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A Decomposition Algorithm for Attitude Control of the Remotely Operated Vehicle at Large Pitch and Roll Angles

The paper deals with the issue of developing an attitude control system for a remotely operated vehicle (ROV) for large inclination angles (pitch, roll). The construction of the orientation control system is considered based on the traditional
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     Abstract  —   The paper deals with the issue of developing an attitude control system for a remotely operated vehicle (ROV) for large inclination angles (pitch, roll). The construction of the orientation control system is considered based on the traditional approach using Euler angles (yaw, pitch and roll). A transfer matrix general form of the ROV attitude control system was obtained. It was shown that with the increase of inclination angles the transfer matrix of the system becomes multivariable. Algorithms of disturbances compensation and a methodology for parameters selection for decomposition algorithm were proposed. The obtained results were verified during the in situ tests of the ROV “Iznos” , developed in Bauman Moscow State Technical University (BMSTU). The proposed decomposition algorithms allow to expand the ranges of working angles and to improve the quality of the control system performance without significant re-engineering of its structure. I.   I  NTRODUCTION   Usually, remotely operated vehicles (ROVs) are operated at zero values of inclination angles or do not impose strict requirements to control system at large deviations of pitch and roll angles. Traditionally, a control system of an angular orientation of such vehicles is considered as a control system of separate channels (yaw, pitch and roll) without accounting cross-links between channels [1], [2] which is sufficient for such operating modes. However, for some problems operability of ROV is required in a whole range of orientation angles. Such tasks include: reconnaissance and identification of mine-like objects, maneuvering and reconnaissance in restricted spaces [3], reconnaissance of nuclear reactors cooling system tunnels, exploratory search of ship hulls, marine propellers, clear-water afterbodies, etc.  Despite the relevance of ROV motion modes at large  pitch and roll angles and existence of ROVs that support these modes (SEA WASP, Double Eagle, Subrov, Sabertooth [4] and Sea Owl, V8 SII, MARES [3]), there are no open access studies dedicated to traditional attitude control system operation at large inclination angles. 1   Bauman Moscow State Technical University, 5, 2-nd Baumanskaya street, Moscow, 105005, Russia (phone +79055111391) ea9055111391@gmail.com 2   V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65, Profsoyuznaya street, Moscow, 117997, Russia (phone: +7495-334-7641) vnchest@rambler.ru 3   Bauman Moscow State Technical University, 5, 2-nd Baumanskaya street, Moscow, 105005, Russia (phone +74992636772) kropotov_a@rambler.ru A traditional approach to a ROV orientation control implies usage of Euler angles and has a number of specific features and limitations :    singularities of kinematic equations at pitch angles equal to ±90°  [5]-[7];    ambiguity of orientation angles determination at  pitch angles equal to ±90°  [7];    control system synthesis is held for separate channels, considering pitch and roll angles close to zero [1], [2], [8];    an increase of mutual influence between control channels of course, pitch and roll angles with the increase of inclination angles and, as a result, control system processes degradation [3], [9];    changing parameters of separate channels for controlling yaw, pitch and roll angles with increasing of the lasts, which leads to degradation of processes in a control system and limiting permissible tilt angles at which a control system is operative. At the same time the problem of development of an attitude control system with operating modes at large pitch and roll angles is solved for missiles, flight and space vehicles [6], [9]-[15]. Such control systems use quaternions and direction cosine matrices that don’t have singularities, as opposed to traditional ROV attitude control systems. However, Euler angles are used at most existing ROVs orientation and attitude control systems. A lot of companies that develop ROVs have gained great experience in designing control and measuring systems using these  parameters. Moreover, solving