Design, Hydrodynamic Analysis, and Testing of a Bio-inspired Movable Bow Mechanism for the Hybrid-driven Underwater Glider

Abstract

Hybrid-driven Underwater Glider (HUG) is a new type of underwater vehicle which integrates the functions of an Autonomous Underwater Glider (AUG) and an Autonomous Unmanned Vehicle (AUV). Although HUG has the characteristics of long endurance distance, its maneuverability still has room to be improved. This work introduces a new movement form of the neck of the underwater creature into HUG and proposes a parallel mechanism to adjust the attitude angle and displacement of the HUG’s bow, which can improve the steering maneuverability. Firstly, the influence of bow movement and rotation on the hydrodynamic force and flow field of the whole machine is analyzed by using the Computational Fluid Dynamics (CFD) method. The degree of freedom, attitude control range and movement amount of the Movable Bow Mechanism (MBM) are obtained, and then the design of MBM is completed based on these constraints. Secondly, the kinematic and dynamic models of MBM are established based on the closed vector method and the Lagrange equation, respectively, which are fully verified by comparing the results of simulation in Matlab and Adams software, then a Radial Basis Function (RBF) neural network adaptive sliding mode controller is designed to improve the dynamic response effect of the output parameters of MBM. Finally, a prototype of MBM is manufactured and assembled. The kinematic, dynamics model and controller are verified by experiments, which provides a basis for applying MBM in HUGs.

1 Introduction

Autonomous Underwater Glider (AUG) is an unmanned submersible vehicle that realizes a sawtooth trajectory by adjusting its net buoyancy [1]. It has the characteristics of long endurance and low noise. However, net buoyancy as the driving power makes AUG less maneuverable and difficult to operate flexibly in terrible ocean environments [2].

Maneuverability is an important indicator of underwater vehicles. In order to improve the maneuverability of AUG, many scholars have researched Hybrid-driven Underwater Gliders (HUGs) by adding a propeller [3]. Currently, the adjustment of heading angle and attitude of HUG mostly adopts the way of controlling vertical tail rudder and rotating battery system around the fuselage axis. The turning radius of HUG is generally in the order of tens of meters to hundreds of meters. We would admit that even though its maneuverability is better than AUG, it leaves plenty of room for improvement compared with underwater creatures.

Compared with the 40% propulsion efficiency of traditional underwater vehicles by a propeller, the propulsion efficiency of fish by swinging caudal fins can reach 80–90% [4, 5]. Many excellent studies [6,7,8,9] have been done to improve the maneuverability of the underwater vehicle by adding the swing tail movement, flapping wing, etc. Sun et al. [10] proposed a novel two-degree-of-freedom multi-mode controllable wing mechanism to improve the maneuverability of HUG, which imitated the humpback whale’s pectoral fin motion, such as wing folding and wing flapping, and successfully developed a prototype of HUG with that controllable wing mechanism and verified its performance by the pool test, which can turn with a near-zero radius. Iliya Mitin [11] describes a bioinspired propulsion system based on an elastic chord. Dependencies of fish velocity on the dynamic characteristics of tail oscillations were analyzed using the Computational Fluid Dynamics (CFD) method and verified by the pool test the pool test. Research in the field of bionics mainly focuses on the bionic design and fluid characteristics of caudal fins and pectoral fins to improve the maneuverability of underwater vehicles, but there are relatively few studies on the vehicle’ maneuverability upgrade by controlling the movement of the bow. Creatures such as penguins, sea lions, dolphins, etc., have high movement maneuverability in water owing to high flexibility and large swing angle space of the head and neck. Compared with underwater creatures that can barely move their heads, flexible bow movement is conducive to predation and high mobility movement [12]. Penguins can turn with a radius of only 10–30% of their body length by taking advantage of the head-neck orientation [13, 14]. However, there are few studies on the design and fluid characteristic analysis of underwater vehicles based on the characteristics of biological necks. The study on the relationship between the attitude of the bow and the hydrodynamics of the underwater vehicle and the parameters design of the Movable Bow Mechanism (MBM) have important theoretical and engineering application value for the innovative design of the underwater vehicles [15].

Wu et al. [17] introduced a single joint that functions as the neck for enhancing turning ability, the MBM can achieve ± 50◦ yawing motion and its feasibility is verified by pool experiment. Wang [18] designed a bionic dolphin robot with MBM and preliminarily analyzed the influence of the motion parameters of the bow on the straight, yaw, and pitch submarine. However, the design of MBM was not given, and the key performance parameters, such as turning radius and motion trajectory, were not analyzed. Liu et al. [19] developed a dolphin-like underwater robot by integrating MBM based on the cable drive with the tandem tail fin propulsion mechanism and verified the “C-type start” movement mode and control method of the bionic dolphin by experiment. However, the MBM driven by wire has poor stiffness, leading to low attitude control accuracy. The influence relationship between the mechanism parameters and the motion of the mechanism has not been analyzed. Yang et al. [20] developed a bionic dolphin underwater robot with a movable bow based on a serial mechanism and built the dynamics model of the bionic dolphin robot, and then planned the motion trajectory of the bionic dolphin robot in the bow deflection mode by using the dynamic model. However, the influence relationship between the attitude angle of bow and the fluid characteristics of the robot is not given which is the basis of the whole machine performance control, and the rigidity of the serial mechanism is poor which is quite different from the parallel driven structure of the biological muscle group.

To sum up, this study introduces the neck movement of underwater creatures into HUG to improve its manoeuvrability, such as penguins, sea lions, dolphins, and other creatures. By analyzing the physiological structure of underwater creatures’ neck movement driven by the parallel muscle groups, a new type of parallel mechanism is proposed to realize the motion of the bow. The regularity of the parameters of MBM on the hydrodynamic and fluid evolution of the whole machine is analyzed, and the relationship between the attitude angle of the bow and the swimming trajectory of HUG with a Movable Bow (HUGMB) is established. Based on the dynamic model of the mechanism, a Radial Basis Function (RBF) neural network adaptive sliding mode controller is designed. Finally, a prototype of MBM is manufactured and assembled, and experiments are conducted to verify the kinematic model. This study provides a new idea to improve the maneuvering performance of HUG.

2 Analysis of Hydrodynamic Characteristics

The bow movement of HUG can cause differences in pressure and changes in hydrodynamic parameters [21, 22] as the bow movement breaks the flow field symmetry on the surface of the vehicle. The range of MBM’s attitude can be acquired according to the hydrodynamic force obtained by CFD. Firstly, the relationship between the attitude angle of the bow and the hydrodynamics of HUGMB is analyzed in this section. In addition, the influence of the bow movement on the flow field around HUGMB is also explored by using the overlapping mesh technology of CFD. The reason why the hydrodynamic force of HUGMB changes with the bow movement is revealed from the perspective of vorticity evolution. Finally, based on the dynamic mesh method, the relationship between the bow parameters and the turning radius of HUGMB is analyzed according to the simulation trajectory of HUGMB under different conditions (Fig. 1).

Fig. 1

Penguin skeleton [16]

In order to facilitate the analysis and calculation, the shape of HUGMB is divided into three parts: the bow, the fuselage, and the wing. The shapes of both the bow and the main body are revolving. The wings are flat. The definition of HUGMB’s shape parameter is shown in Fig. 2. The values of shape parameters are shown in Table 1. Figure 3 shows the Three-Dimensional (3D) model of the HUGMB and defines the attitude angles of the bow, which are described by α and β. The coordinate system o₀-x₀y₀z₀ are defined as shown in Fig. 14. V is velocity vector of the HUGMB and the attitude vector k points from o₀ to the vertex M of the bow. Angle of Attack (AOA) is the angle between the projection of V on the o₀-y₀z₀ plane and the o₀-z₀x₀ plane, α is the angle between k and the o₀-z₀x₀ plane and β is the angle between the projection of k on the o₀-z₀x₀ plane and the o₀-y₀z₀ plane.

Fig. 2

Shape parameter definition of HUGMB

Table 1 Shape parameters of HUGMB

Fig. 3

Simplified calculation model of HUGMB

It is assumed that seawater is an ideal viscous fluid and HUGMB is a rigid body when HUGMB moves underwater. The Reynolds number of HUGMB is 8.2 × 10⁵. Therefore, the gliding motion of HUGMB is defined as the turbulent flow of incompressible viscous fluid, its governing equation can be expressed as follows [23]:

$$\left\{ \begin{gathered} \nabla {\mathbf{u}} = 0 \\ \frac{{\partial {\mathbf{u}}}}{\partial t} + {\mathbf{u}} \cdot \nabla {\mathbf{u}} = \frac{ - \nabla p}{p} + v\nabla^{2} {\mathbf{u}} \\ \end{gathered} \right.$$

(2-1)

The governing equation is solved in the ANASYS Fluent based on pressure, and the numerical discrete method adopts the Second Order Upwind Scheme. The SST k − ω turbulence model is used in this paper [24,25,26].

2.1 Hydrodynamic Simulation

In this section, the hydrodynamics of HUGMB at different AOAs and the attitude angles of the bow is analyzed by using CFD. The computational domain is shown in Fig. 4. The number of cells is 65 × 10⁴. The inlet of the computational domain is set as the velocity inlet with a velocity of 0.5 m/s, the outlet as the outflow, the wall as the symmetry, and the surface of HUGMB as the no-slip wall. The results of the grid independence verification are shown in Table 2, taking the AOA is 3° and α, β are both 0 as an example to ensure the accuracy of the calculation results. The maximum relative error is less than 1%, which meets the evaluation criteria of grid independence [27]. The parameter settings of different simulation cases are shown in Table 3.

Fig. 4

The computational domain for calculations of hydrodynamics force and moment of HUGMB

Table 2 Grid independence verification results

Table 3 Parameter Settings for simulation

The simulation results of the hydrodynamics of HUGMB are shown in Fig. 5. As shown in Fig. 5a, as β increases, the yaw moment increases and the relationship are approximately linear when the AOA is the same. However, the increase of α does not bring a big change in the yaw moment. Therefore, α has little effect on the yaw moment. In addition, The three curves in Fig. 5b basically coincide, proving that the AOA has almost no effect on the yaw moment and does not affect the relationship between β and yaw moment. The pitch moment is proportional to α when the AOA is the same, and the influence of β on the pitch moment is small, as shown in Fig. 5c. The three curves in Fig. 5d are almost parallel with the same spacing, which proves that the relationship between the AOA and the pitch moment is linear, and the AOA does not affect the relationship between α and the pitch moment. The faces in Fig. 5e are essentially flat surfaces, and the three curves in Fig. 5f are almost parallel, and the spacing is large; therefore, the effect of the attitude angle of the bow on the lift-to-drag ratio is obviously weaker than that of the AOA. By comparing the results in Fig. 5, we can know that: (1) the influences of α and β on the hydrodynamic force and moment of HUGMB are obviously independent. When both the value of α and β are changed, the hydrodynamic force is basically equal to the linear superposition of the respective influences of α and β. (2) the attitude angle of the bow can significantly affect the external moment of HUGMB almost without changing the lift-to-drag ratio.

Fig. 5

Simulation results of the hydrodynamics of HUGMB: (a) relation between yaw moment and α, β, AOA and (b) relation between yaw moment and β when α is 0 and (c) relation between pitch moment and α, β, AOA and (d) relation between pitch moment and α when β is 0 and (e) relation between the lift-to-drag ratio and α, β, AOA and (f) relation between the lift-to-drag ratio and β when α is 0.

2.2 Fluid Simulation During Bow Movement

According to Sect. 2.1, the influences of $\alpha$ and $\beta$ on hydrodynamics are obviously independent; therefore, the fluid characteristics of HUGMB in the Two-Dimensional (2D) plane can be used to reflect the changing performance of that in the 3D state to a certain extent. The computational domain on the o₀-z₀x₀ plane for fluid simulation during bow movement is shown in Fig. 6, its size is 20 m × 6 m. The settings of the computational domain are the same as those in Fig. 4. The number of cells is 6 × 10⁴, and grid refinement has little effect on fluid simulation after verifying the grid independence.

Fig. 6

The computational domain for fluid simulation during bow movement

The pressure distributions around the HUGMB when the bow is at different attitude angles with a velocity of 0.5 m/s is shown in Fig. 7. When the value of β is 0°, the pressure distribution on both sides of HUGMB is basically symmetric with respect to the central axis of the HUGMB, therefore, the corresponding value of yaw moment in Fig. 5 is almost 0. When the value of β is not 0°, the area that the bow deflected has a negative pressure, while the area in the opposite direction has a positive pressure. The pressure area is directly proportional to the attitude angle of the bow. The asymmetry of the pressure distribution near the bow can provide a yaw moment for the HUGMB.

Fig. 7

Pressure contour map of HUGMB with different attitude angles of the bow on the o₀-z₀x₀ plane. (a) β = 0°, (b) β = 10°, (c) β =− 10°, (d) β = 20°, (e) β  = − 20°, (f) β  = 30°, (g) β  = − 30°

The vorticity distribution around HUGMB in a motion period of the bow after the flow field is stabilized is shown in Fig. 8a. The vorticity fields on both sides of HUGMB are symmetric when the bow is in the initial position at 0 T, and there are vortices in opposite directions behind the wing. At the 1/4 T, the bow is at the left limit position. The vorticity distribution around HUGMB at the 1/4 T is described in detail in Fig. 8b. The deflection of the bow breaks the symmetry of the vorticity field on both sides. A small area of vortex appears in front of the right-wing, while no vortex is generated in front of the left-wing; the asymmetry of the vorticity distribution provides a yaw moment for HUGMB. Correspondingly, at the 3/4 T, the bow is at the right limit position, and a small area of the vortex also appeared in front of the left-wing, and its direction was opposite to that of the right small area vortex in the 1/4 T. In the 1/2 T and the T, the bow is in the initial position, and there is no similar vortex generation in front of the wing. This proves that the bow’s motion can generate a vortex, which provides a yaw moment for HUGMB while it sheds into the wake flow. As shown in Fig. 8c, as the vortex generated by the bow’s motion continues to shed into the wake flow, the Karman vortex street phenomenon appears at the wake flow of HUGMB, and its influence is to generate alternative lateral forces on the glider perpendicular to the direction of motion, which provides HUGMB with force needed to change its momentum [28].

Fig. 8

Vorticity contour map of HUGMB on the o₀-z₀x₀ plane in a motion period of the bow

2.3 Trajectory Simulation

The same as the previous section, in this section, we still only study the relationship between the attitude angle of the bow of HUGMB and the turning radius of it in the 2D plane. Taking the attitude angle of − 30°as an example, the computational domain on the o₀-z₀x₀ plane for trajectory simulation is shown in Fig. 9, its size is 18 m × 18 m. The settings of the computational domain are the same as those in Fig. 4. The number of cells is 18 × 10⁴, and grid refinement has little effect on fluid simulation after verifying the grid independence.

