A practical robust yaw servo architecture of ROVs by multi-vector propulsion and nonlinear controller

Abstract

Multi-vector arrangement is a novel propulsion architecture for remotely operated vehicles (ROV) because of its high manoeuvrability and efficiency, but the influence on the ROV dynamics and attitude servo control has not yet been clearly evaluated. This study fully investigated the kinematic behaviours of a hexagonal multi-vector propulsion ROV with communication delay constraint and reduced the complex model for precision control system design. An enhanced model-based PI robust controller (EMPRC) based on the nominal model is proposed to solve the nonlinear hydrodynamics and communication problems with high performance yaw control, whose stability is also analysed. The conventional proportional-integral-derivative (PID) and integral separation PID are used in the experiments for comparison. The results indicate that the proposed EMPRC can effectively track the desired attitude and reject the external disturbances, while the conventional ones are limited by the nonlinear dynamics and communication delays. The improvement is 3x on average in terms of overshoot, settling time and anti-disturbance recovery time compared to conventional algorithms and proves this proposed novel EMPRC is a practical solution for multi-vector propulsion ROVs.

Keywords

EMPRC anti-disturbance multi-vector propulsion ROV

Introduction

Discovering and developing the oceans have become vital strategies for many coastal countries in the world to expand the horizon (García-Valdovinos et al., 2014)), since the ocean covers $71 %$ of the earth’s surface and potentially provides enormous resources and space for the human race. Remotely operated vehicles (ROV) play very important roles in the ocean engineering due to their portable size, high reliability and excellent manoeuvrability (García et al., 2015)). They have been widely utilized in ocean scientific research for data acquisition as well as to discover the resources and repair works in the engineering practice (Azis et al., 2012)).

The attitude control is a vital issue for ROVs since a steady attitude is highly recommended for many scenarios such as underwater oil pipeline inspection and seabed equipment maintenance (Sahu and Subudhi, 2017; Wu et al., 2019). This is very challenging, because ROVs typically travel in complicated ocean environments with time-varying and uneven currents (Fan and Woolsey, 2013). Within this extreme underwater nonlinear environment, it is critical to design a vehicle that adapts to a nonlinear environment (Vlahakis and Halikias, 2019).

The control system design approach tends to be an effective shortcut to improve the performance without modifying the hardware (Eidsvik and Schjlberg, 2016; Li et al., 2017). For any attitude control task, the first solution that comes to mind would be the proportional-integral-derivative (PID), which has advantages of high efficiency, small computation load and easy debugging for engineers. However, the conventional PID failed to meet the specifications of high-performance ROV attitude control because of the nonlinear hydrodynamics, and scholars proposed an advanced PID-type controller to overcome this issue. An experimental comparative performance of a PD controller and a nonlinear feedback adaptive algorithm to regulate the depth of an autonomous underwater vehicle (AUV) was presented in Maalouf et al. (2013). Bijani and Khosravi (2018) presented a new strategy for tuning the coefficients of a PID controller regarding $H \infty$ robust performance and stability constraints based on the constrained artificial bee colony (CABC) algorithm, which is one of the most recently introduced optimization algorithms with the advantages of good robustness, fast convergence and high flexibility and fewer setting parameters. Bijani et al. (2016) provided a straightforward method using evolutionary optimization algorithms to design an appropriate autopilot for non-minimum phase missiles. A control scheme based on PID technique for auto depth control and the equations for roll pitch stability is proposed by Tehrani et al. (2010). A nonlinear model-aided PD was proposed by Ramesh et al. (2013). Manzanilla et al. (2017) studied a control algorithm based on back-stepping technique with adaptive and integral properties for positioning and trajectory tracking of work-class ROVs. Rahimian and Tavazoei (2012) proposed an adjustment method that provides designers with a stable PI controller through using stable region centroids in controller’s parameter space. A new strategy was designed by Bagheri et al. (2010) to adjust PID coefficients for robust performance. In Li et al. (2018), a simple finite-time attitude stabilization controller was investigated to enforce a closed-loop attitude control system that can converge to origin in finite time by using estimated information from the proposed observer.

Other than PID-like methods, other controllers have been investigated, such as robust adaptive PID, adaptive control, sliding mode, etc. (Javadi-Moghaddam and Bagheri, 2010); Li et al., 2012; Molero et al., 2011). A systematically MIMO nonlinear robust controller designed based on adaptive backstepping technique and sliding mode control for ROV was introduced by Zhu and Gu (2011). Campos et al. (2017) accomplished two nonlinear controllers based on saturation functions with varying parameters for ROV with good experimental validation. Methodology designed for ROVs by Medina et al. (2016) and the design simulation was evaluated to reduce undesirable effect of the sliding mode chattering for ROV.

Intelligent control systems for ROVs have also attracted the attention of academia, which is advanced by the developed computer and instrumentation technology. Neural networks with PID were investigated in vehicles and nonlinear systems (Marzbanrad et al., 2011; Song et al., 2011) and the intelligent systems were mainly used as online parameter adjustment (Hernández-Alvarado et al., 2016)). The thoughts of intelligent controllers can be summarized as using an online data-driven policy to optimize the control variables. However, the stability of these methods is difficult to guarantee in the complex environment and the computational costs (time and power) are not affordable in the real ROV applications.

The influence of communication time delay characteristics has caused more complexity in identification (Sarhadi et al., 2015).

Structure and hardware design are other important factors for attitude control. There is usually a engineering trade-off between the attitude stability and the manoeuvrability. Multi-vector propulsion arrangement is becoming popular in lightweight ROVs since over-actuated engines provide more power, which can potentially avoid disturbances without decreasing manoeuvrability, but so far, few engineers have fully taken advantage of it. An undesired crucial limiting factor that is usually overlooked in ROV control systems is the communication delay. The fundamental electronics such as microprocessors, data converters and power electronics are typically placed in a well-sealed container with few communication connectors. Frequently programming and debugging the microprocessor is not economical. So implementing the advanced control algorithm in the remote PC side and the microprocessor only reserved for the basic propeller operations would be a common choice, especially if the researchers want to prototype a variety of control algorithms. In this case, the communication delay between the ROV and PC becomes very serious, considering the length of the connection cable can is at the 100-metre level.

Based on the above discussions, we identified that investigating the dynamics of the multi-vector propulsion ROV with communication delay and establishing a practical yaw servo control system can contribute both to the industry and to academia. The validation platform is a lightweight ROV with multi-vector propulsion arrangement designed and manufactured by Ocean University of China.

The contributions of this paper are summarized as follows:

This paper first proposes a comprehensive architecture that takes advantage of multi-vector propulsion arrangement mechanism and its cooperated EMPRC controller to provide improved attitude tracking and anti-disturbance capability.

A practical kinematic model of the multi-vector propulsion arrangement driving system with communication delay is first established. It initiates with the hydrodynamics of the ROV, then proves the nonlinearity cancellation effect by using the multi-propulsion design that improves the inherent motional performance with a time delay factor. The residual non-ideal factors under this hardware configuration are discussed for the control system design.

The EMPRC is designed to cooperate with this unique driving system and optimized for the residual non-ideal components. It describes the system with a nominal model obtained by system identification underwater experiments and a residual hydrodynamic error term. An optimized control law includes the variable gain PI component and robust terms is proposed and analysed to meet the high performance tracking and anti-disturbance requirement.

The enhancement of EMPRC is validated experimentally by comparing with the conventional PID controller and integral-separation PID controller. The results demonstrate that the key metrics have been improved by 3x on average. In detail, the overshoot of EMPRC is reduced by $10 % - 80 %$ and the settling time is shortened by $4 s - 15 s$ in experiments.

Establishment of ROV dynamics model

Analysis of the overall structure of ROV

An open-frame ROV is utilized for experimental analysis in this paper, in which the propellers are multi-vector arranged. One of our aims is to study the advantages of vector symmetric polyhedron mechanical structure for vehicles in dealing with nonlinear environments. In this vein, we first propose and design a ROV with the mechanical structure of vector-symmetric hexagonal. In addition, the excellent performance of ROVer’s mechanical structure is constantly reflected in actual yaw tracking experiments underwater. To further research yaw angle motion of ROVer, this paper first derives the motion equation of ROVer on the horizontal surface. According to Society of Naval Architects and Marine Engineers (SNAME), the static coordinate system $E - ξ η ζ$ and the dynamic coordinate system $O (G) - xyz$ are established according to right-hand rule (Chin et al., 2018). The overall structure sketch is shown in Figure 1.

Figure 1.

The vehicle ROVer, with the static and dynamic coordinate system.

The static coordinate system is based on a point on water surface as the origin. As shown in Figure 1, $E ξ$ , $E ζ$ on the horizontal plane. $E η$ pointing to centre of the earth. Displacement and direction vector of ROV are described as $η$ ,

\begin{matrix} η = (η_{1}, η_{2})^{T} = [(x, y, z), (φ, θ, ψ {)]}^{T} \in R^{6 \times 1} . \end{matrix}

(1)

where $x, y, z$ refer to displacement of ROV in fixed coordinate system. $φ$ (roll), $θ$ (pitch), and $ψ$ (yaw) are Euler angles between fixed coordinate system and the dynamic coordinate system, which are utilized to represent ROV’s attitude. Linear and angular velocity vector as $v$ ,

\begin{matrix} v = (v_{1}, v_{2})^{T} = [(u, v, w), (p, q, r {)]}^{T} \in R^{6 \times 1} . \end{matrix}

(2)

The force and moment vector is $[X, Y, Z, K, M, N]^{T}$ in dynamic coordinate system. R0V’s speeds along the directions of coordinate axes $x$ , $y$ , and $z$ are $u$ , $v$ , and $w$ , respectively, and the acting forces are $X$ , $Y$ , and $Z$ . ROV’s angular velocities around the three coordinate axes are $p$ , $q$ , and $r$ and the moments are $K$ , $M$ , $N$ .

ROVer is actuated by eight propellers for roll, pitch and yaw, and the propellers of that distribution diagram are shown in Figure 2. Four horizontal propellers (T1 to T4) are utilized to control yaw. Undesired interference factors can be mitigated by ROVer’s multi-vector propulsion device with a hexagonal three-dimensional space structure. The linear velocity of the horizontal plane in $x$ and $y$ can be guaranteed to be small when the propeller’s speed is slow.

Figure 2.

Propellers distribution diagram of ROVer.

ROVer’s propellers thrust (moment) vector $T = [T_{h 1}, T_{h 2}, T_{h 3}, T_{h 4}]^{T}$ , and the thrust (moment) vector $[F_{x}, F_{y}, W_{z}]^{T}$ of ROVer in x-axis, y-axis, and z-axis are expressed as:

[\begin{matrix} F_{x} \\ F_{y} \\ W_{z} \end{matrix}] = [\begin{matrix} \cos α & - \cos α & \cos α & - \cos α \\ \sin α & \sin α & - \sin α & - \sin α \\ K & - K & K & - K \end{matrix}] [\begin{matrix} T_{h 1} \\ T_{h 2} \\ T_{h 3} \\ T_{h 4} \end{matrix}],

(3)

where $K = (y_{h} \cos α - x_{h} \sin α)$ . The expression for thrust calculation is: $T_{ih} = (ρ n^{2} D^{4} K_{T})$ , $i = 1, 2, 3, 4$ . where $ρ$ refers to water density, $n$ is propeller’s speed, $D$ is propeller’s diameter, $K_{T}$ is thrust coefficient. $x_{h}$ , $y_{h}$ represent the distance of the horizontal propeller relative to the origin of coordinate system. $α$ refers to the angle between the propeller axis and the horizontal plane. We focus on yaw control and kinematics model of the horizontal plane for ROVer will be established and illustrated.

Dynamics analysis of the yaw by multi-vector propulsion solution

The six degrees-of-freedom (DOF) equation of motion for vehicle is actually cross-coupled. Intuitively, the movement of each degree of freedom is not isolated but related to several other degrees of freedom. Complex kinematic equations are one of the obstacles to the precise control of ROV (Rezazadegan et al., 2015) and can be solved by decomposing or simplifying the system. Assuming the vehicle’s depth is constant and the change in heading makes the centre of gravity for ROV always on the horizontal plane, the coupling of horizontal and vertical plane degrees of freedom is ignored to decompose the horizontal motion of ROVer. The equation of motion on the horizontal plane is expressed as:

{\begin{matrix} X = m [(\overset{\cdot}{u} - vr + wq - x_{g} (q^{2} + r^{2}) + y_{g} (pq - \overset{\cdot}{r}) \\ + z_{g} (pr + \overset{\cdot}{q})] \\ Y = m [(\overset{\cdot}{v} + ur - wp - y_{g} (r^{2} + p^{2}) + z_{g} (qr - \overset{\cdot}{p}) \\ + x_{g} (pq + \overset{\cdot}{r})] \\ N = I_{z} \overset{\cdot}{r} + (I_{y} - I_{x}) pq + m [x_{g} (v + ur - wp) \\ - y_{g} (u + wq - vr)] . \end{matrix}

(4)

ROVer’s position in dynamic coordinate system is represented by $G = [x_{g}, y_{g}, z_{g}]$ . $I_{xg}$ , $I_{yg}$ , $I_{zg}$ are the moments of inertia of the ROV with respect to the x, y, and z axes, respectively. The inertia shift theorem is:

{\begin{matrix} I_{x} & = I_{xg} + m (y_{g}^{2} + z_{g}^{2}) \\ I_{y} & = I_{yg} + m (z_{g}^{2} + x_{g}^{2}) \\ I_{z} & = I_{zg} + m (x_{g}^{2} + y_{g}^{2}) . \end{matrix}

(5)

The physical model parameters and hydrodynamic parameters are listed in Table 1.

Table 1.

Physical parameters of ROVer.

Physical parameters (mark)	Unit
Mass of ROV( $m$ )	$kg$
Length of ROV( $L$ )	$m$
Water density ( $ρ$ )	$kg / m^{3}$
Reference speed( $u_{0}$ )	$m / s$
Moment of inertia( $I_{x, y, z}$ )	$kg . m^{2} / s^{2}$
Hydrodynamic coefficient ( $N_{(.)} / N_{(.) \| . \|}$ )	$kg / m^{2}$

Focusing on ROVer’s yaw control, we ignore the role of coriolis force. Ignore the effects of the off-diagonal elements of ROV’s matrix inertia tensor and the motion of the swaying degrees of freedom. $τ_{pro}$ is the total thrust and torque. The simplified ROV’s motion model is:

\begin{matrix} (M_{RB} + M_{A}) \overset{\cdot}{v} + (D_{L} + D_{Q} | v |) v + g (η) = τ_{pro}, \end{matrix}

(6)

$M_{RB}$ represents the inertia matrix. Additional mass matrix is $M_{A}$ . $D_{L}$ refers to linear matrix of hydrodynamic damping coefficients, and its second-order linear hydrodynamic damping coefficient matrix $D_{Q}$ . $g (η)$ refers to the restoring force matrix.

To calculate the additional mass and the linear dynamic damping coefficient matrix of ROVer, we utilize computational fluid dynamics (CFD) method to calculate hydrodynamic damping coefficient about yaw control. Consider the motion of ROVer in rotation mode and calculate the rotational damping moment. As shown in Figure 3, watershed I is the internal rotating watershed, and sub-watersheds II, III, IV and V are external static watersheds. ROVer’s length is $L$ , and its height is $H$ .

Figure 3.

Structure of yaw rotation calculation domain.

The fitting relationship between the damping force and moment of yaw rotation is:

\begin{matrix} M_{z} = N_{r | r |} {w_{z}}^{2}, \end{matrix}

(7)

where $M_{z}$ is the damping torque received by ROV with the rotational angular velocity $w_{z}$ . The additional mass $N_{\overset{\cdot}{r}}$ in the case of rotation is:

\begin{matrix} N_{\overset{\cdot}{r}} = \frac{| M_{z} - M |}{{\overset{\cdot}{w}}_{z}} . \end{matrix}

(8)

The coordinate conversion relationship between the angular velocity in the ROV dynamic coordinate system and the static coordinate system is:

[\begin{matrix} \overset{\cdot}{φ} \\ \overset{\cdot}{θ} \\ \overset{\cdot}{ψ} \end{matrix}] = [\begin{matrix} 1 & \sin φ \tan θ & \cos φ \tan θ \\ 0 & \cos φ & - \sin φ \\ 0 & \sin φ / \cos θ & \cos φ / \cos θ \end{matrix}] [\begin{matrix} p \\ q \\ r \end{matrix}] .

