Modeling and Nonlinear Optimal Control of N-Rotor VTOL Unmanned Aerial Vehicles

Abstract

Quadcopters, hexa-copters and multi-rotor unmanned aerial vehicles (UAV) in general have become one of the most common types of vertical takeoff and landing (VTOL) aerial vehicles where the thrust vectors of all the rotors are usually parallel. These UAVs are typically under-actuated meaning that the number of actuators is less than the degrees of freedom of the vehicle and because of that they cannot achieve holonomic motion. Recently, new designs for multi-rotor UAVs are proposed where the thrust vectors of the rotors are not necessarily parallel, and the rotors can have specific orientations with respect to the body of the vehicle. These new designs can achieve holonomy by manipulating thrust forces and moments of individual rotors which results in independent control of attitude and position of the UAV. In this paper, modeling of forces and moments of rotors is presented first. Second, translation and rotation model of the vehicle is presented, followed by nonlinear optimal control design for attitude and position of the vehicle. Attitude controller is designed according to the state dependent Riccati equation (SDRE) in nonlinear optimal control. Using input-state feedback linearization technique, we simplify the problem and then an analytical solution-utilizing quaternion parametersfor the SDRE is presented. Similarly, a linear quadratic regulator (LQR) for controlling the speed and position of the vehicle is designed. In addition, using Lyapunov theory, proofs for global asymptotic stability of all controllers are provided. Finally, simulations verifying the results are presented.

Modeling and Control of Aerial Manipulation Systems: From Conventional to Continuum Manipulation

“…In addition, in control methods with quaternions, control law often leads to a linear function of quaternion parameters. Therefore, working with quaternion is easier than Euler angles and takes advantage of its ease and simplicity, we used them in control design [93]. Another advantage of quaternions over Euler angles is that quaternions do not face representation singularity issue.…”

“…The Lagrangian, L, is obtained by formulating the total kinetic and energies, K, and U, respectively, as follows: where 𝒇 𝐵 𝑇 is the input vector of forces given by the UAV motors and 𝒇 𝑡𝑒𝑛 𝑇 represents the generalized force for continuum arm which can be found based on tendon actuation in [32]. Furthermore, 𝑰 𝟐 is identity matrix of size 2 and 𝛀 can be calculated based on motors configurations [93].…”

Coupled Dynamic Modeling and Control of Aerial Continuum Manipulation Systems

“…Angle ϑ depends on the number of motors (n m ) and is given by ϑ = 2π n m . To control the torque applied to the system, motors with odd numbers are rotating clockwise, while motors with even numbers are rotating counterclockwise [44]. Torque produced by each motor can be expressed as τ i = kT i , where T i is the thrust of the i-th motor, and k is the ratio of the torque to thrust produced by each motor.…”

“…To compute the net force/torque acting on the UAV from all thrusters, first each motor's thrust and torque is decomposed into components on the body frame as given in Equation (30). Then the total force/torque for motors can be obtained [44].…”

“…If all cant angles (φ i ) are equal to zero, it means that all motors are perpendicular to the UAV horizontal plane and the generalized model will reduce to a conventional multi-rotors model. To control the torque applied to the system, motors with odd numbers are rotating clockwise, while motors with even numbers are rotating counterclockwise [44]. Torque produced by each motor can be expressed as =…”