Command filtered backstepping sliding mode control for the hose whipping phenomenon in aerial refueling

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Command filtered backstepping sliding mode control for the hose whipping phenomenon in aerial refueling ✩

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a

Qiusheng He , Haitao Wang

b,c,∗

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, Yufeng Chen , Ming Xu , Wanfeng Jin

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a

School of Electronics and Information Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China b Theory Training Department, Harbin Air Force Flight Academy, Harbin 150001, China c Aeronautics and Astronautics Engineering College, Air Force Engineering University, Xi’an 710038, China

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a r t i c l e

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Article history: Received 19 July 2016 Received in revised form 13 April 2017 Accepted 13 April 2017 Available online xxxx Keywords: Aerial refueling Command filtered backstepping Hose–drogue assembly Hose whipping phenomenon Permanent magnet synchronous motor Sliding mode

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Studied in this paper is the design of a command filtered backstepping sliding mode controller for the hose whipping phenomenon in aerial refueling. Firstly, a dynamic model of the variable-length hose– drogue assembly with the restoring force due to bending previously built by the authors is introduced as a control plant. According to the kinematics, the hose length control is transformed to the angular control of the permanent magnet synchronous motor to keep the hose tension stable and suppress the hose whipping. Then, an active control strategy based on command filtered backstepping sliding mode angular control of the permanent magnet synchronous motor is proposed. To maximize the output torque, the space vector modulation method is adopted. Considering the strict-feedback configuration of the angular velocity subsystem, command filtered backstepping is used to eliminate the analytic computation of high order command derivatives required in the previous control system design. In the current voltage subsystem, exponential sliding mode reaching laws of d-axis and q-axis current errors are applied to enhance convergence speed, control accuracy, and robustness. Results show that the control system is simplified substantially and the requirement on the number of sensors is also relaxed. Finally, the superiority of the control laws is analyzed by simulations. © 2017 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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Since the hose–drogue aerial refueling system (HDARS) was invented in 1950s, aerial refueling has played an integral and essential role for a successful execution of modern Air Force military operations [1]. Based on the cost and operational considerations, the hose–drogue-based refueling method is appealing for unmanned aerial vehicle (UAV) operations, especially with recent and continued improvement of autonomous aerial refueling (AAR) using vision-based control [2–4] and navigation techniques [5,6]. However, there have been a growing number of incidents reported involving the hose–drogue system as this platform has been more widely used over the past years [7,8]. For instance, it is reported that the hose–drogue system suffered a 2.5% failure rate [9].

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✩

Foundation item: National Natural Science Foundation of China (No. 61473307); Excellent Doctoral Dissertation Foundation of the AFEU (KGD 081114006); National Natural Science Foundation of China and Shanxi Provincial People’s Government Jointly Funded Project of China for Coal Base and Low Carbon (No. U1510115). Corresponding author at: Theory Training Department, Harbin Air Force Flight Academy, Harbin 150001, China. E-mail address: [email protected] (H. Wang).

*

http://dx.doi.org/10.1016/j.ast.2017.04.020 1270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

A skillful piloting technique is still the key to success in the hose– drogue-based refueling process for both manned and unmanned receiver aircrafts. A sense of urgency of the pilot or an unreasonable design of UAV control system usually results in a higher closure speed, which further increases the chance of a catastrophic failure. It is also important to note that not all catastrophic failures are the result of excessive closure speed by the receiver. In fact, if the tensator (spring-loaded take-up device) is disabled during coupling, it is virtually impossible for a receiver to dock without experiencing the so-called hose whipping phenomenon. This phenomenon generates high loads on the hose and probe. Whereas these extreme loads only persist for a fraction of a second, they result in critical damage to the hose assembly and the receiver probe, and they eventually lead to a potentially catastrophic accident. This dynamic event is responsible for a large fraction of the documented aerial refueling catastrophic failures [10]. Consequently, the hose whipping has seriously limited success rates and security of aerial refueling. Fig. 1 shows a severe example of hose whipping during coupling [11]. The hose whipping phenomenon is a highly complex, coupled, and extreme dynamic behavior of the hose subject to nonlinear hose elastodynamic characteristics, hose rewind control, receiver

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Fig. 1. Time history of the hose whipping phenomenon during coupling.

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approach motion profile, tanker wake, and so on [12]. Understanding the dynamic characteristics of the HDARS and designing an effective hose control strategy are essential to suppress the hose whipping. Unfortunately, only a limited number of references are available. To model the hose–drogue assembly, the early studies [13,14] analyzed the effects of external disturbances on the hose–drogue assembly motion prior to probe contact, such as vertical gusts, wing vibration, and vortex. Ref. [15] obtained abundant aerodynamic data specifically required in dynamic modeling of the hose– drogue assembly by wind tunnel and flight tests. The dynamic characteristics of KC-10 hose whipping phenomenon was investigated in Refs. [7,10,16,17]. Ref. [18] analyzed the post-contact phase using a hose–drogue dynamic model simplified in the twodimensional vertical plane mentioning that a severe hose whipping during coupling may occur in the absence of the reel take-up system. The loads on the probe in the hose whipping phenomenon were investigated by numerical simulations in Ref. [12]. Refs. [9] and [19] presented a good literature review for aerial towed system modeling. To depict the dynamics of the hose–drogue assembly, they proposed a new locking-free, curved-beam, finite element formulation based on mechanics of materials that overcomes the difficulties of classical cable theory in treating large rotations and deformations of the hose–drogue assembly, and then analyzed the dynamic characteristics of the hose–drogue assembly. Ref. [6] proposed a lumped-mass, finite-segment approach based on multi-body dynamic modeling framework by extending the previous work in the marine engineering community [20]. The hose–drogue assembly is modeled by a series of constantlength links connected with frictionless ball-and-socket joints. The link masses and all external forces are lumped at the connecting joints. In Refs. [21–24], a dynamic model of the hose–drogue assembly is built by the authors based on Ref. [8] with all links having variable length, other than constant length as Ref. [6], and also not only the first link as Ref. [25]. The effect of the hose restoring force due to bending [10] is also considered in modeling process. The model can reflect the complete payout/retrieval dynamics, and support exploiting hose whipping suppression methods. To maintain the stability of the hose tension and suppress the hose whipping, the refueling pod is usually equipped with a tensator [10]. However, the HDARS still suffered a 2.5% failure rate as mentioned above when the hose slackened and lost the stability. Vassberg et al. [10,16] confirmed that the tensator take-up speed lagging behind the closure speed was responsible for the failure. Ro et al. [25] tried to improve the control laws of the tensator by modifying the mechanism architectures. However the tensator-based control strategy is still a passive method in nature, and mechanism rebuilding of the tensator is quite difficult. Recently, the integration of a permanent magnet synchronous motor (PMSM) and high-precision position sensors into the refueling

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Fig. 2. The configuration, modeling assumptions, and definitions of coordinate frames of a HDARS.

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pod [26] provides another chance for active high-performance hose whipping suppression methods. A new type of refueling pod was invented, in which the reel is driven by the PMSM [27]. Adopting this new thought, an integral sliding mode backstepping active control strategy for the hose whipping was proposed by the authors [22]. Although this control method can actively match the receiver coupling motion profile and suppress the hose whipping effectively, the control laws must require analytic expressions of high ...order derivatives of the rotor angular position command (ϑ¨ c and ϑ c , as listed in Ref. [22]) due to using standard backstepping. In fact, it is well known that analytic computation of command derivatives will become increasingly complex as the order of the system increases, moreover high order derivatives of the command may not be achievable due to the discontinuity of the physical system. Fortunately, Command Filtered Backstepping proposed in Refs. [28–30] can overcome these disadvantages effectively. By extending our previous work of Ref. [22], the objective of this paper is to design an active suppress strategy of the hose whipping for probe–drogue-based autonomous aerial refueling. Section 2 reviews the dynamic model of the variable-length hose– drogue assembly with the hose restoring force due to bending, which is built in Refs. [21–24] and used as a realistic controlled plant in the controller design. PMSM command filtered backstepping sliding mode position control for the hose whipping suppression and the global stability of the control system in the working region are presented in Sections 3 and 4. Simulation results are given in Section 5.

