A Self-tuning Algorithm for Z-axis Micro Rate Integrating Gyroscopes

Vibratory micro rate integrating gyroscopes (MRIG) are conventionally modeled using the position and velocity of the gyroscope, as the state space variables. This work describes the dynamic analysis of a z-axis MRIG using an alternate set of dynamic variables, which are the angular momentum, the inner product between position and linear momentum vectors, the Lagrangian and total energy of the system. This alternate description is used to derive the conditions for the ideal gyroscope to operate at the correct precession rate with minimal quadrature. A self-tuning algorithm is presented, which compensates the gyroscope’s damping and sitffness mismatches, while allowing the gyroscope to precess at the correct precession rate with minimal quadrature and reasonable control effort. INTRODUCTION Z-axis micro rate integrating gyroscopes are sensing devices that are used to measure the angular rotation of the base on which they are mounted. The principle behind the working of any gyroscope is popularly known as the ‘gyroscopic effect’. In the case of vibratory type mechanical gyroscopes, the vibrating mass is set to oscillate along a line and as the base rotates the inertia of the vibrating mass holds back the line of vibration with respect to the inertial frame of reference (X ′−Y ′ coordinate frame), resulting in an apparent precession of the gyro relative to the rotating frame of reference (x′− y′). It is this precession of the line of vibration, as seen from the (x′− y′) rotating frame that enables us to measure the angle of rotation of the rotating base. From a resonator point of view which is the case here, the resonator acts as ∗Address all correspondence to this author. the replacement to the 2-dof system described above. The N = 2 mode of the gyroscope (the first in-extensional mode of the gyroscope) has two distinct normal modes. The vibration in one of the normal modes is transferred to the other normal mode due to the Coriolis acceleration that couples the two normal modes in the presence of a nontrivial angular rotation rate. In practise, such an ideal device is difficult to build because of manufacturing imperfections which induce frequency mismatches between the two modes, non-diagonal stiffness terms and damping terms (both diagonal and non-diagonal). These mismatches degrade the performance and cause a non ideal elliptical motion, generally known as quadrature error, as well as an erroneous precession rate [1, 2]. Force balancing adaptive control schemes have been proposed in [6][8] to cancel the effect of mismatches and attain ideal gyro operation. Previous works like [1] and [2], discuss the complications that arise as a result of such mismatches. The schemes developed so far [3][8] cannot simultaneously compensate and attain the correct precession rate, which remains a hurdle in terms of the precession measurement being interrupted. Here we present a self-tuning control strategy that asymptotically achieves mismatch compensation and the correct precession rate with a improved convergence rate. In the next section, we introduce the 2-dof dynamic model of a gyroscope. To the best of our knowledge, none of the previous work [1, 3, 5] has described the dynamics of a z-axis gyroscope in terms of the angular momentum, the quantity which is the inner product of position and linear momentum vectors, the Lagrangian and the energy, which will be respectively denoted by the symbols h, p, L and e in this paper. This description in terms of the variables h, p, L and e is pivotal in identifying the ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference DSCC2012-MOVIC2012 1 Copyright © 2012 by ASME DSCC2012-MOVIC2012-8773 October 17-19, 2012, Fort Lauderdale, Florida, USA Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 01/11/2014 Terms of Use: http://asme.org/terms Figure 1. SCHEMATIC OF A GYROSCOPE correct measure of quadrature (phase difference in the motion of the two independent modes). We then utilize a similarity transformation to express the dynamics in terms of a second set of dynamic variables h̃, p̃, L̃ and e.