"详尽研究欠驱动船路径跟踪的LOS算法"

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1.2. ACADEMIC CONTROL APPR OACHES 3

Figure 1.2: The fully actuated supply vessel Nordmand Mermaid. Courtesy of

DeepOcean AS, http://www.deepocean.no.

• Economical justiﬁcations (stable requirements for transit operations un-

justify the cost of obtaining full actuation).

• Speed eﬀects (some fully actuated vessels become underactuated when

trav elling at high speed).

• Safety issues (actuator failure render vessels underactuated).

1.2 Academic con trol approac hes

The literature dealing with the con t rol of underactuatedshipsisfairlyextensive

and ever growing. Underactuated ship con trol seems to ha ve become a ﬁeld in

which considerable research eﬀort has been put down over the last 10 years, and

still is. The concepts appr oac h ed for solving the con t rol problem seems to be

divided into two notably diﬀerent strategies:

1. Full 3 DOF state control. By using only 2 control inputs, full 3 DOF

state control is attempted. Both stabilization and trajectory tracking

control is reported in the literature. The stabilization control problem

is gover n ed by the limitation stated in (Brockett 1983). Since only the

stabilization problem falls victim to Brockett’s condition, trac king time-

varying trajectories can actually be achieved by utilizing smooth, time-

invarian t state feedback controllers. The disadvan tage is that a single

controller cannot simultaneously be used for tracking of a time-varying

trajectory and stabilization to a point.

2. Reduced 2 DOF state control. By lo wering the output s pace from full 3

DOF to a reduced 2 DOF inspired by a physical interpretation of the

4 INTRODUCTION

system at hand, it is possible to control 2 DOF directly whilst achieving

indirect control of the unactuated DOF in a manner ensuring that the

vessel behaves as desired. Since this approach is not directly attempting

underactuated con trol of a ship, but to control an underactuated ship by

aredeﬁnition of the output space, it is not victimized by the condition of

Brockett. Hence ordinary smooth, static state feedback control laws can

be employed for both trajectory tracking and stabilization to a point.

These two strategies, their achievements, advantages and disadvantages will

be discussed in the rest of the section. Subsequently, a section giving a glimpse

of the control schemes presently in use by the industry is included. Some con-

cluding remarks sums up what has been presented. The Chapter ﬁnishes oﬀ by

giving an overview of the contributions this thesis has to oﬀer the problem of

con trolling underactuated ships.

1.2.1 Full state control

By far the largest con trol communit y is engaged in trying to solve the problem

of controlling 3 DOFs simultaneously by utilizing only 2 independen t contro ls.

As already emphasized, this means that a smooth, time-invariant, state feed-

back contro l algorithm cannot be utilized for stabilization to a point. One of

the earliest approaches to solving this proble m for underactuated ships is re-

ported in P ettersen & Egeland (1996), inspired by work already performed on

underactuated spacecraft and underwater vehicles. Only contr ol force in surge

and control momen t in yaw is assumed. The paper considers stabilizing avessel

exponentially to the origin by utilizing a continous, periodically time-varying

state feedback law. The approac h of introducing explicit time dependence in

the feedback law is originally proposed by Samson (1992) for the control of

mobile robots. Later, proposals of using discontinous, time-in variant state feed-

back laws is giv en (Reyhanoglu 1996). These schemes ac hiev e convergence to

the desired equilibrium point, but stability in the Lyapunov sense has not been

established. The discontin uity approach will not be treated in what follows.

Returning to the approach at hand, Pettersen & Egeland (1996) performs sig-

niﬁcant simpliﬁcations of the vessel model by assuming that both the inertia and

damping matrices are diagonal, e.g. assuming fore/aft-symmetry of the vessel.

A local stability result is achieved. Because of periodic, time-varying surge and

yaw reference velocities, the vessel approach es the origin in a very oscillat ory

manner. Hence the ac hieved result must be considered as being of purely aca-

demic intere st rather than practically rea lizable. The proposed control law is

later applied for the dynamical positioning of a scaled model ship of an oﬀshore

supply vess el. This is reported in (P ettersen & Fossen 2000). The controller is

extended by including integral action in an attempt to remove stationary errors

due to constant disturbances. However, both stationary errors and severe oscil-

lations of the ship wa s observed in the experiments. The authors conclude that

the main reason is the constant environmental forces acting on the ship. Hence,

the control law did not supply the desired local behaviour.

1.2. ACADEMIC CONTROL APPR OACHES 5

Later, in (Pettersen & Nijmeijer 2000), a new approa ch for stabilizing the

position and orien tation of a ship subject to environmental forces is discussed.

Semi-global practic al asymptotic stability is achieved, i.e. convergence to an

arbitrarily small neighborhood of the origin. The approach does not succeed

in removing the stationary oscillations, but it is reported that the amplitude

of these oscillations can be aﬀected. The pr oposed control law does not allow

the ship to be turned directly up against the environmental disturbance, thus

demanding the fulﬁllment of a persistency of excitation condition for being able

to function. The control law also called for a more high-frequent control than

was necessary in (Pettersen & Fossen 2000) and convergence of the disturbance

estimate w as reported to be slow. Finally, global stabilization is reported in

(Mazenc, Pettersen & Nijmeijer 2002). Global uniform asymptotic stabilization

is achiev e d b y means of a backstepping approach as given in (Krsti

c, Kanel-

lakopoulos & Kokotovi

c 1995).

A tracking con trol la w is developed for underactuated ships in (Pettersen

& Nijmeijer 1998). The control law is constructed by performing a change of

coordinates and utilizing integrator backstepping. For the controller to work,

the reference yaw angular velocit y has to be persistently exciting (PE), i.e. non-

zero. Hence, straight-line trajectories cannot be tracked by this control law.

Semi-glob al exponential tracking is achieved. Finally, in the work presented in

(Lefeber 2000), global tracking is reported for the ﬁrst time. The trac king error

dynamics is proven globally K-exponentially stable. The controller is developed

by means of a cascaded approach, and a much simpler structure than the one

derived in (P ettersen & Nijmeijer 1998) is obtained. Nevertheless, the reference

yaw velocity is still required to be non-zero. This work is also reported in

(Lefeber, Pettersen & Nijmeijer 2003).

In the work of Jiang (2002), another global tracking controller for underactu-

ated ships is derived. Unlik e the result by Lefeber (2000), an explicit Lyapunov

function is produced, suiting the need for robust and adaptiv e designs to fol-

low. Diagonal inertia and damping matrices are assume d, and a persistency

of excitation demand is placed on the reference yaw angular velocity. Global

asymptotic tracking is achieved. In (Do, Pan & Jiang 2002a), Lyapunov’s direct

method and backstepping is utilized to achieve glob al asymptotic tracking with-

out the demand that the reference yaw velocity be persistently exciting. Hence,

straight-line trajectories are also feasible. This is seemingly the ﬁrst time such

a result is reported in the literature. However, the controller is not applicable

for solving the stabilization problem, since such a problem cannot be solved b y

smooth, time-invariant state feedback. If global exponential tracking is to be

achieved by this controller, a PE condition mu st be placed on the reference yaw

velocity, rendering straight-line trajectories infeasible. Exactly the same results

are reported in (Do, Jiang & Pan 2002b).

The task of making an underactuated ship follow a straight-line is treated in

(Pettersen & Lefeber 2001). The longitudinal DOF is neglected in the analysis,

assumed to be independently controlled. This is made possible by the assump-

tion of port-starboard symmetry for the ship, th us decoupling the ship dynamics

into a longitudinal and lateral part. Hence, only the lateral subsystem is con-

6 INTRODUCTION

sidered. The aim is to minimize the cross-trac k error to the straight-line path

betweentwoway-points. ALine-Of-Sightalgorithmisusedtoobtaintheref-

erence yaw angle. The result is global asymptotic stabilization of the hea d ing

and cross-trac k error of the ship. How ever, only one straight line segment is

considered and no coherent strategy for following a path made up of several

straight-line segments is proposed. Environmental disturbances are not consid-

ered in the paper, but (Do, Jiang & Pan 2002c) takes such disturbances into

consideration. The tracking errors are shown to be globally ultimately b ounded,

conv erging to zero when no disturbances aﬀect the vessel. Both papers con-

sider straight line control only since they neglect the longitudinal DOF in the

analysis, hence the controller s are not applicable for trac king other trajectories.

To sum up the results achieved b y the community which adresses full state

control,globalstabilization to the origin has been attained by utilizing smooth,

time-varying state feedback. Such stabilization cannot be achieved without

placing persistency of excitation (PE) conditions on the reference velocities.

