Back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling

Accepted Manuscript Back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling

Zikang Su, Honglun Wang, Peng Yao, Yu Huang, Yong Qin

PII: DOI: Reference:

S1270-9638(16)30256-5 http://dx.doi.org/10.1016/j.ast.2016.11.028 AESCTE 3845

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

12 July 2016 11 November 2016 28 November 2016

Please cite this article in press as: Z. Su et al., Back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling, Aerosp. Sci. Technol. (2016), http://dx.doi.org/10.1016/j.ast.2016.11.028

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Back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling Zikang Sua,b,c, Honglun Wanga,c,*, Peng Yaoa,c, Yu Huanga,c, Yong Qina,c a

School of Automation Science and Electrical Engineering, Beihang University, 100191, Beijing, China;

b

Honors College of Beihang University, 100191, Beijing, China;

c

The Science and Technology on Aircraft Control Laboratory, Beihang University, 100191, Beijing, China.

Abstract This paper proposes a novel back-stepping based flight controller for the receiver in Autonomous Aerial Refueling (AAR) by the combination of active disturbance rejection control (ADRC) and preview control methodology. Firstly, the proposed flight control law design is divided into five loops by the back-stepping technique. The 6 DOF model for aircraft is written into several strict-feedback nonlinear forms through tactful transformation, considering that the path dynamics are originally non-affine nonlinear forms and are intractable for controller design. Secondly, the influences of the unknown flow perturbations on a receiver in each loop are viewed as the components of the “total disturbances” which are estimated and compensated by extended state observer (ESO). Thirdly, the preview control methodology is introduced into the position loop design as the system dynamic of the receiver is much slower than drogue. The position loop determines the proper reference path angle by using a fuzzy-logic ADRC controller in which previewing error and current tracking error are both taken into consideration. And a novel adaptive look-ahead distance scheme based on the fuzzy logic control (FLC) is proposed for the selection of the preview point. Finally, extensive simulations and comparisons on the 6 DOF receiver model are carried out to demonstrate the effectiveness of the proposed flight controller. Keywords: autonomous aerial refueling; receiver flight control; back-stepping; active disturbance rejection control; preview control; fuzzy logic control

1. Introduction Autonomous Aerial Refueling (AAR) [1], which refuels other aircrafts in the air, is an effective method of increasing the endurance and region of the aircrafts. It has drawn more and more significant interests from the research and development community [2-4], especially for the purpose of enabling unmanned aerial vehicles with this critical capability [5]. There are two ways of refueling [1]: flying boom method and probe-and-drogue method. In either case, it would be better if the receiver aircraft were automatically controlled for aerial refueling. In this paper, we focus on the probe-drogue refueling (PDR) [6,7], as shown in Fig.1. In the PDR, the receiver aircraft is required to fly precisely to track the wobbly drogue rapidly and accurately [1-5]. However, considering the particularity of the environment and mission in the PDR, the following problems should not be ignored during the receiver flight controller design. i.

The strong impact of the multiple flow perturbations with varying magnitude and direction

* Corresponding author. Tel.: +86-10-82317546. E-mail address: [email protected]

1

on the receiver aircraft. The receiver is subject to not only the prevailing wind and turbulence but also an additional wind field induced by the wake of the tanker [1,8,9]. ii.

The system dynamic response of the receiver is much slower than the drogue. The lighter drogue, which can be easily affected by the multiple flows behind the tanker, swings fleetly. And that makes it intractable for the receiver to track due to its slower dynamic response. V

tanker

OT XT YT

ZT

OW

hose

XW

ZW

probe

Og

pod

Xg

drogue Zg

reciever

YW

Yg

Fig.1 The configuration of a hose-drogue aerial refueling system

Although there have already been some previous works [10-18] discussing flight controller design for the receiver aircraft, unfortunately, few kinds of literature focus on the above problems during the controller design. Actually, the flight controller of the receiver is required to be designed to ensure the probe on the receiver tracks the wobbly drogue rapidly and accurately under the influence of the prevailing wind, tanker’s trailing vortex and atmospheric perturbation [1]. However, the existing literature mostly design receiver trajectory tracking controller with the Linear Quadratic Regulator (LQR) theory [1,10-12,16-18]. But LQR theory uses the linearized nominal plant model to design the controller, and the unknown flow perturbations are not considered during the controller designing. Its disturbance rejection mechanism is passive, and the disturbance rejection ability only depends on the nominally designed controller. What's more, the receiver’s slower dynamic response problem also does not be taken into consideration in the existing methodology. It definitely makes it difficult to track the swift movement of the drogue, and the tracking lag will arise. Another linear model based method known as L1 adaptive control methodology is adopted in AAR, but the same problem above will also be faced. The nonlinear dynamic inversion (NDI) which is featured as a nonlinear control method is also tried to be applied to the receiver tracking control in AAR together with some uncertainty compensation technique [41-44]. However, these existing NDI based tracking controllers either in the attitude control loop or in the path control loop. And to our knowledge, few papers designed the AAR tracking controller in both path and attitude loop using these nonlinear model based methods such as the NDI, because the 6-DOF nonlinear model of the receiver will be non-affine and particularly complex when the influence of the multiple flow disturbances is considered [8]. All of the above issues pose challenges on the specific mission of receiver flight controller from theoretical and practical perspectives. Unlike the existing methodology for receiver flight controller design, a new anti-disturbance receiver flight controller needs to be designed for PDR. Recently, active disturbance rejection control (ADRC) has been well developed for nonlinear uncertain system control problems [23-31]. ADRC takes the system nonlinearities, uncertainties, and external disturbances as the “total disturbance” which is observed and compensated by extended state observer (ESO). The ESO is relatively independently of a mathematical model of the plant, is simpler to implement, and can offer better performance [25]. The nominal performance of the closed-loop system can be recovered if the “total 2

uncertainty” is timely compensated via ESO [26]. It should be noted that the existing standard ADRC is only available for integral chain systems that satisfy the so-called “matching conditions” [23-25], such as the motion control system [23] and actuated MEMS device [31]. However, as the path dynamic equations of a fixed wing aircraft are usually formulated as a non-affine nonlinear system in flight-path axis [32], a problem arises naturally, how to extend the existing ADRC technique to the translation kinematics and dynamics controllers design by back-stepping? To our best knowledge, few efforts have been contributed to this issue. Inspired by the function of ESO above, the path dynamic equations are transformed to the strict-feedback nonlinear subsystems in this paper, and the introduced items in the transformed equations are all taken as parts of the “total disturbance” which are then observed and compensated by ESO. Then, by the back-stepping technique, the receiver flight control system can be designed through five independent control loops: position loop, flight path loop, attitude loop (flow angles or Euler angles scheme), angular rate loop and ground velocity loop. And it seems that the first problem mentioned above can be resolved by this way. For the second problem above, we adopted the preview control methodology [33] during the flight controller design. The motivation behind the wide applied preview control methodology is that it utilizes the future information of disturbances or references to improve the disturbance rejection or tracking quality [34]. That is, as an extra degree of freedom (DOF) available in controller design, preview control can obtain better performance beyond only typical feedback control design. Preview

control has attracted much attention for its various applications, such as autonomous vehicle guidance [35], manufacturing control [36-37] and flight control [38]. It should be noted that preview control requires the signals to be tracked or rejected are available a priori by a certain amount of time [34]. But the drogue’s motion seems to be unpredictable because the drogue is affected by its current position and the varying flows. Actually, we use an approximate predict method, which is sufficiently precise in a certain amount of time, to obtain the drogue’s trajectory in a certain future time. As it well known that the look-ahead distance has a significant impact on the control performance, we proposed a fuzzy logic controller to get an adaptive variable look-ahead distance, according to the curvature of the drogue’s trajectory and the preview error. Inspired by the above analysis, we investigate in this paper the feasibility of a novel back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling. Our main contributions in this paper lie in following aspects: (i) The flow perturbations’ influence on the receiver and the receiver’s slow system dynamic response characteristic are both considered during the flight controller design, and a novel back-stepping based anti-disturbance flight controller with preview methodology is proposed for PDR. (ii) The 6 DOF receiver model is written into several strict-feedback nonlinear forms, and an integrative back-stepping ADRC based flight controller, which considers anti-disturbance ability to flow perturbations, is firstly designed throng five control loops. (iii) Preview control methodology is introduced into the position controller design to suppress the receiver’s tracking lag due to its slower dynamic response, and a novel adaptive look-ahead distance scheme based on the fuzzy logic control is proposed for the preview point selection. The paper is organized as follows. The 6 DOF rigid model of the receiver and problem formulation are presented in Sec.2. In Sec.3, the detailed overall procedure for the receiver’s integrated flight controller design based on ADRC is illustrated. And the detailed overall procedure of the 3

where ρ is the atmosphere density, V is the airspeed, ȣ is a virtual control vector, and Q is the dynamic pressure. Note that X i ,i = 1...4 and Vk are taken as controlled states as we use the flow angles control scheme in this paper. We rewrite Eq. (1-4) in the following strict-feedback form as Eq. (6) and Eq. (7). Vk = fVk + BVk δ T

(6)

 = F (X ,V ) + B (V )X = F + B X ; ­X 1 1 2 k 1 k 2 1 1 2 ° ° X 2 = F2 (X 2 , X3 , Vk ) + B 2 (X 2 , X3 , Vk , Q )ȣ = F2 + B 2 ȣ, ° 2 2 T ® X3 = [ υ1 + υ1 β atan(υ1 / υ2 )] °  ° X3 = F3 (X 2 , X3 , X 4 ) + B 3 (X3 )X 4 = F3 + B 3 X 4 ; °X  = F (X ) + B (X , Q )U = F + B U ; act 4 4 4 2 4 4 act ¯ 4

(7)

where the explicit forms for fVk , BVk , Fi , Bi , i = 1,..., 4 are given in Appendix B. Based on the above analysis and definition, the control task can be given to designing a robust integrated flight controller for receiver described by (6-7). 3. Back-stepping ADRC based flight controller for receiver Consider the actual receiver flight in AAR process, we give the following assumptions here before designing the flight controller. Assumption 1. All the flight state variables ( X i , i = 1,..., 4 and Vk ) can be obtained through direct or indirect measuring means. Assumption 2. All the “total disturbances” ( Fi , i = 1,..., 4 and fVk ) are differentiable, and their differentials F i , i = 1,..., 4 , fVk are bounded. Based on the time-scale separation and singular perturbation theory, the proposed integrated receiver flight controller can be divided into five loops: ground velocity loop, position loop, flight path loop, attitude loop, as well as angular rate loop, where the five loops admit ADRC configuration primarily, as shown in Fig.2. Before introducing the scheme, we remark that asterisk denotes the reference command, as shown in Fig.2. Prevailing wind

Preview control strategy

Tanker vortex Turbulence

e pP Drogue Position

VK*

X *1

e pC

Multiple flow perturbations Position loop controller (ADRC)

ˆ ,F ˆ X 1 1

X*2

Flight path X * 3 loop controller (ADRC)

Attitude loop X * 4 controller (ADRC)

Angular rate loop controller (ADRC)

ˆ ,F ˆ X 4 4

ˆ ,F ˆ X 3 3

ˆ ˆ ,F X 2 2

Uact

δT

LESO

LESO

LESO LESO Ground Velocity controller (ADRC)

VˆK , fˆVK

LESO

Fig.2.The architecture of the proposed flight controls scheme for receiver 5

6-DOF dynamic s for Receiver

The disturbances caused by unknown flow perturbations are taken as component of “total disturbance”, and are observed and compensated by ESO. Then, based on back-stepping method, the position and flight path control laws are designed to suppress the tracking errors in the presence of unknown flow perturbations using ADRC. Then, the posture controller, which contains the attitude loop controller and angle rate loop controller, is designed to accurately follow the reference attitude command X *3 via ADRC, as shown in Fig. 2. Finally, the desired ground velocity is ensured by an independent control loop with ADRC. The preview control strategy, as shown in Fig.2, is introduced in the position controller to suppress the tracking lag of the designed closed-loop system above. The drogue’s motion and the receiver’s states are predicted until the preview point, then the preview control error and the preview command signal’s derivative are obtained for the position controller. The position loop determines the proper reference path angle by using a hybrid fuzzy logic ADRC controller in which previewing error and current tracking error are both taken into consideration. Then we propose a novel adaptive look-ahead distance scheme based on the FLC for the proper selection of the preview point, due to the look-ahead distance effects the tracking performance greatly. 3.1 Anti-disturbance flight controller design via ADRC The non-affine nonlinear forms of translational dynamics (1-2) make it intractable to design a position controller and flight path controller for a receiver using the existing ADRC. Therefore, its equivalent strict-feedback form that appears as the former two equations in (7) will be adopted to design the position controller and flight path controller instead of (1-2). Define the virtual control vectors and tracking error vectors as following: ­ X*2 = u1 , X*3 = u 2 , X*4 = u3 ° * ˆ °epC = X1 − X 1 ° * ˆ °e 2 = X 2 − X 2 (8) ® * ˆ °e3 = X3 − X 3 ° * ˆ °e 2 = X 4 − X 4 °e = V * − Vˆ K k ¯ Vk where u1 , u 2 , u3 are the virtual controls of position, flight path, and attitude loop, respectively. ˆ , i = 1,..., 4 and Vˆ are the estimations of X , i = 1,..., 4 and V by ESO [23-25]. X i k i k

Remark 1: In this paper, the states and disturbances in each loop are estimated by a second order linear ESO (LESO) [26]. And it can be proved that if above Assumption 1 holds, the estimation error will finally converge to a small enough domain in a finite time by a proper selection of the design parameters [24,27-29]. Detailed descriptions and convergence theorem can be found in these Refs. According to the design principle of ADRC [24-30], and with the consideration of effective suppression for tracking errors, the tracking error vector z1 and the command signal’s derivative are both taken into consideration in position control law:

(

* u1 = B1−1 −Fˆ1 + k 1epC + X 1

6

)

(9)

where B1−1 is the inversion of B1 , and Fˆ1 is the estimation of F1 by LESO (10). ˆ ˆ ˆ X 1 = F1 + ȕ11 ( X 1 - X 1 ) + B1u1  ˆ ) Fˆ1 = ȕ12 ( X 1 - X 1

(10)

where ȕ11 = [2ω1 , 2ω1 ]T and ȕ12 = [ω12 , ω12 ]T are the observer gain parameters. Analogously, the control law for flight path loop can be designed as Eq. (11) according to ADRC principle.

(

* u 2 = B 2 −1 −Fˆ 2 + e 2 z 2 + X 2

)

(11)

where B2 −1 is the inversion of B 2 , and Fˆ 2 is the estimation of F2 by LESO (12). ˆ ˆ ˆ X 2 = F2 + ȕ 21 ( X 2 - X 2 ) + B 2 u 2  ˆ ) Fˆ 2 = ȕ22 ( X 2 - X 2

(12)

where ȕ21 = [2ω2 , 2ω2 ]T and ȕ22 = [ω22 , ω22 ]T are the observer gain parameters. In this paper, the coordinated turn is adopted, and the command signal for sideslip angle can be set as β * = 0 . Hence, the controller for attitude loop is designed as Eq. (13).

(

* u 3 = B 3-1 -Fˆ 3 + k 3 e 3 + X 3

)

(13)

where B 3 -1 is the inversion of B 3 , and Fˆ 3 is the estimation of F3 by LESO (14). ˆ ˆ ˆ X 3 = F3 + ȕ 31 (X 3 - X 3 ) + B 3 u 3  ˆ ) Fˆ 3 = ȕ 32 (X 3 - X 3

(14)

where ȕ 31 = [2ω3 , 2ω3 ]T and ȕ 32 = [ω32 , ω32 ]T are the observer gain parameters. Angular rate controller is designed as Eq. (15).

(

* Uact = B 4-1 -Fˆ 4 + k 4 e4 + X 4

)

(15)

where B 4 -1 is the inversion of B4 , and Fˆ 4 is the estimation of F4 by LESO (16). ˆ ˆ ˆ X 4 = F4 + ȕ 41 (X 4 - X 4 ) + B 4 U act ˆ ˆ ) F4 = ȕ 42 (X4 - X 4

(16)

where ȕ41 = [2ω4 , 2ω4 ]T and ȕ42 = [ω42 , ω42 ]T are the observer gain parameters. Finally, the independently designed ground velocity control law:

δ T = BV

k

−1

( − fˆ

Vk

+ kVk eVk + VK*

)

where BVk −1 is the inversion of BVk , and fˆVk is the estimation of fVk by LESO (18). 7

(17)

Vˆk = fˆVk + βVk 1 (Vk -Vˆk ) + BV k δ T

(18)

 fˆVk = βVk 2 (Vk -Vˆk )

where βVk 1 = 2ωVk and βVk 2 = ωV2 are the observer gain parameters. k 3.2 Preview control strategy for position controller The drogue that dragged behind the tanker moves fleetly and disorderly under the influence of multiple flows, especially the turbulence. On the other hand, the receiver’s system dynamic response is slower than the drogue’s motion due to its specific maneuvering characteristics. Consequently, the tracking lag will definitely arise, and it will directly affect the safety and success of the AAR. Thus, there must be some strategies to suppress this tracking lag during the controller design. Actually, in the manned aircraft aerial refueling, the pilot not only focus on the drogue’s current position but also focus on the drogue’s possible future position in a short future time [1]. That means the pilot actually doesn’t track the irregularly moving drogue only by using the drogue’s current position as the target. Instead, the pilot will predict the possible positions of the receiver and drogue based on their current positions and moving trends and composite the current and future information to track the irregularly moving drogue. Actually, this processes is very similar to the preview control strategy which is widely researched in the vehicle-driving control area [34-36]. Therefore, inspired by above analysis, we try to introduce the preview control strategy to imitate the pilot’s mature operation for a better tracking performance in UAV AAR. The brief schematic diagram of this preview control idea is shown in Fig.3. In Fig. 3, the yellow cycles denote the current positions of probe and drogue ( X1C , XpC ), the red cycles denote the preview points ( X1P , XpP ), ΔTP = NTs is the preview time, N is the look-ahead distance (preview steps from current), Ts is the sampling time, || epC || and || epP || are absolute values of the actual current tracking error and preview error. Preview Point Current Preview

