# A Strategy of Robust Control for the Dynamics of an Unmanned Surface Vehicle under Marine Waves and Currents

Authors: Manuel J. Velueta; Jose L. Rullan; Jose A. Ruz-Hernandez; H. Alazki
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This work addresses one of the most common problems for mobile robotics (autonomous navigation) but is applied to the dynamical model of a catamaran of small dimensions for monitoring and data acquisition applications. In this work, we present the study of the dynamics of a USV (Unmanned Surface Vehicle) under the presence of two simulated environmental perturbations: marine induced waves and currents. The mathematical model of the vehicle is studied and the equations that describe the behavior of environmental perturbations are also described. A numerical simulation of the model considering the effects of these perturbations is carried out in three degrees of freedom. Also, a strategy of robust control-based Sliding Mode Control (SMC) is developed for counteracting the effects of the perturbations over the trajectory of the USV.


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## Body

## 1. Introduction

Mobile robotics has been widely studied for a long time ago; very important advances in technology and science have allowed that autonomous navigation could be possible today. It was in the 70s when the first robot teleoperated was launched to the moon, and since then the advances for mobile robots are still developing and testing. The dynamic robots can be found in many areas such as industries, military, health systems, and security industry as many others; even in domestic environments, small robots clean the houses autonomously and have an auto turn-off feature that allows the energy saving when they finish their job.A variety of common surfaces as offices and homes, to the air, snow, and water, are the fields in which the robots can be displaced. The marine robotics has not been explored in depth, since at most of the algorithms it has been developed for terrestrial and aerial robots. A type of marine robots is the Unmanned Surface Vehicles (USVs) which are usually smaller boats used for testing control algorithms or practical applications like surveillance or data acquisition of water environments [1]. In [2] a control strategy for maneuvering of ships was tested for a model ship in a marine control laboratory; the results show the effectiveness of the proposed control strategy, but they are limited because it only tests the performance of the controller under uncertainties in the model; no external perturbations are added to the experiment. In [3] a novel autonomous system for water monitoring is presented; this system is able to have online updates on their missions through a network of sensors deployed in the area of interest.Dynamical modeling of USVs has been behavior since 1991; when in [4] one of the first works that deal with this task was presented, an autopilot based on linear and nonlinear control is also developed. The dynamical model is based on kinematic and kinetic studies through the Newton-Euler approach; this allows working with forces and moments acting over the vehicle in each degree of freedom.The environment in which a USV is working is always affecting its behavior because disturbances whose nature does not have a predictable behavior can be found. Several numerical simulations to test if a control strategy satisfies a task can be carried out and the mathematical models that describe the behavior of the disturbances must be known for being included in simulation and in the mathematical model of the USV. Some works like [2] define these environmental disturbances as uncertainties in the mathematical model and adaptive control is used for the geometrical control task addressed. Other authors that develop works like [5] use onboard sensors for measuring and estimating the environmental disturbances and the states of the vehicle through a nonlinear passive observer, specifically the observer, is used to estimate the wave velocity and the vehicle’s velocity relative to the wave.Station-keeping is also a very interesting problem in which the USV or ship must keep his place while the effects of the environmental disturbances are acting over the vehicle. Works like [6] develop LQR controllers and observers for this station-keeping task. Other works like [7] present the implementation of various controllers for the station-keeping problem of a USV under environmental perturbations; the dynamic of the perturbations is not considered in the model of the USV; however, they include a wind feed-forward control for counteracting the effects. In [8] an antidisturbance controller based on a disturbance observer (DO) is developed and tested through simulations and comparisons; the proposed method can be used for different scenarios and control tasks. The author uses the observer to estimate the lumped disturbance composed of parametric model uncertainties, modeling errors, and unknown environmental forces. The control law is based on the DO and a K matrix of gains for the controller has to be found. Also a neurodynamic optimization to solve and obtain the control inputs for the command governor is described and applied.A work for the path-following problem in which the marine currents are considered in the dynamical model, is presented in [9]; the effects of the disturbances are considered at the kinematic and kinetic level; the current velocities are constants and an illustrative example is given to verify the proposed method; also in [10], a path-following control design method is presented for autonomous underwater vehicles (AUV) subject to velocity and input constrains. The authors use a classic PI control technique for control the displacement of the AUV.In this paper, we take the existent mathematical model and the knowledge of the dynamics of two perturbations: marine waves and currents for simulating a mathematical model of a proposed USV for exploration of coastal zones. According to [11] the parameters of the perturbations can be generated by adding some randomness to the perturbations instead of having only constant values in the simulation. These two environmental perturbations show, in a simulation, a more realistic behavior that can be found in a real environment; this model under the perturbations open the possibilities for test this controller in a real prototype and helps to take into account the settings that must be taken in the real test. The combination of the model that includes the perturbation of waves and the generation of parameters via Gauss-Markov processes together with a control strategy SMC has not been developed by other authors. The model of the USV is described; also the equations to include in simulation corresponding to environmental perturbations (marine waves and currents) are presented; by last a robust control technique is used to reject the effects of the perturbations over the performance of the vehicle. The numerical simulation results and some conclusions of this paper are presented in the last section.


