Development of a Six-Degree-of-Freedom Deep-Sea Water-Hydraulic Manipulator

Abstract

With the advancement of deep-sea exploration, the demand for underwater manipulators capable of long-duration heavy-duty operations has intensified. Water-hydraulic systems exhibit less viscosity variation with increasing depth than oil-based systems, offering better adaptability to deep-sea conditions. Using seawater as the driving medium inherently eliminates issues such as oil contamination by water, frequent maintenance limiting underwater operation time, and environmental pollution caused by oil leaks. This paper introduces a deep-sea manipulator directly driven by seawater from the deep-sea environment. To address the challenges of weak lubrication and high corrosion associated with water hydraulics, a reciprocating plunger seal was adopted, and a water-hydraulic actuator was developed. The installation positions of actuator hinges and maximum output force requirements were optimized using particle swarm optimization (PSO), effectively reducing the manipulator’s self-weight. Through kinematic and inverse kinematic analyses and joint performance tests, a six-degree-of-freedom water-hydraulic manipulator was designed with a maximum reach of 2.5 m, a lifting capacity of 5000 N, and end-effector positioning accuracy within 18 mm.

1. Introduction

Underwater manipulators, particularly hydraulically driven manipulators, are widely used in deep-sea operations and ocean exploration due to their high power density [1,2]. However, the viscosity of mineral oil changes significantly with the sea depth, causing hydraulic oil to become more viscous under deep-sea pressure, leading to the severe performance degradation of the manipulator [3,4]. Additionally, the reciprocating and rotational movements of oil-hydraulic cylinders in deep-sea environments result in oil–water cross-contamination, causing water ingress into the hydraulic system and limiting the duration of continuous heavy-duty underwater operations [5,6]. The oil leaking into the environment also poses a pollution risk. In contrast, a new seawater hydraulic drive system can fundamentally avoid these issues [7]. Unlike oil-hydraulic systems, the water-hydraulic manipulator extracts seawater directly from the surrounding environment as the driving medium, eliminating the need for an oil tank for pressure compensation and supply [8,9]. This reduces the complexity and weight of the manipulator system. Moreover, the physical properties of water are less affected by sea depth compared to mineral oil, resulting in better environmental adaptability and more stable performance in deep-sea conditions [10].

Oil-hydraulic systems primarily drive heavy-duty deep-sea manipulators, and several commercial companies have developed relatively mature product lines. Examples include the ORION7R, RigMaster, and TITAN4 series from Schilling Robotics (USA); the HLK-MB4, EH4, and CRA6 series from Hydro-Lek (UK); the Magnum3-7F from International Submarine Engineering (ISE, Canada); and the Raptor from Kraft TeleRobotics (USA). These systems have achieved full-depth coverage and are widely used in deep-sea exploration. However, the existing oil-hydraulic systems face significant limitations for deep-sea development, where manipulators are required to perform long-duration, heavy-duty operations continuously in deep-sea environments. These limitations hinder the ability of manipulators to sustain extended underwater operations. In contrast, water-hydraulic manipulators that use seawater as the driving medium offer superior environmental adaptability and are better suited for prolonged continuous operations. However, transitioning the driving medium of underwater manipulators from oil to water introduces new challenges in sealing, lubrication, and control [11]. With recent advancements in engineering materials, tribology, and advanced manufacturing, research institutes and companies in countries such as the United States, Japan, Germany, Finland, and China have been conducting foundational research on water-hydraulic technology [12]. They have developed relatively mature water-hydraulic components, including water-hydraulic pumps, vane motors, and control valves [13]. However, a fully developed water-hydraulic manipulator has yet to emerge [14,15]. In 1991, H. Yoshinada from Japan’s Komatsu Ltd. proposed a seawater hydraulic-driven manipulator system [16]. Around 2000, in response to the zero-pollution requirements of the International Thermonuclear Experimental Reactor (ITER), the French CEA laboratory designed the Maestro, a six-degree-of-freedom general-purpose water-hydraulic manipulator for the disassembly of reactor components and pipeline cutting [17,18]. The University of Lappeenranta in Germany developed the Penta-WH, a six-degree-of-freedom spherical coordinate parallel robot for reactor assembly and repair [19]. Tampere University of Technology developed the WHMAN [20], a six-degree-of-freedom water-hydraulic manipulator for nuclear fuel loading and equipment maintenance. However, these three water-hydraulic manipulators have only been used on land. In 2015, Hassan et al. from Malaysia developed a prototype of an underwater manipulator driven by a water-hydraulic system and studied its sustainable energy efficiency and kinematics [21]. In 2022, Beijing University of Technology initiated research on water-hydraulic-driven flexible gripper manipulators, and some seawater hydraulic components have been developed, including a water-hydraulic flexible gripper and a swash-plate seawater hydraulic motor [22]. In 2022, Dalian Maritime University developed a fabric-reinforced soft manipulator driven by a water-hydraulic system; however, such water-hydraulic manipulators are challenging to use for heavy-duty operations in deep-sea environments [23].

In summary, oil-hydraulic systems primarily drive existing heavy-duty deep-sea manipulators. Water-hydraulic manipulators have limited applications in the land-based nuclear industry, while water-hydraulic soft manipulators suitable for underwater use are only effective for light-load operations and lack development in heavy-duty deep-sea applications. Therefore, to address the demand for long-term, maintenance-free, heavy-duty operations in deep-sea development that traditional oil-hydraulic technologies struggle to meet, research on deep-sea water-hydraulic manipulators has been conducted based on the team’s previous work on full-depth seawater pumps [24], ultra-high-pressure balancing valves [25], and other studies. This research fundamentally addresses the issue of oil–water cross-contamination, leading to the development of a relatively mature deep-sea water-hydraulic manipulator to provide technical support for the long-duration, long-term operations of deep-sea engineering machinery.

The remainder of this paper is organized as follows: Section 2 covers the system analysis and essential actuator design of the water-hydraulic manipulator. It analyzes the composition and characteristics of the deep-sea water-hydraulic manipulator system, addressing the challenges of sealing and poor lubrication when using water-hydraulic drives for the manipulator’s actuators. The design includes the use of a reciprocating plunger structure, a water-hydraulic linear cylinder for small-angle rotation output under high loads, a water-hydraulic gear-rack swing cylinder for large-angle rotation output, and a radial plunger motor for continuous-rotation output, fulfilling the motion requirements of each manipulator joint. Section 3 focuses on the overall structural optimization of the manipulator. It addresses the increased non-stroke length of the linear cylinder caused by the oil-free compensation sealing method of the water-hydraulic manipulator’s position feedback sensors, which leads to increased joint output torque demand. A particle swarm optimization algorithm is used to optimize the hinge positions of the first three joints of the manipulator, achieving minimal and stable output force, reducing component dimensions, and applying topology optimization to reduce the weight of the manipulator’s main arm. Section 4 provides an analysis of the motion control of the manipulator. It includes a kinematic analysis to verify the manipulator’s performance, inverse kinematics solutions to achieve end-effector pose control, and pressure and position tracking tests on the developed manipulator actuators to validate the joint motion accuracy, fitting the control precision of the manipulator. Section 5 summarizes the main work in this paper.

2. Water-Hydraulic Manipulator’s Overall Performance and Structural Design

2.1. Working Principle

Compared to oil-hydraulic-driven manipulators, water-hydraulic-driven systems can directly extract seawater from the deep-sea environment as the driving medium [26]. As a result, the submersibles carrying these manipulators do not require additional oil tanks, reducing the system’s overall weight and enabling small underwater robots to be equipped with such manipulators. However, due to the presence of chloride ions and other ions in natural seawater, which cause strong corrosion, and the low viscosity and impurities in seawater, conventional oil-hydraulic controls and actuator components cannot simply switch the driving medium to achieve water-hydraulic operation. Therefore, it is necessary to design power sources, control components, and actuator components suitable for seawater hydraulic driving based on the characteristics of the water medium.

As shown in Figure 1, the water-hydraulic manipulator system mainly consists of three parts: the water pump (power component), the hydraulic valve group (control component), and the slave arm (actuation component). The water pump is located inside the submersible. It is driven by an electric motor, which extracts seawater from the surroundings and outputs high-pressure seawater to provide driving pressure for the hydraulic system of the manipulator. The hydraulic control valve group primarily consists of directional valves, throttle valves, and bidirectional hydraulic locks. Control programs that send commands to the hydraulic control valve group control the transmission of water pressure within the manipulator’s internal pipelines. This changes the pressure inside the driving chamber of the joint actuators, enabling the rotational movements of the manipulator’s joints and driving the end-effector claw to the working position.

2.2. Components of the Manipulator

Figure 2 shows the structural components of the six-degree-of-freedom water-hydraulic manipulator. The water-hydraulic manipulator mainly consists of linkage mechanisms and actuating mechanisms. The linkage mechanisms primarily include the base, upper arm, and forearm. The actuating mechanisms mainly include Joint 1’s water-hydraulic linear cylinder, Joint 2 water-hydraulic linear cylinder, Joint 3 water-hydraulic linear cylinder, Joint 4 water-hydraulic rack-and-pinion swing cylinder, Joint 5 water-hydraulic rack-and-pinion swing cylinder, Joint 6 water-hydraulic motor, and Joint 7 parallel gripper. Considering the effects of corrosion by water and the electrochemical reaction of metals underwater, all components of the manipulator are made of TC4 material. For critical force transmission structures, such as the gear shafts in the rotary cylinders and the push rods in the linear cylinders, higher-strength 17-4PH stainless steel material is used. The system employs carbon-fiber-reinforced PEEK to reduce friction between metal components.

