Sage Journals: Discover world-class research

Abstract

In this article, the three-dimensional trajectory tracking control of an autonomous underwater vehicle is addressed. The vehicle is assumed to be underactuated and the system parameters and the external disturbances are unknown. First, the five degrees of freedom kinematics and dynamics model of underactuated autonomous underwater vehicle are acquired. Following this, reduced-order linear extended state observers are designed to estimate and compensate for the uncertainties that exist in the model and the external disturbances. A backstepping active disturbance rejection control method is designed with the help of a time-varying barrier Lyapunov function to constrain the position tracking error. Furthermore, the controller system can be proved to be stable by employing the Lyapunov stability theory. Finally, the simulation and comparative analyses demonstrate the usefulness and robustness of the proposed controller in the presence of internal parameter uncertainties and external time-varying disturbances.

Keywords

ADRC backstepping technique barrier Lyapunov function trajectory tracking control underactuated AUV

Introduction

Autonomous underwater vehicles (AUVs) are widely used in marine scientific investigation, marine mineral exploration, and oceanographic mapping.¹ So the need for AUVs has become increasingly apparent and the research on the trajectory tracking control of AUVs becomes more important. Considering reducing the actuator cost and weight or increasing the reliability of the system in case of actuator failure, most of AUVs have fewer actuators than the number of degrees of freedom.^2,3 Nowadays, motion control of underactuated AUVs has been absorbing significant attention of researchers mainly in nonlinear control.

In past decades, lots of methods have been proposed to track AUVs, such as sliding mode control,^4,5 robust control,⁶ neural network control,⁷ and so on. There are also several methods combining the backstepping technique and other methods that are proposed to solve trajectory tracking problem of underactuated AUVs in a variety of complex environments. A current observer-based backstepping controller is developed to achieve trajectory tracking control of underactuated AUVs in the presence of unknown current disturbances.⁸ The combination of backstepping technique and adaptive sliding mode control enhances the robustness of an AUV in the presence of model parameter uncertainties and external environmental disturbances.⁹ Backstepping technique and bio-inspired models are used to increase system robustness and avoid the singularity problem in backstepping control of virtual velocity error.¹⁰

Active disturbance rejection control (ADRC) was originally proposed by Han.¹¹ It’s worth noting that nonlinear tracking differentiator (TD) and extended state observer (ESO) are important parts of ADRC. The uniqueness of ADRC is that it treats all factors affecting the plant, such as system nonlinearities, uncertainties, and external disturbances as total disturbances to be observed and compensated by ESO and it has been applied in almost all domains of control engineering.¹² ADRC has been adopted to solve the path following control problem of underactuated AUVs,¹³ but it has rarely been used in underactuated AUVs’ trajectory tracking control.

On the other hand, in some cases, it is necessary to constrain the position tracking error of an underactuated AUV within certain given boundary function all the time for safety reasons. It has been proved that the barrier Lyapunov function method^14

–17 and prescribed performance control method^18,19 are effective solutions to prevent constraint violation. A barrier Lyapunov function is incorporated with the backstepping control scheme to handle the position tracking error constraint.²⁰ Prescribed performance functions have been adopted to constrain the position and orientation errors of an underactuated AUV which can ensure both the prescribed transient and steady-state performance constraints.^21,22 But in these studies, point-to-point navigation is used and it is worth noting that the desired yaw angle is not continuously differentiable when the position error equals zero. To avoid this problem, the position tracking error converges to a constant instead of zero.

Underactuated AUVs lack sway and heave propellers, so the transverse and vertical position errors of trajectory tracking are usually eliminated by controlling the yaw and pitch angles, that is also why the backstepping technique is often employed in the trajectory tracking control of underactuated AUVs. However, backstepping controller has two obvious disadvantages. The first one is that the unknown transverse and vertical disturbances are usually omitted when deducing the virtual control variables of yaw angular and pitch angular velocities. As a result, the unknown transverse and vertical disturbances cannot be compensated in time, which always affects the result of trajectory tracking. The other one is that the problem of “explosion of terms” caused by the operation of differentiation always exists. Based on the above analysis, that reduced-order linear extended state observers (RLESOs) are used to estimate and compensate for the total disturbances which can improve the performance of backstepping controller and enhance the robustness of underactuated AUVs. Besides, TD can be used to provide the filtered version of the input signal and its differentiation. So TDs are employed to solve the problem of “explosion of terms.” What’s more, by barrier Lyapunov function, the controller can prevent the violation of the time-varying position error constraint and render the position tracking error converges to zero. In this work, the proposed controller can solve the three-dimensional trajectory tracking control problem of underactuated AUVs in the presence of internal parameter uncertainties and external unknown disturbances.

The rest of this article is organized as follows. The mathematical model of an underactuated AUV system is presented in the second section. In the third section, RLESOs are designed to estimate the total disturbances. The backstepping ADRC controller design procedure for the three-dimensional trajectory tracking of an underactuated AUV is illustrated in the fourth section. The simulation and comparative analyses are provided in the fifth section and some conclusions of this article are brought forward in the sixth section.

Problem formulation

This section presents the five degrees of freedom kinematics and dynamics model of an underactuated AUV and then formulates the problem of trajectory tracking.

AUV modeling

We define the earth-fixed coordinate system {E} and the body-fixed coordinate system {B} of the AUV as shown in Figure 1, where $ξ$ , $η$ , and $ζ$ represent the inertial coordinates of the vehicle in the earth-fixed frame; u, v, and w are the surge, sway, and heave velocities, respectively, defined in the body-fixed frame; p, q, and r represent roll angular velocity, pitch angular velocity, and yaw angular velocity; $φ$ , $θ$ , and $ψ$ are roll angle, pitch angle, and yaw angle.

Figure 1.

AUV coordinate system. AUV: autonomous underwater vehicle.

The underactuated AUV is assumed to satisfy the following assumptions^23,24: (1) the center of gravity coincides with the center of buoyancy, (2) the mass distribution is homogeneous, and (3) the hydrodynamic drag terms of order higher than two and roll motion are neglected. Then the five degrees of freedom (DOF) AUV kinematics equations can be written as

[\begin{matrix} \dot{ξ} \\ \dot{η} \\ \dot{ζ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}] = [\begin{matrix} cos ψ cos θ & - sin ψ & sin θ cos ψ & 0 & 0 \\ sin ψ cos θ & cos ψ & sin θ sin ψ & 0 & 0 \\ - sin θ & 0 & cos θ & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 / cos θ \end{matrix}] [\begin{matrix} u \\ v \\ w \\ q \\ r \end{matrix}]

The five DOF underactuated AUV dynamics equations, considering the internal parameter uncertainties and external environmental disturbances, can be expressed by the following differential equations

\begin{matrix} \dot{u} = \frac{m_{22} + Δ m_{22}}{m_{11} + Δ m_{11}} v r - \frac{m_{33} + Δ m_{33}}{m_{11} + Δ m_{11}} w q - \frac{X_{u} + Δ X_{u}}{m_{11} + Δ m_{11}} u_{r} - \frac{X_{| u | u} + Δ X_{| u | u}}{m_{11} + Δ m_{11}} | u_{r} | u_{r} + \frac{τ_{u} + d_{1}}{m_{11} + Δ m_{11}} \\ = \frac{m_{22}}{m_{11}} v r - \frac{m_{33}}{m_{11}} w q - \frac{X_{u}}{m_{11}} u - \frac{X_{| u | u}}{m_{11}} | u | u + \frac{τ_{u}}{m_{11}} + D_{u} = f_{u} + \frac{τ_{u}}{m_{11}} + D_{u} \end{matrix}

\begin{matrix} \dot{v} = - \frac{m_{11} + Δ m_{11}}{m_{22} + Δ m_{22}} u r - \frac{Y_{v} + Δ Y_{v}}{m_{22} + Δ m_{22}} v_{r} - \frac{Y_{| v | v} + Δ Y_{| v | v}}{m_{22} + Δ m_{22}} | v_{r} | v_{r} + \frac{d_{2}}{m_{22} + Δ m_{22}} \\ = - \frac{m_{11}}{m_{22}} u r - \frac{Y_{v}}{m_{22}} v - \frac{Y_{| v | v}}{m_{22}} | v | v + D_{v} = f_{v} + D_{v} \end{matrix}

\begin{matrix} \dot{w} = \frac{m_{11} + Δ m_{11}}{m_{33} + Δ m_{33}} u q - \frac{Z_{w} + Δ Z_{w}}{m_{33} + Δ m_{33}} w_{r} - \frac{Z_{| w | w} + Δ Z_{| w | w}}{m_{33} + Δ m_{33}} | w_{r} | w_{r} + \frac{d_{3}}{m_{33} + Δ m_{33}} \\ = \frac{m_{11}}{m_{33}} u q - \frac{Z_{w}}{m_{33}} w - \frac{Z_{| w | w}}{m_{33}} | w | w + D_{w} = f_{w} + D_{w} \end{matrix}

\begin{matrix} \dot{q} = \frac{(m_{33} + Δ m_{33}) - (m_{11} + Δ m_{11})}{m_{55} + Δ m_{55}} u_{r} w_{r} - \frac{M_{q} + Δ M_{q}}{m_{55} + Δ m_{55}} q - \frac{M_{| q | q} + Δ M_{| q | q}}{m_{55} + Δ m_{55}} | q | q - \frac{ρ g \nabla \bar{G M_{L}}}{m_{55}} sin θ + \frac{τ_{q} + d_{5}}{m_{55} + Δ m_{55}} \\ = \frac{m_{33} - m_{11}}{m_{55}} u w - \frac{M_{q}}{m_{55}} q - \frac{M_{| q | q}}{m_{55}} | q | q - \frac{ρ g \nabla \bar{G M_{L}}}{m_{55}} sin θ + \frac{τ_{q}}{m_{55}} + D_{q} = f_{q} + \frac{τ_{q}}{m_{55}} + D_{q} \end{matrix}

\begin{matrix} \dot{r} = \frac{(m_{11} + Δ m_{11}) - (m_{22} + Δ m_{22})}{m_{66} + Δ m_{66}} u_{r} v_{r} - \frac{N_{r} + Δ N_{r}}{m_{66} + Δ m_{66}} r - \frac{N_{| r | r} + Δ N_{| r | r}}{m_{66} + Δ m_{66}} | r | r + \frac{τ_{r} + d_{6}}{m_{66} + Δ m_{66}} \\ = \frac{m_{11} - m_{22}}{m_{66}} u v - \frac{N_{r}}{m_{66}} r - \frac{N_{| r | r}}{m_{66}} | r | r + \frac{τ_{r}}{m_{66}} + D_{r} = f_{r} + \frac{τ_{r}}{m_{66}} + D_{r} \end{matrix}

[\begin{matrix} u_{c} \\ v_{c} \\ w_{c} \end{matrix}] = [\begin{matrix} cos ψ cos θ & sin ψ cos θ & - sin θ \\ - sin ψ & cos ψ & 0 \\ sin θ cos ψ & sin θ sin ψ & cos θ \end{matrix}] [\begin{matrix} V_{c x} \\ V_{c y} \\ V_{c z} \end{matrix}]

where $m_{11} = m - X_{\dot{u}}$ , $m_{22} = m - Y_{\dot{v}}$ , $m_{33} = m - Z_{\dot{w}}$ , $m_{55} = I_{y} - M_{\dot{q}}$ , and $m_{66} = I_{z} - N_{\dot{r}}$ ; $u_{r} = u - u_{c}$ , $v_{r} = v - v_{c}$ , and $w_{r} = w - w_{c}$ represent the relative velocities of the vehicle with respect to current in the body-fixed frame and $V_{c x}$ , $V_{c y}$ , and $V_{cz}$ represent the ocean current velocities in the earth-fixed frame; $Δ (•)$ represents the parameter uncertainties in the vehicle model. d ₁, d ₂, $d_{3}$ , $d_{5}$ , and $d_{6}$ are the external environmental disturbances; D_u , D_v , D_w , D_q , and D_r are the total disturbances to be estimated; $τ_{u}$ , $τ_{q}$ , and $τ_{r}$ are considered as the available control inputs.

Assumption 1

The underactuated AUV’s velocities and control inputs are bounded, that is, $| τ_{u} | \leq {\bar{τ}}_{u}$ , $| τ_{q} | \leq {\bar{τ}}_{q}$ , $| τ_{r} | \leq {\bar{τ}}_{r}$ , $| u | \leq \bar{u}$ , $| v | \leq \bar{v}$ , $| w | \leq \bar{w}$ , $| q | \leq \bar{q}$ , and $| r | \leq \bar{r}$ , where ${\bar{τ}}_{u}$ , ${\bar{τ}}_{q}$ , ${\bar{τ}}_{r}$ , $\bar{u}$ , $\bar{v}$ , $\bar{w}$ , $\bar{q}$ , and $\bar{r}$ are the known upper bounds.

Assumption 2

For equation (2), the total disturbances and their derivatives are all bounded. There exists:

| \frac{d^{k} D_{j}}{d t^{k}} | \leq {\bar{D}}_{j}

where ${\bar{D}}_{j} (j = u, v, w, q, r)$ are positive constants and $k = 0, 1$ .

Control objectives

In order to facilitate the formulation, $p = {[ξ (t), η (t), ζ (t)]}^{T}$ is defined as the actual coordinate variable and $p_{d} = {[ξ_{d} (t), η_{d} (t), ζ_{d} (t)]}^{T}$ is a given sufficiently smooth time-varying desired trajectory with its derivative with respect to time bounded.

Considering the kinematics and dynamics equations, design a controller to render the tracking error $‖ p - p_{d} ‖$ converges to a neighborhood of the origin that can be made arbitrarily small.

RLESOs design

According to the design principle of RLESO which is firstly proposed by Huang and Xue,¹² RLESOs for the estimations of the total disturbances are designed as follows

{\begin{cases} {\dot{p}}_{1} = - β_{1} p_{1} - β_{1}^{2} u - β_{1} (f_{u} + τ_{u} / m_{11}) \\ {\hat{D}}_{u} = β_{1} u + p_{1} \end{cases}

{\begin{cases} {\dot{p}}_{2} = - β_{2} p_{2} - β_{2}^{2} v - β_{2} f_{v} \\ {\hat{D}}_{v} = β_{2} v + p_{2} \end{cases}

{\begin{cases} {\dot{p}}_{3} = - β_{3} p_{3} - β_{3}^{2} w - β_{3} f_{w} \\ {\hat{D}}_{v} = β_{3} v + p_{3} \end{cases}

{\begin{cases} {\dot{p}}_{4} = - β_{4} p_{4} - β_{4}^{2} q - β_{4} (f_{q} + τ_{q} / m_{55}) \\ {\hat{D}}_{q} = β_{4} q + p_{4} \end{cases}

{\begin{cases} {\dot{p}}_{5} = - β_{5} p_{5} - β_{5}^{2} r - β_{5} (f_{r} + τ_{r} / m_{66}) \\ {\hat{D}}_{r} = β_{5} r + p_{5} \end{cases}

where ${\hat{D}}_{u}$ , ${\hat{D}}_{v}$ , ${\hat{D}}_{w}$ , ${\hat{D}}_{q}$ , and ${\hat{D}}_{r}$ are the estimations of D_u , D_v , D_w , D_q , and D_r ; $β_{i}$ are the observer gains and p_i ( $i = 1, 2, 3, 4, 5$ ) are the auxiliary states of the observer.

If Assumption 2 is satisfied, we can obtain²⁵:

‖ E ‖ \leq \frac{max ({\bar{D}}_{j})}{min (| β_{i} |)}

where $E = {[{\tilde{D}}_{u}, {\tilde{D}}_{v}, {\tilde{D}}_{w}, {\tilde{D}}_{q}, {\tilde{D}}_{r}]}^{T} = {[D_{u} - {\hat{D}}_{u}, D_{v} - {\hat{D}}_{v}, D_{w} - {\hat{D}}_{w}, D_{q} - {\hat{D}}_{q}, D_{r} - {\hat{D}}_{r}]}^{T}$ represents the estimation error of the RLESOs and can be tuned arbitrarily small by increasing the observer gains $β_{i}$ .

Backstepping ADRC controller design

Coordinate transformation

The desired yaw angle and pitch angle are obtained only based on the reference trajectory

ψ_{d} = {\begin{matrix} arctan (\frac{{\dot{η}}_{d}}{{\dot{ξ}}_{d}}), {\dot{ξ}}_{d} \geq 0 \\ π + arctan (\frac{{\dot{η}}_{d}}{{\dot{ξ}}_{d}}), {\dot{ξ}}_{d} < 0 \end{matrix}

θ_{d} = - arctan (\frac{{\dot{ζ}}_{d}}{v_{t}})

where $v_{p} = \sqrt{{\dot{ξ}}_{d}^{2} + {\dot{η}}_{d}^{2} + {\dot{ζ}}_{d}}$ and $v_{t} = v_{p} cos θ_{d} = \sqrt{{\dot{ξ}}_{d}^{2} + {\dot{η}}_{d}^{2}}$ .

Then the position and attitude error variables x_e , y_e , z_e , $θ_{e}$ , and $ψ_{e}$ are defined as

[\begin{matrix} x_{e} \\ y_{e} \\ z_{e} \\ θ_{e} \\ ψ_{e} \end{matrix}] = [\begin{matrix} cos ψ cos θ & sin ψ cos θ & - sin θ & 0 & 0 \\ - sin ψ & cos ψ & 0 & 0 & 0 \\ sin θ cos ψ & sin θ sin ψ & cos θ & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} ξ - ξ_{d} \\ η - η_{d} \\ ζ - ζ_{d} \\ θ - θ_{d} \\ ψ - ψ_{d} \end{matrix}]

which express the errors in the body-fixed frame. $ξ_{d}$ , $η_{d}$ , $ζ_{d}$ , $θ_{d}$ , and $ψ_{d}$ are the desired position and attitude variables in the earth-fixed frame.

The derivatives of the position and attitude error variables are described as

{\begin{array}{l} {\dot{x}}_{e} = u - v_{p} cos θ_{d} cos θ cos ψ_{e} - v_{p} sin θ_{d} sin θ + r y_{e} - q z_{e} \\ {\dot{y}}_{e} = v + v_{p} cos θ_{d} sin ψ_{e} - r x_{e} - r z_{e} tan θ \\ {\dot{z}}_{e} = w - v_{p} cos θ_{d} sin θ cos ψ_{e} + v_{p} sin θ_{d} cos θ + q x_{e} + r y_{e} tan θ \\ {\dot{θ}}_{e} = q - {\dot{θ}}_{d} \\ {\dot{ψ}}_{e} = r / cos θ - {\dot{ψ}}_{d} \end{array}

Backstepping controller with RLESOs

This section designs a trajectory tracking controller by combining the backstepping technique and RLESOs and analyzes the stability based on the Lyapunov stability theory.

Step 1: A time-varying barrier Lyapunov function is defined as

V_{1} = \frac{1}{2} log (ρ^{2} / (ρ^{2} - ρ_{e}^{2}))

where $ρ$ is the position error constraint function and $ρ_{e}^{2} = x_{e}^{2} + y_{e}^{2} + z_{e}^{2}$ . The time derivative of V ₁ yields

{\dot{V}}_{1} = (ρ_{e} {\dot{ρ}}_{e} - ρ_{e}^{2} \dot{ρ} / ρ) / (ρ^{2} - ρ_{e}^{2}) = (x_{e} {\dot{x}}_{e} + y_{e} {\dot{y}}_{e} + z_{e} {\dot{z}}_{e} - ρ_{e}^{2} \dot{ρ} / ρ) / (ρ^{2} - ρ_{e}^{2})

Choose virtual velocity error variables²⁶

α_{1} = v_{p} cos θ_{d} sin ψ_{e}

α_{2} = v_{p} sin θ_{e}

In order to make ${\dot{V}}_{1}$ negative, we choose u, $α_{1}$ , and $α_{2}$ as virtual controls, and their desired values are as follows

u_{d} = v_{p} cos θ_{d} cos θ cos ψ_{e} + v_{p} sin θ_{d} sin θ - k_{1} x_{e} + x_{e} \dot{ρ} / ρ

α_{1d} = - v - k_{2} y_{e} + y_{e} \dot{ρ} / ρ

α_{2d} = w - v_{p} cos θ_{d} sin θ (cos ψ_{e} - 1) + k_{3} z_{e} - z_{e} \dot{ρ} / ρ

where k ₁, k ₂, and k ₃ are positive constants.

Since the virtual variables u_d , $α_{1 d}$ , and $α_{2 d}$ are not true controls, we define error variables

u_{e} = u - u_{d}

α_{1 e} = α_{1} - α_{1 d}

α_{2 e} = α_{2} - α_{2 d}

Substituting equations (14 ) to (19) into equation (11) yields

{\dot{V}}_{1} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} + u_{e} x_{e} + α_{1 e} y_{e} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2})

Step 2: To stabilize the error variable u_e , define

V_{2} = V_{1} + \frac{1}{2} u_{e}^{2}

Its derivative along with equation (17) and equation (2a) becomes

\begin{matrix} {\dot{V}}_{2} & = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} + α_{1 e} y_{e} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) + u_{e} (\dot{u} - {\dot{u}}_{d} + x_{e} / (ρ^{2} - ρ_{e}^{2})) \\ = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} + α_{1 e} y_{e} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) + u_{e} (f_{u} + τ_{u} / m_{11} + D_{u} - {\dot{u}}_{d} + x_{e} / (ρ^{2} - ρ_{e}^{2})) \end{matrix}

If we choose control input $τ_{u}$ as

τ_{u} = - m_{11} f_{u} - m_{11} {\hat{D}}_{u} + m_{11} {\dot{u}}_{d} - m_{11} x_{e} / (ρ^{2} - ρ_{e}^{2}) - k_{4} m_{11} u_{e}

where k ₄ is a positive constant. Then equation (22) can be rewritten as

{\dot{V}}_{2} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} + α_{1 e} y_{e} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} + u_{e} {\tilde{D}}_{u}

Step 3: To stabilize the error variable $α_{1 e}$ , consider the following Lyapunov function

V_{3} = V_{2} + \frac{1}{2} α_{1 e}^{2}

Then the time derivative of V ₃ becomes

{\dot{V}}_{3} = {\dot{V}}_{2} + α_{1 e} {\dot{α}}_{1 e} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} (y_{e} / (ρ^{2} - ρ_{e}^{2}) + {\dot{α}}_{1 e})

Define

ϖ = - k_{2} y_{e} + y_{e} \dot{ρ} / ρ

It can be known

α_{1 d} = - v + ϖ

{\dot{α}}_{1 d} = - \dot{v} + \dot{ϖ}

The time derivative of equation (18) along with equation (12) and equation (29) becomes

{\dot{α}}_{1 e} = {\dot{α}}_{1} - {\dot{α}}_{1 d} = {\dot{v}}_{t} sin ψ_{e} + v_{t} cos ψ_{e} (r / cos θ - {\dot{ψ}}_{d}) + \dot{v} - \dot{ϖ}

In order to make ${\dot{V}}_{3}$ negative, define $\bar{r} = r cos ψ_{e}$ and its desired value ${\bar{r}}_{d} = r_{d} + {\dot{ψ}}_{d} cos θ (cos ψ_{e} - 1)$ . The desired value of r is as follows

r_{d} = {\dot{ψ}}_{d} cos θ + (- {\dot{v}}_{t} sin ψ_{e} - {\dot{v}}_{1} + \dot{ϖ} - y_{e} / (ρ^{2} - ρ_{e}^{2}) - k_{5} α_{1 e}) cos θ / v_{t}

where ${\dot{v}}_{1} = f_{v} + {\hat{D}}_{v}$ and k ₅ is a positive constant.

Considering that r_d is not a true control input, we define error variables

r_{e} = r - r_{d}

{\bar{r}}_{e} = \bar{r} - {\bar{r}}_{d}

We can obtain

{\bar{r}}_{e} = r_{e} cos ψ_{e} + δ_{1}

where $δ_{1} = (r_{d} - {\dot{ψ}}_{d} cos θ) (cos ψ_{e} - 1)$ .

Substituting equations (30 ) to (34) into equation (26) yields

{\dot{V}}_{3} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} + α_{1 e} v_{t} {\bar{r}}_{e} {cos}^{- 1} θ

Step 4: To stabilize the error variable r_e , define

V_{4} = V_{3} + \frac{1}{2} r_{e}^{2}

Its derivative along with equation (32), equation (35) and equation (2e) becomes

\begin{matrix} {\dot{V}}_{4} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} + α_{1 e} v_{t} {\bar{r}}_{e} {cos}^{- 1} θ + r_{e} (\dot{r} - {\dot{r}}_{d}) \\ = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} \\ + r_{e} (f_{r} + τ_{r} / m_{66} + D_{r} - {\dot{r}}_{d} + α_{1 e} v_{t} cos ψ_{e} {cos}^{- 1} θ) + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ \end{matrix}

In order to make ${\dot{V}}_{4}$ negative, control input $τ_{r}$ is chosen as

τ_{r} = - m_{66} f_{r} - m_{66} {\hat{D}}_{r} + m_{66} {\dot{r}}_{d} - m_{66} α_{1 e} v_{t} cos ψ_{e} {cos}^{- 1} θ - k_{6} m_{66} r_{e}

where k ₆ is a positive constant. Then equation (37) becomes

\begin{matrix} {\dot{V}}_{4} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2} - α_{2 e} z_{e}) / (ρ^{2} - ρ_{e}^{2}) \\ - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} \\ + r_{e} {\tilde{D}}_{r} + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ \end{matrix}

Step 5: To stabilize the error variable $α_{2 e}$ , consider the following Lyapunov function

V_{5} = V_{4} + \frac{1}{2} α_{2 e}^{2}

Then the time derivative of V ₅ becomes

\begin{matrix} {\dot{V}}_{5} = {\dot{V}}_{4} + α_{2 e} {\dot{α}}_{2 e} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2}) / (ρ^{2} - ρ_{e}^{2}) \\ - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} + α_{2 e} ({\dot{α}}_{2 e} - z_{e} / (ρ^{2} - ρ_{e}^{2})) \\ + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} + r_{e} {\tilde{D}}_{r} + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ \end{matrix}

Define

μ = v_{p} cos θ_{d} sin θ (cos ψ_{e} - 1) - k_{3} z_{e} + z_{e} \dot{ρ} / ρ

It can be known

α_{2d} = w - μ

{\dot{α}}_{2 d} = \dot{w} - \dot{μ}

The time derivative of equation (19) along with equations (16) and (44) becomes

{\dot{α}}_{2 e} = {\dot{α}}_{2} - {\dot{α}}_{2 d} = {\dot{v}}_{p} sin θ_{e} + v_{p} cos θ_{e} (q - {\dot{θ}}_{d}) - \dot{w} + \dot{μ}

In order to make ${\dot{V}}_{5}$ negative, define $\bar{q} = q cos θ_{e}$ and its desired value ${\bar{q}}_{d} = q_{d} + {\dot{θ}}_{d} (cos θ_{e} - 1)$ . The desired value of q is as follows

q_{d} = {\dot{θ}}_{d} + (- {\dot{v}}_{p} sin θ_{e} + {\dot{w}}_{1} - \dot{μ} + z_{e} / (ρ^{2} - ρ_{e}^{2}) - k_{7} α_{2 e}) / v_{p}

where ${\dot{w}}_{1} = f_{w} + {\hat{D}}_{w}$ and k ₇ is a positive constant.

Considering that q_d is not a true control input, we define error variables

q_{e} = q - q_{d}

{\bar{q}}_{e} = \bar{q} - {\bar{q}}_{d}

We can get

{\bar{q}}_{e} = q_{e} cos θ_{e} + δ_{2}

where $δ_{2} = (q_{d} - {\dot{θ}}_{d}) (cos θ_{e} - 1)$

Substituting equations (45 ) to (49) into equation (41) yields

\begin{matrix} {\dot{V}}_{5} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} \\ - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} - k_{7} α_{2 e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} \\ + r_{e} {\tilde{D}}_{r} - α_{2 e} {\tilde{D}}_{w} + α_{2 e} v_{p} {\bar{q}}_{e} + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ \end{matrix}

Step 6: To stabilize the error variable q_e , define

V_{6} = V_{5} + \frac{1}{2} q_{e}^{2}

Differentiating V ₆ with respect to time along with equation (47), equation (50), and equation (2d) yields

\begin{matrix} {\dot{V}}_{6} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} - k_{7} α_{2 e}^{2} + u_{e} {\tilde{D}}_{u} \\ + α_{1 e} {\tilde{D}}_{v} + r_{e} {\tilde{D}}_{r} - α_{2 e} {\tilde{D}}_{w} + α_{2 e} v_{p} {\bar{q}}_{e} + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ + q_{e} (\dot{q} - {\dot{q}}_{d}) \\ = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} - k_{7} α_{2 e}^{2} + u_{e} {\tilde{D}}_{u} + α_{1 e} {\tilde{D}}_{v} \\ + r_{e} {\tilde{D}}_{r} - α_{2 e} {\tilde{D}}_{w} + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ + q_{e} (f_{q} + τ_{q} / m_{55} + D_{q} - {\dot{q}}_{d} + α_{2 e} v_{p} cos θ_{e}) + α_{2 e} v_{p} δ_{2} \end{matrix}

In order to make ${\dot{V}}_{6}$ negative, control input $τ_{q}$ is chosen as

τ_{q} = - m_{55} f_{q} - m_{55} {\hat{D}}_{q} + m_{55} {\dot{q}}_{d} - m_{55} α_{2 e} v_{p} cos θ_{e} - k_{8} m_{55} q_{e}

where k ₈ is a positive constant. Then equation (52) becomes

\begin{matrix} {\dot{V}}_{6} = (- k_{1} x_{e}^{2} - k_{2} y_{e}^{2} - k_{3} z_{e}^{2}) / (ρ^{2} - ρ_{e}^{2}) - k_{4} u_{e}^{2} \\ - k_{5} α_{1 e}^{2} - k_{6} r_{e}^{2} - k_{7} α_{2 e}^{2} - k_{8} q_{e}^{2} + u_{e} {\tilde{D}}_{u} \\ + α_{1 e} {\tilde{D}}_{v} + r_{e} {\tilde{D}}_{r} - α_{2 e} {\tilde{D}}_{w} + q_{e} {\tilde{D}}_{q} \\ + α_{1 e} v_{t} δ_{1} {cos}^{- 1} θ + α_{2 e} v_{p} δ_{2} \end{matrix}

The complete Lyapunov function is as follows

V_{6} = \frac{1}{2} log (ρ^{2} / (ρ^{2} - ρ_{e}^{2})) + \frac{1}{2} u_{e}^{2} + \frac{1}{2} α_{1 e}^{2} + \frac{1}{2} r_{e}^{2} + \frac{1}{2} α_{2 e}^{2} + \frac{1}{2} q_{e}^{2}

Here, if we define

δ = | u_{e} {\tilde{D}}_{u} + α_{1 e} ({\tilde{D}}_{v} + v_{t} δ_{1} {cos}^{- 1} θ) + r_{e} {\tilde{D}}_{r} + α_{2 e} (v_{p} δ_{2} - {\tilde{D}}_{w}) + q_{e} {\tilde{D}}_{q} |

Then with the help of the inequality $log (ρ^{2} / (ρ^{2} - ρ_{e}^{2})) < ρ_{e}^{2} / (ρ^{2} - ρ_{e}^{2})$ , we have

{\dot{V}}_{6} \leq - 2 γ V_{6} + δ

where $γ = min {k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}, k_{7}, k_{8}}$ .

We know from the section “RLESOs design” that $E = {[{\tilde{D}}_{u}, {\tilde{D}}_{v}, {\tilde{D}}_{w}, {\tilde{D}}_{q}, {\tilde{D}}_{r}]}^{T}$ is bounded, so it is easy to know that $δ$ is bounded. According to the comparison principle,² the following inequality can be obtained

V_{6} \leq V_{6} (0) e^{- 2 γ t} + \frac{δ}{2 γ}

Therefore

‖ z (t) ‖ \leq ‖ z (0) ‖ e^{- γ t} + \sqrt{\frac{δ}{γ}}, t \in [0, + \infty)

Equation (59) means that the tracking error signals converge to a compression bounded value near the zero by increasing control gains appropriately and the system is stable.

Backstepping controller with TDs and RLESOs

TD can be used to provide the filtered version of the input signal and its differentiation as fast as possible. And it is well known that the operation of differentiation always causes the problem of “explosion of terms” when the traditional backstepping technique is employed. In order to solve this problem, we add TDs to the above-mentioned controller and the TDs are as follows

{\begin{array}{l} f h = f h a n (u_{c} (k) - u_{d} (k), R_{1}, h) \\ u_{c} (k + 1) = u_{c} (k) + h \cdot {\dot{u}}_{c} (k) \\ {\dot{u}}_{c} (k + 1) = {\dot{u}}_{c} (k) + h \cdot f h \end{array}

60a

{\begin{array}{l} f h = f h a n (r_{c} (k) - r_{d} (k), R_{2}, h) \\ r_{c} (k + 1) = r_{c} (k) + h \cdot {\dot{r}}_{c} (k) \\ {\dot{r}}_{c} (k + 1) = {\dot{r}}_{c} (k) + h \cdot f h \end{array}

60b

{\begin{array}{l} f h = f h a n (q_{c} (k) - q_{d} (k), R_{3}, h) \\ q_{c} (k + 1) = q_{c} (k) + h \cdot {\dot{q}}_{c} (k) \\ {\dot{q}}_{c} (k + 1) = {\dot{q}}_{c} (k) + h \cdot f h \end{array}

60c

where R ₁, R ₂, and R ₃ are the acceleration factors to be adjusted and h is the sampling period. The function $f h a n (\cdot)$ can be found in Han¹¹ and the following statement can be found in Miao et al.¹³

Corollary: Considering the TDs described by equation (60), if the input signals u_d , q_d , and r_d are differentiable and bounded, and if there exist arbitrarily small values a, b, c, $ε_{k} (k = 1, 2, \dots, 6)$ , then the solution of the considered TDs are

{\begin{matrix} lim_{R_{1} \to 0} | u_{c} - u_{d} | \leq ε_{1} \\ lim_{R_{1} \to 0} | {\dot{u}}_{c} - {\dot{u}}_{d} | \leq ε_{2} \end{matrix} \begin{matrix}  \end{matrix} t \in [a, \infty)

61a

{\begin{matrix} lim_{R_{2} \to 0} | r_{c} - r_{d} | \leq ε_{3} \\ lim_{R_{2} \to 0} | {\dot{r}}_{c} - {\dot{r}}_{d} | \leq ε_{4} \end{matrix} \begin{matrix}  \end{matrix} t \in [b, \infty)

61b

{\begin{matrix} lim_{R_{3} \to 0} | q_{c} - q_{d} | \leq ε_{5} \\ lim_{R_{3} \to 0} | {\dot{q}}_{c} - {\dot{q}}_{d} | \leq ε_{6} \end{matrix} \begin{matrix}  \end{matrix} t \in [c, \infty)

61c

Using u_c , q_c , r_c and ${\dot{u}}_{c}$ , ${\dot{q}}_{c}$ , ${\dot{r}}_{c}$ produced by TDs, the controller can be rewritten as

{\begin{array}{l} τ_{u} = - m_{11} f_{u} - m_{11} {\hat{D}}_{u} + m_{11} [{\dot{u}}_{c} + k_{9} (u_{c} - u)] - m_{11} x_{e} / (ρ^{2} - ρ_{e}^{2}) - k_{4} m_{11} u_{e} \\ τ_{q} = - m_{55} f_{q} - m_{55} {\hat{D}}_{q} + m_{55} [{\dot{q}}_{c} + k_{10} (q_{c} - q)] - m_{55} α_{2 e} v_{p} cos θ_{e} - k_{8} m_{55} q_{e} \\ τ_{r} = - m_{66} f_{r} - m_{66} {\hat{D}}_{r} + m_{66} [{\dot{r}}_{c} + k_{11} (r_{c} - r)] - m_{66} α_{1 e} v_{t} cos ψ_{e} {cos}^{- 1} θ - k_{6} m_{66} r_{e} \end{array}

Redefine equation (56) as

\begin{array}{l} δ = | u_{e} [{\tilde{D}}_{u} + ({\dot{u}}_{c} - {\dot{u}}_{d}) + k_{9} (u_{c} - u)] + α_{1 e} ({\tilde{D}}_{v} + v_{t} δ_{1} {cos}^{- 1} θ) + α_{2 e} (v_{p} δ_{2} - {\tilde{D}}_{w}) \\ \begin{matrix} + r_{e} [{\tilde{D}}_{r} + ({\dot{r}}_{c} - {\dot{r}}_{d}) + k_{10} (r_{c} - r)] + q_{e} [{\tilde{D}}_{q} + ({\dot{q}}_{c} - {\dot{q}}_{d}) + k_{11} (q_{c} - q)] \end{matrix} | \end{array}

where $δ$ is still bounded. And the system is still stable.

Simulation and discussion

In this section, the simulation and comparative analyses demonstrate the usefulness and robustness of the proposed controller. The simulation is performed in MATLABR2014a/Simulink and the following two controllers are compared.

Backstepping sliding mode controller.

{\begin{array}{l} τ_{u} = - m_{11} f_{u} + m_{11} {\dot{u}}_{d} - m_{11} x_{e} - k_{4} m_{11} u_{e} - η_{1} s a t (u_{e}, σ_{1}) \\ τ_{q} = - m_{55} f_{q} + m_{55} {\dot{q}}_{d} - m_{55} α_{2 e} v_{p} cos θ_{e} - k_{8} m_{55} q_{e} - η_{2} s a t (q_{e}, σ_{2}) \\ τ_{r} = - m_{66} f_{r} + m_{66} {\dot{r}}_{d} - m_{66} α_{1 e} v_{t} cos ψ_{e} {cos}^{- 1} θ - k_{6} m_{66} r_{e} - η_{3} s a t (r_{e}, σ_{3}) \end{array}

where

s a t (x_{1}, x_{2}) = {\begin{matrix} x_{1} / x_{2} \begin{matrix} | x_{1} | \leq x_{2} \end{matrix} \\ sgn (x_{1}) \begin{matrix} | x_{1} | > x_{2} \end{matrix} \end{matrix}

The control gains are chosen as $k_{1} = 0.2$ , $k_{2} = 0.1$ , $k_{3} = 0.1$ , $k_{4} = 1$ , $k_{5} = 1$ , $k_{6} = 2$ , $k_{7} = 1$ , $k_{8} = 3$ , $η_{1} = 2$ , $η_{2} = 2$ , $η_{3} = 2$ , $σ_{1} = 1$ , $σ_{2} = 1$ , and $σ_{3} = 1$ .

(2) Backstepping ADRC controller combining the backstepping technique, TDs, and RLESOs (equation (62)), is marked as backstepping ADRC in the simulation figures. The time-varying error constraint is chosen as $ρ = 3 exp (- 0.3 t) + 0.5$ and the control gains are chosen as $k_{1} = 0.2$ , $k_{2} = 0.1$ , $k_{3} = 0.1$ , $k_{4} = 1$ , $k_{5} = 1$ , $k_{6} = 2$ , $k_{7} = 1$ , $k_{8} = 3$ , $k_{9} = 1$ , $k_{10} = 1$ , $k_{11} = 10$ , $β_{1} = 50$ , $β_{2} = 50$ , $β_{3} = 70$ , $β_{4} = 40$ , $β_{5} = 70$ , $R_{1} = 15$ , $R_{2} = 8$ , $R_{3} = 9$ , and $h = 0.01$ .

We conduct simulation on the underactuated AUV WeiLong-II (Figure 2) developed by Harbin Engineering University in China. The nominal values of model parameters are $m = 45 kg$ , $I_{y} = 8.09 kg \cdot m^{2}$ , $I_{z} = 8.067 kg \cdot m^{2}$ , $X_{\dot{u}} = - 2.52 kg$ , $Y_{\dot{v}} = - 49.05 kg$ , $Z_{\dot{w}} = - 49.12 kg$ , $M_{\dot{q}} = - 5.43 kg \cdot m^{2}$ , $N_{\dot{r}} = - 5.31 kg \cdot m^{2}$ , $X_{u} = 13.5 kg / s$ , $X_{u | u |} = 6.44 kg / m$ , $Y_{v} = 44.96 kg / s$ , $Y_{v | v |} = 194.77 kg / m$ , $Z_{w} = 46.67 kg / s$ , $Z_{w | w |} = 141.66 kg / m$ , $M_{q} = 23.76 kg \cdot m^{2} / (s \cdot rad)$ , $M_{q | q |} = 23.76 kg \cdot m^{2} / {rad}^{2}$ , $N_{r} = 27.2 kg \cdot m^{2} / (s \cdot rad)$ , and $N_{r | r |} = 4.09 kg \cdot m^{2} / {rad}^{2}$ . We assume that parameter perturbations exist and all the model parameters are increased by 20% with respect to their respective nominal values. The ocean current disturbances are set as $V_{c x} = 0.3 m/s$ , $V_{c y} = 0.2 m/s$ , and $V_{c z} = 0.1 m/s$ . The time-varying disturbances are given as

{\begin{array}{l} d_{1} = 0.4 sin (0.2 t) \begin{matrix} N \end{matrix} \\ d_{2} = 0.4 sin (0.2 t) \begin{matrix} N \end{matrix} \\ d_{3} = 0.4 sin (0.2 t) \begin{matrix} N \end{matrix} \\ d_{5} = 0.4 sin (0.2 t) \begin{matrix} Nm \end{matrix} \\ d_{6} = 0.4 sin (0.2 t) \begin{matrix} Nm \end{matrix} \end{array}

Figure 2.

WL-II underactuated AUV. AUV: autonomous underwater vehicle.

The reference trajectory is given as follows

{\begin{array}{l} ξ_{d} = 10 sin (0.1 t) \begin{matrix} m \end{matrix} \\ η_{d} = 10 cos (0.1 t) \begin{matrix} m \end{matrix} \\ ζ_{d} = - 0.1 t \begin{array}{l} m \end{array} \end{array}

The AUV initial conditions are set as follows: $x_{0} = 2 m$ , $y_{0} = 8 m$ , $z_{0} = 0 m$ , $θ_{0} = 0 rad$ , $ψ_{0} = p i / 3 rad$ , $u_{0} = 0.3 m / s$ , $v_{0} = 0 m / s$ , $w_{0} = 0 m / s$ , $q_{0} = 0 rad / s$ , and $r_{0} = 0 rad / s$ . It can be seen that $ρ_{e} (0) < ρ (0)$ .

The simulation results are shown in Figures 3 to 8. As can be seen in Figure 3, the vehicle actual trajectory fluctuates around the desired trajectory under the backstepping sliding mode controller which is induced by the model uncertainties and disturbances, but the backstepping ADRC controller shows perfect performance. Clearly, the performance of the backstepping ADRC control laws is better than that of the backstepping sliding mode control laws in the presence of both internal parameter uncertainties and external time-varying disturbances.

Figure 3.

(a to c) Desired trajectory and vehicle actual trajectory.

Figure 4.

(a to d) Position tracking errors.

Figure 5.

(a to e) Velocity errors and virtual variable errors.

Figure 6.

(a to c) Responses of actual control inputs.

Figure 7.

(a to e) Estimates and actual of the total disturbances.

Figure 8.

(a to f) u_d ,r_d ,q_d and u_c , ${\dot{u}}_{c}$ ,r_c , ${\dot{r}}_{c}$ ,q_c , ${\dot{q}}_{c}$ produced by TDs. TDs: tracking differentiators.

The position error $ρ_{e}$ and x_e , y_e , and z_e are displayed in Figure 4. It shows the proposed control laws can constrain the position tracking error of an underactuated AUV within the given boundary function all the time by using a time-varying barrier Lyapunov function and make position tracking errors converge to zero while backstepping sliding mode control method results in large tracking error.

The responses of velocity errors and virtual variable errors are shown in Figure 5. The velocity errors and virtual variable errors converge to zero rapidly and smoothly under the proposed controller, but the velocities and virtual variables fluctuate around desired values with the disturbances under the backstepping sliding mode controller. The results clearly demonstrate the effectiveness of the designed RLESOs.

The responses of actual control inputs are displayed in Figure 6.

The actual values of the total disturbances and the estimation values of the RLESOs are shown in Figure 7 which demonstrates that the designed RLESOs can quickly and accurately estimate the total disturbances D_u , D_v , D_w , D_q , and D_r .

In addition, Figure 8 shows ${\dot{u}}_{c}$ , ${\dot{r}}_{c}$ , ${\dot{q}}_{c}$ and u_c , r_c , q_c produced by TDs, which can rapidly converge to u_d , r_d , q_d .

Conclusion

In this work, a backstepping ADRC controller based on the time-varying barrier Lyapunov function is developed to achieve three-dimensional trajectory tracking control of underactuated AUVs in the presence of internal parameter uncertainties and external time-varying disturbances and guarantee the satisfaction of predefined performance requirements. RLESOs are used to estimate and compensate for the total disturbances and TDs are employed to calculate the derivative of virtual control commands. From the simulation results, the proposed controller can remain satisfactory performance of an underactuated AUV in a complex environment and the simulation shows high accurate tracking capacity which demonstrates that using RLESOs can improve the performance of backstepping controller and using TDs can make the calculation of backstepping control laws simplified. Conclusions can be drawn that the backstepping ADRC controller has the ability to reject both internal parameter uncertainties and external time-varying disturbances. Experiments will be conducted to demonstrate the effectiveness of the proposed control laws and the effect of actuator saturation and faults will be studied in the near future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Key Research and Development Plan of China (No. 2017YFC0305700), National Natural Science Foundation of China (No. 51879057, No. U1806228, and No. 51809064), Qingdao National Laboratory for Marine Science and Technology (No. QNLM2016ORP0406), and Fundamental Research Funds for the Central Universities (No. HEUCFG201810 and No. 3072019CFJ0103).

ORCID iD

Ye Li

Yanqing Jiang

References

Pang

Wan

, et al. Stability analysis on speed control system of autonomous underwater vehicle. China Ocean Eng 2009; 23: 345–354.

Repoulias

Papadopoulos

. Planar trajectory planning and tracking control design for underactuated AUVs. Ocean Eng 2007; 34: 1650–1667.

Wang

Liu

, et al. Underactuated robotics: a review. Int J Adv Robot Syst 2019; 16: 1–29.

Elmokadem

Zribi

Youcef-Toumi

, et al. Trajectory tracking sliding mode control of underactuated AUVs. Nonlinear Dyn 2016; 84: 1079–1091.

Patre

Londhe

Waghmare

, et al. Disturbance estimator based non-singular fast fuzzy terminal sliding mode control of an autonomous underwater vehicle. Ocean Eng 2018; 159: 372–387.

Mousavian

Koofigar

HR.

Identification-based robust motion control of an AUV: optimized by particle swarm optimization algorithm. J Intell Robot Syst 2016; 85: 331–352.

Yildirim

. Design of neural network control system for controlling trajectory of autonomous underwater vehicles. Int J Adv Robot Syst 2014; 11: 1–17.

Liang

Hou

, et al. Three-dimensional trajectory tracking control of an underactuated autonomous underwater vehicle based on ocean current observer. Int J Adv Robot Syst 2018; 15: 1–9.

Wang

Qiao

, et al. Dynamical sliding mode control for the trajectory tracking of underactuated unmanned underwater vehicles. Ocean Eng 2015; 105: 54–63.

10.

Zhou

Zhao

, et al. Three-dimensional trajectory tracking for underactuated AUVs with bio-inspired velocity regulation. Int J Nav Archit Ocean Eng 2018; 10: 282–293.

11.

Han

. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009; 56: 900–906.

12.

Huang

Xue

Active disturbance rejection control: methodology and theoretical analysis. ISA Trans 2014; 53: 963–976.

13.

Miao

Wang

Zhao

, et al. Spatial curvilinear path following control of underactuated AUV with multiple uncertainties. ISA Trans 2017; 38: 1786–1796.

14.

Tee

Tay

. Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 2009; 45: 918–927.

15.

Tee

, et al. Control of nonlinear systems with time-varying output constraints. In: IEEE international conference on control and automation, Christchurch, New Zealand, 9–11 December2009, pp. 524–529. IEEE.

16.

Jin

. Adaptive fixed-time control for MIMO nonlinear systems with asymmetric output constraints using universal barrier functions. IEEE Trans Autom Control 2019; 64: 3046–3053.

17.

Nie

Lin

. Robust nonlinear path following control of underactuated MSV with time-varying sideslip compensation in the presence of actuator saturation and error constraint. IEEE Access 2018; 6: 71906–71917.

18.

Bechlioulis

Rovithakis

. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans Autom Control 2008; 53: 2090–2099.

19.

Elhaki

Shojaei

. Neural network-based target tracking control of underactuated autonomous underwater vehicles with a prescribed performance. Ocean Eng 2018; 167: 239–256.

20.

Wang

. Adaptive neural-based finite-time trajectory tracking control for underactuated marine surface vessels with position error constraint. IEEE Access 2019; 7: 16309–16322.

21.

Jia

Zhang

. Adaptive output-feedback control with prescribed performance for trajectory tracking of underactuated surface vessels. ISA Trans 2019; 95: 18–26.

22.

Dai

Wang

, et al. Adaptive neural control of underactuated surface vessels with prescribed performance guarantees. IEEE Trans Neural Netw Learn Syst 2019; 30: 3686–3698.

23.

Fossen

. Guidance and control of ocean vehicle. New Jersey: John Wiley & Sons, 1994.

24.

Pan

. Control of ships and underwater vehicles. Berlin: Springer, 2009.

25.

Shao

Wang

. Back-stepping active disturbance rejection control design for integrated missile guidance and control system via reduced-order ESO. ISA Trans 2015; 57: 10–22.

26.

Wang

Qiao

, et al. Backstepping-based controller for three-dimensional trajectory tracking of underactuated unmanned underwater vehicles. Control Theory Appl 2014; 31: 1589–1596.

Backstepping active disturbance rejection control for trajectory tracking of underactuated autonomous underwater vehicles with position error constraint

Abstract

Keywords

Introduction

Problem formulation

AUV modeling

Assumption 1

Assumption 2

Control objectives

RLESOs design

Backstepping ADRC controller design

Coordinate transformation

Backstepping controller with RLESOs

Backstepping controller with TDs and RLESOs

Simulation and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References