attitude control problem for large pitch and roll angles by using traditional Euler angles can increase operational inclination angles and improve control systems of existing ROVs without significant re-engineering of their measurement and control systems structure. The problem of kinematic equations singularities, as described in [9], can be solved by switching between orientation systems using different sequences of angles of rotation, and, therefore, having different angles at which the mentioned kinematic equations have singularities. An ambiguity of orientation angles determination can be solved algorithmically, as it has been shown in [6]. In addition, control systems that use direction cosines and quaternions, modified Rodrigues parameters, are more sensitive to external disturbances. A Decomposition Algorithm for Attitude Control of the Remotely Operated Vehicle at Large Pitch and Roll Angles   E.A. Gavrilina 1 , V.N. Chestnov 2 , A.N. Kropotov 3 2019 18th European Control Conference (ECC)Napoli, Italy, June 25-28, 2019978-3-907144-00-8 ©2019 EUCA3334    However, the problem of eliminating mutual effects  between channels and parameter variations of separate channels with increasing of ROV inclination angles remains unresolved. The purpose of the current study is to develop a control system for a ROV orientation based on Euler angles that will provide the required quality of a ROV control over the entire range of orientation angles. The study is structured as follows. A ROV mathematical model is presented in Section II. A transfer matrix of the control system with ROV angular orientation and decomposition algorithms are obtained in sections III and IV, respectively. The results of experimental verification of the proposed algorithms on a real ROV are presented in section V. II.   M ATHEMATICAL MODEL   Mathematical model includes a kinematic model, a dynamic model and a propulsion model.  A. Kinematics of ROV motion A ROV spatial motion is considered as a superposition of translational motion of its pole and rotational motion around the pole. In this study, only ROV rotational motion is considered. Therein, the ROV orientation is specified by three successive rotations of the Oxyz coordinate system associated with a ROV relative to Ox g y g z g  ROV intermediate coordinate system at a yaw angle ψ , a pitch angle  υ  and a roll angle γ . Coordinate system Ox g y g z g  is defined as follows: vertex O is aligned with the ROV (pole) center of mass, axis Ox g is directed to the north along the tangent to the meridian, Oz g  - to the east, along the tangent to the  parallel, axis Oy g  - upwards along the vertical of the  position. A pole of Oxyz system related with the ROV, as well as Ox g y g z g , coincides with the ROV center of mass, Ox axis is directed along the longitudinal axis into the ship ’s  head, Oy axis lies in the ROV diametral plane and is directed upwards, Oz axis is directed to the starboard. Kinematic equations of a ROV motion are described by Euler equations: ̇ =  ̇ ̇̇ =       ,  (1) where  = 0  () ()    () () 0 sin() cos ()1 ()cos () ()sin () , ψ ̇,ϑ ̇,γ̇    –   angle velocities by yaw, pitch and roll respectively,   ,   ,      –   ROV rotational velocities about  , ,  axis.   B. Dynamics of ROV motion A complete non-linear model of a ROV dynamics is  presented in studies [2], [8] and has the following form: ̇  ()  ()  () =     where  = [  ,  ,  ]     –   vector of a ROV angular velocity in   coordinate system; -     –   a ROV inertia matrix in the form:  = {      ,      ,      } , where   ,  ,     –   ROV moments of inertia and   ,  ,     –  associated moments of inertia about Ox, Oy, Oz axes respectively [2];   - ()    –   matrix of forces and moments of centrifugal inertial forces and Coriolis inertia forces (for ROV and associated masses): С() =  0 (      )   (      )  (     )   0 (     )  (      )   (      )   0    - ()    –   a matrix of hydrodynamic drag forces: ()=    ,   ,       |  |,     ,   |  |,   where    ,   ,   ,   ,   ,      –    coefficients of a ROV hydrodynamic resistance. - ()    –   vector of hydrostatic forces and moments acting on ROV. A mandatory requirement for the considered ROVs is minimization of metacentric height. For this reason, to simplify the ROV mathematical model, it is assumed that the ROV center of mass coincides with the center of its volume and moments from hydrostatic forces do not affect the ROV. -     –   moment of force vector created by a propulsion system. The linearization of equation (2) is held for the worst case in terms of stability (for  ∗  =  ∗  =  ∗  = 0 ). The obtained equations are transformed according to Laplace and the following transfer functions of a ROV rotational motion are obtained:    =    ()()  =      +   where    =    +  2С   ∗ +С  ,   =  2С   ∗ +С  , ∗  - linearization parameter,  = ,,, = 4,5,6 . C. A propulsion model A propulsion model is considered for the case when a ROV is in a mooring mode. In accordance with the studies [1], [2], [8], dynamics of the propulsion in this operational mode could be described by a first-order aperiodic link:  _  =   _ ()  ()  =   _  _ + ,   where  _ , _    –   time constant and propulsion gain, that control movement on an i-th channel, respectively,      –   voltage applied to propulsion. In accordance with formulas (3), (4) for further calculations, a ROV mathematical model can be taken as a second-order aperiodic link. 3335   III.  TRADITIONAL APPROACH TO ORIENTATION CONTROL   The structural scheme of a ROV attitude control system corresponding to the traditional approach is presented in Fig.1. The control system is constructed as follows: the mismatch between current and specified value of angle   ,  ,   is transferred to the regulator unit, conditionally designated W reg. ψ , W reg.  υ  and W reg. γ , which can be written in accordance with the following expressions:  ˙ ∘  =    ∗  .   ˙ ∘  =    ∗  .   ˙ ∘  =    ∗  .   Figure 1. Traditional sheme of the ROV attitude control system   Predetermined values of angular velocities of yaw, pitch and roll  ̇ ∘ ,    ̇ ∘ , ̇ ∘  are transferred to the P -1  matrix unit, in which, according to the inverse Euler kinematic equations, they are converted into given angular velocities  ∘ , ∘ , ∘  with respect to axes associated with ROV:  ∘  ∘  ∘  =  − ψ ̇ ∘ ϑ ̇ ∘  γ̇ ∘ , (5) where  −  =  () 0 1()() sin() 0()() () 0 , ,    –   current pitch and roll values respectively. In accordance with given angular velocities  ∘ , ∘ , ∘  voltages   ,  ,   are formed, which are fed to the ROV  propulsion-steering system for formation of moments relative to the axes associated with the apparatus. The data measuring system detects angular velocities about the axes associated with the apparatus, which vary in accordance with the following equation:    =    ∘   where    = ,, ,   ,  ,     –   current angular velocities around axes associated with ROV; W   x  , W   y  , W   z     –    part of the transfer functions of the ROV control channel, enclosed between the matrices  P  -1  and  Р  . The current angular velocities enter the matrix P to obtain the angular velocities at  ̇   yaw,  γ̇   roll and ϑ ̇   pitch in accordance with Euler equations (1). Angular velocities are integrated and transferred into CS as feedbacks:    =    ˙      =    ˙     =    ˙   where     –   integration operator. By substituting the equations (1), (5), (6) in (17), we obtain the following transfer matrix of an open-loop system:        = 1 ×   [ .    2 +   2 22с . (  −  )0122 . (  −  ) . (   2 +   2 )0 . (  −   2 −   2 )122 . (  −  ) .   ]   ×          where c  –   cos, s  –   sin. It follows from (8) that, in case of differences in  parameters of transfer functions W   x  , W   y  , W   z  , with increase of  pitch and roll angles of a control system, orientation of a ROV becomes multivariable and has the following features:    Parameters of diagonal elements of a system transfer matrix vary with an increase of a roll angle.    As a pitch angle increases, mutual effects between channels increase.    A roll loop is most susceptible to the disturbing effects of the other channels. IV.   D ECOMPOSITION ALGORITHM    A. Full decomposition In order to solve the problem of interaction of channels of ROV orientation multivariable control system, it is proposed to bring the matrix (8) to a diagonal form [17]:  ur  ur  ur  =      .  0 00    reg.  00 0    reg.          It follows from (8) that this is possible, if the transfer functions of the channels enclosed between the matrices P and P -1  are the same, i.e.    =    =   . This condition can  be fulfilled when designing a symmetric ROV with a symmetric propulsion system. Under the symmetry we mean the uniformity of the apparatus parameters (the uniformity of hydrodynamic characteristics, moments of inertia, thrusts; zero metacentric height, etc.) and its propulsive system for each of the channels of the ROV angular motion around the axes associated with it. As an example of such a ROV, controlled at large angles, the apparatus V8 from the company Ocean Modules can be considered. However, such approach leads to constructive and functional limitations, which may be unacceptable for solving some problems. Another way to bring the transfer matrix (8) to a diagonal form is introduction of a decomposition algorithm. In this regard it is proposed to introduce corrective links to the control law (5). After the appropriate transformations, the 3336    algorithm that provides the conditions for autonomous control system is obtained:  ∘  = γ̇ ∘   ψ ̇ ∘ ()       ∘  = ϑ ̇ ∘  ()       ψ ̇ ∘ ()()    ∘  = ϑ ̇ ∘ ()       ψ ̇ ∘ ()()  With this control law of ROV orientation in space mutual influences between channels are absent, and stability of the separate channels at zero tilt angles ensures stability of the orientation system at any tilt angles. However, it should be noted that in order to implement this control law, the corrective elements need to be basically integro-differentiating links with polynomials of large degrees in the numerators and denominators. In practice, implementation of such a law is difficult, since it requires solving the problem of identification while taking into account changes in the parameters of the transfer functions included in (10). Due to this feature of the control law, let's consider the ways to simplify the compensating algorithm.  B. Simplified decomposition algorithm In order to simplify the form of the decomposition algorithm, let's analyze the operation of the system in the steady state and bring the transfer matrix W(0) to a diagonal form. Then the simplified decomposition law will be:  ∘  = γ̇ ∘   ψ ̇ ∘ ()       ∘  = ϑ ̇ ∘  ()       ψ ̇ ∘ ()()    ∘  = ϑ ̇ ∘ ()       ψ ̇ ∘ ()()  The simplified decomposition algorithm (11) is the law (10) in which the correcting elements are the ratios of the gains of the channels transfer functions (3). The control law (11) does not allow to compensate the dynamic error fully due to interference between the channels, but will improve the accuracy of the control system in the steady state mode. C. Practical recommendations To implement the simplified algorithm, it is necessary to solve the problem of identifying the gain factors of the ROV rotational motion. The solution of this task can be simplified if there is a damping velocity coupling in the circuit. In such a case, the gain of the transfer function concluded between the matrices P and P -1  can be approximately calculated in accordance with the following expression:    =   _   +   _   ~    ,ℎ = ,,        –   coefficient of the damping velocity coupling. At the same time, with some assumption, it can be concluded that the gain of the channel for controlling the ROV rotation is inversely proportional to the coefficient of the damping velocity coupling. In this case, the control law (11) could be written as follows:  ∘  =  ˙ ∘    ˙ ∘ ()       ∘  =  ˙ ∘  ()        ˙ ∘ ()()    ∘  =  ˙ ∘ ()        ˙ ∘ ()()  V.   E XPERIMENTAL RESULTS   The results obtained in this study were verified during full-scale tests of ROV “ Iznos ” , developed in BMSTU Scientific Research Institute of Mechanical Engineering.  A. ROV “  Iznos ”   The ROV layout is presented at Fig. 2. ROV “ Iznos ”  is intended for flaw detection of ship hulls and has a hybrid  propulsion system [18]. The propulsion system includes 8 thrusters and a chassis with wheels. Thrusters are used to control the ROV in water, and also allow to dock the ROV to the vessel hull for work performance. Control of the ROV movement along the vessel hull is performed by wheels. The ROV data-measuring complex includes: attitude heading reference system (based on 3 optic fiber gyroscopes, 3 accelerometers and 3 magnetometers) and a depth sensor. The ROV should take a position with large angles of roll and pitch when docking. ROV control system is built in accordance with the traditional approach presented at Fig. 1. The PI  –  regulator with a damping velocity feedback is selected as a control law. Figure 2.  ROV “  Iznos ”    B. The experiment description The control system operation was investigated as follows. At the initial stage, the control system of separate channels for controlling the yaw, pitch and roll was tuned. In the next stage, simultaneous operation of orientation control channels at various angles of roll and pitch was examined. The pitch value to the control system was set more than 45 ° , at the end of the transition process in the contour of the pitch, the ROV was controlled by the yaw or roll. In the final stage, compensation was introduced into the control algorithm in accordance with the equation (11). For the resulting system, a similar research was conducted on the simultaneous operation of the yaw, pitch and roll control channels. 3337    C. The traditional approach results The results of attitude control system work based on the traditional approach are shown in Fig. 3, 4. The presented results allow to make the following conclusions:    As the pitch angle grows, mutual influence between the channels increases;    Yaw control has the most effect on the roll channel with the ROV large pitch angle. For example, with the pitch of ~ 50 °, the dy namic error in the roll contour from yaw control is 50 °.   Figure 3. The results of the attitude control system work based on the traditional approach Figure 4. The results of the attitude control system work based on the traditional approach  D. Results of the decomposition algorithm Transients in the attitude control system with a decomposition algorithm are shown in Fig. 5. The type of transient processes allows to conclude that the mutual influence between the channels decreased. For example, the dynamic error in the roll contour when the channels work together is reduced tenfold and is equal to5 °.   VI.   C ONCLUSION  A general form of the transfer matrix of the ROV orientation control system for the traditional approach using Euler angles is obtained in this study. The resulting transfer matrix allows to conclude that the ROV orientation control systems are multivariable. In this case, the mutual influence  between the channels is enhanced with increasing of the ROV pitch and roll angles. Figure 5. The results of the attitude control system work based on the traditional approach with simplified decomposition algorithm The decomposition algorithm proposed in the study transforms the control system to an autonomous form. The results of the algorithm experimental development on the real ROV showed that the dynamic error due to mutual influence between the channels decreased tenfold. A CKNOWLEDGMENTS   The authors are grateful to BMSTU Scientific Research Institute of Mechanical Engineering for the possibility of checking theoretical results during the field tests of the ROV “Iznos”.  In addition authors are grateful to BMSTU for sponsorship. R  EFERENCES   1.   Kostenko V.V., Michailov D.N. Development of the Remotely Operated Underwater Vehicle “MAKS - 300”. Underwater Investigation and Robotics , 2012 №1 (13), p  p. 36-46 [in Russian]. 2.   Egorov S.A. ROV Motion Control in the Mode of joint movement with the carrier: Ph.D. thesis. Moscow, Bauman Moscow State Technical University, 2002.  –   361 p. [in Russian]. 3.   Petrich J. Robust Control for an Autonomous Underwater Vehicle that suppresses Pitch and Yaw Coupling / J. Petrich, D. J. Stilwell // Ocean Eng.  –   2011.  –   Vol. 38  –    № 1–   pp. 197  –  204. 4.   Johansson B. Seaeye Sabertooth A Hybrid AUV/ROV Offshore System. OCEANS 2010 Seattle, USA, 2011.  –   pp. 1  –  11. 5.   Stuelpnagel J. On the Parametrization of the Three-Dimensional Rotation Group Society for Industrial and Applied Mathematics. Vol. 6  –    № 4, 1964, pp. 422  –  430. 6.   Grumondz V.T., Polovinkin V.V. Theory of motion of missiles. Mathematikal models and research methods  –   Moscow: Vuzovskaya kniga Publ., 2012.  –   644 p.[in Russian] 7.   Guravlev V. F. Fundamental Principles of Classical Mechanics  –   Moscow: Fizmatlit Publ., 2001. 320 p. [in Russian] 8.   Fossen, T.I. Guidance and Control of Ocean Vehicles. John Wiley & Sons Ltd, 1994.  –   480 p. 9.   O. E. Fjellstad and T. I. Fossen, "Singularity-free tracking of unmanned underwater vehicles in 6 DOF," in Proceedings of the 33rd IEEE Conference on Decision and Control, Vol. 2, 1994, pp. 1128-1133. 10.   Zubov, N.E., Mikrin, E.A., Misrikhanov, M.S. Output control of the longitudinal motion of a flying vehicle. Journal of Computer and Systems Sciences International. 2015, vol. 54, issue 5, pp. 825-837. DOI: 10.1134/S1064230715040140 11.   Fortescue P.W., Swinerd G.G . Attitude control in spacecraft system engineering. Chichester, UK.: John Wiley & Sons, Ltd, 2011. 12.   B. Wie and P. M. Barba. "Quaternion feedback for spacecraft large angle maneuvers", Journal of Guidance, Control, and Dynamics, Vol. 8, No. 3 (1985), pp. 360-365. 3338
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