There is currently a growing interest in developing devices that can be used to exploit energy from oceans. In the recent past, the search for oil and gas at ever-greater depths has led to the evolution of devices with which these resources are extracted. These devices range from those that simply rest on the seabed to those that are fully floating and anchored to it. This trend can be considered as the basis needed to understand the future evolution of devices for harnessing depth renewable resources. This paper presents a simple dynamic modeling and a nonlinear multivariable control model-based system for a new three-degree-of-freedom underwater generator with which energy from depth marine currents is harnessed when reference trajectory tracking for the emersion maneuvers needed to carry out maintenance tasks is performed. The goodness of both the model and the proposed controller has been demonstrated through the development of various simulations in the MATLAB-Simulink environment. Additionally, the validation of the control algorithms was carried out by using the dynamic model offered by the simulation environment Orcina OrcaFlex (software for the dynamic analysis for offshore marine systems) through the MATLAB-OrcaFlex interface.
1. Introduction
The growing interest in the exploitation of marine renewable energies began several years ago, and various devices with which to harness energy from seas have therefore been conceived or developed (see [1–3]), their main natural energy sources being wind, waves, and marine currents.
One of the most promising sources of marine energy is the exploitation of tidal or oceanic water flows [4–7], and the industry’s effort is currently focused on the so-called first generation devices [8] (fixed to the sea bottom and suitable for sites with depths below 40 m). But there are an increasing number of second-generation devices that have been conceived to be moored to the sea bottom with an expected similar trend to that which has taken place during the development of oil and gas platforms that must access resources at increasingly greater depths [9].
This evolution has led these devices to evolve from being anchored to the seabed to being located in a floating location, and the most appropriate ways and means to perform maintenance tasks have therefore also had to evolve these kinds of devices [10–15].
The successful installation of these kinds of devices for harnessing energy from depth currents can only take place once it has been proved that they are both technically and economically feasible in comparison to other traditional energy sources. One well known way in which to reduce costs is by successively automating more tasks, thus signifying less human intervention or the possibility of using the cheapest general purpose ships rather than high cost special vessels for maintenance purposes. In [16] are studied the automatic maneuvering emersion and immersion of one of these devices.
Nowadays there exist different devices with the following main alternatives for performing maintenance tasks:
There is the use of a servo actuated crabbing based system to move the main generation unit from the support structure. See different devices from [17, 18].
There is the use of elevation and placement by means of floating cranes. See [19–23].
There is the use of a ballast management system to generate vertical forces, thus enabling the device’s emersion and immersion movements to be controlled. See [24–26].
A new family of generators is briefly presented in this work. They use a mooring system based on buoys and wires that allow the device to be located at the desired position on the seabed and positioned at any desired depth of the layer with almost no human resources. By simply disconnecting stern wire that joins the generator to the seabed and a proper management of the ballast water that is strategically located inside the generator, it is possible to perform closed loop emersion and immersion maneuvers with three degrees of freedom (DoFs), two orientation angles and a depth control, in such a way that the process of moving from a submerged and vertically disposed state to a floating and horizontally disposed one and vice versa can be carried out fully automatically without any kind of human intervention.
This paper presents a new dynamic model for this new family of second-generation tidal energy converters [8, 27, 28] for simulation and control design purposes [29, 30]. The proposed model is very simple, fully parameterized, and easily scalable with minimum computer effort. Very simple models have already been used to control complex devices with a good performance and good correspondence among simulated and experimental signals. For example, in [31], a simple model based on only one lumped mass and a special kinematic uncoupling design was used to model a three-degree-of-freedom flexible arm with complex nonlinear and time variant dynamics. This simplified dynamic model was employed as a basis for various controllers, which were used to control this robot with excellent results [32]. Another different system, in this case, a stair-climbing mobility system, was also modeled with only one lumped mass [33] and this simple dynamic model has been used as the basis for various control systems (see, a.e., [34]).
In a similar manner to that shown above, the dynamic model proposed in this paper is based on the computation of the values of only four lumped masses, which are handled solely by buoying forces applied to three star-shaped distributed equilateral torpedoes on the pod (nacelle) of the generator. The model developed exhibits time-varying, nonlinear, and strongly coupled behavior. It was, meanwhile, necessary to design the control law based on the dynamic model developed in such a way that it would have a successful closed loop behavior when the underwater three-DoF tidal energy converter performs emersion and immersion maneuvers with only passive buoyancy forces. In this work, the proposed control system is characterized by its simplicity, computational efficiency, and easy implementation in a microprocessor-/microcontroller-based system. The proposed control law is composed of the following two main terms: (i) a matrix that is responsible for decoupling the open loop dynamics from one degree of freedom with regard to the others and (ii) a feedback controller based on centrifugal and Coriolis force cancellation and a simple multivariable and diagonal PD controller.
The paper is organized as follows: after providing a brief description of the proposed family of generators in Section 2, Section 3 shows how the derivation of the proposed dynamic model of the submerged device was derived when performing coupled three-degree-of-freedom motions. Section 4 proposes the solution adopted to control the system. Section 5 presents the numerical simulations obtained to validate the proposed dynamic model and the proposed control algorithm. In particular, different smooth robotic-based trajectories were used to perform uncoupled simultaneous multiple degree-of-freedom motions. Finally, Section 6 is devoted to the conclusions of the paper, along with proposals for immediate future works.
2. System Description
A brief description of the proposed family of generators is now presented (see [27], a.e.). These generators were conceived to extract energy from marine currents and to fulfill the following four main features: (i) being valid for operation at depths greater than 40 m, (ii) minimum installation support structure and civil works, (iii) being floating and easily transportable device, and (iv) being fully automated for emersion and immersion maneuvers.
Figure 1 shows the general view of one of the proposed devices. It is composed of a three-fixed pitch blade propeller and a central pod (gondola), and the shaft of the propeller is coupled to an electrical generator by means of a multiplier gearbox. Three main radial and symmetrically distributed columns start at the pod and end at three torpedoes of an approximately cylindrical shape. These torpedoes contain the inner ballast system used to apply hydrostatic forces to the generator. Another of the principal missions of these torpedoes, which are aligned to the direction of the current, is to minimize rotations around the main axis of the whole generator by using hydrodynamic force compensation when it is extracting energy and the propeller reaction causes heeling torques. Three structural bars starting at the torpedoes and ending in front of the propeller have been added for mooring purposes.
Main appearance of a generator for unidirectional currents.
The main features of two of the conceived generators are briefly presented in Table 1.
Main features.
A.6.7 model
U1M model (Figure 1)
Power (kW)
600
1,000
Stream highest velocity (m/s)
2.0
1.8
Seabed depth (m)
60/100
80
Propeller diameter (m)
20
32
Minimum blade end depth (m)
15
34
Rotor/gear output (rpm)
12/1500
12/750
Structure material
Steel
Steel
Various anchoring or mooring systems can be used to place the generator in the sea at the desired position. Figure 2 shows a simple mooring system based on wires and buoys that allows the device to be positioned at the desired depth. By simply providing the generator with positive floatability, the wires become under strain and the device can be moored in both the desired position and depth and is ready to harness energy.
Mooring system based on wires and buoys.
When employing the emersion maneuver procedure, the device has to evolve from its vertical and submerged operating position and orientation (usually denominated as posture or attitude) to its maintenance and transportation position and orientation (floating horizontally) (see Figure 3). This movement is achieved in three basic sequential steps: (1) by providing the device with zero buoyancy rather than positive buoyancy; (2) by releasing the stern wire; and (3) by managing the water in the ballast tanks.
Final stages of the emersion maneuver.
3. Dynamic Model
Obtaining a dynamic model of the device shown previously under three DoFs in order to perform submerged motions is an important step for the simulation and design control algorithms needed to perform an emersion maneuver in a fully automatic closed loop mode. The proposed model is based on the following assumptions (see Figure 4 for details):
Only four lumped masses are considered.
Added masses are constant, well known, and those which are lumped are also considered.
Vertical translation (one DoF) dynamics and rotation (two DoFs) dynamics are fully uncoupled.
Viscous friction is modeled as being constant and fully uncoupled.
The device free-surface interaction (considered semisubmergible) is not considered in this paper.
The influences of residual flow of water, waves, or wind effects are considered as external perturbations in the model.
A fixed reference frame S0 is defined as an orthogonal reference frame whose z-axis is vertical. Zero depth (z=0) is computed at the level of the sea (free surface) and the vertical plane (x, z) always contains the center of the generator. Only the depth of the center of the generator is related to this reference frame, and components x and y of the generator with regard to S0 are not considered. The other two degrees of freedom (only rotations around axes xG and yG are possible) are defined with regard to the other intermediate reference frame, SG, whose origin is at the center gravity of the generator and whose plane (xG, yG) is placed on an absolutely horizontal plane (parallel to the sea surface). If the generator is positioned with null orientation (φx=0, θy=0), the xG-axis remains perpendicular to the plane of the generator and the zG-axis is placed vertically and forms a symmetry axis that is aligned with the mass denoted as m3.
Main parameters and magnitudes.
