Adaptive Sliding Mode Relative Motion Control for ... - IEEE Xplore

Received January 18, 2017, accepted February 13, 2017, date of publication March 1, 2017, date of current version May 17, 2017. Digital Object Identifier 10.1109/ACCESS.2017.2671440

Adaptive Sliding Mode Relative Motion Control for Autonomous Carrier Landing of Fixed-Wing Unmanned Aerial Vehicles ZEWEI ZHENG1 , (Member, IEEE), ZHENGHAO JIN2,3 , LIANG SUN1 , (Member, IEEE), AND MING ZHU2 1 Seventh 2 School 3 Tianjin

Research Division, Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China Zhong Wei Aerospace Data System Technology Co., Ltd, Tianjin 300301, China

Corresponding author: Z. Jin ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 61503010, in part by the China Postdoctoral Science Foundation under Grant 2016M590031, in part by the Aeronautical Science Foundation of China under Grant 2016ZA51001, and in part by the Fundamental Research Funds for the Central Universities under Grant YWF-16-GJSYS-02.

ABSTRACT In this paper, relative motion model and control strategy for autonomous fixed-wing unmanned aerial vehicle (UAV) carrier landing are addressed. First, a coupled six-degrees-of-freedom (6-DOF) non-linear relative motion model is established from 6-DOF UAV and carrier models. Second, because of the under-actuated characteristic of two vehicles, the 6-DOF relative motion model is simplified to a four-degreeof-freedom (4-DOF) model to facilitate the control design. Third, an adaptive sliding mode control law is proposed to track desired landing trajectory and maintain constant relative pitch and roll angles. Finally, simulation results demonstrate the effectiveness of the proposed control method. INDEX TERMS Carrier landing control, fixed-wing UAV, adaptive sliding mode, 4-DOF control. I. INTRODUCTION

The autonomous fixed-wing UAV carrier landing control technology is served as a vital premise for UAV carrier and UAV fulfilling the tasks. Meanwhile, automatic landing technology is also so complicated that it can not be conquered easily. In order to reduce the carrier landing deviation, some methods have been applied to model and control design. In term of model, most of landing models are built on the basis of a single UAV model. Some technologies or methods have been applied to control system, such as deck motion compensation technology, power compensation link, automatic throttle control, direct lift control technology. In the early stage of carrier landing control technology, the classical control methods were adopted to automatic landing systems. The automatic landing system was developed for nearly half a century, the basic structure of the system got little change. With the maturity of the classical control theory, especially the theory and method of frequency domain analysis, the stability of automatic landing carrier have a large progress. Moreover, most of the automatic landing controls only concern about attitude loop control. In recent years, based on the single UAV model, many nonlinear control methods were proposed in carrier 5556

landing systems. In the field of control design, the latest research works mainly focus on optimizing parameters [1]–[3], improving the control accuracy and robustness of the dynamic inversion control system [4], [5], improving the accuracy of sensor [6], reducing the noise of radar tracking and radio data link [7]. In addition, the fuzzy control method [8] was introduced into control system. In [9]–[11], it was considered the control of height and pitch angle for landing carrier by designing a fuzzy PID flight control system, but the application of the control law is limited and only developed for the pitch angle under a given height motion. In [12], a dynamic inversion attitude control law was proposed by using 6-DOF aircraft model. From the recently studies, there are little change in the model. However, in the aspect of space docking, some scholars have tried to use the relative motion mode [13] to achieve the relative motion control design. Moreover, it is rare to take the relative motion model into consideration by means of carrier landing control law. The probable reason is that the 6-DOF carrier model [14], [15] is very complicated for control design. Therefore, the corresponding 6-DOF relative motion model control law is hard to design. So simplification is inevitable for fulfilling the desired controller [16].

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VOLUME 5, 2017

Z. Zheng et al.: Adaptive Sliding Mode Relative Motion Control for Autonomous Carrier Landing

This paper mainly illustrates the control problem of the autonomous fixed-wing UAV carrier landing by relative motion model. The main contributions of this paper are stated as follows. Firstly, based on the 6-DOF models of fixed-wing UAV and carrier, a 6-DOF relative motion model is established for the autonomous fixed-wing UAV carrier landing missions. Taking the under-actuated property of the controlled fixed wing aircraft into account, the proposed 6-DOF relative motion model is simplified into a 4-DOF fully actuated relative motion model to facilitate the control design. Secondly, in order to assure the desired landing trajectory, constant relative pitch and roll angles, and lateral position and velocity, a sliding mode control law [17] is designed for autonomous landing missions. Moreover, the uncertainty parameters [18] and unknown external disturbances [19] are considered in control design, then an adaptive control method is combined with sliding mode control approach to compensate mode uncertainties. Thirdly, it is proved that the designed controller ensures that the relative position and attitude errors converge to zero, and the simulation example verifies the feasibility of the proposed method.

FIGURE 1. Structure of the UAV and aicraft carrier.

II. PROBLEM STATEMENT

As shown in Fig. 1, Fg , {og , xg , yg , zg } denotes the Earth reference frame(ERF), where Og is the origin on the surface of the Earth, og xg points to the north, og yg points to the east, and og zg points vertically down along the gravity vector. Fa , {oa , xa , ya , za } denotes the UAV body-fixed reference frame(BRF), where the origin Oa is fixed to the centre of gravity of the UAV, oa xa points to the forward of the UAV, oa ya points to the right of the UAV, oa za completes the right-hand orthogonal coordinate system. Similarly, Fs , {os , xs , ys , zs } denotes the carrier body-fixed reference frame, where the origin os is fixed with the centre of gravity of the carrier, os xs points to the forward of the carrier, os ys points to the right of the carrier, os zs completes the right-hand orthogonal coordinate system. The model of the UAV carrier landing design in this study is formulated based on the models of the UAV and the carrier. Thus, the 6-DOF mathematical model of UAV and aircraft carrier need to be established firstly. Before setting up the model of UAV and carrier, it is assumed that the travel distance of the UAV and carrier is relatively small VOLUME 5, 2017

compared with the dimensions of the Earth. Then, the earth curvature is ignored and the surface of earth is assumed to be flat. A. UAV MOTION MODEL

The kinematics of UAV are described by [20] # " p˙ Ea υa = Rbg E ˙a a 2

(1)

where position pEa = [xa , ya , za ]T and euler angle Θ Ea = [φa , θa , ψa ]T are defined in ERF; velocity va = [ua , va , wa ]T and angular velocity Ω a = [pa , qa , ra ]T are defined in BRF. Rbg is represented by Ra 03×3 Rbg = (2) 03×3 Ka where Ra is the direction cosine matrix of BRF to ERF; K a is the Jacobian matrix, and   a a cθa cψa r12 r13 a a  r22 r23 Ra =  cθa sψa −sθa cθa sφa cθa cφa   1 tθa sφa tθa cφa cφa −sφa  Ka =  0 (3) 0 sφa /cθa cφa /cθa where s(x), c(x) and t(x) are short for sin(x), cos(x), tan(x), a = sθ cψ sφ − sψ cφ , r a = sθ cψ cφ + respectively; r12 a a a a a 13 a a a a a = sθ sψ cφ − sψa sφa , r22 = sθa sψa sφa + cψa cφa , r23 a a a cψa sφa . The dynamics of UAV in Fa are described by ˙ a + S(a )va ] = Fa + d f ma [v (4) ˙ a + S(a )I a a = M a + d τ I a where ma ∈ R is the whole mass of the UAV. The moment of inertia and the products of inertia are described by   Ix 0 −Ixz Iy 0  Ia =  0 (5) −Izx 0 Iz S(a ) is the skew-symmetric matrix for a ∈ R3 ; d f , d τ ∈ R3 are disturbance force and torque respectively; the external force Fa = [Fx , Fy , Fz ]T is defined by Fx = Xu ua + Xw wa + Xq qa + XδT δT + Xδe δe − ma g sin θa Fy = Yv va + Yp pa + Yr ra + Yδa δa + Yδr δr + mma g sin φa cos θa Fz = Zu ua + Zw wa + Zq qa + ZδT δT + Zδe δe + ma g cos φa cos θa

(6)

where δT , δa , δe , δr , and g denotes the thrust, aileron, elevator, rudder, and gravity, respectively; Xi , Yj , Zk (i = u, w, q, r, δT , δa , δe ; j = v, p, r, δa , δr ; k = u, w, q, δT , δe ) 5557


are aerodynamic derivatives. The aerodynamic moment M a = [L, M , N ]T is defined by L = Lv va + Lp pa + Lr ra + Lδa δa + Lδr δr M = Mu ua + Mw wa + Mq qa + MδT δT + Mδe δe N = Nv va + Np pa + Nr ra + Nδa δa + Nδr δr

(7)

and the Li , Mj , Nk (i = v, p, r, δa , δr ; j = u, w, q, δT , δe ; k = v, p, r, δa , δr ) are aerodynamic derivatives. B. CARRIER MOTION MODEL

The kinematics of carrier are described by [14] # " p˙ Es Rs 03×3 vs D E ˙s 03×3 Ks s 2

(8)

where position pEs = [xs , ys , zs ]T and euler angle Θ Es = [φs , θs , ψs ]T are defined in ERF; velocity vs = [us , vs , ws ]T and angular velocity Ω s = [ps , qs , rs ]T are defined in carrier’s BRF. Rs is the direction cosine matrix of carrier’s BRF to ERF; K s is the Jacobian matrix, and   s s cθs cψs r12 r13 s s  r22 r23 Rs =  cθs sψs −sθs cθs sφs cθs cφs   1 tθs sφs tθs cφs cφs −sφs  Ks =  0 (9) 0 sφs /cθs cφs /cθs s s where r12 = sθs cψs sφs − sψs cφs , r13 = sθs cψs cφs + s s sψs sφs ,r22 = sθs sψs sφs +cψs cφs , r23 = sθs sψs cφs −cψs sφs , respectively. The dynamics of carrier are described by [21]–[23]  m22 d11 1   u˙ s = vs rs − us + τu    m m m 11 11 11    m11 d22 1   v˙ s = − us rs − vs +   m m m  22 22 22  w ˙ s = ζ1 sin(ω1 t) + ζ2 sin(ω2 t) (10)   p ˙ = ζ sin(ω t) + ζ sin(ω t) + b  s 3 3 4 4 1     q˙ s = ζ5 sin(ω5 t) + ζ6 sin(ω6 t) + b2      d33 1 m11 − m22   us vs − r+ τr  r˙s = m33 m33 m33

where m11 = ms − Xu˙ , m22 = ms − Yv˙ , m33 = Iz − Nr˙ , d11 = −Xu , d22 = −Yv , d33 = −Nr , mii are the mass and inertia model terms with md = m22 − m11 > 0. The term mii includes additional mass generated by hydraulic pressure forces and torque due to forced harmonic motion of the vessel. The model term dii represents the hydrodynamic damping forces related with the corresponding velocities in general. Xu˙ , Yv˙ , Yr˙ , Yv˙ , Xu , Yv , Nr represent the additional mass constants. The surge force τu and yaw moment τr are provided by two actuators actuated on the carrier. ζ1 , ζ2 , ω1 , ω2 is the vertical motion coefficient in different sea conditions; ζ3 , ζ4 , b1 , ω3 , ω4 is the rolling motion coefficient in different sea conditions; ζ5 , ζ6 , b2 , ω5 , ω6 is the pitching motion coefficient in different sea conditions. 5558

C. RELATIVE MOTION MODEL

The relative position and relative attitude in ERF can be described by [24] E E E pe p p = aE − s E (11) 2Ee 2a 2s Thus, the relative kinematics are presented from the UAV’s kinematics by " # p˙ Ee υe = R (12) bg ˙ Ee e 2 where pEe is the relative position in ERF; 2Ee is the relative attitude in ERF; υ e is the relative velocity in UAV’s BRF; e is the relative angular velocity in UAV’s BRF. The relative velocity and relative angular velocity expressed in BRF are ve va a vs = − Rs (13) e a s where Ras =

Rsa 03×3

03×3 Rsa

(14)

and Rsa = RTa Rs is the direction cosine matrix of carrier’s BRF to UAV’s BRF. Taking the time derivative of the relative velocity and relative angular velocity in (13) results in the 6-DOF relative motion model as υ˙ e v˙ a ˙ as vs − Ras υ˙ s = − R (15) ˙e ˙a ˙s s ˙ e is the relative angular where υ˙ e is the relative acceleration, acceleration. Due to the under-actuated property of the fixed-wing aircraft in the control aspect, this 6-DOF relative motion model should be simplified to 4-DOF relative motion model. Because the lateral displacement ya of the fixed-wing aircraft is associated with the yaw angle ψa , and the vertical velocity wa is related to the forward velocity ua in the autonomous landing missions. Therefore, to ensure the successful landing missions, the 6-DOF relative model is simplified as 4-DOF relative model, such as relative lateral displacement ye , relative vertical displacement ze , relative rolling φe , and pitching attitude θe . Thus, the 4-DOF relative kinematics and dynamics are presented by [18], [19] x˙ 1 = Reb x2 (16) x˙ 2 = f (x) + Bu + d where x1 = [ye , ze , φe , θe ]T , x2 = [ve , we , pe , qe ]T ; Reb is simplified from Rbg to a 4 × 4 rotation matrix,  a  a r22 r23 0 0  cθa sφa cθa sφa 0 0   Reb =   0 0 1 tθa sφa  0 0 0 cφa VOLUME 5, 2017




0

 Z  δT  m B= a  0    Mδ T Iy

Yδr ma

Yδa ma

0

0

Iz Lδr + Ixz Nδr 2 Ix Iz − Ixz

Iz Lδa + Ixz Nδa 2 Ix Iz − Ixz

0

0

T

u = [u1 u2 u3 u4 ]  1  F¯ y  ma     1  e ¯  Fz  f (x) =   − Rs  ma   p˙¯ a  ˙q¯ a

= [ δT δr δa δe ]

0



 Zδe    ma   0    Mδe  Iy

T



   v˙ s vs  w˙ s      − SRe  ws  s p   p˙ s  s q˙ s qs

Res is simplified from Ras to be a 4 × 4 rotation matrix; S and Res can be presented by   s11 s12 0 0  s21 s22 0 0   S= (17)  0 0 s11 s12  0 0 s21 s22   h11 h12 0 0  h21 h22 0 0   (18) Res =   0 0 h11 h12  0 0 h21 h22 with s11 = −qa cθs sφa + (cθa cψa2 sθa + cθa sψa2 sθa )pa + 2ψa ra cψa cθa sθa sφa − sθa cθs ps , s12 = −qa cθs cφs + (cθa sφa cψa + 2cθa ψa sθa cφa )ra + pa (cθa cψa − cθa sψa cψa ), s + sθ sψ r s + cθ cθ sφ ) + s21 = pa (sθa cψa r12 a a 22 a s s s s ) + r (sθ sφ r s − cφ r s ), s = qa (cψa sφa r12 + sψa sφa r22 a a a 22 a 12 22 s + sθ sψ r s + cθ cθ cφ ) + q (cψ sφ r s + pa (sθa cψa r13 a a 23 a s s a a a 13 s ) + r (sθ sφ r s − cφ r s ), and h = cθ cψ r s + sψa sφa r23 a a a 23 a 13 11 a a 12 s s + cθ sψ r s − cθa sψa r22 − sθa cθs sφs , h12 = cθa cψa r13 a a 23 a r s + r a r s + cθ sφ cθ sφ , h sθa cθs cφs , h21 = r12 a a s s 22 = 12 22 22 a s a s r12 r13 + r22 r23 + cθa sφa cθs cφs , respectively. Moreover, F¯ y , F¯ z , p˙¯ a , and q˙¯ a are denoted by

very high. RTK (Real Time Kinematic) technology can meet the requirement of positioning accuracy. Unlike other positioning techniques, the RTK directly obtain the relative position data of the UAV and the aircraft carrier by calculating, rather than their respective position coordinates. So the accuracy of RTK technology can reach centimeter level. This is one of the reasons why this paper uses the relative motion model. Assumption 1: The aerodynamic and control parameters can be presented by εi = εi0 + εi1 , where ε , X , Y , Z , L, M , N ; i , δa , δe , δe , δr , u, v, w, p, q, r. εi0 are known constants and εi1 are unknown and bounded scalars. Assumption 2: The external disturbance d f and d 1 are unknown but bounded by kdk ≤ dm with an unknown constant dm . D. CONTROL OBJECT

The control objective in this work is to drive the UAV tracking the carrier a certain position. The control aim of the autonomous landing missions is that the relative vertical position ze tracks to desired trajectory zce , and the relative lateral position ye , velocity ve , roll angle φe , and pitch angle θe track to yce , vce , φec , θec , respectively. Thus, the control objective aims to design a control input u under Assumptions 1 and 2, such that the controlled tracking with model can guarantee limt→∞ x1 = xc1 , where xc1 = [yce , zce , φec , θec ]T , III. CONTROLLER DESIGN

The relative motion controller is presented based on an adaptive sliding mode design method. Adaptive laws are derived to compensate the parametric uncertainties and restrain the external environment disturbances in model [26]. Define a sliding surface [27] s = x˙˜ 1 + a˜x1 where x˜ 1 = x1 − xc1 , a ≥ 0. Differentiating (20) with (16) leaves s˙ = x¨˜ 1 + ax˙˜ 1 = x¨ 1 + a(˙x1 − x˙ c1 ) e = R˙ b x2 + Re [f (x) + Bu + d] + ax˙˜ 1

F¯ y = (Yv va + Yp pa + Yr ra + mg sin φa cos θa ) − ua ra + pa wa ¯ Fz = (Zu ua + Zw wa + Zq qa + mg cos φa cos θa )

b

− pa va + ua qa 1 p˙¯ a = [(Iz (Lv va + Lp pa + Lr ra ) 2 Ix Iz − Ixz + Ixz (Nv va + Np pa + Nr ra )) + Ixz (Ix + Iz − Iy )pa qa 2 − Iy Iz )qa ra − (Iz2 + Ixz 1 q˙¯ a = [Mu ua + Mw wa + Mq qa + (Iz Iy

− Ix )pa ra + Ixz (ra 2 − pa 2 )]

(21)

According to Assumption 1, we know the term f (x) can be divided into f (x) = f 0 + f 1 and B can be divided into B D B0 + B1 . Then, Eq. (21) becomes e s˙ = R˙ b (Reb )−1 x˙ 1 + Reb [f 0 + f 1 + (B0 + B1 )u Cd] +ax˙˜ 1

(22)

(19)

and d τ , d f ∈ R3 are also simplified as d τ s , d fs ∈ R2 , respectively [25]. Remark 1: In terms of navigation, the accuracy of positioning technology for UAV autonomous carrier landing is VOLUME 5, 2017

(20)

Introducing a linear operator L(a1 ) ∈ R4×15 for any vector a1 = [ua , va , wa , pa , qa , ra ]T results in a matrix (23), as shown at the top of the next page, where s1 = Iz , s2 = I IIxz−I 2 . Then, f 1 = L(a1 )ϑ 1 , where I I −I 2 x z

xz

x z

xz

ϑ1 , [Yv1 , Yp1 , Yr1 , Zu1 , Zw1 , Zq1 , Lv1 , Lp1 , Lr1 , Nv1 , Np1 , Nr1 , Mu1 , Mw1 , Mq1 ]T . Similarly, introducing a linear operator M(a2 ) ∈ R4×10 for any 5559


 va  ma   0 L(a1 ) =    0  0  u2  ma   0 M(a2 ) =    0  0

pa ma 0

ra ma 0

0 0

0

0 ua ma 0

0 wa ma 0

0 qa ma 0

0

0

0

0

0 u4 ma 0

0

0

0

0

0

0

0

0

0

0

0

0 u1 ma 0

s1 u2

s1 u3

s2 u2

s2 u3

0

0

0

0

0

0

0

0 u1 Iy

u3 ma 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

s1 va

s1 pa

s1 ra

s2 va

s2 pa

s2 ra

0

0

0

0

0

0

0 ua Iy

0 wa Iy

vector a2 = [u1 , u2 , u3 , u4 ]T results in a matrix (24), as shown at the top of the next page, Thus, B1 u = M(a2 )ϕ 1 , where ϕ 1 , [Yδr1 , Yδa1 , ZδT 1 , Zδe1 , Lδr1 , Lδa1 , Nδr1 , Nδa1 , Mδr1 , Mδa1 ]T . Then Eq. (22) becomes ˙ eb (Re )−1 x˙ 1 + Re [f 0 + L(a1 )ϑ 1 + B0 u s˙ = R b b + M(a2 )ϕ + d] + ax˙˜ 1 (25) 1

Define estimation errors ϑ˜ 1 = ϑˆ 1 − ϑ 1 ,ϕ˜ 1 = ϕˆ 1 − ϕ 1 , d˜ m = dˆ m − dm , B1 = Bˆ 1 − B˜ 1 , and choose a Lyapunov function 1 1 ˜2 1 ˜T ˜ 1 T V = sT s + ϕ˜ 1 ϕ˜ 1 + d (26) ϑ 1ϑ 1 + 2 2γ1 2γ2 2γ3 m with γi > 0(i = 1, 2, 3). Then, taking time derivative of (26) leads to 1 T 1 1 V˙ = sT s˙ + ϑ˜ 1 ϑˆ˙ 1 + ϕ˜ T1 ϕ˙ˆ 1 + d˜ m dˆ˙ m (27) γ1 γ2 γ3 Substituting (25) into (27) gives V˙ = sT [Reb f 0 + Reb L(a1 )ϑ 1 + Reb B0 u e + Reb M(a2 )ϕ 1 + Reb d + R˙ b (Reb )−1 x˙ 1 + ax˙˜ 1 ] 1 1 T 1 (28) + ϑ˜ 1 ϑˆ˙ 1 + ϕ˜ T1 ϕ˙ˆ 1 + d˜ m d˙ˆ m γ1 γ2 γ3 Since B1 = Bˆ 1 − B˜ 1 = M(a2 )ϕˆ 1 − M(a2 )ϕ˜ 1 , then Eq. (28) becomes ˆ 1 )u V˙ = sT [Reb f 0 + Reb L(a1 )ϑ 1 + Reb (B0 + B e e −1 e e ˙ − R M(a2 )ϕ˜ + R d + R (R ) x˙ 1 + ax˙˜ 1 ] b

1

b

b

1 ˜T ˙ 1 1 ϑ 1 ϑˆ 1 + ϕ˜ T1 ϕ˙ˆ 1 + d˜ m d˙ˆ m (29) γ1 γ2 γ3 Designing the adaptive sliding mode control input as ˆ 1 )]−1 [−Re (f 0 + L(a1 )ϑˆ 1 ) u = [Re (B0 + B b

e − dˆ m kReb ksgn(s) − k1 s − ax˙˜ 1 − R˙ b (Reb )−1 x˙ 1 ]

where sgn(s) = [sgn(s1 ), sgn(s2 ), sgn(s3 ), sgn(s4 sgn(si ) denotes the signum function defined by  si > 0  1, si = 0 i = 1, 2, 3, 4 sgn(si ) = 0,  −1, si < 0 5560

)]T ,

(30) and

  0    0  qa  Iy

(23)

  0    0  u4  Iy

(24)

Bˆ 1 can be presented by  Yˆ δr  0 ma   Zˆ δ  T 0   m ˆ B1 =  a Iz Lˆ δr + Ixz Nˆ δr   0 2  Ix Iz − Ixz  ˆ δT M 0 Iy

