Modeling and anti-swing control for a helicopter slung-load system

Applied Mathematics and Computation 372 (2020) 124990

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Modeling and anti-swing control for a helicopter slung-load system Yong Ren a,∗, Kun Li b, Hui Ye c a

College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, People’s Republic of China b School of Mathematical Sciences, Qufu Normal University, Qufu 273165, People’s Republic of China c School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 2 July 2019 Revised 7 November 2019 Accepted 15 December 2019

Keywords: Helicopter slung-load system Rigid-body model Suppressing swing Positioning control Input constraints

a b s t r a c t In this paper, the problems of suppressing swing and positioning control are investigated for a rigid-body model of helicopter slung-load system subject to input constrains and external disturbances. By utilizing Lagrange’s equations, a rigid-body model of helicopter slung-load system is established. To eliminate the swing and transport the load to target position, a desired trajectory of helicopter is proposed. Thereby, suppress swing and position control problems are transformed to a trajectory tracking control problem. Based on the energy technique, a trajectory tracking control scheme is proposed. Under the constructed control laws, the closed-loop system states can asymptotically converge to system equilibrium points in the presence of external disturbances, meanwhile, input constraints will not be violated. Finally, simulation results illustrate that the developed controllers work well in suppressing swing and positioning control for a rigid-body model of helicopter slung-load system. © 2019 Elsevier Inc. All rights reserved.

1. Introduction With the development of society, helicopter slung-load system which includes helicopter, cable, and load has widely used in military and civilian fields and has received increasing attention. The modeling and near-hover control problems have been investigated for a helicopter with a hanging load in [1]. The method of input shaping has been used to investigate the problem of swinging reduction for helicopter slung load system in [2,3]. In [4], the dynamic characteristics and control performance have been tested for a helicopter with suspending loads by using a small-scale helicopter. According to aforementioned articles, the existing research results for helicopter slung load system can be divided into two parts: one is only to test the dynamic characteristics of the helicopter slung load system; another is to investigate the stability of the helicopter slung load system by employing feed-forward technology. Thus, it is imperative and urgent to propose an effective feedback control scheme to investigate the anti-swing problem. Although the anti-swing control has been studied for a flexible helicopter suspension cable system in [5–8], the attitude angle has not been considered in these papers. In addition, the helicopter slung-load system is an underactuated system. This increases the difficulty of solving the reducing

∗

Corresponding author. E-mail address: [email protected] (Y. Ren).

https://doi.org/10.1016/j.amc.2019.124990 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

swing problem for the helicopter slung-load system. Therefore, to develop an effective control technique is an intractable issue to solve the problem of swing for the underactuated systems. The underactuated systems exist widely in reality, such as underwater robots, pendulum, overhead crane and so on, and many fundamental and indispensable results have been extensively established by the energy-based control method [9,10]. Sliding-mode control method has been utilized to investigate the stabilization problem for underactuated systems in [11–13]. A generalized payload horizontal-displacement signal has been constructed to eliminate the vibration in [14–16]. Moreover, position control is also an significant topic, some effective methods are necessary to be proposed to deal with this issue. Since target position is a constant value, the initial error between initial position and target position will be big when target position is distant from initial position. And to achieve the target of positioning control, the information of initial error is usually inevitable in the proposed control scheme. A big initial error will result in a big oscillation in the initial period of the helicopter motion. To tackle with this problem, a reasonable scheme has been proposed to tackle with this issue in [17–21]. By tracking a suitable desired trajectory, the problem of the big initial error has been avoided. The main idea of this method is to transform the problem of anti-swing and position control to the problem of tracking the desired trajectory. Inspired by this method, since the position control is also considered in this article, the method of tracking the desired trajectory is adopted in this paper. Due to physical limitations of mechanical equipments, between system control input and actual output of actuator exist an error usually. In this paper, the actuator output of helicopter has also a limitation level, thus, input constraint is necessary to consider to enhance robust control performance. For the research of input constraint, many effective techniques have been extensively obtained [22–28]. Considering nonsymmetric input saturation and deadzone, the problem of robust adaptive neural network control has been studied for a general class of multiple-input and multiple-output (MIMO) nonlinear systems in [29], while command filters have been presented to tackle with the input nonlinearities. To tackle efficiently the input saturation, an smooth function has been utilized to approximate the saturation function, then, the error between the smooth function and the saturation function and external disturbance has been considered as a composite disturbance [30]. By this means, the problems of input saturation and external disturbance have been translated to the problem of the composite disturbance for a class of uncertain nonlinear systems. In order to compensate for the effect of input saturation, a novel auxiliary system has been designed in [31,32]. Based on the designed auxiliary system, the control problems for the near space vehicles and helicopter suspension cable system, respectively, have been investigated. In [33], by using the hyperbolic tangent functions and saturation functions the problem of input constraint has been tackled for an Euler-Bernoulli beam system. However, to the best of authors’ knowledge, the helicopter slung-load system control problem is still very open due to some issues that remain unsolved. Among them, the modeling and anti-swing control are not ignorable for the helicopter slung-load system with input constraints. Inspired by the above mentioned results, the problem of modeling and reducing swing are studied for a rigid-body model of helicopter slung-load system in the presence of input constraints and external disturbances. Firstly, the rigid-body model of helicopter slung-load system is established by using the Lagrange’s equations. Then, based on the proposed model, energy-based control method is used to design the trajectory tracking control laws for the helicopter slung-load system with input constraints and external disturbances. Under the proposed control scheme, the asymptotically stability of closed-loop system is guaranteed. The main contributions of the paper are listed as follows: 1. According to the Lagrange’s equations, a nonlinear model is established for a helicopter with a rigid body slung load. 2. The tan (·) function is used to guarantee that input constraints will not be violated. Then, by utilizing the energy-based control method, a trajectory tracking control strategy is proposed to realize the purposes that decrease swing and transport the load to target position. 3. The trajectory tracking control problem is transformed to the asymptotically stable problem by defining a series of state errors. Under the designed control laws, the stability of closed-loop system is analyzed by employing extended Barbalat lemma. 4. Based on the established rigid body helicopter slung load system model, the rationality and effectiveness of the proposed control scheme is verified by using a numerical simulation. The rest of this paper is organized as follows. In Section 2, a rigid-body model system model is proposed for helicopter slung-load system. Section 3 presents the problem formulation and some preliminaries. Controller design and stability analysis are implemented in Section 4. Section 5 validates the obtained results by a simulation, which is followed by the conclusion in Section 6. Notations. R+ denotes the set of positive real numbers; [ · ]T represents the transpose of a vector or a matrix; ln (·) and tanh (·) are the natural logarithmic and hyperbolic tangent function of (·), respectively; ε ∈ L∞ illustrates the infinite norm of ε is bounded; ε ∈ L2 shows the 2−norm of ε is bounded. 2. System modeling The helicopter slung-load system contains three parts which are helicopter, cable, and load. The schematic principle for a rigid-body model of helicopter slung-load system is shown in Fig. 1. Frame I − J is the fixed inertia coordinate frame system, while frame i − j is the body coordinate frame system. G is the center of gravity of the helicopter which is also the origin of the body coordinate frame system. H is the suspension point of the load which locates at the frame j. The horizontal