If we consider now the actuators, which are responsible for creating the buoyancy forces, they are ideally located at the same positions at which each of these lumped masses is considered to be placed. These actuators will produce vertical component forces only that are obtained as differences between buoyancy forces (really produced by actuators) and forces due to gravity. Each of these masses (denoted and numbered with subindex i=1,2,3) is formed of both real mass mGi and added mass mADDi and is(1)mi=mGi+mADDi,(2)fi=ρWVi-mig,where Vi is the volume fraction of the generator corresponding to the ith-mass. Under static equilibrium conditions ρWVi=mi, no forces are produced and the generator will remain in its previous state without any kind of motion. In the opposite sense, if motion of the generator is desired, each actuator will produce an incremental force that has to be computed as an increase in either mass or volume (with the opposite sign). For the sake of clarity, volume increments are considered to be responsible for creating buoyancy forces. Consider(3)fi=+gρWΔVi.The force vector applied to the generator with regard to the fixed reference frame S0 (also with regard to SG) can be expressed in a matrix form as the sum of all the vertical forces produced by actuators:(4)F=FxFyFz=00f1+f2+f3.
3.1. Forces and Torques Involved
Let us consider a local frame S placed at the center of the central cylinder of the generator at which its center of gravity is located. This S frame is defined as follows: the u-axis is perpendicular to the plane of the generator, the w-axis is located in the opposite direction to the lower mass m3 (see Figure 4), and the v-axis is located at the plane of the generator and forms a right-hand frame. The central mass mCG is then located at the following local position:(5)PCLOC=000Twhile the three resting lumped masses are located at the center of gravity of each of the three torpedoes with local coordinates (for the sake of simplicity, notations cα≡cosα and sα≡sinα are used):(6)P1LOC=0-Ls60∘Lc60∘,P2LOC=0Ls60∘Lc60∘,P3LOC=00-L.According to the definition of the second frame, SG, the relation of the orientation of both the S and SG frames can easily be obtained by computing the rotation matrix that relates the local generator orientation with regard to the SG reference system. This rotation is the result of composing the next two elementary rotations with regard to the next axes (first horizontal xG-axis and then horizontal yG-axis basic rotations):(7)R=RY,θy·RX,φx·I3×3,(8)R=cθy0sθy010-sθy0cθy·1000cφx-sφx0sφxcφx=cθysθysφxsθycφx0cφx-sφx-sθycθysφxcθycφx.The positions of the three lumped masses with regard to SG can therefore be written as(9)P1=R·P1LOC=Lsθ-s60∘sφx+c60∘cφx-Ls60∘cφx-c60∘sφxLcθ-s60∘sφx+c60∘cφx,P2=R·P2LOC=Lsθs60∘sφx+c60∘cφxLs60∘cφx-c60∘sφxLcθs60∘sφx+c60∘cφx,P3=R·P3LOC=-LsθcφxLsφx-Lcθcφx.Moreover, the application of each of the forces caused by actuators (see forces f1, f2, and f3 in Figure 4) to their respective centers of gravity on each torpedo leads to a torque vector (with regard to SG), which is defined and obtained as follows:(10)Γ=ΓxΓyΓz=P100f1+P200f2+P300f3.And, by substituting (9) in (10), one obtains(11)Γ=Ls60∘cφx-f1+f2-c60∘sφxf1+f2+sφxf3Lsθys60∘cφxf1-f2-c60∘cφxf1+f2+cφxf30.As expected, from (11), it is proven that it is impossible to cause changes in the horizontal direction (rotations about zG-axis or z-axis) with only vertical forces.
3.2. Force Conversion
In this subsection, a matrix relation between the forces applied to the generator and a new generalized force vector is proposed. This relation makes it possible to decouple the motions of the generator in the sense that different sets of forces can be applied which then produce independent motions for each of the three degrees of freedom of the generator.
By rearranging (4) and (11), the following relation is easily obtained:(12)Ls60∘cφx-f1+f2-c60∘sφxf1+f2+sφxf3Lsθys60∘cφxf1-f2-c60∘cφxf1+f2+cφxf3f1+f2+f3=ΓxΓyFz.By substituting c60∘and s60∘ for their respective values and rearranging (12), the following is attained:(13)-L23cφx+sφxL23cφx-sφxLsφxL2sθy3sφx-cφx-L2sθy3sφx+cφxLsθycφx111·f1f2f3=ΓxΓyFz.Or in a compact form,(14)Λφx,θy·FG=τ,where FG=(f1f2f3)T denotes the buoyancy forces provided by all the actuators, τ=(ΓxΓyFz)T denotes the proposed vector of generalized forces, and Λ denotes the matrix that relates τ to FG, which depends on φx and θy according to (13) and (14).
