Khac Duc Do

Department of Mechanical Engineering, Curtin University, Kent Street, Bentley, WA 6102, Australia

Path-tracking Control of Underactuated Ships Under Tracking Error Constraints

Khac Duc Do*

Department of Mechanical Engineering, Curtin University, Kent Street, Bentley, WA 6102, Australia

This paper presents a constructive design of new controllers that force underactuated ships under constant or slow time-varying sea loads to asymptotically track a parameterized reference path, that guarantees the distance from the ship to the reference path always be within a specified value. The control design is based on a global exponential disturbance observer, a transformation of the ship dynamics to an almost spherical form, an interpretation of the tracking errors in an earth-fixed frame, an introduction of dynamic variables to compensate for relaxation of the reference path generation,p-times differentiable step functions, and backstepping and Lyapunov’s direct methods. The effectiveness of the proposed results is illustrated through simulations.

underactuated ship; path-tracking; error constraint; Lyapunov method; backstepping method

1 Introduction1

The main difficulty with controlling an underactuated ship is that only the yaw and surge axes are directly actuated while the sway axis is not. An application of Brockett’s theorem (Brockett, 1983) shows nonexistence of pure-state feedbacks that are able to asymptotically stabilize an underactuated ship at a fixed point. Thus, the stabilization problem is often solved by either discontinuous or time-varying feedback (e.g., (Reyhanoglu, 1997; Pettersen and Egeland, 1996; Aguiar and Pascoal, 2001; Mazencet al., 2002; Doet al., 2002b)).

A global exponential position tracking system without controlling the ship’s yaw angle was proposed in (Godhavnet al., 1998). In (Pettersen and Nijmeijer, 2001), a high-gain, local exponential tracking result was obtained based on the work in (Jiang and Nijmeijer, 1999). Based on cascade and passivity approaches, several global tracking results were obtained in (Lefeberet al., 2003; Jiang, 2002). Note that in (Jiang, 2002; Lefeberet al., 2003; Pettersen and Nijmeijer, 2001), the yaw velocity was required to be nonzero, i.e., a straight-line cannot be tracked. This restrictive assumption was removed in (Doet al., 2002a; 2002b; Lee and Jiang, 2004), where various relaxations on the reference trajectory and ship dynamics were made, see also (Chwa, 2011) for a solution to the tracking problem with input constraints. An assumption of low speed (nonlinear damping terms are ignored) is usually made in the above works due to the complex generation of the reference trajectories, and difficulties in stability analysis (especially stability analysis of the sway dynamics).

There are three main approaches to path-following control of ships. In the first approach, the Serret-Frenet frame is used to define the path-following (cross-track and yaw angle) errors, then the yaw moment control input is designed to stabilize these errors at the origin (e.g., (Skjetne and Fossen, 2001; Encarna??oet al., 2000; Do and Pan, 2004; Liet al., 2009) for nonlinear (curved) paths, (Pettersen and Lefeber, 2001; Fredriksen and Pettersen, 2006; Moreiraet al., 2007) for linear (straight) paths). This approach results in local results (except for the linear path) due to singularity in the cross-track error dynamics. The second approach defines the path-following objective as one of controlling the vessel so that it is in the tube of nonzero diameter centered on the path, and moves along the path with a desired speed (e.g., (Aicardiet al., 2001; Doet al., 2004; Liet al., 2008)). The control design aims to force the vessel to follow a virtual point moving along the path. This approach requires the vessel not be too close to the path. The third approach (referred to as path-tracking) is based on a combination of trajectory-tracking and path-following in the first approach. In the sense that the lateral path-following error is not always set to zero (to avoid singularity) and that the path parameter is used as an additional control to stabilize the lateral path-following error. Thus, global control results are often obtained (e.g., (Lapierre and Jouvencel, 2008; Do and Pan, 2006; Ghommamet al., 2008)).

