Simulation of Plane Motion of Semiautonomous Underwater Vehicle

Transcription

1 Proceedings o the European Computing Conerence Simulation o Plane Motion o Semiautonomous Underwater Vehicle JERZY GARUS, JÓZEF MAŁECKI Faculty o Mechanical and Electrical Engineering Naval University 8-3 Gdynia, ul. Smidowicza 69 POLAND j.garus@amw.gdynia.pl j.malecki@amw.gdynia.pl Abstract: - he paper addresses nonlinear control o a small torpedo-shape underwater vehicle designed to detection and identiication o dangerous objects located close to a sea bottom. Correct recognition o targets and collection o appropriate data depends on its precise displacement in an undersea space. Hence, a problem o its ability to move along a reerence path and keep a desired orientation is regarded. A non-linear mathematical model describes the vehicle s dynamics. Command signals are generated by an autopilot with uzzy control law implemented. A method o power distribution in a vehicle s propulsion system is also proposed. Some results o computer simulations are provided to demonstrate eectiveness, correctness and useulness o the approach. Key-Words: - Underwater vehicle, control, uzzy logic, thrust allocation Introduction A presented semiautonomous underwater vehicle is used to detect and identiy dangerous objects located close to a sea bottom. It is a small torpedoshaped remotely operated vehicle o our degrees o reedom. Main technical parameters are given in the Appendix A. A typical mission o detection and identiication o undersea targets consists o two periods. he irst one is motion to a target area with a relative speed about 2 m/s obtained by means o changes o speed o thrusters. he tracking is accomplished by the acoustic sensor ixed to the vehicle s body and responds to the ultra short base line navigation system. A diving depth and an altitude are measured simultaneously. During the second period the searching object is ound by means o the vehicle s sonar and cameras which provide required inormation. Behaviour o the vehicle is controlled by a trained pilot located on a board o a surace ship or an oshore structure. He uses navigation data, sonar and television images and is supported by a navigation computer that integrates all available data. In order to acilitate him execution o standard manoeuvres and operations a control system has been designed and implemented. Automatic control o such objects is a diicult problem due to their nonlinear dynamics [2, 4, 5, 6]. Moreover, dynamics can change according to the alteration o coniguration to be suited to the mission. o cope with those diiculties, the control system should be lexible. Basic modules o the proposed control system are depicted in Fig.. he autopilot computes command signals τ d comparing desired vehicle s position, orientation and velocities with their current estimates. Corresponding values o propellers thrust are calculated in the thrust distribution module and transmitted as control input to the propulsion system. Fig.. Main parts o control system. he work consists o the ollowing sections. It starts with a brie description o dynamical and kinematical equations o the underwater vehicle s ISBN:

2 Proceedings o the European Computing Conerence motion. hen a uzzy control law and a power distribution algorithm are presented. Next some simulation results are provided. Conclusions are given in the last section. 2 Equations o motion he ollowing vectors describe motion o marine vessels o six degrees o reedom (DOF) [, 4]: η = v = τ = [ x, y, z, φ, θ, ψ ] [ u, v, w, p, q, r] [ X, Y, Z, K, M, N ] where: η vector o position and orientation in the inertial rame; x, y, z coordinates o position; φ, θ, ψ coordinates o orientation (Euler v () angles); vector o linear and angular velocities in the body-ixed rame; u, v, w linear velocities along longitudinal, transversal and vertical axes; p, q, r angular velocities about longitudinal, transversal and vertical axes; τ vector o orces and moments acting on the vehicle in the body-ixed rame; X, Y, Z orces along longitudinal, transversal and vertical axes; K, M, N moments about longitudinal, transversal and vertical axes. he nonlinear dynamical and kinematical equations o motion in body-ixed rame can be expressed as [4, 5]: where: Mv& + C( v) v + D( v) v + g( η) η& = J( η)v = τ M inertia matrix (including added mass); C(v) matrix o Coriolis and centripetal terms (including added mass); D(v) hydrodynamic damping and lit matrix; g (η) vector o gravitational orces and moments; J (η) velocity transormation matrix between inertial and body ixed rames. (2) 3 Fuzzy control law Adopted rom [3] a uzzy proportional derivative controller, working in coniguration presented in Fig. 2, has been used or control o the vehicle s motion. Fig. 2. Structure o uzzy controller. Membership unctions o uzzy sets o input variables: an error signal e and a derived change in error e as well as an output variable: command signal τ are shown respectively in Fig. 3, where the ollowing notation is used: N negative, Z zero, P positive, S small, M medium and B big. Presented in able rules, taken rom [9, ], have been chosen as the control rules. able. Base o rules. Error signal e NB NM Z PM PB Derived N NB NM NS Z PS change in Z NM NS Z PS PM error e B NS Z PS PM PB Command signal τ 4 Procedure o power distribution A propulsion system o the regarded underwater vehicle consists o ive thrusters. Four o the thrusters, called roll axis thrusters, installed in the stern assure surge, pitch and yaw motions. he ith one, called vertical axes thruster, located in a middle o the body is responsible or heave motion. A general structure o the propulsion system shows Fig. 4. he vertical axis thruster produces propelling orce Z which is equal to a developed thrust orce. he roll axis thrusters are identical and mounted in a vertical plane, symmetrically in relation to the roll axis o the vehicle. I all o them give the same thrusts then produced moments compensate and resultant propelling orce X causes translate motion along the longitudinal axis. I not then apart o translation also rotation about transversal and vertical axes, caused by not equal ISBN:

3 Proceedings o the European Computing Conerence zero resultant propelling moments M and N, is realized. he produced propelling orce X and moments M and N are a linear combination o thrusts developed by the roll axis thrusters. Hence, rom an operating point o view, the control system should have a procedure o power distribution among the roll axis thrusters and it is done in the trust distribution module (see Fig. ). Fig. 4. Structure o propulsion system. Fig. 3. Membership unctions or uzzy sets: error e, derived change in error e and command signal τ. Relationship between the vector o orces and moments τ acting on the vehicle s body and the thrust vector is a complicated nonlinear unction depending on density o water, a tunnel length and cross-sectional area, a thruster s diameter and revolutions and the vehicle s velocity [5]. In many practical applications it is approximated by so called simpliied model, i.e. the system being linear in its input [4], which or the regarded case takes a orm: where: [ τ, τ, τ ] τ = (3) τ = ; X M N thruster coniguration matrix:.... = ; =,, thrust vector. [ ] 2 3, 4 As it is seen the matrix has the ollowing properties: is a row-orthogonal, can be written as a product o two matrices: a diagonal matrix Q and a row-orthogonal matrix W having values ±:. = QW =.9.9 Substitution o (4) into (3) gives: (4) τ = QW (5) Multiplying both sides o (5) by expression is obtained: Q the ollowing Q τ = W (6) ISBN:

4 Proceedings o the European Computing Conerence Ater substitution: and τ X q τ M Q τ S = = q22 τ N q33, W W =, w w =. [ ] the equation (6) can be transormed into a orm: S = W (7) he matrix W is the Walsh matrix [7] and has the ollowing properties: W = W and WW = 4I. Hence, the thrust vector can be expressed as ollows: Q τ = W S = W (8) Simulation study Computer experiments have been made to conirm validity o the proposed control algorithm or the ollowing assumptions:. the nonlinear mathematical model (2) is used to simulate the vehicle s behaviour (see the Appendix B), 2. the uzzy control law is used with membership unctions presented in Fig. 3 to steer the vehicle, 3. the vehicle has to ollow the desired path starting rom the point Pb = ( xb, yb, zb, φ b, θb, ψ b ) and ending at the P = x, y, z, φ, θ, ψ with point ( ) maximum velocity u, 4. vector o position and orientation is measurable, 5. travel time is not ixed, thus the navigation between two points is not constrained by time. Some results o simulations are depicted in Figures 5, 6 and 7. he case study has showed that the proposed autopilot enhanced good pitch and yaw control along o the desired path. he main advantage o the approach is its simplicity and satisactory perormance. pitch theta [deg] M [Nm] yaw psi [deg] time [s] N [Nm] Fig. 5. ime histories o pitch θ and moment about transversal axis M or surge velocity u =.5 m/s time [s] Fig. 6. ime histories o yaw ψ and moment about vertical axis N or surge velocity u =.5 m/s. he quality o control can be improved by adequate choosing o parameters o membership unctions o input and output variables. uning o their values can be done i.e. by Genetic Algorithms [8]. hereore urther investigations are required, especially in case o using described approach to control the vehicle behaviour in more degrees o reedom. 6 Conclusions he problem o simulation o plane motion o the underwater vehicle controlled by the autopilot designed with uzzy set theory has been considered. he nonlinear mathematical model o the underwater system is applied in numerical experiments. Dynamics o the propulsion system is ISBN:

5 Proceedings o the European Computing Conerence No. No. 2 No. 3 No. 4 also regarded and the aine model o the thrusters is used to in a procedure o the power distribution. thrusts [N] time [s] Fig. 7. hrusts o roll axis thrusters. [6] J. Garus, P. Szymak, Application o Fuzzy Control to Steering o Semiautonomous Underwater Vehicle, Proc. o the 7 th WSEAS Int. Con. on Circuits, Systems, Electronics, Control and Signal Processing CSECS 8, Puerto de la Cruz (Spain) 28, pp [7] J. Garus, Optimization o hrust Allocation in Propulsion System o Underwater Vehicle, International Journal o Applied Mathematics and Computer Science, Vol.4, No.4, 24, pp [8] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer- Verlag, Berlin, 994. [9]. J. Ross, Fuzzy Logic with Engineering Applications, John Wiley and Sons, Chichester 25. [] R. R. Yager, D. P. Filev, Essential o Fuzzy Modelling and Control, John Wiley and Sons, Chichester 994. Inserted simulation results illustrate the useulness o the proposed approach to practical usage. Further works will concentrate on identiying the best uzzy structure o the autopilot and making tests o the described algorithms in real undersea conditions. Acknowledgment his work was partially sponsored by the he National Centre or Research and Development, Grant No. O N Reerences: [] R. Bhattacharyya, Dynamics o Marine Vehicles, John Wiley and Sons, Chichester 978. [2] J. Craven, R. Sutton, R. S. Burns, Control Strategies or Unmanned Underwater Vehicles. Journal o Navigation, No.5, 998, pp [3] D. Driankov, H. Hellendoorn, M. Reinrank, An Introduction to Fuzzy Control, Springer- Verlag, Berlin, 993. [4]. I. Fossen, Guidance and Control o Ocean Vehicles, John Wiley and Sons, Chichester 994. [5]. I. Fossen, Marine Control Systems, Marine Cybernetics AS, rondheim 22. Appendices A. echnical speciication o the underwater vehicle External dimensions:. length.4 m; 2. width.36 m; 3. height.36 m; Mass 45. kg; Buoyancy. N to 2. N; Operating depth 2 m; Maximum speed 3 m/s; Range 5 m; Propulsion:. roll axis thrusters our thrusters, 3 blade screw thrusters, electrically driven, each 5 W power; 2. vertical axes thruster single thruster, electrically driven 3 blade screw propeller in tunnel, 5 W power; Mission duration time 3 minutes; Energy source lithium ion accumulator battery; Control remote, computer aided, using single optical ibre o 2 m length. ISBN:

6 Proceedings o the European Computing Conerence B. A parameters o the mathematical model o the underwater vehicle he ollowing parameters o the underwater vehicle s dynamics have been used in the computer simulations: { } M = diag ; ( v) = diag{.5u.9 v.9 w.8 q. 7 r } D ; C ( v) = ; g ( η) = sin( cos( θ cos( θ θ ) )sin( )cos( φ) φ). ISBN: