Showing papers on "Inverse trigonometric functions published in 2022"
TL;DR: In this article , a single-shot high-precision backpropagation (BP) neural network based color fringe projection profilometry (CFPP) in rapid measurement is proposed.
9 citations
TL;DR: In this article , a novel strategy based on a metasurface composed of simple and compact unit cells to achieve ultra-high-speed trigonometric operations under specific input values is theoretically and experimentally demonstrated.
Abstract: Abstract In this paper, a novel strategy based on a metasurface composed of simple and compact unit cells to achieve ultra-high-speed trigonometric operations under specific input values is theoretically and experimentally demonstrated. An electromagnetic wave (EM)-based optical diffractive neural network with only one hidden layer is physically built to perform four trigonometric operations (sine, cosine, tangent, and cotangent functions). Under the unique composite input mode strategy, the designed optical trigonometric operator responds to incident light source modes that represent different trigonometric operations and input values (within one period), and generates correct and clear calculated results in the output layer. Such a wave-based operation is implemented with specific input values, and the proposed concept work may offer breakthrough inspiration to achieve integrable optical computing devices and photonic signal processors with ultra-fast running speeds.
9 citations
TL;DR: In this article , a novel strategy based on a metasurface composed of simple and compact unit cells to achieve ultra-high-speed trigonometric operations under specific input values is theoretically and experimentally demonstrated.
Abstract: Abstract In this paper, a novel strategy based on a metasurface composed of simple and compact unit cells to achieve ultra-high-speed trigonometric operations under specific input values is theoretically and experimentally demonstrated. An electromagnetic wave (EM)-based optical diffractive neural network with only one hidden layer is physically built to perform four trigonometric operations (sine, cosine, tangent, and cotangent functions). Under the unique composite input mode strategy, the designed optical trigonometric operator responds to incident light source modes that represent different trigonometric operations and input values (within one period), and generates correct and clear calculated results in the output layer. Such a wave-based operation is implemented with specific input values, and the proposed concept work may offer breakthrough inspiration to achieve integrable optical computing devices and photonic signal processors with ultra-fast running speeds.
8 citations
TL;DR: In this paper , series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers were derived.
Abstract: In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.
8 citations
TL;DR: In this article , the identification of hysteresis parameters for two different models, operated in quasi-static and dynamic behaviors, was performed via standard Epstein frame on automotive parts for different magnetization levels and frequencies.
Abstract: This paper considers the identification of hysteresis parameters for two different hysteresis models, operated in quasi-static and dynamic behaviors. The extended Jiles-Atherton model via the Bertotti approach and the improved arctangent model are tested and compared. The hysteresis measurement was performed via standard Epstein frame on automotive parts for different magnetization levels and frequencies. The robustness of the Levenberg-Marquardt algorithm is assessed via a comparison of measured, modeled hysteresis loops, and relative permeability profiles, at various frequencies.
7 citations
TL;DR: In this paper , a normalized least mean squares adaptive filtering scheme based on arctangent cost function (Arc-NLMS) was proposed to suppress harmonics and negative sequence components in grid currents.
Abstract: Wind energy conversion systems (WECSs) based on doubly fed induction generator (DFIG) are often connected to local loads, and excess power is supplied to the grid. The nonlinear and unbalanced local loads integrated with WECS, degrade the power quality by injecting harmonics and negative sequence components in grid currents. Another major concern in grid-integrated WECS is related to sudden steep fluctuations in wind and load powers, which bring about frequency and voltage deviations that ultimately affect grid stability. Thus, the objective of this work is twofold. First, to suppress harmonics and negative sequence components in grid currents, using a normalized least mean squares (NLMS) adaptive filtering scheme based on arctangent cost function (Arc-NLMS). The Arc-NLMS adaptive algorithm is robust against model uncertainties and exhibits optimal convergence performance. Second, to guarantee grid security amidst sudden erratic variations in wind speed or load power, through implementation of a power management scheme (PMS) using a battery energy storage. The PMS allocates exponential values to weight components, to diminish the effects of power fluctuations. Based on performance evaluation of the system using a developed laboratory prototype, the PMS proves to be effective in smoothening power fluctuations while the power quality issues related with connection of local nonlinear and unbalanced loads are also alleviated.
