Optimal Pseudospectral Path Planning of Oil Tankers

Transcription

1 Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP Optimal Pseudospectral Path Planning o Oil Tankers M. T. Ghorbani *1, H. Salarieh Department o Mechanical Engineering, Shari University o Technology, Tehran, Iran *1 mt_ghorbani@alum.shari.edu; salarieh@shari.edu Abstract- In the current paper, the problem o path planning o the ESSO, a 19, deadweight oil tanker is considered. The path planning o the tanker is a challenging issue both in narrow channels due to collision possibility and in open waters where minimum uel usage is important. To handle this problem, the vessel and its actuator dynamics in shallow and deep waters, the economy and the waypoint guidance are translated into the optimal control ramework. In order to solve the resulted nonlinear constrained optimal control problem (OCP), the Gauss pseudospectral collocation method (GPCM) is used to transcribe the OCP into a nonlinear programming problem (NLP) by discretization o controls and states. The resulted NLP is then solved by a well-developed algorithm known as SNOPT. The procedure or modeling, compilation and solving o resulted OCP is done in Matlab optimal control sotware known as PROPT. Nonlinear optimal tracking, course-changing autopilot and way point guidance are then designed or the tanker using the proposed method. The obtained results provide uel economy and better maneuvering perormance in a low computational time. Keywords- Optimal Control; Gauss Pseudospectral Collocation Method; Super-tanker; Path Planning; PROPT I. INTRODUCTION Due to the precarious nature o the oil shipment through the narrow channels, careul attention should be paid to this issue. Banks and shallow water are two major risk actors endangering the large tankers ocean voyages [1]. There are many straits all over the world which have width less than 4 kilometres (5 miles). I they are rated based on the essential actors like economic excellence, number o ships, military useulness and geographic vulnerability, undoubtedly, the Strait o Hormuz in the south o Iran is one o the top ive straits which connect Middle East to the north o Arica. It is a narrow, important strategic strait between Persian Gul and the Gul o Oman (see Fig. 1). The strait at its narrowest part is 1 nautical miles (39 km) wide [1]. The deepest part o the strait in the shores o Musandam is a dense traic territory in which the tanker accident risk is high. Among petroleum-exporting areas in Persian Gul, the Strait o Hormuz is the only sea gate to the open water which makes it a choke point. Base on the U.S. Energy Inormation Administration report, averagely in a day, 17 million barrels o crude oil are shipped through the Strait to the Asia, Europe and North America. This amount o transportation includes % o oil traded worldwide and 35% o the world's seaborne oil shipments []. Fig. 1 The Strait o Hormuz [1] Some accidents have occurred in the Strait o Hormuz which results in ecosystem degradation o Pars Sea. As an example, according to the ith American navy announcement at the end o 9, the USS Hartord submarines and the USS New Orleans warship had an accident in Hormuz Strait which caused about 95, litters o diesel uel were released to the water o the strait [3]. These make the Strait o Hormuz one o the most hazardous, potentially dangerous and crowded channels in the

2 Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP world [1, 3]. The main reason or accidents was insuicient stopping or manoeuvring room. Accurate knowledge o manoeuvring behavior o the ships under the eects o channel bank, water depth and environmental conditions such as waves, wind and current, are required to reach navigation saety [4]. There are many variations in the steering characteristics o super-tankers, so the manual handling o super-tankers is too diicult because o their massive size. Thus, the application o autopilots, specially, in narrow coastal waters seems to be necessary to make accurate course changing manoeuvres [4]. The autopilot designing problem or ships can be divided into two sub problems: path planning and path control problem [5]. The process o generating a desired path to do a certain job while considering the dynamics o the vessel, actuator saturation, economy (minimum uel usage) and obstacles avoidance possibilities, is called path planning. The path control reers to the process o matching between the actual and the desired values o position, attitude and velocity o the vessel. There is no doubt that optimal control theory is the most natural ramework or solving path planning problems. Nonlinear optimal control approaches have signiicant beneits in trajectory planning. They are more applicable than other approaches in satisaction o vehicle dynamics and other constraints as well as providing solutions with intrinsic extremal nature [6]. Solving OCPs has been historically considered diicult due to the twin curses o dimensionality and complexity. These diiculties are exacerbated in the presence o state constraints. However, the growth o computational power and impressive advances in solving nonlinear OCPs are changing that paradigm. The computational method adopted in this paper, is a pseudospectral (PS) method. PS methods were used and developed in the 197s to solve partial dierential equations especially in meteorology and luid dynamics. Later, PS methods were considered or solving OCPs [7, 8]. The exponential convergence rate in approximating o analytic unctions as well as Eulerian-like simplicity, are major reasons that make PS methods so much popular [9]. For a certain error bound, they also can produce a smaller-scale optimization problem than other approaches. That is why PS methods are more applicable or complex nonlinear dynamics o super-tankers in comparison with other methods. The whole results provided in this paper are computed by Matlab optimal control sotware known as PROPT [1], in which PS methods are used or solving OCPs. The method is tested on the model o ESSO 19,-dwt tanker which has a highly nonlinear and coupled behavior in the propulsion and heading dynamics. The detailed model o the vessel as well as manoeuvring coeicients and rudder data are presented in [11], [1] and [13]. We show that it is possible to do path planning or super-tankers under the uniied ramework o optimal control and pseudospectral methods. This paper is structured as ollows: at irst, the Gauss pseudospectral method is introduced in its most current orm. Aterwards, the mathematical model o the ESSO super-tanker is presented. The last sections present the simulation results together with some discussions and conclusions. II. GAUSS PSEUDOSPECTRAL COLLOCATION METHOD Consider the ollowing general OCP. Determine the state, x(t), and the control, u(t), that minimize the cost unctional, with given dynamics and constraint equations, i.e. minimize J ( x( t ), t, x( t ), t ) g( x( t), u( t), t) dt the dynamic constraints: x( t) ( x( t), u( t), t), t [ t, t ] S.t. the boundary conditions: h( x( ), t, x( ), t ) the inequality path constraints : C( x( t), u ( t), t). where t is the initial time, t is the ixed or ree inal time, and problem [8]. The GPCM method requires a ixed time interval, such as [ 1, 1]. So the time variable is mapped to the general t t t interval [ 1,1] via the aine transormation:. Now the OCP is rewritten as t t t t t t t [ t, t ]. Equation (1) is known as the continuous Bolza (1) t t J ( x( ),, x( ), t ) g( x( ), u( ); t, t ) d dx t t the dynamic constraints: ( x( ), u( ), ; t, t ) d s.t. the boundary conditions: h( x( ), t, x( ), t ) the inequality path constraints : C( x( ), u( ), ; t, t ). ()

3 Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP In the GPCM, the OCP is discretized at the Legendre-Gauss (LG) discretization points, and then transcribed into an NLP by approximating the controls and states using Lagrange interpolating polynomials. The set o N discretization points includes the initial point 1, K = N interior LG collocation points, deined as the roots o the Kth-degree Legendre polynomial, and the inal point 1. The approximated state, X(τ), is estimated with a basis o K+1 Lagrange interpolating polynomials. The control is estimated with a basis o K Lagrange interpolating polynomials namely U(τ). The orthogonal collocation method also transcribes the continuous dynamics into a set o K algebraic constraints. In addition, the integral term in the cost unctional can be approximated with a Gauss quadrature. The resulted NLP are inally ound as: In Eq. (3), i are the Gauss weights. t t Minimize J ( X( ), t, X( ), t ) g( X( ), U( ), ) X( k), U( k) K i i i i i1 K t t X( ) X( ) i ( X( i ), U( i ), i; t, t ) i1 K ( k j) K K j, j i, t t s.t. X( i ) ( X( k ), U( k ), k ; t K, t ) i ( i j) j, ji h( X( ), t, X( ), t) C( X( k ), U( k ), k ; t, t ), ( k 1,..., K) The solution o Eq. (3) is an approximation to the continuous Bolza problem. In this paper, to solve this NLP, we use SNOPT optimization solver. The solver can handle problems with many thousands o variables and constraints [14]. It helps us to solve the resulted non-convex optimization problem. The GPCM method or solving OCPs has been implemented in a MATLAB based pseudospectral optimal control sotware known as PROPT together with the NLP solver SNOPT. PROPT is a combined modeling, compilation and solver engine or generation o highly complex OCPs. Further inormation about PROPT can be ound in [1]. In order to ind a solution or the OCP o Eq. (3) as eiciently as possible, 9 Legendre-Gauss collocation points are chosen. Since, it is beyond the scope o the current paper to introduce various pseudospectral methods and their accuracy, more detailed inormation can be ound in [8, 15-17]. III. DESCRIPTION OF MATHEMATICAL MODEL OF TANKER DYNAMICS A mathematical model or a 34.8 m long, ESSO 19, dwt oil tanker in surge, sway, and yaw has been introduced in T [1]. The tanker 3-DoF nonlinear dynamics has 8 states: x [ u, v, r, x, y,, n, ] T and control inputs: u [ n c, c ]. u, v and r are the surge, sway velocity and yaw rate in the body-ixed rame respectively; x, y and ψ are the vessel position and orientation in the inertial rame. The propeller shat speed n and the rudder angle δ are the other states. The commanded propeller speed n c and the commanded rudder angle δ c are two inputs to the model. The rudder angle and rate saturation and the propeller shat speed saturation are also combined in this model so that deg,.33deg/ s and n 8rpm. The surge, sway and yaw equations o motion o the vessel are given by the ollowing three equations: Surge: u vr gx ( u, u, v, r, T,, c, ) Sway: v ur gy ( u, u, v, r, T,, c, ) Yaw: ( Lk zz ) r xgur gln ( u, u, v, r, T,, c, ) where g is gravity acceleration, c is the low velocity at rudder, k zz is the non-dimensional gyration radius, x G is the distance o tanker center o gravity rom the origin o coordinate system, L is ship length between perpendiculars, T is the propeller T thrust which is the unction o u and n. Deep and shallow waters are described by water depth parameter: d where t d is Td h ship s drat and h is water depth. Note that changing the value o h aects on which have inluence on speed and steering dynamics o the tanker. In Eq. (4) X, Y and N are non-dimensional nonlinear unctions relating to the surge, sway and yaw dynamics, respectively [1]. Note that the inluence o heave, roll, and pitch motions in the horizontal plane is assumed zero. This assumption is approximately valid, at least or large ships such as supertankers where there is negligible coupling between (3) (4)

