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Modelling and Control of a Flexible Floating Boom: First Approach
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th World The International Federation of Congress Automatic Control Available online at www.sciencedirect.com Proceedings of the the 20th9-14, World Congress Toulouse, France, July 2017 The International Federation of Toulouse, France,Federation July 9-14, 2017 The International of Automatic Automatic Control Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017
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IFAC PapersOnLine 50-1 (2017) 13108–13113
Modelling and Control of a Flexible Floating Boom: Modelling and Control of a Flexible Floating Boom: Modelling aa Flexible First of Approach Modelling and and Control Control Flexible Floating Floating Boom: Boom: First of Approach First Approach Approach G. E. First Gapingsi, R. Korbas, M. Santos
G. E. Gapingsi, R. Korbas, M. Santos G. Korbas, G. E. E. Gapingsi, Gapingsi, R. R. Korbas, M. M. Santos Santos Computer Architecture and Automatic Control Department, Computer Architecture and Automatic Control Department, Computer Science Faculty, Complutense University of Madrid, Computer Architecture and Automatic Control Department, Computer Architecture and Automatic Control Department, Computer Science Faculty, Complutense University of Spain Madrid, C/ Profesor García Santesmases 9, 28040-Madrid, Computer Science Faculty, Complutense University of Madrid, Computer Science Faculty, Complutense University of Madrid, C/ Profesor García Santesmases 9, 28040-Madrid, Spain (e-mail: [email protected], [email protected], [email protected]) C/ Profesor García Santesmases 9, 28040-Madrid, Spain C/ Profesor García Santesmases 9, 28040-Madrid, [email protected]) Spain (e-mail: [email protected], [email protected], (e-mail: (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) [email protected]) Abstract: Floating booms are useful tools in the marine world, especially for marine demarcation where Abstract: booms areofuseful tools or in the marine world, especially for work marine wherea some kind Floating of contamination sea, ocean coastal water is present. In this wedemarcation have developed Abstract: Floating booms are areofuseful useful tools or in the the marine world, especially for work marine demarcation wherea Abstract: booms tools in marine world, especially for marine demarcation where some kind Floating of contamination sea, ocean coastal water is present. In this we(unmanned have developed mathematical model for a flexible floating boom which is hooked on two boats surface some kind of contamination of sea, ocean or coastal water is present. In this work we have developed a some kind of contamination of sea, ocean or coastal water present. In this work we have developed mathematical model for a flexible floating boom which is hooked on two boats (unmanned surface vehicles). It is model meant to bea used for oil spill containment inismarine areas. In order to (unmanned simplify thesurface modela mathematical for flexible floating boom which hooked on two boats mathematical model for a used flexible floating boom which hooked on two boats surface vehicles). It ispossible meant to for simulation, oil spill containment inismarine areas. In order to (unmanned simplify the model and to make itsbe computer we have adopted a quasi-static approximation where we vehicles). It is meant to be used for oil spill containment in marine areas. In order to simplify the model vehicles). It is meant to be used for oil spill containment in marine areas. In order to simplify the model and to make possible its computer simulation, we have adopted a quasi-static approximation where we assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation and to make possible its computer simulation, we have adopted aa quasi-static approximation where we and to make possible its computer simulation, we have adopted quasi-static approximation where we assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation of this mathematical model hasare allowed the analysis of the behavior of the floating boom.The Based on this assume that the inertial forces small compared to the drag forces applied to the boom. simulation assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation of this mathematical model has allowed the analysis of the behavior of the floating boom. Based on this knowledge, we have designed a allowed control system whichofallows us to maintain the flexible floating boom in of this model has the analysis the behavior of floating boom. Based on ofspecific this mathematical mathematical model has theshape. analysis ofallows the purpose, behavior of the the floating boom. Basedwith on this this we have designed a allowed control its system which us to maintain the flexible floating boom in aknowledge, configuration by controlling For that the control approach deals the knowledge, we have designed aa control system which allows us to maintain the flexible floating boom in we have designed control system which allows us to maintain the flexible floating boom in aknowledge, specific configuration by controlling its shape. For that purpose, the control approach deals with the speed of the two ends of the floating boom. The results obtained withthe the control implemented control system are specific configuration by controlling controlling its shape. shape. For that that purpose, approach deals with the the aaencouraging, specific configuration by its For purpose, the control approach deals with speed of the two ends of the floating boom. The results obtained with the implemented control system are considering the model is a simplified discrete one and the control effort is reasonable. speed ends The obtained with implemented system speed of of the the two two ends of of the the floating boom. The results results obtained with the implemented control system are are encouraging, considering thefloating model isboom. a simplified discrete one and thethe control effort is control reasonable. Keywords: Floating boom, Modelling and Simulation, Unmanned Surface Vehicles (USV), Knowledgeencouraging, considering the model is a simplified discrete one and the control effort is reasonable. © 2017, IFACconsidering (International Federation ofsimplified Automaticdiscrete Control)one Hosting by control Elsevier effort Ltd. All rights reserved. encouraging, the model is a and the is reasonable. Keywords: Floating boom, Modelling and Simulation, Unmanned Surface Vehicles (USV), Knowledgebased Control System. Keywords: Floating boom, Modelling Modelling and and Simulation, Simulation, Unmanned Unmanned Surface Surface Vehicles Vehicles (USV), (USV), KnowledgeKnowledgeKeywords: Floating boom, based Control System. based Control System. based Control System. The task is not easy because even if the model and control of The task iscan notbe easy because(Menoyo-Larrazabal even if the model and control of the available and Santos1. INTRODUCTION The USV task is iscan notbe easy because(Menoyo-Larrazabal even if if the the model model and and control of The task not easy because even control of the USV available and Santos1. INTRODUCTION Peñas, 2016), the presence of the flexible floating boom has a USV can be available (Menoyo-Larrazabal and Santos1.floating INTRODUCTION This work deals with1. booms. Fortunately, it may not the the USV can be available (Menoyo-Larrazabal and SantosPeñas, 2016), the presence of the flexible floating boom has a INTRODUCTION significant impact on the dynamics of the boats. And vice 2016), the presence of the flexible floating boom has aa This work deals withbut floating booms. Fortunately, itprovide may nota Peñas, be a common issue, these containment barriers Peñas, 2016), the presence of the flexible floating boom has significant impact on the dynamics of the boats. And vice This work deals with floating booms. Fortunately, it may not versa. When attached at dynamics each end of to the a USV, itAnd deforms This work deals with floating booms. Fortunately, it may not significant impact on the boats. vice be a common issue, but these containment barriers provide a fundamental help forbut marine trash and debris control, marinea versa. significant impact the of boats. When at dynamics eachplane end under to the a USV, itAnd deforms be aa common issue, these containment barriers provide inattached theon horizontal the effects ofvice the be commonhelp issue, these containment barriers provide marinea continuously versa. When attached at each end to aa USV, it deforms fundamental forbut marine trash and protection debris control, demarcation, and therefore ecosystem (Chatterjee, versa. When attached at each end to USV, it deforms continuously in the horizontal plane under the effects of the fundamental help for marine trash and debris control, marine forces the vehicles impart at its endpoints and the drag due to fundamental help for marineecosystem trash and protection debris control, marine in horizontal under the effects of the demarcation, and applications therefore (Chatterjee, 2015). Different include e.g. algal blooms or continuously continuously in the the horizontal plane under thethe effects oftheir the forces the vehicles impart at between its plane endpoints and drag due to demarcation, and therefore ecosystem protection (Chatterjee, the water. The separation the vehicles and demarcation, and therefore ecosystem protection (Chatterjee, 2015). Different applications include e.g. algal blooms or forces the vehicles impart at its endpoints and the drag due to sewer spills, among others. forces the vehicles impart at its endpoints and the drag due to the water. The separation between the vehicles and their 2015). Different applications include e.g. algal blooms or accelerations govern thethe planar deformation of 2015).spills, Different applications include e.g. algal blooms or instantaneous sewer among others. the water. The separation between vehicles and their the water. The separation between the vehicles and their instantaneous accelerations govern the planar deformation of sewer spills, among others. boom (Bhattacharya, 2011). The spill among into theothers. sea may cause serious damages to our the seweroilspills, instantaneous accelerations govern the the planar planar deformation deformation of of instantaneous accelerations govern boom (Bhattacharya, 2011). The oil spill into thetherefore sea mayimportant cause serious damages toreact our the environment. It is to be able to the boom (Bhattacharya, 2011). The oil spill into the sea may cause serious damages to our The structure of the paper is the following. Section 2 presents the boom (Bhattacharya, 2011). The oil spill sea mayimportant cause serious our environment. It isthe therefore to bedamages totoreact efficiently to into such misfortunes. One example ofable an accident 2 presents The structure of the paper is the following. environment. It is therefore important to to react a brief introduction on floating barriers Section and develops the environment. It of is misfortunes. therefore important to be bewaters able to react efficiently tospill such Oneinexample ofable an accident structure of the paper is the following. Section 2 involving a oil contaminant marine was the The structure of the paper is the following. Section 2 presents presents aThe brief introduction on floating barriers and develops the efficiently to such misfortunes. One example of an accident mathematical model of the flexible floating boom. Model efficiently to such misfortunes. One example of an accident involving a spill of oil contaminant in marine waters was the amathematical brief on floating barriers and develops the one in thea spill Gulf of ofoil Mexico in April, 2010 (Upton, 2011). asimulations brief introduction introduction on floating barriers and develops the model of the flexible Model floating boom. involving contaminant in marine waters was the shownofin the section 3. Infloating Section boom. 4, the control involving a spill of contaminant in marine waters was the mathematicalaremodel one in the Gulfare ofoil Mexico in April, 2010in (Upton, 2011). flexible Model Floating booms a widely used solution such situations mathematical model of the flexible floating boom. Model simulations are shown in section 3. In Section 4, the control one in the Gulf of Mexico in April, 2010 (Upton, 2011). system designed for controlling the shape of the boom is one in Gulfare ofexist in April, 2011). are in Section 4, control Floating booms aMexico widely used in (Upton, such situations that willthe probably as long assolution our2010 society depends on simulations simulations are shown shown in section section 3. 3. Inshape Section 4,thethe theboom control system designed for controlling theIn of is Floating booms are a widely used solution in such situations explained. Then the simulation results are discussed. The Floating are a widely usedassolution in such situations on system designed for controlling the shape of the boom is that will booms probably exist as long our society depends petroleum and its derived products (Fingas, 2012). In fact, an system designed for controlling the shape of the boom is explained. Then the simulation results are discussed. The that will probably exist as long as our society depends on paper ends with the conclusions and futureare work. that will probably exist as long as our society depends on explained. Then the simulation results discussed. The petroleum and its derived products (Fingas, 2012). In fact, an up-to-date list of oil spill catastrophes is available at explained. Then the simulation results are discussed. The paper ends with the conclusions and future work. petroleum and its products (Fingas, 2012). In an petroleum and itsofderived derived products (Fingas, 2012). In fact, fact,and an with the and up-to-date list oilit was spill catastrophes is available at paper (Wikipedia, 2016). As reported in (Laffon, Pásaro, paper ends ends the conclusions conclusions and future future work. work. up-to-date list of oil spill catastrophes is available at 2. with MODELLING THE FLOATING BOOM up-to-date list of As oilthe spill catastrophes ismarine available at (Wikipedia, 2016). it was reported in (Laffon, Pásaro, and Valdiglesias, 2016), contamination of waters 2. MODELLING THE FLOATING BOOM (Wikipedia, 2016). 2016). As the it was was reported in in (Laffon, (Laffon, Pásaro, and (Wikipedia, As it reported Pásaro, and Valdiglesias, 2016), contamination of marine waters 2. FLOATING BOOM strongly affects the ecosystem, causing the death of marine As a response to maritimeTHE oil pollution, floating booms are 2. MODELLING MODELLING THE FLOATING BOOM Valdiglesias, 2016), the of marine waters Valdiglesias, 2016), the contamination contamination of marine waters As strongly affects the ecosystem, causing the death of involve marine a response to maritime oil oil pollution, floating easier boomstheir are mammals as well as some birds. Oil spills may even generally used to confine the stains, making strongly affects the ecosystem, causing the death of marine As aa response to maritime oil pollution, floating booms are strongly affects the ecosystem, causing the death of marine As response to maritime oil pollution, floating booms are mammals as well as some birds. Oil spills may even involve generally used to confine the oil stains, making easier their health risksasfor people participating inspills the cleanup operations removal from the surface ofthe theoil water. mammals well as some birds. Oil may even involve generally used to confine stains, making easier their mammals as well as some birds. Oil spills may even involve generally used to confine the oil stains, making easier their removal from the surface of the water. health risks for people participating in the cleanup operations and coastal inhabitants, given the toxicological properties of removal from the surface of the water. health risks for in operations According to the (Oebius, booms and barriers can be health risks for people people participating participating in the the cleanup cleanup operations from surface1999), of the water. and coastal inhabitants, given the toxicological properties of removal the oil components. According regarding to (Oebius,the1999), booms and barriers can be and coastal inhabitants, process of blocking, sweeping, andoil coastal inhabitants, given given the the toxicological toxicological properties properties of of classified the components. According to (Oebius, 1999), booms and barriers can be According to (Oebius, 1999), booms and barriers canthey be classified regarding the process of blocking, sweeping, the oil In work, we have developed a model of a floating boom diverting, or herding of theprocess spilled of floating substances thethis oil components. components. classified regarding the blocking, sweeping, classified regarding the process of blocking, sweeping, In this work, we have developed a model of a floating boom diverting, or herding of the spilled floating substances they dragged by two boats and designed a control system to boom shape carry out. We are mainly interested in two types: In this work, we have developed aa model of floating diverting, or of spilled substances In this work, weboats have developed model of aasystem floating boom diverting, or herding herding of the the spilledinfloating floating substances they they out. We are mainly interested two types: dragged by two and designed a control toSurface shape carry the barrier. The small ships are USV (Unmanned dragged by two boats and designed a control system to shape carry out. We are mainly interested in two types: - Sweeping booms: used to transport a types: contained floating dragged by two boats and designed a control system to shape carry out. We are mainly interested in two the barrier. The small ships are USV (Unmanned Surface Vehicle). The scenario consists ofUSV an oil(Unmanned spill on theSurface water - Sweeping booms: used to transport a contained floating the barrier. The small ships are volume to abooms: different place (semicircular shape). floating the barrier. The smallused ships are Vehicle). The scenario consists ofUSV an the oil(Unmanned spill on theSurface water used to aa contained surface and a barrier to confine spill until the sea -- Sweeping Sweeping used to transport transport contained volume to abooms: different place (semicircular shape). floating Vehicle). The scenario consists of an oil spill on the water Vehicle). The scenarioused consists of an oil spill thethe water surface and a cleaned barrier to confine the spill on until sea volume to aa different place (semicircular shape). surface can be up with other systems. volume to different place (semicircular shape). surface and aa barrier to the and be barrier used used to confine confine the spill spill until until the the sea sea surface can cleaned up with other systems. surface surface can can be be cleaned cleaned up up with with other other systems. systems.
Copyright © 2017, 2017 IFAC 13650 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 13650 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 13650 10.1016/j.ifacol.2017.08.2163 Copyright © 2017 IFAC 13650
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 G.E. Gapingsi et al. / IFAC PapersOnLine 50-1 (2017) 13108–13113
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Herder booms: used to contain and simultaneously concentrate a floating substance that is drifting (circular shape).
Other elements to be considered are the boats towing the boom. Some of the possible effects of the floating boom on the USV dynamics have been studied in (Giron-Sierra et al., 2015). In our case, the flexible floating boom has an infinite number of degrees of freedom. To tackle this problem, it has been assumed, as in (Pereda et al., 2011; Boushaba et al., 2005), that a good approach could be considering it a juxtaposition of n cylindrical rigid bodies which are linked one to another by a rotational degree of freedom. This assumption reduces the problem to modelling a discrete system with a finite number of degrees of freedom. Following (Boushaba et al., 2005), the coordinates of the left end of the floating boom are xL, yL, and i is the angle between the central axis of the i-th cylindrical rigid body and the positive horizontal axis, x, for i = 1,…,n. Therefore, we have a system with n + 2 degrees of freedom (Figure 1, a).
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Let v be the speed of the boom, then v and v are the components in the normal and tangential direction of the cylinder, respectively. Considering ûi the unit vector parallel to the direction of the cylindrical rigid body i, and ûi the normal unit vector, the components of the speed in the normal and parallel directions of the point ri(s) of the rigid body can be written as: 𝑣𝑣𝑖𝑖∥ (𝑠𝑠) = (𝑢𝑢̂𝑖𝑖∥ ⋅ 𝑟𝑟𝑖𝑖̇ (𝑠𝑠)) 𝑢𝑢̂𝑖𝑖∥
(5)
𝑣𝑣𝑖𝑖⊥ (𝑠𝑠) = (𝑢𝑢̂𝑖𝑖⊥ ⋅ 𝑟𝑟𝑖𝑖̇ (𝑠𝑠))𝑢𝑢̂𝑖𝑖⊥
(6)
Therefore, the external dragging forces and moments applied to the rigid body i during the navigation can be calculated as: 𝐹𝐹𝑖𝑖,𝑥𝑥 𝐿𝐿/2 𝐹𝐹𝑖𝑖 (𝑠𝑠) = [ ] = − ∫𝐿𝐿/2 (𝑐𝑐𝑉𝑉 𝑣𝑣̂𝑖𝑖∥ (𝑠𝑠) + 𝑐𝑐𝑆𝑆 𝑣𝑣̂𝑖𝑖⊥ (𝑠𝑠))𝑑𝑑𝑑𝑑 𝐹𝐹𝑖𝑖,𝑦𝑦
(7)
𝜏𝜏𝑖𝑖 = − ∫𝐿𝐿/2 𝑟𝑟𝑖𝑖 (𝑠𝑠)×(𝑐𝑐𝑉𝑉 𝑣𝑣̂𝑖𝑖∥ (𝑠𝑠) + 𝑐𝑐𝑆𝑆 𝑣𝑣̂𝑖𝑖⊥ (𝑠𝑠))𝑑𝑑𝑑𝑑
(8)
𝐿𝐿/2
Here cV, cS are the dragging resistances in the parallel and normal directions, respectively, of the considered rigid body. Therefore, one can determine the generalized forces applied to the floating boom as follows:
xL xR n p j f Lx f Rx y y QxL . L . R Fj . xL (9) j 1 f Ly xL f Ry xR n
f Lx f Rx Fj , x j 1
n
QyL f Ly f Ry Fj ,y Fig. 1. Structure of the flexible boom (a) and (b) section. Let pi be the position of the centre of mass of the i-th rigid body (1), being L the length of any cylinder.
i
(11)
,n
N 1 1 mL2 2 2 i ) K ( m pi 2 12 j 1 2
xL i 1 sin( j ) L j 1 sin( i ) L j i 1 L 2 s cos( ) i y cos( ) j i L j 1 (3) On the other hand, let xR, yR, be the components of the right end of the floating boom; due to the continuity condition of the boom the following equation can be derived:
(4)
j 1
The overall kinetic energy of the system is given by:
(2)
cos( i ) d pi ( s ) s dt sin( i )
xR xL n cos( j ) y y L sin( j ) R L j 1
p j
In the previous equations, (FLx,Fly) and (FRx,FRy), the components of the dragging forces are applied to the left and right ends of the flexible floating boom, respectively.
The velocity of this point ri(s) is therefore given by: ri ( s )
n
Qí f Rx L sin(i ) f Rx L sin(i ) i Fj . i 1, 2,
xL L cos(i ) j 1 cos(j ) (1) pi i 1 L . j sin( ) sin( ) y 2 i L Any point ri(s) at a distance s from pi, can be defined by: cos(i ) ri ( s) pi ( s) s . sin(i )
(10)
j 1
(12)
Using all previous expressions, the differential equations describing the dynamics of the flexible floating boom can be derived using the Lagrange formalism (13).
d K K Qql 0 dt ql ql
(13)
With ql = {xL,yL,θi,i=1,...,n} and Qql being the generalized forces from equations (9), (10) and (11). These equations can be numerically integrated to obtain the complete dynamics of the system.
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3. SIMULATION OF THE MODEL
0 , , , , 0 2 2 2
For simulation purposes and without loss of generality, let’s consider: n = 5, m = 5 kg (the mass of each cylindrical rigid body), L = 5 m (the length of each rigid body), cV = 0.05 and cS = 0. In a real scenario, the number of segments could be bigger but as the section length varies between 5-200 m, more short sections would only mean more flexibility. Indeed, Sun, Ye, & Fei, (2013) work with a real 46-meterlong five-boom system.
T
t=0.000 s 16 14 12
Y(m)
10
Even so, we have 7 strongly coupled differential equations to solve and 14 initial conditions to analyse. We have first used Mathematica to simplify the equations and then modified them to simulate the system with Matlab.
8 6 4 2 0
We have also assumed the quasi-static approximation. Thus, the inertial forces applied to the floating boom are considered to be very small compared to the dragging forces. This assumption is in fact realistic in case of low speed navigation. This is represented by the following equation: Qθi = 0 i 1, 2,
(14)
,n
On the other hand, the time derivative of the equation (4) gives:
xR xL i 1 sin( j ) y y L j cos( j ) 0 R L j 1
-10
-5
0
5
X (m)
Fig. 2. Initial shape of the boom (5 segments). In the following figures, we have represented the shape of the flexible floating boom at different values of time (t=0s, 12s, 27s, 42s and 57s) for different velocity vectors. These samples were chosen randomly. In Figure 3, we can notice a slow variation of the shape of the boom. Figure 4 shows a similar behaviour but in this case the shape of the boom changes faster and the final position may damage the boom. Nevertheless, the results obtained for velocity vector c) (Figure 5) after 40 seconds are unsatisfactory (since it almost tends to be linear, it will not be useful for oil spill confinement anymore).
(15)
If we consider the following state vectors: P xL , y L , xR , y R
T
1 , 2 , 3 , 4 , 5
-2
T
t=12.0 s
16
Then equations (14) and (15) can be rewritten as: A.P B. 0
20
12
(16)
t=27.0 s
22
14
18
10 Y(m)
Y(m)
16
Where A is a 7×4 matrix and B is a 7x5 matrix, both depending on {xL,yL,θi, i=1,2,3,4,5}.
8
14
6 12
4
10
2
Therefore the dynamic of the quasi-static model that gives the evolution of the shape of the boom is:
0
-2
0
2
4
(17)
10
8
10
12
14
Y(m)
Y(m)
24
24
26
34 32 30
20 18
28 16 20
22
24
26
28
30 X(m)
32
34
36
38
30
40
35
40 X(m)
45
50
Fig. 3. Results of the boom simulation for velocity a). t=12.0s
t=42.0 s
48
17 16
64 62
44 60
12 11
42
Y(m)
Y(m)
Y(m)
13
40
P 1 4.t 1 4.t
38
9 8 7 -2
54 52
36
0
2
4
6 X(m)
8
10
12
14
30
58 56
10
T
t=57.0 s
66
46
15
The initial value of the vector θ (initial shape of the boom) is (Figure 2):
22
36
22
T
P 1 1 1 1
20
t=57.0 s
38
26
P 0.8364 0.5477 0.8364 0.5477 T
18 X(m)
14
c)
16
t=42.0 s
28
We have used Matlab/Simulink for the simulations, and the following values of the speed vector:
b)
8
12
30
where B+ is the Moore-Penrose pseudo inverse of the matrix B. Hence, having a good knowledge of the vector of speeds P at a given time, one can determine the shape of the flexible floating boom by integrating the non-lineal system of differential equations (17)
a)
8
X(m)
32
B . A.P
6
32
34
36
38
40 X(m)
42
44
46
48
50 50
52
54
56
58 X(m)
60
Fig. 4. Results of the boom simulation for velocity b).
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64
66
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 G.E. Gapingsi et al. / IFAC PapersOnLine 50-1 (2017) 13108–13113
7206
Y(m)
322
Y(m)
1620
324
Y(m)
7208
1622
326
1618
318
1614
316
1612
0
2
4
6 X(m)
8
10
12
14
1610 12
7204 7202
1616
320
The controlled variable, yi, is the vector θ, which describes the shape of the flexible floating boom whereas the control variable ui is the speed vector P . Besides, the controller system is also in charge of the reduction of the interactions between the controlled variables.
t=57.0 s 7210
1624
328
314 -2
t=27.0 s
1626
t=12.0 s
330
7200 7198
14
16
18
20
22
24
26
28
30
7196 40
42
44
X(m)
46
48
50 52 X(m)
54
56
58
60
62
Fig. 5. Results of the boom simulation for velocity c). Actually, such results are expected since the model is built considering a quasi-static approximation which works well for low speed navigation. Figure 6 shows the evolution of the shape of the boom. The end points (red) are the boats. It is possible to see that it always keeps a reasonable shape when the velocity is low. According to this behaviour, this model can be used but we still need to control the shape of the flexible floating boom.
35 30 25 Y(m)
Assuming that θr is the reference signal, the control objective is to determine the values of the speed vector P which makes the output of the system equal to the reference. The following control law has been applied: P A .B.K ( s).( r )
(18)
where A+ is now the Moore-Penrose pseudo-inverse of the matrix A and K(s) = Gc(s) is a 5×5 diagonal matrix which elements are PI regulators, as shown in equation (19): 1 Kii ( s) k Pi k Ii s
(19)
The simulation results of the system response with this control law are shown in the following figures, where the initial configuration of the floating boom is given by (Figure 8a):
t=1, 3, 6.............57 s
40
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3 0 , , , ,0 4 2 4
20 15 10 5
T
and the reference is set to (Figure 8b):
0 -5
0
10
20
30
40
5 r , , 0, , 4 6 2
50
X(m)
T
Fig. 6. Shape evolution of the boom for speed vector a). 4. DESIGNING A CONTROL SYSTEM TO SHAPE THE BOOM
Forma deseada
Forma de la barrera deseada
14
5 4
12
3
10
2
In our case, we have opted for a centralized control strategy (Figure 7), where G and Gc are the transfer function matrix and the decentralized controller matrix, respectively, with compatible dimensions.
8 Y(m)
A control strategy is needed in order to shape the boom according to the size and shape of the oil spill. Different techniques have been applied, from artificial intelligence methods (Iglesias, Castro, and Fraguela, 2010), to solutions based on multi-robot systems (Kim et al., 2012; Zahugi, Shanta, and Prasad, 2012; Giron-Sierra et al. 2015). A hierarchical control system is reported in (Arrichiello et al., 2010). Another interesting control approach is described in (Zheng, Negenborn, and Lodewijks, 2014), where first the oil spill is located using a predictive model, and then the control system implements a path following strategy for the two boats.
6
1 0
4
-1
2
-2 -3
0 -2 -12
-4 -6 -10
-8
-6
-4
-2
0
-4
-2
2
0 X(m)
2
4
6
Fig. 8. Initial shape (a) and desired final shape (b) of the flexible floating boom. In this case the aim is to contain and concentrate the oil spill shaping the flexible boom with a close configuration, as a herder boom. Taking, 1 K ( s ) k P k I .I 5 s
Where I5 is the identity matrix, the problem of designing the controllers has been reduced to finding the values (kP, kI) which allows us to achieve the control objectives. By trial and error, the following values have been selected. kP = 0.6005;
Fig. 7. Architecture of the centralized control system.
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kI = 1.0502
Proceedings of the 20th IFAC World Congress 13112 G.E. Gapingsi et al. / IFAC PapersOnLine 50-1 (2017) 13108–13113 Toulouse, France, July 9-14, 2017
The results for the variables (rad) are shown in figure 9 (the five angles, where the reference angle is drawn in blue and the controlled variable in red. As it can be seen, the designed controllers stabilize the systems after around 45 seconds, where in all the cases the system response reaches the reference.
Finally, figure 11 shows how the shape of the flexible floating boom varies from its initial configuration to the desired one. At t= 66,9124 s the shape is exactly the reference one. t=0.00 s
t=1.5018 s
12
12
Actual
10
10
8
Deseada 8
6 Y(m)
Y(m)
6 4
4 2
2 0
0
-2
-2
-4
-4 -10
-8
-6
-4
-2 X(m)
0
2
4
6
8
t=25.7731 s
-5 X(m)
0
5
t=55.2579 s
3
4
2 1 Y(m)
2 Y(m)
-10
4
6
0
-2
0 -1 -2 -3
-4
-4
-6 -8
-15
-6
-4
-2
0 X(m)
2
4
-5 -6
6
-4
-2
0 X(m)
2
4
6
Fig. 11. Variations of the shape of the flexible floating boom during the control process. A new scenario has been tested, going from the initial shape 𝝅𝝅 𝝅𝝅 𝝅𝝅 𝝅𝝅 𝝅𝝅 given by 𝜽𝜽𝟎𝟎 = [ 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 ] (Figure 12, a) to the configuration 𝜽𝜽𝒓𝒓 = [𝜋𝜋,
𝟐𝟐𝝅𝝅 𝝅𝝅 𝟑𝟑
𝝅𝝅
, , 0, − ] (Figure 12, b). 𝟐𝟐
𝟑𝟑
Fig. 9. System response for each controlled variable The speeds (m/s) necessary to achieve these results are represented in figure 10. The evolution of the speed for the left end of the boom (10c and 10d) seems to be lower than the one corresponding to the right end (10a and 10b).
Fig. 12. Initial and final configuration of the flexible floating boom. In this case the two ships are dragging the spill without having it completely confined. The big difference between the initial and the desired final configuration of the boom makes the control slower (settle time around 180 sec), as it is shown in figure 13a for one of the angles, θ 5, and the control action (Figure 13b) Fig. 10. Speed (control variables). 13654
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Fig. 13. System response for controlled variable θ5 (a) and VxL speed (b). The results obtained from these simulations are encouraging. The behaviour of the controlled flexible floating boom allows us to use it to confine the oil spill. 5. CONCLUSIONS AND FUTURE WORKS In this work, a computer model of a flexible floating boom has been developed. Mathematical equations of the initial model of this non-linear strongly coupled system have been simplified in order to make it feasible from the computational point of view. We have simulated a discrete model under quasi-static assumptions. In addition, we have designed a centralized control system to control the shape of the boom. Simulations of the flexible boom pulled by two Unmanned Surface Vehicles are encouraging since it reaches the desired shape in around one minute. Given the complexity of the system, this first control approach is promising. As future works, a more realistic model could be developed without considering the quasi-static approximation, thus taking into account the term in the second order derivation of the Lagrange equation. Besides, perturbations such as windy conditions could be included, although this effect will have stronger influence on the pitch acceleration of the boat more than on the floating boom because of its flexible structure. We think that increasing the number of the segments of the flexible boom will not give any additional knowledge. Additionally, the model would require huge computing resources. ACKNOWLEDGEMENTS The authors would like to thank the Spanish Ministry of Economy and Competitiveness for support under project DPI2013-46665-C2-1-R. REFERENCES Arrichiello, F., Heidarsson, H., Chiaverini, S., & Sukhatme, G. S. (2010, May). Cooperative caging using autonomous aquatic surface vehicles. In Robotics and automation (ICRA), 2010 IEEE international conference on (pp. 4763-4769). IEEE. Bhattacharya, S., Heidarsson, H., Sukhatme, G. S., & Kumar, V. (2011, May). Cooperative control of autonomous surface vehicles for oil skimming and cleanup. In Robotics and automation (ICRA), 2011 IEEE international conference on (pp. 2374-2379). IEEE.
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Boushaba, F., Nouchi, S., Boulerhcha, M., & Muttin, F. (2005). Structural modelling of oil spill containment booms by the finite element method. In International Oil Spill Conference 1, 595-599. American Petroleum Institute. Chatterjee, S. (2015). Oil spill cleanup: Role of environmental biotechnology. Applied Environmental Biotechnology: Present Scenario and Future Trends. Springer India, 129-143. Fingas, M. (2012). The basics of oil spill cleanup. CRC Press. Giron-Sierra, J. M., Gheorghita, A. T., Angulo, G., & Jimenez, J. F. (2015). Preparing the automatic spill recovery by two unmanned boats towing a boom: Development with scale experiments. Ocean Engineering, 95, 23-33. Iglesias, G., Castro, A., & Fraguela, J. A. (2010). Artificial intelligence applied to floating boom behavior under waves and currents. Ocean Engineering, 37(17), 15131521. Kim, Y. H., Lee, S. W., Yang, H. S., & Shell, D. A. (2012). Toward autonomous robotic containment booms: visual servoing for robust inter-vehicle docking of surface vehicles. Intelligent Service Robotics, 5(1), 1-18. Laffon, B., Pásaro, E., & Valdiglesias, V. (2016). Effects of exposure to oil spills on human health: Updated review. Journal of Toxicology and Environmental Health, Part B, 1-24. Menoyo-Larrazabal, J., & Santos-Peñas, M. (2016). Intelligent rudder control of an unmanned surface vessel. Expert Systems with Applications, 55, 106-117. Oebius, H. U. (1999). Physical properties and processes that influence the clean up of oil spills in the marine environment. Spill Science & Technology Bulletin, 5(3), 177-289. Pereda, F. J., de Marina, H. G., Giron-Sierra, J. M., & Jimenez, J. (2011). Towards automatic oil spill confinement with autonomous marine surface vehicles. In OCEANS 2011 IEEE-Spain. Sun, X., Ye, H., & Fei, S. (2013). A closed-loop detection and open-loop control strategy for booms of truckmounted concrete pump. Automation in Construction, 31, 265-273. Upton H.F. (2011). The Deepwater Horizon oil spill and the Gulf of Mexico fishing industry. Congressional Research Service. R41640, 14 p. Wikipedia. https://en.wikipedia.org/wiki/List_of_oil_spills (2016). Zahugi, E. M. H., Shanta, M. M., & Prasad, T. V. (2012). Design of multi-robot system for cleaning up marine oil spill. International Journal of Advanced Information Technology, (IJAIT) Vol, 2. Zheng, H., Negenborn, R. R., & Lodewijks, G. (2014, August). Trajectory tracking of autonomous vessels using model predictive control. In 19th IFAC World Congress (IFAC WC’14).
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Modelling and Control of a Flexible Floating Boom: Modelling and Control of a Flexible Floating Boom: Modelling aa Flexible First of Approach Modelling and and Control Control Flexible Floating Floating Boom: Boom: First of Approach First Approach Approach G. E. First Gapingsi, R. Korbas, M. Santos
G. E. Gapingsi, R. Korbas, M. Santos G. Korbas, G. E. E. Gapingsi, Gapingsi, R. R. Korbas, M. M. Santos Santos Computer Architecture and Automatic Control Department, Computer Architecture and Automatic Control Department, Computer Science Faculty, Complutense University of Madrid, Computer Architecture and Automatic Control Department, Computer Architecture and Automatic Control Department, Computer Science Faculty, Complutense University of Spain Madrid, C/ Profesor García Santesmases 9, 28040-Madrid, Computer Science Faculty, Complutense University of Madrid, Computer Science Faculty, Complutense University of Madrid, C/ Profesor García Santesmases 9, 28040-Madrid, Spain (e-mail: [email protected], [email protected], [email protected]) C/ Profesor García Santesmases 9, 28040-Madrid, Spain C/ Profesor García Santesmases 9, 28040-Madrid, [email protected]) Spain (e-mail: [email protected], [email protected], (e-mail: (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) [email protected]) Abstract: Floating booms are useful tools in the marine world, especially for marine demarcation where Abstract: booms areofuseful tools or in the marine world, especially for work marine wherea some kind Floating of contamination sea, ocean coastal water is present. In this wedemarcation have developed Abstract: Floating booms are areofuseful useful tools or in the the marine world, especially for work marine demarcation wherea Abstract: booms tools in marine world, especially for marine demarcation where some kind Floating of contamination sea, ocean coastal water is present. In this we(unmanned have developed mathematical model for a flexible floating boom which is hooked on two boats surface some kind of contamination of sea, ocean or coastal water is present. In this work we have developed a some kind of contamination of sea, ocean or coastal water present. In this work we have developed mathematical model for a flexible floating boom which is hooked on two boats (unmanned surface vehicles). It is model meant to bea used for oil spill containment inismarine areas. In order to (unmanned simplify thesurface modela mathematical for flexible floating boom which hooked on two boats mathematical model for a used flexible floating boom which hooked on two boats surface vehicles). It ispossible meant to for simulation, oil spill containment inismarine areas. In order to (unmanned simplify the model and to make itsbe computer we have adopted a quasi-static approximation where we vehicles). It is meant to be used for oil spill containment in marine areas. In order to simplify the model vehicles). It is meant to be used for oil spill containment in marine areas. In order to simplify the model and to make possible its computer simulation, we have adopted a quasi-static approximation where we assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation and to make possible its computer simulation, we have adopted aa quasi-static approximation where we and to make possible its computer simulation, we have adopted quasi-static approximation where we assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation of this mathematical model hasare allowed the analysis of the behavior of the floating boom.The Based on this assume that the inertial forces small compared to the drag forces applied to the boom. simulation assume that the inertial forces are small compared to the drag forces applied to the boom. The simulation of this mathematical model has allowed the analysis of the behavior of the floating boom. Based on this knowledge, we have designed a allowed control system whichofallows us to maintain the flexible floating boom in of this model has the analysis the behavior of floating boom. Based on ofspecific this mathematical mathematical model has theshape. analysis ofallows the purpose, behavior of the the floating boom. Basedwith on this this we have designed a allowed control its system which us to maintain the flexible floating boom in aknowledge, configuration by controlling For that the control approach deals the knowledge, we have designed aa control system which allows us to maintain the flexible floating boom in we have designed control system which allows us to maintain the flexible floating boom in aknowledge, specific configuration by controlling its shape. For that purpose, the control approach deals with the speed of the two ends of the floating boom. The results obtained withthe the control implemented control system are specific configuration by controlling controlling its shape. shape. For that that purpose, approach deals with the the aaencouraging, specific configuration by its For purpose, the control approach deals with speed of the two ends of the floating boom. The results obtained with the implemented control system are considering the model is a simplified discrete one and the control effort is reasonable. speed ends The obtained with implemented system speed of of the the two two ends of of the the floating boom. The results results obtained with the implemented control system are are encouraging, considering thefloating model isboom. a simplified discrete one and thethe control effort is control reasonable. Keywords: Floating boom, Modelling and Simulation, Unmanned Surface Vehicles (USV), Knowledgeencouraging, considering the model is a simplified discrete one and the control effort is reasonable. © 2017, IFACconsidering (International Federation ofsimplified Automaticdiscrete Control)one Hosting by control Elsevier effort Ltd. All rights reserved. encouraging, the model is a and the is reasonable. Keywords: Floating boom, Modelling and Simulation, Unmanned Surface Vehicles (USV), Knowledgebased Control System. Keywords: Floating boom, Modelling Modelling and and Simulation, Simulation, Unmanned Unmanned Surface Surface Vehicles Vehicles (USV), (USV), KnowledgeKnowledgeKeywords: Floating boom, based Control System. based Control System. based Control System. The task is not easy because even if the model and control of The task iscan notbe easy because(Menoyo-Larrazabal even if the model and control of the available and Santos1. INTRODUCTION The USV task is iscan notbe easy because(Menoyo-Larrazabal even if if the the model model and and control of The task not easy because even control of the USV available and Santos1. INTRODUCTION Peñas, 2016), the presence of the flexible floating boom has a USV can be available (Menoyo-Larrazabal and Santos1.floating INTRODUCTION This work deals with1. booms. Fortunately, it may not the the USV can be available (Menoyo-Larrazabal and SantosPeñas, 2016), the presence of the flexible floating boom has a INTRODUCTION significant impact on the dynamics of the boats. And vice 2016), the presence of the flexible floating boom has aa This work deals withbut floating booms. Fortunately, itprovide may nota Peñas, be a common issue, these containment barriers Peñas, 2016), the presence of the flexible floating boom has significant impact on the dynamics of the boats. And vice This work deals with floating booms. Fortunately, it may not versa. When attached at dynamics each end of to the a USV, itAnd deforms This work deals with floating booms. Fortunately, it may not significant impact on the boats. vice be a common issue, but these containment barriers provide a fundamental help forbut marine trash and debris control, marinea versa. significant impact the of boats. When at dynamics eachplane end under to the a USV, itAnd deforms be aa common issue, these containment barriers provide inattached theon horizontal the effects ofvice the be commonhelp issue, these containment barriers provide marinea continuously versa. When attached at each end to aa USV, it deforms fundamental forbut marine trash and protection debris control, demarcation, and therefore ecosystem (Chatterjee, versa. When attached at each end to USV, it deforms continuously in the horizontal plane under the effects of the fundamental help for marine trash and debris control, marine forces the vehicles impart at its endpoints and the drag due to fundamental help for marineecosystem trash and protection debris control, marine in horizontal under the effects of the demarcation, and applications therefore (Chatterjee, 2015). Different include e.g. algal blooms or continuously continuously in the the horizontal plane under thethe effects oftheir the forces the vehicles impart at between its plane endpoints and drag due to demarcation, and therefore ecosystem protection (Chatterjee, the water. The separation the vehicles and demarcation, and therefore ecosystem protection (Chatterjee, 2015). Different applications include e.g. algal blooms or forces the vehicles impart at its endpoints and the drag due to sewer spills, among others. forces the vehicles impart at its endpoints and the drag due to the water. The separation between the vehicles and their 2015). Different applications include e.g. algal blooms or accelerations govern thethe planar deformation of 2015).spills, Different applications include e.g. algal blooms or instantaneous sewer among others. the water. The separation between vehicles and their the water. The separation between the vehicles and their instantaneous accelerations govern the planar deformation of sewer spills, among others. boom (Bhattacharya, 2011). The spill among into theothers. sea may cause serious damages to our the seweroilspills, instantaneous accelerations govern the the planar planar deformation deformation of of instantaneous accelerations govern boom (Bhattacharya, 2011). The oil spill into thetherefore sea mayimportant cause serious damages toreact our the environment. It is to be able to the boom (Bhattacharya, 2011). The oil spill into the sea may cause serious damages to our The structure of the paper is the following. Section 2 presents the boom (Bhattacharya, 2011). The oil spill sea mayimportant cause serious our environment. It isthe therefore to bedamages totoreact efficiently to into such misfortunes. One example ofable an accident 2 presents The structure of the paper is the following. environment. It is therefore important to to react a brief introduction on floating barriers Section and develops the environment. It of is misfortunes. therefore important to be bewaters able to react efficiently tospill such Oneinexample ofable an accident structure of the paper is the following. Section 2 involving a oil contaminant marine was the The structure of the paper is the following. Section 2 presents presents aThe brief introduction on floating barriers and develops the efficiently to such misfortunes. One example of an accident mathematical model of the flexible floating boom. Model efficiently to such misfortunes. One example of an accident involving a spill of oil contaminant in marine waters was the amathematical brief on floating barriers and develops the one in thea spill Gulf of ofoil Mexico in April, 2010 (Upton, 2011). asimulations brief introduction introduction on floating barriers and develops the model of the flexible Model floating boom. involving contaminant in marine waters was the shownofin the section 3. Infloating Section boom. 4, the control involving a spill of contaminant in marine waters was the mathematicalaremodel one in the Gulfare ofoil Mexico in April, 2010in (Upton, 2011). flexible Model Floating booms a widely used solution such situations mathematical model of the flexible floating boom. Model simulations are shown in section 3. In Section 4, the control one in the Gulf of Mexico in April, 2010 (Upton, 2011). system designed for controlling the shape of the boom is one in Gulfare ofexist in April, 2011). are in Section 4, control Floating booms aMexico widely used in (Upton, such situations that willthe probably as long assolution our2010 society depends on simulations simulations are shown shown in section section 3. 3. Inshape Section 4,thethe theboom control system designed for controlling theIn of is Floating booms are a widely used solution in such situations explained. Then the simulation results are discussed. The Floating are a widely usedassolution in such situations on system designed for controlling the shape of the boom is that will booms probably exist as long our society depends petroleum and its derived products (Fingas, 2012). In fact, an system designed for controlling the shape of the boom is explained. Then the simulation results are discussed. The that will probably exist as long as our society depends on paper ends with the conclusions and futureare work. that will probably exist as long as our society depends on explained. Then the simulation results discussed. The petroleum and its derived products (Fingas, 2012). In fact, an up-to-date list of oil spill catastrophes is available at explained. Then the simulation results are discussed. The paper ends with the conclusions and future work. petroleum and its products (Fingas, 2012). In an petroleum and itsofderived derived products (Fingas, 2012). In fact, fact,and an with the and up-to-date list oilit was spill catastrophes is available at paper (Wikipedia, 2016). As reported in (Laffon, Pásaro, paper ends ends the conclusions conclusions and future future work. work. up-to-date list of oil spill catastrophes is available at 2. with MODELLING THE FLOATING BOOM up-to-date list of As oilthe spill catastrophes ismarine available at (Wikipedia, 2016). it was reported in (Laffon, Pásaro, and Valdiglesias, 2016), contamination of waters 2. MODELLING THE FLOATING BOOM (Wikipedia, 2016). 2016). As the it was was reported in in (Laffon, (Laffon, Pásaro, and (Wikipedia, As it reported Pásaro, and Valdiglesias, 2016), contamination of marine waters 2. FLOATING BOOM strongly affects the ecosystem, causing the death of marine As a response to maritimeTHE oil pollution, floating booms are 2. MODELLING MODELLING THE FLOATING BOOM Valdiglesias, 2016), the of marine waters Valdiglesias, 2016), the contamination contamination of marine waters As strongly affects the ecosystem, causing the death of involve marine a response to maritime oil oil pollution, floating easier boomstheir are mammals as well as some birds. Oil spills may even generally used to confine the stains, making strongly affects the ecosystem, causing the death of marine As aa response to maritime oil pollution, floating booms are strongly affects the ecosystem, causing the death of marine As response to maritime oil pollution, floating booms are mammals as well as some birds. Oil spills may even involve generally used to confine the oil stains, making easier their health risksasfor people participating inspills the cleanup operations removal from the surface ofthe theoil water. mammals well as some birds. Oil may even involve generally used to confine stains, making easier their mammals as well as some birds. Oil spills may even involve generally used to confine the oil stains, making easier their removal from the surface of the water. health risks for people participating in the cleanup operations and coastal inhabitants, given the toxicological properties of removal from the surface of the water. health risks for in operations According to the (Oebius, booms and barriers can be health risks for people people participating participating in the the cleanup cleanup operations from surface1999), of the water. and coastal inhabitants, given the toxicological properties of removal the oil components. According regarding to (Oebius,the1999), booms and barriers can be and coastal inhabitants, process of blocking, sweeping, andoil coastal inhabitants, given given the the toxicological toxicological properties properties of of classified the components. According to (Oebius, 1999), booms and barriers can be According to (Oebius, 1999), booms and barriers canthey be classified regarding the process of blocking, sweeping, the oil In work, we have developed a model of a floating boom diverting, or herding of theprocess spilled of floating substances thethis oil components. components. classified regarding the blocking, sweeping, classified regarding the process of blocking, sweeping, In this work, we have developed a model of a floating boom diverting, or herding of the spilled floating substances they dragged by two boats and designed a control system to boom shape carry out. We are mainly interested in two types: In this work, we have developed aa model of floating diverting, or of spilled substances In this work, weboats have developed model of aasystem floating boom diverting, or herding herding of the the spilledinfloating floating substances they they out. We are mainly interested two types: dragged by two and designed a control toSurface shape carry the barrier. The small ships are USV (Unmanned dragged by two boats and designed a control system to shape carry out. We are mainly interested in two types: - Sweeping booms: used to transport a types: contained floating dragged by two boats and designed a control system to shape carry out. We are mainly interested in two the barrier. The small ships are USV (Unmanned Surface Vehicle). The scenario consists ofUSV an oil(Unmanned spill on theSurface water - Sweeping booms: used to transport a contained floating the barrier. The small ships are volume to abooms: different place (semicircular shape). floating the barrier. The smallused ships are Vehicle). The scenario consists ofUSV an the oil(Unmanned spill on theSurface water used to aa contained surface and a barrier to confine spill until the sea -- Sweeping Sweeping used to transport transport contained volume to abooms: different place (semicircular shape). floating Vehicle). The scenario consists of an oil spill on the water Vehicle). The scenarioused consists of an oil spill thethe water surface and a cleaned barrier to confine the spill on until sea volume to aa different place (semicircular shape). surface can be up with other systems. volume to different place (semicircular shape). surface and aa barrier to the and be barrier used used to confine confine the spill spill until until the the sea sea surface can cleaned up with other systems. surface surface can can be be cleaned cleaned up up with with other other systems. systems.
Copyright © 2017, 2017 IFAC 13650 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 13650 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 13650 10.1016/j.ifacol.2017.08.2163 Copyright © 2017 IFAC 13650
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 G.E. Gapingsi et al. / IFAC PapersOnLine 50-1 (2017) 13108–13113
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Herder booms: used to contain and simultaneously concentrate a floating substance that is drifting (circular shape).
Other elements to be considered are the boats towing the boom. Some of the possible effects of the floating boom on the USV dynamics have been studied in (Giron-Sierra et al., 2015). In our case, the flexible floating boom has an infinite number of degrees of freedom. To tackle this problem, it has been assumed, as in (Pereda et al., 2011; Boushaba et al., 2005), that a good approach could be considering it a juxtaposition of n cylindrical rigid bodies which are linked one to another by a rotational degree of freedom. This assumption reduces the problem to modelling a discrete system with a finite number of degrees of freedom. Following (Boushaba et al., 2005), the coordinates of the left end of the floating boom are xL, yL, and i is the angle between the central axis of the i-th cylindrical rigid body and the positive horizontal axis, x, for i = 1,…,n. Therefore, we have a system with n + 2 degrees of freedom (Figure 1, a).
