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Nonlinear algorithm with adaptive properties to stabilize an underwater vehicle: real-time experiments
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of The International Federation of Automatic Automatic Control Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 6857–6862
Nonlinear algorithm with adaptive properties to Nonlinear algorithm with adaptive properties to Nonlinear with Nonlinear algorithm with adaptive adaptive properties to stabilizealgorithm an underwater vehicle: properties real-time to stabilize an underwater vehicle: real-time stabilize an underwater vehicle: real-time stabilize an underwater vehicle: real-time experiments experiments experiments experiments
A. Manzanilla ∗∗ P. Castillo ∗∗ R. Lozano ∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗ A. Manzanilla P. Castillo Lozano ∗ P. ∗∗ R. ∗∗ A. Manzanilla Castillo R. Lozano A. Manzanilla P. Castillo R. Lozano ∗ UMI LAFMIA CNRS 3175, CINVESTAV-IPN, Mexico, D. F. (e-mail: ∗∗∗ UMI 3175, [email protected]) UMI LAFMIA LAFMIA CNRS CNRS 3175, CINVESTAV-IPN, CINVESTAV-IPN, Mexico, Mexico, D. D. F. F. (e-mail: (e-mail: 3175, CINVESTAV-IPN, Mexico, D. F. (e-mail: ∗∗ UMI LAFMIA CNRS [email protected]) Sorbonne Universit´ e s, Universit´ e de Technologie de Compi` e gne, CNRS, [email protected]) [email protected]) ∗∗ ∗∗ Universit´ es, Universit´ Technologie de Compi` eegne, CNRS, UMR 7253, Heudiasyc, Avenue deeee de Landshut, CS 60319, 60203 Compi` egne ∗∗ Sorbonne Sorbonne Universit´ s, de Technologie de CNRS, Sorbonne Universit´ee57 s, Universit´ Universit´ de Technologie de Compi` Compi` egne, gne, CNRS, UMR 7253, Heudiasyc, 57 Avenue de Landshut, CS 60319, 60203 Compi` egne cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) UMR 7253, Heudiasyc, 57 Avenue de Landshut, CS 60319, 60203 Compi` UMR 7253,cedex, Heudiasyc, 57(e-mail: Avenue de Landshut, CS 60319, 60203 Compi`eegne gne France. (castillo, rlozano)@hds.utc.fr) cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) Abstract: Mini underwater vehicles becomes popular and interesting for the control community. Abstract: Mini underwater vehicles becomes popular and interesting for the control community. Navigate it is a challenge requires control approaches to improve Abstract: Mini underwater becomes popular and interesting for control Abstract:them Miniautonomously, underwater vehicles vehicles becomes that popular and different interesting for the the control community. community. Navigate them autonomously, it is a challenge that requires different control approaches to performance due to external and unknown perturbations that could be present. Our interest is to improve propose Navigate them autonomously, it is a challenge that requires different control approaches to improve Navigate them autonomously, itunknown is a challenge that requires different controlOur approaches to improve due to external and perturbations that could be present. interest is to propose aperformance robust nonlinear controller to stabilize an underwater robot. The mini submarine used in this work performance due to external and unknown perturbations that could be present. Our interest is to propose performance due tocontroller external and unknown perturbations that could bemini present. Our interest is tothis propose aa robust nonlinear to stabilize an underwater robot. The submarine used in has some mechanical properties that increases its potential to apply it in several applications. Solving robust nonlinear nonlinear controller controller to to stabilize stabilize an an underwater underwater robot. robot. The The mini mini submarine submarine used used in in this this work work ahas robust work some mechanical properties that increases its potential to apply it in several applications. Solving this problem requires, in a first time, the development and analysis of its dynamic model in order to has some mechanical properties that increases its potential to apply it in several applications. Solving has some mechanical properties that increases its potential to apply it in several applications. Solving this problem requires, in aacontrol first time, the development and analysis of its dynamic model in order to design a control law. The algorithm is based on the backstepping technique with adaptive and this problem requires, in first time, the development and analysis of its dynamic model in order to this problem requires, in acontrol first time, the development and analysis of itstechnique dynamic model in orderand to design a control law. The algorithm is based on the backstepping with adaptive integral properties. The stability analysis is proved using Lyapunov functions. Graphs from simulations design a control law. The control algorithm is based on the backstepping technique with adaptive and design a control law. The control algorithm is based on the backstepping technique with adaptive and integral properties. stability analysis proved using Lyapunov from simulations results introduce theThe good behavior of the is system even infunctions. presence Graphs of unknown external integral properties. The stability analysis is proved Lyapunov functions. Graphs from simulations integral properties. The stability analysis isclosed-loop proved using using Lyapunov functions. Graphs fromand simulations results introduce the good behavior of the closed-loop system even in presence of unknown and external perturbations. Experimental results confirm the well performance when applying the controller in real results introduce the good behavior of the closed-loop system even in presence of unknown and results introduce the good behavior of the closed-loop system even in presence of unknown and external external perturbations. Experimental results confirm the well performance when applying the controller in real time. perturbations. Experimental results confirm the well performance when applying the controller perturbations. Experimental results confirm the well performance when applying the controller in in real real time. time. time. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Underwater vehicle, Backstepping approach, nonlinear control, adaptive control, external Keywords: Underwater Underwater vehicle, Backstepping Backstepping approach, nonlinear nonlinear control, adaptive adaptive control, external external perturbations, integral action. Keywords: vehicle, Keywords: Underwater vehicle, Backstepping approach, approach, nonlinear control, control, adaptive control, control, external perturbations, integral action. action. perturbations, integral perturbations, integral action. work. An adaptive control for stabilizing depth and pitch states 1. INTRODUCTION work. An adaptive adaptive control controlisfor for stabilizing depth and and pitch pitch states 1. INTRODUCTION INTRODUCTION of an underwater proposed in Maalouf al. (2015). work. An stabilizing depth states 1. work. An adaptivevehicle controlisfor stabilizing depth andet states 1. INTRODUCTION of an analgorithm underwater vehicle proposed inand Maalouf etpitch al. (2015). (2015). The was tested in simulation implemented in real of underwater vehicle is proposed in Maalouf et al. an underwater vehicle is proposed in Maalouf et al. (2015). The ocean has large living and mineral resources, but their of The algorithm algorithm was tested in in simulation simulation and and implemented implemented in in real real time with good results. The was tested The ocean ocean has has large null living and mineral resources, buthuman their The algorithm was tested in simulation and implemented in real exploration is almost dueand to the limitations of thebut The large living mineral resources, their time with good results. The ocean has large living and mineral resources, but their time results. exploration isconditions almost null null due due to to the limitations of thepopular human Vehicle time with with good goodcontrol results. body and theis by nature. The of most algorithm is designed in Zanoli and Conte exploration almost limitations the human exploration isconditions almost nullimposed due to the the limitations of thepopular human Vehicle depth body and the imposed by nature. The most depth control algorithm isin designed infuzzy Zanoli and Conte Conte robot to explore the ocean is the Unmanned Underwater Ve(2003). The algorithms are basedis PD andin control techbody and the conditions imposed by nature. The most popular Vehicle depth control designed Zanoli and body and the conditions imposed by nature. The most popular Vehicle depthalgorithms control algorithm algorithm isin designed infuzzy Zanoli and Conte robot to explore the ocean is the Unmanned Underwater Ve(2003). The are based PD and control techhicles (UUVs) which provided with sensors, manipulators and niques, simulations and experimental results introduce the good robot to explore the ocean is the Unmanned Underwater Ve(2003). The algorithms are based in PD and fuzzy control techrobot to explore the ocean is the Unmanned Underwater and Ve- (2003). The algorithms based in PD and fuzzy control techhicles (UUVs) which provided with sensors, manipulators niques, simulations simulations and are experimental results introduce the good good thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly control algorithm is hicles (UUVs) which provided with sensors, manipulators and niques, and experimental results introduce the hicles (UUVs) which provided with sensors, manipulators and niques, simulations and experimental resultscontrol introduce the good thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly algorithm is sible (Yuh (2000)). Autonomous Underwater Vehicles (AUVs) presented in Jaskot et al. (2014) where the authors conceived thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly control algorithm is thrusters fail to reach places thatUnderwater before it was simply(AUVs) impos- behavior ofinthe controllers. A particularly control algorithm is sible (Yuh (2000)). Autonomous Vehicles presented Jaskot et al. (2014) where the authors conceived and Remotely Operated VehiclesUnderwater (ROVs) have become(AUVs) a tool apresented simple control scheme to(2014) stabilize a ROV, nevertheless these sible (Yuh (2000)). Autonomous Vehicles in Jaskot et al. where the authors conceived sible (Yuh (2000)). Autonomous Underwater Vehicles (AUVs) presented in Jaskot et al. (2014) where the authors conceived and researchers Remotely Operated Operated Vehicles (ROVs) application have become become tool simple control control scheme to stabilize ROV, neverthelessa these these for and industry in different areas as algorithms use only data to coming from accelerometers, presand Remotely Vehicles (ROVs) have aaa tool aaa simple scheme stabilize aaa ROV, nevertheless and Remotely Operated Vehicles (ROVs) application have become tool simple control scheme stabilize ROV, neverthelessa these for researchers researchers and industry industry in different different areas as sure algorithms use only data to coming from accelerometers, presinspection, mapping, etc. sensor and a magnetometer. A robust diving control of an for and in application areas as algorithms use only data coming from accelerometers, a presfor researchers and industry in different application areas as algorithms use only data coming A from accelerometers, a of presinspection, mapping, etc. sure sensor and a magnetometer. robust diving control an AUV is developed in Lapierre (2009). Here the authors use the inspection, mapping, etc. sure sensor and a magnetometer. A robust diving control of an inspection, mapping, sensor and a magnetometer. A robust diving controluse ofthe an Several works can be etc. found in the literature presenting various sure AUV is developed in Lapierre (2009). Here the authors Lyapunov theory and backstepping techniques toauthors designuse a conAUV is developed in Lapierre (2009). Here the the Several works works can be be found in the the literature presenting various AUV is developed in Lapierre (2009). Here the authors use the control techniques to regulate position and orientation of an Several can found in literature presenting various Lyapunov theory and backstepping techniques to to designMotion conSeveral works can be found in the literature presenting various law to theory stabilize thebackstepping depth of an underwater vehicle. Lyapunov and techniques design aaa concontrolNevertheless, techniques todue regulate position and orientation of an an trol Lyapunov theory and techniques to designMotion conAUV. sensorsposition cost to and locate an underwater control techniques to regulate orientation of trol law lawof toan stabilize thebackstepping depth of an an underwater underwater vehicle. control techniques to regulate position and orientation of an control AUV using a neuronal network adaptive controller trol to stabilize the depth of vehicle. Motion AUV. Nevertheless, due sensors cost to locate an underwater trol law to stabilize the depth of an underwater vehicle. Motion vehicle some works are only validated in simulation and few in AUV. Nevertheless, due sensors cost locate control of an an AUV AUV using neuronal network adaptive controller AUV. dueonly sensors cost to to locate an an underwater underwater proposed Li etusing al. (2002). The network algorithm is basedcontroller on a LP control of aaa neuronal adaptive vehicleNevertheless, some works works are validated in simulation and few in in is control of anin AUV using neuronal network adaptive controller experimental platforms. For validated example, in in simulation Medina et and al. (2016) vehicle some are only few is proposed in Li et al. (2002). The algorithm is based on aaonly LP vehicle some works are only validated in simulation and few in neuronal network and a sliding mode control. nevertheless, is proposed in Li et al. (2002). The algorithm is based LP experimental platforms. For example, in Medina et al. (2016) proposed in Li et al.a (2002). The algorithm is based on on aonly LP aexperimental controller sliding modeFor to example, adjust thein yaw movement is pro- is platforms. Medina et al. (2016) neuronal network and sliding mode control. nevertheless, experimental platforms. For example, in Medina et al. (2016) simulations are presented in this work. A special underwater neuronal network and mode control. nevertheless, only controller sliding mode to adjust adjust the yaw yaw movement movement is etproproneuronal network and aa sliding sliding mode control. nevertheless, only posed, similarly a fuzzy control is introduced in Gonz´ a lez al. aaa controller sliding mode to the is simulations are presented in this work. A special underwater controller sliding mode to adjust the yaw movement is etprois conceived, presented andwork. controlled (using a PD) in simulations are in A underwater posed, similarly fuzzy control is introduced introduced inetGonz´ Gonz´ lez al. robot simulations are presented presented in this this work. A special special underwater (2015) to stabilize also this angle. In Tehraniin al. (2010) the posed, similarly aaa fuzzy control is aaalez et al. robot is et conceived, presented and controlled (using a PD) in ´ posed, similarly fuzzy control is introduced in Gonz´ lez et al. robot is conceived, presented and controlled (using in Alvarez al. (2009) where the authors propose a ROV with (2015) to tostudy stabilize also this this angle. for In Tehrani Tehrani etnamed al. (2010) (2010) the robot is conceived, presented and controlled (using aa PD) PD) in authors the depth problem a ROVet DENA. (2015) stabilize also angle. In al. the ´ Alvarez et al. (2009) where the authors propose a ROV with (2015) to stabilize also this angle. In Tehrani et al. (2010) the ´ Stewart parallel structure (REMO I) to change its aageometry to Alvarez et al. (2009) where the authors propose ROV with authors study the the depthto problem problem for adepth, ROV the named DENA. ´ Alvarez et al. (2009) where the authors propose ROV with A PID control is used stabilize the roll and the authors study depth for a ROV named DENA. Stewart parallel parallel structure (REMO I) to to change change its opted geometry to authors study the depthto problem for ROV the named DENA. the navigation. In(REMO recent years, has been to mix Stewart structure I) its geometry to A PID PIDangle control is used used stabilize the adepth, depth, roll and and the improve pitch of this vehicle. An experimental comparative perStewart parallel structure (REMO I) to it change its opted geometry to A control is to stabilize the the roll the improve the navigation. In recent years, it has been to mix A PID control is used to stabilize the depth, the roll and the control methods for some nonlinear electromechanical systems the navigation. In recent years, it has been opted to mix pitch angle angleofof of thiscontroller vehicle. An An experimental comparative per- improve formance a PD and a nonlinear feedback adaptive improve the navigation. In recent years, it has been opted to mix pitch this vehicle. experimental comparative percontrol methods for some nonlinear electromechanical systems pitch angleofof thiscontroller vehicle. An comparative per- with uncertainties the parameters with the idea to improve methods some nonlinear systems formance PD andexperimental a nonlinear nonlinear feedback adaptive control methods for forin nonlinear electromechanical electromechanical systems algorithm toaaa regulate the depth an AUV is presented in control formance of PD and feedback adaptive with uncertainties uncertainties insome thetoparameters parameters withsee the Koshkouei idea to to improve improve formance of PD controller controller and aaof nonlinear feedback adaptive robustness with respect disturbances, et al. with in the with the idea algorithm to regulate the depth of an AUV is presented in Maalouf et al. (2013). The goal in this paper is to introduce the with uncertainties in the parameters with the idea to improve algorithm to regulate the depth of an AUV is presented in robustness with respect to Khodayari disturbances, see Koshkouei et al. al. algorithm to regulate the goal depthin of anpaper AUV is introduce presentedthe in (2004), Hong et al. (2010), and Balochian (2015). robustness with respect to disturbances, see Koshkouei et Maalouf et al. (2013). The this is to behavior inal. closed-loop system and validate robustness of the robustness with respect to Khodayari disturbances, see Koshkouei et al. Maalouf et (2013). The goal in this paper is to introduce (2004), Hong et al. (2010), and Balochian (2015). Maalouf et al. (2013). The goal in this paper is to introduce the (2004), Hong et al. (2010), Khodayari and Balochian (2015). behavior in in closed-loop closed-loop system and validate robustness of the the (2004), Hong et al.paper (2010), Khodayari Balochian algorithm (2015). controllers. heading control of and a ROV using robustness nonlinear modelbehavior system validate of goal of this is to introduceand a nonlinear behavior in A closed-loop system validate robustness of the The controllers. A heading control of and ainROV ROV using nonlinear modelThe goal of this paper is to introduce a nonlinear algorithm aided PD algorithms is presented Ramesh et al. (2013). Drags controllers. A heading control of a using nonlinear modelwith integral an adaptive properties to stabilize an underwater The goal of paper to aa nonlinear algorithm controllers. A heading of ainROV using The of this this paper is isproperties to introduce introduce nonlinear algorithm aided PD PD algorithms algorithms is control presented Ramesh et nonlinear al. (2013). (2013).modelDrags with goal integral an adaptive adaptive to stabilize stabilize an underwater underwater coefficients and added mass are in studied and identified in this with aided is presented Ramesh et al. Drags integral an properties to an aided PD algorithms is presented in Ramesh et al. (2013). Drags with integral an adaptive properties to stabilize an underwater coefficients and added mass are studied and identified in this coefficients coefficients and and added added mass mass are are studied studied and and identified identified in in this this
Copyright © 2017, 2017 IFAC 7061Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 7061 Copyright © 2017 IFAC 7061 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 7061Control. 10.1016/j.ifacol.2017.08.1207
Proceedings of the 20th IFAC World Congress 6858 A. Manzanilla et al. / IFAC PapersOnLine 50-1 (2017) 6857–6862 Toulouse, France, July 9-14, 2017
vehicle. The design and stability analysis of the controller is based on the Lyapunov theory and backstepping technique. The notion for adding an integral component is to obtain robustness in the system against unknown and external disturbances. The adaptive property is used to estimate the unknown and nonlinear disturbances in the model as the hydrodynamic parameters. The outline of the paper is the following: The control scheme, the stability analysis and the nonlinear controllers for the underwater vehicle are described in section 3. Simulations are carried out and main graphs illustrating the good behavior are introduced in section 4. An underwater prototype is built to validate in real time the proposed algorithms, its main characteristics are presented in section 5. Finally, experimental graphs showing the good performance when applying the control scheme are depicted in section 6 and conclusions about this work are discussed in section 7. 2. PROBLEM STATEMENT The dynamic equations of motion of an UV can be written as following, according to Fossen (1994)-Fossen (2011) and using the notation described by SNAME (1952) M ν˙ + C(ν )ν + D(ν )v + g(η ) = τ + w (1) η˙ = J(η )v where M represents the inertia matrix (including added mass), C(ν ) includes the Coriolis and centripetal terms (including added mass), D(ν ) describes the hydrodynamic damping matrix, g(η ) is the vector of the gravitational and buoyancy forces, τ means the vector of control inputs, w denotes the disturbance vector, J(η ) is the kinematic transformation between the body and the inertial frames, η describes the position and orientation of the vehicle and ν represents the linear and angular velocities in the body frame. In addition, τ , η and ν are defined as τ = [X ,Y, Z, K, M, N]T , η = [x, y, z, φ , θ , ψ ]T and ν = [u, v, w, p, q, r]T , where X ,Y, Z, K, M, N denote external forces and moments about the body frame, x, y, z, φ , θ , ψ define the position and orientation of the vehicle in the inertial frame and u, v, w, p, q, r represents the linear and angular velocities in the body frame.
only four degree of freedom (x, z, θ , ψ ) and four control inputs, where fi is the force of the propeller i = 1 : 4, see Fig. 1. The body frame coordinates are referenced as xB , yB , zB , considering the origin of that the body frame is set in the center of gravity and is collinear with the main axes of the reference frame, besides the center of buoyancy is collinear along the zB axis, see Fig. 2.