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2. Model of a HDARS and control objective transformation

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2.1. Model of the hose–drogue assembly controlled by a PMSM

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A HDARS includes three subsystems, namely, a reel mechanism, a refueling hose, and a drogue. It is assumed that the reel is driven by a PMSM to deploy or retrieve the hose through a reducer. As illustrated in Fig. 2, the hose–drogue assembly is discretized as a link-connected system, where the hose consists of a chain of variable-length links connected with frictionless joints. The drogue is treated as a mass point at the hose end. The model of the hose–drogue assembly is introduced in detail in our previous work [21–24], which is used as the controlled plant. Here O w X w Y w Z w is defined as a towing point frame. The axes of O w X w Y w Z w are parallel to the tanker trajectory frame. O n X n Y n Z n is the north-east-down frame. r k is the position vector of joint k in O w X w Y w Z w . pk is the position vector from joint k − 1 to joint k. θk1 and θk2 are angles of the link k relative to the planes O w X w Y w and O w X w Z w respectively. L H is the distance between the drogue and the outlet of the pod before coupling. V tanker is the velocity of the tanker.

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Assuming there are no magnetic saturation, hysteresis, eddy current loss, and friction of the reducer, excluding sinusoidal magnetic field distribution, the mathematical model of the PMSM [22, 24,31] can be described in the d–q coordinate system as follows

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Ls

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(1)

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where ω is the rotor angular velocity, ϑ is the rotor angular position, R is the stator resistance, L s is the stator inductance, P is the number of pole pairs, ψ f is the stator flux linkage, J is the moment of inertia, B is the viscous friction coefficient, T L is the load torque. i d , i q and ud , u q are d-axis and q-axis currents and voltages, respectively [22,24,31].

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2.2. Control objective transformation

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Fig. 3. The control objective transformation principle.

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The most effective way to suppress the hose whipping is to stabilize the hose tension by real-time active hose reeling in/out on the basis of the relative position between the tanker and the receiver. According to the kinematics, a change in hose length L can be expressed as a function of the rotor angular position of PMSM ϑ as follows [22,24]

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L = ϑ r /λ,

(2)

where r is the radius of the reel, λ is the reduction ratio of the reducer. L is also a function of relative distance between the probe and the pod during coupling, which is calculated as [22,24]



L = η x2 +  y 2 +  z2 − L 0 ,

(3)

where L 0 is the hose length before coupling. As shown in Fig. 2, η = L 0 / L H . η reflects the degree of hose slack. x,  y, z are three spatial components of relative distance between probe and pod respectively. It is assumed herein that x,  y,  z are measured exactly without sensor error and noise. Therefore, the ideal hose length reference trajectory is generated based on the initial hose length directly before coupling and the relative distance between the probe and the pod to maintain a constant hose slack. From Equ. (2) and Equ. (3), length control of the hose can be converted into the rotor angular position control of the PMSM as follows:

 



ϑc = λ L 0 x2 +  y 2 +  z2 / L H − L 0 /r .

(4)

Then keeping the hose tension stable and suppressing the hose whipping can be achieved by the PMSM position control through Equ. (4). The control objective transformation principle of the proposed control system as shown in Equ. (4) is illustrated in Fig. 3.

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3. PMSM command filtered backstepping sliding mode position control

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strict-feedback system with i q as its input. Command Filtered Backstepping [28,29] is used to eliminate the analytic computation of command derivatives. For Subsystem 2, namely the later two equations of Equ. (1), an exponential reaching law is used to construct the sliding mode surface equations of d-axis and q-axis current errors to ensure convergence speed and robustness. To maximize the output torque, the space vector modulation (SVM) method [31] is commonly used for a PMSM. The simplest and most efficient way in the SVM methods is set

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id,c = 0,

(6)

where i d,c is the d-axis current command. Under this condition, the electric torque of the PMSM T e can be expressed as

T e = 3P ψ f i q /2.

