Motion Envelope of a Polymorphic Underwater Vehicle During Its Folding Process

Abstract

This study investigates a polymorphic underwater vehicle designed to combine long-range cruising with stable underwater operation, reducing dependence on surface support vessels. By introducing a foldable polymorphic structure, the vehicle can switch configurations, including serial and parallel. However, underwater environments often contain obstacles, and the vehicle may collide with them during the folding process. To prevent collisions between the vehicle and surrounding obstacles during the folding process, this paper investigates the motion envelope of the vehicle and examines how motion parameters and mass distribution influence the motion envelope. In this work, the polymorphic underwater vehicle is modeled as a multibody system operating under a neutrally buoyant condition. Based on space robot modeling methodologies and the linear and angular momentum theorems, the equations of motion of the polymorphic underwater vehicle are derived and verified using the Adams software 2020. In summary, the present study establishes a clear relationship between motion parameters, mass distribution, hydrodynamic effects, and the resulting motion envelope of a polymorphic underwater vehicle. The results show that the attitude of the vehicle during the folding process is uniquely determined by the joint angles, and a larger relative speed between the outer and inner folding motions produces a more compact attitude during the folding process. Mass distribution further influences the motion envelope of the vehicle: concentrating mass toward the center of the vehicle shifts the overall motion envelope upward, whereas concentrating mass toward both ends of the vehicle shifts it downward. In addition, hydrodynamic forces introduce an upward velocity component of the vehicle in the vertical direction during the folding process, which leads to an upward shift in the overall center of mass of the vehicle.

1. Introduction

With the development of marine resources and the growing demand for deep-sea environmental monitoring, underwater operations have become a critical component in advancing marine resource exploitation [1]. Contemporary research increasingly emphasizes unmanned underwater platforms to enhance operational autonomy and efficiency in deep-sea environments. Zhang et al. [2] proposed a full-coverage path-planning algorithm for multiple autonomous underwater vehicles. Liu et al. [3] numerically investigated the interactive hydrodynamic performance of two adjacent unmanned underwater vehicles. Conventional underwater operational platforms, such as remotely operated vehicles (ROV) and manned submersibles, rely on surface support vessels and trained operators, which leads to high deployment costs and limited capability in complex missions [4]. In recent years, driven by advances in battery endurance, onboard computing, and artificial intelligence, work-class autonomous underwater vehicles (AUV) have progressively emerged as a promising alternative to traditional systems and autonomous underwater vehicles are expected to play an increasingly important role in deep-sea operations [5]. Recent advancements in AUV architectures have demonstrated significant progress in payload capacity and endurance; however, a trade-off between hydrodynamic efficiency and operational stability remains a persistent challenge [6]. However, most existing work-class AUVs adopt open-frame architectures; while such designs enhance operational stability, they cannot satisfy the requirements of long-duration, high-efficiency, long-range cruising, thereby still necessitating support from surface vessels [7]. To address this limitation, this study proposes a polymorphic underwater vehicle that integrates long-range cruising and underwater operation capabilities, as shown in Figure 1. The polymorphic underwater vehicle proposed in this paper features two configurations, namely the serial configuration and the parallel configuration: in the serial configuration, the included angles between adjacent links are zero, whereas in the parallel configuration, these angles between adjacent links are 90°. The vehicle can switch its configuration according to mission requirements, enabling long-duration remote cruising in a low-drag serial configuration while providing a stable underwater operational platform in a parallel configuration. This design can not only reduce operational costs but also improve the stealth of the operational process, making it particularly suitable for special environments such as polar regions.

Real underwater environments often contain complex obstacles such as reefs, shipwrecks, subsea mountainous terrain, and man-made installations [8]. To prevent collisions between the polymorphic vehicle and surrounding obstacles, this paper investigates the motion envelope of the vehicle during the folding process and proposes a design scheme for the vehicle to avoid collisions with nearby obstacles in water. During the folding process, the vehicle can be modeled as a multibody system formed by two buoyancy compartments, one payload compartment, and two connecting rods that are mutually hinged, with coupled motions among the components. In addition, owing to its neutrally buoyant condition, the vehicle remains suspended in the water during the folding process, which further increases the complexity of the equations of motion.

