Adaptive recurrent neural network motion control for observation class remotely operated vehicle manipulator system with modeling uncertainty

Abstract

Precise motion control of remotely operated vehicles plays an important role in a great number of submarine missions. However, the high-performance operations are difficult to realize due to the uncertainty in system modeling with self-disturbance. On the basis of the multibody system dynamics, self-disturbances from the tether and manipulator have been systematically analyzed in order to transform them into observed forces. A novel S surface–based adaptive recurrent wavelet neural network control system has been proposed on the nonlinear control of underwater vehicles, with its recurrent wavelet neural network structure designed for the approximation of the uncertain dynamics. Moreover, a robust function has been proposed to improve system robustness and convergence. The comparison shows that the remotely operated vehicle operation performance including the three-dimensional path following and vehicle-manipulator coordinate control has been greatly improved.

Keywords

Remotely operated vehicle three-dimensional path following multibody system adaptive recurrent neural network

Introduction

Remotely operated vehicles (ROVs) have been applied worldwide for various operations such as environmental survey, harbor inspection, petroleum hunting, and scientific investigation and even wrecks recovering.¹ Remotely operated vehicle manipulator system (ROVMS) plays an important role in a number of submarine missions for marine science, oil and gas exploration, and salvage.² In these application fields, the motions of ROVMS are usually teleoperated from a human pilot with a master manipulator on the surface control platform, the ROV is applied as a mobile platform for its slave manipulator operation.³ However, the teleoperated master–slave manipulation systems often puzzled with simultaneously control on tens of degree of freedoms (DOFs) particularly because of the system interactions between the vehicle and manipulators.⁴ In the last decades, increasing research has been focused on precise motion control of underwater vehicle and manipulator systems.⁵ Although some projects endeavor more autonomous operation on the intervention autonomous underwater vehicle (I-AUV),⁶ ROVMS with long duration of operation time, and equipment carrying flexibility, still can perform more accurate tasks.⁷

A teleoperated observation class ROVMS is usually an open frame one with one manipulator at the front and weight about 100 kg, such as American Seaeye Falcon ROV⁸ and French ECA H300 ROV. Although the vehicle is fully actuated,⁹ observation class ROVMS carries a series of difficulties in precise motion control and operation due to current disturbances, coupled nonlinearities from the tether and manipulator.¹⁰ Conventional control techniques which assume that the dynamic model is linear with unknown parameters usually result in great position errors.¹¹ Although self-learning-based nonlinear controllers are well qualified for position control,¹² disturbance-induced tether drag forces have to be investigated and overcome for the path following in complex environment.

In the adaptive and robust control methodologies of nonlinear systems, dynamic modeling is often applied in the controller to compensate and observe nonlinear effects to obtain robust control results. Sun et al.¹³ introduced an adaptive and robust controller against parametric uncertainties and to realize the disturbance suppressions. Chen et al.¹⁴ developed an adaptive robust control algorithm with the online tuning of the unknown weights and system parameters to achieve accurate tracking performance for linear motors. Yao et al.¹⁵ proposed a nonlinear robust controller with extended-state-observer-based output feedback in order to guarantee a prescribed tracking transient performance and final tracking accuracy. In the consideration with the role of tether drag forces¹⁶ and manipulator disturbances¹⁷ in the operation control, some researchers have made estimations and developed model-based control methodologies¹⁸ through dynamic modeling.¹⁹ A decoupled proportional–derivative (PD) controller with computational tether model was developed by Prabhakar and Buckham²⁰ based on finite element technique. In the consideration with ocean flow, MA Jordan²¹ proposed an adaptive control scheme with the tether dynamics model under quasi-stationary state. But input signal should be rich enough to excite all parameters in the system so that the adaptation algorithm can obtain the true values of the estimated parameters. A Bagheri et al.²² have developed an adaptive neural network controller for four DOFs control of ROV on the basis of the dynamic behavior effects of the communication cable. Han and Chung²³ have proposed an optimal proportional–integral–derivative (PID) merged robust adaptive control scheme for underwater vehicle manipulator system (UVMS) with restoring forces compensation. Mohan and Kim²⁴ provided a generalized framework for indirect adaptive control of UVMS, with the consideration of the dynamic coupling between the vehicle and the manipulator. In the underwater operations, however, some of the hydrodynamic parameters in the models are very difficult to determine,¹ modeling uncertainties should be further considered to eliminate modeling errors. Moreover, the tethered ROVMS should be considered as an integrated rigid and flexible multibody system in the modeling and control.

For the approximation of unknown robotic dynamics, neural networks have been successfully used. Le et al.²⁵ proposed a feed forward neural network in the combination with an error estimator in a chattering free neural sliding mode controller. Zhang and Chu²⁶ used the local recurrent neural network to estimate the unknown model of the underwater vehicle and thrust online in the adaptive sliding mode controller. Van Cuong and Nan²⁷ applied a radial basis function network to improve the adaptive trajectory tracking control performance against the large uncertainty of the system. However, it would take a long time for the weight values of the forward neural network to reach convergence. In the combination with the learning and decomposition capability of the wavelet function, wavelet neural network (WNN) is able to overcome these drawbacks.²⁸ The contributions of this study are described as follows:

This study has constructed dynamic models for the ROVMS on the basis of a multibody system and systematically analyzed the operation disturbances from tether and manipulator.

A novel S surface–based adaptive recurrent wavelet neural network (SARWNN) controller system has been proposed to approximate dynamic uncertainty and realize nonlinear control for the ROVMS. Moreover, a robust function has been proposed to improve the system robustness and convergence.

Tank experiments have verified the proposed controller’s validity; in the implementation of ROVMS operation control the proposed controller is more advanced in the tethered-vehicle-manipulator system coordination.

The rest of this study is organized as follows: section “Dynamic modeling of ROVMS” investigates the ROVMS dynamic model with disturbance. An SARWNN control system is proposed in section “Adaptive recurrent WNN controller.” Tank experiments are discussed and analyzed in section “Experiments.” Conclusion is made in section “Conclusion.”

Dynamic modeling of ROVMS

Modeling construction of ROVMS multibody system

A teleoperated observation class ROVMS is usually an open-framed one with a manipulator and tether. The manipulator with multiple links is responsible for underwater intervention operations while the tether is necessary for communication and energy supply. For the dynamics of the vehicle, we can obtain from the theorem of momentum²⁹

m \frac{d V}{dt} = F

(1)

where m is the mass, $V$ and $F$ are the vectors of vehicle velocity and external forces, respectively. However, the ROVMS is a multibody system including a flexible neutral buoyancy tether and multiple-joint manipulator. For the node construction of ROVMS multibody system, each node is assigned to a single body.³⁰ In Figure 1, the node construction of ROMS platform includes three nodes sets. The flexible nodes set includes flexible tether node (7), the surface end node (6), and vehicle end node (8) of the flexible tether; the vehicle fixed nodes set include the nodes fixed on the ROV body such as geometric center node (9), manipulator fixed point node (11), camera node (10); the active and rigid nodes set includes each DOF of the manipulator as nodes (1)–(5) respectively.

Figure 1.

Nodes construction for the tethered ROVMS.

As a hybrid rigid and flexible multibody system, the applied forces on the tether can be transformed into the disturbances on the ROVMS. The dynamic of a flexible tether moving in fluid can be described through a finite element equation²²

m_{t} a = W_{t} + F_{fluid} + \frac{\partial T}{\partial s}

(2)

where s is the upstretched Lagrangian coordinate, T is the tension force exerted at s, $a$ is the inertial acceleration, $F_{fluid}$ is the fluid dynamic force for per unit length, $m_{t}$ is the tether mass of per unit length, $W_{t} = (ρ_{t} - ρ_{w}) A_{t} g$ is the immersed weight of per unit length, $ρ_{w}$ is the sea water density, g is the gravitational acceleration, $ρ_{t}$ is the tether density, and $A_{t}$ is the cross-sectional area of the tether.

For the kinematics coordinate transformation from vehicle to tether, $O x_{0} y_{0} z_{0}$ , $O x_{v} y_{v} z_{v}$ , and $O x_{R} y_{R} z_{R}$ are set as the tether coordinate systems, its vehicle end coordinate system, and its surface end coordinate system, respectively.

In the tether coordinate system, $[\begin{matrix} x_{t} & y_{t} & z_{t} \end{matrix}]$ is set at the point along with the tether so that $x_{t}$ is tangent to the tether in the direction of increasing arc length from the tow-point, and $y_{t}$ is in the plane of surface coordinate $(x_{0}, y_{0})$ . The tether coordinate can be acquired through coordinate rotations in the following order: (1) rotate $z_{0}$ through counter-clockwise angle $α$ about the $y_{0}$ axis so as to bring the $x_{0}$ axis into the plane of $(x_{t}, y_{t})$ ; (2) rotate the new $x_{0}$ axis through counter-clockwise angle $π / 2$ to bring the $z_{0}$ axis into coincidence with $z_{t}$ ; (3) rotate $z_{t}$ through $β$ to bring $(x_{0}, y_{0})$ into coincidence with $(x_{t}, y_{t})$ . Therefore, the relationship between surface fixed coordinate and tether fixed coordinate can be expressed as

[\begin{matrix} x_{t} & y_{t} & z_{t} \end{matrix}] = [\begin{matrix} x_{0} & y_{0} & z_{0} \end{matrix}] \cdot B (α, β)

(3)

And the relationship between vehicle fixed coordinate and tether fixed coordinate can be expressed as

[\begin{matrix} x_{t} & y_{t} & z_{t} \end{matrix}] = [\begin{matrix} x_{v} & y_{v} & z_{v} \end{matrix}] \cdot H^{T} (φ, θ, ψ) \cdot B (α, β)

(4)

where $φ,$ $θ,$ and $ψ$ are the vehicle roll, pitch, and heading angle, respectively. If we use c to denote cos and s to denote sin, one obtains

H = [\begin{matrix} c ψ c θ & c ψ s θ s ϕ - s ψ c ϕ & c ψ s θ c ϕ + s ψ s ϕ \\ s ψ c θ & s ψ s θ \sin ϕ + c ψ c ϕ & s ψ s θ c ϕ - c ψ s ϕ \\ - s θ & c θ s ϕ & c θ c ϕ \end{matrix}]

(5)

and

B (α, β) = (\begin{matrix} c α c β & - c α s β & s α \\ - s α c β & s α s β & c α \\ - s β & - c β & 0 \end{matrix})

(6)

For the manipulator of ROVMS, the absolute velocity vector of each node can be obtained through forward propagation from vehicle to the manipulator operation end, while the interaction disturbances from manipulator to the vehicle will be obtained through backward propagation from the operation forces exerted on the manipulator operation end to the vehicle.

Through forward propagation, the velocity of each node of manipulator can be obtained as

V_{i} = V_{i - 1} + W_{i} \times r_{i}

(7)

If we set earth fixed coordinate system as $O x_{a} y_{a} z_{a}$ , vehicle fixed coordinate system as $O x_{V} y_{V} z_{V}$ , the ith manipulator link coordinate system as $O x_{i} y_{i} z_{i}$ ; the angular transformation matrix from earth to the vehicle is

{}^{a}{R_{v}} = [\begin{matrix} c ψ c θ & c ψ s θ s ϕ - s ψ c ϕ & c ψ s θ c ϕ + s ψ s ϕ \\ s ψ c θ & s ψ s θ \sin ϕ + c ψ c ϕ & s ψ s θ c ϕ - c ψ s ϕ \\ - s θ & c θ s ϕ & c θ c ϕ \end{matrix}]

(8)

According to Denavit–Hartenberg (D-H) convention, the angular transformation matrix from the ith link to the i + 1th link is

{}^{i}{R_{i + 1}} = [\begin{matrix} c θ_{i} & - s θ_{i} c α_{i} & s θ_{i} s α_{i} \\ s θ_{i} & c θ_{i} c α_{i} & - c θ_{i} s α_{i} \\ 0 & s α_{i} & c α_{i} \end{matrix}]

(9)

Thus, the inertial forces and moments applying on the ith node are, respectively

\begin{array}{l} {{}^{i}F}_{i} = M_{i} [{{}^{i}a}_{i} + {}^{i}{\overset{\cdot}{ω}}_{i} \times {{}^{i}r}_{c i} + {{}^{i}ω}_{i} \times ({{}^{i}ω}_{i} \times {{}^{i}r}_{c i})] a n d \\ {{}^{i}Q}_{i} = {{}^{i}Ω}_{i} {}^{i}{\overset{\cdot}{ω}}_{i} + {{}^{i}ω}_{i} \times ({{}^{i}Ω}_{i} {{}^{i}ω}_{i}) \end{array}

(10)

where $M_{i}$ is the mass matrix with added mass of the ith link, ${}^{i}{Ω_{i}}$ is the inertia matrix with added inertia with respect to the center of mass, ${}^{i}{a_{i}}$ is the linear acceleration of the ith link, ${}^{i}{r_{ci}}$ is the vector of link i mass center from the origin of frame i in the frame of the ith link, ${}^{i}{ω_{i}}$ is the angular velocity of the ith link relative to the joint on the origin of the ith link.

Therefore, the dynamic model of underwater vehicle can be represented as

M (q) \overset{\cdot\cdot}{q} + C (\overset{\cdot}{q}) \overset{\cdot}{q} + D (\overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + τ_{disturb} = U

(11)

where $M (q) = M_{RB} (q) + M_{A} (q)$ represents the mass and inertial matrix of the ROVMS, including the rigid-body matrix $M_{RB} (q)$ and the added mass matrix $M_{A} (q)$ , $q$ represents the motion vector relative to the earth fixed coordinate system, $D (\overset{\cdot}{q}) \overset{\cdot}{q}$ represents the viscous damping forces, $C (\overset{\cdot}{q}) = C_{RB} (\overset{\cdot}{q}) + C_{A} (\overset{\cdot}{q})$ represents the Coriolis and centripetal matrix of the rigid body and added mass, $G (q)$ represents the summation of the gravitational and buoyancy forces, $τ_{disturb}$ represents the disturbance on the underwater vehicle, and $U$ represents the control forces.

