# The Hydrodynamic Coefficient Analysis and Motion Control of the Lingyun Moveable Lander

Authors: Guoxin Li; Shaowei Zhang
Journal: Geofluids
 (2021)

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## Abstract

A moveable lander has the advantages of low cost and strong controllability and is gradually becoming an effective autonomous ocean observation platform. In this study, the hydrodynamic property of the Lingyun moveable lander, which has completed experiments in the Mariana Trench in 2020, is analyzed with the semiempirical method and computational fluid dynamic (CFD) method. We calculate the inertial hydrodynamic coefficients and viscous hydrodynamic coefficients of the lander. The results show that the CFD can provide the hydrodynamic property for the moveable lander’s design. The dynamic equations and kinematic equations are completely constructed combined with the hydrodynamic coefficients. Subsequently, this paper utilized the PID control method andS control method to control the motions of the lander. The simulation results show that the methods accurately follow the preplanned path.


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## Body

## 1. Introduction

At present, autonomous underwater vehicle (AUV) has been widely utilized in ocean exploration, marine science, marine sampling, and other fields [1]. Most of the existing AUVs work in shallow seas, such as for environmental mapping [2], the measurement of the synoptic observational field of current ocean surface radar [3], sample and data gathering [4–6], seabed inspection [7], and real-time water quality analysis [8]. The Lingyun moveable lander can provide assistance and guarantee for modular seabed landers operating at deep sea.The hydrodynamic coefficient is significant to moveable lander design, motion control, and cruise plan [9]. The moveable lander is subjected to complex hydrodynamic conditions, strong coupling, and strong nonlinearity, since there are many influencing factors, such as shapes, current ocean conditions, and vehicle motion conditions. However, no similar-shape moveable lander has been applied to the deep sea, and let it alone dive down to the Mariana Trench. This study is based on this kind of situation. The methods of hydrodynamic coefficient analysis include the empirical method, system identification, free model test, captive model test, and computational fluid dynamic (CFD). The empirical method can quickly calculate the hydrodynamic coefficients, but its accuracy depends on the accumulation of experience and the shape of AUV [10, 11]. Free model tests [12, 13] and captive model tests [14] can get many hydrodynamic coefficients directly, but these experiments are expensive, and the test equipment, such as large water tank and towing devices, is demanding, compared to the CFD method. Commonly, the captive model tests include a tow-tank experiment, rotating arm test, circular motion test, and planar motion mechanism (PMM). When we carry out a tow-tank test, the model is tilted into the pool by an angle (drift angle) concerning the direction of the drag, which can be used to measure velocity coefficients, et cetera. However, it is impossible to accurately measure the angular velocity coefficients and angular acceleration coefficients [15–18]. The rotating arm test makes it easy to measure individual rotate derivatives. However, the test equipment is large in scale and extremely expensive. The PMM test [19–21] can measure various hydrodynamic coefficients, but it is limited by the length of the towing tank. Using finite element software to calculate the hydrodynamic coefficient is a practical, convenient, and low-cost method. Moreover, with the development of computer technology, the calculation result is getting faster in speed and better in precision. When we calculate the coefficient with the CFD method, the model needs to be accurately constructed. The next step is the meshing, followed by the boundary condition setting. The last step is the coefficient fitting [22]. In [23], the hydrodynamic coefficients of the TUNA-SAND AUV are calculated with the CFD method. Researchers analyze the influence of surface waves on the hydrodynamic performance of the AUV in [24]. The hydrodynamic coefficients of a moving body with NACA0012 hydrofoil utilizing CFD are calculated, and the relationship between oscillation frequency, amplitude, and hydrodynamic coefficients is discussed in [25].In order to precisely control a moveable lander, its dynamic model needs to be established to meet hydrodynamic simulation and real-time control. As shown in Figure1, the Lingyun, which is similar to the shape of the autonomous underwater vehicle in [26], is a four-degree-of-freedom moveable lander that can control the motion of surge, roll, heave, and yaw. The related parameters of the Lingyun moveable lander are shown in Table 1.Figure 1
The Lingyun: (a) 3D model of the Lingyun; (b) Lingyun in the Mariana Trench.
(a)(b)Table 1
Parameters of the Lingyun.
ParameterSymbolUnitValueLengthLm1.25HeightHm0.93WidthWm0.47 (excluding thrusters)Superficial areaAm24.411727VolumeVm30.5066747Massmkg519.798The moment of inertia of theX axisIxkg·m230.717279The moment of inertia of theY axisIykg·m247.148837The moment of inertia of theZ axisIzkg·m265.763013Diameter of the propellerDmm134After calculating all the hydrodynamic coefficients, we establish the kinematic and dynamic model of the Lingyun lander. Then, the motion simulation is carried out by the PID control method andS control method [27].


