# Adaptive Integral Sliding Mode Based Course Keeping Control of Unmanned Surface Vehicle

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## Abstract

**:**

## 1. Introduction

**Waypoint control:**In this strategy, Line of Sight (LOS) based approach is adopted to follow a certain waypoints, generated heuristically, in the required maritime environment.**Path following control:**In this strategy, a path generated through path planning algorithms is used as a reference, to be followed with no temporal constraints. Here, USV should converge and follow the desired path without any time constraints and simultaneously satisfies its assigned velocity profile.**Trajectory tracking:**In this strategy, temporal constraints are enforced upon the path generated using path planners. This is predominantly used with fully actuated marine vehicles reasoned with better maneuvering capabilities.

#### 1.1. State of the Art

**course keeping control**is highly non-linear in nature and has been studied from a perspective of observed disturbance control using sliding mode control (SMC) approach. The SMC problem for USVs, subjected to, higher order non linear operational disturbances, have been studied with varying control approaches like sliding mode [3,4,5,6]; fuzzy sliding mode [7]; proportional derivative fuzzy [8]; backstepping [9,10,11,12]; backstepping with adaptive radial basis function neural network [13]; sine function-based non-linear feedback [14]; hyperbolic tangent based nonlinear control [15]; sigmoid based nonlinear control [16]; function adaptive neural path following control [17]; model predictive control [18,19]; event-triggered control approach [20] and non-linear feedback power functions [21].

#### 1.2. Major Contributions

- A number of simulation studies in the manuscript demonstrate that the proposed adaptive control approach can be reconfigured for various input trajectories and marine environmental disturbances, without requiring parametric adjustment.
- The cut-off frequency of the system response is an indication of the bound to be assigned to the disturbance derivative in the algorithm. This relationship is based on low-pass filtering properties associated with the second order adaptive linear dynamics generated at the sliding variable. As a result, frequencies over ${\omega}_{c}$ do not affect the sliding variable response. In practice, this feature offers some advantages when estimating the maximum value of the disturbance derivative is a challenging task.
- The proposed adaptive profile generates low/high gains based on the absolute error. As a result, the control input is not saturated when there is a large error (gain is small) and the response at steady state becomes fast disturbance compensation (gain is large).
- Based on the adaptive placement of two poles relating to a second order dynamical system with critical damping, we can generate an overdamped response that avoids the occurrence of considerable overshoots.
- By avoiding the need of the derivative of the fractional power terms with respect to time, the singularity problem associated with terminal sliding mode solutions can be avoided. Thus, the high sensitive performance around the equilibrium point generated by set value or fractional order functions can be reduced.

## 2. Problem Statement

**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 gravity.

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

- The tests includes results without disturbances ($d\left(t\right)=0$) and with disturbances ($d\left(t\right)\ne 0$).
- The algorithm parameters are configured in the case of the step input reference without disturbances, such that all solutions provide the same value of the MIA index at the end of the test time.
- After that, the algorithms parameters are fixed and tested in the case of step with disturbances and in the case of the sinusoidal input reference. In this way we check the robustness of the solutions with respect to its capacity of adaptation to different scenarios from a specific parameter configuration.

**Theorem**

**1.**

**Proof.**

## 3. Adaptive Integral Sliding Mode Surface Control Design

**Theorem**

**2.**

**Proof.**

**Case 1: $\left|z\right|\le \mu \wedge \alpha ={\alpha}_{min}\wedge \alpha ={\alpha}_{max}$**Assuming the worst case scenario, that is, when $z>\mu \wedge \alpha ={\alpha}_{min}$, substitution of $\dot{\alpha}=0$ implies that the second order dynamics related to $s\left(t\right)$ are$$\begin{array}{cc}\hfill \ddot{s}\left(t\right)+{\alpha}_{min}\dot{s}\left(t\right)+\frac{{\alpha}_{min}^{2}}{4}s\left(t\right)+\dot{d}\left(t\right)& =0\hfill \end{array}$$Let’s define a sliding vector state $\eta \left(t\right)$ as$$\begin{array}{cc}\hfill \eta \left(t\right)& ={\left[\begin{array}{cc}s\left(t\right)& \dot{s}\left(t\right)\end{array}\right]}^{T}\hfill \end{array}$$Dynamics of $\eta \left(t\right)$ are given by$$\begin{array}{cc}\hfill \dot{\eta}\left(t\right)& =A\eta \left(t\right)+F\left(t\right)\hfill \end{array}$$$$\begin{array}{c}\hfill A=\left(\begin{array}{cc}0& 1\\ -{\alpha}_{min}& -\frac{{\alpha}_{min}^{2}}{4}\end{array}\right)\end{array}$$$$\begin{array}{cc}\hfill F\left(t\right)& =\left(\begin{array}{c}0\\ \dot{d}\left(t\right)\end{array}\right)\hfill \end{array}$$Lets’ define$$\begin{array}{cc}\hfill {p}_{1}& =\frac{{\alpha}_{min}}{8}+\frac{2({\alpha}_{min}+1)}{{\alpha}_{min}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {p}_{2}& =\frac{0.5}{{\alpha}_{min}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {p}_{3}& =\frac{2({\alpha}_{min}+1)}{{\alpha}_{min}^{3}}\hfill \end{array}$$$$\begin{array}{cc}\hfill P& =\left(\begin{array}{cc}{p}_{1}& {p}_{2}\\ {p}_{2}& {p}_{3}\end{array}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \left|P\right|& =\frac{{\lambda}^{4}+16{\lambda}^{2}+32\lambda +16}{4{\lambda}^{5}}>0\hfill \end{array}$$It can be shown that$$\begin{array}{cc}\hfill PA+{A}^{T}P& =-Q\hfill \end{array}$$$$\begin{array}{cc}\hfill V\left(\eta \right)& =\frac{1}{2}{\eta}^{T}P\eta \hfill \end{array}$$$$\begin{array}{cc}\hfill \dot{V}\left(\eta \right)& =-{\eta}^{T}Q\eta +{F}^{T}P\eta \hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =-{\eta}^{T}Q\eta -\dot{d}\left(t\right)({p}_{2}s\left(t\right)+{p}_{3}\dot{s}\left(t\right))\hfill \end{array}$$Let’s define$$\begin{array}{cc}\hfill {p}^{*}& =\sqrt{2}max({p}_{2},{p}_{3})\hfill \end{array}$$Therefore$$\begin{array}{cc}\hfill |\dot{d}\left(t\right)({p}_{2}s\left(t\right)+{p}_{3}\dot{s}\left(t\right))|& \le {p}^{*}\Delta \left|\right|\eta \left|\right|\hfill \end{array}$$Applying (51) and assumption 2 it is obtained$$\begin{array}{cc}\hfill \dot{V}\left(\eta \right)& <-{\gamma}_{Q}^{min}{\left|\right|\eta \left|\right|}^{2}+{p}^{*}\Delta \left|\right|\eta \left|\right|\hfill \end{array}$$Note that ${\gamma}_{Q}^{min}=1$, thus the closed set ${\Omega}_{\eta}$, which includes the origin, defined as$$\begin{array}{cc}\hfill {\Omega}_{\eta}& =\{\eta \left(t\right)\in {\mathbb{R}}^{2}:|\left|\eta \right||\le {p}^{*}\Delta \}\hfill \end{array}$$- −
- $sign\left(s\dot{s}\right)<0$: implies that $\left|s\right(t\left)\right|\to 0$, that is, the dynamics is stable an converges to the origin, with a value of $\alpha $ adjusted to keep this condition.
- −
- $sign\left(s\dot{s}\right)\ge 0$: implies that $\left|s\right(t\left)\right|$ grows inside ${\Omega}_{\eta}$, that is, with an upper bound, or its value is stationary. According to (23) and (1), there exist an instant where the condition $sign\left(z\right)sign\left(s\right)>0$ is met, which implies that $\alpha $ grows, that is, condition $\dot{\alpha}\ne 0$ is achieved. Therefore, $\alpha $ grows only when it is needed to keep the sliding mode condition at steady state, which is related to the performance given by the value of ${\lambda}_{max}$.

**Case 2: $\left|z\right|>\mu \wedge {\alpha}_{min}<\alpha <{\alpha}_{max}$**Substitution of $\dot{\alpha}\ne 0$ from (24) implies that the second order dynamics equation related to $s\left(t\right)$ is$$\begin{array}{cc}\hfill \ddot{s}\left(t\right)+\alpha \dot{s}\left(t\right)+\frac{{\alpha}^{2}}{4}s\left(t\right)+\kappa {\left|z\right|}^{\zeta +1}sign\left(s\right)+\dot{d}\left(t\right)& =0\hfill \end{array}$$Applying assumption 2 and condition $\left|z\right|>\mu $ implies that$$\begin{array}{cc}\hfill {\kappa \left|z\right|}^{\zeta +1}sign\left(s\right)+\dot{d}& ={\rho}_{z}\left(t\right)s\hfill \end{array}$$$$\begin{array}{cc}\hfill {\Omega}_{z}& =\left\{z\right(t)\in \mathbb{R}:|z\left(t\right)|<\mu \}\hfill \end{array}$$$$\begin{array}{cc}\hfill {s}_{1,2}^{*}& =\frac{\alpha}{2}\pm j\sqrt{{\rho}_{z}\left(t\right)}\hfill \end{array}$$A condition of the following form can be used to avoid chattering (high frequency oscillations caused by a large imaginary value in the pole position as a result of overestimation) at the steady-state response.$$\begin{array}{c}\hfill {\left|\kappa \right|z|}^{\zeta +1}sign\left(s\right)+\dot{d}{\left(t\right)|\le \kappa |z|}^{\zeta +1}+\Delta <{\rho}_{z}^{max}\end{array}$$This provides an upper bound of the perturbation generated at the dynamics with respect to the solution with $\kappa =0$ and $d\left(t\right)=0$. In order to estimate the correlation between the sampling time $\tau $ and the natural frequency $\sqrt{{\rho}_{z}^{max}}$ (in $\frac{rad}{s}$), we must verify that the frequency given by the Nyquist-Shannon sampling theorem (the maximum operating frequency for a system with sampling time $\tau $) does not create a change in sign in $s\left(t\right)$ at the limit condition $\left|z\right(t\left)\right|=\mu $, that is$$\begin{array}{c}\hfill \sqrt{{\rho}_{z}^{max}}\le \frac{\pi}{\tau}\mu \end{array}$$Applying condition (58) an upper bound for $\kappa $ as a function of $\Delta $, $\tau $ and the absolute value of $z\left(t\right)$ is obtained as$$\begin{array}{cc}\hfill \kappa & \le \frac{{\left(\frac{\pi \mu}{\tau}\right)}^{2}-\Delta}{{\left|z\right|}^{\zeta +1}}\hfill \end{array}$$This constraint provides a limit on the application of the method that employs the bound $\Delta $ of the disturbance derivative, the sampling time $\tau $ and, taking into consideration relation (27), the required precision $\mu $.Inside ${\Omega}_{z}$ we have that$$\begin{array}{cc}\hfill |s\left(t\right)+\frac{\alpha}{2}\overline{s}\left(t\right)|& <\mu \hfill \end{array}$$$$\begin{array}{cc}\hfill |\overline{s}\left(t\right)|& <\frac{\mu}{|sin(\vartheta \left)\right|}\hfill \end{array}$$$$\begin{array}{cc}\hfill \left|s\right(t\left)\right|& <\frac{\mu}{|cos(\vartheta \left)\right|}\hfill \end{array}$$$$\begin{array}{cc}\hfill |\dot{e}\left(t\right)+\lambda e\left(t\right)|& <\frac{\mu}{|cos(\vartheta \left)\right|}\hfill \end{array}$$Following the previous approach implies that:$$\begin{array}{cc}\hfill \left|e\right(t\left)\right|& <\frac{\mu}{|cos(\vartheta \left)\right||sin(\theta \left)\right|}\hfill \end{array}$$$$\begin{array}{cc}\hfill |\dot{e}\left(t\right)|& <\frac{\mu}{|cos(\vartheta \left)\right||cos(\theta \left)\right|}\hfill \end{array}$$

## 4. Numerical Simulations

**Assumption**

**4.**

#### 4.1. Constant Yaw Reference

- Consider a settling time ${t}_{s}=150s$, a maximum desired yaw rate ${r}_{max}=\frac{0.70\pi}{180}$ degrees per second and a required precision $\mu =1.0\times {10}^{-6}$.
- The value of $\alpha \left(0\right)$ is obtained assuming an exponential convergence of the error from initial condition $e\left(0\right)$ to desired precision $\mu $ with a desired settling time ${t}_{s}$$$\begin{array}{cc}\hfill \alpha \left(0\right)& =-1.25\frac{log\left(\frac{\mu}{\left|e\right(0\left)\right|}\right)}{{t}_{s}}=0.0756\hfill \end{array}$$The values of ${\alpha}_{min}$ and ${\alpha}_{max}$ are selected as$$\begin{array}{cc}\hfill {\alpha}_{min}& =\alpha \left(0\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\alpha}_{max}& =5({\alpha}_{min}+\Delta )\hfill \end{array}$$
- The value of ${\lambda}_{min}$ is related to the initial conditions of the problem and the maximum desired yaw rate as$$\begin{array}{cc}\hfill {\lambda}_{min}& =\frac{{r}_{max}}{\left|e\right(0\left)\right|}=0.014\hfill \end{array}$$$$\begin{array}{cc}\hfill {\lambda}_{max}& =2.0{\lambda}_{min}=0.028\hfill \end{array}$$
- The value of $\kappa $ must be higher than $\Delta $ in order to obtain a small value for $\mu $. Due ti the low-pass filtering properties of (54), the value of $\Delta $ can be further refined by estimating the cut-off frequency ${\omega}_{c}\left(t\right)$ of the second order system related to $s\left(t\right)$$$\begin{array}{cc}\hfill {\omega}_{c}\left(t\right)& =\frac{\alpha \left(t\right)}{2}\hfill \end{array}$$Therefore $\kappa $ is calculated as an adaptive gain that takes account of ${\omega}_{c}$ and the desired precision$$\begin{array}{cc}\hfill \kappa \left(t\right)& =\frac{\alpha \left(t\right)}{2{\mu}^{\frac{1}{\zeta +1}}}\hfill \end{array}$$
- Simulations are used to set the values of ${\zeta}_{min}$ and ${\zeta}_{max}$ such that the value of the performance index MIA is equal to the value achieved with benchmark chosen controllers at the conclusion of the test period.$$\begin{array}{cc}\hfill {\zeta}_{min}& =0.800\hfill \\ \hfill {\zeta}_{max}& =1.685\hfill \end{array}$$This condition generates an adequate adaption of the value of $\zeta $ that allows to obtain the desired power factor profile with respect to the absolute value of $e\left(t\right)$.