Target drogue position

probe reciever

Preview trajectory

Current drogue position

Fig.3 The schematic diagram of the preview strategy for PDR

Actually, the AAR is usually selected to be conducted in a relative fair-weather, so the intensity of the turbulence is generally no more than moderate turbulence [1]. Hence, the turbulence usually can be considered smaller than the sum of prevailing wind and tanker vortex, and the mean value of turbulence is zero by the Dryden model [8]. Besides, the prevailing wind and tanker vortex are specified if the drogue’s position and the refueling environment are specified. Consequently, with the 8

nonlinear hose-drogue model [7], the drogue’s future trajectory in a priori by a certain small amount of time ΔTP can be approximately predicted based on the current states of the hose-drogue assembly with the assumption that the turbulence is zero (as the mean value of turbulence is zero and the sampling time Ts is very small). Analogouslyˈthe future states of the receiver at the future time ΔTP can also be approximately predicted based on its current states with the assumption that the turbulence is zero. The architecture of the previewing strategy in Fig.2 can be shown as Fig.4.

Fig.4 The architecture of the preview strategy for position controller

Based on this idea, we introduce the preview error and the preview command signal’s derivative into the position controller. We define the preview error

epP = XdP − X1P where XdP = [ ydP

(19)

zdP ]T is the drogue’s predicted position vector at the preview point under the

look-ahead distance ΔTP ( ΔTP seconds from the current time), and X1P = [ y pP

z pP ]T is the probe

position vector at the same preview point. On the basis of the ADRC position controller (9), we give the hybrid preview ADRC controller for position loop:

(

 * +k X  u1 = B1−1 −Fˆ1 + k 1 ( (1 − k P )epC + k P epP ) + (1 − k P ) X P dP 1

)

(20)

 is the derivative of X , and it can be obtained via a tracking differentor. where X dP dP As it’s well known that the look-ahead distance effects the performance of the closed-loop to some extent, the problem that how to obtain a proper look-ahead distance adaptively with the preview (future) information still remains, despite the position is given in Eq. (20). An adaptive look-ahead distance which can be adjusted according to the change trend of the drogue trajectory and the possible tracking deviation is especially in AAR, due to the rapid and irregular drogue movement, slower receiver dynamic response, and strict requirements of the tracking error. As the drogue’s movement may instantaneously change, the trajectory may be relatively smooth in somewhere and may also be quiet irregular in the other place. Obviously, a fixed look-ahead distance cannot be always suited throughout the tracking due to different drogue trajectories and tracking status of the receiver. For instance, if a large look-ahead distance is selected abrupt changing stage of the drogue trajectory, the prediction for the receiver and drogue will certainly be inaccurate, and this will further case the poor tracking performance or even failure during AAR. It should be noted that a small look-ahead distance is 9

needed at sometimes, but a large one is better at some other times. Therefore, we propose an adaptive look-ahead distance scheme based on the fuzzy logic control for the selection of the preview point. Actually, it also should be noted that the preview accuracy will recede if N is too large, and on the other hand, the capacity of preview cannot be brought out if N is too small. Hence, a tradeoff for the selection of N is required. Thus, the Linguistic quantification used to specify a set of rules for the fuzzy logic controller is characterized by the following typical situations from the physical point of view in AAR:

i.

If | ρ dP | is small in the preview, the drogue trajectory changes smaller, and a larger N is preferred to bring out the capacity of preview strategy; otherwise, when | ρ dP | is large, the drogue trajectory changes larger, and a smaller N is preferred to ensure the preview capacity and stability of the closed-loop;

ii.

If the modulus of preview error || e pP || is smaller, a larger N can be selected to enlarge the effect of the preview in the controller (20); otherwise, a smaller N can be selected to enlarge the effect of the feedback in (20) and ensure the stability of the closed-loop.

According to the preview control theory [34-36], the preview error and the curvature of the drogue trajectory are two key factors to determine a proper look-ahead distance. And for convenience, we take the modulus of preview error e pP and the curvature of the commanded trajectory ρ dP as two inputs of the fuzzy logic controller, and the look-ahead distance N is taken as the only output. As shown in Fig.5, the shape of membership function for all the three variables is the isosceles triangle, in which six linguistic variables are defined, including Negative Big “NB” , Negative Medium “NM” , Negative Small “NS”, Zero “ܼ” , Positive Small “PS” , Positive Medium “PM” , and Positive Big “PB” . For this controller, with two inputs and seven linguistic values for each of these, there are a total of 7 2 = 49 rules. To establish the fuzzy controller rules, Mamdani is selected as the type of fuzzy

Degree of membership Degree of membership Degree of membership

inference system, the typical form of the fuzzy rules can be described as: IF x is G and y is H , Then z = C . 1NB

NM

NS

ZO

PS

PM

PB

0.5 0 0

1NB

0.1

0.2

NM

0.3

NS

0.4 0.5 Curvature | ρ| ZO

0.6

PS

0.7

PM

0.8

PB

0.5 0 0

1NB

0.1

NM

0.2

0.3 0.4 0.5 PreveiwError || epP || NS

ZO

PS

0.6

0.7

PM

0.8

PB

0.5 0 15

20 25 Look-ahead distance N

Fig.5 Membership functions for inputs and output

10

30

Based on above principles which are designed based on the actual physical process in AAR, the fuzzy

rules are given in Table 1 through simulations, and the output of fuzzy rules is shown in Fig. 6. It can be obviously seen form the above updated table and figure that the look-ahead value N depends on not only the preview error

e pP but also the curvature of the commanded trajectory ρ dP . And the

look-ahead value N can be adaptively adjusted with the forthcoming change of the ρ dP (reflects the drogue’s trajectory change) and the e pP

(reflects the imminent tracking error between the receiver

and the drogue). And by this way, the controller for each loop is designed (position (20), flight path (11), attitude (13), angular rate (15), and the ground velocity (17)). And it can be easily found that the control law for each loop is analytical, which is more convenient for the completion of the real system. Table 1 Rules for the fuzzy controller Curvature of the commanded trajectory | ρ dP |

Look-ahead distance N

Preview error || e pP ||

NB

NM

NS

ܼ

PS

PM

PB

NB

PB

PM

PM

PS

PS

PS

Z

NM

PM

PM

PS

PS

PS

Z

Z

NS

PM

PS

PS

PS

Z

Z

NS

Z

PS

PS

PS

Z

Z

NS

NS

PS

PS

PS

Z

Z

NS

NS

NM

PM

PS

Z

Z

NS

NS

NM

NM

PB

Z

Z

NS

NS

NM

NM

NB

28 Look-ahead distance N

26 24 22 20 18 16 0 0.2 0.4 0.6 0.8

0.8

0.6

0.4 Curvature |ρ |

PreveiwError ||epP||

Fig.6 Membership functions for inputs and output

11

0.2

0

4. Simulation results and comparison To verify the validity of proposed back-stepping based anti-disturbance flight controller with preview methodology in AAR, extensive simulations have been carried out based on the 6 DOF model of receiver described by (6) - (7), and the receiver in [18] and the parameters of the hose-drogue assembly in [41] are adopted here. In addition, the nonlinear dynamic inversion (NDI) control scheme (all the five loops are designed with NDI) and the conventional ADRC control scheme (all the five loops are designed with ADRC) are chosen to make comparisons. For all numerical simulations presented below, the initial parameters and controller parameters are provided in Table 2. Table 2 Simulation parameters of the compared control schemes

Flight condition Altitude: 7010 m; and tracking error limitations NDI

Ground velocity: 200 m/s;

tracking error requirement: 0 m ≤ R0= e y 2 + ez 2 ≤ 0.3 m

Controller gain for each loop:

(all the five loops

k 1 = d ia g (1, 1) , k P = d ia g ( 0 .7 , .0 7 ) k 2 = d ia g ( 2 , 2 )ˈk V k = 1 .2 ˈ

are designed with

k 3 = d ia g ( 3, 3, 3 ) , k 4 = d ia g ( 8 , 8 , 8 ) ,

NDI))

Controller gain for each loop: Conventional

k 1 = d ia g (1, 1) , k P = d ia g ( 0 .7 , .0 7 )

k 2 = d ia g ( 2 , 2 )ˈk V k = 1 .2 ˈ

ADRC

k 3 = d ia g ( 3, 3, 3 ) , k 4 = d ia g ( 8 , 8 , 8 ) ,

(all the five loops

rV k = 5 , r1 = d ia g (1, 1) , r 2 = d ia g ( 4 , 4 ) , h = 0 .0 2 s ,

are designed with TD for attitude each loop:

r 3 = d ia g (1 0 , 1 0 , 1 0 ) , r 4 = d ia g ( 2 0 , 2 0 ) , h 0 = 3 h

ADRC))

LESO parameter for each loop:

ω V = 1 0, ω 1 = 5, ω 2 = 1 0, k

ω 3 = 20,ω 4 = 40,

Controller gain for preview control in (20): Proposed Control Parameters for fuzzy logic controller: scheme

k P = d ia g ( 0 .7 , .0 7 )

0 ≤| ρ dP |≤ 0.8 , 0 ≤|| e pP ||≤ 0.8 ,

15 ≤ N ≤ 30 Other parameters are kept the same as Conventional ADRC

In simulation study, the light and moderate turbulence refueling environment are considered in this paper: Case 1: Suppose that the AAR is conducted in the light turbulence, as shown in Fig.7 (a). The sums of the multiple flow perturbations that receiver exposed is shown in Fig. 7 (b).

12

(a). Turbulence

(b). Sum of flows 1 W x(m/s)

Turbx(m/s)

1 0 -1

0

50 t(s)

-1

100

0

0

50 t(s)

100

0

50 t(s)

100

0

50 t(s)

100

-4

3 W z(m/s)

Turbz(m/s)

50 t(s)

-3

-5

100

1 0

-1

0

-2 W y(m/s)

Turby(m/s)

1

-1

0

0

50 t(s)

2 1 0

100

Fig. 7 Airflows in Case 1

Assume that the receiver starts to track the drogue from t = 30 s . In this case, the drogue’s trajectory from the 30s to 100s by the nonlinear hose-drogue model [7] is shown in Fig.8. The preview distance is shown in Fig. 9, as the intensity of the turbulence is light, the value of the curvatures along the drogue trajectory relatively centralized, so the preview distance from the fuzzy logic controller are also relatively centralized along the trajectory. And the 6 DOF nonlinear closed-loop simulation for receiver tracking and relevant results are shown in Fig. 10 to Fig. 14. In order to demonstrate the advantages of the proposed controller, the results for the position tracking, position tracking errors together with their integral of time-weighted absolute error (ITAE) are presented primarily in Fig. 10 and Fig.11. The tracking for the ground velocity by ADRC and the proposed method are basically the same due to they adopt the same controller. But there are still some obvious differences when the NDI is adopted, this relative poorer tracking performance will be more obvious in the position tracking, which can be easily seen in tracking results in Fig10 and the position tracking errors in Fig.11. The ITAE figures for the tracking errors are also be given in Fig.11 to further show the tracking performance of these methods. The proposed “ADRC + Preview” scheme show the best tracking performance. The ITAE index from 30 s to 100 s for the lateral and vertical position error are always smaller than other two methods’. And they are ensured below 160, which are all much smaller than others’. Drogue trajectory in Case 1 -6999.45 -6999.5 -6999.55

Start

z(m)

-6999.6 -6999.65 -6999.7 -6999.75 End

-6999.8 -6999.85

-18.1

-18

-17.9 y(m)