All actuators of the manipulator are driven by water transmission, directly sourcing water from the external environment. This eliminates the need to carry oil sources and redundant hydraulic system compensator structures, reducing the complexity and weight of the manipulator. Joints 1, 2, and 3 are driven by water-hydraulic linear cylinders, achieving heavy-load, small-angle rotation outputs with rotation ranges of 120° for Joints 1 and 2 and 135° for Joint 3. Joints 4 and 5 are driven by water-hydraulic rack-and-pinion rotary cylinders, with rotation ranges of 270° and 180°, respectively. Joint 6 utilizes a custom-designed radial piston water-hydraulic motor for continuous rotational drive. The end-effector features a parallel gripper for grasping actions. The manipulator provides six degrees of freedom for movement, combined with a parallel gripper, totaling seven functional capabilities. The manipulator’s six degrees of freedom and the parallel gripper comprise seven functions that can meet the demands of most deep-sea operational conditions. The leading performance indicators of the manipulator are shown in

Table 1. Performance specification of the manipulator.

Actuator Function	Value	Unit
Depth rating	3000	m
Working pressure	21	MPa
Weight in air	120	kg
Lift at full extension	500	kg
Lift at full arm	150	kg
Maximum reach	2500	mm
Elbow torque	400	Nm
Motor torque	300	Nm
Gripper opening	260	mm

.

2.3. Design of Actuation Components

Given that the driving medium for the manipulator is water, which has weak lubrication and low viscosity, the actuators face significant challenges related to sealing and wear [27]. These issues severely reduce the actuators’ lifespan and impact the manipulator’s continuous operation time. Therefore, selecting a reliable water-hydraulic transmission sealing method is crucial. Comparative analysis indicates that the piston-rod-type reciprocating seal is the simplest and most reliable sealing method. Thus, all actuators in the water-hydraulic manipulator utilize a piston rod structure with reciprocating seals using Glyd rings. Experiments have shown that copper-powder-reinforced PTFE, commonly used in oil-hydraulic systems, undergoes electrochemical corrosion with TC4 material, damaging the surface of the drive cylinder’s inner wall and causing seal leakage. In contrast, carbon-fiber-reinforced PTFE is stable, has high surface hardness, and performs well in the presence of fine particulate impurities, making it suitable for water-hydraulic transmission sealing [28,29].

The manipulator uses linear cylinders to drive a three-hinge-point amplitude variation mechanism to fulfill the high-torque, small-angle output requirements for non-continuous rotational actions. For low-torque, large-angle output requirements, the vane-type rotary cylinder with vane seals shows low sealing efficiency and high-pressure leakage with a water-hydraulic drive. If using helical gear transmission, the helical rotary cylinder requires high-gear surface machining precision and is prone to jamming under poor lubrication conditions [30]. Instead, the rack-and-pinion rotary cylinder, which uses the reciprocating motion of a rack to drive the gear shaft rotation, offers reliable sealing and simple transmission, making it suitable for the large-angle output elements of the manipulator. For continuous rotational actions, the radial piston motor, which uses pistons to drive the shaft in constant rotation with a reciprocating sealing form, offers simple sealing and is selected as the continuous rotational output element for the manipulator.

As the most common hydraulic actuator, the linear cylinder features a simple transmission structure, high stability, and good sealing properties. Unlike oil-hydraulic drives, sealing for water as a driving medium must be considered, along with the impact of environmental pressure on the linear cylinder in deep-sea conditions and the realization of position feedback functionality for the manipulator. The overall model of the water-hydraulic linear cylinder designed in this paper is shown in the following figure (Figure 3). This water-hydraulic cylinder is divided into three chambers. Chamber A is the sealed chamber for the magnetostrictive sensor, while chambers B and C are the working chambers of the water-hydraulic cylinder, each sealed with O-rings in between. When high-pressure water enters the working chambers through the ports, a pressure difference is maintained between chambers B and C due to the differential area of the piston rod, which generates pressure to push the piston assembly and output thrust. To achieve control of the linear cylinder position, a magnetostrictive sensor is used for oil-free compensation sealing, increasing the non-stroke length of the linear cylinder. As a result, the hinge positions of the hydraulic-driven manipulator are no longer suitable and need to be redesigned.

Output thrust F_p is defined as follows:

$$F_p={\displaystyle \frac{\pi D^2}{4}}\cdot \left(P_1-P_2\right)\cdot \eta$$

(1)

where F_p is the maximum output thrust of the hydraulic cylinder’s rodless chamber, N; P₁ is the supply pressure in the rodless chamber; P₂ is the return pressure in the rod chamber; η is the working efficiency of the hydraulic cylinder, typically taken as 0.95; and D is the internal diameter of the hydraulic cylinder, m.

The structure of the rack-and-pinion rotary hydraulic cylinder is shown in Figure 4. It primarily consists of a gear shaft and a rack. The cavities on both sides of the rack are connected to the system’s hydraulic pipelines, with a piston head installed at the end of the rack. In operation, the hydraulic differential at the piston ends drives the rack, transferring hydraulic pressure to the gear shaft, thus achieving torque output. To address the critical gear transmission pair in the rack-and-pinion rotary cylinder, an independent lubrication compensation chamber is added to the gear meshing surface for independent lubrication compensation. The gear surface is treated with TiN coating to enhance the surface strength and reduce the meshing wear.

The output torque M and swing angle θ of the gear shaft are given by

$$M=\pi (P_1-P_2){d_1}^2{d}_2/4$$

(2)

$$\theta =360°\times L_s/mz$$

(3)

where P₁ and P₂ are the pressures at the two ends of the rack piston, respectively, d₁ is the diameter of the piston, d₂ is the pitch circle diameter, L is the stroke of the piston inside the cylinder, m is the module of the gear and rack, and z is the number of teeth on the gear.

As shown in Figure 4, the radial piston motor primarily operates by the reciprocating motion of pistons, driving an eccentric shaft to rotate continuously. The engine controls the piston force output by integrating the distribution valve with the pistons according to the rotation angle. A needle roller structure reduces friction between the pistons and the output shaft. A floating connection design is also implemented for the pistons to eliminate lateral forces, enhancing torque output stability. The instantaneous output torque for a 10-piston motor is given by

$$M({\phi }_i)=P_0{e}_0\left[\sin {\phi }_i\cos {\theta }_i+\cos {\phi }_i\sin {\theta }_i\right]$$

(4)

where ${\phi }_i$ is the rotation angle of the crankshaft, P is the hydraulic driving force, e is the eccentric distance between the piston center and the crankshaft, and ${\theta }_i$ is the swing angle of the swing cylinder.

3. Optimization of Manipulator Structure

For the manipulator’s heavy-load, small-angle rotation requirements, this paper employs a three-hinge amplitude variation mechanism driven by a linear cylinder. The arrangement of the hinge positions directly impacts the output force requirements of the water-hydraulic linear cylinder. Improper hinge positioning can significantly increase the output force of the actuating mechanism, severely affecting the stability of the manipulator’s movements. Additionally, it dramatically increases the maximum output force design requirement of the linear cylinder, increasing the actuating mechanism’s size. In this section, we analyze the forces acting on the manipulator under load conditions and construct a force model for the linear cylinder and the three hinge points based on the geometric constraints of the manipulator’s movements. Utilizing particle swarm optimization (PSO), we determine the optimal positions for the hinge points. Additionally, we conduct a topology optimization of the rigid body of the manipulator’s linkage to reduce the overall weight of the manipulator while maintaining structural stiffness.

3.1. Heavy-Load Hinge Force Analysis and Model Construction

3.1.1. Shoulder Joint Swing-Hinge Analysis

For the shoulder structure of the deep-sea manipulator, the connection with the base forms a horizontal swing joint (Manipulator Joint 1), which primarily enables the overall rotation of the manipulator. This driving joint mainly overcomes the frictional force of the disk caused by the manipulator’s weight and load while also considering the impact of water resistance torque in the deep-sea environment on its torque [31,32].

Considering that the shoulder horizontal swing joint drives the upper arm in a nearly uniform motion in actual operation, the added water inertia torque can be neglected. When the manipulator is in a fully extended state, the maximum value of the water resistance torque is

$$M_{d\max }=C_D\rho r{\displaystyle {\int }_0^{L_o}{v}_i}\left|v_i\right|ldl$$

(5)

where C_D is the water resistance coefficient, taken as 1.5; ρ is the fluid density, with water taken as 1000 kg/m³; $v_i$ is the velocity vector of the link due to the corresponding joint rotation; r is the radius of the simplified cylindrical body of each manipulator link, taken as 0.125 m; l is the distance from the point of rotation to an end in the cylinder; and L₀ is the full extension length of the manipulator.

It is assumed that the total mass of the manipulator’s arm and load, m, is 270 kg, and the full extension length L_o is 2.5 m. The shoulder joint and base are connected through carbon-fiber-reinforced PEEK spacers, with a safety factor considered, and the friction coefficient μ is set to 0.5. g is the gravitational acceleration. The radius R₁ of the friction disk for approximate disk friction is set to 0.15 m. According to the disk friction force formula and Morison’s empirical formula, the total load torque M₁ on the shoulder swing joint when the manipulator is in a fully extended state can be obtained as follows:

$$M_1=M_f+\max (M_d)={\displaystyle \frac{2}{3}}(\mu mg{R}_1)+C_D\rho r{\displaystyle {\int }_0^{L_o}{v}_i}\left|v_i\right|ldl$$

(6)

Taking the midpoint of the main arm as the initial position of the manipulator, the initial position is set at ∠A₁B₁C₁ = ${\phi }_1,$ and α₁ is the swing angle of the shoulder joint within the range of rotation, with an angle range of [−60°, 60°]. When the swing angle of the shoulder joint changes, in the top view, the left swing angle of the main arm is set as positive, and the right swing angle is unfavorable. Assuming that the time t for the main arm to rotate 120° is 10 s, its motion is considered uniform. Thus, the horizontal swing angular velocity ${\alpha }^{\prime }$ of the shoulder is π/15 rad/s.

$$\angle A_1{O}_1{B}_1={\phi }_1+{\alpha }_1$$

(7)

$${\alpha }_1={{\alpha }^{\prime }}_1·t-{\displaystyle \frac{\pi }{3}}(0\le t\le 10)$$

(8)

As shown in Figure 5, L₁ is the distance from point O₁ to A₁B₁. From the moment balance at point O₁, we have

$$F_1={\displaystyle \frac{\left[\frac{2}{3}(\mu mg{R}_1)+C_D\rho r{\displaystyle {\int }_0^L{v}_i}\left|v_i\right|ldl\right]·A_1{B}_1}{O_1{A}_1·O_1{B}_1·\sin \left({\phi }_1+{{\alpha }^{\prime }}_1·t-\frac{\pi }{3}\right)}}$$

(9)

When the positions of hinge points A, B, and O are determined, the function expression of the output force F₁ of the water-hydraulic cylinder as a function of the swing angle α₁ can be differentiated with respect to time. The derivative function of the output force with respect to time can be obtained, so the minimum objective function of the derivative of the output force of the shoulder rotary drive joint water-hydraulic cylinder is

$${F^{\prime }}_1({\alpha }_1)={\displaystyle \frac{d{F}_1}{d{\alpha }_1}}·{\displaystyle \frac{d{\alpha }_1}{dt}}={\displaystyle \frac{\left[\frac{2}{3}(\mu mg{R}_1)+C_D\rho r{\displaystyle {\int }_0^L{v}_i}\left|v_i\right|ldl\right]·A_1{B}_1·\left[-\cos \left({\alpha }_1+{\phi }_1\right)\right]}{O_1{A}_1·O_1{B}_1·{\left[\sin \left({\alpha }_1+{\phi }_1\right)\right]}^2}}·{{\alpha }^{\prime }}_1$$

(10)

3.1.2. Shoulder Joint Pitch-Hinge Analysis

For the pitching motion of the manipulator’s shoulder, the pitch angle range is 120°, achieved by the extension and retraction of Joint 2 connecting the main arm to the base. The pitching motion of the shoulder joint is primarily influenced by the manipulator’s weight, the end load, and the hydrodynamic resistance torque M_d generated during the arm’s movement in the deep-sea environment. As shown in Figure 5, by selecting hinge point O₂ as the moment balance point, the total load torque M₂ for the pitching swing joint can be obtained:

$$M_2=G_0(L_{G1}\cos {\alpha }_2+e_1\sin {\alpha }_2)+G(L_{G0}\cos {\alpha }_2+e_1\sin {\alpha }_2)+M_{d\max }$$

(11)

where O₂ is the rotational connection point of the upper arm and the shoulder joint; A₂ is the hinge point where the base of the water-hydraulic cylinder connects to the shoulder joint; B₂ is the hinge point where the piston rod end connects to the upper-arm linkage; e₁ is the vertical distance from point O₂ to the centerline of the arm; $\phi$₂ is the initial position angle of the joint hinge point; ${\alpha }_2$ is the angle between the main arm of the manipulator and the base plane, with an angle range of [−30°, 90°]; F₂ is the output force of the water-hydraulic cylinder driving the upper-arm pitch; L_G1 is the projected distance along the arm centerline from the main arm’s center of gravity to point O₂ in the fully extended horizontal state of the manipulator, taken as 1.1 m; L_G0 is the maximum projected distance along the arm centerline from the end load to point O₂, taken as 2.5 m; L₂ is the distance from point O₂ to line A₂B₂; G₀ is the weight of the manipulator, taken as 120 kg; G is the weight of the end load, taken as 150 kg; and M_d is the hydrodynamic resistance torque.

$$\angle A_2{O}_2{B}_2={\phi }_2+{\alpha }_2$$

(12)

$${\alpha }_2={{\alpha }^{\prime }}_2·t-{\displaystyle \frac{\pi }{6}}(0\le t\le 12)$$

(13)

where ω is the angular velocity of the pitching motion of the manipulator’s upper-arm joint, with a value of ω = π/15 rad/s, and t is the total swing time, set to 12 s.

By taking moments about point O₂ and using the moment balance equation, we obtain

$$F_2={\displaystyle \frac{M_2}{L_2}}={\displaystyle \frac{\left[G_0(L_{G1}\cos {\alpha }_2+e_1\sin {\alpha }_2)+G(L_{G0}\cos {\alpha }_2+e_1\sin {\alpha }_2)+C_D\rho r{\displaystyle {\int }_0^{L_o}{v}_i}\left|v_i\right|ldl\right]·A_2{B}_2}{O_2{A}_2·O_2{B}_2\sin ({\alpha }_2+{\phi }_2)}}$$

(14)

3.1.3. Elbow Joint Pitch-Hinge Analysis

The function of the three-hinge shoulder joint is for the pitching motion of the forearm of the manipulator, with a pitch angle range of 120°, achieved through the extension and retraction of Joint 3. As shown in Figure 5, taking moments about point O₃, we obtain the following from the moment balance equation:

$$F_3={\displaystyle \frac{M_3}{L_3}}={\displaystyle \frac{G_1(L_{G2}\cos \beta +e_2\sin \beta )+G(L_G\cos \beta +e_2\sin \beta )}{L_3}}$$

(15)

In the equation, the total load torque M₃ for the pitching swing joint is calculated based on the weight of the forearm and the end load. L₃ is the distance from point O₃ to line A₃B₃. e₂ is the distance from point O₂ to the centerline of the secondary arm. L_G2 is the projected distance along the centerline from the secondary arm’s center of gravity to O₂. G₁ is the weight of the secondary arm. In the initial position, ∠A₃B₃C₃ = ${\phi }_3$. The angle variation range of the load angle ${\alpha }_3$ for Joint 3 is [15°, 135°], and its expression is as follows:

$$\begin{array}{l}\angle A_3{O}_3{B}_3={\phi }_3+{\alpha }_3\\ {\alpha }_3={{\alpha }^{\prime }}_3·t+{\displaystyle \frac{\pi }{12}}(0\le t\le 12)\end{array}$$

(16)

where ${\alpha }_3^{\prime }$ is the angular velocity of the forearm pitching swing joint, with a value of ω = π/18  rad/s, and t is the load space swing time, set to 12 s.

By traversing the angular positions of the main-arm pitch joint, the most adverse condition for forearm pitching is identified, ensuring that the load angle of Joint 3 is reachable in any primary-arm position. The pitch angle β of the forearm’s centerline relative to the base plane can be obtained from the following equation:

$$\begin{array}{l}\beta =\pi -{\alpha }_3-{\alpha }_2\\ {\alpha }_2={\displaystyle \frac{\pi }{18}}·t-{\displaystyle \frac{\pi }{6}}(0\le t\le 12)\end{array}$$

(17)

Solving for L₃ is similar to that for the shoulder swing joint. Ultimately, by combining the relevant equations, we obtain the following:

$$F_3={\displaystyle \frac{M_3}{L_3}}={\displaystyle \frac{\left[G_1(L_{G2}\cos \beta +e_2\sin \beta )+G(L_G\cos \beta +e_2\sin \beta )\right]·A_3{B}_3}{O_3{A}_3·O_3{B}_3·\sin \left({\phi }_3+{\alpha }_3\right)}}$$

(18)

3.2. Particle Swarm Optimization

3.2.1. Boundary Conditions

Taking the pitching motion of the main arm as an example, the optimization constraints for the manipulator’s main-arm pitch joint hinge points include hinge boundary constraints, geometric constraints, structural size constraints of the shoulder joint and upper-arm linkage, and the extension ratio, structural size, and stability constraints of the main-arm pitch-driving water-hydraulic cylinder.

