A Time Delay Controller included terminal sliding

Ocean Engineering 107 (2015) 97–107

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

A Time Delay Controller included terminal sliding mode and fuzzy gain tuning for Underwater Vehicle-Manipulator Systems Hossein Nejatbakhsh Esfahani a, Vahid Azimirad a,n, Mohammad Danesh b a b

School of Engineering Emerging Technologies, University of Tabriz, Tabriz, Iran Department of Mechanical Engineering, Isfahan University of Technology, 84156-83111 Isfahan, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 12 January 2013 Accepted 22 July 2015

An improved Time Delay Control (TDC) method for an Underwater Vehicle-Manipulator System (UVMS) is proposed. The proposed controller consists of three terms: a time-delay-estimation term that cancels nonlinearities of the UVMS dynamics, a Terminal Sliding Mode (TSM) term that provides a fast response and a PID term that reduces the tracking error. In addition the proposed controller uses fuzzy rules to adaptively tune the gains of TDC and PID terms. A full dynamic of UVMS is presented and some simulation studies are done. Simulation results demonstrate the effectiveness of proposed method. The proposed controller not only provides the high performance in trajectory tracking, but it also has an acceptable robustness in presence of the external disturbance and unknown forces/torques in comparison with conventional sliding mode controller. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Underwater Vehicle-Manipulator System Time Delay Control Terminal sliding mode Fuzzy logic

1. Introduction Recently underwater manipulation has been developed because of needs in marine industries. Underwater VehicleManipulator Systems (UVMS) are used in oceanic environments for tasks such as welding, nondestructive test of the marine structures or underwater oil/gas pipes and ocean exploration. The UVMS consists of a mobile platform such as Autonomous Underwater Vehicle (AUV) and one or more manipulators. In the presence of hydrodynamic disturbances and the dynamic coupling between manipulator and AUV, the trajectory control of the UVMS is complicated. The hydrodynamic effects on the underwater manipulators have been studied recently through aproximated generalized torques and an indirect adaptive control method (Jun et al., 2011; Mohan and Kim, 2012), but they do not guarantee the robustness for systems like UVMS with high uncertainties. In the oceanic conditions, the real hydrodynamic disturbances acting on the UVMS are difficult to measure or estimate accurately. Therefore, robust, precise and fast convergence control scheme should be developed for the appropriate operation of UVMS. Sarkar and Podder (2001) proposed a method to generate the desired trajectory for reducing drag force acting on the whole system. An adaptive tracking control based on the virtual decomposition has also been proposed (Antonelli et al., 2004). Sun and Cheah (2004) designed an adaptive set point controller based on

n

Corresponding author. Tel.: þ 98 413 339 3854; fax: þ 98 413 329 4626. E-mail address: [email protected] (V. Azimirad).

http://dx.doi.org/10.1016/j.oceaneng.2015.07.043 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

Lyapunov's direct method and Lassalle's invariance principle. Some researchers have proposed methods based on estimation of the dynamics models of the UVMS (Canudas et al., 2000; Antonelli, 2003). Some researches about control of UVMS focused on fuzzy controllers. For avoiding the problems of gain tuning, Xu et al. (2006) designed a sliding mode fuzzy controller, in which inputs of the fuzzy controller are error and derivative of error, respectively. Furthermore, a PD sliding surface is used. Piltan et al. (2011a) showed that a PID sliding surface has a better performance in comparison with PD sliding surface. Amer et al. (2012) designed an adaptive fuzzy sliding mode controller which was consisting of three main parts: the equivalent control of SMC as a model-based control term; a new estimation technique to estimate the perturbation from the dynamics of the sliding surface; and a twosupervisory fuzzy self-tuning controller. A mathematical tunable gain model free PID-like sliding mode fuzzy controller (GTSMFC) is designed by Piltan et al. (2011b) to achieve the best performance. A simplified fuzzy logic controller for an underwater vehicle is proposed by Ishaque et al. (2011). They showed that Mamdani and Sugeno type fuzzy logic controller give identical response to the same input sets. However their method is not robust in presence of ocean disturbances. Esfahani et al. (2014) studied the sliding mode PID fuzzy controller for underwater manipulators without considering time delay terms. Also, Fuzzy Logic as an incorporated section of main controller is used in control of some underwater robotics manipulators (Sebastian and Vázquez, 2006; Sebastian and Sotelo, 2007; Song and Smith, 2006). However, being sensitive to disturbances

