Dynamics Modeling and Analysis of an Underwater Glider with Dual-Eccentric Attitude Regulating Mechanism Using Dual Quaternions

Abstract

The underwater glider has difficulty accessing the complex and narrow hadal trench for observation, which is affected by its limited regulation capability of pitch angle (−45°~45°). In this study, a compact attitude regulating mechanism is proposed to extend the regulation range of pitch angle from −90°to 90° and to install it on the hadal-class underwater glider Petrel-X^PLUS. Subsequently, the dynamics model of Petrel-X^PLUS is established using dual quaternions to solve the “gimbal lock” problem caused by the increased pitch angle range. Within the extended pitch range, the motion modes of the glider are enriched into long-range, virtual mooring, and Lagrangian float modes for long-range, small-area, and current-following observation missions, respectively, and are analyzed using the established dynamics model. Moreover, a ballast method was used to modify the pitch angle range and initial equilibrium state of a constructed underwater glider. Finally, Petrel-X^PLUS achieved a pitch angle regulation range of −90°~90° in a water pool experiment and completed three consecutive profiles in a sea trial in the Challenger Deep, Mariana Trench, with all depths over 10,000 m, of which the maximum depth was 10,619 m. The proposed mechanism and methods can also be applied to other submersibles to facilitate ocean observations.

1. Introduction

As the deepest and most inaccessible area of the ocean, the hadal trenches remain a mystery to date. The hadal class submersibles are mainly human-operated vehicles [1,2], unmanned underwater vehicles with cables [3,4], and landers [5] on the seabed, which have provided valuable observation data for hadal science. However, with the development of hadal science, it is considered necessary to develop new platforms for long-term and extensive observations in hadal trenches to improve the current situation of lacking and fragmented available observation data [6]. Underwater gliders, with their advantages of low cost, long range, long endurance and independence, are considered to be possibly one of the viable methods for effective data sampling to the extensive measurements of the abyss [7]. Currently, some studies [8,9,10] have attempted to extend the application of underwater gliders to hadal trenches, yet their observation depths are still insufficient to cover the hadal zones. The funnel-shaped and narrow terrain of hadal trenches [11] poses great challenges to the glider for effective access, which requires the glider to provide more optional sailing trajectories.

The attitude regulating mechanism (ARM) controls the horizontal distance of an underwater glider at a target dive depth by changing the pitch angle and adjusts the heading direction by changing the roll angle, which further controls the actual sailing trajectory in the ocean. There are currently three typical legacy ARMs: (1) the pitch and roll angle are regulated by the movement and rotation of an eccentric battery pack along the glider’s longitudinal axis, respectively [8,12,13,14,15]; (2) the pitch and roll angle are regulated by the movement of one battery pack and the rotation of the other battery pack, respectively [16]; (3) the pitch and heading angle are regulated by the movement of an eccentric battery pack and vertical rudder, respectively [17]. However, the regulation of pitch angle in the above mechanisms are all achieved by moving an eccentric battery pack along the longitudinal axis, resulting in a typical range of −45°~45° due to the space restraints of the pressure housing. This regulation range imposes limitations on the glider’s trajectory, which need to be eliminated by a compact mechanism design that extends the pitch angle range from −90° to 90°. Currently, Zhu et al. [18] designed a new type of ARM with multiple degrees of freedom to realize the vertical motion of an underwater glider with two eccentric battery packs. However, there is an overlap of the working space when the two battery packs rotate independently, thus, adversely affecting the roll angle regulation. Cao et al. [19] designed a special ARM that can convert an underwater glider with a sawtooth motion to an ARGO float with a vertical motion. However, the regulation range of pitch angle by the mechanism is not continuous in the range of −90°~90°. Therefore, it is necessary to design a new ARM to achieve continuous regulation of the pitch angle in the range of −90°~90°.

Moreover, many studies have been conducted on the dynamics modeling of underwater gliders using the Newton–Euler method [20], Kane’s method [21], Lagrange’s equation [22], the Gibbs–Appell equation [23], or differential geometry [24]. Furthermore, based on the established models, research on underwater glider performance analysis and optimization [10,25,26], parameter identification [27,28], motion control [29,30], path planning [31,32], and formation [33,34] have been carried out. These studies have contributed to the development of underwater gliders in ocean observations, which hold promise for their integration into the Global Ocean Observing System [35]. In the above models, Euler angles are usually used to describe the attitude of underwater gliders. It is intuitive and reasonable because the pitch angle of a legacy glider is usually in the range of −45°~45°. However, after pitch angle is extended to −90°~90°, the “gimbal lock” problem occurs when the pitch angle is equal to ±90°.

In this study, a compact dual-eccentric attitude regulating mechanism (DARM) is developed to increase the regulation range of the pitch angle. The mechanism has two battery packs with equal mass and eccentric distance and regulates pitch angle by the forward and backward movement of the two battery packs as a whole and further enhances the regulation of the pitch angle by the equal angle reverse rotation of the two battery packs. DARM is mounted on Petrel-X^PLUS, a hadal-class underwater glider for large spatial–temporal monitoring in hadal trenches. There are four advantages of DARM: (1) the pitch angle of an underwater glider can be regulated in the continuous range of −90°~90°; (2) the mechanism can directly replace the ARM of a legacy glider without additional installation space; (3) it retains the pitch angle regulation pattern by the ARM in a legacy glider; (4) the motion modes of the underwater glider are enriched into a long-range mode (LR mode), virtual mooring mode (VM mode), and Lagrangian float mode (LF mode).

Dual quaternions can avoid the “gimbal lock” problem in describing the pose of a rigid body and are more computationally efficient owing to the avoidance of trigonometric calculations. They also have more compact and uniform forms than the way to model the dynamics of underwater gliders by the attitude representation with quaternions [36]. Therefore, dual quaternions are used to describe the pose of Petrel-X^PLUS in establishing its dynamics model. Based on the established model, the performances of Petrel-X^PLUS in the three typical motion modes were further analyzed. Moreover, the influence of ballast mass and mounting position on the regulation range of the pitch angle is analyzed based on a ballast method, which provides guidance for the modification of the pitch angle regulation range and the calibration of the initial equilibrium state. Finally, a water pool experiment verified the pitch angle regulation capability of DARM, and a sea trial of Petrel-X^PLUS proved the validity of the dynamics model by comparing the sea trial navigation data with the simulation data.

The remainder of this paper is organized as follows. Section 2 analyzes the limitations of the ARM in legacy gliders and introduces the design and operation processes of DARM. Section 3 establishes the dynamics model of Petrel-X^PLUS using dual quaternions. Section 4 introduces the simulation analysis of Petrel-X^PLUS in the three typical motion modes based on the established model. Section 5 analyzes the effects of ballast mass and mounting position on the pitch angle regulation range and the initial equilibrium state based on a ballast method. In Section 6, the water pool experiment and sea trial of Petrel-X^PLUS are presented. Finally, Section 7 summarizes the main contributions and gives the conclusions.

2. DARM Design

In this study, the compact dual-eccentric attitude regulating mechanism (DARM) was innovatively designed, as shown in Figure 1. The DARM uses a translation mechanism and two rotation mechanisms to control two equal eccentric battery packs to move as a whole and rotate independently of each other (Figure 1b,c). The translation mechanism consists of a screw motor that drives the two battery packs forward and backward along the square tube, with a linear potentiometer monitoring the position for feedback control. The rotation mechanism is shown in the red dashed box in Figure 1b. A rotating motor (with a power gear) mounted on a fixed plate drives a battery pack (with a driven gear) to rotate around the round tube. A detection gear is also mounted on the end of the angular sensor to monitor the rotation angle for feedback control. The round tube is mounted outside the fixed square tube and can be moved forward and backward along the square tube, ensuring that the rotation and movement of battery packs are independent. Therefore, θ can be regulated in the legacy pattern by moving the two battery packs together forward and backward. It can be further regulated in the DARM pattern by the equal-angle reverse rotation of the two battery packs when the regulation requirement cannot be met through the legacy pattern. The regulation process of DARM is shown in Figure 2, with the legacy pattern in the red box and the DARM pattern in the blue dashed box. The significant benefits of DARM through the sequential regulation of the two patterns are as follows:

• The regulation of θ can be extended to a continuous range of −90°~90°;
• The mechanism inherits the regulation pattern of a legacy glider, including pitch and heading regulation.
• DARM can be directly replace the ARM in the legacy glider without additional installation space.

