# Sensor Development and Diagnostics Lab

The **long term goal** of this research is to establish *new and innovative approaches to advance the leading edge of technology for the MEMS gyros to meet navigation grade performance*. To achieve this goal, in both cases, my research team is following these **short term objectives**: develop nonlinear mathematical models utilizing differential equations**; **fabricate macro-scale proofs of concept for nonlinear analysis and to demonstrate the underlying concepts; use perturbation and averaging techniques to identify various nonlinear behaviors, stability and nonlinear resonances associated with the proposed gyros; confirm these results using simulation of the mathematical models; confirm results experimentally (macro-models); design and fabrication of the MEMS gyros; address challenges arising from miniaturization; experimental characterization of noise, bias, drift, calibration and thermal compensation; verify MEMS performance experimentally; utilize the new gyros in INS applications discussed earlier.

IMUs have great potential for indoor motion tracking systems in manufacturing [3], human supporting systems [4], [5], and rehabilitation [6] areas. The position error simulations of different grades used in stationary IMUs [7,8] with small biases show growth in position error to 450mm after 7 seconds with a commercial grade MEMS IMU, 10 seconds with a tactical grade IMU, and 90 seconds with a navigation grade IMU [9]. Hence, position tracking systems using an IMU alone are not reliable for an extended period of time because of the exponential growth in errors associated with integration of IMU accelerometer and gyro signals [3, 10]. At best an IMU may be used as an orientation sensor utilizing accelerometers and magnetometers – because accelerometer can find the tilt angle without any integration step [10]. As a result, MEMS IMUs are usually hybridized with another sensor to estimate position [3, 11, 12]. There has been a great deal of research to reduce errors associated with the MEMS accelerometers [2]. The performance of the MEMS gyros, however, remain to be a critical issue before they can be utilized in applications requiring high precision, and gyro performance is hindered by errors, temperature dependency, gain error, noise and bias, input voltage and aging [10-12]. A MEMS gyro is usually designed as an electromechanically driven resonator [2]. Most commercial MEMS gyros are composed of two degrees of freedom – forming drive and sense modes. Upon periodic excitation of the drive mode mainly electrostatically), an external rotation (with rotation axis normal to that of the drive mode) results in a Coriolis force that transfers the vibratory energy from the drive to the sense mode. These modes are normally tuned to same resonance frequencies, to maximize their corresponding vibration amplitudes [13]. ** Realistically, because of manufacturing imperfections this is not achievable, and “significant amplification gains are lost” **[14]. New, innovative approaches are required to meet navigation grade (< 1 deg/hr) performance to advance the leading edge of technology [1]. Utilizing nonlinear dynamics in design process can significantly improve performance of gyros [14].

**Development of a New Nonlinear Gyro** – As referenced in literature [13], I was first, in two key publications [15 and 16], to utilize the nonlinear phenomenon of internal resonance to control the vibration of an oscillatory system. More specifically, in [15], I proposed a control strategy for controlling the free vibrations of a second-order system that is coupled to a controller (itself a second-order system) via quadratic nonlinear terms. Upon proper tuning of the controller’s natural frequency to a 2:1 ratio, the nonlinear terms act as an energy bridge, causing the vibrations of the plant to be transferred to the controller. In [16] we extended this concept to forced vibration, which takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of 2:1. *When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.* The governing equations in this case are very similar to those of a gyroscope, and the quadratic nonlinearities used are identical to the Coriolis terms in the gyro models. Hence, *my immediate goal will be to develop *** a new gyro design, based on the nonlinear internal resonance and saturation phenomenon.** Using the concepts developed in [16], in an electrostatically actuated gyro, the higher frequency drive mode (with very small excitation amplitude), can excite the lower frequency sense mode at high amplitudes through the Coriolis coupling. The magnitude of the sense mode is amplified through enhancing the quadratic nonlinear coupling through design. Having high amplitude/lower frequency sense mode oscillations will enhance the gyro’s signal to noise ratio and will reduce some of the errors, as described earlier. As discussed in [17] even when the drive mode is subjected to bounded-random excitations this nonlinear phenomenon is still attainable.

**Development of a parametric resonance gyro**(PhD 2&5) –

**Parametric resonance has recently (and uniquely) been applied to preliminary micro gyro designs [18-22]. Parametrically excited systems are described by differential equations, similar to Mathieu’s equation, where the excitation is a time varying coefficient of system parameters such as stiffness – which may include linear or nonlinear terms. Since there are no closed form solutions to nonlinear Mathieu’s equation, perturbation techniques are used to define regions of instability. The gyroscope designed here represents two nonlinear Duffing’s resonators that are parametrically excited through electrostatic actuation of nonlinear restoring force, and are coupled through the Coriolis force. Utilizing this intriguing concept the resulting system is robust under frequency mistuning [21], and the system allows for amplification of Coriolis force that is capable of exciting the sense mode far from its natural frequency. This behavior implies a system with large amplitude, broad frequency range of operation and less sensitivity to parameter variations, and the resulting design shows significant improvements on noise, drift, and scale-factor to meet tactical grade requirements [20, 21]. While the preliminary results are compelling, the dynamics of the system is still not fully understood, and there is still a great deal of room and opportunity to explore this concept [22].**

**In both cases, **in years 1 and 2 we will study these concepts on a macro system (PhD 1&2) while designing the MEMS gyros by year 3. We will utilize perturbation and averaging techniques to establish the performance characteristics, and regions of instability. These classes of nonlinear problems will exhibit chaotic behavior that is not desirable and must be characterized and avoided. In years 3-5 we will address fabrication, performance characterization (including measuring sensor inherent noise, drift, scale factor etc.), design modifications, calibration, thermal compensation and other challenges that are associated with MEMS gyros, as described earlier. It may help to **alleviate potential concerns **(about my limited strength in MEMS fabrication) by mentioning that I am in very close working relation with SFU School of Engineering colleagues at the Institute for Micromachine and Microfabrication Research (IMMR). The organizational support offered by the IMMR will be a tremendous asset to the development of these novel sensors. In years 3-5 the PhD 4 & 5, addressing the MEMS design and fabrications, will benefit from full technical and co-supervisory support. With the goal of achieving at least tactile gyro performance, in Year 5, PhDs integrate these sensors to TEHME 1 INS applications.

**IMPACT**

Through development of two gyros based on internal resonance and parametric resonance, I intend to make significant contributions to meet the compelling need for advancement of leading edge of technology for the MEMS gyros towards meeting navigation grade performance. Through this effort, we will reduce the need for additional sensors in INS, enhance the performance – or reduce the complexity – of filters and our dependency on black box tools.

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