Nonlinear passive weather optimal positioning ... - Semantic Scholar
Automatica 37 (2001) 701}715
Nonlinear passive weather optimal positioning control (WOPC) system for ships and rigs: experimental results夽 Thor I. Fossen *, Jann Peter Strand Department of Engineering Cybernetics, The Norwegian Institute of Technology, Norwegian University of Science and Technology, N-7491 Trondheim, Norway ABB Industri AS, Marine Division, Hasleveien 50, N-0501 Oslo, Norway Received 2 February 1999; revised 6 April 2000; received in "nal form 2 November 2000
A new concept for nonlinear positioning control of marine vessels is presented and documented experimentally. The vessel is controlled such that the resulting wind, wave and current force does not produce a yaw moment without measuring the environmental disturbances.
Abstract Nonlinear weather optimal control of ships exposed to environmental disturbances is discussed. Emphasis is placed on weather vaning, that is the ship should be orientated such that the moment in yaw due to mean current, wind and wave forces is zero and at the same time maintain a "xed position and heading. The main problem in doing this is that the environmental forces acting on the ship are impossible to measure with su$cient accuracy. A new concept for weather optimal position control (WOPC) based on nonlinear control theory is invented and presented for the "rst time in this paper. Experiments with a model ship are used to document the performance of the controller. The proposed controller and the concept of WOPC were patented by the authors in cooperation with ABB Industri AS, Oslo in December 1998 under patent no. NO308.334. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear control; Adaptive control; Passive; Marine systems; Ship control; Disturbance rejection
1. Introduction Conventional DP systems for ships and free-#oating rigs are usually designed for station-keeping by specifying a desired constant position (x , y ) and a desired constant heading angle , see Balchen, Jenssen, and Sv lid (1976, 1980), Grimble, Patton, and Wise (1980), Fossen (1994) for instance. In order to minimize the ship fuel consumption, the desired heading should in many operations be chosen such that the yaw moment is zero. For vessels with port/starboard symmetry this means
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Saberi under the direction of Editor Hassan K. Khalil. This work was partially supported by ABB Industri AS, Marine Division and the Norwegian Research Council. * Corresponding author. Tel.: #47-73-594-361; fax: #47-73-594399. E-mail addresses: [email protected] (T. I. Fossen), jann-peter. [email protected] (J. P. Strand).
that the mean environmental force due to wind, waves and currents attack through the center line of the vessel. Then the ship must be rotated until the yaw moment is zero. Unfortunately, it is impossible to measure or compute the direction of the mean environmental force due to wind, waves and currents with su$cient accuracy. Hence, the desired heading is usually taken to be the measurement of the mean wind direction which can be easily measured. In rough weather, however, this can result in large o!-sets from the true mean direction of the total environmental force. The main reason for this is the unmeasured current force component and the waves do not coincide with the wind direction. Hence, the DP system can be operated under highly non-optimal conditions if fuel saving is the issue. A small o!-set in the optimal heading angle will result in a large use of thrust. One attractive method for computing the weather optimal heading is to monitor the resulting thruster forces in the x- and y-directions. Hence, the bow of the ship can be turned in one direction until the thruster force in the y-direction approaches zero. This method is
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 0 6 - 1
702
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
appealing but the main catch in doing this is that the resulting thruster forces in the x- and y-directions have to be computed since there are no sensors doing this job directly. The sensors only measure the angular speed and pitch angle of the propellers. Hence, the thrust for each propeller must be computed by using a model of the thruster characteristic resulting in a pretty rough estimate of the total thruster force in each direction. Another principle (Pinkster & Nienhuis, 1996) is to control the x- and y-positions using a PID feedback controller while only derivative action is used for the yaw angle. This principle requires that the rotation point of the vessel is located in a certain distance fore of the centre of gravity, or even fore of the bow, and it also puts restrictions on the thruster con"guration and the number of thrusters installed. This method has shown to be applicable for large tankers since the moment arm between the center of gravity and the rotation point, e.g. located in the bow is large enough for these type of vessels. The main contribution of this paper is a new concept for weather optimal positioning control (WOPC) of ships and free-#oating rigs. Based on the results in this paper the proposed method was patented by the authors in cooperation with ABB Industri AS, Oslo in 1998 under patent no. NO308.334. The control objective is that the vessel heading should adjust automatically to the mean environmental disturbances (wind, waves and currents) such that a minimum amount of energy is used in order to save fuel and reduce NO /CO without using any V V environmental sensors. This is particularly useful for o!shore vessel, e.g. #oating production, storage and o!loading (FPSO) systems like drilling rigs and ships which can be located at the same position for years. Also dynamically positioned supply vessels which must keep their position for days while waiting for loading/o!loading have a great WOPC fuel saving potential. Nonlinear and adaptive backstepping designs are used to derive the WOPC system. Lyapunov theory is used to guarantee exponential stability of the closed-loop system. Adaptive backstepping showed to be a useful tool when designing the integral controller. The concept of WOPC can also be implemented by using other control methods, e.g. feedback linearization, model-based predictive control (MPC), nonlinear PID-control, etc. The key feature of the WOPC is that no information about the environmental disturbances is required. This is important since the mean environmental disturbances acting on a #oating vessel cannot be accurately measured or computed. It is shown that the ship can be exponentially stabilized on a circle arc with constant radius by letting the bow of the ship point towards the origin of the circle. In order to maintain a "xed position at the same time, a circle center control law is designed. Moreover, the circle center is translated on-line such that the Cartesian position is constant while the bow of the ship is automatically turned up against the mean environmental
Fig. 1. Weather optimal positioning principle: equivalent to a pendulum in the gravity "eld, where gravity is the unmeasured quantity.
force (weather vaning). This approach is motivated by a pendulum in the gravity "eld where gravity is the unmeasured quantity (Fig. 1). The circular motion of the controlled ship where the mean environmental force can be interpreted as an unknown force "eld can be programmed to behave like a pendulum in the gravity "eld. Experimental results with a model ship where a ducted fan was used for wind generation are also reported. 2. Mathematical modeling 2.1. Kinematics Let the Earth-"xed position (x, y)31 (Euclidean space of dimension two) and heading 3S (torus of dimension one, i.e. the circle) of the vessel relative to an Earth-"xed frame X > Z be expressed in vector form # # # by "[x, y, ]231;S and let the vessel-"xed vessel velocities be represented by the state vector "[u, v, r]231. These three modes are referred to as the surge, sway and yaw modes of a ship. The origin of the vessel-"xed frame X>Z is located at the vessel centre line in a distance x from the center of gravity. In general, the % transformation between the vessel- and Earth-"xed coordinate frames can be described by the three Euler angles;
(roll), (pitch) and (yaw). The horizontal motion of the ship is described by one Euler angle only, that is the yaw angle . 2.1.1. Cartesian coordinates Let R() : 1PSO(3) denote the rotation matrix:
cos !sin 0
R()" sin 0
cos 0
0 , 1
(1)
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
where R() is nonsingular for all 31 and that R\()"R2(). The transformation between the vesseland the Earth-"xed velocity vectors is (Fossen, 1994): "R().
(2)
2.1.2. Polar coordinates The Cartesian coordinates (x, y) is related to the polar coordinates by x"x # cos , y"y # sin , (3) where (x , y )31 is the origin of a circle with radius 31 and polar angle 3S: > "((x!x )#(y!y ), "atan2((y!y ), (x!x )). Time di!erentiation of (3), yields
(4)
x "x # cos ! sin , y "y # sin # cos . De"ne the state vectors
(6)
(5)
(7)
where 31 is a control vector of forces and moment. Unmodeled external forces and moment due to wind, currents and waves are lumped together into a vessel"xed disturbance vector w31, see Section 2.3 for a more detailed description. The matrices M, C() and D() represent inertia including hydrodynamic added inertia, Coriolis-centripetal and damping forces (see Fossen & Fjellstad, 1995). When desinging the WOPC system, it is advantageous to represent the ship dynamics in polar coordinates. Therefore, the following state transformation is introduced. 2.2.1. Ship model transformation The ship model (13) can be represented by polar coordinates by using (11) and substituting "¹\(x)x #R2¸p ,
(14)
"¹\(x)xK #¹Q \(x)x #R2¸pK #RQ 2¸p
(15)
such that: M#C()#D()" #w, U '0,
p O[x , y ]2, xO[, , ]2. (8) From (6) and (7) a new kinematic relationship can be derived as
M (x)xK #C (, x)x #D (, x)x V V V
"R()H()x #¸p , where
where
(9)
1
0
0
1 0
H()" 0 0 , ¸" 0 1 . 0
0
1
(10)
0 0
703
"¹\2 #¹\2q(, x, p ,pK )#¹\2w,
(16)
M (x)"¹\2(x)M¹\(x), V
(17)
C (, x)"¹\2(x)(C()!M¹\(x)¹Q (x))¹\(x), V
(18)
D (, x)"¹\2(x)D()¹\(x), V
(19)
Substituting the Cartesian kinematics (2) into (9) yields the following di!erential equation for the polar coordinates:
q(, x, p , pK )"MR2()¸pK #MRQ 2()¸p
x "¹(x)!¹(x)R2()¸p , where
(11)
Here M (x)"M2(x)'0, C (, x) and D (, x)'0 can be V V V V shown to satisfy
¹(x)OH\()R2()R()"H\()R2(!).
