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Abstract

This paper mainly studies the stability of a sea cucumber adsorption robot (SCAR) under external disturbances and parameter uncertainty. As a fishing robot with a suction effect, a dynamic model of its vertical operation is established, taking into account the mechanical structure of its suction port and the surrounding flow conditions. Specifically, robust model predictive control (RMPC) is adopted to ensure that the device maintains excellent stability in complex underwater environments. Considering the simplification of the model structure and real-time control, a discrete nominal model with the introduction of a feedback correction mechanism is designed to be close to the actual system. In the process of control design, the upper bound of robot speed and propeller saturation have been taken into account, and the optimization functions to reduce input consumption of the propellers are also provided at the same time. Furthermore, the feasibility of the recursive structure and the stability of the closed-loop structure are proved. Importantly, the external factors including disturbance sources obtained from pool experiments are subjected to our consideration. In total, the simulation results and comparative analysis are evidenced by the practicality of the designed vehicle and the effectiveness of the established methodology.

Keywords

SCAR RMPC dynamic model external disturbance

Introduction

The oceans occupy Earth’s largest area of space, and scholars have never stopped exploring it.¹ As a flexible and efficient underwater device, the remotely operated vehicle (ROV) plays a crucial role in the exploration and development of marine resources.² With the development of science and technology, the use of underwater robots instead of manual fishing of bottom-seeded marine organisms has become a research hotspot in the field of fishery, with the most nutritious sea cucumber as the main one.^3,4 Restricted by the characteristics of its living environment, it is necessary to require the sea cucumber adsorption robot (SCAR) have better stability and anti-disturbance.⁵

Accurate dynamics models are established as the basis for the system design as well as key issues such as anti-disturbance control.⁶ In Ref.⁷, the model of the designed fishing robot is established through dynamic analysis. However, the force situation of the robot system structure is not considered. Reference⁸ adopts visual positioning-based target acquisition for suction and capture devices, and models them based on the feedback information, but the existence of delay errors cannot be ignored. Reference⁹ has designed a new type of controller based on self-adjusting adaptive fuzzy logic, which can be modeled through fuzzy logic networks without accurately understanding system dynamics. However, the proposed controller requires speed measurement except for position measurements.

In addition, it is indispensable to realize the control of complex SCAR systems. As a classical control method, PID is widely used to combine with new control strategies. Reference¹⁰ proposes an optimal bio-inspired PID controller to make vehicles robust to external disturbances. However, this offline control method only improves the stability of the system by adjusting parameters and reducing interference. Reference¹¹ combines radial basis function neural networks and adaptive control techniques into sliding mode control, making the system tracking error converge to zero in a finite time. Nevertheless, when the control trajectory of sliding membrane variable structure control reaches the vicinity of the sliding membrane surface, chattering often occurs, which increases the system adjustment time. References^12,13 applies model model-free adaptive control method to ROV depth control, but due to the high computational cost, the nonlinear system model of the robot is replaced by a dynamic linearized time-varying model. A disturbance rejection based on an adaptive neural network controller is designed by Ref.¹⁴ to control underwater robots with disturbance, but the selection of the controller parameters should be done very carefully. Otherwise, it may lead to an enormous higher value of the model uncertainties.¹⁵

Although the above methods achieve motion control of the target object, it is still necessary to comprehensively consider the dynamically changing characteristics such as parameters perturbations and external disturbances that the system may be subjected to under different operating environments.^16,17 Model predictive control (MPC) provides an online solution for optimal control strategies of MIMO-constrained systems such as underwater robots.¹⁸ It relies heavily on the controlled system model to predict future states, which makes it vulnerable to unknown external disturbances and modeling errors.¹⁹ Large external disturbances and model uncertainties can reduce or even destabilize the controlled system.²⁰ Therefore, it is crucial to adopt a robust control method with strong robustness to deal with the effects of coupled nonlinearities in the control model of the underwater vehicle as well as the system’s own immunity to disturbances.

In summary, since the suction and capture mechanism of SCAR is the key mechanism to achieve the acquisition and delivery of sea cucumbers, combining its working principle and structural design, establishing a dynamic model with realistic and high capture rate is the main challenge. In addition, based on the fact that SCAR operates in a complex underwater environment, as well as the nonlinearity and high motion redundancy of the system model, it is also one of the difficulties to realize the problem of anti-disturbance under limited control inputs and state constraints.

To address the above challenges, this paper establishes a mathematical model to describe the system behavior based on the dynamic characteristics of SCAR, which is tailored to its own structure. The designed robust model predictive control (RMPC) method calculates the state change of the system in the future time through the measured value of the system state at the current moment and the estimated value of the control input. Optimization of the control inputs is achieved and a control strategy is implemented to regulate the behavior of the system. Finally, the sequence of control inputs is continuously and iteratively updated to adapt to the dynamic changes and external perturbations of the system. Specifically, the excellent contributions in this work can be written as:

Innovatively, an open-shelf sea cucumber adsorption robot (SCAR) with a capture function is designed, and based on that the system dynamics and kinematics model are established. Besides, the mechanical structure model of the robot suction and trap structure are considered, as well as the force analysis is conducted. This makes the model of the entire SCAR system more accurate and improves control accuracy.

External disturbances and parameter uncertainties are introduced into the system model and controlled by RMPC to achieve better results. Additionally, the system input and state constraints to achieve as little energy consumption as possible are considered.

The pool experiment obtained disturbance data under real operating conditions, the feasibility and effectiveness of RMPC are verified on this basis. The results demonstrate that the overshoot of RMRC is reduced by $10 % - 30 %$ and the settling time is shortened by $10 s - 17 s$ in experiments. And a mass of simulation experiments have revealed its prominent control effect and robust performance.

The rest of the paper is organized as follows: The dynamic model of the SCAR system as well as the kinematic model are introduced, taking into account the structural model of system uptake. The design approach of RMPC and the analysis of its feasibility and stability are described. Finally, the simulation results are confirmed and the paper is summarized.