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(as one of the main parameters in oceanic conditions), these methods should be used only in laboratorial conditions. In other words they are not robust and so these controllers could not be used in oceanic conditions. Multiple Impedance Control (MIC) is proposed for dual arm UVMS in Farivarnejad and Moosavian (2014). This controller is an effective controller, especially when the end-effector interacts with the environment but has not proper robustness in the presence of hydrodynamic disturbances. An optimized robust controller was proposed in Thangavel et al. (2010) based on bond graph method. Several researchers have studied Time Delay Control (TDC) for different applications (Park and Kim, 2011; Cho et al., 2005; Lim et al., 2010; Xu and Yang, 2011; Han et al., 2011). But this method is not applied on UVMS. Although lots of works are done in this area, still there is not complete model of UVMS considering full dynamic terms and proper controller for it. Generally, none of previous works consider full dynamic of UVMS, Time Delay Controller and terminal sliding mode. In this study, an improved robust TDC is designed. It proposes a new method for control of UVMS through adding two concepts, a PID-Sliding Mode Controller (SMC) and a fuzzy controller for gain tuning of SMC. The aim is to reduce tracking error and increase the speed of converging to the desired path in comparison with the conventional TDC. This paper is outlined as follows. In Section 2, general form of UVMS is modeled considering full dynamic terms. The conventional TDC is presented in Section 3. An improved robust and accurate TDC is proposed in Section 4. The computer simulation results are shown in Section 5 and finally, the conclusion is presented in Section 6.

Joint

a

α

α

d

θ

1 2 3 4 5

0 0 0 dv ll

 90 þ 90 0 þ 90  90

 90 þ 90 0 þ 90  90

0 0

qx qy qz q1 q2

UVMS is an underwater vehicle with a robotic manipulator mounted on it. Fig. 1 illustrates a two-Degrees Of Freedom (2 DOF) manipulator mounted on an underwater vehicle. The coupled effect between the manipulator and the vehicle should be considered in the modeling. The motions of underwater vehicle consist of three linear and three revolute motions. Three linear motions are along the x, y and z axes that are named surge, sway and heave respectively. Additionally, three revolute motions are about x, y and z axes that are named roll, pitch and yaw, respectively. The DenaviatHartenburg parameters for UVMS with considering only revolute motions are shown in Table 1. Fig. 2 illustrates a 7 DOF Underwater Vehicle-Manipulator System.

2.1. Energy of UVMS

lv 0 0

Fig. 2. UVMS with only revolute motions.

Kinetic energy of the UVMS due to rigid body is expressed as Eq. (2) 1 1 1 1 T RB ¼ md x_ 2c1 þ ml x_ 2c2 þ I 1 q_ 21 þ I 2 q_ 22 2 2 2 2 1 1 1 1 þ I z q_ 2z þ I y q_ 2y þ I x q_ 2x þ mv ðx_ 2v þ y_ 2v þ z_ 2v Þ 2 2 2 2

2. Dynamics of Underwater Vehicle-Manipulator System

ð2Þ

In the above equation, xc1 and xc2 are mass centers of the first link (dv ) and the second link (ll ) of manipulator, respectively. x_ v ,y_ v and z_ v are the linear velocities of the spherical vehicle along the x, y and z axes, respectively and I 1 , I 2 are the moment of inertia of the manipulator. Also, I x , I y and I z are the moments of inertia of spherical vehicle that are expressed as Eq. (3) (Sun and Cheah, 2003) Ix ¼ Iy ¼ Iz ¼

8 πρl5v 15

ð3Þ

where ρ is the water density and lv is the radius of spherical vehicle. Kinetic energy of the spherical vehicle due to added mass is 1 T A ¼ vTv M Av vv ; vv ¼ ½x_ v ; y_ v ; z_ v ; q_ z ; q_ y ; q_ x T 2

ð4Þ

where M Av is the inertia matrix of the spherical vehicle that is expressed as following (Sun and Cheah, 2003):

Added mass of an Underwater Vehicle-Manipulator System is included in the system dynamics in the form of kinetic energy. Neglecting the kinetic energy of the manipulator due to added mass, total kinetic energy of the UVMS is written as Eq. (1) T ¼ T RB þ T A

Table 1 Denaviat-Hartenburge parameters of the UVMS.

ð1Þ

2 3 2 3 2 3 M Av ¼ diagð πρlv ; πρlv ; πρlv ; 0; 0; 0Þ 3 3 3

ð5Þ

2.2. Lagrange formulation Consider the UVMS as an 8 DOF dynamic system. The Lagrange equation of motion in the matrix form is (Korayem et al., 2010)

 d ∂T ∂T  ¼ Q ¼ ½F 1 ; F 2 ; F 3 ; τ4 ; τ5 ; τ6 ; τ7 ; τ8 T ð6Þ dt ∂q_ ∂q

Fig. 1. Underwater Vehicle-Manipulator System (UVMS).

In which, T is the total kinematic energy of the system which is obtained from Eq. (1) and Q is the vector of generalized forces and torques applied to the UVMS. Also, q is vector of generalized positions and q_ is the vector of generalized velocities. The vector of

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99

Fig. 3. Block Diagram of the Time Delay Control System.

Fig. 6. Output membership function.

Table 2 Fuzzy rule base.

S_

S NB

NS

ZE

PS

PB

N Z P

B B B

B M S

M S M

S M B

B B B

Fig. 4. The Block Diagram of the Proposed Control System.

Fig. 7. Control surface of kf uzz .

Fig. 8. UVMS with 4 DOF. Fig. 5. Input membership function, (a) Input variable s(t) and (b) Input variable . s(t).

generalized positions is shown as Eq. (7) q ¼ ½xv ; yv ; zv ; qz ; qy ; qx ; q1 ; q2 T

ð7Þ

Substituting the above expression in Eq. (6), the dynamic equations of motion which includes rigid body and added mass is obtained as follows: _ Q ¼ MðqÞq€ þ Cðq; qÞ

ð8Þ

2 of the manipulator (Shinohara, 2011): Z dv ρ D1 ¼ cd dia1 v21 dx1 2 0

ρ

D2 ¼ cd dia2 2

Z 0

ll

v22 dx2

ð9Þ

ð10Þ

Where dia1 , dia2 and cd are the diameter of link 1, the diameter of link 2 and the drag coefficient, respectively. v1 ; v2 are translational velocities of links. The second part (Dv) is the drag force applied on the underwater vehicle (Sun and Cheah, 2003) Dv ¼ diagðdt j x_ v j ; dt j y_ v j ; dt j z_ v j ; d1 j q_ z j þ d2 ; d1 j q_ y j þ d2 ;

2.3. Drag force

d1 j q_ x j þ d2 Þ½x_ v ; y_ v ; z_ v ; q_ z ; q_ y ; q_ x T

Drag force in UVMS is divided into two parts. The first part is derived from following equations with regarding to link 1 and link

Where dt , d1 and d2 are translational quadratic damping factor, angular quadratic damping factor and angular linear damping

ð11Þ

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Table 3 Parameters of the UVMS. Added mass for 4 DOF UVMS Constant coefficient for drag force equations mv mass of spherical vehicle ml mass of link lv radius of spherical vehicle ll length of link E1 ¼ 2 kg; E2 ¼ 2 kg E3 ¼ 1 kg; E4 ¼ 1 kg

F 1 ¼ 5; F 2 ¼ 5

F 3 ¼ 1; F 4 ¼ 1

100 (kg)

factor respectively. Therefore, the total drag force applied on UVMS is conveyed as Eq. (12)  T Dðq; q_ Þ ¼ Dv D1 D2 ð12Þ 2.4. Gravitational and buoyant forces Buoyancy force is equal to the weight of the fluid displaced by link/body and acts through the center of buoyancy of the link/ vehicle (Fossen, 1994). It is in the opposite direction of gravitational force (Eq. 13) F U ðqÞ ¼ GðqÞ  BðqÞ ¼ mg  ρgv

ð13Þ

Where v is the volume of the link/vehicle. Therefore, this force affects the heave motion of vehicle and link 2. Through calculating the force (F U ðqÞ) in all directions, the potential energy (u) is achieved. Therefore the matrix hðqÞis expressed as Eq. (14) d ∂u ∂u ∂u hðqÞ ¼  ð Þ þ ¼ dt ∂q_ ∂q ∂q

ð14Þ

in which hðqÞ ¼ ½h1 ; h2 ; h3 ; h4 ; h5 ; h6 ; h7 ; h8 T

ð15Þ

In the Eq. (15), all of the components are zero except h3 and h8 . h3 ¼ gmv  g ρvv

ð16Þ

h8 ¼  0:5ll ml g cos ðq2 þ qy Þ þ 0:5g cos ðq2 þ qy Þρvl ll

ð17Þ

In which, vv and vl are volume of the spherical vehicle and the cylindrical link, respectively. 2.5. Final dynamic equation of UVMS By using Eqs. (8), (12) and (14) the final form of the dynamic equations of the UVMS is derived as _ þ Dðq; q_ Þ þ hðqÞ Q ¼ MðqÞq€ þCðq; qÞ

ð18Þ

3. Time Delay Controller Suppose the reference input trajectory is denoted by qd , then the control objective is to track qd . Eqs. (19–21) define the tracking error and its first and second derivatives as e ¼ qd  q

ð19Þ

e_ ¼ q_ d  q_

ð20Þ

e€ ¼ q€ d  q€

ð21Þ

in which, q is the position vector. In Eq. (18) let MðqÞ be approximated with MðqÞ as a constant matrix. Hence, Eq. (18) is rewritten as Eq. 22 € τ ¼ Mq€ þ Hðq; q_ ; qÞ

ð22Þ

Where _ þ hðqÞ þ Dðq; q_ Þ H ¼ ðM  MÞq€ þCðq; qÞ

ð23Þ

20 (kg)

1 (m)

1 (m)

According to the computed torque method, the input is expressed as ^ _ q€ Þ τ ¼ MU þ Hðq; q;

ð24Þ

U ¼ q€ d þkD e_ þ kp e

ð25Þ

^ kD and kp are the estimated value of H, the derivative and where H, the proportional gain matrices, respectively. Substituting Eqs. (24), (25) into Eq. (22), the target error dynamic equation is e€ þ kD e_ þkp e ¼ 0

ð26Þ

In the TDC approach, with assuming that the time delay L is very small,HðtÞ could be approximated as Eq. (27) (Cho and Chang, 2005) ^ HðtÞ ¼ Hðt LÞ ¼ HðtÞ ¼ τðt  LÞ  M q€ ðt  LÞ

ð27Þ

Combining Eqs. (24), (25) and (27), the TDC law is obtained as Eq. (28)

τ ¼ τðt  LÞ  Mq€ ðt  LÞ þ Mðq€ d þ kD e_ þ kp eÞ

ð28Þ

€  LÞ is given by numerical Incidentally, the past acceleration qðt differentiation as Eq. (29) q€ ðt  LÞ ¼

qðtÞ  2qðt  LÞ þ qðt  2LÞ

ð29Þ

L2

According to Eq. (23), to guarantee the stability of control system, the constant matrix M should be designed as follows: M ¼ γ I; 0 o γ o 2α

ð30Þ

where α is the minimum bound of eigenvalues (λi ) of M(q) for all q. Fig. 3 shows the TDC for UVMS.

4. Proposed controller To improve the performance of the proposed controller, two terms are added to the obtained the control law in Eq. (28). The first term is Terminal Sliding Mode (TSM) which increases rate of the convergence to zero for tracking error (Jin et al., 2011). Furthermore, sliding mode gain and PID gains are tuned through fuzzy logic. The second term has the features of PID controller that leads to decreasing of the tracking error in transient and steady state. Therefore Eq. (28) is rewritten as follows:
 s τ ¼ τðt  LÞ  Mq€ ðt  LÞ þ Mðq€ d þ kD e_ þ kp eÞ þ k1 sat þ k2 s k1 ¼ N 1 kf uzz ; k2 ¼ N2 kf uzz ; s ¼ e_ þ λ1 eþ

λ

2 1

2

φ

Z e

ð31Þ

The saturation function for chattering reduction is used in control torques. In the above equation, the term k1 satðs=φÞ performs as the TSM. The term k2 s operates as a PID controller because sliding surface is selected in the form of proportional, integral and derivative of the error. k1and k2 are the control gains which are set according to the output gain of fuzzy controller (kf uzz ).N1 ; N 2 are the fuzzy gains normalizing factors. Fig. 4 illustrates modified TDC controller Block Diagram.

H. Nejatbakhsh Esfahani et al. / Ocean Engineering 107 (2015) 97–107

4.1. Fuzzy gain tuning of TSM and PID In the conventional TSM and PID, control gains depend on uncertainties. This dependence is a defect for complex dynamics system. To solve this problem a fuzzy controller is designed here. Furthermore using fuzzy logic, gains are tuned based on the

101

distance of the states to the sliding surface. The main advantage of fuzzy control is that the tracking error and control effort are reduced. The configuration of our TSM–Fuzzy–PID Control (TSMF– PID) scheme is shown in Fig. 4; it includes a sliding surface and a two-input single-output fuzzy controller in which Mamdani's fuzzy algorithm is used. It is commonly is used in systems with

Fig. 9. Simulation Results of SMC (saturation function), (a) Trajectory tracking, (b) Tracking error and (c) Input forces and torques.

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Fig. 9. (continued)

two inputs and one output. In fact, the Mamdani controller is closer to approximation expression for implementation of complex nonlinear controllers. kf uzz is the output of the fuzzy controller, which is determined by inference on input linguistic variables sðtÞ and s_ ðtÞ. The membership function of input linguistic variables and the membership functions of output linguistic variable are shown in Figs. 5 and 6, respectively. sðtÞ, s_ ðtÞ and kf uzz are decomposed into five, three and three fuzzy partitions, respectively. The fuzzy controller consists of four steps: fuzzification, rules evaluation, aggregation and defuzzification. The fuzzy rule base is given in Table 2 in which the following symbols have been used: NB: negative big; NS: negative small; ZE: zero; PS: positive small; PB: positive big; N: negative; Z: zero; P: positive; M: medium; B: big; S: small. These linguistic fuzzy rules are defined heuristically in the following form: RðlÞ : If sðtÞ is Al1 and s_ ðtÞ is Al2 then; kf uzz is Bl Where Al1 and Al2 are the labels of the input fuzzy sets and Bl is the labels of the output fuzzy sets. l¼ 1, 2…, 15 denotes the number of the fuzzy rules. Fuzzy implication is modeled by Mamdani's minimum operator, the conjunction operator is Min, the t-norm from compositional rule is Min and for the aggregation of the rules the Max operator is used. In this paper the centroid deffuzification method is used and calculated by the following equation: n P

z ¼ i ¼n1 P

ci μAi ðxÞμBi ðyÞ

i¼1

μAi ðxÞμBi ðyÞ

ð32Þ

The degree of membership function is chosen for the input and output variables based on (S.E. Shaffei, 2010). Increasing PID gains results in: (a) reducing the “reaching time to sliding surface” and (b) increment of the oscillations in the input torque around the

sliding surface. Therefore, if the gains are tuned based on the distance of the states to the sliding surface, more acceptable performance will be achieved. Hence, the degree of membership function for the input and output is taken to be variable. Variable gain may be a disadvantage because of more computational cost in comparison to fixed gain (which could be solved by using high performance computers). Furthermore, to increase the precision of tracking and sensitivity, the membership functions are considered thinner around zero point. The fuzzy control surface of the output kf uzz is shown in Fig. 7.

5. Simulation results In this section, the simulation results of the proposed controller, which is performed on a 4 DOF UVMS, are presented. In this case study for surge motion (xv ) and heave motion (zv ), sine and cosine desired trajectories with zero initial conditions are chosen respectively. The trajectory generates a circular path in x–z plane for vehicle. The desired orientation of vehicle in pitch direction (qy ) and desired path of q2 is taken to be as Eq. (33) qy ¼ q2 ¼ π ð1  e  t=2 Þ

ð33Þ

The vector of positions is shown in Eq. (34) q ¼ ½xv ; zv ; qy ; q2 T

ð34Þ

Fig. 8 illustrates 4 DOF Underwater Vehicle-Manipulator System with vector of positions expressed in Eq. (34). Table 3 shows the parameters of the UVMS which is used in case studies. In this case study (a) the conventional sliding mode controller, (b) proposed controller without external disturbance and (c) proposed controller in presence of external disturbances and uncertainties are compared through simulation. Fig. 9 shows the simulation results of SMC with saturation function and Fig. 10

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shows the simulation results of proposed controller in a normal condition. In addition, robustness of the proposed controller with variable parameters in presence of external disturbance is tested (the simulation results of it are shown in Fig. 11). The values of fuzzy gains normalizing factors are equal to 10 (N 1 ; N 2 ¼ 10). The

103

parameters of the proposed control system in Eq. (31) are considered as below M ¼ diagð0:7; 0:7; 0:1; 0:1Þ kD ¼ diagð20; 20; 20; 20Þ

Fig. 10. Simulation results of proposed controller with L ¼ 0.001 s, (a) Trajectory tracking, (b) Tracking error, (c) Input forces and torques and (d) Fuzzy gains.

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Fig. 10. (continued)

kp ¼ diagð5; 5; 5; 5Þ L ¼ 0:001 s

ð35Þ

The simulation results of the SMC are shown in Fig. 9, while trajectory tracking of pitch motion of a divergence has occurred in

the time interval 2 to 5 s. Additionally, through this controller, the system is not able to track the desired path of joint, and after 3 s, the path is diverging from input reference. The simulation results of the improved TDC (the trajectory, tracking errors, torque profiles of joints and fuzzy gains) without

H. Nejatbakhsh Esfahani et al. / Ocean Engineering 107 (2015) 97–107

external disturbance are shown in Fig. 10. Primary fluctuations in the plots are results of the low frequency chattering about of sliding surface. This phenomenon is due to two factors in the control law (according to Eq. (31)): (a) the PID sliding surface and

105

(b) fuzzy gains in the fuzzy subsystem. The new proposed controller has a fast response in reaching mode2 due to using PID λ R sliding surface in TSM (in which integral term 21 e is effective in creating the primary fluctuations). Furthermore, fuzzy gain tuning

Fig. 11. Simulation results of proposed controller with external disturbances and uncertainties (L ¼ 0.007 s) (a) Trajectory tracking, (b) Tracking error (c) Input forces and torques and (d) Fuzzy gains.

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Fig. 11. (continued)

is an effective method for adjusting gains of SMC which results in faster response. However, the faster response results in primary fluctuation (that is a normal behavior). Finally, the simulation results of proposed controller using another time delay L¼ 0.007 s and external disturbances are shown in Fig. 11. As example, following functions are considered for disturbances:

1- Unknown disturbance forces/torques, Q d ¼ sin ð8π tÞ 2- Variable mass of coupled link, ml ¼ 20 þ 5 cos ðtÞ At the first case, unknown disturbance forces/torques are added to Eq. (18) as follows: Q ¼ MðqÞq€ þ Cðq; q_ Þ þ Dðq; q_ Þ þ hðqÞ þQ d

ð36Þ

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107

6. Discussion

References

As it is shown in Fig. 9, the SMC based system is unable to track the desired path. According to highly nonlinear terms of hydrodynamic forces, conventional SMC is not suitable for use in UVMS. In addition, the controller diverges temporarily to track the path regarding to pitch. These problems do not exist in the new proposed controller and through it, the system is able to track all of the desired paths with acceptable precision (Fig. 10). Furthermore, some unknown torques and variable mass of coupled link are applied to dynamics model of UVMS as external disturbances. As it is shown in Fig. 11, the proposed method leads to proper results of trajectory tracking and it has an acceptable robustness. However, the larger time delay causes a lower speed in trajectory tracking (and increasing tracking time for specified error). The proposed controller can be performed for larger time delay by changing of elements in MðqÞ as a constant matrix in Eq. (22) that it may be weak point of this method. In this paper, sampling times L¼ 0.001 s and L¼0.007 s are studied as example (it can be larger by using adaptive method for estimation of elements of the matrix MðqÞ). PID sliding surface have a better performance in comparison with PD sliding surface. The proposed controller includes a free chattering TSM and PID with a fuzzy tunable gain. The main idea is that the robustness property of SMC and good response characteristics of PID are combined with fuzzy tuning gain approach to achieve more acceptable performance. The Proportional-Integral switching part of PID is chosen to ensure the stability of the overall dynamics during the entire period i.e. the reaching phase and the sliding phase. Also, to further penalize tracking error, integral part may be used in the sliding surface part. Although PID controller acts better than PD in terms of steady state error, PD–SMC has more steady chattering behavior compared with the PID–SMC (Piltan et al. (2011a)).

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7. Conclusion A time delay estimation based general framework for trajectory tracking control of Underwater Vehicle-Manipulator System is presented. It consists of three elements: (a) “time delay estimation” element that cancels nonlinearities of UVMS dynamics, (b) TSM controller that provided a fast convergence, and (c) PID element that reduced tracking error. In the conventional TSM and PID controllers, gains depend on uncertainties. This dependence is a defect for complex dynamics system. To solve it a fuzzy controller is designed to adaptively tune the gains of TSM and PID elements. Some simulation studies are done to compare it with conventional SMC. The new controller assures fast convergence and accurate trajectory tracking, which facilitates an effective control. In addition, its robustness is tested in presence of some uncertainties and disturbances. The simulation results showed that this procedure has significant properties in trajectory tracking with acceptable precision. Some future works in this area are as follow: 1- Developing the control strategy in which an adaptive method for estimation of inertial matrix MðqÞ is used. 2- Considering wave effects as a disturbance for UVMS (we will evaluate control performance in presence water wave effects in the shallow waters). UVMS is a large scale system with complex nonlinear dynamics. Rather than its application on earth, in near future, it will be useful for investigations on other planets in which there are possible fluids. However, space application of these systems needs much more research in this area.