### 1.1. Kinematic and Dynamic Model

Kinematic and dynamic models are the theoretical basis for navigation and control. In order to establish the dynamic model of a movable lander, it is necessary to obtain its hydrodynamic characteristics and carry out experimental research on numerical simulation pools based on the research experience of AUV. Through the experimental design, the parameters of the experimental conditions are reasonably selected, the experimental data are fitted by the least squares, and the viscous hydrodynamic coefficients of the moveable lander are obtained.The study of the motion of moveable landers generally uses two coordinate systems: body-fixed coordinate system (dynamic coordinate system) for studying the hydrodynamic characteristics and the inertial coordinate system (earth-fixed coordinate system) for describing the motion trajectory and position of the lander.Recommended by the International Towing Tank Conference (ITTC) and the Society of Naval Architects and Marine Engineers (SNAME), two coordinate systems are established, including the earth-fixed coordinate system and the body-fixed coordinate system, as shown in Figure2. The inertial coordinate system E−ξηψ is fixed to the earth. Origin E is located at the sea surface or sea at a certain point; the Eξ axis remains horizontal and takes the forward direction of the lander in the positive direction; the Eη axis is located on the horizontal plane where the Eξ axis is located; according to the right-hand law, the axis rotates 90 degrees clockwise; the Eζ axis perpendicular to the ξEη plane in the center of the earth direction is positive. The body-fixed coordinate system G−XYZ is a right-angle coordinate system based on the lander’s center of gravity G as the coordinate origin, and it is fixed to the lander. The X, Y, and Z axes are the intersections of horizontal, vertical, and midprofiles that pass through point G, respectively.Figure 2
Frames and elementary lander motions.The linear velocity and angle velocity of the lander are represented byu,v,w,p,q,r. The accelerations and angular accelerations of the moveable lander are represented by u̇,v̇,ẇ,ṗ,q̇,ṙ. Forces and moments are represented by X,Y,Z,K,M,N. The dynamic parameters of the moveable lander are defined as shown in Table 2:Table 2
Definition of dynamic parameters of a moveable lander.
AxisPositionForceMomentLinear velocityAngular velocityAttitude angleX axisxXKupφY axisyYMvqθZ axiszZNwrψThe control position and attitude of the moveable lander can be determined by the earth-fixed coordinate valuesξ0,η0,ζ0 of the origin of the body-fixed coordinate system and the attitude angles φ,θ,ψ of the body-fixed coordinate system (motion coordinate system) relative to the earth-fixed coordinate. φ is the heeling angle, tilted to the right as a positive direction. θ is the trim angle, which indicates the positive direction when the stern is tilted. ψ is the heading angle, where the right is turned positive.The motion equation should be established in the earth-fixed coordinate system and then converted to the body-fixed coordinate system to derive the motion equation of a moveable lander under the body-fixed coordinate system. The six-degree-of-freedom motion equation of a moveable lander is represented as the following vector form:(1)η̇=Jηυ,η=ξηζϕθφT,υ=uvwpqrT,where
(2)Jη=cosφcosθ−sinφcosϕ+cosφsinθsinϕsinφsinϕ+cosφcosϕsinθ000sinφcosθcosφcosϕ+sinϕsinθsinφ−cosφsinϕ+sinθsinφcosϕ000−sinθcosθsinϕcosθcosϕ0000001sinϕtanθcosϕtanθ0000cosϕ−sinϕ0000sinϕcosθcosϕ/cosθ.A moveable lander is regarded as a rigid body, based on the theoretical mechanics’ speed synthesis theorem, and on the basis of the establishment of the coordinate system, the kinematic model of a moveable lander is introduced by momentum definition:(3)mu̇−vr+wq−xGq2+r2+yGpq−ṙ+zGpr+q̇=X,mv̇−wp+ur−yGr2+p2+zGqr−ṗ+xGqp+ṙ=Y,mẇ−uq+vp−zGp2+q2+xGrp−q̇+yGrq+ṗ=Z,Ixṗ+Iz−Iyqr+myGẇ+pv−qu−zGv̇+ru−pw=K,Iyq̇+Ix−Izrp+mzGu̇+wq−vr−xGẇ+pv−uq=M,Izṙ+Iy−Ixpq+mxGv̇+ur−pw−yGu̇+qw−vr=N,where
(4)X=XI+Xvis+gX+τX,Y=YI+Yvis+gY+τY,Z=ZI+Zvis+gZ+τZ,K=KI+Kvis+gK+τK,M=MI+Mvis+gM+τM,N=NI+Nvis+gN+τN,where m is the mass of the moveable lander; xG,yG,zG are the coordinates of the center of gravities of the moveable lander; Ix,Iy,Iz are the moments of inertia of the moveable lander’s mass m with respect to the GX,GY,GZ axes; XI,YI,ZI,KI,MI,NI are the inertial hydrodynamic forces and moments to which the moveable lander was subjected; Xvis,Yvis,Zvis,Kvis,Mvis,Nvis are the viscous hydrodynamic forces and moments to which the moveable lander was subjected; τX,τY,τZ,τK,τM,τN are the thrusts provided by the moveable lander thrusters; and gX,gY,gZ,gK,gM,gN are the static forces and moments.


