MODELLING AND CONTROL OF UNDERACTUATED SYSTEMS APPLIED TO A UAV WITH A CABLE-SUSPENDED LOAD

Abstract

Underactuated Mechanical System (UMS) is a mechanical system that is not able to command an arbitrary instantaneous acceleration for one or more of its degrees of freedom from some or all its initial states. One of the common nonlinear applications of UMSs is the quadcopter which is an unmanned aerial vehicle (UAV) with four inputs and six coupled degrees of freedom (6DOF). Owing to their agility and easy configuration, quadcopters are recently used in many civil and military applications like imaging, security, agriculture, load transportation, etc.

In this work, a quadcopter with a cable-suspended load was studied where an eight degree of freedom (8DOF) nonlinear mathematical model was derived from the basic principles and simulated using MATLAB/SIMULINK. The system consists of two rigid bodies; a six degree of freedom (6DOF) quadcopter with its center of gravity assumed to be at a general position shifted from its geometric centroid and a two degree of freedom (2DOF) cable-suspended load (spherical pendulum) hanged at a general location on the quadcopter.

It is well known that the controllers designed based on Linear Quadratic Regulator (LQR), which is considered as an optimal control, are robust and produce low steady state error if compared to the classical Proportional-Integral-Derivative (PID) controllers. For this reason, the controller designed in this thesis was based on LQR for both stability and tracking where the nonlinear mathematical models were linearized and the gains were calculated upon trial and error method for the LQR performance index weighing parameters.

In this thesis, the controller was designed using the system linearized model but tested and simulated using the nonlinear mathematical model. A fixed gain controller proved insu cient to achieve the system stability and that’s due to the wide range of change of the spherical pendulum azimuth angle in the purview of the system operation which implied that there was no definite linearized model could represent the system correctly during all the operation range. Thus, there was a need for an adaptive controller (Adaptive LQR) which changes its gains as the spherical pendulum azimuth angle changes. First a look-up table was calculated where there was a set of gains for every pendulum azimuth angle but it was very large in size and impractical in implementation, therefore a curve fitting was performed between the controller gains and the azimuth angle of the spherical pendulum which required to only store the fit functions’ coe cients instead of the large sets of values of the lookup table with comparatively memory size needed approximately 1:5. The designed controller achieved, according to the given assumptions, satisfactory performance in both stability and tracking.