(12)
Note that the conversion between Cartesian and polar coordinates is only a local diweomorphism, since the radius must be kept larger than a minimum value, i.e. ' '0 in order to avoid the singular point "0.
z2(M Q !2C )z"0 ∀z, x. V V
(20)
(21)
The expression for ¹Q can be written as ¹Q (x)"HQ \()R2(!)!H\()RQ 2(!) "! ()!(!Q )H\()SR(!),
(22)
where
2.2. Ship dynamics The low-frequency motion of a large class of surface ships can be described by the following model (Fossen, 1994): M#C()#D()" #w,
#[C()#D()]R2()¸p .
(13)
()"diag 0, 1/, 1 , RQ ()"R()S,
0
1 0
S" !1 0 0 "!S2. 0
(23)
0 0
(24)
704
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
From Fig. 2, the body-"xed environmental load vector w31 can be expressed as
w () F cos( !) w" w () " . F sin( !) w () l F sin( !)!l F cos( !) V W (25) Notice that the environmental loads vary with the heading angle of the ship. Moreover F "(w #w , (26)
"#tan\(w /w ). (27) The environmental forces X and > with attack point U U (l , l ) are shown in Fig. 2. It should be noted that the V W attach point l "l () and l "l () will also change V V W W with the yaw angle . This relationship will be a complicated function of the hull and superstructure geometries. Fig. 2. Environmental force F decomposed into the components w and w .
2.3. Disturbance modelling The steady-state low-frequency motion of the ship and also the ship's equilibrium position depend on the unknown environmental loads acting on the vessel. Let the environmental loads due to wind, waves and currents be represented by E a slowly varying mean force F which attacks the ship in a point (l , l ) in body-"xed coordinates. V W E a slowly varying mean direction , relative to the Earth-"xed frame (see Fig. 2). The slowly varying terms include second-order waveinduced disturbances (wave drift), currents and mean wind forces. The "rst-order wave-induced forces (oscillatory wave-induced motion) is assumed to be "ltered out of the measurements by using a wave xlter (see Fossen & Strand, 1999; Strand & Fossen, 1999). Since there are no sensors which can be used to measure (F , ) and (l , l ) with su$cient accuracy, it is V W impossible to use feedforward from the environmental disturbances. This motivates the following assumption. A1: The unknown mean environmental force F and its direction are assumed to be constant or at least slowly varying. A2: The unknown attack point (l , l ) is constant for each V W constant F . Discussion. These are good assumptions since the ship control system is only supposed to counteract the slowly varying motion components of the environmental disturbances.
3. Weather optimal control objectives The weather optimal control objectives make use of the following de"nitions: De5nition 1 (=eather optimal heading). The weather optimal heading angle is given by the equilibrium state where the yaw moment, w ( )"0, at the same time as the bow of the ship is turned up against the weather (mean environmental disturbances), that is w ( )"0. This implies that the moment arms l ( )" constant V and l ( )"0, and: W w ( ) !F w( )" w ( ) " 0 . w ( ) 0 Hence, the mean environmental force attacks the ship in the bow (minimum drag coe$cient for the resulting current and wind loads).
