# The Butterfly Gyro (35889)

### The Butterfly Gyro (35889)

Dag Billger, The Imego Institute, Arvid Hedvalls Backe 4, SE-411 33, Göteborg, Sweden, Tel. +46 317 501 853, Fax. +46 317 501 801, Email: dag.billger@imego.com

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 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 of importance 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 from time-varying nodal forces applied at the centres of the excitation electrodes (see Fig. 2). The amplitude and frequency of each force is equal to 0.055 micro-Newtons and 2384 Hz, respectively. The Dirichlet boundary conditions have been applied to all degree 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 centres of the detection electrodes (see Fig. 3).

The structural modelM d^{2}x/dt^{2} + E dx/dt + K x = B

y = C x

contains the mass M and stifness 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 1e-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 dynamic model has been converted to Matrix Market format by means of mor4fem. The statistics for the matrices is 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 |

Table 2. Outputs for the Butterfly Gyro Model

# | 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) |

Download matrices in Matrix Market format in File 1. The size of the file is 7808482 bytes. The matrix name is used as an extension of the matrix file. File *.C.names contains a list of ouput names written consecutively.

The model reduction of the gyroscope model by means of mor4fem is described in Ref [1].

[1] Jan Lienemann, Dag Billger, Evgenii B. Rudnyi, Andreas Greiner, and Jan G. Korvink, MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices, the Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA