Design of X-rudder autonomous underwater vehicle’s quadruple-rudder allocation with Lévy flight character

Abstract

This article addresses one quadruple-rudder allocation method for an autonomous underwater vehicle (AUV) equipped with X rudder, in which all of the rudders can be operated independently. By considering X rudder’s character, one X-rudder AUV’s control information frame is designed. It contains the anti-normalization method based on virtual rudders and one quadruple-rudder allocation with Lévy flight character. This quadruple-rudder allocation method has the advantages of Lévy flight and avoids the shaking problem. One contrast simulation of a lawn mower path following mission in three-dimensional (3D) space is performed. The results of simulation show that the quadruple-rudder allocation with Lévy flight character can offer accurate and reliable control ability. Besides that, compared to the quadruple-rudder allocation based on pseudoinverse and fixed point iteration, the method designed with Lévy flight character can achieve the same mission with less X rudder’s operation.

Keywords

Autonomous underwater vehicle X rudder quadruple-rudder allocation Lévy flight multi-objective optimization

Introduction

Autonomous underwater vehicles (AUVs) can work under water without requiring inputs from operators. AUVs are playing more and more important roles in exploring and developing the ocean, such as bottom topographic survey and the exploring of submarine hydrothermal sulfide.^1,2 AUVs can also extend men’s ability and do the work that is much more difficult for humans, such as detecting mines in harbour, checking oil pipes’ safety and finding crashed planes’ wreckage in deep sea.^{3
–5} The multiple kinds of work for AUVs come from the researchers’ study. They never stop trying to push forward the limit of AUV’s control ability.

Some researchers try to design new control methods. Kumar et al. proved the modified gain extended Kalman filter in AUV’s obstacle avoidance and navigation.⁶ Elmokadem et al. proposed one trajectory tracking control using the sliding mode control technique for underactuated AUV.⁷ Wan et al. designed one bottom following method based on active disturbance rejection control for AUV in complex environment.⁸ Wang et al. adjusted the force allocation strategy and motion control method to make one AUV finish tasks while the dynamic positioning function is reserved.⁹ Podder et al. designed an allocation of thruster forces for one AUV to produce desired force when AUV is in trouble.¹⁰ Besides, some of the researchers try to design new motion operators. Pyo et al. specially designed a hovering-type AUV with symmetric body structure and thruster for a lawn mower trajectory.¹¹ Ma et al. developed a cownose ray-inspired robotic fish that can be propelled by oscillation and chordwise twisting pectoral fins.¹² As mentioned before, X rudder is one of the new motion operators.

Traditional cross rudder is usually composed of two parts, one is horizontal rudder and the other is vertical rudder. They are usually placed at the stern of one vehicle. As to X rudder, it usually forms the shape of X at the bow or stern of one vehicle. According to the available literature, X rudder can be classified into three kinds as follows:

Fixed X rudder: The rudders of this kind of X rudder cannot be operated. Liu et al. designed one AUV equipped with this kind of X rudder at stern.¹³ The AUV’s motion control is fulfilled by the vertical and horizontal tunnel propellers and four main thrusters. This X rudder is fixed and its function is just to offer the extra stability when AUV moves.

Diagonal-linkage X rudder: This kind of X rudder is driven in the unit of a diagonal rudder pair. Yoshida et al. designed one AUV called Yumeiruk equipped with two X rudders.¹⁴ One is at the bow and the other one is at the stern. The AUV can use them to cause the corresponding moments to control yaw and pitch. In March 2013, the 15-days sea trial of the AUV was carried out in the Sagami Bay. The diagonal-linkage X rudder is used during altitude changes.

Independent X rudder: This kind of X rudder can be operated independently. According to the available literature, it has not been used by AUV. However, it has been studied by some countries’ navy and been equipped on some submarines, such as Gotland class from Sweden, Type 212A from Germany and Soryu class from Japan.

The available literature about X rudder control is very rare and similar. They just mention some advantages of X rudder on submarines compared to traditional cross rudder or the relationship between X rudder and traditional cross rudder.¹⁵ However, it does not mention any details about auto control of X rudder, especially the quadruple-rudder allocation method. So our research pays more attention to the independent X rudder and the situation of four rudders being responsible for AUV’s motion control.

In this article, the whole auto control information framework of X-rudder AUV and one quadruple-rudder allocation with Lévy flight character are designed. By considering X rudder’s character and practical situation, the information framework includes the anti-normalization part based on virtual rudders and rudder allocation part. The anti-normalization method based on virtual rudders is used to solve the abstract situation of X rudder’s control. The quadruple-rudder allocation with Lévy flight character belongs to the rudder allocation part. The simulation test shows that the quadruple-rudder allocation with Lévy flight character has efficient and accurate responses to corresponding commands.

The remainder of this article is organized as follows. In ‘Model description of an X-rudder AUV’ section, X-rudder AUV’s kinematics equations, dynamics equations and corresponding parameters are offered. In ‘Quadruple-rudder allocation based on pseudoinverse and fixed point iteration’ section, the control information frame of X-rudder AUV is designed. It mainly contains anti-normalization and quadruple-rudder allocation based on pseudoinverse and fixed point iteration. In ‘Lévy flight quadruple-rudder allocation and its optimization’ section, one quadruple-rudder allocation with Lévy flight character is designed. It contains the thought of Lévy flight and multi-objective optimization. In ‘Simulation and results’ section, the results and analysis of comparative simulation in a 3D lawn mower path following mission are presented.