De5nition 2 (=eather optimal positioning). Weather optimal positioning (station-keeping) is de"ned as the equilibrium state where w ( )"!F , w ( )"w ( )"l ( )"0, W (28) (weather optimal heading) and the position (x, y)" (x , y ) is kept constant. These de"nitions motivates the following two control objectives: O1. Weather optimal heading control (WOHC). This is obtained by restricting the ship to move on a circle with constant radius " and at the same time
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
705
Fig. 3. Principle for weather optimal heading control (WOHC).
force the ship's bow to point towards the center of the circle until the weather optimal heading angle, " , is reached (see Fig. 3, analogy to a pendulum in gravity "eld). The position, (x, y)"(x # cos , y # sin ), will vary until the weather optimal heading angle is reached. This is obtained by specifying the control objective in polar coordinates according to "constant, "0, "#.
(29)
Discussion. The requirement "constant im plies that the ship moves on a circle with constant radius. The second requirement, "0, implies that the tangential speed, , is kept small while the last requirement, "#, ensures that the ship's bow points towards the center of the circle. O2. Weather optimal positioning control (WOPC). In order to maintain a "xed Earth-"xed position (x, y)"(x , y ), the circle center p "[x , y ]2 must be moved simultaneously as control objective O1 is satis"ed. This is referred to as circle center control. A nonlinear and adaptive backstepping controller which satis"es O1 and O2 is derived in the next section. 4. Nonlinear and adaptive control design In this section, the WOPC positioning controller is derived by using the polar coordinate representation.
The backstepping design methodology (Krstic, Kanellakopoulos, & Kokotovic, 1995) with extension to integral control (Loria, Fossen, & Teel, 1999) is used to derive the feedback controller. It is assumed that all states can be measured by using conventional sensor technology and a satellite navigation system, e.g. DGPS. An attractive navigation system is strapdown DGPS/INS (Vik, 2000). In the case, where only DGPS position measurements and a gyro compass are available the body-"xed velocities can be estimated by using an extended Kalman "lter (Fossen, 1994) or the nonlinear and passive observer of Fossen and Strand (1999). The combination of the nonlinear observer and the full state feedback WOPC controller can be further analyzed through a nonlinear separation principle by using the approach of Loria, Fossen, and Panteley (2000). The WOPC controller will be derived in three successive steps: (1) Nonlinear backstepping (PD-control): the ship is forced to move on a circle arc with desired radius , with minimum tangential velocity and desired heading . (2) Adaptive backstepping (PID-control): this is necessary to compensate for the unknown environmental force F . (3) Translational control of the circle center: the circle center (x , y ) is translated such that the ship main tains a constant position (x , y ) even though it is moving on a virtual circle arc. Hence, the Captain of the ship will only notice that the ship is rotating
706
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
a yaw angle about a constant position (x , y ) until the weather optimal heading is reached.
By using the property (21) and choosing the nonlinear PD-control law as ¹\2 "M xK #C x #D x !K z V V V
4.1. Nonlinear backstepping (PD-control)
!K z !¹\2q( ) ),
A general positioning controller is derived by using vectorial backstepping (Fossen & Gr~vlen, 1998). The tracking objective is speci"ed in polar coordinates using a smooth reference trajectory x "[ , , ]23C where x ,x ,xK 3L . Since the transformed system (16) is of order 2, backstepping is performed in two vectorial steps resulting in a nonlinear PD-control law. First, we de"ne a virtual reference trajectory as x Ox !z , (30) where z "x!x is the Earth-"xed tracking error and '0 is a diagonal design matrix. Furthermore, let z denote a measure of tracking de"ned according to z Ox !x "z #z . (31) From (31), the following expressions are obtained x "z #x , xK "z #xK . (32) This implies that the vessel model (16) can be expressed in terms of z , x and xK as M z #C z #D z "¹\2 #¹\2q( ) ) V V V !M xK !C x !D x #¹\2w. (33) V V V Step 1: Let z be the "rst error variable, which from (31) has the dynamics: z "!z #z . (34) A Lyapunov function candidate (LFC) for the "rst step is < "z2 K z , (35)
Nonlinear passive weather optimal positioning control (WOPC) system for ships and rigs: experimental results夽 Thor I. Fossen *, Jann Peter Strand Department of Engineering Cybernetics, The Norwegian Institute of Technology, Norwegian University of Science and Technology, N-7491 Trondheim, Norway ABB Industri AS, Marine Division, Hasleveien 50, N-0501 Oslo, Norway Received 2 February 1999; revised 6 April 2000; received in "nal form 2 November 2000
A new concept for nonlinear positioning control of marine vessels is presented and documented experimentally. The vessel is controlled such that the resulting wind, wave and current force does not produce a yaw moment without measuring the environmental disturbances.