(9)

The relationship between $\overset{\cdot}{φ}$ and $r$ can be deduced. Assuming that the origin $O$ of the $O (G) - xyz$ coordinate system coincides with the centre of gravity $G$ of ROVer (Tran et al. (2015)). Ignore the effects of other uncertain small disturbances and currents in the equation. The yaw moment equation of ROVer is as follows:

\begin{matrix} I_{z} \overset{\cdot}{r} + (I_{y} - I_{x}) pq = \frac{1}{2} ρ L^{5} [N_{\overset{\cdot}{r}} \overset{\cdot}{r} + N_{pq} pq + N_{qr} qr \\ + N_{r | r |} r | r |] + \frac{1}{2} ρ L^{4} [N_{\overset{\cdot}{v}} \overset{\cdot}{v} + N_{ur} ur + N_{vq} vq \\ + N_{wp} wp + N_{v | r |} v | r |] + \frac{1}{2} ρ L^{3} [N_{uv} uv \\ + N_{vw} vw + N_{v | v |} v | v |] \\ + \frac{1}{2} ρ L^{3} u^{2} N_{ψ} ψ + W_{z} . \end{matrix}

(10)

According to arrangement of propellers on ROVer, the residual linear velocity generated by course control of propeller is negligible when the speed is small in the experiment. i.e. $u = v = w = 0$ . Ignore hydrodynamic items with smaller values. (10) is converted to:

\begin{matrix} (I_{z} - \frac{1}{2} ρ L^{5} N_{\overset{\cdot}{r}}) \overset{\cdot}{r} & = \frac{1}{2} ρ L^{5} [N_{pq} pq + N_{qr} qr + N_{r | r |} r | r |] \\ + W_{z} - (I_{y} - I_{x}) pq . \end{matrix}

(11)

Obviously, incomplete equilibrium will inevitably occur during the movement. Angular velocities $p$ , $q$ appear on the horizontal plane. To respond to this situation, we classify ROV as a large dynamic interference term. From (9), the relationship between angular velocity and pitch angle is as follows:

\overset{\cdot}{ψ} = r \cos φ / \cos θ + q \sin φ / \cos θ .

(12)

The transmission mechanism is typically done through a cable and results in a time delay $τ$ between $T_{hi}$ and $W_{z}$ . The actual torque $W_{i}$ is calculated by the host and transmitted to the vehicle. In addition, due to the existence of mechanical kinematics and hardware structure errors of actual system, delays occur in the control process. We assume that time delay coefficient is $τ$ ,

{\overset{\cdot}{W}}_{z} = τ W_{z} + W_{i} .

(13)

Based on the above, we select ${[\begin{matrix} ψ & r & W_{z} \end{matrix}]}^{T}$ as state variables. Combined with (11), (12), (13), we reintegrated and simplified the equation. Obviously, in a large dynamic environment, a nonlinear damping term $| r |$ represents dead zone characteristic in the system. The equivalent gain of nonlinear characteristics of the system is changed in real time, resulting in stability and precision. The presence of external disturbances causes yaw to change as well. Many nonlinear coupling relationships lead to poor control of traditional PID.

The nonlinear damping $| r |$ is determined to a constant value $ε$ in a small dynamic environment. $θ = φ = 0$ . The linear model in small dynamics can be expressed as:

{\begin{matrix} (I_{z} - \frac{1}{2} ρ L^{5} N_{\overset{\cdot}{r}}) \overset{\cdot}{r} = \frac{1}{2} ε ρ L^{5} N_{r | r |} r + W_{z} \\ \overset{\cdot}{ψ} = r \\ {\overset{\cdot}{W}}_{z} = τ W_{z} + W_{i} . \end{matrix}

(14)

For small dynamic linear (12), PID can play an effective control role. It is difficult for PID to achieve the control requirements of accurate and fast when the speed of the ROV is too fast or the external interference is too large. The model of ROVer can be built as a standard state space model:

{\begin{matrix} \overset{\cdot}{x} (t) = A x (t) + B u (t) \\ y = C x (t) . \end{matrix}

(15)

Observation vector $y = ψ$ . Ignoring the influence of vertical motion of ROVer and hydrodynamic parameters with smaller values. Model parameters of ROVer obtained by system identification through pool experiment are described in detail in the experimental section. The detailed information of state space model is described by:

{\begin{matrix} A & = [\begin{matrix} 0 & \frac{ε ρ L^{5} N_{r | r |}}{2 I_{z} - ρ L^{5} N_{r}} & \frac{2}{2 I_{z} - ρ L^{5} N_{r}} \\ 0 & 1 & 0 \\ 0 & 0 & τ \end{matrix}] \\ B u & = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{matrix}] \frac{T_{hi}}{y_{h} \cos α - x_{h} \sin α} \\ C & = [\begin{matrix} 1 & 0 & 0 \end{matrix}] . \end{matrix}

(16)

With nonlinear enhancement of system under large dynamics, it is difficult to achieve stable and reliable control of PID for ROV. The multi-vector propulsion arrangement of ROV makes the system more stable and offsets some of the unknown forces. However, it is not enough to rely solely on mechanical mechanisms to achieve excellent control performance. The design of a stable and efficient controller is essential.

Control system design

Three control methods are utilized for yaw tracking. These methods are described in this section. For the horizontal motion of ROVer, two state variables are our main consideration, yaw $ψ$ and angular velocity $r$ . $x_{d}$ is defined as desired angle. $e = x_{d} - ψ$ refers to the yaw tracking error. Errors are defined in this paper as follows:

{\begin{matrix} \overset{\cdot}{e} = {\overset{\cdot}{x}}_{d} - \overset{\cdot}{ψ} \\ \overset{\cdot\cdot}{e} = {\overset{\cdot\cdot}{x}}_{d} - \overset{\cdot\cdot}{ψ} . \end{matrix}

(17)

Classical PID controller

PID’s control law can be expressed in following basic form (Astrom 2005):

\begin{matrix} u (t) = K_{p} [e + \frac{1}{T_{1}} \int_{0}^{t} edt + T_{d} \frac{de}{dt}], \end{matrix}

(18)

where $K_{p}$ refers to proportional gain coefficient. $T_{1}$ refers to time integral constant. $T_{d}$ refers to differential integral constant. PID provides precise control in a linear system with accurate system structural parameters (e.g. 14). But a system in reality is usually located in a dynamic environment. For (10), (11), PID cannot obtain better control effect.

Integral separation PID controller

The purpose of introducing the integral link in PID is mainly to eliminate static difference and improve precision. However, during the period of tracking of yaw in the beginning or when there is a large disturbance in environment, accumulation of a large amount of angular errors generated in a short time causes the output to overshoot, which damages the propeller. The specific implementation steps of integral separation PID are as follows:

Reasonably set the value $ε > 0$ ;

When $| e | > ε$ , utilize PD mode to solve overshoot problem and obtain faster response;

When $| e | \leq ε$ , utilize PID mode to ensure control precision of system.

The control law of integral separation PID is as follows:

\begin{matrix} u (k) = K_{p} (k) + f (β) K_{i} \sum_{j = 0}^{k} e (j) T + K_{d} (e (k) - e (k - 1)) / T, \end{matrix}

(19)

where $T$ represents sampling time. $f (β)$ refers to switching factor. The rule is as follows:

\begin{matrix} f (β) = {\begin{matrix} 0, & | e | > ε \\ 1, & | e | \leq ε . \end{matrix} \end{matrix}

(20)

Enhanced model-based PI robust controller

According to models (14) and (15), the system is further described as follows:

{\begin{matrix} \overset{\cdot}{ψ} = r \\ J \overset{\cdot}{r} + Cr = d_{2} + u_{e}, \end{matrix}

(21)

where the state variabls are ${[\begin{matrix} ψ & r \end{matrix}]}^{T}$ . $J$ refers to the system moment of inertia. $C$ refers to the viscosity coefficient. $d_{2}$ refers to uncertainty disturbance on control input $u_{e}$ . $d_{2}$ includes system delay interference in (13).

We define error function $s = \overset{\cdot}{e} + ce$ , where $c > 0$ . We define ${\overset{\cdot}{x}}_{r} = s + r$ .

{\begin{matrix} {\overset{\cdot}{x}}_{r} = {\overset{\cdot}{x}}_{d} + ce \\ {\overset{\cdot\cdot}{x}}_{r} = {\overset{\cdot\cdot}{x}}_{d} + c \overset{\cdot}{e} . \end{matrix}

(22)

Derived from (21).

u_{e} = J \overset{\cdot}{r} + Cr - d_{2} .

(23)

According to model-based PI robust control method, control law is designed as:

u_{e} = u_{m} + k_{p} s + k_{i} \int_{0}^{t} s d σ + u_{r}, k_{p} > 0, k_{i} > 0,

(24)

$u_{m}$ refers to the model-based control. $u_{r}$ refers to the robust term. where

{\begin{matrix} u_{m} = J {\overset{\cdot\cdot}{x}}_{r} + C {\overset{\cdot}{x}}_{r} \\ u_{r} = k_{r} sgn (s), k_{r} \geq | d_{\max} |, \end{matrix}

(25)

Through a large number of experiments, we have summed up an effective experience that system control parameters should be proportional to real error, which improves the anti-disturbance. We have established a function in control system for precise control on ROVer. The formula is expressed as

{\begin{matrix} {K'}_{p} = \frac{1}{λ} {(\overset{\cdot}{e} + ce)}^{2} k_{p} = \frac{1}{λ} s^{2} k_{p}, λ > s_{\max}^{2} \\ {K'}_{i} = \frac{1}{λ^{2}} {(\overset{\cdot}{e} + ce)}^{2} k_{i} = \frac{1}{λ^{2}} s^{2} k_{i}, \end{matrix}

(26)

where $λ$ refers to tuning parameter. Error function $s$ is bounded certain in practical control system and $s_{\max}^{2}$ is measurable. Appropriate values can be easily measured during the experiment.

Combining with (24) and (26), EMPRC’s control law can be expressed as:

u_{e} = u_{m} + K'_{p} s + K'_{i} \int_{0}^{t} s d σ + u_{r} .

(27)

Where $u_{e}$ refers to control input of ROVer’s propeller. Specifically, the control amount calculated by EMPRC can be quantized and converted into an analog amount which is recognized by the driving mechanism of ROVs.

Combine (21) with (25):

\begin{matrix} J \overset{\cdot}{r} + Cr & = J {\overset{\cdot\cdot}{x}}_{r} + C {\overset{\cdot}{x}}_{r} + {K'}_{p} s + {K'}_{i} \int_{0}^{t} s d σ \\ + k_{r} sgn (s) + d_{2} . \end{matrix}

(28)

We further reduce (28) to get (29)

\begin{matrix} J \overset{\cdot}{s} + Cs & = - {K'}_{p} s - {K'}_{i} \int_{0}^{t} s d σ \\ - k_{r} sgn (s) - d_{2} . \end{matrix}

(29)

