A proposed method of processing the outputs of multiple gyroscopes to increase the accuracy of rate (that is, angular- velocity) readings has been developed theoretically and demonstrated by computer simulation. Although the method is applicable, in principle, to any gyroscopes, it is intended especially for application to gyroscopes that are parts of microelectromechanical systems (MEMS). The method is based on the concept that the collective performance of multiple, relatively inexpensive, nominally identical devices can be better than that of one of the devices considered by itself. The method would make it possible to synthesize the readings of a single, more accurate gyroscope (a “virtual gyroscope”) from the outputs of a large number of microscopic gyroscopes fabricated together on a single MEMS chip. The big advantage would be that the combination of the MEMS gyroscope array and the processing circuitry needed to implement the method would be smaller, lighter in weight, and less power-hungry, relative to a conventional gyroscope of equal accuracy.

The method (see figure) is one of combining and filtering the digitized outputs of multiple gyroscopes to obtain minimum-variance estimates of rate. In the combining-and-filtering operations, measurement data from the gyroscopes would be weighted and smoothed with respect to each other according to the gain matrix of a minimum-variance filter. According to Kalman-filter theory, the gain matrix of the minimum-variance filter is uniquely specified by the filter covariance, which propagates according to a matrix Riccati equation. The present method incorporates an exact analytical solution of this equation.The analytical solution reveals a wealth of theoretical properties and enables the consideration of several practical implementations. Among the most notable theoretical properties are the following:

- Even though the terms of the Riccati equation grow in an unbounded fashion, the Kalman gain can be shown to approach a steady-state matrix. This result is fortuitous because it simplifies implementation and can serve as a basis of practical schemes for realizing the optimal filter by use of a constant- gain matrix.
- The analytical solution enables the development of a complete statistical theory that characterizes the drift of the virtual gyroscope and provides theoretical limits of improvement obtainable by use of an ensemble of correlated sensors as a single virtual sensor.
- The minimum-variance gain matrix has been analyzed in detail. The structure of the gain matrix indicates the presence of a marginally unstable pole, which would make implementation impossible if it were not properly understood and compensated. In addition, a simple algebraic method for computing the optimal gain matrix has been developed, making it possible to avoid the Riccati equation completely.
- The notion of statistical commonmode rejection (CMR) has been characterized mathematically. For statistically uncorrelated gyroscopes, it is shown that the component drift variances add like parallel resistors (e.g., in units of rad
^{2}/sec^{3}). For identical devices, this means that the combined drift is reduced by a factor of*1√N*compared to the individual gyroscope drift, where N is the number of gyroscopes being combined. - Potential improvement in drift is much more impressive when the devices are correlated. For correlated gyroscopes, an exact mathematical expression is developed for the combined drift. The expression indicates that the internal noise sources count to the extent that they infect multiple devices coherently (e.g., with common sign), and can be removed to the extent that they infect multiple devices incoherently (e.g., with randomized signs). Noise eliminated by correlations between devices can potentially reduce the virtual gyroscope drift far beyond the
*1√N*factor attainable using uncorrelated devices.

*This work was done by David S. Bayard of Caltech for NASA’s Jet Propulsion Laboratory.*

*In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to*

*Intellectual Assets Office JPL Mail Stop 202-233 4800 Oak Grove Drive Pasadena, CA 91109 (818) 354-2240 E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.*

*Refer to NPO-30533, volume and number of this NASA Tech Briefs issue, and the page number.*