Disturbance from tether

For observation class ROVMS, disturbances from the tether are one of the most important nonlinear disturbances particularly when the current is relatively strong.³¹

7The disturbance forces from the tether can be decomposed into the surface section, underwater section, ship end, and ROV end. For the tether ship end, we have $v_{iyt} = v_{izt} = 0$ at point $O_{0}$ . The three axis drag forces at point $O_{0}$ on the tether are $H_{0 x}$ , $H_{0 y}$ , and $H_{0 z}$ , respectively. Thus, the tether tension force at point $O_{0}$ is

| T_{0} | = \sqrt{H_{0 x}^{2} + H_{0 y}^{2} + H_{0 z}^{2}}

(12)

For the tether in the water

{\begin{matrix} T_{j + 1} \frac{d ζ_{j}}{dp} - T_{j} \frac{d ζ_{j}}{dp} + F_{j, water} = m \frac{d v_{jt}}{dt} \frac{d ζ_{j}}{dp} \\ T_{j} \frac{d ξ_{j}}{dp} - T_{j + 1} \frac{d ξ_{j}}{dp} - W_{j, water} = m \frac{d v_{jt}}{dt} \frac{d ξ_{j}}{dp} \\ F_{j, water} = \frac{1}{2} d ξ C_{w} ρ_{w} A | v_{j, rel} | v_{j, rel} \end{matrix}

(13)

where $W_{j, water} = (ρ_{t} - ρ_{w}) Ag$ , $ρ_{w}$ is the density of water, $T_{j}$ and $T_{j + 1}$ are the two tension forces at the two ends of the jth micro-unit length underwater, $F_{j, water}$ is the wave and current disturbances force on the jth micro-unit length tether in the local frame $[x_{t}, y_{t}, z_{t}]$ , $C_{w} = [C_{xt}, C_{yt}, C_{zt}]$ is the drag coefficient in the water, $v_{j, rel} = [\begin{matrix} v_{j, xrel} & v_{j, yrel} & v_{j, zrel} \end{matrix}]^{T}$ , and $v_{j, rel}$ is the relative velocity between the ith micro-tether unit and the water.

For the ROV end of tether, we assume $F_{Rx}$ and $F_{Ry}$ as the horizontal force and $F_{Rz}$ as the vertical force at point $O_{R}$ on ROV. The tether tension force at point R is

{\begin{matrix} [\begin{matrix} F_{Rx} \\ F_{Ry} \\ F_{Rz} \end{matrix}] = H^{T} B (α (R, t), β (R, t)) [\begin{matrix} T_{R} \\ 0 \\ 0 \end{matrix}] \\ | T_{R} | = \sqrt{F_{Rx}^{2} + F_{Ry}^{2} + F_{Rz}^{2}} \end{matrix}

(14)

Therefore, in the local frame $O x_{t} y_{t} z_{t}$ the tether tension force can be calculated as

T_{R} \cdot H^{T} (φ, θ, ψ) \cdot B (α, β) + F_{water} - [\begin{matrix} 0 & 0 & W_{t} \end{matrix}] - T_{0} \cdot H (φ, θ, ψ) H^{T} (φ, θ, ψ) \cdot B (α, β) = \int_{O_{c}}^{sur} m a_{i} + \int_{sur}^{R} m a_{j}

(15)

where $a_{i} = [\begin{matrix} a_{i, Hx} & a_{i, Hy} & a_{i, Hz} \end{matrix}]$ and $a_{j} = [\begin{matrix} a_{j, Hx} & a_{j, Hy} & a_{j, Hz} \end{matrix}]$ are the corresponding horizontal and vertical accelerations of the micro-units tether in air and water, respectively, in the tether local frame, $T_{R} = [\begin{matrix} F_{Rx} & F_{Ry} & F_{Rz} \end{matrix}]$ in the ROV end frame, $T_{0} = [\begin{matrix} H_{0 x} & H_{0 y} & H_{0 z} \end{matrix}]$ in the mother ship end frame. Therefore, the tether induced moments are

M_{t} = r_{t} \times T_{R} = [\begin{matrix} r_{ty} F_{Rz} - r_{tz} F_{Ry} \\ r_{tz} F_{Rx} - r_{tx} F_{Rz} \\ r_{tx} F_{Ry} - r_{ty} F_{Rx} \end{matrix}] = [\begin{matrix} M_{tx} \\ M_{ty} \\ M_{tz} \end{matrix}]

(16)

where the position of the tow-point in the ROV end frame is $r_{t} = [\begin{matrix} r_{tx} & r_{ty} & r_{tz} \end{matrix}]$ . Therefore, one obtains

τ_{disturb}^{teth} = {[F_{Rx}, F_{Ry}, F_{Rz}, M_{tx}, M_{ty}, M_{tz}]}^{T}

(17)

Manipulator disturbance analysis

During the manipulation process of the underwater vehicle manipulator system, interaction forces between underwater vehicle and manipulator include restoring forces resulted from the change of gravity center and buoyancy center and the coupling forces between underwater vehicle and manipulator.³²

Thus, the restoring forces resulted from the change of gravity center and buoyancy center during the manipulation can be expressed as

G (q) = [\begin{matrix} {}^{v}{f_{g}^{v}} + {}^{v}{f_{B}^{v}} + \sum_{i = 1}^{n} ({}^{v}{f_{g}^{i}} + {}^{v}{f_{B}^{i}}) \\ {}^{v}{r_{g}^{v}} \times {}^{v}{f_{g}^{v}} + {}^{v}{r_{B}^{v}} \times {}^{0}{f_{B}^{v}} + \sum_{i = 1}^{n} ({}^{v}{r_{g}^{i}} \times {}^{v}{f_{g}^{i}} + {}^{v}{r_{B}^{i}} \times {}^{v}{f_{B}^{i}}) \end{matrix}]

(18)

where ${}^{v}{f_{g}^{i}} = {}^{n}{R_{v}^{T}} {}^{n}{f_{g}^{i}}$ and ${}^{v}{f_{B}^{i}} = {}^{n}{R_{v}^{T}} {}^{n}{f_{B}^{i}}$ represent the restoring force of the ith link relative to underwater vehicle, where

{}^{v}{f_{g}^{v}} = {}^{n}{R_{v}^{T}} {}^{n}{f_{g}^{v}} and {}^{v}{f_{B}^{v}} = {}^{n}{R_{v}^{T}} {}^{n}{f_{B}^{v}}

(19)

{}^{n}{f_{g}^{v}} = {[\begin{matrix} 0 & 0 & m_{v} g \end{matrix}]}^{T} and {}^{n}{f_{B}^{v}} = {[\begin{matrix} 0 & 0 & B_{v} \end{matrix}]}^{T}

(20)

denote the restoring force of underwater vehicle, ${}^{v}{r_{g}^{v}}$ and ${}^{v}{r_{B}^{v}}$ are the underwater vehicle CB and CG positions vectors relative to underwater vehicle geometric center, respectively, ${}^{v}{r_{g}^{i}}$ and ${}^{v}{r_{B}^{i}}$ are the ith link CB and CG positions vectors relative to the underwater vehicle geometric center, respectively.

