Orthogonal Error Research Articles

ParkUs: A Novel Vehicle Parking Detection System

Finding on-street parking in congested urban areas is a challenging chore that most drivers worldwide dislike. Previous vehicle traffic studies have estimated that around thirty percent of vehicles travelling in inner city areas are made up of drivers searching for a vacant parking space. While there are hardware sensor based solutions to monitor on-street parking occupancy in real-time, instrumenting and maintaining such a city wide system is a substantial investment. In this paper, a novel vehicle parking activity detection method, called ParkUs, is introduced and tested with the aim to eventually reduce vacant car parking space search times. The system utilises accelerometer and magnetometer sensors found in all smartphones in order to detect parking activity within a city environment. Moreover, it uses a novel sensor fusion feature called the Orthogonality Error Estimate (OEE). We show that the OEE is an excellent indicator as it’s capable of detecting parking activities with high accuracy and low energy consumption. One of the envisioned applications of the ParkUs system will be to provide all drivers with guidelines on where they are most likely to find vacant parking spaces within a city. Therefore, reducing the time required to find a vacant parking space and subsequently vehicle congestion and emissions within the city.

Far-field outdoor experimental demonstration of down-looking synthetic aperture ladar

A specific system structure of down-looking synthetic aperture imaging ladar (SAIL) is given, and a far-field experiment over 6 km of down-looking SAIL under this system design is carried out. The down-looking SAIL can overcome the influence of atmospheric turbulence to a great extent. By applying this system design, it also has advantages in self-compensating phase modulation. A fine image is obtained after aligning in the orthogonal direction and phase error compensation in the travel direction based on a dominant scatterer. The achieved imaging resolutions in the two dimensions are both better than 5 cm.

Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk

We study approximation properties of weighted $L^2$-orthogonal projectors onto the space of polynomials of degree less than or equal to $N$ on the unit disk where the weight is of the generalized Gegenbauer form $x \mapsto (1-|x|^2)^\alpha$. The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted $L^2$ norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases and otherwise strongly hinted at by reported numerical tests. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and non-uniformly weighted Sobolev spaces involved, a Markov-type inequality, connection coefficients between Zernike polynomials and relations between the Fourier-Zernike expansions of a function and its derivatives.

Convex relaxation for IMSE optimal design in random-field models

The construction of optimal designs for random-field interpolation models via convex design theory is considered. The definition of an Integrated Mean-Squared Error (IMSE) criterion yields a particular Karhunen–Loève expansion of the underlying random field. After spectral truncation, the model can be interpreted as a Bayesian (or regularised) linear model based on eigenfunctions of this Karhunen–Loève expansion, and can be further approximated by a linear model involving orthogonal observation errors. Using the continuous relaxation of approximate design theory, the search of an IMSE optimal design can then be turned into a Bayesian A-optimal design problem, which can be efficiently solved by convex optimisation. A careful analysis of this approach is presented, also including the situation where the model contains a linear parametric trend, which requires specific treatments. Several approaches are proposed, one of them enforcing orthogonality between the trend functions and the complementary random field. Convex optimisation, based on a quadrature approximation of the IMSE criterion and a discretisation of the design space, yields an optimal design in the form of a probability measure with finite support. A greedy extraction procedure of the exchange type is proposed for the selection of observation locations within this support, the size of the extracted design being controlled by the level of spectral truncation. The performance of the approach is investigated on a series of examples indicating that designs with high IMSE efficiency are easily obtained.

As Good as GOLD: Gram-Schmidt Orthogonalization by Another Name.

Generalized orthogonal linear derivative (GOLD) estimates were proposed to correct a problem of correlated estimation errors in generalized local linear approximation (GLLA). This paper shows that GOLD estimates are related to GLLA estimates by the Gram-Schmidt orthogonalization process. Analytical work suggests that GLLA estimates are derivatives of an approximating polynomial and GOLD estimates are linear combinations of these derivatives. A series of simulation studies then further investigates and tests the analytical properties derived. The first study shows that when approximating or smoothing noisy data, GLLA outperforms GOLD, but when interpolating noisy data GOLD outperforms GLLA. The second study shows that when data are not noisy, GLLA always outperforms GOLD in terms of derivative estimation. Thus, when data can be smoothed or are not noisy, GLLA is preferred whereas when they cannot then GOLD is preferred. The last studies show situations where GOLD can produce biased estimates. In spite of these possible shortcomings of GOLD to produce accurate and unbiased estimates, GOLD may still provide adequate or improved model estimation because of its orthogonal error structure. However, GOLD should not be used purely for derivative estimation because the error covariance structure is irrelevant in this case. Future research should attempt to find orthogonal polynomial derivative estimators that produce accurate and unbiased derivatives with an orthogonal error structure.

Stability and Performance of Various Singular Value QR Implementations on Multicore CPU with a GPU

Singular Value QR (SVQR) can orthonormalize a set of dense vectors with the minimum communication (one global reduction between the parallel processing units, and BLAS-3 to perform most of its local computation). As a result, compared to other orthogonalization schemes, SVQR obtains superior performance on many of the current computers, where the communication has become significantly more expensive compared to the arithmetic operations. In this article, we study the stability and performance of various SVQR implementations on multicore CPUs with a GPU. Our focus is on the dense triangular solve, which performs half of the total floating-point operations of SVQR. As a part of this study, we examine an adaptive mixed-precision variant of SVQR, which decides if a lower-precision arithmetic can be used for the triangular solution at runtime without increasing the order of its orthogonality error (though its backward error is significantly greater). If the greater backward error can be tolerated, then our performance results with an NVIDIA Kepler GPU show that the mixed-precision SVQR can obtain a speedup of up to 1.36 over the standard SVQR.

Error analysis of analytical coarse alignment formulations for stationary SINS

This paper presents a comprehensive error analysis of coarse alignment formulations for stationary strapdown inertial navigation systems (SINS). Analytical expressions for the residual normality, orthogonality, alignment, and Euler angle errors are systematically derived, allowing us to establish comparisons in terms of rapidity, accuracy, and autonomy requirements. For the purpose of the error analysis, geographic latitude and local gravity acceleration uncertainties are considered, in addition to the inertial sensor uncertainties. As the main contribution of this paper, an improved coarse alignment method is proposed, which is based on orthonormality constraints existing between the Euler angles and the attitude matrix. In contrast to the traditional methods, the proposed formulation does not imply normality and orthogonality errors, besides producing time-independent alignment and Euler angle errors. The latter is seen to be particularly interesting for situations wherein, due to time constraints, posterior fine alignment and orthonormalization procedures cannot be implemented. The superiority of the proposed formulation is validated through simulated and experimental tests, regardless of the SINS initial orientation. The accuracy degradation produced by each alignment formulation in the navigation stage is used as the performance index.

A total error-based multiquadric method for surface modeling of digital elevation models

Digital elevation model (DEM) source data are subject to both horizontal and vertical errors owing to improper instrument operation, physical limitations of sensors, and bad weather conditions. These factors may bring a negative effect on some DEM-based applications requiring low levels of positional errors. Although classical smoothing interpolation methods have the ability to handle vertical errors, they are prone to omit horizontal errors. Based on the statistical concept of the total least squares method, a total error-based multiquadric (MQ-T) method is proposed in this paper to reduce the effects of both horizontal and vertical errors in the context of DEM construction. In nature, the classical multiquadric (MQ) method is a vertical error regression procedure, whereas MQ-T is an orthogonal error regression model. Two examples, including a numerical test and a real-world example, are employed in a comparative performance analysis of MQ-T for surface modeling of DEMs. The numerical test indicates that MQ-T performs better than the classical MQ in terms of root mean square error. The real-world example of DEM construction with sample points derived from a total station instrument demonstrates that regardless of the sample interval and DEM resolution, MQ-T is more accurate than classical interpolation methods including inverse distance weighting, ordinary kriging, and Australian National University DEM. Therefore, MQ-T can be considered as an alternative interpolator for surface modeling with sample points subject to both horizontal and vertical errors.

Physics-driven Shape Variation Modelling at Early Design Stage

Modern markets are becoming increasingly competitive emphasizing the importance of achieving Right First Time (RFT) during the early design stage as a key enabler facilitating cost and time-to-launch (or time-to-market) reduction. One of the leading challenges to deliver RFT is the lack of effective methods to model product errors at early design stage. Usually, the assembly process is designed under the assumption of ideal (nominal) products. On the contrary, it has been demonstrated that product errors (both geometrical and dimensional) affect the performance of the final assembly. To facilitate easy decision making at early design stage, new methods and models are required to support design engineers. In this study, a framework has been proposed for early design support to generate product variation.International standard provides guidelines for product control and inspection (ISO-GPS or ASME-GD&T); however, the integration of tolerance standard into nominal sized CAD models is not yet achieved. Current, Computer Aided Tolerancing (CAT) tools mainly capable to model orientation and position tolerance specifications, whereas part shape errors are omitted. This paper presents an innovative physics-driven simulation framework to model shape errors of compliant sheet metal parts at early design stage.The modelling framework consists of three important stages: (i) initial shape error prediction using physic-based simulation, such as, stamping process simulation; (ii) individual orthogonal shape error modes/patterns identification based on decomposition techniques, such as, Geometric Modal Analysis (GMA); and, (iii) simulation of shape error variation classes by assigning distribution to each orthogonal shape error modes. The proposed approach enables to generate shape errors at early design stage of assembly process which can be utilized to optimize the assembly process, including fixture design and joining process parameters. An industrial automotive component illustrates the proposed methodology.

激光直写机床系统正交性误差影响下计算全息图相位分析

激光直写机床系统正交性误差影响下计算全息图相位分析

Calibration of low-cost triaxial inertial sensors

Accelerometers (ACCs) and gyroscopes (gyros) are commonly known as inertial sensors and their orthogonal triads generally form an inertial measurement unit (IMU) used as a core means of a navigation system. Before the navigation system is to be used, it is necessary to perform its calibration. A typical process of the IMU calibration usually estimates scale-factors, orthogonality or misalignment errors, and offsets of both triads. These parameters compose the so-called sensor error model (SEM). The process of obtaining accurate information that describes the motion performed within the calibration generally requires a costly and specialized means [1], [2]. Therefore, much effort has been put into cost-effective calibration using an optical motion tracking system [3]–[6], or transferring the calibration into a state estimation problem [7]. In the ACC case, most of the current calibration methods utilize the fact that ACCs are affected by gravity when they are under static conditions. Therefore, we proceed with calibration performed under static conditions, which utilizes the knowledge of the gravity magnitude and ACC output measurements collected at predetermined orientations and performs ACC SEM estimation using nonlinear optimization [6], [8]–[10]. In the gyro case, calibration based on the Earth's rate might be inapplicable, for instance due to the fact that Earth's rate is under or around the resolution of the gyro, and thus other means to apply and measure angular rates need to be used. This situation commonly arises in the case of low-cost MEMS (Micro-Electro-Mechanical System) based gyros. Thus, expensive mechanical platforms are often inevitable [11]–[14].

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

SummaryProper range and precision analysis play an important role in the development of fixed‐point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image‐processing applications. These algorithms have not been attempted much in fixed‐point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round‐off errors. For any real‐time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed‐point formats to decide on integer wordlengths during float‐to‐fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed‐point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed‐point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed‐point Lanczos algorithm in three different formats has been compared on the basis of signal‐to‐quantization‐noise‐ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed‐point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean‐square‐error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed‐point formats produce vectors with lesser round‐off error than constant format. All the three fixed‐point formats of Lanczos algorithm have been synthesized on Virtex 7 field‐programmable gate array using Vivado high‐level synthesis design tool. A comparative study of resource usage and power consumption is carried out. Copyright © 2015 John Wiley & Sons, Ltd.