(1)

Boundary Constraints

The initial positions of each hinge point are chosen to be consistent with the hinge points of Schilling Robotics’ commercial manipulator, TITAN 4. To cover a working space of 2.5 m, the manipulator’s main arm and secondary arm must be longer than 2.5 m when fully extended. Assuming the main and secondary arms are 1.4 m long and the maximum included angle is 135°, the maximum coverage range is 2.58 m, which meets the target requirement according to the cosine theorem. As shown in Figure 6, a coordinate system is established with the intersection of the shoulder joint rotation axis and the base plane as the origin, the X-axis parallel to the base plane, and the Y-axis perpendicular to the base plane. This coordinate system allows for the selection of seven parameters (X_A, Y_A, X_B, Y_B, X_O, Y_O, e) as optimization design variables, expressed as the following vector:

$$X={\left[X_A,Y_A,X_B,Y_B,X_O,Y_O,e\right]}^{\mathrm{T}}={\left[X_1,X_2,X_3,X_4,X_5,X_6,X_7\right]}^{\mathrm{T}}$$

(19)

We extend the boundary range based on the initial hinge-point coordinates to find the optimal solution. According to the physical structure and motion range of the manipulator’s main arm and shoulder joint, point O₂ must remain on the right side of the Y-axis and ensure sufficient shoulder joint strength. The minimum value for point X_O2 is set to 100 mm, and the maximum value, referencing the TITAN 4 hinge-point position, is set to 160 mm. Point A₂, positioned to the right of point O₂, must have a minimum value more significant than the maximum range of point O₂, with the range for point A₂ on the X-axis set between 160 mm and 200 mm. The main arm, composed of two cylinders, must ensure that point B₂ on the X-axis does not exceed half of the main arm’s length. Considering the minimum distance from the starting point of the main arm to the shoulder joint, X_B2 is limited to less than 800 mm. The eccentric distance, e, determined by the manipulator’s lever arm, load characteristics, and pipeline installation space, ranges from 10 mm to 35 mm. Based on the structural dimensions of the manipulator’s shoulder joint and main-arm linkage, the value ranges for the seven optimization design variables are set, and their boundary constraints are established as follows:

$$X_{i\min }\le X_i\le X_{i\max }(i=1,2,3,4,5,6,7)$$

(20)

The initial hinge-point parameters for the main-arm pitch joint are set, with their respective values and ranges shown in

Table 2. Optimization design variable ranges of three primary-arm pitch hinge points (mm).

Design Variable	X1	X2	X3	X4	X5	X6	X7
Minimum	160	42	650	100	100	120	10
Titan 4 initial value	186	48.5	718	134	143.5	194	15
Maximum	200	60	760	165	160	200	35

.

(2)

Geometric Constraints

During the pitching motion of the upper-arm joint, the positions of hinge points O₂ and A₂ remain unchanged, while point B₂ changes with the swing angle ${\alpha }_2$ of the main arm. The triangle ΔOAB formed by these three hinge points must satisfy primary geometric conditions. As the primary-arm swing angle changes over time within the range [${\phi }_2-30,{\phi }_2+90°$], the triangles ΔOAB_s and ΔOAB_e formed at the start and end positions must satisfy

$$\{\begin{array}{l}OA+A{B}_s-O{B}_s>0\\ OA+O{B}_s-A{B}_s>0\\ A{B}_s+O{B}_s-OA>0\end{array}$$

(21)

$$\{\begin{array}{l}OA+A{B}_e-O{B}_e>0\\ OA+O{B}_e-A{B}_e>0\\ A{B}_e+O{B}_e-OA>0\end{array}$$

(22)

During the motion, to ensure that the primary-arm swing is reachable at both the start and end positions without encountering dead points, and considering the interference of the piston rod diameter of Joint 2 and the rotation axis diameter of the upper arm and shoulder joint, the distance $L_2$ from point O to line AB must always be greater than 42 mm. Therefore, there are angle constraints on the initial position ${\phi }_2$ and geometric constraints on the distance $L_2$.

During the motion, to ensure that the initial and final positions of the main-arm swing are reachable without encountering dead points, and considering the interference impact of Joint 2′s water-hydraulic cylinder piston rod diameter of 40 mm and the rotation axis diameter of the main arm and shoulder joint of 26 mm, the distance L₂ from point O₂ to line A₂B₂ must always be greater than 33 mm. Therefore, there are angular constraints on the initial position ${\phi }_2$ and geometric constraints on the distance L₂:

$${\displaystyle \frac{\pi }{6}}<{\phi }_2<{\displaystyle \frac{\pi }{2}},L_2\ge 33mm$$

(23)

(3)

Structural Size, Extension Ratio, and Stability Constraints of the Water-Hydraulic Cylinder

As shown in Figure 3, the distance between the bottom of the linear cylinder piston movement and the center of the hinge point is the non-stroke dimension of the linear cylinder, denoted by Δ. Constraints are required to ensure sufficient installation space for the main-arm pitch joint driving the water-hydraulic cylinder within the upper-arm linkage. The fully retracted length of the hydraulic cylinder is denoted by AB_s, and the fully extended length by AB_e. The stroke length S of the hydraulic cylinder is AB_e − AB_s, with a non-stroke length Δ that must satisfy the following constraints:

$$A{B}_e-A{B}_s\le A{B}_s-\Delta$$

(24)

AB_s ≥ 534 mm and AB_e ≥ 799 mm based on the actual design dimensions of Joint 2’s water-hydraulic linear cylinder. Considering the installation method of the MH-type magnetostrictive sensor, Δ is set to 270 mm.

When the piston rod of the main-arm pitching water-hydraulic cylinder is fully extended, the support length L_B must satisfy L_B ≥ (10~15)d. Being a “slender” compression rod, the buckling stability of the piston rod must be constrained:

$$F_{2\max }\left({\alpha }_2\right)⩽F_r/n_{\mathrm{k}}$$

(25)

where F_r is the critical buckling compressive force, and n_k is the safety factor, taken as 3.5.

$$F_r={\displaystyle \frac{{\pi }^{3}E{d}^4\times {10}^6}{64(1+a)(1+b)K^2{L}_{\mathrm{B}}^2}}$$

(26)

where E is the elastic modulus of the material, with 17-4PH stainless steel used for the piston rod, $E=2.02\times {10}^5$ MPa; d is the diameter of the piston rod, taken as 0.04 m; a is the material defect coefficient, taken as 1/12; b is the non-uniformity coefficient of the piston rod cross-section, typically taken as 1/13; K is the installation and guiding coefficient of the hydraulic cylinder, taken as 2 for sleeve hinge connections; and L_B is the support length of the water-hydraulic cylinder, which is the fully extended length AB_e.

(4)

Structural Size Constraints

Due to the dimensional constraints of the manipulator’s shoulder joint, the positional relationship between points O₂ and A₂ must be restricted, with the coordinates of the two points satisfying the following constraints:

$${65}^{°}\le \arctan \left({\displaystyle \frac{\left|Y_{O2}-Y_{A2}\right|}{\left|X_{O2}-X_{A2}\right|}}\right)\le {90}^{°}$$

(27)

3.2.2. Objective Function

To ensure smooth and reliable movement of the manipulator, it is essential to minimize abrupt changes in the motion of each joint actuator and reduce the magnitude of the output force at extreme positions. Variations in the output force manifest abrupt changes in the linear cylinder’s motion; hence, rapid changes in the output force of the linear cylinder should be avoided. High output force demands from the linear cylinder would result in larger actuator structural dimensions. A significant difference between the maximum and minimum output forces within the linear cylinder’s range of motion can lead to considerable system pressure fluctuations. Therefore, optimizing the manipulator’s joints driven by linear cylinders aims to minimize the maximum output force of the actuating elements, reduce the rate of change, and decrease the amplitude of fluctuations. Employing a particle swarm optimization (PSO) algorithm, the coordinates of the three pivotal hinge points O, A, and B and the offset distance between the arm centerline and the hinge connection line are used as optimization parameters. The optimal hinge positions of the manipulator are determined by the iterative and collaborative capabilities of particle swarm optimization to search for optimal solutions in a nonlinear and complex space [33].

To ensure that the manipulator’s motion is stable and reliable, it is essential to minimize the maximum output force of the actuators, reduce the rate of change, and limit the fluctuation amplitude. Therefore, the particle swarm optimization (PSO) algorithm is employed. The optimization parameters include the coordinate positions of the three amplitude variation hinge points O, A, and B and the offset distance of the arm centerline from the hinge line. By leveraging the iterative and collaborative search functions of the PSO algorithm among particle individuals, the optimal hinge positions for the manipulator are determined.

Equations (9), (14), and (18) show that when a set of design variables, X, is determined, the function curve of the output force F varying with the rotation angle can be obtained. As shown in Equation (28), $f_1\left(X\right)$ represents the maximum output force of the water-hydraulic linear cylinder. $f_2\left(X\right)$ is the rate of change in the output force obtained by differentiating the output force F with respect to the rotation angle. $f_3\left(X\right)$ is the difference between the linear cylinder’s maximum and minimum output forces throughout its motion.

$$\{\begin{array}{l}{f}_1(X)=F_{\max }(\alpha )\\ f_2(X)={F^{\prime }}_{\max }(\alpha )\\ f_3(X)=F_{\max }(\alpha )-F_{\min }(\alpha )\end{array}$$

(28)

Since hinge-point optimization involves three objective functions, optimizing the hinge positions for the shoulder swing joint is a multi-objective optimization problem. During the optimization process, each objective function tends to its optimal value, often restricting each other, making it difficult to achieve an absolute optimal solution simultaneously. Therefore, this paper uses a linear weighting method, multiplying each objective function by its respective weight coefficient and adding them together to construct a single objective function for optimization.

$$f(X)=w_1{F}_{\max }(\alpha )+1000w_2{F^{\prime }}_{\max }(\alpha )+w_3\left[F_{\max }(\alpha )-F_{\min }(\alpha )\right]$$

(29)

In the equation, w₁, w₂, and w₃ are the weight coefficients of each single objective function. An increase in the maximum output force requirement will necessitate a larger linear cylinder to meet the output force demand, directly affecting the structural dimensions of the linear cylinder. The increase in structural size will, in turn, increase the weight of the manipulator. A lower rate of change and a smaller amplitude of fluctuations aim to reduce system pressure fluctuations and output force variations. This optimization primarily focuses on reducing the manipulator’s overall weight while avoiding significant impacts on control due to system pressure fluctuations and output force changes. Therefore, the weight coefficient for the maximum output force index is set to w₁ = 0.8, and the weight coefficients for the rate of change and the amplitude of fluctuations are set to the same value, w₂ = w₃ = 0.1. To ensure that $f_2\left(X\right)$ is of the same order of magnitude as the other two objective functions, its value is multiplied by 1000.

As shown in Figure 7, the hinge-point coordinates at the shoulder joint of the manipulator are initially set to random values, with constraint conditions enforced through a penalty fitness method. Considering the optimization variables for the main-arm pitch joint, the particle dimension of the particle swarm optimization (PSO) algorithm is set to 7, with a particle population size of 500 and 1000 iterations. The acceleration constants are set to 0.5 and 1.5. The adaptive mutation of particle positions is introduced, and the influence of particle inertia weight on optimization is considered. The upper limit of the inertia weight is set to 0.9 and the lower limit to 0.4, gradually decreasing with the increase in iterations.

3.3. Optimization Results Analysis

PSO was performed on the hinge positions of the three-hinge amplitude variation mechanism for the manipulator’s shoulder joint rotation, shoulder joint pitching, and elbow joint pitching movements. The following figure shows the maximum output force curve and hinge length of the linear cylinder after 1500 iterations of the PSO algorithm, with adaptive inertia weight and mutation terms added to the initial hinge points.

Figure 8 shows that before optimization, the maximum output force for driving the main-arm rotary joint (Joint 1) was $f_1\left(X\right)$ = 8330 N with an output force fluctuation amplitude of $f_3\left(X\right)$ = 6228 N. After optimization, the maximum driving force was reduced to 4714 N, a decrease of 43.4%, and the output force fluctuation amplitude was reduced to 2471 N, a decrease of 60.3%.

Figure 9 shows that before optimization, the maximum output force for driving the main-arm pitch joint (Joint 2) was $f_1\left(X\right)$ = 36,176 N, with an output force fluctuation amplitude of $f_3\left(X\right)$ = 27,580 N. After optimization, the maximum driving force was reduced to 33,902 N, a decrease of 6.3%, and the output force fluctuation amplitude was reduced to 24,405 N, a decrease of 11.5%.

For the pitch motion of the elbow joint, the force on the elbow joint varies with different pitch postures of the main arm. By using α₂ and α₃ as dual independent variables, the maximum output force and tensile force of Joint 3’s linear cylinder are determined. An exhaustive search of the entire arm’s pitch and elbow joint’s pitch motion found that when α₂ = 37.9°, Joint 3’s linear cylinder experiences the maximum thrust. When α₂ = −22.1°, Joint 3’s linear cylinder experiences the maximum tensile force. Figure 8 compares the maximum thrust and tensile force of Joint 3 with extreme postures of the manipulator with the output force for the initial posture of the manipulator. It can be seen that within the range of [0°~15°], there is a sudden change in the output force of Joint 3. Through comprehensive analysis, it was found that the manipulator is in a near-closed state, causing abnormal force on the elbow joint. Therefore, the pitch angle range of [15°~135°] is selected for the elbow joint of the manipulator to avoid the region of sudden force change.

Figure 10 shows that before optimization, the maximum output force for driving the pitch joint of the auxiliary arm (Joint 2) was $f_1\left(X\right)$ = 84,835 N, with an output force fluctuation amplitude of $f_3\left(X\right)$ = 169,397 N. After optimization, the maximum driving force was reduced to 39,218 N, a decrease of 53.8%, and the output force fluctuation amplitude was reduced to 63,100 N, a decrease of 62.8%.

The optimized hinge positions are shown in

Table 3. Optimized hinge-point values for the main-arm pitch joint (mm).

Hinge-Point Position Parameters		X1	X2	X3	X4	X5	X6	X7
Shoulder joint rotation	Before optimization	22.5	30.5	155	384.5	94	417.5	-
	After optimization	24.5	22.5	153.2	410.7	89.7	424.1	-
Shoulder joint pitch	Before optimization	186	48.5	718	134	143.5	194	15
	After optimization	160	42	760	165	116.9	179.2	24.2
Elbow joint pitch	Before optimization	132.8	826.6	128.6	1042.6	117.4	1091.9	135.8
	After optimization	106.3	990	141.3	1356	105.3	1489	147.5

. Based on this, the design of the manipulator’s large-load, small-angle joint hinge is completed. Additionally, the design of the linear cylinder is determined by the requirements of the maximum output force and the telescopic displacement length.

3.4. Topology Optimization of the Upper-Arm Structure

During operations, underwater manipulators are often mounted on manned submersibles (ROVs) or unmanned submersibles (AUVs). Increasing the manipulator’s weight requires the submersible to increase its weight by several times to offset the impact. Therefore, the manipulator’s weight must be minimized as much as possible while meeting operational requirements [34]. However, when the manipulator is engaged in load-bearing tasks, the weight of the end load can cause structural deformation, leading to positional deviations at the end-effector. The manipulator must have sufficient rigidity to ensure stability and accuracy during end-effector operations. This paper constructs a stress model for the manipulator’s main arm and employs structural topology optimization methods to reduce the weight. A cantilever beam topology optimization approach for the forearm is used to ensure that the end deformation remains within controllable limits [35].

3.4.1. Force Analysis and Model Construction

After optimizing and determining the optimal hinge-point positions, during the manipulator’s grasping operations, the main arm is subjected to both the thrust at the hinge connection of Joint 2′s linear cylinder and the push or pull force at the hinge point of Joint 3. Setting the shoulder joint connection as a fixed cylindrical support, we analyze the forces at the remaining three hinge points of the main arm. We identify the extreme condition where the end displacement is maximized during various operations under the maximum load. The topology optimization of the main-arm structure is performed for this condition to ensure that the manipulator maintains sufficient stiffness under extreme working conditions, control the end-load deformation displacement, and guarantee operational precision.

After optimizing the hinge positions and determining their locations, it is assumed that, during the manipulator’s gripping operation, the connection between the main arm and the shoulder joint is a fixed cylindrical support. Currently, the main arm is subjected to the thrust from the hinge connection of Joint 2′s linear cylinder and the push or pull force from the hinge of Joint 3. This creates a bending moment on the main arm of the manipulator. The main arm of the manipulator is simplified to a linkage, with the starting point as O₂ and the endpoint as O₃, to analyze the impact of the forces acting on the main arm on the deformation at the end point O₃. Neglecting the effect of the end swing angle on the overall mechanism, the angle α₂ between the main arm and the horizontal plane, as well as the angle α₃ between the secondary arm and the main arm, determines the different working postures of the manipulator. Different working postures result in varying forces on the main arm. By changing α₂ and α₃, the posture that produces the maximum bending moment at point O₂ under the maximum load during the gripping operation is identified. The main-arm structure is then topologically optimized for this posture to ensure sufficient stiffness under extreme working conditions, control the deformation displacement of the end load, and maintain the precision of the operation.

As shown in Figure 11, the angle between Joint 2 and the main arm at point B₂ is ${\theta }_1$, the angle between Joint 3 and the main arm at point A₃ is ${\theta }_2$, and the angle between Joint 3 and the centerline of the forearm is γ. These three angles can be expressed as

$${\theta }_1=\arcsin ({\displaystyle \frac{O_2{A}_2\sin ({\alpha }_2+{\phi }_2)}{A_2{B}_2}}),{\theta }_2=\arcsin ({\displaystyle \frac{O_3{B}_3\sin ({\alpha }_3+{\phi }_3)}{A_3{B}_3}})$$

(30)

$$\gamma ={180}^{\circ }-{\theta }_2-{\alpha }_3$$

(31)

The axial tensile force of the forearm is

$$F_4=F_3\cos \gamma +G\cos \beta$$

(32)

Therefore, the torque M generated by the forces from each hinge-point position and the self-weight of the main arm under maximum load conditions is calculated as follows:

$$M=O_2{B}_2\cdot F_2\sin {\theta }_1+O_2{A}_3\cdot F_3\sin {\theta }_2+O_2{O}_3\cdot F_4\sin {\alpha }_3$$

(33)

In the equation, O₂B₂ is 0.643 m, O₂A₃ is 0.81 m, and O₂O₃ is 1.31 m. Based on the ranges of α₁ and α₂, a plot was generated in MATLAB, resulting in the figure below.

As shown in Figure 12, the maximum value is obtained when α₂ = 39.1° and α₃ = 135° are at their respective critical angles. At this point, the hinge point O₂ of the main arm endures a maximum bending moment of 21,349 N·m, significantly impacting the end position. Based on this condition, the topology optimization of the main arm is carried out.

3.4.2. Topology Optimization

The initial design of the manipulator’s primary-arm model involves constructing the main arm based on optimized hinge positions and performing a thinning design on the side walls. The entire arm is made of TC4 material with a density of 4.44 g/cm³, an elastic modulus of 110 GPa, and a Poisson’s ratio of 0.34. The final weight of the main arm is 25.5 kg. As shown in Figure 13, using ANSYS software, a cylindrical support is applied at hinge point O₂, connecting the main arm and the shoulder joint, in the static structural analysis settings. Forces are applied at hinge points B₂, A₃, and O₃ when α₂ = 39.1° and α₃ = 135°. The extreme condition forces F₂, F_3, and F₄ are calculated using Equations (14), (18), and (32). Under these conditions, the manipulator experiences the maximum bending moment relative to hinge point O₂. In the static structural simulation, the initial model without topology optimization has a maximum end displacement of 38.47 mm.

Topology optimization was conducted in ANSYS, with the optimization region designed as a rectangular area based on the dimensions of the main arm, measuring 150 mm in height, 150 mm in width, and 1600 mm in length, excluding the hinge points where load constraints are applied [36]. The level-set topology optimization method was used, with an initial weight of 120.5 kg. The objective was to minimize compliance while retaining 20% of the mass. The optimization results are shown in Figure 13b. As shown in Figure 14a, by varying the mass retention ratio of the main arm and repeatedly iterating the topology optimization, the relationship between the maximum deformation and weight of the main arm was obtained.

3.4.3. Optimization Results Analysis

A topological analysis of the main-arm structure of the manipulator was performed, and a structure suitable for operating conditions was designed. The main-arm structure was optimized for extreme load conditions to distribute and transmit loads better, avoiding local overloads. This optimization enhances the structural stability of the manipulator while meeting the overall load precision requirements, providing robust structural support for high-precision deep-sea operations. The final optimization results show that when the initial weight is relatively high, reducing the weight has a minor impact on the maximum deformation. However, when the weight is lower, weight optimization exacerbates the deformation of the main arm. As shown in Figure 14, the analysis of the relationship between the topologically optimized mass and maximum end deformation reveals a relationship approximating an inverse function. For precision operations, maintaining the maximum deformation of the end load within 15 mm is considered appropriate. Therefore, when the weight is set at 19.6 kg, the maximum deformation is 14.48 mm. Compared to the non-topologically optimized model, the optimized main-arm weight decreased from 25.5 kg to 19.6 kg, a reduction of 23%. The maximum deformation of the main arm decreased from 38.47 mm to 14.48 mm, a reduction of 62.4%. The optimized manipulator reduces weight and enhances the main arm’s stiffness, improving the manipulator’s load-bearing capacity and meeting the usage conditions under extreme working conditions.

4. Manipulator Performance and Kinematic Analysis

4.1. Overall Performance Analysis

To meet the requirements for the complex deep-sea working environment, avoid the impact of seabed obstacles, and reduce singularities in the joint space, the deep-sea water-hydraulic-driven manipulator is designed as a six-degree-of-freedom, seven-function RRRRRCG manipulator. Large-load, small-angle outputs are achieved through a linear cylinder with a three-hinge amplitude variation mechanism. In contrast, small-load, large-angle outputs are realized through a rack-and-pinion rotary cylinder. The schematic diagram of the manipulator structure and linkage coordinate system, established using the improved D-H parameter method [37], is shown in Figure 15.

In this study, the base is designated Link 0, and the gripper is designated Link 6. The linkage coordinate system and the corresponding linkage parameters are shown in

Table 4. D-H parameters of the deep-sea water-hydraulic manipulator.

Joint Number	${\mathit{a}}_{\mathit{i}-1}$/mm	${\mathit{\alpha }}_{\mathit{i}-1}$/(°)	${\mathit{d}}_{\mathit{i}}$/mm	${\mathit{\theta }}_{\mathit{i}}$/(°)	Joint Angle Range/(°)	Link Parameters
1	0	0	0	θ1	−60~+60	-
2	$a_1$	90	0	θ2	−30~+90	$a_1$ = 108 mm
3	$a_2$	0	0	θ3	−75~+45	$a_2$ = 1310 mm
4	$a_3$	90	d4	θ4	−135~+135	$a_3$ = 180 mm
5	0	−90	0	θ5	−90~+90	$d_4$ = 750 mm
6	0	90	d6	θ6	−180~+180	$d_6$ = 515 mm

.

Using the homogeneous transformation matrix of the linkage coordinate system, the transformation between each link ${}_i{}^{i-1}T$ can be obtained. The coordinate position and orientation of the manipulator’s end-effector center relative to the base can be calculated using MATLAB (R2024a)’s symbolic computation.

4.1.1. Forward Kinematic Analysis

Using the homogeneous transformation matrix of the linkage coordinate system, the transformation between each link ${}_i{}^{i-1}T$ can be obtained. By multiplying the transformation matrices of each link, the pose matrix of the end-effector ${}_6{}^{0}T$ is obtained. The position of the end-effector relative to the base coordinate can be expressed by the rotation transformation matrix R and the translation vector P:

$${}_6{}^{0}T={}_1{}^{0}T{}_2{}^{1}T\cdots {}_6{}^{5}T=\left[\begin{array}{cccc}nx& ox& ax& px\\ n_y& o_y& a_y& p_y\\ n_z& o_z& a_z& p_z\\ 0& 0& 0& 1\end{array}\right]$$

(34)

where [n_x, n_y, n_z]^T is the first column of the rotation matrix R, representing the x-axis direction of the transformed coordinate system in the original coordinate system; [o_x, o_y, o_z]^T is the second column of the rotation matrix R, representing the y-axis direction of the transformed coordinate system in the original coordinate system; [a_x, a_y, a_z]^T is the third column of the rotation matrix R, representing the z-axis direction of the transformed coordinate system in the original coordinate system; and [p_x, p_y, p_z]^T is the translation vector P, representing the position of the transformed origin in the original coordinate system.

The position of the end-effector is expressed as follows, where S₁ = sinθ₁, C₁ = cosθ₁, S₁ = sin(θ₁ + θ₂), and similarly for the other terms:

$$P_x=d_4{C}_1{S}_{23}+a_3{C}_1{C}_{23}+d_6\left(S_5{S}_1{S}_4+C_1{C}_4{C}_{23}{S}_5\right)+d_6{C}_5{C}_1{C}_{23}+a_1{C}_1+a_2{C}_1{C}_2$$

(35)

$$P_y=d_4{S}_1{S}_{23}+a_3{S}_1{C}_{23}-d_6\left(S_5{C}_1{S}_4+S_1{C}_4{C}_{23}{C}_5\right)-d_6{C}_5{S}_1{S}_{23}+a_1{S}_1+a_2{C}_2{S}_1$$

(36)

$$P_z=a_3{S}_{23}-d_4{C}_{23}+a_2{S}_2-d_6(C_5{C}_{23}-C_4{C}_5{S}_{23})$$

(37)

In the case where the structural parameters of the manipulator are determined, the position of the end-effector relative to the base is only related to the angles of each joint. Using the pose matrix of the end-effector established by forward kinematics and the D-H parameters of each link of the manipulator, the workspace of the manipulator was simulated using the analytical method. By setting the number of sampling points to 1,000,000 and running the simulation in Matlab, the operational space of the manipulator’s end-effector was obtained, as shown in Figure 16.

These predictions are crucial for understanding the capabilities and limitations of the manipulator in various planes, thereby allowing for the effective planning of operational tasks in nuclear industry environments. Analyzing Figure 16, we can derive the following observations from the operational space results obtained using the Monte Carlo method:

(1) In the XOY working plane projection, the maximum coverage length in the X-direction ranges from −0.268 m to 2.63 m, and the maximum coverage on both sides in the Y-direction is 2.33 m.

(2) From the XOZ plane projection, it can be seen that the coverage below the manipulator base is relatively complete, with the maximum coverage length in the Z-direction ranging from −1.94 m to 2.48 m.

(3) From the YOZ plane projection, the overall coverage area of the manipulator’s working direction is relatively complete, with a coverage rate greater than 75%.

These results indicate that the manipulator can achieve the expected workspace coverage required for operations, ensuring it can effectively perform tasks within its designed range.

4.1.2. Inverse Kinematic Analysis

Based on Pieper’s criterion, an inverse kinematic model of the manipulator is established using the analytical method. By solving the forward kinematic equation and multiplying both sides of Equation (34) by the matrix ${}_1{}^0{T}^{-1}$, we obtain the following form:

$${{}_1{}^{0}T}^{-1}\left[\begin{array}{cccc}{n}_x& o_x& a_x& p_x\\ n_y& o_y& a_y& p_y\\ n_z& o_z& a_z& p_z\\ 0& 0& 0& 1\end{array}\right]={}_2{}^{1}T{}_3{}^{2}T\cdots {}_6{}^{5}T$$

(38)

Expanding the left side of the expression results in

$$\left[\begin{array}{cccc}{C}_1{n}_{\mathrm{x}}+S_1{n}_y& C_1{o}_{\mathrm{x}}+S_1{o}_y& C_1{a}_{\mathrm{x}}+S_1{a}_y& C_1{p}_{\mathrm{x}}+S_1{p}_y\\ -S_1{n}_{\mathrm{x}}+C_1{n}_y& -S_1{o}_{\mathrm{x}}+C_1{o}_y& -S_1{a}_{\mathrm{x}}+C_1{a}_y& -S_1{p}_{\mathrm{x}}+C_1{p}_y\\ n_z& o_z& a_z& p_z\\ 0& 0& 0& 1\end{array}\right]$$

(39)

Expanding the right side of the expression leads to

$$\left[\begin{array}{cccc}-C_6(S_5{S}_{23}-C_4{C}_5{C}_{23})-S_4{S}_6{C}_{23}& S_6(S_{23}{S}_5-C_{23}{C}_4{C}_5)-C_{23}{C}_6{C}_4& C_{23}{C}_4{S}_5+S_{23}{C}_5& a_1+a_3{C}_{23}+d_4{S}_{23}+a_2{C}_2+d_6(C_5{S}_{23}+C_4{S}_5{C}_{23})\\ -C_4{S}_6-C_5{C}_6{S}_4& C_5{S}_4{S}_6-C_4{C}_6& -S_4{S}_5& -d_6{S}_4{S}_5\\ C_6(S_5{C}_{23}+C_4{C}_5{S}_{23})-S_4{S}_6{S}_{23}& -S_6(C_{23}{S}_5+S_{23}{C}_4{C}_5)-S_{23}{C}_6{C}_4& S_{23}{C}_4{S}_5-C_{23}{C}_5& a_3{S}_{23}-d_4{C}_{23}+a_2{S}_2-d_6(C_5{C}_{23}+C_4{S}_5{S}_{23})\\ 0& 0& 0& 1\end{array}\right]$$

(40)

Since the matrices are equal, the corresponding elements on both sides must be equal. From this, the joint angles can be calculated as follows:

$${\theta }_1=a\tan 2(0,\pm 1)-a\tan 2\left(d_6{a}_y-p_y,p_x-d_6{a}_x\right)$$

(41)

$${\theta }_2=a\tan 2(j/\rho ,\pm \sqrt{1-{\left(j/\rho \right)}^2})-a\tan 2\left(i,h\right)$$

(42)

where $j=a_2+a_3{C}_3+d_4{S}_3,\rho \sin \varphi =i,\rho \cos \varphi =h$.

$${\theta }_3=a\tan 2(k/\rho ,\pm \sqrt{1-{\left(k/\rho \right)}^2})-a\tan 2\left(2a_2{a}_3,2a_2{d}_4\right)$$

(43)

where $h=p_z-d_6{a}_z, q=a_3^2+d_4^2+a_2^2, i=P_x{C}_1+P_y{S}_1-a_1-d_6{a}_x{C}_1-d_6{a}_y{S}_1, k=h^2+i^2-q$.

$${\theta }_4=a\tan 2({\displaystyle \frac{a_x{S}_1-a_y{C}_1}{S_5}},{\displaystyle \frac{a_z{S}_{23}+a_x{C}_{23}{C}_1+a_y{C}_{23}{S}_1}{S_5}})$$

(44)

$${\theta }_5=a\tan 2(\pm \sqrt{1-m^2},m)$$

(45)

where $m=-a_z{C}_{23}+a_x{S}_{23}{C}_1+a_{\nu }{S}_{23}{S}_1$.

$${\theta }_6=a\tan 2({\displaystyle \frac{-o_z{C}_{23}+o_x{S}_{23}{C}_1+o_y{S}_{23}{S}_1}{S_5}},{\displaystyle \frac{n_z{C}_{23}-n_x{S}_{23}{C}_1-n_y{S}_{23}{S}_1}{S_5}})$$

(46)

Multiple sets of angle solutions exist for the same manipulator end-effector position. This paper filters feasible solutions through angular space constraints to reduce the impact of sizable joint angle changes on end-effector precision. The pose matrix of the expected trajectory points is compared with the homogeneous transformation matrices of multiple feasible solutions through column decomposition. This comparison yields weighted variations in postures n, o, and a and position p. The importance of angle variation and posture variation is set during the sorting phase. The solutions are sorted using the sort function to output the optimal solution with the smallest overall difference from the expected trajectory points [38].

$$\{\begin{array}{l}{x}_i=x_c+r\cdot cos({\displaystyle \frac{2\pi i}{N}})\\ y_i=y_c+r\cdot sin({\displaystyle \frac{2\pi i}{N}})\\ z_i=z_c\end{array},i=0,1,2,\dots ,N-1$$

(47)

where the Cartesian spatial coordinates of the circle’s center are set to [x_c, y_c, z_c], with values of [1, 0, 0]; the radius r is set to 0.5 m; and the number of path points N is set to 1000.

This paper uses a circular path in space as the target trajectory for the end-effector. By solving the spatial parameters of the trajectory circle and adding pose constraints, trajectory points are generated. The optimal joint angles for each set of trajectory points are obtained through an inverse kinematic model, and the angle matrix is substituted into the forward kinematic model to verify the feasibility of the optimal solution in Cartesian space path planning. Given an initial set of angles [0°, 20°, 15°, 22°, 16°, 12°], the corresponding position of the manipulator’s end-effector is [1.5, 0, 0], and the motion time is 20 s. The trajectory circle’s center coordinates, radius, normal vector, and plane parameters are obtained through spatial geometric formulas. The rotation axis vector for the end-effector’s motion along the arc is further derived, resulting in the homogeneous transformation matrix for each trajectory point corresponding to the end-effector.

As shown in Figure 17a, the optimal joint angles corresponding to the trajectory points are obtained using MATLAB, and the joint angle curves of the manipulator are plotted. The joint angle variations are relatively smooth, and the end-effector’s trajectory-following performance is good. As shown in Figure 17b–d, for the high-load joint, the linear cylinder’s stroke length and output force vary relatively smoothly, verifying the feasibility of the optimal inverse kinematics solution method.

4.2. Manipulator Hydrodynamic Analysis

The deep-sea water-hydraulic manipulator, a complex nonlinear system with coupling variables across its links, requires the determination of the driving torques at each joint for the dynamic control of the system during end-effector motion. Compared to surface environments, the manipulator in deep-sea environments is affected by buoyancy, water resistance, and other factors, necessitating a hydrodynamic model analysis.

As shown in Equation (48), the Euler torque balance equation describes the relationship between the torques acting on the links, their angular velocities, and angular accelerations:

$$\tau ={\rm I}\cdot \alpha +\omega \times ({\rm I}\cdot \omega )$$

(48)

where I is the inertia tensor, α is the angular acceleration, and ω is the angular velocity.

During the manipulator’s motion, points on the links rotate around their respective joints. Using the recursive formulas for velocity and acceleration between the links, the angular velocity, angular acceleration, and linear velocity and acceleration of the center of mass of each link can be progressively calculated starting from the base. From Equation (48), the linear velocity $v_i$ of each point on the link can be determined as a function of the distance l from the rotational axis:

$$v_i=\omega l$$

(49)

where l is the distance from the pivot point to a certain point on the cylinder, ranging from 0 to L, and L is the length of the link, which, in a manipulator, refers to the length of the upper or secondary arm.

The main forces acting on the link underwater include buoyancy and fluid resistance. According to the Morison equation, the fluid resistance on the link consists of drag force and inertia force. Based on the assumptions for the hydrodynamic environment of the manipulator, the deep-sea working environment is considered an infinite flow field, and the link is simplified to a regular cylinder with negligible effect on the flow field. During the link’s motion, it is primarily subjected to buoyancy and fluid resistance arising from the relative motion between the link and the fluid. When the manipulator operates underwater, assuming that the fluid velocity at any water point is zero, the fluid resistance moment experienced is

$${\tau }_h={\tau }_d+{\tau }_m=C_D\rho r{\displaystyle {\int }_0^L{v}_i\left|v_i\right|ldl}+C_M\rho \pi r^2{\displaystyle {\int }_0^L\frac{\partial v_i}{\partial t}}ldl$$

(50)

where ${\tau }_h$ represents the generalized water resistance torque acting on the manipulator system; ${\tau }_d$ represents the drag torque generated by the fluid; ${\tau }_m$ represents the inertia torque generated by the fluid; C_D is the drag coefficient; C_M is the inertia coefficient; $\rho$ is the density of water; r is the radius of the simplified cylindrical body of each link; and $v_i$ represents the velocity vector of the link relative to the fluid’s velocity component.

Since the direction of buoyancy is always opposite to gravity and the center of buoyancy coincides with the center of mass of the link, this paper considers the combined effect of the gravity and buoyancy forces acting on the manipulator, resulting in a net force of

$$F_f=\Delta \rho V_{i}g$$

(51)

where $\mathsf{\Delta }\rho$ represents the density difference between the manipulator link material and water; $V_i$ is the volume of the link; and g is the gravitational acceleration. Based on the manipulator’s Newton–Euler dynamic model, considering the effects of hydrodynamics and incorporating the buoyancy and hydrodynamic terms into the equations, the overall underwater dynamics equation of the manipulator is obtained as follows:

$$M(\theta )\ddot{\theta }+v(\theta ,\dot{\theta })+g(\theta )+F_f+{\tau }_h=\tau$$

(52)

where $M\left(\theta \right)$ represents the mass matrix of the manipulator; $v\left(\theta ,\dot{\theta }\right)$ is the Coriolis and centrifugal force vector matrix; $g\left(\theta \right)$ is the gravity vector matrix of the manipulator; and τ is the driving torque.

Based on the inverse kinematics solution of the target trajectory, the rotational speeds of each manipulator link were calculated in MATLAB. The main material of the manipulator is TC4, with a density of 4.44 g/cm³. According to the manipulator model, the centroids of each link were determined, with the main arm simplified as a cylindrical body with a radius R = 0.15 m and the secondary arm simplified as a cylindrical body with a radius r = 0.13 m. The effects of underwater buoyancy and hydrodynamic resistance torque on the manipulator are considered separately. For the water-hydraulic manipulator discussed in this paper, the first three joints generate the most significant disturbances on the links; therefore, this section only analyzes the impacts on Joints 1, 2, and 3 in the underwater environment.

As seen from Equation (50), when the structural dimensions of the manipulator are fixed, the velocity significantly affects the fluid resistance of the manipulator. The water-hydraulic manipulator designed in this paper is a low-speed, heavy-duty manipulator. As shown in Figure 18, when completing the set target trajectory within 20 s, the maximum water resistance torque is 3.11 N·m at the 15th second during the pitch movement of Joint 2. Currently, the torque output of Joint 2 in air is 1218 N·m, which is only 0.25% of its value. Considering the effect of buoyancy, the output is 821 N·m, only 0.36% of its value, and the impact of underwater buoyancy accounts for 32.56% of the torque output of the manipulator in air. Therefore, during the low-speed operation of the underwater manipulator described in this paper, the proportion of water resistance torque is minimal. Thus, fluid resistance is temporarily not considered in the end-effector position tracking error simulation.

