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Journal of the Franklin Institute 356 (2019) 3831–3848 www.elsevier.com/locate/jfranklin

Integrated guidance and control strategy for homing of unmanned underwater vehicles Zheping Yan, Man Wang∗, Jian Xu College of Automation, Harbin Engineering University, Harbin, Heilongjiang 150001, China Received 2 May 2018; received in revised form 14 November 2018; accepted 29 November 2018 Available online 13 March 2019

Abstract This paper presents a novel integrated guidance and control strategy for homing of unmanned underwater vehicles (UUVs) in 5-degree-of-freedom (DOF), where the vehicles are assumed to be underactuated at high speed and required to move towards the final docking path. During the initial homing stage, the guidance system is first designed by geometrical analysis method to generate a feasible reference trajectory. Then, in the backstepping framework, the proposed trajectory tracking controller can achieve all the tracking errors in the closed-loop system convergence to a small neighbourhood of zero. It means that the vehicle’s dynamics are consistent with the reference trajectory derived in the previous step. To demonstrate the effectiveness of the proposed guidance and control strategy, the complete stability analysis used Lyapunov’s method is given in the paper, and simulation results of all initial conditions are presented and discussed. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

1. Introduction The topics of navigation, guidance and control of unmanned underwater vehicles (UUVs) have received considerable attention in the past decades due to theoretical challenges and significant applications in the field of ocean engineering. Therefore, the large number of control strategies have been developed to achieve practical tasks, for example, homing and docking [1–3], path following [4–6], and trajectory tracking [7–11]. Indeed, most of them rely on ∗

Corresponding author. E-mail address: [email protected] (M. Wang).

https://doi.org/10.1016/j.jfranklin.2018.11.042 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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the assumptions that the vehicles are fully actuated or the dynamics are 3-degree-of-freedom (DOF) horizontal models. In addition, in these existing works, how to generate a satisfactory reference trajectory is not described in detail. In this paper, we are interested in the problem that usually denominated as homing in the literature, where the vehicle can autonomously approach the vicinity of the base station but without aforementioned assumptions. During the homing and docking stages, the main challenge is how to design a guidance and control scheme to drive the UUV to approach the base along a particular path, with an appointed direction of arrival. The earlier works in the literature on this problem can be found in [12,13], which presented the homing and docking controllers for an underactuated UUV based on an electromagnetic homing system and the ultra-short baseline (USBL) sensor, respectively. More recently, a time differences of arrival based homing strategy and a two-step control approach were proposed in [1,2], respectively. However, the main limitation of these methods is that the performance of the designed tracking controller may be deterioration due to the possible faults of sensors. Fortunately, this drawback is well worked out in [14,15] by providing sensor fault detection and higher order sliding mode observers to the system. On the other hand, the concept of a pure pursuit guidance algorithm has been reported in [16–18]. It should be pointed out that all aforementioned references share a common strategy: the UUV’s dynamics should be consistent with the reference guidance trajectory that includes the time evolutions of the position, attitude, as well as the velocities. Therefore, in practical applications, how to generate a suitable reference trajectory that does not rely on the sensors and vehicle models has become a significant challenge. To the best of author’s knowledge, it is the first attempt to present a guidance system for homing tracking of an underactuated UUV by using geometrical analysis method without any vehicle kinematics and dynamics. In addition, it is the fact that most UUVs are underactuated, especially when they execute maneuvers [19]. The available control inputs can be provided only by stern propellers, steering and diving rudders [7,20]. As a result, the absence of actuation in sway and heave directions poses significant challenges on the control system design side in path following and trajectory tracking scenarios [21]. It also motivates author’s research investigation and attracts considerable attention from the control community. Among these interesting works, backstepping control has become one of the most commonly used methods to deal with highly nonlinear systems. For example, planar backstepping controller has been developed for underactuated UUVs in [4,8]. To extend the applications to three-dimensional space, one-step ahead backstepping method in [22] and observer backstepping technique in [23] have been investigated, respectively. However, these results assume that the reference trajectory satisfies the persistent excitation condition, which means that the UUV cannot track a straight-line trajectory as the reference yaw velocity does not converge to zero. Therefore, it significantly restricts the class of reference trajectories to be used for practical applications, which also motivates our current research investigation. Furthermore, to provide the robustness and adaption of the vehicles, many works on the trajectory tracking control of underactuated UUVs with different techniques, such as sliding mode [24–26], neural network [7,27,28], robust and adaptive control [29–31], and model-based control [23,32,33], are reported in the recent literatures. In [24], a second-order sliding mode controller is proposed, which is effective in compensating for the modelling uncertainties and rejecting the unknown disturbances. In [7], a neuro-adaptive tracking controller is presented by combining the input-output feedback linearization, neural network approximation and adaptive techniques. However, the parametric uncertainties and environmental disturbances in these papers are assumed to be constant and bounded. To enhance the applicability of control strategies in real case, the nonlinear stochastic system for

