturbance to the input, PFC might lead poor control performance of y although the augmented output ya can be controlled well. In this subsection, the design of PFC is discussed taking the effect of the disturbance into account. As the update law gain γ is usually set large, the gain K can be assumed to be almost constant after the very initial

d r

+ −

K

+ +

y G

GP

+ +

Fig. 4. Control loop : General form transient response. With a constant feedback gain K, the system considered can be shown as in Fig.4 generally. NP Denote G = N D and GP = DP respectively. The sensitivity function from d to y, which is denoted by Td,y is given as (DP + NP K)N , Td,y = DDP + K(DNP + DP N ) where DP and NP are chosen as DNP + DP N becomes stable i.e. ASPR condition is satisfied. Assume D(0) = 0 and d = 1s (G has a pole at origin and constant disturbance is added). Because of the definition of GP , the system becomes stable with sufficiently large K, so K is considered to be large. With a large feedback gain K, Td,y can be approximated as NP N . Td,y ≈ DNP + DP N From this Td,y ≈ GP (0) which leads that the steady output error y(∞) is GP (0). In order to attenuate the steady error, |NP (0)| must be small enough.

If GP (s) is given by a first order filter, or it is expressed as q s+p , then |q| should be as small as possible. In this case, Td,y can be further approximated as NP = GP , Td,y ≈ DP

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which implies that the sensitivity is governed by PFC itself. Generally, PFC should have small gain and fast response in order to attenuate the disturbance. When GP has higher order than 1, NP can be chosen to have 0 for the constant term to satisfy 0 steady state error. On the other hand, the selection of GP becomes difficult since the response of GP gets much complicated than a first order filter. Now, let’s focus on the VTAV case. As the nominal model for one axis of VTAV can be given as s12 , the nominal Td,y for this system becomes q Td,y = 2 , qs + s + p and the final value of the step response is pq . Because of the stability requirement, a and q must be positive. 3.3 Role of filter F (s) Let introduce the unmodeled dynamics given as a first α in order to show the effect of order filter, i.e. ∆ = s+α the filter F (s) inserted in the loop. This uncertainty can represent the dynamics of engine to rotate a fan inside the duct. Let denote the perturbed sensitivity function with uncertainty dynamics as T˜d,y .

The system is stable if q is sufficiently small again. As discussed above, the steady output error becomes small as q diminishes, and in this case, q can be small even under ∆ of a first order filter. It is worth noting that the system with a filter is ASPR if the system is nominal, but that it is no more ASPR when it contains uncertainty ∆. Therefore PFC GP is needed, which leads difficulties in disturbance attenuation of the original output y. Instead of tuning PFC carefully for ASPR-ness and the disturbance attenuation simultaneously, which is usually time consuming, the proposed approach provides rather simple since PFC has only two parameters to tune. 4. SIMULATION In order to evaluate the proposed controller, numerical simulations have been conducted. Simple PID controller was also tested as a comparison. Parameters of the plant and the controller are summarized in Table 1 and 2. Table 1. Parameters of VTAV Nominal 1 , Iz = 325 m: 4.5kg, Ix = Iy = 25 0.2 √ α0 = α1 = α2 = 0.5, a = , b = 0.2 l = 2a 3 Jd = 1e − 5 Ω = 10

Without the in-the-loop filter, or F (s) = 1 in Fig.2, T˜d,y becomes qα , T˜d,y = 3 qs + qαs2 + αs + pα

Parameter perturbation α0 = 0.5 → 0.6, α1 = 0.5 → 0.3

and α > p must hold in order to guarantee the stability. When α is small, or the uncertain dynamics has large time constant, poles get close to the roots of qs2 + qαs + α = 0

Initial state φ = 0.2, θ = −0.1

Unmodeled dynamics 2 ∆ = s+2

Table 2. Parameters of Controller

which are −

α √ ± α 2

p

Nominal KP = 5, KI = 0.5, KD = 0.5 γ = 1000, σ = 1e − 3 Initial gain: Kφ (0) = Kθ (0) = 50 0.1 PFC: s+10 Trim inputs: ω0t = ω1t = ω2t = 5.4249

q 2 α − 4q . 2q

As α becomes smaller, roots approach the imaginary axis and the system takes long time to converge, or it may become unstable if q is too small, which means that q can not be arbitrary small. Therefore, the steady state error may cause a difficulty.

Perturbed case Trim inputs: ω0t = 6.4249, ω1t = 4.4249

On the other hand, if F (s) is inserted in the loop, the plant with F (s) can be represented as in (4). Then, the nominal 4.1 Nominal case sensitivity function becomes qKD s2 + qKP s + qKI Fig.5 shows the response of attitude φ and θ. For the , Td,y = (q + KD )s3 + (KP + pKD )s2 + (KI + pKP )s + pKI comparison, the response by the PID controller with the same gain of PID compensator for the proposed controller and the system is stable if q is sufficiently small. As in the is also shown. Both controller could stabilize the system, but the response of the proposed one was much faster than above case, the perturbed sensitivity becomes PID since the adaptive gain was around 50, which means 2 αq(KD s + KP s + KI ) T˜d,y = , SAC gains were about 50 times larger than those of PID. DT (s) 4.2 With uncertainties

where DT (s) = qs4 + α(q + KD )s3 + α(KP + pKD )s2 +α(KI + pKP )s + αpKI .

