A novel bilateral impedance controls for underwater tele-operation systems

Applied Soft Computing Journal 91 (2020) 106194

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

A novel bilateral impedance controls for underwater tele-operation systems ∗

Ting Wang a , Yujie Li b , , Jianjun Zhang c , Yuan Zhang d a

30, Puzhu South Road, Pukou District, College of Electrical Engineering And Control Science, Nanjing Tech University, Nanjing, Jiangsu, China School of Information Engineering, Yangzhou University, 225127, China c School of Electrical Engineering and Automation, Henan Polytechnic University, 454003, China d 30, Puzhu South Road, Pukou District, College of Material Science And Engineering, Nanjing Tech University, Nanjing, Jiangsu, China b

article

info

Article history: Received 30 October 2019 Received in revised form 10 February 2020 Accepted 17 February 2020 Available online 24 February 2020 Keywords: Underwater tele-operation Time delay Adaptive neural fuzzy inference system Disturbance observer-based Impedance control

a b s t r a c t Owing to characteristics of the flow and the variability, it is extremely difficult to achieve the stability and the transparency of the underwater tele-operation system. In practice, the accurate force may not easily be acquired due to model uncertainties, the time delay and external disturbances. In order to enhance the stability and the transparency of the underwater tele-operation, an adaptive neural fuzzy inference system disturbance observer-based impedance control is proposed to both the master side and slave side. The learning algorithm of the adaptive neural fuzzy inference system network and the disturbance observer may simultaneously suppress model uncertainties of the nonlinear system and disturbances of external underwater environment. Concerning the time delay, the stability is analyzed by Lyapunov theorem. Numerical simulations are performed and results demonstrate the effective performance of the proposed method. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Tele-operation systems are successfully applied to many hazardous and complicate environment, such as space explorations [1], surgical operations [2], the rehabilitation training and so on. Especially, in the bilateral underwater tele-operation system, via operating the master manipulator, the human operator may manipulate the slave manipulator to accomplish many tasks, such as underwater salvage [3], sample collection [4], pipeline assembly [5], cable cutting [6], underwater exploration [7] and so on. Aiming at executing accurate tasks, the human operator needs to acquire the actual reflected force of the slave manipulator so as to adjust the motion in the master side. In this case, both the stability of the motion control and the transparency of the reflected force must be achieved during the operation. Comparing with the motion control, it is more difficult to achieve the practical reflected force as well as to enhance the transparency of the tele-operation environment. Different from general teleoperation systems, enhancing the transparency of the underwater tele-operation system is extremely hard due to the dynamic and the complexity of the flow underwater environment [8–11]. ∗ Corresponding author. E-mail addresses: [email protected] (T. Wang), [email protected] (Y. Li), [email protected] (J. Zhang), [email protected] (Y. Zhang). https://doi.org/10.1016/j.asoc.2020.106194 1568-4946/© 2020 Elsevier B.V. All rights reserved.

In the absence of force sensors, for the purpose of increasing the transparency, some researchers attempt to estimate the environmental force. In [12], a novel control method is proposed on the basis of the force estimation on both the master and slave robots for the purpose of substituting the force sensors. Concerning the time delay, position tracking experiments are performed to verify the proposed method. A sliding mode algorithm is used to estimate external forces for a bilateral teleoperation system [13]. The time variant environmental forces are discussed and estimated in [14]. Based on the estimation of bounded forces, a sliding mode control is used to the linear bilateral tele-operation system. Subsequently, researchers investigate a PD controller to the nonlinear bilateral tele-operation system applying a force estimation method in [15]. An improved extended active observer (IEAOB) and extended active observer (EAOB) for bilateral tele-operation system are introduced to the force estimation system. Mojtaba and his colleagues tune an adaptive law with bilateral impedance control in order to adjust the impedance model parameters during the tele-surgery process [2]. The advantage is that the tele-surgery process needs neither force sensor’s measurements nor the estimation of heart motion. After the stability analysis, experimental results demonstrate that the proposed controller may enhance the safety of patients and compensate the motion of the beating heart. Besides the force estimation methods, researchers generally use admittance control, compliant force control, impedance control methods to enhance the transparency of tele-operation systems. Moreover, the impedance control methods stand out for