Measurement Uncertainty Sources Analysis and Verification of Time-Grating with Variable Coupling Coefficient

In order to improve the precision of Time-grating with Variable Coupling Coefficient, the electrical traveling wave equation was derived in detail base on the further study of working principle of Time-grating sensor. Combined with the processing and assembling methods of sensor, the uncertainty sources of sensor standing wave, synthesis traveling wave signal and the produce reasons of higher harmonic were analyzed by using uncertainty theory. The produce reasons of higher harmonic were verified with spectral analysis of the original sensor error data, and then the primary and secondary uncertainty sources affected the sensor measurement uncertainty sources were got, which were time and spatial orthogonal error, amplitude error of two roads signal and stator and rotor air gap uneven. The conclusion made in this paper can be used to conduct the structure optimal design, sensor error correction and uncertainty evaluation.

Onboard System Identification for Improved Flight Control of UAS

A new technique is proposed to select and estimate the significant aerodynamic parameters of micro unmanned aerial systems from ight data to improve the dynamical qualities of an indirect adaptive ight control system. The aerodynamic variables are estimated in the frequency domain using the angle of attack and sideslip air-ow angles which in turn are estimated using an extended Kalman filter. Parameter estimation and selection procedures of significant aerodynamic parameters are based on linear regression model structures with forward orthogonal least square (OLS) and error reduction ratio (ERR) methods. When combined, the methods can be applied to create an indirect adaptive ight control system. This new approach is verified by comparing the results with those obtained from conventional sensor data, including air ow angle measurements. Performance comparison of the system identification methods show that the proposed technique can obtain the same quality of ight performance as if the airow measurements were available. The new methods are demonstrated in simulation of a benchmark ight performance experiment on an Aerosonde UAV.

Mixed-Precision Cholesky QR Factorization and Its Case Studies on Multicore CPU with Multiple GPUs

To orthonormalize the columns of a dense matrix, the Cholesky QR (CholQR) requires only one global reduction between the parallel processing units and performs most of its computation using BLAS-3 kernels. As a result, compared to other orthogonalization algorithms, CholQR obtains superior performance on many of the current computer architectures, where the communication is becoming increasingly expensive compared to the arithmetic operations. This is especially true when the input matrix is tall-skinny. Unfortunately, the orthogonality error of CholQR depends quadratically on the condition number of the input matrix, and it is numerically unstable when the matrix is ill-conditioned. To enhance the stability of CholQR, we recently used mixed-precision arithmetic; the input and output matrices are in the working precision, but some of its intermediate results are accumulated in the doubled precision. In this paper, we analyze the numerical properties of this mixed-precision CholQR. Our analysis shows that by selectively using the doubled precision, the orthogonality error of the mixed-precision CholQR only depends linearly on the condition number of the input matrix. We provide numerical results to demonstrate the improved numerical stability of the mixed-precision CholQR in practice. We then study its performance. When the target hardware does not support the desired higher precision, software emulation is needed. For example, using software-emulated double-double precision for the working 64-bit double precision, the mixed-precision CholQR requires about $8.5\times$ more floating-point instructions than that required by the standard CholQR. On the other hand, the increase in the communication cost using the double-double precision is less significant, and our performance results on multicore CPU with a different graphics processing unit (GPU) demonstrate that the overhead of using the double-double arithmetic is decreasing on a newer architecture, where the computation is becoming less expensive compared to the communication. As a result, with a latest NVIDIA GPU, the mixed-precision CholQR was only $1.4\times$ slower than the standard CholQR. Finally, we present case studies of using the mixed-precision CholQR within communication-avoiding variants of Krylov subspace projection methods for solving a nonsymmetric linear system of equations and for solving a symmetric eigenvalue problem, on a multicore CPU with multiple GPUs. These case studies demonstrate that by using the higher precision for this small but critical segment of the Krylov methods, we can improve not only the overall numerical stability of the solvers but also, in some cases, their performance.

从入射面检测锥体棱镜直角误差的方法

从入射面检测锥体棱镜直角误差的方法

Calibration of the straightness and orthogonality error of a laser feedback high-precision stage using self-calibration methods

An ultra high-precision 3-DOF air-bearing stage is developed and calibrated in this study. The stage was developed for the transportation of a glass or wafer with x and y following errors in the nanometer regime. To apply the proposed stage to display or semiconductor fabrication equipment, x and y straightness errors should be at the sub-micron level and the x–y orthogonality error should be in the region of several arcseconds with strokes of several hundreds of mm. Our system was designed to move a 400 mm stroke on the x axis and a 700 mm stroke on the y axis. To do this, 1000 mm and 550 mm bar-type mirrors were adopted for real time Δx and Δy laser measurements and feedback control. In this system, with the laser wavelength variation and instability being kept to a minimum through environmental control, the straightness and orthogonality become purely dependent upon the surface shape of the bar mirrors. Compensation for the distortion of the bar mirrors is accomplished using a self-calibration method. The successful application of the method nearly eliminated the straightness and orthogonality errors of the stage, allowing their specifications to be fully satisfied. As a result, the straightness and orthogonality errors of the stage were successfully decreased from 4.4 μm to 0.8 μm and from 0.04° to 2.48 arcsec, respectively.

Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

Abstract We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.

Dimensionality reduction of collective motion by principal manifolds

While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods is not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the mapping from the original high-dimensional space to the embedding space. Here, we propose an alternative approach that demands a two-dimensional embedding which topologically summarizes the high-dimensional data. In this sense, our approach is closely related to the construction of one-dimensional principal curves that minimize orthogonal error to data points subject to smoothness constraints. Specifically, we construct a two-dimensional principal manifold directly in the high-dimensional space using cubic smoothing splines, and define the embedding coordinates in terms of geodesic distances. Thus, the mapping from the high-dimensional data to the manifold is defined in terms of local coordinates. Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets. The principal manifold finding algorithm is applied to configurations obtained from a dynamical system of multiple agents simulating a complex maneuver called predator mobbing, and the resulting two-dimensional embedding is compared with that of a well-established nonlinear dimensionality reduction method.

Fast estimation of coil factor and orthogonality errors in tri-axial magnetic coil systems

We investigate methods used for calibrating magnetic sensors and tri-axial magnetic coil systems that are used to generate the calibration field for magnetometer sensor calibration. Since existing magnetic coil calibration techniques found in the literature either did not fully account for secondary errors such as orthogonality, did not account for time-varying noise levels during measurement, or were too time-consuming, existing magnetometer calibration algorithms were evaluated for use in calibrating coils. The TWOSTEP and Geometric Approach algorithms were extensively tested using synthetic data. The TWOSTEP algorithm was selected to be adapted to enable its use to rapidly calibrate tri-axial magnetic coils using an automated procedure to determine coil factors (i.e. the proportionality constants relating the excitation currents to the induced magnetic fields) and coil orthogonality alignment errors. Experimental measurements on an 8 m length, 2 m diameter tri-axial coil system, consisting of two orthogonal sets of cos(θ) coils and a solenoidal coil, show that the automated procedure enables an accurate tri-axial coil calibration to be conducted with 90 s of measurement data, sampled at 6.25 kHz. Furthermore, this method was shown to significantly reduce the difference between the predicted and generated magnetic field—as measured by a calibrated reference magnetometer, compared to using coil factors derived theoretically from geometric coil parameters.