Parametric uncertainties Under the parametric perturbation (α0 , α1 , ω0t and ω1t ), robustness of the proposed system was also examined. 2610

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

0.2

0.2 φ

0.1 0

0.1 0

θ

−0.10

φ θ

2

4 6 Proposed

8

−0.10

10

4 0.2

φ θ

−4

0

−8

−0.10

10

−120

30

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20

PID

40 60 PID

80

100

Fig. 7. Parametric uncertainty case: φ and θ response

Fig. 5. Nominal case: φ and θ response 8

8

4

4

0

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−40

0.2 0.6 1.0 0.4 0.8 Inputs of the proposed controller

−40

53 Kφ

51 2

4 6 Adaptive gain K

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20 10 Adaptive gain K

0

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4

0.1

2 0

0 −0.10

Kθ

150

Kθ 0

20 30 10 Inputs of the proposed controller

250

52

50

30

0

φ θ

0.1

10 20 Proposed

20

10

−2

30

−40

PID Fig. 6. Nominal case: Inputs: rotor speed ratio δω and adaptive gain K, and PID inputs (comparison) Proposed method could keep stable and fast convergence as the nominal case. PID also showed the stability of the system, but it took quite a long time to converge and large deviation at the beginning. Additional dynamics Besides the parametric uncertain2 ties shown above, a first order additional dynamics s+2 is inserted as ∆.

20

40 60 PID

80

100

Fig. 8. Parametric uncertainty case: Inputs: rotor speed ratio δω and adaptive gain K, and PID inputs (comparison)

Fig.9 shows the response of the proposed system and PID controller. The proposed one could keep the desired feature, but the response of PID controller was no more stable. Control signals of the proposed system varied severely at the beginning, but it got smooth once the gain K reached large enough.

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Since only first order dynamics were considered as ∆, PID type filter was adopted in this case. The proposed approach can be easily extended to utilize higher order filters if the system may contain more complicated uncertainties. This is from the fact that the order of DP N should be larger than or equal to DNP in order to satisfy ASPRness as far as the original zeros of the plant are stable because the filter introduces additional zeros to the plant that makes the order of N larger.

0.2 0.1 φ 0 θ −0.1 0

10 20 Proposed

30

The selection of the in-the-loop filter F (s) needs to be selected appropriately because the steady error could not be attenuated effectively when PD type filter was utilized instead of PID type.

400 200

φ

0

θ

5. CONCLUSION

−200

−400 0

4

2

The paper proposes a combination of a filter and PFC in the framework of SAC to achieve robust attitude control of VTAV. The approach eases PFC design with the help of the filter, and numerical results validated the approach. Further discussion of in-the-loop filter design and experimental validations will be the future work.

6

PID Fig. 9. Unmodeled dynamics case: φ and θ response

Appendix A. DYNAMICS AND DECOUPLING MATRIX

10

Details of the dynamics (1) and (2) are written as follow: ! ! x Sψ Sφ + C ψ Sθ C φ d2 m 2 y = (f0 + f1 + f2 ) −Cψ Sφ + Sψ Sθ Cφ dt z Cθ Cφ ! 0 0 , − mg  ! !  θ˙ψ(I ˙ y − Iz ) −b(f1 − f2 ) φ d2 ˙ z − Ix )  + −l0 f0 +a(f1 + f2 ) J 2 θ =  φ˙ ψ(I dt ˙ ˙ 0 ψ φθ(Ix − Iy )   −(ω0 + ω1 + ω2 )θ˙  +Jd (ω0 + ω1 + ω2 )φ˙  . 0 fi is modeled by fi = αi ωi2 (i = 1, 2, 3), and C⋆ and S⋆ show simplified notation of cosine and sine functions respectively. αi shows the i-th engine’s aerodynamic coefficient, and it is natural to assume that α1 = α2 = α3 for the nominal model. The nominal value is denoted by α.

0 −10

0

20 30 10 Inputs of the proposed controller Kφ

250

Kθ 150 50

0

10 20 Adaptive gain K

30

200 100 0 −100 0

4 2 6 Inputs of the conventional PID controller

Fig. 10. Unmodeled dynamics case: Inputs: rotor speed ratio δω and adaptive gain K, PID inputs(comparison) 4.3 Discussions It is apparent from the above results that the proposed controller outperform the conventional approach. As it has been mentioned that PFC may lead “offset” , the effect of the proposed filter provides satisfactory contribution in the controller performance.

Linearized model is expressed as the following: ! ! ! Ix 0 0 −b(f1 − f2 ) φ d2 0 Iy 0 θ = −l0 f0 + a(f1 + f2 ) 2 0 0 m dt f0 + f1 + f2 z ! ! ˙  0 0 −W 0 φ 0 +Jd W 0 0  θ˙  − , mg 0 0 0 z˙

and the decoupling matrix D is defined by   b b −1 0 −  Ix Ix     l0 a a  D = 2α  −  .  Iy Iy Iy   1 1 1 

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m

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

Denote f0 , f1 and f2 at the equilibrium as f0eq , f1eq and f2eq respectively, and they must hold followings. f1eq − f2eq = 0

−l0 f0 + a(f1 + f2 ) = 0

f0 + f1 + f2 − mg = 0 Recalling the model of the aerodynamic force shown in Sec.2, linearized aerodynamic force model can be written as fi = fieq + 2αi ωit δωi , where ωit shows the angular velocity of the rotor i at the equilibrium. Substituting these relationships into the above model, the following model is obtained. ! ˙  ! ! φ δω 0 −W 0 φ 2 0 d  δω W 0 0 θ = B + J 1 θ˙  , d dt2 z δω2 0 0 0 z˙ which corresponds to (3).

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