Abstract Nonlinear weather optimal control of ships exposed to environmental disturbances is discussed. Emphasis is placed on weather vaning, that is the ship should be orientated such that the moment in yaw due to mean current, wind and wave forces is zero and at the same time maintain a "xed position and heading. The main problem in doing this is that the environmental forces acting on the ship are impossible to measure with su$cient accuracy. A new concept for weather optimal position control (WOPC) based on nonlinear control theory is invented and presented for the "rst time in this paper. Experiments with a model ship are used to document the performance of the controller. The proposed controller and the concept of WOPC were patented by the authors in cooperation with ABB Industri AS, Oslo in December 1998 under patent no. NO308.334. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear control; Adaptive control; Passive; Marine systems; Ship control; Disturbance rejection
1. Introduction Conventional DP systems for ships and free-#oating rigs are usually designed for station-keeping by specifying a desired constant position (x , y ) and a desired constant heading angle , see Balchen, Jenssen, and Sv lid (1976, 1980), Grimble, Patton, and Wise (1980), Fossen (1994) for instance. In order to minimize the ship fuel consumption, the desired heading should in many operations be chosen such that the yaw moment is zero. For vessels with port/starboard symmetry this means
夽
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Saberi under the direction of Editor Hassan K. Khalil. This work was partially supported by ABB Industri AS, Marine Division and the Norwegian Research Council. * Corresponding author. Tel.: #47-73-594-361; fax: #47-73-594399. E-mail addresses: [email protected] (T. I. Fossen), jann-peter. [email protected] (J. P. Strand).
that the mean environmental force due to wind, waves and currents attack through the center line of the vessel. Then the ship must be rotated until the yaw moment is zero. Unfortunately, it is impossible to measure or compute the direction of the mean environmental force due to wind, waves and currents with su$cient accuracy. Hence, the desired heading is usually taken to be the measurement of the mean wind direction which can be easily measured. In rough weather, however, this can result in large o!-sets from the true mean direction of the total environmental force. The main reason for this is the unmeasured current force component and the waves do not coincide with the wind direction. Hence, the DP system can be operated under highly non-optimal conditions if fuel saving is the issue. A small o!-set in the optimal heading angle will result in a large use of thrust. One attractive method for computing the weather optimal heading is to monitor the resulting thruster forces in the x- and y-directions. Hence, the bow of the ship can be turned in one direction until the thruster force in the y-direction approaches zero. This method is
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 0 6 - 1
702
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
appealing but the main catch in doing this is that the resulting thruster forces in the x- and y-directions have to be computed since there are no sensors doing this job directly. The sensors only measure the angular speed and pitch angle of the propellers. Hence, the thrust for each propeller must be computed by using a model of the thruster characteristic resulting in a pretty rough estimate of the total thruster force in each direction. Another principle (Pinkster & Nienhuis, 1996) is to control the x- and y-positions using a PID feedback controller while only derivative action is used for the yaw angle. This principle requires that the rotation point of the vessel is located in a certain distance fore of the centre of gravity, or even fore of the bow, and it also puts restrictions on the thruster con"guration and the number of thrusters installed. This method has shown to be applicable for large tankers since the moment arm between the center of gravity and the rotation point, e.g. located in the bow is large enough for these type of vessels. The main contribution of this paper is a new concept for weather optimal positioning control (WOPC) of ships and free-#oating rigs. Based on the results in this paper the proposed method was patented by the authors in cooperation with ABB Industri AS, Oslo in 1998 under patent no. NO308.334. The control objective is that the vessel heading should adjust automatically to the mean environmental disturbances (wind, waves and currents) such that a minimum amount of energy is used in order to save fuel and reduce NO /CO without using any V V environmental sensors. This is particularly useful for o!shore vessel, e.g. #oating production, storage