## 2. USV Model

The modeling task of marine vehicles is based on classical mechanics involving statics and dynamics principles and is derived in a local reference frame to take advantage of the vehicle’s geometrical properties. The model is based on the second Newton’s Law in terms of conservation of linear and angular moment; in [4] the reader can find a very detailed lecture of the modeling tasks.Let us define the following vectors:(1)η=η1η2T=xyzϕθψT(2)v=uvwpqrTin which η is the vehicle’s position and attitude vector with coordinates on the inertial frame (earth fixed) and v is the linear and angular velocity vector measured in the referential frame (body fixed). The complete model involving kinematic and kinetic studies is given by(3)η˙=Jη2v(4)Mv˙+Cvv+Dvv=τE+τwhere τ is the vector of external forces and moments acting on the rigid body and τE are the forces and moments of environmental disturbances. J(η2) is a diagonal matrix composed of two rotation matrices J1(η2) and J2η2 defined as(5)J1η2≔cψcθ-sψcϕ+cψsθsϕsψsϕ+cψcϕcθsψcθcψcϕ+sϕsθsψ-cψsϕ+sθsψcϕ-sθcθsϕcθcϕJ1η2-1=J1η2T∈OS3where OS(3): Orthogonal Special group (three dimension)and(6)J2η2≔1sϕtθcϕtθ0cϕ-sϕ0sϕcθcϕcθsγ=sin⁡γ,cγ=cos⁡γ,tγ=tan⁡γ⁡In Figure 1 the inertial and the referential frame can be observed. r is a vector that joins the gravity center with the origin of the inertial frame, rG is the vector of coordinates of the gravity center with respect to the referential frame, and rO is a vector between the origins of the inertial and referential frame.Figure 1
Frames: inertial (earth fixed) and referential (body fixed).Other parameters in the complete model areM=MRB+MA and C(v)=CRB(v)+CA(v) where MRB and CRB(v) are the inertia and Coriolis and centripetal force matrices and MA and v are the added mass matrices due to the inertia of the liquid surrounding the rigid body. Dv is the matrix of hydrodynamic drag that includes the linear and quadratic terms and finally τE are the forces and moments due to external disturbances. The matrices MRB, MA, CRB(v), CA(v), and Dv have the structure presented in [4]; the reader can refer to that bibliography to see in detail.When the vehicle is moving in an ideal fluid, thenMRB=MRBT>0 and MA=MAT. Cv is a skew-symmetric matrix and Dv is nonsymmetric and strictly positive matrix. In the matrix J2η2 it can be noted that there is a singularity at θ±90°. For ships, this is not a problem because the working space for the Pitch angle is much less.


## 3. Environmental Perturbations

In this section, the mathematical representation of three main disturbances found in water areas is introduced. These disturbances are wind, marine waves, and currents. The first two are commonly added to the dynamical model via superposition, that is,(7)τE=τEwi+τEwawhere τEwi and τEwa are the forces and moments of wind and waves, respectively, and for the currents, a new dynamical model in terms of relative velocities is presented in the last part of this section.