13

-17.8

-17.7

Fig. 8 Drogue trajectory in Case 1(30s-100s) Preview distance in Case 1 30 28

N

26 24 22 20 30

40

50

60

70

80

90

100

t(s)

Fig. 9 Preview distance in Case 1 Drogue NDI ADRC ADRC+Preview

VK(m/s)

200.01 200.005 200 199.995 30

40

50

60

70

80

90

100

70

80

90

100

70

80

90

100

t(s) -17.8 y(m)

-17.9 -18 -18.1 30

40

50

60 t(s)

z(m)

-6999.4 -6999.6 -6999.8 30

40

50

60 t(s)

Fig. 10 Simulation results for ground velocity and position tracking in Case 1

14

x 10

-3

8

0

-5

-10 30

40

50

60

0.2

70 t(s) 0.15 0.1 0.05

4 2 0 30

40

50

60

70

80

90

100

70

80

90

100

70

80

90

100

t(s) 300

ITAE-ey

ey(m)

NDI ADRC ADRC+Preview 80 90 100

97 98 99 100

0.1 0 -0.1 -0.2 30

40

50

60

70 t(s)

0.2 0.1

80

90

-0.1 50

60

70

40

50

60

300

0

40

100

t(s)

0.16 0.14 0.12 0.1 0.08 0.06 91 92

-0.2 30

200

0 30

100

ITAE-ez

ez(m)

NDI ADRC ADRC+Preview

6 ITAE-eVk

eVK(m/s)

5

80

90

200

100

0 30

100

t(s)

40

50

60 t(s)

Fig. 11 Simulation results for tracking error of ground velocity and position in Case 1

Besides, the drogue tracking statistical results in YOZ plane are given in Fig.12. The red cycle represents the AAR tracking error requirement boundary R0=0.3 m , the green cycle is the reachable tracking error boundary for each method under this light turbulence. It’s obvious that the three methods track the drogue trajectory well and the tracking errors are all held within a sufficiently small range R0 ≤ 0.3 m under light turbulence. Even so, one can still find some advantages of the latter two methods. The “ADRC + Preview” can ensure the tracking error within the reachable error cycle R ≤ 0.1 m , and the ADRC ensures R ≤ 0.13 m , but the NDI’s reachable cycle are larger as R ≤ 0.2 m . On the other hand, with preview control strategy introduced, it can be found that the proposed method performs better in tracking the rapidly changed drogue trajectory compared to ADRC method, as Fig.10 shows. The tracking trajectory with the previewing strategy is closer to the drogue trajectory, which can be easily got in Fig.12. Actually, the tracking trajectory with the preview strategy is shifted forward to some extent on the basis of ADRC. And it can be easily found that its tracking errors are all smaller than ADRC’s. Thus, the preview control strategy shows its good performance to suppress the influence of the receiver’s slower dynamic response. Other results for the receiver’s states, such as the path angles, flow angles, and the position angles, are shown in Fig.13. And compared to ADRC, the forward leadings by the proposed method can also be found in the receiver’s states. The results for the control inputs can be seen in Fig.14.

15

NDI

ADRC

0.3

0.3

0.3

R0=0.3

R0=0.3 0.2

0.1

0.1

0.1

0 -0.1

ez (m)

0.2

ez (m)

0.2

0 -0.1

-0.2

R0=0.3

0 -0.1 R=0.1

R=0.13

-0.2

-0.2

R=0.2 -0.3

-0.3 -0.2

0 ey(m)

0.2

-0.3 -0.2 -0.1

0 0.1 ey(m)

0.2

-0.2 -0.1

0 0.1 ey(m)

Fig. 12 Statistical results for tracking errors in Case 1

NDI ADRC ADRC+Preview

0.05

0.1 γ(°)

χ(°)

0.1

0 -0.05

40

60 t(s)

80

0 -0.1

100

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

40

60 t(s)

80

100

0.2 β(°)

α(°)

2.2 2.1

0

2 1.9

40

60 t(s)

80

-0.2

100

4 1.2 φ(°)

μ(°)

2 0

1

-2 -4

40

60 t(s)

80

0.8

100

2 ψ(°)

2 θ(°)

ez (m)

ADRC+Preview

1.5

0 -2

1

40

60 t(s)

80

100

Fig. 13 Simulation results for angles in Case 1

16

0.2

ADRC ADRC+Preview NDI

4

δE

δA

2

-1

0

-1.5

-2 40

60 t(s)

80

-2

100

4

60 t(s)

80

100

40

60 t(s)

80

100

0.62

2

0.61 δT

δR

40

0

0.6

-2 -4

40

60 t(s)

80

0.59

100

Fig. 14 Simulation results for control inputs in Case 1

Case 2: Suppose that the AAR is conducted in the moderate turbulence, as shown in Fig.15 (a). The sum of the multiple flow perturbations that receiver exposed is shown in Fig. 15 (b). (a). Turbulence

(b). Sum of flows 2 W x (m/s)

Turbx (m/s)

2 0 -2

0

50 t(s)

-2

100

0

0

50 t(s)

100

0

50 t(s)

100

0

50 t(s)

100

-4

3 W z (m/s)

Turbz (m/s)

50 t(s)

-3

-5

100

2 0

-2

0

-2 W y (m/s)

Turby (m/s)

2

-2

0

0

50 t(s)

2 1 0

100

Fig. 15 Airflows in Case 2

Assume that the receiver starts to track the drogue from t = 30 s . In this case, the drogue’s trajectory from the 30s to 100s by the nonlinear hose-drogue model is shown in Fig.16, it can be found that the drogue moves faster and its motion range is bigger under moderate turbulence. The preview distance is shown in Fig. 17. As the intensity of the turbulence is moderate, the preview distance from the fuzzy logic controller adaptively changes along the trajectory. The 6 DOF nonlinear closed-loop simulation for receiver tracking and relevant results are shown in Fig. 18 to Fig. 22. It can also be seen that the tracking for the ground velocity by the three methods are basically similar and obvious differences exist in trajectory tracking. The anti-disturbance ability of ADRC still works, but the trajectory tracking errors increase as the intensity of the turbulence increases, especially the error ez , which has already violated the tracking error limitations. While, the proposed method still works and ensures the required tracking error limitations even under moderate turbulence. For instance, the maximum of ez has been nearly decreased by 25% on the ADRC method via the preview control strategy. And the forward leadings by 17

the proposed method can also be found not only in the receiver’s positions but also in other states. The

proposed “ADRC + Preview” scheme show the best tracking performance. The ITAE index is always smaller than other two methods’. The ITAE index from 30 s to 100 s for the lateral is ensured below 300 and vertical position error is ensured below 370. Besides, from the drogue tracking statistical results in YOZ plane shown in Fig.20, one can see that the NDI cannot control the tracking errors within the required cycle R0 ≤ 0.3 m under moderate turbulence. And although the tracking errors by ADRC are much smaller than NDI, the ADRC also cannot ensure the tracking errors are always among required cycle as the errors may cross the required red boundary sometimes. Only the “ADRC + Preview” scheme can still ensure the tracking error within the desired area. The results for the control inputs can be seen in Fig.22, and their amplitudes are larger than Case 1 because larger controls are needed to track the faster-moving drogue with the intensity of the

turbulence increases. The control inputs of smaller than the other two because of its relative slower dynamic response to the command. Thus, the preview control strateg

18

VK(m/s)

Drogue NDI ADRC ADRC+Preview

200.02 200 199.98 30

40

50

60

70

80

90

100

70

80

90

100

70

80

90

100

t(s)

y(m)

-17.6 -17.8 -18 -18.2 30

40

50

60 t(s)

z(m)

-6999 -6999.5 -7000 30

40

50

60 t(s)

Fig. 18 Simulation results for ground velocity and position tracking in Case 2

NDI ADRC ADRC+Preview

eVK(m/s)

0.02 0.01 0

15 ITAE-eVk

0.03

NDI ADRC ADRC+Preview

10 5

-0.01 40

60 t(s)

0.4

80 0.3 0.25 0.2 0.15

ey (m)

0.2

0

100

60

80

100

80

100

300 200 100

40

60

80

0

100

40

60

t(s) 1

0.4

0.5

0.2

t(s) 800 600 ITAE-ez

ez (m)

100

400

66 67 68

-0.2

83.58484.585 0 -0.5

80

500

0

-0.4

40

t(s)

ITAE-ey

-0.02

400 200

40

60

80

0

100

t(s)

40

60 t(s)

Fig. 19 Simulation results for tracking error of ground velocity and position in Case 2

19

NDI

ADRC

0.5

0.4

0.4

0.3

ADRC+Preview 0.3 0.2

0.3 0.2

0 R0=0.3

-0.1

0.1

ez (m)

ez (m)

0.1

0.1

0 -0.1

0 R0=0.3 -0.1

R0=0.3

-0.2

-0.2 -0.2

-0.3

-0.3 -0.3

-0.4 -0.2

0 ey(m)

0.2

-0.2 -0.1

0 0.1 ey(m)

0.2

-0.2 -0.1

0 0.1 ey(m)

Fig. 20 Statistical results for tracking errors in Case 2 NDI ADRC ADRC+Preview γ(° )

χ(° )

0.1 0 -0.1

40

60 t(s)

80

0.2 0

-0.2

100

40

60 t(s)

80

100

40

60 t(s)

80

100

2.5 β (° )

0.2 α (° )

ez (m)

0.2

2

0 -0.2

1.5

40

60 t(s)

80

100

20

0.2

ADRC ADRC+Preview NDI

2 0 δE

δA

10 0

-2 -10 60 t(s)

80

-4

100

40

0.64

20

0.62 δT

δR

40

0 -20 -40

40

60 t(s)

80

100

40

60 t(s)

80

100

0.6 0.58

40

60 t(s)

80

100

Fig. 22 Simulation results for control inputs in Case 2

5. Conclusion In this paper, a novel back-stepping based anti-disturbance flight controller with preview methodology for autonomous aerial refueling. The 6 DOF model for aircraft is written into several strict-feedback nonlinear forms through tactful transformation, especially the dynamics of the path dynamics equations whose original non-affine nonlinear forms are tough for controller design by conventional back-stepping technique. The influences of the unknown flow perturbations on the receiver in each loop are viewed as the component of the ‘‘total disturbances’’ which are estimated and compensated with ESO. Besides, the preview control strategy is adopted in the position controller due to the slower system dynamic response of the receiver. Simultaneously, a novel adaptive look-ahead distance scheme based on the fuzzy logic control, which considers both previewing error and curvature and the reference trajectory, is proposed for the selection of the preview point. Comparison and extensive simulations are carried out to effectively verify the superiority and feasibility of the proposed method under different turbulence. Since the dynamic model of the preview error is very important for the tracking performance, the dynamic model of the preview error and the dynamic model based preview control scheme for AAR will be possible areas of future research. Appendix A. Aerodynamic force and moment T = Tmaxδ T , ρ = ρ 0 e

−k z

,Q =

1 ρV 2 , W = [Wx Wy Wz ]T , 2

ª cL ,0 + cαL α + cLα α 2 + cLq cqrel / ( 2V ) + cLδ e δ e º ªL º « » « D » = QS « c + cα α + cα 2 α 2 » ,0 D D D « » « » δ δ β ¬«C ¼» « cC,0 + cC β + cCa δ a + cCr δ r » ¬ ¼, 2

ª cL0 + cαL α º ª cL º « » = QS « cD0 + cαDα » = QS «« cD »» «c0 + c β β » «¬ cC »¼ ¬ C C ¼ 21

ª$ «% « «¬ &

ª c $ ,0 + c $δ a δ a + c $δ r δ r + c $β β + c $p bp rel / (2V) + c $r brrel / (2V) º º » δ q » = QSl « c « % ,0 + c %e δ e + c % cq rel / (2V) » » « » δa δr β p r »¼ «¬ c & ,0 + c & δ a + c & δ r + c & β + c & bp rel / (2V) + c & brrel / (2V) »¼

ª c $0 + c $δ a δ a + c$δ r δ r º ª c$ « 0 » « δe = QSl « c % + c% δ e » = QSl « c % « 0 » «c δa δr ¬ & ¬ c& + c& δ a + c& δ r ¼

,

º » » » ¼

­ prel = p − peff ° ®qrel = q − qeff ° ¯rrel = r − reff

where W is the sum flow vector in inertial frame, V is the airspeed, L , D and C are the lift, drag and lateral force, $ , % , & are the moments along the axis of body frame, cL , cD , cC are the lift, drag and lateral force coefficients, c$ , c% , c& are the rolling, yawing and pitching moment coefficients,

Tmax is the maximum available thrust. δ a , δ e , δ r are control actuators (aileron, rudder, and elevator deflection), and δ T denote the throttle setting. peff , qeff , reff are rotational winds relative to the inertial frame [18]. α K , β K are the path flow angles (angle of attack and sideslip angle in flight path frame) [39,40], α w , β w are the appendant flow angles caused by flow perturbations. B. Variables and Vectors in equations (6-7)

fVk = ( − D − C β w + Lα w − mg sin γ ) / m, BVk = Tmax cos(α + σ ) cos β / m

ªVk ªV ( cos γ sin χ − χ ) º F1 = F1 (X 2 ,Vk ) = « k » , B1 = B1 (Vk ) = « 0 ¬ ¬ −Vk ( sin γ − γ ) ¼

0 º , −Vk »¼

ª § T ( −βk cos μ + (α k + σ ) sin μ ) + ·º 1 « ¨ ¸» « ( mVk cos γ ) ¨ ( C − Dβw ) cos μ + ( QScL0 − Dα w ) sin μ ¸ » © ¹» , F2 = F2 (X2 , X3 ,Vk , Q) = « « § ·» − − + + + T sin cos mg cos β μ α σ μ γ ( ) ( ) k k « 1 ¨ ¸» « ( −mVk ) ¨ ( C − Dβ ) sin μ − ( QSc0 − Dα ) cos μ ¸» w L w © ¹¼ ¬

B 2 = B 2 (X 2 , X3 , Vk , Q ) =

1 ªQScαL / cos γ « mVk ¬ 0

0 º », QScCβ ¼

ª º ( −γ cos μ − χ sin μ cos γ ) / cos β « »,    F3 = F3 (X 2 , X 3 , X 4 ) = « −γ sin μ + χ cos μ cos γ » «(γ sin β cos μ + χ ( sin γ cos β + sin β sin μ cos γ ) ) / cos β » ¬ ¼

ª − cos α tan β sin α B3 = B3 (X 3 ) = « « ¬« cos α / cos β

1 − sin α tan β º », 0 cos α » 0 sin α / cos β ¼»

22

ª 1 « 2 I I « x z − I xz « F4 = F4 (X 4 ) = « « « 1 « 2 I I ¬« x z − I xz

º ª ( I y I z − I z2 − I xz2 ) rq + ( I x I xz − I y I xz − I z I xz ) pq + I z QSlc$0 + I xz QSlc&0 º » ¬ ¼ » », 1 2 2 0 ª¬ ( I z − I x ) pr − I xz p + I xz r + QSlc% º¼ » Iy » » ª ( I x2 − I x I y + I xz2 ) pq − ( I x I xz − I y I xz − I z I xz ) rq + I xzQSlc$0 + I xQSlc&0 º » ¬ ¼» ¼

ª I z c$δ a + I xz cδ&a « 2 « I x I z − I xz « B 4 = B 4 (X 2 , Q ) = QSl « 0 « « δa δa « I xz c$ + I x c& « I x I z − I xz2 ¬

0 δe c% Iy

0

I z c$δ r + I xz cδ&r º » I x I z − I xz2 » » » 0 » δr δr » I xz c$ + I x c& » I x I z − I xz2 »¼

Acknowledgment This research has been funded in part by the National Natural Science Foundations of China under Grant 61175084 and 61473012, in part by Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT 13004. The authors would also like to thank the reviewers and the editor for their comments and suggestions that helped to improve the paper significantly. References [1] Peter R. Thomas, Ujjar Bhandari, Steve Bullock, Thomas S. Richardson, Jonathan L. du Bois. Advances in air to air refueling. Progress in Aerospace Sciences 71 (2014) 14-35. [2] Zikang Su, Honglun Wang, Xingling Shao, et al. A robust back-stepping based trajectory tracking controller for the tanker with strict posture constraints under unknown flow perturbations. Aerospace Science and Technology 56 (2016): 34-45. [3] Hensen J H, Murray J E, Campos N V. The NASA Dryden AAR Project: A Flight Test Approach to an Aerial Refueling System. AIAA Atmospheric Flight Mechanics Conference & Exhibit, AIAA 2004-4939, Aug 2004. [4] Dibley R, Allen M, Nabaa N. Autonomous Airborne Refueling Demonstration Phase I Flight-Test Results. AIAA Atmospheric Flight Mechanics Conference and Exhibit, Aug 20-23, 2007. [5] J. P. Nalepka and J. L. Hinchman. Automated aerial refueling: Extending the effectiveness of unmanned air vehicles, in AIAA Modeling and Simulation Technologies Conference and Exhibit, San Francisco, California, August 15-18 2005. [6] Fravolini ML, Ficola A, Campa G, Napolitano MR, Seanor B. Modeling and control issues for autonomous aerial refueling for UAVs using a probe-drogue refueling system. Aerospace Science and Technology 8(2004):611-618. [7] K. Ro, T. Kuk, J.W. Kamman, Modeling and Simulation of Hose-Paradrogue Aerial Refueling Systems, Journal of Guidance, Control, and Dynamics 33(1) (2010):53-63. [8] Atilla Dogan, Sriram Venkataramanan. Modeling of Aerodynamic Coupling Between Aircraft in Close Proximity. Journal of Aircraft 42(4) (2005):941-955. [9] Dai Xunhua, Wei Zibo, Quan Quan. Modeling and simulation of bow wave effect in probe and drogue aerial refueling. Chinese Journal of Aeronautics 29(2) (2016): 448-461.

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