Similar to the polymorphic underwater vehicle investigated in this study, space robots remain in a free-floating condition throughout on-orbit operations; therefore, their kinematic and dynamic modeling methodologies provide valuable references for the present research. Vafa and Dubowsky [9] proposed the Virtual Manipulator approach, in which a space robot is assumed to be free of external forces and is mapped to an idealized virtual robot with a fixed base, thereby simplifying the complex dynamic coupling between the manipulator and its carrier. However, the Virtual Manipulator approach is applicable only to dynamic modeling in a microgravity environment. To further simplify the dynamic modeling of space robots, Liang et al. [10] proposed a Dynamically Equivalent Manipulator (DEM) model, which establishes an equivalent mapping between a free-floating space robot and a conventional fixed-base robot, making the modeling procedure more akin to that of traditional manipulators. In addition, Ma et al. [11] presented a dynamic modeling method for redundant space robots and derived the equations of motion using multibody system dynamics. Based on the momentum theorem, Zhang and Wen [12] proposed a motion planning strategy to achieve end-effector trajectory tracking for space robots. Umetani and Yoshida [13] introduced the Generalized Jacobian Matrix (GJM) for resolved motion rate control of space manipulators and established a direct relationship between end-effector motion and joint motion. Nenchev et al. [14] further analyzed a redundant free-flying manipulator system, extending the application of the GJM framework to redundant free-floating systems. Mukherjee and Nakamura [15,16], Zhou and Luo [17], and Dong et al. [18] also derived the GJM through different derivation procedures. Soleymani and Kiani [19] developed a comprehensive dynamic model and configuration-control method for a free-floating soft manipulator-spacecraft system. In a related study, they also investigated dynamic modeling and control of planar soft space robotic manipulators, further extending free-floating manipulator modeling to soft robotic systems [20]. Since the computation of the GJM is independent of momentum conservation, in recent years several researchers have extended the GJM to nonconservative systems. Eslami and Babazadeh [21,22] incorporated the variation in system momentum to obtain a modified dynamic formulation for space robots. Rybus et al. [23] compensated for the momentum variation in a space robot via the virtual work principle. Similarly, the polymorphic underwater vehicle proposed in this paper also operates in a free-floating condition. Moreover, the momentum of the vehicle system is no longer conserved with the presence of hydrodynamic forces [24]. Although previous studies have provided useful modeling methods for free-floating multibody systems, the motion envelope of a polymorphic underwater vehicle during the folding process has not been sufficiently investigated. In particular, how the motion parameters, mass distribution, and hydrodynamic forces influence the physical boundary of the vehicle during the transition between the serial configuration and the parallel configuration still requires further study [25]. This is necessary because the space occupied by the vehicle during the folding process cannot be determined only from the serial configuration or the parallel configuration but depends on the variation in the physical boundary of the vehicle throughout the folding process. Therefore, this paper investigates the motion envelope of the vehicle during the folding process. The framework of the GJM is applicable to the vehicle considered in this paper. Accordingly, the equations of motion of the polymorphic underwater vehicle are established based on the linear and angular momentum theorems while using the derivation route of GJM. The equations of motion are then verified using the Adams software, and the influences of motion parameters, mass distribution, and hydrodynamic forces on the motion envelope of the vehicle are analyzed.

This paper is structured as follows. Section 2 presents the equations of motion of the polymorphic underwater vehicle and verifies the equations of motion. Section 3 investigates how the motion parameters and mass distribution of the polymorphic underwater vehicle influence the motion envelope of the vehicle under different hydrodynamic coefficients. Finally, the conclusions are presented in Section 4.

2. Equations of Motion of the Polymorphic Underwater Vehicle

In this section, the equations of motion of the polymorphic underwater vehicle are established and then verified using the Adams software.

2.1. Dynamic Modeling of the Polymorphic Underwater Vehicle

The polymorphic underwater vehicle consists of one payload compartment, two buoyancy compartments, and several connecting rods and can be regarded as a multibody system. To simplify the establishment of the equations of motion, based on Refs. [26,27,28] the following assumptions are made in this paper:

• The two outermost connecting rods of the polymorphic underwater vehicle are neglected, and the entire system is simplified into a planar five-link mechanism, with the leftmost link of the polymorphic underwater vehicle defined as the base link, and the base link is free to translate and rotate;
• All joints of the polymorphic underwater vehicle are single-DOF revolute joints, and the joints rotate at constant angular velocity during the folding process;
• The two buoyancy compartments at the ends of the polymorphic underwater vehicle fold inward at the same speed, and the attitudes of the links are symmetric to the plumb line passing through the overall center of mass of the polymorphic underwater vehicle. Owing to the symmetry of the physical structure and folding motion of the vehicle about the Y-axis, the hydrodynamic moments induced by the hydrodynamic forces acting on the symmetric links are equal in magnitude and opposite in direction with respect to the overall center of mass of the vehicle. Therefore, the resultant hydrodynamic moment about the overall center of mass of the vehicle is zero, and the angular momentum of the polymorphic underwater vehicle about its center of mass is conserved.
• In the serial configuration, the line connecting the center of mass of the polymorphic underwater vehicle and the center of buoyancy of the polymorphic underwater vehicle is perpendicular to the still-water surface and the center of mass of the polymorphic underwater vehicle lies below the center of buoyancy of the polymorphic underwater vehicle, such that the polymorphic underwater vehicle is in a level floating attitude;
• In the vertical direction, the buoyancy affecting the polymorphic underwater vehicle balances the gravity affecting the polymorphic underwater vehicle, and no external forces act on the polymorphic underwater vehicle other than hydrodynamic forces;
• The polymorphic underwater vehicle is initially at rest in a level floating attitude;
• Since the parallel sections of the payload compartment and the buoyancy compartments account for the majority of the length of the hull, each compartment is simplified as a cylinder.

Figure 2 illustrates the simplified polymorphic underwater vehicle, modeled as a 5R serial mechanical arm. CM denotes the overall center of mass of the polymorphic underwater vehicle. O_I-X_IY_I is the inertial frame, whose origin O_I is located at the overall center of mass of the polymorphic underwater vehicle position when the vehicle is in the level floating attitude, and whose Y_I-axis is aligned with the plumb line. Starting from the base link of the vehicle, the links are consecutively denoted as L₀, L₁, L₂, L₃, and L₄, and the joints of the vehicle are denoted as J₁, J₂, J₃, and J₄, respectively. Since the buoyancy compartments at the ends of the vehicle fold inward at the same speed, the angular momentum of the polymorphic underwater vehicle about its center of mass is conserved and no rotation occurs to its center of mass; therefore, the attitudes of L₀ and L₄, as well as those of L₁ and L₃, are symmetric about the plumb line passing through the overall center of mass of the polymorphic underwater vehicle, and L₂ remains horizontal during the folding process. Furthermore, because the net external force in the X-direction is zero, the overall center of mass of the polymorphic underwater vehicle does not translate along the X-direction; hence, the overall center of mass of the polymorphic underwater vehicle always lies on the Y_I-axis of the inertial frame. The vectors r_i (i = 0, 1, 2, 3, 4) denote the position vectors of the centers of mass of the link, and r_g denotes the position vector of the overall centers of mass of the polymorphic underwater vehicle. According to the operating principle of the polymorphic underwater vehicle, prior to folding the vehicle is in a level floating attitude in the serial configuration, and the joint angles are defined to be zero in this attitude; during the folding process, each joint rotates continuously in a prescribed direction until reaching an absolute angle of 90°.

