# LAMDA Controller Applied to the Trajectory Tracking of an Aerial Manipulator
###### Abstract  
In this work, a novel LAMDA (Learning Algorithm for Multivariable Data Analysis) control strategy for trajectory tracking for an aerial manipulator is presented. Four control strategies are developed: Kinematic, Inverse Dynamics, Sliding Mode (SMC), and LAMDA. These are compared with each other in order to verify their performance to fulfill the control objective. Experimental tests were also carried out to validate the developed controllers. In addition, a study of stability has been also performed for all the controllers. The results obtained by the LAMDA controller demonstrated the good performance of the controller in the aerial manipulator robot. To the best of our knowledge, this is the first time a LAMDA controller has been applied to an aerial robotic manipulator. aerial manipulator; inverse dynamics; kinematic; LAMDA; trajectory tracking; SMC; fuzzy; multivariable +
Footnote †: journal: Applied to the Trajectory Tracking of an Aerial Manipulator
unmanned aerial manipulators (UAM) in a leader-follower configuration that performed a cooperative manipulation task taking into account the dynamics of the system. In the study of [PERSON] et al. [12] the dynamics were compensated with an inverse dynamics controller for trajectory tracking. Different control algorithms have been developed to improve the performance of aerial manipulators robots. For example, a controller based on neural networks was implemented for trajectory tracking with the dynamic model [13]. In the same context, a controller based on null space to coordinate a group of robots was presented in [14]. An adaptive position control was implemented by [PERSON] and [PERSON] [15] aimed at compensating for certain perturbations in the modeling of the aerial manipulator, whereas a feedback control strategy was presented by [PERSON] et al. [16] allowing both the position and orientation of the end-effector to follow a desired trajectory considering the manipulator dynamics. The abovementioned research considers the control of an aerial manipulator from the point of view of its dynamics. In order to carry this out, the search for new controllers that can include the dynamics of aerial manipulators robots, meet the control objectives, and improve the performance of the task is vital for researchers. Artificial intelligence (AI) approaches help to improve the behavior of the controllers. Among others, the fuzzy logic allows obtaining good results when the plant model is inaccurate or is not known in detail, which usually arises in the field of robotics. One of these methods is LAMDA (Learning Algorithm for Multivariable Data Analysis), which has been designed originally for classification and clustering tasks [17; 18; 19; 20]. For example, a LAMDA fuzzy classification technique has been implemented to interpret the behavior of a drinking water treatment plant [21]. Recently, the use of the LAMDA features (in the detection of functional states of systems) has been proposed to make it work as a controller. For example, a class-based LAMDA controller was proposed and validated in the regulation of humidity and temperature of a complex Heating, Ventilation, and Air-Conditioning (HVAC) system in [22], obtaining excellent performance when considering that the mathematical model of the plant was variable or inexact. In [23], an adaptive LAMDA has been proposed and formalized for the control and modeling of systems, modifying the LAMDA structure with the addition of layers operating similar to neural networks, but with the advantage of having a fixed number of layers whose calibration is not trivial. This controller has proven to be adequate where the system dynamics are uncertain and complex. Finally, previous work [24] focuses on the formalization of a LAMDA algorithm for control based on the fundamentals of the Lyapunov theory and the Sliding-Mode Control (SMC), to guarantee stability and robustness of the overall system. The method is called LAMDA-SMC (LSMC) and takes advantage of LAMDA features to design a chattering-free controller.