Stability analysis

According to the control law (27), we take the Lyapunov function as

V = \frac{1}{2} J s^{2} + \frac{1}{2} K'_{i} {(\int_{0}^{t} s d σ)}^{2},

(30)

then

\begin{matrix} \overset{\cdot}{V} = s (J \overset{\cdot}{s} + {K'}_{i} \int_{0}^{t} s d σ + \overset{\cdot}{s} \frac{k_{i}}{λ^{2}} {(\int_{0}^{t} s d σ)}^{2}) \\ = (1 + \frac{k_{i}}{J λ^{2}} (\int_{0}^{t} s d σ^{2})) - C s^{2} \\ - \frac{k_{p}}{λ} s^{4} - k_{r} sgn (s) s - d_{2} s) - s^{3} \frac{k_{i}}{J λ^{4}} {(\int_{0}^{t} s d σ)}^{3} \\ = - kQ - W, \end{matrix}

(31)

where

{\begin{matrix} k = (1 + \frac{k_{i}}{J λ^{2}} {(\int_{0}^{t} s d σ)}^{2}) \\ Q = (C s^{2} + \frac{k_{p}}{λ} s^{4} + k_{r} sgn (s) s + d_{2} s) \\ W = s^{3} \frac{k_{i}}{J λ^{4}} {(\int_{0}^{t} s d σ)}^{3} . \end{matrix}

(32)

The analysis by summarizing (31) is as follows:

For $s > 0$ , there exist $k > 0, Q > 0, W > 0$ . Thus $\overset{\cdot}{V} < 0$ .

For $s < 0$ , there exist $k > 0, Q > 0, W < 0$ , where $λ > s_{\max}^{2}$ was mentioned in the previous subsection. Considering the value of $λ$ in practical system is larger. $W$ is ignored due to the numerical weight is small in (31). The equation of $\overset{\cdot}{V}$ is negative.

For $s = 0$ , there exist $k = 1, Q = 0, W = 0$ . When $\overset{\cdot}{V} \equiv 0$ , then $s \equiv 0$ . According to the principle of lasalle invariance, ontrol system utilized in this paper is progressively stable.

Using a strict Lyapunov function to prove stability of a closed-loop control system is critical (Li et al., 2019)). In summary, $\overset{\cdot}{V}$ is negative. Control law of ROVer is designed to meet requirements of stability.

Experiments

In this section, we present experimental comparisons of controllers. All pool experiments in this paper were performed inside Ocean University of China. The proposed EMPRC is compared with integral separation PID and PID to validate its performance and feasibility.

Hardware platform

The self-developed ROV has a portable, expanded modular structure, whose size is $55 \times 45 \times 29 cm$ , and $7.95 kg$ weight. Its shape has a high degree of plasticity due to the structure and materials that can be arbitrarily constructed, with its overall structure a hexagon. An ARM architecture embedded controller is applied in the internal system to provide basic motor control and communication functions. Underwater communication connection between host computer and vehicle is implemented through serial port to network with a mature power-based carrier technology. Software program architecture on this ROV is based on the development of UCOSiii operating system with the advantages of unlimited number of tasks, high CPU utilization and scheduling tasks by priority. The host computer software is programmed in LabVIEW platform, offering the advantages of graphical programming and debugging on line, which includes communication interaction, data storage and algorithm control and provides practical experimental platform. Control algorithms are expressed as programming language embedded in the host computer framework.

As shown in Figure 4, the complete control system of ROV consists of five layers. The display layer refers to human-computer interaction interface, including underwater image and visualization status, etc. The management layer refers to monitoring of various states. The role of the task layer is more prominent, which includes control algorithms, system parameter configuration, data storage and other specific operations. The purpose of the data processing layer is to analyse communication packets and generation protocols between host computer and vehicle. All the data information and control commands obtained by the controller are transmitted at a frequency of 10 Hz between the host computer and the ROV through power carrier communication.

Figure 4.

System block diagram of ROV.

The type of propeller arrangement is divided into vertical and horizontal, which completes six degrees of freedom of motion. The 45-degree tilting placement on horizontal propellers mitigates the effects of unknown forces. The internal driver of the propeller is a brushless motor controlled by pulse width modulation (PWM). The control amount calculated by the controller is converted to a motor identifiable signal and transmitted to the vehicle. The actual control signal of electronic speed control (ESC) of brushless motor is input signal of Figure 7 plus 1500 – the propeller does not work when the PWM is 1500. Figure 5 shows the vehicle that is conducting a yaw tracking experiment in the pool.

Figure 5.

Experiment of yaw tracking in the pool.

System identification

Our purpose of identification experiment is not only to verify model derivation of ROV, but also to obtain a nominal model for control algorithm to further design. To better obtain the reference model which is closer to the actual ROV’s model, we performed a number of pool experiments on it. System identification helps to deal with complex systems that are difficult for us to analyse (Ljung, 2001). In recent years, system identification is applied in many experimental processes and performed by calculating the vehicle’s start-up response process and comparing control signal with actual output. The steps for system identification implementation are as follows:

Providing a variety of sine wave control signals at different frequencies to stimulate the vehicle. No human commands and remote control operations are applied to ensure the system identification accuracy.

Recording the input/output signals of the vehicle and generating a data matrix that includes time stamp, input signals and yaw.

Exporting the data for system identification.

The input and output data obtained in the experiment are shown in Figure 6. The details of state space equation of the reference model are described by the following:

{\begin{matrix} A & = [\begin{matrix} - 1.101 & - 0.2947 & - 2.439 \\ 0.5 & 0 & 0 \\ 0 & 0.5 & 0 \end{matrix}] \\ B & = [\begin{matrix} 0.5 & 0.5 & 0.5 & 0.5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ C & = [\begin{matrix} 0 & 0.4266 & - 0.1646 \end{matrix}] . \end{matrix}

(33)

Figure 6.

Experimental data for system identification.