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a fully-actuated UUV is derived in [29]. It can also be achieved in [30,31] by using adaptive fuzzy control. Other classic control schemes, such as a Lie group variational integrator and the finite-time tracking controller, are presented in [34,35], respectively. This paper aims at developing a novel integrated guidance and control strategy to solve the homing problem, where the underactuated UUV in three-dimensional space can autonomously approach the base station along the final docking path. The main contributions of the present work are concluded as follows: (1) in the initial homing stage, the reference guidance trajectory is derived by using geometrical method, which provides the commands of the position, attitude and velocity for the following works; (2) in the trajectory tracking, the UUV moving in three-dimensional space at high speed is assumed to be underactuated. The vehicle only has three available control inputs provided by stern propellers, steering and diving rudders but five degrees of freedom to be controlled. How to achieve a novel trajectory tracking controller for this underactuated UUV is presented with the aid of backstepping. The main difference between our approach and others is in the fact that in the proposed algorithm, the dynamical couplings and underactuated characteristic of the UUV are directly taken into account. In addition, a complete stability analysis based Lyapunov’s method is given. All the tracking errors in the closed-loop system are guaranteed to converge to a small neighbourhood of zero. In contrast to previous work, this paper considers more nonlinear and coupled behaviors of the vehicle such that the guidance and control laws are suitable for the three-dimensional case. As such, no further stability issues need to be addressed, and simulation results for all initial conditions are presented to show the satisfactory performance of the proposed controller. The rest of the paper is organized as follows: Section 2 describes the model of an underactuated UUV and the homing problem at hand. Section 3 presents the design of the guidance laws in detail, while in Section 4, the controller design and stability analysis for the trajectory tracking are presented. Simulation results and some concluding remarks are given in Sections 5 and 6, respectively.

2. Problem statement Consider an unmanned underwater vehicle moving in three-dimensional space, which is expected to autonomously return and dock to its base station. As shown in Fig. 1, there exist two transponders equally spaced from the final docking position D(xD , yD , zD ). One of the transponders is called the left transponder, whereas the other is the right transponder, and their inertial positions are denoted as L(xL0 , yL0 , zL0 ) and R(xR0 , yR0 , zR0 ), respectively. To facilitate the docking, the transponders and the base station in the mission scenario are in the horizontal plane, i.e. zD = zL0 = zR0 . All the homing and docking tasks are divided into two stages, which is similar to the previous work [2]. However, it is carried out in threedimensional space and the vehicle is assumed to be underactuated in 5-DOF. In the initial homing stage, the vehicle is expected to drive at a certain cruise speed toward the reference path that is orthogonal to the straight line defined by the two transponders. The main goals are to shorten the distance to the docking position and achieve the final docking approach in the horizontal plane. Then in the final docking approach, the vehicle at a much lower speed is assumed to be fully actuated, and the model uncertainties and environmental disturbances are essential to be considered. The final docking is the further work to be addressed, in this paper how to design an integrated guidance and control strategy for the initial homing is the primary work.

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X

Fig. 1. The homing and docking scenario of an UUV in three-dimensional space.

Fig. 2. Motion variables for an underactuated UUV.

As shown in Fig. 2, let η=[x, y, z, θ , ψ ]T be the 5-DOF position and attitude of the UUV in the earth-fixed frame, and let v = [u, υ, w, q, r ]T be the corresponding linear and angular velocities in the body-fixed frame. The dynamical model of an underactuated UUV in 5-DOF can be written as [20] ⎧ x˙ = u cos(θ ) cos(ψ ) − υ sin (ψ ) + w sin (θ ) cos(ψ ) ⎪ ⎪ ⎪ ⎪ ⎨y˙ = u cos(θ ) sin (ψ ) + υ cos(ψ ) + w sin (θ ) sin (ψ ) z˙ = −u sin (θ ) + w cos(θ ) ⎪ ⎪ θ˙ = q ⎪ ⎪ ⎩˙ ψ = r/ cos(θ )

(1)

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⎧ m22 τu m33 d11 ⎪ ⎪ u˙ = υr − wq − u+ ⎪ ⎪ m11 m11 m11 m11 ⎪ ⎪ ⎪ m11 d22 ⎪ ⎪ ⎪ υ˙ = − ur − υ ⎪ ⎪ m22 m22 ⎪ ⎨ m11 d33 w˙ = uq − w (2) m33 m33 ⎪ ⎪ ⎪ ⎪ τq m33 − m11 d 55 ρg∇ GML sin (θ ) ⎪ ⎪ q˙ = uw − q− + ⎪ ⎪ ⎪ m m m m 55 55 55 55 ⎪ ⎪ m11 − m22 d66 ⎪ τr ⎪ ⎩r˙ = uυ − r + m66 m66 m66 where mii and dii , i = 1, 2, . . . , 6, are the mass and inertia parameters, and damping coefficients, respectively. τu , τq and τr are the control inputs, and other symbols can be referred to the literature. Assumption 1. The UUV is neutrally buoyant and the xz plane symmetric. The roll motion is passively stabilized through internal/external roll actuators or by gravity, and thus, it can be neglected. Remark 1. Assumption 1 is a common assumption in the maneuvering control of slender body UUVs. They can be found in many literatures, e.g. [7,20,27]. Assumption 2. The pitch angle of the vehicle θ is assumed that |θ (t )| < π /2, ∀t ≥ 0. Remark 2. Notice that the last equation in Eq. (1) is not defined when the pitch angle is equal to ±π /2. In practice, this problem is unlikely to happen due to the metacentric restoring forces. One way to avoid the singularities is to use a four-parameter description known as the quaternion [20]. In this paper, the Euler parameters are used because of their physical representation and computational efficiency. Remark 3. As the absence of available actuators in sway and heave directions, the second and third equations in Eq. (2) are not controlled directly. Thus, the UUV is an underactuated dynamical system. 3. Guidance system for trajectory generation In the simplest form, open-loop guidance systems for UUVs are used to generate a reference trajectory for time-varying trajectory tracking or, alternatively, a path for time-invariant path following. This paper focuses on the initial homing problem that the vehicle is expected to move at a certain cruise speed towards the final docking path. Therefore, the first problem to be addressed is how to generate a feasible trajectory. As shown in Fig. 1, the straight line AB generated by the guidance systems is the reference trajectory for initial homing of an underactuated UUV. The following presents the design process of the guidance system used geometrical method in detail. 3.1. The construction of the straight line AB In order to derive the guidance laws notice that, when the vehicle is moving along the final docking trajectory that is orthogonal to the straight line defined by the two transponders, the distance between the vehicle and each transponder is identical. In addition, the docking

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position and two transponders are in the horizontal plane, i.e. zD = zL0 = zR0 . Then, the function of the straight line AB can be derived by the following equations (x − xL0 )2 + (y − yL0 )2 + (z − zL0 )2 = (x − xR0 )2 + (y − yR0 )2 + (z − zR0 )2 (3) z = zL0 = zR0 If yL0 = yR0 , the function of the straight line AB is as y = kx + b0 where ⎧ xR0 − xL0 ⎪ ⎪ ⎨k = − yR0 − yL0 2 2 2 2 x + yR0 − xL0 − yL0 ⎪ ⎪ ⎩b0 = R0 2(yR0 − yL0 )

(4)

(5)

If yL0 = yR0 , one obtains xL0 + x R0 (6) 2 Remark 4. It should be pointed out that the reference trajectory for initial homing is determined by the positions of two transponders fixed in the mission scenario. Indeed, this idea is first to be adopted to solve the homing and tracking problems. In other existing literature, the reference trajectory is assumed to be known or generated by virtual vehicle that relies on the vehicle’s kinematics and dynamics. x=

3.2. The definitions of the points A and B In the following, how to obtain the inertia positions of the points A and B will be given. Firstly, the initial position of the UUV is defined by the point C(x0 , y0 , z0 ), and A is the projection of the point C on the straight line AB. Then, the different expressions of A will be derived in the following three scenarios. If yL0 = yR0 , the position of A is as ⎧ xL0 + xR0 ⎪ ⎨xA = 2 (7) y = y0 ⎪ ⎩ A zA = zD If k = 0, that is xL0 = xR0 , the result is ⎧ ⎪ ⎨xA = xy0 + y L0 R0 yA = ⎪ 2 ⎩ zA = zD In other conditions, the expression of A is derived by the following equations ⎧ 2 2 2 x 2 + yR0 − xL0 − yL0 x − xL0 ⎪ ⎨y = − R0 x + R0 yR0 − yL0 2(yR0 − yL0 ) ⎪ ⎩y = yR0 − yL0 (x − x0 ) + y0 xR0 − xL0

(8)

(9)

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Based on Eq. (9), one obtains ⎧ k(b0 − y0 ) − x0 ⎪ ⎪ ⎪x A = − ⎨ k2 + 1 1 yA = − ( xA − x0 ) + y0 ⎪ ⎪ ⎪ k ⎩ zA = zD

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(10)

Next, the position of the point B that is at a positive distance R from the docking base will be presented. Accordingly, if the straight line AB is expressed by Eq. (6), the position of B is as ⎧ xL0 + xR0 ⎪ ⎨x B = 2 (11) y = yL0 − R or yL0 + R ⎪ ⎩ B z B = zD On the other hand, the position of B is derived by the following equations y = kx +b0 (x − xD )2 + (y − yD )2 = R2

(12)

Based on Eq. (12), it is derived ax + bx + c = 0 2

where ⎧ ⎨a = k 2 + 1 b = 2(k b0 − k yD − xD ) ⎩ c = xD2 + (b0 − yD )2 − R2 Then it yields √ −b ± b2 − 4ac xB = 2a ⎩ yB = k xB + b0

(13)

(14)

⎧ ⎨

(15)

The result is finally determined by the comparison of the distance between A and B according to the fact that it is closer to A. 3.3. The generation of a reference trajectory In the initial homing stage, the underactuated UUV is expected to move at a certain cruise speed along the reference trajectory. Here the desired velocity is denoted by Vd , and the function of the line segment AB has been derived in above section. A is the starting point of the reference trajectory, and the point B is the finish. Then, the reference guidance trajectory is generated as follows ⎧ xr = xA + Vd t cos(ϕ) ⎪ ⎪ ⎪ ⎪ ⎨yr = yA + Vd t sin (ϕ) zr = zD (16) ⎪ ⎪ θ = 0 ⎪ r ⎪ ⎩ ψr = ϕ

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π

, i f yL0 = yR0 2 . Notice that, the reference trajectory generated by the arctan (k), others guidance system is a time-varying straight line with a constant velocity. Therefore, the next work will present a novel trajectory tracking algorithm for an underactuated UUV moving in three-dimensional space. where ϕ =

4. Controller design and stability analysis 4.1. Trajectory tracking controller design This section details the trajectory tracking control design for an underactuated UUV in three-dimensional space. Essentially, due to the underactuated property of the UUV, the sway and heave velocities cannot be directly controlled by the available control inputs. Therefore, using the coupled dynamics, the angular velocities of the vehicle are designed to ensure that the sway and heave velocities converge to the desired values. In the end, the control force and torque are determined to achieve the velocity tracking, which in turn guarantees that all the tracking errors converge to a small neighbour of zero. In order to derive the control laws, firstly, it is necessary to define the position and attitude error variables as follows ⎡ ⎤ ⎡ ⎤⎡ ⎤ xe cos(θ ) cos(ψ ) cos(θ ) sin (ψ ) − sin (θ ) 0 0 x − xr ⎢ye ⎥ ⎢ − sin (ψ ) ⎢ ⎥ cos(ψ ) 0 0 0⎥ ⎢ ⎥ ⎢ ⎥⎢y − y r ⎥ ⎢ze ⎥ = ⎢ sin (θ ) cos(ψ ) sin (θ ) sin (ψ ) ⎥ ⎢ cos(θ ) 0 0 ⎥⎢z − z r ⎥ (17) ⎢ ⎥ ⎢ ⎥ ⎣θe ⎦ ⎣ 0 0 0 1 0⎦⎣θ − θr ⎦ ψe 0 0 0 0 1 ψ − ψr Taking the time derivative of Eq. (17), it yields ⎧ x˙e = u − qze + r ye − x˙r cos(θ ) cos(ψ ) − y˙r cos(θ ) sin (ψ ) + z˙r sin (θ ) ⎪ ⎪ ⎪ ⎪y˙e = υ − r (xe + ze tan (θ ) ) + x˙r sin (ψ ) − y˙r cos(ψ ) ⎨ z˙e = w + qxe + r ye tan (θ ) − x˙r sin (θ ) cos(ψ ) − y˙r sin (θ ) sin (ψ ) − z˙r cos(θ ) ⎪ ⎪ θ˙e = q − θ˙r ⎪ ⎪ ⎩˙ ψe = r/cos(θ ) − ψ˙ r

(18)

Step 1. Consider the Lyapunov candidate function V1 =

1 2 xe + ye2 + ze2 2

(19)

The time derivative of Eq. (19) along Eq. (18) yields V˙1 = xe (u − x˙r cos(θ ) cos(ψ ) − y˙r cos(θ ) sin (ψ ) + z˙r sin (θ )) +ye (υ + x˙r sin (ψ ) − y˙r cos(ψ ) ) +ze (w − x˙r sin (θ ) cos(ψ ) − y˙r sin (θ ) sin (ψ ) − z˙r cos(θ )) Following the backstepping, the traditional desired vehicle velocity defined by ⎧ ⎨ud = x˙r cos(θ ) cos(ψ ) + y˙r cos(θ ) sin (ψ ) − z˙r sin (θ ) − k1 xe υd = −x˙r sin (ψ ) + y˙r cos(ψ ) − k2 ye ⎩ wd = x˙r sin (θ ) cos(ψ ) + y˙r sin (θ ) sin (ψ ) + z˙r cos(θ ) − k3 ze

(20)

(21)

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where ki , i = 1, 2, 3, are positive constants. According to the generated guidance trajectory in Eq. (16), one obtains ⎧ ⎨ud = Vd cos(θ ) cos(ψe ) − k1 xe υd = −Vd sin (ψe ) − k2 ye (22) ⎩ wd = Vd sin (θ ) cos(ψe ) − k3 ze Clearly, if the linear velocities u, υ and w of the UUV respectively coincide with ud , υd and wd , the position errors xe , ye and ze converge to zero, as one gets V˙1 = −k1 xe2 − k2 ye2 − k3 ze2 ≤ 0. However, due to the underactuated property of the vehicle, the sway and heave velocities cannot be directly controlled in practice. Therefore, aforementioned controller design is unable to achieve the trajectory tracking for an underactuated UUV in three-dimensional space. Using the coupled dynamics, here we define two virtual velocity variables α and β. Then, combining with the control of the yaw angle ψ and pitch angle θ , the global tracking can be guaranteed. Firstly, the novelty velocity variables are defined by α=Vd sin (ψe ) (23) β = Vd sin (θ ) Then, the desired velocity commands can be chosen as ⎧ ⎨ud = Vd cos(θ ) cos(ψe ) − k1 xe αd = −υ − k2 ye ⎩ βd = w − Vd sin (θ )(cos(ψe ) − 1) + k3 ze

(24)

To avoid the controller complexity in the traditional backstepping, which induced by differentiating the desired velocity commands, let the virtual control variables ud , αd , and βd pass the following first-order filter: ⎧ ⎨ku u˙c + uc = ud , uc (0) = ud (0) kα α˙ c + αc = αd , αc (0) = αd (0) (25) ⎩ ˙ kβ βc + βc = βd , βc (0) = βd (0) where ki , i = u, α, β, are positive design parameters. Then, the new velocity errors are introduced as ⎧ ⎨u e = u − u c , e u = u c − u d αe = α − αc , eα = αc − αd (26) ⎩ βe = β − βc , eβ = βc − βd Taking the time derivative of V1 in Eq. (19), it yields V˙1 = −k1 xe2 − k2 ye2 − k3 ze2 + xe (ue + eu ) + ye (αe + eα ) − ze (βe + eβ )

(27)

Remark 5. Notice that the sway and heave velocities of the UUV are not directly controlled. However, using the coupled dynamics, they can be regulated by steering and diving rudders. In particular, the virtual velocity variables α and β are defined by Eq. (23) such that the position tracking errors ye and ze are uniformly ultimately bounded (UUB) by the control of the orientation θ and ψ. In addition, the dynamic surface control (DSC) is introduced such that the virtual variables can be computed by the first-order filters. It not only simplifies the control structure with less computational effort but also benefits for its practical implementation. Step 2. Considering that α and β are virtual velocity variables, the following control design aims at stabilizing the tracking errors αe and βe as well as the attitude errors θe and ψe by

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the available angular velocities q and r. The second Lyapunov control function V2 is chosen as 1 1 1 1 V2 = V1 + θe2 + ψe2 + αe2 + βe2 2 2 2 2

(28)

Taking the time derivative of V2 along Eq. (23), it is derived V˙2 = −k1 xe2 − k2 ye2 − k3 ze2 + xe (ue + eu ) + ye (αe + eα ) − ze (βe + eβ ) r r +θe q + ψe + αe Vd cos(ψe ) − α˙ c cos(θ ) cos(θ )

˙ +βe Vd cos(θ )q − βc Then, the desired angular velocities are designed as qd = − k4 (θe + Vd βe cos(θ ) ) rd = −k5 (ψe + Vd αe cos(ψe ) ) cos(θ )

(29)

(30)

where k4 and k5 are positive constants to be chosen later. As similar process in above, let the virtual control commands qd and rd pass the first-order filter: kq q˙c + qc = qd , qc (0) = qd (0) (31) kr r˙c + rc = rd , rc (0) = rd (0) And the angular velocity errors are defined as qe = q − qc , eq = qc − qd r e = r − r c , er = r c − r d

(32)

Then, the time derivative of V2 in Eq. (29) along Eq. (32) yields V˙2 = −k1 xe2 − k2 ye2 − k3 ze2 − k4 (θe + Vd βe cos(θ ) )2 − k5 (ψe + Vd αe cos(ψe ) )2

+xe (ue + eu ) + ye (αe + eα ) − ze (βe + eβ ) + (θe + Vd βe cos(θ ) ) qe + eq (ψe + Vd αe cos(ψe ) )(re + er ) + − αe α˙ c − βc β˙c cos(θ )

(33)

Step 3. In above, the kinematic controller is achieved. Next, using the desired commands of the velocities, the dynamic controller will be derived in details. First, consider the Lyapunov function candidate as follows 1 1 1 V3 = V2 + ue2 + qe2 + re2 2 2 2 Combining with the UUV dynamic equations in Eq. (2), one gets ⎧ ⎪ ⎪u˙e = m22 υr − m33 wq − d11 u + τu − u˙c ⎪ ⎪ ⎪ m11 m11 m11 m11 ⎪ ⎨ τq m33 − m11 d 55 ρg∇ GML sin (θ ) q˙e = uw − q− + − q˙c ⎪ m m m m ⎪ 55 55 55 55 ⎪ ⎪ m11 − m22 τr d66 ⎪ ⎪ ⎩r˙e = uυ − r+ − r˙c m66 m66 m66

(34)

(35)

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The dynamic controller is designed as follows ⎧ τu = m11 (u˙c − xe ) − m22 υr + m33 wq + d11 u − m11 k6 ue ⎪ ⎪ ⎨ τq = m55 (q˙c − θe − Vd βe cos(θ ) ) −(m33 − m11 )uw + d 55 q + ρg∇ GML sin (θ ) − m55 k7 qe ψe + Vd αe cos(ψe ) ⎪ ⎪ − (m11 − m22 )uυ + d66 r − m66 k8 re ⎩τr = m66 r˙c − cos(θ ) (36) where ki , i = 6, 7, 8, are positive control gains. Then, taking the time derivative of V3 along Eqs. (35) and (36), it yields V˙3 = −k1 xe2 − k2 ye2 − k3 ze2 − k4 (θe + Vd βe cos(θ ) )2 −k5 (ψe + Vd αe cos(ψe ) )2 − k6 ue2 − k7 qe2 − k8 re2 + δ

(37)

where δ = xe eu + ye eα − ze eβ + αe (ye − α˙ c ) − βe (ze + β˙c ) (ψe + Vd αe cos(ψe ) )er +(θe + Vd βe cos(θ ) )eq + cos(θ )

(38)

4.2. Stability analysis The controller design for the trajectory tracking of an underactuated UUV in threedimensional space has been completed in above section. The main results can be summarized in the following theorem. Theorem 1. Consider an underactuated 5-DOF UUV moving in three-dimensional space, with the kinematics and dynamics given by the Eqs. (1) and (2). In practical applications, the underactuated UUV is in a mission scenario as described in Section 2. Then, with the reference guidance trajectory Eq. (16), and the control laws (24), (30) and (36), the vehicle can autonomously approach the vicinity of the base station along the final docking path. Moreover, if the control gains are chosen to satisfy the conditions in Eq. (44), all signals in the closed-loop system are bounded and the tracking errors can be guaranteed to converge to a small neighbourhood of zero. Proof. In above Section 4.1, the first-order filter is introduced to avoid differentiating the desired velocity commands. It simplifies the complexity of controllers but also results in boundary layer errors eζ , ζ = u, α, β, q, r. Based on the definitions of the boundary layer error, one obtains eζ e˙ζ = ζ˙c − ζ˙d = − − ζ˙d (39) kζ where ζd denotes the desired velocity command, which is a continuous closed-loop signal. Then, using Young’s inequality e2ζ 1 1 ˙ eζ e˙ζ = − − eζ ζd ≤ − − 1 e2ζ + ζ˙d2 (40) kζ kζ 4 Therefore, the boundary layer errors are uniformly ultimately bounded as the control gain kζ satisfies k1ζ − 1 > 0. Finally, the complete stability analysis for overall system will be

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presented. Choose the following Lyapunov function candidate 1 1 1 1 1 V4 = V3 + e2u + e2α + e2β + e2q + e2r 2 2 2 2 2 Taking the time derivative of V4 , it is derived

(41)

V˙4 = −k1 xe2 − k2 ye2 − k3 ze2 − k4 (θe + Vd βe cos(θ ) )2 −k5 (ψe + Vd αe cos(ψe ) )2 − k6 ue2 − k7 qe2 − k8 re2 e2β e2q e2 e2 e2 − u− α − − − r +δ ku kα kβ kq kr ˙ −eu u˙d − eα α˙ d − eβ βd − eq q˙d − er r˙d

(42)

Due to the actual control constraints on the thrusters and rudders of the vehicle, there exists a positive constant λ0 satisfies the condition | cos1(θ ) | ≤ λ0 . In addition, it is noted that all the desired velocity commands are bounded closed-loop signals, which can thus ensure the boundedness of their time derivatives. Therefore, the following inequalities hold ⎧ ⎨|α˙ c | = λ1 , β˙c = λ2 , |u˙d | = λ3 , |q˙d | = λ4 , (43) ⎩ |r˙d | = λ5 , |α˙ d | = λ6 , β˙d = λ7 where λi , i = 1, 2, . . . , 7, are positive constants. Using Young’s inequality, i.e., ab ≤ μ 2 b , with μ > 0, one obtains 2

1 2 a 2μ

+

1 2 μ 2 1 2 μ 2 1 2 μ 2 e + x , ye eα ≤ e + ye , z e eβ ≤ e + ze 2μ u 2 e 2μ α 2 2μ β 2 1 2 μ 2 1 2 μ 2 1 2 μ 2 α + ye , αe α˙ c ≤ α + λ1 , ze βe ≤ β + ze ye αe ≤ 2μ e 2 2μ e 2 2μ e 2 2 V μ 1 2 μ 2 1 2 μ 2 βe + λ2 , θe eq ≤ eq + θe , Vd βe cos(θ )eq ≤ d e2q + βe2 βe β˙c ≤ 2μ 2 2μ 2 2μ 2 2 2 2 λV ψe er μ Vd αe cos(ψe )er μ λ ≤ 0 e2r + ψe2 , ≤ 0 d e2r + αe2 cos(θ ) 2μ 2 cos(θ ) 2μ 2 1 2 1 2 2 k ψ V α cos ( ψe ) 5 e d e βe + μθe2 , αe + μψe2 2k4 θeVd βe cos(θ ) ≤ k4Vd ≤ λ0 k5Vd μ cos(θ ) μ 1 2 μ 2 1 2 μ 2 1 2 μ 2 e + λ , eq q˙d ≤ e + λ , er r˙d ≤ e + λ5 eu u˙d ≤ 2μ u 2 3 2μ q 2 4 2μ r 2 μ 1 2 μ 2 1 e + λ6 , eβ β˙d ≤ e2 + λ27 eα α˙ d ≤ 2μ α 2 2μ β 2 xe eu ≤

Then, taking the time derivative of V4 , it yields V˙4 ≤ −k¯ 1 xe2 − k¯ 2 ye2 − k¯ 3 ze2 − k¯ 41 θe2 − k¯ 42 βe2 − k¯ 51 ψe2 − k¯ 52 αe2 −k6 ue2 − k7 qe2 − k8 re2 − k9 e2u − k10 e2q − k11 e2r − k12 e2α − k13 e2β + σ where μ k¯ 1 = k1 − > 0, k¯ 2 = k2 − μ > 0, k¯ 3 = k3 − μ > 0 2

(44)

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μ 1 + k4Vd μ k¯ 41 = k4 − − k4Vd μ > 0, k¯ 42 = k4 (Vd cos(θ ) )2 − − >0 2 μ 2 2 Vd cos(ψe ) μ 1 + k5 λ0Vd μ − k¯ 51 = k5 − − k5Vd λ0 μ > 0, k¯ 52 = k5 − >0 2 cos(θ ) μ 2

2 2 2 1 + λ0 1 + Vd 2 + Vd 1 1 1 1 > 0, k11 = >0 k9 = − > 0, k10 = − − ku μ kq 2μ kr 2μ 1 1 μ 2 1 1 − > 0, k13 = − > 0, σ = λ kα μ kβ μ 2 i=1 i 7

k12 =

Set κ = min{k¯ i , k j }, then from Eq. (44), it is derived that V˙4 ≤ −2κV4 + σ

(45)

By solving the inequality (45), one obtains V4 (t ) ≤ V4 (0)e−2κt +

σ , ∀t > 0. 2κ

(46)

The result means that all the tracking errors in closed-loop system exponentially converge to a small neighbourhood of origin, i.e., uniformly ultimately bounded (UUB). Towards this end, the completed stability analysis is given. Remark 6. The controller parameters k¯ i and k j may be tuned to adjust the convergence rate σ κ and the size of ultimate bound 2κ . Based on above presented Lyapunov stability analysis, σ ¯ the larger values of gains ki and k j increase κ and decrease the size of ultimate bound 2κ , which leads to better tracking accuracy. However, too larger gains may result in the control signals chattering even unstable. So it should be noted that the control parameters will be properly chosen by trial and error. The whole process is similar as the gain tuning in PID control. 5. Simulation results This section presents the numerical simulations performed on an underactuated UUV in 5-DOF to illustrate the satisfactory performance of the proposed integrated guidance and tracking control scheme. All the simulations are carried out by using MATLAB/Simulink. The parameters of the kinematics Eq. (1) and dynamics Eq. (2) of the vehicle, as in [36], are given by m11 = 215(kg ), m22 = 265(kg ), m33 = 265(kg ), m55 = 80(kgm2 ), m66 = 80(kgm2 ), d11 = 70 + 100|u|(kgs−1 ), d22 = 100 + 200|υ|(kgs−1 ), d33 = 100 + 200|w|(kgs−1 ), d55 = 50 + 100|q|(kgm2 s−1 ), d66 = 50 + 100|r|(kgm2 s−1 ). The reference trajectory is generated using the guidance system in Eq. (16), where the desired surge velocity ud = Vd = 2m/s, the sway and heave velocities υd = wd = 0. Initial position and attitude errors are assumed to exist in several scenarios as shown in Table 1, and all the initial velocities of the vehicle are (u(0), υ(0), w(0), q(0), r(0)) = (0, 0, 0, 0, 0 ). When the distance between the vehicle and the docking position reaches R = 25(m), the initial homing stage is over and the final docking is start, which is the further work to be addressed. Here, the reference trajectories of scenarios 1 and 2 are: (1) a straight line is parallel to the XE axis as the positions of the two transponders are xL0 = xR0 ; (2) a straight line is parallel to

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Table 1 Initial positions and attitudes of the underactuated UUV and two transponders.

Scenario 1 Scenario 2 Scenario 3

UUV initial positions and attitudes

Positions of the left transponder

Positions of the right transponder

[−100,1,12,0,0] [−1,−115,12,0,π/4] [−115,−116,12,0,0]

[0,2,10] [−2,10,10] [2,5,10]

[0,−2,10] [2,10,10] [5,2,10]

Fig. 3. The tracking results for homing of an underactuated UUV in scenario 1: (a) Reference and actual trajectories. (b) Actual velocities. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

the YE axis as the condition is yL0 = yR0 . In the case of scenario 3, the reference trajectory is the line segment AB, where the slope k is nonzero. The controller is designed as in Eqs. (24), (30) and (36), where the control parameters are chosen as k1 = k2 = k3 = 0.3, k4 = k5 = 10, k6 = 1, k7 = k8 = 10, ki = 0.05 where i = u, α, β, q, r. It is essential to point out that these control parameters can be easily done using the MATLAB commands by trial and error. This process is similar as the gain tuning in PID control. On the other hand, it will be shown below that even with different initial conditions, satisfactory performance could be maintained under the same control gains, which also states the robustness of the proposed algorithm. Fig. 3 shows the tracking results of the vehicle in the mission scenario 1, where the positions of the two transponders are marked with red crosses. A, B and D are marked with the blue and red points, respectively. It can be clearly seen that the underactuated UUV can approach the vicinity of the base station along the reference guidance trajectory AB. The linear and angular velocities of the vehicle are shown in Fig. 3(b), where the surge velocity is as the desired value ud = Vd = 2(m/s ), and other velocities all converge to zero. The position and attitude tracking errors in Fig. 4(a) converge fast to zero, whereas the control inputs are shown in Fig. 4(b). It is relevant to point out that the Theorem 1 guarantees that all the tracking errors in the closed-loop system converge to a small neighbourhood of zero, but it does not provide convergence speeds. Hence, simulations must be carried out to tune the control gains such that the proposed controller presents the satisfactory performance, which is a common solution in nonlinear control problems.

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Fig. 4. The tracking results of the trajectory tracking: (a) The position and attitude tracking errors. (b) The control inputs of the vehicle.

Fig. 5. The tracking results for homing of an underactuated UUV in scenario 2: (a) Reference and actual trajectories. (b) Actual velocities.

As shown in Fig. 5(a), the underactuated UUV moves in three-dimensional space towards the final docking path even with the initial position and attitude errors. And the actual velocities of the vehicle are shown in Fig. 5(b). At the beginning the velocity response is very fast due to the large initial position errors. The results of the tracking errors and control inputs of the vehicle are also seen in Fig. 6. Changes of the available control forces and torques confirm a short time of the system response. In addition, it should be noted that the reference guidance trajectories of the scenarios 1 and 2 are parallel to the axis XE and YE , respectively. In Fig. 7, a general reference guidance trajectory for homing of an underactuated UUV is given, where the vehicle at a certain cruise speed towards the base station. As observed in Fig. 8, the proposed controller can also guarantee that all the tracking errors converge to a small neighbourhood of zero, whereas the bounded control inputs of the vehicle are shown in Fig. 8(b). From all above simulations, we can conclude that the proposed integrated guidance and control scheme is effectiveness, which can work out the initial homing problem of an underactuated UUV in three-dimensional space.

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Fig. 6. The tracking results of the trajectory tracking: (a) The position and attitude tracking errors. (b) The control inputs of the vehicle.

Fig. 7. The tracking results for homing of an underactuated UUV in scenario 3: (a) Reference and actual trajectories. (b) Actual velocities.

Fig. 8. The tracking results of the trajectory tracking: (a) The position and attitude tracking errors. (b) The control inputs of the vehicle.

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6. Conclusion In this paper, a novel integrated guidance and tracking control scheme is developed for homing of an underactuated UUV in 5-DOF, where the vehicle can autonomously approach the vicinity of the base station along the final docking path. During the initial homing stage, the reference guidance trajectory is generated using geometrical method that does not dependent on the vehicle model and any assumptions. Then in the trajectory tracking, three-dimensional controller is presented with the aid of backstepping, where the vehicle’s dynamics are all consistent with the reference “state-space” trajectory including the time evolutions of position, attitude, as well as the velocities. In addition, all the tracking errors in the closed-loop systems are demonstrated to be uniformly ultimately bounded (UUB). And the simulations for all initial conditions are also given to show the effectiveness of the control strategy. In the further work, the final docking problem should be addressed that may take the environmental disturbances and model uncertainties into account. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant 51179038, 51105088, 51309067, 51409055, and partly supported by the Fundamental Research Funds for the Central Universities under Grant HEUCFX160402. References [1] P. Batista, C. Silvestre, P. Oliveira, A time differences of arrival-based homing strategy for autonomous underwater vehicles, Int. J. Robust Nonlinear Control 20 (2010) 1758–1773. [2] P. Batista, C. Silvestre, P. Oliveira, A two-step control approach for docking of autonomous underwater vehicles, Int. J. Robust Nonlinear Control 25 (2015) 1528–1547. [3] D.W. George, R.R. Andre, C. Jason, et al., A Concept for docking a UUV with a slowly moving submarine under waves, IEEE J. Ocean. Eng. 41 (2) (2016) 471–498. [4] L. Lionel, J. Bruno, Robust nonlinear path-following control of an AUV, IEEE J. Ocean. Eng. 33 (2) (2008) 89–102. [5] T.I. Fossen, K.Y. Pettersen, R. Galeazzi, Line-of-sight path following for Dubins paths with adaptive sideslip compensation of drift forces, IEEE Trans. Control Syst. Technol. 23 (2) (2015) 820–827. [6] S. Chao, S. Yang, B. Brad, Integrated path planning and tracking control of an AUV: a unified receding horizon optimization approach, IEEE/ASME Trans. Mechatron. 22 (3) (2017) 1163–1173. [7] S. Khoshnam, M.A. Mohammad, On the neuro-adaptive feedback linearising control of underactuated autonomous underwater vehicles in three-dimensional space, IET Control Theory Appl. 9 (8) (2015) 1264–1273. [8] X. Jian, W. Man, Q. Lei, Dynamical sliding mode control for the trajectory tracking of underactuated unmanned underwater vehicles, Ocean Eng. 105 (2015) 54–63. [9] L. Qiao, W.D. Zhang, Adaptive non-singular integral terminal sliding mode tracking control for autonomous underwater vehicles, IET Control Theory Appl. 11 (8) (2017) 1293–1306. [10] D. Alejandro, G.R. Jose, P. Tristan, Trajectory tracking passivity-based control for marine vehicles subject to disturbances, J. Frankl. Inst. 354 (2017) 2167–2182. [11] P. Herman, W. Adamski, Nonlinear trajectory tracking controller for a class of robotic vehicles, J. Frankl. Inst. 354 (2017) 5145–5161. [12] M. Feezor, F. Sorrell, P. Blankinship, J. Bellingham, Autonomous underwater vehicle homing/docking via electromagnetic guidance, IEEE J. Ocean. Eng. 26 (4) (2001) 515–521. [13] P. Batista, C. Silvestre, P. Oliveira, A sensor based controller for homing of underactuated AUVs, IEEE Trans. Robot. 25 (3) (2009) 701–716. [14] S. Aouaouda, M. Chadli, P. Shi, H.R. Karimi, Discrete-time H- /H∞ sensor fault detection observer design for nonlinear systems with parameter uncertainty, Int. J. Robust Nonlinear Control 25 (2015) 339–361.

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