In (14), vector τ is conceived as the uncoupled generalized forces vector applied to the center of the generator, so the first component will produce a torque that is aligned to the xG-axis and the second component will produce a torque that is aligned to the yG-axis while the third component will produce a force that is aligned to the z-axis.
3.3. Dynamic Submodel for Only Vertical Movement
The dynamics of an underwater body that is considered to be a rigid body when it performs vertical motions is well known (see [35–37], a.e.). As was seen in the previous section, depth z is the only variable of position that can be controlled. Using the same notation given in the previous subsection and under hypothesis shown above, the next dynamic model is proposed for only vertical motions (the effects resulting from free-surface interaction are not computed):(15)Fz=mG+mADD·z¨+z˙·υz·z˙︸Fzvz˙,where mG=mCG+m1G+m2G+m3G denotes the total mass of the generator and is decomposed into the mass of the central cylinder mCG and the three masses of each torpedo. For the sake of simplicity, the term mADD denotes the total added mass resulting from the motion in a viscous fluid (see [38]) which is considered to be constant and well known. A quadratic and speed opposite friction term is computed as Fzvz˙=z˙·υz·z˙, where υz is also considered to be constant.
3.4. Dynamic Submodel for Only Rotation Movements
In this subsection, the Lagrange formulation allows us to obtain the equations of motion for the generator provided with only rotation movements. The two rotation angles φx and θy around their respective xG-axis and yG-axis are taken as generalized rotation coordinates qR=(φxθy)T for rotation purposes solely. For the compotation of the Lagrange function, the kinetic and potential energies are computed beforehand.
3.4.1. Local Inertia Matrix
The inertia matrix in local coordinates is obtained from the lumped mass distribution employed. Because the general rotation motion is of dimension 3, the inertia matrix results are of dimensions 3×3:(16)JLOC=IxxPxyPxzPyxIyyPyzPzxPzyIzzwhich satisfies that it is a symmetrical and positive definite matrix. This implies that eigJLOC>0, Pxy=Pyx, Pxz=Pzx, and Pyz=Pzy. In our case, the inertia matrix incorporates the effects of the added masses and takes the form (details of how the coefficients in this inertia matrix were obtained can be found in Appendix A):(17)JLOC=m1+m2+m3000m1+m24+m33m2-m1403m2-m143m1+m24Â·L2.As in the case shown above, each of the masses considered is computed as the sum of its real mass fraction and its added mass, as seen in (1). It will be observed that if m1=m2, the generator becomes fully statically and dynamically equilibrated with all of its null value inertia products (Pxy=Pxz=Pyz=0).
3.4.2. Computation of the Rotation Kinetic Energy
The kinetic energy of the generator when only rotations over first the xG-axis and then the yG-axis are considered is given by(18)K=12ΩxΩyΩz=0·RTφx,θy·Ixx000IyyPyz0PyzIzz·Rφx,θy·ΩxΩyΩz=0with Ωx, Ωy, and Ωz being the absolute rotational velocities of the generator with respect to SG.
By substituting (8) and (17), one obtains(19)K=12Ωx2Ixxc2θy+Izzs2θy+2ΩxΩyIxx-Izzsθycθysφx-Pyzsθycφx+Ωy2Ixxs2θy+Izzc2θys2φx+Iyyc2φx+2Pyzcθysφxcφx.
3.4.3. Computation of the Rotation Potential Energy
In order to control the generator, it is necessary to obtain operation conditions that force the generator to be at an equilibrium point. Neutral buoyancy is therefore required (mG=ρwV), and the total potential energy thus has a zero value under this condition(20)U=0.
3.4.4. Lagrange Formulation
The kinetic and the potential energy computed previously are used, and the Lagrange function of the generator with only rotation motions with regard to the xG-axis and yG-axis is(21)L=K-U=K.Lagrange’s equations are then expressed by(22)ddt∂L∂q˙R-∂L∂qR=τ,where τ is the sum of external torques, which are the control torques (ΓxΓy)T and the friction terms that have been modeled as a function Fvq˙R with υR=(υφxυθy)T. The following expression is obtained (see the coefficients of MR and CR in Appendix B): (23)MRqR·q¨R+CRqR,q˙R=ΓxΓy-q˙RT·υR·q˙R︸Fvq˙Rin which qR denotes the generalized rotation coordinates, qR=(φxθy)T with regard to SG frame, MR are the inertia terms which are qR dependent, positive definite, and symmetrical, and CR includes both the centrifugal and Coriolis effects. A term υR=(υφxυθy)T allows the friction terms to be modeled in a similar way to that occurring in (15).
