Robustness margin for attitude control of a four rotor mini-rotorcraft: Case of study David Lara a, * , Gerardo Romero a , Anand Sanchez b , Rogelio Lozano b , Alfredo Guerrero b a Universidad Autónoma de Tamaulipas, Posgrado-UAMRR, PO Box 1460 CP 88779 Reynosa Tamps, Mexico b Université de Technologie de Compiègne, HEUDIASYC-UMR-CNRS-6599, BP 20529, Compiègne 60200, France article info Article history: Received 26 November 2007 Accepted 4 November 2009 Keywords: Robust control Mechanical systems-robotics Time-delay systems Robustness margin Parametric uncertainty abstract In this paper we present new results to compute the robustness margin of an attitude control algorithm for a quad-rotor mini-rotorcraft known as X4-flyer. The maximum parametric uncertainty is calculated when multivariable PD controller is used to stabilize the attitude of the aerial vehicle. This work is based on the value set characterization approach of the mathematical model for the closed-loop control system, which is represented by an interval plant with time delay. The zero exclusion principle is used to com- pute the robustness margin of the closed-loop system. This approach transforms the original robust sta- bility problem into simple problem of graphic inspection, where we only needed to verify if a graph on the complex plane does not contain the origin of the plane. Additionally, different experiments were made in order to validate theoretical results; these experimental results will be presented at the end of this work. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, Unmanned Aerial Vehicles (UAVs) are used in differ- ent applications either for military or civil purposes such as forest fire detection [12], oil pipeline inspection, high voltage lines failure detection and surveillance [10,17]. In recent years the interest on the UAV’s control problem has increased due to the different ways it can be applied. In this work we study a rotary wing UAV called X4-flyer which is a four-rotor helicopter or quad-rotor as the one shown in Fig. 1. Many papers considering this type of configuration have been published. For example, in [9] the author presents a the- oretical analysis of the dynamics of the quad-rotor in order to de- velop a mathematical model used to implement a simulation of a control system for stable hovering and indoor flight. In [4] a control strategy for trajectory tracking or point to point steering based on receding horizon is proposed. In [8] an H 1 control is proposed and this control is implemented using a PC platform of a rotorcraft that freely rotates. In [5] backsteeping and sliding mode techniques are applied to an indoor micro quad-rotor; their results are validated in a PC-based bench test platform. The attitude stabilization prob- lem is treated in [16] where the authors use a quaternion based mathematical model to propose a controller based upon the com- pensation of the Coriolis and gyroscopic torques using a PD feed- back structure; publication’s results were presented in both simulation and experimentally using an embedded control system where the quad-rotor is only allowed to rotate freely around a fixed pivot. Other interesting control technique based in nested saturations is found in [6,7] where the dynamic model of the quad-rotor was obtained using the approach of Euler–Lagrange. In this paper, the authors propose a controller based on the Lyapunov analysis of the closed-loop system and real time algo- rithms were implemented in a PC-based platform as well. The purpose of this work is to use the value set methodology to compute the robustness margin in relation to parametric uncer- tainty in the synthesis of a PD type controller used to stabilize the attitude of the quad-rotor. The analysis of the controller is performed using numerical simulations and is validated with experimental results. This paper is structured as follows. Section 2, presents a nonlin- ear mathematical model. Section 3, provides the mathematical preliminaries for the value set methodology. Section 4, deals with the problem formulation and the controller design. Section 5, pre- sents the computation of the robustness margin using the pro- posed method to the quad-rotor. Also the performance of the controller is shown in numerical simulations. Finally, in Section 6 some experimental results are presented as well as the conclusions of this work. 2. Four rotor rotorcraft attitude dynamical model The X4-flyer used in this work is an electric four motor rotor- craft with each motor attached to a rigid cross frame as shown in Fig. 1. It is a vertical takeoff and landing vehicle (VTOL) able to move omnidirectionally with the ability to fly in a stationary way like the conventional helicopter. In this type of flying machine the front and rear rotors rotate counter-clockwise while the left and right rotors rotate clockwise canceling gyroscopic effects and 0957-4158/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2009.11.002 * Corresponding author. Tel.: +52 899 9213300; fax: +52 899 9213301. E-mail address: dlara@uat.edu.mx (D. Lara). Mechatronics 20 (2010) 143–152 Contents lists available at ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronics