Robust fuzzy 3D path following for autonomous underwater vehicle subject to uncertainties

Author’s Accepted Manuscript Robust fuzzy 3D path following for autonomous underwater vehicle subject to uncertainties Xianbo Xiang, Caoyang Yu, Qin Zhang

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To appear in: Computers and Operation Research Received date: 17 November 2015 Revised date: 6 May 2016 Accepted date: 20 September 2016 Cite this article as: Xianbo Xiang, Caoyang Yu and Qin Zhang, Robust fuzzy 3D path following for autonomous underwater vehicle subject to uncertainties, Computers and Operation Research, http://dx.doi.org/10.1016/j.cor.2016.09.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust fuzzy 3D path following for autonomous underwater vehicle subject to uncertainties Xianbo Xiang a,∗, Caoyang Yu a , Qin Zhang b a School b

of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, 1037, Luoyu Road, 430074, Wuhan, China State Key Lab of Digital Manufacturing, Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

Abstract This paper addresses a three-dimensional (3D) path following control problem for underactuated autonomous underwater vehicle (AUV) subject to both internal and external uncertainties. A two-layered framework synthesizing the 3D guidance law and heuristic fuzzy control is proposed to achieve robust adaptive following along a predefined path. In the first layer, a 3D guidance controller for underactuated AUV is presented to guarantee the stability of path following in the kinematics stage. In the second layer, a heuristic adaptive fuzzy algorithm based on the guidance command and feedback linearization Proportional-Integral-Derivative (PID) controller is developed in the dynamics stage to account for the nonlinear dynamics and system uncertainties, including inaccuracy modelling parameters and time-varying environmental disturbances. Furthermore, the sensitivity analysis of the heuristic fuzzy controller is presented. Against most existing methods for 3D path following, the proposed robust fuzzy control scheme reduces the design and implementation costs of complicated dynamics controller, and relaxes the knowledge of the accuracy dynamics modelling and environmental disturbances. Finally, numerical simulation results validate the effectiveness of the proposed control framework and illustrate the outperformance of the proposed controller as well. Keywords: Autonomous underwater vehicle (AUV), Path following, 3D guidance, Robust fuzzy control.

1. Introduction Metaheuristics have a substantial history in fine-tuning learning algorithms for robots and make a significant impact on the robotics application, in which fuzzy logic control is one of the main approaches to achieve artificial intelligent functionalities and capabilities [1, 2, 3, 4]. Numerous applications of metaheuristics algorithms in robotic systems include mobile wheeled robots [5], emergency service robot [6], mobile robotic manipulators [7], unmanned aerial vehicles [8], and autonomous surface vehicles [9], just to name a few. In recent decades, emerging applications of a new underwater robotic system, named as underwater robotic vehicle (AUV) [10, 11, 12], rise up for replacing human beings on board in the unpredictable and dangerous undersea environment, to explore and exploit abundant ocean resources, for instance, 3D seafloor imaging [13], geomorphologic mapping [14], rapid environmental assessment [15], and underwater pipeline inspection [16]. Beyond achieving various underwater applications, the motion control of AUV is one of the essential problems in attaining underwater operational objectives [17, 18]. However, the modelling parameters of the underwater vehicle is difficult to be accurately acquired and the vehicle is also vulnerable by environmental disturbances including ocean currents and waves [19, 20]. These internal and external uncertainties along with ∗ Corresponding author at: School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, China. Tel.: +8627-87543157; Fax: +86-27-87542146. Email: [email protected]

the nonlinearity of the vehicle dynamics render the AUV control problem difficult. Furthermore, most of AUVs are designed with underactuated configurations [21], which have more degrees of freedom (DOF) to be controlled than the number of the independent control inputs, in order to keep actuator efficiency while traveling at high speed [22]. In this sense, control of underactuated AUVs has become one of the most challenging tasks in the robotic community. Metaheuristics algorithms including neural network and genetic algorithm have been applied to the field of AUV motion control [23, 24, 25]. Obviously, neural network requires off-line learning in advance or on-line learning to achieve weight adjustments. The main challenge in this process is to calculate the optimal weight changes from the system input and output as well as the reference trajectory for the system. Furthermore, ensuring the stability of such controllers is a challenging work where substantial research still remains to be done [23]. For genetic algorithm, it needs to accomplish complex encoding and decoding calculation, which has high computational complexity of non-dominated sorting and is lack of elitism [26, 27]. However, fuzzy logic control can provide a formal methodology to represent and implement human heuristic knowledge on how to control a system subject to uncertainties, which makes it become a popular approach to attain artificial intelligent functionalities and capabilities. Considering the control problem of underwater vehicle moving in the vertical or horizontal two-dimensional (2D) plane, an adaptive fuzzy sliding mode controller against modelling imprecisions and external disturbances is proposed

Preprint submitted to Computers & Operations Research, SI: Methaheuristics & robotics

September 22, 2016

stage to account for the modelling nonlinearity, and a heuristic fuzzy logic method is adopted to adaptively tune the PID gains online and offer the robustness with respect to the system uncertainties, including inaccuracy model parameters and time-varying environmental disturbances. Against some conventional path following schemes based on backstepping, RISE or cascade techniques, the proposed two-layered control framework reduces the design and implementation costs for 3D robust path following controller, and relaxes the prior knowledge of the accurate dynamics modelling and environmental disturbances. Furthermore, by adopting the sensitivity analysis and comparison with the numerical simulation results of several existed control algorithms, the effectiveness of the proposed twolayer control framework is validated and the outperformance of the proposed controller is well illustrated. The rest of the paper is organized as follows. The dynamics model of AUV is firstly described and the 3D path following problem is stated in the next section. In Section 3, a two-layered control framework is proposed, in which an adaptive fuzzy PID controller synthesizing 3D guidance laws is designed for robust path following of underactuated AUVs subject to uncertainties. Subsequently, the performance of the proposed controller is illustrated through numerical simulation results in Section 4, and Section 5 makes conclusions for completed work and prospects to warrant further research. Note that we summarize all the abbreviations and the corresponding full names in Table 1.