Model description of an X-rudder AUV

Coordinate systems and kinematic and dynamic models

The test X-rudder AUV’s appearance and corresponding coordinate systems are shown in Figure 1. Two coordinates are designed. One is a body-fixed frame $O - x y z ({B})$ and the other one is an inertial reference frame $E - ξ η ζ ({N})$ . The AUV’s motion actuators are composed of one main thruster and one X rudder, whose four rudders can move independently. The rudders are laid out at the stern.

Figure 1.

X-rudder AUV and coordinate systems. AUV: autonomous underwater vehicle.

The kinematics and dynamics equations of an AUV in 3D space can be described as follows¹⁶

\dot{η} = T V

M \dot{V} + C (V) V + D (V) V + g + f = τ

where $T = diag (T_{1}, T_{2})$

T_{1} = [\begin{array}{l} cos ψ cos θ & cos ψ sin θ sin φ - sin ψ cos φ & cos ψ sin θ cos φ + sin ψ sin φ \\ sin ψ cos θ & sin ψ sin θ sin φ + cos ψ cos φ & sin ψ sin θ cos φ - cos ψ sin φ \\ - sin θ & cos θ sin φ & cos θ cos φ \end{array}]

T_{2} = [\begin{matrix} 1 & tan θ sin φ & tan θ cos φ \\ 0 & cos φ & - sin φ \\ 0 & sin φ / cos θ & cos φ / cos θ \end{matrix}]

$η = {[ξ, η, ζ, φ, θ, ψ]}^{T}$ represents AUV’s position in ${N}, V = {[u, v, w, p, q, r]}^{T}$ represents AUV’s speed component in {B}, M accounts for the mass and added mass coefficients, C ( V ) accounts for the body moments of inertia and added moments of inertia coefficients, D ( V ) represents the drag force and moments, g accounts for forces and moments caused by gravitation and buoyancy, f represents disturbance in environment and $τ = {[X, Y, Z, K, M, N]}^{T}$ represents the motion executors’ force to AUV.

X rudder’s layout and its main parameters

X rudder’s layout and the rudder numbers are shown in Figure 2. Serial numbers are used to simplify the description of X rudder’s control. The rudder’s angle is positive, while the rudder can cause the corresponding force direction shown in Figure 2.

Figure 2.

Layout of X rudder.

Equation (3) shows the relationship between τ and X-rudder AUV’s motion executers

{\begin{array}{l} X = X_{δ_{1} δ_{1}} \times u^{2} \times δ_{1}^{2} + X_{δ_{2} δ_{2}} \times u^{2} \times δ_{2}^{2} + X_{δ_{3} δ_{3}} \times u^{2} \times δ_{3}^{2} + X_{δ_{4} δ_{4}} \times u^{2} \times δ_{4}^{2} + X_{T} \\ Y = Y_{δ_{1}} \times u^{2} \times δ_{1} + Y_{δ_{2}} \times u^{2} \times δ_{2} + Y_{δ_{3}} \times u^{2} \times δ_{3} + Y_{δ_{4}} \times u^{2} \times δ_{4} \\ Z = Z_{δ_{1}} \times u^{2} \times δ_{1} + Z_{δ_{2}} \times u^{2} \times δ_{2} + Y_{δ_{3}} \times u^{2} \times δ_{3} + Y_{δ_{4}} \times u^{2} \times δ_{4} \\ K = K_{δ_{1}} \times u^{2} \times δ_{1} + K_{δ_{2}} \times u^{2} \times δ_{2} + K_{δ_{3}} \times u^{2} \times δ_{3} + K_{δ_{4}} \times u^{2} \times δ_{4} \\ M = M_{δ_{1}} \times u^{2} \times δ_{1} + M_{δ_{2}} \times u^{2} \times δ_{2} + M_{δ_{3}} \times u^{2} \times δ_{3} + M_{δ_{4}} \times u^{2} \times δ_{4} \\ N = N_{δ_{1}} \times u^{2} \times δ_{1} + N_{δ_{2}} \times u^{2} \times δ_{2} + N_{δ_{3}} \times u^{2} \times δ_{3} + N_{δ_{4}} \times u^{2} \times δ_{4} \end{array}

where $X_{δ_{*} δ_{*}}, Y_{δ_{*}}, Z_{δ_{*}}, K_{δ_{*}}, M_{δ_{*}} and N_{δ_{*}}$ are the corresponding parameters related to the fluid viscosity force of * rudder in X rudder. δ_* represents rudder *’s angle. X_T is the force of the main thruster. Equation (3) shows that the main thruster can only influence X in τ . However, the X rudder can influence all of the elements in τ .

As to the test X-rudder AUV in this article, its mass is 1420 kg and its length is 6 m. Its moments of inertia of the coordinate axes x, y and z in {B} are 17, 1835 and 1800 kg m², respectively. Some other main parameters about X rudder are shown in Table 1. They are calculated by computation fluid dynamics methods and they will be used in the simulation tests.

Table 1.

Main parameters of the X rudder.

$X_{δ_{1} δ_{1}}$	−5.0303 × 10⁻⁴	$Y_{δ_{1}}$	−3.8375 × 10⁻⁴	$Z_{δ_{1}}$	−6.6442 × 10⁻⁴	$K_{δ_{1}}$	−1.5876 × 10⁻⁴	$M_{δ_{1}}$	−5.6597 × 10⁻⁵	$N_{δ_{1}}$	5.2033 × 10⁻⁵
$X_{δ_{2} δ_{2}}$	−5.6451 × 10⁻⁴	$Y_{δ_{2}}$	−4.6028 × 10⁻⁴	$Z_{δ_{2}}$	−7.3760 × 10⁻⁴	$K_{δ_{2}}$	2.0030 × 10⁻⁴	$M_{δ_{2}}$	−6.8421 × 10⁻⁵	$N_{δ_{2}}$	6.7854 × 10⁻⁵
$X_{δ_{3} δ_{3}}$	−5.6451 × 10⁻⁴	$Y_{δ_{3}}$	4.6028 × 10⁻⁴	$Z_{δ_{3}}$	−7.3760 × 10⁻⁴	$K_{δ_{3}}$	−2.0030 × 10⁻⁴	$M_{δ_{3}}$	−6.8421 × 10⁻⁵	$N_{δ_{3}}$	−6.7854 × 10⁻⁵
$X_{δ_{4} δ_{4}}$	−5.0303 × 10⁻⁴	$Y_{δ_{4}}$	3.8375 × 10⁻⁴	$Z_{δ_{4}}$	−6.6442 × 10⁻⁴	$K_{δ_{4}}$	1.5876 × 10⁻⁴	$M_{δ_{4}}$	−5.6597 × 10⁻⁵	$N_{δ_{4}}$	−5.2033 × 10⁻⁵

Quadruple-rudder allocation based on pseudoinverse and fixed point iteration

The X rudder control information frame

If the control law is based on error and the rate of error, cross-rudder AUV’s and X-rudder AUV’s control information frames can be described as shown in Figure 3. In Figure 3, the function of {B} to {N} is used to transform the values in coordinate system {B} to coordinate system {N}. It is obvious that the rudder allocation is one unique part in X-rudder AUV’s control, which cross-rudder AUV does not have. However, the anti-normalization in X rudder control should be analysed first not only because this part is different from traditional cross rudder control but also this is the foundation of rudder allocation part in X rudder control.

Figure 3.

Cross-rudder AUV’s and X-rudder AUV control information frames. AUV: autonomous underwater vehicle.

Anti-normalization

In traditional cross-rudder AUV’s control, the anti-normalization part can be explained by the following

{\begin{array}{l} δ_{v} = {CTR}_{yaw_cross} \times δ_{v_max} \\ δ_{h} = {CTR}_{pitch_cross} \times δ_{h_max} \\ - 1 \leq {CTR}_{yaw_cross}, {CTR}_{pitch_cross} \leq + 1 \end{array}

where CTR_{yaw_cross} and CTR_{pitch_cross} are the outputs of vertical rudder’s and horizontal rudder’s control laws, respectively, δ_{v_max} and δ_{h_max} are the maximum values of the corresponding cross rudder and δ_v and δ_h are the vertical and horizontal rudders, respectively. They can affect yaw and pitch separately.

However, according to equation (3), any X rudder’s operation can cause changes to all the values in matrix τ . As to one AUV equipped with rudders, which are the main motion operators to change [K, M, N], the following anti-normalization for X-rudder AUV based on virtual rudders is designed

δ_{VR} = M_{CTR} δ_{AN} M_{I}

where $M_{CTR} = diag ({CTR}_{roll}, {CTR}_{pitch}, {CTR}_{yaw}), - 1 \leq {CTR}_{roll}, {CTR}_{pitch}, {CTR}_{yaw} \leq + 1$ . CTR_roll, CTR_pitch and CTR_yaw represent the outputs of corresponding control laws of roll, pitch and yaw, respectively. δ_AN is one anti-normalization angle, and according to test results, it can be set between 75% and 150% of the maximum values of rudders in X rudder. $M_{I} = [I_{roll}, I_{pitch}, I_{yaw}]$ . I_roll, I_pitch and I_yaw are adjustable parameters. They represent the control priority in roll, pitch and yaw, that is, the greater element in M _I means the greater priority of the corresponding motion control. $δ_{VR} = {[δ_{VR_roll}, δ_{VR_pitch}, δ_{VR_yaw}]}^{T}$ . δ_{VR_roll}, δ_{VR_pitch} and δ_{VR_yaw} are the virtual rudders, which are designed to control roll, pitch and yaw.

However, virtual rudders are not the direct causes of the changes of [K, M, N] in τ . The direct causes are the changes of X rudders. So the rudder allocation part in Figure 3 is needed to build the relationship between virtual rudders and X rudders.

Rudder allocation

As to one independent X-rudder AUV, one of its great unique characters is its rudders’ operation. These rudders’ operation is very flexible because there are many combination kinds. For example, the X-rudder AUV’s motion control can be achieved by double-rudder, triple-rudder or quadruple-rudder allocation. According to our research, double-rudder allocation and triple-rudder allocation can be designed as equations (6) and (7) referring to under-actuated AUV’s control and full-actuated AUV’s control¹⁷

{\begin{array}{l} M_{δ_{1}} \times δ_{1} + M_{δ_{3}} \times δ_{3} = δ_{VR_pitch} \\ \begin{array}{l} N_{δ_{1}} \times δ_{1} + N_{δ_{3}} \times δ_{3} = δ_{VR_yaw} \\ \underline{δ} \leq δ_{1}, δ_{3} \leq \bar{δ} \end{array} \end{array}

{\begin{array}{l} K_{δ_{1}} \times δ_{1} + K_{δ_{2}} \times δ_{2} + K_{δ_{3}} \times δ_{3} = δ_{VR_roll} \\ M_{δ_{1}} \times δ_{1} + M_{δ_{2}} \times δ_{2} + M_{δ_{3}} \times δ_{3} = δ_{VR_pitch} \\ N_{δ_{1}} \times δ_{1} + N_{δ_{2}} \times δ_{2} + N_{δ_{3}} \times δ_{3} = δ_{VR_yaw} \\ \underline{δ} \leq δ_{1}, δ_{2}, δ_{3} \leq \bar{δ} \end{array}

where K_*, M_* and N_* are shown in equation (3). δ_{VR_roll}, δ_{VR_pitch} and δ_{VR_yaw} are shown in equation (5). δ_* represents rudder *, which is responsible for X-rudder AUV’s control. $\underline{δ} and \bar{δ}$ represent the minimal and maximal rudder. Equation (6) represents one double-rudder allocation using Rudder 1 and Rudder 2 as example. Equation (7) represents one triple-rudder allocation using Rudder 1, Rudder 2 and Rudder 3 as example.

Compared with double-rudder allocation and triple-rudder allocation, quadruple-rudder allocation is more complex and confusing. To the X-rudder AUV’s control, quadruple-rudder allocation should be paid more attention here and its control method should be efficient and accurate.

Quadruple-rudder allocation

Problem description and pseudoinverse method

The quadruple-rudder allocation can be described as follows

{\begin{array}{l} K_{δ_{1}} \times δ_{1} + K_{δ_{2}} \times δ_{2} + K_{δ_{3}} \times δ_{3} + K_{δ_{4}} \times δ_{4} = δ_{VR_roll} \\ M_{δ_{1}} \times δ_{1} + M_{δ_{2}} \times δ_{2} + M_{δ_{3}} \times δ_{3} + M_{δ_{4}} \times δ_{4} = δ_{VR_pitch} \\ N_{δ_{1}} \times δ_{1} + N_{δ_{2}} \times δ_{2} + N_{δ_{3}} \times δ_{3} + N_{δ_{4}} \times δ_{4} = δ_{VR_yaw} \\ \underline{δ} \leq δ_{1}, δ_{2}, δ_{3}, δ_{4} \leq \bar{δ} \end{array}

Equation (8) can be simplified as follows

{\begin{cases} B u_{δ} = δ_{VR} \\ \underline{δ} \leq δ_{1}, δ_{2}, δ_{3}, δ_{4} \leq \bar{δ} \end{cases}

where $B = [K_{δ_{1}} K_{δ_{2}} K_{δ_{3}} K_{δ_{4}}; M_{δ_{1}} M_{δ_{2}} M_{δ_{3}} M_{δ_{4}}; N_{δ_{1}} N_{δ_{2}} N_{δ_{3}} N_{δ_{4}}] and u_{δ} = {[δ_{1}, δ_{2}, δ_{3}, δ_{4}]}^{T}$ . K_*, M_* and N_* are shown in equation (3). δ_VR is mentioned in equation (5). $\underline{δ} and \bar{δ}$ are shown in equation (6). $\tilde{B}$ is B ’s augmented matrix, and it is shown as follows

\tilde{B} = [K_{δ_{1}} K_{δ_{2}} K_{δ_{3}} K_{δ_{4}} δ_{VR_roll}; M_{δ_{1}} M_{δ_{2}} M_{δ_{3}} M_{δ_{4}} δ_{VR_pitch}; N_{δ_{1}} N_{δ_{2}} N_{δ_{3}} N_{δ_{4}} δ_{VR_yaw}]

Before solving equation (9), Rank( B ) and $Rank (\tilde{B})$ should be calculated first, which are the rank of matrix B and the rank of matrix $\tilde{B}$ . According to the determinant theorem, the solutions for the linear equations are as follows:¹⁸

If $Rank (B) \neq Rank (\tilde{B})$ , there will be no solution for equation (9), and the value of δ_AN in equation (5) should be changed until $Rank (B) = Rank (\tilde{B})$ . As how to change the value of δ_AN, there are many ways, such as the method based on experience or some particular equations, as follows:

If $Rank (B) = Rank (\tilde{B}) = 4$ , there will be only one solution for equation (9).

If $Rank (B) = Rank (\tilde{B}) < 4$ , there will be innumerable solutions for equation (9).

When solving the problem as equation (9), the method of pseudoinverse¹⁹ shown in equation (10) can be used

{\begin{cases} u_{δ} = B^{+} δ_{VR} \\ B^{+} = B^{T} {(B B^{T})}^{- 1} \\ \underline{δ} \leq δ_{1}, δ_{2}, δ_{3}, δ_{4} \leq \bar{δ} \end{cases}

where B ⁺ is the pseudoinverse matrix of B . ${(B B^{T})}^{- 1}$ is the inverse matrix of BB ^T . If the solution of pseudoinverse is out of the range $[\underline{δ}, \bar{δ}]$ , the truncation method, which is usually used in the traditional cross-rudder AUV control, can be used to set the corresponding maximum or the minimum rudder value.

If $Rank (B) = Rank (\tilde{B}) = 4$ , the solution of pseudoinverse according to equation (10) is the answer of X rudder’s value. However, if $Rank (B) = Rank (\tilde{B}) < 4$ , the solution of pseudoinverse is just one possible answer of X rudder’s value. Therefore, one method to find the best answer is needed. Before that, the influence of X rudder’s operation should be analysed first.

X-rudder resistance and fixed point iteration

Any movement of X rudder can change φ, θ and ψ directly, besides that, the movement of X rudder can also cause resistance against AUV’s forward movement. The resistance of one rudder’s operation can be described by the following

{\begin{cases} F_{re} = (ρ / 2) \times C_{D} \times v_{f}^{2} \times S_{rudder} \times | sin δ | \\ Re = \int_{0}^{T} F_{re} u d t \end{cases}

where F_re is the resistance force caused by one rudder’s operation. ρ, C_D, v_f, S_rudder and δ represent density of environment, drag coefficient,²⁰ flow velocity to rudder, area of the rudder and rudder angle, respectively. Re is the energy cost by resistance, T is the time length of AUV’s mission and u is AUV’s velocity. The sum of X rudder’s Re can be called as Re_Total, which can be calculated by the following

{Re}_{Total} = \sum_{i = 1}^{4} {Re}_{i}

where i represents the rudder’s number in X rudder and Re_Total is the whole energy consumption generated by the X rudder’s movements.

In this article, the X rudder’s appearances are the same and X rudder’s design is based on NACA0012.²¹ The whole C_D curve and a part of C_D curve from 0° to 35° are shown in Figures 4 and 5, respectively. Curve in Figure 5 can be described as follows

C_{D} = {\begin{array}{l} 0, | δ | < 5 \times p i / 180 \\ 1.4083 \times {| δ |}^{2} + 0.4832 \times | δ | - 0.0457, 5 \times p i / 180 \leq | δ | \leq 35 \times p i / 180 \end{array}

Figure 4.

C_D of NACA0012.

Figure 5.

A part of C_D and fitting curve.

where δ is rudder’s angle in the unit of rad. 1 rad = 180/π deg.

According to equations (11)–(13), the less operation of rudders, the less energy the X-rudder AUV consumes. As to the solution of equation (10) in situation $Rank (B) = Rank (\tilde{B}) < 4$ , in order to find out the precise answer, the criterion of the least energy²² shown in equation (14) is introduced

min J = (1 - ε) {‖ W_{v} (B u_{δ} - δ_{V R}) ‖}_{2}^{2} + ε {‖ W_{u} (u_{δ} - u_{δ d}) ‖}_{2}^{2}, \underline{δ} \leq u_{δ} \leq \bar{δ}, 0 < | ε | < 1

where W _v and W _u are diagonal weight matrixes, and their dimension is 4 × 4, ε is an adjustable parameter, u _δd is u _δ ’s expected value. If u _δd = 0, it means that the rudders need to make the minimum movements and equation (14) can be changed as follows

\begin{array}{l} min J = (1 - ε) {‖ W_{v} (B u_{δ} - δ_{V R}) ‖}_{2}^{2} + ε {‖ W_{u} u_{δ} ‖}_{2}^{2} \\ = (1 - ε) {(B u_{δ} - δ_{V R})}^{T} Q_{1} (B u_{δ} - δ_{V R}) + ε u_{δ}^{T} Q_{2} u_{δ} \end{array}

where $Q_{1} = W_{v}^{T} W_{v} and Q_{2} = W_{u}^{T} W_{u}$ .

If u _δ satisfies equation (15), it is the best answer needed. According to the contraction mapping theorem in fixed point iteration, equation (16) is designed to solve the problem of equation (15) ²³

u_{δ}_{n + 1} = sat [(1 - ε) ω B^{T} Q_{1} δ_{V R} - (ω H - I) u_{δ n}]

where $H = (1 - ε) B^{T} Q_{1} B + ε Q_{2}, ω = 1 / {‖ H ‖}_{2}, I = d i a g (1, 1, 1, 1)$ and u _δn and u _δn+1 mean u _δ in the nth iteration and the (n + 1)th iteration. The definition of sat can be explained by the following

sat (x) = {\begin{array}{l} \underline{δ}, x \leq \underline{δ} \\ x, \underline{δ} < x < \bar{δ} \\ \bar{δ}, x \geq \bar{δ} \end{array}

When $Rank (B) = Rank (\tilde{B}) < 4$ in equation (10), the solution of pseudoinverse can be the original iteration point u _δ0 in this fixed point iteration. Besides that, the criterion of the iteration’s end can be designed as follows

| J (u_{δ k + 1}) - J (u_{δ k}) | \leq J_{end}

where J_end is an adjustable parameter. Equation (18) shows the situation of the kth iteration and $u_{δ k + 1}$ is the final answer needed.

The whole process of quadruple-rudder allocation based on pseudoinverse and fixed point iteration is shown in Figure 6.

Figure 6.

The information framework of quadruple-rudder allocation based on pseudoinverse and fixed point iteration.

Lévy flight quadruple-rudder allocation and its optimization

Lévy flight quadruple-rudder allocation

According to the analysis in ‘Quadruple-rudder allocation’ section, when $Rank (B) = Rank (\tilde{B}) < 4$ in equation (9), the criterion of the iteration’s end is designed as equation (18), which is decided by J of the adjacent fixed point iteration’s results. Once the adjacent fixed point iteration’s results satisfy equation (18), it means that the iteration has been stopped. However, the corresponding u _δ from the result of fixed point iteration may be one locally optimal solution. In order to avoid the locally optimal solution, Lévy flight may be on solution to solve that.

Lévy flight is named by French mathematician Paul Lévy. It is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions.²⁴ In X rudder control, Lévy flight is used to complete the global search part and fixed point iteration in ‘Quadruple-rudder allocation’ section is used to complete the local search part during solving the problem as equation (15). Lévy flight in X rudder control can be described as equation (19). The pseudoinverse results in ‘Quadruple-rudder allocation’ section can be used as one initial point, and the end of Lévy flight can be decided by the flight’s times $N_{L} \in N_{+}$

u_{δ k + 1} = u_{δ k} + α \times L (λ), α > 0

where u _δk and u _{δk + 1} are the kth and (k + 1)th Lévy flight result, α is one adjustable parameter, which can change the flight step length, and L (λ) is the matrix of Lévy flight random search step, which can be calculated by the following²⁵

{\begin{cases} L (λ) = (φ \times {μ) / | v |^{1 / β} \\ φ = {((Γ (1 + β) \times sin ((π \times β) / 2)) / (Γ ((1 + β) / 2) \times β \times 2^{(β - 1) / 2}))}^{1 / β} \\ 1 \leq β \leq 2 \end{cases}

where μ and v are two random matrixes, and they belong to standardized normal distribution. Their dimensions are the same as u _δ ’s dimension. Γ(*) represents Gamma function, which can be defined by the following

Γ (x) = \int_{0}^{+ \infty} t^{x - 1} e^{- t} d t

Every time Lévy flight is done, there will be several new points and the points number is the same as N_L. Plus the initial point in Lévy flight, there can be N_L + 1 points, and they can be the original iteration points in fixed point iteration mentioned in ‘Quadruple-rudder allocation’ section. Once all the fixed point iterations are done, all the u _δ ’s J can be calculated according to equation (14). u _δ with the minimum J is the final result needed in this process. In order to expand the Lévy flight influence, it can be done by increasing some other initial points for Lévy flight. Equation (22) shows the way to get one extra u _δ for Lévy flight

u_{δ extra} = δ_{min} + r a n d (0, 1) • (δ_{max} - δ_{min})

where u _δextra is the initial point needed, δ_min and δ_max are the matrixes composed by the corresponding X rudder’s minimum and maximum values, and rand (0,1) is one random matrix composed by the value from 0 to 1, and its dimension is the same as u _δ ’s. The number of extra initial point for Lévy flight is decided by the core computer’s calculation ability on X-rudder AUV.

Here, one simulation of Lévy flight quadruple-rudder allocation is made without the fixed point iteration part. One assumption is made as $- 1 \leq δ_{1}, δ_{2}, δ_{3}, δ_{4} \leq + 1 and - 1 \leq δ_{VR_roll}, δ_{VR_pitch}, δ_{VR_yaw} \leq + 1$ . The assumptions will simplify the description of this analysis, and the units of variables will not be mentioned in this section because the analysis results will not be affected by them. As u _δ is four dimensional and it is difficult to show directly, the direct relationship in 3D space according to equation (9) is built. The relationship between reasonable solution set of u _δ and δ_VR can be shown in Figure 7, which is generated by MATLAB 2010b. In Figure 7, the reasonable solution set of u _δ in space δ_VR is represented by Ω _δ .

Figure 7.

The space of u _δ ’s reasonable solution set.

Four initial points are designed. They are (0,0,0,0) calculated by pseudoinverse and three other points are calculated by equation (22). The flight’s time is five. Table 2 and Figure 8 show the detailed process of Lévy flight in 3D.

Table 2.

u _δ ’s distribution during Lévy flight.

u _δ in 4D	u _δ transformed into 3D						Colour in Figure 8
u _δ in 4D	0 Lévy	1st Lévy	2nd Lévy	3rd Lévy	4th Lévy	5th Lévy	Colour in Figure 8
(0,0,0,0)	(0,0,0)	(−0.1635, 0.369, 0.2399)	(−0.1631, 0.3684, 0.2392)	(−0.1633, 0.3676, 0.2385)	(−0.164, 0.3688, 0.2327)	(−0.1636, 0.3695, 0.233)	Blue
(0.6294, 0.8116, −0.746, 0.8268)	(0.2694, −0.5028, 0.4115)	(0.1413, 0.0139, −0.1721)	(0.1406, 0.0207, −0.1724)	(0.1412, 0.0219, −0.1724)	(0.1419, 0.0212, −0.171)	(0.1424, 0.0226, −0.169)	Yellow
(0.2647, −0.8049, −0.443, 0.0938)	(−0.0608, 0.3078, −0.0348)	(0.1457, −0.3066, 0.3135)	(0.1458, −0.3003, 0.3208)	(0.1452, −0.2988, 0.3225)	(0.1448, −0.2988, 0.3214)	(0.1427, −0.3056, 0.3157)	Pink
(0.915, 0.9298, −0.6848, 0.9412)	(0.2551, −0.7008, 0.4695)	(−0.1292, −0.274, −0.0238)	(−0.132, −0.2686, −0.0197)	(−0.1318, −0.2669, −0.0187)	(−0.1314, −0.2663, −0.0167)	(−0.1316, −0.2655, −0.0207)	Green

3D: three dimensional.

Figure 8.

Lévy flight and partial enlarged detail.

In Figure 8, it can be found that the step length and the direction between the Lévy flight points are not the same. With the help of Lévy flight, the initial point can move in random direction. Usually it moves by small steps and occasionally by one big step. This character can prevent the repeated research in one small field and avoid the locally optimal solution compared to the quadruple-rudder allocation mentioned in ‘Quadruple-rudder allocation’ section.

According to the analysis in this section, the information frame of Lévy flight quadruple-rudder allocation can be built as Figure 9.

Figure 9.

Control information frame of Lévy flight quadruple-rudder allocation.

In Figure 9, u _δ0, $u_{δ 0_extra_1}, u_{δ 0_extra_2}, …, u_{δ 0_extra_n}$ are the initial points for each Lévy flight. $u_{δ 0_extra_1}, …, u_{δ 0_extra_n}$ are calculated by equation (22). With the help of ‘Quadruple-rudder allocation’ section, there will be many u _δ , and the u _δ with the minimum J is the final answer.

