Christoph Gugg

An inverse problem approach to approximating sensor data in cyber physical systems

This paper presents a new matrix algebraic approach to the direct solution of inverse boundary value problems (IBVP). The synthesis of admissible functions and differentiating matrices is given particular attention. All the necessary mathematical elements are derived from basic principles. The method yields a linear operator for the solution of IBVPs. A single matrix multiplication is required at run-time to determine the solution. The number of FLOPS required is constant and known a-priory making the solution suitable for use in embedded real-time systems. The concept of discrete basis function design is introduced for the first time. The method enables the design of special discrete basis functions which yield optimal noise performance and numerical efficiency for specific tasks. All the methods are also verified in a laboratory test system and compared with results from an optical reference measurement.

Large scale optical position sensitive detector

This paper presents the development of a large scale optical position sensitive detector. The device is designed for the precise guidance of machines with respect to a reference laser plane in large working areas. The 1D detector has a measurement range of 1 [m] and, with the present implementation, a position measurement standard deviation of s <; ±0.6 [mm] in a 95% confidence interval. With this length it is orders of magnitude larger than all presently available position sensitive detectors. The instrument is based on a multi-camera image processing concept. An aluminum bar serves as the target for the laser. The targets surface is specially prepared to ensure optimal scattering of the laser light. Presently, four cameras with overlapping fields of view are deployed to observe the scattered light. Additional optical components reduce the susceptibility to extraneous light sources. Each camera is calibrated using Gram polynomials and the data from the four cameras is fused to give a consistent measurement over the complete measurement range. The linear nature of the computations algebraic framework offers the advantage that the error propagation can be computed analytically. Weighted polynomial approximation determines the calibration coefficients and weighted polynomial interpolation is used to determine the measurement results. Complete testing of the instrument is presented, whereby cross validation ensures the correct determination of errors. A Kolmogorov-Smirnov test is performed to determine the statistical nature of the measurement errors.

Analytic Multisource Data Fusion and its Application to a Large-Scale Optical PSD

This paper introduces an analytic algebraic framework for multisource data fusion using covariance weighted discrete orthogonal polynomials. The approach is implemented and tested in a prototype for a large-scale optical position sensitive detector (PSD). The device is designed for the precise guidance of machines with respect to a reference laser plane in large working areas. The 1-D detector has a measurement range of 1 m and, with the present implementation, a position measurement standard deviation of s <; ±0.6 mm in a 95% confidence interval at a distance of 300 m. With this length, it is orders of magnitude larger than all presently available PSDs. The instruments concept is based on a multicamera image processing setup, enabling a relatively compact hardware design. An aluminum bar serves as the target for the laser. The targets surface is specially prepared to ensure optimal scattering of the laser light. At present, four cameras with wide-angle lenses and overlapping fields of view monitor the scattered light; however, the theoretical framework supports the fusion of data from an arbitrary number of sensors. Additional optical components reduce the susceptibility to ambient light sources. Each camera is calibrated using Gram basis functions and the data from the four cameras are fused to give a consistent measurement over the complete measurement range. The linear nature of the computation offers the advantage that the error propagation can be derived analytically. Weighted polynomial approximation determines the calibration coefficients and weighted polynomial interpolation is used to obtain the measurement results. Complete testing of the instrument is presented, whereby cross validation ensures the correct quantification of errors. A Kolmogorov-Smirnov test is performed to prove the Gaussian nature of the measurement data and its error.

Structural deformation measurement via efficient tensor polynomial calibrated electro-active glass targets

This paper describes the physical setup and mathematical modelling of a device for the measurement of structural deformations over large scales, e.g., a mining shaft. Image processing techniques are used to determine the deformation by measuring the position of a target relative to a reference laser beam. A particular novelty is the incorporation of electro-active glass; the polymer dispersion liquid crystal shutters enable the simultaneous calibration of any number of consecutive measurement units without manual intervention, i.e., the process is fully automatic. It is necessary to compensate for optical distortion if high accuracy is to be achieved in a compact hardware design where lenses with short focal lengths are used. Wide-angle lenses exhibit significant distortion, which are typically characterized using Zernike polynomials. Radial distortion models assume that the lens is rotationally symmetric; such models are insufficient in the application at hand. This paper presents a new coordinate mapping procedure based on a tensor product of discrete orthogonal polynomials. Both lens distortion and the projection are compensated by a single linear transformation. Once calibrated, to acquire the measurement data, it is necessary to localize a single laser spot in the image. For this purpose, complete interpolation and rectification of the image is not required; hence, we have developed a new hierarchical approach based on a quad-tree subdivision. Cross-validation tests verify the validity, demonstrating that the proposed method accurately models both the optical distortion as well as the projection. The achievable accuracy is e ≤ ±0.01 [mm] in a field of view of 150 [mm] x 150 [mm] at a distance of the laser source of 120 [m]. Finally, a Kolmogorov Smirnov test shows that the error distribution in localizing a laser spot is Gaussian. Consequently, due to the linearity of the proposed method, this also applies for the algorithms output. Therefore, first-order covariance propagation provides an accurate estimate of the measurement uncertainty, which is essential for any measurement device.

Robust machine vision based displacement analysis for tunnel boring machines

Underground construction projects require accurate techniques for machine guidance control. This paper presents an advanced design of a fully integrated active laser target (ALT), which is employed as an optical displacement and orientation sensor. The instrument is mounted on a tunnel boring machine (TBM) and delivers the TBMs pitch and yaw angles to a remote host system. The optical arrangement of the device includes a semitransparent glass target and an opaque aluminum target, whereby measures are taken to increase the sensitivity of red light. A red reference laser beam is projected onto both parallel targets; the positions of the laser spots are observed by two Power over Ethernet (PoE) enabled Gigabit Ethernet (GigE) cameras. A robust plane-to-plane mapping using projective transformation is presented together with a cross validation procedure, which evaluates the quality of the calibration. An a-priori estimation of measurement uncertainty can be given. The system level calibration process yields two sets of transformation coefficients, such that the distortion associated with the optical components and the inexactness of the mechanical construction are effectively canceled out. Multiple images are acquired for a single measurement and analyzed statistically to deliver a statement of measurement uncertainty in order to compensate for mechanical vibrations during the machines operation.

Mathematical model and software architecture for the solution of inverse problems involving sensor arrays

This paper presents a mathematical model and partitioning for a software system for the solution of inverse problems involving arrays of sensors. Handling the data from the array of sensors as vectors and matrices, while defining the inverse problem as a least squares computation with linear constraints, leads naturally to the use of matrix algebra for the solution of the system of equations. The matrix algebra is also well suited for the use of automatic code generation to support the rapid development of embedded code. The full functionality of the proposed methods was demonstrated with an inclinometer based system for the monitoring of structural deformation.

Virtual calibration environment for a-priori estimation of measurement uncertainty

During product engineering of a measuring instrument, the question is which measures are necessary to achieve the highest possible measurement accuracy. In this context, a measuring instruments target uncertainty is an essential part of its requirement specifications, because it is an indicator for the measurements overall quality. This paper introduces an algebraic framework to determine the confidence and prediction intervals of a calibration curve; the matrix based framework greatly simplifies the associated proofs and implementation details. The regression analysis for discrete orthogonal polynomials is derived, and new formulae for the confidence and prediction intervals are presented for the first time. The orthogonal basis functions are numerically more stable and yield more accurate results than the traditional polynomial Vandermonde basis; the methods are thereby directly compared. The new virtual environment for measurement and calibration of cyber-physical systems is well suited for establishing the error propagation chain through an entire measurement system, including complicated tasks such as data fusion. As an example, an adaptable virtual lens model for an optical measurement system is established via a reference measurement. If the same hardware setup is used in different systems, the uncertainty can be estimated a-priori to an individual systems calibration, making it suitable for industrial applications. With this model it is possible to determine the number of required calibration nodes for system level calibration in order to achieve a predefined measurement uncertainty. Hence, with this approach, systematic errors can be greatly reduced and the remaining random error is described by a probabilistic model. Verification is performed via numerical experiments using a non-parametric Kolmogorov-Smirnov test and Monte Carlo simulation.

Conference Record - IEEE Instrumentation and Measurement Technology Conference

This paper presents a new matrix algebraic approach to the direct solution of inverse boundary value problems (IBVP). The synthesis of admissible functions and differentiating matrices is given particular attention. All the necessary mathematical elements are derived from basic principles. The method yields a linear operator for the solution of IBVPs. A single matrix multiplication is required at run-time to determine the solution. The number of FLOPS required is constant and known a-priory making the solution suitable for use in embedded real-time systems. The concept of discrete basis function design is introduced for the first time. The method enables the design of special discrete basis functions which yield optimal noise performance and numerical efficiency for specific tasks. All the methods are also verified in a laboratory test system and compared with results from an optical reference measurement.

An autonomous water sampling and monitoring device for deployment in harsh underground environment

A better understanding of hydrological processes is required in order to ensure fresh water supply for future generations. In this paper, a concept describing a transportable water sampling device is presented. The sampler is placed inside a mountains karst water region and autonomously collects discrete water samples over a period of one year for later isotope analysis. Furthermore, environmental properties are monitored and logged. A complete system design is presented, including mechanical and electronic components which fulfil the demanding system constraints. The device operates under water and must withstand a hydrostatic pressure of up to 20 [bar]. Because of the difficult deployment it has to have a compact and light design. The exceptional requirements on the device led to the development of a new sampling process and custom-built components. Due to the flexibility and economical production some components were produced by rapid prototyping. The validation confirmed that the device is able to improve the research on the flow of water in karst aquifers. It has also significant potential for further developments and a broad range of applications.

WiP Abstract: Model Based Design for the Real-Time Solution of Inverse Problems in Cyber-Physical Systems

Measurement data is continuously processed by cyber-physical systems in condition monitoring applications, e.g. sensor networks with inclinometers. The presented numerical ordinary differential equation (ODE) solver runs on independent and decentralized embedded systems thanks to its real-time capability and computational efficiency.