Underactuated tracking control of underwater vehicles using control moment gyros

Abstract

This article studies the trajectory tracking control of underactuated underwater vehicles using control moment gyros through a biologically inspired approach based on homeomorphism transformation and Lyapunov functions in the horizontal plane. First, a series of assumptions and simplifications need to be made to build the kinematic and dynamic equations of the underwater vehicle under a single-frame pyramid configuration structured with four control moment gyros. Second, the error dynamics analysis of the submarine based on the control moment gyros is derived from the equations, and a tracking control algorithm is proposed to demonstrate the feasibility and stabilization of this tracking control scheme from theoretical analysis. Finally, the numerical simulation results are given for verifying the effectiveness and feasibility of the rendered control law.

Introduction

There is no doubt that the development and protection of the ocean cannot be separated from research on marine engineering equipment. Study of underwater robots is an active field and may have a far-reaching impact on human beings. ¹ As one kind of important underwater robots, autonomous underwater vehicles (AUVs) should be able to be moved freely and manipulated conveniently. As internal control actuator devices, control moment gyros (CMGs) have been widely used in spacecraft and satellites. ^2,3 Despite their singularities and complicated nature, CMG systems can provide higher torque and large momentum to meet the typical requirements of the reactions of wheels. ⁴ In particular, whether the speed of the vehicle is high or low, CMGs system can be appropriate for the attitude control, especially under zero momentum condition. In other words, the CMG-based underwater vehicle possesses a capability of zero-radius gyration under stationary state, so it can make a difference for maneuverability in narrow underwater space compared to the traditional underwater robots. ⁵ Since Blair Thornton proposed Zero-G class AUVs using CMGs, a new class of research on AUVs that are able to accomplish attitude control unrestrictedly to some extent has been opened. ^5,6 Using CMGs as a new type of underwater actuator can be used to develop a novel control scheme, with a zero-turning-radius circle manipulated on the surface of a sphere based on the internal momentum exchange principle. However, the author did not consider the issue of trajectory tracking and formation control for AUVs using the CMGs.

As a fact and also implied by Brockett’s theorem (Brockett 1983), ⁷ any smooth or continuous time-invariant feedback control does not make the solution of underactuated vehicle dynamics asymptotically stable, the stabilization of planar motion and tracking control has gradually become an active field. ⁸ Unlike traditional underwater actuators such as thrusters, fins, and rudders, the CMG-based underwater vehicle can preserve the hydrodynamic integrity of its hull and be free of erosion or damage caused by harsh underwater environments or small creatures. Based on this, it makes the response speed independent of the interaction between fluid and its attitude control actuator. The above properties will make the CMG-based underwater vehicle possess wider application fields such as marine rescues, military scouting, and special operations. Recently, finite-time state feedback and output feedback for stabilization have been studied from different perspectives. ⁹ Moreover, since homogeneity, σ-transformation and backstepping, and other methods have gradually been introduced in the study of the stabilization control problem, such as in observer-based backstepping control ^10,11 and neural networks, ^{12 –14} nonlinear model predictive control ¹⁵ has begun to include state and input constraints, adaptive fuzzy logic systems, ¹⁶ and sliding mode control. ^{17 –19} Sliding mode approach for path tracking of the unmanned agricultural tractor has been studied under restricted slipping effects and control saturation. ²⁰ In addition, the finite-time trajectory tracking of an autonomous surface vehicle ^21,22 has been achieved by adding a power integrator and a homogeneous-like controller in the research of Hong et al., ²³ therein achieving finite-time tracking using the homogeneous theory. It should be noted that for the trajectory-tracking problem of mobile robot, in his study Sun ²⁴ utilizes a dynamic extension approach of the differential geometry control theory to fulfill the exact feedback linearization on the kinematic error equation. A distributed finite-time consensus tracking control for second-order multiagent systems has been developed based on a new class of observer-based control algorithms in the literature by He and Wang. ²⁵ In order to reinforce the control of inexact model, a kernel-based regression learning method is applied, in the literature by Lou and Guo, ²⁶ to estimate the inaccurate system parameters. In addition, incomplete symmetry underactuated USV tracking control ²⁷ has been developed using a cascade system. Unfortunately, most underactuated systems are developed using a sufficiently simplified model, and chatting phenomena may not be mitigated by the proposed control strategy.

Motivated by the above results, this article addresses a biologically inspired model used in the construction of a Lyapunov function approach for underactuated multidimensional systems, describing underwater vehicles using CMGs for trajectory tracking on the horizontal plane. Accordingly, this vehicle, containing a propulsion propeller coupled with CMGs system, should have the ability to control the three-axis attitude and move forward. Predictably, its underwater trajectory tracking is an interesting problem worthy to be discussed and studied. The control law involved in the design and analysis is based on backstepping, Lyapunov functions, and shunting neurodynamics models inspired by Hodgkin and Huxley’s membrane equation, ²⁸ therein using the biological neural system to replace the virtual control inputs to avoid differentiating the desired items. Nevertheless, it should be noted that the actual control strategies in the horizontal or vertical plane are not highly different.

The reminder of this article is organized as follows. In the “Preliminaries and problem formulation” section, we recall some assumptions on the models of the underactuated equation and use the homeomorphic transformation to modify the dynamical system. The “Trajectory tracking control law” section is devoted to designing a reasonable control scheme using the shunting model based on Lyapunov functions for this problem. In the “Simulation studies” section, the simulation results are presented to demonstrate the effectiveness and robustness of the proposed controller. Finally, conclusions and further considerations for the trajectory tracking control of this underwater vehicle using CMGs are provided in the “Conclusion” section.

Preliminaries and problem formulation

In this section, we consider the underwater vehicle using CMGs that is presented in Figure 1, where the square pyramid configuration is composed of the four-single gimbal control moment gyro (SGCMG) cluster and four CMGs located at the side-center position of a square pyramid, with gimbal axes perpendicular to the bottom face (Figure 2). It is important to note that Kurokawa studied the momentum envelope of the pyramid CMGs system by differential geometric theory and found that a third of its internal singular surface is passable. ²⁹ The kinematic and dynamic models of the underwater vehicle using the CMGs moving in the horizontal plane are presented, and a homeomorphic transformation is used to achieve a better control design and objective.

Figure 1.

Small underwater vehicle using CMGs. CMG: control moment gyro.

Figure 2.

Four-CMG cluster in a pyramid configuration. CMG: control moment gyro.

Through the Newton–Euler formulation and laws of conservation of linear and angular momentum, a dynamic model describing underactuated underwater vehicles using CMGs can be built. Decoupling the complex nonlinear system, coinciding with the kinematics, we can obtain the dynamical system in the horizontal plane

$\{\begin{array}{l}\dot{\eta }=\text{R}(\psi )\nu \hfill \\ \text{M}\dot{\nu }+C(\nu )\nu +D(\nu )\nu +{\tau }_{\text{dis}}=\tau \hfill \end{array}$ 1

where $\eta =[x,y,\psi {]}^T\in {ℝ}^3$ and $\nu =[u,v,r{]}^T\in {ℝ}^3$ , and $\tau =[{\tau }_1\mathrm{,0},{\tau }_3{]}^T\in {ℝ}^3$ is the control input and R(ψ) is a rotation matrix given by

$\text{R}(\psi )=\left[\begin{array}{ccc}\text{cos}\psi & -\text{sin}\psi & 0\\ \text{sin}\psi & \text{cos}\psi & 0\\ 0& 0& 1\end{array}\right]$ 2

with the following properties: ${\text{R}}^T(\psi )\text{R}(\psi )=$ I, $\parallel \text{R}(\psi )\parallel =1$ , and $\dot{\text{R}}(\psi )=\text{R}(\psi )S(r)$ . Here,

$S(r)=\left[\begin{array}{ccc}0& -r& 0\\ r& 0& 0\\ 0& 0& 0\end{array}\right]$ 3

Take P = [x, y] ^T as the vertical position in the inertial frame and ψ as the heading angle measured from the inertial axes. u, v, and r describe the surge, sway, and yaw velocities in the body-fixed frame, respectively; M and C(ν) are, respectively, the inertia matrix and the Coriolis and centripetal matrix (including the added mass); D(ν) is the drag matrix; and τ _dis is the disturbance vector. Notice that τ ₁ is the control input actuated by the propeller, and τ ₃ is the yaw moment provided by the CMGs. The following assumptions are made for the underwater vehicle to simplify the dynamical system and support the control scheme; in addition, some work must be performed beforehand to design the controller.

Assumption 1

The disturbance vector τ _dis is ignored, namely, τ _dis = 0.

Then, the simplified dynamical system (equation (1)), based on the above assumption and studies, ^4,6 can be written as

$\dot{u}=\frac{1}{m_{11}}(X_{u\left|u\right|}u|u|+X_{vr}vr+X_{rr}{r}^2)+\frac{1}{m_{11}}{\tau }_1$ 4 $\dot{v}=\frac{m_{33}}{\Delta }f(v)-\frac{m_{23}}{\Delta }f(r)-\frac{m_{23}}{\Delta }{\tau }_3$ 5 $\dot{r}=\frac{m_{22}}{\Delta }f(r)-\frac{m_{32}}{\Delta }f(v)+\frac{m_{22}}{\Delta }{\tau }_3$ 6

where $\Delta =m_{22}{m}_{33}-m_{23}{m}_{32}$ and $f(v)=Y_{v\left|v\right|}v\left|v\right|+Y_{r\left|r\right|}r\left|r\right|+Y_{ur}ur+Y_{uv}uv$ ,    $f(r)=N_{v\left|v\right|}v\left|v\right|+N_{r\left|r\right|}r\left|r\right|+N_{ur}ur+N_{uv}uv$ . Here, $m_{11}=m-X_{\dot{u}}$ , $m_{22}=m-Y_{\dot{v}}$ , $m_{23}=-Y_{\dot{r}}$ , $m_{32}=-N_{\dot{v}}$ , $m_{33}=I_z-N_{\dot{r}}$ , ${\tau }_3=-h\text{sin}\beta {\dot{\delta }}_4\text{cos}{\delta }_4$ , m is the mass of the vehicle, I_z is the moment of inertia about the yaw rotation, β = 54.73^° denotes the pyramid skew angle, ³⁰ h = 20 Nms is the magnitude of the angular momentum of each gyro rotor, and δ ₄ is the fourth CMG rotation angle about the axle h ₄ (see Figure 1).

Assumption 2

Similar to Behal et al. (2001) ³¹ , there exist bounded constraints, such as $\left|{\tau }_1\right|\le {\overline{\tau }}_1$ , $\left|{\tau }_3\right|\le {\overline{\tau }}_3$ , $\left|u\right|\le \overline{u}$ , $\left|v\right|\le \overline{v}$ , $\left|r\right|\le \overline{r}$ , and heading angle ψ ∈ [0, 2π], for the control inputs and velocities to ensure an effective control strategy. Here, ${\overline{\tau }}_1$ , ${\overline{\tau }}_3$ , $\overline{u}$ , $\overline{v}$ , and $\overline{r}$ are positive constants.

Consider a desired sufficiently smooth trajectory path $P_d=[x_d,y_d{]}^T$ and the tracking errors ${[x_e,y_e]}^T=[x-x_d,y-y_d{]}^T$ , ${\psi }_e=\psi -{\psi }_d$ . In addition, define ${\phi }_e=\psi -{\varphi }_d$ , where ${\varphi }_d=\text{arctan}(\frac{{\dot{y}}_d}{{\dot{x}}_d})$ , and from the relationship between the inertial and body-fixed frames, there exists the following coordinate transformation

$\left[\begin{array}{c}{e}_x\\ e_y\end{array}\right]=\left[\begin{array}{cc}\text{cos}\psi & \text{sin}\psi \\ -\text{sin}\psi & \text{cos}\psi \end{array}\right]\left[\begin{array}{c}{x}_e\\ y_e\end{array}\right]$ 7

Use the properties above equation (3) to take the time derivative of equation (7) and considering the kinematical systems, we can obtain

$\left[\begin{array}{c}{\dot{e}}_x\\ {\dot{e}}_y\end{array}\right]=\left[\begin{array}{c}u+r{e}_y-v_p\text{cos}{\phi }_e\\ v-r{e}_x+v_p\text{sin}{\phi }_e\end{array}\right]$ 8

Here, we have the following equivalence relations

$\{\begin{array}{c}{v}_p\text{cos}{\phi }_e=u_d\text{cos}{\psi }_e+v_d\text{sin}{\psi }_e\\ v_p\text{sin}{\phi }_e=u_d\text{sin}{\psi }_e-v_d\text{cos}{\psi }_e\end{array}$ 9

Therefore, the tracking error vector ${[x_e,y_e]}^T=0$ under the inertial frame is equivalent to the error vector under the body-fixed frame ${[e_x,e_y]}^T=0$ . For the sake of designing a control scheme to make the vector become zero, similar to Pan et al., ¹³ an assumption about the desired path needs to be made.

Assumption 3

The desired path $P_d:[0,\infty ]\to {ℝ}^2$ is sufficiently smooth, and its time derivatives are bounded.

For the complex system in equations (4) to (6), notice that the control inputs τ ₁ and τ ₃ are contained in the three equations. When considering the dynamical error analysis, this may cause numerous issues. Consider the following homeomorphic transformation

$\{\begin{array}{l}{z}_1=x+\alpha (\text{cos}\psi -1)\hfill \\ z_2=y+\alpha \text{sin}\psi \hfill \\ z_3=\psi \hfill \\ z_4=u\hfill \\ z_5=v+\alpha r\hfill \\ z_6=r\hfill \end{array}$ 10

where $\alpha =\frac{m_{23}}{m_{22}}$ , and the control inputs and the variation in the sway velocity are as follows:

$\{\begin{array}{l}{f}_1=\frac{1}{m_{11}}(\overline{f}(u)+{\tau }_1)\hfill \\ f_2=\frac{1}{m_{22}}\overline{f}(v)\hfill \\ f_3=\frac{m_{22}}{\Delta }\overline{f}(r)-\frac{m_{32}}{\Delta }\overline{f}(v)+\frac{m_{22}}{\Delta }{\tau }_3\hfill \end{array}$ 11

where

$\overline{f}(u)=X_{u\left|u\right|}u\left|u\right|+X_{vr}(v-\alpha r)r+X_{rr}{r}^2$ $\overline{f}(v)=Y_{v\left|v\right|}(v-\alpha r)|v-\alpha r|+Y_{r\left|r\right|}r\left|r\right|+Y_{ur}ur+Y_{uv}u(v-\alpha r)$ $\overline{f}(r)=N_{v\left|v\right|}(v-\alpha r)|v-\alpha r|+N_{r\left|r\right|}r\left|r\right|+N_{ur}ur+N_{uv}u(v-\alpha r)$

For convenience and consistency, we use the previous notation. Then, the system equation (1) becomes

$\{\begin{array}{l}\dot{x}=u\text{cos}\psi -v\text{sin}\psi ,\hfill \\ \dot{y}=u\text{sin}\psi +v\text{cos}\psi ,\hfill \\ \dot{\psi }=r,\hfill \\ \dot{u}=f_1,\hfill \\ \dot{v}=f_2,\hfill \\ \dot{r}=f_3\hfill \end{array}$ 12

Trajectory tracking control law

To continue the following discussion, we shall investigate the model of shunting neural dynamics proposed by Hodgkin and Huxley ²⁸ , which is recommended by Grossberg to understand real time adaptive behaviors and is applied in biological and machine vision, sensor motor control, and other areas. ^13,32,33 A typical shunting model can be written as

${\dot{\xi }}_i=-(A+B{S}_i^{+}(t)+C{S}_i^{-}(t)){\xi }_i+B{S}_i^{+}(t)-C{S}_i^{-}(t)$ 13

where ξ_i is the membrane potential of the ith neuron. The nonnegative constants A, B, and C represent the passive decay rate, and the variables $S_i^{+}(t)$ and $S_i^{-}(t)$ are called the excitatory input and the inhibitory input, respectively. This article combines the shunting model (13) with the backstepping method to design a reasonable control scheme for the new system (12).

Virtual controller design for the kinematics

The major change in the system equations (1) and (12) is that the control input τ ₃ is no longer in the differential equation concerning the sway velocity v, and the control objectives, namely, the surge and yaw velocities u and r, are tightly coupled with the sway velocity; therefore, the main goal is to converge the tracking error vector ${[e_x,e_y]}^T$ and φ_e to a small neighborhood of the origin.

First, consider the body-fixed tracking error vectors e_x and e_y , that is, making the actual path P track the desired path P_d ; then, a positive-definite Lyapunov function is defined as

$V_1=\frac{1}{2}(e_x^2+e_y^2)$ 14

Consider equation (8) and obtain the time derivative ${\dot{V}}_1$

$\begin{array}{c}{\dot{V}}_1=e_x(u+r{e}_y-v_p\text{cos}{\phi }_e)+e_y(v-r{e}_x+v_p\text{sin}{\phi }_e)\\ =e_x(u-v_p\text{cos}{\phi }_e)+e_y(v+v_p\text{sin}{\phi }_e)\end{array}$ 15

To make the Lyapunov function V ₁ be negative, set w = v_p sinφ_e , and the virtual control inputs are assumed to be u and w. Then, the desired virtual control w_d and ${\tilde{u}}_d$ are designed as follows:

$\{\begin{array}{l}{\tilde{u}}_d=-l_1{e}_x+v_p\text{cos}{\phi }_e\hfill \\ w_d=-l_2{e}_y-v\hfill \end{array}$ 16

where l ₁ and l ₂ are positive constants to be selected. Then, based on the shunting model equation (13), we obtain a new control w_f to avoid differentiating w_d

${\dot{w}}_f=-(A_1+B_{1}f(w_d)+C_{1}g(w_d))w_f+B_{1}f(w_d)-C_{1}g(w_d)$ 17

where A ₁ is a nonnegative constant and B ₁ and C ₁ are the upper and lower bounds of w_f , respectively. The functions f(x) and g(x) are defined as f(x) = max{x, 0} and g(x) = max{−x, 0}. Then, we obtain new error variables δ_w and ε_w :

${\delta }_w=w_f-w_d\mathrm{,} {\epsilon }_w=w-w_f$

Substituting w = w_d + (δ_w + ε_w ) into equation (15) yields

${\dot{V}}_1=-l_1{e}_x^2-l_2{e}_y^2+{\delta }_w{e}_y+{\epsilon }_w{e}_y$ 18

Second, consider the orientation error φ_e and the virtual input error δ_w . Regard the yaw velocity r as a new virtual control input and take the time derivative of ε_w . We obtain

$\begin{array}{c}{\dot{\epsilon }}_w=\dot{w}-{\dot{w}}_f\\ ={\dot{v}}_p\text{sin}{\phi }_e+v_p(r-{\dot{\varphi }}_d)\hspace{0.17em}\text{cos}{\phi }_e-{\dot{w}}_f\end{array}$ 19

Then, we define the second Lyapunov function V ₂ as

$V_2=V_1+\frac{1}{2}({\epsilon }_w^2+{\phi }_e^2)$ 20

Differentiating the function V ₂ yields

$\begin{array}{c}{\dot{V}}_2={\dot{V}}_1+{\epsilon }_w{\dot{\epsilon }}_w+{\phi }_e{\dot{\phi }}_e\\ =-l_1{e}_x^2-l_2{e}_y^2+{\delta }_w{e}_y+{\epsilon }_w(e_y+{\dot{v}}_p\text{sin}{\phi }_e-{\dot{w}}_f)+(r-{\dot{\varphi }}_d)({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)\mathrm{.}\end{array}$ 21

Introduce the virtual control yaw velocity r, and assume its desired control input as

${\tilde{r}}_d=-l_3({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)+{\dot{\varphi }}_d$ 22

${\dot{V}}_2$ becomes

${\dot{V}}_2=-l_1{e}_x^2-l_2{e}_y^2-l_3{({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)}^2+{\delta }_w{e}_y+{\epsilon }_w(e_y+{\dot{v}}_p\text{sin}{\phi }_e-{\dot{w}}_f)$ 23

Then, the virtual kinematical control inputs ${\tilde{u}}_d$ in (16) and ${\tilde{r}}_d$ in (22) are used in the following actual dynamical control scheme.

Virtual controller design for dynamics

Based on the virtual controllers u and r, the following section will discuss the actual control inputs τ ₁ and τ ₃ in the dynamics and will realize the design of the trajectory tracking control.

Similar to the control w using the shunting model (17), we obtain new controls u_f and r_f , and the shunting models are

$\begin{array}{l}{\dot{u}}_f=-(A_2+B_{2}f({\tilde{u}}_d)+C_{2}g({\tilde{u}}_d))u_f+B_{2}f({\tilde{u}}_d)-C_{2}g({\tilde{u}}_d)\\ {\dot{r}}_f=-(A_3+B_{3}f({\tilde{r}}_d)+C_{3}g({\tilde{r}}_d))r_f+B_{3}f({\tilde{r}}_d)-C_{3}g({\tilde{r}}_d)\end{array}$ 24

The new error variables are

$\{\begin{array}{c}{\delta }_u=u_f-{\tilde{u}}_d\mathrm{,} {\epsilon }_u=u-u_f\\ {\delta }_r=r_f-{\tilde{r}}_d\mathrm{,} {\epsilon }_r=r-r_f\end{array}$ 25

where the definitions of the constraints A_i , B_i , and C_i (i = 2, 3) are the same as in equation (17). Then, consider the errors above and substitute $u={\tilde{u}}_d+({\delta }_u+{\epsilon }_u)$ and $r={\tilde{r}}_d+({\delta }_r+{\epsilon }_r)$ into equations (15) and (21). The derivative of the function V ₂ becomes

$\begin{array}{c}{\dot{V}}_2=-l_1{e}_x^2-l_2{e}_y^2-l_3{({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)}^2+{\delta }_w{e}_y+e_x({\delta }_u+{\epsilon }_u)\\ + {\epsilon }_w(e_y+{\dot{v}}_p\text{sin}{\phi }_e-{\dot{w}}_f)+({\delta }_r+{\epsilon }_r)({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)\end{array}$ 26

Noting the dynamic equation (12) and differentiating the error variables in (25) yield

$\{\begin{array}{l}{\dot{\epsilon }}_u=f_1-{\dot{u}}_f\hfill \\ {\dot{\epsilon }}_r=f_3-{\dot{r}}_f\hfill \end{array}$ 27

Finally, take the error variables ε_u and ε_r into account and define a new Lyapunov function as follows

$V_3=V_2+\frac{m_{11}}{2}{\epsilon }_u^2+\frac{\Delta }{2m_{22}}{\epsilon }_r^2$ 28

where f ₁, f ₂, and Δ are defined in equation (11). Differentiating V ₃ with respect to time yields

$\begin{array}{c}{\dot{V}}_3={\dot{V}}_2+m_{11}{\epsilon }_u{\dot{\epsilon }}_u+\frac{\Delta }{m_{22}}{\epsilon }_r{\dot{\epsilon }}_r\\ =-l_1{e}_x^2-l_2{e}_y^2-l_3{({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)}^2+{\epsilon }_u({\tau }_1+e_x+\overline{f}(u)-m_{11}{\dot{u}}_f)\\ +{\epsilon }_r({\tau }_3+{\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e+\overline{f}(r)-\frac{m_{32}}{m_{22}}\overline{f}(v)-\frac{\Delta }{m_{22}}{\dot{r}}_f)+\Theta \end{array}$ 29

where

$\Theta ={\delta }_w{e}_y+{\delta }_u{e}_x+{\delta }_w(e_y+{\dot{v}}_p\text{sin}{\phi }_e-{\dot{w}}_f)+{\delta }_r({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)$ 30

and $\overline{f}(u),\overline{f}(v),\hspace{0.17em}\text{and}\hspace{0.17em}\overline{f}(r)$ are also defined in equation (11). Then, the control inputs τ ₁ and τ ₃ are chosen as

${\tau }_1=-l_4{\epsilon }_u-e_x-\overline{f}(u)+m_{11}{\dot{u}}_f$ 31 ${\tau }_3=-l_5{\epsilon }_r-{\phi }_e-{\epsilon }_w{v}_p\text{cos}{\phi }_e-\overline{f}(r)+\frac{m_{32}}{m_{22}}\overline{f}(v)+\frac{\Delta }{m_{22}}{\dot{r}}_f$ 32

where the parameters l ₄ and l ₅ are positive constants. Then, equation (29) can be rewritten as

${\dot{V}}_3=-l_1{e}_x^2-l_2{e}_y^2-l_3{({\phi }_e+{\epsilon }_w{v}_p\text{cos}{\phi }_e)}^2-l_4{\epsilon }_u^2-l_5{\epsilon }_r^2+\Theta$ 33

Remark 1

If the proposed controllers in equation (31) and (32) are used in the transformed system (12), the trajectory tracking errors will convergent quickly to a small neighborhood of the origin, and its theoretical verification can be performed based on theorem 1 ¹³ ; that is the coupled and transformed system can be tracked by the proposed control theoretically. As a main limitation, the resulting controller relies on the improved knowledge of the dynamical system.

Simulation studies

To demonstrate the effectiveness of the designed controller for the tracking control of underwater vehicles in the horizontal plane, simulation studies are conducted on the vehicle using the pyramid configuration CMGs, where the parameters are mainly from Yang et al., ³⁴ as listed in Table 1.

Table 1.

Main parameters of the underwater vehicle.

m	31.4 kg	Iz	3.45 kg m2	$X_{\dot{u}}$	−0.93 kg
$Y_{\dot{v}}$	−35.5 kg	$Y_{\dot{r}}$	1.93 kg m rad−2	$N_{\dot{v}}$	1.93 kg m
$N_{\dot{r}}$	−4.88 kg m2 rad−1	X u|u|	−1.62 kg m−1	Xvr	35.5 kg rad−1
Xrr	−1.93 kg m rad−1	Y v|v|	−1310 kg m−1	Y r|r|	$-6.32\hspace{0.17em}\text{kg}\hspace{0.17em}\text{m}\hspace{0.17em}{\text{rad}}^{-2^2}$
Yur	5.22 kg rad−1	Yuv	−28.6 kg m−1	N v|v|	−3.18 kg
N r|r|	−94 kg m2 rad−2	Nur	−2 kg m rad−1	Nuv	−24 kg

Our actual control objective is to track the desired trajectory [η_d , ν_d ] ^T given by

$\{\begin{array}{l}{\dot{\eta }}_d=\text{R}({\psi }_d){\nu }_d\hfill \\ \text{M}{\dot{\nu }}_d+C({\nu }_d){\nu }_d+D({\nu }_d){\nu }_d={\tau }_d\hfill \end{array}$ 34

where ${\tau }_d=[10,0,\frac{\pi t}{2}{]}^T$ , the disturbance vector τ _dis = 0, and the initial conditions of the underwater vehicle are set as ${\eta }_d(0)=[3,2,\frac{\pi }{4}{]}^T$ , ${\nu }_d{=[1,2,0]}^T$ , $\eta (0)=[2,1,\frac{\pi }{2}{]}^T$ , and $\nu {(0)=[0,0,0]}^T$ . Correspondingly, the controller parameters are chosen as l ₁ = 2.3, l ₂ = 1, l ₃ = 12, l ₄ = l ₅ = 0.01, A ₁ = A ₂ = A ₃ = 5, B ₁ = B ₂ = B ₃ = 6, and C_i = B_i , (i =1, 2, 3). Considering the actual error dynamics and the equivalent relation in equation (9), the error yaw angle ψ_e replaces φ_e in the simulation to avoid oscillations resulting from the inverse operation of the trigonometric functions. Then, from Figure 3, which plots the desired and actual trajectories, we can observe that the proposed control scheme is computationally efficient. The scheme also presents a high convergence rate and state performance in Figure 4, and in Figure 5 some oscillations appear in the initial stage. The actual system states η and ν and their desired targets η_d and ν_d are shown in the image, which shows that the designed controller achieves a good performance for the tracking control of positions and velocities simultaneously. The corresponding control force τ ₁ and torque τ ₃ are shown in Figure 6, which demonstrates that chattering is found in the control action. In addition, from the value of the torque input τ ₃, we can calculate the desired fourth angle δ ₄ of the gyro rotor; then, the actual control input can be realized.

Figure 3.

Desired and actual trajectories in the xy plane within 20 s.

Figure 4.

Desired and actual states x, y, and ψ.

Figure 5.

Desired and actual states u, v, and r.

Figure 6.

Control force τ ₁ and torque τ ₃.

Conclusion

In this article, a controller based on a biologically inspired model and Lyapunov functions acting on a transformed nominal underactuated system is proposed. The state of the underwater vehicle in the horizontal plane can be estimated by the designed state observer. Moreover, if the system is further modified or simplified, we can design a more intelligent learning algorithm using neural networks to achieve a better performance of the unknown or uncertain variables of the dynamics. It should also be noted that this control scheme can be transplanted in parallel to tracking for the vertical plane. In addition, this trajectory tracking control law is proposed to obtain the control signal of the thruster directly and the torque input to calculate the expected angle input of the fourth gyro rotor. The stability can be guaranteed by Lyapunov’s stability theory. Because of the fact that the virtual kinematic inputs are composed of the tracking guidance laws, and because the actual boundedness of the inputs is needed, the simulation results demonstrate reasonable robustness against parameter perturbation. Furthermore, if the trajectory tracking control for six--degree-of-freedom motion is considered, then more simplifications may be needed for the coupled non-affine nonlinear dynamical system, and other measures need to be taken to address both the unknown current or disturbance and the problem of how to design finite-time state feedback tracking controllers.