Analog Integrated Circuits and Signal Processing, 29, 85–94, 2001  C 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Mathematical Modelling on the Quadrature Error of Low-rate Microgyroscope for Aerospace Applications BAO Y. YEH, 1 YUNG C. LIANG 1* AND FRANCIS E. H. TAY 2 1 Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge 119 260, Singapore 2 Department of Mechanical Engineering, National University of Singapore, Kent Ridge 119 260, Singapore Abstract. This paper presents the mathematical modelling on the quadrature error of a microgyroscope due to the imbalance of the asymmetric spring flexures. Quadrature error occurs when the proof mass of a microgyroscope oscillates along an axis that is not exactly parallel to the lateral-axis. The imbalance due to manufacturing variation can cause the proof mass to rotate when a force acts on the proof mass. The proposed mathematical model was verified by the finite element software IntelliCAD, and found to have a good agreement on angles of rotation of comb fingers. The mathematical model provides a new avenue in evaluating the quadrature error system and in saving the overall simulation time. Key Words: microsystem modelling, quadrature error modelling, low-rate microgyroscope 1. Introduction Z-axis vibratory rate microgyroscope is used to mea- sure Z-axis angular rate  z for aerospace and weapon systems. The structure of the vibratory fishhook micro- gyroscope is best illustrated by the spring mass system shown in Fig. 1. The micro-electro-mechanical structure comprises three major elements: namely the fishhook flexure sys- tem, the comb drive used to provide a force in the X direction to sustain oscillation [1] and the sensing ca- pacitance used to detect deflections along the Y-axis. The arched fishhook flexure provides an almost equal compliance in both the sensing and actuation axes, i.e., X and Y axes. For a Z-axis vibratory gyroscope, ro- tating proof mass generates the Coriolis acceleration. The proof mass oscillates along the X-axis, the ref- erence frame rotates about the Z-axis and the Coriolis acceleration is detected as deflections along the Y-axis. By monitoring the change in the upper and lower ca- pacitor banks, information on the rotation rate can then be obtained. ∗ Address correspondence to: Yung C. Liang, Tel.: +65 8742175, Fax: +65 7791103. E-mail: chii@nus.edu.sg In microgyroscope, the measured quantity is the Coriolis acceleration. The magnitude of Coriolis ac- celeration in a microgyroscope at low rotation rate is very small. Any off-axis oscillation of the proof mass to generate Coriolis acceleration will constitute an unde- sired motion that is 90 ◦ out of phase with the measured signal, thus called quadrature error. Quadrature error is common to occur in all vibratory rate microgyroscopes. William Tang [1] reported in his thesis on levitation force which causes an out-of-plane Z-axis oscillation while W. A. Clark [2] reported an in-plane Y-axis oscil- lation that is 90 ◦ out of phase with the Coriolis acceler- ation. The analytical modelling of cross-axis coupling of crab leg U-spring and serpentine spring was made in [3]. The work on modelling and analysis of quadrature error is quite necessary because it is linked to the os- cillatory deflection along the same (Y) axis of Coriolis acceleration and it has direct influence on the accuracy of rate measurement. For aerospace applications, a very low-rate micro gyroscope is required to infer the orientation of a mov- ing object. The requirements for inertial navigation in aerospace applications demand high accuracy. Further- more, the angular rate is very small causing Coriolis acceleration to be very small. Therefore, for a low