Fuel-efficient rudder and propeller control allocation for ... - IEEE Xplore

850

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 6, NOVEMBER 2003

Fuel-Efficient Rudder and Propeller Control Allocation for Marine Craft: Experiments With a Model Ship Karl-Petter Lindegaard, Member, IEEE, and Thor I. Fossen, Senior Member, IEEE

Abstract—We derived a control allocation algorithm for lowspeed marine craft using propellers and rudders. Active use of rudders has advantages in a low-speed operation by decreasing the need for propeller power and fuel. However, at low speed, a rudder is effective only for positive thrust. This complicates the thrust allocation problem which can no longer be solved by convex quadratic programming. In fact, the existence of local minima introduces discontinuities in the commanded thruster signals even if the desired control force is continuous. Discontinuous signals cause excessive wear on the thruster system and must be avoided. This paper suggests an analytic, 2-norm optimal method that can ensure continuity of the solutions. Being analytic, however, its limitation is the capability of handling only configurations where one single thrust device is subject to sector constraints at a time. Experiments with a model ship illustrate the potential for fuel saving. For this particular vessel, the energy consumption was halved. An output feedback tracking control law with integral action was simultaneously derived and analyzed. Semiglobal ship controllers like this one rely on the yaw rate being bounded, and an admittedly conservative method for determining this upper bound was proposed. Index Terms—Asymptotic stability, fuel optimal control, marine vehicle control, nonlinear systems, output feedback.

I. INTRODUCTION

A

UTOMATIC control of ships has been studied for almost a century. In 1911, Elmer Sperry constructed the first automatic ship steering mechanism called “Metal Mike.” Today, the range of marine vessels is expanded to a huge diversity of vehicles such as remotely operated vehicles (ROVs) and semisubmersible rigs. Automatic control systems for heading and depth control, way-point tracking control, fin and rudder-roll damping, dynamic positioning (DP), position mooring, etc., are commercial products [1]. A commercial control system is usually designed as three interconnected subsystems for Guidance, Navigation, and Control (GNC). The control system block contains a feedback control law and a control allocation (CA) algorithm needed for optimal distribution of controller commands to the different actuators. Manuscript received February 20, 2002; revised January 27, 2003. Manuscript received in final form March 18, 2003. Recommended by Associate Editor J. M. Buffington. This work was supported by the Norwegian Research Council through the Strategic University Program, Marine Cybernetics. K.-P. Lindegaard was with Kongsberg Simrad AS, N-1375 Billingstad, Norway. He is now with ABB Corporate Research, N-1375 Billingstad, Norway (e-mail: [email protected]). T. I. Fossen is with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2003.815613

Fig. 1. Overview of an automatic ship control system with a high level controller.

The main contribution of this paper is a thrust and rudder (control) allocation algorithm for marine vessels. This is a challenging problem since a ship at zero speed with a main propeller and a single rudder produces lift (sideforce) for positive propeller revolutions only. Rudder forces are approximately zero when the propeller is reversed. The rudder force is also limited to a small sector due to rudder angle saturation. Optimal thrust allocation in station-keeping has been addressed previously [2]–[6], while simultaneous thrust and rudder allocation for ships at zero speed has received less attention. Since operating a rudder servo is relatively inexpensive compared to operating a propeller, this gives a large fuel saving potential. This solution is well known in practice since experienced captains use a nonzero rudder angle (for positive thrust) in station-keeping and docking to produce additional forces in the transverse direction. In this paper, a constrained rudder/propeller control allocation algorithm is derived in conjuncture with a Lyapunov stable output feedback controller for low-speed tracking. For ships with more control inputs than controllable degrees of freedom (DOF), it is possible to find an “optimal” distribuprovided that the demand is attaintion of control forces able. Here is the number of actuators/propulsive devices which can be operated individually. The commanded generalized force is generated by a higher-level con(forces and moments) troller, for instance a dynamic positioning system. In this context a higher-level controller may be any arbitrary automatic control law (Fig. 1) or, alternatively, a joystick operator manually commanding surge and sway forces as well as a yaw moment (Fig. 2). Regardless of the source of commanded forces, the CA algorithm is responsible for calculating an “optimal” solution of actuator set-points that at all times satisfy the presumed attainable commanded generalized force signal.

1063-6536/03$17.00 © 2003 IEEE

LINDEGAARD AND FOSSEN: FUEL-EFFICIENT RUDDER AND PROPELLER CONTROL ALLOCATION

Fig. 2.

Overview of a manual joystick control system.

Fig. 3.

The thrust allocation problem seen as a two-step process.

In general, CA is a dynamic nonlinear optimization problem since there are sector limitations for operation of the rudders and the rotatable thrusters (azimuth thrusters). This problem can be solved using nonlinear optimization techniques [7], e.g., quadratic programming (QP), see Lindfors [3]. An alternative method using the singular value decomposition and a filtering scheme to control the azimuth directions has been proposed by Sørdalen [4] who also presented results from sea trials [5]. A similar technique using the damped least squares algorithm has been reported in [6]. Compared to advanced controller and filter design, the allocation part of ship control has received less attention [4]. This seems unfortunate, because the performance of an automatic control system will suffer if the CA module is poorly designed. Moreover, the design of the allocation module is the decisive element determining the costs, in terms of energy consumption, of obtaining the desired level of performance. A more efficient CA algorithm requires less power, reduces fuel consumption, and decreases gas emissions. While the mentioned CA schemes for ships perform an unconstrained optimization, the aerospace community has addressed constrained CA methods for quite some time. The two most frequently reported strategies are Durham’s tailor-made generalized inverses [8]–[13] and actuator daisy chaining [14]–[16]. The set of admissible controls is usually an -dimensional rectangle, as in the tailor-made inverses, and the controls can be selected independently of each other. These methods do not apply to ships with rotatable thrusters or rudders. Neither is the set of admissible controls necessarily convex, nor will the controls be independent. Another difference between an aircraft and a ship is how a universal cost criterion should be defined. For a ship at low speed, it is an obvious goal to minimize the total thrust force (and thereby fuel consumption) while for a cruising airplane the cost in terms of energy of using the passive control surfaces is of minor importance, although drag minimization has been considered [17]. The control allocation problem arises in cases where there are more actuators than DOFs, or more precisely when the decision variables outnumber the equality constraints. An algorithm can be divided into two steps as described in Fig. 3. In the first

851

step, called force allocation, the commanded generalized force is distributed out to each one of the available actuators. The decision made at this stage determines how “good” the algorithm is. The second step deals with finding which actuator set-points will generate the desired forces . This step is called “inverse mapping” because it must solve the inverse of the acis tuator models. The vessel’s velocity relative to the fluid considered an auxiliary signal that must be accounted for since rudder forces greatly depend on surge velocity. Evidently, the thrust models used in the second step influence the domain of . For low-speed applications, as considered in this paper, the ship’s relative velocity may be neglected. The thrust allocation problem is an optimization problem with several requirements to consider. For instance, the algorithm must take into account the capacity of each thrust device and it must avoid “singular solutions” [4], [18] that produce unacceptably large thrust magnitudes. Furthermore, finding a suitable solution must take a minimum of time and computational effort because the implementation runs on a real-time platform often with limited resources. More sophisticated algorithms should include support for thrusters with a preferred thrusting direction (many thrust devices are more effective when producing positive thrust than negative) and also take into account the dynamic constraints of the propulsion system, e.g., a rotatable thruster cannot rotate arbitrarily fast (rate saturation). Those issues will, however, not be addressed here. A topic related to saturation handling is to the determine attainable thrust set [8]–[10], that is all combinations of that the thruster system can provide. Limitations will be imposed by the maximum capacity of each device (box constraints), but also by the amount of available power on the power buses on which the thrusters are connected. If rate limitations are to be handled, the current states influence the attainable thrust, too. By analyzing the attainable set before sending the demand to the force allocation module (Fig. 3), or at least taking it into account in the allocation itself, we are given the opportunity to analyze and modify the demand in light of the limitations. The desired should be checked for attainability. If it is not attainable, explicit action must be taken because if the actuators are individually limited after the inverse mapping, overall control of the resulting thrust may be lost. In such cases one approach could be to give preference to some of the controllable DOFs in the -vector and to modify the other ones such that eventually becomes attainable. In aerospace applications the set of attainable thrust is convex [9], [10] simplifying such priority considerations significantly. This is, however, not the case for ships with sector restricted rotatable thrusters or propeller-rudder pairs. Because of this nonconvexity, it will be beyond the scope of this paper to pursue this topic further. Instead we assume that is always attainable. II. PROBLEM STATEMENT A. Notation and Definitions Consider a vessel with actuators. Actuators denote rotatable or fixed propellers, waterjets, and other devices capable of producing a propulsive force. A main propeller equipped with a rudder is considered a rotatable device. Passive devices such as standalone rudders and flaps are to be considered to be actuators

852


at nonzero velocities, while at velocities near zero they should not. rotatable and fixed force devices Assume there are . Notice that a rotatable actuator has such that two controls, one for the direction and one for force amplitude. and For the rotatable and fixed actuators this yields number of decision variables respectively. In sum there decision variables. Let denote the number of are the vessel is said to be DOFs under consideration, and if overactuated. From now on we shall consider surge, sway and . yaw only, thus Each thruster is located at

For rotatable thrusters , and for fixed we have . thrusters is confined to a domain of Most likely an extended thrust (or ), e.g., an azimuth thruster could be restricted to thrust be only inside a predefined sector. Let each extended thrust , that is for we have confined to the domain , and for . Using the concept of extended thrust, the generalized thrust vector becomes (8) where

consists of

and

(1)

(9)

with respect to the origin of the vessel-fixed coordinate system in a direction . The -axis points and can produce a force forward and the -axis to the right (starboard). The difference and between the rotatable thrusters numbered is that the direction can fixed thrusters be manipulated. be the maximum thrust force such that the Let defined as normalized thrust