Fig. 2. Gravity and buoyancy forces acting in the underwater robot. Solving (1) for the translational movement, it follows that X = (m − Xu˙ )u˙ + Xv˙v˙ − (myg − Xr˙ )˙r − (Zw˙ w + mw)q
(2)
+(Yv˙ v + mw)r − (Xu)u + fB sin θ Z = (m − Zw˙ )w˙ + (myg − Z p˙ ) p˙ − (mxg − Zq˙ )q˙ − (Yv˙ v + mv)p (3) +(Xu˙ u − (mu))q − (Zw)w − fB cos θ cos φ where m denotes the mass of the vehicle, Xi ,Yi , Zi , Ki , Mi , Ni are the linear damping coefficients; i : u, v, w, p, q, r and Xk˙ , Yk˙ , Zk˙ , Kk˙ , Mk˙ , Nk˙ represent the hydrodynamic added mass coefficients; k˙ : u, ˙ v, ˙ w, ˙ p, ˙ q, ˙ r˙. In addition, xg , yg , zg are elements of the rg which defines the distance from the origin OB of the body frame to the vehicle’s center of gravity (CG), fW introduces the gravity and fB the buoyancy forces, see Fossen (1994). And for the rotational dynamics, it yields M = (−mxg − Zq˙ )w˙ − (Ixy − Kq˙ ) p˙ + (Iyy − Mq˙ )q˙
(4)
+(Zw˙ w + mw)u − (Xu˙u + mu)w + (Nr˙r − Izz r)p 2.1 The underwater vehicle at UMI laboratory
−(K p˙ p − Ix p)r + (Mq )q − ZB cos θ sin φ
The underwater robot at the UMI laboratory is composed by four rotors that produce forces and torques to move it. By design the underwater robot has symmetry in two of its planes and is stable mechanically in roll angle, i.e., φ ≈ 0, and this implies that the lateral displacement is so small. As consequence, it has
N = (−myg − Xr˙ )u˙ + (mxg − Yr˙ )v˙ + (Izz − Nr˙ )˙r −(Yv˙ v − mv)u + (Xu˙u + mu)v − (Mq˙q − Iyyq)p +(K p˙ p − Ixx p)q − (Nr )r (5) where Ixx , Iyy and Izz are the moments of inertia and Ixy is the product of inertia. 2.2 Analysis of the dynamic model
Fig. 1. Forces acting in the submarine.
For control designing some hypothesis need to be stated: the vehicle will move at low speed and with this assumption, some parameters from damping and added mass can be estimated using SolidWorks and CFD Autodesk. The vehicle has two planes of symmetry and as a consequence only parameters in the main diagonal from the matrix equations (damping and added mass) are stronger and computed. Table 1 introduces main parameters considering the geometry of the vehicle and the properties of the materials used in the developing. The others parameters are too small and for our study, these parameters are considered as nonlinear uncertainties in the model. 7062
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uncertainties in the model such that Xu u + Xr˙ r˙ ≤ δux . Thus, it follows that I¯u u˙ ≈ (kwu q − mr)w + ux + δux (11)
Table 1. UV parameters Estimated prototype parameters m 33.99 kg Ixx 0.99 N*m Iyy -0.02 N*m Izz -0.02 N*m Xu˙ 0.963 kg Yv˙ 0.367 kg Zw˙ 0.367 kg Kp˙ 0.002 N*m Mq˙ 0.002 N*m Nr˙ 0.002 N*m
Finally, for the altitude z, I¯z w˙ =Zw w + Zq˙ q˙ + I¯uuq + uz (12) ¯ where Iz = m−Zw˙ . Define again, Zw w+Zq˙ q˙ ≤ δuz as a nonlinear uncertainty parameter. Thus (13) I¯z w˙ ≈ uz + I¯uuq + δuz
In addition, by design the roll angle is stable mechanically and as a consequence too small, i.e., φ ≈ 0. This will implies that the lateral displacement is also small and could be neglected, hence, v, v, ˙ p, p˙ ≈ 0. Note from Fig. 2 that the origin of the body frame OB coincides with the center of gravity (CG), then xg , yg , zg ≈ 0. From the earlier assumptions and taking into account the characteristics of the underwater robot, the yaw dynamics can be written as −Xr˙ u˙ + Izzr˙ − Nr˙ r˙ − Nr r = uψ where N is rewrite as uψ . Note that Izz and Nr˙ can be estimated with SolidWorks considering the geometry of the vehicle (more details see Cruz-Villar et al. (2008)), and therefore they could be considered as constants and written as I¯r = Izz − Nr˙ . Therefore the yaw dynamics yields I¯r r˙ = uψ + Nr r + Xr˙ u˙ (6) Nr is a very small damping parameter very difficult to estimate while Xr˙ at low speed is also very small (they are not main parameters in the damping and added mass matrices). The hydrodynamic analysis is out of the scope of this paper. In this paper, the goal is to stabilize the underwater vehicle even in presence of these nonlinear uncertainties in the model or unknown and external perturbations. Thus, without loss of generality, we can write Nr r + Xr˙ u˙ < δψ , where δψ can be considered as an unknown and external perturbation or a nonlinear uncertainty in the model that depends of the states of the vehicle. Then, rewriting (6) I¯r r˙ ≈ uψ + δψ (7) Now, the θ -dynamics is analyzed, yields to uθ = − Zq˙ w˙ + (Iyy − Mq˙ )q˙ + (Zw˙ w + mw)u − (Xu˙ u + mu)w + Mqq with M is rewriting as uθ , Iyy and Mq˙ can be computed, then I¯q = Iyy − Mq˙ and it follows that I¯q q˙ = −Mq q + Zq˙ w˙ − kwu wu + kuw uw + uθ , where kwu = Zw˙ + m and kuw = Xu˙ + m. Therefore I¯q q˙ = − Mq q + Zq˙ w˙ + k¯ uw uw + uθ (8) ¯ with kuw = kuw − kwu = Xu˙ − Zw˙ . From Table 1, it follows that k¯ uw = 0, 596. Nevertheless, Zq˙ and Mq are considered small enough and grow if the vehicle’s velocity increases. They are considered in our study as unknown and external perturbations as −Mq θ˙ + Zq˙ q ≤ δθ . Hence (9) I¯q q˙ ≈ uθ + k¯ uwuw + δθ Rewriting the longitudinal movement and considering that the vehicle has neutral buoyancy fB ≈ 0, it yields I¯u u˙ = Xu u + Xr˙r˙ + (kwu q − mr)w + ux (10) with X = ux and I¯u = m − Xu˙ . Xr˙ and Xu are too small due to the slow vehicle’s velocity and will be considered as unknown
3. NONLINEAR INTEGRAL AND ADAPTIVE CONTROLLER Notice from the vehicle mathematical equations that there are nonlinearities in the model and also some parameters are unknown and depend of its states. The goal in this section is to propose a general nonlinear controller to stabilize our vehicle. Let be the following system Iα1 α¨ = uα + Iα2 γλ + δα (14) where α , γ , λ are states, Iαi denotes a parameter of the system, δα represents an unknown/external perturbation/uncertainties and uα is the control input. Define the error as eα = α − αd and propose s1 = 0t eα d τ , then s˙1 = eα . Define the following positive definite function as 1 1 V1 = (ks1 s21 ) + e2α 2 2 where ks1 is a positive constant. Then V˙1 = ks1 s1 s˙1 + eα e˙α . It is proposed ρ 1 = α˙ d − ks1 s1 − kα eα with kα > 0 is constant. Then, α˙d = ρ 1 + ks1 s1 + kα eα . This implies that e˙α = α˙ − ρ 1 − ks1 s1 − kα eα and therefore V˙1 = −kα e2α + eα (α˙ − ρ 1).
Define e˙¯α = α˙ − ρ 1. Thus, V1 = −kα e2α + eα e˙¯α . It is proposed 1 V2 = V1 + e˙¯2α 2 then V˙2 = V˙1 + e˙¯α e¨¯α . Deriving e˙¯α with respect to time e¨¯α = α¨ − ρ˙ 1 . Rewriting V˙2 and introducing (14), it yields uα + Iα2 γλ + δα − ρ˙ 1 V˙2 = −kα e2α + e˙¯α eα + Iα1
(15)
It is proposed uα with the form (16) uα = Iα1 (ρ˙ 1 − eα − ke˙¯α e˙¯α ) − Iα2 γλ − δˆα where ke˙¯α is a positive constant and δˆα defines the estimate of δα . Rewriting (15) and defining e = δα − δˆα . It follows that δα
1 (17) e˙¯α eδα Iα1 Considering that the perturbation is changing slowly, this means that k could be considered as constant for some instants, i.e. δα = cte. This will implies that e˙δα = −δ˙ˆα . V˙2 =
−kα e2α
− ke˙¯α e˙¯2α +
Proposing the following candidate Lyapunov function 1 2 VL = V2 + e 2kδ δα
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˙ where kδ is a positive constant. Then V˙L = V˙2 + kδ−1 eδα (−δˆα ). It is proposed 1 ˙ δˆα = (18) k e˙¯α Iα1 δ Then V˙L = −kα eα2 − ke˙¯ e˙¯2α + 1 e˙¯α eδα − 1 e˙¯α eδα , and thereα
fore
Iα1
Iα1
(19) V˙L = −kα eα2 − ke˙¯α e˙¯α2 ≤ 0 ∀ t ≥ 0 Thus, (19) is semidefinite negative and implies that is locally bounded. 3.1 Nonlinear algorithms for the underwater vehicle The idea now is to use the previous controller to stabilize the underwater robot. From (7), (9), (11) and (13) notice that the states are in the body frame. The goal is to validate the strategies in real time, then, it is most common to use sensors referred to the Euler angles (for the orientation) and to the inertial frame (for the position). From (1), it follows that for the orientation ν2 = J2−1 (η2 )η˙ 2 , with ν2 = (p, q, r)T and η2 = (φ , θ , ψ )T . Hence, q = θ˙ cφ + ψ˙ cθ sφ and r = −θ˙ sφ + ψ˙ cθ cφ , where cα , sα denote cos(α ) and sin(α ) respectively. Remember that from design φ˙ , φ ≈ 0, then, q = θ˙ and r = ψ˙ cθ . Thus (7) becomes Iψ1 ψ¨ = uψ + Iψ2 ψ˙ θ˙ + δψ (20) ¯ ¯ with Iψ1 = Ir cθ and Iψ2 = Ir sθ . It will be proven later that Iψ1 �= 0, i.e. cθ �= 0. Remark that previous equation has similar form that (14) with α = ψ , γ = ψ˙ and λ = θ˙ . Then, all the control procedure can be applied to obtain the control law to stabilize the yaw dynamics. Thus the control becomes, (21) uψ = Iψ (ρ˙ 1 − eψ − k ˙ e¯˙ψ ) − Iψ ψ˙ θ˙ − δˆψ 1
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ψ and θ were previously stabilized i.e. ψ , θ → 0. Then, the previous equations become u˙ ≈ x¨ and w˙ ≈ z¨. Hence, rewriting (24) (25) I¯u x¨ = ux + δux and the controller yields ux = I¯u (ρ˙ x1 − ex − ke˙¯x e¯˙x ) − δˆx (26) ˙ with Iψ2 = 0, ex = x − xd ; ρx1 = x˙d − ksx sx − kx ex ; δˆx = I¯1 kδx e˙¯x u and s˙x = ex . The previous implies that x, ¨ x˙ → 0 and x → xd . Finally, rewriting (13) and taking into account that x˙θ˙ → 0, then (27) I¯z z¨ = uz + δuz with uz = I¯z (ρ˙ z1 − ez − ke˙¯z e˙¯z ) − δˆz (28) ˙ where ez = z − zd ; ρz1 = z˙d − ksz sz − kz ez ; δˆz = I1¯ kδz e˙¯z and z s˙z = ez . The previous implies that z¨, z˙ → 0 and z → zd .
4. NUMERICAL VALIDATION Numerical simulation for controlling the UV are performed to illustrate the effectiveness of the proposed controller. By length paper limitations only simulations for the yaw state are presented. Note that same control structure is used to stabilize the other states. The simulation introduces the yaw control performance when following the sinusoidal reference: (29) ψd = 45 ∗ sin(0.1 ∗ t) + 45 ∗ cos(0.2 ∗ t) + 100 and with a variable perturbation (|δψ | + white noise). The gains used in simulations are: ksψ = 0.09, kψ = 12, ke˙¯ψ = 15, kδψ = 0.012, while the initial conditions are ψ (0) = ψ˙ (0) = 0. Results illustrating this case can be seen in Figure 3.
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˙ with eψ = ψ − ψd ; ρψ1 = ψ˙ d − ksψ sψ − kψ eψ ; δˆψ = Iψ1 kδψ e˙¯ψ 1 and s˙ψ = eψ . The previous implies that ψ¨ , ψ˙ → 0 and ψ → ψd .
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Rewriting (9) using the q = θ˙ , we have I¯q θ¨ = uθ + k¯ uw uw + δθ Applying the control procedure, the controller is uθ = I¯q (ρ˙ θ1 − eθ − ke˙¯ e¯˙θ ) − k¯ uw uw − δˆθ
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5. PROTOTYPE
algorithms. The gains used in experimental tests for yaw are: ksψ = 0.0009, kψ = 14.11, ke˙¯ψ = 12.6, kδψ = 0.000015, for pitch are: ksθ = 0.0000989, kθ = 14.15, ke˙¯θ = 11.75, kδθ = 0.00015 and for z are: ksz = 0.00099, kz = 2, ke˙¯z = 2.5, kδz = 0.00011. 2
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The AUV UMI is a underwater mini-vehicle developed in the French-Mexican laboratory on computer science and automation membership to the UMR 3175 CNRS. The structure is made of nylon and polyethylene, the prototype has four motors that generate the movements of translation and rotation of the vehicle. Its dimensions are 63 × 63 × 26 cm with a weight of 33.99 kg, a picture of the underwater vehicle can be seen in Fig. 4.
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In Fig. 5, the main components of the embedded system are illustrated. The NUC computer acquires all signals and computes all the algorithms at rate of 12 Hz, while the perception of the robot’s environment is made by several sensors as the pressure sensor measures the depth of the vehicle, the IMU that provides guidance and angular velocity.
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The Arduino-One is implemented as board communication between the speed controllers and the embedded computer. The speed controllers MD-03 are responsible for generating the appropriate signal to manipulate the magnitude and direction of the force generated by the thrusters Seabotix BTD150. The cameras are used to acquire image information, however, their use is optional and depends on the task assigned to the vehicle.
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Fig. 6. Depth, pitch and yaw response for the first real- time experiment. A second experiment was carried out, the objective here is stabilize the underwater robot at θd = 0; ψd = 0 deg and zd = 1 m, and later to move the vehicle in the longitudinal axis, x, in manual mode to increase the hydrodynamic parameters, see Fig. 7. The Fig. 8 introduces this experimental results, the experiment was realized in a pool when people was swimming producing currents (external perturbations) and increasing also the hydrodynamic parameters. Notice in these figures how the yaw and pitch angles are perturbed but the controller maintains all the system stable. A video of this experiment can be seen at: https://youtu.be/Jf-MvKTCIJQ.
Fig. 5. Embedded system 6. EXPERIMENTAL RESULTS The first test were made to stabilize all the system, nevertheless at the moment, it is not possible to measure the x position and its longitudinal velocity in our underwater prototype. Then, only the yaw and pitch angles and the depth can be validated. The goal of this experiment is to dive the vehicle at depth of 1m with a θd = 0 ◦ and follow a sinusoidal reference (ψd = 50 ∗ sin(0.1 ∗ t) + 100)◦in the yaw angle. Fig. 6 introduces the experimental performance when applying the proposed controllers. Notice here the good behavior of the control
7. CONCLUSION A nonlinear model of an underwater prototype vehicle was obtained and analyzed to develop a control algorithm. The submarine configuration considered the lateral movement mechanically stabilized. A nonlinear control law with integral action and an adaptive parameter was proposed to stabilized the underwater vehicle. This controller was obtained for a general form and then applied to the prototype.
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REFERENCES θ
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Fig. 8. Yaw, pitch and z responses when the vehicle is moving in the x axis. Several simulations were carried-out to illustrate the well performance of the closed-loop system. In addition an underwater prototype was built to validate control algorithms. Experimental results verify the well behavior of the controllers when stabilizing the yaw, pitch and depth states. Future work includes to validate the proposed algorithm in presence of aggressive unknown perturbations as water currents. In addition, we are working to add a sensor in the prototype to measure the longitudinal axis and to realize tasks in complete autonomous mode.
´ Alvarez, C., Saltaren, R., Aracil, R., and Garc´ıa, C. (2009). Concepci´on, desarrollo y avances en el control de navegaci´on de robots submarinos paralelos: El robot remo-i. RIAI, 6(3), 92–100. Cruz-Villar, C., Parra-Vega, V., and Rodriguez-Angeles, A. (2008). Project-oriented low cost autonomous underwater vehicle with servo-visual control for mechatronics curricula. In Mechatronics and Machine Vision in Practice, 41–50. Springer. Fossen, T.I. (1994). Guidance and control of ocean vehicles. John Wiley & Sons Inc. Fossen, T.I. (2011). Handbook of marine craft hydrodynamics and motion control. John Wiley & Sons. Gonz´alez, J., Gom´ariz, S., and Batlle, C. (2015). Control difuso para el seguimiento de gui˜nada del auv cormor´an. RIAI, 12(2), 166–176. Hong, E.Y., Soon, H.G., and Chitre, M. (2010). Depth control of an autonomous underwater vehicle, starfish. In OCEANS 2010 IEEE-Sydney, 1–6. IEEE. ´ Jaskot, K., Sciegienka, P., and Nawrat, A.M. (2014). The prototype of an unmanned underwater vehicle–stability and maintain a specified course. In Innovative Control Systems for Tracked Vehicle Platforms, 73–100. Springer. Khodayari, M.H. and Balochian, S. (2015). Modeling and control of autonomous underwater vehicle (auv) in heading and depth attitude via self-adaptive fuzzy pid controller. Journal of Marine Science and Technology, 20(3), 559–578. Koshkouei, A.J., Zinober, A.S., and Burnham, K.J. (2004). Adaptive sliding mode backstepping control of nonlinear systems with unmatched uncertainty. Asian Journal of Control, 6(4), 447–453. Lapierre, L. (2009). Robust diving control of an auv. Ocean Engineering, 36(1), 92–104. Li, J.H., Lee, P.M., and Lee, S.J. (2002). Motion control of an auv using a neural network adaptive controller. In Underwater Technology Symposium, 2002, 217–221. IEEE. Maalouf, D., Chemori, A., and Creuze, V. (2015). L1 adaptive depth and pitch control of an underwater vehicle with realtime experiments. Ocean Engineering, 98, 66–77. Maalouf, D., Tamanaja, I., Campos, E., Chemori, A., Creuze, V., Torres, J., and Lozano, R. (2013). From pd to nonlinear adaptive depth-control of a tethered autonomous underwater vehicle. IFAC Proceedings Volumes, 46(2), 743–748. Medina, Y.V., Tkachova, A.F., Santana, L.H., and Entenza, P.J.P. (2016). Yaw controller in sliding mode for underwater autonomous vehicle. IEEE Latin America Transactions, 14(3), 1213–1220. Ramesh, R., Ramadass, N., Sathianarayanan, D., Vedachalam, N., and Ramadass, G. (2013). Heading control of rov rosub6000 using non-linear model-aided pd approach. Int J Emerg Technol Adv Eng, 3(4). SNAME (1952). Nomenclature for treating the motion of a submerged body through a fluid jr. New York: Technical and Research Bulletin, 1–5. Tehrani, N.H., Heidari, M., Zakeri, Y., and Ghaisari, J. (2010). Development, depth control and stability analysis of an underwater remotely operated vehicle (rov). In ICCA, 814–819. Yuh, J. (2000). Design and control of autonomous underwater robots: A survey. Autonomous Robots, 8(1), 7–24. Zanoli, S.M. and Conte, G. (2003). Remotely operated vehicle depth control. Control Engineering Practice, 11(4), 453– 459.
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Nonlinear algorithm with adaptive properties to Nonlinear algorithm with adaptive properties to Nonlinear with Nonlinear algorithm with adaptive adaptive properties to stabilizealgorithm an underwater vehicle: properties real-time to stabilize an underwater vehicle: real-time stabilize an underwater vehicle: real-time stabilize an underwater vehicle: real-time experiments experiments experiments experiments
A. Manzanilla ∗∗ P. Castillo ∗∗ R. Lozano ∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗ A. Manzanilla P. Castillo Lozano ∗ P. ∗∗ R. ∗∗ A. Manzanilla Castillo R. Lozano A. Manzanilla P. Castillo R. Lozano ∗ UMI LAFMIA CNRS 3175, CINVESTAV-IPN, Mexico, D. F. (e-mail: ∗∗∗ UMI 3175, [email protected]) UMI LAFMIA LAFMIA CNRS CNRS 3175, CINVESTAV-IPN, CINVESTAV-IPN, Mexico, Mexico, D. D. F. F. (e-mail: (e-mail: 3175, CINVESTAV-IPN, Mexico, D. F. (e-mail: ∗∗ UMI LAFMIA CNRS [email protected]) Sorbonne Universit´ e s, Universit´ e de Technologie de Compi` e gne, CNRS, [email protected]) [email protected]) ∗∗ ∗∗ Universit´ es, Universit´ Technologie de Compi` eegne, CNRS, UMR 7253, Heudiasyc, Avenue deeee de Landshut, CS 60319, 60203 Compi` egne ∗∗ Sorbonne Sorbonne Universit´ s, de Technologie de CNRS, Sorbonne Universit´ee57 s, Universit´ Universit´ de Technologie de Compi` Compi` egne, gne, CNRS, UMR 7253, Heudiasyc, 57 Avenue de Landshut, CS 60319, 60203 Compi` egne cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) UMR 7253, Heudiasyc, 57 Avenue de Landshut, CS 60319, 60203 Compi` UMR 7253,cedex, Heudiasyc, 57(e-mail: Avenue de Landshut, CS 60319, 60203 Compi`eegne gne France. (castillo, rlozano)@hds.utc.fr) cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) cedex, France. (e-mail: (castillo, rlozano)@hds.utc.fr) Abstract: Mini underwater vehicles becomes popular and interesting for the control community. Abstract: Mini underwater vehicles becomes popular and interesting for the control community. Navigate it is a challenge requires control approaches to improve Abstract: Mini underwater becomes popular and interesting for control Abstract:them Miniautonomously, underwater vehicles vehicles becomes that popular and different interesting for the the control community. community. Navigate them autonomously, it is a challenge that requires different control approaches to performance due to external and unknown perturbations that could be present. Our interest is to improve propose Navigate them autonomously, it is a challenge that requires different control approaches to improve Navigate them autonomously, itunknown is a challenge that requires different controlOur approaches to improve due to external and perturbations that could be present. interest is to propose aperformance robust nonlinear controller to stabilize an underwater robot. The mini submarine used in this work performance due to external and unknown perturbations that could be present. Our interest is to propose performance due tocontroller external and unknown perturbations that could bemini present. Our interest is tothis propose aa robust nonlinear to stabilize an underwater robot. The submarine used in has some mechanical properties that increases its potential to apply it in several applications. Solving robust nonlinear nonlinear controller controller to to stabilize stabilize an an underwater underwater robot. robot. The The mini mini submarine submarine used used in in this this work work ahas robust work some mechanical properties that increases its potential to apply it in several applications. Solving this problem requires, in a first time, the development and analysis of its dynamic model in order to has some mechanical properties that increases its potential to apply it in several applications. Solving has some mechanical properties that increases its potential to apply it in several applications. Solving this problem requires, in aacontrol first time, the development and analysis of its dynamic model in order to design a control law. The algorithm is based on the backstepping technique with adaptive and this problem requires, in first time, the development and analysis of its dynamic model in order to this problem requires, in acontrol first time, the development and analysis of itstechnique dynamic model in orderand to design a control law. The algorithm is based on the backstepping with adaptive integral properties. The stability analysis is proved using Lyapunov functions. Graphs from simulations design a control law. The control algorithm is based on the backstepping technique with adaptive and design a control law. The control algorithm is based on the backstepping technique with adaptive and integral properties. stability analysis proved using Lyapunov from simulations results introduce theThe good behavior of the is system even infunctions. presence Graphs of unknown external integral properties. The stability analysis is proved Lyapunov functions. Graphs from simulations integral properties. The stability analysis isclosed-loop proved using using Lyapunov functions. Graphs fromand simulations results introduce the good behavior of the closed-loop system even in presence of unknown and external perturbations. Experimental results confirm the well performance when applying the controller in real results introduce the good behavior of the closed-loop system even in presence of unknown and results introduce the good behavior of the closed-loop system even in presence of unknown and external external perturbations. Experimental results confirm the well performance when applying the controller in real time. perturbations. Experimental results confirm the well performance when applying the controller perturbations. Experimental results confirm the well performance when applying the controller in in real real time. time. time. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Underwater vehicle, Backstepping approach, nonlinear control, adaptive control, external Keywords: Underwater Underwater vehicle, Backstepping Backstepping approach, nonlinear nonlinear control, adaptive adaptive control, external external perturbations, integral action. Keywords: vehicle, Keywords: Underwater vehicle, Backstepping approach, approach, nonlinear control, control, adaptive control, control, external perturbations, integral action. action. perturbations, integral perturbations, integral action. work. An adaptive control for stabilizing depth and pitch states 1. INTRODUCTION work. An adaptive adaptive control controlisfor for stabilizing depth and and pitch pitch states 1. INTRODUCTION INTRODUCTION of an underwater proposed in Maalouf al. (2015). work. An stabilizing depth states 1. work. An adaptivevehicle controlisfor stabilizing depth andet states 1. INTRODUCTION of an analgorithm underwater vehicle proposed inand Maalouf etpitch al. (2015). (2015). The was tested in simulation implemented in real of underwater vehicle is proposed in Maalouf et al. an underwater vehicle is proposed in Maalouf et al. (2015). The ocean has large living and mineral resources, but their of The algorithm algorithm was tested in in simulation simulation and and implemented implemented in in real real time with good results. The was tested The ocean ocean has has large null living and mineral resources, buthuman their The algorithm was tested in simulation and implemented in real exploration is almost dueand to the limitations of thebut The large living mineral resources, their time with good results. The ocean has large living and mineral resources, but their time results. exploration isconditions almost null null due due to to the limitations of thepopular human Vehicle time with with good goodcontrol results. body and theis by nature. The of most algorithm is designed in Zanoli and Conte exploration almost limitations the human exploration isconditions almost nullimposed due to the the limitations of thepopular human Vehicle depth body and the imposed by nature. The most depth control algorithm isin designed infuzzy Zanoli and Conte Conte robot to explore the ocean is the Unmanned Underwater Ve(2003). The algorithms are basedis PD andin control techbody and the conditions imposed by nature. The most popular Vehicle depth control designed Zanoli and body and the conditions imposed by nature. The most popular Vehicle depthalgorithms control algorithm algorithm isin designed infuzzy Zanoli and Conte robot to explore the ocean is the Unmanned Underwater Ve(2003). The are based PD and control techhicles (UUVs) which provided with sensors, manipulators and niques, simulations and experimental results introduce the good robot to explore the ocean is the Unmanned Underwater Ve(2003). The algorithms are based in PD and fuzzy control techrobot to explore the ocean is the Unmanned Underwater and Ve- (2003). The algorithms based in PD and fuzzy control techhicles (UUVs) which provided with sensors, manipulators niques, simulations simulations and are experimental results introduce the good good thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly control algorithm is hicles (UUVs) which provided with sensors, manipulators and niques, and experimental results introduce the hicles (UUVs) which provided with sensors, manipulators and niques, simulations and experimental resultscontrol introduce the good thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly algorithm is sible (Yuh (2000)). Autonomous Underwater Vehicles (AUVs) presented in Jaskot et al. (2014) where the authors conceived thrusters fail to reach places that before it was simply imposbehavior of the controllers. A particularly control algorithm is thrusters fail to reach places thatUnderwater before it was simply(AUVs) impos- behavior ofinthe controllers. A particularly control algorithm is sible (Yuh (2000)). Autonomous Vehicles presented Jaskot et al. (2014) where the authors conceived and Remotely Operated VehiclesUnderwater (ROVs) have become(AUVs) a tool apresented simple control scheme to(2014) stabilize a ROV, nevertheless these sible (Yuh (2000)). Autonomous Vehicles in Jaskot et al. where the authors conceived sible (Yuh (2000)). Autonomous Underwater Vehicles (AUVs) presented in Jaskot et al. (2014) where the authors conceived and researchers Remotely Operated Operated Vehicles (ROVs) application have become become tool simple control control scheme to stabilize ROV, neverthelessa these these for and industry in different areas as algorithms use only data to coming from accelerometers, presand Remotely Vehicles (ROVs) have aaa tool aaa simple scheme stabilize aaa ROV, nevertheless and Remotely Operated Vehicles (ROVs) application have become tool simple control scheme stabilize ROV, neverthelessa these for researchers researchers and industry industry in different different areas as sure algorithms use only data to coming from accelerometers, presinspection, mapping, etc. sensor and a magnetometer. A robust diving control of an for and in application areas as algorithms use only data coming from accelerometers, a presfor researchers and industry in different application areas as algorithms use only data coming A from accelerometers, a of presinspection, mapping, etc. sure sensor and a