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

3

Fig. 1. A suspension cable system of helicopter.

location of helicopter is x(t), vh (t) is the velocity of helicopter, respectively. θ (t) is the pitch angle. The suspended load has swing angle α (t) relative to vertical. In the following section, we will establish the nonlinear governing equations of helicopter slung-load. The following assumptions are necessary for the process of modeling helicopter slung-load. Assumption 1 [3]. The horizontal motion of helicopter is only considered, moreover, the height of gravity of the helicopter remains the same value in the whole transportation process. Assumption 2 [3]. The horizontal swing of load is only considered during transportation process. ˙ < Assumption 3 [4]. The pitch angle θ and horizontal swing angle α satisfy the following conditions: −π /2 < θ < π /2, |θ| υ , and −π /2 < α < π /2. Assumption 4 [4]. The rope is massless and rigid. Moreover, the rope length is kept constant during transportation process. The coordinate of the suspension point in the body coordinate frame can be described as



rh =

rh1



(1)

rh2

with rh1 and rh2 being the component values in the frame i and the frame j, respectively. Thus, the distance of the load suspension point below the center of gravity of helicopter can be represented as

ls =



rh21 + rh22 .

(2)

Lagrange’s equations will be used to derive the equations of motion of the coupling model. The generalized coordinates x, θ , and α . Lagrange’s equations are given by Meirovitch [34]:

d dt

 ∂T  ∂ q˙ n

−

∂T ∂V + = Qn , n = 1, 2, 3 ∂ qn ∂ qn

(3)

where qn is the generalized coordinates, T is the total kinetic energy of the system; V is the total potential energy of the system; and Qn is the generalized forces (torques). The kinetic energy of the helicopter-load system is given by

Wk =

1 1 1 Mv2h + IG θ˙ 2 + mvTl vl 2 2 2

(4)

where M and m are the masses of helicopter and load, respectively; vh and IG are the velocity in the inertial coordinate system and the moment of inertia of helicopter, respectively; vl is the velocity of load in the inertial coordinate system. The velocity of helicopter in the inertial coordinate frame is given by

vh = x˙ .

(5)

The rotation matrix from the body coordinate frame to the inertial coordinate frame can be derived as



R=



cos θ

cos(α − θ )

sin θ

− sin(α − θ )

.

(6)

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Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

The velocity of load in the body coordinate is given by

vl = [vl1 , vl2 ]T

(7)

where

vl1 = ls θ˙ + R · [x˙ , l α˙ ]T = ls θ˙ + x˙ cos θ + l α˙ cos(α − θ ),

(8)

vl2 = R · [x˙ , l α˙ ]T = x˙ sin θ − l α˙ sin(α − θ ).

(9)

Invoking (5) and (7), the kinetic energy of the helicopter-load system can be rewritten as

Wk =

 1 1 1  Mx˙ 2 + (IG + mls2 )θ˙ 2 + m x˙ 2 + (l α˙ )2 + 2l α˙ x˙ cos α + 2ls θ˙ x˙ cos θ + 2lls α˙ θ˙ cos(α − θ ). 2 2 2

(10)

The potential energy of the helicopter-load system is given by

Wp = mgl (1 − cos α ) + mgls (1 − cos θ ).

(11)

According to (3), we have





d ∂ Wk = (M + m )x¨ + mls θ¨ cos θ + ml α¨ cos α − ml α˙ 2 sin α , dt ∂ x˙   d ∂ Wk = (IG + mls2 )θ¨ + mls x¨ cos θ − mls x˙ θ˙ sin θ + mlls α¨ cos(α − θ ) − mlds α˙ 2 sin(α − θ ) + mlds α˙ θ˙ sin(α − θ ), dt ∂ θ˙ d dt

 ∂W  k

∂ α˙

= ml 2 α¨ + ml x¨ cos α − ml x˙ α˙ sin α + mlls θ¨ cos(α − θ ) − mlds α˙ θ˙ sin(α − θ ) + mlls θ˙ 2 sin(α − θ ).

∂ Wk = 0, ∂x ∂ Wk = −mls θ˙ x˙ sin θ + mlls θ˙ α˙ sin(α − θ ), ∂θ ∂ Wk = −ml x˙ α˙ sin α − mlls θ˙ α˙ sin(α − θ ), ∂α

(12)

(13)

and

∂ Wp = 0, ∂x ∂ Wp = mgls sin θ , ∂θ ∂ Wp = mgl sin α . ∂α

(14)

According to (3), quoting (12)–(14), the coupled dynamic equation of the helicopter horizontal motion, helicopter attitude motion, and the load swing can be formulated as

J (q )q¨ + C (q, q˙ )q˙ + G(q ) = Q where



T α ,

θ

q= x

⎡

(15)

M+m

J (q ) = ⎣mls cos θ ml cos α

⎡

0

C (q, q˙ ) = ⎣0 0

mls cos θ

ml cos α

⎤

IG + mls2

mlls cos(α − θ )⎦,

mlls cos(α − θ )

2

−mls θ˙ sin θ 0 mlls θ˙ sin(α − θ )

ml

−ml α˙ sin α

⎤

−mlls α˙ sin(α − θ )⎦, 0

(16)

(17)

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990



G (q ) = 0

mgls sin θ



Q = F − cx x˙ + d

mgl sin α

T

τ − cθ θ˙ + τd

,

−cα˙

5

(18)

T

(19)

where F is the generalized force for the generalized coordinate x; τ is the generalized torque for the generalized coordinate θ ; cx x˙ is the generalized resistance force for the generalized coordinate x; d is the external disturbance force for the generalized coordinate x; cθ θ˙ is the generalized pitch resistance torque for the generalized coordinate θ ; τ d is the generalized external disturbance for the generalized coordinate θ ; cα˙ is the resistance torque which acting on the load while cx , cθ , and c are the damping coefficients. Moreover, the helicopter slung-load system model (15) satisfies the following two properties. Property 1 [35]. The inertia matrix J(q) is a positive defined matrix. Property 2 [35]. The matrix J˙(q )/2 − C (q, q˙ ) is a skew symmetric, i.e., for all suitable dimensional vector ζ , the following equation holds:

ζ T (J˙(q )/2 − C (q, q˙ ))ζ = 0.

(20)

Remark 1. The boundedness of θ˙ in Assumption 3 will be proved in the proof of Theorem 1. Remark 2. The specific proof processes for Property 1 and Property 2 are easy to be obtained, thus, for the sake of simplicity, they are omitted here. 3. Problem statement and preliminaries In order to facilitate the control design process, (15) is expanded to be the following forms:

(M + m )x¨ + mls θ¨ cos θ + ml α¨ cos α − mls θ˙ 2 sin θ − ml α˙ 2 sin α = F − cx x˙ + d,

(21)

mls x¨ cos θ + (IG + mls2 )θ¨ + mlls α¨ cos(α − θ ) − mlls α˙ 2 sin(α − θ ) + mgls sin θ = τ − cθ θ˙ + τd ,

(22)

ml x¨ cos α + mlls θ¨ cos(α − θ ) + ml 2 α¨ + mlls θ˙ 2 sin(α − θ ) + mgl sin α + cα˙ = 0.

(23)

To realize the targets that decrease the swing for the helicopter-load system and track the target trajectory accurately, the desired trajectory r needs to satisfy the following conditions [36]: (1)

lim r = rd

(24)

t→∞

where rd is the target position. (2) The r, r˙ , r¨, r(3) are bounded, i.e., there exist positive constants κ 1 , κ 2 , κ 3 , and κ 4 such that

|r| ≤ κ1 ; 0 ≤ r˙ ≤ κ2 , (0 < r˙ (0 ) ≤ κ2 ); |r¨| ≤ κ3 ; |r (3) | ≤ κ4 ,

(25)

in addition,

lim r˙ = 0, lim r¨ = 0.

t→∞

(26)

t→∞

Similarly, a suitable desired trajectory is designed to be the same as in [36]:

r=

rd ρ1 cosh(ρ2 t − ρ3 ) + ln 2 2 cosh(ρ2 t − ρ3 − rd /ρ1 )



(27)

where ρ 1 , ρ 2 , and ρ 3 are the designed positive constants. Moreover, the following assumptions and lemma are necessary to stability analysis. Assumption 5. The desired trajectory r does not violate the actuator output constraint values. Assumption 6. The disturbance d and disturbance torque τ d are bounded, i.e., there exist known d¯, d¯1 , τ¯d , and τ¯d1 such that |d| ≤ d¯, |τd | ≤ τ¯d , |d˙| ≤ d¯1 , |τ˙ d | ≤ τ¯d1 , limt→∞ d = 0, and limt→∞ τd = 0; moreover, there are known positive constants π1 > d¯ and π2 > τ¯d render that [5]:

|de˙ x | ≤ π1 e˙ x tanh(e˙ x ),

(28)

|τd e˙ θ | ≤ π2 e˙ θ tanh(e˙ θ ).

(29)

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Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

Lemma 1. For ∀ς ∈ R+ , the following inequation holds:

ς − tanh ς ≥ 0,

(30)

moreover, if and only if ς = 0, equal sign of (30) holds. Proof. Let = ς − tanh ς , considering e2ς + e−2ς ≥ 2, the derivative of ϖ with respect to ς can be written as

= 1 −

4

( e ς + e −ς ) 2

e2ς + e−2ς − 2 ( e ς + e −ς ) 2 ≥ 0,

=

where = dd ς .

In addition, if and only if ς = 0, = R+ . Thus, this concludes the proof. 

(31) e2ς +e−2ς −2 e ς + e −ς

= 0. Thus, the minimum value of ϖ is min = 0 − tanh 0 = 0, for ∀ ς ∈

Remark 3. For the second term of the desired trajectory r, when t → ∞, we have

e(ρ2 t−ρ3 ) + e(−ρ2 t+ρ3 ) ln (ρ t−ρ −r /ρ ) 2 3 1 + e (−ρ2 t+ρ3 +rd /ρ1 ) d 2 e ( ρ t−ρ3 ) 2 ρ1 e = lim ln (ρ t−ρ −r /ρ ) t→∞ 2 e 2 3 d 1 r = d. 2

lim

ρ1

t→∞

(32)

Hence, limt→∞ r = rd . Moreover, since the initial position of helicopter is usually set to zero, in order to avoid a big initial error, the initial value of r should be zero or approximate to zero. In the format of r, the bigger the value of ρ 3 , the more closer to zero of the initial value of r. However, a too big value of ρ 3 will render that the desired trajectory r converges to rd is going to take a long period of time. Therefore, it is necessary to make compromises between the value of ρ 3 and the time that the desired trajectory r converges to rd . The other related instructions for the desired trajectory r has been provided in [36]. Remark 4. According to the forms of the generalized resistance force and the generalized pitch resistance torque, it is obvious that if x˙ → 0 and θ˙ → 0, then cx x˙ → 0 and cθ θ˙ → 0. In addition, for the derivatives of external disturbance d˙ and τ˙ d , the bounds d¯1 and τ¯d1 can be unknown. Remark 5. Different from the form of disturbance in [37,38], the assumption of boundedness for external disturbances d and τ d are reasonable, for example, unit impulse disturbance and gust wind. In addition, there exist external disturbances d and τ d render that limt→∞ d = 0 and limt→∞ τd = 0, for example, gust wind. Hence, the assumptions for the disturbances in the Assumption 6 are reasonable. 4. Control design and stability analysis In this section, the control objectives are to suppress the swing and ensure that the trajectory of helicopter x can track the desired trajectory r for the helicopter slung-load system in the presence of input constraints and external disturbances. Energy-based control technique is used to analyze the stability of the closed-loop system. In order to illustrate the process of robust adaptive control design clearly, to construct energy shaping function conveniently, define the error signals as follows:

ex = x − r, eθ = θ , eα = α .

(33)

Based on (33), to realize the above control targets, the control scheme is proposed as:

F = −k1 tanh(ex ) + (M + m )r¨ + cx r˙ − λ1 tanh(e˙ x ) − π1 tanh(e˙ x )

(34)

τ = −k2 tanh(eθ ) + mls r¨ cos θ − λ2 tanh(e˙ θ ) − π2 tanh(e˙ θ )

(35)

where k1 , k2 , λ1 , λ2 , π 1 , and π 2 are the designed positive constants. Consider input constraints, i. e.,

|F | ≤ F¯ , |τ | ≤ τ¯ with F¯ and τ¯ being the positive constants.

(36)

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

7

According to |tanh (·)| < 1, (25), Assumption 3, and Assumption 5, the constraint condition (36) will not be violated by regulating k1 , k2 , λ1 , λ2 , π 1 , and π 2 . In addition, when the energy of system will attenuate to zero, system state error signals will converge to zero. Furthermore, taking the control targets into account, an energy shaping function can be constructed as follows:

V =

1 T e˙ J (q )e˙ + k1 ln[cosh(ex )] + k2 ln[cosh(eθ )] + mgls (1 − cos θ ) + mgl (1 − cos α ) 2

(37)

where e = [ex , eθ , eα ]T . Remark 6. The actuator output constraints (36) can be ensured by regulating the design parameters k1 , k2 , λ1 , λ2 , π 1 , and π 2. Remark 7. The design parameters of the control laws (34) and (35) should take system model parameters and the input constraint condition (36) into consideration, i.e., the designs for k1 , k2 , λ1 , λ2 , π 1 , and π 2 are motivated by the constraint condition and experience. Theorem 1. Consider the helicopter slung-load system in the presence of input constraints and external disturbances which is described by (21), (22), and (23). Under the proposed control laws (34) and (35), ex , eθ , and eα satisfy the following conditions:

lim ex = 0, lim eθ = 0, lim eα = 0. t→∞ t→∞

(38)

t→∞

Proof. Let Vo =

1 T ˙ ˙ 2 e J ( q )e

and R = [r, 0, 0]T , invoking (15) and (16), the derivative of Vo can be written as

1 T e˙ J˙(q )e˙ 2 1 = e˙ T J (q )(q¨ − R¨ ) + e˙ T J˙(q )e˙ 2

V˙ o = e˙ T J (q )e¨ +

1 T e˙ J˙(q )e˙ 2     1 = e˙ T Q − C (q, q˙ )R˙ − G(q ) − e˙ T C (q, q˙ ) − J˙(q ) e˙ − e˙ T J (q )R¨ . 2 = e˙ T [Q − C (q, q˙ )q˙ − G(q )] − e˙ T J (q )R¨ +

(39)

According to the Property 2, quoting (16)–(19) and (33), the (39) can be rewritten as

V˙ o = e˙ T [Q − C (q, q˙ )R˙ − G(q )] − e˙ T J (q )R¨ = [F − cx x˙ + d]e˙ x + [τ − cθ e˙ θ + τd − mgls sin θ ]e˙ θ − [cα + mgl sin α ]e˙ α − [e˙ x (M + m ) + mls e˙ θ cos θ + ml e˙ α cos α ]r¨ = [F − (M + m )r¨ − cx x˙ + d]e˙ x + [τ − mls r¨ cos θ + τd ]e˙ θ − mgl e˙ α sin α − mgls e˙ θ sin θ − ml r¨e˙ α cos α − cθ e˙ 2θ − ce˙ 2α .

(40) Invoking (40) and taking the derivative of V given in (37) with respect to time, it follows that

V˙ = V˙ o + k1 tanh(ex )e˙ x + k2 tanh(eθ )e˙ θ + mgl θ˙ sin θ + mgl α˙ sin α

= [F + k1 tanh(ex ) − (M + m )r¨ − cx x˙ + d]e˙ x + [τ + k2 tanh(eθ ) − mls r¨ cos θ + τd ]e˙ θ − ml r¨e˙ α cos α − cθ e˙ 2θ − ce˙ 2α . (41)

Substituting (34) and (35) into (41), it can be derived that

V˙ = −λ1 tanh(e˙ x )e˙ x − λ2 tanh(e˙ θ )e˙ θ − ml r¨e˙ α cos α − cx e˙ 2x − cθ e˙ 2θ − ce˙ 2α . m2 l 2 2 ¨ 2c r

Considering −ml r¨α˙ cos α ≤

+

c 2 2 α˙ ,

(42) can be rewritten as

V˙ ≤ −λ1 tanh(e˙ x )e˙ x − λ2 tanh(e˙ θ )e˙ θ +

c m2 l 2 2 r¨ − cx e˙ 2x − cθ e˙ 2θ − e˙ 2α 2c 2

where c is a designed positive constant. Integrating (43) yields

V (t ) ≤ V (0 ) − λ1 −

cx 2



t

0



0

t

tanh(e˙ x )e˙ x dω − λ2

c e˙ 2x dω − θ 2



t 0

(42)

e˙ 2θ dω −

c 2





t

0 t 0

tanh(e˙ θ )e˙ θ dω + e˙ 2α dω.

m2 l 2 2c

(43)

 0

t

r¨2 dω (44)

Then, considering the fact that limt→∞ r¨ = 0, it obtains

V (t ) ≤ V (0 ) +

m2 l 2 U¯ ∈ L∞ 2c

where U¯ = supt∈[0,+∞ ) {

t 0

r¨2 dω}.

(45)

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Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

Quoting Lemma 1 and considering that there are the same plus-or-minus sign between

−λ1 = −λ1 −λ2 = −λ1 −λ2 = −λ1 −λ2 ≤ −λ1



t



0



0



0



0



0



0



0

t

t

t

t

t

t

t 0

tanh(e˙ x )e˙ x dω − λ2



t 0

∗

and tanh (∗ ) yields

tanh(e˙ θ )e˙ θ dω

tanh(e˙ x )(e˙ x − tanh(e˙ x ) + tanh(e˙ x ))dω tanh(e˙ θ )(e˙ θ − tanh(e˙ θ ) + tanh(e˙ θ ))dω

| tanh(e˙ x )|[e˙ x − tanh(e˙ x )]sign(e˙ x ) + tanh2 (e˙ x )dω | tanh(e˙ θ )|[e˙ θ − tanh(e˙ θ )]sign(e˙ θ ) + tanh2 (e˙ θ )dω | tanh(e˙ x )|(|e˙ x | − | tanh(e˙ x )| ) + tanh2 (e˙ x )dω | tanh(e˙ θ )|(|e˙ θ | − | tanh(e˙ θ )| ) + tanh2 (e˙ θ )dω tanh (e˙ x )dω − λ2 2

 0

t

tanh (e˙ θ )dω. 2

(46)

Considering (44)–(46), it has

λ1

 0

t

tanh (e˙ x )dω + λ2 2

≤ V (0 ) − V (t ) +



t 0

tanh (e˙ θ )dω + 2

cx 2

 0

t

c e˙ 2x dω + θ 2



t 0

e˙ 2θ dω +

c 2



t 0

e˙ 2α dω

m2 l 2 U¯ ∈ L∞ . 2c

(47)

Thus,

e˙ x , e˙ θ , e˙ α ∈ L2 .

(48)

Then, from (33) and limt→∞ r˙ = 0, the following conclusion can be derived:

x˙ , θ˙ ∈ L2 .

(49)

In addition, according to (37) and (45), the following conclusion holds:

ex , eθ , e˙ x , e˙ θ , e˙ α ∈ L∞ ,

(50)

then, invoking (25) yields

x, x˙ , θ˙ , α˙ ∈ L∞ . Multiplying (21) by IG +

(51) mls2

on both sides yields

(IG + mls2 )(M + m )x¨ + mls (IG + mls2 )θ¨ cos θ + ml (IG + mls2 )α¨ cos α − mls (IG + mls2 )θ˙ 2 sin θ − ml (IG + mls2 )α˙ 2 sin α = (F + d − cx x˙ )(IG + mls2 ).

(52)

Multiplying (22) by mls cos θ on both sides yields

m2 ls2 x¨ cos2 θ + mls (IG + mls2 )θ¨ cos θ + m2 l ls2 α¨ cos(α − θ ) cos θ − m2 l ls2 α˙ 2 sin(α − θ ) cos θ + m2 gls2 sin θ cos θ = mls (τ + τd − cθ θ˙ ) cos θ .

(53)

Invoking (52) and (53) yields

[IG (M + m ) + Mmls2 + m2 ls2 sin

2

θ ]x¨ + [mlIG cos α − m2 l ls2 sin(α − θ ) sin θ ]α¨

− mls (IG + mls2 )θ˙ 2 sin θ − [mlIG sin α + m2 lls2 cos(α − θ ) sin θ ]α˙ 2 − m2 gls2 sin θ cos θ = (F + d − cx x˙ )(IG + mls2 ) − mls (τ + τd − cθ θ˙ ) cos θ .

(54)

Similarly, multiplying (21) by l cos(α − θ ) on both sides yields

(M + m )l x¨ cos(α − θ ) + mls l θ¨ cos θ cos(α − θ ) + ml 2 α¨ cos α cos(α − θ ) − mlls θ˙ 2 sin θ cos(α − θ ) − ml 2 α˙ 2 sin α cos(α − θ ) = (F + d − cx x˙ )l cos(α − θ ).

(55)

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

9

Multiplying (23) by cos θ on both sides yields

ml x¨ cos α cos θ + mlls θ¨ cos(α − θ ) cos θ + ml 2 α¨ cos θ + mlls θ˙ 2 sin(α − θ ) cos θ + mgl sin α cos θ − cα˙ cos θ = 0.

(56)

Combining (55) and (56), it has

[Ml cos(α − θ ) + ml sin α sin θ ]x¨ − ml 2 sin α sin(α − θ )α¨ − mlls θ˙ 2 sin α − ml 2 α˙ 2 sin α cos(α − θ ) − mgl sin α cos θ + cα˙ cos θ = (F + d − cx x˙ )l cos(α − θ ). Then, multiplying (54) by

ml 2

(57)

sin α sin(α − θ ) on both sides yields

ml sin α sin(α − θ )[IG (M + m ) + Mmls2 + m2 ls2 sin 2



= −ml sin α sin(α − θ ) mlIG cos α − m 2



2

l ls2

2

θ ]x¨



sin(α − θ ) sin θ α¨ + m2 l 2 ls (IG + mls2 )θ˙ 2 sin θ sin α sin(α − θ )



+ ml 2 sin α sin(α − θ ) mlIG sin α + m2 l ls2 cos(α − θ ) sin θ α˙ 2 + m3 gl 2 ls2 sin α sin(α − θ ) sin θ cos θ + (F + d − cx x˙ )(IG + mls2 )ml 2 sin α sin(α − θ ) − m2 l 2 ls (τ + τd − cθ θ˙ ) sin α sin(α − θ ) cos θ . and multiplying (57) by ml IG cos α

[ml IG cos α − m

2

l ls2

− m2 l ls2

(58)

sin(α − θ ) sin θ on both sides yields

sin(α − θ ) sin θ ][Ml cos(α − θ ) + ml sin α sin θ ]x¨

= ml [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ] sin α sin(α − θ )α¨ + ml ls [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ]θ˙ 2 sin α 2

+ ml 2 [mlIG cos α − m2 lls2 sin(α − θ ) sin θ ]α˙ 2 sin α cos(α − θ ) + mgl [ml IG cos α − m2 l ls2 × sin(α − θ ) sin θ ] sin α cos θ − c[mlIG cos α − m2 lls2 sin(α − θ ) sin θ ]α˙ cos θ + (F + d − cx x˙ )l [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ] cos(α − θ ).

(59)

Considering (58) and (59), invoking (34) and (35), it obtains

ml 2 sin α sin(α − θ )[IG (M + m ) + Mmls2 + m2 ls2 sin

2



θ ] + [mlIG cos α − m2 l ls2 sin(α − θ ) sin θ ]

× [Ml cos(α − θ ) + ml sin α sin θ ] x¨

= m2 l 2 ls (IG + mls2 ) sin θ sin α sin(α − θ ) + ml ls [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ] sin α

 θ˙ 2

+ ml 2 sin α sin(α − θ )[mlIG sin α + m2 l ls2 cos(α − θ ) sin θ ] + ml 2 [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ]



× sin α cos(α − θ ) α˙ 2 + m3 gl 2 ls2 sin α sin(α − θ ) sin θ cos θ + mgl [ml IG cos α − m2 l ls2 sin(α − θ ) sin θ ] × sin α cos θ + [−k1 tanh(ex ) + (M + m )r¨ + cx r˙ − λ1 tanh(e˙ x ) − cx x˙ + d − π1 tanh(e˙ x )]

× (IG + mls2 )ml 2 sin α sin(α − θ ) + [ml 2 IG cos α − m2 l 2 ls2 sin(α − θ ) sin θ ] cos(α − θ )



− [−k2 tanh(eθ ) + mls r¨ cos θ − λ2 tanh(e˙ θ ) − cθ θ˙ + τd − π2 tanh(e˙ θ )]m2 l 2 ls sin α sin(α − θ ) cos θ = m2 l 2 ls IG θ˙ 2 cos(α − θ ) sin α cos θ + m2 l 3 IG α˙ 2 sin α cos θ + [(M + m )r¨ − cx e˙ x − λ1 tanh(e˙ x ) + d − π1 tanh(e˙ x )] 2 × [ml 2 IG + m2 l 2 ls2 sin (α − θ )] cos θ − [mls r¨ cos θ − cθ θ˙ − λ2 tanh(e˙ θ ) + τd − π2 tanh(e˙ θ )]

1 2 2 2 m gl IG sin(2α ) cos θ − k1 tanh(ex )[ml 2 IG + m2 l 2 ls2 sin (α − θ )] cos θ 2 + k2 m2 l 2 ls tanh(eθ ) sin α sin(α − θ ) cos θ . × m2 l 2 ls sin α sin(α − θ ) cos θ +

(60)

The polynomial ml 2 sin α sin(α − θ )[IG (M + m ) + Mmls2 + m2 ls2 sin2 θ ] + [mlIG cos α − m2 l ls2 sin(α − θ ) sin θ ][Ml cos(α − θ ) + ml sin α sin θ ] can be rewritten as follows:

ml 2 sin α sin(α − θ )[IG (M + m ) + Mmls2 + m2 ls2 sin

2

θ ] + [mlIG cos α − m2 l ls2 sin(α − θ ) sin θ ]

× [Ml cos(α − θ ) + ml sin α sin θ ] = ml 2 IG (M + m ) sin α sin(α − θ ) + Mm2 l 2 ls2 sin α sin(α − θ ) + m3 l 2 ls2 sin α sin(α − θ ) sin

2

+ Mml IG cos α cos(α − θ ) + m l IG sin α sin θ cos α − 2

−

m3 l 2 ls2

2 2

sin(α − θ ) sin α sin

2

= Mml IG cos θ + m l IG sin 2

2 2

2

Mm2 l 2 ls2

θ

sin(α − θ ) cos(α − θ ) sin θ

θ

α cos θ + Mm2 l 2 ls2 sin2 (α − θ ) cos θ .

(61)

10

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

Then, according to (61) and Assumption 1, it follows that

ml 2 sin α sin(α − θ )[IG (M + m ) + Mmls2 + m2 ls2 sin

2

θ ] + [mlIG cos α − m2 l ls2 sin(α − θ ) sin θ ]

× [Ml cos(α − θ ) + ml sin α sin θ ] = Mml 2 IG cos θ + m2 l 2 IG sin

2

α cos θ + Mm2 l 2 ls2 sin2 (α − θ ) cos θ > 0.

(62)

Let  = Mml 2 IG cos θ + m2 l 2 IG sin2 α cos θ + Mm2 l 2 ls2 sin2 (α − θ ) cos θ and x¨ = χ1 + χ2 , where

χ1 = m2 l 2 ls IG θ˙ 2 cos(α − θ ) sin α cos θ + m2 l 3 IG α˙ 2 sin α cos θ + [cx e˙ x − (λ1 + π1 ) tanh(e˙ x )]     2 × ml 2 IG + m2 l 2 ls2 sin (α − θ ) cos θ − cθ θ˙ − (λ2 + λ2 ) tanh(e˙ θ ) m2 l 2 ls sin α sin(α − θ ) cos θ ,

(63)

  1 2 2 2 m gl IG sin(2α ) cos θ + [(M + m )r¨ − k1 tanh(ex ) + d] ml 2 IG + m2 l 2 ls2 sin (α − θ ) cos θ 2 − [mls r¨ cos θ − k2 tanh(eθ ) + τd ]m2 l 2 ls sin α sin(α − θ ) cos θ .

(64)

χ2 = Therefore,

x¨ =

χ1 χ2 + .  

(65)

Invoking (21)–(23), eliminating the terms which include θ¨ and α¨ , considering (25), (51) and (65), it can be acquired that

x¨ ∈ L∞ .

(66)

Similarly, θ¨ and α¨ satisfy the following conditions:

θ¨ , α¨ ∈ L∞ .

(67)

Thus, according to (48)–(50), it has

e˙ x , θ˙ , α˙ ∈ L2 ∩ L∞ .

(68)

Using Barbalat lemma, invoking (66)–(68), it can be derived that

lim (e˙ x , θ˙ , α˙ ) → (0, 0, 0 ),

(69)

lim (e˙ x , e˙ θ , α˙ ) → (0, 0, 0 ).

(70)

t→∞

i.e., t→∞

Considering (63) and (69), it obtains

lim

t→∞

Define

=

χ1 = 0. 