### 1.2. The Analysis of Hydrodynamic Coefficients

There are two significances in studying hydrodynamic characteristics. On the one hand, the stability of a moveable lander is studied from the perspective of maneuverability. On the other hand, the influence of hydrodynamics needs to be considered in the design of the control system.Generally speaking, the hydrodynamic characteristics and the hydrodynamic coefficients are related to the following factors of a moveable lander:(1)
The geometric shape of the moveable lander(2)
The motion state of the moveable lander referring to the motion parameters such as the linear velocity, angular velocity, and angular acceleration(3)
The property of the flow field, including the physical characteristics of the flow field and the geometric characteristics of the flow field(4)
Maneuver dynamical characteristicsIn general, hydrodynamics is related to the characteristics of the moveable lander, the motion characteristics, and the fluid characteristics. If the structure of the moveable lander has been determined and is operating in an infinite stream field, hydrodynamic forceFF is only related to the moveable lander’s motion characteristics. It can be represented in the following form:
(5)FF=fV,V̇,Ω,Ω̇.Equation (5) is a general expression of hydrodynamics.Taylor’s stage expansion is a valuable tool in studying hydrodynamic calculations. To simplify the problem, we assume that a moveable lander is moving in an infinite stream field, and the hydrodynamic effect is independent of its position in the flow field. Generally speaking, the hydrodynamics in its motion property is only related to the state of motion at that time, but not to the history of motion. Based on this assumption, only velocity and acceleration items are retained in hydrodynamic expansion, regardless of the derivatives of the second order and above of velocity to time.Inertial hydrodynamic coefficients are generally obtained according to empirical methods. According to the theory of potential flow, the fluid inertial force is linearly related to acceleration, and considering the left and right symmetry of the Lingyun, the expression of inertial force acting on the Lingyun is as follows:(6)FI=XIYIZIKIMINIT=Ru̇v̇ẇṗq̇ṙT,where
(7)R=12ρL3Xu̇′00000012ρL3Yv̇′012ρL4Kv̇′012ρL4Nv̇′0012ρL3Zẇ′012ρL4Mẇ′0012ρL4Yṗ′012ρL5Kṗ′012ρL5Nṗ′0012ρL4Zq̇′012ρL5Mq̇′0012ρL4Yṙ′012ρL5Kṙ′012ρL5Nṙ′,Kv̇′=Yṗ′,Nv̇′=Yṙ′,Mẇ′=Zq̇′,Nṗ′=Kṙ′,K11=0.2Xu̇′=−π/3B/LH/LK11=‐5.8589×10‐2,K22=1.3Yv̇′=−π/3B/LH/LK22=‐0.38083,K33=0.4Zẇ′=−π/3B/LH/LK33=‐0.11718,K55=0.8Mq̇′=−π/60B/LH/L1+B2/L2K55=‐1.33745×10‐2,K66=0.25Nṙ′=−π/60B/LH/L1+B2/L2K66=‐5.6888×10‐3.The viscous hydrodynamic coefficients of the moveable lander are related to its speed, i.e.,Fvis=fu,v,w,p,q,r. In the Taylor expansion of viscous hydrodynamics, the hydrodynamic items related only to linear velocity (u,v,w) are positional forces, whereas only the hydrodynamics related to angular velocity (p,q,r) is rotational forces, and the others are coupled hydrodynamics. Positional forces are generally obtained by the wind tunnel test or tow-tank test, while rotational force and coupling hydrodynamic power are obtained mainly through the rotating arm test. When we select the CFD method to obtain the hydrodynamic coefficients, an appropriate virtual boundary is established to convert the bypass flow problem into an internal flow problem, and the RANS equation is utilized to solve in the space region formed by the virtual boundary and the moveable lander.In this study, the CFD method is utilized to calculate the hydrodynamic coefficients, and the finite element software is used for simulation calculations. We mesh the lander in the virtual water tank at first in the preprocess software. The mesh is shown in Figure3. In the simulation of the tow-tank test, the number of mesh grids obtained by the section is about 1.02 million. Moreover, in the simulation of the rotating arm test, the number of mesh grids obtained by the section is about 1.05 million. There are many settings in the preprocess, including setting grid boundaries, inlet, outlet, and other related parameters, such as setting dynamic viscosity to 0.001219 kg/(m·s), setting density to1025.9 kg/m3, setting molar mass to 18.02 kg/kmol, setting specific heat capacity to 0.932 J/(kg·K), setting the inlet speed to 1 knot, and setting the relative pressure of the outlet to 0 Pa. The calculation and coefficient solution are based on the foregoing situation. Finally, in postprocessing, the calculated force and torque of 6 degrees of freedom, the velocity contour, the pressure contour, the streamline, et cetera can be obtained, and the relevant results are shown in Figures 4–7.Figure 3
Mesh: (a) mesh for the tow-tank test; (b) mesh for the rotating arm test.
(a)(b)Figure 4
Viscous hydrodynamic contour: (a) forceX contour; (b) force Y contour.
(a)(b)Figure 5
Contour and velocity streamline: (a) forceZ contour; (b) velocity stream line.
(a)(b)Figure 6
Pressure contour: (a) in entire simulation area; (b) in the Lingyun.
(a)(b)Figure 7
Velocity contour: (a) vertical plane; (b) horizontal plane.
(a)(b)The experiment is divided into three stages: the first stage calculates the navigational resistance of the moveable lander, the second stage is a single-plane test, and the third stage calculates the force in the coupling state and calculates the linear coupling derivative of hydrodynamics.Positional forces are calculated using thek−ε turbulence model, at the same inward flow speed U, the hydrodynamics with respect to different attack and drift angles are calculated, and the corresponding positional force coefficients are obtained by linear regression. In the simulation tow-tank test, the basin setting is shown in Figure 3(a): 5L long, 6W wide, and 6D high (L, W, and D are the length, width, and height of the Lingyun, respectively). The vertical and moveable lander axis of the basin forms a certain angle. It means that the inward flow and the direction of the moveable lander movement form a certain angle of attack angle α or drift angle β.For simulating the rotating arm test to calculate the rotational forces and coupled hydrodynamics of the moveable lander, the simulation computing environment is established and is shown in Figure3(b), which is rotated by a rectangular circle of 6W×6H around the center axis, and the arc length of the center arc of the fluid domain is 5L. The speed of the dynamic coordinate origin of the moveable lander is U, the angle of attack is α, and the drift angle is β; then, the conversion matrix T of the flow coordinate system to the body-fixed coordinate system is
(8)T=cosαcosβcosαsinβ−sinα−sinβcosβ0sinαcosβsinαsinβcosα.When the moveable lander rotates around theζ-axis of the earth-fixed coordinate system at an angular velocity ω, the components p,q,r of the angular velocity in the body-fixed coordinate system are
(9)pqr=T00ω=−ωsinα0ωcosα.The situation is similar when the moveable lander rotates around thex-axis and y-axis at the angular velocity ω.Viscous hydrodynamics includes the linear, nonlinear, and coupling hydrodynamics caused by the interaction of horizontal and vertical surface motion. The highest order is taken as second order, and the conditions of all computational simulations in this study are shown in Table3.Table 3
Calculation statistics table.
Test itemsConditionsNumbersDrag calculationU=0.25,0.5,0.75,1,1.25,1.5,1.75,2knotsα=β=08Positional force calculationU=1knot; β=0,α=−12,−11⋯⋯11,12U=1knot; α=0,β=−12,−11⋯⋯11,1248Coupled force calculationU=1knot; α=−3,3,6,9, β=−3,3,6,916Rotational force calculation (horizontal plane)U=1knot; R=5,10mα=0, β=−3,0,3,6,9,12β=0, α=−3,3,6,9,1244Rotational force calculation (vertical plane)U=1knot; R=5,10mα=0, β=−3,0,3,6,9,12β=0, α=−3,3,6,9,1244Total160The six-degree-of-freedom viscous hydrodynamics is shown as formulas (10), (11), (12), (13), (14), and (15). The simplified hydrodynamic coefficients on the six degrees of freedom fitted from the calculations results are shown in Table 4 (inertial hydrodynamic coefficients are included in the table). The viscous hydrodynamic coefficients are all dimensionless numbers, and for the viscous hydrodynamic coefficients, the dimensionless coefficient to u2,v2,w2 is 1/2ρL2, the dimensionless coefficient to q2,r2,rp is 1/2ρL4, and the dimensionless coefficient to vr,wq is 1/2ρL3, and for the viscous hydrodynamic moment coefficients, the dimensionless coefficient to u2,v2,w2 is 1/2ρL3, the dimensionless coefficient to q2,r2,rp is 1/2ρL5, and the dimensionless coefficient to vr,wq is 1/2ρL4. The relationship between angular velocity and forces (moments) in the rotating arm test is shown in Figure 8. The relationship between drag and velocity is shown in Figure 9.
(10)Xvis=Xuuu2+Xvvv2+Xwww2+Xrrr2+Xqqq2+Xvrvr+Xwqwq+Xprpr,(11)Yvis=Yvv+Yrr+Yrrrr+Ypp+Ypppp+Yvwvw+Yvvvv2+w2+Yvvvvv2+w2v+Yvqvq+Ywpwp+Ypqpq+Ywrwr+Yqrqr+Yww,(12)Zvis=Zww+Zqq+Zqqqq+Zwww2+Zvvv2+Zrrr2+Zppp2+Zwwww+Zwv2+w2wv2+w2+Zwwv2+w2qwwv2+w2q+Zwqwq+Zvpvp+Zprpr+Zvrvr+Zww,(13)Kvis=Kuu+Kvv+Kww+Kqq+Kuuu2+Krrr2+Kpppp+Kvvvv+Kvqvq+Kwpwp+Kwrwr+Kpqpq+Kuvuv+Kqqq2,(14)Mvis=Mww+Mqq+Mqqqq+Mrrr2+Mppp2+Mwv2+w2wv2+w2+Mwwv2+w2rwwv2+w2r+Mvrvr+Muu+Mvv,(15)Nvis=Nvv+Nrr+Nrrrr+Npp+Npppp+Nvwvw+Nvv2+w2vv2+w2+Nvvv2+w2rvvv2+w2r+Nvqvq+Nwpwp+Npqpq+Nwrwr+Nww.Table 4
Simplified hydrodynamic coefficient.
CoefValueCoefValueCoefValueXu̇−5.8589e−02Ywp1.0303904e+01Kpp−2.80136e+00Xuu−1.35758e−01Ypq5.5370142e+00Kwp7.22954e−01Xvv−8.89852e−01Ywr2.5674126e+01Kpq2.16510e−01Xww−1.00505e+00Yqr−4.9189501e+01Mq̇−1.33745e−02Xqq−1.82638e+00Zẇ−1.1718e−01Mqq1.33698e−01Xrr1.15448e+00Zqq−2.3971221e+00Mpp−8.35750e−01Xpr−2.50100e+00Zww1.4742334e+00Nṙ−5.6888e−03Yv̇−3.8083e−01Zpp4.8484061e+00Npp−8.41556e+00Yp3.70578e+04Zwv2+w2−1.3851717e+00Nvw1.00346e−01Ypp−8.47231e+00Zwq1.5969039e−01Nv/vv2+w2r3.95702e−02Yvv−7.85474e−01Zvp1.1612743e+00Nvq−5.01860e−01Yvv−1.00552e+00Zpr1.0412445e+00Nwp1.63202e+00Yvq2.35154e+01Zw−6.6430789e+00Nwr−1.64769e−01Figure 8
Relationship between angular velocity and forces and moments in the rotating arm test: (a) betweenFx and ω (β=0) in the horizontal plane; (b) between Fy and ω (β=0); (c) between N and ω (β=0); (d) between Fx and ω (α=0) in the horizontal plane; (e) between Fy and ω (α=0); (f) between N and ω (α=0); (g) between Fx and ω (β=0) in the vertical plane; (h) between Fz and ω (β=0); (i) between M and ω (β=0); (j) between Fx and ω (α=0) in the vertical plane; (k) between Fz and ω (α = 0); (l) between M and ω (α=0).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 9
Relationship between the drag force and velocity.According to equations (3), (10), (11), (12), (13), (14), and (15), the simplified six-degree-of-freedom space motion equations of the Lingyun are as follows:Surge motion equation:(16)mu̇−vr+wq−xGq2+r2+yGpq−ṙ+zGpr+q̇=12ρL4Xqq′q2+Xrr′r2+Xpr′pr+12ρL3Xu̇′u̇+12ρL2Xuu′u2+Xvv′v2+Xww′w2−W−Bsinθ+τX.Sway motion equation:(17)mv̇−wp+ur−yGr2+p2+zGqr−ṗ+xGpq+ṙ=Ypp+12ρL4Ypppp+Ypqpq+Yqrqr+12ρL3Yv̇v̇+Yvqvq+Ywpwp+Ywrwr+W−Bcosθsinφ+τY+12ρL2Yvvvvv2+w2v+Yvvvv2+w2.Heave motion equation:(18)mẇ−uq+vp−zGp2+q2+xGrp−q̇+yGrq+ṗ=Zww+12ρL4Zppp2+Zprpr+Zqqqq+12ρL3Zẇẇ+Zvpvp+Zwqwq+12ρL2Zwww2+Zwv2+w2wv2+w2+W−Bcosθcosφ+τZ.Roll motion equation:(19)Ixṗ+Iz−Iyqr+myGẇ−uq+vp−zGv̇−wp+ur=12ρL5Kpppp+Kpqpq+12ρL4Kwpwp+yGW−yCBcosθcosφ−zGW−zCBcosθsinφ+τK.Pitch motion equation:(20)Iyq̇+Ix−Izrp+mzGu̇−vr+wq−xGẇ−uq+vp=12ρL5Mq̇q̇+Mppp2+Mqqqq+xGW−xCBcosθcosφ−zGW−zCBsinθ+τM.Yaw motion equation:(21)Izṙ+Iy−Ixpq+mxGv̇−wp+ur−yGu̇−vr+wq=12ρL5Nṙṙ+Npppp+12ρL3Nvwvw+12ρL4Nwrwr+Nwpwp+Nvqvq+Nvvv2+w2rvvv2+w2r+xGW−xCBcosθsinφ+yGW−yCBsinθ+τN.


