Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model

Abstract

This paper addresses the issue of trajectory tracking control for an autonomous underwater vehicle in the presence of parameter perturbations and disturbances in three-dimensional space. The control scheme is based on a combination of the backstepping method, the adaptive integral sliding mode control scheme, and velocity transformation resulting from the decomposition of the inertia matrix, which is symmetric. In addition, adaptive laws were applied to eliminate the effects of parameter perturbations and external disturbances. The main feature of the proposed approach is that the vehicle model is not fully symmetric but contains quantities due to the shift of the center of mass. Another important feature of the control scheme is the ability to detect some of the consequences caused by reducing the vehicle model by neglecting dynamic couplings. Numerical results on the five degrees of freedom (DOF) vehicle model show the efficiency, effectiveness, and robustness of the developed controller.

1. Introduction

Researching and developing underwater vehicles has attracted the attention of many scientists and researchers. Underwater research is a major challenge because the marine environment is complex and variable; thus, autonomous underwater vehicles (AUVs) can replace humans in performing many difficult and dangerous marine operations. Among this group of vehicles, underactuated marine vehicles are of interest as they have fewer actuators than degrees of freedom. An important reason for investigating underactuated systems is that fully actuated autonomous vehicles usually generate considerable costs and may be impractical given their weight, reliability, complexity, and efficiency. Of the various tasks that can be accomplished with these systems, one is tracking the desired trajectory.

Since general vehicle models have a complex mathematical form, due to the need to take into account both nonlinearities and disturbances from the environment, e.g., [1,2], various simplifications of these models are therefore used to design simplified control schemes. These difficulties make it very common to use reduced models of marine vehicles so that control strategies are simplified.

The basic model of a marine vehicle has six DOF and includes inertial couplings. Relevant trajectory tracking schemes can be found, for example, in [3,4,5]. In [3], the backstepping method and Lyapunov function were used, but the simulation example included a diagonal mass matrix. The strategy in [4] was based on artificial neural networks (ANNs) with an event-triggered control approach. The control scheme in [5] consisted of the model predictive control (MPC) method and an extended active observer. It is also sometimes assumed that the mutual couplings between variables can be ignored because they have small values, making it easier to design a control system. Such a simplification leads to the users obtaining a model with six DOF but with a diagonal inertia matrix. Examples of approaches based on the above assumption are shown in [6] (backstepping, Lyapunov function, fuzzy observer), in [7] (combination of adaptive fuzzy control with other methods), in [8] (backstepping and sliding mode control (SMC)), in [9,10] (neural network), and in [11,12] (backstepping and Lyapunov function). Making the model simpler is mathematically beneficial, but it moves the model further away from the real object. In such a situation, experimental studies on a real vehicle are convincing, e.g., [13].

Another known simplification is to reduce the vehicle model to five DOF so that it contains a diagonal mass matrix. Such a model can be found in the book by Do and Pan [1]. There are a number of trajectory tracking algorithms for a vehicle, as described. Schemes connecting ANNs to other controllers are given in [14] (with SMC), in [15] (with optimal control), in [16] (with prescribed performance), in [17] (with Lyapunov approach), and in [18] (backstepping and event-triggered approach). Other control methods for this class of models are as follows: the dynamic surface control (DSC) technique [19], the integral terminal SMC [20], a combination of a single critic network, adaptive dynamic programming (ADP), and backstepping [21], backstepping with an integral SMC scheme using bioinspired neurodynamics [22], backstepping with a Lyapunov function [23], a bioinspired model [24], backstepping, a Lyapunov function, and active disturbance rejection control (ADRC) [25], a PID regulator with SMC [26], SMC and optimal control [27], a method based on error and velocity transformation techniques enabling the design of a practical prescribed-time controller [28] and a controller that allowed the design of the desired error dynamics combining state variables in the body’s fixed coordinate and world coordinate [29]. Thus, it can be seen that the five DOF model is also used in real vehicles. However, the verification problem is similar to that for control based on six DOF models, namely, if experimental tests are not feasible for any reason, then the effectiveness of the control strategy for the simplified model with five DOF should be validated in the case of a shift in the center of mass. An effort to solve this problem has just been made in this work.

This paper proposes a tracking control for a five DOF underwater vehicle model with couplings due to a shift in the center of mass. First, the symmetric inertia matrix is decomposed to obtain the diagonal form expressed in inertial quasi-velocities (IQVs). Only then is a control strategy designed that combines backstepping and SMC. Controllers that use IQVs are applicable to tracking trajectories set in three-dimensional space for marine vehicles, but only when all input signals (for fully actuated systems) are available. In contrast, for underactuated vehicles, designing control algorithms using IQVs is challenging. The novelty of the proposed approach is that, in the event of the lack of even a single input signal, control using IQVs is not comparable to algorithms for a fully forced system. An additional advantage of control algorithms using IQVs is the ability to obtain insight into vehicle dynamics when changing dynamic parameters and the desired trajectory. Such information may be useful for preparing the correct operating conditions or modifying model parameters at the initial step of vehicle design. The proposed approach has been verified on a model representation of realistic vehicle parameters via numerical simulations. It turned out to be effective in the task of trajectory tracking, confirming the usefulness of using IQVs for control purposes. The simulation tests included one vehicle and two scenarios.

The article’s contributions are as follows:

(1)

Obtaining a five DOF underactuated marine vehicle model with coupling by diagonalization of the inertia matrix. This model is a generalization of the usually considered models. This is an extension of commonly used control schemes for marine vehicles with a diagonal inertia matrix to an algorithm suitable for vehicles with a lack of symmetry.

(2)

The design of a trajectory tracking control scheme for a vehicle moving in three-dimensional space that uses IQVs in traditional control methods. An important feature of the approach offered is the ability to examine the effect of shifting the mass center of the vehicles on the controller’s performance to see some of the consequences of simplifying a model containing dynamic couplings to one that ignores those couplings.

(3)

Simulation verification of the proposed strategy assuming different operating conditions regarding trajectories and couplings. A model of a real vehicle was adopted and, in contrast to some of the known work, the technical possibilities of the actuators were taken into account in order to provide operating conditions close to real ones.

It is not difficult to see that in the literature, trajectory tracking of five DOF vehicles by means of an experiment is rarely found because various problems associated with it have to be solved [29,30]. Much more often, work ends with simulation verification and the vehicle model is described by a diagonal inertia matrix. The proposed approach seems particularly useful at the design stage or in deciding whether to carry out an eventual real-world experiment because it indicates what difficulties may arise from using the controller if the center of mass is not identically located to the geometric center.

The article is organized as follows. Section 2 describes the problem under consideration. Section 3 presents the theorem concerning the proposed strategy. Section 4 contains simulation results confirming the effectiveness of the controller. Section 5 concludes the paper. The theoretical supplements can be found in Appendix A and Appendix B.

2. Description of Problem

2.1. Underactuated Marine Vehicle Model

An underactuated marine vehicle and the frame coordinates are presented in Figure 1.

The position and orientation variables vector is $\eta ={[x,y,z,\theta ,\psi ]}^T$, whereas the kinematic and dynamic models are [2]

$$\begin{array}{ccc}\hfill & & \dot{\eta }=J\left(\eta \right)\nu ,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill & & M\dot{\nu }+C\left(\nu \right)\nu +D\left(\nu \right)\nu +g\left(\eta \right)=\tau +{\tau }_D,\hfill \end{array}$$

(2)

where $\nu ={[u,v,w,q,r]}^T$ means the vector of linear and angular velocities (taking into account surge, sway, heave, pitch, and yaw velocity). The vector $\tau ={[{\tau }_u,0,0,{\tau }_q,{\tau }_r]}^T$ means the set of control input signals. The symbol ${\tau }_D$ means the vector of environmental disturbances together with unknown dynamics (because, in the dynamic equation, the matrix M must be symmetric). Other matrices and vectors are ($c\theta ,c\psi ,s\theta ,s\psi$ are abbreviations of $cos\theta ,cos\psi ,sin\theta ,sin\psi )$

$$\begin{array}{ccc}\hfill & & J\left(\eta \right)=\left[\begin{array}{ccccc}c\theta c\psi & -s\psi & s\theta c\psi & 0& 0\\ c\theta s\psi & c\psi & s\theta s\psi & 0& 0\\ -s\theta & 0& c\theta & 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1/c\theta \end{array}\right],M=\left[\begin{array}{ccccc}{m}_{11}& 0& 0& m_{14}& 0\\ 0& m_{22}& 0& 0& m_{25}\\ 0& 0& m_{33}& m_{34}& 0\\ m_{14}& 0& m_{34}& m_{44}& 0\\ 0& m_{25}& 0& 0& m_{55}\end{array}\right],\hfill \\ & & C\left(\nu \right)=\left[\begin{array}{ccccc}0& 0& 0& c_{14}& c_{15}\\ 0& 0& 0& c_{24}& c_{25}\\ 0& 0& 0& c_{34}& 0\\ -c_{14}& -c_{24}& -c_{34}& 0& 0\\ -c_{15}& -c_{25}& 0& 0& 0\end{array}\right],D\left(\nu \right)=\left[\begin{array}{ccccc}{d}_{11}& 0& 0& d_{14}& 0\\ 0& d_{22}& 0& 0& d_{25}\\ 0& 0& d_{33}& d_{34}& 0\\ d_{41}& 0& d_{43}& d_{44}& 0\\ 0& d_{52}& 0& 0& d_{55}\end{array}\right],\hfill \end{array}$$

(3)

and $g\left(\eta \right)={[0,0,0,g_4,0]}^T$ (for neutrally buoyant vehicle $W=B$ with $g_4=z_{g}Ws\theta +x_{g}Wc\theta$). Moreover, $J\left(\eta \right)$ denotes the transformation matrix, M denotes the mass matrix, including the added masses, $C\left(\nu \right)$ denotes the Coriolis and centripetal forces matrix, and $D\left(\nu \right)$ is the hydrodynamic damping matrix. The disturbances are included in vector ${\tau }_D={[{\tau }_{Du},{\tau }_{Dv},{\tau }_{Dw},{\tau }_{Dq},{\tau }_{Dr}]}^T$. Elements of the matrices are defined in Appendix A. Noting that $C^{\circ }\left(\nu \right)=C\left(\nu \right)\nu$ and $D^{\circ }\left(\nu \right)=D\left(\nu \right)\nu$, one obtains

$$\begin{array}{ccc}\hfill & & M\dot{\nu }+C^{\circ }\left(\nu \right)+D^{\circ }\left(\nu \right)+g\left(\eta \right)=\tau +{\tau }_D,\hfill \end{array}$$

(4)

where $C^{\circ }\left(\nu \right)={[c_1^{\circ },c_2^{\circ },c_3^{\circ },c_4^{\circ },c_5^{\circ }]}^T$ and $D^{\circ }\left(\nu \right)={[d_1^{\circ },d_2^{\circ },d_3^{\circ },d_4^{\circ },d_5^{\circ }]}^T$. Equation (4) can be rewritten in more compact form:

$$\begin{array}{ccc}\hfill & & M\dot{\nu }=F(\nu ,\eta )+\tau +{\tau }_D,\hfill \end{array}$$

(5)

where $F(\nu ,\eta )=-[C^{\circ }\left(\nu \right)+D^{\circ }\left(\nu \right)+g\left(\eta \right)]$.

The paper considers the effect of the center of mass shift on the trajectory tracking task. For this reason, all disturbances due to parameter inaccuracies and environmental influences are collected in an additional component that reflects them. The inertia matrix, in contrast, includes displacements so that their effect on system dynamics during vehicle movement can be estimated.

Equation (4) represents a description of the dynamics that ensure that this condition is met. One of the possible decomposition methods used in this paper is given in [31]. The application of the equations expressed in IQVs for fully actuated marine vehicles can be found, e.g., in [32].

After decomposition of the matrix M, one obtains a diagonal matrix $\mathbb{N}={\widehat{{\rm Y}}}^{T}M\widehat{{\rm Y}}$ ($\widehat{{\rm Y}}$ contains nominal parameters). This also means that $M={\widehat{{\rm Y}}}^{-T}\mathbb{N}{\widehat{{\rm Y}}}^{-1}$ (both matrices are positive definite). All nonlinearities resulting from their omission from the real matrix ${\rm Y}$ occur in the modified perturbation function $f={\tau }_D+\Delta {\rm Y}$, where $f={[f_u,f_v,f_w,f_q,f_r]}^T$. Furthermore, the nominal value of the matrix $\widehat{\mathbb{N}}$ is $\widehat{\mathbb{N}}={\widehat{{\rm Y}}}^T\widehat{M}\widehat{{\rm Y}}$.

The transformed equations in the place of (4) can be written as

$$\begin{array}{ccc}\hfill & & \mathbb{N}\dot{\zeta }=-{\widehat{{\rm Y}}}^T[C^{*}\left(\nu \right)+D^{*}\left(\nu \right)+g\left(\eta \right)]+{\widehat{{\rm Y}}}^T\tau +{\widehat{{\rm Y}}}^{T}f,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill & & \nu =\widehat{{\rm Y}}\zeta ,\hfill \\ \hfill & & \widehat{{\rm Y}}=\left[\begin{array}{ccccc}1& 0& 0& {\widehat{{\rm Y}}}_{14}& 0\\ 0& 1& 0& 0& {\widehat{{\rm Y}}}_{25}\\ 0& 0& 1& {\widehat{{\rm Y}}}_{34}& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],\mathbb{N}=diag\{{\mathbb{N}}_1,{\mathbb{N}}_2,{\mathbb{N}}_3,{\mathbb{N}}_4,{\mathbb{N}}_5\},\hfill \end{array}$$

(7)

where $\zeta ={[{\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5]}^T$ denotes the IQV vector. The remaining quantities are as follows: ${\mathbb{N}}_1=m_{11}$, ${\mathbb{N}}_2=m_{22}$, ${\mathbb{N}}_3=m_{33}$, ${\mathbb{N}}_4=m_{44}-(m_{14}^2/m_{11})-(m_{34}^2/m_{33})$, ${\mathbb{N}}_5=m_{55}-(m_{25}^2/m_{22})$, ${\widehat{{\rm Y}}}_{14}=-({\widehat{m}}_{14}/{\widehat{m}}_{11})$, ${\widehat{{\rm Y}}}_{25}=-({\widehat{m}}_{25}/{\widehat{m}}_{22})$, ${\widehat{{\rm Y}}}_{34}=-({\widehat{m}}_{34}/{\widehat{m}}_{33})$.

The equations of motion replacing (6) have the form

$$\begin{array}{ccc}\hfill & & {\zeta }_1=u-{\widehat{{\rm Y}}}_{14}q,{\zeta }_2=v-{\widehat{{\rm Y}}}_{25}r,{\zeta }_3=w-{\widehat{{\rm Y}}}_{34}q,{\zeta }_4=q,{\zeta }_5=r,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill & & {\mathbb{N}}_1{\dot{\zeta }}_1=F_1\left(\zeta \right)+{\tau }_u+f_u,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill & & {\mathbb{N}}_2{\dot{\zeta }}_2=F_2\left(\zeta \right)+f_v,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill & & {\mathbb{N}}_3{\dot{\zeta }}_3=F_3\left(\zeta \right)+f_w,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill & & {\mathbb{N}}_4{\dot{\zeta }}_4=F_4\left(\zeta \right)+{\tau }_{{\zeta }_4}+f_{{\zeta }_4},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill & & {\mathbb{N}}_5{\dot{\zeta }}_5=F_5\left(\zeta \right)+{\tau }_r+f_{{\zeta }_5},\hfill \end{array}$$

(13)

where

$$\begin{array}{ccc}& & F_1\left(\zeta \right)=-m_{33}wq-m_{34}{q}^2+m_{25}{r}^2+m_{22}vr-d_{11}u-d_{14}q,\hfill \\ & & F_2\left(\zeta \right)=-m{z}_{g}qr-m_{11}ur-d_{22}v-d_{25}r,\hfill \\ & & F_3\left(\zeta \right)=m_{14}{q}^2+m_{11}uq-d_{33}w-d_{34}q,\hfill \\ & & F_4\left(\zeta \right)={\widehat{{\rm Y}}}_{14}{F}_1\left(\zeta \right)+{\widehat{{\rm Y}}}_{34}{F}_3\left(\zeta \right)+F_{41}\left(\zeta \right),\hfill \\ & & F_5\left(\zeta \right)={\widehat{{\rm Y}}}_{25}{F}_2\left(\zeta \right)+F_{51}\left(\zeta \right),\hfill \end{array}$$

$$\begin{array}{ccc}& & F_{41}\left(\zeta \right)=m_{34}uq+(m_{33}-m_{11})uw+m{z}_{g}vr-m_{14}wq-d_{41}u-d_{43}w-d_{44}q-g_4,\hfill \\ & & F_{51}\left(\zeta \right)=-m_{25}ur+(m_{11}-m_{22})uv-d_{52}v-d_{55}r,\hfill \\ & & {\tau }_{{\zeta }_4}={\widehat{{\rm Y}}}_{14}{\tau }_u+{\tau }_q,f_{{\zeta }_4}={\widehat{{\rm Y}}}_{14}{f}_u+{\widehat{{\rm Y}}}_{34}{f}_w+f_q,f_{{\zeta }_5}={\widehat{{\rm Y}}}_{25}{f}_v+f_r.\hfill \end{array}$$

2.2. Preliminaries and Assumptions

The desired trajectory is defined as ${\eta }_d={[x_d,y_d,z_d]}^T$ (smooth and continuous functions), whereas linear position errors are defined as ${[x_e^E,y_e^E,z_e^E]}^T={[x-x_d,y-y_d,z-z_d]}^T$ (in the Earth frame). Moreover, the desired attitude angles are calculated from the relationship

$${\theta }_d=-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}2\left(\frac{{\dot{z}}_d}{\sqrt{{\dot{x}}_d^2+{\dot{y}}_d^2}}\right),{\psi }_d=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}2\left(\frac{{\dot{y}}_d}{{\dot{x}}_d}\right).$$

(14)

Comment. In [22,33], the arctan function was used. Here, however, the arctan2 function was applied as it was used previously, e.g., in [7,18].

Furthermore, the coordinate transformation has the form (e.g., [22,23])

$$\begin{array}{ccc}& & x_e=c\psi c\theta x_e^E+s\psi c\theta y_e^E-s\theta z_e^E,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & y_e=-s\psi x_e^E+c\psi y_e^E,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & z_e=c\psi s\theta x_e^E+s\psi s\theta y_e^E+c\theta z_e^E,\hfill \end{array}$$

(17)

with $x_e$, $y_e$, $z_e$ given in the body frame. After calculation of the time derivative using (1), one obtains (cf. [22,33])

$$\begin{array}{ccc}\hfill & & {\dot{x}}_e=u-u_d(c{\psi }_{e}c\theta c{\theta }_d+s\theta s{\theta }_d)+r{y}_e-q{z}_e,\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & {\dot{y}}_e=v+u_{d}s{\psi }_{e}c{\theta }_d-r(x_e+z_{e}t\theta ),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & {\dot{z}}_e=w-u_d(c{\psi }_{e}s\theta c{\theta }_d-c\theta s{\theta }_d)+q{x}_e+r{y}_{e}t\theta ,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & {\dot{\theta }}_e=q-q_d,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill & & {\dot{\psi }}_e=r/c\theta -r_d/c{\theta }_d,\hfill \end{array}$$

(22)

where $t\theta =tan\theta$ and $u_d=\sqrt{{\dot{x}}_d^2+{\dot{y}}_d^2+{\dot{z}}_d^2}$, $q_d={\dot{\theta }}_d$, and $r_d={\dot{\psi }}_d$.

Assumption 1.

The conditions specified for application of the five DOF model specified, e.g., in [30,34] are met.

Comment. In this work, CG lies in x–z plane, which means that $(x_g,0,z_g)$ instead of $(x_g,0,0)$. This change in assumption is crucial to the proposed control algorithm.

Assumption 2.

The bounded perturbations of the model parameter are defined as $|m_{ij}-{\widehat{m}}_{ij}|\le {\tilde{m}}_{ij}$, $|d_{ij}-{\widehat{d}}_{ij}|\le {\tilde{d}}_{ij}$, $|{\mathbb{N}}_i-{\widehat{\mathbb{N}}}_i|\le {\tilde{N}}_i$, in which $i,j=1,\cdots ,5$. In inequalities, the symbols represent the differences between the actual and nominal values of the parameters, which must be at most equal to the expected upper limit. Furthermore, the desired trajectories can be designed as continuous and differentiable, are bounded, and can be applied to the controller.

Comment. This type of assumption is known from, e.g., [6,35].

Assumption 3.

The functions $f_u,f_v,f_w,f_q,f_r$, as well as first time derivatives, are unknown but limited.

Comment. Such an assumption appears in [6,35,36,37].

Assumption 4.

The lumped uncertainty is limited [35,36]. Moreover, the thrust saturation effect is not dominant or persistent; therefore, ${\tau }_u$, ${\tau }_q$, and ${\tau }_r$ are considered to be bounded.

Assumption 5.

The approximation of the discontinuous signum function by the hyperbolic tangent function is allowed, as in, e.g., [38,39].

2.3. Control Objective

The aim of this work is to design a proposed controller for trajectory tracking that is robust, will guarantee that the tracking error converges to a small region in finite time, and that will ensure all closed-loop signals have limited values. The controller will work effectively when Assumptions 1–5 are satisfied. Due to the assumptions made, it is designed for vehicles moving slowly (less than 1 m/s).

3. Control Scheme for Trajectory Tracking

This section shows how to design the proposed control that allows trajectory tracking in three-dimensional space. It can be given in the following theorem.

Theorem 1. 

Consider the underactuated underwater vehicle described by (1) and (6)–(13). The aim of the control is the stabilization of trajectory tracking errors, i.e., ${lim}_{t\to \infty }{x}_e={\rho }_x$, ${lim}_{t\to \infty }{y}_e={\rho }_y$, ${lim}_{t\to \infty }{z}_e={\rho }_z$, ${lim}_{t\to \infty }{\theta }_e={\rho }_{\theta }$, and ${lim}_{t\to \infty }{\psi }_e={\rho }_{\psi }$, where ${\rho }_x,{\rho }_y,{\rho }_z,{\rho }_{\theta },{\rho }_{\psi }$ are some constants that define the neighborhood of zero. This means that it is possible to guarantee the convergence of tracking errors to this neighborhood. Thus, if Assumptions 1 ÷ 5 are met, then the signals from the controller ${\tau }_u$, ${\tau }_{{\zeta }_4}$ (including ${\tau }_u$, ${\tau }_q$), and ${\tau }_r$ enable the desired trajectory to be traced in three-dimensional space when parameter perturbations and external disturbances occur. The full control strategy includes the following:

(a) 

A kinematic algorithm for stabilizing kinematic errors;

(b) 

A dynamic algorithm consisting of ${\tau }_u$, ${\tau }_{{\zeta }_4}$, and ${\tau }_r$, which will guide the vehicle to the desired trajectory and provide uniformly ultimate boundedness.

Proof of this theorem is given in Appendix B.

4. Numerical Simulations

4.1. Model of the Tested Vehicle and Assumptions

The underwater vehicle ROPOS described in [37,40] was assumed for testing. Its parameters are given in

Table 1. ROPOS ROV parameters [37,40].

Parameter	Value	Parameter	Value
L	1.75 m	$X_u$	725 Ns/m
b	2.6 m	$Y_v$	1240 Ns/m
h	1.45 m	$Z_v$	825 Ns/m
m	2268 kg	$K_p$	3000 Ns/m
$J_x$	1937 Nms2	$M_q$	3000 Ns/m
$J_y$	2883 Nms2	$N_r$	1804 Nms
$J_z$	2457 Nms2	$X_{\left|u\right|u}$	1000 Ns2/m2
$X_{\dot{u}}$	−4380 kg	$Y_{\left|v\right|v}$	525 Ns2/m2
$Y_{\dot{v}}$	−9518 kg	$Z_{\left|w\right|w}$	400 Ns2/m2
$Z_{\dot{w}}$	−4268 kg	$K_{\left|p\right|p}$	100 Ns2
$K_{\dot{p}}$	−5000 kgm2	$M_{\left|q\right|q}$	100 Ns2
$M_{\dot{q}}$	−5000 kgm2	$N_{\left|r\right|r}$	72 Ns2
$N_{\dot{r}}$	−5000 kgm2

. It is a vehicle with a length, width, and height of $1.75$ m, $2.6$ m, and $1.45$ m, respectively. The achievable velocities can be assumed to be $u_{max}=1.25$ m/s and $v_{max}=0.5÷0.6$ m/s.

Limitations on forces and torques due to propulsion forces were taken into account for the tests so as to guarantee a realistic tracking task: $|{\tau }_u|\le 1400$ N, $|{\tau }_q|\le 1768$, and $|{\tau }_r|\le 350$ Nm.

Given a symmetric M matrix, as shown in Appendix A, the values $m_{14}=m_{41}=226.8$ kgm and $m_{25}=m_{52}=113.4$ kgm were assumed (for the original vehicle, they are omitted, but are needed for the simulation to assess the effect of the displacement of the center of mass). These values correspond to shifts in the center of mass of $z_g=0.1$ m and $x_g=0.05$ m.

Numerical calculations were made in the Matlab/Simulink environment assuming the duration of motion $t=200$ s (linear), $t=300$ s (complex), and time step size $\Delta t=0.05$ s applying the Bogacki–Shampine (ODE3) method. The desired trajectories are taken to be

$$\begin{array}{ccc}\hfill & & p_{d1}={[0.3t,0.1t,0.15t,{\theta }_{d1},{\psi }_{d1}]}^T,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill & & p_{d2}={[0.5t+5sin\left(0.025t\right),-5cos\left(0.025t\right),-0.1t,{\theta }_{d2},{\psi }_{d2}]}^T,\hfill \end{array}$$

(24)

with initial points $p_{0d1}={[-4,2,0,0,0]}^T$ (${\psi }_{0d1}=0$) and $p_{0d2}={[-2,1,-0.2,0,0]}^T$.

The following disturbance functions were selected:

$$\begin{array}{ccc}& & \left[\begin{array}{c}{f}_u\left(t\right)\\ f_v\left(t\right)\\ f_w\left(t\right)\\ f_q\left(t\right)\\ f_r\left(t\right)\end{array}\right]=\left[\begin{array}{c}12+sin\left(0.05t\right)+0.5sin\left(0.02t\right)N,\\ 1+0.2sin\left(0.02t\right)-0.3cos\left(0.03t\right)N,\\ 3+0.4sin\left(0.05t\right)-0.3cos\left(0.01t\right)N\\ 0.7+0.4sin\left(0.01t\right)-0.7cos\left(0.05t\right)Nm\\ 1.2+0.4sin\left(0.03t\right)-0.2cos\left(0.02t\right)Nm\end{array}\right]\hfill \end{array}$$

(25)

The designed algorithm allows for any assumptions of inaccuracy in the parameters of the dynamic model. For testing, it was assumed to be a factor of $W=0.2$, which means a 20% parameter error.

The performance evaluation is based on the desired and realized trajectory graph and the time history of important signals and indexes.

The time history presented on graphs concern linear position errors, angular position errors, linear velocities, angular velocities, applied force and torque, kinetic energy time history, quasi-velocity errors (difference between the quasi-velocities) and velocity, i.e., $\Delta {\zeta }_i={\zeta }_i-{\nu }_i$ ($i=1,2,3$).

The following quantities were used to evaluate the numerical results concerning the controller performance:

(1)

Mean integrated absolute error (MIAE) for $a=x_e,y_e,z_e,{\theta }_e,{\psi }_e$;

(2)

Mean integrated absolute control (MIAC) for ${\tau }_u,{\tau }_q,{\tau }_r$;

(3)

Root mean square of the tracking error (RMS), i.e., $RMS=\sqrt{\frac{1}{t_f-t_0}{\int }_{t_0}^{t_f}{∥e\left(t\right)∥}^{2}dt}$ where

$∥e_L\left(t\right)∥=\sqrt{{\left(x_e\right)}^2+{\left(y_e\right)}^2+{\left(z_e\right)}^2}$ ($x_e$, $y_e$, $z_e$ mean the linear position errors in the body frame) and $∥e_A\left(t\right)∥=\sqrt{{\left({\theta }_e\right)}^2+{\left({\psi }_e\right)}^2}$ (${\theta }_e$, ${\psi }_e$ mean the angular position errors);

(4)

Mean kinetic energy $\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(K\right)$, $K={\sum }_{i=1}^3{K}_i={\sum }_{i=1}^5\frac{1}{2}{N}_i{\zeta }_i^2$ (K denotes the kinetic energy).

Controllers’ parameter selection method. The selection of parameters used in the tests was based on the heuristic method described in [41], which worked well when parameters could be ranked according to their role in the controller. This intuitive method of taking into account the vehicles’ dynamics can also be used for five DOF vehicles.

4.2. Algorithm Test with Weak Couplings

4.2.1. Results for Quasi-Velocity (QV) Algorithm

The following control parameters were selected for the linear trajectory:

$$\begin{array}{ccc}\hfill & & k_x=0.2,k_{\theta }=1.0,k_{\psi }=0.8,{\lambda }_1={\lambda }_2={\lambda }_3=0.1,{\Gamma }_1={\Gamma }_2={\Gamma }_3=0.1,\hfill \\ & & {\beta }_1=8,{\beta }_2=0.2,{\beta }_3=8.\hfill \end{array}$$

(26)

Linear trajectory. The linear trajectory is tracked correctly, as can be observed in Figure 2a. Also, linear and angular errors, as shown in Figure 2b,c, decrease to their final values after about 80 s.

It is evident from Figure 2d,e that the speed u has a value that is much lower than that allowed for the ROPOS vehicle. The movement mainly takes place because of this velocity, since others have much smaller values and angular velocities have negligible values. The vehicle mainly performs forward motion, as is clear from Figure 2f, since the values of the surge force ${\tau }_u$ are the largest. Kinetic energy (including coupling) is primarily consumed by forward motion, as can be observed from Figure 2g. Figure 2h confirms that the dynamic couplings are very small because the linear velocity deformation values are not significant (the angular velocities are not deformed because the corresponding quasi-velocities are equal to them). For linear trajectories and weak couplings, the settling time for linear position errors is about 60 s, and for angular position errors, it is about 80 s.

Complex trajectory. The control parameters were changed due to the value of couplings, i.e.,

$$\begin{array}{ccc}\hfill & & k_x=0.2,k_{\theta }=k_{\psi }=1.0,{\lambda }_1={\lambda }_2={\lambda }_3=0.1,{\Gamma }_1={\Gamma }_2={\Gamma }_3=1.0,\hfill \\ & & {\beta }_1={\beta }_2={\beta }_3=0.6.\hfill \end{array}$$

(27)

Figure 3a clearly shows that the trajectory is tracking correctly, which is also reflected by the results in Figure 3b (linear position errors). However, it is apparent from Figure 3c that the angular position errors change slightly (it is worth noting that the angular velocities are variable in character, as is evident from their calculation).

Figure 3d,e show that the linear and angular velocities are also allowed, although with slightly higher values. The movement is primarily at forward velocity u. The drive of the vehicle is mainly by means of pitch torque, and is therefore different from the linear trajectory realization. The surge force is also important, as indicated by its values in Figure 3f. However, the kinetic energy consumption is more than double that of the previous case, as illustrated in Figure 3g. It is worth noting (Figure 3h) that the deformation of linear velocities is also small, although their changes are initially oscillatory. This proves that the coupling effect value is detectable regardless of the realized trajectory. For complex trajectories and weak couplings (until a change in the direction of motion), the settling time for linear position errors is about 50 s, and for angular position errors, it is about 80 s. Thus, these times are comparable for linear and complex trajectories.

4.2.2. Results for Algorithm without Couplings

The control algorithm, which omitted couplings and was taken for comparison, was designated CL (short for classical algorithm). This control scheme is known from [22], but here the neurodynamic model is excluded, so it is closer to the concept of the same authors published in [42]. The purpose of the comparative tests is to point out differences in the performance of the control algorithm based on the inertia matrix model, which takes into account the shift of the center of mass. Since the proposed control scheme uses a combination of the backstepping technique and the adaptive integral SMC method, for comparison, a paper using the same control methods but with a diagonal inertia matrix model was selected. In this way, the effect of dynamic couplings on the controller’s work and performance can be estimated. No other baseline was considered, which would have included a similar approach but with, for example, additional methods or components.

Linear trajectory. In order to show the differences in the signals obtained from the QV and CL controllers, the same set of parameters was assumed as in (26). Figure 4 demonstrates analogous results to those in Figure 2. The goal here is to make signal changes visible when the control algorithm does not take into account weak dynamic couplings resulting from small center of mass shifts. From Figure 4a, it can be observed that not all linear positions are tracked accurately. The large tracking error $z_e$ is due to the fact that there is no forcing signal acting on the variable z.

The high error values of the angular position ${\theta }_e$ (shown in Figure 4b) can also be explained by the lack of forcing of the variable z because the error is due to the backward and forward pitches. The results in Figure 4c show that the torque ${\tau }_q$ changes significantly as the vehicle moves. This proves that the controller is trying to reduce the variations caused by the dynamical couplings, but since they are constantly present, the values of this signal must also fluctuate. For linear trajectories and weak couplings, the settling time for linear position errors is about 50 s ($x_e,y_e$), and for angular position errors, it is about 50 s (${\psi }_e$). The errors of the other positions do not settle, which indicates that the control algorithm is not working properly.

The aim of the next test was to obtain acceptable linear trajectory tracking. The set found was as follows:

$$\begin{array}{ccc}\hfill & & k_x=0.2,k_{\theta }=4.0,k_{\psi }=6.0,{\lambda }_1=0.3,{\lambda }_2={\lambda }_3=0.5,{\Gamma }_1=1.0,{\Gamma }_2={\Gamma }_3=0.5,\hfill \\ & & {\beta }_1=5.0,{\beta }_2={\beta }_3=10.\hfill \end{array}$$

(28)

The need to use larger gain values for the kinematic $k_{\theta }=4.0,k_{\psi }=6.0$, and dynamic controller ${\lambda }_1,{\lambda }_2,{\lambda }_3$, ${\Gamma }_1,{\Gamma }_2,{\Gamma }_3$ is due to the fact that, when using similar values to the QV scheme, similar results will not be obtained. The reason for this is that the CL controller attempts to adapt its operation to the dynamic couplings in the system. It also means that assuming even 20 percent of the unknown knowledge of the model parameters is insufficient for the CL controller to work properly.

Figure 5a shows that trajectory tracking is inaccurate and the duration of obtaining the end value increases. This is also confirmed in Figure 5b, and the reason for the inaccuracy is still the error value $z_e$ (uncontrolled variable z). It may be observed that ${\theta }_e$ has a larger end error than the QV controller (Figure 5c). As is shown in Figure 5d,e, the linear velocities are similar to those of the QV controller, but the initial angular velocities have higher values. When the vehicle starts to move, the force ${\tau }_u$ and torque ${\tau }_q$ have large values, which indicates a large impulse in this phase of movement (Figure 5f).

For linear trajectories, weak couplings, and new control parameters set, the settling time for linear position errors is about 200 s for $y_e$ (and about 20 s and 80 s for $x_e$ and $z_e$, respectively), and for angular position errors, it is about 80 s. The error values are higher than they are for the QV controller.

Complex trajectory. This test allowed us to check how weak dynamic couplings affect the operation of CL controller when the trajectory is changed. It turned out that the parameter set (28) also allows the tracking of a complex trajectory.

By comparing Figure 3 and Figure 6, it can be noticed that, in the case of weak coupling, the CL algorithm is able to track the complex trajectory, but with a level of precision that may not be sufficient. Trajectory tracking is less exact than it is with the QV controller because the errors of the linear variables have larger values (cf. Figure 3a,b and Figure 6a,b). As can be noted from Figure 6d–f, the values of speed, force, and torque show that the correct vehicle response is achieved. From a comparison of the results obtained, it is clear that the main problem is tracking linear positions when center of mass position shifts are not considered. It is worth noting, however, that it is necessary to use larger values of ${\lambda }_1,{\lambda }_2,{\lambda }_3$ (relating to the sliding surfaces), smaller ${\Gamma }_2,{\Gamma }_3$ (relating to inaccurate knowledge of the model parameters), and larger $k_{\theta },k_{\psi }$ (relating to the kinematic controller). These modifications are due to changes in vehicle dynamics by omitting couplings.

For complex trajectories, weak couplings, and new gains set (until a change in the direction of motion), the settling time for linear position errors is about 120 s, and for angular position errors, it is about 80 s. In this case, there is an improvement in the performance of the CL controller, but the linear position errors are larger than they are for the QV controller.

4.3. Algorithm Test with Stronger Couplings

This section presents the test results of the QV controller under stronger dynamic couplings than before. Shifts in the center of mass have been increased to $x_g=-0.1$ m and $z_g=0.2$ m to show changes in the signals obtained from the system. Only the tracking of the more difficult trajectory for this task was investigated.

Complex trajectory. For the correct operation of the control scheme, the same set of parameters was sufficient, i.e., (27). From a comparison of Figure 3 and Figure 7, little change can be observed in the signals investigated. The similarities are in Figure 3a–e,g and Figure 7a–e,g, respectively. The ${\tau }_q$ signal has the opposite sign than before, although its values are comparable in both cases (Figure 3f and Figure 7f).

It should be noted that, despite the change in the sign of $x_g$ and the increase in $z_g$, the values of the control parameters may have remained the same, which may indicate their robustness to some changes in vehicle dynamics. The velocity deformations in Figure 3h and Figure 7h clearly indicate that the coupling effects have grown (as indicated by the value of the signals at the start of the movement). For complex trajectories and strong couplings (until a change in the direction of motion), the settling time for linear position errors is about 60 s, and for angular position errors, it is about 80 s. Thus, these times are comparable for complex trajectories and weak couplings.

4.4. Results Using Indexes

The tests performed were confirmed by the results collected in

Table 2. Performance for ROPOS.

		QV Controller			CL Controller
Index		Linear t.	Complex t.(1)	Complex t.(2)	Linear t.	Complex t.(1)
MIAE	$x_e$	0.3740	0.2981	0.2973	0.2026	0.3253
	$y_e$	0.1443	0.2393	0.2517	0.6107	0.4834
	$z_e$	0.1023	0.0356	0.0388	0.4713	0.4290
	${\theta }_e$	0.0436	0.0190	0.0145	0.0179	0.0134
	${\psi }_e$	0.0351	0.0478	0.0518	0.0351	0.0423
mean	$x_e$	−0.3411	−0.2666	−0.2658	−0.1616	−0.1147
	$y_e$	0.1379	0.2134	0.1941	0.6108	0.2509
	$z_e$	−0.0407	0.0266	0.0276	0.4713	0.4286
	${\theta }_e$	0.0097	0.0084	0.0081	0.0052	0.0076
	${\psi }_e$	−0.0297	−0.0221	−0.0221	−0.0351	−0.0206
std	$x_e$	0.9636	0.9303	0.9285	0.7007	0.9168
	$y_e$	0.3763	0.7915	0.7991	0.5722	0.9093
	$z_e$	0.1846	0.0564	0.0535	0.1706	0.1491
	${\theta }_e$	0.0854	0.0339	0.0224	0.0403	0.0205
	${\psi }_e$	0.0828	0.0967	0.1002	0.0428	0.0808
MIAC	${\tau }_u$	403.58	699.42	699.19	401.48	700.47
	${\tau }_q$	149.48	1568.8	1249.7	87.612	1567.5
	${\tau }_r$	7.2623	47.137	50.617	11.758	48.028
RMS	$\left|\right|e_L\left|\right|$	1.1130	1.2686	1.2685	1.2109	1.3950
RMS	$\left|\right|e_A\left|\right|$	0.1229	0.1051	0.1054	0.0684	0.0861
mean	K	460.55	1019.6	1019.1	459.50	1022.9

.

For a linear trajectory, it is enough to compare the values of the corresponding MIAE, mean, std, and RMS. On the other hand, the values of the applied forces and torques and the kinetic energy are comparable, which indicates that the controller effort for both algorithms (QV and CL) is also comparable. If either a linear or complex (1) trajectory is tracked, smaller tracking errors $y_e$, $z_e$ were obtained for the QV controller (than for CL), and therefore, in those directions for which there is no control signal. On the other hand, when there are control inputs, lower error values ($x_e$, ${\theta }_e$, ${\psi }_e$) were obtained for the CL algorithm. The total RMS value of linear errors is lower for the QV controller, while the RMS for angular errors is lower for the CL controller. However, by comparing the time histories of these quantities (shown previously), it can be seen that the CL scheme is less effective than the QV.

For the complex trajectory in case (1), as shown in Table 2, the linear position errors (according to MIAE) are also smaller for the QV controller compared to CL, although the angular errors have slightly larger values (but not significantly). The mean values and standard deviations do not support this for $x_e$, but overall, the RMS confirms the higher tracking accuracy of linear variables for the QV controller. The kinetic energy consumption is slightly lower for it. There is also little difference in the average values of force and torques (MIAC index). For the complex trajectory (2), performance improvements were obtained for the QV controller. Some values of the MIAE, mean, std, MIAC, and RMS indexes were reduced. The movement took place with less control input and the kinetic energy consumption was slightly lower (it is worth noting that these increased the gains responsible for reducing the effects of inaccurately known model parameters, as for the CL controller). The omission of indexes for the CL controller in the case of the complex trajectory was due to the fact that previously worse results than those for the QV control scheme related to tracking accuracy were obtained.

It is also worth noting that although some index values are smaller for the CL controller, this does not mean that it performs better than the QV-based controller. For linear position errors, the performance of the QV controller is definitely better. On the other hand, for angular position errors, the differences in numerical values are small (also their maximum values), and from the figures, it can be seen that the final values are not necessarily worse.

4.5. Analysis of Results

The control scheme was tested for a five DOF model used in ROPOS underwater vehicle marine research. The aim was to verify the effect of dynamic coupling resulting from the displacement of the center of mass relative to the geometric center. It was also taken as an assumption that verification should apply to different trajectories (linear and complex) and be performed taking into account the real technical possibilities and not only the performance of the control algorithms.

Position error convergence times of 50 to 100 s were obtained for the proposed algorithm. However, it must be taken into account that the initial position points were 2 to 6 m. Such distances, taking into account the weight of the vehicle (2268 kg) and the drag of movement in the water, significantly increase the time to reach the final positions in a steady-state motion. The convergence time depends on the initial distance points from the desired trajectory, the inertial parameters of the vehicle, and the assumed motion disturbance functions. Control algorithms for the ROPOS vehicle were tested, for example, in [37,40]. This work also took into account the limitations of the thrusters. From [37], the error convergence was achieved after about 50 s, but for an initial point less than 5 cm away from the reference trajectory. In [40], the error convergence was achieved after 25 s, but the starting distances from the desired trajectory were an order of magnitude shorter than assumed in this paper.

From the analysis of the chosen signals, the following was concluded:

1.

The performance of the QV controller was correct (i.e., it ensured that the task was completed in a finite period of time) for two trajectories and for selected slow, time-varying disturbances.

2.

Using the QV algorithm, it is possible to control the vehicle for both weak couplings and stronger ones resulting from the lack of full symmetry of the vehicle.

3.

The simulation test based on QV provides an insight into the impact of dynamic changes in the vehicle model resulting from a shift in the center of mass and makes it possible to determine whether the control strategy in this case is effective for different trajectories.

The abovementioned observations were confirmed by an analysis of the selected indexes. This analysis shows that tracking a complex trajectory needs more kinetic energy than tracking a linear trajectory.

Remarks on the comparison of the QV algorithm with another algorithm of the same type (based on a similar control strategy and denoted as CL):

1.

The QV algorithm provided acceptable performance for the assumed linear and complex trajectories, as well as for specified center-of-mass displacements.

2.

This scheme can be used for comparative tests with other controls designed for trajectory tracking containing both diagonal and symmetric inertia matrix models because, after reducing the elements containing couplings, a controller for systems without couplings will be obtained. This can provide information on the effects of shifting the center of mass.

3.

The QV controller allows one to determine the reduced kinetic energy of each quasi-velocity and therefore includes the coupling energy or deformation of linear velocities arising from the presence of couplings. No similar information is available with the CL controller or it will be incomplete, as in the case of kinetic energy.

4.

The CL algorithm, i.e., the one in which the inertia matrix is diagonal, is able to track a complex trajectory if the dynamic couplings are weak. However, the tracking accuracy is not sufficient for the linear position. In addition, due to inaccurate information on the model parameters, higher gain values are required for the sliding surfaces and for the kinematic controller. The selection of controller parameters may not be as easy as it is for the QV algorithm, which can be explained by the lack of inclusion of dynamic couplings in the control process.

5.

The CL algorithm tested included an option for robustness to dynamic modeling errors. It was possible to obtain worse tracking errors for linear variables than for the QV scheme; therefore, the algorithm tried to overcome this problem, although with insufficient results. Without this component in the control algorithm, it would be impossible to even obtain such performance. Therefore, it is very important to take into account the possibility of changing the model parameters.

6.

Using the QV algorithm, other information about changes in vehicle dynamics (velocity deformation errors) during movement can also be obtained.

5. Conclusions

The paper proposes a control scheme that is suitable for marine vehicles whose model includes dynamic couplings. The method implied transforming a model with a symmetric inertia matrix using a velocity transformation into a model where this matrix was diagonal. A control strategy consisting of two controllers, namely kinematic and dynamic, was then developed. The presented approach is particularly useful when the position of the center of mass differs from the geometric center and competing algorithms do not deliver satisfactory results. Simulation verification was carried out on a five DOF model of an ROPOS vehicle used in marine research with conditions that correspond to real operating conditions. Finally, an analysis of the results was given.

The QV control scheme made it possible to achieve acceptable performance with both weak and strong couplings for two different trajectories and under the assumption of engine limitation (according to vehicle specifications obtained from the literature). From the tests performed, it can be concluded that including the couplings arising from the shift in the center of mass in the inertia matrix is sensible because it leads to better results than when they are omitted. If the perturbation functions have small values (e.g., in the order of several N or Nm), they can represent small uncertainties in the model and environment perturbations, but inaccuracies due to a lack of symmetry should be included in the inertia matrix. Using a model with a symmetric inertia matrix resulted in better controller performance than if the model with a diagonal inertia matrix was applied.

The proposed control method is particularly useful at the pre-experimental stage (design, decision to possibly perform an experiment on a real object), as it allows the effect of dynamic couplings on the performance of the controller to be estimated.

In the future, more detailed research should point out the implications of vehicle model reduction for controller performance and include tests to determine whether the algorithm under study (known or designed) is effective when dynamic couplings are omitted from the dynamics model.