magnetometer. robust diving control an AUV is developed in Lapierre (2009). Here the authors use the inspection, mapping, etc. sure sensor and a magnetometer. A robust diving control of an inspection, mapping, sensor and a magnetometer. A robust diving controluse ofthe an Several works can be etc. found in the literature presenting various sure AUV is developed in Lapierre (2009). Here the authors Lyapunov theory and backstepping techniques toauthors designuse a conAUV is developed in Lapierre (2009). Here the the Several works works can be be found in the the literature presenting various AUV is developed in Lapierre (2009). Here the authors use the control techniques to regulate position and orientation of an Several can found in literature presenting various Lyapunov theory and backstepping techniques to to designMotion conSeveral works can be found in the literature presenting various law to theory stabilize thebackstepping depth of an underwater vehicle. Lyapunov and techniques design aaa concontrolNevertheless, techniques todue regulate position and orientation of an an trol Lyapunov theory and techniques to designMotion conAUV. sensorsposition cost to and locate an underwater control techniques to regulate orientation of trol law lawof toan stabilize thebackstepping depth of an an underwater underwater vehicle. control techniques to regulate position and orientation of an control AUV using a neuronal network adaptive controller trol to stabilize the depth of vehicle. Motion AUV. Nevertheless, due sensors cost to locate an underwater trol law to stabilize the depth of an underwater vehicle. Motion vehicle some works are only validated in simulation and few in AUV. Nevertheless, due sensors cost locate control of an an AUV AUV using neuronal network adaptive controller AUV. dueonly sensors cost to to locate an an underwater underwater proposed Li etusing al. (2002). The network algorithm is basedcontroller on a LP control of aaa neuronal adaptive vehicleNevertheless, some works works are validated in simulation and few in in is control of anin AUV using neuronal network adaptive controller experimental platforms. For validated example, in in simulation Medina et and al. (2016) vehicle some are only few is proposed in Li et al. (2002). The algorithm is based on aaonly LP vehicle some works are only validated in simulation and few in neuronal network and a sliding mode control. nevertheless, is proposed in Li et al. (2002). The algorithm is based LP experimental platforms. For example, in Medina et al. (2016) proposed in Li et al.a (2002). The algorithm is based on on aonly LP aexperimental controller sliding modeFor to example, adjust thein yaw movement is pro- is platforms. Medina et al. (2016) neuronal network and sliding mode control. nevertheless, experimental platforms. For example, in Medina et al. (2016) simulations are presented in this work. A special underwater neuronal network and mode control. nevertheless, only controller sliding mode to adjust adjust the yaw yaw movement movement is etproproneuronal network and aa sliding sliding mode control. nevertheless, only posed, similarly a fuzzy control is introduced in Gonz´ a lez al. aaa controller sliding mode to the is simulations are presented in this work. A special underwater controller sliding mode to adjust the yaw movement is etprois conceived, presented andwork. controlled (using a PD) in simulations are in A underwater posed, similarly fuzzy control is introduced introduced inetGonz´ Gonz´ lez al. robot simulations are presented presented in this this work. A special special underwater (2015) to stabilize also this angle. In Tehraniin al. (2010) the posed, similarly aaa fuzzy control is aaalez et al. robot is et conceived, presented and controlled (using a PD) in ´ posed, similarly fuzzy control is introduced in Gonz´ lez et al. robot is conceived, presented and controlled (using in Alvarez al. (2009) where the authors propose a ROV with (2015) to tostudy stabilize also this this angle. for In Tehrani Tehrani etnamed al. (2010) (2010) the robot is conceived, presented and controlled (using aa PD) PD) in authors the depth problem a ROVet DENA. (2015) stabilize also angle. In al. the ´ Alvarez et al. (2009) where the authors propose a ROV with (2015) to stabilize also this angle. In Tehrani et al. (2010) the ´ Stewart parallel structure (REMO I) to change its aageometry to Alvarez et al. (2009) where the authors propose ROV with authors study the the depthto problem problem for adepth, ROV the named DENA. ´ Alvarez et al. (2009) where the authors propose ROV with A PID control is used stabilize the roll and the authors study depth for a ROV named DENA. Stewart parallel parallel structure (REMO I) to to change change its opted geometry to authors study the depthto problem for ROV the named DENA. the navigation. In(REMO recent years, has been to mix Stewart structure I) its geometry to A PID PIDangle control is used used stabilize the adepth, depth, roll and and the improve pitch of this vehicle. An experimental comparative perStewart parallel structure (REMO I) to it change its opted geometry to A control is to stabilize the the roll the improve the navigation. In recent years, it has been to mix A PID control is used to stabilize the depth, the roll and the control methods for some nonlinear electromechanical systems the navigation. In recent years, it has been opted to mix pitch angle angleofof of thiscontroller vehicle. An An experimental comparative per- improve formance a PD and a nonlinear feedback adaptive improve the navigation. In recent years, it has been opted to mix pitch this vehicle. experimental comparative percontrol methods for some nonlinear electromechanical systems pitch angleofof thiscontroller vehicle. An comparative per- with uncertainties the parameters with the idea to improve methods some nonlinear systems formance PD andexperimental a nonlinear nonlinear feedback adaptive control methods for forin nonlinear electromechanical electromechanical systems algorithm toaaa regulate the depth an AUV is presented in control formance of PD and feedback adaptive with uncertainties uncertainties insome thetoparameters parameters withsee the Koshkouei idea to to improve improve formance of PD controller controller and aaof nonlinear feedback adaptive robustness with respect disturbances, et al. with in the with the idea algorithm to regulate the depth of an AUV is presented in Maalouf et al. (2013). The goal in this paper is to introduce the with uncertainties in the parameters with the idea to improve algorithm to regulate the depth of an AUV is presented in robustness with respect to Khodayari disturbances, see Koshkouei et al. al. algorithm to regulate the goal depthin of anpaper AUV is introduce presentedthe in (2004), Hong et al. (2010), and Balochian (2015). robustness with respect to disturbances, see Koshkouei et Maalouf et al. (2013). The this is to behavior inal. closed-loop system and validate robustness of the robustness with respect to Khodayari disturbances, see Koshkouei et al. Maalouf et (2013). The goal in this paper is to introduce (2004), Hong et al. (2010), and Balochian (2015). Maalouf et al. (2013). The goal in this paper is to introduce the (2004), Hong et al. (2010), Khodayari and Balochian (2015). behavior in in closed-loop closed-loop system and validate robustness of the the (2004), Hong et al.paper (2010), Khodayari Balochian algorithm (2015). controllers. heading control of and a ROV using robustness nonlinear modelbehavior system validate of goal of this is to introduceand a nonlinear behavior in A closed-loop system validate robustness of the The controllers. A heading control of and ainROV ROV using nonlinear modelThe goal of this paper is to introduce a nonlinear algorithm aided PD algorithms is presented Ramesh et al. (2013). Drags controllers. A heading control of a using nonlinear modelwith integral an adaptive properties to stabilize an underwater The goal of paper to aa nonlinear algorithm controllers. A heading of ainROV using The of this this paper is isproperties to introduce introduce nonlinear algorithm aided PD PD algorithms algorithms is control presented Ramesh et nonlinear al. (2013). (2013).modelDrags with goal integral an adaptive adaptive to stabilize stabilize an underwater underwater coefficients and added mass are in studied and identified in this with aided is presented Ramesh et al. Drags integral an properties to an aided PD algorithms is presented in Ramesh et al. (2013). Drags with integral an adaptive properties to stabilize an underwater coefficients and added mass are studied and identified in this coefficients coefficients and and added added mass mass are are studied studied and and identified identified in in this this
Copyright © 2017, 2017 IFAC 7061Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 7061 Copyright © 2017 IFAC 7061 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 7061Control. 10.1016/j.ifacol.2017.08.1207
Proceedings of the 20th IFAC World Congress 6858 A. Manzanilla et al. / IFAC PapersOnLine 50-1 (2017) 6857–6862 Toulouse, France, July 9-14, 2017
vehicle. The design and stability analysis of the controller is based on the Lyapunov theory and backstepping technique. The notion for adding an integral component is to obtain robustness in the system against unknown and external disturbances. The adaptive property is used to estimate the unknown and nonlinear disturbances in the model as the hydrodynamic parameters. The outline of the paper is the following: The control scheme, the stability analysis and the nonlinear controllers for the underwater vehicle are described in section 3. Simulations are carried out and main graphs illustrating the good behavior are introduced in section 4. An underwater prototype is built to validate in real time the proposed algorithms, its main characteristics are presented in section 5. Finally, experimental graphs showing the good performance when applying the control scheme are depicted in section 6 and conclusions about this work are discussed in section 7. 2. PROBLEM STATEMENT The dynamic equations of motion of an UV can be written as following, according to Fossen (1994)-Fossen (2011) and using the notation described by SNAME (1952) M ν˙ + C(ν )ν + D(ν )v + g(η ) = τ + w (1) η˙ = J(η )v where M represents the inertia matrix (including added mass), C(ν ) includes the Coriolis and centripetal terms (including added mass), D(ν ) describes the hydrodynamic damping matrix, g(η ) is the vector of the gravitational and buoyancy forces, τ means the vector of control inputs, w denotes the disturbance vector, J(η ) is the kinematic transformation between the body and the inertial frames, η describes the position and orientation of the vehicle and ν represents the linear and angular velocities in the body frame. In addition, τ , η and ν are defined as τ = [X ,Y, Z, K, M, N]T , η = [x, y, z, φ , θ , ψ ]T and ν = [u, v, w, p, q, r]T , where X ,Y, Z, K, M, N denote external forces and moments about the body frame, x, y, z, φ , θ , ψ define the position and orientation of the vehicle in the inertial frame and u, v, w, p, q, r represents the linear and angular velocities in the body frame.
