Two-time-scale analysis of a ring of coupled vibratory gyroscopes.

A coupling inertial navigation sensor (INS) system may proven to be beneficial for performance improvement, especially when the manufacturing yield is very low for meeting the specification requirement of various applications. For instance, navigation grade sensors using the current fabrication process would yield one in every few hundreds which would meet the specification requirement after careful selection process and testing. We propose to couple these sensors by putting together the "low grade" sensors in a small array of particular coupling topology to explore their stability properties of known parameter variations produced during the fabrication process. By coupling them in a particular way one may improve the system stability to effect the performance of the INS. Thus in this work we present a coupled inertial navigation sensor (CINS) system consisting of a ring of vibratory gyroscopes coupled through the driving axis of each individual gyroscope. Numerical simulations show that under certain conditions, which depend mainly on the coupling strength, the dynamics of the individual gyroscopes will synchronize with one another. The same simulations also show an optimal network size at which the effects of noise can be minimized, thus yielding a reduction in the phase drift. We quantify the reduction in the phase drift and perform an asymptotic analysis of the motion equations to determine the conditions for the existence of the synchronized state. The analysis yields an analytical expression for a critical coupling strength at which different nonzero mean oscillations merge in a pitchfork bifurcation; passed this critical coupling the synchronized state becomes locally asymptotically stable. The Liapunov-Schmidt (LS) reduction is then applied to determine the stability properties of the synchronized solution and to further show that the pitchfork bifurcation can be subcritical or supercritical, depending on the coefficient of the nonlinear terms in the equations of motion.