Adaptive Integral Sliding Mode based Path Following Control of Unmanned Surface Vehicle

This paper investigates the path following control problem for a unmanned surface vehicle (USV) in the presence of unknown disturbances and system uncertainties. The simulation study combines two different types of sliding mode surface based control approaches due to its precise tracking and robustness against disturbances and uncertainty. Firstly, an adaptive linear sliding mode surface algorithm is applied, to keep the yaw error within the desired boundaries and then an adaptive integral non-linear sliding mode surface is explored to keep an account of the sliding mode condition. Additionally, a method to reconfigure the input parameters in order to keep settling time, yaw rate restriction and desired precision within boundary conditions is presented. The main strengths of proposed approach is simplicity, robustness with respect to external disturbances and high adaptability to static and dynamics reference courses without the need of parameter reconfiguration.


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With the growing advancement in the sensor technology and navigation aids, USVs

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The motion of the USV is shown in Figure 1, where a six degrees of freedom (DOF) model is presented. The earth fixed Oo is an inertial reference frame fixed to the earth's surface and the body fixed with origin O is a moving coordinate frame that it is fixed to the craft as in given in [1]. It is assumed an homogeneous mass distributed and xz-plane symmetrical, such that origin of the body fixed reference frame is chosen to be coincident with the center of the gravity. If we consider the path following problem the dynamics of heave, roll, and pitch can be neglected, so that the reduced model dynamics are given as where m is the mass, u is the surge velocity, v the sway velocity, r the yaw rate, I z the 105 rotational inertia with respect to z axis, x c is the x coordinate of the vehicle center in the 106 fixed body reference frame and X,Y and N are the external forces and moments with 107 respect to the surge, sway and yaw, respectively.
Assumption of constant forward speed and using the ship's Norrbin nonlinear mathematical model, see [43], implies that the steering equations of motion can be obtained asψ where, ψ(t) is the yaw (orientation) angle, r(t) is the yaw rate, δ(t) is the rudder angle (the control variable to be designed) and d(t) is an unknown term to be compensated that includes parametric uncertainty and external disturbances (wind, waves, mobile loads). The dynamics functions are given as where (K, T) are hydrodynamic coefficients and (a 1 , a 2 ) are Norrbin coefficients.

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In the path following problem it is required that the yaw angle ψ follows a refer-

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A dynamic reference model is used, in this work, to generate the desired course (ψ r (t),ψ r (t),ψ r (t)).

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The objective is to design a control law that creates overdamped responses with minimal overshooting (undershooting) and robustness properties for response of the yaw error, which is defined as In order to check the control performance of the proposed controller for the path following problem, we consider the following performance analysis indices mentioned in [12,14].
where τ is the sampling time used in the simulation. scenarios from a specific parameter configuration.

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The following theorem is introduced in order to analyse the stability properties of 132 the AISM proposed solution.

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The dynamics of (10) when z 2 = 0 arė If systems (12) and (11) are globally uniformly asymptotically stable (GUAS) and we know a C 1 Lyapunov function V(t, z 1 ), two class-K ∞ functions ϕ 1 and ϕ 2 , a class-K ϕ 3 function and a positive semidefinite function W(z 1 ) such that ∂V ∂t Besides, for each fixed z 2 there exists a continuous function ζ : Then we can conclude that the cascade system (10) and (11) is GUAS.

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Derivation of e(t) in (6) leads tȯ An adaptive sliding surface s(t) variable is defined as with λ(e) a real positive time varying parameter. Consider the integral terms(t)s Let's choose the control law as with λ(e) defined as , the variable z(t), related a new sliding surface, defined as and with the parameters α(s,s), γ(α) and δ(e) given aṡ Derivation of γ(α) and λ(e) are given aṡ The control algorithm is designed by an appropriate selection of the parameters 140 λ max , λ min , α(0), κ, δ max and δ min , as it will be introduced in the numerical simulations 141 section.

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Theorem 2. Consider the ship course dynamics described in (4) that complies with assumption 1. The application of the control law (21) to dynamic system (4) implies that the closed compact set Ω e defined as is GUAS with µ, θ and ϑ given as Proof. Application of control law (21) to dynamic system (4) creates the following cascade system.ė The dynamics ofė(t) when s(t) = 0 (dynamics of the yaw error at the sliding condition) areė with λ > 0. Therefore system (35) is GUAS, with exponential convergence.

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This implies that (34) is GUAS with respect to the closed set Ω z . Note that dynamics 145 in (36) can be viewed as a second order linear dynamics with adaptive critical damping 146 (exponential convergence related to the fastest response with no overshooting), being 147 perturbed by the overestimation ρ z s caused by the compensation of the unknown term.

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Inside Ω z we have that which geometrically entails: with ϑ defined in (32).

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Applying Theorem 1 with  In this section we introduce numerical simulations of the path following problem 159 with parameters given in Table 1 and being executed under the following assumption.    degrees per second and a required precision ϵ = 1.0e − 3.

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• The value of α(0) is obtained assuming an exponential convergence of the error from initial condition e(0) to desired precision ϵ with a desired settling time t s • The value of λ min is related to the initial conditions of the problem and the maximum desired yaw rate as and λ max is calculated as • The value of κ must be higher thanḋ max in order to obtain a small value for µ. Because of the low-pass filtering properties of (36), the value ofḋ max can be further refined by estimating the cross over frequency ω c (t) of the second order system related to s(t) Therefore κ is calculated as an adaptive gain that takes account of ω c and the desired precision • The values of δ min and δ max are related by means of the condition δ max = 2.0δ min . The value of δ min is adjusted with simulations such that the value of the performance index MIA is equal, at the end of test time, to the value obtained with benchmark selected controllers. This choice leads to the following numerical values of δ min and δ max δ min = 1.76952 (48) This condition generates an adequate adaption of the value of δ that allows to obtain 172 the desired low/high gain profile with respect to the absolute value of e(t).

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States and control effort are provided in Figure 2 where it is observed that all the  Table 4.    In this case, as in [14], the yaw reference to follow is a sinusoidal signal defined as where the initial yaw angle is States and control effort are provided in Figure 7 where it is clear that AISM is capa-191 ble to follow the yaw reference with no appreciable delay keeping the desired settling