Modelling system and approach of control

The quadrotor is classified as one of the most complex robotic systems due to the number of physical effects, forces and moments that affect its dynamics such as aerodynamic effects, gravity, gyroscopic effects, friction and moments of inertia (Samir Bouabdallah, 2007b; Hwangbo, Sa, Siegwart, & Hutter, 2017; Powers, Mellinger, & Kumar, 2015; Richter, Bry, & Roy, 2016). This type of rotorcraft achieves stable hovering and precise trajectory tracking by balancing the forces produced by the four rotors.

Description and Movement of Quadrotor 

A quadrotor is an aerial mobile robot with four rotors defined in space by six degrees of freedom. The four rotors are placed at the ends of a cross, and the electronics parts are placed in the center. Each two opposite propellers rotate in the same direction, and the other two propellers rotate in the other direction in order to prevent the quadrotor from spinning around 𝑧 axis. Moving the quadrotor is performed by varying motor speeds, it can be moved up / down, tilted left / right (roll) or forward / backward (pitch) or rotate around itself (yaw). The quadrotor has six degrees of freedom, three translational movements and three rotational movements. The six degrees must be controlled using only four actuators, therefore the quadrotor is known to be as an under-actuated system. In conventional helicopters, when the main rotor rotates, it produces a reactive torque that would cause the helicopter’s body to turn in the opposite direction. This is usually balanced by adding a tail rotor that produces a thrust into a lateral direction. However, this rotor with its associated power supply does not provide thrust. In the quadrotor, basic movements are achieved by varying the speed of each rotor thereby changing the thrust produced. The quadrotor inclines towards the slower rotor direction, which then makes translation along the corresponding axis. Therefore, the motion is coupled, meaning that the quadrotor cannot perform the translation without rolling or pitching, which means that a change in the speed of a rotor translates into a movement.

In order to hover, all the thrust forces should work in the same speed , increasing thrust leads to upward movement along 𝑧 axis with a magnitude exactly opposite to the gravitational force. Moreover, the thrust force created by each rotor must be equal to prevent the vehicle from tilting. Therefore, the thrust produced in each rotor must be identical. The upward and downward movement is obtained by the variation of the speed of rotation of the motors (consequently the thrust produced), if the thrust force is greater than the weight of the quadrotor, the movement is ascending, and if the lift force is lower than the weight of quadrotor movement is descending.

The rotation around 𝑧 axis is known as Yaw   by applying a speed difference between rotors  and rotors  a torque is generated around 𝑧 axis, either clockwise or counterclockwise. The direction of the thrust force does not shift during movement. Increasing thrust forces on one pair of rotors must be equal to the decrease of the other pair of rotors and so the total thrust force remains the same.

Mechanical Model of the Quadrotor

The quadrotor is a complex mechanical system, it goes under many internal and external physical effects in the aerodynamic and the mechanic domains. The model of the quadrotor should consider all important forces and moments including the gyroscopic effects. The model developed in this thesis assumes the following (Samir Bouabdallah, 2007b; S. Bouabdallah & Siegwart, 2007):

• The structure of the quadrotor is rigid and symmetrical, which induces that the matrix of inertia is diagonal.
• The center of gravity and the body fixed frame origin are aligned.
• The propellers are supposed to be rigid so as to neglect the effect of their deformation during the rotation.
• Thrust and drag forces are proportional to the square of the speed of rotation (𝜔²), which is an approximation that is very close to the aerodynamic behavior.

This chapter demonstrates the modelling of the quadrotor flying robots. The quadrotor is subject to different forces and moments. This system is a six degrees of freedom robot and it is controlled by the velocity of four rotors. By varying the rotational speeds of these rotors, the quadrotor can make different translational and rotational movements. The complexity, nonlinearity and the interaction between system states can be seen clearly.

The obtained dynamic model allows analysis, simulation and control design of high level controllers in four different articles. Two articles are nonlinear control systems, “Vision based Leader Follower Approach for Uncertain Quadrotor Dynamics Using Feedback Linearization Sliding Mode Control (FLSM)” and “Position and Attitude tracking of Uncertain Quadrotor UAV based on Non-Singular Terminal Super-Twisting Algorithm”. The other two articles are perturbation compensation systems, “Hierarchical Perturbation Compensation System with Exponential Reaching Law Sliding Mode Controller” and “Three Loop Uncertainty Compensation System with Exponential Reaching Law Sliding Mode Controller”.