Autonomous surveillance maneuvers for marine vessels in specially designated waters with a focus on the consecutive port and starboard turning cases

2.1 Underactuated 3DOF ship model

In this section, we organize the mathematical modeling for autonomous surveillance maneuvers based on an underactuated 3DOF model including surge, sway, and yaw motions. To improve error performance, we modify the damping terms in the vector representation. A well-known backstepping skill is considered to stabilize the error dynamics via virtual input. Then, an adaptive law treats both the unknown parameters and disturbances.

As shown in Figure 1, one can see two frames: body-fixed o_bx_by_bz_b(moving coordinate) and earth-fixed O_EX_EY_EZ_E.

Figure 1:

General coordinates (Body-fixed and Earth-fixed)

The origin o_b of the body-fixed frame is located at the center of gravity (CG). The body axes x_b, y_b, and z_b are chosen to coincide with the principal axes of inertia eh [12]. The (x, y, z) and its time derivatives in the frame are equivalent to the position and translational movement, whereas (ϕ, θ, ψ) and their derivatives indicate the rotational change. No heave (w), pitch (q), or roll (p) are presented under the assumption that these motions are small.

Then, assuming that the marine vessels have a homogeneous mass distribution and symmetry in the xz-plane (port/STBD symmetry) such that I_xy = I_yz = 0 [12], the kinematics can be summarized as a mass-damper-spring system, as shown in Equations (1) and (2).

where η and ν are the orientation/position vector and linear/angular velocity vector, respectively; M denotes the system inertia matrix for kinetic rigid-body mass (M_b) and for hydrodynamic added mass (M_a); C is the Coriolis-centripetal matrix including the added mass; D is the linear viscous damping matrix generated by skin friction, wave drift, and vortex shedding; D_n(v) is the nonlinear damping matrix of quadratic and high-order; g(η) is the hydrostatic term of the gravitational forces and moments. It will be ignored in this study, however; and A_ii (added mass) and B_ii (linear frequency-dependent damping) are the potential coefficients (i = 1, 2, 6) at the center of flotation. The hydrodynamic terms can be computed based on the potential theory, where it is assumed that the flow is constant, irrotational, and incompressible; T_surge, T_sway, and T_yaw are the time constants in the surge, sway, and yaw related to the linear viscous damping; τ_u and τ_r are the two actuators for surge and yaw; d_wi(i = 1, 2, 3) is the environmental disturbance in the directions of the surge, sway, and yaw, sequentially [12], an underactuated 3DOF marine vessel can be rearranged as

where

m₁₁=m-$$ X_{\dot{u}}$$, m₂₂=m-$$ Y_{\dot{v}}$$, m₃₃=I_z-$$ N_{\dot{r}}$$
W_u=[m₂₂ / m₁₁, d_u1 / m₁₁, d_u2 / m₁₁, d_u3 / m₁₁, d_u4 / m₁₁]
W_v=[m₁₁ / m₂₂, d_v1 / m₂₂, d_v2 / m₂₂, d_v3 / m₃₃, d_v4 / m₃₃]
W_r=[(m₁₁-m₂₂)/m₃₃, d_r1 / m₃₃, d_r2 / m₃₃, d_r3 / m₃₃, d_r4 / m₃₃]
f_u=[υr, -u, -u|u|, -u³, -u⁵]^T
f_v=[ur, -v, -v|v|, -v³, -v⁵]^T
f_r=[uυ, -r, -r|r|, -r³, -r⁵]^T

where $$ f_u\left(\dot{\eta },\eta \right)\in {\mathfrak{R}}^{n_u}$$, $$ f_{\nu }\left(\dot{\eta },\eta \right)\in {\mathfrak{R}}^{n_{\nu }}$$, and $$ f_r\left(\dot{\eta },\eta \right)\in {\mathfrak{R}}^{n_r}$$ are the known vectors; $$ W_u\in {\mathfrak{R}}^{n_u}$$, $$ W_{\nu }\in {\mathfrak{R}}^{n_{\nu }}$$, and $$ W_r\in {\mathfrak{R}}^{n_r}$$ represent the unknown vectors with the dimensions n_u, n_v, and n_r ; d_u2, d_u3, d_u4, d_v2, d_v3, d_v4, d_r2, d_r3, and d_r4 denote the unknown nonlinear damping [3][7]. As stated in the Introduction, we suggest a novel damping-like quintic polynomial form. Thus, we can expect a better performance of the error dynamics.

Remark 1. There are various types of environmental disturbances acting on marine vessels, such as longitudinal/beam waves, winds, ocean currents, and hydrodynamic interaction forces between two marine vessels. See details in previous reports [16]–[21].

Assumption 1. The environmental disturbances can be observed in specially designated waters. However, the accurate amounts of disturbance are unknown and immeasurable. Thus, it is reasonable to assume that the time-varying disturbances d_wi are bounded by the unknown constants d_{wu max}, d_{wv max}, and d_{wr max}, that is, |d_wu| ≤ d_{wu max}, |d_wv| ≤ d_{wv max}, |dwr| ≤ d_{wr max} [10].

Assumption 2. The sway velocity (v) is stable under passive boundedness. For the definition of the passive bound, see [10].

2.2 Control design

The basic framework of the control design is shown in Figure 2. One can see a virtual ship in front of it. The ideal (virtual) ship is defined as follows:

Figure 2:

Framework of model-based path following control between the virtual and own ship

where x_d, y_d, ψ_d, u_d, and r_d are the desired positions, yaw angle, forward speed, and turning rate of the virtual ship, respectively. Sailing routes can be generated using r_d .

Assumption 3. The surveillance routes are generated by virtual marine vessels that are similar to their own marine vessels, such that η_d = [x_d, y_d, ψ_d]^T and its first/second derivatives $$ {\dot{\eta }}_d$$, $$ {\ddot{\eta }}_d$$ are all bounded [10].

As stated in [7][10], let the error variables be x_e = x_d - x, y_e = y_d - y , and ψ_e = ψ_r - ψ , the azimuth angle ψ_r(ψ_r ∈(−π,π]) of the ship can be defined as

where $$ z_e \left(=\sqrt{x_e^2+y_e^2}\right)$$ is the error in the diagonal line and sgn(⋅) denotes a sign function with sgn(0) = 1 .

Then, let the error variables be x_e = z_ecos(ψ_r) and y_e= z_ecos(ψ_r), and the derivatives of tracking errors z_e and ψ_e are calculated as

To succeed in stabilization and tracking, backstepping can be designed recursively by applying the state variable as virtual control for surge ( α_u ) and yaw ( α_r ), as well as actual control inputs for surge ( τ_u ) and yaw ( τ_r ) are chosen as

where k_u1, k_r1, k_u2, and k_r2≥0 are the design parameters and z_m is a positive value. The virtual ship is situated in front of the ship by z_e-z_m; The function ϑ(⋅) satisfies Lemma 1 [10].

Lemma 1. The δ > 0 , there exists a smooth function ϑ(⋅) , such that ϑ(0)=0

Remark 2. As stated in [10], ϑ(ς)=(1/4δ)ς satisfying Lemma 1 is used in this study.

Finally, the parameter updating law and estimated disturbances are given as

where $$ {\widehat{W}}_i$$ is an estimate of W_i(i=u,r ); γ_wi1 ∈ ℜ^n_i×n_i (i=u,r ) is the positive definite matrix; γ_wi2>0 , γ_dwi1 , γ_dwi2 ≥0 (i=u,r ) is the weighting factors; W_u0 , W_r0 , d_{wu max 0}, and d_{wr max 0} are the initial design parameters [7][10].

Theorem 1. Consider a marine vessel in Equations (3) and (4) with Assumptions 1 to 3, the control law in Equations (13) and (14), and the adaptive law in Equations (16) to (19). All the signals in the closed-loop system are satisfied as being semiglobally uniformly ultimately bounded (SGUUB) [7].