for the depth control of remotely operated underwater vehicles in [28]. In [29], a simple fuzzy controller taking the pitch angle error and error difference as its inputs, and the actuator voltage as its output is designed to accomplish fixed-depth control. A simplified depth control scheme to design a single input fuzzy logic controller for an underwater vehicle subject to ocean wave disturbances is presented in [30]. In [31], a sliding mode fuzzy controller for the guidance and control of an AUV is designed for way-points tracking and square path tracking under ocean currents disturbance in 2D horizontal plane. Yet, it is a challenging step to design the AUV controller from the decoupling horizontal or vertical motion in 2D space to the coupling 3D motion. For 3D trajectory tracking, an on-line neuro-fuzzy controller is proposed for an ODIN AUV to follow a desired straight-line trajectory from one way-point to another in 3D space [32]. An adaptive fuzzy PID controller based on nonlinear structure is proposed for an AUV with two decoupled and independent control planes to track the two channels of heading and depth, respectively [33]. By adopting some complex control design methods, a bunch of 3D path following controller for underactuated AUVs is proposed, in which the backstepping technique is one of the classic approaches, for instance, nonlinear 3D path following control for underactauted AUVs is first proposed by using backstepping techniques to modify the kinematics controller to explicitly accommodate the accuracy vehicle dynamics [34]. Line-of-sight (LOS) guidance based path following control is proposed for both fully actuated and underactuated AUVs by using the backstepping technique [35]. Through combining a nonlinear backstepping based control law with adaptive switching supervisory strategy, 3D path following for underactuated autonomous vehicles is achieved in the presence of large modelling parametric uncertainty [36]. A robust adaptive control strategy adopts nonlinear backstepping techniques and Lipschitz continuous projection algorithms to force an underactuated AUV to follow a predefined path despite of the presence of environmental disturbances and vehicle’s unknown physical parameters [37]. Recently, a novel control structure based on continuous robust integral of the sign of the error (RISE) is developed for a fully-actuated AUV following a desired path to compensate for system uncertainties and sufficiently smooth bounded exogenous disturbances in [38], and a new path following controller for underwater robots is proposed by using the Lagrange multipliers and the cascade interconnection stability [39]. Nevertheless, the 3D path following algorithms based on backstepping, RISE or cascade control techniques, combined with nonlinear robust adaptive methods to account for uncertainties, are quite complicate and difficult to be implemented. In this paper, an effective two-layered control framework by resorting to the 3D guidance law and heuristic fuzzy logic is proposed, in order to achieve simple and easily applicable 3D path following for underactuated AUVs subject to uncertainties. In the first layer, a 3D guidance controller is presented to guarantee the path following stability of underactuated AUVs in the kinematics stage. By synthesizing the guidance commands in the second layer, a simplified PID controller based on feedback linearization techniques is developed in the dynamics

Table 1: Abbreviation and full name

Abbreviation 3D AUV PID DOF 2D LOS RISE NB NM NS ZE PS PM PB MSE MAE TMF GMF

Full name Three-Dimensional Autonomous Underwater Vehicle Proportional-Integral-Derivative Degree of Freedom Two-Dimensional Line of Sight Robust Integral of the Sign of the Error Negative Large Negative Medium Negative Small Zero Positive Small Positive Medium Positive Large Mean Square Error Mean Absolute Error Triangular Membership Function Gaussian Membership Function

2. Problem Statement This section presents the dynamics model of an underactuated AUV moving in 3D space, and formulates the motion control problem of 3D path following for underactuated AUV subject to internal and external uncertainties. 2

2.1. AUV reference frames As illustrated in Fig.1, in order to explicitly describe an AUV moving in 3D space, the following coordinate frames and corresponding notations will be used throughout the paper. {I}: Inertial frame (earth fixed) {B}: Body frame (body fixed) {W}: Flow frame (body fixed, x-axis along the direction of the AUV composite speed [40]) {F}: Path frame (attached to the reference point P p on the path, x-axis along the tangent direction of the path [41]). The position vector of the AUV p = [x, y, z]  and the attitude vector Θ = [φ, θ, ψ] are described in frame {I}. The translational speed vector v = [u, v, w]  and angular speed vector ω = [p, q, r] are described in frame √ {B}. The AUV composite speed U = [U, 0, 0] with U = u2 + v2 + w2 is described in frame {W}. The composite speed of the reference point P p on the path Up = [U p , 0, 0] is described in frame {F}.

Figure 2: 3D path following

the kinematic transformation, inertia, Coriolis, damping matrices and the restoring vector, respectively. Assuming that the weight of the body and buoyancy force are equal and the center of buoyancy is located in the vertical plane, and neglecting the nonlinear hydrodynamic damping terms and the roll motion, the dynamics model of AUVs in (1) can be simplified into the following kinematics and kinetics equations [42] [43]. 1. Kinematics equations ⎧ ⎪ x˙ = u cos(ψ) cos(θ) − v sin(ψ) + w cos(ψ) sin(θ) ⎪ ⎪ ⎪ ⎪ ⎪ y ˙ = u sin(ψ) cos(θ) + v cos(ψ) + w sin(ψ) sin(θ) ⎪ ⎪ ⎨ z ˙ = −u sin(θ) + w cos(θ) (2) ⎪ ⎪ ⎪ ⎪ ⎪ θ˙ = q ⎪ ⎪ ⎪ ⎩ ψ˙ = r/ cos(θ) 2. Kinetics equations

Figure 1: AUV reference frames

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2.2. AUV modelling The type of AUV considered in this paper is underactuated, as there are no actuators in the sway and heave directions. In general, most of the underwater vehicles are designed to guarantee the hydrodynamic restoring force to be large enough in the roll direction, such that the vehicle can be controlled without the roll torque control effort [37]. Hence, the available control inputs for the vehicle in question are the force τ u in the surge direction, and torques τ q , τr in the pitch and yaw directions. The kinematics and kinetics of the underactuated AUV can be described as follows:  η˙ = J(Θ)ν (1) M˙ν = −C(ν)ν − D(ν)ν − g(Θ) + τ + τw

m33 d11 22 u˙ = m m11 vr − m11 wq − m11 u + τwv m11 d22 v˙ = − m22 ur − m22 v + m22 d33 τww 11 w˙ = m m33 uq − m33 w + m33

q˙ = r˙ =

τu +τwu m11

WGML sin(θ) m33 −m11 d55 m55 uw − m55 q − m55 d66 τr +τwr m11 −m22 m66 uv − m66 r + m66

(3) +

τq +τwq m55

where W denotes the gravity of the AUV, m (·) express hydrodynamic derivatives of the system, d (·) capture hydrodynamic damping effects, and GM L = zg − zb with zg and zb denoting the coordinates of the center of mass and buoyancy, respectively. 2.3. Problem formulation As illustrated in Fig.2, an underactuated AUV follows a predefined 3D spatial path p p (s) = [x p (s), y p (s), z p (s)] ∈ R3 , which is parameterized by the variable s. In fact, the path is also can be considered as being generated and parameterized by a virtual moving point P p . The path frame {F} can be built according to the virtual reference point P p on the path. Let P be the center of mass of the AUV, and define ε = [x e , ye , ze ] as the path following error vector between P p and P in frame {F}, where xe , ye , ze denotes the along-track error, the cross-track error and the vertical-track error in the x-axis, y-axis and z-axis of frame {F}, respectively.

where η = [p, Θ] denotes the posture of the AUV in 3D space, ν = [v, ω] denotes the AUV speed, τ = [τ u , 0, 0, 0, τq, τr ] denotes the control force and torques acting on the underactuated AUV, and τw = [τwu , τwv , τww , τwp , τwq , τwr ] denotes the time-varying environmental disturbance forces and torques acting on the vehicle. The terms J(Θ), M, C, D and g(Θ) denotes 3

ing error vector ε can be expanded as

The problem of robust 3D path following control for an underactuated AUV can be formulated as follows: Given a predefined path p p (s) in the 3D space, develop robust feedback control laws for the force and torques acting on an underactuated AUV subject to uncertainties, such that its center of mass P asymptotically converges to the virtual reference point P p and moves long the spatial path p p (s), while its composite speed U aligns with the tangent vector of the path and tracks a desired profile U d at the same time.

⎧ ⎪ xe = cos(ψ p ) cos(θ p )(x − x p ) + sin(ψ p ) cos(θ p )(y − y p ) ⎪ ⎪ ⎪ ⎪ ⎪ − sin(θ p )(z − z p ) ⎪ ⎪ ⎨ y = − sin(ψ p )(x − x p ) + cos(ψ p )(y − y p ) ⎪ e ⎪ ⎪ ⎪ ⎪ z = cos(ψ ⎪ e p ) sin(θ p )(x − x p ) + sin(ψ p ) sin(θ p )(y − y p ) ⎪ ⎪ ⎩ + cos(θ p )(z − z p ) (7) Consider the following positive definite and radially unbounded control Lyapunov function: Vε =

3. 3D Robust Path Following Control The two-layered control framework proposed for robust 3D path following of an underactuated AUV is depicted in Fig.3. It consists of a 3D guidance-based layer in the kinematics stage and a robust fuzzy PID layer in the dynamics stage. In the first layer, the desired surge speed u d , azimuth angle χd , and elevation angle υ d for an underactuated AUV are obtained with the assistant of 3D guidance laws. By setting the 3D guidance commands of desired speed/orientations derived in the first layer as the input of the second layer, a feedback linearization PID controller is designed to account for the complicated modelling nonlinearity, and then a heuristic fuzzy logic method is adopted to adaptively tune the PID gains online and offer the robustness to derive the proper force and torques acting on the underactuated AUV subject to inaccurate model parameters and time-varying environmental disturbances.

V˙ ε =xe (Ud cos(ψr ) cos(θr ) − Up ) + ye Ud sin(ψr ) cos(θr ) − ze Ud sin(θr )

U p = Ud cos(ψr ) cos(θr ) + k x xe where the LOS guidance angles are set as 

⎧ −ky ye ⎪ ⎪ ⎪ = arctan ψ r ⎨ Δy

⎪ ⎪ ⎪ ⎩ θr = arctan kz ze

(10)

(11)

Δz

with the guidance variables Δ y > 0 and Δz > 0. The control gains k x , ky , kz are positive constants. By replacing the auxiliary control law in (10) and the guidance law in (11) into (9), it concludes that V˙ ε is negative definite, such that the 3D path following error tends to zero asymptotically in the kinematics stage. Subsequently, by considering the rotation relationships among inertial frame {I}, flow frame {W} and path frame {F}, the desired azimuth angle υ d and elevation angle χ d of the desired speed vector of the AUV to be oriented in 3D path following can be derived as follows:

(4)



Similarly, define the angles θ p , ψ p for the virtual reference point P p on the predefined path p p (s) as

υd = arcsin(sin(θ p ) cos(θr ) cos(ψr ) + cos(θ p ) sin(θr )) χd = atan2(χdy , χdx )

(12)

where χdy = cos(ψ p ) sin(ψr ) cos(θr ) − sin(θ p ) sin(θr ) sin(ψ p ) + sin(ψ p ) sin(ψr ) cos(θ p ) sin(θr ), χdx = − sin(ψ p ) sin(ψr ) cos(θr ) − sin(θ p ) sin(θr ) sin(ψ p ) + sin(ψ p ) cos(ψr ) sin(θ p ) cos(θr ). The detailed derivation can be attained through related references therein [41] [44] [22]. Considering  the composite speed of the virtual reference

(5)

x p +y p

where xp = dx p /ds, yp = dy p /ds, and zp = dz p /ds. Hence, the tracking error vector ε of 3D path following control can be built in path frame {F} as follows  ε = xe , ye , ze  = RF (p − p p )

(9)

where U d is the desired composite speed of the AUV. Let the auxiliary control input of the virtual moving reference point P p be chosen as

In this part, 3D guidance laws for path following control of an underactuated AUV in 3D space is presented in the kinematics layer, which is instrumental in designing a simple fuzzy PID controller in the dynamics layer to achieve robust adaptive 3D path following. Let the orientation of the underactuated AUV be characterized by the azimuth angle χ and elevation angle υ. There is

⎧ ⎪ ⎪ ψ p = atan2 yp , xp ⎪ ⎪   ⎨ −zp ⎪ ⎪ ⎪ ⎪ ⎩ θ p = arctan √  2  2

(8)

The derivative of (8) can be directly computed as

3.1. 3D guidance for path following

⎧ ⎪ χ = atan2 (˙y, x˙)  ⎪ ⎪ ⎨ ⎪ −˙z ⎪ ⎪ ⎩ υ = arctan √ x˙2 +˙y2

1  ε ε 2

point U p = xp 2 + yp 2 + zp 2 s˙ , it yields from (10) that an extra DOF coming from the virtual reference point to collaborate with the path following controller is

(6)

s˙ =

where RF := Rz (ψ p )Ry (θ p ) denotes the rotation matrix from path frame {F} to inertial frame {I}. Therefore, the path follow4

Ud cos(ψr ) cos(θr ) + k x xe  xp 2 + yp 2 + zp 2

(13)

Figure 3: Control framework of 3D path following

However, the dynamics equation of the underactuated AUV in (3) has a significant nonlinear characteristics, which hinders the path following control design to render the control efforts in the dynamics layer. Herein, feedback linearization technique is firstly adopted to cancel the nonlinear terms in the system dynamics, and then an adaptive PID controller is designed to offer the robustness with respect to the system uncertainties, including inaccurate model parameters and time-varying environmental disturbances. First, based on the guidance commands in (12) and (15), we can define the PID tracking error vector e PID = [eu , eθ , eψ ] as follows ⎧ eu (k) = ud (k) − u(k) ⎪ ⎪ ⎪ ⎪ ⎨ eθ (k) = υd (k) − υ(k) (16) ⎪ ⎪ ⎪ ⎪ ⎩ e (k) = χ (k) − χ(k) ψ d

In fact, this extra DOF for controller design reveals the essential characteristics of path following control which is different from that of the trajectory tracking control [45]. Based on the flow frame {W} in Fig.2, the relationship between the composite speed U and the surge, sway and heave speed is written as ⎧ ⎪ u = U cos α cos β ⎪ ⎪ ⎨ v = U sin β (14) ⎪ ⎪ ⎪ ⎩ w = U sin α cos β where α is the angle of attack and β is the sideslip angle. According to (14), given the desired composite speed U d , the desired surge speed u d can be written as u ud = U d √ 2 u + v2 + w2

(15)

and the corresponding error ratio vector ec PID = [ecu , ecθ , ecψ ] can be written as ⎧ eu (k) − eu (k − 1) ⎪ ⎪ ⎪ ecu (k) = ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ eθ (k) − eθ (k − 1) ⎨ (17) ecθ (k) = ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ec (k) = eψ (k) − eψ (k − 1) ⎩ ψ T where k is the sampling index and T is the sampling period in the PID control loop. Consequently, the 3D path following control law in the dynamics stage for an underactuated AUV with model (3) can be achieved via fuzzy PID control method based on feedback linearization techniques as follows: ⎧ ˜ 22 vr + m ˜ 33 wq + d˜11 u τu = −m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +m ˜ 11 (k pu eu + kiu Σeu + kdu ecu ) ⎪ ⎪ ⎪ ⎪ ˜ sin(θ) ⎪ ˜ GM ⎨ τq = −(m ˜ 33 − m ˜ 11 )uw + d˜55 q + W L (18) ⎪ ⎪ ⎪ +m ˜ 55 (k pq eθ + kiq Σeθ + kdq ecθ ) ⎪ ⎪ ⎪ ⎪ ⎪ τr = (m ⎪ ˜ 22 − m ˜ 11 )uv + d˜66 r ⎪ ⎪ ⎩ +m ˜ 66 (k pr eψ + kir Σeψ + kdr ecψ )

Hence, the guidance commands in (12) and (15) for the desired speed/orientations, and the auxiliary control input for the virtual reference point in (13) are derived in order to guarantee the path following convergence of the underactuated AUV in the first layer. These command signals will be used to design the actual control efforts including the surge force and yaw/pitch torques in the following second layer. 3.2. Fuzzy PID controller for 3D path following The classic PID controller has been widely used in various control plants due to its simple control design and easy implementation, the parameters of classic PID controller are stationary and not adaptively tuned online for the nonlinear system with unpredictable parameter variations and external perturbations. Therefore, it is necessary to tune the parameters of the PID controller automatically for the nonlinear plants, such as industrial robots, ships, and AUVs. As aforementioned, fuzzy logic control can represent and implement human heuristic knowledge on how to control a system subject to uncertainties, which is a convenient method for constructing robust adaptive controllers by using heuristic information obtained from experiences. In this sense, the advantages of fuzzy logic and PID controller can be incorporated into a single intelligent controller, called fuzzy PID controller in order to achieve robust performance for the control systems [46].

˜ means the modelling parameters known in-prior are where (·) inaccurate. By replacing (18) into (3), it reveals that the resulted error dynamics consist of a 2nd-order linear system if the modelling parameters are accurate. 5

In order to account for the inaccurate modelling parameters and environmental disturbances, fuzzy logic is adopted in (18) such that the control gains, k p∗ , ki∗ , kd∗ , of the path following PID controller can be self-tuned around the initial values, k p∗0 , ki∗0 , kd∗0 , as follows ⎧ k p∗ = k p∗0 + Δk p∗ (e∗ , ec∗ ) ⎪ ⎪ ⎪ ⎪ ⎨ ki∗ = ki∗0 + Δki∗ (e∗ , ec∗ ) ⎪ ⎪ ⎪ ⎪ ⎩ k = k + Δk (e , ec ) d∗ d∗0 d∗ ∗ ∗

3.3.2. Fuzzification Let each input of the controller have seven types of fuzzy subsets, namely NB, NM, NS, ZE, PS, PM and PB, representing negative large, negative medium, negative small, zero, positive small, positive medium, and positive large, respectively. As illustrated in Fig.5, all the fuzzy subsets for the tracking error e∗ and the corresponding error ratio ec ∗ are designed through classic triangular membership function (TMF) in fuzzy logic.

(19)

where Δk p∗ , Δki∗ , and Δkd∗ with ∗ ∈ {u, q, r} are outputs of the designed fuzzy logic module, called adaptive incremental gains. In fact, the adaptive PID control gains based on fuzzy logic offer the robustness to the AUV control system subject to internal and external uncertainties. The fuzzy PID controller will be implemented in the following sections through three steps, i.e., fuzzification, fuzzy inference and defuzzification. 3.3. Implementation of fuzzy PID controller The fuzzy logic system is an inference system to mimic the human thinking, which consists of a fuzzifier, some fuzzy IFTHEN rules, a fuzzy inference engine and a defuzzifier. Fuzzifier converts each piece of input data degrees of membership by a lookup in one or several membership functions. Fuzzification matches the input data with the conditions of the fuzzy rules. In the fuzzy inference engine, the corresponding rules are activated and all the activations are synthesized by using classic max-min operations. Finally, the output value is calculated in the defuzzifier [47].

