Dynamic robust path-following control of UVMS subject to manipulator disturbance

This study addresses path-following control issue of underwater vehicle-manipulator systems (UVMS) in the presence of manipulator disturbances. It considers disturbance forces arising from manipulator position changes, including restorative forces and coupling forces, to achieve precise UVMS path-following control. By analysing the variations in the center of mass and buoyant center of manipulator links during motion, the corresponding restorative torque is deduced, and disturbance forces are calculated. Additionally, joint coupling forces generated during manipulator startup, braking, and rotation are taken into account. These aspects collectively yield the expression for disturbance forces. Subsequently, adaptive sliding mode control are employed to achieve accurate control of the UVMS system. This study effectively addresses the issue of UVMS body control under manipulator disturbances. By analysing the motion of manipulator links, the disturbance forces exerted by the manipulator on the UVMS are identified, offering insights for similar control challenges arising from manipulator disturbances. Simulation results are conducted to show the effectiveness of the proposed control method.


Introduction
Unmanned Underwater Vehicle-Manipulator System (UVMS) holds significant value in modern marine engineering and research, finding extensive applications in deep-sea exploration, marine resource development, environmental monitoring, underwater rescue, and scientific research.For instance, the "Deep Sea Warrior" [1] as shown in fig 1, a collaborative development between the Shenyang Institute of Automation of the Chinese Academy of Sciences and the China Shipbuilding Industry 702 Institute, features a manned submersible equipped with a deep-sea hydraulic manipulator system.This system has successfully completed multiple submersion missions and seabed operations, effectively showcasing the crucial role of UVMS in marine science and engineering.
Nevertheless, practical applications of UVMS encounter challenges, particularly disturbances caused by the changes in manipulator arm positions, which result in restorative forces and coupling forces.This study draws inspiration from successful cases like the "Deep Sea Warrior" and employs a combination of backstepping control with adaptive sliding mode control to address these challenges.By analyzing the variations in the center of mass and buoyant center of manipulator links, as well as the joint coupling forces under different states, the study effectively resolves the issues of pathfollowing control of UVMS in the presence of manipulator disturbances.
Through continuous in-depth research and technological optimization, UVMS is poised to further propel the advancement of marine science and engineering, opening up new horizons for future marine exploration and resource development.Underwater Vehicle Manipulator System (UVMS) consists of two parts: the main body and the manipulator arm, which need to collaborate to efficiently execute tasks.The manipulator arm is connected to the UVMS main body.When the manipulator arm performs tasks, its restoring forces, torques, and coupling forces and torques affect the Remotely Operated Vehicle (ROV) main body, known as internal disturbance forces.This can influence the path tracking of UVMS during underwater missions, especially when the manipulator arm is sampling or performing operations.To address this issue, it is necessary to design coordinated control algorithms.
There are several common coordinated control algorithms: (1) Inverse Control Method: This method relies on Lyapunov functions to achieve widespread stability and is suitable for strict feedback and nonlinear systems.C. Yang et al proposed an adaptive backstepping terminal sliding mode control based on recurrent neural networks (RNN) [2].Wei Chen designed an adaptive inverse controller based on a nonlinear observer (NDO) for the depth tracking control of the UVMS main body [3].
(2) Adaptive Sliding Mode Control: This method estimates uncertain parameters of the controlled object based on sensor data and combines sliding mode control to achieve control for nonlinear systems.Cunha et al. designed an output feedback controller based on sliding mode model reference adaptive control for ROV, which can effectively suppress uncertain disturbances [4].Researchers at Shanghai Jiao Tong University applied adaptive sliding mode control to underwater gliders, successfully achieving accurate trajectory tracking [5].Ha designed an adaptive robust coordinated controller with RBF compensation terms and, based on that, designed an adaptive sliding mode controller with local model approximation for the UVMS main body, achieving precise control [6].
(3) Intelligent Control Methods: These methods don't require precise mathematical models of the system and perform well in handling parameter uncertainty.Xu used a genetic algorithm to optimize manipulator arm deployment parameters to minimize the impact of swinging forces and torques between the manipulator arm and the ROV main body during underwater robot motion [7].Tony Salloom used adaptive neural networks to estimate uncertainty in the system dynamics to achieve more robust control [8].
In conclusion, different methods have shown advantages in various application scenarios for controlling UVMS.However, the intrinsic nonlinearity and high dynamic of arm often cause the whole system extremely sensitive to external and internal disturbances.In order to tackle this issue, we propose to use a heuristic strategy which includes an adaptive sliding mode controller for achieving robust path-following of UVMS.By using this strategy, we anticipate leveraging the robust disturbance rejection and enhancing UVMS control performance simultaneously.

UVMS Body Dynamics
The modeling of the UVMS body adopts the η -ν Fossen formula recommended by the Society of Naval Architects and Marine Engineers (SNAME) [9], and the formula is as follows: ( ) In this context, [ , , , , , ] v u v w p q r = represents the velocity vector in the body coordinate system (corresponding to surge, sway, heave, roll, pitch, and yaw), while Equation (1) represents the relationship between the velocity in the body coordinate system and the velocity in the Earth-fixed coordinate system [12], where J is the rotation matrix connecting them; Equation ( 2) is the Fossen formula, where each letter holds significant meanings.M is the inertia matrix of the vessel, which is symmetric and strictly positive definite [13].It is solely related to the inherent parameters of the UVMS body, forming a constant matrix.It comprises two parts: The skew-symmetric Coriolis matrix is denoted as ( )

=+
, where   represents the Coriolis moment matrix of the rigid body system, and A C represents the added Coriolis coefficient matrix [14]; The vessel's damping matrix consists of the linear damping matrix D and the nonlinear damping matrix n D .When the vessel's velocity is slow, the nonlinear damping term can be neglected.
Assuming the origin of the vehicle is located at the center of mass of the vessel, based on these assumptions, a restoring moment ( ) g  is derived, composed of gravity and buoyancy forces.This moment is only dependent on the vessel's attitude.

UVMS manipulator arm dynamic modeling
The aforementioned dynamic models are fundamentally similar to those of traditional AUVs and ROVs.However, compared to other conventional underwater vehicles, the motion of the UVMS manipulator can easily introduce substantial disturbances to the vessel's body.This includes restorative forces (torques) arising from changes in the manipulator's center of gravity and buoyant center during the manipulator's positional adjustments, as well as coupling torques generated by the instantaneous start, braking, and constant-speed rotation of the manipulator's individual joints.Therefore, a thorough analysis of UVMS's dynamic characteristics is essential to mitigate the disturbance effects caused by its own operations.dis  represents the disturbance caused by the manipulator on the vessel's body, composed of the restorative force (torque) term r  and the coupling force (torque) term c  .Taking into account changes in the center of gravity and buoyant center of each link in the manipulator during its motion, the restorative force (torque) term exerted by the manipulator on the vessel r  ,can be expressed as follows: iV g r and iV b r denote the positions of the center of gravity and buoyant center of the i-th generalized link of the manipulator in the vessel's coordinate system.In addition to the disturbances caused by the restorative forces (torques) of the manipulator, the coupling forces (torques) c  resulting from the velocities and accelerations of the manipulator's individual joints can also induce significant disturbances on the vessel.On one hand, using the Newton-Euler algorithm to establish the dynamic model, the inertial force experienced by the i-th generalized link can be represented as follows: ( ) In Equation ( 4), On the other hand, the hydrodynamic forces i i F and moments i i M experienced by the i-th generalized link of the manipulator during underwater motion can be calculated using the following expressions: 0.5 ( ) Here, D C represents the drag force coefficient, and s D represents the damping force coefficient.
According to the inward recursion method, the forces and moments experienced by the i-th generalized link of the manipulator can be expressed as follows: In Equation ( 7), i i f and i i n represent the constraint forces (torques) exerted on the i-th generalized link at its i-th joint.Based on the transformation matrix between coordinate systems, one can obtain: Combining equations ( 4), (7), and (8) in a sequential inward recursion manner, the coupled forces and moments exerted by the manipulator on the vessel can be obtained as: Additionally, by combining the Fossen formula in equation ( 2), along with the rotation matrices between the body coordinate system and the Earth-fixed coordinate system, the entire Fossen formula can be transformed into the underwater robot's dynamic model in the Earth-fixed coordinate system: ( ) ( ) ( ) ( ) ( ) Where: ) ICFOST-2023 Journal of Physics: Conference Series 2718 (2024) 012056

