# LAMDA Controller Applied to the Trajectory Tracking of an Aerial Manipulator
## 5 Discussion and Conclusions  
In this work, experimental tests were carried out for four controllers: Kinematic, Inverse Dynamics, SMC Sliding Mode, and LAMDA, aimed at trajectory tracking by an aerial manipulator robot. The experimental results obtained from the Kinematic controller showed higher errors with respect to the other controllers. This fact was expected since the Kinematic controller does not compensate the dynamics of the system. Furthermore, the presence of wind causes the controller to not exhibit a good performance, as shown in Figure 18. The Inverse Dynamics, SMC, and LAMDA controllers exhibited better results with respect to the Kinematic controller, showing a good performance. The corresponding stability analysis of the proposed controllers have also been presented. The errors obtained in the latter controllers were as expected according to their nature. Furthermore, the presence of wind generates errors in the trajectory tracking. The proposed LAMDA controller applied to trajectory tracking for the aerial manipulator robot demonstrated its stability in experimental tests under disturbance conditions. Therefore, the LAMDA controller shows a better performance than the Kinematic controller and a similar performance to the rest of the controllers developed in this work, as it can be seen in the experimental tests through the ISE. Although LAMDA has similar results to the Inverse Dynamics and SMC controllers, its advantage over them is that LAMDA does not depend on the model of the aerial manipulator robot. Thus, it is not necessary to determine and obtain the parameters of the aerial manipulator robot for the implementation of the algorithm, which is usually complex and requires validation prior to use in the different control algorithms that are based on the model. Due to the nature of its structure, the LAMDA controller is based on a set of fixed hidden layers and is considered as an artificial intelligence method, whose  
Figure 18: ISE factor for the different control techniques used in the experiments. computational performance is not as high compared to similar algorithms [24], as well as the fact that the mathematics turns out to be not so complex. The LAMDA controller is a technique that has not been used previously for robotic systems such as aerial manipulators robots, although this type of controller has been implemented in other applications such as data analysis, data classification, system identification, and process regulation. LAMDA is robust against disturbances due to its design based on the fundamentals of SMC, considering the stability of Lyapunov. LAMDA has presented good behavior in the presence of wind whose velocity was around 8.3 km/h, so there is evidence that promising results could be obtained with the application of this controller in the area of robotics, specifically for aerial manipulator robots. The objective of this work is to demonstrate that the LAMDA controller was experimentally tested in applications not common to which it has been used habitually. Thus, for the first time, it has been used in aerial manipulators robots for trajectory tracking, the results of which confirm that LAMDA controllers can be brought into the field of robotics with promising results. Future work deals with the implementation of this controller for cooperative systems, as well as the application of adaptive fuzzy logic to the LAMDA controller to compensate systems with delays or disturbances that are usually present in robotic systems. Conceptualization, G.M.A., L.M., P.L. and G.P-N.; methodology, G.M.A., L.M. and P.L. ; software, G.M.A., P.L. and V.H.A. ; validation, G.M.A., P.L. and V.H.A. ; formal analysis, G.M.A., L.M., and P.L. ; investigation, G.M.A. ; resources, G.M.A. and L.M. ; data curation, G.P-N.; writing--original draft preparation, G.M.A. and L.M. ; writing--review and editing, P.L., G.P-N. and V.H.A. ; visualization, G.M.A. ; supervision, G.P-N. and P.L. ; project administration, G.P-N. and P.L., funding acquisition, G.M.A. and P.L.