Investigation of the Asymmetric Features of X-Rudder Underwater Vehicle Vertical Maneuvring and Novel Motion Prediction Technology

Abstract

An X-rudder underwater vehicle’s hydrodynamic force acting on its rudder will display asymmetrical characteristics during vertical movement that are absent from a cross-rudder vehicle. This paper presents a novel hydrodynamic expression method based on rotational hydrodynamic transformation through a detailed analysis of the local flow characteristics around the tail attachment during the vertical plane maneuvering of the X-rudder vehicle, given that the conventional Taylor expansion-based hydrodynamic expression method is unable to characterize this asymmetric characteristic. With the help of this technique, a novel expression that can precisely describe the asymmetric hydrodynamic properties during the X-rudder vehicle’s underwater vertical plane maneuvering is created. This paper next concentrates on common vertical plane maneuvering motion situations and performs simulation predictions using both new and conventional expressions based on Taylor expansion. The asymmetric characteristics of the X-rudder underwater vehicle in vertical plane maneuvering have been experimentally confirmed, and the asymmetric characteristics become more pronounced as the speed increases, according to the results, which are compared with those of tests using self-driving models. Overall, the new model accurately describes the asymmetric features of the X-rudder vehicle’s vertical maneuvering motion and correlates well with the experimental findings.

1. Introduction

During the underwater vertical plane maneuvering of an X-rudder vehicle, the hydrodynamic forces acting on the rudders manifest asymmetric characteristics absent in cross-rudder vehicles. This asymmetry gives rise to a motion phenomenon wherein the X-rudder vehicle demonstrates a tendency for “easier pitch-down than pitch-up” during certain vertical plane maneuvers. This approach has been widely applied to simulate and predict underwater maneuvering motions of various vehicles for a long time, playing a pivotal role in evaluating underwater maneuvering performance and analyzing motion characteristics of vehicles [1,2,3,4].

The most authoritative simulation mathematical model for predicting underwater maneuvering motions of vehicles is the Gertler equation proposed by the U.S. Naval Ship Research and Design Center in 1967. Its hydrodynamic forces are characterized by Taylor series expansion, while rudder forces are described by a first-order linear formulation, making it suitable for simulation predictions under conventional small angle-of-attack maneuvering scenarios [5]. Considering the prominent flow separation on the leeward side of underwater vehicles and severe asymmetric vortex shedding under large angle-of-attack conditions, rendering the continuity assumption invalid, the Taylor expansion fails to accurately characterize the complex hydrodynamic characteristics in such scenarios [6,7,8,9]. In 1979, the U.S. Naval Ship Research and Design Center proposed the revised Gertler equation [10], which incorporates an integral term for cross-flow drag and a time term for vortex shedding. Thus, this revision enables its application to extremely large angle-of-attack motions, such as unpowered surfacing [11,12,13]. With the exception of variations in the representation of hydrodynamic coefficients and the order of hydrodynamics (certain nonlinear and coupled hydrodynamic orders have reached six), the Russian simulation mathematical model [14] is formally consistent with the American version. Furthermore, Russia characterizes rudder force using a nonlinear formulation of the third-order term.

For X-rudder vehicles, the prevailing approach is to adopt the framework of the Gertler equation and integrate X-rudder force predictions, formulating the four rudder forces of X-rudders with linear expressions to replace the yaw rudder and tail elevator terms in the original equation [15,16,17,18]. Existing experiments have demonstrated that this method exhibits satisfactory prediction accuracy for low-speed maneuvers with small rudder angles. However, in high-speed vertical plane maneuvers involving large rudder angles, the traditional first-order linear rudder force formulation fails to characterize the asymmetric features of X-rudder forces, leading to notable discrepancies between simulation predictions and model test results.

This study undertakes an in-depth analysis of the asymmetric characteristics in vertical plane maneuvers of X-rudder vehicles. Building on this analysis, an innovative hydrodynamic formulation based on rotational hydrodynamic transformation is proposed to accurately capture the asymmetric hydrodynamic behaviors during underwater vertical plane maneuvers. Subsequent simulations are conducted for vertical plane diving-surfacing maneuvers and rudder jam recovery maneuvers, employing the traditional Taylor expansion-based formulation and the new rotational hydrodynamic transformation-based formulation in sequence, with results compared against self-propelled model test data [19]. The test results validate the presence of asymmetric characteristics in X-rudder vertical plane maneuvers, which become increasingly pronounced with higher speeds. The new formulation effectively reflects this feature, demonstrating excellent agreement with test results, whereas the traditional formulation, owing to its failure to account for this asymmetry, yields substantial deviations [20,21].

2. Investigation of the Asymmetric Properties and Hydrodynamic Expression Method in X-Rudder Navigation Vehicle Vertical Maneuvring

2.1. Application of Traditional Hydrodynamic Expression Methods in X-Rudder Vehicles and Existing Problems

When the hydrodynamic expression method based on traditional Taylor expansion is used to characterize the hydrodynamic characteristics of vertical plane maneuvers of X-rudder vehicles, the coordinate origin is located at the buoyancy center of the vehicle (in some vehicles, it is set at the center of gravity or the central axis where the center of gravity is located). The expressions of hydrodynamic forces and moments in vertical plane maneuvers can be divided into three parts: position hydrodynamic forces, rotational hydrodynamic forces, and rudder force-related hydrodynamic forces, all characterized with the buoyancy center as the coordinate origin. Specific dimensionless expressions are as follows, where Z′ and M′ denote the dimensionless coefficients of vertical force and pitching moment, respectively. Their expressions are as follows: $Z^{\prime }=Z/0.5·\rho ·V^2·L^2$, $M^{\prime }=M/0.5·\rho ·V^2·L^3$. Where Z and M represent the vertical force and pitching moment of the vehicle, respectively. ρ is the fluid density, L is the total length of the vehicle, and V is the vehicle’s speed.

Position hydrodynamic force:

$${\mathrm{Z}}^{\prime }=Z_w^{\prime }{u}^{\prime }{w}^{\prime }+Z_{\left|w\right|}^{\prime }{u}^{\prime }\left|w^{\prime }\right|+Z_{ww}^{\prime }{w}^{\prime }{w}^{\prime }+Z_{w\left|w\right|}^{\prime }{w}^{\prime }\left|w^{\prime }\right|$$

(1)

$${\mathrm{M}}^{\prime }=M_w^{\prime }{u}^{\prime }{w}^{\prime }+M_{\left|w\right|}^{\prime }{u}^{\prime }\left|w^{\prime }\right|+M_{ww}^{\prime }{w}^{\prime }{w}^{\prime }+M_{w\left|w\right|}^{\prime }{w}^{\prime }\left|w^{\prime }\right|$$

(2)

Rotational hydrodynamic force:

$${\mathrm{Z}}^{\prime }=Z_q^{\prime }{u}^{\prime }{q}^{\prime }+Z_{q\left|q\right|}^{\prime }{q}^{\prime }\left|q^{\prime }\right|+Z_{w\left|q\right|}^{\prime }{w}^{\prime }\left|q^{\prime }\right|$$

(3)

$${\mathrm{M}}^{\prime }=M_q^{\prime }{u}^{\prime }{q}^{\prime }+M_{q\left|q\right|}^{\prime }{q}^{\prime }\left|q^{\prime }\right|+M_{w\left|q\right|}^{\prime }{w}^{\prime }\left|q^{\prime }\right|$$

(4)

Rudder force:

$${\mathrm{Z}}^{\prime }={\sum }_{i=1}^4(Z_{{\delta }_i}^{\prime }{\delta }_i+Z_{{\left|q\right|\delta }_i}^{\prime }{\left|q^{\prime }\right|\delta }_i)$$

(5)

$${\mathrm{M}}^{\prime }={\sum }_{i=1}^4({\displaystyle M_{{\delta }_i}^{\prime }}{\delta }_i+M_{{\left|q\right|\delta }_i}^{\prime }{\left|q^{\prime }\right|\delta }_i)$$

(6)

In the formula, u′ represents the dimensionless axial velocity; w′ represents the dimensionless vertical velocity; q′ represents the dimensionless pitch angular velocity; δ_i is the rudder angle of the X-rudder, where i = 1∼4;

X-rudders are generally distributed orthogonally at 45° to the central axis, and the rudder surface parameters are consistent. The rudder derivatives corresponding to each rudder blade (such as $Z_{{\delta }_i}^{\prime }$, $Z_{{\left|q\right|\delta }_i}^{\prime }$, $M_{{\delta }_i}^{\prime }$ and $M_{{\left|q\right|\delta }_i}^{\prime }$) theoretically match those of rudder No. 1. Therefore, the above rudder force expressions can be written as:

$${\mathrm{Z}}^{\prime }\left(\mathsf{\delta }\right)={\sum }_{i=1}^4(Z_{{\delta }_1}^{\prime }{\delta }_i+Z_{{\left|q\right|\delta }_1}^{\prime }{\left|q^{\prime }\right|\delta }_i)$$

(7)

$${\mathrm{M}}^{\prime }\left(\mathsf{\delta }\right)={\sum }_{i=1}^4(M_{{\delta }_1}^{\prime }{\delta }_i+M_{{\left|q\right|\delta }_1}^{\prime }{\left|q^{\prime }\right|\delta }_i)$$

(8)

In vertical plane maneuvers, cross-rudder vehicles typically employ only two coupled rudder surfaces of the tail elevator for steering control, whereas X-rudder vehicles are equipped with four independent rudder surfaces for vertical plane steering (as depicted in Figure 1). Consequently, compared to cross-rudder vehicles, X-rudder vehicles exhibit unique control scenarios absent in cross-rudder configurations, as illustrated in Figure 1.

Figure 1. Schematic diagrams of cross-rudder and X-rudder.

When two rudder angles δ_i and δ_j with equal magnitude but opposite directions exist simultaneously, the rotational flow forms local rudder angles δ_Ri and δ_Rj at the tail rudders (as shown in Figure 2). Due to the significant difference between them, the rudder effectiveness of δ_i and δ_j varies greatly, generating remarkable asymmetric characteristics ($\left|Z’\left({\delta }_i\right)\right|$ ≠ $\left|Z’\left({\delta }_j\right)\right|$, $\left|M’\left({\delta }_i\right)\right|$ ≠ $\left|M’\left({\delta }_j\right)\right|$). However, when using the traditional Taylor expansion-based rudder force expressions (5) and (6) for characterization, $\left|Z’\left({\delta }_i\right)\right|$ = $\left|Z’\left({\delta }_j\right)\right|$ and $\left|M’\left({\delta }_i\right)\right|$ = $\left|M’\left({\delta }_j\right)\right|$, failing to reflect the asymmetric characteristics.

Figure 2. Schematic of local flow around X-rudder tail under rudder jam condition.

Further analysis reveals that when the vehicle’s dive rudder δ_i jams (as shown in the dive schematic of Figure 3, where δ_j = 0), the vehicle will dive with its nose submerged. When applying an equal opposite rudder angle δ_j for recovery, although δ_i = δ_j, the rudder force generated by δ_j cannot balance that of δ_i due to the local rudder angle δ_Rj < δ_Ri. This leads to difficulty in pitching up, creating the motion phenomenon of “easier to pitch down than up”.

Figure 3. Schematic of local tail flow at the same angle of attack.

Similarly, during vertical plane maneuvers of X-rudder vehicles, when rotational flows with equal inflow angles of attack α ($w^{\prime }=\mathrm{s}\mathrm{i}\mathrm{n} \alpha$) pass through the buoyancy center of the vehicle (seen in Figure 3), the corresponding local tail inflow angles of attack α_R at the tail appendages exhibit significant differences. This thereby causes remarkable asymmetric characteristics in rotational hydrodynamic forces, with obvious increments in rotational hydrodynamic forces and $∆Z^{\prime }\left(q\right){|}_{\alpha }\ne -∆Z^{\prime }\left(q\right){|}_{-\alpha }$, $∆M^{\prime }\left(q\right){|}_{\alpha }\ne ∆M^{\prime }\left(q\right){|}_{-\alpha }$.

When characterized by the traditional Taylor expansion-based hydrodynamic expressions (3) and (4); however, $∆Z^{\prime }\left(q\right){|}_{\alpha }=-∆Z^{\prime }\left(q\right){|}_{-\alpha }$ and $∆M^{\prime }\left(q\right){|}_{\alpha }=∆M^{\prime }\left(q\right){|}_{-\alpha }$, failing to reflect the asymmetric characteristics of rotational hydrodynamic forces.

From the above analysis, it can be seen that since the traditional hydrodynamic expression method based on Taylor expansion takes the buoyancy center of the vehicle as the coordinate origin, it fails to truly reflect the local flow around characteristics of the tail appendages during vertical plane maneuvers, and cannot characterize the asymmetric characteristics of hydrodynamic forces in vertical plane maneuvering motions. The detailed information concerning the flow around the rear part of the underwater vehicle can be seen in Appendix A.

2.2. Investigation of the X-Rudder Undersea Vehicle’s Vertical Maneuvering Hydrodynamics Using Rotational Hydrodynamic Transformation

The coupling hydrodynamics of vertical angle of attack and angular velocity, rudder angle and angular velocity, respectively, are represented by $M_{\left|w\right|q}^{\prime }$ and $M_{\left|q\right|\delta }^{\prime }$. Traditional mathematical models use Taylor expansion to characterize the coupling effects of vertical attitude angle, rudder angle, and angular velocity. According to this statement, the rotating hydrodynamic force is increased regardless of the angle of attack, whether it is positive or negative. The angular velocity, whether rotating downward or upward, increases the rudder force and is unable to describe the asymmetrical features of vertical maneuvering motion.

This study builds a new mathematical model for vertical plane maneuvering motion and suggests a hydrodynamic expression based on rotational hydrodynamic transformation, taking into account that conventional hydrodynamic expressions are unable to describe the asymmetric properties in maneuvering.

According to the analysis results in Section 2.1, this model consists of the following two parts:

(1)

A hydrodynamic expression for the coupling of angle of attack and angular velocity of underwater vehicles is established based on the local angle of attack α_R at the tail attachment. This expression can be divided into two parts: the variation with the angle of attack at the coordinate origin and the variation with the local incoming flow angle of the tail attachment.

$${ Z}^{\prime }\left(q^{\prime },\alpha \right)=Z^{\prime }\left({\alpha }_R\right)+Z^{\prime }\left(\alpha \right)$$

(9)

$${ M}^{\prime }\left(q^{\prime },\alpha \right)=M^{\prime }\left({\alpha }_R\right)+M^{\prime }\left(\alpha \right)$$

(10)

(2)

The hydrodynamic expression of the coupling between the rudder angle and angular velocity of the X-rudder underwater vehicle can also be expressed as the variation of the local incoming flow angle (local rudder angle) with the rudder surface and the variation of the rudder angle, depending on the local rudder angle δ_R of the tail attachment body.

$${ Z}^{\prime }\left(q^{\prime },{\delta }_i\right)=Z^{\prime }\left({\delta }_{Ri}\right)+Z^{\prime }\left({\delta }_i\right)$$

(11)

$${ M}^{\prime }\left(q^{\prime },{\delta }_i\right)=M^{\prime }\left({\delta }_{Ri}\right)+M^{\prime }\left({\delta }_i\right)$$

(12)

Referring to reference [7], the α_R and δ_R involved in Equations (9) and (10) can be expressed. In this study, this formulation is simplified.

$${\alpha }_R=arctan\left(x\prime q\prime \right)·\epsilon +\alpha$$

(13)

$${\delta }_{Ri}=arctan(x_i^{\prime }q\prime )·{\epsilon }_i+{\delta }_i$$

(14)

In the formula, x′ and x_i′, respectively, represent the dimensionless positions of the tail attachment and the ith X-rudder; q′ represents dimensionless pitch velocity; ε and ε_i′ represent the local angle of attack correction coefficients for the tail attachment and the ith rudder, respectively.

The formula’s dimensionless coordinates for the tail attachment and the i^th X-rudder are denoted by x′ and x_i′, respectively; dimensionless pitch velocity is represented by q′, and the local angle of attack correction coefficients for the tail attachment and the ith i^th rudder are denoted by ε and ε_i′, respectively.

3. Mathematical Modeling of Vertical Maneuvering Motion of X-Rudder Underwater Vehicle Based on Rotational Hydrodynamic Transformation Expression

This section mainly introduces the new mathematical expressions of the X-rudder vehicle based on the transformation of rotational hydrodynamic forces. Section 3.1 introduces the research object of this paper, and gives the main dimension parameters and coordinate system definition. Section 3.2 introduces the hydrodynamic transformation expression of the coupling between the rudder angle and the angular velocity of the vehicle. Section 3.3 gives the hydrodynamic transformation expression of the coupling between the angle of attack and the angular velocity of the vehicle.

3.1. Coordinate System

This article’s study object is a navigation vehicle with an X-shaped, completely dynamic rudder that has a total length of 6.9 m and an underwater displacement of 2.0 tons. It is made up of a main body, a horizontal rudder at the bow, an X-shaped rudder at the tail, and a propeller.

This article describes the movements of the navigating body using a fixed coordinate system (E-ξηζ) and a ship accompanying coordinate system (O-xyz). The origin E is situated at a certain place on the sea level, and the coordinate system (E-ξηζ) is fixed to the Earth. The primary heading of the navigation vehicle is the positive direction of the ξ axis, and the E-ξη plane is situated at sea level. Perpendicular to the primary heading downward, the ξ axis has a positive direction, and the right-hand system is used to establish this direction. With O situated in the underwater vehicle’s floating center and the O-xz plane situated in its longitudinal section, the (O-xyz) coordinate system is fixedly attached to the underwater vehicle. The right-hand system determines that the y-axis is positive, the z-axis is positive when pointing downhill, and the x-axis is in line with the vehicle’s baseline and points positively towards the head of the vehicle. Figure 4 illustrates the positive directions of the coordinate axes, angles, velocities, forces, and moments in the O-xyz system as well as the link between the two coordinate systems.