#### 4.2. Sinusoidal Yaw Reference

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

USV | Unmanned Surface Vehicle |

IMO | International Maritime Organisation |

LOS | Line of Sight |

SMC | Sliding Mode Control |

AISM | Adaptive Integral Sliding Mode |

GUAS | globally uniformly asymptotically stable |

MAE | Mean Absolute Error |

MIA | Mean Integral Absolute |

MTV | Mean Total Variation |

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**Figure 1.**6 DOF motion representation with North-East-Down coordinate system (green) and body fixed reference frame (black).

**Figure 3.**Constant yaw reference test with $d\left(t\right)=0$. States and control. Cyan line: Reference; Red line: Concise backstepping (Zhang et al.); Blue line: Synergetic (Muhammad et al.); Black line: Adaptive sliding mode (González-Prieto et al.).

**Figure 4.**Constant yaw reference test with $d\left(t\right)=0$. Performance indices evolution. Cyan line: Reference; Red line: Concise backstepping (Zhang et al.); Blue line: Synergetic (Muhammad et al.); Black line: Adaptive sliding mode (González-Prieto et al.).

**Figure 6.**Constant yaw reference test with $d\left(t\right)\ne 0$. States and control. Cyan line: Reference; Red line: Concise backstepping (Zhang et al.); Blue line: Synergetic (Muhammad et al.); Black line: Adaptive sliding mode (González-Prieto et al.).

**Figure 7.**Constant yaw reference test with $d\left(t\right)\ne 0$. Sliding mode variable $s\left(t\right)$ and external disturbance $d\left(t\right)$.

**Figure 9.**Sinusoidal yaw reference test with $d\left(t\right)=0$. States and control. Cyan line: Reference; Red line: Concise backstepping (Zhang et al.); Blue line: Synergetic (Muhammad et al.); Black line: Adaptive sliding mode (González-Prieto et al.).

**Figure 11.**Sinusoidal yaw reference test with $d\left(t\right)\ne 0$. States and control. Cyan line: Reference; Red line: Concise backstepping (Zhang et al.); Blue line: Synergetic (Muhammad et al.); Black line: Adaptive sliding mode (González-Prieto et al.).

**Figure 12.**Sinusoidal yaw reference test with $d\left(t\right)\ne 0$. Sliding mode variable $s\left(t\right)$ and external disturbance $d\left(t\right)$.

**Figure 13.**Sinusoidal yaw reference test with $d\left(t\right)\ne 0$. Adaptive parameters evolution.

Parameter | Value |
---|---|

K | 0.21 |

T | 107.76 |

${a}_{1}$ | 13.17 |

${a}_{2}$ | 16,323.46 |

Parameter | Value |
---|---|

${k}_{1}$ | 0.0017 |

$\omega $ | 0.6000 |

Parameter | Value |
---|---|

${a}_{1}$ | 0.090 |

${a}_{2}$ | 1.891 |

${T}_{1}$ | 28.000 |

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González-Prieto, J.A.; Pérez-Collazo, C.; Singh, Y. Adaptive Integral Sliding Mode Based Course Keeping Control of Unmanned Surface Vehicle. *J. Mar. Sci. Eng.* **2022**, *10*, 68.
https://doi.org/10.3390/jmse10010068

**AMA Style**

González-Prieto JA, Pérez-Collazo C, Singh Y. Adaptive Integral Sliding Mode Based Course Keeping Control of Unmanned Surface Vehicle. *Journal of Marine Science and Engineering*. 2022; 10(1):68.
https://doi.org/10.3390/jmse10010068

**Chicago/Turabian Style**

González-Prieto, José Antonio, Carlos Pérez-Collazo, and Yogang Singh. 2022. "Adaptive Integral Sliding Mode Based Course Keeping Control of Unmanned Surface Vehicle" *Journal of Marine Science and Engineering* 10, no. 1: 68.
https://doi.org/10.3390/jmse10010068