Figure 5: Membership function of input variable e∗ and ec∗

3.3.3. Fuzzy inference The self-tuning rules for the control gains of the proposed fuzzy PID controller are established in Table 2, Table 3, Table 4 for Δk p , Δki , Δkd , respectively. For each table, the first column represents the different fuzzy subsets of the tracking error vector ePID , the first row represents the different fuzzy subsets of the tracking error ratio vector ec PID , and other cells in the table are outputs of IF-THEN rules in different cases, namely Δk p , Δki , and Δkd . In this paper, the output rules of the fuzzy controller in the pitch and yaw directions are set as the same in a simplified manner, where the adopted fuzzy inference methods are min implication and max aggregation.

3.3.1. Fuzzy Structure As described in Fig.3, the PID control gains are adaptively tuned around an initial value based on the fuzzy inference, which provides a heuristic nonlinear mapping from the tracking error vector and the corresponding error ratio vector to the updated control gains [48]. The fuzzy controller adopts doubleinput and three-output inference structure via Mamdani model consisting of triangular fuzzification, min implication, max aggregation and centroid defuzzification as shown in Fig.4, where the double inputs are the tracking error e ∗ in (16) and the corresponding error ratio ec ∗ in (17), and the three outputs are selftuning control gains Δk p∗ , Δki∗ , and Δkd∗ in (19).

Δk p NB NM NS ZE PS PM PB

Table 2: Fuzzy control rules for Δkp

NB PB PB PB PB PS PS ZE

NM PB PB PM PM PS ZE ZE

NS PM PM PM PS ZE NS NM

ZE PM PS PS ZE NS NM NM

PS PS PS ZE NS NS NM NM

PM ZE ZE NS NM ZM NM NB

PB ZE NS NS NM NM NB NB

Heuristically, when the absolute value of the tracking error is extremely large, we should choose large k p and small kd in order to prevent the error from persistent increasing, and meanwhile set ki be zero to avoid the wind-up effect resulting from the integrator; when the absolute value of the tracking error is

Figure 4: Mamdani model structure

6

efforts in the previous step is driving the guidance error towards zeros sharply, i.e., AUV is approaching quickly to the desired path and is likely to cross over the desired path if the control effort still keeps. In this case, the corresponding increments NS, PS, and ZE should be added to k p , ki , and kd , respectively, which means that the updated gain k p will slightly decrease (NS) to prevent the tracking system from overshoot, k i will slightly increase (PS) to have a better steady-state error of the tracking system and kd will keep the last value (ZE) in order to avoid the potential oscillation.

Table 3: Fuzzy control rules for Δki

Δki NB NM NS ZE PS PM PB

NB NB NB NB NM NM ZE ZE

Δkd NB NM NS ZE PS PM PB

NB PS PS ZE ZE ZE PS PS

NM NB NB NM NM NS ZE ZE

NS NM NM NS NS ZE PS PS

ZE NM NS NS ZE PS PS PM

PS NS NS ZE PS PS PM PM

PM ZE ZE PS PM PM PB PB

PB ZE ZE PS PM PB PB PB

PM NM NS NS NS ZE PB PS

PB PS ZE ZE ZE ZE PB PB

3.3.4. Defuzzification During the defuzzification procedure, the unambiguous output of the fuzzy controller can be calculated by adopting the conventional centroid strategy. As illustrated in Fig.6(a), the incremental gains Δk p , Δki , and Δkd are different from each other by taking the different rules in Table 2, Table 3 and Table 4.

Table 4: Fuzzy control rules for Δkd

NM NS NS NS NS ZE NS PM

NS NB NB NM NS ZE PS PM

ZE NB NM NM NS ZE PS PM

PS NB NM NS NS ZE PS PS

medium, we should set k p be small so as to avoid overshoot, and choose proper-size k i and kd to guarantee response speed of the system; when the absolute value of the tracking error is small, we should set k p and ki be large to guarantee good stability of the PID control system. In addition, while the absolute value of error ratio is small/large, k d should be set large/small to keep the system from oscillation and improve the anti-perturbation of the system. In this way, the formulation of 49 inference rules in the table for 3D path following control of the AUV can be interpreted as follows: (1) If the speed/orientation tracking error e ∗ and the corresponding error ratio ec ∗ are NB and NB, implying that, under the previous control efforts, the tracking error between actual speed/orientation signals and guidance speed/orientation commands is sharply deviating, and the AUV continues to move far away from the desired path. In this case, the proper increments PB, NB, and PS should be added to k p , ki , and kd , respectively, such that the updated gain k p largely increases (PB) and kd slightly increases (PS) in order to quickly increase the control efforts and prevent the tracking error from increase continuously. The self-tuned gain k i tends to zero because of a large decrease (NB) in order to avoid integral saturation effects of the actuators. (2) If the speed/orientation tracking error e ∗ and the corresponding error ratio ec ∗ are PM and NS, implying that the control force and torques of the fuzzy PID controller in the previous step rendered the guidance error approaching to zero, i.e., AUV is moving towards the desired path. In this case, the corresponding increments NS, PS, and PS should be added to k p , ki , and kd , respectively, which means the updated gain k p will slightly decrease (NS), ki and kd will slightly increase (PS) in order to prevent the system from overshoot while keeping a good path following performance. (3) If the speed/orientation tracking error e ∗ and the corresponding error ratio ec ∗ are NS and PB, implying the control

Figure 6: Fuzzy control surface of the incremental gains

It clearly shows that these three incremental gains are all represented by the nonlinear functions with respect to the tracking error and the corresponding error ratio, which indicates that the self-tuned gains of the PID controller will change according to the fuzzy rules by considering both the tracking error and the corresponding error ratio. The adaptability of the control gains based on the fuzzy logic rules derived from the human knowledge and experience, will offer the fuzzy PID controller robustness performance for the 3D path following system subject to uncertainties. 3.4. Sensitivity analysis In order to validate the control performance of the sevenvalue TMF fuzzy controller, the sensitivity analysis and comparison of fuzzy PID controller response are carried out under the case of TMF versus Gaussian membership function(GMF), and the case of seven-value fuzzy subsets versus five-value fuzzy subsets. All the output surfaces are shown in Fig.6. We 7

chose four groups of valid inputs to analyze the sensitivity of the fuzzy controller as listed in Table 5.

Table 6: Parameters of the AUV dynamics model

Parameter W/N GML /m m11 /kg m22 /kg m33 /kg m55 /kgm2 m66 /kgm2 d11 /kgs−1 d22 /kgs−1 d33 /kgs−1 d55 /kgm2s−1 d66 /kgm2s−1

Table 5: Evaluation and comparison of fuzzy controller

Case

Condition

1

e=0.07 ec=-0.1

2

e=-0.07 ec=-0.1

3

e=-0.07 ec=0.1

4

e=0.07 ec=0.1

Gain Δk p Δki Δkd Δk p Δki Δkd Δk p Δki Δkd Δk p Δki Δkd

7-valued TMF 1.31 0.00062 1.22 1.96 −0.00111 0.113 1.69 −0.00062 0.113 1.17 0.00111 1.22

7-valued GMF 0.85 0.00574 0.45 1.29 0.00383 0.209 1.15 0.00421 0.209 0.77 0.00613 0.45

5-valued TMF 1.28 0.00072 0.82 1.87 −0.00118 0.109 1.72 −0.00072 0.109 1.15 −0.00118 0.84

Nominal value 10690.9 0.0065 1116 2133 2133 4061 4061 25.5 138 138 490 490

Case 1 value 5880.0 0.0036 613.8 1173.2 1173.2 2233.6 2233.6 14.0 75.9 75.9 269.5 269.5

Case 1 -% 55 55 55 55 55 55 55 55 55 55 55 55

Case 2 value 5880.0 0.0036 948.6 2133 1706.4 2233.6 5888.5 14.0 138 138 392 392

Case 2 -% 55 55 85 100 80 100 145 55 100 100 80 80

• PID 2: conventional PID controller in (24) with inaccurate dynamics parameters

Note that the inputs e and ec are selected near the origin since at this point their effect will dominate the system response including the overshoot, steady-state error and capability to reject disturbances. First, consider the case 1 with e = 0.07 and ec = −0.1. It implies that the control efforts in the previous step is driving the speed/orientation error towards zero asymptotically. The differential gain k d of the seven-value TMF fuzzy controller has a larger increment in order to prevent the system from overshoot. In case 2 where e = −0.07 and ec = −0.1, it implies that the control efforts in the previous step is forcing the speed/orientation error far away from zero in the negative direction. The proportional gain k p of the seven-value TMF fuzzy controller has a larger increment in order to prevent the system from divergence. Similarly, better adjustment in case 3 and case 4 is shown for the seven-value TMF fuzzy controller, which indicates that the seven-value TMF fuzzy controller has better control performance. Hence, the seven-value TMF is chosen for the proposed fuzzy PID controller in this paper.

• Backstepping 1: Backstepping-based controller in (25) with accurate dynamics parameters • Backstepping 2: Backstepping-based controller in (26) with inaccurate dynamics parameters where the backstepping-based controller is adopted in classic literatures about AUV path following, i.e., [34, 40, 41], and the PID controller is widely used in practical engineering applications. 4.1. Example 1: 3D tanh path following Suppose that an underactuated AUV in the mission is required to follow a tanh path which is parameterized as follows ⎧ xd = s ⎪ ⎪ ⎪ ⎪ ⎨ yd = 50 tanh(0.02(s − 2)) ⎪ ⎪ ⎪ ⎪ ⎩z =0 d

(20)

The initial posture of the AUV is [x(0), y(0), z(0), θ(0), ψ(0)] =[-8m, 2m, 4m, 0rad, π/16rad]  and the initial speed is [u(0), v(0), w(0), q(0), r(0)] =[1.5m/s, 0m/s, 0m/s, 0rad/s, 0rad/s]  . The desired composite speed is set as Ud =2.0m/s. In this simulation, we assume that the dynamic parameters are chosen from case 1 and the environmental disturbances are

4. Numerical simulations for 3D path following In order to illustrate the performance of the proposed control framework in this paper, numerical simulations are carried out by taking a classic AUV dynamics model in [42]. The hydrodynamic parameters with accurate values (i.e., nominal value) and inaccurate values in case 1 and case 2 are listed in Table 6. The desired paths in the simulation are 3D tanh-like path and ellipse, which are derivations of the typical straight line and helix adopted in [34, 36, 37, 41]. In fact, the tanh or ellipse path has the time-varying curvature or torsion, which will place more demands on the AUV controller in the algorithm tests. In order to show the robust performance of the proposed fuzzy PID controller, the following four controllers are chosen as benchmarks and comparative counterparts:

τwu = 0.02m11 d(t), τwv = 0.05m22d(t), τww = 0.05m33 d(t), τwq = 0.02m55 d(t), τwr = 0.2m66 d(t) (21) with d(t) = 1 + 0.1 sin(0.2t), t ∈ [160, 350]s. It means the timevarying disturbances occur at 160s and disappear after 350s. Fig.7 and Fig.8 show that the reference path and actual paths of the underactuated AUV adopting five different controllers in 3D space and x-y projection plane, respectively. It can be seen that the AUV starting from the initial position marked by a red diamond finally converges to and follows the desired tanh

• PID 1: conventional PID controller in (23) with accurate dynamics parameters 8

Figure 7: 3D tanh path following

Figure 9: Tanh path following errors

Figure 8: 3D tanh path following projection

Figure 10: Control inputs of tanh path following

path in 3D space by adopting these controllers, but their performances are different from each other. During the first 160s, the tracking paths in Fig.7 for PID 1, Backstepping 1 and Fuzzy PID controllers smoothly converge to the desired path without overshoot, which indicates that the fuzzy PID controller with inaccurate model parameters has a good performance on 3D tanh path following as the classic PID controller and Backstepping-based controller with accurate model parameters. While the paths for PID 2 and Backstepping 2 controllers with inaccurate model parameters converges to the desired path with oscillations, which means that the proposed fuzzy PID controller has a better performance than the PID controller and Backstepping-based controller on dealing with the inaccuracy of the modelling parameters. Once time-varying disturbances in (21) occur from 160s to 350s, there exist both the static inaccuracy of the model and time-varying environmental disturbances for the controllers. Fig.7 and Fig.8 show that the tracking path of Backstepping 2 significantly deviates from the desired path compared with other controllers, which might be resulted by the strong sensitivity and vulnerability to the time-varying disturbances due to the complex computation of the Backstepping-based con-

0.5 q[rad/s]

u[m/s]

3 2 1

0

200

0 −0.5

400

0

200

t[s] 0.5 r[rad/s]

v[m/s]

1 0 −1

0

200

400

0 −0.5

0

200

t[s] 1

4 U[m/s]

w[m/s]

400 t[s]

0 −1

400 t[s]

0

400

200 t[s]

PID 1

PID 2

2 0

0

400

200 t[s]

Backstepping1

Backstepping2

FuzzyPID

Figure 11: Linear and angular speeds of an AUV in tanh path following

troller. Furthermore, as depicted in Fig.9 and Fig.10, the path following errors, force and torques of the fuzzy PID controller are smoother than those of the classic PID controller and the controller based on backstepping techniques subject to environmental disturbances. 9

At time instant t = 350s, the time-varying disturbances vanish which also has a significant impact on the control of the AUV. As shown in Fig.9, there are obvious tracking errors appearing in the cross-track and vertical-track directions. Yet, the tracking errors of the fuzzy PID controller with inaccurate model are also smoother than those of PID 2 and Backstepping 2, and even converge to zero faster than those of PID 1 and Backstepping 1 with accurate model parameters. In addition, the speed change of the fuzzy PID controller is less fluctuant than those of PID 2 and Backstepping 2 in Fig.11. This qualitative analysis also suggests that the fuzzy PID controller has better robustness against disturbances due to the favorable capability of adaptively tuning control gains. Quantitative comparisons including the mean square error (MSE) and mean absolute error (MAE) of the path following control are listed in Table 7. Note that we take the PID 1 controller with accurate dynamics parameters as an idea paradigm, yet it reveals that the fuzzy PID controller subject to both the inaccurate dynamics parameters and the environmental disturbances has the similar performance with the PID 1 controller. Moreover, it shows that the fuzzy PID controller adopted in this paper has better steady and average tracking performances than those of the PID 2 controller and Backstepping 2 controller in the terms of both internal and external uncertainty rejection.

Figure 13: Incremental gain of fuzzy PID controller in the pitch direction

Table 7: Quantitative error comparisons in straight line following

Controller PID 1 PID 2 Backstepping 2 Fuzzy PID

MSE/m2 0.9709 1.0425 2.1170 1.0179

MAE/m 0.3406 0.4416 1.4311 0.4260 Figure 14: Incremental gain of fuzzy PID controller in the yaw direction

By considering both of the qualitative and quantitative analysis of the simulation results, we can conclude that the numerical case study on the tanh path following validates the effectiveness of the proposed control framework, and illustrates the outperformance of the proposed fuzzy PID controller with respect to other benchmark algorithms (PID 2/Backstepping 2) as well.

In Fig.12, Fig.13 and Fig.14, it clearly shows how the control gains of the fuzzy PID controller are adaptively tuned along with the varied tracking error and the error ratio resulted by initial posture differences, inaccurate model parameters and environmental disturbances, such that the controller renders appropriate control efforts to force the surge speed, elevation and azimuth angles to attain the desired speed/orientations and achieve the goal of 3D path following. In the simulation, the initial constant gains in the PID controller are set by combining the critical proportion method with the empirical method. Due to the differences in the restoring moment, uncertainties and initial posture in the pitch and yaw directions, it results in slight differences in the amplitude and frequency of adaptive tuning changes in the pitch and yaw directions. 4.2. Example 2: 3D ellipse path following In this example, the AUV is required to follow a spatial ellipse path which is parameterized as follows ⎧ xd = 60 cos(0.2618s) ⎪ ⎪ ⎪ ⎪ ⎨ yd = 60 sin(0.2618s) ⎪ ⎪ ⎪ ⎪ ⎩ z = sin(0.2618s) + 5 d

Figure 12: Incremental gain of fuzzy PID controller in the surge direction

10

(22)

Different from the tanh path in (20), not only x d and yd , but also zd of the 3D ellipse path in (22) keep changing with time, which demands higher capability of the path following controller. Assume that the desired composite speed is set as Ud =2.0m/s. The initial posture of the AUV is [x(0), y(0), z(0), θ(0), ψ(0)] =[64m, 3m, 0m, 0rad, 3π/4rad] and the initial speed of the AUV is [u(0), v(0), w(0), q(0), r(0)]  =[1.5m/s, 0m/s, 0m/s, 0rad/s, 0rad/s] . Moreover, we assume that the dynamic parameters are chosen from case 2, and the environmental disturbances are described as follows:

In Fig.17, it shows the linear speeds, angular speeds and composite speed of the AUV during the path following, respectively. By examining Fig.16 and Fig.17, it concludes that the fuzzy PID controller has better robustness against disturbances than other counterpart controllers for 3D ellipse path following due to the slighter fluctuation of the tracking errors and smoother changes of speeds. In Fig.17, the sway speed and yaw speed at the final stage are equal to -0.52m/s and 0.03rad/s for ellipse path following instead of zero in Fig.11 for tanh path following, since the AUV only needs the surge speed to follow a nearly straight-line path with zero curvature for tanh path following but it also needs the non-zero sway speed to follow the ellipse path with non-zero curvature. It results in u < U in the final stable stage of the ellipse path following as shown in Fig.17, but the composite speed U has been stabilized at the desired speed U d =2.0m/s. Yet, the surge speed u is equal to the composite speed U d of the AUV as shown in Fig.11, which conforms to the zero sway/heave speed during the tanh path following with zero curvature at the fixed depth as time goes on.

τwu = −0.5m11 d(t), τwv = 0.02m22d(t), τww = 0.02m33d(t), τwq = −0.15m55d(t), τwr = −0.15m66d(t) with d(t) = 1 + 0.1 sin(0.2t), t ∈ [70, 140]s. The actual tracking paths of the underactuated AUV which is requested to follow an ellipse in 3D space are illustrated in Fig.15, where the AUV starting from an initial position finally converges to the desired ellipse path by using PID 1, PID 2, Backstepping 1 and fuzzy PID controller, respectively. The path following errors x e , ye and ze are depicted in Fig.16, where the errors of the fuzzy PID controller have smaller deviations while the disturbances occur and then have faster convergent speed after the disturbances vanish than those of the PID 2 controller and Backstepping 1 controller. In addition, the average errors are listed in Table 8. The qualitative analysis of the path following errors suggests that the fuzzy PID controller has better robustness and tracking performance against inaccurate model parameters and time-varying environmental disturbances.

It is worthwhile to notice that the fully-actuated AUV can directly generate the sway/heave force through the lateral/vertical thruster, but the underactuated AUV without the lateral/vertical thruster must resort to the yaw/pitch torque to indirectly contribute the control effects in the un-actuated sway/heave directions, in order to achieve a non-zero sway/heave speed and enable the underactuated AUV to follow a non-zero curvature path. As clearly depicted in Fig.18, the torque input τ r for the underactuated AUV keeps changing along with the tracking error and error ratio in the first 140s to deal with the timevarying environmental disturbances, and then it is stabilized at -1007Nm in order to afford the control effort to attain a non-zero sway speed during the ellipse path following without external disturbances, despite the absence of the lateral thrusters in the sway direction.

Table 8: Quantitative error comparisons in helix following

Controller PID 1 PID 2 Backstepping 1 Fuzzy PID

MSE/m2 1.1141 1.1128 1.7685 1.0976

MAE/m 0.4564 0.4871 1.1581 0.4665

Figure 16: 3D ellipse path following errors Figure 15: 3D ellipse path following

11

q[rad/s]

u[m/s]

2 0

0

50

100 150 t[s]

0 −1

200

r[rad/s]

v[m/s]

0

0

50

100 150 t[s]

200

−2

100 150 t[s]

6. Acknowledgements

200

The first author was supported by the European Marie Curie ESR Fellowship for three years. This work was partially supported by the EU FP6 FreeSubNet project [under grant 036186], National Natural Science Foundation (NNSF) of China [under Grant 51209100 and Grant 51579111].

200

0

50

100 150 t[s]

0

50

100 150 200 t[s] FuzzyPID

4 U[m/s]

w[m/s]

50

0

2 0 −2

0

2

2

−2

metaheuristics-based approaches, and further test the proposed algorithms for the AUV prototype in open water.

1

4

0

50

100 150 200 t[s] PID 1 PID 2

2 0

Backstepping1

Appendix In the appendix, it lists four controllers adopted in this paper to make comparative simulations with respect to the proposed robust fuzzy PID controller.

Figure 17: Linear and angular speed of an AUV in 3D ellipse path following

• PID 1 controller with accurate model ⎧ ⎪ τu = −m22 vr + m33 wq + d11 u ⎪ ⎪ ⎪ ⎪ ⎪ + m11 (k pu0 eu + kiu0 Σeu + kdu0 ecu ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ τq = −(m33 − m11 )uw + d55 q + WGML sin(θ) ⎪ ⎪ ⎪ + m55 (k pq0 eθ + kiq0 Σeθ + kdq0 ecθ ) ⎪ ⎪ ⎪ ⎪ ⎪ τ = (m ⎪ r 22 − m11 )uv + d66 r ⎪ ⎪ ⎩ + m66 (k pr0 eψ + kir0 Σeψ + kdr0 ecψ )

(23)

• PID 2 controller with inaccurate model ⎧ ˜ 22 vr + m ˜ 33 wq + d˜11 u τu = −m ⎪ ⎪ ⎪ ⎪ ⎪ +m ˜ 11 (k pu0 eu + kiu0 Σeu + kdu0 ecu ) ⎪ ⎪ ⎪ ⎪ ⎪ ˜ sin(θ) ⎪ ˜ GM ⎨ τq = −(m ˜ 33 − m ˜ 11 )uw + d˜55 q + W L ⎪ ⎪ ⎪ +m ˜ 55 (k pq0 eθ + kiq0 Σeθ + kdq0 ecθ ) ⎪ ⎪ ⎪ ⎪ ⎪ τr = (m ˜ 22 − m ˜ 11 )uv + d˜66 r ⎪ ⎪ ⎪ ⎩ +m ˜ 66 (k pr0 eψ + kir0 Σeψ + kdr0 ecψ )

Figure 18: Control inputs of 3D ellipse path following

5. Conclusion

(24)

• Backstepping 1 controller with accurate model

In this paper, a two-layered control framework is proposed to address the problem of 3D path following control for underactuated AUVs subject to system uncertainties. Against some complicated path following controllers, an adaptive fuzzy PID controller is adopted to simplify the path following control design by synthesizing the 3D guidance law for underactuated vehicles. The control gains of the PID controller can be adaptively tuned through the heuristic fuzzy logic which offers the robustness of the overall control system to the inaccurate model parameters and time-varying environmental disturbances. The sensitivity analysis of the adaptive fuzzy PID controller is carried out under various scenarios including different membership functions and fuzzy subsets. Numerical simulation results show the effectiveness of the proposed control framework, and illustrate the outperformance of the proposed controller compared with other counterpart methods through a quantitative error analysis in terms of MSE and MAE statistics. Future work will investigate the controllability and reachability of the whole control system, address the problem of optimizing the self-tuned control gains by integrating advanced

⎧ ⎪ τu = m11 u˙ d + m33 wqd − m22 vrd + d11 ud − k1 (u − ud ) ⎪ ⎪ ⎪ ⎪ ⎪ τ ˙ d − m33 wud + m11 uwd + d55 qd ⎪ q = m55 q ⎪ ⎪ ⎨ + WGM L sin(θ) − k2 (υ − υd ) − k3 (q − qd ) ⎪ ⎪ ⎪ (χ−χd ) ⎪ ⎪ τr = m66 r˙d + m22 vud − m11 uvd + d66 rd − k4cos(θ) ⎪ ⎪ ⎪ ⎪ ⎩ − k5 (r − rd )

(25)

• Backstepping 2 controller with inaccurate model ⎧ ˜ 11 u˙ d + m ˜ 33 wqd − m ˜ 22 vrd + d˜11 ud − k1 (u − ud ) τu = m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τq = m ˜ 55 q˙ d − m ˜ 33 wud + m ˜ 11 uwd + d˜55 qd ⎪ ⎪ ⎪ ⎨ ˜ ˜ GML sin(θ) − k2 (υ − υd ) − k3 (q − qd ) +W ⎪ ⎪ ⎪ ⎪ (χ−χd ) ⎪ ⎪ ˜ 66 r˙d + m ˜ 22 vud − m ˜ 11 uvd + d˜66 rd − k4cos(θ) τr = m ⎪ ⎪ ⎪ ⎩ − k5 (r − rd )

(26)

where ki > 0 with i ∈ [1, 5] are constant control gains in backstepping-based controller. [1] I. Boussaid, J. Lepagnot, P. Siarry, A survey on optimization metaheuristics, Information Sciences 237 (2013) 82 – 117.

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