3.5. Proposed Dynamic Model
The complete proposed dynamic model can be obtained by joining (13), (15), and (23) and expressed in a compact form as(24)Mq·q¨+Cq,q˙=Λq·FG-q˙T·υ·q˙,where the complete vector of generalized coordinates q is(25)q=qRzand matrices M, C, and υ are(26)Mq=MRqR02×101×2mG+mADD,(27)Cq,q˙=CRqR,q˙R0,(28)υ=υR=υφxυθyυz.Figure 5 depicts the proposed dynamic model, along with the two submodels presented above. The set of input signals are the buoyancy forces produced by actuators FG=(f1f2f3)T while the output measurable signals are the two-degree-of-freedom orientation angles together with the generator depth denoted previously as a vector of the generalized coordinates q=(φxθyz)T.
Proposed dynamic model.
4. Uncoupled Multivariable PD Control
The three-degree-of-freedom model proposed clearly exhibits a nonlinear, time variant, and multivariable uncoupled behavior. A new and simple control scheme based on the following three stages is proposed: (i) an uncoupled term that allows the generalized control torques to be converted into forces applied to each of the torpedoes (by means of actuators), (ii) a nonlinear uncoupling model-based term that ideally cancels out the nonlinearities of the system caused by centrifugal and Coriolis torques, and (iii) a multivariable diagonal proportional-derivative (PD) controller.
4.1. Uncoupling Matrix
The determinant of the matrix Λq defined in (13) and (14) is(29)Λq=332L2sθy.This determinant makes matrix Λq nonsingular, with the exception of θy=0 which corresponds to the normal operation orientation. In other words, if the generator is placed in a fully vertically position, it is not possible to turn around the yG-axis using vertical forces alone. In order to avoid this singularity, the position of the third actuator can be displaced a small distance δx along the local u-axis, thus resulting in its new local position at (see Figure 6)(30)P3LOC=δx>00-L.For relatively small values of δx the angle of the new plane that conforms to the three points of application of forces with regard to the plane of the generator is (see Figure 6)(31)Δθy=tan-12δx3L≈2δx3Land the range of the angle θy is -π/2+2δx/3L≤θy≤-2δx/3L rather than -π/2≤θy≤0, thus avoiding the singularity of matrix Λq in (29). For the angle φx is considered range -π/2≤φx≤π/2.
Singularity avoidance.
Then, after avoiding the singular values of the angle θy, the inverse of matrix Λq can be obtained symbolically, yielding(32)Λ-1q=-3cφx+sφx3L-cφx-3sφx3Lsθy133cφx-sφx3L-cφx+3sφx3Lsθy132sφx3L2cφx3Lsθy13.
4.2. Proposed Control System
The existence of the inverse of matrix Λq into (29), (32) makes it possible to handle the generator in an uncoupled mode and even in open loop, by simply obtaining the forces to be applied by actuators FG from a set of desired and uncoupled generalized forces τ for nonsingular values of θy. The existence of external disturbances and model uncertainties between the real matrix Λq and the computed matrix Λ^q requires the addition of closed loop controllers.
Equations (24) and (32) provide the following control system, which is proposed for a given desired reference qd=(φxdθydzd)T and which can be time dependent:(33)FG=Λ^q-1·C^q,q˙-KDq˙+KPqd-q,where Λ^q and C^q,q˙ denote their respective computed matrices as a function of the measured output signals q. The complete dynamics of the system controller, under the assumption of perfect cancellation, that is, C^q,q˙=Cq,q˙ and Λ^q=Λq, results in(34)Mq·q¨+KD·q˙+KP·q=KP·qd.Since the inertia matrix Mq is defined according to (26) and the coefficients shown in Appendix A, it is a bounded symmetric and positive definite matrix which satisfies that λminI3×3≤Mq≤λMAXI3×3 for all q. Upon considering a mean inertia matrix as (35)M¯=MqMAX+Mqmin2the closed loop controlled system with mean matrix M- exhibits fully decoupled multivariable linear dynamics defined by(36)q¨+M¯-1·KD·q˙+M¯-1·KP·q=M¯-1·KP·qdand three independent single input/single output (SISO) PD controllers with derivative in the feedback loops can be tuned by choosing diagonal matrices of gains KD=diag(kDφxkDθykDz) and KP=diag(kPφxkPθykPz) that allow closed loop poles to be placed where desired. Equation (36) can then be analyzed as three SISO independent closed loop systems for any of the three degrees of freedom of the generator.
Figure 7 depicts the proposed control scheme. The uncoupling matrix and the linearization term that compensates the centrifugal and Coriolis torques are clearly shown, together with an uncoupled three-dimensional diagonal PD controller with derivative component from only the output signal as proposed in (33).
Main parameters for simulation (per each of the cylindrical torpedoes).
Nominal value
Units
Number of torpedoes
3
Torpedo angle distribution
0, 2π/3, 4π/3
rad
Length
0.608
m
Diameter
0.200
m
Volume
0.191
m^{3}
Real mass
19.1
kg
Added mass
15.3
kg
Friction term
79.4
N⋅s^{2}⋅m^{−2}
G
9.81
m⋅s^{−2}
ρw
1,000
Kg⋅m^{−3}
L
0.400
m
The main reason for using a laboratory prototype is the high economic cost of a real scale device. The research group has successfully tested a 1 : 10 scale prototype [39] in a real protected sea environment. From these trials, the high acceleration values observed in open loop in some of the time intervals during the execution of the emersion maneuvers performed (see Figure 22 in [39]) could damage the rotor blades making necessary the study of the proposed closed loop control.
The first set of numerical simulations was performed on step reference signals with nominal parameters of the added mass and the friction term in order to validate both the desired transient and steady state responses and there were no coupled dynamics among the independent movements (φx, θy, and z) that the generator can perform. Figures 8 to 10 illustrate the time responses of each of the independent closed loop systems of each of the degrees of freedom of the generator when the following step reference signals were used (units in rad for angles, m for depth, resp.):(37)φxdfrom0toπ3att=10s,θydfrom-0.05to-π2att=20s,zdfrom-20to-1att=20s.The PD controller gains for each of the loops were tuned using the nominal value of matrix M¯ in order to obtain closed loop double poles placed at p1,2=-0.3 rad/s and a unit gain that will guarantee critically damped second-order dynamics. For a given fully uncoupled open loop dynamic (second-order and type one system) of the form Gi(s)=Ai/(s·s+Bi) and a desired closed loop dynamic of the form Mi(s)=p1,22/(s2+2p1,2s+p1,22), for i=φx,θy,z, the gains of the PD controllers are easily obtained as kPi=p1,22/Ai and kDi=(2p1,2-Bi)/Ai. Actuator saturation is not considered here. Gaussian noises from a normal distribution N0,3e-3 were added to each of the output signals, and first-order filters were used to obtain the time derivative of the output signals as Ld/dt=s≈s/(0.001s+1), where L denotes the Laplace transform.
Step response of φx(t) from 0 to π/3 rad with transition time at t=10 s.
Figures 8–10 depict that the system performs extremely well with uncoupled dynamics, unit gain, and the desired setting time response with no overshoot (the effect of the θyd transition at t=20s and the real θy response at t≥20s is not perceived in the time evolution of the response of φx).
Step response of θy(t) from −0.05 to -π/2 rad with transition time at t=20 s.
Step response of z(t) from -20 to -1 m with transition time at t=20 s.
The second set of simulations were carried out assuming nonnominal terms (increments of ±20% of the nominal added mass term and null Coriolis and centripetal compensation term). Additionally, the actuators were saturated to ±5N, thus simulating a more realistic scenario. Due to the feedback robustness provided by the PD controller (see [40], a.e.), the time responses were very similar to the nominal responses shown in Figures 8–10. Figures 11–13 depict the evolution of the errors between the nonnominal and the nominal responses. In these figures is clearly observed the coupling effect at instants t=10 s and t=20 s when transitions occur due to step signals.
Error between φx time responses with nonnominal mADD and null Coriolis and centripetal compensation terms.
Error between θy time responses with nonnominal mADD and null Coriolis and centripetal compensation terms.
Error between z time responses with nonnominal mADD and null Coriolis and centripetal compensation terms.
The third set of simulations was carried out using synchronous smooth time trigonometric sixth-order S-trajectories (see [41, 42]) with starting and ending times of t0=20 to tf=120s, respectively, for the orientation references, and t0=20 to tf=220s for the depth reference with the maximum velocities shown below(38)φxdfrom0toπ3radφ˙xdMAX=0.0125rad/s,θydfrom-0.05to-π2radθ˙ydMAX=0.0175rad/s,zdfrom-20to-1mz˙dMAX=0.125m/s.Figures 13–16 illustrate that the generator exhibits a good correspondence with both the desired responses and the simulated responses obtained.
Reference S-trajectory φdx(t) and time response φx(t).
Reference S-trajectory θdy(t) and time response θy(t).
Reference S-trajectory zd(t) and time response z(t).
Finally, the fourth set of simulations corresponds to a complete sequence of sixth-order S-trajectories to achieve an emersion maneuver. Figures 17 to 19 depict both the references and the time responses obtained for each of the independent movements of the generator. The desired sequence is defined below (units in rad for angles, m for depth):(39)fromqd0=0-0.05-20toqd1=0-0.05-20t0=0tot1=25s,fromqd1=0-0.05-20toqd2=π3-0.05-5t1=25tot2=75s,fromqd2=π3-0.05-5toqd3=π3-0.05-5t2=75tot3=175s,fromqd3=π3-0.05-5toqd4=π3-0.05-π2-0.1t3=175tot4=225s,fromqd4=π3-0.05-π2-0.1toqdf=π3-0.05-π2-0.1t4=225totf=275s.Figures 17–19 illustrate the excellent response of the system during a complete emersion sequence. It will be observed that the initial posture qd0 is equivalent to that shown in Figure 2, while the final posture qdf is equivalent to that shown in Figure 3 (right), corresponding with the normal submerged operation posture for energy harnessing and the normal floating posture for maintenance tasks. Very similar results were obtained by integrating the dynamics of the laboratory prototype into the simulation environment Orcina OrcaFlex [43]. OrcaFlex is one of the world’s leading software packages for the design and analysis of a wide range of marine systems (riser systems, mooring systems, installation planning, towed systems, marine renewables, etc.) and is considered validated and certified software for anchor marine systems. The control signals given in (33) were computed in the runtime of Matlab which is connected to OrcaFlex through an external DLL function which is responsible for obtaining the magnitudes from the simulated responses and providing the control signals from and to OrcaFlex with a time sampling of 10 ms. Figure 20 displays a visual sequence of the response of the 3-DoF prototype when following the references given in (39).
Emersion maneuver reference φdx(t) and time response φx(t).
Emersion maneuver reference θdy(t) and time response θy(t).
Emersion maneuver reference zd(t) and time response z(t).
Graphical view of the closed loop simulation with Matlab-OrcaFlex.
6. Conclusions
This paper proposes a new dynamics model that can be used to control a family of submarine electrical generators that was conceived to harness energy from marine currents. The submarine generator is provided with three degrees of freedom by which it is possible to perform closed loop emersion and immersion maneuvers. The dynamic model proposed is based solely on the definition of four lumped masses placed on a plane and a matrix that permits the forces from the three buoyancy actuators to be converted into two torques and a vertical force that are responsible for the rotations and the vertical displacement of the whole system. The model developed exhibits a nonlinear, strongly coupled, and time-varying behavior between buoyancy forces produced by the three actuators and the posture magnitudes measured using a depth sensor and a three-dimensional inclinometer on which the angle rotation around the z-axis is ignored. As a first conclusion, we should note that the dynamic model developed in this research is sufficiently precise to describe the underwater 3-DoF tidal energy converter motions and sufficiently simple to be used in the control law design.
The proposed control scheme with which to control the generator using only depth and orientation measurements is, meanwhile, based on three stages: (a) the definition of an uncoupling matrix is used to decouple the generator motions for nonsingular values of orientation; this matrix can be easily computed in real-time and allows the generator to be handled in an uncoupled manner, even in open loop; (b) it is a nonlinear uncoupling-based model for the compensation of the centrifugal and Coriolis torques; and (c) it is a proportional-derivative (PD) control action. The closed loop dynamics was chosen by simply designing proportional and derivative matrix gains. The proposed control allows the generator to perform fully automatic closed loop emersion/immersion maneuvers from a submerged position with vertical orientation to a floating posture with a horizontal orientation by using time-varying smooth reference trajectories. A last obtained conclusion is that the control law implemented in this work demonstrates that it is a simple control strategy, which is computationally efficient and easily implementable in a microprocessor-/microcontroller-based system.
Finally, based on more than satisfactory numerical simulations achieved in this research, various experimental branches are now the focus of our attention and are detailed as follows. Our intention is (a) to finalize the construction of a real small-sized laboratory prototype; (b) to use real-time experiments to validate the proposed research; and (c) to study different control strategies and new trajectory generations in order to improve the quality of the closed loop emersion/immersion maneuvers. These will be the main topics of our future research.