Fig. 9

The computational domain for trajectory simulation

The initial velocity of the foreground mesh is set as − 0.5 m/s, whose direction is the forward direction of HUGMB. The vorticity of HUGMB at the 30 s is shown in Fig. 10, using an attitude angle is − 30°as an example. It can be seen that the trajectory of HUGMB’s wake flow approximates a circle, and there exists obvious periodic vorticity generation and disappearance around the body. The deflection of the bow can effectively control the course of HUGMB. The trajectories corresponding to different attitude angles of the bow are plotted in Fig. 11a. The turning radius can be effectively controlled by adjusting the attitude angle of the bow, which proves that the bow movement can improve the maneuverability of the glider. According to the result between the attitude angle of the bow and theturning radius shown in Fig. 11b, we can conclude that the turning radius is inversely proportional to the attitude angle of the bow, there’s a linear relationship between them, and the minimum turning radius is 3.616 m, which is 2.26 times the total length of the HUGMB model used in the simulations.

Fig. 10

Vorticity contour map of HUGMB when it is in the 30s

Fig.11

Effect of the attitude angle of the bow on the moving trajectories: (a) the moving trajectories of HUGMB at various β from − 10° to − 30° and (b) fitting of β and turning radius

In addition to the attitude angles of the bow, the gliding speed (v) and the bow-to-body ratio (L_B/L) are the main factors affecting the turning radius [14]. The moving trajectories of HUGMB corresponding to different initial speeds are plotted in Fig. 12a when β is − 30°. The simulation results show that when the speed is low, the turning radius decreases as the speed increases, but the numerical value does not change much, which proves that speed has less effect on the turning radius at low-speed conditions. The trajectories of HUGMB with different bow-to-body ratios are plotted in Fig. 12b when β is − 30°. It can be seen that the turning performs better as the proportion of the bow to the total length grows larger, the minimum turning radius is 0.392 m, and the bow is about 1/4 of the total length of the simulation model.

Fig. 12

Moving trajectories of HUGMB with: (a) different speeds and (b) different bow-to-body ratios when β is − 30°

In summary, HUGMB can effectively control its hydrodynamic parameters and improve its maneuverability by adjusting the bow attitude. In addition, HUGMB can obtain a larger external torque and a smaller turning radius by changing the amplitude of the attitude angle of the bow and the bow-to-body ratio, which provides a new idea to improve the maneuverability of HUG.

3 Design and Motion Analysis of MBM

It is concluded that the maneuverability of HUGMB can be improved by adjusting the attitude angles and length of the bow after analysing the hydrodynamic characteristics of HUGMB. Therefore, an MBM that can realize two-rotations and one-shift motion is proposed in this section, the degree-of-freedom, attitude control range and movement amount of which are obtained based on the simulation results in Sect. 2, and then the design of MBM was completed based on these constraints.

3.1 Mechanism Description

Based on the physiological structure of the parallel muscle groups in the necks of creatures such as penguins, this section proposes a new type of MBM to realize the precise control of the bow motion of HUGMB and to improve its maneuverability. The 3D model of MBM is shown in Fig. 13, and the schematic diagram of it is shown in Fig. 14. The base of the mechanism is connected to the shroud through three motion branches, which include two RSS motion branches (2RSS) orthogonally distributed along the circumference and a PU motion branch (PU) located on the central axis of the glider. The 2RSS branches are both driven by the rotating pair R and transmitted by two spherical pairs S. The 2RSS branches can realize the weak coupling motion based on the structural characteristics of the parallel mechanism and then realize the precise control of two attitude angles of the bow. The PU branch is driven by prismatic pair P, which can make the bow telescopic and expand the movement space. The mechanism can realize two rotations and one shift through 2RSS branches and PU branch, thereby controlling the spatial posture of the bow.

Fig. 13

The 3D Model of MBM

Fig. 14

Schematic diagram of MBM

3.2 Kinematic Analysis

The coordinate system $o_{0} - x_{0} y_{0} z_{0}$ is shown in Fig. 14, established to describe the motion of MBM: the z₀-axis coincides with the central axis of the glider, and its direction points to the hooke hinge U; the x₀-axis is perpendicular to the central axis of the glider, and its direction points to R₁; the y₀-axis is perpendicular to the central axis of the glider, and its direction points to R₂; the origin $o_{0}$ is located at the intersection of the three axes. The following variables are defined: $s$ is the movement amount of P; $\theta_{1} ,$$\theta_{2}$ are the rotation angles of R₁, R₂ respectively, whose direction conforms to the right-hand rule; the definitions of α and β are the same as in Fig. 3. The structural parameters are defined in Fig. 15. $S$ is the length of $o_{0} {\text{R}}_{1}$ and $o_{0} {\text{R}}_{2}$; $D$ is the length of $o_{0} {\text{U}}$, which is the associated variable of $s$; $L_{1}$ is the length of S_ox1R₁ and S_oy1R₂, $L_{2}$ is the length of S_ox1S_ox2 and S_oy1S_oy2, $L_{3}$ and $L_{4}$ are used to describe the distance of US_ox2 and US_oy2. When the mechanism is at the initial position S_ox1R₁ and S_oy1R₂ are perpendicular to z₀-axis. Meanwhile, the angle values of $\theta_{1}$ and $\theta_{2}$ are both 0°.

Fig. 15

Structural parameters of MBM on the o₀-z₀x₀ plane

3.2.1 Inverse Kinematics

>The inverse kinematics equation of MBM based on the closed vector method is established in this part. According to the coordinate system and parameter definitions, the deflection matrices of R1 and R2 can be written:

$$\begin{gathered} {\varvec{f}}_{{{\varvec{R}}{\mathbf{1}}}} = {\varvec{f}}_{{\varvec{R}}} \left( {y_{0} ,\theta_{1} } \right) \\ = \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } & 0 & {\sin \theta_{1} } \\ 0 & 1 & 0 \\ { - \sin \theta_{1} } & 0 & {\cos \theta_{1} } \\ \end{array} } \right] \\ \end{gathered}$$

(3-1)

$$\begin{gathered} {\varvec{f}}_{{{\varvec{R2}}}} = {\varvec{f}}_{{\varvec{R}}} \left( {x_{0} ,\theta_{2} } \right) \\ = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \theta_{2} } & { - \sin \theta_{2} } \\ 0 & {\sin \theta_{2} } & {\cos \theta_{2} } \\ \end{array} } \right] \\ \end{gathered}$$

(3-2)

The coordinates of ${\text{S}}_{1}$ can be written by f_R1 and f_R2:

$${\text{S}}_{ox1} = \left( {\begin{array}{*{20}c} {L_{1} \cos \theta_{1} + S} & 0 & { - L_{1} \sin \theta_{1} } \\ \end{array} } \right)$$

(3-3)

$${\text{S}}_{oy1} = \left( {\begin{array}{*{20}c} 0 & {L_{1} \cos \theta_{2} + S} & {L_{1} \sin \theta_{2} } \\ \end{array} } \right)$$

(3-4)

Similarly, the deflection matrix f_R of the bow is as follows:

$$\begin{gathered} {\varvec{f}}_{{\varvec{R}}} = {\varvec{f}}_{{\varvec{R}}} \left( {x_{0} ,\alpha } \right)f_{R} \left( {y_{0} ,\beta } \right) \\ = \left[ {\begin{array}{*{20}c} {\cos \beta } & 0 & {\sin \beta } \\ {\sin \alpha \sin \beta } & {\cos \alpha } & { - \sin \alpha \cos \beta } \\ { - \cos \alpha \sin \beta } & {\sin \alpha } & {\cos \alpha \cos \beta } \\ \end{array} } \right] \\ \end{gathered}$$

(3-5)

The coordinates of ${\text{S}}_{2}$ can be written by f_R:

$${\text{S}}_{ox2} = \left( {\begin{array}{*{20}c} {L_{3} \cos \beta + L_{4} \sin \beta } \\ {L_{3} \sin \alpha \sin \beta - L_{4} \sin \alpha \cos \beta } \\ { - L_{3} \cos \alpha \sin \beta + L_{4} \cos \alpha \cos \beta + D} \\ \end{array} } \right)^{T}$$