Simulation of depth control is used to test the whole control method. X-rudder AUV’s origin depth is 10 m, expected depths are 40 and 80 m separately, origin yaw angle is 0°, and it will keep this yaw angle during the whole test. The classical PD control is chosen as the control law here.²⁶ AUV’s test speed is 3 m/s. There are one initial Lévy flight point calculated by pseudoinverse and three extra initial Lévy flight points calculated by equation (22). The flight’s times are five. The maximum angular velocity of X rudder is 1°/s. The angle range of X rudder varies from −35° to 35°. Figure 10 shows the control results.

Figure 10.

Depth control results by Lévy flight quadruple-rudder allocation. (a) Depth curve. (b) Pitch angle curve. (c) X rudder curve of 80 m control.

The Lévy flight quadruple-rudder allocation designed in this section can achieve the depth control; however, Figure 10(c) shows that the X rudder shakes strongly, which should be avoided in AUV’s control. As to the situation of X rudder curve, the rudder curve of 40 m control is similar to the curve of 80 m control. That situation is obviously related to Lévy flight’s random step character.

Optimization for Lévy flight quadruple-rudder allocation

The thought of multi-objective optimization is used to balance the influence of Lévy flight’s random step character, which is used to avoid locally optimal solution and is also one main cause of the X rudder shaking situation. Even though multi-objective optimization cannot make every objective get the best answers, it can balance every objectives in the best way. Considering the X rudder shaking situation shown in Figure 10(c), one new objective is designed, and it is shown as follows

min (| u_{δ L_current} - u_{δ L_former} |)

where u _{δL_current} is the current result of Lévy flight quadruple-rudder allocation, and it contains every rudder’s value in X rudder. u _{δL_former} is u _{δL_current}’s last time result. Equation (23) means that the adjacent results of Lévy flight quadruple-rudder allocation need the minimum changes in X rudder. So, the quadruple-rudder allocation problem can be described as follows

{\begin{cases} min J = (1 - ε) {‖ W_{v} (B u_{δ L_current} - δ_{V R}) ‖}_{2}^{2} + ε {‖ W_{u} u_{δ L_current} ‖}_{2}^{2} \\ min (| u_{δ L_current} - u_{δ L_former} |) \\ \underline{δ} \leq u_{δ L_current} \leq \bar{δ} \end{cases}

There are many methods to solve the multi-objective optimization, and one method called transformation²⁷ is chosen here. Its basic thought is to transform the multi-objective optimization to single-objective optimization according to the objective’s form. In order to achieve this goal, equation (23) is transformed into equation (25)

| u_{δ L_current} - u_{δ L_former} | \leq δ_{L min}, δ_{L min} > 0

where δ _Lmin is one matrix of adjustable parameters and its dimension is the same as u _δ . δ _Lmin is used to limit the changes of X rudder in Lévy flight quadruple-rudder allocation. So, the function of equation (25) is the same as the function of equation (23). With the help of equation (25), the multi-objective optimization problem shown in equation (24) can be changed to one single-objective optimization problem shown in equation (26)

{\begin{cases} min J = (1 - ε) ‖ W_{v} (B u_{δ L_current} - δ_{VR}) ‖_{2}^{2} + ε ‖ W_{u} u_{δ L_current} ‖_{2}^{2} \\ max (\underline{δ}, u_{δ L_former} - δ_{L min}) \leq u_{δ L_current} \leq min (\bar{δ}, u_{δ L_former} + δ_{L min}), δ_{L min} > 0 \end{cases}

Equation (26) can be solved by the method mentioned in ‘Quadruple-rudder allocation’ section. This optimized allocation can be called as quadruple-rudder allocation with Lévy flight character.

The simulations of 40 m depth control and 80 m depth control are used to test this quadruple-rudder allocation with Lévy flight character. X-rudder AUV’s origin situation, initial Lévy flight point’s situation, corresponding parameters and motion executor’s situation are the same as the depth control test in ‘Lévy flight quadruple-rudder allocation’ section. $δ_{L min} = {[0.25, 0.25, 0.25, 0.25]}^{T}$ . Figure 11 shows the control results.

Figure 11.

Depth control results by quadruple-rudder allocation with Lévy flight character. (a) Depth curve. (b) Pitch angle curve. (c) X rudder curve of 80 m control.

Compared with Figure 10(c) and 11(c), it can be found that the X rudder’s shaking situation is greatly restrained. Quadruple-rudder allocation with Lévy flight character is working.

Simulation and results

In order to show the control ability of quadruple-rudder allocation with Lévy flight character, which will be called QR with Lévy flight character in the following part, one comparison test with the quadruple-rudder allocation without Lévy flight character discussed in ‘X rudder resistance and fixed point iteration’ section is made, which will be called former QR in the following part. The whole process of following one lawn mower path in three dimension environment is simulated, and the process can be mainly separated into two steps. The first step is the control from X-rudder AUV’s start point (0,0,10) to expected depth (30 m) by certain yaw angle (30°). The second step is X-rudder AUV following the certain path after reaching the desired depth. The expected path is the connection of P₁ (850,1700,30), P₂ (3250,4100,30), P₃ (4450,2900,30), P₄ (2050,500,30), P₅ (3250,−700,30), P₆ (5650, 1700,30), P₇ (6850,500,30) and P₈ (4450,−1900,30). The control law is based on traditional PD control, and the path following strategy is designed by the line of sight guidance method.²⁸

The basic line of sight guidance method can be explained by Figure 12.

