Hydrodynamic response on trajectory tracking of a tethered underwater robot system under hybrid control algorithms of umbilical cable and propellers

Abstract

A hydrodynamic control model is established in attention to coupling relationships among cable, propellers and robot body of a tethered underwater robot system. In this model the governing equations of umbilical cable and the robot are firstly introduced, then supplementary conditions are coupled into the equations to forming the dynamic mathematical model; finally a hybrid control strategy based on feed-forward negative feedback method for the cable and PID rule for the propellers are integrated in the mathematical model for composing the whole hydrodynamic control model. Both the mathematical model and the control algorithms are proved to be effective and reliable through comparing simulation with the experimental data in existed references. Based on the numerical model constructed in this paper, trajectory tracking of a tethered underwater robot system in different motion combinations are numerically simulated through computational fluid dynamics method. In the numerical simulations, finite difference method is used to solving the kinematic parameters of the mathematical model, while finite volume method is applied on calculating the hydrodynamic forces under a hybrid control manipulations. The robot motion in vertical direction is determined primary by feed-forward negative feedback strategy of adjusting the cable length, while the horizontal movement of the robot is controlled mainly through PID algorithm; The hydrodynamic loading on the robot body are influenced by the flow fields around the robot.

Article Highlights

Established the coupling equations with considering dynamic behaviors among propellers, umbilical cable and robot body.

Discovered the conversion relation which unrelated to objective factors between the rotating speeds of propellers and the corresponding thrusts.

Analyzed the changing rules of hydrodynamic loading on the underwater robot under a hybrid control strategy.

Introduction

During the underwater manipulations of an underwater robot, control strategies and the corresponding hydrodynamic performance are key items. While the strong coupling characteristics among the robot main body, cable and the control thruster making the mathematical model of the robot system very complicated.

To accurately comprehend and predict the hydrodynamic characteristics of a tethered underwater robot under a series of control actions, the description of mathematical model and the applications of calculation methods are crucial. Vu [1] proposed a new mathematical model on the hydrodynamics of an underwater vehicle (UV) with combining the movements of an UV rigid body and the flexible motion of umbilical cable (UC). Singh [2] studied the underwater glider dynamics by fabricating the glider and its variable buoyancy engine, then tested its trajectory in water and compared it with numerical obtained trajectory in the vertical plane through a Computational Fluid Dynamics (CFD) approach. Ferreira [3] inspected underwater structures by using the combination of back stepping and the Control Lyapunov Function (CLF) strategies for trajectory tracking problem of a Hybrid Remotely Operated Vehicle (HROV). Hajivand [4] obtained hydrodynamic damping coefficients of a full-scale autonomous underwater vehicle (AUV) through oblique towing test (OTT) and developed a mathematical model to calculate hydrodynamic force in order to predict autonomous underwater vehicle maneuverability. Gwangsoo [5] presented a new methodology to determine hydrodynamic derivatives of a two-fish underwater vehicle using CFD approach. Yue [6] proposed a dynamic equation to describe the relationship between force and velocity for the second-generation Spherical Underwater Robot (SUR-II) and verified the theoretical calculations and simulation results through experiments. Bevilacqua [7] studied the dynamic behaviors of the umbilical cable of remotely operated underwater vehicles (ROV) through numerical simulations with considering the hydrodynamic effects. It can be seen that there is neither research considering the whole coupling relations among robot body, umbilical cable and propellers of an underwater robot system, nor control method is introduced into the mathematical model of a tethered underwater robot system to form an integrated hydrodynamic control model.

Currently new types of control algorithms for applying on trajectory tracking of underwater robot and investigating the hydrodynamics are becoming research focuses. Jordan [8] presented an algorithm for on-line estimation of the nonlinear hydrodynamics on a spherical vehicle with numerical simulations. Guerrero [9] designed an adaptive high order sliding mode controller for trajectory tracking of autonomous underwater vehicle (AUV) and the controller preserves the advantages of robust control. Park [10] applied Time Delay Control for trajectory tracking experiments of underwater robot; the proposed control law had not requiring nonlinear plant dynamics was highly robust against nonlinearities, uncertainties and disturbances. From above many researchers have studied single control method for analyzing hydrodynamic characteristics which is effective and reasonable, hybrid control method of two control instructions is also valuable.

Although there are so many studies on trajectory control and hydrodynamic features of underwater robots, the mathematical models are simplified more or less. Based on this, a composite dynamic control model of tethered underwater robot system is put forward in this research. The trajectory tracking control movements of the robot are manipulated through a hybrid control actions of adjusting the cable length and modulating the propellers revolution speeds; meanwhile the coupling relations among robot body, propellers and the cable are taken into consideration. Firstly both the cable governing equations and the robot motion equations are given, then the velocity relationships at two ends of the cable which described at the towing ship and the underwater robot are added into above forming a whole hydrodynamic model of an underwater robot system. Finally a controller of adjusting the cable length through feed-forward negative feedback control method and the other one of regulating the propellers revolution speeds by PID algorithm are merged into the hydrodynamic model combining an integral hydrodynamic control model. In numerical simulation, the control laws of modifying the cable length and the propellers revolution speeds are executed to guide the robot moving in a stable posture along the specified routes. Multi slip grid technology is applied on solving the turbulence equations numerically, the thrust from propellers and the hydrodynamic loading on the robot body are calculated with CFD code Fluent. The hydrodynamic responses of the robot system under the hybrid control strategy are observed and analyzed.

Hydrodynamic mathematical model

The foundation of comprehending hydrodynamic properties is to establish an accurate and reasonable hydrodynamic mathematical model. In this paper the tethered underwater robot system is consist of three parts: umbilical cable, the robot main body in the shape of a torpedo and three propellers among which two fixed symmetrically at both sides of the robot body and one erected inside center of the body cavity. There are three different coordinate systems used for establishing the mathematical model, that is: the towing ship fixed coordinate system, the local coordinates of the cable and the underwater robot as shown in Fig. 1. In this research italic represents scalar, bold italic means vector and uppercase with bold is matrix; variables and matrices with dots on the top means the differentiation of time.

Fig. 1 [Images not available. See PDF.]

Tethered underwater robot system

Dynamic equations for the cable and underwater robot

The governing equations for the cable are written at each node of the cable local coordinate (t, n, b)

1

$${\mathbf{MY}}^{\prime }=\mathbf{N}\stackrel{·}{\mathbf{Y}}+\mathbf{Q}$$

2

$$\mathbf{Y}={\left(T,v_{\text{t}},v_{\text{n}},v_{\text{b}},\alpha ,\beta \right)}^{\text{T}},{\mathbf{Y}}^{\prime }=\frac{\partial \mathbf{Y}}{\partial s},\stackrel{·}{\mathbf{Y}}=\frac{\partial \mathbf{Y}}{\partial t}$$

In Eqs. (1, 2), M, N and Q are coefficient matrices related to the initial state of umbilical cable. T expresses the cable tension and v_t, v_n, v_b are translational velocity components of the cable in the cable local coordinate; α, β are the directional angles between the cable local frame and the towing ship frame. s is the cable length at the initial moment. More details can be found in reference [11].

Motion equations of the underwater robot about surge, sway, heave, roll, pitch and yaw are written as [12]

3

$$m\left[\dot{u}-vr+wq-x_G\left(q^2+r^2\right)+y_G\left(pq-\dot{r}\right)+z_G\left(pr+\dot{q}\right)\right]=X$$

4

$$m\left[\dot{v}+ur-wp+x_G\left(pq+\dot{r}\right)-y_G\left(p^2+r^2\right)+z_G\left(qr-\dot{p}\right)\right]=Y$$

5

$$m\left[\dot{w}-uq+vp+x_G\left(pr-\dot{q}\right)-y_G\left(qr+\dot{p}\right)-z_G\left(p^2+q^2\right)\right]=Z$$

6

$$\begin{array}{c}{I}_x\dot{p}+\left(I_z-I_y\right)qr+I_{\mathit{xy}}\left(pr-\dot{q}\right)-I_{\mathit{yz}}\left(q^2-r^2\right)-I_{\mathit{xz}}\left(pq+\dot{r}\right)+\\ m\left[y_G\left(\dot{w}-uq+vp\right)-z_G\left(\dot{v}+ur-wp\right)\right]=H\\ \end{array}$$

7

$$\begin{array}{c}{I}_y\dot{q}+\left(I_x-I_z\right)pr-I_{\mathit{xy}}\left(qr-\dot{p}\right)+I_{\mathit{yz}}\left(pq-\dot{r}\right)+I_{\mathit{xz}}\left(p^2-r^2\right)-\\ m\left[x_G\left(\dot{w}-uq+vp\right)-z_G\left(\dot{u}-vr=wq\right)\right]=J\\ \end{array}$$

8

$$\begin{array}{c}{I}_z\dot{r}+\left(I_y-I_x\right)pr-I_{\mathit{xy}}\left(p^2-q^2\right)-I_{\mathit{yz}}\left(pr+\dot{q}\right)+I_{\mathit{xz}}\left(qr-\dot{p}\right)+\\ m\left[x_G\left(\dot{v}+ur-wp\right)-y_G\left(\dot{u}-vr+wp\right)\right]=K\\ \end{array}$$

In Eqs. (3–8), (x_G, y_G, z_G) denote the coordinates of the robot gravity center in the robot-fixed coordinate system; I_x, I_y, I_z, I_xy, I_yz and I_xz are the mass moment of inertia and the polar moments of inertia of the underwater robot seperately. u,v,w and p,q,r are the three-dimensional linear velocity and angular velocity components of the robot respectively. The left side of Eqs. (3–8) describe the inertial forces and inertial moments, while the right side (X, Y, Z) and (H, J, K) present the combined external forces containing hydrostatic recovery force F_R, rope tension T, hydrodynamic force F_H, control force from propellers F_C and the related torques acting on the robot, i.e.

9

$${\mathbf{F}}_0={\mathbf{F}}_{\text{R}}+\mathbf{T}+{\mathbf{F}}_{\text{H}}+{\mathbf{F}}_{\text{C}}$$

10

$${\mathbf{M}}_0={\mathbf{M}}_{\text{R}}+{\mathbf{M}}_{\text{T}}+{\mathbf{M}}_{\text{H}}+{\mathbf{M}}_{\text{C}}$$wherein F₀ represent (X, Y, Z)^T in Eqs. (3–8) and M₀ means (H, J, K)^T.

Supplementary equations

In order to couple the cable governing equations with the underwater robot motion equations to form a whole system, some parameters relationship of specific positions are taken into consideration, i.e. the three-dimensional linear speeds of umbilical cable at the upper end should be coincident with the towing ship after certain transformation, and the same patterns could be inferred at the towed point between the underwater robot and umbilical cable. Thus, the linear speeds relations between towing ship and umbilical cable, and between underwater robot and the cable can be described as

11

$${\mathbf{V}}_{\text{C}}={\mathbf{DV}}_{\text{S}}$$

12

$$\left[{\mathbf{V}}_0+\mathbf{\omega }\times {\mathbf{r}}_T\right]={\mathbf{EDV}}_a$$

V_C = (v_t, v_n, v_b)^T and V_S = (v_X, v_Y, v_Z)^T in Eq. (11) are the translational velocity components of umbilical cable and the tow ship separately, D is the converted matrix between cable local coordinate system and the tow ship fixed coordinate frame.

In Eq. (12), V₀ = [u, v, w]^T and ω = [p, q, r]^T are the linear and angular speeds of the underwater robot respectively in the robot local coordinates; r_T = (x_T, y_T, z_T)^T is the coordinates of conjunction point in the robot local frame. V_a is the connection point speed between underwater robot and umbilical cable in the cable local coordinates; E is the transform matrix between robot local coordinate system and the towing ship fixed coordinates [11].

