We describe a modification of a previously described measurement–analysis protocol to determine the intrinsic properties of triaxial accelerometers by using a measurement protocol based on angular stepwise rotation in the Earth’s gravitational field. This study was conducted with MEMS triaxial accelerometers that were co-integrated in four consumer-grade wireless microsensors. The measurements were carried out on low-cost rotation tables in different laboratories in different countries to simulate the reproducibility environment encountered in inter-comparisons of calibration capabilities. We used a previously described calibration–uncertainty metric to independently characterize the overall uncertainty of the calibration and analysis process. The intrinsic property analysis suggested, and the uncertainty metric confirmed, an unacceptably large error in one combination of microsystem and low-cost rotation table. A simple modification of the analysis protocol provided a substantial improvement in the reproducibility of the protocol with all combinations of microsystem and rotation table. Later, measurements with a high-performance triaxial accelerometer using a significantly more expensive rotation table carried out at one location further validated the usefulness of this modification. The results reported here also demonstrate the existence of unidentified defects in one microsystem and one low-cost rotation table that interact with each other in ways not currently understood to produce anomalously large errors with the old protocol but not with the new protocol.

We extend the use of the intrinsic properties calibration method for triaxial accelerometers that we reported previously from discrete angular steps to using a constant rotation rate to produce a time varying sinusoidal excitation in the earth's gravitational field. We show that this extension yields the low frequency calibration response of the device under test. Whereas traditional vibration-based methods using shakers generally exhibit an increased measurement uncertainty with decreased excitation frequency, we show that this approach does not. We report results obtained from a commercial triaxial digital accelerometer from DC up to a 0.5 Hz rotation rate. The maximum rotation rate that we report is limited by our rotation stage; but we expect that the method can be extended to higher rotation rates with an upper limit constrained by what can be tolerated as a maximum centripetal acceleration.

We simulated the effects of gimbal-alignment errors and rotational step-size errors on measurements of the sensitivity matrix and intrinsic properties of a triaxial accelerometer. We restricted the study to measurements carried out on a two-axis calibration system using a previously described measurement and analysis protocol. As well as imperfections in the calibration system, we simulated imperfect orthogonality of the accelerometer axes and non-identical sensitivity of the individual accelerometers in an otherwise perfect triaxial accelerometer, but we left characterization of other accelerometer imperfections such as non-linearity for future study. Within this framework, sensitivity-matrix errors are caused by imperfections in the construction and installation of the accelerometer calibration system, but not by the accelerometer imperfections included in the simulations. We use the results of this study to assign type B uncertainties to the components of the sensitivity matrix and related intrinsic properties due to imperfections in the measurement system. For calibrations using a reasonably well manufactured and installed multi-axis rotation stage such as that studied in this paper, we estimated upper bounds to the standard uncertainties of the order of 1 ×10−5 , 2 ×10−5 , 2 ×10−4 , and 5 ×10−5 for the intrinsic sensitivities, diagonal elements of the sensitivity matrix, off-diagonal elements of the sensitivity matrix, and zero-acceleration offsets, relative to a sensitivity-matrix element of 1, respectively, and 5 ×10−3 degrees for the intrinsic angles

A type-A uncertainty analysis of the magnitude of the local gravitational field can be used to validate type-B uncertainty estimates of calibrations of triaxial accelerometers by rotation in the gravitational field. We demonstrate the application of this method to validate a type-B uncertainty estimate at the sub 0.1% level of the errors introduced by a high-accuracy, two-axis, accelerometer-characterization system.

A standard paradigm of localization microscopy involves extension from two to three dimensions by engineering information into emitter images, and approximation of errors resulting from the field dependence of optical aberrations. We invert this standard paradigm, introducing the concept of fully exploiting the latent information of intrinsic aberrations by comprehensive calibration of an ordinary microscope, enabling accurate localization of single emitters in three dimensions throughout an ultrawide and deep field. To complete the extraction of spatial information from microscale bodies ranging from imaging substrates to microsystem technologies, we introduce a synergistic concept of the rigid transformation of the positions of multiple emitters in three dimensions, improving precision, testing accuracy, and yielding measurements in six degrees of freedom. Our study illuminates the challenge of aberration effects in localization microscopy, redefines the challenge as an opportunity for accurate, precise, and complete localization, and elucidates the performance and reliability of a complex microelectromechanical system.

We show that the calibration of tri-axis accelerometers based on the device’s intrinsic properties alleviates the uncertainty due to mounting misalignment in comparison to the use of the sensitivity matrix. The intrinsic properties of a tri-axis accelerometer are based on a (u, v, w) coordinate system that represent the direction of maximum sensitivities of each of the three accelerometers (U, V, W) and are assumed not to be perfectly orthogonal to each other. The calibration procedure requires rotation of the device in the gravitational field around each of the Cartesian coordinate (x, y, z) axes. One component in driving down the uncertainty of laboratory comparisons and calibration repeats relates to misalignment in mounting the device onto the calibration instrument. We show that the uncertainty of the cross-axis terms of the sensitivity matrix is a dominating factor affecting uncertainty down to a 0.01° misalignment at a 100 µV noise level. The misalignment component can be exacerbated when calibrating modern microelectromechanical systems (MEMS)-based accelerometers, which are typically a few millimeters in dimension.

We propose a concept of the use of dynamic uncertainty to improve the accuracy and robustness of digital sensor networks. The digital smart transducer is integrated with a microcontroller that reads the binary data from the calibrated sensor, performs an analysis to determine the measurand and its instantaneous uncertainty, and communicates these values in SI units with a time stamp to the host computer. We postulate that communicating the measurement with its uncertainty from the sensor node could eliminate the need for the host computer to maintain a database of the sensor models and performance parameters while at the same time improving accuracy and maintainability. We demonstrate this protocol using calibrated triaxial accelerometers as a test case. This protocol could eliminate the need for the host computer(s) to keep track of individual sensor performance parameters and facilitate the functionality to seamlessly upgrade sensors in the network.

We make use of the intrinsic aberrations of an optical microscope to track single particles in three dimensions, and we combine information from multiple particles on a rigid body of a microelectromechanical system to measure its motion in six degrees of freedom. Our tracking method provides an extraordinary amount of information from an ordinary imaging system, revealing unintentional motion of the microsystem due to fabrication tolerance and nanoscale clearance between parts in sliding contact. Our work facilitates quantification and study of the actuation performance and reliability of complex microsystems.

We report on a new approach to characterize the performance of a laser Doppler vibrometer (LDV). The method uses two acousto-optic modulators (AOMs) to frequency shift the light from an LDV by a known quantity to create a synthetic velocity shift that is traceable to a frequency reference. Results are presented for discrete velocity shifts and for sinusoidal velocity shifts that would be equivalent to what would be observed in an ideal accelerometer vibration calibration. The method also enables the user to sweep the synthetic vibration excitation frequency to characterize the bandwidth of an LDV together with its associated electronics.

A new method is described to provide a primary calibration of shock measurements produced by a shock measurement system consisting of pendulum excitation and laser Doppler velocimetry. The method uses the laser Doppler velocimeter to determine the total distance traveled by a rigid block that slides along a Teflon (fluorocarbon) channel after being struck by a pendulum head, and the resulting distance is compared to the distance measured by an SI-traceable length measurement. The instantaneous velocity of the block is measured by the velocimeter and is used to calculate the displacement of the block by integrating the velocity data. The result is compared to the displacement measured using calibrated rulers and calipers. The method was applied to an independently calibrated commercial velocimeter for impact accelerations ranging from 2000 to 30,000 m/s2. The results of the independent mechanical-displacement measurements agreed with those from the commercial velocimeter to within ±0.3 %, with better agreement above accelerations of order 10,000 m/s2 to within ±0.1 %. A conservative, upper-bound, uncertainty analysis included the effects of noise and other random errors, as well as type B estimates for systematic errors from occasional momentary demodulation failures (dropouts), use of a different number of rulers before and after shock distance measurement, and the relative frequency response of the velocimeter.

A standard paradigm of localization microscopy involves extension from two to three dimensions by engineering information into emitter images, and approximation of errors resulting from field dependence of optical aberrations. We invert this standard paradigm, introducing the concept of fully exploiting the latent information of intrinsic aberrations by complete calibration of an ordinary microscope, enabling accurate localization of single emitters in three dimensions across an ultrawide and deep field. To complete the extraction of spatial information from microscale bodies ranging from imaging substrates to microsystem technologies, we introduce a synergistic concept of the rigid transformation of the positions of multiple emitters in three dimensions, improving precision and yielding measurements in six degrees of freedom. Our study provides new insight into the challenge of aberration effects in localization microscopy, redefines the challenge as an opportunity for accurate, precise, and complete localization, and elucidates the performance and reliability of a complex microsystem.

Microelectromechanical systems (MEMS) that require contact of moving parts to implement complex functions exhibit limits to their performance and reliability. Here, we advance our particle tracking method to measure MEMS motion in operando at nanometer, microradian, and millisecond scales. We test a torsional ratcheting actuator and observe dynamic behavior ranging from nearly perfect repeatability, to transient feedback and stiction, to terminal failure. This new measurement capability will help to understand and improve MEMS motion.

The common assumption that precision is the limit of accuracy in localization microscopy and the typical absence of comprehensive calibration of optical microscopes lead to a widespread issue—overconfidence in measurement results with nanoscale statistical uncertainties that can be invalid due to microscale systematic errors. In this article, we report a comprehensive solution to this underappreciated problem. We develop arrays of subresolution apertures into the first reference materials that enable localization errors approaching the atomic scale across a submillimeter field. We present novel methods for calibrating our microscope system using aperture arrays and develop aberration corrections that reach the precision limit of our reference materials. We correct and register localization data from multiple colors and test different sources of light emission with equal accuracy, indicating the general applicability of our reference materials and calibration methods. In a first application of our new measurement capability, we introduce the concept of critical-dimension localization microscopy, facilitating tests of nanofabrication processes and quality control of aperture arrays. In a second application, we apply these stable reference materials to answer open questions about the apparent instability of fluorescent nanoparticles that commonly serve as fiducial markers. Our study establishes a foundation for subnanometer localization accuracy in widefield optical microscopy.

We fabricate and test subresolution aperture arrays as calibration devices for optical localization microscopy. An array pitch with a relative uncertainty of approximately three parts in ten thousand enables localization with subnanometer accuracy.

Three-axis MEMS accelerometers are typically characterized in terms of their cross-sensitivity matrix. We present an analysis and protocol to characterize them in terms of their intrinsic properties, which we define as the responsivity of each accelerometer along its axix of maximum response in a gravitational field and the angles between each of these axes. An analysis and test protocol is developed to determine these properties using a gimbal to roate each axis of the device orthogonally to the gravitational field. We propose tat this approach is better suited for laboratory inter-comparisons for SI relaization of acceleration units.

Cross-sensitivity matrices are used to translate the response of three-axis accelerometers into components of acceleration along the axes of a specified coordinate system. For inertial three-axis accelerometers, this coordinate system is often defined by the axes of a gimbal-based instrument that exposes the device to different acceleration inputs as the gimbal is rotated in the local gravitational field. Therefore, the cross-sensitivity matrix for a given three-axis accelerometer is not unique. Instead, it depends upon the orientation of the device when mounted on the gimbal. We define nine intrinsic parameters of three-axis accelerometers and describe how to measure them directly and how to calculate them from independently determined cross-sensitivity matrices. We propose that comparisons of the intrinsic parameters of three axis accelerometers that were calculated from independently determined cross-sensitivity matrices can be useful for comparisons of the cross-sensitivity-matrix measurement capability of different institutions because the intrinsic parameters will separate the accelerator-gimbal alignment differences among the participating institutions from the purely gimbal-related differences, such as gimbal-axis orthogonality errors, z-axis gravitational-field alignment errors, and angle-setting or angle-measurement errors.

Mechanical linkages are fundamentally important for the transfer of motion through assemblies of parts to perform work. Whereas their behavior in macroscale systems is well understood, there are open questions regarding the performance and reliability of linkages with moving parts in contact within microscale systems. Measurement challenges impede experimental studies to answer such questions. In this study, we develop a novel combination of optical microscopy methods that enable the first quantitative measurements at nanometer and microradian scales of the transfer of motion through a microelectromechanical linkage. We track surface features and fluorescent nanoparticles as optical indicators of the motion of the underlying parts of the microsystem. Empirical models allow precise characterization of the electrothermal actuation of the linkage. The transfer of motion between translating and rotating links can be nearly ideal, depending on the operating conditions. The coupling and decoupling of the links agree with an ideal kinematic model to within approximately 5%, and the rotational output is perfectly repeatable to within approximately 20 microradians. However, stiction can result in nonideal kinematics, and input noise on the scale of a few millivolts produces an asymmetric interaction of electrical noise and mechanical play that results in the nondeterministic transfer of motion. Our study establishes a new approach towards testing the performance and reliability of the transfer of motion through assemblies of microscale parts, opening the door to future studies of complex microsystems.

The concept of localization precision, which is essential to localization microscopy, is formally extended from optical point sources to microscopic rigid bodies. Measurement functions are presented to calculate the planar pose and motion of microscopic rigid bodies from localization microscopy data. Physical lower bounds on the associated uncertainties - termed centroid precision and orientation precision - are derived analytically in terms of the characteristics of the optical measurement system and validated numerically by Monte Carlo simulations. The practical utility of these expressions is demonstrated experimentally by an analysis of the motion of a microelectromechanical goniometer indicated by a sparse constellation of fluorescent nanoparticles. Centroid precision and orientation precision, as developed here, are useful concepts due to the generality of the expressions and the widespread interest in localization microscopy for super-resolution imaging and particle tracking.

This report studies the effect of frequency offset, quantization error, random additive noise, and random phase jitter on the results of sine fitting and performing Discreet Fourier Transforms (DFT) of measurements of sinewaves with Data Acquisition Systems (DAS).Quantization error is the only effect of non-linearity that is considered.The majority of the simulations are carried out for full-scale signals on a 16 bit DAS, but the simulations do not simulate the effect of clipping on portions of noisy signals that require more than 16 bits.Instead, the signals are rounded in a way that provides the minimum number of bits required to represent the integer part of the signal.The results of this approach are easier to interpret and simple corrections for this shortcoming are given.Only quantization error is studied as a function of available bits, ranging from 8-to 64-bits in increments of 8 bits.Among other results, it is shown that probability plots of the residuals of fits of sinewaves to simulated distorted sinewaves can distinguish among random additive noise, random jitter, and frequency offset error when the effect of quantization noise is small compared to the effects of these other sources of distortion.