(3-6)

$${\text{S}}_{oy2} = \left( {\begin{array}{*{20}c} {L_{4} \sin \beta } \\ {L_{3} \cos \alpha - L_{4} \sin \alpha \cos \beta } \\ {L_{3} \sin \alpha + L_{4} \cos \alpha \cos \beta + D} \\ \end{array} } \right)^{T}$$

(3-7)

The following constraint equation can be obtained from the structural geometric relationship of MBM:

$$\left| {{\text{S}}_{ox1} {\text{S}}_{ox2} } \right| = L_{2}$$

(3-8)

$$\left| {{\text{S}}_{oy1} {\text{S}}_{oy2} } \right| = L_{2}$$

(3-9)

The coordinates of S_ox1, S_ox2, S_oy1 and S_oy2 are substituted into Eqs. (3-8) and (3–9), respectively, and we can get the following:

$$\begin{gathered} L_{2}^{2} = \left( {L_{1} \cos \theta_{1} + S - L_{3} \cos \beta - L_{4} \sin \beta } \right)^{2} + \hfill \\ \left( {L_{3} \sin \alpha \sin \beta - L_{4} \sin \alpha \cos \beta } \right)^{2} + \hfill \\ \left( { - L_{1} \sin \theta_{1} + L_{3} \cos \alpha \sin \beta - L_{4} \cos \alpha \cos \beta - D} \right)^{2} \hfill \\ \end{gathered}$$

(3-10)

$$\begin{gathered} L_{2}^{2} = \left( {L_{4} \sin \beta } \right)^{2} + \hfill \\ \left( {L_{1} \cos \theta_{2} + S - L_{3} \cos \alpha + L_{4} \sin \alpha \cos \beta } \right)^{2} + \hfill \\ \left( {L_{1} \sin \theta_{2} - L_{3} \sin \alpha - L_{4} \cos \alpha \cos \beta - D} \right)^{2} \hfill \\ \end{gathered}$$

(3-11)

Given the bow parameters α, β, and s, the inputs $\theta_{1}$ and $\theta_{2}$ of MBM can be obtained by using Eqs. (3-–10) and (3-11).

3.2.2 Forward Kinematics

R₁S_ox1S_ox2 is defined as Branch-1 and R₂ S_oy1S_oy2 as Branch-2. Due to the weak coupling between the two RSS branches, Branch 1 is taken in o₀-z₀x₀ plane as an example to calculate the forward kinematics. As shown in Fig. 15, θ₁ is the angle input of Branch-1, D is a variable associated with s, so the four angles φ₁, φ₂, φ₃ and φ₄ are the functions of s and θ₁, the geometric constraints on the relevant parameters are:

$$\left[ {\begin{array}{*{20}c} {D_{1} } \\ {D_{2} } \\ {D_{3} } \\ {\varphi_{1} } \\ {\varphi_{2} } \\ {\varphi_{3} } \\ {\varphi_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sqrt {S^{2} + D^{2} } } \\ {\sqrt {\left( {D + L_{1} \sin \theta_{1} } \right)^{2} + \left( {S + L_{1} \cos \theta_{1} } \right)^{2} } } \\ {\sqrt {L_{3}^{2} + L_{4}^{2} } } \\ {\arctan \left( \frac{S}{D} \right)} \\ {\arccos \left( {\frac{{D_{1}^{2} + D_{2}^{2} - L_{1}^{2} }}{{2D_{1} D_{2} }}} \right)} \\ {\arccos \left( {\frac{{D_{2}^{2} + D_{3}^{2} - L_{2}^{2} }}{{2D_{2} D_{3} }}} \right)} \\ {\arctan \left( {\frac{{L_{3} }}{{L_{4} }}} \right)} \\ \end{array} } \right]$$

(3-12)

The attitude angle can be expressed as:

$$\beta = 180^\circ - \left( {\varphi_{1} + \varphi_{2} + \varphi_{3} + \varphi_{4} } \right)$$

(3-13)

3.2.3 Mechanism Driving Angle Range Analysis

The RSS branch can be regarded as a plane four-bar mechanism when the weak coupling between the two RSS branches is not considered. The driving angle θ of the mechanism based on mechanical indicators such as dead-center position and transmission angle is analyzed in this part.

(1) Dead-center.

The mechanism is at the Dead-center position when ${\text{S}}_{1} {\text{R}}_{1}$ and ${\text{S}}_{2} {\text{R}}_{1}$ are collinear. At this time, ${\text{S}}_{1}$ is located between ${\text{S}}_{2} {\text{R}}_{1}$ or ${\text{S}}_{1}$ is located on the extension line of ${\text{S}}_{2} {\text{R}}_{1}$, then the driving angle θ_d corresponding to the Dead-center position is:

$$\theta_{{\text{d}}} { = }\left\{ \begin{gathered} \arctan \left(\frac{D}{S}\right) + \arccos (\frac{{D_{1}^{2} + (L_{2} + L_{1} )^{2} - D_{3}^{2} }}{{2D_{1} (L_{2} + L_{1} )}}) - 180^\circ \left( {{\text{case}}1} \right) \hfill \\ \arctan \left(\frac{D}{S}\right) + \arccos \left(\frac{{D_{1}^{2} + (L_{2} - L_{1} )^{2} - D_{3}^{2} }}{{2D_{1} (L_{2} - L_{1} )}}\right)\left( {{\text{case}}2} \right) \hfill \\ \end{gathered} \right.$$

(3-14)

The designed range of $s$ is − 10 mm to 10 mm, which is determined by the mechanism layout. The relationship between $\theta_{{\text{d}}}$ and $s$ can be obtained from the Eq. (3-14). As shown in Fig. 16, the minimum safe driving angles of case1 and case2 are − 79.2° and 128.9°, respectively.

Fig. 16

Relationship of $\theta_{{\text{d}}}$with s

(2) Transmission angle.

According to the definition of the minimum transmission angle, when ${\text{S}}_{1} {\text{R}}_{1}$ and ${\text{UR}}_{1}$ are collinear, the mechanism has two situations at the minimum transmission angle position:

• A) ${\text{S}}_{1}$ is between ${\text{UR}}_{1}$

It can be seen from Fig. 15 that the value of the driving angle θ_γ corresponding to the minimum transmission angle position equals to the complementary angles of $\angle {\text{UR}}_{1} o_{.0}$, so θ_γ has a negative correlation with D. And, when s = 10, D is the largest, θ_γ is − 116° at this time, the corresponding minimum transmission angle $\gamma$ is 32°, which does not meet the $\gamma_{\min } \ge 40^\circ$ [29]. θ_γ can be designed to be − 102°according to $\gamma_{\min } = 40^\circ$.

• B) ${\text{S}}_{1}$ is on the extension line of UR₁

It can be seen from Fig. 15 that the value of θ_γ equals to $\angle {\text{UR}}_{1} o_{.0}$, so θ_γ has a positive correlation with D. When s = − 10, D is the smallest, θ_γ is 51° at this time, the corresponding minimum transmission angle $\gamma$ is 66°, which meets the $\gamma_{\min } \ge 40^\circ$.

In order to ensure that the servo motor can avoid the dead-center position, θ should be between − 79.2° ~ 128.9°. In order to ensure that the minimum transmission angle meets the design requirements, θ should be between − 102° ~ 51°. Furthermore, due to the limitation of the mechanism layout, θ should be less than 50° to avoid interference between the rudder arm and the base. Above all, θ should be − 79.2° ~ 50°.

3.2.4 Position Space

As shown in Fig. 13, point M is the front shroud apex. The position space that M can achieve is closely related to the hydrodynamic performance, and the coordinates of point M are:

$$\left[ {\begin{array}{*{20}c} {x_{m} } \\ {y_{m} } \\ {z_{m} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {L_{{\text{s}}} \sin \beta } \\ { - L_{{\text{s}}} \sin \alpha \cos \beta } \\ {L_{{\text{s}}} \cos \alpha \cos \beta + D} \\ \end{array} } \right]$$

(3-15)

Among them, L_s is the distance between M and ${\text{U}}$ when the mechanism is in the initial position.

The position of M is determined by the three variables of α, β and s according to Eq. (3-15). The accessible position space of M is shown in Fig. 17. It is easy to see that the position space increases significantly along a curved surface as α and β increase when $s$ fixed. When α, β and s increase simultaneously, the position space along the z₀-axis increases significantly, and the overall position space expands from surface to body. So, $s$ has a great influence on the axial expansion and contraction of the bow. Figure 18 shows how the different parameters affect the position space in detail. The forward increasement of $s$ can increase the forward position space, but it reduces the negative position space. If M needs to reach the extreme negative position, the value of $s$ has to increase negatively, which leads to the fact that the position space is not spatially symmetric. The workspace (attitude space, position space) of the bow is completely symmetric only when $s$ is 0.

Fig. 17

Position space of M

Fig. 18

Position space of M corresponding to (a) $s = 10$and (b) $s = 0$ and (c) $s = - 10$

3.2.5 Velocity Model

The value of $s$ only needs to ensure that $\theta_{1} ,\theta_{2}$ is in the safe range without multi-frequency and long-term changes for different attitude requirements; therefore, $s$ is used as a fixed structural parameter in velocity analysis and dynamic modeling.

The matrix describing the relationship between the servo motor angular velocity $\dot{\theta }_{1} ,\dot{\theta }_{2}$ and the bow attitude angular velocity $\dot{\alpha },\dot{\beta }$ can be obtained by differentiating the inverse solution Eqs. (3-10) and (3-11):

$$\left( {\begin{array}{*{20}c} {\dot{\theta }_{1} } \\ {\dot{\theta }_{2} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{ - \left( {A_{v} B_{v} + D_{v} F_{v} } \right)}}{{I_{v} }}} & {\frac{{ - \left( {A_{v} C_{v} + D_{v} E_{v} + G_{v} H_{v} } \right)}}{{I_{v} }}} \\ {\frac{{\left( {J_{v} K_{v} + M_{v} L_{v} } \right)}}{{O_{v} }}} & {\frac{{ - N_{v} }}{{O_{v} }}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\dot{\alpha }} \\ {\dot{\beta }} \\ \end{array} } \right)$$

(3-16)

where,

$$\left[ {\begin{array}{*{20}c} {A_{v} } \\ {B_{v} } \\ {C_{v} } \\ {D_{v} } \\ {E_{v} } \\ {F_{v} } \\ {G_{v} } \\ {H_{v} } \\ {I_{v} } \\ {J_{v} } \\ {K_{v} } \\ {L_{v} } \\ {M_{v} } \\ {N_{v} } \\ {O_{v} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {L_{4} \cos \beta \sin \alpha - L_{3} \sin \alpha \sin \beta } \\ {L_{3} \cos \alpha \sin \beta - L_{4} \cos \alpha \cos \beta } \\ {L_{3} \cos \beta \sin \alpha + L_{4} \sin \alpha \sin \beta } \\ {L_{1} \sin \theta_{1} - L_{3} \cos \alpha \sin \beta + L_{4} \cos \alpha \cos \beta + D} \\ {L_{3} \cos \alpha \cos \beta + L_{4} \cos \alpha \cos \beta } \\ {L_{4} \cos \beta \sin \alpha - L_{3} \sin \alpha \sin \beta } \\ {L_{1} \cos \theta_{1} - L_{3} \cos \beta - L_{4} \sin \beta + S} \\ {L_{4} \cos \beta - L_{3} \sin \beta } \\ {L_{1} G\sin \theta_{1} - L_{1} D\cos \theta_{1} } \\ {L_{1} \cos \theta_{2} - L_{3} \cos \alpha + L_{4} \cos \beta \sin \alpha + S} \\ { - L_{3} \sin \alpha - L_{4} \cos \alpha \cos \beta } \\ { - L_{3} \cos \alpha + L_{4} \cos \beta \sin \alpha } \\ {L_{3} \sin \alpha - L_{1} \sin \theta_{2} + L_{4} \cos \alpha \cos \beta + D} \\ {L_{4}^{2} \cos \beta \sin \beta - J_{v} L_{4} \sin \alpha \sin \beta - M_{v} L_{4} \cos \alpha \sin \beta } \\ { - J_{v} L_{1} \sin \theta_{2} - M_{v} L_{1} \cos \theta_{2} } \\ \end{array} } \right]$$

Let I = $\left( {\begin{array}{*{20}c} {\frac{{ - \left( {A_{v} B_{v} + D_{v} F_{v} } \right)}}{{I_{v} }}} & {\frac{{ - \left( {A_{v} C_{v} + D_{v} E_{v} + G_{v} H_{v} } \right)}}{{I_{v} }}} \\ {\frac{{\left( {J_{v} K_{v} + M_{v} L_{v} } \right)}}{{O_{v} }}} & {\frac{{ - N_{v} }}{{O_{v} }}} \\ \end{array} } \right)$, then the bow attitude angular velocity $\dot{\alpha }$ and $\dot{\beta }$ can be obtained by Eq. (3-17):

$$\left( {\begin{array}{*{20}c} {\dot{\alpha }} \\ {\dot{\beta }} \\ \end{array} } \right) = I^{ - 1} \left( {\begin{array}{*{20}c} {\dot{\theta }_{1} } \\ {\dot{\theta }_{2} } \\ \end{array} } \right)$$

(3-17)

3.3 Dynamic Model

The Lagrange equation [30] is used to build the dynamic model of the moveable bow mechanism in this section:

$$Q = \frac{d}{dt}\left( {\frac{{\partial E_{k} }}{{\partial \dot{q}}}} \right) - \frac{{\partial E_{k} }}{\partial q} + \frac{{\partial E_{p} }}{\partial q}$$

(3-18)

where, Q represents the generalized force of the mechanism, q represents the generalized coordinate of the mechanism, E_k represents the total kinetic energy of the mechanism, and E_p represents the total potential energy of the mechanism.

The mechanism’s angle parameters on the o₀-z₀x₀ plane are shown in Fig. 19. ϕ_ij is the angle between the j-th rod and the line connecting the UR_i in Branch-i, θ₁₂ is the angle between rod 2 and the x₀-axis in Branch-1, and θ₂₂ is the angle between rod 2 and y₀-axis in Branch-2. Let $\delta_{1} = \beta$, $\delta_{2} = \alpha$, then:

$$\left[ {\begin{array}{*{20}c} {\phi_{i1} } \\ {\phi_{i3} } \\ {\phi_{i2} } \\ {\theta_{i2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{\pi }{2} + \varphi_{1} - ( - 1)^{i} \theta_{i} } \\ {\varphi_{1} + \varphi_{4} - ( - 1)^{i} \delta_{i} } \\ {\arctan \frac{{D_{3} \sin \phi_{i3} - L_{1} \sin \phi_{i1} }}{{D_{1} + D_{3} \cos \phi_{i3} - L_{1} \cos \phi_{i1} }}} \\ {\frac{\pi }{2} + \varphi_{1} - \phi_{i2} } \\ \end{array} } \right]$$

(3-19)

Fig. 19

Angle parameters of MBM on the o₀-z₀x₀ plane

3.3.1 Total Mechanism Energy

The total kinetic energy of the mechanism can be expressed by:

$$E_{k} = \sum\limits_{i}^{2} {\sum\limits_{j}^{3} {\left(\frac{1}{2}m_{j} v_{{_{\sigma ij} }}^{2} + \frac{1}{2}J_{j} \omega_{{_{ij} }}^{2} \right)} }$$

(3-20)

In Eq. (3-20), v_σij and ω_ij represent the centroid velocity and angular velocity of the j-th rod in in Branch-i, respectively.