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Let v be the speed of the boom, then v and v are the components in the normal and tangential direction of the cylinder, respectively. Considering ûi the unit vector parallel to the direction of the cylindrical rigid body i, and ûi the normal unit vector, the components of the speed in the normal and parallel directions of the point ri(s) of the rigid body can be written as: 𝑣𝑣𝑖𝑖∥ (𝑠𝑠) = (𝑢𝑢̂𝑖𝑖∥ ⋅ 𝑟𝑟𝑖𝑖̇ (𝑠𝑠)) 𝑢𝑢̂𝑖𝑖∥
(5)
𝑣𝑣𝑖𝑖⊥ (𝑠𝑠) = (𝑢𝑢̂𝑖𝑖⊥ ⋅ 𝑟𝑟𝑖𝑖̇ (𝑠𝑠))𝑢𝑢̂𝑖𝑖⊥
(6)
Therefore, the external dragging forces and moments applied to the rigid body i during the navigation can be calculated as: 𝐹𝐹𝑖𝑖,𝑥𝑥 𝐿𝐿/2 𝐹𝐹𝑖𝑖 (𝑠𝑠) = [ ] = − ∫𝐿𝐿/2 (𝑐𝑐𝑉𝑉 𝑣𝑣̂𝑖𝑖∥ (𝑠𝑠) + 𝑐𝑐𝑆𝑆 𝑣𝑣̂𝑖𝑖⊥ (𝑠𝑠))𝑑𝑑𝑑𝑑 𝐹𝐹𝑖𝑖,𝑦𝑦
(7)
𝜏𝜏𝑖𝑖 = − ∫𝐿𝐿/2 𝑟𝑟𝑖𝑖 (𝑠𝑠)×(𝑐𝑐𝑉𝑉 𝑣𝑣̂𝑖𝑖∥ (𝑠𝑠) + 𝑐𝑐𝑆𝑆 𝑣𝑣̂𝑖𝑖⊥ (𝑠𝑠))𝑑𝑑𝑑𝑑
(8)
𝐿𝐿/2
Here cV, cS are the dragging resistances in the parallel and normal directions, respectively, of the considered rigid body. Therefore, one can determine the generalized forces applied to the floating boom as follows:
xL xR n p j f Lx f Rx y y QxL . L . R Fj . xL (9) j 1 f Ly xL f Ry xR n
f Lx f Rx Fj , x j 1
n
QyL f Ly f Ry Fj ,y Fig. 1. Structure of the flexible boom (a) and (b) section. Let pi be the position of the centre of mass of the i-th rigid body (1), being L the length of any cylinder.
i
(11)
,n
N 1 1 mL2 2 2 i ) K ( m pi 2 12 j 1 2
xL i 1 sin( j ) L j 1 sin( i ) L j i 1 L 2 s cos( ) i y cos( ) j i L j 1 (3) On the other hand, let xR, yR, be the components of the right end of the floating boom; due to the continuity condition of the boom the following equation can be derived:
(4)
j 1
The overall kinetic energy of the system is given by:
(2)
cos( i ) d pi ( s ) s dt sin( i )
xR xL n cos( j ) y y L sin( j ) R L j 1
p j
In the previous equations, (FLx,Fly) and (FRx,FRy), the components of the dragging forces are applied to the left and right ends of the flexible floating boom, respectively.
The velocity of this point ri(s) is therefore given by: ri ( s )
n
Qí f Rx L sin(i ) f Rx L sin(i ) i Fj . i 1, 2,
xL L cos(i ) j 1 cos(j ) (1) pi i 1 L . j sin( ) sin( ) y 2 i L Any point ri(s) at a distance s from pi, can be defined by: cos(i ) ri ( s) pi ( s) s . sin(i )
(10)
j 1
(12)
Using all previous expressions, the differential equations describing the dynamics of the flexible floating boom can be derived using the Lagrange formalism (13).
d K K Qql 0 dt ql ql
(13)
With ql = {xL,yL,θi,i=1,...,n} and Qql being the generalized forces from equations (9), (10) and (11). These equations can be numerically integrated to obtain the complete dynamics of the system.
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3. SIMULATION OF THE MODEL
0 , , , , 0 2 2 2
For simulation purposes and without loss of generality, let’s consider: n = 5, m = 5 kg (the mass of each cylindrical rigid body), L = 5 m (the length of each rigid body), cV = 0.05 and cS = 0. In a real scenario, the number of segments could be bigger but as the section length varies between 5-200 m, more short sections would only mean more flexibility. Indeed, Sun, Ye, & Fei, (2013) work with a real 46-meterlong five-boom system.
T
t=0.000 s 16 14 12
Y(m)
10
Even so, we have 7 strongly coupled differential equations to solve and 14 initial conditions to analyse. We have first used Mathematica to simplify the equations and then modified them to simulate the system with Matlab.
8 6 4 2 0
We have also assumed the quasi-static approximation. Thus, the inertial forces applied to the floating boom are considered to be very small compared to the dragging forces. This assumption is in fact realistic in case of low speed navigation. This is represented by the following equation: Qθi = 0 i 1, 2,
(14)
,n
On the other hand, the time derivative of the equation (4) gives:
xR xL i 1 sin( j ) y y L j cos( j ) 0 R L j 1
-10
-5
0
5
X (m)
Fig. 2. Initial shape of the boom (5 segments). In the following figures, we have represented the shape of the flexible floating boom at different values of time (t=0s, 12s, 27s, 42s and 57s) for different velocity vectors. These samples were chosen randomly. In Figure 3, we can notice a slow variation of the shape of the boom. Figure 4 shows a similar behaviour but in this case the shape of the boom changes faster and the final position may damage the boom. Nevertheless, the results obtained for velocity vector c) (Figure 5) after 40 seconds are unsatisfactory (since it almost tends to be linear, it will not be useful for oil spill confinement anymore).
(15)
If we consider the following state vectors: P xL , y L , xR , y R
T
1 , 2 , 3 , 4 , 5
-2
T
t=12.0 s
16
Then equations (14) and (15) can be rewritten as: A.P B. 0
20
12
(16)
t=27.0 s
22
14
18
10 Y(m)
Y(m)
16
Where A is a 7×4 matrix and B is a 7x5 matrix, both depending on {xL,yL,θi, i=1,2,3,4,5}.
8
14
6 12
4
10
2
Therefore the dynamic of the quasi-static model that gives the evolution of the shape of the boom is:
0
-2
0
2
4
(17)
10
8
10
12
14
Y(m)
Y(m)
24
24
26
34 32 30
20 18
28 16 20
22
24
26
28
30 X(m)
32
34
36
38
30
40
35
40 X(m)
45
50
Fig. 3. Results of the boom simulation for velocity a). t=12.0s
t=42.0 s
48
17 16
64 62
44 60
12 11
42
Y(m)
Y(m)
Y(m)
13
40
P 1 4.t 1 4.t
38
9 8 7 -2
54 52
36
0
2
4
6 X(m)
8
10
12
14
30
58 56
10
T
t=57.0 s
66
46
15
The initial value of the vector θ (initial shape of the boom) is (Figure 2):
22
36
22
T
P 1 1 1 1
20
t=57.0 s
38
26
P 0.8364 0.5477 0.8364 0.5477 T
18 X(m)
14
c)
16
t=42.0 s
28
We have used Matlab/Simulink for the simulations, and the following values of the speed vector:
b)
8
12
30
where B+ is the Moore-Penrose pseudo inverse of the matrix B. Hence, having a good knowledge of the vector of speeds P at a given time, one can determine the shape of the flexible floating boom by integrating the non-lineal system of differential equations (17)
a)
8
X(m)
32
B . A.P
6
32
34
36
38
40 X(m)
42
44
46
48
50 50
52
54
56
58 X(m)
60
Fig. 4. Results of the boom simulation for velocity b).
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7206
Y(m)
322
Y(m)
1620
324
Y(m)
7208
1622
326
1618
318
1614
316
1612
0
2
4
6 X(m)
8
10
12
14
1610 12
7204 7202
1616
320
The controlled variable, yi, is the vector θ, which describes the shape of the flexible floating boom whereas the control variable ui is the speed vector P . Besides, the controller system is also in charge of the reduction of the interactions between the controlled variables.
t=57.0 s 7210
1624
328
314 -2
t=27.0 s
1626
t=12.0 s
330
7200 7198
14
16
18
20
22
24
26
28
30
7196 40
42
44
X(m)
46
48
50 52 X(m)
54
56
58
60
62
Fig. 5. Results of the boom simulation for velocity c). Actually, such results are expected since the model is built considering a quasi-static approximation which works well for low speed navigation. Figure 6 shows the evolution of the shape of the boom. The end points (red) are the boats. It is possible to see that it always keeps a reasonable shape when the velocity is low. According to this behaviour, this model can be used but we still need to control the shape of the flexible floating boom.
35 30 25 Y(m)
Assuming that θr is the reference signal, the control objective is to determine the values of the speed vector P which makes the output of the system equal to the reference. The following control law has been applied: P A .B.K ( s).( r )
(18)
where A+ is now the Moore-Penrose pseudo-inverse of the matrix A and K(s) = Gc(s) is a 5×5 diagonal matrix which elements are PI regulators, as shown in equation (19): 1 Kii ( s) k Pi k Ii s
(19)
The simulation results of the system response with this control law are shown in the following figures, where the initial configuration of the floating boom is given by (Figure 8a):
t=1, 3, 6.............57 s
40
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3 0 , , , ,0 4 2 4
20 15 10 5
T
and the reference is set to (Figure 8b):
0 -5
0
10
20
30
40
5 r , , 0, , 4 6 2
50
X(m)
T
Fig. 6. Shape evolution of the boom for speed vector a). 4. DESIGNING A CONTROL SYSTEM TO SHAPE THE BOOM
Forma deseada
Forma de la barrera deseada
14
5 4
12
3
10
2
In our case, we have opted for a centralized control strategy (Figure 7), where G and Gc are the transfer function matrix and the decentralized controller matrix, respectively, with compatible dimensions.
8 Y(m)
A control strategy is needed in order to shape the boom according to the size and shape of the oil spill. Different techniques have been applied, from artificial intelligence methods (Iglesias, Castro, and Fraguela, 2010), to solutions based on multi-robot systems (Kim et al., 2012; Zahugi, Shanta, and Prasad, 2012; Giron-Sierra et al. 2015). A hierarchical control system is reported in (Arrichiello et al., 2010). Another interesting control approach is described in (Zheng, Negenborn, and Lodewijks, 2014), where first the oil spill is located using a predictive model, and then the control system implements a path following strategy for the two boats.
6
1 0
4
-1
2
-2 -3
0 -2 -12
-4 -6 -10
-8
-6
-4
-2
0
-4
-2
2
0 X(m)
2
4
6
Fig. 8. Initial shape (a) and desired final shape (b) of the flexible floating boom. In this case the aim is to contain and concentrate the oil spill shaping the flexible boom with a close configuration, as a herder boom. Taking, 1 K ( s ) k P k I .I 5 s
Where I5 is the identity matrix, the problem of designing the controllers has been reduced to finding the values (kP, kI) which allows us to achieve the control objectives. By trial and error, the following values have been selected. kP = 0.6005;
Fig. 7. Architecture of the centralized control system.
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The results for the variables (rad) are shown in figure 9 (the five angles, where the reference angle is drawn in blue and the controlled variable in red. As it can be seen, the designed controllers stabilize the systems after around 45 seconds, where in all the cases the system response reaches the reference.
Finally, figure 11 shows how the shape of the flexible floating boom varies from its initial configuration to the desired one. At t= 66,9124 s the shape is exactly the reference one. t=0.00 s
t=1.5018 s
12
12
Actual
10
10
8
Deseada 8
6 Y(m)
Y(m)
6 4
4 2
2 0
0
-2
-2
-4
-4 -10
-8
-6
-4
-2 X(m)
0
2
4
6
8
t=25.7731 s
-5 X(m)
0
5
t=55.2579 s
3
4
2 1 Y(m)
2 Y(m)
-10
4
6
0
-2
0 -1 -2 -3
-4
-4
-6 -8
-15
-6
-4
-2
0 X(m)
2
4
-5 -6
6
-4
-2
0 X(m)
2
4
6
Fig. 11. Variations of the shape of the flexible floating boom during the control process. A new scenario has been tested, going from the initial shape 𝝅𝝅 𝝅𝝅 𝝅𝝅 𝝅𝝅 𝝅𝝅 given by 𝜽𝜽𝟎𝟎 = [ 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 , 𝟐𝟐 ] (Figure 12, a) to the configuration 𝜽𝜽𝒓𝒓 = [𝜋𝜋,
𝟐𝟐𝝅𝝅 𝝅𝝅 𝟑𝟑
𝝅𝝅
, , 0, − ] (Figure 12, b). 𝟐𝟐
𝟑𝟑
Fig. 9. System response for each controlled variable The speeds (m/s) necessary to achieve these results are represented in figure 10. The evolution of the speed for the left end of the boom (10c and 10d) seems to be lower than the one corresponding to the right end (10a and 10b).
Fig. 12. Initial and final configuration of the flexible floating boom. In this case the two ships are dragging the spill without having it completely confined. The big difference between the initial and the desired final configuration of the boom makes the control slower (settle time around 180 sec), as it is shown in figure 13a for one of the angles, θ 5, and the control action (Figure 13b) Fig. 10. Speed (control variables). 13654
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Fig. 13. System response for controlled variable θ5 (a) and VxL speed (b). The results obtained from these simulations are encouraging. The behaviour of the controlled flexible floating boom allows us to use it to confine the oil spill. 5. CONCLUSIONS AND FUTURE WORKS In this work, a computer model of a flexible floating boom has been developed. Mathematical equations of the initial model of this non-linear strongly coupled system have been simplified in order to make it feasible from the computational point of view. We have simulated a discrete model under quasi-static assumptions. In addition, we have designed a centralized control system to control the shape of the boom. Simulations of the flexible boom pulled by two Unmanned Surface Vehicles are encouraging since it reaches the desired shape in around one minute. Given the complexity of the system, this first control approach is promising. As future works, a more realistic model could be developed without considering the quasi-static approximation, thus taking into account the term in the second order derivation of the Lagrange equation. Besides, perturbations such as windy conditions could be included, although this effect will have stronger influence on the pitch acceleration of the boat more than on the floating boom because of its flexible structure. We think that increasing the number of the segments of the flexible boom will not give any additional knowledge. Additionally, the model would require huge computing resources. ACKNOWLEDGEMENTS The authors would like to thank the Spanish Ministry of Economy and Competitiveness for support under project DPI2013-46665-C2-1-R. REFERENCES Arrichiello, F., Heidarsson, H., Chiaverini, S., & Sukhatme, G. S. (2010, May). Cooperative caging using autonomous aquatic surface vehicles. In Robotics and automation (ICRA), 2010 IEEE international conference on (pp. 4763-4769). IEEE. Bhattacharya, S., Heidarsson, H., Sukhatme, G. S., & Kumar, V. (2011, May). Cooperative control of autonomous surface vehicles for oil skimming and cleanup. In Robotics and automation (ICRA), 2011 IEEE international conference on (pp. 2374-2379). IEEE.
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Boushaba, F., Nouchi, S., Boulerhcha, M., & Muttin, F. (2005). Structural modelling of oil spill containment booms by the finite element method. In International Oil Spill Conference 1, 595-599. American Petroleum Institute. Chatterjee, S. (2015). Oil spill cleanup: Role of environmental biotechnology. Applied Environmental Biotechnology: Present Scenario and Future Trends. Springer India, 129-143. Fingas, M. (2012). The basics of oil spill cleanup. CRC Press. Giron-Sierra, J. M., Gheorghita, A. T., Angulo, G., & Jimenez, J. F. (2015). Preparing the automatic spill recovery by two unmanned boats towing a boom: Development with scale experiments. Ocean Engineering, 95, 23-33. Iglesias, G., Castro, A., & Fraguela, J. A. (2010). Artificial intelligence applied to floating boom behavior under waves and currents. Ocean Engineering, 37(17), 15131521. Kim, Y. H., Lee, S. W., Yang, H. S., & Shell, D. A. (2012). Toward autonomous robotic containment booms: visual servoing for robust inter-vehicle docking of surface vehicles. Intelligent Service Robotics, 5(1), 1-18. Laffon, B., Pásaro, E., & Valdiglesias, V. (2016). Effects of exposure to oil spills on human health: Updated review. Journal of Toxicology and Environmental Health, Part B, 1-24. Menoyo-Larrazabal, J., & Santos-Peñas, M. (2016). Intelligent rudder control of an unmanned surface vessel. Expert Systems with Applications, 55, 106-117. Oebius, H. U. (1999). Physical properties and processes that influence the clean up of oil spills in the marine environment. Spill Science & Technology Bulletin, 5(3), 177-289. Pereda, F. J., de Marina, H. G., Giron-Sierra, J. M., & Jimenez, J. (2011). Towards automatic oil spill confinement with autonomous marine surface vehicles. In OCEANS 2011 IEEE-Spain. Sun, X., Ye, H., & Fei, S. (2013). A closed-loop detection and open-loop control strategy for booms of truckmounted concrete pump. Automation in Construction, 31, 265-273. Upton H.F. (2011). The Deepwater Horizon oil spill and the Gulf of Mexico fishing industry. Congressional Research Service. R41640, 14 p. Wikipedia. https://en.wikipedia.org/wiki/List_of_oil_spills (2016). Zahugi, E. M. H., Shanta, M. M., & Prasad, T. V. (2012). Design of multi-robot system for cleaning up marine oil spill. International Journal of Advanced Information Technology, (IJAIT) Vol, 2. Zheng, H., Negenborn, R. R., & Lodewijks, G. (2014, August). Trajectory tracking of autonomous vessels using model predictive control. In 19th IFAC World Congress (IFAC WC’14).
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