In addition, the relationship between pitch, roll, yaw and their corresponding angular velocities ω = [p, q, r].^T can be derived through the transformation of the cable directional angles α and β as follow [13]

13

$$\left(\begin{array}{c}\stackrel{·}{\theta }\\ \stackrel{·}{\varphi }\\ \stackrel{·}{\psi }\end{array}\right)=\left[\begin{array}{ccc}1& sin\beta tan\alpha & cos\beta tan\alpha \\ 0& cos\beta & -sin\beta \\ 0& sin\beta /cos\alpha & cos\beta /cos\alpha \end{array}\right]\left(\begin{array}{c}p\\ q\\ r\end{array}\right)$$θ, φ and ψ are the pitch, roll, and yaw of the underwater robot in towing ship coordinates. $\stackrel{·}{\theta },\stackrel{·}{\varphi },\stackrel{·}{\psi }$ are the rate of change over time of pitch, roll and yaw angles.

Solving strategies

Finite difference method for solving equations

The cable Eqs. (1) and (2) are approximated through finite differences central in space and time. In order to solve the dynamic equations of the underwater robot system numerically, the umbilical cable is assumed cutting into a number of continuous length units Δs_j and simulation time is divided into a range of time sequence. The cable segment unit number is named 0 ~ N_P and there are six dependent variables ${\mathbf{Y}}_j={\left(T,v_{\text{t}},v_{\text{n}},v_{\text{b}},\alpha ,\beta \right)}_j^{\text{T}}$ (j = 0 ~ N_P) in each segment unit of the cable, so the governing equations of umbilical cable at the midpoint of a segment unit Δs_j and at the midway of a time interval Δtⁱ between old and new time steps can be described as [12]

14

$$\begin{array}{c}\left[{\mathbf{M}}_{j+1}^{i+1}+{\mathbf{M}}_j^{i+1}\right]\frac{{\mathbf{Y}}_{j+1}^{i+1}-{\mathbf{Y}}_j^{i+1}}{\mathrm{\Delta }{s}_j}+\left[{\mathbf{M}}_{j+1}^i+{\mathbf{M}}_j^i\right]\frac{{\mathbf{Y}}_{j+1}^i-{\mathbf{Y}}_j^i}{\mathrm{\Delta }{s}_j}\\ =\left[{\mathbf{N}}_{j+1}^{i+1}+{\mathbf{N}}_j^{i+1}\right]\frac{{\mathbf{Y}}_{j+1}^{i+1}-{\mathbf{Y}}_{j+1}^i}{\mathrm{\Delta }{t}^i}+\left[{\mathbf{N}}_j^{i+1}+{\mathbf{N}}_j^i\right]\frac{{\mathbf{Y}}_j^{i+1}-{\mathbf{Y}}_j^i}{\mathrm{\Delta }{t}^i}+{\mathbf{Q}}_{j+1}^{i+1}+{\mathbf{Q}}_j^{i+1}+{\mathbf{Q}}_{j+1}^i+{\mathbf{Q}}_j^i\\ \end{array}$$

There are 6(N_P + 1) unknown variables Y_j at all nodes of the cable but only 6N_P finite difference Eq. (14). For the complete umbilical cable system, there exists velocity coupling relations with the towing ship and the underwater robot because the upper end and the lower end of the cable are connected to the towing ship and the underwater robot respectively, so Eqs. (11) and (12) can be incorporated into Eq. (14); while additional 9 unknown variables V₀ = [u, v, w]^T, ω = [p, q, r]^T and pitch angel θ, roll angel φ and yaw angel ψ of the underwater robot are introduced into Eq. (14). For the sake of making the dynamic equations of umbilical cable and the underwater robot closed, Eq. (13) is supplemented. Through this process the cable governing equations and the robot motion equations are coupling to become a whole system and the simultaneous dynamic equations of this system are solvable.

Finite volume method for calculating hydrodynamics

In Eqs. (9, 10), the resilience in still water F_R is arising when the centers of robot gravity and buoyancy are not on the same vertical line; Tension of the cable T is calculated through solving the hydrodynamic mathematical model at every time step. The hydrodynamic loadings F_H on the robot main body and the control force from propellers F_C in Eq. (9) are calculated with CFD software Fluent through discretizing governing equations and turbulence model by using finite volume method (FVM). The fluid in the working domain of the propellers is supposed as incompressible, the continuity equations and momentum equations of flow field around are [14]

15

$$\frac{\partial u_x^{}}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}=0$$

16

$$\left\{\begin{array}{c}\rho \left(\frac{\partial u_x}{\partial t}+u_x\frac{\partial u_x}{\partial x}+u_y\frac{\partial u_x}{\partial y}+u_z\frac{\partial u_x}{\partial z}\right)=-\frac{\partial p_a}{\partial x}+\upsilon \rho \left(\frac{{\partial }^2{u}_x}{\partial x^2}+\frac{{\partial }^2{u}_x}{\partial y^2}+\frac{{\partial }^2{u}_x}{\partial z^2}\right)+\rho g\\ \rho \left(\frac{\partial u_y}{\partial t}+u_x\frac{\partial u_y}{\partial x}+u_y\frac{\partial u_y}{\partial y}+u_z\frac{\partial u_z}{\partial z}\right)=-\frac{\partial p_a}{\partial y}+\upsilon \rho \left(\frac{{\partial }^2{u}_y}{\partial x^2}+\frac{{\partial }^2{u}_y}{\partial y^2}+\frac{{\partial }^2{u}_y}{\partial z^2}\right)+\rho g\\ \rho \left(\frac{\partial u_z}{\partial t}+u_x\frac{\partial u_z}{\partial x}+u_y\frac{\partial u_z}{\partial y}+u_z\frac{\partial u_z}{\partial z}\right)=-\frac{\partial p_a}{\partial z}+\upsilon \rho \left(\frac{{\partial }^2{u}_z}{\partial x^2}+\frac{{\partial }^2{u}_z}{\partial y^2}+\frac{{\partial }^2{u}_z}{\partial z^2}\right)+\rho g\\ \end{array}\right)$$

In Eqs. (15, 16): x, y, z are the spatial coordinates of flow field; u_x, u_y, u_z are velocity components of fluid; p_a, ρ, υ are the pressure, density and motion viscosity of flow field respectively.

The calculation in this paper is moderate and the geometric model is relative easy, while accuracy of calculation resultant is high, so the typical k-ε turbulence model is adopted which assumed turbulent viscosity is related to turbulent kinetic energy and dissipation rate. The turbulence pulsation kinetic energy k equation and turbulence dissipation rate ε equation are

17

$$\frac{\partial }{\partial t}\left(\rho k\right)+\frac{\partial }{\partial x_i}\left(\rho k{u}_i\right)=\frac{\partial }{\partial x_j}\left[\left({\mu }_1+\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+{\mu }_t\frac{\partial u_j}{\partial x_i}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)-\rho \epsilon$$

18

$$\frac{\partial }{\partial t}\left(\rho \epsilon \right)+\frac{\partial }{\partial x_i}\left(\rho \epsilon u_i\right)=\frac{\partial }{\partial x_j}\left[\left({\mu }_1+\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_1{\mu }_t\frac{\epsilon }{k}\frac{\partial u_j}{\partial x_i}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)-C_2\rho \frac{{\epsilon }^2}{k}$$in which μ₁ is the viscosity coefficient of turbulence vortex;$C_1,C_2,{\sigma }_k,{\sigma }_{\epsilon }$ are empirical constants.

Multi slip grid technology for the geometric model

To achieve the goal of the research in this paper, divisions for geometrical model and computational domains are first required, then boundary conditions for these computational domains are determined; finally multi slip grid technology is applied to describe coupled rotational motions of propellers with the combinational motions of the robot through FVM. For numerical computations on the hydrodynamic behaviors of propellers and the robot under different control operations, four types of computational domains are constructed which have already done by previous work of the author [15].

Control strategy and simulation process

Due to the hydrodynamic response of tethered underwater robot system undergoes complex changes with the movement of the robot, rotation of propellers and the action of retrieving and releasing umbilical cable, it is difficult to monitor the hydrodynamic load acting on the underwater robot main body and the thrust of propellers in actual experiments. Meanwhile any changing in hydrodynamic load can also causing changes of the external forces of the robot which lead to producing deviations in the control process. Therefore, most researchers designed controllers for underwater robot system focusing on control parameters and the accuracy, without conducting systematic analysis of hydrodynamic response.

There existing closed relationships between motion control and hydrodynamic behaviors of tethered underwater robot system. Predicting the hydrodynamic behaviors of tethered underwater robot system accurately is significant for the control performance. Until now the effectiveness means of investigating hydrodynamic characteristics are numerical simulations.

For analyzing the hydrodynamic characteristics of the underwater robot about trajectory tracking under special control operations, a hybrid control strategy containing feed-forward negative feedback method to adjust cable length and the PID algorithm for modulating the propellers revolution speeds is adopted in this paper.

Feed-forward negative feedback manoeuvre for the cable

Feed-forward negative feedback control can compensate for interference to a certain extent with reducing the accuracy requirements of feed-forward control and ensuring the stable operation of the system. There are significant delay in the deployment and retraction of umbilical cable and the disturbance of cable tension are measurable and uncontrollable. Therefore it is appropriate to adopt a feed-forward negative feedback method for adjusting the umbilical cable length.

The control operation of modifying the umbilical cable length based on feed-forward negative feedback control is conducted according to the coordinates deviation of the underwater robot between instantaneous coordinate and the predesignated one in every time step. First a feed-forward program is executed at the beginning of every time step according to the prescribed trajectory deviation of two continuous time steps; then the solving program of motion equations for the robot system is calculated, finally negative feedback control module is operated based on the trajectory displacement variation between instantaneous coordinates and the predesignated one updated in this time step. The mathematical formulation about feed-forward negative feedback algorithm for adjusting the umbilical cable length is give as

19

$$dl_{\text{f}}^{}(nt)=d{l}_{\text{f}}(nt-1)+\lambda [C_{\text{p}}(nt)-C_{\text{p}}(nt-1)]$$

20

$$dl_{\text{b}}^{}(nt)=d{l}_{\text{f}}(nt)+(-1)\times \xi [e(nt-1)-e(nt-2)]$$where dl_f (nt) is the cable length unit at time step nt obtained by feed-forward approach and the subscript b in dl_b (nt) means negative feedback method. C_p (nt) is the predesignated trajectory coordinates, e is the displacement variation through the comparison of instantaneous coordinates with the predesignated; λ and ξ are the correction coefficients.

PID mechanism for the control propellers

In addition to modifying the umbilical cable length, the incremental PID technology is applied on regulating the propellers revolution speeds as well.The PID algorithm is suitable for real-time control scenarios because it can quickly response to changes in control process and make sure the system to reach a preset target in a short period of time. Meanwhile PID controller can accurately track error signals with good robustness and easy to implement, it has been widely used in propeller controllers for the underwater robot system with high flexibility.

In this paper the tethered underwater robot system is mainly consist of three parts: umbilical cable, the robot main body in a shape of a torpedo and three propellers among which two fixed at both sides of the robot body symmetrically and one erected inside center of the body cavity as shown in Fig. 2.

Fig. 2 [Images not available. See PDF.]

