## 2 Trajectory Tracking
Knowing that:  
\[L_{1}\ =\ 2R\sin\eta \tag{2}\]  
where the lateral acceleration calculated by Equation (1) equals the centripetal acceleration required to follow instantaneously a circular segment and \(a_{s}\) is the property used to track a circle of any radius R. This characteristic predisposes the guidance logic to work faithfully with a curved path. Moreover, as shown elsewhere [24], the method shows better capabilities than the PID controller when used on the UAV in the presence of wind. In this research, the method will be examined on the unmanned surface vessel (USV) using state-of-the-art geodetic grade measurements. The reference position of the USV was measured using independently acquired GPS navigation data (GPS RTK (Real Time Kinematic) geodetic receiver). The original Equations (1) and (2) were modified [41] for USV navigation purposes. The implementation required the addition of two parameters (\(L_{1}\) Damping Factor and \(L_{1}\) Period). Therefore, for the new \(L_{1}\):  
\[L_{1t}=\frac{1}{\pi}\zeta Tv \tag{3}\]  
\(\zeta\) is the \(L_{1}\) Damping Factor, \(T\) is the \(L_{1}\) Period (s), and \(v\) is the speed of the unit. Finally, substituting Equation (3) into Equation (1) can be written as:  
\[a_{st}=\frac{\mathrm{k}\,v^{2}}{L_{1t}}\sin(\eta_{1}-\eta_{2}) \tag{4}\]  
where \(k\) is the \(L_{1}\) Control Gain, defined as:  
\[\mathrm{k}=4\zeta^{2} \tag{5}\]  
The cross-track error (XTE), corresponding to \(L_{\perp}\) on Figure 3 and defined as a distance between actual USV position perpendicular to the intended (desired) track, can be approximated as a second order differential equation [42]:  
\[\ddot{d}+2\zeta a_{n}\dot{d}+\omega_{n}^{2}d\ =\ 0 \tag{6}\]  
Figure 2: Guidance logic trajectory tracking. where the natural frequency \(\omega_{n}\) (natural frequency (eigenfrequency) is the frequency at which a structure or system have the tendency to oscillate in the absence of any driving or damping forces), is related to \(T\):  
\[\omega_{n}\ =\ \ \frac{2\pi}{T} \tag{7}\]  
The dynamics of Equation (6) indicate that changing the values of \(\zeta\) and \(\omega_{n}\) adjusts the control response of XTE. The \(\zeta\) and \(\omega_{n}\) values are specified by the user and, for this research, were adjusted experimentally after long and extensive trials. The parameter tuning requires some user experience regarding the USV's parameters and the response observations during the trials. As a general rule for tuning, the following instructions are given. The \(L_{1}\) period \(T\) is given in seconds with a range from 1 to 60 (increments of 1) for the \(L_{1}\) tracking loop and is the primary control parameter for aggressive turns in auto mode (Figure 1). This parameter should be larger for less responsive USV platforms. For smaller and more maneuverable USVs a lesser value can be set. The starting value was adjusted experimentally as 20 s. The \(L_{1}\) control damping ratio \(\zeta\) with range from 0.6 to 1 (increments of 0.05) should be increased if the USV overshoots the track being followed. Significant changes made to the original Equation (1) enable the length \(L_{1}\) to be calculated dynamically by the navigation loop depending on the USV ground speed changes and enable the user to specify a constant period for the tracking loop. The L1 Control Gain was changed from a fixed value of 2 (Equation (1)) to be calculated based on the \(\zeta\) value set by the user. This enables additional damping to be specified to compensate for delays in the velocity measurement and for the USV frame to respond. Figure 4 presents all possible tracking movements. The current tracking mode depends on the area where the USV is located in relation to the route plan. Figure 4 presents a simple survey plan based on 6 WPs. When the vehicle is located in \(USV_{1}\), the first WP is acknowledged to be tracked and vehicle i proceeds to point A. The point is considered to be reached when the unit is within the \(WP_{Radius}\) parameter.