Butterfly Gyroscope

Butterfly Gyroscope

Background
Benchmark ID	butterflyGyroscope_n17361m1q12
Category	oberwolfach
System-Class	LTI-SOS
Parameters
nstates	17361
ninputs	1
noutputs	12
nparameters	0
components	B, C, E, K, M
Copyright
License	NA
Creator	Christian Himpe
Editor	Christian Himpe Petar Mlinarić Manuela Hund
Location
NA

Description

Figure 1

Figure 2

Figure 3

The butterfly gyro is developed at the Imego Institute in an ongoing project with Saab Bofors Dynamics AB. The butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications. The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market. This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model-based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element. The project has also included the manufacturing of an ASIC, named µSIC, that has been especially designed for the sensor (see Fig. 1).

The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element. The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see Fig. 2 and Fig. 3); this is the reason the gyro is called the butterfly. Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted. This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs. The excitation electrodes are the smaller dashed areas shown in Fig. 2. The detection electrodes correspond to the four larger ones. By applying DC-biased AC voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane. This is the excitation mode. As the structure rotates about the axis of sensitivity (see Fig. 2), each of the masses will be affected by a Coriolis acceleration. This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane. This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes.

When planning for and making decisions on future improvements of the butterfly, it is important to improve the efficiency of the gyro simulations. Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues. Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases, and the effect of force-feedback.

The use of model order reduction indeed decreases runtimes for repeated simulations. Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation. This will prove helpful in testing the model-based Kalman signal processing algorithms that are being designed for the butterfly gyro.

The structural model of the gyroscope has been done in ANSYS using quadratic tetrahedral elements (SOLID187, see Fig. 3). The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches. It includes the pure structural mechanics problem only. The load vector is composed of time-varying nodal forces applied at the centers of the excitation electrodes (see Fig. 2). The amplitude and frequency of each force are equal to $0.055 \, \mu \text{N}$ and $2384 \, \text{Hz}$, respectively. The Dirichlet boundary conditions have been applied to all degrees of freedom of the nodes belonging to the top and bottom surfaces of the frame. The output nodes are listed in Table 2 and correspond to the centers of the detection electrodes (see Fig. 3). The structural model

$\begin{align} M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\ y(t) & = C x(t), \end{align}$

contains the mass $M$ and stiffness matrices $K$. The damping matrix $E$ can be modeled as $E = \alpha M + \beta K$, where the typical values of $\alpha$ and $\beta$ are $0$ and $10^{-6}$ respectively. The nature of the damping matrix is in reality more complex (squeeze film damping, thermo-elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test. $B$ is the load vector, $C$ is the output matrix.

The statistics for the matrices are shown in Table 1.

Table 1: System matrices for the gyroscope.

Matrix	m	n	nnz	Is Symmetric?
M	17361	17361	178896	Yes
K	17361	17361	519260	Yes
B	17361	1	8	No
C	12	17361	12	No

The outputs are detailed in Table 2.

Table 2. Outputs for the Butterfly Gyro Model.

Index	Code	Comment
1-3	det1m_Ux, det1m_Uy, det1m_Uz	Displacements of detection electrode 1, (bottom left large electrode of Fig. 2)
4-6	det1p_Ux, det1p_Uy, det1p_Uz	Displacements of detection electrode 2, (bottom right large electrode of Fig. 2)
7-9	det2m_Ux, det2m_Uy, det2m_Uz	Displacements of detection electrode 3, (top left large electrode of Fig. 2)
10-12	det2p_Ux, det2p_Uy, det2p_Uz	Displacements of detection electrode 4, (top right large electrode of Fig. 2)

The model reduction of the gyroscope model by means of mor4fem is described in .

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 35889, see: .

Data

Download matrices in Matrix Market format:

• ButterflyGyro-dim1e5-gyro.tar.gz (7.4 MB)

The matrix name is used as an extension of the matrix file. File *.C.names contains a list of output names written consecutively.

Dimensions

System structure:

$\begin{align} M \ddot{x}(t) + E \dot{x}(t) + K x(t) & = B u(t) \\ y(t) & = C x(t) \end{align}$

System dimensions:

$M \in \mathbb{R}^{17361 \times 17361}$, $E \in \mathbb{R}^{17361 \times 17361}$, $K \in \mathbb{R}^{17361 \times 17361}$, $B \in \mathbb{R}^{17361 \times 1}$, $C \in \mathbb{R}^{12 \times 17361}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

Oberwolfach Benchmark Collection, Butterfly Gyroscope. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Butterfly_Gyroscope

@MISC{morwiki_gyro,
  author =       {{Oberwolfach Benchmark Collection}},
  title =        {Butterfly Gyroscope},
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Butterfly_Gyroscope},
  year =         2004
}

• For the background on the benchmark:

@INPROCEEDINGS{morBil05,
  author =        {D. Billger},
  title =         {The Butterfly Gyro},
  booktitle =     {Dimension Reduction of Large-Scale Systems},
  publisher =     {Springer-Verlag, Berlin/Heidelberg, Germany},
  year =          2005,
  volume =        45,
  pages =         {349--352},
  series =        {Lecture Notes in Computational Science and Engineering},
  doi =           {10.1007/3-540-27909-1_18}
}