(71)

χ2 , 

(72)

then, it obtains

˙ =

χ˙ 2  + ˙ χ2 2

(73)

where

χ˙ 2  + ˙ χ2





1 2 2 ˙ 4 m gl IG θ sin(2α ) sin θ + (M + m )r (3) − k1 e + d˙ 2 (e x + e−ex )2    2 × ml 2 IG + m2 l 2 ls2 sin (α − θ ) cos θ + [(M + m )r¨ − k1 tanh(ex ) + d] −ml 2 IG θ˙ sin θ + 2m2 l 2 ls2 sin(α − θ )

= m2 gl 2 IG α˙ cos(2α ) cos θ −





× cos(α − θ )(α˙ − θ˙ ) − m2 l 2 ls2 θ˙ sin θ sin (α − θ ) − m2 l 2 ls mls r (3) cos θ − mls r¨ sin θ θ˙ − k2 2

( e eθ

4 + τ˙ d + e−eθ )2

× sin α sin(α − θ ) cos θ − m l ls [mls r¨ cos θ − k2 tanh(eθ ) + τd ]α˙ cos α sin(α − θ ) cos θ 2 2

− m2 l 2 ls [mls r¨ cos θ − k2 tanh(eθ ) + τd ](α˙ − θ˙ ) sin α cos(α − θ ) + m2 l 2 ls [mls r¨ cos θ − k2 tanh(eθ ) + τd ]θ˙



× sin α sin(α − θ ) sin θ × Mml 2 IG cos θ + m2 l 2 IG sin

2

α cos θ + Mm2 l 2 ls2 sin2 (α − θ ) cos θ





Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

+

− Mml 2 IG θ˙ sin θ + 2m2 l 2 IG α˙ sin α cos α cos θ − m2 l 2 IG θ˙ sin

2

2 × sin(α − θ ) cos(α − θ ) cos θ − Mm2 l 2 ls2 θ˙ sin (α − θ ) sin θ

×

1 2



11

α sin θ + 2Mm2 l 2 ls2 (α˙ − θ˙ )

m2 gl 2 IG sin(2α ) cos θ + [(M + m )r¨ − k1 tanh(ex ) + d][ml 2 IG + m2 l 2 ls2 sin (α − θ )] cos θ 2



+ [mls r¨ cos θ − k2 tanh(eθ ) + τd ]m2 l 2 ls sin α sin(α − θ ) cos θ .

(74)

Invoking (25), (51), (73), and (74), considering Assumption 3 and  ∈ L∞ , it follows that

˙ ∈ L∞ .

(75)

By utilizing extended Barbalat lemma [39], considering (71) and (75), the following conclusion can be obtained:

lim x¨ = 0.

(76)

t→∞

Using the same analysis process, it can be derived that

lim θ¨ = 0, lim α¨ = 0.

t→∞

t→∞

(77)

According to (23), (69), (76), and (77), it follows that

lim sin α = 0.

(78)

t→∞

Moreover, since α ∈ (− π2 , π2 ), quoting (77), it has

lim α = 0.

(79)

t→∞

Based on (21) and (34), it has

lim ex = 0.

(80)

t→∞

Invoking (54)and (80), considering Assumption 3, the following conclusion can be addressed as

lim sin(2θ ) = 0.

(81)

t→∞

Considering Assumption 3 and (81), it obtains

lim θ = 0.

(82)

t→∞

Combining (33), (79), (80), and (82), it can be drawn that

lim (ex , eθ , eα ) = (0, 0, 0 ).

t→∞

This concludes the proof.

(83)



5. Numerical simulation A helicopter slung-load system with input constraints and external disturbances is considered in this section; the external disturbance d and the external disturbance torque τ d are given as follows:

d = [0.6 + 0.2 sin(0.4t ) + 0.4 sin(0.2t )]e−0.01t × 104 N,

(84)

τd = [0.6 + 0.2 sin(0.4t ) + 0.4 sin(0.2t )]e−0.01t × 103 N · m.

(85)

The parameters of the suspension cable system are listed in Table 1 [40]. To achieve the targets of positioning control and reducing the oscillation for the helicopter slung-load system, the parameters are given as follows: the desired trajectory r can be described as (27) with ρ1 = 2.5, ρ2 = 2, ρ3 = 0.01, and rd = 100. The initial conditions are chosen as x(0 ) = 0, x˙ (0 ) = 0, θ (0 ) = 0, θ˙ (0 ) = 0, α (0 ) = 0, and α˙ (0 ) = 0. The damping coefficients cx = 8400N · s/m, cθ = 1000N · s · m/rad, and c = 120N · s · m/rad [40]. The input constraint values F¯ = 3.0 × 105 and τ¯ = 4.1 × 105 . Next, the design parameters are k1 = 6.0 × 104 , k2 = 1.1 × 104 , λ1 = 6.0 × 104 , λ2 = 3.0 × 105 , π1 = 2.2 × 104 , and π2 = 2.05 × 103 . Combining the Figs. 2 and 3 with the aforementioned design parameters and system parameters, it has |F | ≤ 2.677 × 105 < F¯ and |τ | ≤ 4.0305 × 105 < τ¯ . Thus, choose suitable design parameters k1 , k2 , λ1 , λ2 , π 1 , and π 2 can such that |F | ≤ F¯ and |τ | ≤ τ¯ . Figs. 4–9 are the responses of the helicopter slung-load system described by (15) under control laws (34) and (35). The position signal x and position track error signal ex are shown in Fig. 4 and Fig. 5, respectively. The fluctuation of Fig. 5 is

12

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

5 4 3 2 1 0

0

50

100

150

Time (s) Fig. 2. The first-order derivative of reference signal r˙ for x.

5

0

−5

0

50

100

150

Time (s) Fig. 3. The second-order derivative of reference signal r˙ for x.

120 100 80 60 40 20 0

0

50

Time (s)

100

150

Fig. 4. The position of helicopter under controllers (34) and (35).

1.5 1 0.5 0 −0.5 −1 −1.5

0

50

100

150

Time (s) Fig. 5. The position of helicopter under controllers (34) and (35).

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

Table 1 Parameters of the helicopter slung-load system. Symbol

Description

Value

M m IG l ls

Mass of the helicopter Mass of the load Moment of inertia of helicopter Length of the cable Distance between load suspension point and the center of gravity of helicopter Acceleration of gravity

1.074 × 104 kg 6.0 × 103 kg 5.37 × 103 kg · m2 15m

g

3m 9.8m/s2

15 10 5 0 −5 −10 −15

0

50

100

150

Time (s) Fig. 6. The pitch angle under controllers (34) and (35).

30 20 10 0 −10 −20 −30

0

50

100

150

Time (s) Fig. 7. The swing angle under controllers (34) and (35).

5

2.5

x 10

2 1.5 1 0.5 0 −0.5 −1

0

50

100 Time (s)

Fig. 8. The control input (34).

150

13

14

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990 5

1

x 10

0.5

0

−0.5

−1

0

50

100

150

Time (s) Fig. 9. The control input (35).

120 100 80 60 40 20 0

0

50

100

150

Time (s) Fig. 10. The position of helicopter under controllers (86) and (87).

2

1

0

−1

−2

0

50

100

150

Time (s) Fig. 11. The position of helicopter under controllers (34) and (35).

resulted from the oscillation of slung-load. It illustrates that the helicopter can track reference signal r(t) accurately; The ranges of the pitch angle and the swing angle are depicted in Figs. 6 and 7, respectively. It shows that the proposed control scheme is effective in dealing with the problems of positioning control and restraining swing for the helicopter slung-load system in the presence of input constraints and external disturbances. Figs. 8 and 9 denote the control inputs (34) and (35), respectively. Obviously, according to Figs. 8 and 9, the input constraints can not be violated. In order to illustrate the effectiveness of the proposed control strategy, the PD control laws are considered as follows:

F = −η1 ex − η2 e˙ x ,

(86)

τ = −η3 eθ − η4 e˙ θ .

(87)

Under the control scheme with η1 = 6.0 × 104 , η2 = 1.0 × 103 , η3 = 1.0 × 105 , and η4 = 2.0 × 105 , the responses of the helicopter slung-load system are described as Figs. 10–15. From Figs. 4, 5, 10, and 11, it shows that system state x can reach the

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990

15

30

20

10

0

−10

−20

0

50

100

150

Time (s) Fig. 12. The pitch angle under controllers (86) and (87).

40 30 20 10 0 −10 −20 −30 −40

0

50

100

150

Time (s) Fig. 13. The swing angle under controllers (86) and (87).

5

1.5

x 10

1 0.5 0 −0.5 −1

0

50

100

150

Time (s) Fig. 14. The PD control input (86).

target position quickly under both control strategies. However, according to Figs. 5 and 11, it is obvious that the convergence time of ex will be longer by utilizing the controllers (86) and (87). Meanwhile, compare Figs. 6 and 7 with Figs. 12 and 13, it derives that the control performances for eθ and α by using the control schemes (34) and (35) are much better than using the control laws (86) and (87). Moreover, according to Figs. 8, 9, 14, and 15, the energy loss by using the proposed control scheme which is described by (34) and (35) is less than using the control strategy which is presented by (86) and (87). It can be seen from Figs. 14 and 15 that there exist some oscillations in the period of 100–150s compared with Figs. 8 and 9. This phenomena in the Figs. 14 and 15 are resulted from the residual swing of load by using PD control method. It means that the designed control scheme which describes by (34) and (35) is more effective compare with using the control laws (86) and (87).

16

Y. Ren, K. Li and H. Ye / Applied Mathematics and Computation 372 (2020) 124990 5

1

x 10

0.5 0 −0.5 −1 −1.5

0

50

100

150

Time (s) Fig. 15. The PD control input (87).

Thus, the rationality and validity of the proposed control scheme is validated by using Figs. 4–15, meanwhile, under the introduced control technique, the input constraints can be avoided. Remark 8. According to Figs. 2–5, the desired trajectory r satisfies the conditions (24)–(26). 6. Conclusion In this paper, the motion trajectory tracking control scheme has been proposed for a rigid-body model of helicopter slung-load system in the presence of input constraints and external disturbances. A novel rigid-body model of helicopter slung-load system has been obtained by using Lagrange’s equations. Then, a desired trajectory of helicopter motion has been used to restrain swing and position control. Considering input constraints, the function of tanh (·) has been utilized to dispose this problem by selecting suitable design parameters. Then, by applying the energy-based control approach, the trajectory tracking controllers have been designed for the helicopter slung-load system with input constraints and external disturbances. The stability analysis of closed-loop system has been performed under the proposed control scheme. Finally, simulation results have validated the feasibility and effectiveness of the developed track control laws in handling with load swing attenuation and position control for the rigid-body model of helicopter slung-load system. In the future research, the problems of modeling and anti-swing will be considered for the full state model of the helicopter slung-load system. In addition, motivated by [41,42], the obtained results can be extended to networked-based case. 7. Acknowledgment This work was supported in part by the China Postdoctoral Science Foundation under Grant 2019M662313, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 19KJB510023, and the National Natural Science Foundation of China under Grant 61903232. References [1] N.K. Gupta, A.E. Bryson, Near-hover control of a helicopter with a hanging load[j], J. Aircr. 13 (3) (1976) 217–222. [2] M. Bisgaard, A. l. Cour-Harboy, J. D. Bendtseny, Input shaping for helicopter slung load swing reduction[c], Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii (2008) 18–21. [3] C.J. Adams, J.J. Potter, W. Singhose, Input-shaping and model-following control of a helicopter carrying a suspended load[j], J. Guid. Control Dyn. 38 (1) (2015) 94–105. [4] J.J. Potter, C.J. Adams, W. Singhose, A planar experimental remote-controlled helicopter with a suspended load[j], IEEE/ASME Trans. Mechatron. 20 (5) (2015) 2496–2503. [5] Y. Ren, M. Chen, P. Shi, Robust adaptive constrained boundary control for a suspension cable system of a helicopter[j], Int. J. Adapt. Control Signal Process 32 (1) (2018) 50–68. [6] Y. Ren, M. Chen, Anti-swing control for a suspension cable system of a helicopter with cable swing constraint and unknown dead-zone[j], Neurocomputing 356 (2019) 257–267. [7] Y. Ren, M. Chen, J. Liu, Unilateral boundary control for a suspension cable system of a helicopter with horizontal motion[j], IET Control Theory Appl. 13 (4) (2019) 467–476. [8] M. Chen, Y. Ren, J. Liu, Antidisturbance control for a suspension cable system of helicopter subject to input nonlinearities[j], IEEE Trans. Syst. Man. Cybern. Syst. 48 (12) (2018) 2292–2304. [9] Y. Ren, W.W. Sun, Robust adaptive control for robotic systems with input time-varying delay using hamiltonian method[j], IEEE/CAA J. Autom. Sin. 5 (4) (2018) 852–859. [10] Y. Wang, S.S. Ge, Augmented hamiltonian formulation and energybased control design of uncertain mechanical systems[j], IEEE Trans. Control Syst. Technol. 16 (2) (2008) 202–213. [11] W. Wang, J. Yi, D. Zhao, D. Liu, Design of a stable sliding-mode controller for a class of second-order underactuated systems[j], IEE Proc. Control Theory Appl. 151 (6) (2004) 683–690. [12] R. Xu, U. Özgüner, Sliding mode control of a class of underactuated systems[j], Automatica 44 (1) (2008) 233–241.

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