(10)

(11) (2) satisfies

for all . Define

as a subset of (3)

and . Then The sum of generalized forces on the vessel

) whenNotice that is a constant matrix of rank (here is a diagever the craft is overactuated. The matrix along the diagonal matrix with corresponding values of onal. Define (12)

is (4)

A control vector linear constraint

(extended thrust) is called feasible if the (13)

lies in its domain . and each and every Let the columns in the matrix be an orthonormal basis of . Then, . The null-space the null-space and row-space of , denoted , are orthogonal subspaces . of The generalized inverse of a matrix , using a symmetric and , is defined as positive definite weight matrix

where

(14)

(5) such that the unconstrained solution The vessel is said to be in a singular configuration if looses rank due to a poor selection of . denote the control vector where the normalized Let for each thruster has been decomposed in the horithrust zontal plane according to (6) be a proThe decomposition is called extended thrust. Let of thruster jection that extracts the extended thrust vector from according to (7)

(15) is the optimal solution, in a weighed 2-norm sense, that is (16) . Moreover, , we get the pseudoinverse If (17) such that instead of

. Without loss of generality, we use in calculating the unconstrained solution.


853

Fig. 4. The physically attainable thrust region (grid) for a propeller-rudder pair. An inscribed circular sector and the negative x-axis together serve as the . feasible region

D

B. Problem Introduction The main objective of this paper was finding a feasible control vector (extended thrust) that is optimal with respect to some cost function quadratic in . Quadratic cost functions are attractive candidates because are described by linear inequalities when the domains the problem is a general quadratic program (QP). Quadratic programs can be solved in a finite number of iterations and are consequently quite predictable which again makes them well suited for real-time implementation. Unfortunately, a probecause peller-rudder device requires a nonconvex domain these propulsive devices are unable to produce lateral forces without simultaneously generating significant longitudinal forces. It is reasonable to assume that the rudder is capable of producing lift for positive thrust force (forward). For negative longitudinal force the rudder can be regarded as inactive, thus the attainable set shrinks to a thin line. Typically the physically attainable thrust region resembles a twisted circular sector, represented by the grid in Fig. 4. Notice that the negative -axis is also a part of the attainable set. For convenience, the grid is approximated by an inscribed circular sector so that the feasible is given by this sector element and the negative -axis. set Even though a sequential QP (SQP) or a sequential linearly constrained (SLC) method does find a solution, careless utilization of any numerical method can lead to unintended results: 1) An SQP method solves a sequence of QP problems and is therefore more computationally demanding than a single QP. Moreover, if the iteration sequence is interrupted or preempted due to CPU time limitations, the current solution may not be optimal. 2) The nonconvex inequality constraints for the propellerrudder couple introduce a discontinuity in the mapping if the optimal solution is to be used at all from times. However, infinitesimal changes in should never lead to discontinuous extended thrust . This second situation is illustrated in Fig. 5: The optimal un, is marked with constrained solution for thruster , that is a cross and the dotted cost curves illustrate the cost involved

Fig. 5. Illustration of a discontinuous mapping from to extended thrust vector u.

getting the solution into the feasible domain . The two circles mark feasible solutions with identical cost. Since the unlies slightly above the equicost line constrained solution crosses (dashed) the uppermost solution is selected. Once the equicost, the lower solution becomes optimal. Hence, the mapping is discontinuous and this will again cause undesired propeller and rudder angle chattering. It is therefore of paramount importance to avoid introducing this discontinuity. III. FORCE ALLOCATION The proposed algorithm is analytic and requires no iterations whatsoever. This means that some assumptions have to be made. 1) Only one thruster, thruster , with sector constraints is supported. 2) Thrust force magnitude constraints (box constraints) are not considered. Due to the feasible region being nonconvex, problems may arise if re-allocation is applied uncritically should one or more of the thrust devices saturate. Reallocating by looping through all possible configurations [19] can also be time-consuming. Still, our method is 2-norm optimal, it does avoid rudder-chattering and it does produce continuous solutions. The two latter properties have, as far as we know, not been considered elsewhere, but any CA algorithm for ships should, in our opinion, support them. Assumption 1 is the key to handle sector constraints analytically and to ensure continuity. The disadvantage is that for configurations where several thrusters are subject to sector constraints, some sub-optimal modification must be made, for instance by handling one thruster at a time and keeping the others at a fixed angle. This particular approach was pursued here since our ship model was equipped with two propeller-rudder pairs. As a consequence, one rudder only could be used at any one time. Thruster saturation is not covered according to Assumption 2. Traditionally, iterative solvers distribute possible overshoot

854


from the saturated thrusters onto the remaining ones provided is attainable. Without saturation handling, that the demand even however, we are not guaranteed to obtain the desired when it is attainable. But as long as is comparatively small with respect to the attainable set, no thruster should saturate and the box constraints can be disregarded. It should be noted that explicit, noniterative, off-line pre-computed schemes do handle box-constraints [20], but they do not support nonconvex constraints. The algorithm for finding a feasible extended thrust vector is solved in two steps in order to maximize computational performance. First, an unconstrained solution is found. If this 2-norm optimal solution lies outside the feasible domain , we need to determine the minimum accumulated cost translating toward the domain . the solution from A. Unconstrained

