# Research on Anti-Interference Nonlinear Adaptive Control Method for Autonomous Underwater Gliders

Authors: Wen-Qing Zhang; Liang-Long Da; De-Yan Tian; Qin-Dong Sun; Wen-Long Wang; Wen-Jing Chen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the characteristics of underactuation are analyzed according to the working principle of underwater glider, a complete 6DOF model of underwater glider is established, and the error equation of 3D path tracking is given. Based on the line-of-sight navigation method in three-dimensional space, the nonlinear adaptive control strategy is designed, and the roll angle control law, pitch angle control law, and depth control law of underwater glider are given. The tracking error of underwater glider is proved to be convergent by Lyapunov stability theorem. Finally, the results of marine experiments verify that the nonlinear controller designed in this paper can overcome the influence of constant current disturbance under certain conditions and has good robustness and good tracking effect under different ocean current sizes and course angles. Of course, one control method cannot be applicable to all sea conditions. When the angle between the size and direction of the ocean current exceeds a certain value, the tracking control of the underwater glider will show obvious divergence. Therefore, the stability range against the interference of the sea current is given in this paper.


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## Body

## 1. Introduction

Underwater gliders have the advantages of long endurance and low energy consumption and can be applied to a variety of complex sea conditions. More and more underwater unmanned mobile platforms such as underwater gliders, wave gliders, and underwater unmanned vehicles are used to carry out tasks. Research on nonlinear and underactuated characteristics of underwater unmanned mobile platform and related control methods has become a hot topic in the control field [1, 2].In recent years, the working mode of autonomous ocean vehicles has developed from the single operation mode to the cooperative formation operation mode [3]. Multiautonomous ocean vehicles can cooperate with each other to accomplish tasks that a single individual cannot accomplish. Therefore, the research on cooperative path tracking control of multiple autonomous marine vehicles has become a hot spot. In Dan Wang’s doctoral thesis [4] of Dalian Maritime University, aiming at the problem of cooperative path tracking control about autonomous marine vehicles with uncertainties and environmental disturbances, combined with dynamic surface control technology, the cooperative path tracking controllers of fully driven autonomous marine vehicles and underactuated autonomous marine vehicles are designed, respectively. A cooperative path tracking control strategy based on the adaptive dynamic surface method is proposed. Simulation results show the effectiveness of the proposed control algorithm. Zhang studied the cooperative tracking control of multiple UUVs to a given path under the constraint of master-slave formation. Based on the detailed analysis and summary of Lyapunov system theory and backstepping design theory, combined with the master-slave formation strategy, the cooperative formation strategy of master-slave multiunmanned underwater vehicles is designed according to the idea that UUV tracking error finally converges to zero. The cooperative tracking of the given desired path by multiple unmanned underwater vehicles under the constraint of master-slave formation is realized. And the designed cooperative path tracking controller is simulated and verified [5]. To solve the problem of 3D path tracking control for underactuated unmanned underwater vehicles, Hongjian et al. [6] designed a linear control term stabilized position tracking system in the form of tracking error feedback gain. Then, a dynamic controller is designed based on the backstepping method, which eliminates some nonlinear terms, simplifies the form of virtual control variables, and ensures the consistent and ultimately bounded state of the closed-loop tracking error system.There are many research studies on unmanned platforms with strong AUV maneuverability in domestic and foreign literatures [7–10], considering the essential difference between the underactuation of underwater gliders and AUV [11–15]. In this paper, a mathematical model of underactuated underwater glider is established and a nonlinear controller is designed under the condition of ocean current interference. Based on the line-of-sight navigation algorithm, the expected pitch angle and yaw angle are designed, and the axial thrust, pitch moment, and yaw moment are obtained. A path tracking strategy [16–18] for the underactuated autonomous underwater vehicle is proposed.