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T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

its superior performances. Many researchers focus on designing various impedance control methods so as to increase the transparency for various tele-operation systems [16]. An improved wave-based bilateral tele-rehabilitation scheme is presented in [17]. For the robot manipulator assisted telerehabilitation system, the motion-based adaptive impedance controls are exerted on both the master and slave robot manipulators. The stability of the tele-rehabilitation system with time delay is analyzed ensuring the stable human–robot interaction and compensating the position drift. Numerical simulations are performed to verify the proposed bilateral impedance control. In [18], in order to raise the patient-therapist interaction, Mojtaba and his colleagues design a nonlinear model reference adaptive bilateral impedance controller for a multi-DOF upper limb tele-rehabilitation system. The stability is analyzed with model uncertainties of the nonlinear tele-operation system and experiments are performed to verify the efficiency of their proposed methods. In [19], researchers study a novel bilateral adaptive control for the underwater tele-operation manipulator. A boundgain-forgetting (BGF) based model reference adaptive impedance control is used to track the force of the slave manipulator. Both numerical simulations and real experiments show the efficiency of the controller. For the special underwater environment, the tele-operation system is extremely difficult to design and realize due to many type of disturbances. Meanwhile, it is vitally important to ensure the transparency and the stability of both the master and the slave manipulators in a tele-operation system. In order to solve the problem, we study the reflected force from the slave side in the complex underwater tele-operation system in this paper. Furthermore, due to the dynamic, variable, and fluid characteristics of the underwater environment, we propose a novel bilateral impedance control for the underwater tele-operation system. In the novel bilateral impedance control, an adaptive neural fuzzy inference system disturbance observer-based algorithm is introduced to the bilateral impedance control method which has advantages to restrain the internal model uncertainties as well as to inhibit the external disturbances. The rest of the paper is organized as follows. The problem formulation is briefly illustrated in Section 2. The bilateral underwater teleoperation dynamic model is introduced in Section 3. In Section 4, the bilateral impedance control of the underwater tele-operation system is explained in detail with stabilized analysis. Numerical simulations are showed in Section 5. Some conclusions are given in the conclusion part. 2. The problem formulation Normally, in the underwater tele-operation system, the human operator is located in the manned submarine above the water, as shown in Fig. 1. Driving the master manipulator, the human operator may control the underwater slave manipulator under to achieve the desired operations. Normally, an umbilical cord is used to supply the power and data to the master side. The communication between the master side and the slave side is implemented by cables or wireless between the surface and the deep of the water [20]. The communication among three parts (the human operator, the master and the slave) may occur the time delay in the complex underwater environment. Due to the fluid, dynamic and disturbances in the water, the master has to obtain the actual force of the slave manipulator so as to execute appropriate operations for the sake of the safety and the economic. Therefore, the stability and the transparency are essential for the bilateral control of the underwater teleoperation system. In addition, the effects of the time delay must be considered since the communication between the ROV and the underwater manipulator is generally difficult.

Fig. 1. The schematic of the bilateral underwater tele-operation system.

The communication channel with time delay can be described as a Lebesgue measurable function T (t) ∈ Rn (t) which is bounded D′ ≤ T (t) ≤ D, t ≥ 0 and differentiable T˙ (t) ≤ d, ∀t > 0, where D′ , D and d are all constants. According to the above assumption, states of the master and the slave after time delays are respectively defined as follows,

{

q˙ md = q˙ m (t − T (t))(1 − T˙ (t)) = qmd (1 − d) q¨ sd = q¨ s (t − T (t))(1 − T˙ (t))2 = qsd (1 − d)2 ,