Underwater robot (1, 2-Duct Propeller1 and 2; 3-Propeller 3)

This controller is conducted after adjusting the umbilical cable length for making the entirety control performance more precisely. The formulation for modulating the propellers revolution speeds is given as

21

$$R(nt)=K_{\text{P}}\left[e(nt)+\frac{T_0}{T_{\text{I}}}\sum _{i=0}^{\mathit{nt}}e(i)+T_{\text{D}}\frac{e(nt)-e(nt-1)}{T_0}\right]$$

Based on iterative format

22

$$R(nt-1)=K_{\text{P}}e(nt-1)+K_{\text{I}}\sum _{i=0}^{nt-1}e(i)+K_{\text{D}}\left[e(nt-1)-e(nt-2)\right]$$where R is the propellers revolution speeds; nt is the number of time steps; e is the instantaneous displacement variation of the underwater robot, T_I and T_D are the integral time and differential time separately, T₀ is the time step size; K_P is the proportional coefficient, K_I and K_D are the integral time constant and differential time constant of PID algorithm

23

$$K_{\text{I}}=\frac{K_{\text{P}}{T}_0}{T_{\text{I}}},K_{\text{D}}=\frac{K_{\text{P}}{T}_{\text{D}}}{T_0}$$

Equation (21) minus Eq. (22) gives

24

$$\mathrm{\Delta }R(nt)=K_{\text{p}}\mathrm{\Delta }e(nt)+K_{\text{I}}\mathrm{\Delta }e(nt)+K_{\text{D}}\left[\mathrm{\Delta }e(nt)-\mathrm{\Delta }e(nt-1)\right]$$

The displacement variation of the underwater robot comparing instantaneous coordinates with the predesignated one at every time step is defined as e(nt); the changing of propellers revolution speeds is described as ΔR. The value of parameters K_P, K_I and K_D are taking different values at different control operations.

The foregoing feed-forward negative feedback method and PID algorithm forming the control module for the tethered underwater robot system in horizontal and vertical motion.

The underwater robot moved under the action of external forces which contain hydrostatic recovery force F_R, rope tension T, hydrodynamics F_H, control force from propellers F_C and related torques as described in Eqs. (9, 10). The cable tension T and the propellers thrust F_C were arising when adjusting the umbilical cable length in Eqs. (19, 20) and regulating the revolution speeds of propellers in Eq. (24). Through this the external force on the underwater robot could be changed and the robot was moving along with the desired trajectory in a stable attitude.

Simulation effectiveness validation of the propellers

In order to validating the effectiveness of the CFD simulation method used for the propellers, grid independence and time step size independence are significant. The geometric model of Propellers 1 and 2 shown in Fig. 3 were divided into different grid numbers as shown in Table 1. The hydrodynamic force of blade and duct were observed in Fig. 4a. Besides, time step size was set 0.01 s, 0.02 s, 0.04 s, 0.1 s and 0.2 s respectively. The hydrodynamic force investigations about blade and duct are shown in Fig. 4b.

Fig. 3 [Images not available. See PDF.]

Geometric model of duct propeller

Table 1. Numbers of grid

Category	Case
	1	2	3	4
Duct	22,397	67,532	173,294	509,672
Blade	27,764	54,059	161,239	485,693
Total	50,161	121,591	334,533	995,365

Fig. 4 [Images not available. See PDF.]

Effectiveness validation

The calculation results of hydrodynamic force tended to be stable when the total numbers of grid exceeded 80 thousands in Fig. 4a; and the hydrodynamic results of blade and duct are both very stable under different time step sizes.

Numerical solution procedure

For observing the control performance and hydrodynamic behavior of the robot system under control operations aforementioned, the control algorithms and the CFD technique are incorporated into the mathematical control model of the robot system forming three solving sections, that is.

Section (a): The solving program of motion equations about the tethered underwater robot system proposed in this research. The core point of this program is solving dynamic parameters of umbilical cable in Eq. (1) through finite difference approximation [16] with Eq. (13). More details are described in reference [17].

Section (b): A hybrid control strategy composed of feed-forward negative feedback method for adjusting the umbilical cable length as described in Eqs. (19, 20) and the incremental PID mechanism for modulating the revolution speeds of propellers as described in Eq. (24). The calculation process of this section is shown in Fig. 5.

Fig. 5 [Images not available. See PDF.]

Calculation process of Section (b)

In which Section (c) is.

Section (c): The algorithm to solve numerically the hydrodynamic loads F_H and M_H on the underwater robot body, the thrusts F_C and their corresponding moments M_C issued from the control propellers as described in Eqs. (9, 10) with multi slip grid technology at every time step [18].

The implement of data transferring and sharing among Sections (a), (b) and (c) through software interface files, the entire numerical simulation steps of the three sections are outlined below:

1. Enter basic physical and geometric parameters of the robot system as well as the initial conditions which containing the umbilical cable, the robot main body and the control propellers; set the incipient time step number of numerical simulation nt = 0.

2. Iterative calculation under a stable flow field condition for obtaining the original hydrodynamic parameters of the robot and the cable; bring in initial motion parameter values of the cable and the robot body acquired from steady solution. At this moment the time step number nt = n.

3. Execute the control actuation of adjusting the cable length with feed-forward strategy in Section (b), then comparing the cable length changing with the predesignated one.

4. Resetting the correction factor λ in Eq. (19) and go back to step (3) if the error between the cable length and the input disturbance bigger than a given threshold value ε₀, otherwise go to step 5.

5. Instantaneous coordinates and the motion parameters of the robot are calculated in Section (a); perform the negative feedback control to adjust the cable length of Section (b) according to the instantaneous displacement variation in step 3 for reducing displacement deviation further.

6. Solving the hydrodynamic loads F_H and M_H on the robot main body in Section (c), then the instantaneous position coordinate and dynamic motion parameters are updated in Section (a) with considering F_H and M_H.

7. Produce the displacement deviation at the time step number nt = n by comparing instantaneous position coordinate from step 6 with the predesignated one.

8. Go back to step 5 if the coordinate deviation in step 7 is bigger than the assigned relative error ε₁; otherwise go to step 9.

9. Execute the PID algorithm of modulating the propellers revolution speeds in Section (b) according to the displacement variation from step 8.

10. Calculate the propellers thrusts F_C and M_C in Eqs.(9) ~ (10) through CFD code Fluent in Section (c) based on the control manipulation of propellers in step 9.

11. The instantaneous position coordinates are closer to the predesignated one from the thrusts of control propellers about horizontal and vertical directions through PID algorithm and the dynamic motion parameters of the robot system are updated in Section (a).

12. Go back to step 5 if the displacement variation updated from step 11 exceeding a given error ε₂; or else continue the program.

13. Iteration calculation increasing one step: n = n + 1,$t_{n+1}^s=t_n^s+\mathrm{\Delta }t$, if n < N go back to step 3; or output results and end the process; where N is the total number of time steps, Δt is time step size.

The comprehensive procedure of numerical simulation is shown in Fig. 6.

Fig. 6 [Images not available. See PDF.]

Flowchart of numerical simulation

Feasibility validations of mathematical model and control algorithms

In order to verify the availability of the mathematical model and the PID algorithms for the simulation of trajectory tracking as well as hydrodynamic response, the existing experimental data about floating up and diving down control manipulation of a tethered underwater towed vehicle [19] and another trajectory tracking test of a spherical underwater robot with PID algorithm [20] are selected as references. Firstly the geometric models were established with the same sizes and shapes according to references [19, 20] separately. Secondly associating above geometric models with the hydrodynamic mathematical model and PID strategy proposed in this research respectively. Finally simulation experiments were operated under the same lab experimental conditions in references [19, 20] successively. Comparing the numerical results with the laboratory experiments data under the same conditions are presented in this part. It has been proved that the proposed mathematical model and control strategy using for hydrodynamics and control performance simulation analysis of a tethered underwater robot system are reliable and effective from the results below.

Verification of the hydrodynamic control mathematical model

The underwater towed vehicle used in the laboratory for depth control test is a wing-shaped body with an adjustable depressing wing which driven by a servomechanism and serves an active force to drive the vehicle floating up and diving down in the vertical direction. At the ends of the wing there are end plates equipped for keeping the span being relative short and increasing its effective aspect ratio. The overall geometry size is 920mm in length, 944mm in width and 700mm in height. The water pool dimension for floating up and diving down control testing of the towed vehicle is 120m (length) × 8m (width) × 4.4m (height) and the rest part of this section had been accomplished before with more details could be read in references [21].

Validation of PID algorithm for control and hydrodynamics simulation

The relative geometric model is established and simulated numerically under the same condition through position control experiment of a spherical underwater robot [20]. In this experiment the PID controller and the State Feedback controller are applied on following the designated trajectory coordinates in horizontal and vertical directions as well as evaluating the control performance, the actuators of both controllers are propellers.

The geometric model was established according to the reference [20] and associated with the dynamic control mathematical model proposed in this paper. Simulation experiment under the same condition of Suarez [20] is operated and the results comparing between simulation and experiment in horizontal direction (X axis) are shown as Figs. 7, 8 and Table 2.

Fig. 7 [Images not available. See PDF.]

Simulation result in horizontal direction (X axis)

Fig. 8 [Images not available. See PDF.]

Experiment in horizontal direction (X axis) [20]

Table 2. System response in horizontal direction (X axis) [20]

Parameters	PID (simulation)	PID (experiment)	FL (experiment)	Unit
OS%	19.36	13.16	11.98	[%]
Tr	9.95	6.83	3.77	[s]
Tf	5.86	3.17	2.18	[s]
TS	32.18	24.94	7.02	[s]
eSS	0.0069	0.0083	0.0056	[m]

The simulation result of PID algorithm for adjusting rotational speeds of propellers to follow the designated trajectory is similar to the experimental one. The parameter values of system overshoot OS%, rising time T_r, falling time T_f and the regulating time T_s in simulation are all bigger than those in experiment because of the delay about the response of control system causing by the flow field changing in simulation, while in experiment no changing of flow field is taken into account. The error of steady state (e_SS) in simulation is smaller than experiment because there is no manual interference and instrument error in simulation.

Result comparing between simulation and experiment in vertical direction (Z axis) are shown as Figs. 9, 10 and Table 3.

Fig. 9 [Images not available. See PDF.]

Simulation in vertical direction (Z axis)

Fig. 10 [Images not available. See PDF.]

Experiment in vertical direction (Z axis) [20]

Table 3. System response in vertical direction (Z axis) [20]

Parameters	PID (Simulation)	PID (Experiment)	FL (Experiment)	Unit
OS%	2.62	2.96	5.92	[%]
Tr	2.55	2.74	2.27	[s]
Tf	2.78	1.70	2.52	[s]
TS	13.80	17.80	7.93	[s]
eSS	0.0049	0.0063	0.0020	[m]

The changing regular (Figs. 9, 10) for vertical direction (Z axis) of control performance in simulation is close to the experimental results and there is little difference for steady state error (e_SS) between simulation and experiment which indicates the effectiveness and reliability of numerical simulation.

The results comparison between simulation and experiments in this part shows that the control process and the system response to the designated trajectory are consistent, which validating the applicable of the hydrodynamic mathematical model and the suitable of PID algorithm proposed in this paper for hydrodynamic and control simulation of a tethered underwater robot system.

Numerical simulations about trajectory tracking of the robot system

From the results comparison above, it can be indicated that the hydrodynamic control model integrated in this research could be used for hydrodynamic behaviors analysis and reasonable results can be obtained in numerical simulation of trajectory tracking under given control manipulations.

In this part, a hybrid control strategy of feed-forward negative feedback method and PID algorithm is used in trajectory tracking simulation with the proposed mathematical model for investigating the hydrodynamic features and control performance of a tethered underwater robot system. In simulating calculation adjusting the umbilical cable length and modulating the propellers revolution speeds to generate appropriate thrusts as the hybrid control strategy for driving the underwater robot to move along a designated trajectory forming the integrated control objective task, the control performance and hydrodynamic characteristics of the robot under these manipulations are acquired and analyzed. The basic composition and appearance outline of the underwater robot are shown in Fig. 2, and the essential parameters of the underwater robot geometric model are shown in Table 4. In every time step of numerical calculation the dynamic parameters of the robot system such as the robot position coordinate, the cable tension and thrusts of control propellers are solved through the processes from Sections (a) to (c).

Table 4. Basic parameters of the underwater robot

Portion	Parameter	Value
Umbilical able	Diameter (mm)	25
	Length (mm)	8000
	Mass distribution (kg/m)	0.3
Robot body	Main body size (mm)	800 in Length × 300 in Diameter
	Mass (kg)	11
	Gravity center coordinates (m)	xG = 0 yG = 0 zG = 0.06
	Buoyancy center coordinates (m)	xB = 0 yB = 0 zB = 0
	Connected point coordinates between robot and cable (m)	xT = 0 yT = 0 zT = -0.13
Control propellers	Type	Ka 4–70/19A
	Pitch ratio (P/D)	1
	Diameter (mm)	180 for Propellers 1 and 2 150 for Propeller 3
	Dynamic reference point coordinates about thrusts of three propellers (m)	Propeller 1
		x1 = 0 y1 = -0.29 z1 = 0
		Propeller 2
		x2 = 0 y2 = 0.29 z2 = 0
		Propeller 3
		x3 = 0 y3 = 0 z3 = 0
	Thrust directions of propellers in their positive revolution	Minus X axis for Propeller 1 and 2
		Minus Z axis for Propeller 3

The trajectory tracking simulation is executed through CFD software Fluent in an underwater environment of current velocity setting 0.2m/s towards the positive X-axis with applying standard k-ε turbulence model as described in Sect. 2.3.2. Total simulation time is 12 s and the time step size is 0.1s. The physical time of numerical simulation is about 14 h, the simulation hardware environment is Inter(R) Xeon(R) CPU E5-2620 v3 at 2.4 GHz.

Iterative calculation starts conducting when the underwater robot maintains a stable state under given inflow conditions and trajectory tracking control simulations are denoted at the connecting position coordinate between umbilical cable and robot body in the towing ship fixed coordinate frame. Basic on this, two different combination motions superimposing horizontal and vertical directions are used as the predesignated trajectories. Their corresponding mathematical expressions are written as follows.

Motion combination I

25

$$X=\left\{\begin{array}{cc}1.2-\frac{0.4}{3}t\hfill & \left(0<t\le 9\right)\\ 0\hfill & \left(9<t\le 12\right)\end{array}\right),Z=\left\{\begin{array}{cc}7.90\hfill & (0<t\le 3)\\ 7.90-0.3\times \left(t-3\right)\hfill & \left(3<t\le 6\right)\\ 7\text{.00}\hfill & \left(6<t\le 9\right)\\ 7.00+0.3\times \left(t-9\right)\hfill & \left(9<t\le 12\right)\end{array}\right)$$

Motion combination II

26

$$X=\left\{\begin{array}{cc}1.20\hfill & 0\le t<2\\ 1.20-\frac{{\left(\text{t}-2\right)}^2}{2}\times 0.1\hfill & 2\le t<4\\ 1.00-0.2\times \left(t-4\right)\hfill & 4\le t<8\\ 0.20+\frac{{\left(t-8\right)}^2}{2}\times \left(-0.1\right)\hfill & 8\le t<10\\ 0\hfill & 10\le t\le 12\end{array}\right),Z=\left\{\begin{array}{cc}7.90-0.3t\hfill & 0\le t<4\\ 6.70\hfill & 4\le t<8\\ 6.70+0.3\times \left(t-8\right)\hfill & 8\le t\le 12\end{array}\right)$$

The predesignated trajectories of both motion combinations I and II are within the vertical plane composed of X and Z axes in which the Z-axis positive direction is pointing downward. The trajectories tracking about two motion combinations and their corresponding errors are shown in Figs. 11 and 12 separately.

Fig. 11 [Images not available. See PDF.]

Running orbits of prescribed trajectories and simulated ones

Fig. 12 [Images not available. See PDF.]

Absolute errors between prescribed trajectories and simulated ones

The floating up and diving down trajectories of the robot along Z axis presented in Fig. 9 are mainly controlled through adjusting the umbilical cable length together with auxiliary propulsion from Propeller 3, while the moving forward and backward trajectories along X axis are maneuvered primary by the thrusts of Propellers 1 and 2 with adjusting the umbilical cable length as assistant operation.

The trajectory tracking control simulations are under water environment with a coming fluid velocity of 0.2m/s which direction toward negative X axis. The duration of each time step are maintaining the constant 0.1s and the entire time for the two simulations are both 12s with dividing into several control stages as written in Eqs. (25, 26). The physical time of numerical simulations are both nearly 14 h.

Comparing between predesignated trajectories with the simulated ones under control manipulations and the absolute errors are acquired in Fig. 12. It can be seen from Figs. 11, 12 that the simulated trajectories are comparatively matching to the predesignated ones with maximum absolute errors of two control operations are 0.1m and 0.05m respectively. The control performance in horizontal motions are better than that in vertical motions, that reveals the trajectory variations are originated principally from the control manipulation of adjusting the cable length.

Figures 13, 14, 15 illustrate the changing processes about the propellers revolution speeds, the propellers thrusts and the velocity components in X and Z axes directions of the robot over time. The changing in length of the cable is shown in Fig. 16. During the control simulation, very similar hydrodynamic response characteristics are observed in time records histories at every motion stage.

Fig. 13 [Images not available. See PDF.]

Revolution speeds of propellers

Fig. 14 [Images not available. See PDF.]

Thrusts of propellers

Fig. 15 [Images not available. See PDF.]

Velocity components in X and Z axes of the robot

Fig. 16 [Images not available. See PDF.]

Length changing of umbilical cable

It can be inferred from Figs. 13, 14, 15, 16 that the hydrodynamic properties of the underwater robot system among the cable, robot body and control propellers are diverse from each other under the control operation. While there is relative clear relationships about instantaneous kinematic parameters among the cable, robot body and propellers which are analyzed and discussed in the following together with their hydrodynamic characteristics:

1. The similarity of changing rules about dynamic parameters over time in Figs. 13, 14, 15 indicates that some certain relations among the elements of velocity components on the robot body and the propellers revolution speeds as well as thrusts from propellers are indirectly existing. From the simulation results it can be found that the thrust from Propeller 3 is getting greater along with the revolution speed of Propeller 3 becoming faster, the same changing rules the velocity component in vertical direction of the robot is; and the values changing tendency over time are coincident in these figures.

2. From Figs. 13, 14, 15 it should be noticed that there is strongly similar changing rules among the horizontal velocity components of the robot along X axis, the revolution speeds of control Propellers 1, 2 and their corresponding thrusts over time. While for the vertical trajectory tracking changing along with time it is obvious that the same changing patterns are found among the vertical velocity components of the robot (in Z direction of Fig. 15), revolution speed and thrust of Propellers 3 which inferred that the determining elements for the robot velocity components over time is the control manipulations of the propellers.

3. It can be concluded from the comparison between Figs. 11 and 16 that the dominant determinant for the underwater robot trajectory tracking control is the manipulation of adjusting cable length, which can be proved through the fact that the cable length changing along with time in Fig. 16 is quite similar to the robot trajectory tracking motion over time in Fig. 11.

4. During different motion control stages over time, a high correlation of kinematic coefficients about the robot body, umbilical cable and the propellers are inferred distinctly from Figs. 14, 15, 16. While in the vertical trajectory tracking control simulation during the rough time intervals 3 ~ 6s in Fig. 11a and 0–4 s in Fig. 11b respectively, the length changing of umbilical cable are both shortened in Fig. 16. At this period of time the robot is driven to move upward in the negative Z axis as shown in Figs. 1, 2 with the translational velocity components opposite from the positive ordinate direction as shown in Fig. 15; meanwhile the thrust direction of Propeller 3 is pointing downward as shown in Fig. 14. It can be seen from this stage of control operations that the thrust of Propeller 3 is opposite to the cable tension for producing a constrain force in order to balance the cable tension through which maintaining the robot moving smoothly and preventing excessive displacement deviation during the vertical trajectory tracking control. Analogous changing rules about the directions of propellers revolution speeds, thrusts and robot instantaneous vertical velocity components happened in the time intervals between 9 and 12 s for motion combination I and 8–12 s for motion combination II separately; however neither thrust of Propeller 3 nor vertical speed of the robot are engendered during the time intervals between 6 and 9 s for motion combination I and 4–8 s for motion combination II, respectively. The changing characteristics of the robot system presenting in this phase indicate that adjusting the cable length is dominant measure while the thrust of Propeller 3 is an assistant means for the robot tracking along with the predesignated vertical trajectory effectively.

5. The changing scopes about the robot velocity component in X axis, revolution speeds of Propellers 1 and 2 and their thrusts are relative small from the Figs. 13, 14, 15 because the horizontal velocity of the robot in Fig. 15 is a combined effect result of the hydrodynamic loads on the robot body under coming velocity condition, thrusts of Propellers 1 and 2 and the cable tension. The major role of Propellers 1 and 2 is manufacturing appropriate thrusts based on PID algorithm to drive the robot moving forward or backward along the predesignated trajectory.

Figure 17 illustrates the hydrodynamic loads about the robot body during the numerical simulations of the underwater robot moving along the predesignated trajectories under the control manipulations.

Fig. 17 [Images not available. See PDF.]

Hydrodynamic loads on the robot main body

According to the comparisons among Figs. 13, 14, 15, 16, 17, it is obvious that during the trajectory tracking control simulations the hydrodynamic loads on the robot body in X axis direction is basically consistent with the revolution speeds changing rules of Propellers 1 and 2, which indicates the hydrodynamic loads on robot body in X direction is determined by the rotating flow fields from Propellers 1 and 2, the robot velocity component in X direction has little effect. While the hydrodynamic loading on robot main body along Z axis direction is nearly the same with the changing rules of the robot velocity component in Z direction, it reveals that during the vertical movements the hydrodynamic loading on robot main body along Z axis direction is decided by the robot velocity component in Z direction which mainly controlled through adjusting umbilical cable length, and Propeller 3 as an auxiliary mean has insignificant influence.

Conclusion

A six degrees of freedom hydrodynamic and control model for trajectory tracking simulations about tethered underwater robot system is proposed. In this research a hybrid control strategy composed of feed-forward negative feedback control method for adjusting the cable length and PID mechanism for modulating the propellers revolution speeds is adopted to control the underwater robot moving along the predesignated trajectories. The hydrodynamic characteristics and control performance of the robot under the hybrid algorithms are investigated numerically with the proposed hydrodynamic and control mathematical model.

In this paper the trajectory tracking simulation results reveal that a superimposed control manipulation of adjusting the cable length in company with modulating the propellers revolution speeds for a tethered underwater robot system is reasonable and effective. There exists quite specific dynamic varying relationships among the robot velocity components, revolution speeds of propellers, kinetic parameters of umbilical cable and the hydrodynamic loading on robot main body.

Research presented in this paper may provide theoretical guidance for towing pool control experiments of tethered underwater robot and the coupling dynamic responses between umbilical cable and robot in the future research.

Author contributions

Jiaming Wu provided theoretical guidance and Dongjun Chen wrote the main manuscript text. All authors reviewed the manuscript.

Funding

This work was supported by the [National Natural Science Foundation of China] (Grant Numbers 51979110) and [Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions] (Grant Number 23KJA560008).

Data availability

The data are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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