This is because Brockett (1983) states that any smooth, time-invariant state

feedbac k law is unable to achieve such stabilization for an underactuated dy-

namical system. When it comes to tr acking, global exponential tracking has

been achie ved, but with the demand that the reference yaw angular velocity be

persistently exciting. Hence, straight-line trajectories are ruled out. Ho wever ,

global asymptotic tracking has been obtained without demanding persistency

of excitation of the reference yaw velocity. The drawba ck is that the controller

achieving this cannot simultaneously be used for solving the so-called parking

problem, i.e. stabilization to a point, because it doesn’t satisfy the condition

of Brockett. In addition, successful attempts ha ve been made for construct-

ing controllers being able to trac k straight line trajectories. However, only the

lateral subsystem is considered in the analysis, so the controllers are not ap-

plicable for other geometrical topologies. Common to all the results gained by

this community is that the ship’s inertia and damping matrices are assumed

to be purely diago nal, i.e. fore/aft-symmetry is presume d. This is not a good

assumption for most ships. Consequently the results curren tly appear to be

mostly academically relevant, having not y et matured to a level applicable for

practical situations and practical implementation.

1.2.2 Reduced state control

By lowering the output space from full 3 DOF to a reduced 2 DOF, ensuring

that the control problem is no longer of an underactuated character, Broc ketts

condition no longer apply. Hence, smooth, time-in variant state feedback can

be utilized for controlling the output space. Ho wever, for this approach to be

successful, the strategy employed must ensur e indirect co ntrol of the unactu-

ated DOF such that it behaves as desired. Such strategies must be based on

a thorough ph ysical understanding of the system at hand. Consequently, un-

deractuated control of underactuated ships is not the issue here, but simply

the control of underactuated ships. The research community emplo ying this

procedure is far smaller than the community working at solving the full state

1.2. ACADEMIC CONTROL APPR OACHES 7

cont rol problem, e.g. the true underactuated control problem. Publications are

sporadic and no coordinated eﬀort seems to be put into the work. Actually, the

term research comm unity is misleading in this sense. Nevertheless, promising

results have been achieved which justiﬁes further research to be carried out.

A work which can be characterized as belonging to this community is given in

(Godhavn 1996). The strategy is to disregard the heading in the control design.

Hence, what is attempted is controlling the position of the vessel along a desired

position trajectory by using a single screw propeller and a rudder. This means

that 2 DOF s is contro lled with 2 controls, thu s the con trol problem at hand is

not underactuated. Global exponential stability of the position error dynam ics

is achieved by using a backstepping approach. The subject of generating both

geometrically and dynamically feasible trajectories for the ship to track is also

considered. In his work, Godhavn assumes diagonal inertia and damping matri-

ces, but it is poin ted out that the control scheme is not implementab le by the

industry as is. A more detailed model has to be considered for this to happen.

To be noted is that no proof of what happens to the uncontrolled yaw (heading)

angle is given, only that the yaw rate is bounded b y assuming ph ys ical saturation

constraints on the inputs. This is actually a cruicial point when assessing the

feasibility of the approac h . If the position tracking entails that the ship at some

instance spins 180 degrees around and tracks the desired position backwards,

the eﬀort must be considered worthless from a practical point of view. Actually,

as is later discovered, this phenomenon happens occasionally. (Lefeber 2000)

uses this as an argument for justifying the consideration of full state con trol. Fi-

nally, Godhavn proposes to com b ine his tracking controller with the stabilizing

controller developed by Pettersen & Egeland (1996) to obtain both trajectory

tracking and point stabilization. Hence, it seems that Godhavn is not aware

that his smooth, time-invariant state feedback controller actually can be used

for both purposes since it does not attempt to control all DOFs simultaneously.

Accordingly, it does not fall victim to Broc kett’s stabilization condition. Fur-

ther details about this work can be found in (Godhavn 1997), where also the

adaptation of a constant environmental disturbance is considered in the con trol

design. (Godhavn, Fossen & Berge 1998) also elaborates on the position control

concept.

Later, in (Fossen, Godha vn, Berge & Lindegaard 1998), the 180 degree spin

problem is solv ed by requiring that the actuator providing the swa y force is

mounted at the bow of the ship. The idea behind this strategy was originally

proposed by Pinkster & Nienhuis (1986), where the motivation is to make it

possible to dynam ically position large tankers using a simple thruster layout

and a minimum amount of installed power. The proposed DP system does not

control the vessel heading directly, but the control scheme and actuator con-

ﬁguration suggested makes the ship behave as a weather vane. An additional

improvement by Fossen et al. (1998) when compared to (Godhavn 1996), which

also (Pinkster & Nienhuis 1986) employed, is that the position error is trans-

formed into a vessel-ﬁxed reference frame, hence it varies with the heading of

the ship.

Yet an amendment is obtained by the fact that the inertia and damping

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