Proof. See detailed in Appendix.

3.1 Surveillance routes and disturbances

Figure 3 shows the specially designated waters (SDW) near Mokpo. After the Sewol ferry disaster occurred, the KCG established SDW near the Mokpo and Tong-Yeong coastal seas. In addition, the SAR convention already specified sufficient maneuvers in search and rescue areas rather than limited zones. The International Convention for the Safety of Life at Sea, 1974) specifies the practical and necessary arrangements for surveillance of search and rescue services considering the density of seagoing traffic and navigational dangers. However, environmental situations such as islands, wrecks, passing ships, and fishing distribution are not considered. This study focuses on surveillance routes consisting of consecutive ports and STBD turning cases with sufficient sailing times rather than traffic distribution.

Figure 3:

Specially designated waters (SDW) near Mokpo

As seen in Figure 3, the surveillance maneuvers will be done by zig-zag pattern (A-B-C-A-D-B-C-A points). First, the rate of turn (ROT, r_d ) of the virtual ship generates the surveillance routes according to Equation (20). In order to understand the analysis of the results, we marked the turn numbers such as 1) to 8) in the turning states. It should be noted the proposed algorithm maintains the ship’s courses using ROT only, which is impractical for the current merchant ships depending on the rudder utilization.

where positive and negative values indicate predominant port and STBD turning, respectively. One can see the downsized sailing routes can be observed using Equation (20).

In addition, considering Remark 1 and Assumption 1, we assume that the environmental disturbances in the directions of surge, sway, and yaw are given as

Remark 3. The estimation of noisy sinusoidal signals is a significant problem for control system. It is known that the periodic disturbances can be expressed as the sum of bias (or offset), amplitude, frequency, and phase (or randomness). In practice, waves acting on marine vessels are nonstationary and unknown in advance. In particular, it is difficult to estimate the frequencies of unknown disturbances [22]. However, it is notable that the proposed control requires only boundaries of periodic disturbances, especially an upper bound. This study chooses the values in Equation (21) for a simple occurrence [23].

The initial conditions are [ x(0) , y(0) , ψ(0) , u(0) , v(0) , r(0) ] = [−50 m, 18 m, −0.15 rad, 0, 0, 0 rad/s]. The other parameters for the simulation are taken from [7][10]. The own ship is a patrol boat governed by Australian custom [1][15].

3.2 Results analysis

Autonomous surveillance maneuvers are performed within a sufficient time (1600 s). The consecutive port and STBD turning under severe disturbances may cause calculation problems in the underactuated system. In practice, officers on the bridge contemplate port turning in navigational situations such as head-on, crossing, and overtaking. In this simulation, we assume that there was no traffic distribution.

Table 1 shows the quantitative results of the maximum position errors (MPE) between surveillance routes (dashed blue) and trajectories (dashed red) during the port and STBD turning cases. From the initial position in Figure 4, own marine vessel navigates in the direction of the arrow until the vessel returns to the initial position. We numbered the each turning state in Equation (20). The MPE values in Table 1 were obtained using Equation (20). For example, a case of turn number 4) results in the two values such as 0.19 m for outer circle and 0.16 m for inner circle. The MPE of both port and STBD do not exceed 0.2 m in any turning situations. There are no significant differences between port and STBD turning maneuvers.

Table 1:

MPE during port and STBD turning cases

Figure 4:

Path trajectories of a patrol boat (red line) and its virtual one (blue) including consecutive port and STBD turning

Figure 4 depicts the path trajectories of a patrol boat (red dash) and its virtual one (blue dash and dot) including consecutive port and STBD turning. In the figure, the surveillance course is represented by the blue dash and dot while the trajectories obtained by the proposed algorithm are represented by the red dash. Thus, the red line always should be situated on the blue line as long as possible, regardless of environmental disturbances. Starting from the initial position (A point) in Figure 3, the own marine vessel bound for the east will turn to STBD in the first turning state r_d=−exp(0.005t/ 300) in 60 s, as seen in Equation (20). Then, a dubbin-type path can be seen from STBD turning (r_d = −0.05 ), and the ship stays her course. The ROT remains zero so that the patrol boat can navigate southwest. After arriving near point, the vessel turns to the port (r_d = 0.05t / 600 ) bound for the northeast. Likewise, the same step is performed until circular surveillance is carried out near point C. After surveillance from C to A, the vessel moves to point D. Similarly, consecutive STBD and port turning are performed until the marine vessel returns to its initial position.

Figure 5 illustrates a magnification of a part of the return position. The return position error (RPE) is almost within 2 m of the final destination. In addition, the RPE are 0.15–0.2 m between the surveillance routes and actual trajectories.

Figure 5:

Enlarged graph of Figure 4

Contrary to Figure 4, the position ( x_e, y_e, z_e ) and velocity errors ( u_e, v_e ) between one’s own vessel and the virtual vessel continue to fluctuate. This implies that the trajectories converge to an invariant set rather than an equilibrium. To analyze the performance of position and velocity errors, we compare the results of the proposed algorithm modified with quintic polynomial damping in Figures 6 and 7 for position errors and Figures 10 and 11 for velocity errors. The corresponding cases described in existing papers [6]-[10] without higher nonlinear damping are depicted in Figures 8, 9, 12, and 13. In addition, enlarged graphs such as Figures 7 and 11 for the current results by the authors and Figures 9 and 13 for the existing results by others [6]-[10] are presented. In addition to the transient state, the quantitative results of the position errors did not exceed 2 m in both the current and existing results.

Figure 6:

Position errors ( xe, ye, ze ) of an own ship model by proposed algorithm

Figure 7:

Enlarged graph of Figure 6

Figure 8:

Position errors ( xe, ye, ze ) of the existing dynamics in the research by others [6]-[10]

Figure 9:

Enlarged graph of Figure 8

Figure 10:

Velocity errors ( ue, ve ) of an own ship model by the proposed algorithm

Figure 11:

Enlarged graph of Figure 10

Figure 12:

Velocity errors ( ue, ve ) of the existing dynamics in the research by others [6]-[10]

Figure 13:

Enlarged graph of Figure 12

For example, the maximum error of existing results in x_e is 1.8 m in Figure 9 whereas the current result is 0.7–0.8 m in Figure 7 for steady-state responses. In particular, the improvement in error performance is dominant in u_e by the suggestion of quintic polynomial damping. The surge velocity errors are suppressed in Figure 11, whereas the corresponding existing cases fluctuate, as shown in Figure 13. Overall, the application of quintic polynomial damping results in lower and periodic responses in the error performance.

Some error variables can be adjusted by changing the control parameters. The position errors reveal that the marine vessel follows behind the virtual vessel within 1 m despite multi-sinusoidal disturbances in the surge, sway, and yaw directions.

Then, the actual corresponding inputs for surge (τ_u) and yaw (τ_r) are shown in Figures 14 and 16. The enlarged graph at the start of control activation can be seen in Figures 15 and 17. This study improves the error performance in u_e considerably. However, the high peak in surge control load in Figure 14 and its corresponding chattering in the transient state (see Figure 11) are somewhat disappointing because the path following control uses constant forward speed at start of simulation. This means that a careful trade-off is necessary to obtain a better performance and lower chattering. Otherwise, the sliding mode control with adaptive super-twisting is recommendable to treat such a problem [17][18]. Finally, it can be seen that the goals of consecutive port and STBD turning maneuvers depending on ROT are reached for autonomous surveillance of marine vessels in the SDW.

Figure 14:

Surge control input (τu) by the proposed algorithm

Figure 15:

Enlarged graph of Figure 14 at the start of the control action

Figure 16:

The yaw control input (τr) by the proposed algorithm

Figure 17:

Enlarged graph of Figure 16 at the start of the control action