Modeling of the Haltere-A Natural Micro-Scale Vibratory Gyroscope

Abstract

Vibratory gyroscopes have gained immense popularity in the microsystem technology because of their suitability to planar fabrication techniques. With considerable effort in design and fabrication, MEMS (Micro-electro-mechanical-system) vibratory gyroscopes have started pervading consumer electronics apart from their well known applications in aerospace and defence systems. Vibratory gyroscopes operate on the Coriolis principle for sensing rates of rotation of the r tating body. They typically employ capacitive or piezoresistive sensing for detecting the Coriolis force induced motion which is, in turn, used to determine the impressed rate of rotation. Interestingly, Nature also uses vibratory gyroscopes in its designs. Over several years, it has evolved an incredibly elegant design for vibratory gyroscopes in the form of dipteran halteres. Dipterans are known to receive mechanosensory feedback on their aerial rotations from halteres for their flight navigation. Insect biologists have also studied this sensor and continue to be fascinated by the intricate mechanism employed to sense the rate of rotation. In most Diptera, including the soldier fly, Hermetia illucens, the halteres are simple cantilever like structures with an end mass that probably evolved from the hind wings of the ancestral four-winged insect form. The halteres along with their connecting joint with the fly’s body constitute a mechanism that is used for muscle-actuated oscillations of the halteres along the actuation direction. These oscillations occur in the actuation plane such that any rotation of the insect body, induces Coriolis force on the halteres causing their plane of vibration to shift laterally by a small degree. This induced deflection along the sensing plane (out of the haltere’s actuation plane) results in strain variation at the base of the haltere shaft, which is sensed by the campaniform sensilla. The goal of the current study is to understand the strain sensing mechanism of the haltere, the nature of boundary attachments of the haltere with the fly’s body, the reasons of asymmetrical geometry of the haltere, and the interaction between both wings and the contralateral wing and haltere. In order to understand the haltere’s strain sensing mechanism, we estimate the strain pattern at the haltere base induced due to rotations about the body’s pitch, roll, and yaw axes. We model the haltere as a cantilever structure (cylindrical stalk with a spherical end knob) with experimentally determined material properties from nanoindentation and carry out analytical and numerical (finite element) analysis to estimate strains in the haltere due to Coriolis forces and inertia forces resulting from various body rotations. From the strain pattern, we establish a correlation between the location of maximum strain and the position of the campaniform sensilla and propose strain sensing mechanisms. The haltere is connected to the meta thoracic region of the fly’s body by a complicated hinge mechanism that actuates the haltere into angular oscillations with a large amplitude of 170 ◦ in the actuation plane and very small oscillation in the sensing plane. We aim to understand the reason behind the dissimilar boundary attachments along the two directions. We carry out bending experiments using micro Newton force sensor and estimate the stiffness along the actuation and sensing directions. We observe that the haltere behaves as a rigid body in the actuation direction and a flexible body in the sensing direction. We find the haltere to be a resonating structure with two different kinds of boundary attachments in the actuation and sensing directions. We create a finite element model of the haltere joint based on the optical and scanning microscope images, approximate material properties, and stiffness properties obtained from the bending experiments. We subsequently validate the model with experimental results. The haltere geometry has asymmetry along the length and the cross-section. This specific design of the haltere is in contrast to the the existing MEMS vibratory gyroscope, where the elastic beams supporting the proof mass are typically designed with symmetric cross-sections so that there is a mode matching between the actuation and the sensing vibrations. The mode matching provides high sensitivity and low bandwidth. Hence, we are interested in understanding the mechanical significance of the haltere’s asymmetry. First, we estimate the location of the maximum stress by using the actual geometry of the haltere. Next, by using the stiffness determined from bending experiments and mass properties from the geometric model, we find the natural frequencies along both actuation and sensing directions. We compare these findings with existing MEMS vibratory gyroscopes. The dipteran halteres always vibrate at the wing beat frequency. Each wing maintains 180 ◦ phase difference with its contralateral haltere and the opposite wing. Both wings and the contralateral wing-haltere mechanism exhibit coupled oscillatory motion through passive linkages. These linkages modulate the frequency and maintain the out- of-phase relationship. We explore the dynamics behind the out-of-phase behaviour and the frequency modulation of the wing-wing and wing-haltere coupled oscillatory motion. We observe that the linear coupled oscillatory model can explain the out-of-phase relationship between the two wings. However, a nonlinear coupled oscillator model is required to explain both frequency synchronization and frequency modulation of the wing with the haltere. We also carry out a finite element analysis of the wing-haltere mechanism and show that the out-of-phase motion between the wing and the haltere is due to the passive mechanical linkage of finite strength and high actuation force. The results of this study reveal the mechanics of the haltere as a rate sensing gyroscope and show the basis of the Nature’s design of this elegant sensor. This study brings out two specific features— the large amplitude actuated oscillations and the asymmetric geometry of the haltere structure— that are not found in current vibratory gyroscope designs. We hope that our findings inspire new designs of MEMS gyroscopes that have elegance and simplicity of the haltere along with the desired performance.