UVMS Path-Following Control
Path following control is an essential technique in the motion control of underwater vehicles, serving as a significant metric for evaluating the maneuvering performance of ships and underwater vehicles.
Path following involves guiding an underwater vehicle from a specified initial state to traverse and follow a designated smooth geometric path in space under the continuous influence of a pathfollowing controller [15].It's important to note that the desired path parameters are not directly related to time, which means that path following emphasizes geometric errors in terms of position and does not have strict temporal constraints.
The UVMS path-following controller comprises two closed-loop control loops: a dynamic model loop and a kinematic model loop [16].The actuator calculates thrust parameters based on the controller's output and provides power to the underwater vehicle's dynamic model to induce motion.The kinematic model continuously computes the vehicle's motion parameters, completing the entire motion loop.As established in sections 2.1 and 2.2, the spatial motion equations of the UVMS under study consist of a set of complex nonlinear differential equations.The essence of the entire control system lies in a second-order nonlinear uncertain system.The controller's input parameters are the desired position ( ) , , , , , , , , ,  ==   , forming a set of 12 state parameters.
In summary, the trajectory tracking problem for the UVMS can be described as follows: Given the desired path and attitude angles, the controller employs navigation system parameters and vehicle characteristics to calculate the necessary forces (moments) for the actuators based on the model.This enables the underwater vehicle to maintain its attitude and navigate along the desired trajectory.
Developing a path tracking controller based on adaptive sliding mode control, which is a control strategy used for handling nonlinear and uncertain systems, combining the principles of sliding mode control and adaptive control, with the aim of achieving stable control performance.
Firstly, defining the variables: 1 ) , , , , , To establish a path-following controller based on adaptive sliding mode control, a strategy designed to handle nonlinearity and uncertainty, a sliding mode surface is defined for this purpose.
Firstly, define the sliding mode surface as follows: id sv =− (10) In the above equations:, s : Sliding surface, used to represent the error for achieving control objectives.
v : Velocity vector in the body coordinate system, including forward speed u, lateral speed v, vertical speed w, roll rate p, pitch rate q, and yaw rate r. .
The sliding surfaces represents the error between actual velocity and desired velocity, and it is controlled to achieve the control objectives.
Next, design the adaptive control law: Where, 1  and 2  are positive definite matrices, and Finally, the output thrust (torque) 2 t  of the controller can be expressed as: In summary, the path-following controller based on adaptive sliding mode control can utilize the introduced sliding surface and adaptive mechanism to address nonlinearities and uncertainties, thereby achieving the tracking of desired positions and orientations.

3D curve path-following control in space
Let the desired three-dimensional trajectory be: 10sin( 402) 10sin( 402) 10 0.1 The simulation starts with an initial state of

Gazebo validation
In the simulation environment, the movement of the robotic arm acts as a disturbance to the UVMS vehicle.Therefore, after the UVMS reaches the operating location, it is necessary to ensure that the movement of the UVMS vehicle remains within a certain range while the robotic arm is in operation.
Assuming the initial position of the UVMS vehicle is (0m, 0m, -5m), after the controller's influence, the UVMS vehicle can maintain its position near the initial location.As shown in fig 8, The UVMS has moved and come to the designated initial position, and the position fluctuations are shown in fig 9 provides a numerical analysis of the positional error as it approaches the target point.By comparing these two figures, it's evident that the position fluctuations of the UVMS satisfy the precision requirements.
The following is a depiction of the UVMS path-following algorithm in the three-dimensional simulation software Gazebo:

Summary
This paper addresses the challenges of path-following control for underwater robot UVMS and proposes innovative solutions.In terms of approach control, a multi-input multi-output adaptive sliding mode control method is introduced.By combining the rapid response of sliding mode control with adaptive parameter adjustments, UVMS achieves precise path-following in complex underwater environments.Nonlinear characteristics are countered by progressively constructing virtual control inputs and state variables, ensuring accurate tracking of desired positions and orientations.
The paper also deeply investigates disturbances caused by the manipulator during operations.By analyzing disturbance sources and proposing solutions, this method has been successfully applied to UVMS manipulator control, effectively reducing the impact of disturbances on control system stability and accuracy.This innovative solution provides a more reliable operational assurance for the practical application of underwater robots.

Figure 1 .
Figure 1.Deep Sea Warrior and Euler angles[10] relative to the Earth-fixed coordinate system (representing east-west displacement, north-south displacement[11], vertical displacement, roll angle, pitch angle, and yaw angle).The vector [ , , , , , ] and moments exerted on the vehicle's body, and   signifies the disturbance moments induced on the vessel due to the manipulator's motion.
inertia matrix of the rigid body system RB M , and the added mass coefficient matrix A M .

F
the matrix expressions for the gravitational and buoyant forces of the i-th generalized link of the manipulator in the vessel's ICFOST-2023 Journal of Physics: Conference Series 2718 forms of gravitational and buoyant forces for the i-th generalized link of the manipulator in the inertial reference frame.Here, represent the mass and buoyant force of the i-th generalized link of the manipulator, respectively.
the mass and added mass of the i-th generalized link of the manipulator, respectively.i I and ai I are the inertia matrices and added inertia matrices of the i-th generalized link of the manipulator.

1d:
First derivative of the desired position 1d

.
The specific simulation results for tracking a three-dimensional trajectory in space are as follows:

Figure 8 .
Figure 8. Verification of the UVMS path-following algorithm