### 1.3. Control Allocation

The thruster configuration of the Lingyun is shown in Figures10 and 11. There are two horizontal thrusters utilized to control the surge motion and the yaw motion, and two vertical thrusts are utilized to control the heave motion and the roll motion. The control effectiveness matrix must first be determined to achieve precise control of the motion of four degrees of freedom by controlling the forces (thruster’s revolutions) on each thruster. Position vectors for the thruster configuration of the Lingyun are shown in Table 5. Orientation vectors for the thruster configuration of the Lingyun are shown in Table 6.Figure 10
Diagram of thruster configuration: (a) rear view; (b) top view.
(a)(b)Figure 11
Diagram of thruster configuration: (a) rear view; (b) top view.
(a)(b)Table 5
Position vectors for thruster configuration of the Lingyun.
Horizontal thrustersVertical thrustersr1HTr2HTr1VTr2VT−rhx−rhy0−rhxrhy00−rvy−rvz0rvy−rvzTable 6
Orientation vectors for thruster configuration of the Lingyun.
Horizontal thrustersVertical thrusterse1HTe2HTe1VTe2VT10010000−100−1The thrust and torque vectors generated by the thrusterTih can be expressed as
(22)τi=TiQi=TieiTiri×ei=iexieyiezri×eixri×eiyri×eizTi.By superimposition on a separate thrustτi, i=1~4, the total thruster force and torque vector are obtained τ:
(23)τ=τXτYτZτKτMτN=∑i=14τi=e1xe2xe3xe4xe1ye2ye3ye4ye1ze2ze3ze4zr1×e1xr2×e2xr3×e3xr4×e4xr1×e1yr2×e2yr3×e3yr4×e4yr1×e1zr2×e2zr3×e3zr4×e4z⏟TT1T2T3T4⏟f=Tf.In the formula above,T is the configuration matrix of the thruster and f is the control force vector. Replace the relevant parameter in the above formula with Ti=Kiui, and the following formula can be obtained:
(24)τ=e1xe2xe3xe4xe1ye2ye3ye4ye1ze2ze3ze4zr1×e1xr2×e2xr3×e3xr4×e4xr1×e1yr2×e2yr3×e3yr4×e4yr1×e1zr2×e2zr3×e3zr4×e4z⏟TK10000K20000K30000K4⏟Ku1u2u3u4⏟u=TKu,where K is the force coefficient matrix and u is the control vector.
(25)B=TK,whereB is the control effectiveness matrix. Thus, the thrust (moment) expression can be written as
(26)τ=Bu.Rows where all terms are zero inB indicate that the associated degrees of freedom cannot be directly controlled by the thruster configured by the Lingyun.
(27)r1×e1=00rhy,r2×e2=00−rhy,r3×e3=rvy00,r4×e4=−rvy00.To sum up, we get the control effectiveness matrixB:
(28)B=KT=K1100000000−1−100rvy−rvy0000rhy−rhy00.After removing uncontrollable degrees of freedom, we get(29)τ=τXτZτKτN⏟τ=KK0000−K−K00Krvy−KrvyKrhy−Krhy00⏟Bu1u2u3u4⏟u,where B is a simplified control matrix.After the dimensionless processing, the control effectiveness matrix can be obtained as(30)B¯=12120000−12−120012−1212−1200.