Establishment and analysis of SCAR mathematical model

When the SCAR moves underwater, it will be disturbed by the external marine environment in addition to its complex nonlinearity and the mutual coupling between degrees of freedom.²¹ These factors will act on the robot’s body, have a certain impact on its motion posture, and position, resulting in lower underwater motion performance and fishing efficiency. Therefore, it is necessary to further explore the movement law of the robot under the influence of external forces through mathematical models.²² And further lay the foundation for the control research on the stability of system in the following paper.

Kinematic model

In order to analyze the motion attitude and anti-disturbance capability of the SCAR, a coordinate system is established to describe the robot’s motion based on this. According to the system recommended by the International Towing Tank Conference (ITTC) and the Society of Naval Architects and Marine Engineers (SNAME),²³ the paper establishes the relevant coordinate system as shown in Figure 1.

Figure 1.

The overall structure of the SCAR with corresponding coordinates.

According to the right-hand rule, two kinds of coordinates are established: One is a fixed coordinate system $E - ξ η ζ$ , which is fixed to a point on the earth and does not move with the SCAR. The other is the motion coordinate system $O (G) - xyz$ , which is fixed to a point of SCAR and moves together with SCAR. The relative position and angle of the SCAR moving coordinate system and fixed coordinate system can be used to describe the specific position and real-time attitude of the robot. The coordinates ( $x$ , $y$ , $z$ ) of the origin $O$ of the motion coordinate system in the fixed coordinate system are the positions of SCAR in the water body. The angular coordinates of each axis of the moving coordinate system relative to the fixed coordinate system ( $φ$ , $θ$ , $ψ$ ) represent the attitude information of SCAR in the water body. They respectively represent the roll angle, pitch angle, and heading angle of the robot body, which are positive when they rotate to the right.

The linear velocity and angular velocity of the origin $O$ of the SCAR motion coordinate system can be expressed in the form of a matrix under a fixed coordinate system as: $V = (V_{1}, V_{2})^{T}$ , where $V_{1} = (u, v, w)^{T}$ is the linear velocity matrix of SCAR motion, they represent the longitudinal, transverse and vertical linear velocity of SCAR respectively. And $V_{2} = (p, q, r)^{T}$ is the angular velocity matrix of robot motion, they represent the roll, pitch, and heading angular velocity of SCAR respectively. The force of SCAR in the moving coordinate system can be expressed in the form of matrix: $F = (F_{1}, F_{2})^{T}$ , where $F_{1} = (X, Y, Z)^{T}$ is the force matrix, and $F_{2} = (K, M, N)^{T}$ is the matrix of moment of robot. The $X, Y, Z$ represents the lateral force, longitudinal force, and vertical force of SCAR respectively, and the $K, M, N$ represents the rolling moment, pitching moment, and yaw moment of SCAR respectively.

The SCAR has certain linear and angular velocities relative to the fixed coordinate system during the motion, and the spatial position of the SCAR depends on the three coordinate components of the origin $G$ of the kinematic coordinate system in the fixed coordinate system, and the attitude of the SCAR depends on the angle between the three axes of the kinematic coordinate system and the three axes of the fixed coordinate system. The fixed coordinate system can coincide with the motion coordinate system after three rotations, so the coordinate conversion matrix can convert the motion coordinate system to the fixed coordinate system and obtain the specific coordinate conversion relationship between the linear velocity in the motion coordinate system and the fixed coordinate system of SCAR as follows:

\begin{matrix} [\begin{matrix} \overset{\cdot}{x} \\ \overset{\cdot}{y} \\ \overset{\cdot}{z} \end{matrix}] = \\ [\begin{matrix} \cos θ \cos ψ & - \cos φ \sin ψ + \sin θ \sin φ \cos ψ \\ \cos θ \sin ψ & \cos φ \sin ψ + \sin θ \sin φ \cos ψ \\ - \sin θ & \sin φ \cos θ \end{matrix} \begin{matrix} \sin ψ \sin φ + \sin θ \cos ψ \cos φ \\ - \sin ψ \sin φ + \sin θ \cos ψ \cos φ \\ \cos φ \cos θ \end{matrix}] V_{1} \end{matrix}

(1)

Similarly, according to the relationship between Euler angular velocity and the angular velocity of SCAR motion coordinate system, the angular velocity conversion matrix can be obtained. Through coordinate transformation, the specific coordinate transformation relationship between the angular velocity in the SCAR motion coordinate system and the fixed coordinate system is obtained as follows:

[\begin{matrix} \overset{\cdot}{φ} \\ \overset{\cdot}{θ} \\ \overset{\cdot}{ψ} \end{matrix}] = [\begin{matrix} 1 & \sin φ \tan θ & \cos φ \tan θ \\ 0 & \cos φ & - \sin φ \\ 0 & \sin φ \cos θ & \cos φ \cos θ \end{matrix}] V_{2}

(2)

Dynamic model and force analysis

Due to SCAR is a complex nonlinear system, we calculate the relationship between its expected motion and the force exerted by the actuator is calculated through the dynamic model of the robot. This paper considers the mathematical model of the structure and hydrodynamic complexity of the suction and capture device, effectively coupling the components of the underwater suction and capture device, making it a unified system.

According to the Refs.^24,25, the underwater robot motion is a 6-DOF space motion. The spatial motion can be represented by linear motion and rotational motion of three coordinate axes ( $x$ , $y$ , $z$ ) in dynamic coordinates. The coordinates of point $O$ and point $G$ of gravity center of SCAR body are usually ( $x_{G}$ , $y_{G}$ , $z_{G}$ ). With²³ and in combination with the vertically downward suction and capture the state of the robot during the fixed-point adsorption of sea cucumber. The expressions of its forces and moments along the $O - Z$ axis of the ground coordinate system can be given under the action of the vertical thrusters:

\begin{array}{l} X = m [(\dot{u} - v r + w q - x_{G} (q^{2} + r^{2}) + y_{G} (p q - \dot{r}) + z_{G} (p r + \dot{q})] \\ Z = m [(\dot{w} - u q + v q + x_{G} (p r - \dot{q}) + y_{G} (q r + \dot{p}) - z_{G} (p^{2} + q^{2})] \\ M = I_{y} \dot{q} + (I_{x} - I_{z}) p r + m [x_{G} (\dot{w} - u q + v p) - z_{G} (\dot{u} - v r + w p)] \end{array}

(3)

In addition to the physical meaning of the parameters mentioned above, $m$ is the mass of SCAR, $I_{x, y, z}$ is the inertia moment of the corresponding coordinate axis around which the vehicle rotates. It can be seen that the model of the underwater vehicle is indeed cross-coupled. Traditional control methods or linear control theory make it difficult to accurately express the state changes of the controlled object. The physical model parameters and hydrodynamic parameters of underwater vehicles can be obtained by using ANSYS and SolidWorks.