(7)

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Define tracking errors of each subsystem of the PMSM as follows

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⎫ ϑ˜ = ϑ − ϑc , ϑ¯ = ϑ˜ − ξ1 ⎪ ⎪ ω˜ = ω − ωc , ω¯ = ω˜ − ξ2 ⎬ , ˜i q = i q − i q,c ⎪ ⎪ ⎭ ˜id = id − id,c

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(8)

For Subsystem 1, according to Command Filtered Backstepping, assume that the angular position command ϑc and its first derivative ϑ˙ c are available, and then define the pseudocontrol signal α and i q0,c as

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− k2 ω˜ + ω˙ c − ϑ¯ ,

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Through the transformation above, the control objective is to achieve angular position-tracking as follows

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˜id is the tracking error of id . ξ1 and ξ2 will be defined later.



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where ϑ˜ is the tracking error of ϑ . ϑ¯ is the compensated tracking ˜ is the tracking error of ω . ω¯ is the compensated error for ϑ˜ . ω ˜ . ωc is the rotor angular velocity command. tracking error for ω ˜i q is the tracking error of i q . i q,c is the q-axis current command.

α = −k1 ϑ˜ + ϑ˙ c ,

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(10)

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t →∞

(5)

where ϑc is the angular position command. As listed in Equ. (1), the structure of PMSM could be treated as two subsystems. Subsystem 1, including the first two equations, could be regarded as a 2nd order single-input–single-output

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(11)

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(12)

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To eliminate the chattering phenomenon of sliding mode control, the symbolic function Equ. (16) is substituted by the nonlinear function as follows [31]

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Fig. 4. The filter that generates the command and command derivative.

where ξ1 (0) = ξ2 (0) = 0. Define the pseudocontrol signal

ωc0 = α − ξ2 .

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(13)

ωc0 and iq0,c through a filter, as shown in Fig. 4, to produce ˙ c , ωc , i q,c , and ˙i q,c . virtual control signals ω Here ωn > 0 and ζ ∈ (0, 1] are designer specified constants. Pass

Considering mechanical inertia, the transfer function from t 1 to the load torque T L can be written as

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ωc0 as

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s˙ 1 = −a1 s1 − ρ1 sgn(s1 ),

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where s1 = c 1 ˜i q . c 1 > 0, a1 > 0, and constants.

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sgn(x) =

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(15)

ρ1 > 0 are designer specified

x≥0 x < 0.

(16)

˙i q = ˙˜i q + ˙i q,c = s˙ 1 /c 1 + ˙i q,c .

(17)

Substituting Equ. (17) into Equ. (1), the real q-axis voltage input is expressed as

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u q = Riq + PLs ω id + P ψ f ω + L s (˙s1 /c 1 + ˙i q,c ).

(18)

Similarly, the sliding mode surface equation of the d-axis current error is designed using the exponential reaching law as

s˙ 2 = −a2 s2 − ρ2 sgn(s2 ),

(19)

where s2 = c 2 ˜i d . c 2 > 0, a2 > 0, and ρ2 > 0 are designer specified constants. From Equ. (8) and Equ. (19), ˙i d is calculated as

˙id = ˙˜id + ˙id,c = s˙ 2 /c 2 .

(20)

Substituting Equ. (20) into Equ. (1), the real d-axis voltage input is expressed as

ud = Rid − PLs ω i q + L s s˙ 2 /c 2 .

(22)

where σi > 0 is a designer specified constant. Compared with the controller presented previously in Ref. [22], the control method proposed here to suppress the hose whipping ... does not require ϑ¨ c and ϑ c , and the system structure is more concise. So it is easier to realize in the practical engineering. Fig. 5 depicts the signal flow in block diagram form of the proposed control system. Here CF represents a command filter as defined in Fig. 4, SM represents a sliding mode surface equation as defined in Equ. (15) and Equ. (19), and α is computed according to Equ. (9). 4. Global stability of the control system in the working region For the system described by Equ. (1), using control laws of Equ. (18) and Equ. (21), the global asymptotical stability in the working region can be guaranteed. The stability analysis of the proposed control system will utilize the dynamics of the tracking ˜ , and the dynamics of the compensated tracking errors errors ϑ˜ , ω ϑ¯ , ω¯ . These equations are derived as follows. According to Equs. (1), (8), and (9), the differential equations of ϑ˜ and ω˜ are derived as

ϑ˙˜ = ϑ˙ − ϑ˙ c = −k1 ϑ˜ + (ω − α ),  3P ψ f  ω˙˜ = ω˙ − ω˙ c = −k2 ω˜ − ϑ¯ + i q − i q0,c .

(21)

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2J

ϑ˙¯ = ϑ˙˜ − ξ˙1 = −k1 ϑ¯ + ω¯ , ¯ ω¯˙ = ω˜˙ − ξ˙2 = −k2 ω¯ − ϑ.

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(23)

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(24)

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According to Equs. (8), (11), (12), (23), and (24), the differential ¯ are derived as equations of ϑ¯ and ω

From Equ. (8) and Equ. (15), ˙i q is calculated as

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s i / | s i | + σi ,

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(25)

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(26)

The stability of the proposed control system is analyzed by considering the following Lyapunov function V

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2

¯ 2 + s21 + s22 . V = ϑ¯ 2 + ω

(27)

The derivative of Equ. (27) can be derived as follows by using Equs. (15), (19), (25), and (26)

¯ ω˙¯ + s1 s˙ 1 + s2 s˙ 2 V˙ = ϑ¯ ϑ˙¯ + ω ¯ k1 ϑ¯ + ω¯ ) + ω¯ (−k2 ω¯ = ϑ(−

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¯ − a1 s21 − ϑ)

− ρ1 s1 sgn(s1 ) − a2 s22 − ρ2 s2 sgn(s2 ) = −k1 ϑ¯ 2 − k2 ω¯ 2 − a1 s21 − ρ1 |s1 | − a2 s22 − ρ2 |s2 | ≤ 0.

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(28)

It is straightforward to show that Equ. (28) is nonpositive definite, namely the global asymptotical stability of the proposed control system is ensured.

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Fig. 5. Block diagram of the proposed control system.

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Fig. 6. Two types of relative coupling profile. (a) Relative acceleration. (b) Relative velocity. (c) Relative displacement.

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5. Simulation results With a 39.88 m wing span and a 2.85 m distance between the pod and the top of the right wing, the tanker wake is modeled using a Helmholtz horseshoe-vortex model [32]. All parameters used in simulations are set as follows. (1) Standard atmosphere plus Dryden wind turbulence is adopted in simulations, and flight altitude and airspeed are set equal to 7620 m and 200 m/s respectively. (2) Parameters of the hose–drogue assembly: μ = 4.11 kg/m, do = 0.0673 m, di = 0.0508 m, ct ,k = 0.0052, cn,k = 0.2182, mdrogue = 29.5 kg, ddrogue = 0.61 m, c drogue = 0.831, and N = 24. Here definitions of variables could be found in Refs. [21–24].

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(3) Parameters of PMSM, reducer and reel: R = 1.65 , L s = 0.0092 H, P = 4, ψ f = 0.175 Wb, J = 0.001 kg·m2 , B = 4.831 × 10−5 N·m·s, i = 10, and r = 0.06 m. (4) Parameters of command filtered backstepping sliding mode control laws: k1 = 10, k2 = 25, c 1 = c 2 = 0.1, a1 = a2 = 5, ρ1 = ρ2 = 8.5, σi = 0.1, ζ1 = 0.9, ωn1 = 40, ζ2 = 0.8, and ωn2 = 180.

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5.1. Dynamics of the hose whipping phenomenon