Based on the simulation result of hydrodynamic characteristics in Sect. 2.1 and the mathematical model of kinematics in Sect. 3.2, it can be seen that the hydrodynamic effects of the attitude angle of the bow corresponding to the two RSS branches are basically independent, and the motion coupling between the two RSS branches is also very small, so Eq. (3-20) is rewritten into the following form:

$$E_{k} = \frac{1}{2}\sum\limits_{i}^{2} {\left( {J_{ei} \omega_{{_{i1} }}^{2} } \right)}$$

(3-21)

where, $J_{ei} = \sum\limits_{j}^{3} {\left( {m_{j} \left( {\frac{{v_{\sigma ij} }}{{\omega_{i1} }}} \right)^{2} + J_{j} \left( {\frac{{\omega_{ij} }}{{\omega_{i1} }}} \right)^{2} } \right)}$.

We call it the equivalent moment of inertia, and its specific expression is as follows:

$$J_{ei} = c_{i0} k_{i1} + c_{i1} k_{i2} + c_{i2} k_{i2}^{2} + c_{i3} k_{i3}^{2}$$

(3-22)

where, $\begin{gathered} c_{i0} = m_{1} L_{\sigma 1}^{2} + m_{2} L_{1}^{2} + J_{1} , \hfill \\ c_{i1} = { - }m_{2} L_{1} L_{\sigma 2} \cos \left( {\theta_{i} - ( - 1)^{i} \theta_{i2} } \right), \hfill \\ c_{i2} = m_{2} L_{\sigma 2}^{2} + J_{2} , \hfill \\ c_{13} = m_{3} \left( {L_{\sigma 3} \cos \alpha } \right)^{2} + J_{3} + m_{2} \left( {L_{\sigma 3} L_{\sigma 2} \cos \beta /L_{2} } \right)^{2} , \hfill \\ c_{23} = m_{3} L_{\sigma 3}^{2} + J_{3} + m_{2} \left( {L_{\sigma 3} L_{\sigma 2} \cos \alpha /L_{2} } \right)^{2} , \hfill \\ k_{i1} = \frac{{\omega_{i1} }}{{\omega_{i1} }} = 1, \hfill \\ k_{i2} = \frac{{\dot{\theta }_{i2} }}{{\omega_{i1} }} = ( - 1)^{i + 1} \frac{{L_{1} \sin \left( {\phi_{i1} - \phi_{i3} } \right)}}{{L_{2} \sin \left( {\phi_{i2} - \phi_{i3} } \right)}}, \hfill \\ k_{i3} = \frac{{\dot{\delta }_{i} }}{{\omega_{i1} }} = \frac{{L_{1} \sin \left( {\phi_{i1} - \phi_{i2} } \right)}}{{D_{3} \sin \left( {\phi_{i3} - \phi_{i2} } \right)}} \hfill \\ \end{gathered}$

The total potential energy of the mechanism can be expressed by Eq. (3-23):

$$\begin{gathered} E_{p} = (m_{1} \left( {S + L_{\sigma 1} \cos \theta_{2} } \right) \hfill \\ + m_{2} \left( {S + L_{1} \cos \theta_{2} + L_{\sigma 2} \cos \theta_{22} } \right) \hfill \\ - m_{3} L_{\sigma 3} \sin \alpha )g \hfill \\ \end{gathered}$$

(3-23)

3.3.2 Mechanism Generalized Force

The generalized moment can be obtained from the principle of virtual work:

$$Q_{1} \delta \theta_{1} = M_{1} \delta \theta_{1} + M_{x} \delta \alpha + M_{y} \delta \beta$$

(3-24)

$$Q_{2} \delta \theta_{2} = M_{2} \delta \theta_{2} + M_{x} \delta \alpha + M_{y} \delta \beta$$

(3-25)

The matrix form of the generalized moments is as follows:

$$\left[ {\begin{array}{*{20}c} {Q_{1} } \\ {Q_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {M_{1} } \\ {M_{2} } \\ \end{array} } \right] + \left( {I^{ - 1} } \right)^{T} \left[ {\begin{array}{*{20}c} {M_{x} } \\ {M_{y} } \\ \end{array} } \right]$$

(3-26)

3.3.3 Mathematical Model of Dynamics

In order to get each term of the Lagrange equation, Eq. (3-21) is differentiated:

$$\left[ {\begin{array}{*{20}c} {\frac{{\partial E_{k} }}{{\partial \theta_{i} }}} \\ {\frac{{\partial E_{k} }}{{\partial \dot{\theta }_{i} }}} \\ {\frac{d}{dt}\left( {\frac{{\partial E_{k} }}{{\partial \dot{\theta }_{i} }}} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{1}{2}\frac{{dJ_{ei} }}{{d\theta_{i} }}\dot{\theta }_{i}^{2} } \\ {J_{ei} \dot{\theta }_{i} } \\ {J_{ei} \ddot{\theta }_{i} + \frac{{dJ_{ei} }}{{d\theta_{i} }}\dot{\theta }_{i}^{2} } \\ \end{array} } \right]$$

(3-27)

where, $\theta_{i} = q_{i} ,\dot{\theta }_{i} = \dot{q}_{i} = \omega_{i1}$.

The expression of $\frac{{dJ_{ei} }}{{d\theta_{i} }}$ can be obtained by differentiating Eq. (3-22):

$$\begin{gathered} \frac{{dJ_{ei} }}{{d\theta_{i} }} = 2c_{i3} k_{i3} \frac{{dk_{i3} }}{{d\theta_{i} }} + \frac{{dc_{i3} }}{{d\theta_{i} }}k_{i3}^{2} + 2c_{i2} k_{i2} \frac{{dk_{i2} }}{{d\theta_{i} }} - \hfill \\ m_{2} L_{1} L_{\sigma 2} \left[ {\frac{{dk_{i2} }}{{d\theta_{i} }}\cos \left( {\theta_{i} + \left( { - 1} \right)^{i + 1} \theta_{i2} } \right)} \right. - \hfill \\ \left. {k_{i2} \sin \left( {\theta_{i} + \left( { - 1} \right)^{i + 1} \theta_{i2} } \right)(1 + \left( { - 1} \right)^{i + 1} k_{i2} )} \right] \hfill \\ \end{gathered}$$

(3-28)

where, $\left[ {\begin{array}{*{20}c} {\frac{{dc_{i3} }}{{d\theta_{i} }}} \\ {\frac{{dk_{i2} }}{{d\theta_{i} }}} \\ {\frac{{dk_{i3} }}{{d\theta_{i} }}} \\ {M_{i1} } \\ {M_{i2} } \\ {M_{i3} } \\ {M_{i4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2k_{i3} m_{2} \left( {L_{\sigma 3} L_{\sigma 2} /L_{2} } \right)^{2} \sin \varphi_{i} \cos \varphi_{i} } \\ {\frac{{L_{1} \left( {M_{i1} + M_{i2} } \right)}}{{L_{2} \sin^{2} \left( {\phi_{i2} - \phi_{i3} } \right)}}} \\ {\frac{{L_{1} \left( {M_{i3} + M_{i4} } \right)}}{{D_{3} \sin^{2} \left( {\phi_{i2} - \phi_{i3} } \right)}}} \\ {(k_{i3} - 1)\sin (\phi_{i2} - \phi_{i3} )\cos (\phi_{i3} - \phi_{i1} )} \\ {(k_{i3} - k_{i2} )\sin (\phi_{i3} - \phi_{i1} )\cos (\phi_{i2} - \phi_{i3} )} \\ {(k_{i2} - 1)\sin (\phi_{i2} - \phi_{i3} )\cos (\phi_{i2} - \phi_{i1} )} \\ {(k_{i3} - k_{i2} )\sin (\phi_{i2} - \phi_{i1} )\cos (\phi_{i2} - \phi_{i3} )} \\ \end{array} } \right]$