## 2. Underwater Glider Model Building

The structure composition of the underwater glider platform [19, 20] is shown in Figure 1. The glider’s workflow can be divided into four stages [21, 22]: surface preparation, gliding down, gliding up, and waiting on the surface. At the beginning of its voyage, an underwater glider floats on the surface of the water under the effect of net buoyancy. After receiving the control command through satellite communication or wireless communication on deck, the glider relies on the buoyancy regulating unit to return oil from the external oil sac to the internal oil cylinder, which reduces the drainage volume of the glider itself and makes the buoyancy less than gravity, thus sinking. At the same time, the attitude adjustment mechanism changes the position of the center of gravity by moving the internal weight so that the glider can reach the desired attitude angle. During the dive, it glides forward and down, using hydrodynamic forces on its horizontal and vertical fins, while mission sensors measure data. After reaching the predetermined working depth, the glider drains the oil from the internal cylinder to the external oil bag through the buoyancy adjusting unit, which increases the glider's own drainage volume and makes the buoyancy greater than gravity, thus realizing the transformation of the system movement from descending to ascending process. At the same time, the attitude adjustment mechanism also works to make it reach the attitude angle required for upward glide. During the gliding process [23, 24], the underwater glider adjusts its attitude in real time through the attitude adjustment mechanism and carries out stable gliding movement according to the set gliding angle and heading angle. If the glider needs to turn, the position of the center of gravity of the platform can be changed by the attitude adjustment mechanism, and the course angle can be deflected by the roll torque and hydrodynamic torque. After completing the task and returning to the water surface [25, 26], the underwater glider will make the communication antenna reach out of the water according to the preset program, carry out satellite positioning, and transmit the measured data to the control center through the satellite. At the same time, it will receive new control instructions and carry out the next cycle of work.Figure 1
Schematic diagram of underwater glider.The six-degree-of-freedom (6DOF) mathematical model of the underactuated underwater glider [27] is as follows:(1)X˙=JΩυ,Mυ˙+Cυυ+Dυυ+gX=τ,where JΩ=J1ΩO3×3O3×3J2Ω, J2Ω=1sinφtanθcosφtanθ0cosφ−sinφ0sinφ/cosθcosφ/cosθ, and J1Ω=cosψcosθcosψsinθsinφ−sinψcosφcosψsinθcosφ+sinψsinφsinψcosθsinψsinθsinφ+cosψcosφsinψsinθcosφ−cosψsinφ−sinθcosθsinφcosθcosφ. X=x,y,z,φ,θ,ψT and Ω=φ,θ,ψTare the position and angle in inertial coordinate system, and υ=u,v,w,p,q,rT indicates the speed in the moving coordinate system and meets the following conditions:(2)M=MT>0,M˙=0,Cυ=−CTυ,XTCυX=0,XTDυX>0,∀υ∈R6,∀X∈R6\0.The design goal of the control system in this paper is to design a feedback control law for the buoyancy, rolling moment, and pitching moment of the oil bag under the condition of unknown constant current interference and uncertain model parameters so that AUG positionx,y,zcan accurately track the desired trajectory xd,yd,zd, and all closed-loop system state variables are bounded. Figure 2 is the principle scheme of nonlinear control of underactuated underwater glider.Figure 2
Principle scheme of nonlinear control of underactuated AUG.


## 3. Nonlinear Controller Design

Define tracking error as follows:(3)e=e1,e2,e3T=QTX−Xd,where X=xt,yt,ztT∈R3 indicates the actual position of AUG, Xd=xds,yds,zdsT∈R3 is the desired path, and γds=arctanyds/xds.(4)ρds=arctan−zdsxd2s+yd2s,Q=cosγdcosρd−sinγdcosγdsinρdsinγdcosρdcosγdsinγdsinρd−sinρd0cosρd,where s is the expected path operator, and the corresponding relationship of the above symbols is as follows:(5)xd′s=dxdds,yd′s=dydds,zd′s=dzdds,x˙ds=∂xd∂ss˙,y˙ds=∂yd∂ss˙,z˙ds=∂zd∂ss˙.DefineXd,Yd,Zd as the desired path coordinate system; let ρr represent the angle between plane Xd,Yd and AUG actual speed X˙ and γr represent the angle between coordinate axesXd to X˙ projected on plane Xd,Yd.Then, the position error dynamics are(6)e˙1=e2γ˙dcosρd−e3ρ˙d+Hcosγrcosρr−Hd,e˙2=−e1γ˙dcosρd−e3γ˙dsinρd+Hsinγrcosρd,e˙3=e1ρ˙d+e2γ˙dsinρd−Hsinρd,where Hd=x˙d2+y˙d2+z˙d2 and H=x˙2+y˙2+z˙2.Firstly, according to the three-dimensional LOS line-of-sight navigation algorithm, we get(7)s˙=Hcosγrcosρr+δ1e1xd′2+yd′2+zd′2,(8)γr=arctan−e2Δ1,(9)υr=arctane3Δ2,where Δ1>0, Δ2=δ2e22+Δ12>0, δ1>0, and δ2>0 are the design parameters.(10)υδ=arcsinsinρdcosρrcosγr+cosρdsinρr,γδ=arctancosγdsinγrcosρr−sinρdsinρrsinγd+sinγdcosγrcosρdcosρr−sinγdsinγrcosρr−sinρdsinρrcosγd+cosγdcosγrcosρdcosρr.It can be known that the desired pitch angle and yaw angle are as follows:(11)θref=υδ−arctan−wu2+v2,ψref=γδ−arctanvu.Letws,θs,ψs be the filtered expected value of wref,θref,ψref. Next, design actually controls the buoyancy τw, rolling moment τp, and pitching momentτq of the input oil bladder.Define(12)K1=k11k12T=φ−φsθ−θsT,(13)K2=υ−a=u−a1v−a2w−a3p−a4q−a5r−a6T.The virtual input is designed as follows:(14)a3=ws,a4=cosφφ˙s−k11+sinφcosθθ˙s−k12,a5=−sinφφ˙s−k11+cosφcosθθ˙s−k12.Take the first derivative of both sides of equations (12) and (13) to obtain(15)K˙1=k˙11k˙12T=φ˙−φ˙sθ˙−θ˙sT,K˙2=J3pqT−φ˙sθ˙sT=J3GK2+a−φ˙sθ˙sT,where J3=cosφ−sinφsinφ/cosθcosφ/cosθ and G=000100000010.Then, we can get(16)K˙1=−K1+J3GK2.Find the first derivative of both sides and combine the dynamic equation of the AUG model to get(17)MK˙2=Mυ˙−a˙=τ−CυK2+a−DυK2+a−gX−Ma˙=τ−Cυ+DυK2−Cυ+Dυa−gX−Ma˙.The design actual control input is(18)τw=−b3k23+M−Xu˙a˙1+M−Zw˙wa5+Yv˙−Mva6+Xu+Xuuua1,τp=−b4k24−k11cosφ−k12sinφcosθ+Iy−Mq˙a˙5+Zw˙−Mwa1+M−Xu˙ua3+Nr˙−Izra4+Ix−Kp˙pa6+Mq+Mqqqa5,τq=−b5k25+k11sinφ−k12cosφcosθ+Iz−Nr˙a˙6+Yv˙−Mva1+Xu˙−Mua2+Iy−Mq˙qa4+Kp˙−Ixpa5+Nr+Nrrra6.