Figure 1. The dual-eccentric attitude regulating mechanism: (a) Installation position on Petrel-X^PLUS; (b) 3D model; (c) DARM in testing.

Figure 1. The dual-eccentric attitude regulating mechanism: (a) Installation position on Petrel-X^PLUS; (b) 3D model; (c) DARM in testing.

Figure 2. The regulation process of θ by DARM. The red dashed box shows the legacy pattern, while the blue dashed box shows the DARM pattern.

Figure 2. The regulation process of θ by DARM. The red dashed box shows the legacy pattern, while the blue dashed box shows the DARM pattern.

The DARM divides the motion of Petrel-X^PLUS into three typical modes: LR mode (−45° ≤ θ ≤ −15° in descent, 15° ≤ θ ≤ 45° in ascent), VM mode (−80° ≤ θ ≤ −45° in descent, 45° ≤ θ ≤ 80° in ascent), and LF mode(−80° ≤ θ ≤ −90° in descent, 80° ≤ θ ≤ 90° in ascent) by regulating θ to control the ratio of horizontal distance to target dive depth, as shown in Figure 3. LR mode is a motion mode in which the underwater glider glides with small pitch angle for long range ocean monitoring. In this mode, the horizontal distance is greater than the dive depth so that the glider can perform long-range monitoring such as surveys along the hadal trench axis. VM mode is a motion mode in which the underwater glider performs observations in a designated small area. In this mode, the horizontal distance is less than the dive depth, which allows the glider to perform station-keeping observations in a small, designated area of the hadal trench. LR mode is a motion mode in which the underwater glider dives or floats at a near vertical pitch angle. In this mode, the horizontal distance is almost negligible compared with the dive depth; therefore, the glider can perform monitoring tasks by following the movement of ocean currents. The three modes have typical pitch angle ranges, and DARM achieves coverage of the above modes by regulating the pitch angle to the corresponding angle range. Moreover, within the range of the glider stall (−15° ≤ θ ≤ 15°), the glider can be operated in a hybrid-driven mode by installing a propeller to perform monitoring missions near the seafloor.

3. Dynamics Modeling

3.1. Mathematics Overview

Quaternions were introduced by Hamilton in 1843 [36] and are the extension of complex numbers to four-dimensional space with one real element and three imaginary elements. A quaternion can be defined as:

$$\{\begin{array}{l}\mathit{q}\in ℍ\simeq {ℝ}^4,\mathit{q}=q_0+q_1\mathit{i}+q_2\mathit{j}+q_3\mathit{k}\\ q_0,q_1,q_2,q_3\in ℝ,{\mathit{i}}^2={\mathit{j}}^2={\mathit{k}}^2=\mathit{i}\mathit{j}\mathit{k}=-1\end{array}$$

(1)

where i, j, and k are mutually orthogonal imaginary unit vectors and ℍ is the set of quaternions. An alternative representation is to simplify the vector expression of the quaternion using ordered pairs:

$$\mathit{q}=\left(q_0,\overline{\mathit{q}}\right);\overline{\mathit{q}}={\left[\begin{array}{ccc}{q}_1& q_2& q_3\end{array}\right]}^T$$

(2)

where q₀ is the scalar part of the quaternion and $\overline{\mathit{q}}$ is the imaginary vector part. The basic operations of quaternions are shown in

Table A1. The basic operations of quaternions.

Operations	Formula
Addition	$\mathit{p}+\mathit{q}=\left(p_0+q_0\right)+\epsilon \left(\overline{\mathit{p}}+\overline{\mathit{q}}\right)$
Multiplication	$\mathit{p}\mathit{q}=\left(p_0{q}_0-\overline{\mathit{p}}\overline{\mathit{q}},p_0\overline{\mathit{q}}+q_0\overline{\mathit{p}}+\overline{\mathit{p}}\times \overline{\mathit{q}}\right)$
Multiplication by a scale	$\lambda \mathit{q}=\mathit{q}\lambda =\left(\lambda q_0,\lambda \overline{\mathit{q}}\right)$
Conjugation	${\mathit{q}}^{*}=\left(q_0,-\overline{\mathit{q}}\right)$
Dot product	$\mathit{p}\cdot \mathit{q}=\left({\mathit{p}}^{*}\mathit{q}+{\mathit{q}}^{*}\mathit{p}\right)/2=\left(p_0{q}_0+\overline{\mathit{p}}\cdot \overline{\mathit{q}},\overline{0}\right)$
Cross product	$\mathit{p}\times \mathit{q}=\left(0,q_0\overline{\mathit{p}}+p_0\overline{\mathit{q}}+\overline{\mathit{p}}\times \overline{\mathit{q}}\right)$
Norm	$\Vert \mathit{q}\Vert =\sqrt{\mathit{q}{\mathit{q}}^{*}}=\sqrt{\mathit{q}\cdot \mathit{q}}=\sqrt{\left(q_0{q}_0+\overline{\mathit{q}}\cdot \overline{\mathit{q}},\overline{0}\right)}$
Scalar part	$sc\left(\mathit{q}\right)=\left(q_0,\overline{0}\right)$
Vector part	$vec\left(\mathit{q}\right)=\left(0,\overline{\mathit{q}}\right)$

. Quaternions can only represent the attitude of a rigid body and are considered to be the best general solution to represent rotation [37]. However, dual quaternions provide a way to represent both rotational and translational motions at the same time, yielding a more uniform, simple result than quaternions.

The dual quaternions were introduced by Clifford in 1873 [36], which combines the dual numbers and quaternions in a multidimensional space, and a dual quaternion can be defined as:

$$\{\begin{array}{l}\hat{\mathit{q}}\in {ℍ}_d\cong {ℝ}^8,\hat{\mathit{q}}={\mathit{q}}_{\mathit{r}}+\epsilon {\mathit{q}}_{\mathit{d}}\\ {\mathit{q}}_{\mathit{r}},{\mathit{q}}_{\mathit{d}}\in ℍ,\epsilon \ne 0, {\epsilon }^n=0,n\ge 2,n\in \mathbb{Q}\end{array}$$

(3)

where ℍ_d denotes the set of dual quaternions, the quaternions q_r and q_d are called the real and imaginary parts of the dual quaternion, respectively, and ɛ is the dual operator. The main dual quaternion algebra is listed in

Table 1. The basic operations of dual quaternions.

Operation	Formula
Addition	$\hat{\mathit{p}}+\hat{\mathit{q}}=\left({\mathit{p}}_{\mathit{r}}+{\mathit{q}}_{\mathit{r}}\right)+\epsilon \left({\mathit{p}}_{\mathit{d}}+{\mathit{q}}_{\mathit{d}}\right)$
Multiplication	$\hat{\mathit{p}}\hat{\mathit{q}}={\mathit{p}}_{\mathit{r}}{\mathit{q}}_{\mathit{r}}+\epsilon \left({\overline{\mathit{p}}}_{\mathit{r}}{\overline{\mathit{q}}}_{\mathit{d}}+{\overline{\mathit{p}}}_{\mathit{d}}{\overline{\mathit{q}}}_{\mathit{r}}\right)$
Multiplication by a scale	$\lambda \hat{\mathit{q}}=\lambda {\mathit{q}}_{\mathit{r}}+\epsilon \left(\lambda {\mathit{q}}_{\mathit{d}}\right)$
Conjugation	${\hat{\mathit{q}}}^{*}={\mathit{q}}_{\mathit{r}}^{*}+\epsilon {\mathit{q}}_{\mathit{d}}^{*}$
Swap	${\hat{\mathit{q}}}^{\mathit{S}}={\mathit{q}}_{\mathit{d}}+\epsilon {\mathit{q}}_{\mathit{r}}$
Dot product	$\hat{\mathit{p}}\cdot \hat{\mathit{q}}={\mathit{p}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{r}}+\epsilon \left({\mathit{p}}_{\mathit{d}}\cdot {\mathit{q}}_{\mathit{r}}+{\mathit{p}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{d}}\right)$
Cross product	$\hat{\mathit{p}}\times \hat{\mathit{q}}={\mathit{p}}_{\mathit{r}}\times {\mathit{q}}_{\mathit{r}}+\epsilon \left({\mathit{p}}_{\mathit{d}}\times {\mathit{q}}_{\mathit{r}}+{\mathit{p}}_{\mathit{r}}\times {\mathit{q}}_{\mathit{d}}\right)$
Dual norm	$\Vert \hat{\mathit{q}}\Vert =\sqrt{\hat{\mathit{q}}{\hat{\mathit{q}}}^{*}}=\sqrt{\hat{\mathit{q}}\cdot \hat{\mathit{q}}}=\sqrt{\left({\mathit{q}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{r}}\right)+\epsilon \left(2{\mathit{q}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{d}}\right)}$
Scalar part	$sc\left(\hat{\mathit{q}}\right)=sc\left({\mathit{q}}_{\mathit{r}}\right)+\epsilon sc\left({\mathit{q}}_{\mathit{d}}\right)$
Vector part	$vec\left(\hat{\mathit{q}}\right)=vec\left({\mathit{q}}_{\mathit{r}}\right)+\epsilon vec\left({\mathit{q}}_{\mathit{d}}\right)$

.

Moreover, the multiplication of a matrix N ∈ ℝ^4×4 with a quaternion $\mathit{q}=\left({\mathit{q}}_0,\overline{\mathit{q}}\right)$ and a matrix M ∈ ℝ^8×8 with a dual quaternion $\hat{\mathit{q}} = {\mathit{q}}_{\mathit{r}}{+\mathit{\epsilon }\mathit{q}}_{\mathit{d}}$ are noteworthy and are defined as:

$$\{\begin{array}{l}N\ast \mathit{q}=\left(N_{11}{q}_0+N_{12}\overline{\mathit{q}},N_{21}{q}_0+N_{22}\overline{\mathit{q}}\right)\\ M\ast \hat{\mathit{q}}=\left(M_{11}\ast {\mathit{q}}_{\mathit{r}}+M_{12}\ast {\mathit{q}}_{\mathit{d}}\right)+\epsilon \left(M_{21}\ast {\mathit{q}}_{\mathit{r}}+M_{22}\ast {\mathit{q}}_{\mathit{d}}\right)\end{array}$$

(4)

where

$$\{\begin{array}{l}N=\left[\begin{array}{cc}{N}_{11}& N_{12}\\ N_{21}& N_{22}\end{array}\right],N_{11}\in ℝ,N_{12}\in {ℝ}^{1\times 3},N_{21}\in {ℝ}^{3\times 1},N_{22}\in {ℝ}^{3\times 3}\\ M=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right],M_{11},M_{12},M_{21},M_{22}\in {ℝ}^{4\times 4}\end{array}$$

(5)

The skew-symmetric operator $[·{]}^{\times }$ for the cross product of a dual quaternion $\hat{\mathit{q}}$ is defined as:

$${\left[\hat{\mathit{q}}\right]}^{\times }=\left[\begin{array}{cc}{\left[{\mathit{q}}_{\mathit{r}}\right]}^{\times }& 0_{4*4}\\ {\left[{\mathit{q}}_{\mathit{d}}\right]}^{\times }& {\left[{\mathit{q}}_{\mathit{r}}\right]}^{\times }\end{array}\right]$$

(6)

where ${\left[\mathit{q}\right]}^{\times }$ is the skew-symmetric operator for the cross product of a quaternion q and is expressed as:

$${\left[\mathit{q}\right]}^{\times }=\left[\begin{array}{cc}0& 0_{1\times 3}\\ 0_{3\times 1}& {\overline{\mathit{q}}}^{\times }\end{array}\right], {\left[\overline{\mathit{q}}\right]}^{\times }=\left[\begin{array}{ccc}0& -q_3& q_2\\ q_3& 0& -q_1\\ -q_2& q_1& 0\end{array}\right]$$

(7)

3.2. Coordinate Systems and Kinematic Equations

As shown in Figure 4, the inertial frame (I-frame), the body frame (B-frame), and the flow frame (F-frame) are established for describing the motion of Petrel-X^PLUS. The I-frame (O_e-x_ey_ez_e) is a North–East–Down coordinate system defined relative to the Earth’s reference ellipsoid, whose origin point O_e is usually chosen as the dive point on the sea surface during deployment. The B-frame (O_b-x_by_bz_b) is a moving coordinate system, the origin of which is usually fixed to the center of buoyancy of the glider body. The x_b-axis is the longitudinal axis of Petrel-X^PLUS and directs to the head, the y_b-axis is defined as the transversal axis of the glider and points to starboard, and the z_b-axis is the normal axis of the glider whose direction is from top to bottom. The F-frame (O_f-x_fy_fz_f) is obtained by rotating the B-frame to analyze and calculate the hydrodynamic force. The attack angle α and slip angle β are the attitude angles of the underwater glider relative to the fluid.

Here, a unit dual quaternion ${\widehat{q}}_{B/I}$ is defined to represent the position and orientation of the B-frame with respect to the I-frame, and the subscript B/I represents the conversion from the I-frame to the B-frame.

$$\{\begin{array}{l}{\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}={\mathit{q}}_{\mathit{B}/\mathit{I},\mathit{r}}+\epsilon {\mathit{q}}_{\mathit{B}/\mathit{I},\mathit{d}}={\mathit{q}}_{\mathit{B}/\mathit{I}}+\epsilon \frac{1}{2}{\mathit{r}}_{\mathit{B}/\mathit{I}}^{\mathit{I}}{\mathit{q}}_{\mathit{B}/\mathit{I}}\\ {\mathit{q}}_{\mathit{B}/\mathit{I},\mathit{r}}=q_0+q_1\mathit{i}+q_2\mathit{j}+q_3\mathit{k}\\ {\mathit{q}}_{\mathit{B}/\mathit{I},\mathit{d}}=q_4+q_5\mathit{i}+q_6\mathit{j}+q_7\mathit{k}\end{array}$$

(8)

where ${\mathit{r}}_{\mathit{B}/\mathit{I}}^{\mathit{I}}$ is a unit quaternion and denotes the translation vector from the origin of the I-frame to the origin of the B-frame expressed in the I-frame, ${\mathit{r}}_{\mathit{B}/\mathit{I}}^{\mathit{I}}{=(0, \overline{\mathit{r}}}_{\mathit{B}/\mathit{I}}^{\mathit{I}}{), \overline{\mathit{r}}}_{\mathit{B}/\mathit{I}}^{\mathit{I}}{=[X, Y, Z]}^T$; q_B/I is a unit quaternion and is defined as a representation of the orientation of the B-frame with respect to the I-frame. The orientation of Petrel-X^PLUS in the I-frame can be denoted as $\overline{\mathit{\eta }}={[\phi , \theta , \psi ]}^T$.