The force and moment of each part on the left side of the (3) are analyzed, mainly including the static force (moment) generated by the combined action of gravity $G$ and buoyancy $B$ of the SCAR body in the water body; thrust (torque) $T_{i}$ generated by thruster; hydrodynamic force (torque) $F_{h}$ generated by SCAR moving in the fluid. As well as the reaction force $F_{s}$ produced by the special suction and capture device of the robot in motion. Because of the cable length of SCAR designed in this paper is long enough and light, the cable force on the SCAR can be ignored.²⁶ In order to establish an accurate SCAR motion equation, the above forces and moments must be accurately analyzed and calculated. The thruster distribution of SCAR adopts a symmetrical and regular arrangement. According to this arrangement, the main thruster is fixed on both sides of the frame, which reduces the kinetic energy consumption efficiency of the thruster during operation.²⁷

Due to the attitude stability of its suction and capture process is mainly studied in this paper. Therefore, the vertical thruster ( $T_{1} ~ T_{4}$ ) is mainly used to study the change of each state during the downward suction process of the robot. Therefore, the thrust (torque) generated by the vertical thruster can be expressed as:

[\begin{matrix} X_{T} \\ Z_{T} \\ M_{T} \end{matrix}] = [\begin{matrix} 0 \\ T_{1} + T_{2} + T_{3} + T_{4} \\ (T_{1} + T_{2} - T_{3} - T_{4}) x_{v} \end{matrix}]

(4)

where $x_{v}$ represents the distance between the vertical thruster relative to the origin of the motion coordinate system. The thrust of the feeder can be obtained by $T_{i} = ρ n^{2} D^{4} K_{T}$ , where $ρ$ refers to water density, $n$ is propeller’s speed, $D$ is propeller’s diameter, $K_{T}$ is thrust coefficient.²⁸ Besides, according to the force analysis, the static force generated by SCAR under the combined action of gravity $G$ and buoyancy $B$ can be expressed as follows:

\begin{matrix} G + B = [\begin{matrix} - (G - B) \sin θ \\ (G - B) \cos θ \cos ψ \\ - (z_{G} G - z_{c} B) \sin θ - (x_{G} G - x_{c} B) \cos θ \cos ψ \end{matrix}] \end{matrix}

(5)

For the SCAR in this paper, in addition to the mechanical structure of the conventional underwater robot. It is also necessary to study the motion of the whole robot by analyzing the mechanics of its suction and capture device. As the suction and capture port is a cylindrical structure when the current flows through it, there will be different flow separation phenomena on its surface.²⁹ The flow separation will not only increase the resistance of the cylinder but also interfere with the flow separation behind different cylinders, which will affect the vortex shedding period and the transverse pulsating load generated in the wake.³⁰ This paper considers the relationship between the force and the system state vector under the horizontal and vertical forces respectively. The main flow around SCAR is shown in Figure 2.

Figure 2.

Velocity cloud of incoming flow around the ROV suction trap in the corresponding horizontal and vertical direction, respectively: (a) horizontal direction and (b) vertical direction.

Here, the inflow velocity of the robot is 3 knot, 1.5 m/s in total. It can be seen from the above figure that the rear part of the suction trap is connected to the robot body, which makes the fluid asymmetric when passing. In this case, the flow around a circular cylinder belongs to the flow around a circular cylinder. It is mainly composed of uniform flow, dipole flow, and point vortex, and the superimposed complex potential at $z$ is:

M (z) = v_{\propto} z + \frac{M}{2 π} z + \frac{Ω}{2 π i} l n z

(6)

where $v_{\propto}$ is the velocity of uniform flow, $M = 2 π v_{\propto} {r_{0}}^{2}$ is the dipole moment of dipole flow, $r_{0}$ is the cylinder radius, and $Ω$ is the point vortex intensity. For $M (z)$ , we use polar coordinates to separate the real and imaginary parts of the complex potential of circular flow around a cylinder:

\begin{matrix} W_{z} = v_{\propto} rcos θ (1 + \frac{{r_{0}}^{2}}{r^{2}}) + \frac{Ω}{2 π} θ + i [v_{\propto} rsin θ (1 - \frac{{r_{0}}^{2}}{r^{2}}) - \frac{Ω}{2 π} lnr] \end{matrix}

(7)

The partial derivative of the real part of $W_{z}$ can obtain the velocity component of $p (r, θ)$ at any point in the flow field as follows:

{\begin{matrix} v_{r} = v_{\propto} (1 - \frac{{r_{0}}^{2}}{r^{2}}) \cos θ \\ v_{θ} = - v_{\propto} (1 + \frac{{r_{0}}^{2}}{r^{2}}) \sin θ + \frac{Ω}{2 π r} \end{matrix}

(8)

By listing the Bernoulli Equation at infinity and at a point on the cylindrical surface of the suction trap, we can get:

P = P_{\propto} - \frac{1}{2} ρ {v_{\propto}}^{2} - \frac{1}{2} ρ (- {v_{r}}^{2} + {v_{θ}}^{2})

(9)