## 4. Stability Analysis

Theorem 1.
Assuming that the desired path is smooth and satisfiesηd,η˙d∈L∞ and Ωd,Ω˙d,Ω¨d∈L∞, using the controller designed in this paper, the tracking error e1,e2,e3 will eventually tend to zero; all signals in equation V1 below are uniformly and ultimately bounded.To prove: the Lyapunov function [28] is defined as follows:(19)V1=12K1TB1K1+12K2TMK2+12∑i=13ΩiTΓi−1Ω˜i+X˜1TPX˜1,where B1 = diagb1,b2 > 0 is the control parameter, and Γi=diagγij, X˜1=υ˜η˜T.Derive the two sides of the above formula, substitute them into formulas (16), (17), (21), and (25), and sort them out as follows:(20)V˙1=K1TB1K˙1+K2TMK˙2+∑i=13ΩiTΓi−1Ω˜˙i+X˜˙1TPX˜1+X˜1TPX˜˙1.LetA=−K3I−K40,(21)V˙1=−K1TB1K1−K2TDK2−K2TCK2+K2TGTJ2TZ1B1+τ−g−Da+Dυ−Ca+Cυ−Ma˙+∑i=13ΩiTΓi−1Ω˜˙i+X˜˙1TATPX˜1+X˜1TPAX˜˙1.According to formula (2), K2TCK2=0.Combined with formula (18), there is a positive definite matrix P so that ATP+PA=−Q, where Qis a Hermite matrix, so that(22)V˙1≤−K1TB1K1−K2TD+B2K2−X˜1TQX˜1≤−λminB1K12−λminD+B2K22−λminQX˜12.Then,(23)V˙1≤−μV1+δ,where(24)B2=diagb1b2b3b4b5b6,μ=min1,2λminB1,2λminD+B2λmaxM,λminQλmaxP,δ=12∑i=13Ω˜iTΓi−1Ω˜i,V1t≤V1t0e−μt−t0+δμ.Furtherly, we get(25)min12,12λminM,12γij−1,λminPXt2≤X∗t02max12,12λminM,12γ1j−1,12γ2j−1,12γ3j−1,λminPe−μt−t0+δμ,where Xt=Z1Z2Ω˜1Ω˜2Ω˜3X˜1T and(26)X∗t0=Z1t0,Z2t0,Ω˜1Tt0,Ω˜2Tt0,Ω˜3Tt0,X˜1t0T.Define Lyapunov function as follows:(27)V2=12e12+12e22+12e32,V˙2=e1e˙1+e2e˙2+e3e˙3.Substitute into formulas (6) and (7) to get(28)V˙2=−δ1e12+e2Hsinγ−γdcosρ−ρd−e3Hsinρ−ρd=−δ1e12+e2Hsink12+γrcosk11+ρr−e3Hsink11+ρr.Substitute into formulas (8) and (9) and sort it out as follows:(29)V˙2=−δ1e12−δ2HΔ22+e32e22cosk12cos11−e2Δ1sink12cosk11−HΔ22+e32Δ12+e22e2e3Δ1sink12sin11−e22e3sink11cosk12−HΔ22+e32Δ2e3sink11+e32cosk11.Asud≥ud,min>0, there is a finite time t1≥t0≥0 which makesH≥Hmin>0, ∀t≥t1. And there is a finite timet2≥t1 which makesk1≤k1,ss, where V˙2k1,ss<0, defined on the interval t1,t2, V˙2≥0, H,cosk11,sink11,sink12,cosk12∈L∞. Therefore, for e1t0e2t0e3t0 from any point, e1t0, e2t0, and e3t0, when t⟶∞, e1,e2,e3⟶0.


## 5. Experimental Verification

The physical parameters of the underwater glider platform in the numerical simulation experiments and sea trials are shown in Table1. The correctness and effectiveness of the path tracking control method studied above are verified.Table 1
Lots of AUG model parameters.
VariablesValuesM69kgmp18 kgmr18 kgL2.17 mIx0.54kg⋅m2Iy19.75kg⋅m2Iz19.82kg⋅m2Xu˙1.3kgYv˙79.66kgZw˙60.53kgKp˙0kg⋅m2Mq˙21.83kg⋅m2Nn˙23.56kg⋅m2Xu−50kg/sYv−100kg/sZw−100kg/sYv−80kg/sZw−80kg/sKp−20kg⋅m2/sMq−35kg⋅m2/sNr−35kg⋅m2/sXuu−12.37kg/mYvv19.35kg/mZww0kg/mKpp0kg⋅m2Mqq21.83kg⋅m2Nrr23.56kg⋅m2


### 5.1. Interference Experiments with Different Sea Conditions

The nonlinear adaptive control algorithm designed in this paper has been programmed into embedded program and loaded into the embedded control system of underwater glider. The embedded control system uses STM32F407 as the core processor, integrates FPU and DSP instructions, and the main frequency of the system is 72 MHz, which can carry out 32-bit floating point operation. C/OS-II real-time operating system is adopted, which can realize high speed operation and large capacity storage. Figure3 shows the motion control test process of an underwater glider in the South China Sea.Figure 3
Underwater glider tests in the South China Sea.Underwater glider test parameter setting: the coordinate system adopts the southeast geocoordinate system, the test time is 30 h, and the control frequency is 10 Hz; the path is designed as a straight line, the length of the path is about 30 km, and the path node set is set as follows:xd=0,5,10,15,20,25 and yd=0,0,0,0,0,0, and the unit is km. Single platform control parameters are as follows: roll angle control kp=1200, ki=0, and kd=−22; trim angle control kp=220, ki=10, and kd=−30; and depth control kp=1500, ki=10, and kd=−150. Parameters of nonlinear adaptive control module are as follows: K1=I2×2, K2=I6×6, Δ1=3.2, δ1=1.2, and δ2=2.0, where I is the identity matrix.


#### 5.1.1. The Size of the Ocean Current is 1.0 kN, and It Runs under the Condition of 8° Angle with the Direction of the Set Path

Under the condition that the path of the glider is a straight line, the size of the ocean current is 1.0 kN, and the angle between the direction of the ocean current and the direction of the path is 8°; the glider is operated by the nonlinear adaptive control method proposed in the paper.Figure4 shows the tracking process of the actual path and the expected path of the underwater glider. It can be seen from the figure that, in the case of unknown ocean current interference and unknown model parameters in the model, the controller can make the underwater glider track the expected trajectory stably, and the tracking error is within 10 m, achieving the expected effect.Figure 4
Stable running process of underwater glider tracking control.During the underwater glider test, the maximum gliding depth was set as 1000 m. The gliding process of profile is shown in Figure5. It can be seen that the underwater glider carried out repeated gliding movement at a depth of 0–1000 m, and the depth control process was stable.Figure 5
Depth profile control of underwater glider.As shown in Figure6, it can be seen that the trim angle of the underwater glider is stable tracking set value. The diving value is −30° and the floating value is +30°, which achieves the expected effect.Figure 6
Trim angle control process of underwater glider.As shown in Figure7, it can be seen that the roll angle of the underwater glider is stable and tracks the set value, and the actual roll angle control achieves the desired effect.Figure 7
Roll angle control process of underwater glider.


#### 5.1.2. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 10°

Under the condition that all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set to be 1.0 kN and the included angle with the path direction is 10°. The running results are shown in Figure8. The tracking control of the glider is in a critical stable state with the tracking error within 50 m.Figure 8
Critical stabilization process of underwater glider tracking control.


#### 5.1.3. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 12°

When all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set at 1.0 kN and the included angle with the path is 12°. The running results are shown in Figure9, and the tracking control of the glider is gradually divergent.Figure 9
Divergence process of underwater glider tracking control.


### 5.2. Comparison Experiment of Control Effect

On the basis of the test in 5.1, all test parameters and control parameters of the underwater glider are kept unchanged. The control effect of the traditional PID control algorithm and the nonlinear adaptive control algorithm designed in this paper is compared.


#### 5.2.1. Traditional Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. During the test, the expected yaw angle of the glider was set at 10°, the current size was 1.0 kN, and the included angle between the glider and the set heading was 8°. The test results are obtained in Figure 10. It can be found that the yaw angle of the underwater glider is controlled around 10°, and the trim angle has obvious oscillation within the range of up and down fluctuation of 2°. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the current amplitude or the angle between the current direction and the set course increases, the divergence of yaw angle control will appear. Therefore, the stable range of the traditional control method against the ocean current interference at this time is within the size of the ocean current 1.0 kN and within the angle of 8° with the set course.Figure 10
Pitch angle control process with the nonlinear active disturbance rejection control algorithm.


#### 5.2.2. Nonlinear Adaptive Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. Parameters of nonlinear adaptive control module are as follows: K1=I2×2, K2=I6×6, Δ1=3.2, δ1=1.2, and δ2=2.0, where I is the identity matrix. The experimental conditions and ocean current disturbance are consistent with those of traditional methods. The yaw angle of the underwater glider was set as expected value of 10°, the current size was 1.0 kN, and the included angle with the set heading was 8°. The test results are obtained in Figure 11. It can be found that the yaw angle of the underwater glider is controlled in the vicinity of 10° and within the range of fluctuation of 0.5°, which reaches the control expectation. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the size of ocean current is 1.0 kN and the angle between it and the set course is increased to 10°, the critical oscillation phenomenon will appear in the yaw angle control. Therefore, the stable range of the nonlinear adaptive control method designed in this paper against the ocean current interference is within 1.0 kN of the ocean current size and within 10° of the angle with the set course. The stability range of the nonlinear adaptive control method is larger than that of the traditional method, and the control effect is more obvious. However, for the step response index of the control algorithm, the steady-state running time is 0.5 s. Since the control frequency of the underwater glider is slower than that of 10 Hz, there is no obvious difference in the response time between the two control methods, and both of them can meet the dynamic course adjustment process of the underwater glider.Figure 11
Pitch angle control process with the PID control algorithm.