7 citations
TL;DR: In this paper , an adaptive dynamic surface control approach is proposed for uncertain strict feedback systems (SFSs) to guarantee both the prescribed transient tracking performance and the asymptotic tracking while realizing the accurate parameter estimation.
Abstract: In this article, an adaptive dynamic surface control approach is proposed for uncertain strict‐feedback systems (SFSs) to guarantee both the prescribed transient tracking performance and the asymptotic tracking while realizing the accurate parameter estimation. It is assumed that SFSs are subject to linearly parametric uncertainties in both the drift terms and the control coefficients. A new inequality on the arctangent function, which can be widely used in the robust or the adaptive control designs, is established. Owing to this inequality, nonlinear robust filters with arctangent functions are designed and embedded into the backstepping control algorithm to avoid the “differential explosion” problem. Moreover, an improved forgetting‐factor‐based parameter estimation error reconstruction mechanism is proposed. And the obtained parameter estimation errors are integrated into the adaptive laws to achieve the accurate parameter estimation. Furthermore, the projection operator is applied to avoid the singularity problem of the control law. Besides, an asymmetric error transformation is introduced to restrict the tracking error within the prescribed performance envelope. It is proved that the tracking error and the parameter estimation errors asymptotically converge to zero and the tracking error satisfies the prescribed transient performance. Finally, the effectiveness of the proposed control approach is validated in terms of the single‐link manipulator actuated by a brush DC motor.
6 citations
TL;DR: In this paper , a cost function framework for developing robust adaptive filtering by embedding the standard cost function into the arctangent framework is proposed. But the performance of the proposed family of algorithms is tested through simulation studies in system identification scenarios.
Abstract: This brief introduces a novel cost function framework for developing robust algorithms for adaptive filtering by embedding the standard cost function into the arctangent framework. This proposed framework is called the arctangent cost function framework. Based on this, we propose an arctangent family of robust algorithms for adaptive filtering. The performance of the proposed family of algorithms is tested through simulation studies in system identification scenarios that confirm the enhanced performance achieved by the arctangent family of algorithms over standard algorithms.
6 citations
TL;DR: In this paper , the Riemann-Liovuille fractional integral and the standard integral were obtained in terms of the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary.
Abstract: Sharp bounds for cosh(x)x,sinh(x)x, and sin(x)x were obtained, as well as one new bound for ex+arctan(x)x. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann–Liovuille fractional integral and in terms of the standard integral.
4 citations
TL;DR: In this paper , a method based on trigonometric expansion properties of the hyperbolic function for hardware implementation which can be easily tuned for different accuracy and precision requirements is presented. But, it is not suitable for DNNs that use different precision in different layers.
Abstract: Hyperbolic tangent and Sigmoid functions are used as non-linear activation units in the artificial and deep neural networks. Since, these networks are computationally expensive, customized accelerators are designed for achieving the required performance at lower cost and power. The activation function and MAC units are the key building blocks of these neural networks. A low complexity and accurate hardware implementation of the activation function is required to meet the performance and area targets of such neural network accelerators. Moreover, a scalable implementation is required as the recent studies show that the DNNs may use different precision in different layers. This paper presents a novel method based on trigonometric expansion properties of the hyperbolic function for hardware implementation which can be easily tuned for different accuracy and precision requirements.
4 citations
TL;DR: In this paper , closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3.
Abstract: Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.
TL;DR: The residual correction method (RCM) developed here can produce simple and accurate approximations of certain nonlinear functions with minimal multiply–add operations and are candidate algorithms that can be used to stabilize the attitude control of robots and drones, which require real-time processing.
Abstract: In modern computers, complicated signal processing is highly optimized with the use of compilers and high-speed processing using floating-point units (FPUs); therefore, programmers have little opportunity to care about each process. However, a highly accurate approximation can be processed in a small number of computation cycles, which may be useful when embedded in a field-programmable gate array (FPGA) or micro controller unit (MCU), or when performing many large-scale operations on a graphics processing unit (GPU). It is necessary to devise algorithms to obtain the desired calculated values without an accelerator or compiler assistance. The residual correction method (RCM) developed here can produce simple and accurate approximations of certain nonlinear functions with minimal multiply–add operations. In this study, we designed an algorithm for the approximate computation of trigonometric and inverse trigonometric functions, which are nonlinear elementary functions, to achieve their fast and accurate computation. A fast first approximation and a more accurate second approximation of each function were created using RCM with a less than 0.001 error using multiply–add operations only. This achievement is particularly useful for MCUs, which have a low power consumption but limited computational power, and the proposed approximations are candidate algorithms that can be used to stabilize the attitude control of robots and drones, which require real-time processing.
TL;DR: In this article , a collision-free planning and control framework for a biomimetic underwater vehicle (BUV) in dynamic environments is presented, which consists of obstacle avoidance planning, arctangent nonsingularity terminal sliding mode (ANTSM) control, and fuzzy inference.
Abstract: In this article, a collision-free planning and control framework for a biomimetic underwater vehicle (BUV) in dynamic environments is presented. It consists of obstacle avoidance planning, arctangent nonsingularity terminal sliding mode (ANTSM) control, and fuzzy inference. A fuzzy artificial potential field with a velocity component is designed for obstacle avoidance planning. An ANTSM controller with an arctangent function is proposed to guarantee a shorter convergence time of system states. The stability of the system is analyzed by the Lyapunov theory. A fuzzy inference module is given to construct the nonlinear relationship between the control parameters of the flippers and force/torque. Finally, comparative simulations, robot operating system-based simulations, and underwater obstacle avoidance experiments of the BUV in a swimming pool are conducted to validate the performance of the proposed collision-free planning and control algorithm.
TL;DR: In this article, a family of two-parameter functions are used to asymptoticly approximate the inverse tangent function, which can recover some of previous results, and it can also recover the inverted tangent functions itself.
Abstract: In this paper, we present a family of two-parameter functions, which are used to asymptoticly approximating the inverse tangent function. It can recover some of previous results, and it can also recover the inverse tangent function itself. Numerical examples show that the new inequalities can achieve much better approximation performance that those of prevailing methods.
TL;DR: In this article , the authors give two Taylor expansions of arctan(x + ω), where ω represents a finite increment of x, and discover several remarkable infinite series from these expansions by special substitutions.
Abstract: In this work, we give two new Taylor expansions of arctan(x + ω), where ω represents a finite increment of x. We discover several remarkable infinite series from these expansions by special substitutions. Some of these infinite series give BBP-type formulae.
TL;DR: Two techniques to propose a new metaheuristic algorithm based on Hybrid Modified Sine Cosine Algorithm with Cuckoo Search Algorithm using Inverse Filtering (IF) and clipping method to achieve better results are implemented to improve the search of local optimal solutions systematically.