The symbols used in the following are defined uniformly below:

• L₀: Base link of the simplified polymorphic underwater vehicle;
• L_i (i = 0, 1, 2, 3, 4): Link i of the simplified polymorphic underwater vehicle;
• J_i (i = 0, 1, 2, 3, 4): Revolute joint i, connecting link L_i−1 and link L_i;
• O_I-X_IY_I: Inertial frame used in the planar model;
• m_i (i = 0, 1, 2, 3, 4): Mass of link L_i;
• r_i (i = 0, 1, 2, 3, 4): Position vector of the center of mass of link L_i in the inertial frame.
• r_g: Position vector of the overall center of mass of the polymorphic underwater vehicle in the inertial frame;
• r_0g: Position vector from the center of mass of the base link to the overall center of mass;
• z_i (i = 0, 1, 2, 3, 4): Position vector of joint J_i in the inertial frame;
• d_kk: Position vector from the center of mass of link L_k to joint J_k;
• v_i (i = 0, 1, 2, 3, 4): Linear velocity vector of the center of mass of link L_i in the inertial frame;
• ω_i (i = 0, 1, 2, 3, 4): Absolute angular velocity vector of link L_i in the inertial frame;
• ω_i (i = 0, 1, 2, 3, 4): Scalar angular velocity of joint J_i;
• ⁱ⁻¹ω_i: Angular velocity vector of link L_i relative to link L_i−1;
• e_i (i = 0, 1, 2, 3, 4): Unit vector along the axis direction of joint J_i in the inertial frame;
• I_zz,i: Moment of inertia of link L_i about the z-axis passing through its center of mass;
• Ψ_m: Joint angular velocity vector.

The position of the center of mass of each link can be expressed in terms of the center of mass position of the base link as:

$${\mathbf{r}}_i={\mathbf{r}}_{\mathit{0}}+{\displaystyle \sum _{k=1}^i\left({\mathbf{d}}_{(k-\mathit{1})k}-{\mathbf{d}}_{kk}\right)}$$

(1)

where d_kk denotes the position vector from the center of mass of link L_k to joint J_k. Differentiating Equation (1) with respect to time yields the linear velocity vectors of the centers of mass of the link as:

$${\mathbf{v}}_i={\mathbf{v}}_0+{\mathbf{\omega }}_0\times \left({\mathbf{r}}_i-{\mathbf{r}}_0\right)+{\displaystyle \sum _{k=1}^i\left[{\mathbf{e}}_k\times \left({\mathbf{r}}_i-{\mathbf{z}}_k\right)\right]}\hspace{0.17em}{\omega }_k$$

(2)

where e_i is the unit vector along the axis direction of joint J_i in the inertial frame and z_i denotes the position vector of the joint in the inertial frame. The position of the overall center of mass of the polymorphic underwater vehicle is given by:

$${\mathbf{r}}_{\mathrm{g}}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{i=0}^4{m}_i{\mathbf{r}}_i}$$

(3)

where M is the total mass of the polymorphic underwater vehicle and m_i is the mass of link L_i. According to Equations (1) and (3), the position of the center of mass of the base link can be written as:

$${\mathbf{r}}_0={\mathbf{r}}_{\mathrm{g}}-{\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=1}^4\left({\displaystyle \sum _{j=k}^4{m}_j}\right)}\left({\mathbf{d}}_{(k-1)k}-{\mathbf{d}}_{kk}\right)$$

(4)

Origin O_I is located at the overall center of mass of the polymorphic underwater vehicle position when the vehicle is in the level floating attitude, i.e., r_g = 0.

Let the absolute angular velocity vector of the base link L₀ in the inertial frame be denoted as ω₀. Then, the absolute angular velocity vector of each link can be expressed as:

$${\mathbf{\omega }}_i={\mathbf{\omega }}_0+{}^0\mathbf{\omega }_i$$

(5)

where ⁰ω_i denotes the angular velocity vector of L_i relative to L₀ in the inertial frame. The angular velocity vector of joint J_i, denoted as ⁱ⁻¹ω_i, can be written as:

$${}^{i-1}\mathbf{\omega }_i={\mathbf{e}}_i\cdot {\omega }_i$$

(6)

where ω_i denotes the angular velocity of joint J_i. The angular velocity vector of link L_i relative to link L₀ is:

$${}^0\mathbf{\omega }_i={\displaystyle {\displaystyle \sum _{k=1}^i}{}^{k-1}\mathbf{\omega }_k}={\displaystyle {\displaystyle \sum _{k=1}^i}{\mathbf{e}}_k·{\omega }_k}$$

(7)

Substituting Equation (7) into Equation (5) yields the absolute angular velocity vector of link L_i in the inertial frame as:

$${\mathbf{\omega }}_i={\mathbf{\omega }}_0+{\displaystyle \sum _{k=1}^i{\mathbf{e}}_k·{\omega }_k}$$

(8)

The total linear momentum of the polymorphic underwater vehicle can be expressed as:

$${\mathbf{P}}_{\mathrm{lin}}={\displaystyle \sum _{i=0}^4{m}_i{\mathbf{v}}_i}$$

(9)

Substituting Equation (2) into Equation (9) yields:

$${\mathbf{P}}_{\mathrm{lin}}={\mathbf{H}}_1\left[\begin{array}{c}{\mathbf{v}}_0\\ {\mathbf{\omega }}_0\end{array}\right]+{\mathbf{H}}_2{\mathbf{\psi }}_{\mathbf{m}}$$

(10)

where

$$\begin{gathered} {\mathbf{H}}_1=\left[\begin{array}{cc}Mdiag(3),& -M{[{\mathbf{r}}_{0\mathrm{g}}]}_{\times }\end{array}\right] \\ {\mathbf{H}}_2\left(:,k\right)={\displaystyle \sum _{i=k}^4{m}_i{\mathbf{H}}_{2i}}{\mathbf{e}}_k\times ({\mathbf{r}}_i-{\mathbf{z}}_k),\hspace{0.25em}\hspace{1em}k=1,2,3,4 \\ {\mathbf{\psi }}_{\mathbf{m}}={[{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4]}^T \end{gathered}$$

where []_× denotes the skew-symmetric matrix, and diag(3) denotes the 3 × 3 identity matrix. The total angular momentum of the polymorphic underwater vehicle about its overall center of mass is given by:

$${\mathbf{P}}_{\mathrm{ang}}={\displaystyle \sum _{i=0}^4\left(I_i{\mathbf{\omega }}_i+({\mathbf{r}}_i-{\mathbf{r}}_{\mathrm{g}})\times m_i{\mathbf{v}}_i\right)}$$

(11)

where I_i is the inertia tensor of link L_i about its center of mass, expressed in the inertial frame. Substituting Equations (2) and (8) into Equation (11), and rewriting in matrix form, gives:

$${\mathbf{P}}_{\mathrm{ang}}=\left[\begin{array}{cc}{\mathbf{0}}_{3\times 3}& {\mathbf{k}}_{\mathsf{\omega }}\end{array}\right]\left[\begin{array}{c}{\mathbf{v}}_0\\ {\mathbf{\omega }}_0\end{array}\right]+{\mathbf{k}}_{\mathrm{q}}{\mathbf{\psi }}_{\mathbf{m}}$$

(12)

where

$$\begin{gathered} {\mathbf{k}}_{\mathsf{\omega }}={\displaystyle \sum _{i=0}^4\left(I_i+m_i{[{\mathit{r}}_{i\mathrm{g}}]}_{\times }^T{[{\mathit{r}}_{i\mathrm{g}}]}_{\times }\right)} \\ {\mathbf{k}}_{\mathrm{q}}\left(:,k\right)={\displaystyle \sum _{i=k}^4\left[I_k{\mathbf{e}}_k+m_i{[{\mathbf{r}}_{i\mathrm{g}}]}_{\times }\left({\mathbf{e}}_k\times ({\mathbf{r}}_i-{\mathbf{z}}_k)\right)\right]},\hspace{0.25em}\hspace{1em}k=1,2,3,4 \end{gathered}$$

Because the polymorphic underwater vehicle is initially at rest, P_ang is identically zero. Combining Equations (10) and (12) leads to the equations of motion of the polymorphic underwater vehicle:

$${\mathbf{r}}_i={\mathbf{r}}_0+{\displaystyle \sum _{k=1}^i\left({\mathbf{d}}_{(k-1)k}-{\mathbf{d}}_{kk}\right)}$$

(13)

Let

$${\mathbf{H}}_b=\left[\begin{array}{cc}Mdiag(3)& -M{[{\mathbf{r}}_{0\mathrm{g}}]}_{\times }\\ {\mathbf{0}}_{3\times 3}& {\mathbf{k}}_{\mathsf{\omega }}\end{array}\right],{\mathbf{H}}_{\mathbf{m}}=\left[\begin{array}{c}{\mathbf{H}}_2\\ {\mathbf{k}}_{\mathrm{q}}\end{array}\right]$$

where H_b is the inertia matrix of the base, and H_m is the coupled inertia matrix between the base link and the other links. Equation (13) can then be simplified to:

$$\left[\begin{array}{c}{\mathbf{P}}_{\mathrm{lin}}\\ {\mathbf{P}}_{\mathrm{ang}}\end{array}\right]={\mathbf{H}}_{\mathbf{b}}\left[\begin{array}{c}{\mathbf{v}}_0\\ {\mathbf{\omega }}_0\end{array}\right]+{\mathbf{H}}_{\mathbf{m}}{\mathbf{\psi }}_{\mathbf{m}}=\left[\begin{array}{c}{\mathbf{F}}_{\mathrm{w}}dt\\ \mathbf{0}\end{array}\right]$$

(14)

Under the symmetry assumption, the resultant hydrodynamic moment about the overall center of mass of the vehicle is zero. Therefore, hydrodynamic forces cause the variation in the linear momentum of the vehicle, while no additional hydrodynamic moment term is introduced into the angular momentum equation.

Because H_b is block upper triangular,

$$det({\mathbf{H}}_b)=M^{3}det({\mathbf{K}}_{\omega })$$

(15)

The parallel-axis terms are positive semidefinite, and the rotated inertia tensors remain positive definite. For the three main cylindrical compartments, the minimum principal inertia is greater than zero. Therefore,

$${\lambda }_{\min }({\mathbf{K}}_{\mathsf{\omega }})>0$$

(16)

Thus, K_ω is positive definite and H_b remains invertible for the whole folding process, including parallel configuration and serial configuration.

Since the base inertia matrix H_b is invertible [29], the linear and angular velocities of the base link can be obtained as:

$$\left[\begin{array}{c}{\mathbf{v}}_0\\ {\mathbf{\omega }}_0\end{array}\right]={{\mathbf{H}}_b}^{-1}\left(\left[\begin{array}{c}{\mathbf{F}}_{w}dt\\ \mathbf{0}\end{array}\right]-{\mathbf{H}}_{\mathbf{m}}{\mathbf{\psi }}_{\mathbf{m}}\right)$$

(17)

where F_w denotes the hydrodynamic forces, and t denotes time. The linear and angular velocities of the base link are computed in Matlab R2022a based on the derived equations, using the time step of 0.008 s.

The hydrodynamic forces in Equation (17) can be evaluated using the Morison equation [30,31,32]. The hydrodynamic forces acting on link L_i can be expressed as:

$${\mathbf{F}}_{\mathrm{w}}={\displaystyle \frac{1}{2}}\rho A{C}_d\left|\mathbf{v}\right|\mathbf{v}+C_m\rho V\dot{\mathbf{v}}$$

(18)

where V is the link volume and A is the projected (frontal) area of the link normal to the incoming flow. C_d and C_m are the drag coefficient and the added-mass coefficient, respectively. For a cylindrical structure, according to the literature [33,34,35], C_d = 0.92 and C_m = 0.85 are adopted in this paper.

The geometric dimensions, masses, and other parameters of the polymorphic underwater vehicle are listed in

Table 1. Parameters of the polymorphic underwater vehicle.

Component	Mass (mi/kg)	Length (Li/m)	Radius (m)	Moment of Inertia (Izz,i/kg·m2)
L0	127.17	1.8	0.15	35.05
L1	0	0.6	0.02	0
L2	127.17	1.8	0.15	35.05
L3	0	0.6	0.02	0
L4	127.17	1.8	0.15	35.05

; it should be noted that the density of each link is uniformly distributed, i.e., the center of mass of the link is located at the midpoint of its geometric length.

2.2. Verification of the Equations of Motion

In this study, the equations of motion are verified using the Adams software. A simplified planar five-link model is constructed in the Adams software, and revolute joint constraints are imposed at the interfaces between adjacent links to emulate the physical joints; the resulting initial model in the Adams software is shown in Figure 3.

The joints of the vehicle are actuated via the driving functions in the Adams software to rotate from the stationary serial configuration with an angular acceleration of 0.2 rad/s². The time span of the simulation in Adams is set to 4 s with the same time step of 0.008 s as in Matlab to ensure a consistent comparison. As shown in Figure 4, the maximum absolute differences in the X- and Y-directions are 0.00158642 m and 0.00156269 m, respectively, while the corresponding RMSE values are 0.000794364 m and 0.000914487 m. It shows the coordinates of the position of the center of mass of L₀ obtained, under the condition that both the drag coefficient and the added-mass coefficient are set to zero, by solving the equations of motion in Matlab and by simulating in the Adams software, respectively. As can be seen from Figure 4, the two results coincide exactly, which verifies the equations of motion in the previous section. However, since both the drag coefficient and the added-mass coefficient are set to zero in this verification, further validation using real underwater experimental data is still needed.

3. Results and Discussion

In this section, the variation in the motion envelope of the polymorphic underwater vehicle with motion parameters (e.g., angular velocity and angular velocity ratio) and mass distribution is first analyzed in the absence of hydrodynamic forces. Then, hydrodynamic forces are applied to the vehicle compartments to investigate the motion characteristics of the polymorphic underwater vehicle in realistic water. Owing to the symmetry of the structure and motion of the polymorphic underwater vehicle, the following analysis only considers the extremum of the motion envelope of the polymorphic underwater vehicle within the half-plane X > 0 of the global frame O_I-X_IY_I. In the following analysis, each joint of the polymorphic underwater vehicle is assumed to rotate at a constant angular velocity. Here we explain why the condition without hydrodynamic forces is analyzed first. When hydrodynamic forces are neglected, the analysis can focus on how motion parameters and mass distribution influence the motion envelope of the vehicle. This analysis also provides a theoretical reference for interpreting the extra influences of hydrodynamic forces on the motion envelope of the vehicle. Moreover, when the joint angular velocities are sufficiently small, the hydrodynamic forces acting on the vehicle can be neglected.

3.1. The Definition of the Motion Envelope of the Polymorphic Underwater Vehicle

Figure 5a shows the attitude of the polymorphic underwater vehicle at a certain instant during the folding process, where X_max denotes the maximum of the physical boundary of the vehicle structure in the X-direction, and Y_max and Y_min denote the maximum and minimum, respectively, of the physical boundary of the vehicle structure in the Y-direction. Figure 5b depicts the motion envelope of the polymorphic underwater vehicle during the folding process, where EX_max denotes the maximum of the motion envelope in the X-direction, and EY_max and EY_min denote the maximum and minimum, respectively, of the motion envelope of the polymorphic underwater vehicle in the Y-direction.The red dashed line in Figure 5b represents the schematic curve of the motion envelope of the vehicle.

3.2. Motion Envelope of the Polymorphic Underwater Vehicle Without Hydrodynamic Forces

In this section, the variations in the motion envelope of the polymorphic underwater vehicle with the vehicle motion parameters and mass distribution are analyzed.

3.2.1. Influence of the Angular Velocity of the Joint on the Physical Boundary of the Vehicle

Figure 6 shows how the physical boundary of the vehicle varies with the folding completion percentage of joint J₃ during the folding process. In this section, assume that joints J₃ and J₄ rotate at the same speed and the angular velocities of joint J₃ are 0.25 rad/s and 0.5 rad/s, respectively. Since joints J₃ and J₄ have identical angular velocities and the same target angle, joint J₄ completes folding when joint J₃ completes folding; that is, when the folding completion percentage of joint J₃ reaches 100%, the entire polymorphic underwater vehicle has completed the folding process.

As shown in Figure 6a, with increasing folding completion percentage of joint J₃, the physical boundary of the polymorphic underwater vehicle in the X-direction first shrinks markedly and then changes gradually, whereas the physical boundary of the polymorphic underwater vehicle in the Y-direction overall expands simultaneously toward both sides and then shrinks simultaneously, with the critical point of these trend changes occurring at a folding completion percentage of 50% for joint J₃. Figure 7 illustrates the attitudes of the polymorphic underwater vehicle at different folding completion percentage of joint J₃. As the folding completion percentage of joint J₃ increases from 0 to 50%, although the long ends of links L₃ and L₄ jointly determine the physical boundary of the polymorphic underwater vehicle, the long end of link L₄ plays a dominant role because its length is much greater than that of link L₃. During this stage, the long end of link L₄ folds from a horizontal attitude to a vertical attitude, resulting in a pronounced shrinkage of the physical boundary of the polymorphic underwater vehicle in the X-direction and a pronounced expansion of the physical boundary of the polymorphic underwater vehicle in the Y-direction. In the absence of hydrodynamic forces, the position of the overall center of mass of the polymorphic underwater vehicle remains unchanged; therefore, the physical boundary of the polymorphic underwater vehicle expands simultaneously in both directions along the Y-axis. As the folding completion percentages of joint J₃ increases from 50% to 100%, the link L₃ and the short end of link L₄ jointly determine the physical boundary of the polymorphic underwater vehicle in the X-direction; because the link L₃ and the short end of link L₄ have comparable lengths and exhibit opposite trends in their influence on the physical boundary of the polymorphic underwater vehicle in the X-direction, the variation in the boundary in the X-direction is not significant. During this stage, the long end of link L₄ still dominates the variation in the physical boundary of the polymorphic underwater vehicle in the Y-direction, and as it folds from a vertical attitude back to a horizontal attitude, the boundary in the Y-direction gradually contracts. As can be seen from Figure 6b, after folding process is completed, the upper and lower physical boundary of the polymorphic underwater vehicle in the Y-direction do not coincide with the height of the overall center of mass of the vehicle; this is because, once folding process is completed, the vehicle is in the parallel configuration, and the buoyancy compartments and the payload compartment lie on horizontal planes at different elevations, such that the upper physical boundary of the polymorphic underwater vehicle in the Y-direction is higher than the center of mass of the vehicle while the lower Y-boundary is lower than the center of mass of the vehicle.

From the above analysis, the maximum of the physical boundary of the vehicle in the Y-direction is determined by the long segment of link L₄, and it attains its maximum when link L₄ reaches the vertical attitude, i.e., when the folding completion percentage of joint J₃ equals 50%. As shown in Figure 6b, the time at which the physical boundary of the vehicle in the Y-direction reaches its minimum is later than that at which it reaches its maximum. The minimum of the physical boundary of the vehicle in the Y-direction is determined by the height of the center of mass of link L₂. Figure 8 shows the variation in the velocity of the center of mass of link L₂ in the Y-direction with the folding completion percentage of joint J₃; it can be seen that when the folding completion percentage of joint J₃ equals 50%, the center of mass of link L₂ still has a downward velocity component in the Y-direction. As marked in the figure, when the folding completion percentage of joint J3 reaches 60%, the velocity of the center of mass of link L₂ in the Y-direction crosses zero and the physical boundary of the vehicle in the Y-direction reaches its minimum. When the folding completion percentage of joint J₃ lies between 50% and 60%, as the joints continue to rotate, the long segment of link L₄ folds inward and the maximum of the physical boundary of the vehicle in the Y-direction decreases accordingly; however, the mass of L₄ located in the upper half-plane increases, and to maintain an unchanged the position of overall center of mass of the polymorphic underwater vehicle, link L₂ continues to move downward, i.e., the minimum of the physical boundary of the vehicle in the Y-direction decreases. This, therefore, causes the vehicle to reach the minimum of the physical boundary of the vehicle in the Y-direction later than the maximum of the physical boundary of the vehicle in the Y-direction.

As can be seen from Figure 5, under different joint speed conditions, the variation trends of the physical boundary of the vehicle with the folding completion percentage of joint J₃ are completely consistent. This is because the attitude of the vehicle at any instant is determined solely by the joint angles and the position of the overall center of mass remains constant; therefore, the positions of the links depend only on the folding completion percentage of the joints.

3.2.2. Influence of Angular Velocity Ratio of Joints on the Physical Boundary of the Vehicle

Figure 9 shows the variation in the physical boundary of the vehicle with the folding completion percentage of joint J₃ during the folding process with the angular velocity of joint J₃ fixed at 0.01 rad/s, when the angular velocity ratio of joint J₄ to joint J₃ is 0.5 and 2, respectively. Since joints J₃ and J₄ rotate at different angular velocities while sharing the same target angle of 90°, if one joint reaches 90° earlier, it stops rotating immediately and remains stationary until the other joint also reaches 90°. Consistent with the previously observed phenomenon, Figure 9a shows that as the folding completion percentage of joint J₃ increases, the physical boundary of the vehicle in the X-direction shrinks markedly at first and then varies gradually, whereas Figure 9b indicates that the physical boundary in the Y-direction expands simultaneously toward both sides and then shrinks simultaneously. When the angular velocity ratio of joint J₄ to joint J₃ is less than 1, after the end of the short segment of link L₄ becomes vertically aligned with the end of link L₂, the end of link L₂ determines the physical boundary of the vehicle in the X-direction; because link L₂ does not translate in the X-direction, the physical boundary of the vehicle in the X-direction remains unchanged. In Figure 9b, all curves exhibit an inflection point with an abrupt change in slope. This occurs because joints J₃ and J₄ have different angular velocities but the same target angle: at a certain instant, one joint reaches the target angle and stops rotating while the other continues to rotate at a constant speed; owing to angular momentum conservation of the polymorphic underwater vehicle about its center of mass, the links undergo an abrupt change in velocity, leading to a sudden change in the slope of the physical boundary of the vehicle in the Y-direction. A comparison between Figure 6 and Figure 9 indicates that the larger the angular velocity ratio of joint J₄ to joint J₃, the smaller the critical folding completion percentage of joint J₃ separating the marked shrinkage and the gradual variation in the X-direction, because link L₄ folds from a horizontal attitude to a vertical attitude more rapidly.

3.2.3. Influence of Angular Velocity Ratio of Joints on the Motion Envelope of the Vehicle

The above analysis indicates that the angular velocity ratio of joints has a significant influence on the physical boundary of the vehicle at each instant. Since the motion envelope is the union of the instantaneous boundaries over time, the angular velocity ratio of joints therefore substantially affects the motion envelope of the vehicle. It should be noted that, for any angular velocity ratio, the maximum of the motion envelope of the vehicle in the X-direction is always equal to the coordinate of the end of the link L₄ in the initial state; hence, the angular velocity ratio does not affect the maximum of the motion envelope of the vehicle in the X-direction. Accordingly, the following analysis focuses only on the influence of the angular velocity ratio of joints on the motion envelope of the vehicle in the Y-direction. Figure 10 shows the variation in the extremum of the motion envelope of the vehicle in the Y-direction with the angular velocity ratio, with the angular velocity of joint J₃ fixed at 0.01 rad/s. It can be observed that the maximum of the motion envelope of the vehicle in the Y-direction decreases as the angular velocity ratio increases, whereas the minimum increases rapidly and then decreases slowly as the angular velocity ratio increases. As discussed above, the upper physical boundary of the vehicle in the Y-direction reaches its extremum when link L₄ is in a vertical attitude. In Figure 10, as the angular velocity of joint J₄ increases, when link L₄ reaches the vertical attitude, the folding completion percentage of joint J₃ decreases and the projected length of link L₃ in the Y-direction becomes smaller, leading to a reduction in the maximum of the motion envelope of the vehicle in the Y-direction; since the height of the overall center of mass of the vehicle remains constant, the minimum of the motion envelope of the vehicle in the Y-direction increases accordingly. In Figure 10a, as the angular velocity ratio approaches 0, the angular velocity of joint J₄ is much slower than that of joint J₃; before joint J₃ reaches 90°, links L₃ and L₄ are nearly collinear. In this case, the link attitudes are as shown in Figure 10b, and the upper and lower physical boundary of the vehicle in the Y-direction can be obtained by simple geometric calculations under the premise that the height of the overall center of mass of the vehicle remains constant, yielding 1.3 and −0.8, respectively; these values coincide with the extremum shown in the figure, further verifying the equations of motion. In Figure 10, the minimum of the motion envelope of the vehicle in the Y-direction exhibits an inflection point near an angular velocity ratio of 2.0. This is because, when the angular velocity of joint J₄ is small relative to that of joint J₃, the extremum of the lower physical boundary of the vehicle in the Y-direction is determined by the height of the center of mass of the link L₂; whereas when the angular velocity of joint J4 is large relative to that of joint J₃, the extremum of the lower physical boundary of the vehicle in the Y-direction is determined by the height of the outer end of the short segment of link L₄. It can be seen that adopting a larger angular velocity ratio results in a more compact vehicle attitude during the folding process.

3.2.4. Influence of Mass Distribution on the Physical Boundary of the Vehicle

According to the equations of motion of the vehicle, the mass distribution of the links is also a key factor affecting the physical boundary of the vehicle. Since link L₃ serves only as a connector, its mass is assumed to be zero; therefore, this paper investigates only the influence of mass distribution between links L₂ and L₄ on the physical boundary of the vehicle. To maintain the neutrally buoyant condition, the total mass of the vehicle is kept constant. Figure 11 presents how the physical boundary of the vehicle varies with the folding completion percentage of joint J₃ during the folding process for the cases where the mass is concentrated at the center of the vehicle and at both ends of the vehicle, respectively, with all joint angular accelerations set to 0.25 rad/s²; specifically, for the mass concentrated at center of the vehicle condition, 2m₄/m₂ = 0.1, whereas for the mass concentrated at both ends of the vehicle condition, 2m₄/m₂ = 10.

As can be seen from Figure 11a, the mass distribution of the vehicle does not influence the physical boundary of the vehicle in the X-direction. This is because the joint angles determine the vehicle attitude, which is symmetric about the Y-axis; under the prescribed angular velocities of joints, the mass distribution does not influence the vehicle attitude at each instant, and thus does not influence the physical boundary of the vehicle in the X-direction, but it does influence the physical boundary of the vehicle in the Y-direction. As shown in Figure 11b, when the mass of the vehicle is concentrated at both ends, the physical boundary of the vehicle shifts downward as a whole. As an extreme case of mass concentrated at both ends of the vehicle, if the mass of link L₂ is assumed to be zero, then the mass of the vehicle is concentrated on links L₀ and L₄, and the overall center of mass of the vehicle is always at the same height as the centers of mass of links L₀ and L₄. During the folding process, since the height of the overall center of mass of the vehicle remains constant and the attitudes of links L₀ and L₄ remain symmetric about the Y-axis, the heights of the centers of mass of L₀ and L₄ remain constant at their minimum, and consequently the limiting height of link L₂ is the lowest. When the mass of the vehicle is concentrated at the center, the physical boundary of the vehicle shifts upward as a whole. As an extreme case of mass concentrated at the center of the vehicle, if the masses of links L₀ and L₄ are assumed to be zero, then the mass of the vehicle is concentrated on link L₂, and the overall center of mass of the vehicle is always at the same height as the center of mass of link L₂. During the folding process, since the height of the overall center of mass of the vehicle remains constant, the height of the center of mass of link L₂ remains constant at its maximum, and consequently the limiting heights of links L₀ and L₄ are the highest.