#### 5.2.3. Stable Range of Nonlinear Adaptive Control Method

The nonlinear adaptive control method for underwater vehicle interference designed in this paper cannot be applied to all interference situations. Of course, there is no control algorithm that can be applied to all interference situations. The applicable scope of the control method of this paper is shown in Figure12. Figure 12 shows the stability analysis of the formation of underwater gliders under the interference of ocean currents.Figure 12
Stability analysis of formation cooperative control under ocean current disturbance.


## 5.1. Interference Experiments with Different Sea Conditions

The nonlinear adaptive control algorithm designed in this paper has been programmed into embedded program and loaded into the embedded control system of underwater glider. The embedded control system uses STM32F407 as the core processor, integrates FPU and DSP instructions, and the main frequency of the system is 72 MHz, which can carry out 32-bit floating point operation. C/OS-II real-time operating system is adopted, which can realize high speed operation and large capacity storage. Figure3 shows the motion control test process of an underwater glider in the South China Sea.Figure 3
Underwater glider tests in the South China Sea.Underwater glider test parameter setting: the coordinate system adopts the southeast geocoordinate system, the test time is 30 h, and the control frequency is 10 Hz; the path is designed as a straight line, the length of the path is about 30 km, and the path node set is set as follows:xd=0,5,10,15,20,25 and yd=0,0,0,0,0,0, and the unit is km. Single platform control parameters are as follows: roll angle control kp=1200, ki=0, and kd=−22; trim angle control kp=220, ki=10, and kd=−30; and depth control kp=1500, ki=10, and kd=−150. Parameters of nonlinear adaptive control module are as follows: K1=I2×2, K2=I6×6, Δ1=3.2, δ1=1.2, and δ2=2.0, where I is the identity matrix.


### 5.1.1. The Size of the Ocean Current is 1.0 kN, and It Runs under the Condition of 8° Angle with the Direction of the Set Path

Under the condition that the path of the glider is a straight line, the size of the ocean current is 1.0 kN, and the angle between the direction of the ocean current and the direction of the path is 8°; the glider is operated by the nonlinear adaptive control method proposed in the paper.Figure4 shows the tracking process of the actual path and the expected path of the underwater glider. It can be seen from the figure that, in the case of unknown ocean current interference and unknown model parameters in the model, the controller can make the underwater glider track the expected trajectory stably, and the tracking error is within 10 m, achieving the expected effect.Figure 4
Stable running process of underwater glider tracking control.During the underwater glider test, the maximum gliding depth was set as 1000 m. The gliding process of profile is shown in Figure5. It can be seen that the underwater glider carried out repeated gliding movement at a depth of 0–1000 m, and the depth control process was stable.Figure 5
Depth profile control of underwater glider.As shown in Figure6, it can be seen that the trim angle of the underwater glider is stable tracking set value. The diving value is −30° and the floating value is +30°, which achieves the expected effect.Figure 6
Trim angle control process of underwater glider.As shown in Figure7, it can be seen that the roll angle of the underwater glider is stable and tracks the set value, and the actual roll angle control achieves the desired effect.Figure 7
Roll angle control process of underwater glider.


### 5.1.2. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 10°

Under the condition that all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set to be 1.0 kN and the included angle with the path direction is 10°. The running results are shown in Figure8. The tracking control of the glider is in a critical stable state with the tracking error within 50 m.Figure 8
Critical stabilization process of underwater glider tracking control.


### 5.1.3. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 12°

When all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set at 1.0 kN and the included angle with the path is 12°. The running results are shown in Figure9, and the tracking control of the glider is gradually divergent.Figure 9
Divergence process of underwater glider tracking control.


## 5.1.1. The Size of the Ocean Current is 1.0 kN, and It Runs under the Condition of 8° Angle with the Direction of the Set Path

Under the condition that the path of the glider is a straight line, the size of the ocean current is 1.0 kN, and the angle between the direction of the ocean current and the direction of the path is 8°; the glider is operated by the nonlinear adaptive control method proposed in the paper.Figure4 shows the tracking process of the actual path and the expected path of the underwater glider. It can be seen from the figure that, in the case of unknown ocean current interference and unknown model parameters in the model, the controller can make the underwater glider track the expected trajectory stably, and the tracking error is within 10 m, achieving the expected effect.Figure 4
Stable running process of underwater glider tracking control.During the underwater glider test, the maximum gliding depth was set as 1000 m. The gliding process of profile is shown in Figure5. It can be seen that the underwater glider carried out repeated gliding movement at a depth of 0–1000 m, and the depth control process was stable.Figure 5
Depth profile control of underwater glider.As shown in Figure6, it can be seen that the trim angle of the underwater glider is stable tracking set value. The diving value is −30° and the floating value is +30°, which achieves the expected effect.Figure 6
Trim angle control process of underwater glider.As shown in Figure7, it can be seen that the roll angle of the underwater glider is stable and tracks the set value, and the actual roll angle control achieves the desired effect.Figure 7
Roll angle control process of underwater glider.


## 5.1.2. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 10°

Under the condition that all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set to be 1.0 kN and the included angle with the path direction is 10°. The running results are shown in Figure8. The tracking control of the glider is in a critical stable state with the tracking error within 50 m.Figure 8
Critical stabilization process of underwater glider tracking control.


## 5.1.3. The Size of the Ocean Current is 1.0 kN, and the Angle with the Path Direction is 12°

When all test parameters and parameters of the glider remain unchanged, the size of the ocean current is set at 1.0 kN and the included angle with the path is 12°. The running results are shown in Figure9, and the tracking control of the glider is gradually divergent.Figure 9
Divergence process of underwater glider tracking control.


## 5.2. Comparison Experiment of Control Effect

On the basis of the test in 5.1, all test parameters and control parameters of the underwater glider are kept unchanged. The control effect of the traditional PID control algorithm and the nonlinear adaptive control algorithm designed in this paper is compared.


### 5.2.1. Traditional Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. During the test, the expected yaw angle of the glider was set at 10°, the current size was 1.0 kN, and the included angle between the glider and the set heading was 8°. The test results are obtained in Figure 10. It can be found that the yaw angle of the underwater glider is controlled around 10°, and the trim angle has obvious oscillation within the range of up and down fluctuation of 2°. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the current amplitude or the angle between the current direction and the set course increases, the divergence of yaw angle control will appear. Therefore, the stable range of the traditional control method against the ocean current interference at this time is within the size of the ocean current 1.0 kN and within the angle of 8° with the set course.Figure 10
Pitch angle control process with the nonlinear active disturbance rejection control algorithm.


### 5.2.2. Nonlinear Adaptive Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. Parameters of nonlinear adaptive control module are as follows: K1=I2×2, K2=I6×6, Δ1=3.2, δ1=1.2, and δ2=2.0, where I is the identity matrix. The experimental conditions and ocean current disturbance are consistent with those of traditional methods. The yaw angle of the underwater glider was set as expected value of 10°, the current size was 1.0 kN, and the included angle with the set heading was 8°. The test results are obtained in Figure 11. It can be found that the yaw angle of the underwater glider is controlled in the vicinity of 10° and within the range of fluctuation of 0.5°, which reaches the control expectation. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the size of ocean current is 1.0 kN and the angle between it and the set course is increased to 10°, the critical oscillation phenomenon will appear in the yaw angle control. Therefore, the stable range of the nonlinear adaptive control method designed in this paper against the ocean current interference is within 1.0 kN of the ocean current size and within 10° of the angle with the set course. The stability range of the nonlinear adaptive control method is larger than that of the traditional method, and the control effect is more obvious. However, for the step response index of the control algorithm, the steady-state running time is 0.5 s. Since the control frequency of the underwater glider is slower than that of 10 Hz, there is no obvious difference in the response time between the two control methods, and both of them can meet the dynamic course adjustment process of the underwater glider.Figure 11
Pitch angle control process with the PID control algorithm.


### 5.2.3. Stable Range of Nonlinear Adaptive Control Method

The nonlinear adaptive control method for underwater vehicle interference designed in this paper cannot be applied to all interference situations. Of course, there is no control algorithm that can be applied to all interference situations. The applicable scope of the control method of this paper is shown in Figure12. Figure 12 shows the stability analysis of the formation of underwater gliders under the interference of ocean currents.Figure 12
Stability analysis of formation cooperative control under ocean current disturbance.


## 5.2.1. Traditional Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. During the test, the expected yaw angle of the glider was set at 10°, the current size was 1.0 kN, and the included angle between the glider and the set heading was 8°. The test results are obtained in Figure 10. It can be found that the yaw angle of the underwater glider is controlled around 10°, and the trim angle has obvious oscillation within the range of up and down fluctuation of 2°. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the current amplitude or the angle between the current direction and the set course increases, the divergence of yaw angle control will appear. Therefore, the stable range of the traditional control method against the ocean current interference at this time is within the size of the ocean current 1.0 kN and within the angle of 8° with the set course.Figure 10
Pitch angle control process with the nonlinear active disturbance rejection control algorithm.


## 5.2.2. Nonlinear Adaptive Control Method

Single platform control parameters are as follows: roll angle control:kp=1200, ki=0, and kd=−22; trim angle control: kp=220, ki=10, and kd=−30; and depth control: kp=1500, ki=10, and kd=−150. Parameters of nonlinear adaptive control module are as follows: K1=I2×2, K2=I6×6, Δ1=3.2, δ1=1.2, and δ2=2.0, where I is the identity matrix. The experimental conditions and ocean current disturbance are consistent with those of traditional methods. The yaw angle of the underwater glider was set as expected value of 10°, the current size was 1.0 kN, and the included angle with the set heading was 8°. The test results are obtained in Figure 11. It can be found that the yaw angle of the underwater glider is controlled in the vicinity of 10° and within the range of fluctuation of 0.5°, which reaches the control expectation. When the current amplitude decreases or the angle between the current direction and the set course decreases, the yaw angle control converges stably. When the size of ocean current is 1.0 kN and the angle between it and the set course is increased to 10°, the critical oscillation phenomenon will appear in the yaw angle control. Therefore, the stable range of the nonlinear adaptive control method designed in this paper against the ocean current interference is within 1.0 kN of the ocean current size and within 10° of the angle with the set course. The stability range of the nonlinear adaptive control method is larger than that of the traditional method, and the control effect is more obvious. However, for the step response index of the control algorithm, the steady-state running time is 0.5 s. Since the control frequency of the underwater glider is slower than that of 10 Hz, there is no obvious difference in the response time between the two control methods, and both of them can meet the dynamic course adjustment process of the underwater glider.Figure 11
Pitch angle control process with the PID control algorithm.


## 5.2.3. Stable Range of Nonlinear Adaptive Control Method

The nonlinear adaptive control method for underwater vehicle interference designed in this paper cannot be applied to all interference situations. Of course, there is no control algorithm that can be applied to all interference situations. The applicable scope of the control method of this paper is shown in Figure12. Figure 12 shows the stability analysis of the formation of underwater gliders under the interference of ocean currents.Figure 12
Stability analysis of formation cooperative control under ocean current disturbance.


## 6. Conclusions

In this paper, the path tracking problem of underactuated underwater glider under the condition of ocean current interference is studied. Combined with the line-of-sight navigation method, an underactuated path tracking model of underwater glider is established and a nonlinear adaptive controller is designed. The roll angle control law, pitch angle control law, and depth control law of underwater glider are given. Through a large number of sea experiments, the control effect of underwater glider path following under different ocean current sizes and directions is analyzed. (1) Under the condition that the size of the ocean current is 1.0 kN and the angle with the latitude direction is 8°, the tracking control of the underwater glider runs stably. (2) Under the condition that the ocean current size is 1.0 kN and the angle with the latitude direction is 10°, the tracking control of the underwater glider is critical stable. (3) The tracking control of the glider gradually diverges under the condition that the ocean current size is 1.0 kN and the angle with the latitude direction is 12°. (4) Compared with the traditional control algorithm, the nonlinear adaptive control algorithm designed in this paper has better control effect under the same sea conditions. It can be seen that when the size and direction of ocean current meet certain conditions, the controller proposed in this paper can overcome the influence of constant ocean current disturbance and unknown model parameters and has good robustness and good tracking effect. Of course, one control method cannot be applicable to all sea conditions. When the angle between ocean current size and direction exceeds a certain value, the tracking control of the underwater glider will show obvious divergence. Therefore, the stability range against ocean current interference is given in this paper (Figure12).



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*Source: 9998061-2021-08-10.xml*