Abstract: : The essential purpose of radar is to detect a target of interest and provide information concerning the target’s location, motion, size, and other parameters. The knowledge about the pulse trains’ properties shows that a class of signals is mainly well suited to digital processing of increasing practical importance. A low autocorrelation binary sequence (LABS) is a complex combinatorial problem. The main problems of LABS are low Merit Factor (MF) and shorter length sequences. Besides, the maximum possible MF equals 12.3248 as infinity length is unable to be achieved. Therefore, this study implemented two techniques to propose a new metaheuristic algorithm based on Hybrid Modified Sine Cosine Algorithm with Cuckoo Search Algorithm (HMSCACSA) using Inverse Filtering (IF) and clipping method to achieve better results. The proposed algorithms, LABS-IF and HMSCACSA-IF, achieved better results with two large MFs equal to 12.12 and 12.6678 for lengths 231 and 237, respectively, where the optimal solutions belong to the skew-symmetric sequences. The MF outperformed up to 24.335% and 2.708% against the state-of-the-art LABS heuristic algorithm, xLastovka, and Golay, respectively. These results indicated that the proposed algorithm’s simulation had quality solutions in terms of fast convergence curve with better optimal means, and standard deviation. the subsequent search for odd sequences will be continued with a larger length to get the difference increment with higher MF than the HMSCACSA-IF algorithm. The significant limitations of LABS are solved through IF and clipping methods to find some new, best-known skew-symmetrical solutions with MF and EL values. The results obtained have significantly improved the results compared to the recent state-of-the-art algorithms, where the MF and EL of the proposed algorithms are better than xLastovka, 1bCAN, and Shotgun Hill-Climbing with their largest binary sequences length up to 267, 1000 and 300, respectively. These proposed methods are uniquely postulated to improve the search of local optimal solutions systematically.
TL;DR: In this paper , the quotient of inverse trigonometric and inverse hyperbolic functions such as sin−1xsinh− 1x and tanh−1xtan−1X was shown to have polynomial bounds using even quadratic functions.
Abstract: We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as sin−1xsinh−1x and tanh−1xtan−1x. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form eax2. Graph validation is also performed.
TL;DR: In this article , an investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, is undertaken.
Abstract: An investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.
TL;DR: In this paper , the authors obtained new natural approaches of Shafer-Fink inequality for arc sine function and the square of arc sines by using the power series expansions of certain functions, which generalize and strengthen those in the existing literature.
Abstract: In this paper, we obtain some new natural approaches of Shafer-Fink inequality for arc sine function and the square of arc sine function by using the power series expansions of certain functions, which generalize and strengthen those in the existing literature.
TL;DR: The arctangent entropy has the best segmentation effect on light colored character images and is compared with the Tsallis entropy, Kapur entropy, Renyi entropy, Minimum error threshold (MET) and Iterative threshold (IT).
TL;DR: In this article , three classes of improper integrals involving higher powers of arctanh, arcta, and arcsin are examined using the recursive approach and several explicit formulae are established, which evaluate these integrals in terms of π, ln2, the Riemann zeta function, and the Dirichlet beta function.
Abstract: Three classes of improper integrals involving higher powers of arctanh, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of π, ln2, the Riemann zeta function, and the Dirichlet beta function.
01 Apr 2022
TL;DR: In this paper , the angular error of rotary and linear encoders with sine/cosine output signals in quadrature that are distorted by superimposed Fourier series is analyzed.
Abstract: We present a rigorous analytical method for harmonic analysis of the angular error of rotary and linear encoders with sine/cosine output signals in quadrature that are distorted by superimposed Fourier series. To calculate the angle from measured sine and cosine encoder channels in quadrature, the arctangent function is commonly used. The hence non-linear relation between raw signals and calculated angle—often thought of as a black box—complicates the estimation of the angular error and its harmonic decomposition. By means of a Taylor series expansion of the harmonic amplitudes, our method allows for quantification of the impact of harmonic signal distortions on the angular error in terms of harmonic order, magnitude and phase, including an upper bound on the remaining error term—without numerical evaluation of the arctangent function. The same approximation is achieved with an intuitive geometric approximation in the complex plane, validating the results. Interaction effects between harmonics in the signals are considered by higher-order Taylor expansion. The approximations show an excellent agreement with the exact calculation in numerical examples even in case of large distortion amplitudes, leading to practicable estimates for the angular error decomposition.
