A generalized mathematical model to analyze the nonlinear behavior of a controlled gyroscope in gimbals

Abstract

In this paper, the effect of the parameter variation on the stability and dynamic behavior of a gyroscope in gimbals with a feedback control system, formed by a Proportional + Integral $+$ Derivative (PID) controller and a DC motor with an ideal train gear is researched. The generalized mathematical model of the gyro is obtained from the Euler-Lagrange equations by using the nutation theory of the gyroscope. The use of approximated models of the control system are deduced from the mathematical model of the gyro, taking into account that the integral action of the PID controller is constrained and that the inductance of the DC motor may be negligible. The analysis and choice of appropriate state variables to simulate the dynamic behavior of different models of the gyro are also considered. The paper shows that from the analysis of the equilibrium points, a Bogdanov Takens and a Poincare-Andronov-Hopf bifurcation can appear. These bifurcations are analyzed from the calculation of a parameter which is known as the first Lyapunov value, showing that it is possible to deduce a procedure to find out when a complicated model can be substituted by a simpler one. In particular, the possibility of self-oscillating and chaotic behavior for different models of the system by using the first Lyapunov value as a function of the parameters of the PID controller is researched. Numerical simulations have been performed to evaluate the analytical results and the mathematical discussions.