Figure 4. Diagram of coordinate system.

Figure 5 illustrates how the X-rudder angle is defined. They are rudder 1^# in the upper right corner, rudder 2^# in the upper left corner, rudder 3^# in the lower left corner, and rudder 4^# in the lower right corner when viewed from the tail to the head. Turning the X-rudder produces a hydrodynamic force that is positive in the vertical direction and negative in the opposite direction. It is defined that the vertical component of the hydrodynamic force generated by the forward horizontal rudder after steering is positive when upward, and negative otherwise.

Figure 5. Definition of X-rudder angle symbol.

3.2. Hydrodynamic Transformation Formulation for Rudder Force Correlation

Using the rudder 1^# as an example, the hydrodynamic force of varying rudder angle and angular velocity of the navigating body were determined using the CFD numerical prediction method. The dimensionless angular velocity range is −0.6~0.6, while the rudder angle range is −20~20°. Figure 6 displays the numerical calculation result curve.

Figure 6. Hydrodynamic coefficients of underwater vehicle under various values of rudder angle and variable angular velocity (rudder 1^#).

The coupled hydrodynamic force depicted in Figure 6 can be described as the changes in local rudder angle Z′(δ_R1), M′(δ_R1), and rudder angle Z′(δ₁), M′(δ₁) (as shown in Figure 7 and Figure 8) using the transformation expression of formulas (3) to (4). Since the data clearly demonstrate a linear relationship between Z′(δ₁), M′(δ₁), and Z′(δ₁), M′(δ₁), linear regression can be carried out using the least squares approach. The reason why linear fitting is adopted in this paper is mainly that the shape presented by the hydrodynamic curve has good linearity. The hydrodynamic forces involved in the mathematical model of maneuvering motion all refer to the overall hydrodynamic forces received. How to regress the hydrodynamic coefficients depends on the characteristics of the hydrodynamic curve. If the hydrodynamic curve shows relatively significant linear characteristics, the linear fitting method can be used for fitting. Otherwise, it cannot be used.

Figure 7. Hydrodynamic variation with local rudder angle (rudder 1^#).

Figure 8. Hydrodynamic variation with rudder angle (rudder 1^#).

The coupled hydrodynamic forces of the 2^# to 4^# rudders in terms of angle and angular velocity are precisely the same as those of the 1^# rudder since their parameters and axial positions are identical to those of the first rudder. In order to obtain the changes Z′(δ_R2~4), M′ (δ_R2~4) related to the local rudder angle and the changes Z′(δ_R2~4), M′ (δ_R2~4) related to the rudder angle, as illustrated in Figure 9 and Figure 10, the regression process requires subtracting the hydrodynamic forces in the pure rotational state (with a rudder angle of 0). The least squares approach can also be used for linear regression.

Figure 9. The variation of rotational hydrodynamic force for rudder No. 2 with respect to local rudder angle.

Figure 10. Rotational hydrodynamic force variation of rudder No. 2 with rudder angle.

3.3. Expression of Coupling Hydrodynamic Transformation Between Angle of Attack and Angular Velocity of a Sailing Vehicle

Figure 11 displays the hydrodynamic analysis of the underwater vehicle’s changeable angle of attack and angular velocity. The dimensionless angular velocity range was −0.6~0.6, while the angle of attack range was −12~12°. In the pure turning state, where the angle of attack was zero, the hydrodynamic analysis was subtracted.

Figure 11. Variation with local angle of attack in rotational hydrodynamics (characterized with rotation speed).

Based on the transformation formulations in Equations (9) and (10), the coupled hydrodynamic forces depicted in Figure 11 are decomposed into their respective variations with respect to the local angle of attack ${ Z}^{\prime }\left({\alpha }_R\right)$, ${ M}^{\prime }\left({\alpha }_R\right)$ and rudder angle ${ Z}^{\prime }\left(\alpha \right)$, ${ M}^{\prime }\left(\alpha \right)$, as illustrated in Figure 12 and Figure 13. The results demonstrate that both pairs ${ Z}^{\prime }\left({\alpha }_R\right)$, ${ M}^{\prime }\left({\alpha }_R\right)$ and ${ Z}^{\prime }\left(\alpha \right)$, ${ M}^{\prime }\left(\alpha \right)$ exhibit markedly nonlinear relationships. Thus, second-order term expressions are employed for regression fitting via the least squares method, yielding the derivatives of and with respect to $\alpha$, $\alpha \left|\alpha \right|$, ${\alpha }_R$, ${\alpha }_R,\left|{\alpha }_R\right|$, specifically $Z_{\alpha }^{\prime }$, $Z_{\alpha \left|\alpha \right|}^{\prime }$, $Z_{{\alpha }_R}^{\prime }$, $Z_{{\alpha }_R\left|{\alpha }_R\right|}^{\prime }$, $M_{\alpha }^{\prime }$, $M_{\alpha \left|\alpha \right|}^{\prime }$, $M_{{\alpha }_R}^{\prime }$, and $M_{{\alpha }_R\left|{\alpha }_R\right|}^{\prime }$.

Figure 12. Variation with local angle of attack in rotational hydrodynamics (characterized with local attack angle).

Figure 13. The variation with angle of attack in rotating hydrodynamics.

3.4. Hydrodynamic Formulations for Vertical-Plane Maneuvering of X-Rudder Vehicles via Rotational Hydrodynamic Transformation

Hydrodynamic force expressions related to angle of attack and angular velocity:

$${\mathrm{Z}}^{\prime }\left({\mathrm{q}}^{\prime },\alpha \right)=Z_{{\alpha }_R}^{\prime }{\alpha }_R+Z_{{\alpha }_R\left|{\alpha }_R\right|}^{\prime }{\alpha }_R\left|{\alpha }_R\right|+Z_{\alpha }^{\prime }\alpha +Z_{\alpha \left|\alpha \right|}^{\prime }\alpha \left|\alpha \right|$$

(15)

$${\mathrm{M}}^{\prime }\left({\mathrm{q}}^{\prime },\alpha \right)=M_{{\alpha }_R}^{\prime }{\alpha }_R+M_{{\alpha }_R\left|{\alpha }_R\right|}^{\prime }{\alpha }_R\left|{\alpha }_R\right|+M_{\alpha }^{\prime }\alpha +M_{\alpha \left|\alpha \right|}^{\prime }\alpha \left|\alpha \right|$$

(16)

Since the rudder force represents the variation caused by steering, its expression should be formulated as follows:

$${\mathrm{Z}}^{\prime }\left(\mathsf{\delta }\right)=Z_{{\delta }_{R1}}^{\prime }{\delta }_{R1}+Z_{{\delta }_1}^{\prime }{\delta }_1+{\sum }_{i=1}^4(Z_{{\delta }_2}^{\prime }{\delta }_{Ri}+Z_{{\delta }_{R2}}^{\prime }{\delta }_i)$$

(17)

$${\mathrm{M}}^{\prime }\left(\mathsf{\delta }\right)=M_{{\delta }_{R1}}^{\prime }{\delta }_{R1}+M_{{\delta }_1}^{\prime }{\delta }_1+{\sum }_{i=1}^4(M_{{\delta }_2}^{\prime }{\delta }_{Ri}+M_{{\delta }_{R2}}^{\prime }{\delta }_i)$$

(18)

4. Simulation and Verification of Vertical Maneuvering Motion of X-Rudder Underwater Vehicle Based on Rotational Hydrodynamic Transformation Expression

The results of self-driving model testing were compared and validated with the simulation forecast of typical vertical maneuvering motion based on the newly developed model of vertical maneuvering motion of the X-rudder underwater vehicle. Simulated forecasting using Taylor expansion based on conventional mathematical models was also conducted for comparison’s sake. The experiment was carried out in the large sea-keeping/maneuverability water tank of our unit. The main dimension parameters of the water tank are as follows: length 170 m, width 47 m, water depth 6 m, and the water temperature during the experiment was 15 °C. As it is an indoor water tank and the edge of the tank is equipped with wave-absorbing banks, environmental disturbances affecting the test results were minimized, ensuring the still water environment required for the experiment. During the experiment, the model was positioned at 2.5 m, with distances from both the water surface and the tank bottom exceeding four times the diameter, so that the influences of the water surface and tank bottom could be ignored. The self-propelled model test used a high-precision gyroscope to measure the three-directional attitude angles and angular velocities of the vehicle (angle deviation less than ±0.1°, angular velocity deviation less than ±0.01°/s), a depth sensor to obtain the navigation depth of the vehicle (measurement range: 0~50 m, measurement accuracy 0.25%), a Doppler velocity log (DVL) to test the speed of the vehicle (measurement range 0~9 m/s, test accuracy 0.2%), and the rudder angle was obtained through real-time feedback from the built-in encoder of the rudder actuator (rudder actuator control accuracy ±0.1°, steering rate adjustable from 0~40°/s). A CRIO controller was used to execute the motion control, data acquisition, and storage of the self-propelled model test. The limitations of the self-propelled model test are as follows. Restricted to the time schedule, repeated tests were not carried out. However, based on the preliminary evaluation of the model processing accuracy and the measurement accuracy of testing equipment, the relative uncertainty of a single measurement in the self-propelled model test should be within 2%. Follow-up research in this regard will be strengthened.

4.1. Simulation Prediction and Experimental Verification of Vertical Surface Diving and Floating Maneuvering Motion

Both novel and conventional mathematical models were used to simulate and forecast two different kinds of underwater maneuvering conditions: 1.5 m/s, X-rudder with four rudders operating at 25° and 3.0 m/s, and X-rudder with four rudders operating at 10°. Figure 14 and Figure 15 display the simulation’s longitudinal inclination and depth time history curves.

Figure 14. Comparison between simulation and experimental in submerged and floating maneuvering (U = 1.5 m/s, δ_i = −25°).

Figure 15. Comparison between simulation and experimental in submerged and floating maneuvering (U = 3.0 m/s, δ_i = −10°).

Seen in Table 1 and Table 2, The new model, based on rotational hydrodynamic transformation, has a prediction accuracy of within 7%, demonstrating the effectiveness of the new model’s prediction results, according to a comparison and analysis of the extreme values of longitudinal inclination and depth obtained from the new model, Traditional Expression, and experiments. The Traditional Expression’s prediction deviation is essentially within 5% at a speed of 1.5 m/s, but at a speed of 3.0 m/s, the deviation reaches 15%~20%, indicating that the Traditional Expression has good prediction results at low speeds but has a larger prediction deviation at high speeds.

Table 1. Comparison between simulation and experiment in submersible and floating maneuvers (U = 1.5 m/s, δ_i = −25°).

Value	Peak Value of Pitching Angle (°)		Peak Value of Depth (m)
	Value	Deviation from the Experiment	Value	Deviation from the Experiment
Current Expression	26.61	−6.92%	2.34	−3.72%
Traditional Expression	27.12	−5.12%	2.38	−2.13%
Experiment	28.59	0%	2.43	0%

Table 2. Comparison between simulation and experiment in submersible and floating maneuvers (U = 3.0 m/s, δ_i = −10°).

Value	Peak Value of Pitching Angle (°)		Peak Value of Depth (m)
	Value	Deviation from the Experiment	Value	Deviation from the Experiment
Current Expression	21.13	−2.60%	1.72	−6.23%
Traditional Expression	24.85	14.57%	1.48	−19.21%
Experiment	21.69	0%	1.84	0%

4.2. Simulation Prediction and Experimental Verification of Rudder Jamming Retrieval Maneuvering Motion

The head horizontal rudder and the third and fourth X-rudders are utilized for recovery while the underwater navigation vehicle is traveling at 1.5 m/s, with two rudder jamming conditions of 20° and 3.0 m/s for the 1^# and 2^# rudder jamming, and 10° for the 1^# and 2^# rudder jamming. The head horizontal rudder is adaptively regulated, and the 3^# and 4^# rudders function in reverse with the same rudder angle. Both novel and conventional mathematical models were used in our simulation forecasting, and Figure 16, Figure 17, Figure 18 and Figure 19 display the longitudinal inclination and depth time history curves that were produced by the simulation.

Figure 16. Rudder jamming retrieval simulation forecast (U = 1.5 m/s, δ_1,2 = 20°).

Figure 17. Rudder jamming retrieval simulation forecast (U = 3.0 m/s, δ_1,2 = 10°).

Figure 18. Rudder jamming retrieval simulation forecast (U = 1.5 m/s, δ_1,2 = 10°).

Figure 19. Rudder jamming retrieval simulation forecast (U = 3.0 m/s, δ_1,2 = 20°).

By comparing the results of the rudder jam recovery tests at 1.5 m/s and 3.0 m/s, it can be seen that the vertical plane maneuvering motion of the X-rudder vehicle has significant asymmetric characteristics. Even if the No. 3 and No. 4 rudders are operated in the opposite direction with the same rudder angle and the bow horizontal rudder is fully operated, it is difficult to balance the rudder force of the No. 1 and No. 2 rudders. The longitudinal inclination shows a significant motion phenomenon of “easy to bow down but difficult to raise the head”, and with the increase of the navigation speed, the asymmetric characteristics become more obvious.

The maximum pitching angle predicted by the old model at a speed of 1.5 m/s and 20° for the 1^# and 2^# rudder clamps is in good agreement with the experimental results, according to a comparison and analysis of the new model, Traditional Expression, and experimental results. The over-depth deviation is more than 50%, and the depth and head horizontal rudder time history curves deviate greatly from the experimental findings. Overall, the new model’s predictions for the pitching angle, depth, and head horizontal rudder time history curve correspond well with the experiment; the difference between the pitching angle and super depth is less than 10%, which is better than the results of the conventional model.

The pitching angle, depth, and first horizontal rudder time history curve predicted using Traditional Expressions show significant deviations from the experiment when the rudder jamming angle of 1^# and 2^# reaches 10° (3.0 m/s). Additionally, the maximum pitching angle deviates from the super depth by more than 30% to 50%. Overall, the prediction results from the new model are much better than those from the conventional model. The pitching angle, depth, and head horizontal rudder curve all match well with the experiment, and the variance between the pitching angle and super depth is within 10%.

Furthermore, it is evident from comparing the outcomes of the rudder jamming retrieval tests at 1.5 and 3.0 m/s that the X-rudder underwater vehicle exhibits a notable asymmetrical feature in its vertical maneuvering motion. Additionally, the pitch angle reflects the “easy to bow but difficult to raise” motion phenomenon, which intensifies with increasing speed.

5. Conclusions

In order to extract the asymmetric characteristics of rudder force and rotational hydrodynamic force, this article proposes a hydrodynamic expression method based on the transformation of rotational hydrodynamic force, performs a thorough analysis and investigation of the local flow characteristics of the tail attachment in the vertical plane maneuvering of the X-rudder underwater vehicle, and builds a new model that can accurately represent the asymmetric characteristics of hydrodynamic force in the vertical plane maneuvering of the X-rudder underwater vehicle.

The vertical surface dive and rudder jamming retrieval maneuvering actions are then the main subject of this study, which simulates and predicts using both novel and conventional models based on Taylor expansion. The outcomes are contrasted with those of tests of self-driving models. The findings demonstrate that the X-rudder undersea vehicle’s asymmetrical features during vertical plane maneuvering have been experimentally verified and that these features intensify with speed. Overall, the new model well describes the asymmetric features of the vertical maneuvering motion of the X-rudder underwater vehicle and fits well with the experimental results. The depth and horizontal rudder angle changes at the bow of the classic model deviate significantly from the experimental results, and the discrepancy progressively grows as speed increases.

The authors conceived the idea of revising the original model after learning that during the high-speed large rudder jam tests conducted by relevant institutions, the test results showed significant discrepancies with the simulation predictions from the traditional model based on Taylor expansion, and the test results were more hazardous in the depth direction. As shown in Table 3 and Table 4 of the main text, the depth predicted by the new model is nearly 50% larger than that of the traditional model, which is consistent with the results of the self-propelled model test. In other words, when the traditional model is used for prediction, it cannot reflect the risks in the actual navigation process, while the new model not only improves the prediction accuracy but also helps to timely identify the hazards in the depth direction, thereby enhancing the safe navigation capability of the vehicle. In addition, the more detailed information concerning the difference between our expression and the existing expressions can be seen in Appendix B.

Table 3. Simulation prediction and experimental comparison of rudder jamming maneuvering (U = 1.5 m/s, δ_1,2 = 20°).

Value	Peak Value of Pitching Angle (°)		Peak Value of Depth (m)
	Value	Deviation from the Experiment	Value	Deviation from the Experiment
Current Expression	−3.17	9.49%	0.42	−9.71%
Traditional Expression	−2.85	−1.67%	0.23	−50.75%
Experiment	−2.89	0%	0.47	0%

Table 4. Simulation prediction and experimental comparison of rudder jamming maneuvering (U = 3.0 m/s, δ_1,2 = 10°).

Value	Peak Value of Pitching Angle (°)		Peak Value of Depth (m)
	Value	Deviation from the Experiment	Value	Deviation from the Experiment
Current Expression	−5.65	−9.10%	0.75	−6.55%
Traditional Expression	−4.42	−28.79%	0.40	−49.90%
Experiment	−6.21	0%	0.80	0%