Similarly, Eq. (3-23) can also be differentiated:

$$\begin{gathered} \frac{{dE_{p} }}{{d\theta_{2} }} = m_{1} L_{\sigma 1} \sin \theta_{2} + m_{2} L_{1} \sin \theta_{2} + \hfill \\ m_{2} L_{\sigma 2} k_{22} \sin \theta_{22} + m_{3} L_{\sigma 3} k_{23} \cos \alpha \hfill \\ \end{gathered}$$

(3-29)

Then, the dynamic model of the mechanism in the form of a Lagrange equation can be expressed by:

$$\left[ {\begin{array}{*{20}c} {M_{1} } \\ {M_{2} } \\ \end{array} } \right] + \left( {I^{ - 1} } \right)^{T} \left[ {\begin{array}{*{20}c} {M_{x} } \\ {M_{y} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {J_{e1} \ddot{\theta }_{1} + \frac{1}{2}\frac{{dJ_{e1} }}{{d\theta_{1} }}\dot{\theta }_{1}^{2} } \\ {J_{e2} \ddot{\theta }_{2} + \frac{1}{2}\frac{{dJ_{e2} }}{{d\theta_{2} }}\dot{\theta }_{2}^{2} + \frac{{dE_{p} }}{{d\theta_{2} }}} \\ \end{array} } \right]$$

(3-30)

4 Simulation and Experiment

An RBF neural network adaptive sliding mode controller is built to precisely control the input of the mechanism. The prototype of MBM is manufactured and assembled. The experimental verification of the mathematical model established in Sect. 3 is carried out. In the experiment, an angle sensor is used to detect the motion parameters of the bow. The performance parameters of the angle sensor used in the experiment are shown in Table 4.

Table 4 Performance parameters of the angle sensor

4.1 Simulation of Controller

As shown in Fig. 20, an RBF neural network adaptive sliding mode controller [31] is built in this section using a Simulink block diagram in Matlab. θ_o is target angle and u is the output of the controller. The parameters necessary for the simulation are shown in Table 5.

Fig. 20

RBF neural network adaptive sliding mode controller [25]

Table 5 Basic structural parameters of MBM

Taking Branch-1 as an example, the mechanism’s input is solved based on the kinematic equation, and the functional relationship of $\theta_{1}$ and $\dot{\theta }_{1}$ with respect to time t are fitted, respectively. The functions are the target trajectory of the controller:

$$\left\{ \begin{gathered} \theta_{1} { = }4.475t - 43.98 \hfill \\ \dot{\theta }_{1} { = 0}{\text{.00848}}t^{2} - 0.1239t + 4.688 \hfill \\ \end{gathered} \right.$$

(4-1)

The simulation results of the controller are shown in Fig. 21. It can be seen that the trajectory of $\theta_{1}$ can stably follow the target trajectory with the Root-Mean-Square Error (RMSE) equalling to 0.272. The trajectory of $\dot{\theta }_{1}$ deviates slightly from the target trajectory when $\theta_{1}$ is close to 0, then it follows the target trajectory faster and tends to match after real-time adjustment. The RMSE of $\dot{\theta }_{1}$ is 0.157.

Fig. 21

Simulation results of controller

4.2 Simulation and Experiment of MBM

The parameter settings of three modes of MBM are defined in Table 6. Turning Mode and Diving Mode are single-angle modes, and only Turning Mode is experimentally verified in this paper. The experimental results also apply to Diving Mode, which is based on the characteristics of MBM. Spiraling Mode is the coupling mode, which is experimentally verified to study the kinematic coupling of two attitude angles.

Table 6 Parameter settings of three modes of MBM

4.2.1 Position Model Verification

It can be seen that the attitude angle of bows $\alpha$ and $\beta$ are both functions of $\theta_{1}$, $\theta_{2}$ and $s$ according to the mathematical model in Sect. 3. Combined with the driving angle range of the mechanism determined in Sect. 3.2.3, the relationship between the attitude angle of the bow and the angle of the servo motor is shown in Fig. 22 when $s$ is 0. The mechanism can achieve an attitude change of ± 20° while $s$ is 0 and the angle of the servo motor changes in a range of ± 50°. Phase I of the motion of the mechanism is defined when $s$ is 0 and the attitude angle is in [− 20°, 20°].

Fig. 22

Relationship between β and θ₁ when MBM is in Phase I

The comparison of the attitude trajectory of the bow between the simulation and experiment is shown in Fig. 23 when MBM is in Phase I, using Turning Mode as an example. The simulation and experimental trajectory of β are in good agreement with a peak value of 20.2° and an error of 0.2°. The maximum overshoot of α is about 2.23°. According to the result of the experiment, there are some causes of the error: (1) the servo motor used in the experiment has no angle feedback, and the servo motor R₂ cannot provide accurate online compensation for α; (2) there are assembly errors in the articulated bearing and mechanical structure, so the trajectory of α will be affected when the servo motor R₁ works at a large angle. The error of attitude angle generated by the mechanism itself is called as the static error.

Fig. 23

Simulation and test results of attitude angles when MBM is in Phase I

Phase II of the motion of the mechanism is defined when $s$ is not 0, and the attitude angle is in [− 30°, − 20°) and (20°, 30°]. In this phase, the value of s can significantly change the workspace of the mechanism, as shown in Fig. 18. The relationship between β and s is shown in Fig. 24, and it also applies to the relationship between α and s. It can be seen that β and s are basically proportional to the proportional coefficient of 1.

Fig.24

Relationship between attitude angle and s

The experimental process when MBM is in Turning Mode is shown in Fig. 25a. The MBM’s attitude trajectory in Turning Mode combining Phase I and II is shown in Fig. 25b. When the value of β reaches 20°, $\dot{\beta }$ changes abruptly. The reason is that the motion of the mechanism changes from Phase I to Phase II. The peak value of the trajectory of β is 30.6°, which meets the design requirements. The trajectory of α appears to have periodic double trough fluctuations. The first trough of the trajectory of α appears at the same time as the peak of the trajectory of β, and the overshoot is about 6.43°. The second trough of the trajectory of α appears at the same time as the trough of the trajectory of β, and the overshoot is about 4.48°. The reasons for the error are as follows: (1) the speed consistency of the servo motor R₂ and the moving pair P cannot be guaranteed in the direction of the z₀-axis, so the trajectory of α has over-compensation and under-compensation; (2) the attitude angular velocity of MBM is so large that the attitude sensor has zero drift. The error of attitude angle generated by the continuous motion of MBM is called as the dynamic error. The bow will not move continuously when HUGMB works underwater, so the dynamic error of MBM can be ignored in the actual working conditions.

Fig. 25

(a) Experimental process and (b) the test results of attitude angles when MBM is in Turning Mode

The experimental process when MBM is in Spiraling Mode is shown in Fig. 26a. The MBM’s attitude trajectory in Spiraling Mode combining Phase I and II is shown in Fig. 26b. The trajectory of α basically coincides with the trajectory of β, and their trajectories are basically the same as the trajectory of β in Fig. 25b, which can verify the correctness of the position model in Sect. 3.