On the other hand, the coupling force exerted on the vehicle from manipulator can be derived according to the following recursion

{}^{i}{f_{i, i - 1}} = {}^{i}{f_{i + 1, i}} - {}^{i}{f_{i}^{*}} - {}^{i}{F_{i, i}}

(21)

{}^{i}{n_{i, i - 1}} = {}^{i}{n_{i + 1, i}} + {}^{i}{r_{ci}} \times {}^{i}{f_{i, i - 1}} + ({}^{i}{r_{i}} - {}^{i}{r_{ci}}) \times ({}^{i}{f_{i + 1, i}} + {}^{i}{F_{i, i}}) - {}^{i}{n_{i}^{*}} - {}^{i}{M_{i, i}}

(22)

{}^{i - 1}{f_{i, i - 1}} = {}^{i - 1}{R_{i}} {}^{i}{f_{i, i - 1}}, {}^{i - 1}{n_{i, i - 1}} = {}^{i - 1}{R_{i}} {}^{i}{n_{i, i - 1}}

(23)

where ${}^{i}{M_{i, i}}$ and ${}^{i}{F_{i, i}}$ are the hydrodynamic force and moment vectors exerted on the ith link, ${}^{i}{f_{i, i - 1}}$ and ${}^{i}{n_{i, i - 1}}$ are the constraint reaction moment from the ith link to the i – 1th link, and ${}^{i}{r_{i}}$ is a vector for the revolt joint

{}^{i}{r_{i}} = {[\begin{matrix} a_{i} & d_{i} s α_{i} & d_{i} c α_{i} \end{matrix}]}^{T}

(24)

{}^{i}{r_{c i}} = {[\begin{matrix} a_{ci} & d_{ci} s α_{i} & d_{ci} c α_{i} \end{matrix}]}^{T}

(25)

representing the constant vector of the ith link geometric center in the frame of the ith link. Thus, coupling forces ${}^{v}{f_{1, v}}$ and ${}^{v}{n_{1, v}}$ from the operation end of the manipulator to the vehicle can be backwardly deduced. The disturbances from the manipulator to the vehicle are the summation of restoring forces and coupling forces

τ_{disturb}^{man} = [\begin{matrix} {}^{v}{f_{g}^{v}} + {}^{v}{f_{B}^{v}} + \sum_{i = 1}^{n} ({}^{v}{f_{g}^{i}} + {}^{v}{f_{B}^{i}}) + {}^{v}{f_{1, v}} \\ {}^{v}{r_{g}^{v}} \times {}^{v}{f_{g}^{v}} + {}^{v}{r_{B}^{v}} \times {}^{0}{f_{B}^{v}} + \sum_{i = 1}^{n} ({}^{v}{r_{g}^{i}} \times {}^{v}{f_{g}^{i}} + {}^{v}{r_{B}^{i}} \times {}^{v}{f_{B}^{i}}) + {}^{v}{n_{1, v}} \end{matrix}]

(26)

Therefore, the disturbance forces $τ_{disturb}$ of ROVMS can include those from tether of equation (17) and from manipulator of equation (26)

τ_{disturb} = τ_{disturb}^{teth} + τ_{disturb}^{man}

(27)

Adaptive recurrent WNN controller

Recurrent WNN structure

Although the dynamic model and disturbance of ROVMS can be obtained from section “Dynamic modeling of ROVMS,” there are still some uncertain hydrodynamic coefficiencies and external disturbance to be determined. In the following, a novel recurrent WNN has been proposed to estimate and reduce the influence of modeling uncertainties. Figure 2 illustrates the structure of the recurrent WNN control: a 5-layer block diagram, which incorporates features of the WNN; the recurrent neural network technique and the fuzzy cerebellar model articulation controller. In such structure, the wavelet space can be regarded as a feature space; the features are extracted through weighting the interior states of input signals. The propagation details of the recurrent WNN are proposed in the following:

In the first layer, the input signals are defined as $X^{(1)} = [x_{1}^{(1)}, x_{2}^{(1)}, \dots, x_{n_{1}}^{(1)}]$ , where $n_{1}$ is the number of the input elements. The inputs include control errors, $τ_{disturb}$ of equation (27), current states of manipulator links and vehicle. Current states of manipulator links are obtained from motor encoders. Current states of the vehicle are obtained from the magnetic compass and Doppler velocity meter (DVL). Through magnetic compass, the vehicle can obtain its posture information like yaw, pitch, and roll, while through DVL the vehicle can obtain forward, lateral, and vertical speed. Therefore, one can obtain current vehicle position through Extended Kalman Filter (EKF)-based dead reckoning. The expected states of the vehicle and each link of the manipulator are also input as reference signals. The input signals are transported to the second layer by neurons. Through these neurons, $X^{(1)}$ can be quantized and moved to the next layer.

In the second layer, several elements can be accumulated into a block as a Mexican Hat wavelet membership function. The Mexican Hat wavelet function has been selected as the membership basic function, and it can be expressed as the following formula

μ_{lm}^{(2)} (x_{l}^{(1)}) = [I - d_{lmk}^{2} {(x_{i}^{(1)} - c_{lmk})}^{2}] \exp (- d_{lmk}^{2} {(x_{l}^{(1)} - c_{lmk})}^{2})

(28)

where $d_{lmk}$ is the dilation parameter, $c_{lmk}$ is the translation parameter, $l = 1, 2, \dots, n_{1}$ , $n_{1}$ is the number of input signals, $m = 1, 2, \dots, n_{l}$ , $n_{l}$ is the number of layers, $k = 1, 2, \dots, n_{b}$ , $n_{b}$ is the number of the blocks of each layer. A local feedback unit is defined as input of this layer with a real-time delay section is defined and added to this layer as follows

y_{lmk}^{(2)} (t) = x_{i}^{(1)} (t) + ω_{lmk}^{(2)} μ_{lmk}^{(2)} (x_{i}^{(1)} (t - T))

(29)

where $ω_{lmk}^{(2)}$ is the recurrent weight, $μ_{lm}^{(2)} (x_{i}^{(1)} (t - T))$ represents the time delay T of $μ_{lm}^{(2)} (x_{i}^{(1)} (t))$ .

3. The third layer is the fuzzy rule layer. Each block in this layer performs as a rule. The execution for the neuron is matched with the relevant rule. Thus, the output of this layer is defined as

y_{mk}^{(3)} = \underset{l}{Π} ω_{lmk}^{(3)} μ_{lmk}^{(2)} (y_{lmk}^{(2)})

(30)

where $ω_{lmk}^{(3)}$ represents the weight from the membership layer to the fuzzy rule layer.

4. In the forth layer, each location can be expressed as

W^{(4)} = [\begin{matrix} ω_{11}^{(4)} & ω_{12}^{(4)} & \dots & ω_{p 1}^{(4)} \\ ω_{12}^{(4)} & ω_{22}^{(4)} & \dots & ω_{p 2}^{(4)} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ω_{1 q}^{(4)} & ω_{2 q}^{(4)} & \dots & ω_{pq}^{(4)} \end{matrix}]

(31)

where $W^{(4)}$ represents the connecting weight matrix from the rule layer to the output layer. Its element $ω_{mk}^{(4)}$ is the weight value associate the mth layer and the kth block.

5. In this fifth layer, each output node can be expressed as

y_{o}^{(5)} = \sum_{m = 1}^{n_{l}} \sum_{k = 1}^{n_{b}} ω_{mk}^{(4)} y_{mk}^{(3)} = (A_{w} X^{(1)}) y^{(3)}

(32)

where $y_{o}^{(5)}$ is the approximation value of the model uncertainties and unknown disturbance, $A_{ω}$ represents optimal parameter matrix, and $y^{(3)}$ represents the output matrix of the third layer. For more information about WNNs, see Li.³³

Figure 2.

Structure diagram of recurrent wavelet neural network.

SARWNN controller

In the literatures of model-based adaptive control, position accuracy vector is usually defined as²⁷

s = \overset{\cdot}{e} + λ e

(33)

where e is the control error, $λ = diag (λ_{1}, λ_{2}, \dots, λ_{6})$ is a positive definite matrix, $λ_{i} > 0$ . However, proportional and derivative gained accuracy vector cannot satisfy the control precision of the underwater nonlinear system. Since the thruster-actuated system often causes vehicle delayed response, the commands are expected to be loosely large when the deviation is relatively large but are expected to be strictly small when the deviation is relatively small. Thus, one can obtain more subdivided control quantity when approaching the target position. This advantage is particularly important for ROVMS operation close to the target. Sigmoid function (see Figure 3) covers strictly subdivided values when close to zero, coincidently satisfies with the expectation. The Sigmoid linear function is expressed as

s = \frac{2}{1 + \exp (- k_{p} e)} - 1

(34)

The Sigmoid surface function is expressed as

s = \frac{2}{1 + \exp (- k_{p} e - k_{d} \overset{\cdot}{e})} - 1

(35)

where $k_{p}$ and $k_{d}$ are the proportional and derivative gains.

Figure 3.

Sigmoid function.

On the basis of the Sigmoid function in equation (35), one can obtain the Sigmoid type of compensator as

S = \frac{2 I}{(I + \exp (- K_{p} q_{e} - K_{d} {\overset{\cdot}{q}}_{e}))} - I

(36)

where $I = [1, \dots, 1]^{T} \in R^{6}$ , the position error vector of underwater vehicle is defined as

q_{e} = [e_{x}^{v}, e_{y}^{v}, e_{z}^{v}, e_{φ}^{v}, e_{θ}^{v}, e_{ψ}^{v}]^{T}

(37)

where $e_{x}^{v}$ represents the longitude error of the vehicle, $e_{y}^{v}$ represents the transverse error of the vehicle, $e_{z}^{v}$ represents the vertical error of the vehicle, $e_{φ}^{v}$ represents the pose error in the roll, $e_{θ}^{v}$ represents the pose error in the pitch, $e_{ψ}^{v}$ represents the pose error in the yaw, $q_{d}$ represents the expected motion vector of the vehicle, and $K_{p}$ and $K_{d}$ are the proportional and derivative matrix gains.

By applying equation (36) into equation (11), one obtains

M_{ζ} \overset{\cdot}{S} + (K_{s} + C_{ζ} + D_{ζ}) S = y_{ζ} + τ_{disturb} - U + K_{s} S

(38)

where $y_{ζ}$ is the function of $q$ and $q_{d}$ and $K_{s}$ is the positive constant matrix, in order to guarantee the control system stability, the proposed SARWNN controller is presented as follows

U = {\hat{y}}_{o}^{(5)} + K_{s} S + {\hat{τ}}_{disturb} + {\hat{u}}_{r}

(39)

where ${\hat{y}}_{o}^{(5)}$ is the approximation function of the position error of $y_{o}^{(5)}$ , ${\hat{y}}_{o}^{(5)} = ({\hat{A}}_{w} X^{(1)}) {\hat{y}}^{(3)}$ and ${\hat{u}}_{r}$ is the robust term to eliminate the estimation uncertainty.

If we apply the controller equation (39) into the dynamic model (11), one obtains

M (q) \overset{\cdot}{S} = {\tilde{y}}_{o}^{(5)} - ({\tilde{K}}_{s} + C (\overset{\cdot}{q}) + D (\overset{\cdot}{q})) S + {\tilde{τ}}_{disturb} - {\hat{u}}_{r}

(40)

where $\tilde{π} = π - \hat{π}$ , $π$ represents $τ_{disturb}$ , $y_{o}^{(5)}$ , $u_{r}$ , $D_{1}$ , $C_{1}$ , $W^{(2)}$ , ${\tilde{τ}}_{disturb}$ is the disturbance estimation error, and ${\tilde{y}}_{o}^{(5)}$ is the network approximation error. In order to apply Lyapunov theorem, the proposed recurrent WNN can be partially linearized through Taylor expansion³⁴

{\tilde{y}}^{(3)} = B^{T} {\tilde{D}}_{1} + J^{T} {\tilde{C}}_{1} + L^{T} {\tilde{W}}^{(2)} + Θ ({\tilde{D}}_{1}, {\tilde{C}}_{1}, {\tilde{W}}^{(2)})

(41)

where $C_{1} = [c_{1 mk}^{T}, c_{2 mk}^{T}, \dots, c_{n_{1} mk}^{T}]^{T}$ , $D_{1} = [d_{1 mk}^{T}, d_{2 mk}^{T}, \dots, d_{n_{1} mk}^{T}]^{T}$ , $W^{(2)} = {[{(ω_{1 m k}^{(2)})}^{T}, {(ω_{2 m k}^{(2)})}^{T}, \dots, {(ω_{n_{1} m k}^{(2)})}^{T}]}^{T}$ , and $Θ$ is high-order vector of Taylor expansion series. If we set n_r as the number of fuzzy rules, one obtains

\begin{array}{l} B = [\frac{\partial y_{1}^{(3)}}{\partial D_{1}}, \frac{\partial y_{2}^{(3)}}{\partial D_{1}}, …, \frac{\partial y_{n_{r}}^{(3)}}{\partial D_{1}}], \\ J = [\frac{\partial y_{1}^{(3)}}{\partial C_{1}}, \frac{\partial y_{2}^{(3)}}{\partial C_{1}}, …, \frac{\partial y_{n_{r}}^{(3)}}{\partial C_{1}}], a n d \\ L = [\frac{\partial y_{1}^{(3)}}{\partial W^{(2)}}, \frac{\partial y_{2}^{(3)}}{\partial W^{(2)}}, …, \frac{\partial y_{n_{r}}^{(3)}}{\partial W^{(2)}}] \end{array}

If we define n_d as the number of ROVMS DOF, n_c as the constraint number of ROVMS, n_m = n_d – n_c, one obtains

\begin{matrix} \frac{\partial y_{Ω}^{(3)}}{\partial D_{1}} = [\underset{(n_{m} - 1) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}, \frac{\partial y_{Ω}^{(3)}}{\partial {d_{1 n_{m}}}_{1}}, \dots, \frac{\partial y_{Ω}^{(3)}}{\partial {d_{1 n_{m}}}_{(n_{d} - n_{c})}}, \underset{(n_{r} - n_{m}) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}] \\ \frac{\partial y_{Ω}^{(3)}}{\partial C_{1}} = [\underset{(n_{m} - 1) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}, \frac{\partial y_{Ω}^{(3)}}{\partial c_{1 n_{m} 1}}, \dots, \frac{\partial y_{Ω}^{(3)}}{\partial {c_{1 n_{m}}}_{(n_{d} - n_{c})}}, \underset{(n_{r} - n_{m}) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}] \\ \frac{\partial y_{Ω}^{(3)}}{\partial W^{(2)}} = [\underset{(n_{m} - 1) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}, \frac{\partial y_{Ω}^{(3)}}{\partial ω_{1 n_{m} 1}^{(2)}}, \dots, \frac{\partial y_{Ω}^{(3)}}{\partial {ω_{1 n_{m}}^{(2)}}_{(n_{d} - n_{c})}}, \underset{(n_{r} - n_{m}) (n_{d} - n_{c})}{\underset{︸}{0, \dots, 0}}] \end{matrix}

(42)

From equation (39), one obtains

\begin{matrix} {\tilde{y}}_{o}^{(5)} & + {\tilde{τ}}_{disturb} = X^{(1)} {\tilde{A}}_{ω}^{T} ({\hat{Y}}^{(3)} - B^{T} {\hat{D}}_{1} - J^{T} {\hat{C}}_{1} - L^{T} {\hat{W}}^{(2)}) \\ + X^{(1)} {\hat{A}}_{ω}^{T} (B^{T} {\tilde{D}}_{1} + J^{T} {\tilde{C}}_{1} + L^{T} {\tilde{W}}^{(2)}) + Γ \end{matrix}

(43)

where

\begin{matrix} Γ = X^{(1)} A_{ω}^{T} ({\tilde{Y}}^{(3)} + B^{T} {\hat{D}}_{1} + J^{T} {\hat{C}}_{1} + L^{T} {\hat{W}}^{(2)}) \\ - X^{(1)} {\hat{A}}_{ω}^{T} (B^{T} {\tilde{D}}_{1} + J^{T} {\tilde{C}}_{1} + L^{T} {\tilde{W}}^{(2)}) + {\tilde{τ}}_{disturb} \end{matrix}

(44)

From equations (40) and (43)

\begin{matrix} M (q) \overset{\cdot}{S} = X^{(1)} {\tilde{A}}_{ω}^{T} ({\hat{Y}}^{(3)} - B^{T} {\hat{D}}_{1} - J^{T} {\hat{C}}_{1} - L^{T} {\hat{W}}^{(2)}) \\ + X^{(1)} {\hat{A}}_{ω}^{T} (B^{T} {\tilde{D}}_{1} + J^{T} {\tilde{C}}_{1} + L^{T} {\tilde{W}}^{(2)}) \\ - ({\tilde{K}}_{s} + C (\overset{\cdot}{q}) + D (\overset{\cdot}{q})) S + Γ - {\hat{u}}_{r} \end{matrix}

(45)

If we assume the weights limit bounded as $‖ D_{1} ‖ ⩽ b_{d}$ , $‖ C_{1} ‖ ⩽ b_{c}$ , $‖ A_{ω} ‖ ⩽ A_{lim}$ , $‖ W^{(2)} ‖ ⩽ b_{w 2}$ , $‖ W^{(4)} ‖ ⩽ b_{w 4}$ , one obtains from³⁵

\begin{matrix} ‖ Γ ‖ + \frac{π {(X^{(1)})}^{2} A_{lim}^{2}}{4} + \frac{π D_{lim}^{2}}{4} + \frac{π C_{lim}^{2}}{4} + \frac{π {(W_{lim}^{(2)})}^{2}}{4} \\ ⩽ ‖ X^{(1)} A_{ω}^{T} {\tilde{Y}}^{(3)} + {\tilde{τ}}_{dis} ‖ + ‖ X^{(1)} A_{ω}^{T} B^{T} ‖ ‖ {\hat{D}}_{1} ‖ \\ + ‖ X^{(1)} A_{ω}^{T} J^{T} ‖ ‖ {\hat{C}}_{1} ‖ \\ + ‖ X^{(1)} A_{ω}^{T} L^{T} ‖ ‖ {\hat{W}}^{(2)} ‖ \\ + ‖ X^{(1)} (B^{T} D_{1} + J^{T} C_{1} + L^{T} W^{(2)}) ‖ ‖ A_{ω} ‖ \\ + \frac{π {(X^{(1)})}^{2} A_{lim}^{2}}{4} + \frac{π D_{lim}^{2}}{4} + \frac{π C_{lim}^{2}}{4} \\ + \frac{π {(W_{lim}^{(2)})}^{2}}{4} ⩽ Λ^{T} σ \end{matrix}

(46)

where $Λ = [Λ_{1}, Λ_{2}, \dots, Λ_{5}]^{T}$ , $σ = [1, ‖ W^{(2)} ‖, ‖ D_{1} ‖, ‖ C_{1} ‖, ‖ A_{ω} ‖]$ , are positive constant vector, and $π$ is positive constant. The bounds are $‖ X^{(1)} A_{ω}^{T} {\tilde{Y}}^{(3)} + {\tilde{τ}}_{dis} ‖ + \frac{π {(X^{(1)})}^{2} A_{lim}^{2}}{4} + \frac{π D_{lim}^{2}}{4} + \frac{π C_{lim}^{2}}{4} + \frac{π {(W_{lim}^{(2)})}^{2}}{4}$ , $‖ X^{(1)} A_{ω}^{T} J^{T} ‖$ , $‖ X^{(1)} A_{ω}^{T} B^{T} ‖ ‖ X^{(1)} A_{ω}^{T} L^{T} ‖$ , respectively. The robust term for uncertainty is proposed as

{\hat{u}}_{r} = \frac{S}{‖ S ‖} {\hat{Λ}}^{T} σ + b_{u} \frac{S}{{‖ S ‖}^{2} + δ}

(47)

where $b_{u}$ and $δ$ are positive constants.

Therefore, the learning update law of the network is proposed as

{\begin{matrix} {\overset{\cdot}{\hat{A}}}_{ω} = K_{W 4} ({\hat{Y}}^{(3)} - B^{T} {\hat{D}}_{1} - J^{T} {\hat{C}}_{1} \\ - L^{T} {\hat{W}}^{(2)}) X^{(1)} S^{T} - π K_{W 4} ‖ S ‖ X^{2} {\hat{A}}_{ω} \\ {\overset{\cdot}{\hat{D}}}_{1} = K_{D} B X^{(1)} {\hat{A}}_{ω} S - π K_{D} ‖ S ‖ {\hat{D}}_{1} \\ {\overset{\cdot}{\hat{C}}}_{1} = K_{C} J X^{(1)} {\hat{A}}_{ω} S - π K_{C} ‖ S ‖ {\hat{C}}_{1} \\ {\overset{\cdot}{\hat{W}}}^{(2)} = K_{W 2} H X^{(1)} {\hat{A}}_{ω} S - π K_{W 2} ‖ S ‖ {\hat{W}}^{(2)} \\ \overset{\cdot}{\hat{Λ}} = ‖ S ‖ K_{Λ} σ \end{matrix}

(48)

where $K_{D}$ , $K_{C}$ , $K_{W 4}$ , $K_{W 2}$ , and $K_{Λ}$ are diagonal matrices of positive constant.

Figure 4 illustrates the controller implementation diagram. For the controller feedback, the function of EKF is to estimate covariance matrix of the current position state, on the basis of ROV kinematic models. Since the vehicle velocity and posture information obtained from DVL and magnetic compass could involve sensor noise, the EKF-based dead reckoning block can obtain relatively accurate position output through measurement update and state estimation.

Figure 4.

SARWNN controller implementation diagram.

Controller stability analysis

According to the Lyapunov stability principle, the Lyapunov function candidate is

\begin{matrix} V = \frac{1}{2} SMS + \frac{1}{2} tr ({\tilde{A}}_{ω}^{T} K_{W 4}^{- 1} {\tilde{A}}_{ω}) + \frac{1}{2} tr ({\tilde{D}}_{1}^{T} K_{D}^{- 1} {\tilde{D}}_{1}) \\ + \frac{1}{2} tr ({\tilde{C}}_{1}^{T} K_{C}^{- 1} {\tilde{C}}_{1}) \\ + \frac{1}{2} tr (({\tilde{W}}^{(2)})^{T} K_{W 2}^{- 1} {\tilde{W}}^{(2)}) + \frac{1}{2} tr ({\tilde{Λ}}^{T} K_{Λ}^{- 1} \tilde{Λ}) \end{matrix}