Figure 12.

One situation of line of sight guidance method.

In Figure 12, AUV will try to follow the path from P₁ to P₂. ψ_line, ψ_design and ψ_expect are the corresponding angles shown in Figure 12. ψ_line can be calculated by the coordinates of P₁ and P₂. ψ_design can be calculated by the following

ψ_{design} = arctan (d / L) = arctan (d / (l_{AUV} \times n)), n \in N_{+}

where d is the distance between AUV and P₁P₂, and l_AUV is the length of AUV. So, in Figure 12, expected yaw angle ψ_expect can be calculated as $ψ_{expect} = ψ_{line} - ψ_{design}$ . Actually, Figure 12 shows only one situation, and ψ_expect is different according to the relative position between AUV and the following path.

In simulation, the following assumptions and arrangements are made:

X-rudder AUV is neutrally buoyant.

All the preset points on the path are reachable.

The disturbance is ignored in the environment, so that the flow velocity to rudder is equal to X-rudder AUV’s velocity in equation (11).

The same parameters’ values are the same in QR with Lévy flight character and former QR. In the fixed point iteration part, $W_{v} = diag (1 / 35, 1 / 35, 1 / 35) and W_{u} = diag (1 / 35, 1 / 35, 1 / 35, 1 / 35)$ in equation (15) and $J_{end} = 0.00125$ in equation (18).

In the Lévy flight part, α = 0.01 in equation (19), β = 1.5 in equation (20) and $δ_{L min} = {[0.25, 0.25, 0.25, 0.25]}^{T}$ in equation (25). There are one initial Lévy flight point calculated by pseudoinverse and three extra initial Lévy flight points calculated by equation (22).

X-rudder AUV’s parameters have been mentioned in ‘X rudder’s layout and its main parameters’ section, and its original state is static. Its original yaw angle, pitch angle and X rudder’s angle are 0°. The maximum angular velocity of every rudder in X rudder is 1°/s. The range of X rudder is from −35° to +35°. X-rudder AUV’s expected speed is 3 m/s. The sample interval is 0.5 s. Runtime of the whole situation is 7000 s. The simulation results are shown in Figures 13 –17.

Figure 13.

X-rudder AUV’s path following results. AUV: autonomous underwater vehicle.

Figure 14.

Depth and yaw control results by quadruple-rudder allocation. (a) Depth controlled by former QR. (b) Yaw controlled by former QR. (c) Depth controlled by QR with Lévy flight character. (d) Yaw controlled by QR with Lévy flight character.

Figure 15.

The situation of roll angle.

Figure 16.

Number of iterations during former QR control.

Figure 17.

X-rudder angle curve of quadruple-rudder allocation. (a) X-rudder angle curve of former QR. (b) X-rudder angle curve of QR with Lévy flight character.

Figures 13–14 show that both former QR and QR with Lévy flight character can complete the mission. Their responds to the expected yaw angle are efficient and accurate. Figure 15 shows the changes of X-rudder AUV’s roll angle during the simulation. The maximum absolute value of roll angle controlled by former QR is 1.01° at t = 3580 s and the maximum absolute value of roll angle controlled by QR with Lévy flight character is 1.12° at t = 5066 s. That means X-rudder AUV controlled by both quadruple-rudder allocation methods mentioned in ‘X rudder resistance and fixed point iteration’ section and ‘Optimization for Lévy flight quadruple-rudder allocation’ section can achieve the lawn mower path following mission in a very stable way, and it will be essential for the work of acoustic and visual equipment on X-rudder AUV.

Figure 16 shows the number of iterations situation in former QR. The number of fixed point iteration does not increase forever, which means the fixed point iteration has run well. The situation of fixed point iteration in QR with Lévy flight character will be nearly the same as Figure 16.

Figure 17 shows the X-rudder’s changes during the simulation test. The X-rudder controlled by both methods does not appear the situation of strong shaking. According to Figure 17(a), the control load is mainly on Rudder 1 and Rudder 3 in former QR’s control. However, it seems X rudder’s operation of QR with Lévy flight character is more average than the operation of former QR.

Figure 18 shows the Re_Total’s curve calculated by equation (12) during the whole mission. During the calculation, the area of every rudder in X-rudder AUV is assumed as 1. The unit is not mentioned here, which will not affect the analysis conclusion. When the simulation ends, Re_Total of X rudder controlled by former QR is 1468 and Re_Total of X rudder controlled by QR with Lévy flight character is 964.4. This means that, during this mission, the X rudder’s operation by QR with Lévy flight character saves nearly 34.3% of the energy compared to the operation by former QR. It can help the X-rudder AUV work for longer time and decrease the damage of rudders.

Figure 18.

Results of X rudder’s Re_Total.

Conclusions

For the quadruple-rudder allocation method in X-rudder AUV, one control method associated with Lévy flight character is designed. The quadruple-rudder allocation with Lévy flight character combines the advantages of the pure Lévy flight method, which can solve the locally optimal problem, and it can avoid strong shaking problem in X rudder’s operation with the help of multi-objective optimization. The simulation shows that, in the lawn mower path following mission, this control method can offer the similar control ability, as the method is just based on pseudoinverse and fixed point iteration with less energy-consuming form X rudder’s operation. Even though this quadruple-rudder allocation with Lévy flight character relies on the core computer’s calculation ability heavily, as the development of microcomputer for AUV, it is believed that the quadruple-rudder allocation with Lévy flight character will show greater control ability and play more important role for X-rudder AUV’s control in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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