Back-Stepping Control of Quadrotor: A Dynamically Tuned Higher Order Like Neural Network Approach

Back-Stepping Control of Quadrotor: A Dynamically Tuned Higher Order Like Neural Network Approach

Abhijit Das (The University of Texas at Arlington, USA), Frank L. Lewis (University of Texas at Arlington, USA) and Kamesh Subbarao (The University of Texas at Arlington, USA)
DOI: 10.4018/978-1-61520-711-4.ch020
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Abstract

The dynamics of a quadrotor is a simplified form of helicopter dynamics that exhibit the same basic problems of strong coupling, multi-input/multi-output design, and unknown nonlinearities. The Lagrangian model of a typical quadrotor that involves four inputs and six outputs results in an underactuated system. There are several design techniques are available for nonlinear control of mechanical underactuated system. One of the most popular among them is backstepping. Backstepping is a well known recursive procedure where underactuation characteristic of the system is resolved by defining ‘desired’ virtual control and virtual state variables. Virtual control variables is determined in each recursive step assuming the corresponding subsystem is Lyapunov stable and virtual states are typically the errors of actual and desired virtual control variables. The application of the backstepping even more interesting when a virtual control law is applied to a Lagrangian subsystem. The necessary information to select virtual control and state variables for these systems can be obtained through model identification methods. One of these methods includes Neural Network approximation to identify the unknown parameters of the system. The unknown parameters may include uncertain aerodynamic force and moment coefficients or unmodeled dynamics. These aerodynamic coefficients generally are the functions of higher order state polynomials. In this chapter we will discuss how we can implement linear in parameter first order neural network approximation methods to identify these unknown higher order state polynomials in every recursive step of the backstepping. Thus the first order neural network eventually estimates the higher order state polynomials which is in fact a higher order like neural net (HOLNN). Moreover, when these NN placed into a control loop, they become dynamic NN whose weights are tuned only. Due to the inherent characteristics of the quadrotor, the Lagrangian form for the position dynamics is bilinear in the controls, which is confronted using a bilinear inverse kinematics solution. The result is a controller of intuitively appealing structure having an outer kinematics loop for position control and an inner dynamics loop for attitude control. The stability of the control law is guaranteed by a Lyapunov proof. The control approach described in this chapter is robust since it explicitly deals with unmodeled state dependent disturbances without needing any prior knowledge of the same. A simulation study validates the results such as decoupling, tracking etc obtained in the paper.
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Nomenclature

  • = subscript or superscript index

  • = mass of quadrotor

  • = inertia matrix

  • = auxiliary matrix

  • x,y,z = position of quadrotor in inertial frame

  • = roll angle

  • = pitch angle

  • = yaw angle

  • = generalized state vector representation

  • = position vector

  • = attitude vector

  • = desired position vector

  • = desired attitude vector

  • g = gravity

  • = force from the i’th motor

  • = summation of four motor forces

  • = rotational speed of i’th motor

  • = distance from motor to center of gravity

  • = force-to-moment scaling factor

  • = torque vector

  • = coriolis component

  • = state dependent disturbances including aerodynamic nonlinearities

  • = state dependent disturbances including aerodynamic nonlinearities

  • = tracking errors

  • = sliding mode errors

  • = basis function for NN approximation

  • = ideal NN weights

  • = desired ideal force

  • = positive definite design matrix

  • = positive definite design matrix

  • = controller gain matrix

  • = NN tuning error vector

  • = design constant

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