Curve Interpolation Research Articles

Dynamic performance improvement of direct image-based visual servoing in contour following

Image-based visual servoing (IBVS) has increasingly gained popularity and has been adopted in applications such as industrial robots, quadrotors, and unmanned aerial vehicles. When exploiting IBVS, the image feature velocity command obtained from the visual loop controller is converted to the velocity command of the workspace through the interaction matrix so as to converge image feature error. However, issues such as the noise/disturbance arising from image processing and the smoothness of image feature command are often overlooked in the design of the visual loop controller, especially in a contour following task. In particular, noise in the image feature will contaminate the image feedback signal so that the visual loop performance can be substantially affected. To cope with the aforementioned problem, this article employs the sliding mode controller to suppress the adverse effects caused by image feature noise. Moreover, by exploiting the idea of motion planning, a parametric curve interpolator is developed to generate smooth image feature commands. In addition, a depth observer is also designed to provide the depth information essential in the implementation of the interaction matrix. In order to assess the feasibility of the proposed approach, a two-degrees-of-freedom planar robot that employs an IBVS structure and an eye-to-hand camera configuration is used to conduct a contour following task. Contour following results verify the effectiveness of the proposed approach.

NURBS Curve Interpolation Method with Flexibility and High Accuracy Based on Curvature Constraint and Displacement Compensation

NURBS Curve Interpolation Method with Flexibility and High Accuracy Based on Curvature Constraint and Displacement Compensation

A Primitive Comparison for Traffic-Free Path Planning

Motion planning in on-road urban driving is usually stated as an optimization problem in a multi-dimensional space that presents a high complexity in obtaining a global optimal solution. In that sense, a great amount of different approaches to solve this problem coexist in the literature. However, to the best of our knowledge, there is no prior work studying how to choose the best strategy in this multi-dimensional space. This paper presents an in-depth analysis of interpolation curve planners based on continuous curvature Bezier compositions. To that end, a comparison framework to benchmark different path-planning primitives for on-road urban driving is proposed, and the evaluation of different primitive configurations and optimization techniques for path-planning is carried out. In addition, the results are openly published together with its consequent analysis, based on a set of key performance indicators related to the aforementioned main features.

Non-uniform interpolatory subdivision surface

This paper presents an interpolatory subdivision scheme with non-uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology. This is the first attempt to generalize the non-uniform four point interpolatory curve subdivision to surface with extraordinary points. The scheme is constructed from the inspiration of the relation between the non-uniform four-point interpolatory subdivision scheme and the non-uniform B-spline refinement rule. Numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non-uniform parameter values for non-uniform initial data.

Efficient and Adaptive Methods for Computing Accurate Potential Surfaces for Quantum Nuclear Effects: Applications to Hydrogen-Transfer Reactions.

We present two sampling measures to gauge critical regions of potential energy surfaces. These sampling measures employ (a) the instantaneous quantum wavepacket density, an approximation to the (b) potential surface, its (c) gradients, and (d) a Shannon information theory based expression that estimates the local entropy associated with the quantum wavepacket. These four criteria together enable a directed sampling of potential surfaces that appears to correctly describe the local oscillation frequencies, or the local Nyquist frequency, of a potential surface. The sampling functions are then utilized to derive a tessellation scheme that discretizes the multidimensional space to enable efficient sampling of potential surfaces. The sampled potential surface is then combined with four different interpolation procedures, namely, (a) local Hermite curve interpolation, (b) low-pass filtered Lagrange interpolation, (c) the monomial symmetrization approximation (MSA) developed by Bowman and co-workers, and (d) a modified Shepard algorithm. The sampling procedure and the fitting schemes are used to compute (a) potential surfaces in highly anharmonic hydrogen-bonded systems and (b) study hydrogen-transfer reactions in biogenic volatile organic compounds (isoprene) where the transferring hydrogen atom is found to demonstrate critical quantum nuclear effects. In the case of isoprene, the algorithm discussed here is used to derive multidimensional potential surfaces along a hydrogen-transfer reaction path to gauge the effect of quantum-nuclear degrees of freedom on the hydrogen-transfer process. Based on the decreased computational effort, facilitated by the optimal sampling of the potential surfaces through the use of sampling functions discussed here, and the accuracy of the associated potential surfaces, we believe the method will find great utility in the study of quantum nuclear dynamics problems, of which application to hydrogen-transfer reactions and hydrogen-bonded systems is demonstrated here.

The Sloan Digital Sky Survey Reverberation Mapping Project: Hα and Hβ Reverberation Measurements from First-year Spectroscopy and Photometry

Abstract We present reverberation mapping results from the first year of combined spectroscopic and photometric observations of the Sloan Digital Sky Survey Reverberation Mapping Project. We successfully recover reverberation time delays between the g+i band emission and the broad Hβ emission line for a total of 44 quasars, and for the broad Hα emission line in 18 quasars. Time delays are computed using the JAVELIN and CREAM software and the traditional interpolated cross-correlation function (ICCF): using well-defined criteria, we report measurements of 32 Hβ and 13 Hα lags with JAVELIN, 42 Hβ and 17 Hα lags with CREAM, and 16 Hβ and eight Hα lags with the ICCF. Lag values are generally consistent among the three methods, though we typically measure smaller uncertainties with JAVELIN and CREAM than with the ICCF, given the more physically motivated light curve interpolation and more robust statistical modeling of the former two methods. The median redshift of our Hβ-detected sample of quasars is 0.53, significantly higher than that of the previous reverberation mapping sample. We find that in most objects, the time delay of the Hα emission is consistent with or slightly longer than that of Hβ. We measure black hole masses using our measured time delays and line widths for these quasars. These black hole mass measurements are mostly consistent with expectations based on the local – relationship, and are also consistent with single-epoch black hole mass measurements. This work increases the current sample size of reverberation-mapped active galaxies by about two-thirds and represents the first large sample of reverberation mapping observations beyond the local universe (z < 0.3).

A real-time interpolator for parametric curves

Driven by the ever increasing need for the high-speed high-accuracy machining of freeform surfaces, the interpolators for parametric curves become highly desirable, as they can eliminate the feedrate and acceleration fluctuation due to the discontinuity in the first derivatives along the linear tool path. The interpolation for parametric curves is essentially an optimization problem, and it is extremely difficult to get the time-optimal solution. This paper presents a novel real-time interpolator for parametric curves (RTIPC), which provides a near time-optimal solution. It limits the machine dynamics (axial velocities, axial accelerations and jerk) and contour error through feedrate lookahead and acceleration lookahead operations, meanwhile, the feedrate is maintained as high as possible with minimum fluctuation. The lookahead length is dynamically adjusted to minimize the computation load. And the numerical integration error is considered during the lookahead calculation. Two typical parametric curves are selected for both numerical simulation and experimental validation, a cubic phase plate freeform surface is also machined. The numerical simulation is performed using the software (open access information is in the Acknowledgment section) that implements the proposed RTIPC, the results demonstrate the effectiveness of the RTIPC. The real-time performance of the RTIPC is tested on the in-house developed controller, which shows satisfactory efficiency. Finally, machining trials are carried out in comparison with the industrial standard linear interpolator and the state-of-the-art Position-Velocity-Time (PVT) interpolator, the results show the significant advantages of the RTIPC in coding, productivity and motion smoothness.

A METHOD OF REAL-TIME NURBS INTERPOLATION WITH CONFINED CHORD ERROR FOR CNC SYSTEMS

This paper presents a method of real-time CNC interpolation for free-form NURBS curves. The interpolation algorithm is based on second order Taylor’s expansion with the principle part being a formula for updating the parametric value u after each sampling period. With the updated value of u, a new interpolated point is calculated based on the DeBoor’s algorithm. In this paper, an efficient method of limiting chord error introduced by the interpolation algorithm is also presented with the basic idea of reducing machining feedrate at positions with a radius of curvature smaller than a critical value.

Cubic Hermite interpolation with minimal derivative oscillation

In this paper, a new optimal cubic Hermite interpolation method is presented. The method is to optimize the derivative of the interpolant. The diagonally dominant property of the obtained system of normal equations and the error bound are better than some of the existing cubic interpolants. For parametric curve design, the vector-valued interpolation method is given. Some numerical examples are provided to illustrate the satisfactory shape of the interpolation curves.

Elliptic Curved Component Macro-Programming and Its Application

Most conventional numerical control systems do not have the function of noncircular curve interpolation instruction. Manual programming is extremely challenging, automatic programming by computer-aided manufacturing software is highly sophisticated, and processing parameters cannot be easily modified. Therefore, macro-programs, which possess powerful parametric programming, are applied for the processing of noncircular curved components. The values of arguments were determined using transfer and loop statements (IF and WHILE), and elliptic curved macro-programs were achieved using normal and parameter equations in this study. The elliptic curved components were fitted using micro-sized segments or arcs. The numerical control machining tests verified the validity and viability of the macro-programs, and elliptic curved components were processed. The results indicated that the elliptic curved components processed using macro-programs met the design requirements.

High‐precision estimation of target range, radial velocity, and azimuth in mechanical scanning LFMCW radar

This study investigates the effects of antenna rotation on range, radial velocity, and azimuth estimation during coherent integration in linear frequency-modulated continuous wave (LFMCW) radars. When antenna scans during the coherent integration time, theoretical analysis indicates that the angle sensing with the traditional sum-difference amplitude-comparison (SDAC) monopulse method and the range, radial velocity estimation with interpolation method are different from that in staring mode. Simulation demonstrates that the estimation performance is heavily impaired by the antenna rotation. In this study, a generalised mathematical model of coherent integration for LFMCW radar is established. Furthermore, a high-precision estimation method, which consists of generalised SDAC monopulse method, radial velocity estimator with modified interpolation curves, and range estimator refined by Doppler frequency, is proposed. A complete signal-processing scheme is proposed for the accurate estimation of the target parameter in mechanical scanning LFMCW radar. A sub-optimal method is provided for practical implementation. Experimental and simulation results show that the proposed method achieves high accuracy and robustness.

C2 positivity-preserving rational interpolation splines in one and two dimensions

A class of rational quartic/cubic interpolation spline with two local control parameters is presented, which can be C2 continuous without solving a linear system of consistency equations for the derivative values at the knots. The effects of the local control parameters on generating interpolation curves are illustrated. For C2 interpolation, the given interpolant can locally reproduce quadratic polynomials and has O(h2) or O(h3) convergence. Simple schemes for the C2 interpolant to preserve the shape of 2D positive data are developed. Moreover, based on the Boolean sum of quintic interpolating operators, a class of bi-quintic partially blended rational quartic/cubic interpolation surfaces is also constructed. The given interpolation surface provides four local control parameters and can be C2 continuous without using the second or higher mixed partial derivatives on a rectangular grid. Simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3D positive data set arranged over a rectangular grid.

Crustal structure of the Amazonian Craton and adjacent provinces in Brazil

The study of the crust using receiver functions can provide valuable geological information, such as average crustal composition, its formation dynamics and the tectonic evolution of a region, as well as serve as an initial reference for the generation of seismic wave velocity models to improve earthquake location. To fill in gaps in information on the crust of the Amazonian Craton and adjacent provinces in Brazil, we used receiver functions and H-k stacking to estimate crustal thicknesses and the VP/VS ratios. The results indicate that the crust of the study region is predominantly felsic, with an average VP/VS around 1.73 and an average thickness of 38.2 km, with a range of 27.4–48.6 km. Minimum curvature interpolation of the crustal thickness values has made it possible to delimitate of the Amazonian Craton, which corresponds to the area with an average thickness equal to or greater than 39 km. In addition, it was possible to identify its potential cratonic blocks, as well as the Paranapanema Block of Paraná Basin. The geometry of the craton, defined by its crustal thickness, is corroborated by the distribution of natural seismicity that accompanies its edges. These are related to suture zones between the Amazonian, São Francisco/Congo and Paranapanema paleocontinents. The sedimentary basins that have undergone rifting processes have a thinner crust, usually less than 37 km thick. Due to the great variability of the results, it was not possible to determine a characteristic value of crustal thickness or VP/VS ratio for each structural province located in the study region.

Development of the graphical-analytical method for calculating electric load at civilian objects

The calculation method is proposed for electric load at civilian objects, whose novelty, in contrast to determining in line with the acting standards, is in the improvement of its determining accuracy. This will be made possible by developing chart models of the load of electrical receivers, their synthesis at the inputs of civilian objects. The original information for modeling is obtained in advance by measuring chart parameters of similar functioning systems of power supply. The implementation of the proposed method would bring down the cost of material and energy resources. We constructed theoretical and actual mathematical models of their dynamics using the function «cspline (x,y)» embedded in MathCAD, extending the interpolation curve with a cubic parabola. We evaluated the errors of theoretically calculated, relative to actual, specific electric load at civilian objects, under conditions of the city of Odessa. Using the graphic-analytical method would improve accuracy in the calculation of electric load by 1.5–3.5 times compared with that determined in line with the standards. This could save energy resources, capital expenditures on the construction and operation of power supply systems.

Prediction of subcooled flow boiling pressure drops in small circular tubes

Two-phase pressure drops in a mini-tube, in transition flow and subcooled boiling, are analyzed in the present paper, with the support of an experimental data set provided by ENEA with their BO.E.MI.A. test section. The methodology can be applied to different fluids according to similarity criteria. Single phase, subcooled and saturated conditions have been analyzed. The Reynolds number in the experiments was mainly in the transition zone between laminar and turbulent conditions, therefore a third order interpolation curve of the friction factor has been employed. The methodology is based on the model from Delhaye. The model considers the fluid properties, the energy, mass and momentum conservation equations to predict the ONB and OSV points and a hyperbolic function has been adopted to calculate the non-equilibrium vapor quality in the subcooled boiling region. The best agreement with the ENEA experimental data has been obtained using in the methodology the Chisholm correlation, with 83.59% of the predicted values with an error lower than 30%.

Smooth orientation interpolation using parametric quintic-polynomial-based quaternion spline curve

In this paper, a G 2 continuous quintic-polynomial-based unit quaternion interpolation spline curve with tension parameters is presented to interpolate a given sequence of solid orientations. The curve in unit quaternion space S 3 is an extension of the quintic polynomial interpolation spline curve in Euclidean space. It preserves the interpolatory property and G 2 continuity. Meanwhile, the unit quaternion interpolation spline curve possesses the local shape adjustability due to the presence of tension parameters. The change of one tension parameter will only affect the adjacent two pieces of curves. Compared with the traditional B-spline unit quaternion interpolation curve and v -spline unit quaternion interpolation curve, the proposed curve can automatically interpolate the given data points, without solving the nonlinear system of equations over quaternions to obtain the control points, which greatly improves the computational efficiency. Simulation results demonstrate the effectiveness of the proposed scheme.

A Smartphone-Based Automatic Measurement Method for Colorimetric pH Detection Using a Color Adaptation Algorithm.

This paper proposes a smartphone-based colorimetric pH detection method using a color adaptation algorithm for point-of-care applications. Although a smartphone camera can be utilized to measure the color information of colorimetric sensors, ambient light changes and unknown built-in automatic image correction operations make it difficult to obtain stable color information. This paper utilizes a 3D printed mini light box and performs a calibration procedure with a paper-printed comparison chart and a reference image which overcomes the drawbacks of smartphone cameras and the difficulty in preparing for the calibration procedure. The color adaptation is performed in the CIE 1976 u’v’ color space by using the reference paper in order to stabilize the color variations. Non-rigid u’v’ curve interpolation is used to produce the high-resolution pH estimate. The final pH value is estimated by using the best-matching method to handle the nonlinear curve properties of multiple color patches. The experimental results obtained using a pH indicator paper show that the proposed algorithm provides reasonably good estimation of pH detection. With paper-printed accurate color comparison charts and smart color adaptation techniques, superior estimation is achieved in the smartphone-based colorimetric pH detection system for point-of-care application.

The TPS Direct Transport: A New Method for Transporting Deformations in the Size-and-Shape Space

Modern shape analysis allows the fine comparison of shape changes occurring between different objects. Very often the classic machineries of Generalized Procrustes Analysis and Principal Component Analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks. However, if size and shape data are structured in different groups thus constituting different morphological trajectories, a data centering is needed if one wants to compare solely the deformation representing the trajectories. To do that, inter-individual variation must be filtered out. This maneuver is rarely applied in studies using simulated or real data. A geometrical procedure named Parallel Transport, that can be based on various connection types, is necessary to perform such kind of data centering. Usually, the Levi Civita connection is used for interpolation of curves in a Riemannian space. It can also be used to transport a deformation. We demonstrate that this procedure does not preserve some important characters of the deformation, even in the affine case. We propose a novel procedure called `TPS Direct Transport' which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools. We recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study.

COMPARISON OF PARAMETERIZATION METHODS USED FOR B-SPLINE CURVE INTERPOLATION

In this work we deal with the enterpolation of B-spline curves to fiven data points. B-spline curves are generated and compared with the given data points by various parameterization methods. To perform B-spline curve interpolation on the input data, the parameterization and the node vector must be generated using the input data. For parameterization purposes, uniform, chord length, centripetal, Foley, universal and similar methods have been developed. The uniform method gives good results if the data points are regular. Chord-to-beam parameterization can produce undesirable oscillations in long chords. Therefore, the centripetal method has been developed which operates according to the square root of the chord distance. In this study, these methods were compared with different data sets

Multi-axis synchronous interpolation feed rate adaptive planning with rotational tool center point function under comprehensive constraints

Real-time nonlinear error caused by rotation axis motion in five-axis interpolation can be compensated by rotational tool center point function. It is well known that reduction of the interpolation nonlinear error can be achieved effectively by increasing the interpolation steps, but what the value of the steps’ number should be has rarely been mentioned in the existing literature. Taking a five-axis interpolation machine tool with B & C swing axis as an example, the real-time planning and controlling for the integrated feed rate of a five-axis interpolation is proposed by analyzing mechanism of the nonlinear error in the interpolation with rotational tool center point function. The nonlinear error and its derivative of the swing axis are derived. Then, a S-shape feed rate real-time planning and control method about curve interpolation is present first, and then considering the restrictions of the tool axis motion by the feed axis’ mechanical properties and the cutting characteristics, the most appropriate interpolation step length in a single interpolation cycle is calculated in real-time. It shows that comparing to approach with no rotational tool center point interpolation error control, the interpolation error acquired by rotational tool center point in proposed approach can be further reduced while ensuring the smooth and safe machining of the machine tool, and the surface quality of workpiece is also greatly improved.