A novel approach to 6-DOF adaptive trajectory tracking control of an AUV in the presence of parameter uncertainties

Ocean Engineering 107 (2015) 246–258

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

A novel approach to 6-DOF adaptive trajectory tracking control of an AUV in the presence of parameter uncertainties F. Rezazadegan a,n, K. Shojaei a, F. Sheikholeslam b, A. Chatraei a a b

Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Department of Electrical & Computer Engineering, Isfahan University of Technology, Esfahan, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 6 November 2013 Accepted 21 July 2015

In this paper, the trajectory tracking control of an autonomous underwater vehicle (AUVs) in sixdegrees-of-freedom (6-DOFs) is addressed. It is assumed that the system parameters are unknown and the vehicle is underactuated. An adaptive controller is proposed, based on Lyapunov's direct method and the back-stepping technique, which interestingly guarantees robustness against parameter uncertainties. The desired trajectory can be any sufficiently smooth bounded curve parameterized by time even if consist of straight line. In contrast with the majority of research in this field, the likelihood of actuators' saturation is considered and another adaptive controller is designed to overcome this problem, in which control signals are bounded using saturation functions. The nonlinear adaptive control scheme yields asymptotic convergence of the vehicle to the reference trajectory, in the presence of parametric uncertainties. The stability of the presented control laws is proved in the sense of Lyapunov theory and Barbalat's lemma. Efficiency of presented controller using saturation functions is verified through comparing numerical simulations of both controllers. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Adaptive controller Autonomous underwater vehicle Barbalat's lemma Lyapunov theory Saturation functions

1. Introduction The need of autonomous underwater robots has become increasingly apparent, due to their important role in oil and gas exploration, deep sea inspections, oceanographic mapping, pipeline maintenance, military applications and marine science. Autonomous underwater vehicles (AUV's) let human access to unreachable regions and can simplify the task of acquiring ocean data fast and cost effectively without placing human lives at risk. Trajectory tracking control implies the design of control laws to guide the vehicle for tracking an inertial reference trajectory, a geometric path, in which a time law is specified. Path-following control aims at forcing a vehicle to follow a desired spatial path, without any temporal specifications (Lapierre and Soetanto, 2007). Hence, researchers mostly have studied the path following problem that is easier than trajectory tracking control for practical implementation (Do et al., 2004a, 2004b). An autonomous underwater vehicle does not have actuators in the sway and heave directions, which cause trajectory tracking control to be a more challenging problem in three dimensional

n

Corresponding author. Tel.: þ 61420506691. E-mail addresses: [email protected] (F. Rezazadegan), [email protected] (K. Shojaei), [email protected] (F. Sheikholeslam), [email protected] (A. Chatraei). http://dx.doi.org/10.1016/j.oceaneng.2015.07.040 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

space. Besides, due to the highly nonlinear dynamics of an autonomous underwater vehicle and the difficulty of modeling its environment interaction, control of an AUV in an uncertain unstructured environment, enters multilateral control problems. Researchers have made significant effort in designing autopilots, docking and collision avoidance methods for mobile robots (Teo et al., 2012; Li and Wang, 2013; Rezazadegan et al., 2015). In past decades, variety of design techniques based on sliding mode control, robust control, adaptive control and so forth has been attempted. Sliding mode control is a nonlinear feedback control scheme that is robust to parameter variation and external disturbances, whereas it causes the chattering problem. Healey and Lienard (1993) designed a sliding-mode controller for a six-degrees-offreedom (DOF) underwater vehicle control. They decomposed the system into noninteracting subsystems and grouped certain key functions for the separate functions of steering, diving and speed control. However, uncertainties have not been considered that are strong restricting conditions in many practical applications due to the highly nonlinear unpredictable environments. Wang et al. (2012) proposed a path following controller based on nonlinear iterative sliding mode incremental feedback. Using sliding mode control in combination with iterative method, overshoot has been reduced, though the control system can be utilized only in horizontal plane.

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

Recently, H1 optimal control techniques have also been employed as an effective solution to deal with robust stabilization and tracking problems (Willmann et al., 2007; Kwan et al., 2009; Wang et al., 2010), but these methods can be used only in the presence of certain model for AUVs. In order to deal with the uncertain nonlinear parts of AUV's dynamics, some researchers concentrated their interests on the applications of neural networks, fuzzy control and their combination to the AUV's control problems (Li et al. 2005; Bagheri et al., 2010, Zhang and Chu, 2012; Shahraz and Boozarjomehry, 2009; Ranjbar-Sahraei et al., 2012; Zhang et al., 2009; Wang et al., 2011; Bian et al., 2010). Li and Lee (2005a, 2005b) considered a single hidden layered neural network to estimate the smooth uncertainties of the vehicles' dynamics, where the networks inputs are all the states of AUV. However, it probably increases the number of hidden neurons and weighting parameters and consequently leads to low speed response for system. Fuzzy strategy is used to estimate uncertainty items online, which realizes switch gain fuzzy adaptive adjustment. A robust AUV docking guidance and navigation approach, handling current ocean disturbances has been presented (Teo et al., 2012). In this work, a Tagaki–Sugeno–Kang (TSK) fuzzy inference system (FIS) has assisted the vehicle with high level guidance maneuvers. Although Neural Networks (NN) and Fuzzy Logic Control (FLC) methods are very promising for AUV applications, they need remarkable computational power due to complex decision making processes. Despite of simpler mathematics requirements and higher degree of freedom in tuning control parameters compared to other nonlinear controllers, FLC deals with fuzzification, rule base storage, inference mechanism and defuzzification operations, while more accurate control will be achieved by larger set of rules at the expense of longer computational time. Therefore it may not be practical because there are many implementation aspects must be addressed, such as communication bandwidth, computational capacity, real-time response and onboard battery. In addition, the stability of NN and FLC methods in AUV control cannot be proven mathematically. Specifically if real time selftuning is addressed, neural networks approaches can be impractical because of its unpredictability. Zhang et al. (2009) proposed an adaptive output feedback controller based on dynamic recurrent fuzzy neural network (DRFNN), in which the location information is only needed for controller design and the dynamic uncertain nonlinear mapping is estimated online. Compared to the conventional neural network, DRFNN has improved the tracking performance of AUV due to its less inputs and stronger memory features. However, recurrent neural network has low learning efficiency and poor mapping accuracy, which is very difficult to meet the high-precision trajectory tracking control of underwater robot. Theoretical and experimental results on trajectory tracking for autonomous underactuated marine vehicles show that adaptive Lyapunov-based techniques can overcome most of the limitations mentioned above, while satisfy the mathematical stability proof. In the recent decade, direct, indirect and composite adaptive control systems have been developed widely for tracking problems and estimating unknown parameters in marine vehicles including AUVs (Do et al., 2004a, 2004b; Li and Lee, 2005a, 2005b; Casado and Ferreiro, 2005; Narasimhan and Singh, 2006; Aguiar and Hespanha, 2007; Repoulias and Papadopoulos, 2007; Rezazadegan et al., 2013). To control nonlinear uncertain systems, a back-stepping design technique is presented for achieving more precise trajectory (Krstic et al., 1995). This is a sequential design process applicable to systems with matched and unmatched uncertainties. It provides flexibility in choosing Lyapunov functions and stabilizing virtual control signals at each step of design process for shaping the closed-loop responses. Adaptive the back-stepping method, firstly

247

developed by Kanellakopoulos et al. (1991), has been a powerful design tool to control a class of uncertain nonlinear systems. A back-stepping nonlinear controller with certain parameters has been attempted (Repoulias and Papadopoulos, 2007; Do and Pan, 2009; Do et al., 2002). In the aforementioned papers, both trajectory tracking and path following issues were addressed in the horizontal plane, whereas these approaches can be readily extended to the vertical plane for underwater vehicles. However, since an underwater vehicle usually has available control force in the surge direction and control torques in roll, pitch and yaw but no actuators in sway and heave directions, the problem is much more challenging in the three-dimensional space. It has caused only a few authors to address this control problem. In Li and Lee (2005a, 2005b), an adaptive back-stepping control law has been developed for dive plane control using a single control surface (stern plane) and the simulation results have demonstrated outperformance of this approach in comparison with other previous methods. The effectiveness of adaptive control systems using back-stepping technique has been also verified by recent work (Ciliz, 2007; Rezazadegan and Shojaei, 2013). In the literature, to the best knowledge of authors, there has not been any work in three dimensional space considering 6-degrees-offreedom trajectory tracking control of AUVs, in which parametric uncertainties and saturated actuators are both addressed. In this paper, we consider a six DOF model to design a controller for an AUV in 3D space. The contribution of this paper lies on extending the model proposed in Do and Pan (2009), assuming system parameters are unknown. The desired trajectory does not need to be of a particular type. It can be any kind of sufficiently smooth bounded curve parameterized by time. Additionally, we address the problem of saturated actuators that rarely been considered beforehand. This paper is organized as follows. Section 2 describes the 6 DOF AUV model, control problem and its formulation. In Section 3, according to nonlinear and adaptive control theory, two adaptive back-stepping control approaches are proposed based on the Lyapanov theory, while the second method is designed particularly for solving the problem of actuators saturation using saturation functions. In Sections 4 and 5, the stability analysis of two different control designs is presented using a novel Lyapunov function and Barbalat's lemma. Then in Section 5, the effectiveness and robustness of the proposed approaches in the presence of parameter uncertainty are verified by Matlab simulations. The effect of addressing saturation functions in control approach can be easily found by comparing the numerical simulations in two parts. Finally, we make a brief conclusion in Section 6 of this paper. 2. Vehicle model and control problem For analyzing the motion of marine vehicles in six degrees-offreedom (6 DOF), it is convenient to define two coordinate frames as indicated in Fig. 1. The moving coordinate frame Xo Yo Zo is conveniently fixed to the vehicle, namely the body-fixed reference frame. The origin 0 of the body-fixed frame is usually chosen to coincide with the center of gravity (CG) when the CG is in the principal plane of symmetry or at any other convenient point if this is not the case. For marine vehicles, the body axes Xo, Yo and Zo coincide with the principal axes of inertia, and are usually defined as follows: Xo: Longitudinal axis (directed from aft to fore) Yo: Transverse axis (directed to starboard) Zo: Normal axis (directed from top to bottom).

2.1. AUV equations of motion We adopted the standard notation for motion equations of an AUV; see Do and Pan (2009) and Prestero (2001). Linear velocity

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Earth-Fixed Frame

r cos ðθÞ m22 m33 d11 1 v r w q uþ τu u_ ¼ m11 m11 m11 m11 m11 d22 u r v v_ ¼  m22 m22 m d _ ¼ 11 u q  33 w w m33 m33

ψ_ ¼

z, ψ y, θ

r0

p (Roll)

q (Pitch)

θ_ ¼ q

x, φ

r (Yaw)

m33  m11 d55 ρgΔGML sin ðθÞ 1 u w q þ τq m55 m55 m55 m55 m11  m22 d66 1 u v qþ τr r_ ¼ m66 m66 m66

q_ ¼

u (Surge)

The positive constant terms dii and mii (i ¼1, 2, 3, 4, and 5) denote the hydrodynamic damping and AUV inertia including added mass in surge, sway, heave, pitch and heading, respectively.

Body-Fixed Frame

w

v (Sway)

2.2. Problem statement

Fig. 1. Earth-fixed frame and body-fixed frame for AUV.

v ¼[u v w]T consists of surge, sway and heave, angular velocity ω ¼ [p q r]T consists of roll rate, pitch rate and yaw rate, and attitude η ¼[φ θ ψ]T consists of roll angle, pitch angle and heading angle. Furthermore we assume the center of gravity and the center of buoyancy are located vertically on the ObZb-axis, and there are no couplings (off-diagonal terms) in the matrices M, D, and Dn(v). The mathematical model of an AUV in 6 DOF can be des cribed as:

η_ 1 ¼ J 1 ðη2 Þv1 ; η_ 2 ¼ J 2 ðη2 Þv2

ð1Þ

where η ¼ ½η1 η2 T with η1 ¼ ½x y zT and η2 ¼ ½φ θ ψ T is the position and orientation vector in earth-fixed frame, v ¼ ½v1 v2 T with v1 ¼ ½u v wT and v2 ¼ ½p q rT is the velocity and angular rate vector in body-fixed frame, the positive definite inertia matrix M¼ MRB þMA includes the inertia MRB of the vehicle as a rigid body and the added inertia MA due to the acceleration of the wave, the skew symmetrical matrix C(v) is the matrix of Coriolis and centripetal, the hydrodynamic damping term D(v) takes into account the dissipation of energy due to the friction exerted by the fluid surrounding AUV. For a rigid body moving in an ideal fluid at low speed, M is positive definite and D(v) is real, non-symmetric and strictly positive. The vector g(η) is the gravitational forces and moments vector, τ is the input torque vector, and the transformation matrices J1(η2) and J2(η2) are defined as follows: cðψ ÞcðθÞ 6 J 1 ðη2 Þ ¼ 4 sðψ ÞcðθÞ  sðθÞ

2

1 6 J 2 ðη2 Þ ¼ 4 1 0

The available controls are the surge force τu, pitch moment τq and the yaw moment τr. Since AUVs do not have independent actuators in the sway and heave axes, the vehicle represented by the mathematical model (1) is underactuated. Our objective is to design an adaptive controller (τu, τq, τr) to track the reference trajectory generated by the virtual vehicle model (4), while the vehicle parameters are unknown. x_ d ¼ cos ðψ d Þ cos ðθd Þud  sin ðψ d Þvd þ sin ðθd Þ cos ðψ d Þwd y_ d ¼ sin ðψ d Þ cos ðθd Þud þ cos ðψ d Þvd þ sin ðθd Þ sin ðψ d Þwd z_ d ¼  sin ðθd Þud þ cos ðθd Þwd θ_ ¼ q d

M v_ ¼  CðvÞv  DðvÞv  gðηÞ  τ

2

 sðψ ÞcðφÞ þ sðφÞsðθÞcðψ Þ cðψ ÞcðφÞ þ sðφÞsðθÞsðψ Þ sðφÞcðθÞ

sðφÞtðθÞ cos ðφÞ

sðφÞ=cðθÞ

cðφÞtðθÞ

ð3Þ

sðψ ÞsðφÞ þ sðθÞcðψ ÞcðφÞ

3

 cðψ ÞsðφÞ þ sðθÞsðψ ÞcðφÞ 7 5 cðφÞcðθÞ

3

 sðφÞ 7 5 cðφÞ=cðθÞ

ð2Þ

where cð U Þ ¼ cos ð UÞ; sð UÞ ¼ sin ð UÞ; tð UÞ ¼ tan ð UÞ. Here, we ignore nonlinear hydrodynamic damping terms and roll, and environmental disturbances. This holds when the vessel is operating at low speed and is equipped with independent roll actuators. So, the general mathematical model of a 5 DOF underactuated AUV in surge, sway, heave and heading motion with ignoring roll motion is: x_ ¼ cos ðψ Þ cos ðθÞu  sin ðψ Þv þ sin ðθÞ cos ðψ Þw y_ ¼ sin ðψ Þ cos ðθÞu þ cos ðψ Þv þ sin ðθÞ sin ðψ Þw z_ ¼  sin ðθÞu þ cos ðθÞw

d

rd cos ðθd Þ m22 m33 d11 1 v r  w q  u þ τ u_ d ¼ m11 d d m11 d d m11 d m11 u d m11 d22 v_ d ¼  u r  v m22 d d m22 d m d _ d ¼ 11 ud qd  33 wd w m33 m33

ψ_ d ¼

m33  m11 d55 ρgΔGML sin ðθd Þ 1 ud wd  q  þ τ m55 m55 q d m55 m55 d m11  m22 d66 1 ud vd  r_ d ¼ q þ τ m66 m66 d m66 r d

q_ d ¼

ð4Þ

Then using definition (5) we convert the problem of forcing the underactuated underwater vehicle given in (3) to track the virtual vehicle (4) to stabilizing problem of the system (6). xe ¼ cos ðψ Þ cos ðθÞðx  xd Þ þ sin ðψ Þ cos ðθÞðy  yd Þ  sin ðθÞðz  zd Þ ye ¼  sin ðψ Þ ðx  xd Þ þ cos ðψ Þ ðy yd Þ ze ¼ sin ðθÞ cos ðψ Þðx xd Þ þ sin ðθÞ sin ðψ Þðy yd Þ þ cos ðθÞðz  zd Þ θe ¼ θ  θd ψe ¼ ψ ψd ue ¼ u  ud ; ve ¼ v  vd ; we ¼ w  wd ; qe ¼ q  qd ; r e ¼ r r d

ð5Þ

From differentiation of (5), error system equations: x_ e ¼ ue  ð cos ðθe Þ  1 þ cos ðθÞ cos ðθd Þ ð cos ðψ e Þ  1ÞÞud  cos ðθÞ sin ðψ e Þvd þ ð sin ðθe Þ  cos ðθÞ sin ðθd Þð cos ðψ e Þ  1ÞÞwd þ ðr d þ r e Þye þðqd þ qe Þze y_ e ¼ ve þ cos ðθd Þ sin ðψ e Þ  1ÞÞud  ð cos ðψ e Þ  1ÞÞvd  sin ðθd Þ sin ðψ e Þwd  ðxe þ tan ðθÞze Þðr d þ r e Þ _ e ¼ we  ð sin ðθe Þ þ sin ðθÞ cos ðθd Þ ð cos ðψ e Þ 1ÞÞud w  sin ðθd Þ sin ðψ e Þvd  ð cos ðθe Þ  1 þ sin ðθÞ sin ðθd Þ ð cos ðψ e Þ  1ÞÞwd

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

þ ð sin ðz2 Þϖ 2 þð cos ðz2 Þ  1Þδ2 ze  cos ðθÞ sin ðθd Þ ðð cos ðz1 Þ  1Þϖ 1

þ tan ðθÞ ðr d þ r e Þye þ ðqd þ qe Þxe

θ_ e ¼ qe ψ_ e ¼

re rd þ þ ð cos ðθd Þ ð1  cos ðθe ÞÞ cos ðθÞ cos ðθÞ cos ðθd Þ þ sin ðθd Þ sin ðθe ÞÞ

þ sin ðz1 Þδ1 ye ÞÞϖ  1 wd  cos ðθÞ ð sin ðz1 Þϖ 1 ð6Þ

From Eq. (6), we can directly found that xe, θe, and ψe can be stabilized by ue, qe, and re. There are several options to stabilize ye and ze, i.e. qe, re, ve, we, xe, θe, or ψe. If qe and re are used, the control design will be extremely, complicated since qe and re enter all of the first three equations of (6). In the other side, an undesired feature of vehicle control practice will be happened, if ve and we are used to stabilize ye and ze. In particular, the vehicle will slide in the sway and heave directions. By using xe and ze for stabilization, the reference yaw and heave velocities must then satisfy persistently exciting conditions (see Appendix). Hence we will choose θe and ψe to stabilize the sway error ye and the heave error ze, respectively. Additionally, using the nonlinear coordinate transformations (7) will cause the vehicle to be avoided from whirling around, when ye and ze are large (Do and Pan, (2009)). ! δ1 ye ffi z1 ¼ ψ e þarcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2e þ y2e þ z2e ! δ2 ze ð7Þ z2 ¼ θe arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2e þy2e þ z2e Now with substituting (7) in the system Eqs. (6), error system Eqs. (8) are achieved after long computations and essential simplification. In (8), some new variables like ϖ ; ϖ 1 ; ϖ 2 ; s1 ; s2 ; s3 ; s4 ; s5 ; h1 ; h2 are employed to shorten equations and avoid complexity in understanding the equations for readers. These variables' values are mentioned in Eq. (8). x_ e ¼ ue  ðϖ 2  ϖ þ cos ðθÞ cos ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 ud

þ ð cos ðθÞÞ  1 ϖ  1 ðð1  cos ðz2 ÞÞϖ 2 þ sin ðz2 Þδ2 ze þ tan ðθd Þ ð sin ðz2 Þ þ ð cos ðz2 Þ  1Þδ2 ze ÞÞ s5 ¼  δ2 ϖ 2 1 ðs3 ze ϖ  2 ðxe s1 þ ye s2 þ ze s3 ÞÞ h1 ¼ ðϖ r d ð1 þ cos  1 ðθd Þ cos ðθÞÞ  ð1  cos ðz2 ÞÞϖ 2  sin ðz2 Þδ2 ze  ðð cos ðz2 Þ  1Þ þ sin ðz2 ÞÞ tan ðθd Þ δ2 ze Þ ð cos ðθÞÞ  1 ϖ  1 þ δ1 ϖ  1 ðve  cos ðθd Þδ1 ud ϖ  1 ye  ðϖ 1  ϖ Þϖ  1 vd þ sin ðθd Þδ1 wd ϖ  1 ye  ðxe þ tan ðθÞze Þr d  ye ϖ  2 ðxe ð  ðϖ 2  ϖ þ cos ðθÞ cos ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 ud þ cos ðθÞδ1 vd ϖ  1 ye þ ðδ2 ze  cos ðθÞ sin ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 wd ÞÞ þ ye ðve  cos ðθd Þ  δ1 ud ϖ  1 ye  ðϖ 1  ϖ Þϖ  1 vd þ sin ðθd Þδ1 wd ϖ  1 ye Þ þ ze ðwe ðδ2 ze þ sin ðθÞ cos ðθd Þðϖ 1  ϖ ÞÞϖ  1 ud þ sin ðθÞ

þ sin ðθd Þ δ1 wd ϖ  1 ye Þ

þ sin ðθÞze ÞÞ ð cos ðθÞÞ  1 r e  δ1 ϖ 1 1 ϖ  2 xe ye ue þ h1 þ s4

þ ze ðwe ðδ2 ze þ sin ðθÞ cos ðθd Þ ðϖ 1  ϖ Þϖ  1 ud

z_ 2 ¼ ð1  δ2 ϖ 2 1 xe Þqe  δ2 tan ðθÞϖ 2 1 ye r e

þ cos ðθÞ cos ðθd Þ ðð cos ðz1 Þ  1Þϖ 1 þ sin ðz1 Þδ1 ye ÞÞϖ  1 ud

s4 ¼ δ1 ϖ 1 1 ðs2  ye ϖ  2 ðxe s1 þye s2 þ ze s3 ÞÞ

þ ye ðve  cos ðθd Þδ1 ud ϖ  1 ye  ðϖ 1  ϖ Þϖ  1 vd

z_ 1 ¼ ð1  δ1 ϖ 1 1 ð cos ðθÞxe

s1 ¼  ðð cos ðz2 Þ  1Þϖ 2  sin ðz2 Þδ2 ze

þ ð cos ðz1 Þ 1Þδ1 ye Þϖ  1 vd

 cos ðθÞ sin ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 wd Þ

 ðϖ 1  ϖ ÞÞϖ  1 wd þ tan ðθÞ ðr d þ r e Þye  ðqd þ qe Þxe þ s3

and

þ sin ðz1 Þδ1 ye ÞÞϖ  1 wd  sin ðθÞ ð sin ðz1 Þϖ 1

þ cos ðθÞδ1 vd ϖ  1 ye þ ðδ2 ze

ye  ðϖ 2  ϖ þ sin ðθÞ sin ðθd Þ

where: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϖ ¼ 1 þ x2e þ y2e þ z2e ; ϖ 1 ¼ 1 þ x2e þ ð1  δ21 Þy2e þ z2e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϖ 2 ¼ 1 þ x2e þy2e þð1  δ22 Þz2e

þ sin ðz1 Þδ1 ye ÞÞϖ  1 ud  ðð cos ðz2 Þ  1Þϖ 2  sin ðz2 Þδ2 ze þ sin ðθÞ sin ðθd Þ ðð cos ðz1 Þ  1Þϖ 1

 ze ϖ  2 ðxe ð  ðϖ 2  ϖ þ cos ðθÞ cos ðθd Þðϖ 1  ϖ ÞÞϖ  1 ud

ud þ ðr d þ r e Þye

þ δ2 ϖ 2 1 ϖ  2 xe ze ue þ h2 þ s5 m22 m33 d11 1 v r w q uþ τ  u_ d u_ e ¼ m11 ud m11 m11 m11 m11 d22 ðue r e þ ue r d þ ud r e Þ  ve v_ e ¼  m22 m22 m d _ e ¼ 11 ðue qe þ ue qd þ ud qe Þ  33 we w m33 m33 m33  m11 d55 ρgΔGML sin ðθÞ 1 q_ e ¼ ud wd  q  þ τ  q_ d m55 m55 qd m55 m55 d m11  m22 d66 1 r_ e ¼ u v rþ τ  r_ d m66 rd m66 m66

 sin ðθd Þ ð sin ðz1 Þϖ 1  ð cos ðz1 Þ  1Þδ1 ye Þϖ  1 wd s3 ¼  ð sin ðz2 Þϖ 2 þ ð cos ðz2 Þ 1Þδ2 ze þ sin ðθÞ cos ðθd Þ ðð cos ðz1 Þ  1Þϖ 1

 ϖ  1 wd þ xe qd þ tan ðθÞye r d

þ sin ðθd Þδ1 wd ϖ  1 ye  ðxe þ tan ðθÞze Þ ðr d þ r e Þ þ s2 þ sin ðθÞδ1 vd ϖ

 ðð cos ðz1 Þ  1Þϖ 1 þ sin ðz1 Þδ1 ye Þϖ  1 vd

þ sin ðθÞ δ1 vd ϖ  1 ye ðϖ 2  ϖ þ sin ðθÞ sin ðθd Þ ðϖ 1  ϖ ÞÞ

ye_ ¼ ve  cos ðθd Þδ1 ud ϖ  1 ye  ðϖ 1  ϖ Þϖ  1 vd

1

s2 ¼ cos ðθd Þ ð sin ðz1 Þϖ 1 ð cos ðz1 Þ  1Þδ1 ye Þϖ  1 ud

h2 ¼  δ2 ϖ 2 1 ðwe  ðδ2 ze þ sin ðθÞ cos ðθd Þ ðϖ 1  ϖ ÞÞ ϖ  1 ud

 ðϖ 1  ϖ ÞÞϖ  1 wd þ ðr d þ r e Þye  ðqd þ qe Þze þ s1

1

 ð cos ðz1 Þ 1Þδ1 ye Þϖ  1 vd

 δ1 vd ϖ  1 ye  ðϖ 2  ϖ þ sin ðθÞ sin ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 wd ÞÞÞÞ

þ cos ðθÞδ1 vd ϖ  1 ye þ ðδ2 ze  cos ðθÞ sin ðθd Þ

z_ e ¼ we  ðδ2 ze þ sin ðθÞ cos ðθd Þ ðϖ 1  ϖ ÞÞϖ

249

þ sin ðθÞδ1 vd ϖ  1 ye ðϖ 2  ϖ þ sin ðθÞ sin ðθd Þ ðϖ 1  ϖ ÞÞϖ  1 wd ÞÞÞÞ

ð9Þ

3. Adaptive nonlinear control design

ð8Þ

Firstly, the virtual velocity controls of ue, qe and re are designed to asymptotically stabilize xe, ye, ze, z1, z2, we and ve at the origin. Then based on the back-stepping technique, the controls τu, τq, τr will be designed to make the errors between the virtual velocity controls and their actual values exponentially be vanished. Since ue enters ve and we dynamics, we will design a bounded virtual control of ue, to simplify the stability analysis. The virtual controls of qe and re are chosen such that z1 and z2 dynamics can be stabilized. Assumption 1. The reference signals ud, qd, rd, u_ d , q_ d and r_ d are There exists a strictly positive constant ud min that bounded.  ud ðtÞ Zud min ; 8 t Z 0. This condition is much less restrictive than a persistently exciting condition on the yaw reference and heave velocities satisfy velocity.   The reference    sway  vd ðtÞ o ud ðtÞ; wd ðtÞ o ud ðtÞ; 8 t Z0.

250

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

Assumption2. The reference pitch angle satisfies 0:5π ; 8 t Z 0 because of singularity avoidance.

  θd ðtÞ r



^ 11  m ^ 22 d m d^ 66 d u vþ ðr þ r d Þ þ r^_ d þ r^_ e ^ 66 ^ 66 e m m ^ 11 m z1  ð1  δ1 ϖ 1 1 ð cos ðθÞxe þ sin ðθÞze ÞÞ ^ 66 m cos ðθÞ  þ δ2 tan ðθÞϖ 2 1 ye z2

^ 66  ρ2 r~ e  τ^ r ¼ m

The virtual control errors are defined as: u~ e ¼ ue  ude ; q~ e ¼ qe  qde ; r~ e ¼ re  r de ude ,

qde

where and respectively.

r de

ð10Þ

are the virtual velocity controls of ue, qe and re



^ 55  ρ3 q~ e  τ^ q ¼ m

3.1. Adaptive control law – part 1 In this part, it is assumed that actuators are not saturated. Since the standard application of back-stepping leads to a complex controller, we have introduced virtual control laws without canceling some known terms using saturation functions: 8 > ude ¼ −δ0 ϖ −1 xe þ ðϖ 2 −ϖ þ cos ðθÞ cos ðθd Þ ðϖ 1 −ϖÞÞϖ −1 ud > > > > > − cos ðθÞδ1 vd ϖ −1 ye −ðδ2 ze − cos ðθÞ sin ðθd Þ ðϖ 1 −ϖÞÞϖ −1 wd ; > > > > d d d > > > qe ¼ q1e þ q2e ; > > > ðδ tan ðθÞϖ −1 ye rd1e −δ2 ϖ −1 ϖ −2 xe ze ude −h2 Þ 2 2 > > qd1e ¼ 2 ; > 1−δ2 ϖ −1 xe > 2 < −1 d ð−c z þ δ tan ðθÞϖ y r −s Þ qd2e ¼ 2 2 21−δ ϖ−1 x2 e 1e 5 > e 2 > 2 > > > > r de ¼ r d1e þ r d2e ; > > > > > ðδ ϖ −1 ϖ −2 x y ud −h Þ cos ðθÞ > > ; r d ¼ 1 1 −1 e e e 1 > > > 1e 1−δ1 ϖ1 ð cos ðθÞxe þ sin ðθÞze Þ > > > ð−c1 z1 −s4 Þ cos ðθÞ > ; : r d2e ¼ −1 1−δ1 ϖ 1 ð cos ðθÞxe þ sin ðθÞze Þ

ð11Þ

where

δ0, c1 and c2 being positive constants.

Remark 1. qde and r de exponentially converge to zero when z1 and z2 do and make stability analysis simpler. Remark 2. qde and r de are Lipschitz in (xe, ye, ze, ve, and we) that plays a crucial role in the stability analysis of the closed loop system. It could be shown that the virtual control ude is bounded as:          d 2 2  2  ð12Þ ue  r δ0 þ ð1 þ δ1 þ δ2 Þud  þ δ1 vd  þ ðk2  þ δ1 Þwd  : ¼ ueb 

^ m

^ m

^ 11  ρ1 :u~ e  22 v r þ 33 wq τ^ u ¼ m ^ 11 ^ 11 m m

d d^ 11 d ðu þud Þ þ u^_ d þ u^_ e þ δ1 ϖ 1 1 ϖ  2 xe ye z1 ^ 11 e m  ^ 66 m  δ ϖ  1 ϖ  2 xe ze z2 ^ 11 2 2 m

þ

^ 11 ^ 33  m d^ 55 d ρgΔGML sin ðθÞ m u wþ ðq þ qd Þ þ ^ 55 ^ 55 ^ 55 e m m m

^ 66 d m þ q^_ d þ q^_ e  ð1  δ2 ϖ 2 1 xe Þz2 ^ 55 m



ð13Þ

where ρi, i¼ 1, 2, and 3 are positive constants and notation ð^Þ is used for estimated expressions. Substituting (13) and (11) into (8) the closed loop system Eqs. (8) are achieved that has not written here to avoid confusing expressions. 3.2. Adaptive control law-part 2

Actuator saturation is inevitable in feedback control systems. If it is ignored in the design, a controller may wind up the actuator, possibly resulting in degraded performance or even instability. Although majority of work did not address this issue, higher performance may be expected if a controller is designed a priori considering the saturation effect. As a result, we consider the problem of actuators' saturation in this section and propose an efficient solution for it. We introduce virtual control laws without canceling some known terms using saturation functions: where δ0, c1 and c2 are positive constants. By differentiating (10) along the solutions of (14) and (6), the actual controls τ^ u , τ^ q , and τ^ r by considering parameter uncertainties, without removing the useful damping terms, are chosen as following:  ^ ^ m m ^ 11  ρ1 :tanhðu~ e Þ  22 v r þ 33 wq τ^ u ¼ m ^ 11 ^ 11 m m d d^ 11 þ ðu  tanhðu~ e ÞÞ þ u^_ d þ u^_ e ^ 11 m  ^ 66 m þ δ1 ϖ 1 1 ϖ  2 xe ye z1  δ ϖ  1 ϖ  2 xe z e z 2 ^ 11 2 2 m  ^ m ^ 22 m d^ 66 ^ 66  ρ2 tanhðr~ e Þ  11 τ^ r ¼ m uv þ ðr  tanhðr~ e ÞÞ ^ 66 ^ 66 m m ^ 11 d z1 m ð1  δ1 ϖ 1 1 ð cos ðθÞxe þ sin ðθÞze ÞÞ þ r_^ d þ r_^ e  ^ 66 m cos ðθÞ  þ δ2 tan ðθÞϖ 2 1 ye z2 

^ 55  ρ3 tanhðq~ e Þ  τ^ q ¼ m

^ 11 ^ 33  m d^ 55 m u wþ ðq  tanhðq~ e ÞÞ ^ 55 ^ 55 m m

8 tanhðude Þ > > ¼ −δ0 ϖ −1 xe þ ðϖ 2 −ϖ þ cos ðθÞ cos ðθd Þ ðϖ 1 −ϖÞÞ ϖ −1 ud − cos ðθÞ δ1 vd ϖ −1 ye −ðδ2 ze − cos ðθÞ sin ðθd Þñðϖ 1 −ϖÞÞϖ −1 wd ; > > d > 1− tanhðu e Þtanhðue Þ > > > d > > q ¼ qd1e þ qd2e ; > > > e > > d d d > ; > > r e ¼ r 1e þ r 2e ! >   > > d −2 d > tanhðr Þ 1−δ1 ϖ −1 δ1 ϖ −1 > 1e 1 ð cos ðθÞxe þ sin ðθÞze Þ 1 ϖ xe ye tanhðue Þ >  ¼ −h1 ñ cos ðθÞ ; > > < 1− tanhðr e Þtanhðr d1e Þ 1− tanhðr d1e Þtanhðr d2e Þ 1− tanhðue Þtanhðude Þ   d −1 > > >  tanhðr 2e Þ 1−δ1 ϖ 1 ð cos ðθÞxe þ sin ðθÞze Þ ¼ ð−c1 z1 −s4 Þ cos ðθÞ ; > > d d Þtanhðr d Þ > > 2e > 1− tanhðr e Þtanhðr 2e Þ 1− tanhðr 1e >  > > d −1 d −1 −2 d > tanhðq Þ 1−δ2 ϖ x δ tan ðθÞϖ −1 e > 1e 2 2 ye tanhðr 1e Þ δ2 ϖ 2 ϖ xe ze tanhðue Þ >   ¼ 2 − −h2 ; > > d d d d d > 1− tanhðqe Þtanhðq1e Þ 1− tanhðq1e Þtanhðq2e Þ 1− tanhðreÞtanhðr 1e Þ 1− tanhðueÞtanhðu1e Þ > > >   > > d > tanhðqd2e Þ 1−δ2 ϖ −1 δ2 tan ðθÞϖ −1 > 2 xe 2 ye tanhðr 1e Þ >    > : 1− tanhðq Þtanhðqd Þ 1− tanhðqd Þtanhðqd Þ ¼ 1− tanhðr Þtanhðr d Þ −c2 z2 −s5 e

2e

1e

2e

e

1e

ð14Þ

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

þ

 ^ 66 ρgΔGML sin ðθÞ _^ ^_ d m þ qd þ q  ð1  δ2 ϖ  1 xe Þz2 ^ 55 m

^ 55 m

e

ð15Þ

2

where ρi, i¼ 1, 2, and 3 are positive constants and notation ð^Þ is used for estimated expressions. Substituting (15) and (14) into (8) the closed loop system Eqs. (8) are achieved.

where 1 1 1 σ 1 ¼ minðc1 ; c2 ; ðρ1 þ d11 m11 Þ; ðρ2 þd55 m55 Þ; ðρ3 þ d66 m66 Þ

(

Z Theorem 4.1. Let assume the virtual vehicle model (4) generates reference signals (xd;yd;zd;θd; ψd;vd;wd) and Assumptions 1 and 2 are relaxed. If the state feedback control law (13) is applied to the vehicle system (3), then the tracking errors (xe;ye;ze;θe; ψe;ve;we) asymptotically converge to zero with a suitable choice of the design constants δ0, δ1, and δ2, i.e., the closed loop system (8) is locally asymptotically stable at the origin. In this section, to simplify the stability analysis of this closed loop system, we consider two subsystems ðxe ; ye ; ze ; ve ; we Þ and ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ in an interconnected structure. Therefore we first show the stability of the ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ-subsystem then move toðxe ; ye ; ze ; ve ; we Þ-subsystem. We prove that the control signals defined above are well defined, bounded, and that the closed loop system (8) is asymptotically stable at the origin. 4.1. Stability analysis of ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ-subsystem From the closed loop system equations, it is direct to show that this subsystem is asymptotically stable at the origin by using the following Lyapunov function: m11 2 m66 2 m11 2 m55 2 m66 2 1 X ~ T  1 ~ V1 ¼ z þ z þ u~ þ q~ þ r~ þ θ Γ θi 2 1 2 2 2 e 2 e 2 e 2i ¼ u;q;r i i ð16Þ After substituting mentioned terms and some calculations following result will be achieved: 1 ~ 2 Þu e V_ 1 ¼  c1 z21  c2 z22  ðρ1 þ d11 m11 1 ~  1 ~2  ðρ2 þ d55 m55 Þq e  ðρ3 þd66 m66 Þr e X _T þ ðφT u~ þ φT q~ þ φT r~ Þθ~ þ θ~ Γ  1 θ^ ; 2

u

e

e

q

r

e

i

i

i

ð17Þ

i

i ¼ u;q;r

where θ~ i ¼ θi  θ^ i . For stability proof, according to the Lyapunov theory, V_ 1 should be a negative value. By considering this simple fact, estimation rule can be derived: X _T θ^ i ¼  Γ i ðφTu u~ e þ φTq q~ e þ φTr r~ eÞ; i ¼ u;q; r

h

m211 d22 m11 m211 d33 m11 m22 m22 m33 m33 h d m m m d m m m θTq ¼ ðm33  m11 Þd55 11 55 11 55 22 55 11 55 m11 m22 m22 m33 i d33 m55 d66 m55  m55 m66 ; m33 m66 h d m m m d m m m θTr ¼ ðm22  m11 Þ 11 66 11 66 22 66 11 66 m11 m22 m22 m33 i d33 m66  d66 m66 m11 m33 V_ 1 ¼  c1 z2  c2 z2  ðρ þ d11 m  1 Þu~ 2

θTu ¼ m22 m33 d11

1

2

1 2

11

1

ð20Þ

Then, with integrating from the last equation of (18), we can see: Z þ1  1 ~ 2 V_ 1 dt ¼  c1 z21  c2 z22  ðρ1 þ d11 m11 Þu e 0  1 ~2  1 ~2 ð21Þ  ðρ2 þ d55 m55 Þq e  ðρ3 þd66 m66 Þr e dt

þ1

Since left side of (21) is bounded, it could be written: z1 ; z2 ; u~ e ; q~ e ; r~ e A L2

ð22Þ

By taking a look on the previous observation (20), we can simply show: z1 ; z2 ; u~ e ; q~ e ; r~ e A ðL2 \ L1 Þ

ð23Þ

Also, considering five last equations of closed loop system, it is not hard to illustrate that z_ 1 ; z_ 2 ; u~_ e ; q~_ e ; r~_ e A L1

ð24Þ

Now using (23) and (24) and Barbalat lemma, asymptotic convergence to zero with an appropriate choice of the design constraints δ0, δ1, and δ2 is proved. 4.2. Stability analysis of (xe, ye, ze, ve, we)-subsystem From the closed loop system equations, it is direct to show that this subsystem is asymptotically stable at the origin by using the following the Lyapunov function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 V 2 ¼ 1 þx2e þ y2e þ z2e  1 þ δ3 ðv2e þ w2e Þ ð25Þ 2 where δ3 is a positive constant to be specified later. The time derivative of (25) along with the solutions of the first five equations of closed loop system equations, after a long but simple calculation using completed squares, satisfies: _ 2 r  λx ðtÞϖ  2 x2  λy ðtÞϖ  2 y2  λz ðtÞϖ  2 z2  λv ðtÞv2 V e e e e  λw ðtÞw2e þ ðζ 1 ð U ÞV 2 þ ζ 2 ð U ÞÞe  σ 1ðt  t0Þ

ð26Þ

where ζ1(  ) and ζ2(  ) are some nondecreasing functions of ‖ðz1 ðt 0 Þ; z2 ðt 0 Þ; u~ e ðt 0 Þ; q~ e ðt 0 Þ; r~ e ðt 0 ÞÞ‖, and

λx ðtÞ ¼ δ0 

δ3 m11 jr d jγ 1 m22

 δ3 m11 ue b γ 1 jδ1 r d j 3 2 2 2  þ jδ1 ud j þ δ1 jvd jþ ðδ1 þ δ2 Þjδ1 wd j 2 m22 ð1  2jδ1 jÞ ð1  δ Þ 1

i m66 ;

δ3 m11 jqd jγ 1 δ3 m11 ue b γ 1  m33 m33 ð1  jδ2 jÞ ! jδ2 qd j 3 2  δ u jþ δ j δ v j þ j 2 d d 1 1 2 ð1  δ1 Þ δ3 m11 ue b jδ2 tan ðθÞjγ 1  3  jδ1 r d jÞ þ jδ1 ud j m33 ð1  jδ2 jÞ  2 2 2 þ δ1 jvd j þðδ1 þ δ2 Þjδ1 wd jþ ue b 

e

1 ~ 1 ~ 2  ðρ2 þ d55 m55 Þq e  ðρ3 þd66 m66 Þ re r0

0



Considering (18), we can observe: z1 ; z2 ; u~ e ; q~ e ; r~ e A L1 ) V 1 A L1 θ~ A L ) θ^ A L 1

4. Stability analysis- part 1

251

ð18Þ

so it could be shown: ‖z1 ðtÞ; z2 ðtÞ; u~ e ðtÞ; q~ e ðtÞ; r~ e ðtÞ‖ r ‖ðz1 ðt 0 Þ; z2 ðt 0 Þ; u~ e ðt 0 Þ; q~ e ðt 0 Þ; r~ e ðt 0 ÞÞ‖e  σ 1ðt  t0Þ

ð19Þ

λy ðtÞ ¼ δ1 r d cos ðθd Þ  δ1 jwd j  γ 3  δ21 ðjud j þ jvd j þjwd jÞ  δ3 m11 ue b γ 1  δ1 jud j þ jwd j þ jδ1 vd j  m22 ð1  jδ2 jÞ δ3 m11 ue b γ 1  2 u jδ1 j þ δ1 ð2jud j þjvd j  m22 ð1  2jδ1 jÞ e b  3 3 2 2 þ 2jwd jÞjδ1 wd j þ 2jδ1 vd j þ ðδ1 þ δ2 Þjδ1 ud j

252

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

 jδ1 vd j δ3 m11 jqd jγ 1  2  δ1 ðjud j þ jwd jÞ þ jδ1 vd j 4γ 2 m33 δ3 m11 ue b γ 1  m33 ð1  jδ2 jÞ !  jδ2 r d j 2  δ1 jδ2 jðjud jþ jwd jÞ þ jδ1 δ2 jðjud j þ jvd jÞ þ 2 1  δ2  δ3 m11 ue b jδ1 tan ðθÞjγ 1 2 ue b jδ1 jþ δ1 ð2jud jþ jvd j þ 2jwd jÞ  m33 ð1  jδ2 jÞ  3 2 2 þ jδ1 jð2jvd j þjwd jÞ þ ðδ1 þ δ2 Þjδ1 ud j  δ m jr jγ  λz ðtÞ ¼ δ2 ud  δ1jvd j  γ 3  δ22 jwd j 3 11 d 1 δ22 jud j þ jδ2 wd j m22  δ3 m11 ue b γ 1  3 2 2 2jδ2 wd j þ ðδ2 þ δ1 jδ2 jÞ ðjud j þ jvd jþ jwd j  m33 ð1  jδ2 jÞ δ3 m11 jqd jγ 1 2 ðδ2 jwd j þ δ2 jud jÞ þ jδ1 δ2 jðjud jþ jvd j þ 2jwd jÞÞ  m33  δ3 m11 ue b jδ2 tan ðθÞjγ 1 δ3 m11 ue b γ 1  2 jδ1 qd j þ δ2 þ  2 m33 ð1  jδ2 jÞ m22 ð1  2jδ1 jÞ 1δ 

þ jδ2 j þ jδ1 δ2 jðjud j þ jwd jÞÞ δ d 1 δ3 m11 jr d j 2 λv ðtÞ ¼ 3 22   δ0 þðδ21 þ δ22 Þjud j þ jδ1 vd j 4γ 3 m22 4γ 1 m22  δ3 m11 ue b  2 jδ1 jðue b þ 2:5 þ jδ2 wd j þ δ1 jwd j  m22 ð1 2jδ1 jÞ4γ 1 þ jr d j  ð1 þ tan ðθÞÞÞ þ δ 2

1

5. Stability analysis – part 2 Theorem 5.1. Let assume the virtual vehicle model (4) generates reference signals (xd;yd;zd;θd; ψd;vd;wd) and Assumptions 1. and 2 are relaxed. If the state feedback control law (15) is applied to the vehicle system (3), then the tracking errors (xe;ye;ze;θe; ψe;ve;we) asymptotically converge to zero with a suitable choice of the design constants δ0, δ1, and δ2, i.e., the closed loop system (8) is locally asymptotically stable at the origin. In this section, like Section 4, we consider two subsystems (xe, ye, ze, ve, we) and ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ in an interconnected structure. We prove that the control signals are well defined, bounded, and the closed loop system is asymptotically stable at the origin. We only show the stability of ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ-subsystem while the stability analysis of (xe, ye, ze, ve, we)-subsystem will be same as Section 4.2 and we do not prove it again here. 5.1. Stability analysis of ðz1 ; z2 ; u~ e ; q~ e ; r~ e Þ-subsystem From the closed loop system Eqs. (8), it is shown directly that this subsystem is exponentially stable at the origin by using the following Lyapunov function:

δ

2 2 2 þ 1 ðjud j þ 3jvd j þ 2jwd jÞ

þ ðδ1 þ δ2 Þjδ1 jðjud jþ jwd jÞ þ jδ1 δ2 jðjud j 2

otherwise the vehicle will slide in the sway and heave directions (Do and Pan, 2009).

2

V3 ¼

þ jwd jÞÞ þ jδ1 jðjud jþ 2jvd j þ jwd jÞ 3

δ3 δ22 m11 ue b j tan ðθÞj δ3 δ22 m11 ue b  m22 ð1  jδ2 jÞ8 γ 1 m22 ð1  jδ2 jÞ8γ 1 δ3 d33 1 δ3 m11 jqd j 2 λw ðtÞ ¼   δ0 þ ðδ21 þ δ22 Þjud j 4γ 3 m33 4γ 1 m33 þ jδ2 jÞ 

δ3 m11 ue b  jδ1 jðue b þ 2:5 γ 1 þ jr d j  m33 ð1  2jδ2 jÞ8γ 1 2 2 ð1 þ tan 2 ðθÞÞÞ þ δ2 þ δ1 ðjud jþ 3jvd j þ 2jwd jÞ 2 2 þ ðδ1 þ δ2 Þjδ1 jðjud jþ jwd jÞ þ jδ1 δ2 jðjud j 3 þ jwd jÞÞ þ jδ1 jðjud jþ jvd j þ jδ1 j ðjud j þ jwd jÞ þ jδ1 δ2 jðjud j þ jwd jÞÞ 3 þ jδ1 jðjud j þ jvd j þjwd jÞ þ jδ2 jÞ  δ3 jδ2 jm11 ue b  2  δ þ ðδ21 þ δ22 Þjud j þjδ1 vd j þ ðjδ2 j þ δ22 Þjwd j m33 ð1 jδ2 jÞ 4 γ 1 0 δ3 m11 ue b  jδ2 jð2:5 γ 1 þ tan 2 ðθÞjqd jÞ m33 ð1  jδ2 jÞ 4γ 1

þ δ1 jδ2 jð3jud jþ 2jvd j þ 2jwd jÞ þ δ2 ðjud j þ2jwd jÞ 2

2

þ jδ2 jðjud j þ 2jwd jÞ þ 2jδ1 δ2 jðjud j þ jvd j þjwd jÞÞ 3

where

ð27Þ

γi, i¼1, 2, and 3 are positive constants, and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

θðtÞ ¼ θd ðtÞ þ z2 ðtÞ þ arcsinðδ2 ze ðtÞ= 1 þ x2e ðtÞ þ y2e ðtÞ þ z2e ðtÞÞ

ð28Þ

Now design constants δi; 0 r i r3 are chosen such that λx ðtÞ; λy ðtÞ; λz ðtÞ; λv ðtÞ; λw ðtÞ are always smaller than some positive n n n n n constants λx ; λy ; λz ; λv ; λw and θ being replaced by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

θðt 0 Þ ¼ θd ðt 0 Þ þ z2 ðt 0 Þ þ arcsinðδ2 ze ðtÞ= 1 þ x2e ðtÞ þy2e ðtÞ þ z2e ðtÞÞ

ð30Þ

After substituting the mentioned terms and some calculations, the following result will be achieved:

 2 þ jδ1 vd jþ jδ2 wd j þ δ1 jwd j



m11 2 m66 2 2 z þ z þ m11 ln coshðu~ 2e Þ þ m55 ln coshðq~ e Þ 2 1 2 2 1 X ~ T 1 ~ þ m66 ln coshðr~ 2e Þ þ ϑ Γ ϑi 2i ¼ u;q;r i i

ð29Þ

This will be satisfied by choosing small enough δ1, δ2 that have the same sign with the surge reference velocity ud, setting γi ¼ δ1; 0 r i r 3 and picking large enough δ3. The small values of δ1, δ2 also imply a small value of δ0. This can be physically interpreted because if the damping in the sway and heave dynamics is small, the control gain in the surge dynamics should also be small

2 1 Þtanh ðu~ e Þ V_ 3 ¼ c1 z21 c2 z22 ðρ1 þ d11 m11 2 2 2 1 1  ðρ2 þ d55 m55 Þtanh q~ e  ðρ3 þ d66 m66 Þtanh r~ 2e T X _ þ ðφT u~ þ φT q~ þ φT r~ Þϑ~ þ ϑ~ Γ  1 ϑ^ u

e

q

e

r

e

i

i

i

i

ð31Þ

i ¼ u;q;r

^ . From (31), estimation rule is derived: where ϑ~ i ¼ ϑi  ϑ i X _T ϑ^ i ¼  Γ i ðφTu u~ e þ φTq q~ e þ φTr r~ e Þ; i ¼ u;q;r

i m211 d22 m11 m211 d33 m11 m66 ; m22 m22 m33 m33 h d m m m d m m m ϑTq ¼ ðm33 m11 Þ d55 11 55 11 55 22 55 11 55 m11 m22 m22 m33 i d33 m55 d66 m55  m55 m66 ; m33 m66 h d m m m d m m m ϑTr ¼ ðm22 m11 Þ 11 66 11 66 22 66 11 66 m11 m22 m22 m33 i d33 m66  d66 m66 m11 m33 2 2 1 V_ 3 ¼ c1 z1 c2 z22 ðρ1 þ d11 m11 Þtanh ðu~ e Þ h

ϑTu ¼ m22 m33 d11

2 2 2 1 1  ðρ2 þ d55 m55 Þtanh q~ e  ðρ3 þ d66 m66 Þ tanh r~ 2e r0

(

ð32Þ

Considering (32), we can observe: z1 ; z2 ; tanhðu~ e Þ; tanhðq~ e Þ; tanhðr~ e Þ A L1 ) V 3 A L1 ϑ~ A L1 ) ϑ^ A L1

ð33Þ

Then, by integrating the last equation of (32), it could be shown that the left side of that equation is bounded (see Section 4.1, Eq. 21). Consequently, we can write: z1 ; z2 ; tanhðu~ e Þ; tanhðq~ e Þ; tanhðr~ e Þ A L2

ð34Þ

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

Fig. 2. Reference and vehicle trajectory in three dimensions.

Fig. 3. Reference and vehicle trajectory in two dimensions.

Fig. 4. Tracking position errors (xe, ye and ze).

253

254

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

Fig. 5. Tracking orientation errors (θe, ψe).

Fig. 6. Control surge force, pitch torque and yaw torque (Fu, Tq and Tr).

Fig. 7. Reference and vehicle trajectory in three dimensions.

By considering (33) and (34), we can conclude: z1 ; z2 ; u~ e ; q~ e ; r~ e A ðL2 \ L1 Þ

ð35Þ

F. Rezazadegan et al. / Ocean Engineering 107 (2015) 246–258

255

Fig. 8. Reference and vehicle trajectory in two dimensions.

Fig. 9. Tracking position errors (xe, ye, ze).

Besides, considering five last equations of closed loop system, it is not hard to illustrate that z_ 1 ; z_ 2 ; u_~ e ; q_~ e ; r_~ e A L1

ð36Þ

Finally, using (35), (36) and Barbalat Lemma, asymptotical convergence to zero is proved with an appropriate choice of the design constants δ0, δ1, and δ2.

6. Simulation results In this section, to illustrate the performance of the tracking control algorithm, simulations have been carried out, assuming the vehicle is directly actuated in force in the xB direction and in torque about the yB and zB axes. Reference trajectory generated by (4) is considered as:

τud ¼ 5 d11  ðm22 vd rd  m33 wd qd Þ;

τqd ¼  ðm33  m11 Þud wd  d55 qd 69:42 sin ðθd Þ þ m55 ð  θd þ 0:2  θ_ d Þ

τrd ¼  ðm11  m22 Þud vd for first 150 s, and:

τrd ¼  ðm11  m22 Þud vd þ 5sinð0:0061tÞ  5cosð0:0061tÞ

ð37Þ

for the rest of simulation time. The initial conditions are picked as:

ηd ðt 0 Þ ¼ ½0; 0; 40; 0; 0; 10; 0; 0; 0; 0; ηðt 0 Þ ¼ ½30; 90; 0; 0; 0; 0; 0; 0; 0; 0

ð38Þ

We used the projection algorithm to estimate the uncertain parameters of vehicle. It is one of the useful practical algorithms for parameter estimation and avoids parameter drift instability. The parameters used for producing reference trajectory that are brought in Appendix are taken from Do et al. (2004a, 2004b). In our simulation, the initial values in parameter estimation loop

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Fig. 10. Tracking orientation errors (θe, ψe).

Fig. 11. Control surge force, pitch torque and yaw torque (Fu, Tq and Tr).

were picked as 40% of parameter values of the reference system at first, then value of parameters were updated such that the system error converges to zero. In the simulation process, based on Sections 4 and 5 the control parameters are taken as:

δ0 ¼ 0:8; δ1 ¼ δ2 ¼ 0:4; γ 1 ¼ γ 2 ¼ γ 3 ¼ 2; ρ1 ¼ ρ2 ¼ ρ3 ¼ 5

ð39Þ

Selected parameters, design constants and other external conditions are kept the same for both experiments of Sections 6.1 and 6.2.

6.1. Simulation result of part 1 Results are shown in Figs. 2–6 for controller part. 1. Figs. 2 and 3 illustrate the spatial and planar views of the actual and reference positions when the AUV intended to track a reference trajectory as given in Eq. (37). Position and orientation tracking errors are shown in Figs. 4 and 5. From these figures, it is observed that after 50 s, the AUV tracked the desired trajectory. Fig. 6 demonstrates the control surge force, pitch and yaw torque that are not bounded and can be saturated. A glimpse at figures above (Figs. 2–6) verified that desired trajectory does not need to be of a particular type. In fact, proposed adaptive controller can track any sufficiently smooth curve parameterized by time.

6.2. Simulation result of part 2 Results are shown in Figs. 7–11 for controller part 2. All conditions in terms of reference trajectory generating, initial parameters and so forth are kept same. Figs. 7 and 8 illustrate the spatial and planar views of the actual and reference positions when the AUV is tracking the above-mentioned reference trajectory. Position and orientation tracking errors are shown in Figs. 9 and 10. From these figures, it is observed that after 60 s, the AUV tracked the desired trajectory. Fig. 11 demonstrates the control surge force, pitch and yaw torque that are bounded and avoided to be saturated. Comparing the results of two parts demonstrated that using saturation functions in control design of the part 2 caused control surge force, control pitch and yaw torque have bounded rather than part 1. This resulted actuators being safe and not being damaged, while the errors still small and tracking is also satisfactory.

7. Conclusion In this paper, a 6-DOF trajectory tracking control scheme was presented for an underactuated underwater vehicle, in the presence of parametric uncertainties. An adaptive controller was designed using the back-stepping method and Lyapunov theory. Projection algorithm was used to update the estimation of unknown parameters, while

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parameter drift instability was avoided. Despite of many previous literatures, desired trajectory did not have to be a particular type and could be any sufficiently smooth bounded curve parameterized by time. Simulation results verified the asymptotical convergence of position and orientation tracking errors even if the reference trajectory comprised a straight line. In Section 4.1, it was revealed that in the closedloop system including the adaptive law, the actual trajectory asymptotically converges to the reference trajectory. However, the control signals were not bound and might damage actuators of the vehicle. To solve this problem, an improved controller was developed based on saturation function, in Section 3.2. As a matter of fact, saturation functions were employed to avoid actuators' saturation, to keep the estimations bounded. Hence, the result was global in state space, which was defined by the assumed upper limit on the velocities. Results clearly verified the satisfying performance of our controllers, particularly in the case of using saturation functions that made actuators not be saturated. The only inappropriate point about the proposed controllers is the explosion of complexity problem that is unavoidable in back-stepping designs. Another approach is now under study, relying exclusively on dynamic surface control (DSC) method. In this method a new algorithm for adaptive back-stepping control of nonlinear uncertain systems is proposed, while repeating differentiation of standard back-stepping algorithms is removed using a LPF filter. In order to complete this study, a collision avoidance scenario and environmental disturbances must be explicitly addressed, with the introduction of Ant Colony Algorithm and Neuro-Fuzzy network systems, respectively.

Acknowledgements We thank Mr. Ahmad Dehghanpoor for assistance with proof reading the article and providing language help.

Appendix Table A1 gives AUV parameters which have been employed for simulations. Definition of persistently exciting conditions: Persistently exciting conditions can be stated as following: There exists a positive constant σr, σq such that, for any pair of (t,t0), 0 rt r t 0 o1: 8R < t r 2 ðτÞ dτ Z σ r ðt  t 0 Þ t0 d Rt 2 : t0 qd ðτÞ dτ Z σ q ðt  t 0 Þ

Table A1 Rigid body and hydrodynamic parameters of the AUV. Parameter

Symbol

Value

Unit

Mass Length Mass þ Added mass in surge Mass þ Added mass in sway Mass þ Added mass in heave Inertia þAdded inertia in roll Inertia þAdded inertia in pitch Inertia þAdded inertia in yaw Surge linear drag Sway linear drag Heave linear drag Roll linear drag Pitch linear drag Yaw linear drag

M L m11 ¼ m X u_ m22 ¼ m Y v_ m33 ¼ m Z w_ m44 ¼ I x  K p_ m55 ¼ I y  M q_ m66 ¼ I z  N r_ d11 ¼  X u d22 ¼  Y v d33 ¼  Z w d44 ¼  K p d55 ¼  M q d66 ¼  N r

1089.8 5.56 1116 2133 2133 36.7 4061 4061 25.5 138 138 10 490 490

kg m kg kg kg kg m2 kg m2 kg m2 kg s  1 kg s  1 kg s  1 kg m2 s  1 kg m2 s  1 kg m2 s  1

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The persistently exciting condition implies that the reference trajectory must be curved and indeed excludes a straight-line reference trajectory, hence, it substantially limits the practical use of the control systems. In our method, Assumption 1 is used instead of the above mentioned condition and the reference yaw velocity does not have to satisfy a persistently exciting condition as was often required in the previous literature. Hence, the reference trajectory is allowed to be a curve including a straight line and a circle which generalize our method's applications.

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