only four degree of freedom (x, z, θ , ψ ) and four control inputs, where fi is the force of the propeller i = 1 : 4, see Fig. 1. The body frame coordinates are referenced as xB , yB , zB , considering the origin of that the body frame is set in the center of gravity and is collinear with the main axes of the reference frame, besides the center of buoyancy is collinear along the zB axis, see Fig. 2.
Fig. 2. Gravity and buoyancy forces acting in the underwater robot. Solving (1) for the translational movement, it follows that X = (m − Xu˙ )u˙ + Xv˙v˙ − (myg − Xr˙ )˙r − (Zw˙ w + mw)q
(2)
+(Yv˙ v + mw)r − (Xu)u + fB sin θ Z = (m − Zw˙ )w˙ + (myg − Z p˙ ) p˙ − (mxg − Zq˙ )q˙ − (Yv˙ v + mv)p (3) +(Xu˙ u − (mu))q − (Zw)w − fB cos θ cos φ where m denotes the mass of the vehicle, Xi ,Yi , Zi , Ki , Mi , Ni are the linear damping coefficients; i : u, v, w, p, q, r and Xk˙ , Yk˙ , Zk˙ , Kk˙ , Mk˙ , Nk˙ represent the hydrodynamic added mass coefficients; k˙ : u, ˙ v, ˙ w, ˙ p, ˙ q, ˙ r˙. In addition, xg , yg , zg are elements of the rg which defines the distance from the origin OB of the body frame to the vehicle’s center of gravity (CG), fW introduces the gravity and fB the buoyancy forces, see Fossen (1994). And for the rotational dynamics, it yields M = (−mxg − Zq˙ )w˙ − (Ixy − Kq˙ ) p˙ + (Iyy − Mq˙ )q˙
(4)
+(Zw˙ w + mw)u − (Xu˙u + mu)w + (Nr˙r − Izz r)p 2.1 The underwater vehicle at UMI laboratory
−(K p˙ p − Ix p)r + (Mq )q − ZB cos θ sin φ
The underwater robot at the UMI laboratory is composed by four rotors that produce forces and torques to move it. By design the underwater robot has symmetry in two of its planes and is stable mechanically in roll angle, i.e., φ ≈ 0, and this implies that the lateral displacement is so small. As consequence, it has
N = (−myg − Xr˙ )u˙ + (mxg − Yr˙ )v˙ + (Izz − Nr˙ )˙r −(Yv˙ v − mv)u + (Xu˙u + mu)v − (Mq˙q − Iyyq)p +(K p˙ p − Ixx p)q − (Nr )r (5) where Ixx , Iyy and Izz are the moments of inertia and Ixy is the product of inertia. 2.2 Analysis of the dynamic model
Fig. 1. Forces acting in the submarine.
For control designing some hypothesis need to be stated: the vehicle will move at low speed and with this assumption, some parameters from damping and added mass can be estimated using SolidWorks and CFD Autodesk. The vehicle has two planes of symmetry and as a consequence only parameters in the main diagonal from the matrix equations (damping and added mass) are stronger and computed. Table 1 introduces main parameters considering the geometry of the vehicle and the properties of the materials used in the developing. The others parameters are too small and for our study, these parameters are considered as nonlinear uncertainties in the model. 7062
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uncertainties in the model such that Xu u + Xr˙ r˙ ≤ δux . Thus, it follows that I¯u u˙ ≈ (kwu q − mr)w + ux + δux (11)
Table 1. UV parameters Estimated prototype parameters m 33.99 kg Ixx 0.99 N*m Iyy -0.02 N*m Izz -0.02 N*m Xu˙ 0.963 kg Yv˙ 0.367 kg Zw˙ 0.367 kg Kp˙ 0.002 N*m Mq˙ 0.002 N*m Nr˙ 0.002 N*m
Finally, for the altitude z, I¯z w˙ =Zw w + Zq˙ q˙ + I¯uuq + uz (12) ¯ where Iz = m−Zw˙ . Define again, Zw w+Zq˙ q˙ ≤ δuz as a nonlinear uncertainty parameter. Thus (13) I¯z w˙ ≈ uz + I¯uuq + δuz
In addition, by design the roll angle is stable mechanically and as a consequence too small, i.e., φ ≈ 0. This will implies that the lateral displacement is also small and could be neglected, hence, v, v, ˙ p, p˙ ≈ 0. Note from Fig. 2 that the origin of the body frame OB coincides with the center of gravity (CG), then xg , yg , zg ≈ 0. From the earlier assumptions and taking into account the characteristics of the underwater robot, the yaw dynamics can be written as −Xr˙ u˙ + Izzr˙ − Nr˙ r˙ − Nr r = uψ where N is rewrite as uψ . Note that Izz and Nr˙ can be estimated with SolidWorks considering the geometry of the vehicle (more details see Cruz-Villar et al. (2008)), and therefore they could be considered as constants and written as I¯r = Izz − Nr˙ . Therefore the yaw dynamics yields I¯r r˙ = uψ + Nr r + Xr˙ u˙ (6) Nr is a very small damping parameter very difficult to estimate while Xr˙ at low speed is also very small (they are not main parameters in the damping and added mass matrices). The hydrodynamic analysis is out of the scope of this paper. In this paper, the goal is to stabilize the underwater vehicle even in presence of these nonlinear uncertainties in the model or unknown and external perturbations. Thus, without loss of generality, we can write Nr r + Xr˙ u˙ < δψ , where δψ can be considered as an unknown and external perturbation or a nonlinear uncertainty in the model that depends of the states of the vehicle. Then, rewriting (6) I¯r r˙ ≈ uψ + δψ (7) Now, the θ -dynamics is analyzed, yields to uθ = − Zq˙ w˙ + (Iyy − Mq˙ )q˙ + (Zw˙ w + mw)u − (Xu˙ u + mu)w + Mqq with M is rewriting as uθ , Iyy and Mq˙ can be computed, then I¯q = Iyy − Mq˙ and it follows that I¯q q˙ = −Mq q + Zq˙ w˙ − kwu wu + kuw uw + uθ , where kwu = Zw˙ + m and kuw = Xu˙ + m. Therefore I¯q q˙ = − Mq q + Zq˙ w˙ + k¯ uw uw + uθ (8) ¯ with kuw = kuw − kwu = Xu˙ − Zw˙ . From Table 1, it follows that k¯ uw = 0, 596. Nevertheless, Zq˙ and Mq are considered small enough and grow if the vehicle’s velocity increases. They are considered in our study as unknown and external perturbations as −Mq θ˙ + Zq˙ q ≤ δθ . Hence (9) I¯q q˙ ≈ uθ + k¯ uwuw + δθ Rewriting the longitudinal movement and considering that the vehicle has neutral buoyancy fB ≈ 0, it yields I¯u u˙ = Xu u + Xr˙r˙ + (kwu q − mr)w + ux (10) with X = ux and I¯u = m − Xu˙ . Xr˙ and Xu are too small due to the slow vehicle’s velocity and will be considered as unknown
3. NONLINEAR INTEGRAL AND ADAPTIVE CONTROLLER Notice from the vehicle mathematical equations that there are nonlinearities in the model and also some parameters are unknown and depend of its states. The goal in this section is to propose a general nonlinear controller to stabilize our vehicle. Let be the following system Iα1 α¨ = uα + Iα2 γλ + δα (14) where α , γ , λ are states, Iαi denotes a parameter of the system, δα represents an unknown/external perturbation/uncertainties and uα is the control input. Define the error as eα = α − αd and propose s1 = 0t eα d τ , then s˙1 = eα . Define the following positive definite function as 1 1 V1 = (ks1 s21 ) + e2α 2 2 where ks1 is a positive constant. Then V˙1 = ks1 s1 s˙1 + eα e˙α . It is proposed ρ 1 = α˙ d − ks1 s1 − kα eα with kα > 0 is constant. Then, α˙d = ρ 1 + ks1 s1 + kα eα . This implies that e˙α = α˙ − ρ 1 − ks1 s1 − kα eα and therefore V˙1 = −kα e2α + eα (α˙ − ρ 1).
Define e˙¯α = α˙ − ρ 1. Thus, V1 = −kα e2α + eα e˙¯α . It is proposed 1 V2 = V1 + e˙¯2α 2 then V˙2 = V˙1 + e˙¯α e¨¯α . Deriving e˙¯α with respect to time e¨¯α = α¨ − ρ˙ 1 . Rewriting V˙2 and introducing (14), it yields uα + Iα2 γλ + δα − ρ˙ 1 V˙2 = −kα e2α + e˙¯α eα + Iα1
(15)
It is proposed uα with the form (16) uα = Iα1 (ρ˙ 1 − eα − ke˙¯α e˙¯α ) − Iα2 γλ − δˆα where ke˙¯α is a positive constant and δˆα defines the estimate of δα . Rewriting (15) and defining e = δα − δˆα . It follows that δα
1 (17) e˙¯α eδα Iα1 Considering that the perturbation is changing slowly, this means that k could be considered as constant for some instants, i.e. δα = cte. This will implies that e˙δα = −δ˙ˆα . V˙2 =
−kα e2α
− ke˙¯α e˙¯2α +
Proposing the following candidate Lyapunov function 1 2 VL = V2 + e 2kδ δα
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˙ where kδ is a positive constant. Then V˙L = V˙2 + kδ−1 eδα (−δˆα ). It is proposed 1 ˙ δˆα = (18) k e˙¯α Iα1 δ Then V˙L = −kα eα2 − ke˙¯ e˙¯2α + 1 e˙¯α eδα − 1 e˙¯α eδα , and thereα
fore
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(19) V˙L = −kα eα2 − ke˙¯α e˙¯α2 ≤ 0 ∀ t ≥ 0 Thus, (19) is semidefinite negative and implies that is locally bounded. 3.1 Nonlinear algorithms for the underwater vehicle The idea now is to use the previous controller to stabilize the underwater robot. From (7), (9), (11) and (13) notice that the states are in the body frame. The goal is to validate the strategies in real time, then, it is most common to use sensors referred to the Euler angles (for the orientation) and to the inertial frame (for the position). From (1), it follows that for the orientation ν2 = J2−1 (η2 )η˙ 2 , with ν2 = (p, q, r)T and η2 = (φ , θ , ψ )T . Hence, q = θ˙ cφ + ψ˙ cθ sφ and r = −θ˙ sφ + ψ˙ cθ cφ , where cα , sα denote cos(α ) and sin(α ) respectively. Remember that from design φ˙ , φ ≈ 0, then, q = θ˙ and r = ψ˙ cθ . Thus (7) becomes Iψ1 ψ¨ = uψ + Iψ2 ψ˙ θ˙ + δψ (20) ¯ ¯ with Iψ1 = Ir cθ and Iψ2 = Ir sθ . It will be proven later that Iψ1 �= 0, i.e. cθ �= 0. Remark that previous equation has similar form that (14) with α = ψ , γ = ψ˙ and λ = θ˙ . Then, all the control procedure can be applied to obtain the control law to stabilize the yaw dynamics. Thus the control becomes, (21) uψ = Iψ (ρ˙ 1 − eψ − k ˙ e¯˙ψ ) − Iψ ψ˙ θ˙ − δˆψ 1
ψ
e¯ψ
ψ and θ were previously stabilized i.e. ψ , θ → 0. Then, the previous equations become u˙ ≈ x¨ and w˙ ≈ z¨. Hence, rewriting (24) (25) I¯u x¨ = ux + δux and the controller yields ux = I¯u (ρ˙ x1 − ex − ke˙¯x e¯˙x ) − δˆx (26) ˙ with Iψ2 = 0, ex = x − xd ; ρx1 = x˙d − ksx sx − kx ex ; δˆx = I¯1 kδx e˙¯x u and s˙x = ex . The previous implies that x, ¨ x˙ → 0 and x → xd . Finally, rewriting (13) and taking into account that x˙θ˙ → 0, then (27) I¯z z¨ = uz + δuz with uz = I¯z (ρ˙ z1 − ez − ke˙¯z e˙¯z ) − δˆz (28) ˙ where ez = z − zd ; ρz1 = z˙d − ksz sz − kz ez ; δˆz = I1¯ kδz e˙¯z and z s˙z = ez . The previous implies that z¨, z˙ → 0 and z → zd .
4. NUMERICAL VALIDATION Numerical simulation for controlling the UV are performed to illustrate the effectiveness of the proposed controller. By length paper limitations only simulations for the yaw state are presented. Note that same control structure is used to stabilize the other states. The simulation introduces the yaw control performance when following the sinusoidal reference: (29) ψd = 45 ∗ sin(0.1 ∗ t) + 45 ∗ cos(0.2 ∗ t) + 100 and with a variable perturbation (|δψ | + white noise). The gains used in simulations are: ksψ = 0.09, kψ = 12, ke˙¯ψ = 15, kδψ = 0.012, while the initial conditions are ψ (0) = ψ˙ (0) = 0. Results illustrating this case can be seen in Figure 3.
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˙ with eψ = ψ − ψd ; ρψ1 = ψ˙ d − ksψ sψ − kψ eψ ; δˆψ = Iψ1 kδψ e˙¯ψ 1 and s˙ψ = eψ . The previous implies that ψ¨ , ψ˙ → 0 and ψ → ψd .
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Observe that q and r have been previously stabilized, then from (11) it results (24) I¯u u˙ = ux + δux −1 ˙ From (1), it follows that for the position ν1 = J1 (η2 )η1 , with ν1 = (x, y, z)T and η1 = (u, v, w)T . Hence with roll mechanically stable, it can be considered that φ ≈ 0 and it yields u = xc ˙ ψ cθ + ys ˙ ψ cθ − z˙sθ and w = xc ˙ ψ sθ + ys ˙ ψ sθ + z˙cθ . Considering y˙ ≈ 0, then, u˙ = −x(c ˙ ψ sθ θ˙ + sψ cθ ψ˙ ) + xc ¨ ψ cθ − z˙cθ θ˙ − z¨sθ w˙ = x(c ˙ ψ cθ θ˙ − sψ sθ ψ˙ ) + xc ¨ ψ sθ − z˙sθ θ˙ + z¨cθ
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Rewriting (9) using the q = θ˙ , we have I¯q θ¨ = uθ + k¯ uw uw + δθ Applying the control procedure, the controller is uθ = I¯q (ρ˙ θ1 − eθ − ke˙¯ e¯˙θ ) − k¯ uw uw − δˆθ
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5. PROTOTYPE
algorithms. The gains used in experimental tests for yaw are: ksψ = 0.0009, kψ = 14.11, ke˙¯ψ = 12.6, kδψ = 0.000015, for pitch are: ksθ = 0.0000989, kθ = 14.15, ke˙¯θ = 11.75, kδθ = 0.00015 and for z are: ksz = 0.00099, kz = 2, ke˙¯z = 2.5, kδz = 0.00011. 2
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The AUV UMI is a underwater mini-vehicle developed in the French-Mexican laboratory on computer science and automation membership to the UMR 3175 CNRS. The structure is made of nylon and polyethylene, the prototype has four motors that generate the movements of translation and rotation of the vehicle. Its dimensions are 63 × 63 × 26 cm with a weight of 33.99 kg, a picture of the underwater vehicle can be seen in Fig. 4.
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In Fig. 5, the main components of the embedded system are illustrated. The NUC computer acquires all signals and computes all the algorithms at rate of 12 Hz, while the perception of the robot’s environment is made by several sensors as the pressure sensor measures the depth of the vehicle, the IMU that provides guidance and angular velocity.
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The Arduino-One is implemented as board communication between the speed controllers and the embedded computer. The speed controllers MD-03 are responsible for generating the appropriate signal to manipulate the magnitude and direction of the force generated by the thrusters Seabotix BTD150. The cameras are used to acquire image information, however, their use is optional and depends on the task assigned to the vehicle.
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Fig. 6. Depth, pitch and yaw response for the first real- time experiment. A second experiment was carried out, the objective here is stabilize the underwater robot at θd = 0; ψd = 0 deg and zd = 1 m, and later to move the vehicle in the longitudinal axis, x, in manual mode to increase the hydrodynamic parameters, see Fig. 7. The Fig. 8 introduces this experimental results, the experiment was realized in a pool when people was swimming producing currents (external perturbations) and increasing also the hydrodynamic parameters. Notice in these figures how the yaw and pitch angles are perturbed but the controller maintains all the system stable. A video of this experiment can be seen at: https://youtu.be/Jf-MvKTCIJQ.
Fig. 5. Embedded system 6. EXPERIMENTAL RESULTS The first test were made to stabilize all the system, nevertheless at the moment, it is not possible to measure the x position and its longitudinal velocity in our underwater prototype. Then, only the yaw and pitch angles and the depth can be validated. The goal of this experiment is to dive the vehicle at depth of 1m with a θd = 0 ◦ and follow a sinusoidal reference (ψd = 50 ∗ sin(0.1 ∗ t) + 100)◦in the yaw angle. Fig. 6 introduces the experimental performance when applying the proposed controllers. Notice here the good behavior of the control
7. CONCLUSION A nonlinear model of an underwater prototype vehicle was obtained and analyzed to develop a control algorithm. The submarine configuration considered the lateral movement mechanically stabilized. A nonlinear control law with integral action and an adaptive parameter was proposed to stabilized the underwater vehicle. This controller was obtained for a general form and then applied to the prototype.
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Fig. 8. Yaw, pitch and z responses when the vehicle is moving in the x axis. Several simulations were carried-out to illustrate the well performance of the closed-loop system. In addition an underwater prototype was built to validate control algorithms. Experimental results verify the well behavior of the controllers when stabilizing the yaw, pitch and depth states. Future work includes to validate the proposed algorithm in presence of aggressive unknown perturbations as water currents. In addition, we are working to add a sensor in the prototype to measure the longitudinal axis and to realize tasks in complete autonomous mode.
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