Note that there are two algebraic conditions satisfied by a unit dual quaternion:

$${\mathit{q}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{r}}=1, {\mathit{q}}_{\mathit{r}}\cdot {\mathit{q}}_{\mathit{d}}=0$$

(9)

According to Equation (8), the translational and rotational kinematic equations of the glider in the B-frame with respect to the I-frame using dual quaternion algebra can be expressed as:

$${\stackrel{\stackrel{·}{^}}{\mathit{q}}}_{\mathit{B}/\mathit{I}}=\frac{1}{2}{\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}{\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}$$

(10)

where ${\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}$ is the dual velocity of the glider in the B-frame with respect to the I-frame expressed in the B-frame:

$${\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}={\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+\epsilon {\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}=(0,{\overline{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}})+\epsilon (0,{\overline{\mathit{v}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}})$$

(11)

where ${\overline{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}{=[\mathit{p}, \mathit{q}, \mathit{r}]}^{\mathit{T}}$ and ${\overline{\mathit{v}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}{=[\mathit{u}, \mathit{v}, \mathit{w}]}^{\mathit{T}}$ are the angular and linear velocity vectors in the B-frame, respectively. Since trigonometric calculations are avoided, the attitude problem of singular values when θ = ±90° can be avoided. Although dual quaternions improve the efficiency of operations due to the absence of trigonometric operations, the pose cannot be obtained directly. Therefore, the pose of the glider needs to be calculated from the dual quaternions, and the transformation relation is expressed as:

$$\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]=\left[\begin{array}{c}2(q_0{q}_5-q_1{q}_4+q_2{q}_7-q_3{q}_6)\\ 2(q_0{q}_6-q_1{q}_7+q_3{q}_5-q_2{q}_4)\\ 2(q_0{q}_7-q_2{q}_5+q_1{q}_6-q_3{q}_4)\end{array}\right]$$

(12)

$$\left[\begin{array}{c}\phi \\ \theta \\ \psi \end{array}\right]=\left[\begin{array}{c}\arctan \left[\frac{2(q_2{q}_3+q_0{q}_1)}{q_0^2-q_1^2-q_2^2+q_3^2}\right]\\ \arcsin \left[-2(q_1{q}_3-q_2{q}_0)\right]\\ \arctan \left[\frac{2\left(q_1{q}_2+q_0{q}_3\right)}{q_0^2+q_1^2-q_2^2-q_3^2}\right]\end{array}\right]$$

(13)

Similarly, the unit dual quaternion ${\hat{\mathit{p}}}_{\mathit{B}/\mathit{F}}$ is defined as a conversion vector from the F-frame to the B-frame and expressed as:

$$\{\begin{array}{l}{\hat{\mathit{p}}}_{\mathit{B}/\mathit{F}}={\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{r}}+\epsilon {\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{d}}={\mathit{p}}_{\mathit{B}/\mathit{F}}+\epsilon \frac{1}{2}{\mathit{r}}_{\mathit{B}/\mathit{F}}^{\mathit{F}}{\mathit{p}}_{\mathit{F}/\mathit{B}}\\ {\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{r}}=p_0+p_1\mathit{i}+p_2\mathit{j}+p_3\mathit{k}\\ {\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{d}}=p_4+p_5\mathit{i}+p_6\mathit{j}+p_7\mathit{k}\end{array}$$

(14)

where ${\mathit{p}}_{\mathit{B}/\mathit{F}}$ is a unit quaternion to denote the orientation of the F-frame with respect to the B-frame and unit quaternion ${\mathit{r}}_{\mathit{B}/\mathit{F}}^{\mathit{F}}$ is the translation vector from the origin of the F-frame to the origin of the B-frame expressed in the F-frame. According to the definition of the F-frame, there is no translational motion between the two frames; then, we have:

$$\{\begin{array}{l}{\mathit{r}}_{\mathit{B}/\mathit{F}}^{\mathit{F}}=0; 0=(0,\overline{0}),\overline{0}={[0,0,0]}^T\\ {\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{d}}=\frac{1}{2}{\mathit{r}}_{\mathit{B}/\mathit{F}}^{\mathit{F}}{\mathit{p}}_{\mathit{B}/\mathit{F}}=0;p_4=p_5=p_6=p_7=0\\ {\hat{\mathit{p}}}_{\mathit{B}/\mathit{F}}={\mathit{p}}_{\mathit{B}/\mathit{F},\mathit{r}}+\epsilon 0={\mathit{p}}_{\mathit{B}/\mathit{F}}+\epsilon 0\end{array}$$

(15)

Herein, ${\hat{\mathit{w}}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}{=\mathit{w}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}{+\epsilon \mathit{v}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}$ is defined as the dual velocity of the glider in the F-frame with respect to the I-frame expressed in the F-frame. According to the definition of the F-frame:

$${\mathit{v}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}=\left(0,{\overline{\mathit{v}}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}\right),{\overline{\mathit{v}}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}={\left[U,0,0\right]}^T,U=\sqrt{u^2+v^2+w^2}$$

(16)

In the F-frame, the pose of the glider is represented by the attack angle α and slip angle β, and their expressions are:

$$\{\begin{array}{l}\alpha =-\arctan \left(\frac{w}{u}\right)\\ \beta =-\arcsin \left(\frac{v}{U}\right)\end{array}$$

(17)

3.3. Force Analysis

The external forces acting on the glider include buoyancy, gravity, viscous hydrodynamic force, inertial hydrodynamic force, and the generated moments when the external forces are moved to O_b. In dual quaternion algebra, a dual force $\hat{\mathit{F}}$ is defined to describe the forces and moments in a unified form as $\hat{\mathit{F}}=\mathit{f}+\epsilon \mathit{\tau }$, where $\mathit{f}=(0,\overline{\mathit{f}})$, ${\overline{\mathit{f}}=[\mathit{f}}_1{, \mathit{f}}_2{, \mathit{f}}_3{]}^T$ are the total external force vectors in the B-frame and $\mathit{\tau }=(0, \overline{\mathit{\tau }})$, ${\overline{\mathit{\tau }}=[\tau }_1{, \tau }_2{, \tau }_3{]}^T$ are the total external moment vectors in the B-frame.

3.3.1. Dual Buoyancy ${\hat{\mathit{B}}}_{\mathit{g}}$ and Dual Gravity ${\hat{\mathit{F}}}_{\mathit{g}}$

The buoyancy center deviation resulting from glider oil volume adjustment is neglected because the adjustable oil is a small percentage of the glider volume [38]. In the initial equilibrium state, the buoyancy force is equal in magnitude and opposite in direction to the gravity force. Therefore, the dual buoyancy force ${\hat{\mathit{B}}}_{\mathit{g}}{=\mathit{b}}_{\mathit{g}}+\epsilon 0$, can be expressed as:

$${\hat{\mathit{B}}}_{\mathit{g}}={\hat{\mathit{q}}}_{\mathit{B}/I}^{*}({[0,0,0,-m_{g}g]}^T+\epsilon 0){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}$$

(18)

where g is the gravity acceleration and m_g is the glider mass.

The dual gravity of the glider in the B-frame is defined as ${\hat{\mathit{F}}}_{\mathit{g}}{=\mathit{f}}_{\mathit{g}}{+\epsilon \mathit{\tau }}_{\mathit{g}}$, whose expression is:

$${\hat{\mathit{F}}}_{\mathit{g}}={\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}^{*}({[0,0,0,m_{g}g]}^T+\epsilon {[{\mathit{r}}_{\mathit{g}}]}^{\times }{[0,0,0,m_{g}g]}^T){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}$$

(19)

where r_g is a quaternion representing the position vector from O_b to the center of gravity O_g.

3.3.2. Dual Net Buoyancy $\Delta \hat{\mathit{B}}$

The change in the net buoyancy of the glider includes the adjustment of the variable buoyancy system (Figure 5) and buoyancy loss caused by the marine environment. Herein, $\Delta \hat{\mathit{B}}=\Delta {\mathit{b}+\epsilon \mathit{\tau }}_{\Delta \mathit{b}}$, $\Delta {\hat{\mathit{B}}}_{\mathit{r}}=\Delta {\mathit{b}}_{\mathit{r}}{+\epsilon \mathit{\tau }}_{\Delta \mathit{b}\mathit{r}}$, and $\Delta {\hat{\mathit{B}}}_{\mathit{l}}=\Delta {\mathit{b}}_{\mathit{l}}{+\epsilon \mathit{\tau }}_{\Delta \mathit{b}\mathit{l}}$ are denoted as the dual net buoyancy, dual regulation buoyancy, and dual lost buoyancy, respectively, and expressed as:

$$\{\begin{array}{l}\mathsf{\Delta }\hat{\mathit{B}}=\mathsf{\Delta }{\hat{\mathit{B}}}_{\mathit{r}}+\mathsf{\Delta }{\hat{\mathit{B}}}_{\mathit{l}}\\ \mathsf{\Delta }{\hat{\mathit{B}}}_{\mathit{r}}={\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}^{*}({[0,0,0,{\rho }_{sea}{V}_{oil}g]}^T+\epsilon {[{\mathit{r}}_{\mathit{o}}]}^{\times }{[0,0,0,{\rho }_{sea}{V}_{oil}g]}^T){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}\\ \mathsf{\Delta }{\hat{\mathit{B}}}_{\mathit{l}}={ \mathit{q} ^}_{\mathit{B}/\mathit{I}}^{*}({[0,0,0,B_l(h)]}^T+\epsilon 0){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}\end{array}$$

(20)

where V_oil is the volume of adjustable oil, ρ_sea is the seawater density on the surface, and r_o is a quaternion representing the position vector from O_b to the center of the outer bladder. B_l(h) is the buoyancy variation of the underwater glider under the effect of seawater density, pressure housing compression, and buoyancy compensation (Figure 6), which is fitted by the polynomial as:

$$B_l\left(h\right)=a_{1}h+a_2{h}^2+a_3{h}^3+a_4{h}^4+a_5{h}^5+a_6{h}^6+a_7{h}^7+a_8{h}^8+a_9{h}^9$$

(21)

where the polynomial fit coefficients a₁ = 0.01816; a₂ = −1.98984 × 10⁻⁵; a₃ = 1.17318 × 10⁻⁸; a₄ = −4.11583 × 10⁻¹²; a₅ = 8.86404 × 10⁻¹⁶; a₆ = −1.18291 × 10⁻¹⁹; a₇ = 9.52836 × 10⁻²⁴; a₈ = −4.2411 × 10⁻²⁸; a₉ = 8.00614 × 10⁻³³.

3.3.3. Dual Regulation Force ${\hat{\mathit{F}}}_{\mathit{r}}$

The DARM generates the regulation force and moment to regulate the attitude of the glider by the translation and rotation of the two battery packs (Figure 7). The dual regulation force ${\hat{\mathit{F}}}_{\mathit{r}}{=\mathit{f}}_{\mathit{r}}{+\epsilon \mathit{\tau }}_{\mathit{r}}$ is defined as the representation of the regulation force and moment and expressed as:

$$\{\begin{array}{l}{\hat{\mathit{F}}}_{\mathit{r}}={\hat{\mathit{F}}}_{\mathit{r}1}+{\hat{\mathit{F}}}_{\mathit{r}2}\\ {\hat{\mathit{F}}}_{\mathit{r}1}={\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}^{*}({[0,0,0,m_{r1}g]}^T+\epsilon {[{\mathit{r}}_{\mathit{r}1}]}^{\times }{[0,0,0,m_{r1}g]}^T){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}\\ {\hat{\mathit{F}}}_{\mathit{r}2}={\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}^{*}({[0,0,0,m_{r2}g]}^T+\epsilon {[{\mathit{r}}_{\mathit{r}2}]}^{\times }{[0,0,0,m_{r2}g]}^T){\hat{\mathit{q}}}_{\mathit{B}/\mathit{I}}\end{array}$$

(22)

where ${\hat{\mathit{F}}}_{\mathit{r}1}{=\mathit{f}}_{\mathit{r}1}{+\epsilon \mathit{\tau }}_{\mathit{r}1}$ and ${\hat{\mathit{F}}}_{\mathit{r}2}{=\mathit{f}}_{\mathit{r}2}{+\epsilon \mathit{\tau }}_{\mathit{r}2}$ are the dual regulation forces generated by battery pack 1 and battery pack 2, respectively; m_r1 and m_r2 are the masses of battery pack 1 and battery pack 2, respectively; r_r1 and r_r2 are quaternions representing the position vector from O_b to the mass center of battery pack 1 and battery pack 2, respectively, and are expressed as:

$$\{\begin{array}{l}{\mathit{r}}_{\mathit{r}1}=(0,{\overline{\mathit{r}}}_{\mathit{r}1}),{\overline{\mathit{r}}}_{\mathit{r}1}={\left[l_1,e\cos {\gamma }_1,e\sin {\gamma }_1\right]}^T\\ {\mathit{r}}_{\mathit{r}2}=(0,{\overline{\mathit{r}}}_{\mathit{r}2}),{\overline{\mathit{r}}}_{\mathit{r}2}={\left[l_2,e\cos {\gamma }_2,e\sin {\gamma }_2\right]}^T\end{array}$$

(23)

where l₁ and l₂ are the projections of the position vectors of battery pack 1 and battery pack 2 on the x_b-axis, respectively; e is the eccentric distance of the two battery packs; λ₁ and λ₂ are the rotation angles of battery pack 1 and battery pack 2, which follow the right-hand spiral rule: the thumb direction is along the x_b-axis, positively in the counterclockwise direction and negatively in the clockwise direction. Since the two battery packs are moving on the whole, l₁ and l₂ can be represented by l, l = (l₁ + l₂)/2.

3.3.4. Dual Inertia Force ${\hat{\mathit{F}}}_{\mathit{i}}$

The motion of the glider in the ocean is affected by the surrounding environment to generate inertial hydrodynamic forces and moments acting on O_b. Based on the symmetry of the glider in the transverse and longitudinal planes, the dual inertial hydrodynamic force ${\hat{\mathit{F}}}_{\mathit{i}}{=\mathit{f}}_{\mathit{i}}{+\epsilon \mathit{\tau }}_{\mathit{i}}$ can be expressed as:

$$\{\begin{array}{l}{\hat{\mathit{F}}}_{\mathit{i}}={\mathit{q}}_{\mathit{B}/\mathit{F}}^{*}{\hat{\mathit{F}}}_{\mathit{i}}^{\mathit{F}}{\mathit{q}}_{\mathit{B}/\mathit{F}}\\ {\hat{\mathit{F}}}_{\mathit{i}}^{\mathit{F}}=M_a\star {\left({\stackrel{\dot{^}}{\mathit{w}}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}\right)}^S,M_a=\left[\begin{array}{llllllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\lambda }_{11}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\lambda }_{22}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & {\lambda }_{26}\hfill \\ 0\hfill & 0\hfill & 0\hfill & {\lambda }_{33}\hfill & 0\hfill & 0\hfill & {\lambda }_{35}\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & {\lambda }_{44}\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & {\lambda }_{53}\hfill & 0\hfill & 0\hfill & {\lambda }_{55}\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\lambda }_{62}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & {\lambda }_{66}\hfill \end{array}\right]\end{array}$$

(24)

where ${\hat{\mathit{F}}}_{\mathit{i}}^{\mathit{F}}{=\mathit{f}}_{\mathit{i}}^{\mathit{F}}{+\epsilon \mathit{\tau }}_{\mathit{i}}^{\mathit{F}}$ is the dual inertia force expressed in the F-frame; M_a ∈ ℝ^8×8 is the added mass matrix; λ_ij in M_a is called the coefficient of inertia hydrodynamic or the added mass, which can be calculated by the strip theory and panel method. The calculated added mass of Petrel-X^PLUS is shown in

Table 2. The added mass of Petrel-X^PLUS.

Added Mass	Value	Added Mass	Value
λ11	13.90 kg	λ66	162.81 kg⋅m2
λ22	201.98 kg	λ26	−2.88 kg⋅m
λ33	268.76 kg	λ35	22.15 kg⋅m
λ44	11.58 kg⋅m2	λ53	−2.88 kg⋅m
λ55	106.41 kg⋅m2	λ62	22.15 kg⋅m

.

3.3.5. Dual Viscous Force ${\hat{\mathit{F}}}_{\mathit{v}}$

The surrounding flow field around the underwater glider also generates viscous hydrodynamic forces and moments during its movement, which is defined as the dual viscous force ${\hat{\mathit{F}}}_{\mathit{v}}{=\mathit{f}}_{\mathit{v}}{+\epsilon \mathit{\tau }}_{\mathit{v}}$ and expressed in the B-frame as:

$$\{\begin{array}{l}{\hat{\mathit{F}}}_{\mathit{v}}={\mathit{q}}_{\mathit{B}/\mathit{F}}^{*}{\hat{\mathit{F}}}_{\mathit{v}}^{\mathit{F}}{\mathit{q}}_{\mathit{B}/\mathit{F}}\\ {\hat{\mathit{F}}}_{\mathit{v}}^{\mathit{F}}=\left(\left[\begin{array}{c}0\\ K_{D0}+K_D{\alpha }^2\\ K_S\beta \\ K_{L0}+K_L\alpha \end{array}\right]+\epsilon \left[\begin{array}{c}0\\ K_K\beta +K_{p}p\\ K_{M0}+K_M\alpha +K_{q}q\\ K_N\beta +K_{r}r\end{array}\right]\right){\Vert {\mathit{v}}_{\mathit{F}/\mathit{I}}^{\mathit{F}}\Vert }^2\end{array}$$

(25)

where ${\hat{\mathit{F}}}_{\mathit{v}}^{\mathit{F}}{=\mathit{f}}_{\mathit{v}}^{\mathit{F}}{+\epsilon \mathit{\tau }}_{\mathit{v}}^{\mathit{F}}$ is the dual viscous force expressed in the F-frame; K_D0, K_D, K_S, K_L0, K_L, K_K, K_p, K_M0, K_M, K_q, K_N, and K_r are the hydrodynamic coefficients and can be obtained by CFD simulation, as shown in

Table 3. The added mass of Petrel-X^PLUS.

Coefficient	Value	Coefficient	Value
KD0	12.01 kg/m	Kp	−41.643 kg·s/rad
KD	888.63 kg/m/rad2	KM0	0.03 kg
KS	−231.04 kg/m/rad	KM	−386.01 kg/rad
KL0	−0.381 kg/m	Kq	−677.01 kg⋅s/rad2
KL	954.35 kg/m/rad	KN	100.42 kg/rad
KK	0 kg/rad	Kr	−631.63 kg⋅s/rad2

.

The total external dual force ${\widehat{F}}^{\mathit{B}}$ acting on the glider about O_b can be expressed as:

$${\hat{\mathit{F}}}^{\mathit{B}}={\mathit{b}}_{\mathit{g}}+{\mathit{f}}_{\mathit{g}}+\mathsf{\Delta }\mathit{b}+{\mathit{f}}_r+{\mathit{f}}_{\mathit{i}}+{\mathit{f}}_v+\mathit{\epsilon }({\mathit{\tau }}_{\mathit{g}}+{\mathit{\tau }}_{\mathsf{\Delta }\mathit{b}}+{\mathit{\tau }}_{\mathit{r}}+{\mathit{\tau }}_{\mathit{i}}+{\mathit{\tau }}_{\mathit{v}})$$

(26)

3.4. Dynamic Equations

Herein, the dual momentum of the underwater glider with respect to the I-frame about O_b is defined as ${\hat{\mathit{H}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}$ and is given by:

$${\hat{\mathit{H}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}=M^B*{\left({\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)}^S$$

(27)

where

$$M^B=\left[\begin{array}{cc}{J}_B& m_g{[{\mathit{r}}_{\mathit{g}}]}^{\times }\\ -m_g{[{\mathit{r}}_{\mathit{g}}]}^{\times }& M_g\end{array}\right]=\left[\begin{array}{llllllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & J_{xx}\hfill & J_{xy}\hfill & J_{xz}\hfill & 0\hfill & 0\hfill & -m_g{z}_g\hfill & m_g{y}_g\hfill \\ 0\hfill & J_{yx}\hfill & J_{yy}\hfill & J_{yz}\hfill & 0\hfill & m_g{z}_g\hfill & 0\hfill & -m_g{x}_g\hfill \\ 0\hfill & J_{zx}\hfill & J_{zy}\hfill & J_{zz}\hfill & 0\hfill & -m_g{y}_g\hfill & m_g{x}_g\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & m_g{z}_g\hfill & -m_g{y}_g\hfill & 0\hfill & m_g\hfill & 0\hfill & 0\hfill \\ 0\hfill & -m_g{z}_g\hfill & 0\hfill & m_g{x}_g\hfill & 0\hfill & 0\hfill & m_g\hfill & 0\hfill \\ 0\hfill & m_g{y}_g\hfill & -m_g{x}_g\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & m_g\hfill \end{array}\right]$$

(28)

According to the momentum and momentum moment theorem, the dynamics equation of the glider can be written as:

$$\frac{d{\hat{\mathit{H}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}}{dt}+{\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\hat{\mathit{H}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}={\hat{\mathit{F}}}^{\mathit{B}}$$

(29)

Substituting Equation (27) into Equation (29), the dynamic equations can be further written as:

$${\left({\stackrel{\stackrel{·}{^}}{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)}^S={\left(M^B\right)}^{-1}\star \left({\hat{\mathit{F}}}^{\mathit{B}}-{\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times \left(M^B\star {\left({\hat{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)}^S\right)\right)$$

(30)

where (M^B)⁻¹ is the inverse of M^B, Equation (30) is the unified dynamic equation of the glider represented by dual quaternions.

Bringing Equation (12) and Equations (27)–(29) into Equation (30), one obtains the following:

$$\begin{array}{l}d\left[\left(m{\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}\right)+\epsilon \left(J_B\ast {\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{r}}_{\mathit{g}}\times {\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)\right]/dt+\\ \left({\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+\epsilon {\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)\times [\left(m{\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}\right)+\epsilon \left(J_B\ast {\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{r}}_{\mathit{g}}\times {\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)]\\ ={\mathit{b}}_{\mathit{g}}+{\mathit{f}}_{\mathit{g}}+\mathsf{\Delta }\mathit{b}+{\mathit{f}}_r+{\mathit{f}}_{\mathit{i}}+{\mathit{f}}_v+\epsilon \left({\mathit{\tau }}_{\mathit{g}}+{\mathit{\tau }}_{\mathsf{\Delta }\mathit{b}}+{\mathit{\tau }}_{\mathit{r}}+{\mathit{\tau }}_{\mathit{i}}+{\mathit{\tau }}_{\mathit{v}}\right)\end{array}$$

(31)

Expanding the real and dual parts of Equation (31) yields:

$$\{\begin{array}{l}m{\stackrel{·}{\mathit{v}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\stackrel{·}{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}+m{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times \left({\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}\right)\\ ={\mathit{b}}_{\mathit{g}}+{\mathit{f}}_{\mathit{g}}+\mathsf{\Delta }\mathit{b}+{\mathit{f}}_r+{\mathit{f}}_{\mathit{i}}+{\mathit{f}}_v\\ J_B\ast {\stackrel{·}{\mathit{w}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{r}}_{\mathit{g}}\times {\stackrel{·}{\mathit{v}}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times \left(J_B\ast {\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\right)\\ +m{\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}\times {\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}+m{\mathit{v}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{w}}_{\mathit{B}/\mathit{I}}^{\mathit{B}}\times {\mathit{r}}_{\mathit{g}}={\mathit{\tau }}_{\mathit{g}}+{\mathit{\tau }}_{\mathsf{\Delta }\mathit{b}}+{\mathit{\tau }}_{\mathit{r}}+{\mathit{\tau }}_{\mathit{i}}+{\mathit{\tau }}_{\mathit{v}}\end{array}$$

(32)

Equation (32) is the classical translational and rotational dynamic equation of an underwater glider.

4. Motion Simulation

The underwater glider has two main motions, sawtooth motion and spiral motion. As the main motion, the sawtooth motion of Petrel-X^PLUS is divided into LR mode, VM mode, and LF mode, which combines vertical and horizontal distances allowing the glider to make profile observations. Spiral motion is mainly used for heading adjustment and to a lesser extent for station-keeping observations [39]. Therefore, it is essential to simulate motions using a dynamics model to guide ocean observations. In this study, the motion simulations of Petrel-X^PLUS are performed in MATLAB software, where the ODE45 function is used for the numerical solution of the dynamics model.