and o!loading (FPSO) systems like drilling rigs and ships which can be located at the same position for years. Also dynamically positioned supply vessels which must keep their position for days while waiting for loading/o!loading have a great WOPC fuel saving potential. Nonlinear and adaptive backstepping designs are used to derive the WOPC system. Lyapunov theory is used to guarantee exponential stability of the closed-loop system. Adaptive backstepping showed to be a useful tool when designing the integral controller. The concept of WOPC can also be implemented by using other control methods, e.g. feedback linearization, model-based predictive control (MPC), nonlinear PID-control, etc. The key feature of the WOPC is that no information about the environmental disturbances is required. This is important since the mean environmental disturbances acting on a #oating vessel cannot be accurately measured or computed. It is shown that the ship can be exponentially stabilized on a circle arc with constant radius by letting the bow of the ship point towards the origin of the circle. In order to maintain a "xed position at the same time, a circle center control law is designed. Moreover, the circle center is translated on-line such that the Cartesian position is constant while the bow of the ship is automatically turned up against the mean environmental
Fig. 1. Weather optimal positioning principle: equivalent to a pendulum in the gravity "eld, where gravity is the unmeasured quantity.
force (weather vaning). This approach is motivated by a pendulum in the gravity "eld where gravity is the unmeasured quantity (Fig. 1). The circular motion of the controlled ship where the mean environmental force can be interpreted as an unknown force "eld can be programmed to behave like a pendulum in the gravity "eld. Experimental results with a model ship where a ducted fan was used for wind generation are also reported. 2. Mathematical modeling 2.1. Kinematics Let the Earth-"xed position (x, y)31 (Euclidean space of dimension two) and heading 3S (torus of dimension one, i.e. the circle) of the vessel relative to an Earth-"xed frame X > Z be expressed in vector form # # # by "[x, y, ]231;S and let the vessel-"xed vessel velocities be represented by the state vector "[u, v, r]231. These three modes are referred to as the surge, sway and yaw modes of a ship. The origin of the vessel-"xed frame X>Z is located at the vessel centre line in a distance x from the center of gravity. In general, the % transformation between the vessel- and Earth-"xed coordinate frames can be described by the three Euler angles;
(roll), (pitch) and (yaw). The horizontal motion of the ship is described by one Euler angle only, that is the yaw angle . 2.1.1. Cartesian coordinates Let R() : 1PSO(3) denote the rotation matrix:
cos !sin 0
R()" sin 0
cos 0
0 , 1
(1)
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
where R() is nonsingular for all 31 and that R\()"R2(). The transformation between the vesseland the Earth-"xed velocity vectors is (Fossen, 1994): "R().
(2)
2.1.2. Polar coordinates The Cartesian coordinates (x, y) is related to the polar coordinates by x"x # cos , y"y # sin , (3) where (x , y )31 is the origin of a circle with radius 31 and polar angle 3S: > "((x!x )#(y!y ), "atan2((y!y ), (x!x )). Time di!erentiation of (3), yields
(4)
x "x # cos ! sin , y "y # sin # cos . De"ne the state vectors
(6)
(5)
(7)
where 31 is a control vector of forces and moment. Unmodeled external forces and moment due to wind, currents and waves are lumped together into a vessel"xed disturbance vector w31, see Section 2.3 for a more detailed description. The matrices M, C() and D() represent inertia including hydrodynamic added inertia, Coriolis-centripetal and damping forces (see Fossen & Fjellstad, 1995). When desinging the WOPC system, it is advantageous to represent the ship dynamics in polar coordinates. Therefore, the following state transformation is introduced. 2.2.1. Ship model transformation The ship model (13) can be represented by polar coordinates by using (11) and substituting "¹\(x)x #R2¸p ,
(14)
"¹\(x)xK #¹Q \(x)x #R2¸pK #RQ 2¸p
(15)
such that: M#C()#D()" #w, U '0,
p O[x , y ]2, xO[, , ]2. (8) From (6) and (7) a new kinematic relationship can be derived as
M (x)xK #C (, x)x #D (, x)x V V V
"R()H()x #¸p , where
where
(9)
1
0
0
1 0
H()" 0 0 , ¸" 0 1 . 0
0
1
(10)
0 0
703
"¹\2 #¹\2q(, x, p ,pK )#¹\2w,
(16)
M (x)"¹\2(x)M¹\(x), V
(17)
C (, x)"¹\2(x)(C()!M¹\(x)¹Q (x))¹\(x), V
(18)
D (, x)"¹\2(x)D()¹\(x), V
(19)
Substituting the Cartesian kinematics (2) into (9) yields the following di!erential equation for the polar coordinates:
q(, x, p , pK )"MR2()¸pK #MRQ 2()¸p
x "¹(x)!¹(x)R2()¸p , where
(11)
Here M (x)"M2(x)'0, C (, x) and D (, x)'0 can be V V V V shown to satisfy
¹(x)OH\()R2()R()"H\()R2(!).
(12)
Note that the conversion between Cartesian and polar coordinates is only a local diweomorphism, since the radius must be kept larger than a minimum value, i.e. ' '0 in order to avoid the singular point "0.
z2(M Q !2C )z"0 ∀z, x. V V
(20)
(21)
The expression for ¹Q can be written as ¹Q (x)"HQ \()R2(!)!H\()RQ 2(!) "! ()!(!Q )H\()SR(!),
(22)
where
2.2. Ship dynamics The low-frequency motion of a large class of surface ships can be described by the following model (Fossen, 1994): M#C()#D()" #w,
#[C()#D()]R2()¸p .
(13)
()"diag 0, 1/, 1 , RQ ()"R()S,
0
1 0
S" !1 0 0 "!S2. 0
(23)
0 0
(24)
704
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
From Fig. 2, the body-"xed environmental load vector w31 can be expressed as
w () F cos( !) w" w () " . F sin( !) w () l F sin( !)!l F cos( !) V W (25) Notice that the environmental loads vary with the heading angle of the ship. Moreover F "(w #w , (26)
"#tan\(w /w ). (27) The environmental forces X and > with attack point U U (l , l ) are shown in Fig. 2. It should be noted that the V W attach point l "l () and l "l () will also change V V W W with the yaw angle . This relationship will be a complicated function of the hull and superstructure geometries. Fig. 2. Environmental force F decomposed into the components w and w .
2.3. Disturbance modelling The steady-state low-frequency motion of the ship and also the ship's equilibrium position depend on the unknown environmental loads acting on the vessel. Let the environmental loads due to wind, waves and currents be represented by E a slowly varying mean force F which attacks the ship in a point (l , l ) in body-"xed coordinates. V W E a slowly varying mean direction , relative to the Earth-"xed frame (see Fig. 2). The slowly varying terms include second-order waveinduced disturbances (wave drift), currents and mean wind forces. The "rst-order wave-induced forces (oscillatory wave-induced motion) is assumed to be "ltered out of the measurements by using a wave xlter (see Fossen & Strand, 1999; Strand & Fossen, 1999). Since there are no sensors which can be used to measure (F , ) and (l , l ) with su$cient accuracy, it is V W impossible to use feedforward from the environmental disturbances. This motivates the following assumption. A1: The unknown mean environmental force F and its direction are assumed to be constant or at least slowly varying. A2: The unknown attack point (l , l ) is constant for each V W constant F . Discussion. These are good assumptions since the ship control system is only supposed to counteract the slowly varying motion components of the environmental disturbances.
3. Weather optimal control objectives The weather optimal control objectives make use of the following de"nitions: De5nition 1 (=eather optimal heading). The weather optimal heading angle is given by the equilibrium state where the yaw moment, w ( )"0, at the same time as the bow of the ship is turned up against the weather (mean environmental disturbances), that is w ( )"0. This implies that the moment arms l ( )" constant V and l ( )"0, and: W w ( ) !F w( )" w ( ) " 0 . w ( ) 0 Hence, the mean environmental force attacks the ship in the bow (minimum drag coe$cient for the resulting current and wind loads).
De5nition 2 (=eather optimal positioning). Weather optimal positioning (station-keeping) is de"ned as the equilibrium state where w ( )"!F , w ( )"w ( )"l ( )"0, W (28) (weather optimal heading) and the position (x, y)" (x , y ) is kept constant. These de"nitions motivates the following two control objectives: O1. Weather optimal heading control (WOHC). This is obtained by restricting the ship to move on a circle with constant radius " and at the same time
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
705
Fig. 3. Principle for weather optimal heading control (WOHC).
force the ship's bow to point towards the center of the circle until the weather optimal heading angle, " , is reached (see Fig. 3, analogy to a pendulum in gravity "eld). The position, (x, y)"(x # cos , y # sin ), will vary until the weather optimal heading angle is reached. This is obtained by specifying the control objective in polar coordinates according to "constant, "0, "#.
(29)
Discussion. The requirement "constant im plies that the ship moves on a circle with constant radius. The second requirement, "0, implies that the tangential speed, , is kept small while the last requirement, "#, ensures that the ship's bow points towards the center of the circle. O2. Weather optimal positioning control (WOPC). In order to maintain a "xed Earth-"xed position (x, y)"(x , y ), the circle center p "[x , y ]2 must be moved simultaneously as control objective O1 is satis"ed. This is referred to as circle center control. A nonlinear and adaptive backstepping controller which satis"es O1 and O2 is derived in the next section. 4. Nonlinear and adaptive control design In this section, the WOPC positioning controller is derived by using the polar coordinate representation.
The backstepping design methodology (Krstic, Kanellakopoulos, & Kokotovic, 1995) with extension to integral control (Loria, Fossen, & Teel, 1999) is used to derive the feedback controller. It is assumed that all states can be measured by using conventional sensor technology and a satellite navigation system, e.g. DGPS. An attractive navigation system is strapdown DGPS/INS (Vik, 2000). In the case, where only DGPS position measurements and a gyro compass are available the body-"xed velocities can be estimated by using an extended Kalman "lter (Fossen, 1994) or the nonlinear and passive observer of Fossen and Strand (1999). The combination of the nonlinear observer and the full state feedback WOPC controller can be further analyzed through a nonlinear separation principle by using the approach of Loria, Fossen, and Panteley (2000). The WOPC controller will be derived in three successive steps: (1) Nonlinear backstepping (PD-control): the ship is forced to move on a circle arc with desired radius , with minimum tangential velocity and desired heading . (2) Adaptive backstepping (PID-control): this is necessary to compensate for the unknown environmental force F . (3) Translational control of the circle center: the circle center (x , y ) is translated such that the ship main tains a constant position (x , y ) even though it is moving on a virtual circle arc. Hence, the Captain of the ship will only notice that the ship is rotating
706
T. I. Fossen, J. P. Strand / Automatica 37 (2001) 701}715
a yaw angle about a constant position (x , y ) until the weather optimal heading is reached.
By using the property (21) and choosing the nonlinear PD-control law as ¹\2 "M xK #C x #D x !K z V V V
4.1. Nonlinear backstepping (PD-control)
!K z !¹\2q( ) ),
A general positioning controller is derived by using vectorial backstepping (Fossen & Gr~vlen, 1998). The tracking objective is speci"ed in polar coordinates using a smooth reference trajectory x "[ , , ]23C where x ,x ,xK 3L . Since the transformed system (16) is of order 2, backstepping is performed in two vectorial steps resulting in a nonlinear PD-control law. First, we de"ne a virtual reference trajectory as x Ox !z , (30) where z "x!x is the Earth-"xed tracking error and '0 is a diagonal design matrix. Furthermore, let z denote a measure of tracking de"ned according to z Ox !x "z #z . (31) From (31), the following expressions are obtained x "z #x , xK "z #xK . (32) This implies that the vessel model (16) can be expressed in terms of z , x and xK as M z #C z #D z "¹\2 #¹\2q( ) ) V V V !M xK !C x !D x #¹\2w. (33) V V V Step 1: Let z be the "rst error variable, which from (31) has the dynamics: z "!z #z . (34) A Lyapunov function candidate (LFC) for the "rst step is < "z2 K z , (35)
