## 4 Evaluation
### Numerical Simulation
#### 4.1.2 Comparison of the VFGL with Other Methods  
To illustrate the superiority of the VFGL, several methods were implemented in the simulation as comparison. They are the NLGL, PLOS, PID, GVF, and VFF. Guidance laws for path following, such as the NLGL, PLOS, and PID, did not consider obstacle avoidance. For fair comparison, there are no obstacles in the comparison with these methods. In comparison with the GVF and VFF, a circular obstacle is set on the reference path to evaluate the performance of the methods for obstacle avoidance.  
A. Curve following using different methods  
In this scenario, the velocity of the UAV is limited to 20 m/s. The heading rate is limited within \([-0.2,0.2]\) rad/s. The parameters of the VFGL are \(k_{v}=0.5\), \(c_{v}=2\sqrt{k_{v}}\); the guidance distance of the NLGL is 120; the parameters for the PLOS are \(k_{e}=2.5\) and \(k_{d}=0.05\); and the parameters for the PID are \(k_{p}=0.1\), \(k_{l}=0.005\), and \(k_{D}=0.64\). The reference path includes a straight-line segment and a part of \"8\" shaped curve with a variable curvature. Figure 6 shows the simulation results using the four kinds of methods.  
Figure 6a shows that all of the four methods can guide the UAV to track the reference curve. The differences lie in the speed of convergence and the tracking accuracy--particularly when the curvature of the reference path changes. The cross-track error using these methods is shown in Figure 6b. This shows that the convergence of the VFGL and  
Figure 5: Simulation results of the UAV path following using the VFGL: (**a**) results with different \(c_{v}\); (**b**) results with different \(k_{v}\) and \(c_{v}\).  
Figure 6: Simulation results of the UAV path following using different methods: (**a**) comparison of the UAV trajectories using different methods; (**b**) comparison of the cross track error using different methods.  
PLOS is faster than the NLGL and PID. Using NLGL, PLOS, and PID, there are overshoots before the UAV converges to the reference. The cross-track error of VFGL is the lowest among these methods, especially when the curvature of the reference path changes.  
The comparison in this scenario shows that the proposed VFGL method has better performance on path following than NLGL, PLOS, and PID, especially when the curvature of the reference curve changes.  
B. Obstacle avoidance using different methods  
In this scenario, a UAV is initially following a straight line at a constant speed of 25 m/s. A circular obstacle is set on the reference path. The VFGL, VFF, and GVF guidance were simulated in this scenario. The parameters for the VFGL are \(k_{v}=0.5\), \(c_{v}=2\sqrt{k_{v}}\), \(k_{rep}=10\), \(R_{d}=100\) m, and \(d_{s}=10\) m. The parameters of the simulation in [33] were used for VFF and GVF in this scenario. Details on the VFF and GVF guidance can be found in [29; 33].  
The results of the three guidance methods are shown in Figure 7. Using the VFGL, the UAV is pushed away from the planned path to avoid the obstacle, and after that it converges back to the planned path without any overshoot. Using the VFF method, the UAV moves away from the planned path to avoid the obstacle; after the obstacle avoidance, it can not return to the planned path but takes the shortest distance directly to its goal. GVF leaves the planned path before VFF and VFGL, tracks the outside of the obstacle, and then quickly converges back to the planned path.  
The deviation from the planned path of the GVF is larger than the VFGL. There is overshoot before the UAV converges back to the planned path using the GVF. By comparison, we can find that the VFGL method achieved better path following and obstacle avoidance results. Compared with GVF, another advantage of the proposed VFGL is that it can be used for curve following and obstacle avoidance (see the results of the hardware-in-the-loop simulation).