(1)

where qmd and qsd respectively refer to the delayed joint angles of the master and the slave manipulators due to the communication channel. q˙ md and q˙ sd stand respectively for the delayed joint angular velocities of the master and the slave manipulators. Similarly, q¨ md and q¨ sd represent angular accelerations of the master and the slave manipulators. 3. The dynamic equation and the impedance model of the underwater tele-operation system 3.1. The dynamic equation of the underwater tele-operation system Assuming both the master and the slave side take n-DOF (Degree of Freedom) manipulators, the bilateral dynamic model of the bilateral underwater tele-operation can be written as follow, Mm (qm (t)) q¨ m (t) + Cm (qm (t), q˙ m (t)) q˙ m (t) + Gm (qm ) = T um (t) − Jm (qm )T (fh + Fˆs ),

(2)

Ms (qs (t)) q¨ s (t) + Cm (qs (t), q˙ s (t)) q˙ s (t) + Gs (qs ) = us (t) − JsT (qs )(fe ),

(3)

where Mi ∈ Rm×n , Ci ∈ Rm×n , Gi ∈ Rm×n , i = m, s are respectively inertial matrices, the Coriolis and centrifugal effects matrices and the gravitational matrices of the master and the slave manipulators. qi , q˙ i , q¨ i ∈ Rn×n are angles, angular velocities and angular accelerations of the master and the slave manipulators in the joint space. um and us are control inputs of the mater and the slave manipulators. fh is the master’s force exerted by the human operator. fe is the contact force between the slave and the environment. Fˆs is the force transmitted from the slave side to the master side. The nonsingular and nonredundant Jacobian matrix Ji (qi ) represents the relations between the joint space and the work space of manipulators.

T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

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Fig. 2. The schematic of the bilateral impedance control of the underwater tele-operation system.

3.2. The impedance model of the bilateral underwater tele-operation The impedance model of the bilateral underwater teleoperation system is written as follow, Hm x¨ m (t) + Bm x˙ m (t) + Km xm = fh + Fˆs

(4)

Hs x¨ s (t) + Bm x˙ s (t) + Ks xs = fe ,

(5)

where Hi , Bi and Ki are respectively inertial, damping and stiffness matrices of the master and the slave sides. xi , x˙ i , x¨ i ∈ Rn×n are positions, velocities and accelerations of the master and the slave manipulators in the work space. The manipulators’ states can be transformed from the task space in Eqs. (3) and (4) to the joint space in Eqs. (1) and (2) by following relations, x˙ i = Ji (qi )q˙ i and x¨ i = J˙i (qi )q˙ i + Ji (qi )q¨ i , i = m, s [19]. As same as the general tele-operation system, the underwater tele-operation system has following properties. (1) The inertia matrices Mi and Hi are symmetric positive definite matrices, Mi = MiT , Hi = HiT , which have upper and lower boundedness. That is, Mdown I ≤ |Mi | ≤ Mup I, Hdown I ≤ |Hi | ≤ Hupp I where Mdown , Mup , Hdown , Hup are positive constants. ˙ i (qi ) − 2Ci (qi , q˙ i ) are skew-symmetric matrices. (2) M (3) For all qi (t), q˙ i (t) ∈ Rn×1 , there exists a positive scalar ci which satisfies following relations Ci (qi (t), q˙ i (t)) ≤ c3 |˙qi | (1 + |qi |) ≤ c3 |˙qi |, where c1 , c2 , c3 > 0 and | · | represents the Euclidean matrix norm. (4) The linear parameterizable underwater tele-operation dynamic model (in Eq. (1)) may be rewritten as Mi (qi (t))q¨ i (t)+ Ci (qi (t), q˙ i (t))q˙ i (t) + Gi (qi (t)) = Yi (qi (t), q˙ i (t), q¨ i (t))θi , where Yi (qi (t), q˙ i (t), q¨ i (t)) ∈ Rn×p are a certain function and θi ∈ Rp are physical parameter vectors of the master and the slave manipulators. 4. The bilateral impedance control of the underwater teleoperation system 4.1. The schematic of the bilateral impedance controller The schematic of the bilateral impedance control of the underwater tele-operation system is illustrated in Fig. 2. The human operator executes the master manipulator to achieve the desired trajectory so as to accomplish the task. Herein, it must be noted that the reflected force Fˆs is considered so that the human operator may get the slave force as quickly and accurately as possible in the complex and dynamic underwater environment. In this paper, the reflected force after the communication channel is

defined as Fˆs = kfs q¨ sd . Normally, the communication is worse in the underwater environment than other environment due to the unpredictability, uncontrollability and the large time delay. Therefore, the desired trajectory of the task is programmed by the computer emitting to both the master side and the slave side. The precise parameters of the underwater impedance model are difficult to depict due to the fluidity and the anisotropy. Meanwhile, the fluid acts on the slave manipulator during the whole operating time and idle time. However, in the small fixed local area, the applied force of underwater fluid to the slave manipulator may be approximately regarded as the resistance opposite to its direction of motion. That is, the force of the environment is calculated as fe = kfe q˙ md . The underwater environment is extremely complicate, including many kinds of underwater animals, plants, underwater vehicles and so on. The ocean current, the wake of large ships may increase the difficulty of the underwater operation. Therefore, the underwater environment is extremely complex and full of environmental disturbances. Besides guaranteeing the safety of the operating objects, the impedance control has to be adaptive to the dynamic environment as well as has to be stable to restrain the disturbances. Therefore, a bilateral ANFIS-DOB control is introduced so as to achieve the stability and the adaptivity of the underwater tele-operation system. 4.2. The adaptive neural fuzzy inference system structure The adaptive neural fuzzy inference system (ANFIS) tunes neural networks and fuzzy inference systems. The neural network of the master/slave ANFIS structure is illustrated in Fig. 3. The master ANFIS system has inputs qm , q¨ sd and the unique output qmy . The slave ANFIS system has inputs qs , q¨ md and the unique output qsy . Use the fuzzy inference as , If qi is Ai1 and q¨ sd /˙qmd is Ai2 then qiy is ωci , i = 1, . . . , n, where i is the index of the rule. Aij is a fuzzy set for ith rule and jth variables. ωci is a real number that represents a consequent part. Taking Gaussian function hij = exp

−(qj −aij )

2

2b2ij

as the membership function and T-

norm as the product operator, the output of the neural network is expressed as follow, qiy =

∑n i=1 hi wci ∑ n i=1

hi

h1

= ∑n

i=1 hi

h2

wc1 + ∑n

= W T H(q, aij , bij ),

i=1 hi

hn

wc2 + · · · + ∑n

i=1

hn

wcn (6)

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T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

optimal cluster radius b∗ij , rendering the ANFIS network to satisfy |ˆuil (k) − ul (k)| < ε, bij = b∗ij . The bilateral inverse models of the bilateral ANFIS network are stable. Then, the ANFIS-DOB structure of both the master and the slave side is illustrated in Fig. 5. In Fig. 5, Gi (s) is defined as transition functions of the master and the slave system, and Pi (s), (i = m, s) is assumed as the inverse matrices. Thus, outputs of the master/slave ANFIS-DOB are calculated as follows,

[

]

Y1i (s) Y2i (s)

[

G11i (s) G21i (s)

=

G12i (s) G22i (s)

][

U1i (s) U2i (s)

]

[ +

D1i (s) D2i (s)

]

,

(8)

where Dexi (s) represent external disturbances while Ni (s) denotes ˆ i (s) indicates the estimanoises in the underwater environment. D tions of the external disturbances. Through a low-pass filter Q (s), ˆ fi (s). Then, estimations of the estimation signals are changed to D underwater disturbances and noises may be written as

Fig. 3. The ANFIS structure.

[

]

ˆ 1i (s) D ˆ 2i (s) D

[

Q1i (s)P11 (s) Q2i (s)P21 (s)

[

Qs (s)U1i (s) Qs (s)U2i (s)

= −

Q1i (s)P12i (s) Q2i (s)P22i (s)

][

Y1i (s) Y2i (s)

]

] ,

(9)

and the feedback control law of ANFIS-DOB is expressed as,

[

Fig. 4. The lth channel’s ANFIS pseudo system.

where W = [wc1 , wc2 , . . . , wcn ]T and H(q, aij , bij ) = [ ∑n 1 h

h2

∑n

i=1 hi 1 (q iy 2

, . . . , ∑n

i=1 hi

hn

qdiy )2 ,

,

T

i=1 hi

] . The value function Vz is defined as V (z) =

− in which qiy and qdiy are the neural network output and the desired output. The center and width parameters of the Gaussian and wci may be updated by the nearest neighbor clustering learning algorithm aiming at minimizing the V (z).

]

U1i (s) U2i (s)

[ =

C1i (s) C2i (s)

]

[ −

ˆ 1i (s) D ˆ 2i (s) D

]

.

(10)

The disturbance estimation error of the MIMO ANFIS-DOB is ˆ 1i (s), Ed2i (s) = D2i (s) − Dˆ 2i (s). defined as Edi (s) = D1i (s) − D We suppose that the filter Qi (s) may establish lims→0 Q (s) = 1. The nominal closed-loop system is assumed to be stable. In addition, state values of disturbances are assumed to be stable, limt →∞ d1i (t) = lims→0 sD1 (s) < ∞, limt →∞ d2i (t) = lims→0 sD2 (s) < ∞. Due to above assumptions, it is easy to deduce that the control law Eq. (10) may asymptotically render the disturbance estimation errors tend to 0. The integral system can asymptotically suppress external disturbances in the underwater environment. It may be concluded that disturbances of the system can be inhibited by choosing appropriate Qi (s). The master and the slave manipulator has the same structure so that H, B, K are observational values of the bilateral impedance model Hi , Bi , Ki , i = m, s. The output after the inverse ANFIS system may be written as follow.

˜ T H(x), F = H(x¨ d − k1 e˙ − k2 e) + Bx˙ + Kx + D + η + W

(11)

d

4.3. The bilateral impedance ANFIS-DOB controller The design of the bilateral impedance ANFIS-DOB controller involves two parts, the bilateral ANFIS pseudo systems and the bilateral ANFIS-DOB impedance controller (Fore more details, please refer to [21]). Firstly, the bilateral ANFIS pseudo system is exhibited in Fig. 4. The output of the ANFIS pseudo system is written as,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

yil (k) =fil (yi (k − 1), . . . , yil (k − ny ), uil (k − T ), . . . , uil (k − T − nu ))

ηil (k) =yil (k) + βil (uil (k − T )), ⎪ ⎪ ⎪ ˆ u (k − T ) =gil (ηil (k), ηil (k − 1), . . . , ηil (k − ny ), il ⎪ ⎪ ⎩ uil (k − T − 1), . . . , uil (k − T − nu ))

(7)

where uil (k), yil (k) represent the input and the internal state of the bilateral ANFIS pseudo system. ηil (k) indicates the output. We suppose that the master manipulator is the same as the slave manipulator. Thus, the master system’s order ny and nu are equal to the slave system. According to [22], the ANFIS pseudo system has following properties. If there exists an certain ε , and an input–output data pair (uil , yl), i = m, s, there must have an

where x is the desired angle calculated from path planning, and e = x − xd , e˙ = x˙ − x˙d . k1 and k2 are positive parameters. D is defined as D = Fˆs + fh in the master side, while D is set to D = fe in the slave side. η denotes the ideal approximation error of the inverse ANFIS system. η is assumed to be bounded to η0 , and η0 = sup∥η∥. W ∗ represents the optimal approximation ˆ , where weight of the ANFIS system and its estimation value is W ˜ = W∗ − W ˆ . Since both of W ˆ and W ∗ are bounded, we may W acquire ∥W ∗ ∥F ≤ Wmax , where Wmax is also bounded. Substituting E = [e, e˙ ]T and Eq. (11) into the bilateral impedance model Eqs. (3) and (4), we have [22]

˜ H(x)), E˙ = AE + B(η + W

[

1 ˜ → 0, , B = Hi−1 . It is obvious that W k2 η → 0, as t → ∞, finally resulting E → 0. Thus, the Lyapunov function is constructed as 1 1 ˜ TW ˜ ), γ > 0 . tr(W (13) V = E T PE + 2 2γ where A =

0 k1

(12)

]

The derivation of the Lyapunov function is written as follow. V˙ =

1 2

(E T P E˙ + E˙ T PE) +

1

γ

˙

˜ TW ˜) tr(W

T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

5

Fig. 5. The master and the slave ANFIS-DOB structure.

=

1 2

matrices in the dynamic function are written as follows. Both of the master and the slave side take PID motion controls.

˜ T H(x))) + E T AT + (η [E T P(AE + B(η + W

˙˜ T W ˜ T Φ T (x)BT PE)] + tr(W ˜) +W 1

= − E QE + η B PE + T

T

T

2

1

γ

˙˜ T W ˜ +W ˜ ), tr(γ B PEH (x)W T

T

M11 = (m1 + m2 )l21 + m2 l22 + 2m2 l1 l2 cos(qi2 ), M22 = m2 l22

(14)

˜ BT PE = tr(BT PEH T (x))W ˜, where Q is a positive matrix and H T (x)W AT P + PA = −Q . ˙ˆ ˆ ). SubDesign the adaptive law as W = γ (H(x)E T PB − ∥E ∥W

˙ˆ ˜˙ = −W stituting the adaptive law and W into Eq. (14), we have 1 ˆ TW ˜ ). Due to F-norm’s properties, the V˙ = − 2 E T QE + ∥E ∥tr(W ∑ 2 following relations may be easily acquired, ∥W ∥2F = ij |ω| = ˜ T (W − W ˜ )] ≤ ∥W ˜ ∥F ∥W ∥F − tr(WW T ) = tr(W T W ), and tr [W ˜ ∥2 . Supposing λmin (Q ) and λmax (P) are the minimum and the ∥W F maximum values of eigenvalues of matrices Q and P, the Eq. (14) may be rewritten as 1 ˜ ∥F ∥W ∗ ∥F − ∥ W ˜ ∥2F ) + ηT BT PE V˙ ≤ − E T QE + ∥E ∥(∥W 2 1 2 ˜ ∥F − Wmax )2 − 1 Wmax ≤ −∥E ∥( λmin (Q )∥E ∥ + (∥W − η0 λmax (P)). (15) 2 2 4

λ ∥ ∥ ≥ −η λ

−η λ

Wmax 2 1 2 If 12 min (Q ) E Wmax ) 0 max (P) or ( W F 4 2 1 2 Wmax 0. That is, E 0 max (P) may be established, the V 4 2 1 1 2 2 ( (P) Wmax ) or W F Wmax 0 max (P) λmin (Q ) 0 max 4 4 Wmax is satisfied. It revealed that the proposed method can guar2

η λ

+

∥˜∥ − ≥ ˙ < ∥ ∥ ≥ √ ∥˜∥ ≥ +η λ +

antee the asymptotic stability of the system and the boundedness of the weight. That is, the boundedness of system variables are ensured. Selecting large eigenvalues of the matrix Q results small eigenvalues of the matrix P. The upper bound of the ANFIS system’s modeling error becoming smaller, the Wmax is smaller, so that E is also small. Therefore, the desired positions of the path planning trajectory may be efficiently adjusted by impedance control. 5. Numerical simulations and result

In the numerical simulations, both the master side and the slave side take the same two-link, three joints manipulators, as shown in Fig. 6. The inertia, Coriolis and Centrifugal, Jacobian

M12 = M21 = m2 l22 + m2 l1 l2 cos(qi2 ), C11 = −m2 l1 l2 sin(qi2 )q˙ i2 , C22 = 0 C12 = −m2 l1 l2 sin(qi2 )(q˙ i1 + q˙ i2 ), C21 = m2 l1 l2 sin(qi2 )q˙ i2 g1 = (m1 + m2 )l1 g cos(qi2 ) + m2 l2 g cos(qi1 + qi2 ), g2 = m2 l2 g cos(qi1 + qi2 ) J11 = −l1 sin(qi1 ) − l2 sin(qi1 + qi2 ), J22 = l2 cos(qi1 + qi2 ) J12 = −l2 sin(qi1 + qi2 ), J21 = l1 cos(qi1 ) + l2 cos(qi1 + qi2 )

