This paper presents a signal processing algorithm that computes trajectories of drilled underground holes together with their uncertainties. The algorithm was tested on data obtained by an instrumented drilling rig, with which several holes were drilled. The collected data was preprocessed using hierarchical indexing to ensure easy processing later on. The algorithm uses Bezier curves and differential geometry to reconstruct the three-dimensional curve from its gradients, and describes a matrix-algebra framework for computing uncertainties and covariance matrices for the case of higher-dimensional curves.The novelty of the paper is the developed algebraic framework for the computation of uncertainties, as well as the comparison of the obtained results to two independent reference measurements to ensure accuracy. Furthermore, the developed algebraic framework of the covariance propagation allows for the computation of envelope curves, which shows the low uncertainty of the developed method, due to the high number of data samples used.

This paper presents a new method for modelling periodic signals having an aperiodic trend, using the method of variable projection. It extends the commonly used four parameter sine wave model by permitting the background to be time varying; additionally, any number of harmonics of the periodic portion can be modelled. This paper focuses on using B-Splines to implement a piecewise polynomial model for the aperiodic portion of the signal. A thorough algebraic derivation of the method is presented, as well as a comparison to using global polynomial approximation. It is proven that B-Splines work better for modelling a more complicated aperiodic portion when compared to higher order polynomials. Furthermore, the piecewise polynomial model is capable of modelling the local signal variations produced by the interaction of a control system with a process in industrial applications. An added benefit of using the method of variable projection is the possibility to calculate the covariances of the linear coefficients of the model, enabling the calculation of confidence and prediction intervals. The method is tested on both real measurement data acquired in industrial processes, as well as synthetic data. The method shows promising results for the precise characterization of periodic signals embedded in highly complex aperiodic backgrounds. Finally, snippets of the m-code are provided, together with a toolbox for B-Splines, which permit the implementation of the complete computation.

This paper presents a real-time parameter identification algorithm for periodic signals, based on the recursive variable projection (RVP) algorithm. The recursive implementation enables the tracking of time-varying parameters. The signal model is linear with respect to the amplitude parameters while being nonlinear with respect to the phase and frequency. This feature motivates the use of a variable projection based approach. Its performance is tested using Monte Carlo simulations and the results are compared with those obtained by a multiobjective Gauss-Newton (MGN) algorithm. Furthermore, the RVP algorithm is applied to measurement data acquired by a MEMS accelerometer and it is demonstrated that it can successfully track time-varying linear and nonlinear parameters.

This paper presents a new approach to the detection of discontinuities in the n-th derivative of observational data. This is achieved by performing two polynomial approximations at each interstitial point. The polynomials are coupled by constraining their coefficients to ensure continuity of the model up to the (n − 1)-th derivative; while yielding an estimate for the discontinuity of the n-th derivative. The coefficients of the polynomials correspond directly to the derivatives of the approximations at the interstitial points through the prudent selection of a common coordinate system. The approximation residual and extrapolation errors are investigated as measures for detecting discontinuity. This is necessary since discrete observations of continuous systems are discontinuous at every point. It is proven, using matrix algebra, that positive extrema in the combined approximation-extrapolation error correspond exactly to extrema in the difference of the Taylor coefficients. This provides a relative measure for the severity of the discontinuity in the observational data. The matrix algebraic derivations are provided for all aspects of the methods presented here; this includes a solution for the covariance propagation through the computation. The performance of the method is verified with a Monte Carlo simulation using synthetic piecewise polynomial data with known discontinuities. It is also demonstrated that the discontinuities are suitable as knots for B-spline modelling of data. For completeness, the results of applying the method to sensor data acquired during the monitoring of heavy machinery are presented.