Experimental comparison

In this section, five experiments were designed, including four different yaw tracking and a disturbance experiment. To comprehensively study the relative performance of control algorithms, the conventional PID, integral separation PID and the proposed EMPRC were evaluated in these tests. As the linear nominal model had been already established, the conventional PID was tuned by the MATLAB PID tuner. The well-known Ziegler–Nichols method was borrowed to provide a fair parameter configuration for the separation integral PI to do the comparison (Mudi and Pal, 1999). The key practical indicators are overshoot and settling time. The pool is inside the building to prevent the possible influence of wind.

As discussed in the previous sections, the nonlinear factors are coupled with travelling angle. To prove the above statement, we set a small range yaw (50º) in the first experiment and increased the desired angle to 200º with a step of 50º gradually. The procedure of experiment is as follows:

Turn on the automatic control mode, which is tracking the yaw through the internal controller, where the set angle of yaw is changed online by the host computer.

Change the setting angle to 0º.

Real-time observation and recording of all states of ROV through the upper computer display interface.

The tracking set time is 20s. After ensuring that there is no external environmental influence, wait for water flow to stabilize, then repeat the steps for the next experiment.

As shown in Figure 7, the EMPRC shows the good performance compared to conventional PID and integral separation PID. The EMPRC has no overshoot and the settling time is only 0.5s slower than integral separation PID. The PID has an overshoot of $10 %$ and the settling time is $2 s$ slower compared to EMPRC.

Figure 7.

Comparison of yaw (50º) tracking experiments.

As shown in Figure 8, EMPRC is beginning to fully show its advantage in having the fastest settling time without obvious overshoot.

Figure 8.

Comparison of yaw (100º) tracking experiments.

The third test has a desired tracked yaw of to 150º. Experimental comparison is shown in Figure 9. The third experiment highlights the advantages of EMPRC. As the travelling angle increases, the nonlinearity of the underwater environment was becoming significant. The response of conventional PID shows both an overshoot and oscillation manner with a very long settling time of more than $14 s$ . The performances of integral separation and EMPRC are still comparable, but the converge time of EMPRC is about 50% faster.

Figure 9.

Comparison of yaw (150º) tracking experiments.

To further investigate the performance of the three controllers, the yaw set point was increased to 200º in the fourth experiment, the results of which are shown in Figure 10. The abnormal trajectory of PID is due to the communication delay and the nonlinear hydrodynamics of the system with a nearly $20 s$ settling time. As the PID was designed based on the linear nominal model, it failed to cooperate with the nonlinear behaviours in the practice. The integral separation one performed better and did not lead to an overshoot, but the settling speed was $10 s$ slower than EMPRC.

Figure 10.

Comparison of yaw (200º) tracking experiments with large deviation.

Finally, a disturbance experiment was performed to verify the anti-disturbance on the performance of controllers. The arbitrary disturbance was generated through an idle propeller in the ROV, which can provide a pulsed disturbance signal of the same intensity to evaluate each algorithm in a fair way. Experimental comparisons are shown in Figure 11. The results show that performance of the three controllers varies greatly under the same intensity disturbance conditions. Both the integral separation PID and the PID have an overshoot of more than $25 %$ and settling time of more than $15 s$ . In contrast, the settling time of EMPRC is shortened by $9 s$ , which is about 2x improvement, and the overshoot is reduced by about 5x better than the other two methods.

Figure 11.

Comparison of yaw (220º) tracking experiments witn strong interference.

Conclusions and future works

The rapid yaw tracking and anti-disturbance capability of the proposed EMPRC algorithm have been both theoretically and experimentally verified against the undesired nonlinear terms and communication delay. The proposed EMPRC method is optimized for hardware configuration and eliminates the residual errors. The experimental evaluation proves the effectiveness of this architecture. In terms of limitations, the authors admit that this method requires more parameter tuning procedures and suggest a system identification to conduct a nominal model for it. This solution can be generalized to varieties of ROVs to impact the practical applications.

The suggested future work would be to expand this method to pitch and roll anti-disturbance control systems by considering the mode coupling between the Euler angles. Control allocation is also worth exploring for this multi-propeller structure to follow complex attitude commands.

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 research was funded by Fundamental Research Funds for the Central Universities under Grant 201962012.

ORCID iDs

Lin Li

Renyu Hou

References

Astrom

(2005) Advanced pid control. ISA-The Instrumentation, Systems, and Automation Society 1–17.

Azis

Aras

Rashid

Othman

Abdullah

(2012) Problem identification for underwater remotely operated vehicle (ROV): A case study. Procedia Engineering 41: 554–560.

Bagheri

Karimi

Amanifard

(2010) Tracking performance control of a cable communicated underwater vehicle using adaptive neural network controllers. Applied Soft Computing 10(3): 908–918.

Bijani

Khosravi

(2018) Robust PID controller design based on

h \infty

theory and a novel constrained artificial bee colony algorithm. Transactions of the Institute of Measurement and Control 40(1): 202–209.

Bijani

Khosravi

Sarhadi

(2016) Optimal acceleration autopilot design for non-minimum phase missiles using evolutionary algorithms. International Journal of Bio-Inspired Computation 8(4): 221–227.

Wen

Wenquan

Yongsheng

( 2018) Finite-time Disturbance Observer Based Attitude Stabilization Control under Actuator Failure[C]. Northeastern University, China Automation Society, Information Physics System Control and Decision Professional Committee. Proceedings of the 30th China Control and Decision Conference (1) 2018 : 287–292.

Campos

Chemori

Creuze

Torres

Lozano

(2017) Saturation based nonlinear depth and yaw control of underwater vehicles with stability analysis and real-time experiments. Mechatronics 45: 49–59.

Chin

Lin

(2018) Experimental validation of open-frame rov model for virtual reality simulation and control. Journal of Marine Science & Technology 23(2): 267–287.

Eidsvik

Schjlberg

Ingrid

(2016) Determination of Hydrodynamic Parameters for Remotely Operated Vehicles. In: Asme 2016 35th International Conference on Ocean, Offshore and Arctic Engineering - Busan, South Korea 7: ocean engineering V007T06A025.

10.

Fan

Woolsey

(2013) Underwater vehicle control and estimation in nonuniform currents : 1400–1405.

11.

García

Patrão

Almeida

Pérez

Menezes

Dias

Sanz

(2015) A natural interface for remote operation of underwater robots. IEEE computer graphics and applications 37(1): 34–43.

12.

García-Valdovinos

Salgado-Jiménez

Bandala-Sánchez

Nava-Balanzar

Hernández-Alvarado

Cruz-Ledesma

(2014) Modelling, design and robust control of a remotely operated underwater vehicle. International Journal of Advanced Robotic Systems 11(1): 1–16.

13.

Hernández-Alvarado

García-Valdovinos

Salgado-Jiménez

Gómez-Espinosa

Fonseca-Navarro

(2016) Neural network-based self-tuning pid control for underwater vehicles. Sensors 16(9): 1–18.

14.

Javadi-Moghaddam

Bagheri

(2010) An adaptive neuro-fuzzy sliding mode based genetic algorithm control system for under water remotely operated vehicle. Expert Systems with Applications 37(1): 647–660.

15.

Yang

(2019) Continuous finite-time extended state observer based fault tolerant control for attitude stabilization. Aerospace Science and Technology 84: 204–213.

16.

(2017) Observer-based fault-tolerant attitude control for rigid spacecraft. IEEE Transactions on Aerospace and Electronic Systems 53(5): 2572–2582.

17.

Yang

Ding

Bogdan

(2012) Robust adaptive motion control for underwater remotely operated vehicles with velocity constraints. International Journal of Control, Automation and Systems 10(2): 421–429.

18.

Ljung

(2007) System identification. Theory for the User 15 (4): 221–246.

19.

Maalouf

Tamanaja

Campos

Chemori

Creuze

Torres

Lozano

(2013) From PD to Nonlinear Adaptive Depth-Control of a Tethered Autonomous Underwater Vehicle[J]. IFAC Proceedings Volumes 46(2): 743–748.

20.

Manzanilla

Castillo

Lozano

(2017) Nonlinear algorithm with adaptive properties to stabilize an underwater vehicle: real-time experiments. IFAC-PapersOnLine 50(1): 6857–6862.

21.

Marzbanrad

Eghtesad

Kamali

(2011) A robust adaptive fuzzy sliding mode controller for trajectory tracking of rovs. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference. IEEE, 2863–2870.

22.

Medina

Tkachova

Santana

Entenza

PJP

(2016) Yaw controller in sliding mode for underwater autonomous vehicle. IEEE Latin America Transactions 14(3): 1213–1220.

23.

Molero

Dunia

Cappelletto

Fernandez

(2011) Model predictive control of remotely operated underwater vehicles. Decision and Control and European Control Conference (CDC-ECC). In: 2011 50th IEEE Conference on IEEE. 2058–2063.

24.

Mudi

Pal

(1999) A robust self-tuning scheme for pi-and pd-type fuzzy controllers. IEEE Transactions on Fuzzy Systems 7(1): 2–16.

25.

Rahimian

Tavazoei

(2012) Application of stability region centroids in robust PI stabilization of a class of second-order systems. Transactions of the Institute of Measurement and Control 34(4): 487–498.

26.

Ramesh

Ramadass

Sathianarayanan

Vedachalam

Ramadass

(2013) Heading control of ROV ROSUB6000 using non-linear model-aided PD approach. International Journal of Emerging Technology and Advanced Engineering 3(4): 382–393.

27.

Rezazadegan

Shojaei

Sheikholeslam

Chatraei

(2015) A novel approach to 6-DOF adaptive trajectory tracking control of an AUV in the presence of parameter uncertainties. Ocean Engineering 107: 246–258.

28.

Sahu

Subudhi

(2017) Potential function-based path-following control of an autonomous underwater vehicle in an obstacle-rich environment. Transactions of the Institute of Measurement and Control 39(8): 1236–1252.

29.

Sarhadi

Khosravi

Bijani

(2015) Identification of nonlinear actuators with time delay and rate saturation using meta-heuristic optimization algorithms. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 229(9): 808–817.

30.

Song

Liu

Zou

Zhu

Yin

(2011) Nonlinear underwater robot controller design with adaptive disturbance prediction and smoother. International Journal of Computational Intelligence Systems 4(4): 634–643.

31.

Tehrani

Heidari

Zakeri

Ghaisari

(2010) Development, depth control and stability analysis of an underwater Remotely Operated Vehicle (ROV). In: Proc. 8th IEEE Int Control and Automation (ICCA) Conf IEEE. 814–819.

32.

Tran

Choi

Bae

Cho

(2015) Design, control, and implementation of a new AUV platform with a mass shifter mechanism. International Journal of Precision Engineering & Manufacturing 16(7): 1599–1608.

33.

Vlahakis

Halikias

(2019) Temperature and concentration control of exothermic chemical processes in continuous stirred tank reactors. Transactions of the Institute of Measurement and Control 41(15): 4274–4284.

34.

Wang

Yang

(2019) Experiments on high-performance maneuvers control for a work-class 3000-m remote operated vehicle. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 233(5): 558–569.

35.

Zhu

(2011) A MIMO nonlinear robust controller for work-class ROVs positioning and trajectory tracking control. In: 2011 Chinese Control and Decision Conference (CCDC) IEEE. 2565–2570.