4.1. Sawtooth Motion Simulation

The adjustable oil volume V_oil, battery pack position x_r, and battery pack rotation angles λ₁ and λ₂ are the main parameters that need to be set in a sawtooth motion at a given target depth. In sawtooth motion simulations, V_oil is set to −1.0~1.0 L, i.e., 1.0 L of oil returns in the initial equilibrium state on the sea surface and 2.0 L of oil discharges after reaching the target depth of 10,000 m. In the initial equilibrium state, x_r of the two battery packs is −0.418 m with respect to O_b on the x_b-axis and is in the range from −0.478 m~−0.358 m due to space constraints. In the descent and ascent phases, x_r of the battery packs is set to −0.438 m and −0.388 m in LR mode, −0.358 m and −0.478 m in VM mode, and −0.358 m and −0.478 m in LF mode for θ regulation, respectively. In addition, λ₁ and λ₂ are set to 0° and 0° in LR mode, −30° and 30° in VM mode, and 140° and −140° in LF mode, respectively, and remain constant during the two phases. Note that the process of regulating the control parameters from the initial equilibrium state to the new state is ignored. In the solution process of the ODE45 function, the time step is set to 0.1 s, and the simulation results are shown in Figure 8, Figure 9 and Figure 10.

Figure 8 shows the simulation results of Petrel-X^PLUS in LR mode, which achieved a horizontal distance of 49.7 km in 38.9 h. However, the descent and ascent phases are not symmetrical, and the descent time and horizontal distance are less and shorter, respectively. It is because B_l (Figure 6) is positive from 0 m to 7000 m and negative from 7000 m to 10,000 m, resulting in a faster vertical velocity in descent than in ascent. In addition, affected by the variation in B_l with depth, Petrel-X^PLUS tends to accelerate and then decelerate during the descent phase. At the same time, the opposite trend is observed during the ascent phase (Figure 8b). The θ of the glider is also affected by B_l, causing θ to decrease from −20.6° to −24.8° and then increase to −15.8° during the descent phase and to decrease from 24.5° to 15.7° and then increase to 20.3° during the ascent phase (Figure 8c). In the whole profile, the variation trend of γ is the same as that of θ, while the variation trend of α (Figure 8d) is opposite to that of θ. The above changes in angles are caused by the change in pitching moment due to the velocity variation. A decrease (increase) in velocity will reduce (amplify) the pitching moment, which reduces (amplify) θ during the descent (ascent) phase.

Figure 9 and Figure 10 show the simulation results of Petrel-X^PLUS in VM and LF modes, respectively. In VM mode, the glider achieves a horizontal distance of 8.17 km in 8.89 h. In LF mode, the horizontal distance of the glider can be neglected compared to its dive depth, which resulted in a much shorter time of 7.78 h to complete a profile. The trends of glider velocities and attitude angles variation in these two modes are the same as those in LR mode. However, the attitude angles vary less as the absolute value of θ increases during descent and ascent, showing that a tendency for attitude is more stable in a larger absolute value of θ. In the LF mode simulation, the attitude angles remain almost constant, where θ is almost equal to γ (87.5°) and α is almost 0°.

4.2. Spiral Motion Simulation

Spiral motion is usually presented as an assisted motion for regulating the heading of an underwater glider. Therefore, the spiral motion simulations of Petrel-X^PLUS are based on the sawtooth motion simulations in Section 4.1. Note that Petrel-X^PLUS in LF mode follows the movement of ocean currents in the horizontal direction; thus, the spiral motion is no longer meaningful in this mode. In the LR mode simulation, λ₁ and λ₂ are all set to 45° counterclockwise from the initial 0°. In the VM mode simulation, the two battery packs are all rotated 45° counterclockwise on the basis of the initial λ₁ = −λ₂ = 30°, i.e., finally, the λ₁ and λ₂ are set to 15° and 75°, respectively. Figure 11 and Figure 12 show the results of spiral motion simulations performed within the descent phase of LR mode and VM mode, respectively.

The time to reach a depth of 10,000 m is 16.2 h and 3.77 h in the LR and VM mode simulations, respectively. However, the spiral motion takes longer than the sawtooth motion (LR mode is 13.98 h, VM mode is 3.6 h) due to the rotation of the two battery packs weakens the regulation ability of θ, resulting in a smaller absolute value of θ and longer dive time. The spiral radius is affected by the velocity variation, as the faster the velocity is, the larger the radius, so in both modes, the spiral radius increases first and then decreases (Figure 11f and Figure 12f), which is consistent with the variation described in the literature [11]. The variations of velocities and attitude angles are basically the same as those in the sawtooth motion, and the yaw angle ψ keeps repeating 0°~360° until reaching the target depth. In LR mode, the glider has more sufficient time to perform the spiral motion, resulting in more spirals to the target depth than that in VM mode. Moreover, the rotation radius in VM mode is larger than that in LR mode, owing to the inverse relationship between the radius and θ [10].

5. Ballast Method to Adjust the Pitch Angle Range

The capability of DARM to regulate θ is related to the position of the center of gravity of the glider, which can be adjusted by the ballast method to ensure minor modifications even after the glider is constructed. The ballast method is usually used to balance the glider in a water pool using lead blocks placed in the submerged fairings at the front and rear, as shown in Figure 13. Therefore, in the longitudinal plane, there are two main types of ballast regulation: (1) placing lead blocks inside the front or rear fairings to regulate the position of the center of gravity on x_b-axis x_g and (2) placing lead blocks equally inside the front and rear fairings to regulate the position of the center of gravity on z_b-axis z_g. Note that the buoyancy force of an underwater glider is generally designed to be slightly greater than its gravity to provide a certain net buoyancy for the correction of construction error. Therefore, it is reasonable to assume that the weight of the installed lead blocks is balanced by a net buoyancy of equal magnitude.

The analysis of forces and moments of Petrel-X^PLUS in the longitudinal and transverse planes in a water pool is expressed as follows:

$$\{\begin{array}{l}{\displaystyle \sum F_{X-Z}={\rho }_{oil}{V}_{oe}}g+{\rho }_{oil}{V}_{oi}g+m_{s}g+m_{r}g+m_{ba1}g+m_{ba2}g-m_{g}g-{\rho }_{sea}{V}_{oe}=0\\ {{\displaystyle \sum M}}_{X-Z}=(m_s{x}_s+{\rho }_{oil}{V}_{oe}{x}_{oe}+{\rho }_{oil}{V}_{oi}{x}_{oi}+m_r{x}_r+m_{ba1}{x}_{ba1}+m_{ba2}{x}_{ba2}-{\rho }_{sea}{V}_{oe}{x}_{oe})\cos \theta \\ -(m_s{z}_s\cos \phi +0.5m_r{z}_r\cos ({\lambda }_1-\phi )+0.5m_r{z}_r\cos ({\lambda }_2-\phi ))\sin \theta =0\end{array}$$

(33)

$$\{\begin{array}{l}{\displaystyle \sum F_{Y-Z}={\displaystyle \sum F_{X-Z}}}\\ {\displaystyle \sum M_{Y-Z}}=0.5m_{r}g{z}_r\sin ({\gamma }_1-\phi )+0.5m_{r}g{z}_r\sin ({\gamma }_2-\phi )\\ -(m_{s}g{z}_s+m_{ba1}g{z}_{ba1}+m_{ba2}g{z}_{ba2})\sin \phi =0\end{array}$$

(34)

where V_oi and V_oe are the volumes of oil in the inner tank and external bladder, respectively, and x_oi and x_oe are the positions of the inner tank and external bladder on the x_b-axis, respectively.

Combining Equations (34) and (35), θ can be obtained as:

$$\theta ={\tan }^{-1}(\frac{m_s{x}_s+{\rho }_{oil}{V}_{oe}{x}_{oe}+{\rho }_{oil}{V}_{oi}{x}_{oi}+m_r{x}_r+m_{ba1}{x}_{ba1}+m_{ba2}{x}_{ba2}-{\rho }_{sea}{V}_{oe}{x}_{oe}}{(m_s{z}_s+m_{ba1}{z}_{ba1}+m_{ba2}{z}_{ba2})\cos \phi +0.5m_r{z}_r\cos ({\lambda }_1-\phi )+0.5m_r{z}_r\cos ({\lambda }_2-\phi )})$$

(35)

where

$$\phi ={\tan }^{-1}(\frac{\sin {\lambda }_1+\sin {\lambda }_2}{2(m_s{z}_s+m_{ba1}{z}_{ba1}+m_{ba2}{z}_{ba2})/m_r{z}_r+\cos {\lambda }_1+\cos {\lambda }_2})$$

(36)

The settings of ballast mass for regulating x_g are shown in

Table 4. The mass of ballast for regulating x_g.

Ballast Mass	Value (kg)
mba1/mba2	0.75/0	0.5/0	0.25/0	0/0	0/0.25	0/0.5	0/0.75

, and the positions of the ballast on both sides are ${\overline{\mathit{r}}}_{ba1}$ = [x_ba1, y_ba1, z_ba1]^T = [1.5, 0, 0]^T m and ${\overline{\mathit{r}}}_{ba2}$= [x_ba2, y_ba2, z_ba2]^T = [−1.5, 0, 0]^T m. The center of gravity O_g will be moved forward (backward) along the x_b-axis by placing ballast in the front (rear) fairing, while the two battery packs of DARM need to be moved in the opposite direction to maintain an equilibrium state. Results in a change in the distances from the new equilibrium position of the battery packs to the front limit (x_r = −0.358 m) and to the rear limit (x_r = −0.478 m). Thus, in the legacy pattern (Figure 14b), the curve of θ vs. x_r is changed, where the larger the mass of ballast in the front fairing is, the larger the regulation range of θ in the head down state and the smaller the regulation range of θ in the head up state, while the opposite variation occurs when adding ballast on the other side. Figure 14a,c show the curves of θ vs. λ₁/−λ₂ in the DARM pattern at different ballast masses when the battery packs are located at the rear limit and front limit, respectively. Although the curves of θ vs. λ₁/−λ₂ do not coincide at different ballast masses, λ₁/−λ₂ is the same when θ reaches 90° or −90° (λ₁ = −λ₂ = 143.1°). Therefore, the regulation range of θ in the legacy pattern is changed when placing the ballast in the front or rear fairing, while it is not affected in the DARM pattern. This also demonstrates the advantages of DARM and provides guidance for the modification of the initial equilibrium state.

The ballast mass and position for regulating z_g are symmetric about the z_b-axis and are listed in

Table 5. The mass and position of ballast for regulating z_g.

Ballast Mass mba (kg)	Position on zb-Axis zba (m)
1.0	0.15	0.5	0.05	0	−0.05	−0.1	−0.15
2.0	0.15	0.5	0.05	0	−0.05	−0.1	−0.15
3.0	0.15	0.5	0.05	0	−0.05	−0.1	−0.15

. Figure 15, Figure 16 and Figure 17 show the curves of θ vs. z_ba with ballast mass m_ba equal to 1.0 kg, 2.0 kg, and 3.0 kg, respectively, where z_ba = z_ba1 = z_ba2 and m_ba = m_ba1 + m_ba2, m_ba1 = m_ba2. When m_ba is constant, the lower z_ba, the smaller the regulation range of θ in the legacy pattern, while the initial equilibrium state of the battery packs remains constant (Figure 15b, Figure 16b, and Figure 17b). In the DARM pattern, the regulation range of θ decreases as z_ba varies from 0 m to 0.15 m and increases as z_ba varies from 0 m to −0.15 m (Figure 15a,c, Figure 16a,c, and Figure 17a,c). Moreover, the regulation range of θ increases as m_ba increases from 1.0 kg to 3.0 kg when z_ba∈[−0.15 m, 0] and decreases as m_ba increases from 1.0 kg to 3.0 kg when z_ba∈[0, 0.15] m. The θ fails to reach −90°~90° at the position of z_ba = 0.1 m and 0.15 m in Figure 15a,c and the position of z_ba = 0.05 m, 0.1 m, 0.15 m in Figure 16a,c and Figure 17a,c. For the same mass of ballast, the upper the mounting position, the smaller the λ₁/−λ₂ when θ reaches 90° or −90°, which gives us an insight that ballasts can be placed symmetrically on the top of the front and rear fairings to achieve a θ range of −90°~90° when its regulation range is insufficient in the DARM pattern.

6. Experimental Verification

6.1. Water Pool Experment

In order to verify the capability of DARM for regulating pitch angle, an experiment was carried out in a water pool, as shown in Figure 18a. In the initial equilibrium state of Petrel-X^PLUS, its net buoyancy is 0 N and pitch angle θ is 0° with its body completely submerged in the water. The position and rotation angles of the two battery packs are regulated according to the process in Figure 2. Figure 18b shows that Petrel-X^PLUS has the ability to regulate the pitch angle from −90° to 90°, which satisfies the design target of DARM.

6.2. Sea Trial

In July 2020, Petrel-X^PLUS completed a sea trial in the western depression of the Challenger Deep, Mariana Trench (Figure 19). The complex and narrow topography of the western depression [40] and the North Equatorial Current [41] pose a significant challenge for access to this region. Therefore, Petrel-X^PLUS conducted a station-keeping observation though VM mode targeting the deepest point (142.2047° E, 11.3331° N) in this depression. Finally, three profiles (G1, G2, and G3) exceeding 10,000 m were completed with a maximum depth of 10,619 m (G3) and an average position deviation of 2.018 km on the sea surface. In addition, the motion performance of Petrel-X^PLUS in the LR mode was also tested.

The comparison between the experiments and simulations in LR and VM modes is shown in Figure 20 and Figure 21, respectively. The experimental profile in LR mode dives to a depth of 6889 m with x_r values of −0.433 m (descent), −0.403 m (ascent), and λ₁ = −λ₂ = 0° (descent and ascent). The experimental profile in VM mode dives to a depth of 10,619 m with x_r values of −0.358 m (descent), −0.478 m (ascent), and λ₁ = −λ₂ = 90° (descent and ascent). Note that the non-stationary process due to oil drainage is ignored in the simulation of VM mode, as the duration of the process is approximately 2 h [42]. In general, the results of the simulations and experiments are in good agreement, while the navigation times of the experimental profiles are longer than those of the simulations. This is because the simulations did not consider the disturbance caused by actual ocean currents, which can produce navigation bias and control regulation and, thus, result in a longer navigation time. As shown in Figure 20c, there is a sawtooth variation of θ in LR mode due to the control threshold setting (regulated if the bias of θ exceeds 5° and turned off if the bias of θ is below 5°), which did not occur in VM mode (Figure 21c). Moreover, oil discharge changes the net buoyancy of the glider, resulting in a non-stationary phenomenon of oscillations in vertical velocity and pitch angle.

7. Conclusions

This study is one of our works to extend the observation depth of underwater gliders to hadal trenches. In this work, a compact ARM named DARM is proposed to increase the regulation range of pitch angle from −45°~45° to −90°~90°. DARM also enriches the motion of an underwater glider in three typical modes, LR mode, VM mode, and LF mode, and helps the glider to enter the complex and narrow hadal trench effectively. Subsequently, the dynamics model of a hadal class underwater glider Petrel-X^PLUS equipped with DARM was established using dual quaternions. The dynamics model of the underwater glider established using dual quaternions does not have trigonometric calculation, so this avoids the “gimbal lock” problem in motion simulation and control when the pitch angle is equal to ±90° and provides the basis model for future research on control methods. Moreover, a ballast method for modifying the regulation range of the pitch angle and initial equilibrium state of the underwater glider was obtained. When the regulation range of the pitch angle does not achieve the range of −90°~90°, it can be modified by symmetrically arranging two equal ballasts along the center of buoyancy in the upper part of the front and rear fairings. When the initial position of the battery packs is not at the center of the front and rear limits, it can be modified by arranging the ballast in the front or rear fairing to maintain an equal movement distance between the two limits. Finally, a water pool experiment and a sea trial in the Mariana Trench verified the effectiveness of the proposed methods. During the sea trial, Petrel-X^PLUS performed station-keeping observations in VM mode and completed three consecutive profiles at depths over 10,000 m with a maximum dive depth of 10,619 m and an average deviation distance of 2.018 km from the target point.

The performance in LF mode was not verified in this study, and it will be performed in conjunction with observation missions. In the future, the large spatial–temporal observations of hadal trenches will be the focus of Petrel-X^PLUS, such as monitoring along the hadal trench axis using LR mode, as this could provide real-time, continuous, and extensive observation data to facilitate hadal science.