Figure 12 illustrates the influence of mass distribution on the motion envelope of the vehicle in the Y-direction. It can likewise be seen that, as the mass of the vehicle becomes increasingly concentrated at both ends of the vehicle, the motion envelope of the vehicle in the Y-direction shifts downward as a whole. As an extreme case of mass concentrated at both ends of the vehicle, if the mass of link L₂ is assumed to be zero, then the mass of the vehicle is concentrated on links L₀ and L₄; when link L₄ is in a vertical attitude, the physical boundary of the vehicle reaches the upper boundary, and the height of the upper physical boundary of the vehicle equals one half of the length of link L₄. As an extreme case of mass concentrated at the center of the vehicle, if the masses of links L₀ and L₄ are assumed to be zero, then the mass of the vehicle is concentrated on link L₂, and during the folding process link L₂ remains at its initial height throughout.

Based on the above findings, it can be concluded that, during the folding process, to avoid collisions between the vehicle and surrounding obstacles, it is preferable to concentrate the mass of the vehicle toward the center of the vehicle when obstacles are located below the vehicle, whereas it is preferable to concentrate the mass of the vehicle toward both ends of vehicle when obstacles are located above the vehicle.

3.3. Motion Envelope of the Polymorphic Underwater Vehicle with Hydrodynamic Forces

In this section, hydrodynamic forces are applied to vehicle compartments to investigate the motion characteristics of the polymorphic underwater vehicle in realistic water. Owing to the symmetry of the vehicle’s physical structure and motion speed about the Y-axis, the hydrodynamic forces in the X-direction are zero; therefore, the physical boundary of the vehicle in the X-direction is unaffected. Figure 13a shows how the physical boundary of the vehicle in the Y-direction varies with the folding completion percentage of joint J₃ during the folding process when the angular velocity of joint J₃ is 0.25 rad/s and the angular velocity ratio of joint J₄ to joint J₃ is 1. It can be seen that, after adding hydrodynamic forces, both the upper and lower physical boundaries of the vehicle shift upward. As shown in Figure 13b, the physical boundary of the vehicle in the Y-direction is calculated with only the drag term, only the added-mass term, and both terms, respectively. Figure 14 presents the time histories of the hydrodynamic forces in the Y-direction (F_y) acting on the overall center of mass of the vehicle. During the folding process, the hydrodynamic forces in the Y-direction gradually reverse from the positive Y-direction to the negative Y-direction. According to the impulse–momentum theorem, the integral of the hydrodynamic forces–time curve corresponds to the increment of the total linear momentum of the vehicle. Integrating the curve in Figure 14 indicates that the increment of the total linear momentum of the vehicle during the folding process is 2.28 N·s, which is greater than zero. The corresponding upward velocity increment of the overall center of mass of the vehicle is 0.00598 m/s; therefore, adding hydrodynamic forces, the vehicle always possesses an upward velocity component, which causes an overall upward shift in the physical boundary of the vehicle in the Y-direction.

4. Conclusions

This paper focuses on the motion envelope of a polymorphic underwater vehicle during the folding process, with the aim of avoiding collisions between the vehicle and the surrounding obstacles during the folding process. Based on the linear and angular momentum theorems, the equations of motion of the vehicle are established and verified using the Adams software. In summary, the present study establishes a clear relationship between motion parameters, mass distribution, hydrodynamic forces, and the resulting motion envelope of the vehicle. By clarifying how the physical boundary of the vehicle varies throughout the folding process, this study provides a basis for estimating the space required when the vehicle switches between the serial configuration and the parallel configuration in obstacle-containing underwater environments, which can further support the selection of operation mode and the design of the joint control system in future work. In this work, each joint of the polymorphic underwater vehicle is assumed to rotate at a constant angular velocity. In the absence of hydrodynamic forces, the influences of motion parameters and mass distribution on the motion envelope of the vehicle are investigated. First, the attitude of the vehicle at any instant is uniquely determined by the joint angles; therefore, the variation trend of the physical boundary of the vehicle during folding progress is independent of the angular velocity of the joint. Second, adopting a larger angular velocity ratio of joint J₄ to joint J₃ enables a more compact attitude of the vehicle during the folding process. Then, concentrating the mass toward the center of the vehicle shifts the overall motion envelope of the vehicle upward, whereas concentrating the mass toward both ends of the vehicle shifts the overall motion envelope of the vehicle downward. Finally, when hydrodynamic forces are considered, hydrodynamic forces do not influence the physical boundary of the vehicle in the X-direction, but they introduce an upward velocity component of the vehicle, which causes an overall upward shift in the physical boundary of the vehicle in the Y-direction. Moreover, the variation trends of the physical boundary of the vehicle in the Y-direction with respect to angular velocity, angular velocity ratio, and mass distribution generally remain consistent with those obtained when hydrodynamic forces are neglected. Future work will conduct real underwater experiments to further validate the equations of motion of the polymorphic underwater vehicle and support the selection of operation mode and the design of the joint control system.