4 r(deg/s) v(m/s) u(m/s) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP these modes [1]. The state-space kinematic equations o motion o the tanker are: x u cos v sin y u sin v cos r (5) Also, the dynamics o the rudder and propeller are incorporated by: where T m is time constant. c 1 n ( nc n ) Tm The detailed model o the vessel as well as main particulars, loading condition, rudder and propeller data and maneuvering coeicients are presented in [1, 18]. A. Nonlinear Optimal Tracking IV. SIMULATIONS AND RESULTS Economy (minimum uel usage), saety (corresponding to maneuverability and accuracy), are the major actors which should be considered in the path tracking [4]. These actors must be considered as a perormance criterion unction minimized in an optimal control procedure. Since the ship control coniguration is required to minimize the heading error or a desired heading d and also minimize the propeller shat speed n, or minimum uel consumption, and minimize the rudder delection angle c, the cost unction to be minimized is chosen as: c (6) t J P Q n R dt (7) ( ( d ) c c ) The above cost unction balances accurate tracking against reduced actuator usage because they are both dependent on each other. P, Q and R are the corresponding weighting coeicients. The more the value o P, the more accurate tracking is achieved. On the other hand, i the value o Q in the cost unction increases, the shat speed and the uel consumption will be decreased. Increasing the value o R, the rudder delection will decrease. Note that the rudder angle and its rate limits are considered as inequality constraints ensuring us rom their satisaction. The simulation o optimal path tracking or the tanker model in deep waters with h = 5 m has been perormed in PROPT sotware, assuming that the tanker states are zero and the desired heading d which should be reached is 9 degrees Fig. Kinetic states o tanker dynamics or P = 1, Q, R =

5 n(rpm) (deg) y(m) Heading error(deg) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP Fig. and Fig. 3 depict the tanker kinetic and kinematic states together with actuator dynamics o the tanker in Fig. 4 or P = 1, Q = and R =. The heading angle o the vessel reaches the desired value. The tanker starts with maximum propeller shat speed to reach surge speed o 5 knots. There is no limit o uel usage in the cost unctional in this scenario so maximum shat speed is expecting. To achieve the desired heading, the rudder angle starts at its maximum minus value and ater 3 s, the direction o the rudder is changed to keep the tanker at the desired heading angle. The physical limits or actuators are satisied. Reerence [13] perormed the same simulation or the tanker based on solving a state dependent Riccati equation or SDRE approach. The advantage o the current approach over that o [13] is satisaction o actuator dynamics limits. 1 (deg) x(m) 3 1 Fig. 3 Kinematic component o o tanker dynamics or P = 1, Q, R = -1 - c n n c Fig. 4 Actuator dynamics or or P = 1, Q, R =

6 y(m) Heading error(deg) r(deg/s) v(m/s) u(m/s) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP Another simulation with dierent gains o P = 1, Q = 1 3 and R = 1 3 is perormed and its results are shown in Figs. 5, 6, and 7. As is seen in Fig. 7, increasing the value o propeller shat speed gain in the cost unctional o Eq. (7) results in the reduction o uel consumption. The heading error in Fig. 6 does not become zero, because its gain in the cost unction is less than the gain o propeller shat speed Fig. 5 Kinetic states o tanker dynamics or P = 1, Q, R = (deg) x(m) Fig. 6 Kinematic component o o tanker dynamics or P = 1, Q, R =

7 n(rpm) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP c (deg) n n c Fig. 7 Actuator dynamics or or P = 1, Q, R = 1 3 Results o two simulations indicate that the more the accurate heading control is needed, the more the control eort is required causing the tear and wear o actuators. On the other hand, the heading error will be large, i the control eort is reduced to a minimum as it would be inadequate to reach the desired heading. This simulation was running on a laptop with a Windows 7 Operating System (OS), an Intel Core i5.7 GHz processor, and 4GB o Random Access Memory (RAM). The mean computation time or 9-node solution is approximately 45 seconds meaning that the computational time is low. B. Course Changing Autopilot During course-changing maneuvers it is desirable to speciy the dynamics o desired heading instead o using a constant reerence signal as perormed in the course-keeping case o pervious section. To change the heading angle o the vessel, the helmsman or an autopilot system applies the step input to the rudder [4]. In practice, a desired step response has the orm o a critically damped second-order response [11]. The response has three phases: (1) start o the turn; () stationary turning; and (3) end o the turn [19]. The start o the turn should be gradual in order to prevent saturation in the actuators due to their dynamic limits, and clearly show the intention o the maneuver to other tankers. The stationary phase o the turn, which corresponds to the constant slope (rate o turn) o the step response, in terms o slow and quick turning, is determined rom the user s point o view depending on the traic situation, etc. The turn should end without any overshoot in the heading. The stationary rate o turn is the only setting chosen by the operator or course-changing. Instead o a critically damped second-order response, however, the desired heading will be commanded using time-varying exponential unctions as ollows [4]: 5 t r{1 exp( 1. 1 t )}, t, d 5 t t r t t t exp( 1. 1 ( ) ),, (8) where r is the reerence input or the course-changing heading command, d is the desired smooth course-changing output

8 y(m) Heading error(deg) r(deg/s) v(m/s) u(m/s) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP signal, and t is the total time required to perorm the desired maneuver and return back to zero. Eq. (8) generates a more gradual S-shaped response o the desired course heading compared to a critically damped second-order response, thus providing an ideal command or a real tanker to accomplish. Figs. 8, 9 and 1 show the course-changing perormance o the algorithm or h = 5 m and or a commanded heading o 18 when all initial states are set to zero, with the exception o 8 ms -1 surge speed, and at 8 rpm propeller speed. As is seen in Fig. 9, a gradual S-shaped o tanker course in x-y plane is achieved by the ramework o optimal control Fig. 8 Kinetic states o tanker dynamics or course changing maneuver in deep waters with h = 5 m (deg) x(m) Fig. 9 Kinematic component o tanker dynamics or course changing maneuverer in deep waters with h = 5 m - 1 -

9 r(deg/s) v(m/s) u(m/s) n(rpm) (deg) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP c n Fig. 1 Actuator dynamics or course changing maneuver in deep waters with h = 5 m To examine robustness o the proposed method to model uncertainties, a simulation or course-changing maneuver in shallow waters with h = 5 m is done and presented in Figs. 11, 1, 13. As it is seen in Fig. 11, the water depth parameter, h, mostly aects the sway velocity. However, it does not signiicantly change the desired heading or other states Fig. 11 Kinetic states o tanker dynamics or course changing maneuver in shallow waters with h = 5 m

10 n(rpm) y(m) Heading error(deg) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP (deg) x(m) Fig. 1 Kinematic component o tanker dynamics or course changing maneuver in shallow waters with h = 5 m 1 5 (deg) -5-1 c n 4 C. Way Point Guidance Fig. 13 Actuator dynamics or course changing maneuver in shallow waters with h = 5 m Let the vehicle mission be given by a set o way points. Hence we can deine the Line o Sight (LOS) in terms o a desired heading angle. The desired route is most easily speciied using waypoints ( P1, P,, P N ) with coordinates Pi ( X i, Y i ), i 1,, N. The desired heading can then be obtained rom the relationship Yi y d arctan X i x where the current ship position (x, y) is calculated rom kinematics equations. In a real system, GPS (Global Positioning System) can detect the position o the vessel. (9)

11 n(rpm) (deg) r(deg/s) v(m/s) u(m/s) Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP The autopilot can hence be used in this way to move the ship toward waypoints. The switch between waypoints happens when the tanker lies within a circle o acceptance around the current waypoint. In the current work, the acceptance radius is m. Assuming that the propeller shat speed is 8 rpm, the cost unctional to be minimized is chosen as: t J P R dt (1) ( ( d) c ) The optimal way point guidance in Figs. 14, 15 and 16, show the capability o the path planner or automatic maneuvering o oil tankers in shallow waters (with 5 meters depth), when initial condition o the tanker states is x =[ 8 ms -1 8 rpm] T. The rudder angle in Fig. 15 seems to be oscillatory. This is because the optimal way point guidance imposes more constraints on the problem and the problem becomes more complicated or the numerical solver. In practice, the results o algorithms or control eorts should be passed via a low-pass ilter to eliminate the noise o control inputs Fig. 14 Kinetic states o tanker dynamics or way point guidance 1 5 c n Fig. 15 Actuator dynamics or way point guidance

12 Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP Fig. 16 Vessel course in way point guidance mode V. CONCLUSION In this study, the dynamics o ESSO 19,-dwt tanker is modelled in PROPT sotware which uses a pseudospectral optimal control approach to solve the problem o path planning o the tanker. Nonlinear optimal tracking, course-changing as well as optimal waypoint guidance have been designed, resulting in better manoeuvring perormance and uel economy. Simulation results indicate the practical useulness o the presented path planner. The success o the proposed method should aect the design o ship autopilots as well as research in nonlinear control. Future work on this problem can study the possibility o real time implementation o the proposed method to automatically steer the supertanker in a channel (such as The Strait o Hormuz). REFERENCES [1] Persian, Gul, Studies, and Center. (7/16). Strait o Hormuz. Available: [] U.S., Energy, Inormation, and Administration. (14 January 1). World oil transit chokepoints. Available: [3] Wikipedia. (1). Strait o Hormuz. Available: en.wikipedia.org/wiki/strait_o_hormuz [4] T. Çimen, "On-line nonlinear optimal maneuvering control o large tankers in restricted waterways," presented at the 7th IFAC Conerence on Manoeuvring and Control o Marine Crat (MCMC 6), Lisbon, Portugal, 6. [5] K. Djouani and Y. Hamam, "Minimum time-energy trajectory planning or automatic ship berthing," IEEE Journal o Oceanic Engineering, vol., pp. 4-1, [6] Q. Gong, L. R. Lewis, and I. M. Ross, "Pseudospectral motion planning or autonomous vehicles," Journal o Guidance, Control, and Dynamics, vol. 3, pp , 9. [7] F. Fahroo and I. M. Ross, "Costate estimation by a Legendre pseudospectral method," Journal o Guidance, Control, and Dynamics, vol. 4, pp. 7-77, 1. [8] G. Huntington, "Advancement and analysis o a Gauss pseudospectral transcription or optimal control problems," Ph.D. thesis, Massachusetts Institute o Technology, United States -- Massachusetts, 7. [9] L. N. Treethen, Spectral Methods in Matlab: Society or Industrial and Applied Mathematics,. [1] P. E. Rutquist and M. M. Edvall, "PROPT - Matlab Optimal Control Sotware," Tomlab Optimization, 1. [11] T. I. Fossen, Guidance and control o ocean vehicles: Wiley, [1] T. Çimen, "Development and validation o a mathematical model or control o constrained non-linear oil tanker motion," Mathematical and Computer Modelling o Dynamical Systems, vol. 15, pp , 9. [13] T. Cimen and S. P. Banks, "Nonlinear optimal tracking control with application to super-tankers or autopilot design," Automatica, vol. 4, pp , 4. [14] P. E. Gill, W. Murray, and M. A. Saunders, "SNOPT: An SQP algorithm or large-scale constrained optimization," SIAM Review, vol. 47, pp , 5. [15] D. Benson, "A Gauss pseudospectral transcription or optimal control," Ph.D. thesis, Massachusetts Institute o Technology, United States -- Massachusetts, 5. [16] D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson, and G. T. Huntington, "A uniied ramework or the numerical solution o optimal control problems using pseudospectral methods," Automatica, vol. 46, pp ,

13 Advanced Shipping and Ocean Engineering Dec. 13, Vol. Iss. 4, PP [17] H. Salarieh and M. T. Ghorbani, "Trajectory Optimization or a High Speed Planing Boat Based on Gauss Pseudospectral Method," in The nd International Conerence on Control, Instrumentation and Automation (ICCIA 11), Shiraz University, 11. [18] T. I. Fossen, Handbook o Marine Crat Hydrodynamics and Motion Control: John Wiley & Sons, 11. [19] J. Van Amerongen, "Adaptive steering o ships A model reerence approach," Automatica, vol., pp. 3-14,