### 3.1. Wind

Wind is the movement of air relative to the surface of the earth. It is produced by differences in temperature in different locations and pressure changes; some works like [12, 13] address the task of modeling these disturbances for improving the design of the palettes of wind generators. In [14] a mathematical representation for considering this disturbance for marine vehicles is presented. For a marine vehicle at rest the vector of forces and moments due to the wind, τEwi, is defined as(8)τEwi=12ρwiVw2CXγwAFwCYγwALwCZγwAFwCKγwALwHLwCMγwAFwHFwCNγwALwLwhere ρwi is the air density, Vw is wind velocity, AFw and ALw are the front and lateral areas of the vehicle, respectively, at same time theHFw and HLw are centroids over water line of AFw and ALw, respectively, and γw is the relative wind angle to the front of the vehicle and is given by(9)γw=ψ-βw-πwith βw, wind direction (Figure 2). All wind coefficients CX(γw), CY(γw), CZ(γw), CK(γw), CM(γw), and CN(γw) are obtained via computational simulation or experiments in the wind tunnel [11].Figure 2
Definitions of relative wind angle.On the other hand, for a marine vehicle moving at a forward speed, the vectorτEwi adopts the following form: (10)τEwi=12ρwiVrw2CXγrwAFwCYγrwALwCZγrwAFwCKγrwALwHLwCMγrwAFwHFwCNγrwALwLwhere Vrw is the relative wind velocity and γrw is the relative angle with respect to the displacement of the vehicle, both defined as(11)Vrw=urw2+vrw2γrw=-arctan⁡2urw,vrwwith (12)urw=u-Vwcos⁡βw-ψvrw=v-Vwsin⁡βw-ψ


### 3.2. Marine Waves

Generally, in all the water zones this phenomenon is present, in a lesser or greater extent, and is water undulation produced over the sea due to the action of wind. Generally, the waves are moving over the sea surface at many different speeds and their effect finish at the beaches or coastal zones. There exist various works that address the modeling and generation of waves for different purposes ([2, 15]) and, generally, for energy conversion such as [3, 4]. In this paper, a first-order model of a wave as a perturbation is used.Consider the slope of the wavesi(t) for the i-th component defined as (13)sit=Ai2πλisin⁡weit+ϕiwhere Ai and λi are the amplitude and length of wave, respectively, wei is the encounter frequency, and ϕi is an aleatory phase uniformly distributed and constant with time in [0,2π) corresponding to the component of the i wave. The vector of forces and moments τEwa is given by [16](14)τEwa=∑i=1NρgBLTcos⁡βsit∑i=1N-ρgBLTsin⁡βsit000∑i=1N124ρgBLL2-B2sin⁡2βsi2twhere ρ is water density, g is the gravity, L,B, and T are the length, width, and craft of the vehicle, respectively, and β is the encounter angle of the vehicle. The angle β is defined as the angle formed between the stern line and the incident direction of the waves. Like the wind the effect of this perturbation on the vehicle can be appreciated in the drift when is sailing at low speed.


### 3.3. Marine Currents

Marine currents can be originated by variations in the temperature, salinity, and density between the different water volumes also can be produced by the wind friction at the surface of the water. These disturbances are water displacements produced inside the sea.Letv the generalized velocity vector be replaced with the hydrodynamic terms of the vector of relative velocities: (15)vr=v-vcwhere vc∈R6 is the velocity of the marine currents expressed in the reference frame. The vector vc has the following form: (16)vc=ucvcwc000Twith vcb=ucvcwcT obtained through (17)vcb=J1η2TvcEvcE∈R3 is the current velocity vector measured in the INERTIAL frame.Finally, the dynamic model, including the perturbations due the marine currents, is(18)η˙=Jη2vr+vcE0(19)Mv˙r+Cvrvr+Dvrvr=τE+τThe parameters such as velocities and directions of real environmental disturbances are commonly considered as stationary because the changes in these environments only vary according to the season of the year. In this work, first-orderGauss-Markov processes are used for generating the parameters for the waves, wind, and currents. Let the speed of currents be denoted by Vc, the angle of the currents and wind relative to the inertial frame as αc, and the angle of attack of currents and waves relative to the bow as βc. Then the following equations define the Gauss-Markov processes:(20)V˙c+μ1Vc=w1(21)α˙c+μ1αc=w2(22)β˙c+μ1βc=w3wi (i=1,2,3) are zero mean quasi-Gaussian white noise processes and μi≥0 (i=1,2,3) are constants. The vector vcE for the currents can be obtained through the following relation:(23)vcE=Vccos⁡αccos⁡βcVcsin⁡βcVcsin⁡αccos⁡βcto be included in (17).


## 3.1. Wind

Wind is the movement of air relative to the surface of the earth. It is produced by differences in temperature in different locations and pressure changes; some works like [12, 13] address the task of modeling these disturbances for improving the design of the palettes of wind generators. In [14] a mathematical representation for considering this disturbance for marine vehicles is presented. For a marine vehicle at rest the vector of forces and moments due to the wind, τEwi, is defined as(8)τEwi=12ρwiVw2CXγwAFwCYγwALwCZγwAFwCKγwALwHLwCMγwAFwHFwCNγwALwLwhere ρwi is the air density, Vw is wind velocity, AFw and ALw are the front and lateral areas of the vehicle, respectively, at same time theHFw and HLw are centroids over water line of AFw and ALw, respectively, and γw is the relative wind angle to the front of the vehicle and is given by(9)γw=ψ-βw-πwith βw, wind direction (Figure 2). All wind coefficients CX(γw), CY(γw), CZ(γw), CK(γw), CM(γw), and CN(γw) are obtained via computational simulation or experiments in the wind tunnel [11].Figure 2
Definitions of relative wind angle.On the other hand, for a marine vehicle moving at a forward speed, the vectorτEwi adopts the following form: (10)τEwi=12ρwiVrw2CXγrwAFwCYγrwALwCZγrwAFwCKγrwALwHLwCMγrwAFwHFwCNγrwALwLwhere Vrw is the relative wind velocity and γrw is the relative angle with respect to the displacement of the vehicle, both defined as(11)Vrw=urw2+vrw2γrw=-arctan⁡2urw,vrwwith (12)urw=u-Vwcos⁡βw-ψvrw=v-Vwsin⁡βw-ψ


## 3.2. Marine Waves

Generally, in all the water zones this phenomenon is present, in a lesser or greater extent, and is water undulation produced over the sea due to the action of wind. Generally, the waves are moving over the sea surface at many different speeds and their effect finish at the beaches or coastal zones. There exist various works that address the modeling and generation of waves for different purposes ([2, 15]) and, generally, for energy conversion such as [3, 4]. In this paper, a first-order model of a wave as a perturbation is used.Consider the slope of the wavesi(t) for the i-th component defined as (13)sit=Ai2πλisin⁡weit+ϕiwhere Ai and λi are the amplitude and length of wave, respectively, wei is the encounter frequency, and ϕi is an aleatory phase uniformly distributed and constant with time in [0,2π) corresponding to the component of the i wave. The vector of forces and moments τEwa is given by [16](14)τEwa=∑i=1NρgBLTcos⁡βsit∑i=1N-ρgBLTsin⁡βsit000∑i=1N124ρgBLL2-B2sin⁡2βsi2twhere ρ is water density, g is the gravity, L,B, and T are the length, width, and craft of the vehicle, respectively, and β is the encounter angle of the vehicle. The angle β is defined as the angle formed between the stern line and the incident direction of the waves. Like the wind the effect of this perturbation on the vehicle can be appreciated in the drift when is sailing at low speed.


## 3.3. Marine Currents

Marine currents can be originated by variations in the temperature, salinity, and density between the different water volumes also can be produced by the wind friction at the surface of the water. These disturbances are water displacements produced inside the sea.Letv the generalized velocity vector be replaced with the hydrodynamic terms of the vector of relative velocities: (15)vr=v-vcwhere vc∈R6 is the velocity of the marine currents expressed in the reference frame. The vector vc has the following form: (16)vc=ucvcwc000Twith vcb=ucvcwcT obtained through (17)vcb=J1η2TvcEvcE∈R3 is the current velocity vector measured in the INERTIAL frame.Finally, the dynamic model, including the perturbations due the marine currents, is(18)η˙=Jη2vr+vcE0(19)Mv˙r+Cvrvr+Dvrvr=τE+τThe parameters such as velocities and directions of real environmental disturbances are commonly considered as stationary because the changes in these environments only vary according to the season of the year. In this work, first-orderGauss-Markov processes are used for generating the parameters for the waves, wind, and currents. Let the speed of currents be denoted by Vc, the angle of the currents and wind relative to the inertial frame as αc, and the angle of attack of currents and waves relative to the bow as βc. Then the following equations define the Gauss-Markov processes:(20)V˙c+μ1Vc=w1(21)α˙c+μ1αc=w2(22)β˙c+μ1βc=w3wi (i=1,2,3) are zero mean quasi-Gaussian white noise processes and μi≥0 (i=1,2,3) are constants. The vector vcE for the currents can be obtained through the following relation:(23)vcE=Vccos⁡αccos⁡βcVcsin⁡βcVcsin⁡αccos⁡βcto be included in (17).


## 4. Control Design

An autonomous mobile system (a USV for this case) in turn needs another system that dictates where it should go, with what speed and direction to be able to fulfill its proposed goal of autonomous navigation. These systems are known as Guidance, Navigation and Control systems, GNC (Guidance, Navigation and Control), which contains the three subsystems mentioned for a good performance of the vehicle.It is desirable that the perturbed system (19) is robust to the marine perturbations described previously; it is also convenient that the controller used to be robust to errors in modeling and uncertainty in the parameters. Some works, like [2], use a nonlinear adaptive controller for maneuvering with experiments in a laboratory and also [9] uses the same technique but for a path-following task. In this section, a control law based on Sliding Mode Controller (SMC) [17] is designed.The Control by Sliding Modes is a technique of nonlinear control that has as main properties the accuracy and robustness and is about easy implementation and tuning. SMC systems are designed to bring the states of the system to a particular surface in the state space called the sliding surface. Once the sliding surface has been reached, the controller causes the states to remain in a nearby neighbor of the sliding surface. It is then divided into two parts; the first involves the design of a sliding surface such that the sliding movement satisfies the design specifications. The second part focuses on selecting the control law that will make the switching surface attractive for system states. The design procedure can be found in [18].First, a sliding surface function is introduced and defined as:(24)σe=ce+e˙,c>0where the error and its derivative are defined as(25)e=ηd-η(26)e˙=η˙d-η˙with ηd=xdydzdT and the current states η=xyzT. The main idea of SMC is to drive the sliding surface σ(e) to zero and once reached keep the states in the close neighborhood of it.We start from system (19) and rewrite the equation as(27)v˙r=M-1fvr+τE+M-1τwhere the elements corresponding to the Coriolis matrices and centripetal and damping forces have been grouped in the function(28)fvr=Cvrvr+Dvrvr+τE;the dynamics of the error are described as(29)e˙=η˙d-Jη2vr+vcEη˙e¨=ηd¨-Jη˙2vr+Jη2v˙r-v˙cESince it is treated with a vector of irrational currents, the derivative of the vector(30)v˙cE=0


### 4.1. Stability Analysis

LetV be a candidate Lyapunov function chosen as (31)V=12σ2To obtain an asymptotic stability in σ=0 the following conditions must be satisfied:(1)
V˙<0 for σ≠0(2)
limσ→∞⁡V=∞The derivative of (25) is(32)V˙=σσ˙=σe¨d+e˙d-Jη˙2+Jη2vr-Jη2M-1fvr+τE-Jη2M-1τwith fvr=Cvrvr+Dvrvr, defining(33)ϕη2,vr=e¨d+e˙d-Jη˙2+Jη2vr-Jη2M-1fvr+τE.Equation (32) can be rewritten as(34)V˙=σϕη2,vr-Jη2M-1τassuming (35)τ=-MJTη2uEquation (32) results in (36)V˙=σL+uwhere L=ϕ(η2,vr) is the undesirable dynamic of the system. Selecting the control law based on sliding mode has the following:(37)u=-Ksignσwith (38)signσ=-1ifσ>01ifσ<0∈-1,1ifσ=0and substituting into (36) we have the following expression: (39)V˙≤σL-σK=σL-KTo obtain an asymptotic stability in σ=0, let us choose(40)V˙≤-α2V,α>0Substituting (31) into (2) and combining with (2), to ensuring the asymptotic stability, the gain K can be computed as (41)K=σ+L


## 4.1. Stability Analysis

LetV be a candidate Lyapunov function chosen as (31)V=12σ2To obtain an asymptotic stability in σ=0 the following conditions must be satisfied:(1)
V˙<0 for σ≠0(2)
limσ→∞⁡V=∞The derivative of (25) is(32)V˙=σσ˙=σe¨d+e˙d-Jη˙2+Jη2vr-Jη2M-1fvr+τE-Jη2M-1τwith fvr=Cvrvr+Dvrvr, defining(33)ϕη2,vr=e¨d+e˙d-Jη˙2+Jη2vr-Jη2M-1fvr+τE.Equation (32) can be rewritten as(34)V˙=σϕη2,vr-Jη2M-1τassuming (35)τ=-MJTη2uEquation (32) results in (36)V˙=σL+uwhere L=ϕ(η2,vr) is the undesirable dynamic of the system. Selecting the control law based on sliding mode has the following:(37)u=-Ksignσwith (38)signσ=-1ifσ>01ifσ<0∈-1,1ifσ=0and substituting into (36) we have the following expression: (39)V˙≤σL-σK=σL-KTo obtain an asymptotic stability in σ=0, let us choose(40)V˙≤-α2V,α>0Substituting (31) into (2) and combining with (2), to ensuring the asymptotic stability, the gain K can be computed as (41)K=σ+L


## 5. Simulation Results

We give complete description of how the environmental perturbations are included in the simulation; Figure3 shows a block diagram to help the comprehension.Figure 3
A block diagram of the interaction of environmental perturbations and the model.The parameters used in the simulation are shown below.(42)M=13.11700013.1170003.653,Cv=00-13.117v0013.117u13.117v-13.117u0,Dv=14.1041u-0.52620000.03725v+0.1690000.1663r+0.0913τEwa=2163.105∑i=1Ncos⁡βsit-2163.105∑i=1Nsin⁡βsit2.9016∑i=1Nsin⁡2βsi2tFor the first-orderGauss-Markov processes,μ1=1, and for the sliding surface function the parameter c=1.By a recursive method, the gain K was found at a value of 6. Two different simulations are conducted to evaluate the proposed controller. In the first experiment, the reference states are chosen as(43)ηd=5,5,π4Tand in the second the desired states are (44)ηd=t,10sin⁡0.2t,arctan⁡yd˙xd˙TFor each simulation, the initial states are η(0)=[0,0,0] and the simulation time is 50 seconds. The results of the simulations are shown in a 3D animated world where it is easy to observe the path described by the vehicle under the SMC controller in Figure 4(a). Figure 4(b) shows the results of the first simulation in which the desired position is fixed, and the task is to achieve and maintain that position. Figure 4(c) shows the results of the second simulation; in this case the desired states are variable in time and the proposed controller should accomplish the task.Figure 4
The 3D model for animation in Simulink.
(a)
Prototype of the USV built-in SolidWorks (b)
The path described by the USV in the first simulation (c)
Sinusoidal path described by the USV in the second simulationThe results of the first simulation are shown in Figures5–9. In Figure 5 the comparison between desired states and the system response is shown; it can be seen that the controller fulfills its function of being robust against the perturbations. In Figure 5(a) the states x,y can be observed in individual axis while in Figure 5(b) the heading angle shows the convergence to the desired reference. Figure 6 shows the evolution of errors; they are maintained in a close neighborhood of zero once the desired state is reached (which would physically cause the boat to be seen at a glance but more closely there would be a slight vibration about its movement).Figure 5
Vehicle’s trajectories (individual axis).
(a)
States of the USV displacement (b)
Heading angle states of the USVFigure 6
Evolution of the position and heading errors.Figure 7
Control moment over z-axis.Figure 8
Control forces in X-axis.Figure 9
Control forces in Y-axis.Finally, in Figures7–9, a zoom of the control signals is observed; this effect of the high frequency in the control signals is called chattering and its characteristic and caused by the SMC. In practical applications this control signals cannot be applied directly to the actuators because they can be damaged or deteriorated. This does not mean that SMC cannot be used as a strategy of control; however, filtering techniques can be used for obtaining smooth control signals that can be applied to actuators preventing the damage on them.In Figures10–14 the results of the second simulation can be seen. Figures 10 and 11 show the response of the perturbed system and the errors and they are quite acceptable. The biggest error is the one from the heading angle, but it should be noted that it is too small and does not affect the performance of the controller. One more time Figure 11(a) shows the individual states of the displacement of the USV and Figure 11(b) shows the heading angle in comparison with the reference. A zoom of a portion of the control signals for each state is shown in Figures 12–14 with the chattering effect. In both experiments it can be seen that the proposed controller accomplishes the task in a good way; however the control signals need to be filtered before being applied to the actuators.Figure 10
Evolution of the position and heading errors.Figure 11
Vehicle’s trajectory (individual axis).
(a)
States of the USV displacement (b)
Heading angle states of the USVFigure 12
Control moment over z-axis.Figure 13
Control force in X-axis.Figure 14
Control force in Y-axis.


## 6. Conclusions

Study of the dynamics of a USV with the presence of environmental perturbations was presented in this paper; a robust controller based on SMC technique is developed for the USV under the presence of the perturbations. Also, a vehicle prototype using SolidWorks was constructed and the parameters of the hydrodynamic drag matrix were obtained through simulations in this software. It can be concluded that the presence of environmental perturbations has a strong impact on the dynamics of the vehicle and the SMC is a good choice of controller.



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*Source: 4704567-2019-03-11.xml*