In all of the above works on trajectory-tracking and path-following control of underactuated ships, a hard constraint on the tracking/following errors has never been addressed. This problem is important since in practice it is desired to steer the ship to be within a certain distance from the reference path, especially in narrow waterways. Moreover, various conditions on the control gains and reference paths/trajectories were imposed in the existing mentioned works to ensure boundedness of the sway velocity instead of being directly controlled in the previous control designs. The above issues motivate contributions in this paper on new controllers for asymptotic path-tracking ofunderactuated ships under constant or slow time-varying sea loads, and a hard constraint on position tracking errors. The method does not require the reference path be generated by a virtual ship. The sway velocity is directly controlled during the control design. First, a disturbance observer is proposed to globally exponentially estimate the sea loads. Second, a primary control surge force is designed to transform the ship dynamics to those of an almost spherical ship. Third, the trajectory tracking errors are represented in the earth-fixed frame and are stabilized at the origin by a design of controllers based on backstepping and Lyapunov’s direct methods. A dynamical variable is introduced to the reference yaw angle during the control design to compensate relaxation of the reference path generation.

2 Problem statement

2.1 Equations of motion

Assume that the ship has an xz-plane of symmetry; heave, pitch and roll modes are neglected; the body-fixed frame coordinate origin is set in the center-line of the ship. Then, the mathematical model of an underactuated ship moving in a horizontal plane can be described as (Fossen, 2011):

where η=col(x, y ,ψ) with (x, y) being the (surge, sway) displacements of the center of mass, and ψ being the yaw angle of the ship coordinated in the earth-fixed frame OEXEYE, see Fig. 1;v=col(u, v, r )denotes the surge, sway, and yaw velocities of the ship coordinated in the body-fixed frame ObXbYb; θ=col(θ1,θ2,θ3) denotes the sea loads on the ship along the surge, sway, and yaw axes coordinated in the earth-fixed frame; τ=col(τu,0,τr) denotes the control inputs: the surge force τuand yaw moment τr; and

In (2), (m11,m22) denote the masses including added masses in the surge and sway axes; m33is the inertia including added inertia in the yaw axis; the damping functions f1( u), f2( v) and f3( r) are

with

where n is an integer larger than 1, dui, dviand driwith i=1,2,... denote the damping coefficients in the surge, sway, and yaw axes, we use ·tanh(·/ε0) with ε0being a small positive constant to smoothly approximate |·|.

Fig. 1 Definition of coordinate systems and motion variables

Remark 2.1The mathematical model (1) holds for underactuated ships equipped with two main aft propellers or water jets because the control momentrτ does not directly enter the sway dynamics. For ships equipped with a rudder, the control forcerτ does directly enter the sway dynamics. Moreover, the off-diagonal terms in the matrixMand the coupling terms in the damping functions1()f u,2()f v, and3()f r are neglected because these terms are relatively small in comparison with11(m,22m,33)m, and those terms already included in1(()f u,2()f v,3())f r, respectively. In the case of ships equipped with a rudder and off-diagonal terms not negligible, the coordinate transformations proposed in (Do and Pan, 2005) and (Do, 2010b) obtain a mathematical model similar to (1). Basically, these coordinate transformations ensure that displacements of a point referred to as the ship’s center of oscillation (similar to the case treated in control of aircraft in (Martin et al., 1996) and (Do et al., 2003)) are controlled instead of displacements of the center of mass of the ship.

2.2 Control objective

In this paper, we study a path-tracking control objective under the following assumptions.

Assumption 2.1

1) The reference path G(s)=col(xd(s),yd(s ))with s being the path parameter is four-times differentiable with respect to s and satisfies

Remark 2.2

1) Item 1) of Assumption 2.1 implies that the reference path is sufficiently regular. If the reference path contains several singular points, then we can split it into several nonsingular reference paths and consider each of them separately.

3) Item 3) means that the position of the ship is within the constrained distance from the reference path at the initial time0t. Indeed, if the control problem of forcing the ship to asymptotically track the reference path without a distance constraint from the reference trajectory is of interest, one can setNequal to zero. Moreover, if we are only interested in tracking constraint either along theOEXE-axis orOEYE-axis, the matrixNcan be set equal to diag(N1,0) or diag(0,N2) withN1andN2being positive constants.