(49)

\begin{matrix} {\overset{\cdot}{V}}_{2} = \frac{1}{2} SM \overset{\cdot}{S} + \frac{1}{2} S \overset{\cdot}{M} S - tr ({\tilde{A}}_{ω}^{T} K_{W 4}^{- 1} {\overset{\cdot}{\hat{A}}}_{ω}) - tr ({\tilde{D}}_{1}^{T} K_{D}^{- 1} {\overset{\cdot}{\hat{D}}}_{1}) \\ - tr ({\tilde{C}}_{1}^{T} K_{C}^{- 1} {\overset{\cdot}{\hat{C}}}_{1}) - tr (({\tilde{W}}^{(2)})^{T} K_{W 2}^{- 1} {\overset{\cdot}{\hat{W}}}^{(2)}) - tr ({\tilde{Λ}}^{T} K_{Λ}^{- 1} \overset{\cdot}{\hat{Λ}}) \end{matrix}

(50)

Substituting equation (44) into equation (49), one obtains

\begin{matrix} \overset{\cdot}{V} = - S^{T} K_{s} S - tr ({\tilde{A}}_{ω}^{T} K_{W 4}^{- 1} {\overset{\cdot}{\hat{A}}}_{ω}) \\ + X^{(1)} S^{T} {\tilde{A}}_{ω}^{T} ({\hat{Y}}^{(3)} - B^{T} {\hat{D}}_{1} - J^{T} {\hat{C}}_{1} - L^{T} {\hat{W}}^{(2)}) \\ + X^{(1)} S^{T} {\hat{A}}_{ω}^{T} (B^{T} {\tilde{D}}_{1} + J^{T} {\tilde{C}}_{1} + L^{T} {\tilde{W}}^{(2)}) \\ - tr ({\tilde{D}}_{1}^{T} K_{D}^{- 1} {\overset{\cdot}{\hat{D}}}_{1}) \\ - tr ({\tilde{C}}_{1}^{T} K_{C}^{- 1} {\overset{\cdot}{\hat{C}}}_{1}) - tr (({\tilde{W}}^{(2)})^{T} K_{W 2}^{- 1} {\overset{\cdot}{\hat{W}}}^{(2)}) \\ - tr ({\tilde{Λ}}^{T} K_{Λ}^{- 1} \overset{\cdot}{\hat{Λ}}) + S^{T} Γ - S^{T} {\hat{u}}_{r} \end{matrix}

(51)

From the updating law of equation (47)

\begin{matrix} \overset{\cdot}{V} = - S^{T} K_{s} S + (X^{(1)})^{2} π ‖ S ‖ tr ({\tilde{A}}_{ω}^{T} {\hat{A}}_{ω}) + π ‖ S ‖ tr ({\tilde{D}}_{1}^{T} {\hat{D}}_{1}) \\ + S^{T} Γ + π ‖ S ‖ tr ({\tilde{C}}_{1}^{T} {\hat{C}}_{1}) + π ‖ S ‖ tr (({\tilde{W}}^{(2)})^{T} {\hat{W}}^{(2)}) \\ - S^{T} {\hat{u}}_{r} - {\tilde{Λ}}^{T} ‖ S ‖ σ \end{matrix}

(52)

Using equation (45) and $tr (A_{ω}^{T} A_{ω}) ⩽ b_{d} ‖ A_{ω} ‖ - ‖ A_{ω} ‖^{2}$ , $tr (D^{T} D) ⩽ b_{d} ‖ D ‖ - ‖ D ‖^{2}$ , $tr ((W^{(4)})^{T} W^{(4)}) ⩽ b_{w 4} ‖ W^{(4)} ‖ - ‖ W^{(4)} ‖^{2}$ , $tr (C^{T} C) ⩽ b_{c} ‖ C ‖ - ‖ C ‖^{2}$ , and $tr ((W^{(2)})^{T} W^{(2)}) ⩽ b_{w 2} ‖ W^{(2)} ‖ - ‖ W^{(2)} ‖^{2}$

we have

\begin{matrix} \overset{\cdot}{V} ⩽ - S^{T} K_{s} S + (X^{(1)})^{2} π ‖ S ‖ (b_{d} ‖ A_{ω} ‖ - ‖ A_{ω} ‖^{2}) - S^{T} {\hat{u}}_{r} \\ + π ‖ S ‖ (b_{d} ‖ D_{1} ‖ - ‖ D_{1} ‖^{2}) \\ + π ‖ S ‖ (b_{c} ‖ C ‖ - ‖ C ‖^{2}) - {\tilde{Λ}}^{T} ‖ S ‖ σ + ‖ S ‖ Λ^{T} σ \\ - ‖ S ‖ (\frac{π {(X^{(1)})}^{2} A_{lim}^{2}}{4} + \frac{π D_{lim}^{2}}{4} + \frac{π C_{lim}^{2}}{4} + \frac{π {(W_{lim}^{(2)})}^{2}}{4}) \end{matrix}

(53)

Thus

\begin{matrix} \overset{\cdot}{V} ⩽ - S^{T} K_{s} S + ‖ S ‖ Λ^{T} σ - {\tilde{Λ}}^{T} ‖ S ‖ σ - ‖ S ‖ {\hat{Λ}}^{T} σ \\ = - S^{T} K_{s} S ⩽ 0 \end{matrix}

(54)

Therefore, $\overset{\cdot}{V}$ is a negative semi-definite function, since $V$ is a bounded function. Through Barbalat’s Lemma,³⁶ $- S^{T} K_{s} S$ is bounded, thus the trajectory error will converge into zero as the time converges to infinity.

Experiments

Experimental setup

In order to verify and analyze the above investigations, the experiments have been performed in a 50 m × 30 m × 10 m tank at the Key Laboratory of Science and Technology on Underwater Vehicle in the Harbin Engineering University. In order to simulate real oceanic conditions in the experiments, waves, and current are generated from the multidirectional waves generator and the local current generation device (Figure 5(a)), respectively. The experimental object is an SY-II ROV equipped with 3 function and 2 DOF manipulator, see Figure 5(b). SY-II ROV is an open frame underwater vehicle with PC104 as the core of the vehicle control system. Its neutrally buoyant tether is responsible with power supply and communication cable. The control commands are sent through network communication between the surface computer and the PC104 embedded processor. It is equipped with a depth gauge, a magnetic compass and a DVL as motion sensors, two main thrusters, two lateral thrusters and two vertical thrusters. The flow chart of hardware operation is illustrated in Figure 6. Parameters of its tether, hydrodynamics, and manipulator are illustrated in Figures 1,2, and 3, respectively.

Figure 5.

Experimental setup. (a) Wave generator and current generation device. (b) SY-II ROV.

Figure 6.

The flow chart of hardware operation.

Table 1.

Tethered cable parameters.

Item	$ρ_{t} (kg / m^{3})$	$C_{f} (m^{- 1})$	E (10⁵ MPa)	$C_{mind} (m^{- 1})$	D (m)	$L_{total} (m)$
Value	1000	1.2	0.02	0.4	10⁻²	300

Table 2.

SY-II hydrodynamic parameters obtained from planar plane mechanism experiments in the towing tank.

Dimensionless coefficient	x	y	z	r	q	p
First-order terms	27.65	47.26	44.86	62.63	57.55	141.33

Table 3.

Parameters of the manipulator.

Length of the upper arm (m)	Weight of the upper arm (kg)	Diameter of the upper arm (mm)	Length of the forearm (m)	Weight of the forearm (kg)	Diameter of the forearm (mm)
0.366	4.5	54	0.292	4	54

Hydrodynamic effects on the ROVMS include added mass forces and viscous damping, which are important differences from vehicle-manipulator system on the land. The added mass forces are the reaction forces produced by acceleration movement of the body. They can be numerically calculated in the unbounded flow field through panel method. The viscous damping is more complicated, including the vortex shedding, skin friction, and potential damping. These can be determined from the full size or towing-tank model tests.³⁷ In the following experiments, hydrodynamic effects of the ROV are predicted through proposed recurrent WNN with the consideration of unknown parameters, in order to realize precise control.

