Adaptive Event-Triggered Control Strategy for Ensuring Predefined Three-Dimensional Tracking Performance of Uncertain Nonlinear Underactuated Underwater Vehicles

This paper presents an adaptive event-triggered control strategy for guaranteeing predefined tracking performance of uncertain nonlinear underactuated underwater vehicles (UUVs) in the three-dimensional space. Compared with the related results in the literature, the main contribution of this paper is to develop a nonlinear error transformation approach for ensuring predefined three-dimensional tracking performance under the underactuated property of 6-DOF UUVs and limited network resources. A nonlinear tracking error function is designed using a linear velocity rotation matrix and a time-varying performance function. An adaptive event-triggered control scheme using the nonlinear tracking error function and neural networks is constructed to ensure the practical stability of the closed-loop system with predefined three-dimensional tracking performance. In the proposed control scheme, auxiliary stabilizing signals are designed to resolve the underactuated problem of UUVs. Simulation results are presented to illustrate the effectiveness of the theoretical methodology.


Introduction
Control of nonlinear underwater vehicles is attracting much research attention recently, due to its practical application in submarine survey, exploration, oceanographic mapping, and region search for deep-sea wrecks [1][2][3][4][5][6]. The initial studies have focused on the planar or depth control design of underwater vehicles in the two-dimensional space [7][8][9][10][11]. However, these studies in the two-dimensional space provide limited solutions to various tracking problems in the practical three-dimensional underwater environment. For more practical application, three-dimensional control approaches have been studied for nonlinear underwater vehicles described by 5-degrees-of-freedom (5-DOF) or 6-DOF kinematics and dynamics. In [12], a path following controller was designed for 5-DOF underwater vehicles with ocean current disturbances. A ocean current observer to detect an external current was designed for trajectory tracking of 5-DOF underactuated underwater vehicles (UUVs) [13]. In [14,15], fuzzy-based or neural-network control techniques were developed for uncertain 5-DOF UUVs with external disturbances. To deal with uncertain 6-DOF models with the roll motion, three-dimensional trajectory tracking control designs were developed using several control techniques such as backstepping control [16,17] and sliding mode control [18]. However, in the aforementioned results, the transient and steady-state performance metrics of tracking errors cannot be designed a priori. To preselect the tracking performance metrics of underwater vehicles, the prescribed performance design technique [19] has been combined with the control methodologies of underwater vehicles [20][21][22][23]. In [21], a region tracking controller with predefined transient performance was designed for fully actuated underwater vehicles in presence of an ocean current and a thruster fault. Neuralnetwork-based prescribed performance control designs were investigated for uncertain

Problem Formulation
The kinematics for the position and attitude of an UUV can be described bẏ where η = [x, y, z] ; x, y, and z are the positions of the center of gravity in an inertial coordinate frame, ζ = [φ, θ, ψ] ; φ, θ, and ψ denote roll, pitch, and yaw angles, respectively, υ = [u, v, w] ; u, v, and w are surge, sway, and heave velocities, respectively, and ω = [p, q, r] ; p, q, and r denote the roll, pitch, and yaw angular velocities in the body-fixed frame, respectively. Here, the linear velocity rotation matrix R 1 (ζ) and and the angular velocity transformation matrix R 2 (ζ) are given by with s (·) = sin(·), c (·) = cos(·), and t (·) = tan(·). The structure of the neutrally buoyant UUV concerned in this paper is depicted in Figure 1. The dynamics of the UUV is given by where M = M 1 + M 2 ; M 1 ∈ R 6×6 and M 2 ∈ R 6×6 are the matrix of the rigid-body mass and the added mass, respectively, C(υ, ω) ∈ R 6 is a vector derived by the Coriolis and damping matrices, G(ζ) ∈ R 6 is a vector induced from the gravitation and the buoyancy of the UUV, and τ = [τ X , τ Y , τ Z , τ K , τ M , τ N ] is a vector denoting the control forces and moments. In this paper, the torpedo-shaped UUV model is considered to deal with the tracking problem. The torpedo-shaped UUV cannot move directly to the yor z-direction in the body reference frame and the roll movement is undesirable in the practical UUV [34]. Thus, the underactuated torque vector τ = [τ X , 0, 0, 0, τ M , τ N ] is considered in this paper. The detailed definitions of M, C, and G are presented in Appendix A. For more details for the model of UUVs, see [35,36]. Assumption 1. The nonlinear function vectors C(υ, ω) and G(ζ) are unknown for the control design.

Assumption 2.
The desired three-dimensional trajectory η d ∈ R 3 and its derivativesη d ∈ R 3 andη d ∈ R 3 are bounded.

Problem 1.
Our problem is to design an adaptive event-triggered control law τ for ensuring predefined three-dimensional tracking performance of the uncertain UUV described by (1) and (2) so that the position trajectory η of the UUV follows the desired trajectory η d in the three-dimensional space.

Adaptive Event-Triggered Control with Predefined Three-Dimensional Tracking Performance
In this section, an adaptive event-triggered control methodology using an error transformation function and stabilizing auxiliary signals is established to ensure predefined three-dimensional tracking performance of the UUV. The dynamic surface design procedure using the predefined performance concept is derived step by step.
Step 1: Let us consider the kinematics (1) and define the position errors s = [s 1 , s 2 , s 3 ] = η − η d . Then, to ensure predefined three-dimensional tracking performance under the underactuated property, we define the nonlinearly transformed error surface Γ 1 = [Γ 1,1 , Γ 1,2 , Γ 1,3 ] as , and ρ = [ , 0, 0] denotes the radius of error surface with the design constant . Here, the design constant is selected relatively small compared to the length of the UUV, and Φ i , i = 1, 2, 3, is defined as where 0 < δ 1,i ≤ 1 and 0 < δ 2,i ≤ 1 are design constants, and . From the definition of R 1 (ζ), there exists a constantR 1 such that R 1 (ζ) ≤R 1 . Then, from Γ 1,i ∈ L ∞ , Θ i are bounded where i = 1, 2, 3. Thus, there exist constants Θ i andΘ i such that Θ i < Θ i (t) < Θ i , ∀t ≥ 0. Using the bijective property Φ i : (−δ 1,i , δ 2,i ) → (−∞, ∞) [37], it holds that and ρ are combined with the nonlinear error function vector Φ in order to design the underactuated control scheme with the predefined three-dimensional tracking performance. From Lemma 1, the boundedness of the error surface vector leads to the satisfaction of the inequality −δ 1,i µ i (t) < s i (t) < δ 2,i µ i (t) for all t ≥ 0 where i = 1, 2, 3. That is, the bounds of the transient and steady-state performance of the position errors s i (t) can be predefined by selecting the design parameters δ 1,i , δ 2,i , and functions µ i (t). Thus, the predefined three-dimensional tracking performance is ensured provided that Γ 1,i ∈ L ∞ . Accordingly, the primary focus of this study is to design an adaptive event-triggered control scheme for ensuring the boundedness of Γ 1,i .
The time derivative of Γ is represented bẏ Then, we have Using (6), we obtain thaṫ where H = R −1 1 AR 1 and Here, H m,n means the (m, n) element of the matrix H.
Using the dynamic surface design concept [38], we define the error surface vector e = [e u , e q , e r ] with e u = u −ᾱ u , e q = q −ᾱ q , and e r = r −ᾱ r , and the boundary layer is the virtual control vector and α = [ᾱ u ,ᾱ q ,ᾱ r ] is the filtered signal vector of virtual control laws α u , α q , and α r that is obtained by the first-order low-pass filter where ξ > 0 is the small constant.
Using the error surface vector e and the boundary layer error c, (7) becomeṡ The virtual control vector α = [α u , α q , α r ] is presented as where Substituting (10) to (9) giveṡ We choose a Lyapunov function V 1 = Γ 1 Γ 1 /2. Then, the time derivative of V 1 is represented byV where Γ 1 KΓ 1 = 0 due to the skew symmetric matrix K.
Using (2), the time derivative of Γ 2 is obtained aṡ For the online approximation of unknown nonlinear function vector F( , ζ), radial basis function neural networks [39] are employed. Then, F can be approximated over the compact set Υ as follows wherex = [ , ζ ] ∈ Υ ⊂ R 6 denotes the input vector of radial basis function neural networks, the optimal weighting matrix W * is defined as . . , 6, and ε ∈ R 6 is a reconstruction error vector such as ε ≤ε with an unknown constantε > 0.
By substituting (19) and (20) into (24) and using Lemma 2,V 2 becomeṡ Remark 2. In the dynamics (2) of the torpedo-shaped UUV, the underactuated control torque vector τ = [τ X , 0, 0, 0, τ M , τ N ] should be designed. That is, the first, fifth, and sixth dynamic equations in (2) only have the control torques τ X , τ M , and τ N , respectively. Thus, the the auxiliary stabilizing signals are required for the state equations for v, ω, and p (i.e., the second, third, and fourth dynamic equations in (2)). In this study, the auxiliary stabilizing signals β 1 , β 2 , and β 3 in (18) are presented to design the underactuated control torque vector τ = [τ X , 0, 0, 0, τ M , τ N ] while ensuring the predefined three-dimensional tracking performance and the stability of the closed-loop system. Because of these auxiliary stabilizing signals, the UUV dynamics (2) is stably controlled by using the only three control inputs τ X , τ M , and τ N .
The three-dimensional tracking result is shown in Figure 2. In Figure 2, the UUV follows the desired trajectory with good performance. Figure 3 depicts the position tracking errors. Figure 3 reveals that the time responses of the position errors s i remain within the predefined time-varying performance bounds −δ 1,i µ i and δ 2,i µ i for all t ≥ 0. The mean square errors of the position errors s i at the steady-state response are presented in Table 1 where the steady-state response is set to the position errors s i (t) for t ≥ 10 s. The outputs of the neural networks are displayed in Figure 4. The event-triggered underactuated control inputs for the UUV are shown in Figure 5. Figure 6 shows the triggered time intervals and the cumulative number of events of the proposed event-triggered control laws. The number of total events of the proposed event-triggered tracker is 2164. Thus, the data required for implementing the proposed tracker are only 18.03% of the total sampled data 12,000 during 60 s. This implies that the proposed event-triggered control scheme can save the signal transmission burden. From these results, we can see that predefined three-dimensional tracking performance under the underactuated property of the uncertain nonlinear 6-DOF UUV can be achieved by the proposed adaptive event-triggered tracking methodology. Table 1. Mean square errors of s 1 (t), s 2 (t), and s 3 (t) at the steady-state response.

Conclusions
We presented an adaptive event-triggered control method for ensuring predefined three-dimensional tracking performance of uncertain nonlinear 6-DOF UUVs. The nonlinearly transformed error function and the auxiliary stabilizing signals were derived for achieving predefined three-dimensional tracking performance while overcoming the underactuated problem of the nonlinear 6-DOF dynamics. It was shown that the adaptive event-triggered tracker using neural networks achieves the practical stability and predefined tracking performance of the closed-loop system. The main contribution of this work to the event-triggered control of uncertain nonlinear 6-DOF UUVs is that the predefined three-dimensional tracking performance under the underactuated dynamics can be admitted than the conventional event-triggered techniques of underwater vehicles in the two-dimensional space.
C(υ, ω) =[C 1 , . . . , C 6 ] C 1 =X u|u| u|u| + (X wq − m)wq + (X qq + mx g )q 2 + (X vr + m)vr + (X rr + mx g )r 2 − my g pq − mz g pr C 2 =Y v|v| v|v| + Y r|r| r|r| + my g r 2 + (Y ur − m)ur + Y uv uv + (Y wp + m)wp + (Y pq − mx g )pq + my g p 2 + mz g qr C 3 =Z w|w| w|w| + Z q|q| q|q| + (Z uq + m)uq + (Z vp − m)vp + (Z rp − mx g )rp + Z uw uw + mz g (p 2 + q 2 ) − my g rq C 4 =K p|p| p|p| − (I zz − I yy )qr + my g (uq − vp) − mz g (wp − ur) C 5 =M w|w| w|w| + M q|q| q|q| + (M uq − mx g )uq + (M vp + mx g )vp + M rp − (I xx − I zz ) rp + mz g (vr − wq) + M uw uw C 6 =N v|v| v|v| + N r|r| r|r| + (N ur − mx g )ur + (N wp + mx g )wp + N pq − (I yy − I xx ) pq − my g (vr − wq) + N uv uv where m is the mass of the UUV, x g , y g , and z g mean the center of gravity of the UUV, x b , y b , and z b represent the center of buoyancy of the UUV, the vehicle weight and the vehicle buoyancy of the UUV are defined as W and F, respectively, I xx , I yy , and I zz denote the inertia tensors of the UUV, Xu, Yv, Yṙ, Zẇ, Zq, Kṗ, Mẇ, Mq, Nṙ, and Nv are the added mass parameters of the UUV, X vr , X wq , X qq , X rr , Y ur , Y wp , Y pq , Y uv , Z uq , Z vp , Z rp , Z uw , M uq , M vp , M rp , M uw , N ur , N wp , N pq , and N uv indicate the parameters of the added mass cross term of the UUV, X |u|u is the axial drag parameter of the UUV, and the cross flow drag parameters of the UUV are defined as Y |v|v , Y |r|r , Z |w|w , Z |q|q , K |p|p , M |w|w , M |q|q , N |v|v , and N |r|r .