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5.1.1. Hose whipping phenomenon at various closure profiles In this first case, assume that the hose length is 22.86 m, the probe initially contacts the drogue at t = 290 s, completes docking at t = 290.25 s, and then pushes the drogue forward along two types of relative coupling profiles as illustrated in Fig. 6. The time-history of the hose geometry during two types of coupling is shown in Fig. 7. The hose whipping is not obvious due to the low degree of the hose slack caused by the lower closure speed of profile 1, while the result of profile 2 is just the opposite. The aerodynamic drags and tensions of the 1st, 13th, and 24th segment hose, and hose restoring forces of the 1st, 13th, and 23th joint are shown in Fig. 8. Fig. 9 shows the external loads on the probe. From Fig. 8 and Fig. 9, when the aerodynamic drag on the drogue is completely absorbed by the probe at the instant of docking, there occurs a sudden drop of the hose tension that initially sets the catenary shape of the hose. Aerodynamic drags lag behind the tension. As the probe pushes the drogue forward, the hose trends to slack and sags rapidly due to the gravity. The disturbance starts around the drogue, and then propagates upwind to the tanker under the interaction of the gravity, aerodynamic drag, and hose restoring force due to bending. As a result, any disturbance to the hose will be pushed by the airflow to travel downstream with an increasing magnitude resulting in a violent

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Fig. 7. Hose geometry (22.86 m) history during two types of coupling. (a) Coupling along the profile 1. (b) Coupling along the profile 2.

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Fig. 8. Drags, tensions, and restoring forces of the hose during two types of coupling. (a) The aerodynamic drag of the 1st link. (b) The aerodynamic drag of the 13th link. (c) The aerodynamic drag of the 24th link. (d) The tension of the 1st link. (e) The tension of the 13th link. (f) The tension of the 24th link. (g) The tension of the 1st joint. (h) The tension of the 13th joint. (i) The tension of the 23th joint.

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Fig. 9. External forces on the probe during two types of coupling. (a) The component of the X w direction. (b) The component of the Y w direction. (c) The component of the Z w direction.

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whipping action near the drogue. High-frequency oscillations in the plots of the drags, tensions, and restoring forces are induced by Dryden wind turbulence.

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5.1.2. Hose whipping phenomenon at various hose lengths In this second case, assume that the hose length is 14.33 m, the probe completes coupling along profile 2. During coupling, the time-history of the hose geometry is shown in Fig. 10. The aerodynamic drags, tensions of the 1st, 13th, and 24th segment hose, and hose restoring forces of the 1st, 13th, and 23th joint are shown in Fig. 11. The external loads on the probe are shown in Fig. 12. Comparing Fig. 7(b), Fig. 8, Fig. 9, Fig. 10, Fig. 11, and Fig. 12, making an identical coupling along profile 2, when the hose length is 14.33 m, which is much shorter than 22.86 m, the weight of the hose out of the pod is lighter, the hose restoring force due to bending is larger, the hose lies closer to the core of the tanker wake, the degree of hose slack induced by coupling is much higher, and the hose whipping goes more violent. The extent of the whipping, the tension at the hose end, and the external loads acting on the probe increase sharply out of control in just 4 seconds. The hose thereby loses stability and quickly turns into a chaotic state of motion.

5.1.3. Effects of the tensator control In this third case, let L 0 = 14.33 m. The coupling profile 2 is chosen. Under this condition, the hose whipping is more violent. When the reel is driven by the tensator as presented in Refs. [10] and [25], the time-history of the hose geometry is shown in Fig. 13. The length and speed history of the hose motion driven by the reel are shown in Fig. 14. The aerodynamic drags, tensions of the 1st, 13th, and 24th segment hose, and the hose restoring forces of the 1st, 13th, and 23th joint are shown in Fig. 15. The external loads on the probe are shown in Fig. 16. As shown in Figs. 13–16, the hose whipping controlled by the tensator is not violent, but the inherent simple harmonic oscillation property of the spring is aroused. This property plus hose restoring forces compels the hose to whip persistently. As shown in Fig. 13, the hose always whips around the equilibrium position, and the amplitude decreases slowly. The length/speed of hose reeling, aerodynamic drags, tensions, and restoring forces of the hose also enter a similar state. This simple harmonic oscillation, which is another type hose whipping phenomenon in nature, may cause aerial refueling to fail. The whipping will generate high internal loads on the hose, and exert high loads with a severe non-axial oscillation on the probe. Whereas these extreme loads only persist for a fraction of a second, they can damage both the hose and the probe, and even cause a catastrophic accident. The hose whipping will worsen owing to the interference caused by tanker wake and atmospheric turbulence. To sum up, the longer distance the probe moves forward and the shorter the hose length is, the more violent the hose whipping goes. For this situation, it is significant to develop a new quick and efficient hose whipping suppression strategy to improve the success rate and security of aerial refueling.

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5.2. PMSM command filtered backstepping sliding mode position control for the hose whipping phenomenon

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Let L 0 = 14.33 m. The coupling profile 2 is chosen. Under this condition, the hose whipping is more violent as shown in Section 5.1. In practical coupling flight, the receiver can not exactly travel along the ideal coupling profile as shown in Fig. 6. Its attitude and position must be disturbed by tanker wake and atmospheric turbulence. Then assuming that the position of the probe in vertical direction is disturbed by a sine disturbance such as

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Fig. 10. Hose geometry (14.33 m) history adopting coupling profile 2.

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Fig. 11. Drags, tensions, and restoring forces of the hose adopting coupling profile 2. (a) The aerodynamic drag of the 1st link. (b) The aerodynamic drag of the 13th link. (c) The aerodynamic drag of the 24th link. (d) The tension of the 1st link. (e) The tension of the 13th link. (f) The tension of the 24th link. (g) The tension of the 1st joint. (h) The tension of the 13th joint. (i) The tension of the 23th joint.

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Fig. 12. External forces on the probe adopting coupling profile 2. (a) The component of the X w direction. (b) The component of the Y w direction. (c) The component of the Z w direction.

0.1 sin(π t /5) after t = 300 s, the PMSM command filtered backstepping sliding mode position control for the hose whipping is analyzed as follows. Assume that the proposed control laws here come to work immediately after docking. Then the time-history of the hose geometry is shown in Fig. 17. The aerodynamic drags, tensions of the 1st, 13th, and 24th segment hose, and the hose restoring forces of the 1st, 13th, and 23th joint are shown in Fig. 18. The external loads on the probe are shown in Fig. 19. The length and speed histories of the hose motion controlled by the control laws proposed are shown in Fig. 20. State variables of the PMSM are shown in Fig. 21. The real d-axis and q-axis voltage inputs are shown in Fig. 22. From Fig. 17, as the probe travels forward, the control laws can retract the hose rapidly and precisely, and keep the catenary shape of the hose steady. The hose slack is maintained at the level just before coupling, and the hose whipping does not occur at all. After t = 300 s, the probe, just like a vibration source, compels the drogue to travel up and down around the initial altitude periodically. If without control, the sine wave will travel toward the pod under the influence of tanker wake and atmospheric turbulence, and the sine wave-type hose whipping phenomenon will occur later. With the compensation provided by the control laws, the catenary shape of the hose can always be kept steady. The sine

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Fig. 13. Hose geometry history controlled by the tensator.

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Fig. 14. Length and speed histories of hose reeling controlled by the tensator. (a) The length history of the hose motion. (b) The speed history of the hose motion.

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wave disturbance does not travel along the hose to induce the hose whipping. From Fig. 18 and Fig. 19, during the coupling interval from 290 s to 290.25 s, a sharp drop of the hose tension and a momentary increasing of the axial force on the probe happen after

the aerodynamic drags on the drogue are totally absorbed. Then excessive slack of the hose occurs. From Fig. 20, a large initial length tracking error (as well as the speed tracking error) also occurs owing to the coupling. At t = 290.25 s, the control laws start to retract the hose. Since the control laws completely match profile 2, the hose tension recovers rapidly and becomes stable. After t = 300 s, drags and tensions of the hose, external forces on the probe change into a sine tracking state around the corresponding equilibrium point to compensate the sine wave position disturbance from the probe. Ultimately, the control laws efficiently suppress the hose whipping. Since a certain extent of the hose slack arises before docking and the control laws do not work at that time, as shown in Fig. 21, each subsystem must accommodate a large initial tracking error. In addition, the load torque T L is disturbed strongly under the interaction of retracting operation, tanker wake, and atmospheric turbulence. Despite adverse impacts above, the d-axis and q-axis currents can track the virtual commands rapidly and precisely, which benefits from the responsivity and robustness of the sliding mode surface equations of the d-axis and q-axis current errors. After t = 300 s, the control laws can match the motion of the receiver completely to control state variables into a dynamic tracking state and compensate for the sine wave position disturbance. Comparing Figs. 18 and 19 with Figs. 11 and 12, and with Figs. 15 and 16, it is observed that the internal loads on the hose

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Fig. 15. Drags, tensions, and restoring forces of the hose controlled by the tensator. (a) The aerodynamic drag of the 1st link. (b) The aerodynamic drag of the 13th link. (c) The aerodynamic drag of the 24th link. (d) The tension of the 1st link. (e) The tension of the 13th link. (f) The tension of the 24th link. (g) The tension of the 1st joint. (h) The tension of the 13th joint. (i) The tension of the 23th joint.

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Fig. 16. External forces on the probe controlled by the tensator. (a) The component of the X w direction. (b) The component of the Y w direction. (c) The component of the Z w direction.

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Fig. 21, from t = 290.25 s to t = 291.26 s, as the electric torque T e drives the PMSM to rotate accurately, the initial tracking errors of ω and ϑ are overcome and virtual commands can be tracked within only 0.4 s. After that, the hose is still retracted to the desired length in the presence of the sine-wave disturbance, T e is completely used to counteract T L , and then all variables enter a stable sine tracking state. Fig. 22 verifies that the proposed control laws can ensure that the real control inputs remain stable and bounded. Moreover the chattering phenomenon of sliding mode control is eliminated efficiently in the presence of the sine-wave disturbance. Compared with the tensator control [10,25], the control method proposed here for the hose whipping can guarantee the global stability efficiently, and completely match the coupling profile. Consequently, the hose whipping can be suppressed rapidly and efficiently. Compared with the method in Ref. [22] proposed previously by the authors, the tracking precision of i d , i q , and ω is greatly improved in spite of smaller control gains k1 and k2 .

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6. Conclusions and the loads on the probe controlled by the suppression strategy are much lower than the loads without suppression, and the loads when controlled by the tensator. The excellent tracking performance of d-axis and q-axis currents leads the PMSM to generate a desired electric torque ensuring the dynamic performance of the whole system. As shown in

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The control strategy proposed here on the basis of the relative position between the tanker and the receiver can completely suppress the hose whipping phenomenon, which overcomes the imperfection of the tensator. The analytic computation of command derivatives is eliminated. Global stability, accuracy, and robustness

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Fig. 18. Drags, tensions, and restoring forces of the hose controlled by the control laws proposed. (a) The aerodynamic drag of the 1st link. (b) The aerodynamic drag of the 13th link. (c) The aerodynamic drag of the 24th link. (d) The tension of the 1st link. (e) The tension of the 13th link. (f) The tension of the 24th link. (g) The tension of the 1st joint. (h) The tension of the 13th joint. (i) The tension of the 23th joint.

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Fig. 19. External forces on the probe controlled by the control laws proposed. (a) The component of the X w direction. (b) The component of the Y w direction. (c) The component of the Z w direction.

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Fig. 21. PMSM state variables of each subsystem controlled by the control laws proposed. (a) The rotor angular position ϑ and its tracking command ϑc . (b) The rotor angular velocity ω and its tracking command ωc . (c) The q-axis current i q and its tracking command i qc . (d) The d-axis current i d and its tracking command i dc . (e) The load torque T L and the electric torque T e .

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can be guaranteed in theory. The elimination of the analytic computation of high order command derivatives can greatly relax the requirement on the number of sensors, and simplify the control system relative to the previous control system.

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Conflict of interest statement

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None declared.

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References

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Fig. 22. The real d-axis and q-axis voltage inputs controlled by the control laws proposed. (a) The real d-axis voltage input. (b) The real q-axis voltage input.

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[8] Kuizhi Yue, Liangliang Cheng, Tieying Zhang, Jinzu Ji, Dazhao Yu, Numerical simulation of the aerodynamic influence of an aircraft on the hose-refueling system during aerial refueling operations 49 (2016) 34–40. [9] Z.H. Zhu, S.A. Meguid, Elastodynamic analysis of aerial refueling hose using curved beam element, AIAA J. 44 (7) (2006) 1317–1324.

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