Case study 1: position and path following control

Two experimental tests are made to evaluate the adaptive recurrent WNN controller algorithm, the first is simultaneously depth and heading control, the second is three-dimensional (3D) path following control. In order to simulate the disturbances in complex oceanic environment, we set current speed as 0.1 m/s and at the same time generate JONSWAP spectrum irregular wave (significant wave height 0.1 m, peak spectral period 1.96 s) to prove controller’s robust performance. The current speed can be obtained through ADCP/DVL Navigator, so that the tether drag forces can also be observed. Comparisons between adaptive recurrent WNN controller and S surface controller have been evaluated in the experiments. For the S surface controller, $K_{d}$ was selected as $diag (3, 3, 3, 3, 1, 1)$ to reduce overshoot, $K_{p}$ was selected as $diag (5, 5, 5, 5, 4, 4)$ to improve convergence rate. For the SARWNN controller $K_{S} = diag (3.5, 3.5, 3.5, 3.5, 3.5, 3.5)$ , $K_{D} = K_{C} = diag (9.5)$ , $K_{W 4} = diag (1.2)$ , $K_{W 2} = diag (10.5)$ , and $K_{Λ} = diag (0.01)$ . Among these parameters $K_{S}$ is the one that represents control quantity distribution between the S surface control and recurrent wavelet neural network controller. If $K_{S}$ is greater enough, the proposed controller is close to an S surface controller.

In Figure 6, the ROV made the 3 m depth control and 150° heading control at the same time in disturbance. Comparisons have been made on the value of control parameter $K_{S}$ . From these comparisons, $K_{S} = diag (3.5, 3.5, 3.5, 3.5, 3.5, 3.5)$ tends to be relatively better in the simultaneously depth and heading control in the disturbance environment. During the control operation, the tether drag forces changed along with angular and diving acceleration. As the tether immersed part was lengthening along with ROV descending, the waves and currents action length was increasing at the same time, the tether drag forces became greater and more fluctuant. Both S surface controller and S surface–based adaptive recurrent WNN controller can realize precise control since the tether disturbance was relatively small and invariable. As a result, the proposed controller would make appropriate reaction to approach the target position. During transient time, the tether drag forces caused greater overshoot and longer regulating time in S surface heading control. In the steady state, the upper and lower bounds of depth and heading control are $\pm 0.1$ m and $\pm 3^{\circ}$ , respectively (see Figure 7(b)). The variation of tether drag forces not only caused greater errors but also generated negative pitch moment which caused the ROV depth oscillation for about 100 s.

Figure 7.

Analysis and comparison in the simultaneous depth and heading control in disturbance: (a) simultaneous depth and heading control in disturbance environment, (b) control errors in disturbance in disturbances environment, and (c) tether forces and moment on the ROV end in disturbance environment.

The path following is extremely important for ROV to accomplish precise inspection missions continuously. To testify the controller robustness, the JONSWAP spectrum irregular wave was generated as unknown disturbance during the experiments. In the path following experiments of Figure 8, the vehicle obtains its positions through EKF-based dead reckoning. Through the comparisons between the current position and the desired position, the proposed controller makes the vehicle approach and follows the path precisely. Different control parameters of proposed controller have been discussed on $K_{s}$ , $b_{u}$ , and $δ$ . In Figure 8(a), the ROV made path following on quadrangle, with different parameter of $K_{s}$ . In Figure 8(b), the ROV made path following on “Z” shape curve, with different parameter of $K_{s}$ , $b_{u}$ , and $δ$ . In Figure 8(c), the ROV made 3D screw path following with different parameter of $b_{u}$ and $δ$ . During the path following operation, the proposed S surface–based adaptive recurrent neural network controller followed the desired path more precisely than the S surface controller. From the comparisons of parameters, the best parameter group is $K_{S} = diag (3.5, 3.5, 3.5, 3.5, 3.5, 3.5)$ , $b_{u} = 0.1$ , $δ = 0.05$ , $K_{D} = K_{C} = diag (9.5)$ , $K_{W 4} = diag (1.2)$ , and $K_{W 2} = diag (10.5)$ for path following in disturbances environment. It adapted to angular changes more quickly, particularly at the angular turning point. The S surface controller cannot make a correspondent reaction control force against the control force, which leads to unacceptable following errors. Through the comparisons of different parameters in the robust term, the proposed S surface–based adaptive recurrent neural network controller does not only testify its robustness but also manifest its accurate path following capabilities in disturbance environment.

Figure 8.

Analysis and comparison of 3D path following experiments: (a) path control in disturbance environment, (b) Z shape following in disturbance environment, and (c) 3D screw path following in disturbance environment.

Case study 2: vehicle-manipulator coordinate control

Figures 8 and 9 illustrate the coordinated vehicle and manipulator experiments for the ROV station keeping during operation. The shoulder and elbow joints rotated simultaneously, while the SY-II ROV keeps its attitude and position with the designed controller. This kind of control is very important for the high-efficiency hovering manipulation in remote or autonomous operations. During the experiments, the angular velocities of the joints were set as 0.35 rad/s. The instant acceleration of the joints is 1 rad/s². Disturbances from the manipulator revealed that the coupling forces increased suddenly at the joint acceleration moment, while the restoring forces were dominant in the steady state with the manipulator pitching at uniform speed. The designed SARWNN controller can estimate the disturbances from the manipulator and make compensation at the same time, while the S surface controller can only compensate the vehicle motion according to its deviations. Therefore, the designed SARWNN controller can obtain more steady and accurate vehicle manipulator coordinate motions.

Figure 9.

Vehicle-manipulator coordinate control experiments for the SY-II ROV station keeping operation: (a) observed disturbance including coupling forces and restoring forces, (b) 3D position of the vehicle, and (c) attitude of the manipulator and the vehicle.

Conclusion

The multibody system dynamic model of ROVMS has been established including the vehicle, tether, and manipulator. Disturbances from the tether and manipulator have been systematically analyzed in order to transform disturbances into the observed forces. The manipulator disturbance on the vehicle mainly includes the coupling forces and the restoring forces.

In order to approximate the dynamic model with parametric uncertainties, a recurrent WNN structure has been proposed. In combination with the fuzzy control concept, a novel SARWNN controller has been proposed on the control of ROVMS. The robust function improves the system robustness and convergence, which is testified through irregular waves in the path following experiments. In the 3D path following and vehicle-manipulator coordinate control experiments, the proposed controller greatly outperforms S surface control.

Future work will apply the proposed control approach in the ROV submarine operations in order to improve the precision and speed of human machine coordinate manipulation.

Footnotes

Handling Editor: Jose Ramon Serrano

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 project was supported by National Natural Science Foundation of China (Nos 61633009, 51579053, 5129050, 51779052, and 51779059), Major National Science and Technology Project (2015ZX01041101), the Key Basic Research Project of “Shanghai Science and Technology Innovation Plan” (No. 15JC1403300), and the Field Fund of the 13th Five-Year Plan for the Equipment Pre-research Fund (No. 61403120301). All these supports are highly appreciated.

ORCID iD

Guocheng Zhang

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