Motion Control of a Low-Cost Underwater Vehicle with Three-Position Cross Rudder

Abstract

Underwater vehicles are widely employed as platforms for marine monitoring and operations, with rudders playing a crucial role in controlling their movements. Traditional underwater vehicles utilize servos to steer rudders, which results in high production and maintenance costs. To reduce the manufacturing costs of small underwater vehicles, this paper proposes a design featuring a three-position cross rudder driven by a binary-controlled electromagnet. Additionally, a novel virtual rudder angle controller is developed, which utilizes PWM frequency conversion to perform rudder control, ensuring motion performance comparable to that of conventional underwater vehicles. By establishing kinematic and dynamic models of a typical underwater vehicle (REMUS100), a corresponding virtual rudder angle control strategy is introduced. Based on the binary-controlled three-position cross rudder and virtual rudder angle controller, the effectiveness of the motion control method is verified by numerical simulation. The results confirm that the proposed virtual rudder angle controller successfully executes turning and spatial path-tracking tasks. The binary-controlled three-position cross rudder, together with the virtual rudder angle controller, offers a cost-effective solution for the practical application of small unmanned underwater vehicles, particularly for disposable underwater vehicles that do not require recovery.

1. Introduction

With the increasing exploitation of marine resources, the demand for underwater vehicles has surged [1]. Currently, there are two primary approaches to the development of underwater vehicles [2]. One approach focuses on heavy-duty, intelligent vehicles [3], which aim to expand scientific applications and adapt to more complex underwater environments while maintaining basic motion capabilities. This is achieved by equipping the vehicles with additional sensors, enhancing pressure resistance, and extending operational endurance. The other approach emphasizes lightweight and miniaturized vehicles [4], designed primarily for smaller-scale or single-purpose scientific missions. These vehicles are cost-effective to manufacture and highly flexible in their use. Generally, the control systems of miniaturized underwater vehicles are underactuated [5], making overshoot reduction in motion control a key design consideration [6].

In motion control, the primary elements are typically the propeller, which controls the vehicle’s propulsion, and the rudder, which governs its attitude. Since propeller control is relatively straightforward, whether motion control is based on optimal control [7,8], fuzzy control [9], or adaptive control [10], the main focus of controller design is still on rudder angle control. For small underwater vehicles developed by various research institutions, whether the rudder is X-shaped [11,12] or cross-shaped [13], servos are typically used to control the rudder angle. The advantage of using servos is that they provide flexible rudder angle adjustment, allowing for integration with other systems and facilitating the overall design of the motion control system.

Servos can be categorized into sealed and non-sealed types, depending on their installation location. Sealed servos require dynamic sealing at the rudder drive shaft, leading to higher power consumption during rudder operation. Non-sealed servos, on the other hand, need to address waterproofing and pressure resistance issues. Both types increase the design cost of underwater vehicles; especially for swarm-style low-cost small underwater vehicles that do not require recovery, such high costs are unacceptable. To reduce costs, this paper proposes a cross rudder driven by an electromagnet based on the principle of electromagnetic induction. Compared to traditional servo control, this solution offers simplicity, operational stability, and cost-effectiveness. However, since it only supports binary switching states, it cannot achieve precise rudder angle control like traditional systems. Therefore, this paper also designs a software-based controller to compensate for this limitation.

The structure of the paper is as follows: Section 2 describes the mathematical modeling of the vehicle and the structural design of the electromagnet-driven three-position cross rudder with switching states. Section 3 focuses on the design and theoretical derivation of the virtual rudder angle controller. Section 4 presents the simulation validation of the control scheme, demonstrating the effectiveness of the virtual rudder angle controller. Finally, Section 5 offers conclusions and discusses future research directions.

2. Modeling of Underwater Vehicle

2.1. Coordinate System Selection and Kinematic Modeling

A kinematic model of the underwater vehicle is developed. Based on the motion model symbols and modeling standards recommended by the Society of Naval Architects and Marine Engineers (SNAME) and the International Towing Tank Conference (ITTC), an earth-fixed coordinate system and a body-fixed coordinate system are established, respectively, to describe the position and orientation of the vehicle in space [14], as shown in Figure 1.

The body-fixed coordinate system takes the center of buoyancy of the vehicle as its origin, with the X, Y, and Z axes representing the surge, sway, and heave directions of the vehicle, respectively. This system is used to describe the velocity and forces acting on the vehicle relative to itself. The earth-fixed coordinate system adopts a self-defined anchor point as its origin and is used to describe the position and orientation of the vehicle in space. The general motion of the vehicle in six degrees of freedom can be represented by the six vectors in

$$\begin{gathered} {\mathit{\eta }}_1={\left[\begin{array}{ccc}x& y& z\end{array}\right]}^T {\mathit{\eta }}_2={\left[\begin{array}{ccc}\phi & \theta & \psi \end{array}\right]}^T \\ {\mathit{\upsilon }}_1={\left[\begin{array}{ccc}u& v& w\end{array}\right]}^T {\mathit{\upsilon }}_2={\left[\begin{array}{ccc}p& q& r\end{array}\right]}^T \\ {\mathit{\tau }}_1={\left[\begin{array}{ccc}X& Y& Z\end{array}\right]}^T {\mathit{\tau }}_2={\left[\begin{array}{ccc}K& M& N\end{array}\right]}^T \end{gathered}$$

(1)

where ${\mathit{\eta }}_1$ represents the position of the vehicle’s center of buoyancy relative to the earth-fixed coordinate system, and ${\mathit{\eta }}_2$ denotes the Euler angles of the body-fixed coordinate system relative to the earth-fixed coordinate system, also known as the vehicle’s attitude angles. ${\mathit{\nu }}_1$ represents the linear velocity of the vehicle relative to the body-fixed coordinate system, and ${\mathit{\nu }}_2$ represents the rotational velocity of the vehicle relative to the body-fixed coordinate system. ${\mathit{\tau }}_1$ denotes the forces acting on the vehicle relative to the body-fixed coordinate system, and ${\mathit{\tau }}_2$ denotes the moments acting on the vehicle relative to the body-fixed coordinate system.

By adding the transpose matrix, the transformation relationship between the body-fixed coordinate system and the earth-fixed coordinate system can be obtained [15]. Using Equation (2), the six degrees of freedom of underwater motion of the vehicle can be accurately described, which constitutes the kinematic modeling of the vehicle.

$$\left[\begin{array}{c}{\dot{\mathit{\eta }}}_1\\ {\dot{\mathit{\eta }}}_2\end{array}\right]=\left[\begin{array}{cc}{\mathit{J}}_1& 0\\ 0& {\mathit{J}}_2\end{array}\right]\left[\begin{array}{c}{\mathit{\nu }}_1\\ {\mathit{\nu }}_2\end{array}\right]$$

(2)

where transpose matrix ${\mathit{J}}_1,{\mathit{J}}_2$ can be expressed as Equation (3).

$$\begin{gathered} {\mathit{J}}_1=\left[\begin{array}{ccc}\mathit{cos}\psi \mathit{cos}\theta & -\mathit{sin}\psi \mathit{cos}\phi +\mathit{cos}\psi \mathit{sin}\theta \mathit{sin}\phi & \mathit{sin}\psi \mathit{sin}\phi +\mathit{cos}\psi \mathit{cos}\phi \mathit{sin}\theta \\ \mathit{sin}\psi \mathit{cos}\theta & \mathit{cos}\psi \mathit{cos}\phi +\mathit{sin}\phi \mathit{sin}\theta \mathit{sin}\psi & -\mathit{cos}\psi \mathit{sin}\phi +\mathit{sin}\theta \mathit{sin}\psi \mathit{cos}\phi \\ -\mathit{sin}\theta & \mathit{cos}\theta \mathit{sin}\phi & \mathit{cos}\theta \mathit{cos}\phi \end{array}\right] \\ {\mathit{J}}_2=\left[\begin{array}{ccc}1& \mathit{sin}\phi \mathit{tan}\theta & \mathit{cos}\phi \mathit{tan}\theta \\ 0& \mathit{cos}\phi & -\mathit{sin}\phi \\ 0& \mathit{sin}\phi /\mathit{cos}\theta & \mathit{cos}\phi /\mathit{cos}\theta \end{array}\right] \end{gathered}$$

(3)

2.2. Dynamic Modeling

The dynamic model addresses the forces and moments acting on the vehicle during underwater motion. When the vehicle moves underwater, ignoring the effects of the Earth’s rotation and the deformation of the vehicle itself, its underwater motion can be considered as rigid body motion. Based on Newton’s laws of linear momentum and angular momentum, the dynamic equations of the vehicle can be derived [16], as shown in

$${\mathit{M}}_{\mathit{R}\mathit{B}}\dot{\mathit{V}}+{\mathit{C}}_{\mathit{R}\mathit{B}}\left(\mathit{V}\right)\mathit{V}={\mathit{\tau }}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}+\mathit{\tau }+\mathit{g}\left(\mathit{\eta }\right)$$

(4)

where $\mathit{V}={\left[\begin{array}{cccccc}u& v& w& p& q& r\end{array}\right]}^T$ represents the motion variables of the vehicle in the body-fixed coordinate system, ${\mathit{M}}_{\mathit{R}\mathit{B}}$ is the mass matrix of the vehicle, ${\mathit{C}}_{\mathit{R}\mathit{B}}\left(\mathit{V}\right)$ is the Coriolis and centripetal matrix, ${\mathit{\tau }}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}$ is the hydrodynamic force, $\mathit{\tau }$ is the control force, and $\mathit{g}\left(\mathit{\eta }\right)$ represents the hydrostatic force.

The mass matrix and the Coriolis and centripetal matrix can be simplified based on the actual vehicle (REMUS100), as shown in

$${\mathit{M}}_{\mathit{R}\mathit{B}}=\left[\begin{array}{cc}diag\left(m,m,m\right)& 0\\ 0& {\mathit{I}}_{\mathit{o}}\end{array}\right]$$

(5)

$${\mathit{C}}_{\mathit{R}\mathit{B}}\left(\mathit{V}\right)=\left[\begin{array}{cccccc}0& 0& 0& 0& mw& -mv\\ 0& 0& 0& -mw& 0& mu\\ 0& 0& 0& mv& -mu& 0\\ 0& -mw& mv& 0& -I_{yz}q-I_{xz}p+I_{z}r& I_{yz}r+I_{xy}p-I_{z}q\\ mw& 0& -mu& I_{yz}q+I_{xz}p-I_{z}r& 0& -I_{xz}r-I_{xy}q+I_{x}p\\ -mv& mu& 0& -I_{yz}r-I_{xy}p+I_{y}q& I_{xz}r+I_{xy}q-I_{x}p& 0\end{array}\right]$$

(6)

where $m$ represents the mass of the vehicle, and ${\mathit{I}}_0$ is the moment of inertia matrix of the vehicle relative to the origin of the body-fixed coordinate system, as shown in

$${\mathit{I}}_0=\left[\begin{array}{ccc}{I}_{xx}& -I_{xy}& -I_{xz}\\ -I_{yx}& I_{yy}& -I_{yz}\\ -I_{zx}& -I_{zy}& I_{zz}\end{array}\right]$$

(7)

Since the vehicle nearly has two planes of symmetry, the off-diagonal elements of the moment of inertia matrix can be approximated as zero. The remaining inertia tensor and some of the vehicle mass parameters are shown in

Table 1. Partial vehicle parameters [17].

Notation	Value	Unit
m	30.51	kg
G	299	N
B	299	N
$I_{xx}$	$0.177$	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
$I_{yy}$	$3.45$	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
$I_{zz}$	$3.45$	$\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$
$x_g$	0	mm
$y_g$	0	mm
$z_g$	19.6	mm

; the model parameters are based on REMUS100.

In Table 1, G represents the weight, B represents the buoyancy of the vehicle, and $x_g,y_g,z_g$ represents the position of the center of weight relative to the body-fixed coordinate system.

According to Fossen’s hydrodynamic model, the hydrodynamic forces acting on the vehicle can be divided into the inertial hydrodynamic force and the viscous hydrodynamic force [18], as represented by

$${\mathit{\tau }}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}=-{\mathit{M}}_{\mathit{A}}\dot{\mathit{V}}-{\mathit{C}}_{\mathit{A}}\left(\mathit{V}\right)\mathit{V}-\mathit{D}\left(\left|\mathit{V}\right|\right)\mathit{V}$$

(8)

In Equation (8), ${\mathit{M}}_{\mathit{A}}$ is the added mass matrix, ${\mathit{C}}_{\mathit{A}}\left(\mathit{V}\right)$ is the added Coriolis and centripetal matrix, and $\mathit{D}\left(\left|\mathit{V}\right|\right)$ is the viscous damping matrix.

Since the vehicle discussed in this paper is approximately symmetric about the XY and XZ planes in the body-fixed coordinate system, the inertial hydrodynamic force, the viscous hydrodynamic force, and the hydrostatic force can be simplified as shown in

$${\mathit{M}}_{\mathit{A}}\dot{\mathit{V}}+{\mathit{C}}_{\mathit{A}}\left(\mathit{V}\right)\mathit{V}=\left[\begin{array}{c}{X}_{\dot{u}}\dot{u}+Z_{\dot{w}}wq+Z_{\dot{q}}{q}^2-Y_{\dot{v}}vr-Y_{\dot{r}}{\dot{r}}^2\\ Y_{\dot{v}}\dot{v}+Y_{\dot{r}}\dot{r}+X_{\dot{u}}ur-Z_{\dot{w}}wp-Z_{\dot{q}}pq\\ Z_{\dot{w}}\dot{w}+Z_{\dot{q}}\dot{q}-X_{\dot{u}}uq+Y_{\dot{v}}vp+Y_{\dot{r}}rp\\ K_{\dot{p}}\dot{p}\\ M_{\dot{w}}\dot{w}+M_{\dot{q}}\dot{q}-\left(Z_{\dot{w}}-X_{\dot{u}}\right)uw-Y_{\dot{r}}vp+\left(K_{\dot{p}}-N_{\dot{r}}\right)rp-Z_{\dot{q}}uq\\ N_{\dot{v}}\dot{v}+N_{\dot{r}}\dot{r}-\left(X_{\dot{u}}-Y_{\dot{v}}\right)uv+Z_{\dot{q}}wp-\left(K_{\dot{p}}-M_{\dot{q}}\right)pq+Y_{\dot{r}}ur\end{array}\right]$$

(9)

$$\mathit{D}\left(\left|\mathit{V}\right|\right)\mathit{V}=\left[\begin{array}{cccccc}{X}_{\mathrm{uu}}\left|u\right|u& 0& 0& 0& 0& 0\\ 0& Y_{\mathrm{vv}}\left|v\right|v& 0& 0& 0& 0\\ 0& 0& Z_{\mathrm{ww}}\left|w\right|w& 0& 0& 0\\ 0& 0& 0& K_{\mathrm{pp}}\left|p\right|p& 0& 0\\ 0& 0& 0& 0& M_{\mathrm{qq}}\left|q\right|q& 0\\ 0& 0& 0& 0& 0& N_{\mathrm{rr}}\left|r\right|r\end{array}\right]$$

(10)

$$\mathit{g}\left(\mathit{\eta }\right)=\left[\begin{array}{c}(G-B)\mathit{sin}\theta \\ -(G-B)\mathit{cos}\theta \mathit{sin}\phi \\ -(G-B)\mathit{cos}\theta \mathit{cos}\phi \\ (z_{g}G-z_{b}B)\mathit{cos}\theta \mathit{sin}\phi -(y_{g}G-y_{b}B)\mathit{cos}\theta \mathit{cos}\phi \\ (z_{g}G-z_{b}B)\mathit{sin}\theta +(x_{g}G-x_{b}B)\mathit{cos}\theta \mathit{cos}\phi \\ -(x_{g}G-x_{b}B)\mathit{cos}\theta \mathit{sin}\phi -(y_{g}G-y_{b}B)\mathit{sin}\theta \end{array}\right]$$

(11)

The definition and assignment of hydrodynamic coefficients in Equations (9) and (10) are listed in Section 4.1; $x_b,y_b,z_b$ in Equation (11) represents the position of the center of buoyancy relative to the body-fixed coordinate system, which are equal to zero.

2.3. Thrust Force Model and Design of a Three-Position Discrete Control Cross Rudder

The thrust force $\mathit{\tau }={\left[\begin{array}{cccccc}{X}_{prop}& 0& 0& X_{fin}& Y_{fin}& Z_{fin}\end{array}\right]}^T$ in Equation (3) can be divided into the propulsive force $X_{prop}$ generated by the thruster and the control force exerted by the rudder. The rudder’s control force can further be separated into drag $X_{fin}$ and lift $Y_{fin}$ and $Z_{fin}$. The lift force generated by a single rudder can also be expressed as $L_{fin}.$ Since the propulsive force, the drag of the rudder, and the axial drag acting on the vehicle itself cancel each other out, represented numerically as the vehicle’s axial velocity $u$, the thrust model only needs to account for the lift force and lift moment generated by the rudder, as indicated in

$$\begin{gathered} L_{fin}={\displaystyle \frac{1}{2}}\rho c_L{S}_{fin}{\delta }_e{v}_e^2 \\ {M}_{fin}=x_{fin}{L}_{fin} \end{gathered}$$

(12)

where $M_{fin}$ represents the moment produced by the control surface of the vehicle, $\rho$ represents the fluid density, $c_L$ denotes the lift coefficient of the rudder, $S_{fin}$ is the surface area of the rudder, ${\delta }_e$ is the rudder angle expressed in radians, $v_e$ is the effective velocity of the rudder, and $x_{fin}$ denotes the axial distance between the rudder’s rotational axis and the center of buoyancy.

The system consisting of the thruster and the cross rudder, which handles surge force, pitch and yaw movements, can be described as an underactuated system. Under these conditions, it is unable to independently perform lateral, vertical, and roll movements. Therefore, when the vehicle is operating in a static fluid, the effective speed can be represented by the vehicle’s axial velocity $u$. Based on hydrodynamic theory, the shape of the vehicle and the position of the rudder panels have a nearly linear effect on the effective speed of the rudder, so the rudder’s effective velocity $v_e$ can be expressed as

$$v_e=eu$$

(13)

where $e$ is the coefficient of the rudder′s effective speed.

For the purposes of reliability and cost reduction, this paper designs a binary-controlled three-position cross rudder. The mechanism is based on electromagnets positioned on both sides of the rudder plate, which attract magnetic fins located at the edges of the rudder plate, thereby controlling its direction. As shown in Figure 2, since the electromagnet operates in discrete states, a single rudder plate can assume three positions: positive, neutral, and negative. The absolute values of the rudder angles for the positive and negative positions are 20°, while the rudder angle for the neutral position is 0°. Additionally, the horizontal rudder is defined as positive when directed upwards and negative when directed downwards, whereas the vertical rudder is positive when directed to the right and negative when directed to the left. These directions are based on the YZ plane of the body-fixed coordinate system.

By adjusting the position states of the horizontal and vertical rudders, the cross rudder generates two degrees of freedom moments, enabling pitch and yaw control of the vehicle. However, since the rudder angles are restricted to values of 20°, 0°, and −20°, the control forces and moments produced by the rudders result in only nine fixed combinations during motion control. Consequently, the vehicle can only be maneuvered in nine predetermined directions. Figure 3 illustrates the correspondence between the rudder position states and the control directions. The control directions are defined relative to the YZ plane of the body-fixed coordinate system and are opposite to the directions of the rudder forces.

While designing the vehicle’s motion control scheme, hardware limitations restrict the control of the cross rudder to a binary control method with switching states. This approach inherently introduces unavoidable overshoot in the system, making it necessary to design an additional controller to mitigate this overshoot.

3. Virtual Rudder Angle Controller Design

This section designs a virtual rudder angle controller based on anti-normalization of virtual rudders and PWM frequency conversion theory. This controller is used to regulate the three-position cross rudder with binary control, addressing the overshoot problem in the vehicle’s motion control.

3.1. Theoretical Basis

3.1.1. Anti-Normalization Virtual Rudder

The anti-normalization virtual rudder is a concept used in traditional cross rudder vehicle attitude control [19]. The principle behind the controller design involves inputting the real-time attitude of the vehicle from sensors, applying the control law to determine the required virtual rudder angle, and then outputting weighted parameters to the rudder actuator. This controls the stroke of the rudder linkage to achieve rudder angle control, as demonstrated in

$$\begin{gathered} {\delta }_v=CT{R}_{yaw}·{\delta }_{vMAX} \\ {\delta }_h=CT{R}_{pitch}·{\delta }_{hMAX} \\ -1\le CT{R}_{yaw},CT{R}_{pitch}\le +1 \end{gathered}$$

(14)

where ${\delta }_v$ and ${\delta }_h$ represent the actual rudder angles for the vertical and horizontal rudders, $CT{R}_{yaw}$ and $CT{R}_{pitch}$ are the weighted parameters of the rudder angles output to the actuators for yaw and pitch control, and ${\delta }_{vMAX}$ and ${\delta }_{hMAX}$ refer to the maximum allowable rudder angles for the vertical and horizontal rudders.

The essence of the anti-normalization virtual rudder lies in controlling the torque output of the rudder actuator according to the weighted parameters, ensuring that the real rudder angle matches the virtual rudder angle input by the controller. The controller designed in this paper adopts the concept of weighted parameters from the anti-normalization virtual rudder. The controller input, referred to as the desired rudder angle, replaces the weighted parameters output to the rudder actuators with the PWM duty cycle output to the rudder-driving electromagnet. During each control cycle, the rudder effectiveness corresponding to the desired rudder angle is achieved by adjusting the rudder-driving frequency, which can also be referred to as the virtual rudder angle.

3.1.2. PWM Frequency Conversion

PWM frequency conversion technology is generally used to control the speed and torque of electric motors. By adjusting the duty cycle of the PWM signal, the output voltage and current can be modified, thereby regulating the operating characteristics of the motor [20]. The output duty cycle $DC$ is expressed by

$$DC={\displaystyle \frac{T_h}{T}}\times 100\mathrm{\%}$$

(15)

where $T_h$ represents the duration of the high level within a single cycle, and $T$ represents the duration of one complete cycle.

3.2. Controller Design

The rudder effectiveness can be approximately regarded as the lift generated by the rudder on the vehicle. According to the principle of momentum conservation, the impulse generated by the rudder on the vehicle is the product of the rudder lift and the rudder operation time, which is expressed by

$$J=t{L}_{fin}$$

(16)

where $J$ is the impulse generated by the rudder on the vehicle, and $t$ is the rudder operation time.

When the absolute value of the required rudder angle is less than the maximum rudder angle of the three-position cross rudder, a virtual rudder angle controller can be designed based on the principle of impulse equivalence. Within each PWM cycle, the impulse generated by the rudder under virtual rudder angle frequency control is numerically equal to the impulse generated by the rudder under the real rudder angle. When the cycle period is short, the rudder effectiveness under the control of the virtual rudder angle controller can be considered equal to the rudder effectiveness under the real rudder angle. The controller design is shown in

$$\begin{gathered} J_{\delta }=L_{\delta }T=D{C}_{{\delta }^{\mathrm{*}}}T{L}_{MAX} \\ D{C}_{{\delta }^{\mathrm{*}}}T\ge \Delta \end{gathered}$$

(17)

where $J_{\delta }$ is the impulse generated by the real rudder within a single period, $L_{\delta }$ is the rudder lift at the real rudder angle $\delta$, $D{C}_{{\delta }^{*}}$ is the duty cycle output by the controller at the virtual rudder angle ${\delta }^{*}$, $L_{MAX}$ is the maximum rudder lift, which corresponds to the rudder angle at its maximum value (20° in this paper), $T$ is the cycle period, and $\Delta$ is the time required for the cross rudder to switch one single position.

When the system provides the controller with the desired rudder angle value for the vehicle, the controller converts this virtual rudder angle input into the corresponding PWM duty cycle. This controls the rudder to operate at varying frequencies, achieving the actual rudder angle effect. Consequently, the virtual rudder angle value is numerically equivalent to the actual rudder angle value. The virtual rudder angle value is a virtual quantity in the control system, while the actual rudder angle value is a digital quantity derived from fitting the actual rudder performance.

4. Simulation and Results

This section presents a numerical simulation to validate the motion control effectiveness of the virtual rudder angle controller. It performs comparative simulations of both planar turning radius and spatial path control for an underwater vehicle, contrasting the performance of real and virtual rudders. This approach is aimed at verifying the effectiveness of the virtual rudder angle controller.

4.1. Coefficient Settings

The hydrodynamic coefficients of the underwater vehicle simulation model are defined and assigned as shown in

Table 2. Definition and assignment of hydrodynamic coefficients in the model [17].

Parameter	Description	Value	Unit
$X_{\dot{u}}$	Added Mass	−0.93	$\mathrm{k}\mathrm{g}$
$Y_{\dot{v}}$	Added Mass	−35.5	$\mathrm{k}\mathrm{g}$
$Y_{\dot{r}}$	Added Mass	1.93	$\mathrm{k}\mathrm{g}·\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$
$Z_{\dot{w}}$	Added Mass	−35.5	$\mathrm{k}\mathrm{g}$
$Z_{\dot{q}}$	Added Mass	−1.93	$\mathrm{k}\mathrm{g}·\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$
$K_{\dot{p}}$	Added Mass	−0.0013	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}\mathrm{d}$
$M_{\dot{w}}$	Added Mass	−1.93	$\mathrm{k}\mathrm{g}·\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$
$M_{\dot{q}}$	Added Mass	−4.88	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}\mathrm{d}$
$N_{\dot{v}}$	Added Mass	1.93	$\mathrm{k}\mathrm{g}·\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$
$N_{\dot{r}}$	Added Mass	−4.88	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}\mathrm{d}$
$X_{uu}$	Axial Drag	−1.62	$\mathrm{k}\mathrm{g}/\mathrm{m}$
$Y_{\mathrm{vv}}$	Lateral Drag	−131	$\mathrm{k}\mathrm{g}/\mathrm{m}$
$Z_{\mathrm{ww}}$	Vertical Drag	−131	$\mathrm{k}\mathrm{g}/\mathrm{m}$
$K_{pp}$	Roll Drag	−0.0141	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}{\mathrm{d}}^2$
$M_{qq}$	Pitch Drag	−9.4	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}{\mathrm{d}}^2$
$N_{rr}$	Yaw Drag	−9.4	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2/\mathrm{r}\mathrm{a}{\mathrm{d}}^2$
$\rho$	Fluid Density	1030	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
$S_{fin}$	Rudder Planform Area	6650	$\mathrm{m}{\mathrm{m}}^2$
$x_{fin}$	Moment Arm	0.638	$\mathrm{m}$
$c_L$	Rudder Lift Coefficient	3.12	$\mathrm{n}/\mathrm{a}$
$e$	Effective Speed Coefficient	0.95	$\mathrm{n}/\mathrm{a}$

.

4.2. Planar Turning Simulation Comparison

The most direct method to examine rudder effectiveness is to verify whether the turning radius of the vehicle under real rudder angles and virtual rudder angles is similar [21]. In the simulation model, the axial velocity $u$ is set to 4 knots, and the initial point is (0, 0, 10) in meters. The rudder angles are set to 1°, 5°, 10°, and 15° for the simulations. Then, a virtual rudder angle controller is added to the rudder output, and the corresponding virtual rudder angles are input. Finally, the planar motion trajectories of the vehicle under real rudder control and virtual rudder control are compared, as shown in Figure 4, and the comparison of turning radii is shown in

Table 3. Comparison of turning radius between real rudder and virtual rudder under different rudder angle values.

Rudder Angle Values/°	Average Turning Radius of Real Rudder/m	Average Turning Radius of Virtual Rudder/m
1	9.73	9.74
5	4.38	4.40
10	3.11	3.07
15	2.57	2.55

.

As can be seen from Figure 4 and Table 3, when the real rudder angle is replaced by the virtual rudder angle’s frequency control, the turning radius of the vehicle remains approximately equal. Therefore, it can be concluded that the virtual rudder angle obtained through frequency control by the virtual rudder angle controller is approximately equivalent to the real rudder angle controlled by the servo in terms of rudder effectiveness.

4.3. Space Path Motion Control

The motion control system of the vehicle is divided into two parts: attitude control and trajectory tracking control [22]. Both utilize the traditional PID control method. The control block diagram is shown in Figure 5.

As is shown in Figure 5, the control principle involves generating the desired position ${{\eta }_1}^{*}$ from the set path points, which, together with the actual position ${\eta }_1$, is input into the path controller to produce the desired attitude angles ${\theta }^{*},{\psi }^{*}$ for the vehicle. The compass and gyroscope provide the actual attitude angles, which are compared with the desired attitude angles to generate an attitude angle error. This error is then input into the attitude controller, which outputs the desired rudder angles ${\delta }_v,{\delta }_h$. Finally, the desired rudder angles are fed into the vehicle model to complete the motion control of the vehicle.

As a comparative simulation, a virtual rudder controller is added at the output end of the attitude control. This controller outputs the PWM duty cycle $D{C}_v,D{C}_h$ to vehicle model, as shown in Figure 6.

After completing the construction of the control system, the initial position of the vehicle is set to A0 (0, 0, 180) (m). The path points are sequentially set as A1 (0, 300, 150), A2 (600, 600, 150), A3 (0, 900, 150), and A4 (600, 1200, 150), with an axial velocity of 4 knots. Since the planar turning simulation has confirmed that the rudder effectiveness of the virtual rudder is nearly identical to that of the real rudder, the space path motion control simulation primarily focuses on the impact of the virtual rudder angle controller on the motion control system. In fact, the underwater environment does not affect the comparison simulation between the virtual rudder and the real rudder. Therefore, the simulation sets the ocean current velocity to zero, assuming that motion control is performed in a static fluid.

Simulations are carried out using two different control methods for comparison. During the simulation, the values of the vehicle’s pitch and yaw angles over time are recorded, and the comparison results are shown in Figure 7.

From Figure 7, it can be observed that the virtual rudder and the real rudder have nearly equivalent effects on the vehicle’s attitude control. Compared to the real rudder, the vehicle controlled by the virtual rudder exhibits slight oscillations during motion. This phenomenon is attributed to overshoot caused by the variable frequency control of the rudder surfaces, which is a limitation of the vehicle’s hardware. Reducing the effective area of the rudder surfaces can decrease the lift moment of the vehicle, thereby mitigating these oscillations by reducing the turning speed, but it will also affect the vehicle’s maneuverability, which means the rapidity of motion control. Therefore, it is necessary to strike a balance between the rapidity and stability of motion control based on the actual operating conditions of the vehicle.

In the same comparative simulation mentioned above, Figure 8 shows a comparison of the vehicle’s spatial motion paths. Figure 9 presents projections of the three-dimensional spatial motion path onto the XY plane and YZ plane. Figure 10 illustrates the projections of the vehicle’s positions along the X-axis, Y-axis, and Z-axis over time.

From Figure 8 and Figure 9, it can be observed that both control methods enable the vehicle to follow the planned trajectory effectively, with the actual path closely aligning with the desired path. As the vehicle approaches the path points, an upward overshoot occurs due to its fluid dynamic characteristics [12]. However, the path control quickly corrects this deviation, bringing the vehicle back into close alignment with the desired path.

From the comparison of the two control methods in Figure 10, it can be seen that the virtual rudder is close to the real rudder in terms of control accuracy. However, due to slight oscillations in the vehicle’s motion under virtual rudder control, a time-domain lag is observed.

The comparison of attitude angles and path simulation results further confirms that the virtual rudder, controlled by the variable frequency controller designed in this study, exhibits a similar steering effect to that of the real rudder controlled by the servos. It indicates that the servo replacement scheme proposed in this study is effective.

5. Conclusions

In order to address the high cost associated with using servos for steering in low-cost small underwater vehicles, this paper designs a three-position cross rudder driven by a binary-controlled electromagnet with switching states. Additionally, it introduces a virtual rudder angle controller that combines PWM frequency modulation technology, anti-normalization virtual rudder theory, and momentum conservation theory. By using frequency-modulated rudder driving, it successfully simulates controllable rudder angles, resolving the issue of constant rudder angles when driven by electromagnets. To verify the effectiveness of the design, comparative simulations between real and virtual rudders were conducted, focusing on planar turning radius and spatial path motion control. The simulation results indicate that the virtual rudder angle controller closely approximates the performance of the real rudder in controlling vehicle motion.

The designed three-position cross rudder and virtual rudder angle controller offer a cost-effective servo replacement scheme. This lays a foundation for reducing the costs associated with small underwater vehicles. Upon evaluation, the three-position cross rudder designed in this paper costs between 10% and 30% of what a real servo motor costs, significantly reducing manufacturing expenses. Although the virtual rudder angle controller has proven effective, it exhibits slight oscillations in attitude control compared to the real rudder angle, resulting in a time-domain lag. Thus, applied research will focus on balancing the rapidity and stability of motion control. Furthermore, the influence of the underwater environment on the motion control is ignored in this paper. In future research, a deeper investigation into the underwater environment will be conducted, and structural improvements and controller design will be implemented to realize the practical application of the findings presented in this paper. Additionally, the current dynamic model overlooks the interaction between the propeller and the rudder [23]. Integrating this interaction into the control system is essential for enhancing the system’s control accuracy.