Parameters of manipulators are set as follows. mi1 = mi2 = 1 kg, l1 = l2 = 1 m. The gravitational parameter takes g = 9.8 N/S2 . Parameters of the impedance models are set as H = diag(1, 1), B = diag(10, 10). Initial positions are located at q1 (0) = q2 (0) = π3 , q˙ 1 (0) = q˙ 2 (0) = 0. Parameters of ANFISDOB are chosen as follows, bij = 0.3, ρi = 0.5, ςi = 0.001, 1 1 Qs = [ (0.005s+1)(0 , ], D(s) = [1, 1]. Time .001s+1) (0.003s+1)(0.002s+1)

delays are set as Ti (t) = |Xi (t)|, (i = 1, 2), where Xi is a normal random distribution variable with Tv as mean, and δ as the standard deviation, that is, Xi (·) ∼ N(0.1, 0.01). Setting the force signal as Fh = 10(1 + sin(0.1t)) along the x-direction, results are displayed from Figs. 7 to 10. The position tracking is showed in Fig. 7(a), in which positions of the master manipulator and the slave manipulator are denoted by blue dots and black lines. The desired position acquiring from the path planning module is marked with red lines. From the results, the motion control always get a good tracking results. Results of two joints’ position tracking of the master and the slave side are depicted in Fig. 7(b). Angular velocities of the master and the slave manipulators are showed in Fig. 8(a). In general, no matter what method is used, trajectory tracking can always achieve satisfactory results than the force tracking control. The estimation of external disturbances and adaptive parameters used to suppress the internal model uncertainties are respectively illustrated in Fig. 8(b) and Fig. 9(a). The force tracking is exhibited in Fig. 9(b). Inputs of the master and the slave sides are demonstrated in Fig. 10(a). In Fig. 10(b), we also compare the prosed method (blue dash lines) with other methods, such as PID bilateral impedance control (marked with green lines) and RBF bilateral impedance control (marked with black dots). From

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T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

Fig. 6. The master and the slave ANFIS-DOB structure [19].

Fig. 7. Position tracking results of manipulators.

Fig. 8. Speed tracking results and the approximation of the underwater tele-operation system.

results, the RBF neural network(NN) may update parameters as soon as possible and it has advantage of nonlinear systems, but

it is deficient of the stability with time delay. The PID method are more stable than the NN, but it is difficult to adapt to the

T. Wang, Y. Li, J. Zhang et al. / Applied Soft Computing Journal 91 (2020) 106194

7

Fig. 9. Force tracking results.

Fig. 10. Control inputs and comparisons with other control methods.

complex underwater environment. From results, it shows that the proposed method exhibits its superior performance in three methods. 6. Conclusion The paper introduces an ANFIS-DOB impedance control to an underwater tele-operation system. The purpose is to make sure the synchronization of force tracking without losing the stability of motion. Since the underwater environment is complex and dynamic, the time delay and the reflected force are taking into account in the dynamic model. A prominent advantage of the ANFIS-DOB is that it may suppress internal model uncertainties and external disturbances at the same time. The disturbances generated by the complex dynamic models are constrained and observed via the learning algorithm of the ANFIS. At the same time, the heavy effects of the external perturbations can be greatly reduced by the DOB control. Aiming to ensure the synchronization force tracking of the slave manipulator, the bilateral impedance controller is designed based on the ANFISDOB method. Concerning of the time delay, the stability of the underwater tele-operation system is analyzed by the Lyapunov

function. Numerical simulations are performed to verify the proposed bilateral impedance control. Comparing with other methods, it may be concluded that the bilateral ANFIS-DOB impedance controller are more stable and more efficient to increase the transparency. Therefore, for the underwater tele-operations, the bilateral ANFIS-DOB impedance control may accomplish more accurate tasks. Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2020.106194. CRediT authorship contribution statement Ting Wang: Conceptualization, Methodology, Software, Writing - original draft. Yujie Li: Data curation, Writing - review & editing. Jianjun Zhang: Writing - review & editing, Investigation. Yuan Zhang: Software, Validation.

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