AppendicesA. Coefficients of the Inertia Matrix <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M238">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="bold">J</mml:mi></mml:mrow>
<mml:mrow>
<mml:mtext>LOC</mml:mtext></mml:mrow>
</mml:msub></mml:mrow>
</mml:math></inline-formula>
The local positions of each of the three masses located at their respective center of gravity on each torpedo and which are taken from (6) are used to obtain the equivalent inertia matrix JLOC with regard to the local reference system. It will be noted that the central mass mCG does not produce any inertia value in accordance with its local position given by (5)(A.1)Ixx=∑i=13mi·yi2+zi2=m1+m2+m3·L2,Iyy=∑i=13mi·xi2+zi2=m1+m24+m3·L2,Izz=∑i=13mi·xi2+yi2=3m1+m24·L2,Pxy=Pyx=∑i=13mi·xi·yi=0,Pxz=Pzx=∑i=13mi·xi·zi=0,Pyz=Pzy=∑i=13mi·yi·zi=3m2-m14·L2.Matrix JLOC in (17) is then obtained from (16) using the coefficients obtained above from (A.1).
Equation (23) was obtained from (19)–(22). This appendix shows how the coefficients of matrices M(qR) and C(qR,q˙R) were obtained. Firstly, partial derivatives with regard to angular velocities were obtained. Note that the Ωx and Ωy notations are maintained as angular velocities in these first equations(B.1)∂L∂φ˙x=ΩxIxxc2θy+Izzs2θy+ΩyIxx-Izzsθycθysφx-Pyzsθycφx,∂L∂θ˙y=ΩxIxx-Izzsθycθysφx-Pyzsθycφx+ΩyIxxs2θy+Izzc2θys2φx+Iyyc2φx+2Pyzcθysφxcφx.Partial derivatives with regard to generalized coordinated rotation are then obtained:(B.2)∂L∂φx=ΩxΩyIxx-Izzsθycθycφx+Pyzsθysφx+Ωy2Ixxs2θy-Iyy+Izzc2θysφxcφx+Pyzc2φx-s2φx,∂L∂θy=Ωx2-Ixx+Izzsθycθy+ΩxΩyIxx-Izzc2θy-s2θysφx-Pyzcθycφx+Ωy2Ixx-Izzsθycθys2φx-Pyzsθysφxcφx.From here on, in order to obtain time derivatives of ∂L/∂φ˙x and ∂L/∂θ˙y, angular velocities Ωx and Ωy are substituted for their respective angular rotation time derivatives(B.3)Ω=ΩxΩyΩz=φ˙xθ˙y0,ddt∂L∂φ˙x=φ¨xIxxc2θy+Izzs2θy+θ¨yIxx-Izzsθycθysφx-Pyzsθycφx-Izzcθysθysφx+φ˙xθ˙y-2Ixxcθysθy+2Izzsθycθy+θ˙y2Ixx-Izzc2θy-s2θysφx-Pyzcθycφx+θ˙yφ˙xIxx-Izzsθycθycφx+Pyzsθysφx,ddt∂L∂θ˙y=φ¨xIxx-Izzsθycθysφx-Pyzsθycφx+θ¨yIxxs2θy+Izzc2θys2φx+Iyyc2φx+2Pyzcθysφxcφx+φ˙xθ˙yIxx-Izzc2θy-s2θysφx-Pyzcθycφx+φ˙x2Ixx-Izzsθycθycφx+Pyzsθysφx2θ˙y2Ixx-Izzsθycθys2φx-Pyzsθysφxcφx+2θ˙yφ˙xIxxs2θy+Izzc2θysφxcφx-Iyysφxcφx+Pyzc2φx-s2φxcθy.Finally, after rearranging (B.2) to (B.3), one obtains the coefficients of the left hand of (23) which is now reproduced again:(B.4)MRqR·q¨R+CRqR,q˙R,withqR=φxθy,MR1,1=Ixxc2θy+Izzs2θy,MR1,2=Ixx-Izzsθycθysφx-Pyzsθycφx,MR2,1=Ixx-Izzsθycθysφx-Pyzsθycφx,MR2,2=Ixxs2θy+Izzc2θys2φx+Iyyc2φx+2Pyzcθysφxcφx,CR1=2φ˙xθ˙yIzz-Ixxcθysθy+θ˙y2Ixxc2θy-s2θy1+cφx-Izzc2θy1+cφx-s2θy+θ˙y2-Pyzcθycφx1+cφx-s2φx+Iyysφxcφx,CR2=φ˙x2Ixx-Izzsθycθy1+cφx+2φ˙xθ˙yIxxs2θy+Izzc2θysφxcφx-Iyy-Pyzcθysφxcφx+θ˙y2Ixx-Izzsθycθys2φx-Pyzsθysφxcφx.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work has been supported by Spanish Ministerio de Economía y Competitividad under Research Grants DPI2011-24113 and DPI2014-53499R.
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