### 1.4. Motion Control and Simulation

In order to control the motion of the Lingyun, it is necessary to construct its kinematic and dynamic equations first. Based on the hydrodynamic coefficient calculated in the foregoing section and the analysis of hydrodynamic coefficients, the kinematic and dynamic equations are constructed and differential equations are solved by the fourth-order Runge-Kutta methods to verify the accuracy of the model. When a fixed thrust (moment) vectorτXτYτZτKτMτNT=3003003Tis given to the model, the motion shown in Figure 12 can be obtained. The speed on six degrees of freedom can be quickly converged by the relationship shown in Figures 13 and 14.Figure 12
3D motion simulation.Figure 13
Relationship between velocity, displacement, and motion simulation time: (a) velocityu; (b) velocity v; (c) velocity w; (d) displacement x; (e) displacement y; (f) displacement z.
(a)(b)(c)(d)(e)(f)Figure 14
The relationship between angular velocity, rotation angle, and simulation time: (a) angular velocityu; (b) angular velocity v; (c) angular velocity w; (d) roll angle; (e) pitch angle; (f) heading angle.
(a)(b)(c)(d)(e)(f)The control schematic is shown in Figure15. The PID controller and S controller are designed to control the Lingyun moving along the desired trajectory.Figure 15
The control schematic.The path controller utilizes three control functions to control heading, depth, and speed, respectively. All control parameters are found separately and put together in a vector, as shown in (31). PC is a vector of control variables in each degree of freedom [28].
(31)PC=PC,speed0PC,depth00PC,headingT.For a PID controller, the control law is given as(32)PC=−kpet−ki∫0Teτdτ−kdėt,wheree represents the error and ė represents the differential of the error.For anS controller, the control law is given as
(33)u=2.01.0+e−k1e−k2ė−1.0,espeed=ut−udt,edepth=zt−zdt,eheading=ψt−ψdt,ėheading=ψ̇t,where e and ė are input variables (error and error rate) and k1 and k2 are the control parameters for the corresponding error and derivative of the error, respectively.The control parametersPC are allowed for each thruster by using the thrust allocation matrix B¯†. The allocated thrust parameters, NRPM, are the values set as RPM (revolutions per minute) for each thruster [28].
(34)NRPM=B¯†PC.The control effect obtained by the PID control method is shown in Figure16, and the moveable lander can follow the desired path. As shown in Figure 17, the motion control effect obtained by the S control method is evidently improved compared with that shown in Figure 16, which can follow the desired path well, and the control effect can meet the requirements of engineering operation.Figure 16
PID control trajectory: (a) in the 3D space; (b) in thex-y plane; (c) in the y-z plane; (d) in the x-z plane.
(a)(b)(c)(d)Figure 17
S control trajectory: (a) in the 3D space; (b) in the x-y plane; (c) in the y-z plane; (d) in the x-z plane.
(a)(b)(c)(d)


## 1.1. Kinematic and Dynamic Model

Kinematic and dynamic models are the theoretical basis for navigation and control. In order to establish the dynamic model of a movable lander, it is necessary to obtain its hydrodynamic characteristics and carry out experimental research on numerical simulation pools based on the research experience of AUV. Through the experimental design, the parameters of the experimental conditions are reasonably selected, the experimental data are fitted by the least squares, and the viscous hydrodynamic coefficients of the moveable lander are obtained.The study of the motion of moveable landers generally uses two coordinate systems: body-fixed coordinate system (dynamic coordinate system) for studying the hydrodynamic characteristics and the inertial coordinate system (earth-fixed coordinate system) for describing the motion trajectory and position of the lander.Recommended by the International Towing Tank Conference (ITTC) and the Society of Naval Architects and Marine Engineers (SNAME), two coordinate systems are established, including the earth-fixed coordinate system and the body-fixed coordinate system, as shown in Figure2. The inertial coordinate system E−ξηψ is fixed to the earth. Origin E is located at the sea surface or sea at a certain point; the Eξ axis remains horizontal and takes the forward direction of the lander in the positive direction; the Eη axis is located on the horizontal plane where the Eξ axis is located; according to the right-hand law, the axis rotates 90 degrees clockwise; the Eζ axis perpendicular to the ξEη plane in the center of the earth direction is positive. The body-fixed coordinate system G−XYZ is a right-angle coordinate system based on the lander’s center of gravity G as the coordinate origin, and it is fixed to the lander. The X, Y, and Z axes are the intersections of horizontal, vertical, and midprofiles that pass through point G, respectively.Figure 2
Frames and elementary lander motions.The linear velocity and angle velocity of the lander are represented byu,v,w,p,q,r. The accelerations and angular accelerations of the moveable lander are represented by u̇,v̇,ẇ,ṗ,q̇,ṙ. Forces and moments are represented by X,Y,Z,K,M,N. The dynamic parameters of the moveable lander are defined as shown in Table 2:Table 2
Definition of dynamic parameters of a moveable lander.
AxisPositionForceMomentLinear velocityAngular velocityAttitude angleX axisxXKupφY axisyYMvqθZ axiszZNwrψThe control position and attitude of the moveable lander can be determined by the earth-fixed coordinate valuesξ0,η0,ζ0 of the origin of the body-fixed coordinate system and the attitude angles φ,θ,ψ of the body-fixed coordinate system (motion coordinate system) relative to the earth-fixed coordinate. φ is the heeling angle, tilted to the right as a positive direction. θ is the trim angle, which indicates the positive direction when the stern is tilted. ψ is the heading angle, where the right is turned positive.The motion equation should be established in the earth-fixed coordinate system and then converted to the body-fixed coordinate system to derive the motion equation of a moveable lander under the body-fixed coordinate system. The six-degree-of-freedom motion equation of a moveable lander is represented as the following vector form:(1)η̇=Jηυ,η=ξηζϕθφT,υ=uvwpqrT,where
(2)Jη=cosφcosθ−sinφcosϕ+cosφsinθsinϕsinφsinϕ+cosφcosϕsinθ000sinφcosθcosφcosϕ+sinϕsinθsinφ−cosφsinϕ+sinθsinφcosϕ000−sinθcosθsinϕcosθcosϕ0000001sinϕtanθcosϕtanθ0000cosϕ−sinϕ0000sinϕcosθcosϕ/cosθ.A moveable lander is regarded as a rigid body, based on the theoretical mechanics’ speed synthesis theorem, and on the basis of the establishment of the coordinate system, the kinematic model of a moveable lander is introduced by momentum definition:(3)mu̇−vr+wq−xGq2+r2+yGpq−ṙ+zGpr+q̇=X,mv̇−wp+ur−yGr2+p2+zGqr−ṗ+xGqp+ṙ=Y,mẇ−uq+vp−zGp2+q2+xGrp−q̇+yGrq+ṗ=Z,Ixṗ+Iz−Iyqr+myGẇ+pv−qu−zGv̇+ru−pw=K,Iyq̇+Ix−Izrp+mzGu̇+wq−vr−xGẇ+pv−uq=M,Izṙ+Iy−Ixpq+mxGv̇+ur−pw−yGu̇+qw−vr=N,where
(4)X=XI+Xvis+gX+τX,Y=YI+Yvis+gY+τY,Z=ZI+Zvis+gZ+τZ,K=KI+Kvis+gK+τK,M=MI+Mvis+gM+τM,N=NI+Nvis+gN+τN,where m is the mass of the moveable lander; xG,yG,zG are the coordinates of the center of gravities of the moveable lander; Ix,Iy,Iz are the moments of inertia of the moveable lander’s mass m with respect to the GX,GY,GZ axes; XI,YI,ZI,KI,MI,NI are the inertial hydrodynamic forces and moments to which the moveable lander was subjected; Xvis,Yvis,Zvis,Kvis,Mvis,Nvis are the viscous hydrodynamic forces and moments to which the moveable lander was subjected; τX,τY,τZ,τK,τM,τN are the thrusts provided by the moveable lander thrusters; and gX,gY,gZ,gK,gM,gN are the static forces and moments.


## 1.2. The Analysis of Hydrodynamic Coefficients

There are two significances in studying hydrodynamic characteristics. On the one hand, the stability of a moveable lander is studied from the perspective of maneuverability. On the other hand, the influence of hydrodynamics needs to be considered in the design of the control system.Generally speaking, the hydrodynamic characteristics and the hydrodynamic coefficients are related to the following factors of a moveable lander:(1)
The geometric shape of the moveable lander(2)
The motion state of the moveable lander referring to the motion parameters such as the linear velocity, angular velocity, and angular acceleration(3)
The property of the flow field, including the physical characteristics of the flow field and the geometric characteristics of the flow field(4)
Maneuver dynamical characteristicsIn general, hydrodynamics is related to the characteristics of the moveable lander, the motion characteristics, and the fluid characteristics. If the structure of the moveable lander has been determined and is operating in an infinite stream field, hydrodynamic forceFF is only related to the moveable lander’s motion characteristics. It can be represented in the following form:
(5)FF=fV,V̇,Ω,Ω̇.Equation (5) is a general expression of hydrodynamics.Taylor’s stage expansion is a valuable tool in studying hydrodynamic calculations. To simplify the problem, we assume that a moveable lander is moving in an infinite stream field, and the hydrodynamic effect is independent of its position in the flow field. Generally speaking, the hydrodynamics in its motion property is only related to the state of motion at that time, but not to the history of motion. Based on this assumption, only velocity and acceleration items are retained in hydrodynamic expansion, regardless of the derivatives of the second order and above of velocity to time.Inertial hydrodynamic coefficients are generally obtained according to empirical methods. According to the theory of potential flow, the fluid inertial force is linearly related to acceleration, and considering the left and right symmetry of the Lingyun, the expression of inertial force acting on the Lingyun is as follows:(6)FI=XIYIZIKIMINIT=Ru̇v̇ẇṗq̇ṙT,where
(7)R=12ρL3Xu̇′00000012ρL3Yv̇′012ρL4Kv̇′012ρL4Nv̇′0012ρL3Zẇ′012ρL4Mẇ′0012ρL4Yṗ′012ρL5Kṗ′012ρL5Nṗ′0012ρL4Zq̇′012ρL5Mq̇′0012ρL4Yṙ′012ρL5Kṙ′012ρL5Nṙ′,Kv̇′=Yṗ′,Nv̇′=Yṙ′,Mẇ′=Zq̇′,Nṗ′=Kṙ′,K11=0.2Xu̇′=−π/3B/LH/LK11=‐5.8589×10‐2,K22=1.3Yv̇′=−π/3B/LH/LK22=‐0.38083,K33=0.4Zẇ′=−π/3B/LH/LK33=‐0.11718,K55=0.8Mq̇′=−π/60B/LH/L1+B2/L2K55=‐1.33745×10‐2,K66=0.25Nṙ′=−π/60B/LH/L1+B2/L2K66=‐5.6888×10‐3.The viscous hydrodynamic coefficients of the moveable lander are related to its speed, i.e.,Fvis=fu,v,w,p,q,r. In the Taylor expansion of viscous hydrodynamics, the hydrodynamic items related only to linear velocity (u,v,w) are positional forces, whereas only the hydrodynamics related to angular velocity (p,q,r) is rotational forces, and the others are coupled hydrodynamics. Positional forces are generally obtained by the wind tunnel test or tow-tank test, while rotational force and coupling hydrodynamic power are obtained mainly through the rotating arm test. When we select the CFD method to obtain the hydrodynamic coefficients, an appropriate virtual boundary is established to convert the bypass flow problem into an internal flow problem, and the RANS equation is utilized to solve in the space region formed by the virtual boundary and the moveable lander.In this study, the CFD method is utilized to calculate the hydrodynamic coefficients, and the finite element software is used for simulation calculations. We mesh the lander in the virtual water tank at first in the preprocess software. The mesh is shown in Figure3. In the simulation of the tow-tank test, the number of mesh grids obtained by the section is about 1.02 million. Moreover, in the simulation of the rotating arm test, the number of mesh grids obtained by the section is about 1.05 million. There are many settings in the preprocess, including setting grid boundaries, inlet, outlet, and other related parameters, such as setting dynamic viscosity to 0.001219 kg/(m·s), setting density to1025.9 kg/m3, setting molar mass to 18.02 kg/kmol, setting specific heat capacity to 0.932 J/(kg·K), setting the inlet speed to 1 knot, and setting the relative pressure of the outlet to 0 Pa. The calculation and coefficient solution are based on the foregoing situation. Finally, in postprocessing, the calculated force and torque of 6 degrees of freedom, the velocity contour, the pressure contour, the streamline, et cetera can be obtained, and the relevant results are shown in Figures 4–7.Figure 3
Mesh: (a) mesh for the tow-tank test; (b) mesh for the rotating arm test.
(a)(b)Figure 4
Viscous hydrodynamic contour: (a) forceX contour; (b) force Y contour.
(a)(b)Figure 5
Contour and velocity streamline: (a) forceZ contour; (b) velocity stream line.
(a)(b)Figure 6
Pressure contour: (a) in entire simulation area; (b) in the Lingyun.
(a)(b)Figure 7
Velocity contour: (a) vertical plane; (b) horizontal plane.
(a)(b)The experiment is divided into three stages: the first stage calculates the navigational resistance of the moveable lander, the second stage is a single-plane test, and the third stage calculates the force in the coupling state and calculates the linear coupling derivative of hydrodynamics.Positional forces are calculated using thek−ε turbulence model, at the same inward flow speed U, the hydrodynamics with respect to different attack and drift angles are calculated, and the corresponding positional force coefficients are obtained by linear regression. In the simulation tow-tank test, the basin setting is shown in Figure 3(a): 5L long, 6W wide, and 6D high (L, W, and D are the length, width, and height of the Lingyun, respectively). The vertical and moveable lander axis of the basin forms a certain angle. It means that the inward flow and the direction of the moveable lander movement form a certain angle of attack angle α or drift angle β.For simulating the rotating arm test to calculate the rotational forces and coupled hydrodynamics of the moveable lander, the simulation computing environment is established and is shown in Figure3(b), which is rotated by a rectangular circle of 6W×6H around the center axis, and the arc length of the center arc of the fluid domain is 5L. The speed of the dynamic coordinate origin of the moveable lander is U, the angle of attack is α, and the drift angle is β; then, the conversion matrix T of the flow coordinate system to the body-fixed coordinate system is
(8)T=cosαcosβcosαsinβ−sinα−sinβcosβ0sinαcosβsinαsinβcosα.When the moveable lander rotates around theζ-axis of the earth-fixed coordinate system at an angular velocity ω, the components p,q,r of the angular velocity in the body-fixed coordinate system are
(9)pqr=T00ω=−ωsinα0ωcosα.The situation is similar when the moveable lander rotates around thex-axis and y-axis at the angular velocity ω.Viscous hydrodynamics includes the linear, nonlinear, and coupling hydrodynamics caused by the interaction of horizontal and vertical surface motion. The highest order is taken as second order, and the conditions of all computational simulations in this study are shown in Table3.Table 3
Calculation statistics table.
Test itemsConditionsNumbersDrag calculationU=0.25,0.5,0.75,1,1.25,1.5,1.75,2knotsα=β=08Positional force calculationU=1knot; β=0,α=−12,−11⋯⋯11,12U=1knot; α=0,β=−12,−11⋯⋯11,1248Coupled force calculationU=1knot; α=−3,3,6,9, β=−3,3,6,916Rotational force calculation (horizontal plane)U=1knot; R=5,10mα=0, β=−3,0,3,6,9,12β=0, α=−3,3,6,9,1244Rotational force calculation (vertical plane)U=1knot; R=5,10mα=0, β=−3,0,3,6,9,12β=0, α=−3,3,6,9,1244Total160The six-degree-of-freedom viscous hydrodynamics is shown as formulas (10), (11), (12), (13), (14), and (15). The simplified hydrodynamic coefficients on the six degrees of freedom fitted from the calculations results are shown in Table 4 (inertial hydrodynamic coefficients are included in the table). The viscous hydrodynamic coefficients are all dimensionless numbers, and for the viscous hydrodynamic coefficients, the dimensionless coefficient to u2,v2,w2 is 1/2ρL2, the dimensionless coefficient to q2,r2,rp is 1/2ρL4, and the dimensionless coefficient to vr,wq is 1/2ρL3, and for the viscous hydrodynamic moment coefficients, the dimensionless coefficient to u2,v2,w2 is 1/2ρL3, the dimensionless coefficient to q2,r2,rp is 1/2ρL5, and the dimensionless coefficient to vr,wq is 1/2ρL4. The relationship between angular velocity and forces (moments) in the rotating arm test is shown in Figure 8. The relationship between drag and velocity is shown in Figure 9.
(10)Xvis=Xuuu2+Xvvv2+Xwww2+Xrrr2+Xqqq2+Xvrvr+Xwqwq+Xprpr,(11)Yvis=Yvv+Yrr+Yrrrr+Ypp+Ypppp+Yvwvw+Yvvvv2+w2+Yvvvvv2+w2v+Yvqvq+Ywpwp+Ypqpq+Ywrwr+Yqrqr+Yww,(12)Zvis=Zww+Zqq+Zqqqq+Zwww2+Zvvv2+Zrrr2+Zppp2+Zwwww+Zwv2+w2wv2+w2+Zwwv2+w2qwwv2+w2q+Zwqwq+Zvpvp+Zprpr+Zvrvr+Zww,(13)Kvis=Kuu+Kvv+Kww+Kqq+Kuuu2+Krrr2+Kpppp+Kvvvv+Kvqvq+Kwpwp+Kwrwr+Kpqpq+Kuvuv+Kqqq2,(14)Mvis=Mww+Mqq+Mqqqq+Mrrr2+Mppp2+Mwv2+w2wv2+w2+Mwwv2+w2rwwv2+w2r+Mvrvr+Muu+Mvv,(15)Nvis=Nvv+Nrr+Nrrrr+Npp+Npppp+Nvwvw+Nvv2+w2vv2+w2+Nvvv2+w2rvvv2+w2r+Nvqvq+Nwpwp+Npqpq+Nwrwr+Nww.Table 4
Simplified hydrodynamic coefficient.
CoefValueCoefValueCoefValueXu̇−5.8589e−02Ywp1.0303904e+01Kpp−2.80136e+00Xuu−1.35758e−01Ypq5.5370142e+00Kwp7.22954e−01Xvv−8.89852e−01Ywr2.5674126e+01Kpq2.16510e−01Xww−1.00505e+00Yqr−4.9189501e+01Mq̇−1.33745e−02Xqq−1.82638e+00Zẇ−1.1718e−01Mqq1.33698e−01Xrr1.15448e+00Zqq−2.3971221e+00Mpp−8.35750e−01Xpr−2.50100e+00Zww1.4742334e+00Nṙ−5.6888e−03Yv̇−3.8083e−01Zpp4.8484061e+00Npp−8.41556e+00Yp3.70578e+04Zwv2+w2−1.3851717e+00Nvw1.00346e−01Ypp−8.47231e+00Zwq1.5969039e−01Nv/vv2+w2r3.95702e−02Yvv−7.85474e−01Zvp1.1612743e+00Nvq−5.01860e−01Yvv−1.00552e+00Zpr1.0412445e+00Nwp1.63202e+00Yvq2.35154e+01Zw−6.6430789e+00Nwr−1.64769e−01Figure 8
Relationship between angular velocity and forces and moments in the rotating arm test: (a) betweenFx and ω (β=0) in the horizontal plane; (b) between Fy and ω (β=0); (c) between N and ω (β=0); (d) between Fx and ω (α=0) in the horizontal plane; (e) between Fy and ω (α=0); (f) between N and ω (α=0); (g) between Fx and ω (β=0) in the vertical plane; (h) between Fz and ω (β=0); (i) between M and ω (β=0); (j) between Fx and ω (α=0) in the vertical plane; (k) between Fz and ω (α = 0); (l) between M and ω (α=0).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 9
Relationship between the drag force and velocity.According to equations (3), (10), (11), (12), (13), (14), and (15), the simplified six-degree-of-freedom space motion equations of the Lingyun are as follows:Surge motion equation:(16)mu̇−vr+wq−xGq2+r2+yGpq−ṙ+zGpr+q̇=12ρL4Xqq′q2+Xrr′r2+Xpr′pr+12ρL3Xu̇′u̇+12ρL2Xuu′u2+Xvv′v2+Xww′w2−W−Bsinθ+τX.Sway motion equation:(17)mv̇−wp+ur−yGr2+p2+zGqr−ṗ+xGpq+ṙ=Ypp+12ρL4Ypppp+Ypqpq+Yqrqr+12ρL3Yv̇v̇+Yvqvq+Ywpwp+Ywrwr+W−Bcosθsinφ+τY+12ρL2Yvvvvv2+w2v+Yvvvv2+w2.Heave motion equation:(18)mẇ−uq+vp−zGp2+q2+xGrp−q̇+yGrq+ṗ=Zww+12ρL4Zppp2+Zprpr+Zqqqq+12ρL3Zẇẇ+Zvpvp+Zwqwq+12ρL2Zwww2+Zwv2+w2wv2+w2+W−Bcosθcosφ+τZ.Roll motion equation:(19)Ixṗ+Iz−Iyqr+myGẇ−uq+vp−zGv̇−wp+ur=12ρL5Kpppp+Kpqpq+12ρL4Kwpwp+yGW−yCBcosθcosφ−zGW−zCBcosθsinφ+τK.Pitch motion equation:(20)Iyq̇+Ix−Izrp+mzGu̇−vr+wq−xGẇ−uq+vp=12ρL5Mq̇q̇+Mppp2+Mqqqq+xGW−xCBcosθcosφ−zGW−zCBsinθ+τM.Yaw motion equation:(21)Izṙ+Iy−Ixpq+mxGv̇−wp+ur−yGu̇−vr+wq=12ρL5Nṙṙ+Npppp+12ρL3Nvwvw+12ρL4Nwrwr+Nwpwp+Nvqvq+Nvvv2+w2rvvv2+w2r+xGW−xCBcosθsinφ+yGW−yCBsinθ+τN.


## 1.3. Control Allocation

The thruster configuration of the Lingyun is shown in Figures10 and 11. There are two horizontal thrusters utilized to control the surge motion and the yaw motion, and two vertical thrusts are utilized to control the heave motion and the roll motion. The control effectiveness matrix must first be determined to achieve precise control of the motion of four degrees of freedom by controlling the forces (thruster’s revolutions) on each thruster. Position vectors for the thruster configuration of the Lingyun are shown in Table 5. Orientation vectors for the thruster configuration of the Lingyun are shown in Table 6.Figure 10
Diagram of thruster configuration: (a) rear view; (b) top view.
(a)(b)Figure 11
Diagram of thruster configuration: (a) rear view; (b) top view.
(a)(b)Table 5
Position vectors for thruster configuration of the Lingyun.
Horizontal thrustersVertical thrustersr1HTr2HTr1VTr2VT−rhx−rhy0−rhxrhy00−rvy−rvz0rvy−rvzTable 6
Orientation vectors for thruster configuration of the Lingyun.
Horizontal thrustersVertical thrusterse1HTe2HTe1VTe2VT10010000−100−1The thrust and torque vectors generated by the thrusterTih can be expressed as
(22)τi=TiQi=TieiTiri×ei=iexieyiezri×eixri×eiyri×eizTi.By superimposition on a separate thrustτi, i=1~4, the total thruster force and torque vector are obtained τ:
(23)τ=τXτYτZτKτMτN=∑i=14τi=e1xe2xe3xe4xe1ye2ye3ye4ye1ze2ze3ze4zr1×e1xr2×e2xr3×e3xr4×e4xr1×e1yr2×e2yr3×e3yr4×e4yr1×e1zr2×e2zr3×e3zr4×e4z⏟TT1T2T3T4⏟f=Tf.In the formula above,T is the configuration matrix of the thruster and f is the control force vector. Replace the relevant parameter in the above formula with Ti=Kiui, and the following formula can be obtained:
(24)τ=e1xe2xe3xe4xe1ye2ye3ye4ye1ze2ze3ze4zr1×e1xr2×e2xr3×e3xr4×e4xr1×e1yr2×e2yr3×e3yr4×e4yr1×e1zr2×e2zr3×e3zr4×e4z⏟TK10000K20000K30000K4⏟Ku1u2u3u4⏟u=TKu,where K is the force coefficient matrix and u is the control vector.
(25)B=TK,whereB is the control effectiveness matrix. Thus, the thrust (moment) expression can be written as
(26)τ=Bu.Rows where all terms are zero inB indicate that the associated degrees of freedom cannot be directly controlled by the thruster configured by the Lingyun.
(27)r1×e1=00rhy,r2×e2=00−rhy,r3×e3=rvy00,r4×e4=−rvy00.To sum up, we get the control effectiveness matrixB:
(28)B=KT=K1100000000−1−100rvy−rvy0000rhy−rhy00.After removing uncontrollable degrees of freedom, we get(29)τ=τXτZτKτN⏟τ=KK0000−K−K00Krvy−KrvyKrhy−Krhy00⏟Bu1u2u3u4⏟u,where B is a simplified control matrix.After the dimensionless processing, the control effectiveness matrix can be obtained as(30)B¯=12120000−12−120012−1212−1200.


## 1.4. Motion Control and Simulation

In order to control the motion of the Lingyun, it is necessary to construct its kinematic and dynamic equations first. Based on the hydrodynamic coefficient calculated in the foregoing section and the analysis of hydrodynamic coefficients, the kinematic and dynamic equations are constructed and differential equations are solved by the fourth-order Runge-Kutta methods to verify the accuracy of the model. When a fixed thrust (moment) vectorτXτYτZτKτMτNT=3003003Tis given to the model, the motion shown in Figure 12 can be obtained. The speed on six degrees of freedom can be quickly converged by the relationship shown in Figures 13 and 14.Figure 12
3D motion simulation.Figure 13
Relationship between velocity, displacement, and motion simulation time: (a) velocityu; (b) velocity v; (c) velocity w; (d) displacement x; (e) displacement y; (f) displacement z.
(a)(b)(c)(d)(e)(f)Figure 14
The relationship between angular velocity, rotation angle, and simulation time: (a) angular velocityu; (b) angular velocity v; (c) angular velocity w; (d) roll angle; (e) pitch angle; (f) heading angle.
(a)(b)(c)(d)(e)(f)The control schematic is shown in Figure15. The PID controller and S controller are designed to control the Lingyun moving along the desired trajectory.Figure 15
The control schematic.The path controller utilizes three control functions to control heading, depth, and speed, respectively. All control parameters are found separately and put together in a vector, as shown in (31). PC is a vector of control variables in each degree of freedom [28].
(31)PC=PC,speed0PC,depth00PC,headingT.For a PID controller, the control law is given as(32)PC=−kpet−ki∫0Teτdτ−kdėt,wheree represents the error and ė represents the differential of the error.For anS controller, the control law is given as
(33)u=2.01.0+e−k1e−k2ė−1.0,espeed=ut−udt,edepth=zt−zdt,eheading=ψt−ψdt,ėheading=ψ̇t,where e and ė are input variables (error and error rate) and k1 and k2 are the control parameters for the corresponding error and derivative of the error, respectively.The control parametersPC are allowed for each thruster by using the thrust allocation matrix B¯†. The allocated thrust parameters, NRPM, are the values set as RPM (revolutions per minute) for each thruster [28].
(34)NRPM=B¯†PC.The control effect obtained by the PID control method is shown in Figure16, and the moveable lander can follow the desired path. As shown in Figure 17, the motion control effect obtained by the S control method is evidently improved compared with that shown in Figure 16, which can follow the desired path well, and the control effect can meet the requirements of engineering operation.Figure 16
PID control trajectory: (a) in the 3D space; (b) in thex-y plane; (c) in the y-z plane; (d) in the x-z plane.
(a)(b)(c)(d)Figure 17
S control trajectory: (a) in the 3D space; (b) in the x-y plane; (c) in the y-z plane; (d) in the x-z plane.
(a)(b)(c)(d)


## 2. Conclusions

In this paper, the analysis and calculation of the full set of hydrodynamic coefficients of the Lingyun moveable lander are completed by the CFD method, and the calculation results are meeting the needs of vehicle design modelling establishment. It is proven that the CFD can meet the dynamic analysis of the moveable lander and can simulate the tow-tank tests, rotating arm tests, et cetera. In this paper, kinematic equations and dynamic equations are established. The PID control method andS control method are utilized to simulate the motion of the Lingyun movable lander, and the simulation result is shown in the paper. The CFD method can offer an alternative (where funding allows) method to analyze the hydrodynamic property of the vehicle, which has the advantages of high efficiency, accurate calculation result, and low cost. And the simulation results can provide significant information for the design of the vehicle system.



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