Control Objective 2.1Design the control inputsuτandrτ, and estimate laws for the loads1θ,2θand3θso that the following objectives are achieved:

1) The position tracking errorsqe(t)=col(x(t)-xd(s(t)),y(t)-yd(s(t)))are always within the constrained distance from the reference path, i.e.,

2) The ship asymptotically tracks the reference pathG(s) in the sense that the ship is on the path, and moves forward along the path tangentially with a desired total linear velocity?d(t) coordinated in the earth-fixed frame. The velocity?d(t) is supposed to be sufficiently regular and sufficiently large to handle sea loadsθ1andθ2.

3 Preliminaries

3.1 Smooth saturation function

Definition 3.1The function ()xσis said to be a smooth saturation function if it is smooth and possesses the properties:

1)σ(x)=0, ifx=0,σ(x)x＞0 ifx≠0.

2)σ(-x)=-σ(x) and (x-y)[σ(x)-σ(y)]≥0.

For the vectorx=col(x1,...,xn), the notationσ(x)=col(σ(x1),...,σ(xn))is used to denote the smooth saturation function vector of the vectorx.

3.2p-times differentiable step function

The following lemma gives a method to construct ap-times differentiable step function.

Lemma 3.1Let the scalar functionh(x,a,b) be defined as

withaandbbeing constants such that ＜0＜ab, and the function ()f ybeing defined as follows:

Then the functionh(x,a,b) is ap-times differentiable step function. Moreover, ifg(y) in (9) is replaced byg(y)=e-1/ythen property 4) is replaced by 4’), i.e.,h(x,a,b) is a smooth step function.

Proof. See (Do, 2010a).

4 Exponential disturbance observer

5 Transformation of ship dynamics to an almost spherical form

As discussed in Section 1, we transform the ship dynamics (1) to an almost spherical form by applying the following primary surge force feedback

where

6 Control design

6.1 Step 1

Let us define

We define the tracking errors

Differentiating both sides of (18) along the solutions of (17) and the first two equations of (14) gives

We define

The functionγis designed such that it penalizes the position tracking errors between the reference and actual trajectories of the ship. This function is chosen as follows:

whereσ(·) is the smooth saturation function vector of the vector · defined in Subsection 3.1,K1=diag(k11,k12) withk11andk12being positive constants. The functionβneeds to be nonnegative definite when the ship is within the constrained distance from the reference path, be equal to zero when the position tracking errors are equal to zero, and be equal to infinity when the ship reaches the constrained distance from the reference path. We propose the functionβas follows

Differentiating both sides of (21) along the solutions of (20) and (19) gives

andΛd(t) is a bounded, positive and twice differentiable function of timetand satisfies the following condition:

On the other hand, substituting (27) into (19) gives

6.2 Step 2

Define

whose derivative along the solutions of (37) and (34) is

We will prove in Appendix 9 thatδd(t) is bounded. Substituting (42) into (41) yields

Moreover, substituting (27) and (42) into (19), and (40) and (42) into (37) gives6.3 Step 3

whose derivative along the solutions of (43) and (45) is

Moreover, substituting (48) into (45) yields

The control design has been completed. We present the main results in Theorem 6.1.

1) The closed-loop system consists of (10), (29), (42), (44), and (50) is forward complete.

2) The ship is always within the constrained distance from the reference trajectory, i.e., the inequality (7) holds for allt≥t0≥0. This does not depend on the convergence of the disturbance observer as the stability analysis is carried out for all signals of the closed-loop system.

4) The functionδd(t) generated by (42) is bounded for allt≥t0≥0.

5) The desired total linear velocity coordinated in the earth-fixed frame is obtained by specifying the functionΛd(t) in (29).

Proof. See Appendix A.

7 Simulations

To demonstrate the performance improvement of the proposed path-tracking controller in this paper over the existing results, we perform a simulation on the trajectory-tracking controller proposed in (Doet al., 2002a). We do not provide a simulation on the path-following controllers proposed in (Skjetne and Fossen, 2001; Encarna??oet al., 2000; Do and Pan, 2004; Liet al., 2009), for example, because these controllers are local as mentioned in Section 1. The framework of the trajectory-tracking control design for an underactuated ship is described in Fig. 4, where the control objective is to design the controlsτuandτrto force the real ship to track the virtual ship. In Fig. 4, (xd,yd,ψd) represent position and orientation of the virtual ship with respect to the Earth-fixed frameOEXEYE, (ud,vd,rd) are velocities of the virtual ship with respect to the virtual ship body-fixed frameOdXdYd. The virtual ship “dynamics" are given by

where

Fig. 2 Simulation results with tracking error constraint

Fig. 3 Simulation results without tracking error constraint

Fig. 4 Trajectory-tracking control design framework

Fig. 5 Trajectory-tracking control design results

To generate (xd,yd,ψd), we specifyud=4 m/s and the profile of (xd,yd) in the Earth-fixed frame, i.e., the sinusoidal form of the aforementioned reference path. From these specifications, the reference inputsτudandτrd. These in turn determine the reference trajectory (xd,yd,ψd). The errors in position and orientation between the real and virtual ships projected to the body-fixed frameObXbYbare denoted by (xe,ye,ψe). Thus, the trajectory-tracking control objective becomes the one of stabilizing the errors (xe,ye,ψe) at the origin, see (Doetal., 2002a) or (Lefeberet al., 2003; Doet al., 2002b; Lee and Jiang, 2004; Chwa, 2011) for details of trajectory-tracking control designs. The control gains are tuned so that the transient response time is almost the samewith the one simulated using the controller proposed in this paper for a fair comparison. The simulation results are plotted in Fig. 5, where the position and orientation tracking errors are plotted in Figs. 5(b) and 5(c); the norm of position tracking error is plotted in Fig. 5(d); the velocity tracking errors are plotted in Figs. 5(e) and 5(f); and the control inputs are plotted in Figs. 5(g) and 5(h). It is seen that the trajectory-tracking controller in (Doet al., 2002a) results in fairly large (steady state) tracking errors and more importantly the transient tracking error norm is much larger than the one proposed in this paper (supt≥0de(t)≈32.3 vs supt≥0de(t)≈20.6), see Fig. 5(d) vs Fig. 2(d). The large tracking errors are due to the fact that the controller in (Doet al., 2002a) was designed for the case without disturbance and nonlinear damping terms, with no hard constraint on tracking errors. It is noted that the trajectory-tracking controllers proposed in (Lefeberet al., 2003; Doet al., 2002b; Lee and Jiang, 2004; Chwa, 2011), for example, will give a similar transient response.

8 Conclusions

A constructive design of new controllers has been developed for path-tracking control of underactuated ships under sea loads and tracking error constraints. The keys to the successful control design include 1) a global exponential disturbance observer, 2) transformation of the ship dynamics to those of an almost spherical ship to almost decouple linear and angular motions of the ship, 3) the use of backstepping and Lyapunov’s direct methods to stabilize the tracking errors expressed in the earth-fixed frame, and the introduction of an auxiliary function for compensation of relaxing the reference path generation. Future work will design an inverse optimal path-tracking controller for underactuated ships and path-tracking controllers for underwater vehicles based on the method proposed in this paper and the one in (Do, 2015).

Acknowledgments

The author would like to express his sincere gratitude to the Editor in Chief and the reviewers for their helpful comments. The work presented in this paper was supported in part by the Australian Research Council under grant DP0988424.

Appendix A: Proof of Theorem 6.1

A.1 Forward completeness of the closed-loop system

A.2 Ship within the constrained distance

Since we have already proved that the closed-loop system is forward complete, we now can consider the closed-loop subsystem consisting of (12), (44) and (50) separately from the rest of the closed-loop system. As such, we consider the following Lyapunov function candidate

where

A.3 Asymptotic convergence of tracking errors to zero

A.4 Boundedness ofδd(t)

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10.1007/s11804-015-1329-3

1671-9433(2015)04-0343-12

Received date: 2015-03-27.

Accepted date: 2015-07-17.

Foundation item: Supported in Part by the Australian Research Council Under Grant No. DP0988424.

*Corresponding author Email: duc@curtin.edu.au

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015