4.3. Manipulator Joint Control Precision Testing

4.3.1. Hydraulic Linear Cylinder Test

The position control characteristics of the hydraulic linear cylinder were tested on a test bench. According to the inverse kinematic analysis of the manipulator, the change in the hinge length of the linear cylinder during the circular motion was determined, and the displacement change of the linear cylinder was calculated. In the program, the displacement variation curve was set, and a magnetostrictive displacement sensor and the output length of the linear cylinder provided the position feedback. This determined the follow-up characteristics of the linear cylinder under the hydraulic drive and verified the control precision of the linear cylinder. As shown in Figure 19, the system is pressurized by a seawater pump. The controller controls the direction and speed of the linear cylinder motion by regulating the directional valve, with the throttle valve limiting the flow [39]. A bidirectional hydraulic lock maintains the pressure in the hydraulic chamber of the linear cylinder. Sensors collect position data to provide position feedback for the linear cylinder, thus adjusting the position of the hydraulic cylinder. The experiment, conducted under the actual working conditions of the manipulator, considered external disturbances such as the friction of the seals in the hydraulic cylinder, the leakage of the control valve group, and pressure loss in the system pipelines. The control accuracy of the water-hydraulic linear cylinder was determined.

Figure 20 shows the displacement tracking curves and errors of the manipulator’s water-hydraulic linear cylinders driving Joints 1, 2, and 3. The target curves are based on the circular trajectory of the manipulator, with joint rotation angles determined through inverse kinematics. The first three joints of the manipulator are driven by a three-hinge-point mechanism with linear cylinders, where the angular output errors can be calculated from the linear output errors. Figure 20a,b show that the actual displacements of the water-hydraulic linear cylinders under PID control fluctuate around the target displacements, with a relatively large tracking error, the maximum value being approximately 1.5 mm. Figure 20c shows that the maximum fluctuation of the displacement tracking error for Joint 3 is within 1 mm. The results indicate that for different waveforms, the tracking accuracy of the hydraulic cylinders can be maintained within 1.5 mm, demonstrating good control performance.

4.3.2. Hydraulic Swing Cylinder Test

As shown in Figure 21, the target angle output tracking accuracy test for the water-hydraulic gear-rack swing cylinder was conducted under load conditions. The test system is pressurized by a seawater pump, with the directional valve controlling the rotational direction of the swing cylinder, a throttle valve limiting the flow, and a bidirectional hydraulic lock maintaining the hydraulic chamber pressure of the swing cylinder. The load torque is adjusted and controlled by a magnetic powder brake powered by an electric source. The real-time output angle of the gear-rack swing cylinder is measured by a magnetostrictive displacement sensor integrated within the rack. The relationship between the rack’s displacement and the gear shaft’s rotation provides feedback for the angle output of the swing cylinder. The experiment, conducted under the actual working conditions of the manipulator, considered external disturbances such as the friction of the seals in the hydraulic cylinder, the leakage of the control valve group, and pressure loss in the system pipelines. The control accuracy of the water-hydraulic swing cylinder was determined.

Figure 22 shows the error between the target output angle and the actual angle of the gear-rack swing cylinder under a load. As seen in Figure 22a, the output angle fluctuation range for Joint 4 is relatively small, with the angle varying within the range of −5° to 5°, and the fluctuation in angular error is within 1.5°. Figure 22b shows that the output angle of Joint 5 is more stable, with fluctuations within 1°, indicating high angular accuracy for the manipulator, meeting the output angle requirements for the manipulator joints.

4.4. Manipulator Kinematic Analysis Simulation Verification

The control of the manipulator’s end position is achieved through the actions of its joints. The precision of each joint’s movement directly impacts the overall control accuracy of the manipulator’s end position. For hydraulic manipulators, achieving a completely sealed drive chamber and handling pressure pulsations from the power source and valve leakage is challenging and can affect position control. Compared to oil-hydraulic systems, water-hydraulic control is even more difficult. The joints of the manipulator described in this paper have been developed, and the entire system is assembled. The motion control accuracy of each joint was determined by conducting displacement tracking tests on each water-hydraulic actuator of the manipulator. Considering the cumulative effect of linkage amplification, the cumulative joint movement errors were simulated to determine the manipulator’s end position and analyze its control accuracy.

Based on the set target trajectory, inverse kinematics determines the variation in each joint angle over time when the manipulator performs the specified actions. From the time-dependent changes in joint angles, the displacement–time curve for the linear cylinder output and the angle–time curve for the swing cylinder output are derived. Experiments were conducted on each actuator of the manipulator according to these curves, obtaining the error between the target and actual positions.

For the first three joints of the manipulator, the linear motion of the linear cylinders is converted into the rotational output of the joints. It is necessary to reverse-calculate the relationship between the displacement of the internal piston rod of the linear cylinder and time based on the joint rotation angles. Position tracking errors are determined through water-hydraulic linear cylinder displacement tracking experiments. These errors are added to the piston rod’s target displacement to obtain the linear cylinder’s adjusted output position, considering the characteristics of the water-hydraulic drive. The actual output angle of the joint is then calculated based on the actual output position of the linear cylinder. The experimentally obtained angle output error can be directly incorporated into the swing cylinder. Finally, forward kinematic analysis is performed to determine the end-effector pose of the manipulator, considering the characteristics of the water-hydraulic drive. The results obtained are shown in Figure 23.

As shown in Figure 24a, the control accuracy of the manipulator is fitted based on the control errors caused by factors such as friction, leakage, and pressure loss that affect each joint actuator in the actual working environment. The maximum projection error of the designed water-hydraulic manipulator’s end-effector in the three-plane space is within 15 mm. As shown in Figure 24b, the automatic tracking accuracy of the end-effector in three-dimensional space can reach within 18 mm. The water-hydraulic manipulator arm designed in this study adopts a master–slave control system mounted on a manned submersible in a deep-sea environment and controlled by an operator through visual observation. Therefore, as a remotely operated water-hydraulic underwater manipulator, its maximum projection error in planar space is less than 15 mm. Although slightly inferior to the 10 mm error of oil-hydraulic-driven manipulators, within this error range, the difference between the selected and measured points is considered reasonable, meeting the requirements for deep-sea exploration, sampling, and other tasks [40].

5. Conclusions

This paper proposes a water-hydraulic manipulator suitable for deep-sea operations. Based on the characteristics of water-hydraulic drive systems, the design includes a water-hydraulic linear cylinder, a water-hydraulic gear-rack swing cylinder, and a water-hydraulic radial piston motor. After its design, the mechanical joint mechanism was optimized. The installation hinge points of the linear cylinders, which had increased non-stroke lengths due to oil-free compensation sealing methods with position sensors, were optimized using the particle swarm optimization (PSO) algorithm. This optimization reduced the demand for output force during the overall rotation, main-arm pitching, and secondary-arm pitching movements; minimized output force fluctuations; and improved the manipulator’s angular output stability. Under extreme load conditions, the manipulator’s main arm underwent topology optimization to reduce the structural weight and enhance the manipulator’s power-to-weight ratio. Finally, regarding manipulator control, the workspace reachability was analyzed using forward kinematics, and motion control solutions within the workspace were achieved through inverse kinematics. Experimental analysis of the position tracking characteristics of the developed water-hydraulic linear cylinder and swing cylinder was conducted, and the control accuracy of the designed water-hydraulic manipulator’s end-effector was fitted to within 18 mm. This paper completes the overall design and structural optimization of the water-hydraulic manipulator and provides a preliminary analysis of the manipulator’s motion and control performance. Future work will focus on improving the control precision of the valve-controlled cylinders and incorporating the effects of deep-sea migration characteristics on the manipulator. The main conclusions are as follows:

(1) By analyzing the working principles of the water-hydraulic manipulator and the characteristics of the water-hydraulic drive, a piston motion sealing form was selected, which offers higher reliability for the water-hydraulic drive. Consequently, reliable actuating components for the water-hydraulic manipulator were designed, including a water-hydraulic linear cylinder, gear-rack swing cylinder, and water-hydraulic radial piston motor.

(2) The particle swarm optimization algorithm was used to optimize the manipulator’s hinge positions of Joints 1, 2, and 3. After optimization, the maximum driving force of Joint 1 decreased by 43.4%, and the amplitude of output force fluctuations decreased by 60.3%. The maximum driving force of Joint 2 decreased by 6.3%, and the amplitude of output force fluctuation decreased by 11.5%. The maximum driving force of Joint 3 decreased by 53.8%, and the amplitude of output force fluctuation decreased by 62.8%. After topological optimization, the manipulator’s main arm saw a weight reduction of 23% and a maximum deformation reduction of 62.4%.

(3) Through forward kinematic analysis, the manipulator’s operating range was 2.5 m, with a spatial coverage rate of over 75%. Inverse kinematic analysis solved the joint angles during trajectory motion, achieving manipulator trajectory planning. Displacement tracking tests of the water-hydraulic actuators showed that the position control precision of the actuators was within 1.5 mm. Joint position error fitting revealed that the water-hydraulic manipulator’s overall end position control precision is within 18 mm.