A Hybrid Control Approach for the Swing Free Transportation of a Double Pendulum with a Quadrotor

Abstract

In this article, a control strategy approach is proposed for a system consisting of a quadrotortransporting a double pendulum. In our case, we attempt to achieve a swing free transportationof the pendulum, while the quadrotor closely follows a specific trajectory. This dynamic system ishighly nonlinear, therefore, the fulfillment of this complex task represents a demanding challenge.Moreover, achieving dampening of the double pendulum oscillations while following a precisetrajectory are conflicting goals. We apply a proportional derivative (PD) and a model predictivecontrol (MPC) controllers for this task. Transportation of a multiple pendulum with an aerial robotis a step forward in the state of art towards the study of the transportation of loads with complexdynamics. We provide the modeling of the quadrotor and the double pendulum. For MPC wedefine the cost function that has to be minimized to achieve optimal control. We report encouragingpositive results on a simulated environmentcomparing the performance of our MPC-PD controlcircuit against a PD-PD configuration, achieving a three fold reduction of the double pendulummaximum swinging angle.

The Regulation of an Electric Oven and an Inverted Pendulum

“…For the first-order sliding-mode regulator, Equations ( 7) and (15), in the same way, in a feedback model with a reference value of 600 can be seen in Figures 6 and 7. In a very similar way, Equations (7) and (20) correspond to the response of Figures 6 and 7 for the second-order slidingmode regulator. are applied to the electric oven.…”

“…is the force applied to the mathematical model, which is the regulation variable. In this case, the generalized variables are 𝑥 and 𝜃; therefore, 𝑞 = (𝑥, 𝜃) , and we obtain the elements of each Lagrange equation as follows: The mathematical model of the inverted pendulum is derived from the Euler-Lagrange equations [19][20][21][22][23]. More details of the mathematical model can be found in [19,20,23,24].…”

“…The second-order sliding-mode regulator is composed of both σ and σ , with the purpose of reaching σ → 0 at finite convergence time. The second-order sliding-mode regulator of Equation ( 19) can be rewritten as in Equation (20).…”

“…The linearization method is known as an approximate linearization, which is based on a Taylor series expansion to represent an asymmetrical non-linear mathematical model as a linear mathematical model. This method is also known as Jacobian linearization because of the Jacobian matrices that are used for the linear approximation [19,20,23].…”

“…where N(s) = [19,20,23], the first-order sliding-mode regulator of [7][8][9], and the second-order sliding-mode regulator of [10][11][12] are known; this manuscript considers a comparison of the three types of regulators for regulation in an electric oven and in an inverted pendulum.…”