Second Order Sliding Mode Control Scheme for an Autonomous Underwater Vehicle with Dynamic Region Concept

The main goal in developing closed loop control system for an Autonomous Underwater Vehicle (AUV) is to make a robust vehicle from natural and exogenous perturbations such as wind, wave, and ocean currents. However a well-known robust control, for instance, Sliding Mode Controller (SMC), gives a chattering effect and it influences the stability of an AUV. Furthermore, some researchers combined other controls to get better result but it tends to present long computational time and causes large energy consumption.Thus, this paper proposed a Super Twisting Sliding Mode Controller (STSMC) with dynamic region concept for an AUV. STSMC or a second order SMC is adopted as a robust controller which is free from chattering effect. Meanwhile, the implementation of dynamic region is useful to reduce the energy usage. As a result, the proposed controller obtains global asymptotic stability which is validated by using Lyapunov-like function. Moreover, some simulations present the efficiency of proposed controller. In conclusion, STSMC with region based control is effective to be applied for the robust tracking of an AUV. It contributes to give a fast response when handling the perturbations, short computational time, and low energy demand.


Introduction
The development of control stability for an Autonomous Underwater Vehicle (AUV) has gained much attention from many researchers since several years ago [1].This happened because of the fundamental role of an AUV for replacing human involved in dangerous underwater activities for instance in underwater rescues, military purposes, underwater pipe inspections, oil and gas explorations, and so forth [2].
The robust tracking of an AUV against some natural disturbances is the main problem in this field [3,4].Whereas a precision of tracking desire trajectory is important to be obtained, so that dissipation of energy can be avoided [5].To cope with this problem, a range number of robust control systems have been introduced.Each control had its advantages and disadvantages.For this reason, many researchers coupled two or three different controllers or unified them with another control technique to improve its performance.
One example of conventional robust control was Proportional-Integral-Derivative (PID) controller [6].It was a simple control but had difficulties not only for setting an appropriate value of its gain but also for reaching the expected requirement.Then, researchers combined PID with several methods such as PID with Fuzzy Control (FC), PID with selftuning technique, and PID with Genetic Algorithm (GA) [7][8][9].The combination of PID with several methods focused on how to determine the gain value automatically.The control combination showed some improvements although there was a drawback like needed long computational steps which affected low real time execution.
Then, researchers adapted Linear Parameter Verifying (LPV) control and  ∞ robust technique as the other methods [10,11].However, the result of LPV control depicted good achievement only at minimum of sampling time, while  ∞ robust technique presented smooth performance even though there was misstracking in some places.Furthermore,

Kinematics and Dynamics of a 6-DOF AUV
The new formulation of robust control can be designed by considering the modeling system of a 6-DOF underwater vehicle.It involves a study on kinematics and dynamics system.Kinematics model is concerned with the equilibrium of the body at both rest and moving with certain velocity, while dynamics model is concerned with acceleration of the body motion.The studies of these were mainly discussed in [23].

Kinematic Model.
The kinematics model has a correlation between inertial frame and body-fixed velocity of vehicle.It can be described by using the Jacobian matrix ( 2 ) in the following form [23]: where denotes the orientation of the vehicle which are expressed in the inertial-fixed frame. 1 and  2 are the transformation matrices described in Euler angles formation. Here, the linear and angular velocity vectors, respectively, which are described in terms of the body-fixed frame relative to the earth's fixed frame.The illustration can be seen in Figure 1.

Dynamic Model.
The dynamic equation of underwater vehicle can be expressed in closed form as Equation ( 2) is changed with respect to the Jacobian matrix as the following equation [23]: where   ∈ R 6 is the inertia matrix including the added mass,   (V, ) ∈ R 6 represents the matrix of the Coriolis and centripetal terms including the added mass,   (V, ) ∈ R 6 denotes the hydrodynamic damping and lift force matrix, and   ( 2 ) ∈ R 6 is vector of gravitational force and moment or the restoring force, while  is the vector of generalized forces acting on the vehicle and can be written as the sum of estimated dynamics disturbances (control input).There are numerous parameters which are required to be known.To reduce the complexity of the model, the dynamic equation in (3) preserves the following properties [23].

Super Twisting Sliding Mode Control Scheme with Dynamic Region Concept
The basic idea of super twisting sliding mode control with dynamic region concept is removing chattering effect, building a robust vehicle, and at the same time also saving the energy consumption.Before determining the proposed control, first it is necessary to construct a state space function of 6-DOF dynamic model: From ( 4), it can be reformed to be a matrix as Then, the proposed control is defined as where  = force acting at the center mass of an AUV,  st = super twisting control law, and  eq = equivalent control law.
From the above equation, final force consists of super twisting control ( st ) which is added with equivalent control ( eq ).After that, the equation  st is determined as the following step: where  1 = discontinuous time derivative and  2 = continuous function of sliding variable.The value of derivative  1 is where  denotes a sliding vector and its value is proposed as where η  = virtual velocity (the value is formulated in the  eq process) and  = control parameter.It is necessary to integrate (8) to get  1 .Then, formula of  2 is determined as where  is a constant value.
Next is formulating the region boundary or  eq .This concept begins by replacing the ordinary trajectory with a desire region.The step to determine the specific region can be seen as the following inequality function: where Δ = ( −   ) ∈ R 6 ,   = reference trajectory, and  = total number of objective function.
For instance, the desire region is described as 2D with inequality function given in (12) and shown in Figure 2: where  is a regional bound which consists of   and   .After that, define the formula of potential energy for desire region.This step is useful as an energy consumption evaluation.The inequality function is described as Here,  ep denotes a positive constant.Note that when the AUV enters the bound or (Δ) ≤ 0, the gradient of EP(Δ) becomes smaller.Then, the region error can be evaluated by partial differentiating ( 13) yielding Therefore, where Δ  denotes region error.
Remark 2.   triggers the AUV toward the desire region.
Once the AUV is inside the region, the gradient of potential energy EP(Δ) becomes zero and at the same time Δ  reduces smoothly to zero.Next step determines the value η  based on the region error as Before formulating the final  eq , it is needed to differentiate (9) and then multiply both sides with   as in the following step: From ( 3), the value of   η is equal to  − −(  η +  η +  ).Therefore, (18) changes to Given ṡ = 0, then the final equation of equivalent control is proposed as The value of η  is obtained by differentiating (16) as Remark 3. Equation of equivalent control ( eq ) can keep the state variables on the sliding surface by excluding the uncertainties in the dynamic equation.
The final proposed control equation ( 6) is transformed as Theorem 4. The control law which is expressed in (22) with respect to the dynamic equation of an AUV in (3) guarantees the global asymptotically stability of close-loop control system.
Proof.See Appendix.
All steps of super twisting sliding mode control with region boundaries are resumed in the general scheme as shown in Figure 3.

Simulation Results
The proposed control is applied on the ODIN vehicle (AUV) [24].It is a 6-DOF holonomic AUV which has spherical shape.Further information can be seen in [24].Here, an AUV is ordered to follow a straight red line trajectory as well as region trajectory.The region trajectory is defined as the following inequality: where , , and  and , , and  components represent the region and orientation specification, respectively, while  ∈ R 6 is the error tolerance.Moreover, AUV starts from "start" sign.
In the middle of tracking activity, the AUV is suddenly attacked by some perturbations.The perturbations are given as velocity and their value is deterministic.Technical specifications of simulation are listed as follows: where , , and  are assumed as wind velocity, wave velocity, and ocean current velocity, respectively.Furthermore, the hitting period is about 10 s and exists in between 50 s and 60 s of time tracking.First, proposed controller is employed under line trajectory.The results in Figures 4 and 5 showed that an AUV could track precisely on the line.However, its tracking performance was disturbed after the presence of perturbation.Then, from Figure 6, it can be seen that an AUV needed 15 s to come back to desire trajectory.Meanwhile, the energy usage is presented in Figures 7 and 8.In the second case, line tracking trajectory is replaced by a region boundary.Its tracking performances are plotted in Figures 9 and 10.Although there were some disturbances, the AUV could keep on its position (inside the region).As a result, the amount of energy consumption is lower than in case 1. Detailed information regarding error position and energy consumption can be seen in Figures 11-13.
To point out the differences with others, the proposed control is compared with adaptive control.An AUV with adaptive control requires tracking region trajectory.The existence of perturbations is the same as that in case 1 and case 2. The results are performed in Figures 14-16.Without robust control, an AUV moved out of the boundary and it also spent much energy to recover into desire condition.Graph of energy consumption is shown in Figures 17 and 18.
The total energy demand of each simulation is summarized in Table 1.Here, energy consumption is calculated from the forces and moment of each control.Total forces and moment are obtained by norm calculation for all the time.Less amount of energy consumption is produced under proposed controller.Then, the propulsion of eight thrusters of ODIN can be described from total energy demand.The calculation is presented in the following formula: where  is given in (22) and  denotes thrusters configuration matrix, while  th is vector of thrusters forces.To get thrusters propulsion ( th ), it is necessary to inverse  then multiplied by .Here, the value of  is equal to where  = sin(1/4),  = 0.381 m, and   = 0.508 m representing the distance from center of vehicle to center of vertical thrusters and distance from center of vehicle to center of horizontal thrusters, respectively [25].Finally, the results of eight propulsions are performed in Figure 19 for propulsion in the case of STSMC with line trajectory and Figure 20 for propulsion in the case of STSMC with region trajectory, while Figure 21 is depicted for thruster's propulsion in adaptive control with region trajectory.The figures indicate that more forces are spent by thrusters when some perturbations hit the AUV (started in 50 s).It means that, in those conditions, thrusters tried to keep the AUV's position from its desire trajectory.Compared to all methods, adaptive control required the most thruster forces followed by STSMC with line trajectory.Furthermore, the least thruster's propulsion is needed by proposed controller, STSMC with dynamic region (in Figure 20).Thrusters 2 and 4 generated around 0 N forces although the perturbations disturbed its movement.These results are not performed by another control.

Conclusion
A new robust control scheme based on second order Sliding Mode Controller with dynamic region is proposed in this paper.Super twisting is adapted as a second order sliding mode which is useful to make a robust AUV while region boundaries technique is applied as energy reduction.Some simulations were arranged to observe the effectiveness of proposed controller.In case 1, the AUV can track line trajectory under perturbations even though there are small errors positions.Then in case 2, the proposed control can maintain the AUV's position inside the region even if it is disturbed by the same amount of perturbations.Meanwhile, in the last case, the AUV moves out of the region and produces large error.From all results, it can be concluded that proposed control can save more energy consumption than others although the tracking movements are not precise in the middle of region.The amount of energy can be reduced as long as an AUV is inside the region.Note that   ( Ṁ − 2  ) is a skew symmetric matrix; therefore, its value is equal to zero (Property 2) [23].Next, use the component   ṡ =  −  − (  η  +   η +   η +   ) from ( 18), then combined with the perturbation which is given in [26] to get a globally bounded of (7): ).

Figure 1 :
Figure 1: Body-fixed frame and earth fixed reference frame.

Figure 3 :
Figure 3: Proposed control scheme with dynamic region concept.

Figure 11 :Figure 12 :
Figure 11: Error position of proposed control with region trajectory.

Figure 13 :Figure 14 :
Figure 13: Moment of proposed control with region trajectory.

Figure 15 :Figure 16 :
Figure 15: Linear position of adaptive control with region trajectory.

Figure 17 :Figure 18 :
Figure 17: Forces of adaptive control with region boundaries.

Figure 21 :
Figure 21: Thruster forces using adaptive control with region boundary.(a) Horizontal thrusters and (b) vertical thrusters.