(23) is orthogonal to , that is . Later, in Secwhere tion III-D we focus on determining which boundary vector to use. Define the Lagrangian

(24) where it can be shown that

The first step in the allocation of extended thrust is achieved by simply employing the pseudo-inverse in order to minimize the 2-norm of (18) , Fig. 5 indicates that simply projecting the onto If one is not guaranteed to end up with a minimum cost . We have to consider the actual cost accumulated of traversing . This is the topic of the next subsection. the null-space B. Sector Constraints Assume that thrust device is restricted in a domain as illustrated in Fig. 5 and that the pseudoinverse solution has satisfies the linear equality conalready been found. Since straint (13), we may add any linear combination of the columns where is a vector of appropriate diin denoted mension, and still have a solution satisfying the linear equality constraint (19) The objective is now to find the optimal which renders . We have sible and minimizes

Once enters the feasible region spanned by and (Fig. 5), it is going to be parallel to one linear boundary curve . Consequently, we have reached the minimum when , which can be rephrased as a linear constraint because and using the any is a design parameter. By letting inner product we get the constraint

(25) . In other words, is positive definite as long as must have at least two columns. It must be pointed out that will be utilized and not itself. Since only the inverse we only have to calculate (26) we may disregard the dimension of the null-space causing a sin. gular , this reduces to an unconstrained QP If problem trivially solved using the generalized inverse. The minimum solution is found by minimizing (27) (28) Since the constrained solution is parallel to , by using (23) (29)

feathe optimal solution is

(30) (20) such that and where we have used that since is an orthonormal basis. Consequently, we seek a vector with minimum 2-norm. Consider device number . We have to find an increment such that

For all thrust devices together, the optimal increment obtained by combining (19), (22), and (30)

is (31)

and, hence, for the modified extended thrust, assuming that

(21) Once again the pseudo-inverse is applicable because it minimizes the 2-norm. Hence (22) is the solution we are looking for, but we have yet to determine .

(32) , and are constant matrices. Even if the Note that varies in time, no time-consector bound , or equivalently suming matrix operations need to be performed on-line, because is a nonnegative scalar.


855

A more detailed calculation reveals that

(37) where we notice that the first term is exactly the same as (32), that is when the -frame coincides with the -frame, and that the contribution from the translation appears as an additional linear term. D. The Equicost Line

Fig. 6. Translating the sector constraint of thruster B -frame.

k:

Definition of the

, we have to decide which of the two In case we should apnearest boundaries of the feasible domain are two vecproach, see Fig. 6. Assume that . We seek the unit vector tors defining the boundaries of describing the equicost line. That is, for any point on , say , where is an arbitrary scalar, the costs of apand are identical. proaching From the results in the two previous sections we know that (38)

C. Sector Constraint With Rudder Anti-Chat In order to remove rudder chattering around zero, we propose to translate the sector constraint a distance , or more specifically, slightly along the -axis as shown in Fig. 6. This modification ensures that the rudder will not be used unless the actuator thrust is above some positive threshold. Furthermore, the transition from zero rudder angle to maximum deflection will be continuous, because rudder will be applied gradually as the required thrust along the -axis increases. The distance between the origin of the original -frame and the origin of the translated -frame is denoted when it is decomposed in the -frame. Then, a point in the -frame is in . Thus, the optimal unconthe -frame given as . strained solution decomposed in the -frame is We may now perform the adjustment due to the sector constraint in the -frame instead of in the -frame, because since the coordinate systems are linearly translated and not rotated with respect to each other. The basic results from the previous section can therefore be applied. More specifically, the as given in (25) serves as the weight in the Lagrangian cost (33) is now given in the where it must be noted that minimum cost adjustment is

and

are the optimal steps toward involved are

and

, respectively. The costs

(40) and in order to find the equicost line we require . Hence

or

(41) where

is given by

Expanding this expression using two linear curves where the hyperbola -plane, that is

we obtain the intersects the

(42)

-frame. The

(34) and the optimal constrained solution and -frames are

(39)

Using polar coordinates tions

we get four solu-

-

(43)

(35) (36)

. From this the correct solution is selected, where and . Obnamely the one lying in the sector spanned by , the infeasible serve that for this method to be valid,

given in the

856


Fig. 9. Weighing functions: f (s) = sat(s) (solid), f (s) = tanh(2s) (dashed) and f (s) = erf(2s) (dotted).

Fig. 7.