The pressure distribution $P$ on the cylindrical surface can be obtained by taking the velocity on the cylindrical surface obtained from (7) into (9). Considering the structure of the suction and capture device, we select the most representative horizontal and vertical planes for analysis, ignoring the smaller lateral force-bearing surface. And taking a segment of microelement line $ds$ on the cross-section of the suction and capture device, and the force on it is $ds$ , that is, the components $d F_{h}$ and $d F_{v}$ of d along the horizontal and vertical planes are obtained. We integrate along the cylindrical surface of the whole suction and capture device, and obtain the expression of the corresponding force in each direction:

{\begin{matrix} F_{h} = - \int_{0}^{2 π} {P_{\propto} + \frac{1}{2} ρ [{v_{\propto}}^{2} - {(- 2 v_{\propto} \sin θ + \frac{Ω}{2 π r_{0}})}^{2}]} r_{0} \cos θ d θ \\ F_{v} = - \int_{0}^{2 π} {P_{\propto} + \frac{1}{2} ρ [{v_{\propto}}^{2} - {(- 2 v_{\propto} \sin θ + \frac{Ω}{2 π r_{0}})}^{2}]} r_{0} \sin θ d θ \end{matrix}

(10)

System model

Regardless of the roll of the device, we can obtain the longitudinal and vertical dynamic equations of the SCAR in the $O - Z$ plane by combining formula (3), (4), (5) and (10), relying on the action of the vertical auxiliary thruster, as follows:

\begin{array}{l} m (\dot{u} + z_{G} \dot{q}) = \frac{1}{2} ρ L^{4} [X_{q q^{'}} q^{2} + X_{r r^{'}} r^{2} + X_{r p^{'}} r p] \\ + \frac{1}{2} ρ L^{3} [X_{u^{'}} \dot{u} + X_{v r^{'}} v r + X_{w q^{'}} w q] \\ + \frac{1}{2} ρ L^{2} [X_{u u^{'}} u^{2} + X_{v v^{'}} v^{2} + X_{w w^{'}} w^{2}] \\ - (G - B) s i n θ + F_{h} \end{array}

(11)

and

\begin{matrix} m (\overset{\cdot}{w} + z_{G} \overset{\cdot}{q}) = \frac{1}{2} ρ L^{4} [Z_{qq}' q^{2} + Z_{rr}' r^{2} + Z_{rp}' rp] \\ + \frac{1}{2} ρ L^{3} [Z_{w}' \overset{\cdot}{w} + Z_{vr}' vr + Z_{q}' uq + Z_{w | q |}' w | q |] \\ + \frac{1}{2} ρ L^{2} [Z_{uu}' u^{2} + Z_{w}' uw + Z_{w | w |}' w | w |] \\ + (G - B) \cos θ \cos ψ + Z_{T} + F_{v} \end{matrix}

(12)

This research focuses on the control of the robot’s longitudinal suction process, so there is still a certain steering movement in the system. We decompose space motion into horizontal plane motion and vertical plane motion. The center of gravity $G$ of the underwater robot coincides with the origin coordinate $O$ . The trim moment equation of the underwater vehicle in the motion coordinate system is obtained, which is written as:

\begin{matrix} I_{y} \overset{\cdot}{q} - m (x_{G} \overset{\cdot}{w} - z_{G} \overset{\cdot}{u}) = \frac{1}{2} ρ L^{5} [M_{q}' \overset{\cdot}{q} + M_{pp}' p^{2} + M_{rr}' r^{2} \\ + M_{rp}' rp + M_{q | q |}' q | q |] + \frac{1}{2} ρ L^{4} [M_{w}' \overset{\cdot}{w} + M_{vr}' vr + M_{uq}' uq \\ + M_{q | w |}' q | w |] + \frac{1}{2} ρ L^{3} [M_{u}' u^{2} + M_{w}' uw + M_{v | w |}' v | w |] \\ - (z_{G} G - z_{c} B) \sin θ - (x_{G} G - x_{c} B) \cos θ \cos ψ + M_{T} \end{matrix}

(13)

Since the SCAR is moving vertically downward at a uniform velocity after deployment and before preparation, the cross-coupling terms and smaller hydrodynamic parameters of the horizontal and vertical degrees of freedom can be neglected. Depending on the layout of the robot’s thrusters, the control produces negligible residual angular velocities when the speed of the thrusters is not too fast. Finally, the (11), (12) and (13) can be written in the following discrete form:

x (k + 1) = x (k) + A x (k) dt + B u (k) dt

(14)

where

\begin{matrix} A = \frac{1}{2} \\ [\begin{matrix} ρ L^{2} X_{uu}' u + ρ Ω & ρ L^{2} ({LX}_{wq}' q + X_{ww}' w) \\ ρ L^{2} ({LZ}_{q}' q + Z_{uu}' u) & ρ L^{2} ({LZ}_{w | q |}' | q | + Z_{w | w |}' | w |) \\ ρ L^{3} ({LM}_{uq}' q + M_{u}' u) & ρ L^{3} M_{w}' u \end{matrix} \begin{matrix} ρ L^{4} X_{qq}' q \\ ρ L^{4} Z_{qq}' q \\ ρ L^{3} (L^{2} M_{q | q |}' | q | + M_{q | w |}' | w |) \end{matrix}], \\ B = [\begin{matrix} 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ x_{v} & x_{v} & - x_{v} & - x_{v} \end{matrix}], \end{matrix}

with $x (k) = (u, w, q)$ , $u$ represents the longitudinal linear velocity of the SCAR, the vertical linear velocity of the capture is $w$ , and $q$ represents the pitch angular velocity of the SCAR. The $u (k)$ is the thrust produced by four vertical thrusters ( $T_{1} ~ T_{4}$ ). In addition, the pitch angle can be expressed as $\overset{\cdot}{φ} = p + qsin φ \tan θ + rcos φ \tan θ$ .

In normal operation, the system can ignore all non-vertical parameters coupling and variables, so as to achieve the control effect of small errors. However, in the complex marine environment and the state of frequent flow around, the system stability may be affected. For this reason, we have to consider the uncertainties, nonlinear terms, and other factors in the system. In addition, this paper also adds the relevant underwater disturbance term $w$ , whose specific superposition state and manifestation will be described in detail in a later section.

Design and analysis of RMPC

Owing to the special characteristics of the operation mode and working environment of the device designed in this study, in order to be able to control its smooth operation more accurately. In this section, a RMPC method based on the system architecture is adopted and the corresponding controller is designed. Ultimately, the integration of the control method with SCAR is realized and the robustness of the system against internal uncertainties and external disturbances is improved.

Method

For system (14), due to the nature of the underwater work of the SCAR, the interference in the form of external currents should be taken into account, and we give the effect $Gs (k)$ of the current uncertainty on the system at time step $k$ . In addition, although the relevant physico-dynamic as well as hydrodynamic parameters with be identified through the scheme, there are still uncertainties present in the dynamic parameters, here denoted by $δ$ . Taking into consideration the factors, we discretize and convert it into a general model as follows:

\begin{matrix} x (k + 1) = F (x (k), u (k)) + Gs (k) \\ = f (x (k), u (k)) + δ (x (k), u (k)) + Gs (k) \end{matrix}

(15)

where $x (k) = (u_{k}, w_{k}, q_{k})^{T} \in R^{3}$ is the state variable selected in the process of SCAR motion, $u (k)$ is the control input generated by $T_{1} ~ T_{4}$ , $δ (k)$ represents model uncertainty, $Gs (k)$ represents external disturbance. Here, system (15) should meet their constraints $x \in X$ , $u \in U$ . For the uncertainty in the model, i.e $δ (0, 0) = 0$ , $∥ δ ∥ \leq ρ ∥ x ∥ + κ ∥ x ∥$ . In order to ensure that the uncertainty is manageable, the $ρ > 0$ and $0 \leq κ < 1$ need to be made. According to the actual operation scenario and working requirements of SCAR designed in this paper, the disturbance $s$ here should meet the boundary condition, namely $s \in S$ . In this case, we can consider making $Dw = δ + Gs = {δ + Gs | \forall x \in X, \forall u \in U .}$ . Equation (15) can be rewritten as:

x (k + 1) = f (x (k), u (k)) + Dw (k)

(16)

Furthermore, we consider that several common underwater tasks (e.g. fixed-point adsorption and underwater detection) require the SCAR to move relatively slowly. Under these circumstances, the velocity vector generated by the thrusters can therefore be set as $(V_{h} V_{l})$ , where $V_{h, l} = (u_{h, l}, w_{h, l}, q_{h, l})^{T}$ . These requirements are met by the system’s state constraint set $X$ , which is given by $x \in X$ . It consists of the following constraints:

\begin{matrix} u_{h} - | u_{l} | \geq 0, w_{h} - | w_{l} | \geq 0, q_{h} - | q_{l} | \geq 0 \end{matrix}

(17)

Similarly, for the force or moment $(τ_{T 1} ~ τ_{T 4})$ generated during the operation of the vertical thrusters $T_{1} ~ T_{4}$ , we can limit them to a range as a way to ensure minimum input consumption. The control input constraint set $U$ as follows:

τ_{k} = {[τ_{T 1 k} τ_{T 2 k} τ_{T 3 k} τ_{T 4 k}]}^{T} \in U

(18a)

| τ_{T 1 k} | \leq {\bar{τ}}_{T 1}, | τ_{T 2 k} | \leq {\bar{τ}}_{T 2}, | τ_{T 3 k} | \leq {\bar{τ}}_{T 3}, | τ_{T 4 k} | \leq {\bar{τ}}_{T 4}

(18b)

where we obtain, based on the above constraints, $∥ τ_{Tik} ∥ \leq \bar{U}$ , with $\bar{U} = \sqrt{{\bar{τ}}_{T 1}^{2} + {\bar{τ}}_{T 2}^{2} + {\bar{τ}}_{T 3}^{2} + {\bar{τ}}_{T 4}^{2}}$ and ${\bar{τ}}_{Ti} \geq 0 (i = 1, 2, 3, 4)$ .

It is not difficult to realize that (16) is the actual dynamical equation of the system, which contains the parameter uptake inside the system as well as the vector of perturbations to which it is subjected externally. Therefore, we consider (14) as a nominal model of the system, where no disturbances are taken into account, and transform it into a more general form, denoted as:

\begin{matrix} z (k + 1) = Az (k) + Bv (k) \end{matrix}

(19)

with $z (0) = x (0)$ , the matrices $A, B$ are known with proper dimensions, and $(A, B)$ is controllable and stabilizable. At this time, the state sequence of the system can be expressed as:

\begin{matrix} Z (k) = ϒ z (k) + Φ V (k) \end{matrix}

(20)

where $ϒ = [\begin{matrix} A \\ A^{2} \\ ⋮ \\ A^{N} \end{matrix}]$ , $Φ = [\begin{matrix} B & 0 & . . . & 0 \\ AB & B & . . . & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A^{N - 1} B & A^{N - 2} B & . . . & B \end{matrix}]$ , the corresponding predicted state sequence $Z (k)$ and input state sequence $V (k)$ can be described as: $Z (k) = [\begin{matrix} z^{T} (1 | k) \\ z^{T} (2 | k) \\ ⋮ \\ z^{T} (N | k) \end{matrix}]$ , $V (k) = [\begin{matrix} v^{T} (0 | k) \\ v^{T} (1 | k) \\ ⋮ \\ z^{T} (0 | k) \end{matrix}]$ , where $z^{T} (i | k)$ and $v^{T} (i | k)$ represent the state vector and input vector predicted at the $i$ -th step for the $k$ -th time, respectively, $i = 0, 1, 2, . . . N$ . In order to make the motion of the SCAR more stable and reduce the complexity of calculation, in the actual suction task, it is necessary to reduce the three-axis rotation. Therefore, we make the velocity rotation matrix and coefficient matrix remain unchanged at the forecast stage.

For the nominal system (19), consideration of optimization problem is usually used to solve the problem of adjusting the current and future input of the system to minimize the predicted performance cost.³¹ Thus, the cost function $J$ is constructed, as follows:

\begin{matrix} J (k) = ∥ z (N | k) ∥_{P}^{2} + ∥ v (N - 1 | k) ∥_{R}^{2} \\ + \sum_{i = 1}^{N - 1} (∥ z (i | k) ∥_{Q}^{2} + ∥ v (i - 1 | k) ∥_{R}^{2}) \\ = Z^{T} (k) Q Z (k) + V^{T} (k) R V (k) \\ = {(ϒ z (k) + Φ V (k))}^{T} Q (ϒ z (k) + Φ V (k)) \\ + V^{k} (k) R V (k) \\ = z^{T} (k) ϒ^{T} Q ϒ z (k) + 2 z^{T} (k) ϒ^{T} Q Φ V (k) \\ + V {(k)}^{T} (Φ^{T} Q Φ + R) V (k) \end{matrix}

(21)

where $J (k)$ is a function of V(k), so the optimization problem of the RMPC can be expressed as follows:

V^{*} (k) = \arg \min_{V (k)} J (k)

(22)

s.t.

\begin{matrix} z (i | k) \in Z, v (i - 1 | k) \in V, \forall i = 1, 2, . . ., N . \\ z (N | k) \in Z_{f} \end{matrix}

(23)

where $Q = diag (Q, Q, . . . Q, P)$ , $R = diag (R, R, . . . R)$ , $Q$ and $R$ is the symmetric positive definite unit matrix and a unit vector corresponding to the compact format. And the terminal weights $P$ satisfies $P - {(A - BK)}^{T} P (A - BK) = Q + K^{T} RK$ , and $K$ satisfies the sufficient condition for the stability of discrete systems to ensure that all eigenvalues of the system state expression are located in the unit circle, that is, $| eig (A - BK) | < 1$ to ensure that the sum of infinite terms is finite. The $V^{*} (k)$ is the best input sequence, $Z$ and $V$ are a contractive constraint, the main purpose is to ensure that the two sets are smaller than the sets $X$ and $U$ , so as to ensure the stability of the real model, and $Z_{f}$ is the terminal set, so as to ensure the stability of the nominal model.

As for this contractive constraint, based on the parameters in the state space of the system and combined with the conditions to ensure the stability of SCAR, $| eig (A_{K}) | < 1$ can be obtained, so that there is a constant $N_{c}$ satisfies $A_{K}^{N_{c}} \subset α D W$ , where $A_{K} = (A - BK)$ , $0 \leq α < 1$ . Since SCAR is in the process of being controlled, the whole system tends to a stable state, so that the pitch angle is in the process of gradually decreasing, so $0 < A_{K}^{N_{c}} < 1$ . It then follows that

\begin{matrix} Γ = \sum_{i = 1}^{\infty} A_{K}^{i - 1} D W = Γ_{N_{c}} + \sum_{i = N_{c} + 1}^{\infty} A_{K}^{i - 1} D W \\ = Γ_{N_{c}} + \sum_{i = 1}^{\infty} A_{K}^{i - 1} A_{K}^{N_{c}} D W \end{matrix}

(24)

where $Γ_{N_{c}}$ represents the range of constraints on the system before item $N_{c}$ , so the subsequent summation terms start at 1. Since the sum of each term in $A_{K}^{i - 1}$ is larger than the previous $N_{c}$ , we can get $Γ \subset {(1 - α)}^{- 1} Γ_{N_{c}}$ according to (24).

The nominal model is implemented in a receding horizon scheme, at time $k$ the first optimal control action is implemented: $v (k) = v^{*} (0 | k) = (I_{p \times p}, 0, 0 . . ., 0) V^{*} (k)$ . At time $k + 1$ , repeat the above procedures and implement the first control action. It is clear that the closed-loop nominal system satisfies recursive feasibility and closed-loop stability, that is, $z (k) \to 0$ as $k \to \infty$ , which makes closed-loop nominal system (19) satisfy the feasibility of recursion and closed-loop stability and $z (k)$ , $v (k)$ satisfy their constraints.

Stability and feasibility analysis

The previous section indirectly realizes the optimal cost function and closed-loop stability of the ROV system by designing the nominal model. Here, we consider the error relationship between the real and the nominal system, then design the control law to ensure system stability, and verify method feasibility. The control law behind it is:

\begin{matrix} u (k) = v (k) - K (x (k) - z (k)) \end{matrix}

(25)

where $K$ means feedback gain and satisfies $| eig (A - BK) | < 1$ to ensure that the characteristic value is within the unit circle.

According to the (25) designed in this paper, the D-value between the actual and the nominal system is obtained. At this point, defining the error as: $e (k) = x (k) - z (k)$ , combined with formula (16) and (19), can be obtained by expansion:

\begin{matrix} e (k + 1) = x (k + 1) - z (k + 1) \\ = (A - BK) e (k) + Dw (k) \end{matrix}

(26)

the meaning of specific parameters has been mentioned above. Due to the fact that $(A - BK)$ is within the unit circle, the interference term $Dw (k)$ fluctuates within a bounded range owing to the characteristics of MPC. Obviously, we can infer that $e (k + 1)$ is exponentially stable in the global range, and indirectly obtain that $x (k)$ is bounded.

In the previous description, we have set the initial condition as $z (0) = x (0)$ . Combining with (26) to get:

\begin{matrix} e (k) = \sum_{i = 1}^{k} A_{K}^{i - 1} Dw (k - i) \end{matrix}

(27)

Since $w (k) \in W$ and $| eig A_{K} | < 1$ , it follows that $Dw \in D W$ , and

\begin{matrix} e (k) \in \sum_{i = 1}^{k} A_{K}^{i - 1} D W \subset \sum_{i = 1}^{\infty} A_{K}^{i - 1} D W \overset{Δ}{=} Γ \end{matrix}

(28)

where $Γ = \sum_{i = 1}^{\infty} A_{K}^{i - 1} D W$ is finite, such that $e (k)$ is bounded.

Since $e (k) \in Γ$ , it follows that $Ke (k) \in K Γ$ . In order to guarantee $\forall x \in X, \forall u \in U$ , the closed-loop nominal system should satisfy:

\begin{matrix} z \in Z = X - Γ, v \in V = U - K Γ \end{matrix}

(29)

and the terminal $Z_{f}$ should be an invariant set such that: $z \in Z_{f} \to A_{k} z \in Z_{f}$ and $u = - Kz - Ke \in K Z_{f} + K Γ \subset U$ . Finally it is clear that stability is guaranteed.

Results

In this section, pool experiments are designed to obtain real disturbance sources to validate the reliability of the method and device. Lots of simulations are also performed to verify the feasibility and effectiveness of the RMPC method for SCAR’s stabilization control.

Hardware platform and experiment preparation

The self-developed SCAR adopts a portable and expandable modular structure, with a volume of about $1.2 m \times 1.0 m \times 0.9 m$ and a weight of $64.54 kg$ . Considering the realities of sea cucumber fishing operations and the requirements for material and structural strength, the main frame of the SCAR is made of a non-metallic engineering plastic copolymer POM. In order to ensure the shore station operator monitors and control the underwater operation, the upper computer software development based on the WinForm development framework is carried out. At the same time, using low-coupling, universal layered design ideas, the lower computer software layered framework design, while the corresponding task scheduler is designed for different task characteristics. Finally, for the upper and lower computer data communication, scalable generalized communication protocol design. The overall architecture of the software system is shown in Figure 3.

Figure 3.

System block diagram of SCAR.

According to the different lengths of the suction objects and the effective water flow rate, the inner radius of the suction pipe $r_{0} = 60 mm$ are designed and we selected the optimal tension angle as $90 °$ . To increase the suction force at the mouth of the capture device as well as to counteract the interference torque of a single propeller on the motion of the underwater carrier, the capture source pipeline is designed as a $T$ -type hedge structure. Concretely, the internal driver of the propeller is a brushless motor controlled by pulse width modulation (PWM). The control quantities calculated by the controller are converted into signals which could be recognize by the motor, and are then transmitted to the vehicle. When the PWM is $0$ , the propeller does not operate. Figure 4. shows the experimental vehicle undergoing underwater operations.

Figure 4.

Overall schematic diagram of the operating platforms and experimental equipment.

All the simulations were completed on a computer equipped with Intel Core i7-8th Gen dual-core processor. The hydrodynamic analysis and system simulation were developed and simulated on ANSYS 2019.2 and Matlab R2022a. In addition, some control parameters selected in this paper are for the purpose of achieving an acceptable performance level, but not necessarily the optimal parameters. The specific parameters and values are shown in Table 1. Among them, the inertial hydrodynamic and viscous hydrodynamic parameters are the fitted hydrodynamic parameter values obtained by the SCAR system by performing translational and rotational movements in the simulation environment, so as to ensure the authenticity and validity of the parameters.

Table 1.

Notations in ROV.

Parameters	Value	Parameters	Value
$m$	$64.54 kg$	$X_{wq}^{'}$	$- 36.4 kg / s$
$ρ$	$10^{3} kg / m^{3}$	$Z_{uu}^{'}$	$- 23.1 kg / s$
$L$	$1.05 m$	$Z_{qq}^{'}$	$- 45.2 kg / s$
$x_{v}$	$0.6 m$	$Z_{w \| w \|}^{'}$	$50.2 kg / m$
$X_{ww}^{'}$	$- 32.0 kg / s$	$M_{w}^{'}$	$- 43.0 kg m^{2}$
$X_{qq}^{'}$	$- 41.6 kg / s$	$M_{uq}^{'}$	$- 47.1 kg m^{2}$
$X_{uu}^{'}$	$- 23.7 kg / s$	$M_{q \| q \|}^{'}$	$37.5 kg m^{2} / ra d^{2}$

Obtaining disturbance source experiments

For the traditional methods of disturbance acquisition, scholars generally take the approach of approximate estimation or hypothetical conjecture. To approximate the simulations of external disturbances and fit the SCAR system model, this paper has carried out several underwater experiments in a pool with a volume of $5 m \times 3 m \times 3 m$ . The disturbance in the form of water flow is caused by the reproduction of suction and capture scenarios to further obtain the real disturbance state under the corresponding operating conditions. Since the main operation mode of the SCAR is sucking and catching sea cucumbers, the stability of the pitching attitude is highly required:

A fixed-frequency control signal is provided to stimulate the vehicle to perform a complete cycle of suction and capture action. No manual commands or remote control operations are used to ensure data accuracy;

Recording the vehicle’s I/O signals and generating a data matrix including timestamps, input signals and pitch;

Performing the same test several times in succession to ensure data reliability;

Collating and optimizing the data to derive multiple counts of the entire process of the SCAR from the start of suction and capture to the end of calming.

Since the size of the disturbance formation is also mainly based on the change of pitch angle at the end of the trapping process. Therefore, the disturbance as a whole showed an oscillatory decay state. A total of $5$ complete cycles of the vibration process were carried out, and each cycle lasted about $10 s$ . The final SCAR’s attitude change data are shown in Figure 5.

Figure 5.

Variation of the disturbance to SCAR in a pool experiment.

Comparative analysis of simulation experiments

In order not to lose the generality of the anti-disturbance effect, we first select the common sinusoidal attenuation function to represent the ocean wave disturbance as the disturbance $s$ in order (15) to be: $4 e^{- α t} \sin (π / 8) m / s$ with $α > 0, t > 0$ , and add $10 %$ parameter uncertainty to the $δ$ in the system. Under these conditions, different control methods are employed to verify the effectiveness of the methods and the anti-disturbance effect of the SCAR. Meanwhile, for the fair validity of the comparison method, the relevant parameter values mainly include the following: the weighting matrix $Q = diag (1, 1, 1)$ , $R = diag (0.5, 0.5, 0.5, 0.5)$ , the sampling period $T = 0.5 s$ , the prediction horizon $N = 10$ , the initial velocity $V_{0} = {[0, 0, 0]}^{T}$ , the maximum value $u_{\max} = {[10, 10, 10]}^{T}$ and minimum value $u_{m i n} = {[- 10, - 10, - 10]}^{T}$ of the input constraints, the Lyapunov function parameter $α = 2$ , the feedback gain $K_{b} = diag (1, - 1, 0.5)$ .

Figure 6(a) shows the system response curve under different control methods. In each plot, the blue curve and The dark pink curve show the change of the pitch of the SCAR through the backstepping control (BSC); the green curve is the model prediction control (MPC) method; the blue curve is the SCAR’s attitude change with the robust model prediction control (RMPC), and the red curve is the desired moving pose. It is clear that each control method achieve steady-state control under this disturbance.

Figure 6.

Stable tracking of SCAR system under sinusoidal perturbation with different control methods and performance metrics: (a) comparison diagram of system state response curves under sinusoidal attenuation signal perturbation and (b) comparison of the average tracking error and overshoot of the system under different methods.

In this case, RMPC shows excellent robust to make the adjustment process smoother even it leads to a longer adjustment time of about $5 s$ and an angular deviation of less than $1 °$ . Figure 6(b) shows the root mean square error produced by each control method during the tracking error, and the average system overshoot when the comparison experiments are performed. The percentage improvement of the RMPC method over BSC and MPC in tracking error is $27.45 %$ and $19.10 %$ , and in overshooting amount is $30.86 %$ and $22.78 %$ , respectively.

In addition to the system immunity, for the robot’s input, the weight selection of $J$ in the optimization equation is considered. When we obtain a smaller system input loss, that is, the required $v$ is smaller, the $u$ in the set $U$ is also minimized at this time. BSC and RMPC are compared in this process to analyze the control effect among the system propellers, as shown in Figure 7(a). The system is subject to external disturbance, the force generated by the thruster into the energy process, RMPC significantly consumes less force or moment to maintain the system smoothly, and for stability when BSC has a certain jitter. This state, for the SCAR working in the vertical plane, the corresponding state of each state variable of the system under RMPC control is as follows Figure 7(b).

Figure 7.

Variation of control parameters associated with SCAR under sinusoidal perturbation: (a) the control command for each vertical propeller and (b) state variable response curve.

The RMPC controller responds to more aggressive actions than the BSC controller to obtain the fastest convergence when the thruster starts to work. This is due to the fact that BSC can not make full use of on-board thrust capability with fixed control gain. In addition, the results that the oscillation of RMPC is weaker than that of BSC also evidenced the advantages. As expected, all control commands are kept within the allowable range.

After verifying the feasibility of the method, we then select one cycle of vibration from the above-acquired disturbance source as $s$ . Furthermore, the $15 %$ parameter uncertainty is added on top of the above disturbances. Figure 8(a) illustrates the effect of the system response in the presence of this disturbance source and parameter uncertainty.

Figure 8.

Stable tracking of SCAR system under disturbance source in pool test with different control methods and performance metrics: (a) comparison diagram of system state response curves under disturbance source in pool test and (b) comparison of the average tracking error and overshoot of the system under different methods.

Despite the irregular decay of the real disturbances and the parameters perturbation, each method is able to fluctuate within a one a smaller angle (about $10 °$ ). BSC has a stabilization time of almost $20 s$ more compared to RMPC due to limitations such as the need for derivation of the virtual control law, which allows for a larger amplitude of the system. Figure 8(b) shows that the RMPC method improves the tracking error by $29.76 %$ and $20.29 %$ compared to BSC and MPC, respectively, and improves the overshoot by and $36.33 %$ .

The changes that occur in each thruster under the desired input constraints and the respective state responses can be observed in Figure 9(a), The adjustment time and energy consumption of RMPC are clearly less than that of BSC. At this time, each state variable of the system can be seen in Figure 9(b).

Figure 9.

Variation of control parameters associated with SCAR under sinusoidal perturbation: (a) the control command for each vertical propeller and (b) state variable response curve.

The results show that when the SCAR starts working and the external disturbance conditions are the same, the system response time of RMPC can be $10 s$ faster than the other two methods, the overshoot of the system is reduced by almost $40 %$ . For the inputs generated by the propeller, the BSC method creates a larger problem of oscillations and flutter, which may lead to propeller damage. In addition, the overall system response for each state under the RMPC method also reaches a stabilized value, further illustrating the robustness and superiority of using the RMPC method.

Conclusion

In this paper, a practical underwater device is designed for the main purpose of catching sea cucumbers, and the vertical stability of the system is controlled during operation to eventually achieve a smaller pitch angle variation. Firstly, a matching model based on the specificity of the system is established. Combined with the structure of the SCAR’s suction and capture ports and the characteristics of the water flow, it ensures a more accurate description of the system’s motion state. In addition, compared with the traditional control strategy, the RMPC method used in this system has the advantage of stability. A nominal model design method and cost functions optimization are used to constrain the inputs and states of the SCAR to ensure minimum energy loss. The disturbances and uncertainties in the system are described by the sources of disturbances in the pool experiments, external factors (e.g. self-designed disturbance states), and uncertain parameters. Finally, the effectiveness and robustness of the method are evidenced by a large number of simulation results coupling with comparing and analyzing multiple methods. However, RMPC requires high computational resources as it needs to perform the optimization solution in each sampling cycle. The computational complexity of the controller is further increased especially if the prediction horizon $N$ is large or the system complexity is high. Thus, practical applications must carefully balance the robot’s operational duration and accuracy requirements to prevent instability caused by overreaction to prediction errors.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the Fundamental Research Funds for the National Natural Science Foundation of China (Grant No. 51909252).

ORCID iDs

Changbin Wang

Weicheng Sun

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Stability control of adsorption robot considering external disturbance and parameter uncertainty using RMPC

Abstract

Keywords

Introduction

Establishment and analysis of SCAR mathematical model

Kinematic model

Dynamic model and force analysis

System model

Design and analysis of RMPC

Method

Stability and feasibility analysis

Results

Hardware platform and experiment preparation

Obtaining disturbance source experiments

Comparative analysis of simulation experiments

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

References