Yˆ δa ma

 0   Zˆ δe    ma    0    ˆ δe  M

0 Iz Lˆ δa + Ixz Nˆ δa 2 Ix Iz − Ixz 0

Iy (32)

and diagonal feedback gain matrix k1 = diag[k1 , k2 , k3 , k4 ], k1 = kT1 . Then substituting (30) into (29) gives e V˙ = sT [Reb f 0 + Reb f 1 + Reb d + R˙ b (Reb )−1 x˙ 1 + ax˙˜ 1 − Re f 0 − Re L(a1 )ϑˆ 1 b

b

e − dˆ m kReb ksgn(s) − k1 s − ax˙˜ 1 − R˙ b (Reb )−1 x˙ 1 1 T − Reb M(a2 )ϕ˜ 1 ] + ϑ˜ 1 ϑ˙ˆ 1 γ1 1 1 + ϕ˜ T1 ϕ˙ˆ 1 + d˜ m dˆ˙ m (33) γ2 γ3

Assign the update laws for unknown parameters as  ˙ e   ϑˆ 1 = γ1 (Rb L(a1 ))T s ϕ˙ˆ 1 = γ2 (Reb M(a2 ))T s 

 ˙ˆ d m = γ3 ksk Re 1

(34)

b

Remark 2: Eq. (34) gives adaptive estimation laws for the unknown parameters ϑ 1 , ϕ 1 , dm , the control input u can be derived before updating the estimations of unknown parameters with the given initial estimations. Substituting (34) into (33) and using properties sT sgn(s) = ksk1 result in V˙ ≤ sT [Reb f 0 + Reb L(a1 )ϑ 1 + dm kReb ksgn(s) ˙ eb (Re )−1 x˙ 1 + ax˙˜ 1 − Re f 0 − Re L(a1 )ϑˆ 1 +R b

(31)





b

+

b

0

0

b

b

e − dˆ m kReb ksgn(s) − k1 s − ax˙˜ 1 − R˙ b (Reb )−1 x˙ 1 T

− Reb M(a2 )ϕ˜ 1 ] + ϑ˜ 1 (Reb L(a1 ))T s VOLUME 5, 2017


+ ϕ˜ T1 (Reb M(a2 ))T s + d˜ m ksk1 kReb k ≤

≤

T −s k1 s − s Reb L(a1 )ϑ˜ 1 + ϑ˜ 1 (Reb L(a1 ))T s − sT Reb M(a2 )ϕ˜ 1 + ϕ˜ T1 (Reb M(a2 ))T s − d˜ m ksk1 kReb k + d˜ m ksk1 kReb k −sT k1 s ≤ −µksk2 ≤ 0 T

T

(35)

where µ = λmin (k1 ), and λmin (k1 ) is the minimum eigenvalue of matrix. Remark 3: Because of the dynamic coupling between relative position and relative attitude, the changing of a single control parameter can affect multiple states, such that the varying controller parameter k1 have an influence on the convergence of the relative lateral displacement ye and rolling angle φe . The decrease of k2 will aggravate the shock of vertical velocity curve. The rise or fall of k4 will affect the convergence of the relative pitch angle θe . Theorem 1: Consider the relative motion model (16) under Assumptions 1 and 2 for autonomous landing missions. The adaptive sliding mode controller (30) and corresponding adaptive laws in (34) can ensure that tracking errors of closedloop control system converge to zero and estimation errors of unknown parameters are uniformly bounded. Proof: Since V (t) ≥ 0 and V˙ (t) ≤ 0, then V˙ (t) is monotonically decreasing along the closed-loop control system trajectory and is bounded by zero. Hence, V (t) have a finite limit V (∞) as t → ∞, and satisfies 0 ≤ V (∞) ≤ V (t) ≤ V (0) < ∞, ∀t ≥ 0. Meanwhile, by integrating both sides of Eq. (35), we have Z Z ∞ V (0) 1 ∞ ˙