Fig. 26

(a) Experimental process and (b) the test results of attitude angles when MBM is in Spiraling Mode

In order to reduce the adverse effect of the dynamic error on the attitude angle control ability of the MBM prototype, Fig. 27a only shows the experiment process in Phase I, and it can be seen that the mechanism can achieve nine typical positions. The attitude trajectory corresponding to Fig. 27a is shown in Fig. 27b, MBM can achieve the target action well.

Fig. 27

(a) Experimental process and (b) the test results of attitude angles when MBM is in the experiment of the attitude angle control ability

4.2.2 Velocity Model Verification

The numerical simulation and experimental verification of the velocity model established in Sect. 3.2.5 are carried out in this section, and still, only the mechanism in Phase I is verified to avoid the adverse effects of dynamic errors. The input parameters required for simulation and experiment are shown in Table 7. The comparison of the angular velocity trajectory of MBM in Phase I between simulation and experiment is shown in Fig. 28, using Turning Mode as an example. It can be seen that the simulation and experimental results are in good agreement.

Table 7 Input parameters of the velocity model

Fig. 28

Comparison of angular velocity trajectories

After the verification experiments of the position model and velocity model, the weak coupling of Branch-1 and Branch-2 can be confirmed, which provides a basis for the decoupling hypothesis in dynamic modeling.

4.2.3 Moment Simulation

Matlab is used in this part to solve the input torque of the mechanism based on the dynamics model established in Sect. 3.3. At the same time, a 3D simplified model of the mechanism is built into Adams software. The torque of each bar of the mechanism can be obtained based on the simulation function of Adams software. The maximum external torque values obtained in Sect. 2.1 are taken as the external torque values in the dynamic model, and their values are M_x = 1.3N·m, M_y = 0.8N·m. The parameters required for the simulation are shown in Table 8.

• (1) The uniform velocity working state

Table 8 The input parameters of torque simulation

When $\theta_{1}$ and $\theta_{2}$ move from − 50° to 50° at an angular velocity of $2^\circ \cdot s^{ - 1}$, the changing laws of the driving torques $M_{1}$ and $M_{2}$ are shown in Fig. 29. Among them, the Root Mean Square Error (RMSE) of $M_{1}$ is 0.005, and the RMSE of $M_{2}$ is 0.018.

• (B) The uniform acceleration working state

Fig. 29

The simulation results of (a)$M_{1}$ and (b)$M_{2}$ using Matlab and Adams under uniform velocity

When $\theta_{1}$ and $\theta_{2}$ move from − 50° to 50° at an angular acceleration of $2^\circ \cdot s^{ - 2}$ from a static state, the changing laws of the driving torques $M_{1}$ and $M_{2}$ are shown in Fig. 30. Among them, the RMSE of $M_{1}$ is 0.004, and the RMSE of $M_{2}$ is 0.009.

Fig. 30

The simulation results of (a)$M_{1}$ and (b)$M_{2}$ using Matlab and Adams under uniform acceleration

The curves obtained by the two methods are basically coincident according to the results in Figs. 38 and 39, which can verify the correctness of the dynamic equation established in this paper. The torque curve provides a basis for the selection of the servo motor.

5 Discussions

Penguins can flexibly turn with a radius of 10% to 30% of their body’s length by using the coordination movement of the head, neck and forelimbs. So, this paper design a MBM by using 2RSS-PU parallel mechanism which can mimic the movement of the head and neck of penguins. The research of this paper is not only theoretical research, but also considers engineering applications. The neck of an organism consists of several joints acting together to achieve neck flexion, but there will be a lot of drives, which will reduce the reliability of the whole machine and not conducive to engineering application. Considering rigidity, strong bearing capacity and accurate control, this paper did not fully mimic the multi-jointed bone structure of the neck. Because of this, the MBM designed in this paper can only swing around a certain direction, unable to achieve S-shaped bending, which is the limitations of MBM.

The MBM designed by Wu can achieve a deflection of ± 50° [17], while the MBM in this paper can only achieve a deflection of ± 30°, but the research method in this paper can provide help for related research. The bionic dolphin robot with movable bow designed by Yang et al. uses the bow to reduce the turning radius of the robot to 0.6BL [20]. There is still a gap between the robot's maneuvering performance and that of the dolphin, because of the robot's rigid body. The minimum turning radius obtained by CFD simulation in this paper is only 0.245BL, and the turning radius is related to head-neck ratio and the attitude angles of the bow. Above all, the current research on the movable bow does not systematically analyze the mechanical law of the effect of the head attitude Angle on the evolution of hydrodynamic and flow field. This paper not only reveals the root cause of the influence of the movable bow on the fluid characteristics around the underwater vehicle, but also fits the results of hydrodynamic. The fitting results lay a foundation for the dynamic modeling of HUGMB. We are also carrying out the design and development of HUGMB. The relevant theory and performance analysis show that the movable bow can not only reduce the turning radius of HUGMB, but also increase the gliding pitch angle. The maximum pitch angle of HUGMB can be increased from 43.8° to 61.3°. In addition, the movable bow can also provide compensation for pitch angle, so that the depth change of HUGMB is stable at 0.1 m within 1200 s, realizing the true depth navigation. But HUGMB also has some drawbacks. The MBM requires three drives, this problem increases the HUGMB's power consumption and affects its range. Secondly, the MBM must occupy the front flooded tank space of the HUGMB, which is not conducive to carrying the mission sensor in the front of HUGMB.

The focus of many research is on the motion performance of the robot. Few studies have proposed the structure design and motion performance of the variable bow mechanism. In this paper, a movable bow mechanism suitable for underwater vehicle is proposed and its motion performance is studied, which provides an idea for the overall design of high maneuverability underwater vehicle. The disadvantage is that the motion performance of HUGMB is not studied in this paper, so it is impossible to verify the degree to which the variable bow mechanism improves the motion performance of the underwater vehicle. This work will be carried out in the future. We will continue to pay attention to the above research, such as the development of HUGMB and its motion performance.

6 Conclusions

In order to improve the maneuverability of HUG, this paper introduces the bow motion of underwater creatures into HUG and proposes a 2RSS-PU parallel mechanism with three degrees of freedom to control the attitude angles of the bow. The CFD simulation method is used as an analysis method to study the relationship between the attitude angles of the bow and the fluid force, the flow field around HUGMB, and the parameter requirements of MBM is obtained. Both the kinematic model based on the closed vector method and the dynamic model based on the Lagrange equation of MBM are established. An RBF neural network adaptive sliding mode controller is designed to improve the dynamic response effect of the output parameters of the mechanism. A prototype of MBM is manufactured and assembled. The kinematic, dynamics model and controller are verified by experiments, which provides a basis for applying MBM in HUGs. The main conclusions of this paper are as follows:

• (1) With the increase of the attitude angles of the bow, the asymmetry of vorticity and pressure around HUGMB is gradually obvious. This asymmetry of flow field characteristics is the root reason why the movable bow affects the maneuvering performance of HUGMB and provides HUGMB with the moment required for steering and affects the turning radius. The value of the moment and the turning radius have an approximately linear relationship with the attitude angles of the bow. When the attitude angle of the bow is − 30°, the minimum turning radius is only 0.245BL. Adjusting the attitude angle of bow on underwater vehicle is an effective way to improve the maneuvering performance of underwater vehicle.

• (2) The inverse kinematic equation of the mechanism is obtained based on the closed vector method which is verified by kinematic measurement experiment. The dynamic equation of the mechanism in the Lagrange form is established, which is verified by Adams software with the maxium RMSE of 0.018.

• (3) A prototype of MBM is manufactured and assembled. The position control accuracy of the prototype can reach 0.27°, and the speed control accuracy can reach 0.16°/s based on the RBF neural network adaptive sliding mode controller.