Translation of u

and that

without loss of continuity.

define a (not necessarily orthogonal) basis for

such (45)

yields (46) The distance between the equicost line and

is (47)

so that the ratio (48)

Fig. 8. Detailed picture of restored continuity and definition of basis vectors e and e .

sector has to be less than 180 degrees wide. For propeller/rudder pairs this will always be the case. E. Restore Continuity We have already predicted that crossing the equicost line even if the commanded could introduce a discontinuity in is continuous. In this section, we propose a thrust force remedy for this situation; a solution that does not necessarily minimize the 2-norm. The idea is outlined in Fig. 7: If is situated inside the sector surrounding the equicost line, we to a point closer to the origin (the intersection between drive and ). Unlike the procedure given earlier, we now follow the solid arrows instead of the dotted lines. Consequently, if lies exactly on the equicost, the desired should be zero and continuity will be restored. Fig. 8 gives a more details: Having already calculated the constrained optimal solution , marked with a circle, we now force it toward the origin increasing the cost. onto the unit vectors and . is parallel Decompose to the equicost line and (44)

along the line . A close to zero defines the location of is close to the equicost line, and we need to means that force toward the origin. The strategy is: Introduce a deand a continuous, nondecreasing sign parameter . Besides (34) we now use weighing function : (49) (50) The width of the sector is determined by . Outside of the and . The optimal but disconsector, or for all tinuous solution corresponds to . Fig. 9 shows three weighing function alternatives. F. Proposed Algorithm The force allocation procedure can be summarized by the following sequence of operations . 1) then terminate. 2) If 3) Approaching . . • Assume thruster such that . • • Find equicost line and determine toward which we should go. Call it . boundary of . • 4) If • Restore continuity of the mapping • • •


Fig. 10.

857

and Real Time Workshop. Opal RT-Lab handles the compilation of the code as well as any other communication with the target system such as transmission of various signals during the experiments. A graphical user interface (GUI) was developed under LabView 6.0i, together with a manual joystick control system. Through the GUI the operator may switch between different modes of operation while supervising the experiment. The entire system was constructed with standard off-the-shelf hardware and software to keep the costs down. During start-up, CS2 automatically logs on to the camera PC in order to receive position and attitude data. The six DOF positions and orientations (attitude) are measured with two standard RGB cameras with the aid of two MagTrak ISA-cards. This system first locates the three colored markers on CS2 using Tsai’s algorithm [21]. An alternative version of the QUEST algorithm [22] based on singular value decomposition instead of eigenvalues calculates the ship’s attitude in unit quaternions and converts it to Euler angles, that is the roll, pitch and heading angles. Once the orientation is found, determining the position of the ship is a straightforward operation. It is reasonable to assume that the rudders are effective when applying positive thrust. We are thus allowed to find general thrust vectors inside the shaded areas in Fig. 10.

Outline of Cybership II: Feasible thrust domains.

A. Model Description At low speed a surface vessel can be reasonably described by

(51)

Fig. 11.

Overview of the GNC Laboratory.

• •

.

is a vector of North position, East poHere, sition, and compass heading, respectively. The body-fixed vecontains surge, sway and yaw and locity vector is the applied thruster force. The matrix : describes the rotation between body-fixed and Earth-fixed coordinates:

5) If • . 6) For all thrusters • • . is 1, it is futile to If the dimension of the null-space of perform step 4 (Restore continuity). Likewise, the concept of the equicost line does not apply.

(52) and its time derivative can be written where is the skew-symmetric (53)

IV. EXPERIMENT DESCRIPTION The experiment was performed in the GNC Laboratory at NTNU (Fig. 11) with the model ship Cybership II (CS2). The boat is a scaled replica of an offshore supply vessel and its meters. It is equipped with three overall length is propulsive devices: In the bow there is a small two-bladed RPMcontrolled tunnel-thruster capable to produce a sway force, and at the stern there are two RPM-controlled main propellers with rudders. CS2 is equipped with a 244 MHz Pentium clone (Cyrix MediaGX) PC running QNX, and it is connected to the local LAN through a BreezeCOM 2 Mbit IEEE 802.11 wireless Ethernet link. The control software is developed under Matlab Simulink

contains the mass of the vehicle, that is the sum of derigid-body and hydrodynamic added mass, and scribes linear damping which is assumed to dominate quadratic damping at velocities around zero. Slowly varying disturbances . A more detailed are lumped together in the bias vector description of the low-frequency vessel model has been given elsewhere [1]. B. Thruster Model For fixed pitch propellers the generated thrust force is more or less proportional to the square of the propeller shaft speeds . The low-speed propeller/rudder model for CS2 can conveniently

858


tively, such that they together form a smooth reference trajectory. Introduce the deviations , and . Consider the following tracking control law

TABLE I NOMINAL THRUST PARAMETERS

(58) The last two terms are feed-forward terms. Notice that the gains and are put to the left of making them bodyfixed gains. This is more intuitive than having them on the right should not influence the [25], since the compass heading convergence rates. and define , When we use we get the compact form

TABLE II RUDDER LIFT AND DRAG PARAMETERS

(59) (60) be separated into two parts: The first one is the nominal thrust ) (rudder angles (54) and the second part gives additional rudder lift and drag forces,

(55)

(56) For the main propellers, forces are

, the resulting surge and sway

(57) The thruster and rudder parameters are given in Tables I and II. C. Control Design A simple output feedback control law with true integral action is derived to satisfy concept the of the “separation principle” [23] for Euler–Lagrange systems, that is the state estimator and the controller can be independently configured. As long as each component is asymptotically stable, the system as a whole will be so as well.

(61) -operator constructs a block-diagonal matrix where the with zeros on the off-diagonal blocks. Since the system is controllable, it is possible to assign arbifor all , the trary eigenvalues to , and because are identical to the ones for . eigenvalues of being Hurwitz is necessary but In contrast to linear systems, not sufficient for global exponential stability of (59). . For small , The limiting factor here is the yaw rate , exponential stability of (59) can be shown that is is larger than the physical limit on , easily, and if and (59) is stable in a global sense. Increased gains , results in a larger due to controllability. Theorem 1: Consider the system (51) controlled by (58). for all . The origin of (59) Suppose is is uniformly locally exponentially stable provided are sufficiently small and if and only if as defined by (60) is Hurwitz. If is chosen such that , (59) is said to be larger than any physical upper limit for uniformly globally exponentially stable (UGES). being Hurwitz is obvious. For Proof: Necessity of . By we hereby proving sufficiency, define . Then mean (62) where there exists a

. If and only if such that for a given

D. State Feedback Tracking Control Fossen, Loria, and Teel [24] applied adaptive backstepping to obtain integral action, while in Loria, Fossen, and Panteley [25] the authors used the bias estimate from the observer. Here we integrate the position error directly (true integral action) and exploit the concept of commutating matrices to separate the nonlinear rotations from the otherwise linear dynamics. , : and : deLet : scribe desired positions, velocities and accelerations, respec-

is Hurwitz (63)

Consider the storage function Its time-derivative

.

(64) (65)


is negative definite provided that is small enough. What is the “stability margin,” i.e., how large is permitted to be for a given controller before it de-stabilizes the system? From (64) we note that if (66) then (59) is exponentially stable. As is linear for all and , then in it suffices to show for all . We summarize this as follows: that Corollary 2: For a given , a yaw rate bound guarantees exponential stability of (59) can be found by solving the following generalized eigenvalue problem in the decision variables and

859

Proof: Using output-feedback (69) instead of (58), the response (59) is perturbed by the observer errors . Standard results [29] can be applied if that perturbation term is bounded linearly in or, equivalently, . The proof will then be complete. and which means that By definition and . Consequently, (69) can be written

(70) and the vessel dynamics (the -system) becomes (71)

(67)

. where This formulation is similar to standard results on stability margins for linear parameter varying (LPV) systems, or polytopic linear differential inclusions [26]. Solving for maximum can be done efficiently using standard software packages [27], but this is usually of academic interest only: For well-beresulting from solving (67) is sighaving controllers, the achievable. nificantly larger than the highest To conclude, low-speed tracking control for overactuated ships can be done with linear tools. When the gains have been in the determined, simply add the proper rotations control law to provide validity for all heading angles.

and are defined in (60) and (61). The parameters Here in the interconnection term are (72) (73) (74) Using

we get (75)

and consequently the complete system, the vessel dynamics and observer, can be written on the cascaded form

E. Observer

(76)

A UGES observer estimating low-frequency velocities and positions from position measurements was implemented [28]. More precisely, for the system (51) uniform global exponential convergence of

where the individual terms of the transformed tracking error are system (77) (78)

(68) for all , Because is bounded linearly in and is, in fact, independent of

could be expected. F. Output Feedback Control

(79)

Consider the state-feedback controller (58) and substitute the states and with the estimated counterparts and : V. EXPERIMENTAL RESULTS (69) Stability of the resulting closed loop system is guaranteed by the following theorem. of Theorem 3: Suppose the tracking error the system (51) subject to the state-feedback controller (58) is UGES and that there exists an observer for which the error as given by (68) converges uniformly asymptotically to zero. Then, by output-feedback (69) the tracking error converges uniformly and asymptotically to zero.

The main objective of the experiments was to compare the energy consumption with and without active use of rudders during a positioning operation: Move 30 cm sideways while maintaining a fixed heading. This operation is particularly energy consuming when the rudders are not to be used because the ship is unable to produce lateral forces in the stern. the desired In order to emphasize the differences, at to in position changed from a step-response fashion, see Fig. 12. In full-scale 30 cm corresponds to approximately 20 m. Measured position is shown in red and the observer’s low frequency estimate in blue.

860

Fig. 12.


Operation overview.

Fig. 14.

Propeller shaft speeds

= 0 (dashed),

32 deg (solid).

Fig. 13. Propeller shaft speeds ! for feasible sector widths (dashed), = 10 (dash-dotted), = 20 (dotted) and deg (solid).

!

during the first 15 seconds of operation. = 20 (dotted), =

= 10 (dash-dotted),

= 0 = 32

Inwards rudder deflection was allowed. Consequently, the port propeller/rudder pair was allowed to have a lateral component pointing port while the starboard propeller/rudder had a component pointing in the starboard direction. Two solutions were calculated at each sample. In the first one the port rudder was allowed to be used while the starboard rudder was fixed the starboard rudder at zero degrees, and in the second one was deflected and the port rudder set at zero degrees. Finally, for we picked the solution with minimum 2-norm, that is . The feasible sector was varied in four steps from zero to a maximum of 32 degrees, more specifically (80) In order to attenuate rudder chattering around zero, the following was used for both main propellers (81) Fig. 13 shows the applied propeller revolution speed for all , the dashfour test runs. The dashed lines denote , the dotted lines , and the dotted lines degrees. Fig. 14 is a close-up of solid lines represent Fig. 13. The applied rudder angles are plotted in Fig. 15. Notice

Fig. 15.

Rudder angles.

= 20 (dotted),

= 0 (dashed), = 32 deg (solid).

= 10 (dash-dotted),

that since the rudders were deflected inwards, the port rudder is always negative while starboard is always positive. Absorbed hydrodynamic power is proportional to the cube of the shaft speed. We assumed that the efficiency factor was constant so that applied electrical power was proportional to hydrodynamic power. Fig. 16 show normalized power, in fractions of the maximum equal to 1, for all four test runs. Thus, we may compare individual runs without having an exact power model available. The main propellers’ energy consumption defined as the integral of normalized power is reproduced in Fig. 17, and the normalized power and energy consumption for the bow thruster is plotted in Fig. 18. A. Comments The responses in terms of tracking the desired position were very similar for all four test runs, and consequently the applied


Fig. 16. Normalized power consumption for the main propellers. =0 (dashed), = 10 (dash-dotted), = 20 (dotted), = 32 deg (solid).

861

Fig. 18. Normalized power and energy consumption of the bow thruster. = 0 (dashed), = 10 (dash-dotted), = 20 (dotted), = 32 deg (solid).

VI. CONCLUDING REMARKS We have introduced a force allocation algorithm for a sectorrestricted thrust device such as a propeller-rudder pair. Handling sector constraints is nontrivial, because the optimization problem is no longer convex and local minima exist. The proposed force allocation method is a three-step algorithm that avoids rudder chattering and ensures a continuous mapping between the commanded thrust force and the extended thrust. In order to conduct the experiments, a semiglobal output-feedback tracking control law with integral action was proposed. Experiments with a model ship confirmed that using rudders actively has a great potential in terms of improving energy efficiency. ACKNOWLEDGMENT

Fig. 17. Consumed energy for the main propellers, given as integrated normalized power. = 0 (dashed), = 10 (dash-dotted), = 20 (dotted), = 32 deg (solid).

The authors would like to express their sincere gratitude to friends and colleagues at the Department of Engineering Cybernetics, NTNU, involved in the development of Cybership 2. REFERENCES

thrust forces were almost indistinguishable. Time series plots of the positions and thrust forces are not shown. When the force allocation algorithm is allowed to use the rudders to generate sway forces in the stern, the power and energy consumption for all three propellers decrease significantly. For degrees, the peak in required power for the main propellers is being reduced to only 15% of that required for . The peak of the bow thruster’s power consumption is reduced by around 50%. When the ship approaches its new position, the differences are less pronounced. The rudder anti-chattering option, this is a nonzero , works satisfactorily. If this functionality is turned off, the rudder angle set-points would switch between zero and maximum rudder deflection. Measurement noise residues in spuriously creates an impression of oscillatory rudder behavior (Fig. 15).

[1] T. I. Fossen, Guidance and Control of Ocean Vehicles: John Wiley & Sons, 1994. [2] N. A. Jenssen, “Estimation and control in dynamic positioning of vessels,” Ph.D. dissertation, Dept. of Engineering Cybernetics, Norwegian University of Science and Technology, 1981. [3] I. Lindfors, “Thrust allocation method for the dynamic positioning system,” in Proc. 10th Int. Ship Contr. Syst. Symp. (SCSS’93), Ottawa, Canada, 1993, pp. 3.93–3.106. [4] O. J. Sørdalen, “Optimal thrust allocation for marine vessels,” Contr. Eng. Practice, vol. 5, no. 9, pp. 1223–1231, 1997. [5] , “Full scale sea trials with optimal thrust allocation,” in Proc. 4th IFAC Conf. Manoeuvring and Control of Marine Craft (MCMC’97), Brijuni, Croatia, 1997, pp. 150–155. [6] S. P. Berge and T. I. Fossen, “Robust control allocation of overactuated ships: Experiments with a model ship,” in Proc. 4th IFAC Conf. Manoeuvring and Control of Marine Craft, Brijuni, Croatia, 1997, pp. 166–171. [7] J. Nocedal and S. J. Wright, Numerical Optimization. New York: Springer-Verlag, 1999. [8] W. C. Durham, “Constrained control allocation,” AIAA J. Guidance, Contr. Dynam., vol. 16, no. 4, pp. 717–725, 1993.

862

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

[25]


, “Constrained control allocation: Three moment problem,” AIAA J. Guidance, Contr. Dynam., vol. 17, no. 2, pp. 330–336, 1994. , “Attainable moments for the constrained control allocation problem,” AIAA J. Guidance, Contr. Dynam., vol. 17, no. 6, pp. 1371–1373, 1994. K. A. Bordignon and W. C. Durham, “Closed-form solutions to constrained control allocation problem,” AIAA J. Guidance, Contr. Dynam., vol. 18, no. 5, pp. 1000–1007, 1995. W. C. Durham, “Efficient, near-optimal control allocation,” AIAA J. Guidance, Contr. Dynam., vol. 22, no. 2, pp. 369–372, 1999. , “Computationally efficient control allocation,” AIAA J. Guidance, Contr. Dynam., vol. 24, no. 3, pp. 519–524, 1999. J. M. Berg, K. D. Hammet, C. A. Schwartz, and S. S. Banda, “An analysis of the destabilizing effect of daisy chained rate-limited actuators,” IEEE Trans. Contr. Syst. Technol., vol. 4, pp. 171–176, 1996. J. M. Buffington and D. F. Enns, “Lyapunov stability analysis of daisy chain control allocation,” AIAA J. Guidance, Contr. Dynam., vol. 19, no. 6, pp. 1226–1230, 1996. J. M. Buffington, D. F. Enns, and A. R. Teel, “Control allocation and zero dynamics,” AIAA J. Guidance, Contr. Dynam., vol. 21, no. 3, pp. 458–464, 1998. W. C. Durham, J. G. Bolling, and K. A. Bordignon, “Minimum drag control allocation,” AIAA J. Guidance, Contr. Dynam., vol. 19, no. 1, pp. 30–36, 1996. T. A. Johansen, T. I. Fossen, and S. P. Berge, “Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming,” IEEE Trans. Contr. Syst. Technol., 2003, to be published. K. A. Bordignon and W. C. Durham, “Null-space augmented solutions to constrained control allocation problems,” in Proc. 1995 AIAA Guidance, Navigation and Control Conf. (AIAA-95-3209), Baltimore, MD, 1995. T. A. Johansen, T. I. Fossen, and P. Tøndel, “Efficient optimal constrained control allocation via multi-parametric programming,” AIAA J. Guidance, Contr. Dynam., 2002, submitted for publication. R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automat., vol. 3, pp. 323–344, 1987. M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” AIAA J. Guidance Contr., vol. 4, no. 1, pp. 70–77, 1981. A. Loría and E. Panteley, “A separation principle for a class of Euler–Lagrange systems,” in New Directions in Nonlinear Observer Design, H. Nijmeijer and T. I. Fossen, Eds: Springer, 1999, pp. 229–247. T. I. Fossen, A. Loria, and A. Teel, “A theorem for UGAS and ULES of (passive) nonautonomous systems: Robust control of mechanical systems and ships,” Int. J. Robust and Nonlinear Contr., vol. 11, pp. 95–108, 2001. A. Loría, T. I. Fossen, and E. Panteley, “A separation principle for dynamic positioning of ships: Theoretical and experimental results,” IEEE Trans. Contr. Syst. Technol., vol. 8, pp. 332–343, 2000.

[26] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics SIAM, 1994. [27] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox: The Mathworks Inc., 1995. [28] K.-P. Lindegaard and T. I. Fossen, “A model based wave filter for surface vessels using position, velocity and partial acceleration feedback,” in Proc. 40th IEEE Conf. Decision Contr., Orlando, FL, 2001. [29] E. Panteley and A. Loria, “On global uniform asymptotic stability of nonlinear time-varying nonautonomous systems in cascade,” Syst. Contr. Lett., vol. 33, no. 2, pp. 131–138, 1998.

Karl-Petter Lindegaard (S’99–M’02) was born in Norway in 1972. He received the M.Sc. and Ph.D. degrees in 1997 and 2003, respectively, both in engineering cybernetics from the Norwegian University of Science and Technology (NTNU), Trondheim. He is currently working as a Scientist at ABB Corporate Research, Norway, where he participates in the development of automation systems for advanced oil field production. His research interests include optimization and applied nonlinear control.

Thor I. Fossen (S’88–M’91–SM’00) received the M.Sc. degree in naval architecture in 1987 and the Ph.D. degree in engineering cybernetics in 1991, both from the Norwegian University of Science and Technology (NTNU), Trondheim. During 1989–1990, he pursued postgraduate studies as a Fulbright Scholar in aerodynamics and flight control at the University of Washington, Seattle. He is teaching ship and ROV control systems design, navigation systems, and nonlinear and adaptive control theory. He is the author of the books Guidance and Control of Ocean Vehicles (New York: Wiley, 1994) and Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles (Marine Cybernetics, 2002). He has been involved in the development of dynamic positioning and autopilot systems for ships. He has also been involved in the design of the SeaLaunch trim and heel correction systems. Dr. Fossen received the Automatica Prize Paper Award in 2002 for work on a weather optimal positioning system for marine vessels.