TL;DR: In this article , the authors established a general inequality for the hyperbolic functions, extended the newly established inequality to trigonometric functions, and obtained some new inequalities involving the inverse sine and inverse Hyperbolic sine functions, applying these inequalities to the Neuman-Sándor mean and the first Seiffert mean.
Abstract: In the paper, the authors establish a general inequality for the hyperbolic functions, extend the newly-established inequality to trigonometric functions, obtain some new inequalities involving the inverse sine and inverse hyperbolic sine functions, and apply these inequalities to the Neuman–Sándor mean and the first Seiffert mean.
TL;DR: In this article , two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions were established and applied to obtain some refinement and extension of Mitrinović-Adamović and Lazarević inequalities.
Abstract: In this paper, we establish two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions and apply them to obtain some refinement and extension of Mitrinović–Adamović and Lazarević inequalities. The inequalities obtained in this paper go beyond the conclusions and conjectures in the previous literature. Finally, we apply the main results of this paper to the field of mean value inequality and obtain two new inequalities on Seiffert-like means and classical means.
TL;DR: In this article , an alternating reflection method on conics is presented to evaluate inverse trigonometric and hyperbolic functions, where the conics of the conic are used to evaluate the inverse functions.
Abstract: <abstract><p>An unusual alternating reflection method on conics is presented to evaluate inverse trigonometric and hyperbolic functions.</p></abstract>
TL;DR: The Lorentz triangle search variable step coefficient was proposed based on the broad-spectrum trigonometric functions combined with the LorentZ chaotic mapping strategy to improve the global search ability and local development ability of the AOA.
Abstract: With the increasing complexity and difficulty of numerical optimization problems in the real world, many efficient meta-heuristic optimization methods have been proposed to solve these problems. The arithmetic optimization algorithm (AOA) design is inspired by the distribution behavior of the main arithmetic operators in mathematics, including multiplication (M), division (D), subtraction (S) and addition (A). In order to improve the global search ability and local development ability of the AOA, the Lorentz triangle search variable step coefficient was proposed based on the broad-spectrum trigonometric functions combined with the Lorentz chaotic mapping strategy, which include a total of 24 search functions in four categories, such as regular trigonometric functions, inverse trigonometric functions, hyperbolic trigonometric functions, and inverse hyperbolic trigonometric functions. The position update was used to improve the convergence speed and accuracy of the algorithm. Through test experiments on benchmark functions and comparison with other well-known meta-heuristic algorithms, the superiority of the proposed improved AOA was proved.
TL;DR: In fact, there are at least six additional trigonometric functions considered important enough to grace the pages of texts in centuries past have fallen by the wayside, to be largely forgotten in favour of the modern standard six as discussed by the authors .
Abstract: Ask anyone who has studied mathematics to a moderate level how many trigonometric functions there are and one is likely to be presented with a range of answers depending on what the person being asked is most likely to remember. Perhaps the ‘calculator button’ three of sine, cosine, and tangent will come to mind as these are the three trigonometric functions found on any standard scientific calculator. At a stretch, perhaps the names for their respective reciprocals, cosecant, secant and cotangent, will be recalled. Beyond the modern standard six, looking at calculus or trigonometric texts published prior to 1900 one soon discovers others going by strange names such as versine, haversine, or coversine (see, for example, [1, pp. 53, 63]). There are at least six others with as many as perhaps ten to twelve having received a name at one time or another. Today all these additional trigonometric functions considered important enough to grace the pages of texts in centuries past have fallen by the wayside, to be largely forgotten in favour of the modern standard six. Of course the pedant amongst us would say there is only one trigonometric function, the sine function, which currently stands as the preferred fundamental trigonometric entity, with all others being simple variations of this function, and they would not be incorrect in asserting this. But having the current standard six seems about the right balance between the minimalistic on the one hand and convenience on the other hand.
TL;DR: In this article , a class of generalized trigonometric functions with two parameters is extended to maximal domains on which they are univalent, and some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity.
Abstract: Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity.