Arctangent Function Research Articles

基于三维反正切函数拟合的光斑质心提取算法

The precision and stability of spot centroid location were crucial in many fields. According to the edge blurring principle on the optical system, a new algorithm of spot centroid extraction was proposed. The proposed algorithm, based on the arctangent function, can be used to fit the gray distribution of the light spot by variable substitution, and then the centroid of spot can be obtained. In the process of solution, the Gauss-Newton method was adopted to iterate, and then the least square method was used for optimal estimation. Firstly, the advantage and disadvantages of the proposed algorithm between the traditional algorithms were obvious through comparison analysis, and then the feasibility was validated experimentally. The experimental results show that the location precision of the spot centroid is 0.125 3 pixels, and the angle measurement accuracy of the photoelectric sensor is 0.172 3, superior to the sensor's technical requirement 0.25. The robustness of the proposed algorithm to noise, contrast, aspect ratio and size is superior to the traditional algorithms. The experimental results were stable and reliable, and meet the requirement of the angle sensor.

Improved Wavelet Denoising by Non-Convex Sparse Regularization Under Double Wavelet Domains

This paper presents a double wavelet denoising (DWAD) method, which can preserve more details of an original signal. Although the noise removal method based on wavelet transform has been widely used, it still performs poorly for the signals with a low signal-to-noise ratio (SNR) or frequency overlap. Different from the wavelet denoising methods based on a single basis function, the DWAD considers filtering the wavelet coefficients of the noisy signal by threshold functions under two different wavelet domains, simultaneously. It considers using the difference of wavelet coefficient distribution and forcing the denoised signals under two wavelet domains to be the same to achieve more retention of details. In addition, the arctangent function is employed as a penalty function for wavelet coefficients to induce strong sparse wavelet coefficients. The DWAD is applied to one-dimensional signals and it is found that some wavelet coefficients which are smaller than the threshold could be retained during noise removal. The experiment results show that the average SNR of different noise levels is improved by at least 4.2 and 2.1 dB compared with the classical soft threshold method for the one-dimensional and image signals, respectively. Besides, the DWAD tends to obtain better performance on the details of original signals.

Vision-Based Adaptive Neural Positioning Control of Quadrotor Aerial Robot

In this paper, a new vision-based adaptive control algorithm is proposed for the positioning of a quadrotor aerial robot (QAR) with an onboard pin-hole camera. First, the transformation between the position tracking error and image projection error is constructed through the spherical projection method, and then the regulation of the position error is achieved indirectly by stabilizing the image projection error. To overcome the challenge that the dynamics of QAR is physically underactuated, a backstepping-based approach that synthesizes the Lipschitz condition and natural saturation of the inverse tangent function is proposed. In the proposed adaptive controller, an optimized adaptive neural network (NN) means is designed, where only the square of the NN weight matrix’s maximum singular value, not the weight matrix itself, is estimated. Moreover, to facilitate practical application, a novel inertial matrix estimator is introduced in the tuning laws, so that the accurate QAR rotation inertial information is not required. By Lyapunov theory, it is proved that the image projection error converges to an adjustable region of zero asymptotically. The effectiveness of the proposed algorithm has been confirmed by the experimental results.

New method for edges detection of magnetic sources using logistic function

The tilt angle of the analytic signal amplitude (TA) is defined as the arctangent of the ratio of the first vertical derivative to the total horizontal derivative of the analytic signal amplitude. It is commonly used as a useful tool to estimate edges of magnetic sources because its value is slightly dependence on the direction of magnetization vector, and it is more effective in estimating the edges of the bodies than the analytic signal amplitude and the standard tilt angle. Based on logistic function (L) that has the same shape with the shape of arctangent function, and the derivatives of the analytic signal amplitude, we introduce some new filters which also can reduce the effect of the magnetization direction. Other notable features of these filters are that they produce amplitude maxima over the edges of sources and that they balance anomalies from shallow and deep sources. The feasibility of the proposed filters is demonstrated on noise-free and noisy synthetic magnetic data from two 3D models where the obtained results coincide well with the actual edges. The effectiveness of the filters is also evaluated by comparing it with other edge detection methods. The results also show that our filters are less sensitive to variations in the depth of the source bodies and that a modified logistic function (Lk) can achieve better edge detection results than the analytic signal amplitude (AS), the analytic signal amplitude of the tilt angle (AT), the TA and L filters. The filters are also applied to real magnetic data from an area in south-central Vietnam, and the results demonstrate that the proposed filters is a useful tool for the qualitative interpretation of magnetic data.

Refined Estimates and Generalizations of Inequalities Related to the Arctangent Function and Shafer’s Inequality

We give some sharper refinements and generalizations of inequalities related to Shafer’s inequality for the arctangent function, stated in Theorems 1, 2, and 4 in Mortici and Srivastava, 2014, by C. Mortici and H.M. Srivastava.

A simple magnetization model for giant magnetostrictive actuator used on an electronic controlled injector

Abstract Currently, most of the physics-based magnetization models for giant magnetostrictive actuator are complicated in their mathematical models. These complicated models pose considerable challenges for real-time control. Focus on the above issue, we proposed a fitting magnetization model using Arctangent function, as the inverse of the Arctangent function can solve it very quickly. This function was suitable to model the magnetic behavior of giant magnetostrictive actuator serving in an electronic controlled injector or other types of on-off valves. Applicable range of the model was determined through computing the maximum magnetic field. Then the deviations of the proposed fitting model and Jiles-Atherton model under different parameters were studied to verify the computing effects of the model. Then the effectiveness of the fitting model was verified by experimental results. With a high computing precision and a concise form, the proposed model shows great potentials for the real-time control of giant magnetostrictive actuators.

Active target tracking: A simplified view aligning method for binocular camera model

While monitoring large-scale scenes, visual surveillance systems are often confronted with a dilemma between obtaining efficient coverage of the scene and getting sufficient resolution of the targets of interest. To alleviate the contradiction, a solution based on binocular cameras has been proposed. In the solution, an active pan–tilt–zoom (PTZ) camera and a fixed camera operate in a collaborative way by dynamically aligning their fields of view. However, due to the asymmetric camera distribution structure and fast-changing scenario, the accurate alignment is yet challenging. In this paper, a novel view aligning method is presented which introduces three reasonable and mild simplifications to transform the complex view aligning problem into a simple arctangent control function. These simplifications not only relax the restrictions of target depth and the intrinsic parameters of PTZ camera, but also constrain the relative external parameters with a coaxial structure. Meanwhile, a corresponding calibration method is designed to estimate the model parameters off-line. The proposed method was tested in comprehensive cases with relatively close target, wide-range scene and altitude variation. Both the simulation and real scene experiments demonstrate that our method outperforms other state-of-the-art methods on accuracy and effectiveness for active target tracking tasks.

New maximum entropy-based algorithm for structural design optimization

• A new algorithm based on nonlinear transformation is proposed to solve practical problems of reliability analysis. • Finite boundaries are obtained for numerical integration of the maximum entropy method through the transformation. • Typical examples are studied to confirm the efficiency of the proposed method. A new algorithm based on nonlinear transformation is proposed to improve the classical maximum entropy method and solve practical problems of reliability analysis. There are three steps in the new algorithm. Firstly, the performance function of reliability analysis is normalized, dividing by its value when each input is the mean value of the corresponding random variable. Then the nonlinear transformation of such normalized performance function is completed by using a monotonic nonlinear function with an adjustable parameter. Finally, the predictions of probability density function and/or the failure probability in reliability analysis are achieved by looking the result of transformation as a new form of performance function in the classical procedure of maximum entropy method in which the statistic moments are given through the univariate dimension reduction method. In the proposed method, the uncontrollable error of integration on the infinite interval is removed by transforming it into a bounded one. Three typical nonlinear transformation functions are studied and compared in the numerical examples. Comparing with results from Monte Carlo simulation, it is found that a proper choice of the adjustable parameter can lead to a better result of the prediction of failure probability. It is confirmed in the examples that result from the proposed method with the arctangent transformation function is better than the other transformation functions. The error of prediction of failure probability is controllable if the adjustable parameter is chosen in a given interval, but the suggested value of the adjustable parameter can only be given empirically.

A Continuous Differentiable Wavelet Shrinkage Function for Economic Data Denoising

In economic (financial) time series analysis, prediction plays an important role and the inclusion of noise in the time series data is also a common phenomenon. In particular, stock market data are highly random and non-stationary, thus they contain much noise. Prediction of the noise-free data is quite difficult when noise is present. Therefore, removal of such noise before predicting can significantly improve the prediction accuracy of economic models. Based on this consideration, this paper proposes a new shrinkage (thresholding) function to improve the performance of wavelet shrinkage denoising. The proposed thresholding function is an arctangent function with several parameters to be determined and the optimal parameters are determined by ensuring that the thresholding function satisfies the condition of continuously differentiable. The closing price data with the Shanghai Composite Index from January 1991 to December 2014 are used to illustrate the application of the proposed shrinkage function in denoising the stock data. The experimental results show that compared with the classical shrinkage (hard, soft, and nonnegative garrote) functions, the proposed thresholding function not only has the advantage of continuous derivative, but also has a very competitive denoising performance.

On the higher derivatives of the inverse tangent function

In this paper, we find explicit formulas for higher order derivatives of the inverse tangent function. More precisely, we study polynomials which are induced from the higher-order derivatives of arctan(x). Successively, we give generating functions, recurrence relations and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas and Matching polynomials are established.

Nonsingular EB-ESPRIT for the localization of early reflections in a room

The eigenbeam-ESPRIT (EB-ESPRIT) can directly identify the direction of arrivals (DoAs) of sound sources through the parametric estimation process. This parametric estimation is beneficial for identifying directions of early echoes in an enclosed space, because it does not need an extra peak searching step for finding dominant peak locations from a beamforming power map. Nevertheless, the use of EB-ESPRIT for echo localization has been hindered due to its ill-conditioning problem for sources positioned near the equator of the spherical coordinates. Room reflections in a reverberant room can impinge from any direction, and hence, the ill-conditioning is highly likely to occur. In this work, a nonsingular EB-ESPRIT technique is utilized for the echo localization without the ill-conditioning problem. The technique is based on new recurrence relations of spherical harmonics that express DoA parameters in form of sinusoidal functions. Therefore, the ill-conditioning due to the use of arctangent function can be avoided. The performance is demonstrated using the room impulse responses simulated and measured for a rigid spherical microphone array. Frequency smoothing is applied to localize echoes that are highly coherent to each other, and results demonstrate that accurate localization of echoes is possible irrespective of their incidence angles.

ANFIS-based course-keeping control for ships using nonlinear feedback technique

Course keeping for ships is the core of automatic navigation in marine technology. A nonlinear Nomoto model and a maneuvering model group (MMG) model of Yupeng ship are established and verified by the turning trial at sea, then an adaptive neuro-fuzzy inference system (ANFIS) controller is trained by learning the actual ship trial data. There is a limit to the achievable performance of ANFIS controller as the structure is fixed in the training process, many researchers pursue advanced control strategies to improve performance. In this research, the performance is improved in another way, it modulates control error using proposed nonlinear feedback scheme. The simulation result shows that the settling time of nonlinear controllers decreases considerably, dropping by 62.5% of arc tangent function, dropping by 29.2% of bipolar sigmoid function and dropping by 37.5% of sine function based on nonlinear Nomoto model, and the settling time of nonlinear sine controller decreases by 13.3% based on MMG model. It is a useful research that the control performance is improved by nonlinear feedback technique for project application in marine practice.

Total Energy of Hemi-cylindrical Shells Subjected to Blast Loading: Comparison with Theoretical Method

This study presents the total energy response of the hemi-cylindrical shell with a steel material under the external blast loading by employing both theoretical approach and numerical simulation. Analytical approach demonstrate that the total energy contributed from blast is inverse proportional to the thickness of shell. It is proportional to the inverse tangent function applied by the ratio of denotation coordinate over the radius of shell. Moreover, to verify the theoretical inference, several numerical simulations were carried out. Those result agree well.

Approximations to inverse tangent function

In this paper, we present a sharp Shafer-type inequality for the inverse tangent function. Based on the Padé approximation method, we give approximations to the inverse tangent function. Based on the obtained result, we establish new bounds for arctanx.

Binary whale optimization algorithm: a new metaheuristic approach for profit-based unit commitment problems in competitive electricity markets

ABSTRACTThis article presents a metaheuristic approach, the binary whale optimization algorithm (BWOA), to solve complex, constrained, non-convex, binary-nature profit-based unit commitment (PBUC) optimization problems of a price-taking generation company (GenCo) in the electricity market. To simulate the binary-nature PBUC problem, the continuous, real-value whale position/location is mapped into binary search space through various transfer functions. This article introduces three variants of BWOA using tangential hyperbolic, inverse tangent (arctan) and sigmoidal transfer functions. The effectiveness of the BWOA approaches is examined in test systems with different market mechanisms, i.e. an energy-only market, and energy and reserve market participation with different reserve payment methods. The simulation results are presented, discussed and compared with other existing approaches. The convergence characteristics, solution quality and consistency of the results across different BWOA variants are discussed. The superiority and statistical significance of the proposed approaches with respect to existing approaches is also presented.

A formula for pi involving nested radicals

We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument $x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}$, where \[ {{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{k\,\,\text{square}\,\,\text{roots}} \] is a nested radical consisting of $k$ square roots. The computational test we performed reveals that the proposed formula for pi provides a significant improvement in accuracy as the integer $k$ increases.

Adaptive fault-tolerant attitude control for satellite reorientation under input saturation

This paper studies the rest-to-rest attitude reorientation problem of a three-axis stabilized satellite subject to inertia uncertainties, external disturbances, actuator faults and input saturation. An arc tangent function is first adopted to model the constrained control input, and meanwhile an augmented plant is constructed to facilitate the control law derivation. Then, a novel adaptive fault-tolerant control scheme is proposed by incorporating the prescribed performance control and adaptive estimation techniques into backstepping design. Exploiting the dynamic surface control method, the complexity problem residing in traditional backstepping approaches is effectively averted. It is shown that the control algorithm developed is not only robust against environmental disturbances and adaptive to unknown time-varying inertia properties caused by the mass displacement of large-scale deployable appendages, but also able to steer the attitude reorientation errors along prescribed transient and steady-state behavioral bounds, despite the presence of actuator faults and input saturation. Based on standard Lyapunov synthesis, all signals in the closed-loop system are proved to be semi-globally uniformly ultimately bounded. Finally, simulation experiments carried out on a miniature satellite testify the effectiveness of the proposed control approach.

The Ambiguity of the Inverse Secant

Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.

Fractional linear viscoelastic constitutive relations of anhydride-cured thermosetting rubber-like epoxy asphalt binders

Epoxy asphalt binders (EABs) are thought to be a prior choice for long-life paving materials for their excellent performances. To facilitate design of epoxy asphalt pavement structures, anhydrides-cured EABs’ (AC-EABs’) linear viscoelastic constitutive equations were studied. Results indicated that, for weak-crosslinking AC-EABs, the fractional 2S2P1D model degraded into simpler Huet-Sayegh (HS) model, thus, HS model was selected to describe the constitutive relations at all reduced frequencies. Moreover, with HS model and arc tangent function for the temperature dependence of AC-EABs, properties of their concretes could be mapped from those of AC-EABs based on empirical correlation between binders and concretes.

Direction of arrival estimation using nonsingular spherical ESPRIT.

The eigenbeam-ESPRIT (EB-ESPRIT) is a parametric method that estimates the direction of arrival of source signals using a recurrence relation of spherical harmonics. In EB-ESPRIT, the sound-source elevation angle is estimated from an arctangent function, which diverges near the equator in the spherical coordinate system and inevitably induces an ill-conditioning problem. Here, a nonsingular spherical ESPRIT technique based on sine-based recurrence relations is proposed, in which the elevation angles are estimated from an arcsine function. It is shown that the proposed technique can estimate more sources than the conventional EB-ESPRIT technique while also avoiding ill-conditioning problems.