Showing papers on "Inverse trigonometric functions published in 2010"

FPGA based floating-point library for CORDIC algorithms

Abstract: Computation of floating-point transcendental functions has a relevant importance in a wide variety of scientific applications, where the area cost, error and latency are important requirements to be attended. This paper describes a flexible FPGA implementation of a parameterizable floating-point library for computing sine, cosine, arctangent and exponential functions using the CORDIC algorithm. The novelty of the proposed architecture is that by sharing the same resources the CORDIC algorithm can be used in two operation modes, allowing it to compute the sine, cosine or arctangent functions. Additionally, in case of the exponential function, the architectures change automatically between the CORDIC or a Taylor approach, which helps to improve the precision characteristics of the circuit, specifically for small input values after the argument reduction. Synthesis of the circuits and an experimental analysis of the errors have demonstrated the correctness and effectiveness of the implemented cores and allow the designer to choose, for general-purpose applications, a suitable bit-width representation and number of iterations of the CORDIC algorithm.

Single variable calculus : concepts and contexts

Abstract: Preface. To the Student. Diagnostic Tests. A Preview of Calculus. 1. FUNCTIONS AND MODELS. Four Ways to Represent a Function. Mathematical Models: A Catalog of Essential Functions. New Functions from Old Functions. Graphing Calculators and Computers. Exponential Functions. Inverse Functions and Logarithms. Parametric Curves. Laboratory Project: Running Circles around Circles. Review. Principles of Problem Solving. 2. LIMITS AND DERIVATIVES. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. Continuity. Limits Involving Infinity. Derivatives and Rates of Change. Writing Project: ? ?Early Methods for Finding Tangents. The Derivative as a Function. What Does f Say about f ? Review. Focus on Problem Solving. 3. DIFFERENTIATION RULES. Derivatives of Polynomials and Exponential Functions. Applied Project: Building a Better Roller Coaster. The Product and Quotient Rules. Derivatives of Trigonometric Functions. The Chain Rule. Laboratory Project: BUzier Curves. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Inverse Trigonometric Functions and their Derivatives. Derivatives of Logarithmic Functions. Discovery Project: Hyperbolic Functions. Rates of Change in the Natural and Social Sciences. Linear Approximations and Differentials. Laboratory Project: Taylor Polynomials. Review. Focus on Problem Solving. 4. APPLICATIONS OF DIFFERENTIATION. Related Rates. Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. Derivatives and the Shapes of Curves. Graphing with Calculus and Calculators. Indeterminate Forms and l'Hospital's Rule. Writing Project: The Origins of l'Hospital's Rule. Optimization Problems. Applied Project: The Shape of a Can. Newton's Method. Antiderivatives. Review. Focus on Problem Solving. 5. INTEGRALS. Areas and Distances. The Definite Integral. Evaluating Definite Integrals. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Writing Project: Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Integration by Parts. Additional Techniques of Integration. Integration Using Tables and Computer Algebra Systems. Discovery Project: Patterns in Integrals. Approximate Integration. Improper Integrals. Review. Focus on Problem Solving. 6. APPLICATIONS OF INTEGRATION. More about Areas. Volumes. Discovery Project: Rotating on a Slant. Volumes by Cylindrical Shells. Arc Length. Discovery Project: Arc Length Contest. Average Value of a Function. Applied Project: Where To Sit at the Movies. Applications to Physics and Engineering. Discovery Project: Complementary Coffee Cups. Applications to Economics and Biology. Probability. Review. Focus on Problem Solving. 7. DIFFERENTIAL EQUATIONS. Modeling with Differential Equations. Direction Fields and Euler's Method. Separable Equations. Applied Project: How Fast Does a Tank Drain? Applied Project: Which Is Faster, Going Up or Coming Down? Exponential Growth and Decay. Applied Project: Calculus and Baseball. The Logistic Equation. Predator-Prey Systems. Review. Focus on Problem Solving. 8. INFINTE SEQUENCES AND SERIES. Sequences. Laboratory Project: Logistic Sequences. Series. The Integral and Comparison Tests Estimating Sums. Other Convergence Tests. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series. Laboratory Project: An Elusive Limit. Writing Project: How Newton Discovered the Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Review. Focus on Problem Solving. APPENDIXES. A. Intervals, Inequalities, and Absolute Values. B. Coordinate Geometry. C. Trigonometry. D. Precise Definitions of Limits. E. A Few Proofs. F. Sigma Notation. G. Integration of Rational Functions by Partial Fractions. H. Polar Coordinates. I. Complex Numbers. J. Answers to Odd-Numbered Exercises.

Efficient evaluation methods of elementary functions suitable for SIMD computation

Abstract: Data-parallel architectures like SIMD (Single Instruction Multiple Data) or SIMT (Single Instruction Multiple Thread) have been adopted in many recent CPU and GPU architectures. Although some SIMD and SIMT instruction sets include double-precision arithmetic and bitwise operations, there are no instructions dedicated to evaluating elementary functions like trigonometric functions in double precision. Thus, these functions have to be evaluated one by one using an FPU or using a software library. However, traditional algorithms for evaluating these elementary functions involve heavy use of conditional branches and/or table look-ups, which are not suitable for SIMD computation. In this paper, efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches. We implemented these methods using the Intel SSE2 instruction set to evaluate their accuracy and speed. The results showed that the average error was less than 0.67 ulp, and the maximum error was 6 ulps. The computation speed was faster than the FPUs on Intel Core 2 and Core i7 processors.

Singularity-robust modular inverse kinematics for robotic gesture imitation

Abstract: We present a method of computing the joint angles for an upper body humanoid robot corresponding to task space motion data from a human demonstrator. Using a divide-and-conquer approach, we group the motors into pan-tilt and spherical units, and solve the inverse kinematics in a modular fashion based on the derivative of the inverse tangent function of the relevant task space variables. For robustness to kinematic singularity, we add a regularization parameter that vanishes whenever the task variables are outside a neighborhood of zero. Simulation study on a 7 degree-of-freedom (DOF) robot arm shows a tradeoff of tracking accuracy in a neighborhood of each singularity in favor of robustness, but high accuracy is recovered outside this neighborhood. Experimental implementation of the proposed inverse kinematics on a 17-DOF upper-body humanoid robot shows that user-demonstrated gestures are well-replicated by the robot.

Approximations to the magic formula

Abstract: Pacejka’s tire model is widely used and well-known by the automotive engineering community. The magic formula describes the brake force, side force and self-aligning torque in terms of the longitudinal slip and slip angle, plus several corrections. This paper uses approximation theory to obtain different types of approximations to the magic formula: rational functions (RA) resulting from the Remez algorithm, expansions in a series of Chebyshev polynomials (ACh), a series of Chebyshev rational polynomials (ARChPs), a series of rational orthogonal functions (ORF) and a series of ARChPs that result from grade-1 ORFs. The last expansion shows the fastest convergence and most effective computation. Jacobi rational polynomials can also be obtained to complement this expansion and facilitate fine-tuning in specific areas of the error curve. This work is complemented by obtaining the original rational approximations to the inverse tangent function, which take advantage of the curve symmetry to reduce the computation load and provide models that include the influence of the vertical load. The convergence properties of the development in series and the error values resulting from numeric examples for the three types of stress are shown. The proposed final ARChP expressions show very low error (1%) compared to the original magic formula. They can be computed 20 times faster; they can be evaluated, derived and integrated analytically easily; and their coefficients can be obtained from tests using common least-squares algorithms.

Redundancy Parameterization for Flexible Motion Control

Abstract: Our overall research interest is in synthesizing human like reaching and grasping using anthropomorphic robot hand-arm systems, as well as understanding the principles underlying human control of these actions. When one needs to define the control and task requirements in the Cartesian space, the problem of inverse kinematics needs to be solved. For non-redundant manipulators, a desired end-effector position and orientation can be achieved by a finite number of solutions. For redundant manipulators however, there are in general infinitely many solutions where the cardinality of the solution set must be made finite by imposing certain constraints. In this paper, we consider the Mitsubishi PA10 manipulator which is similar to the human arm, in the sense that both wrist and shoulder joints can be considered to emulate a 3DOF ball joint. We explicitly derive the analytic solution for the inverse kinematics using quaternions. Then, we derive a parameterization in terms of a pure quaternion called the swivel quaternion. The swivel quaternion is similar to the elbow swivel angle used in most approaches, but avoid the computation of inverse trigonometric functions. This parameterization of the self-motion manifold is continuous with any end-effector motion. Given the pose of the end-effector and the swivel quaternion (or swivel angle), the algorithm derives all solution of the inverse kinematics (finite number). We then show how the parameterization of the elbow self-motion can be used for the real-time control of the PA10 manipulator in the presence of obstacles.Copyright © 2010 by ASME

An approach based on AD converted resolver demodulation

Abstract: This paper improves the software approach of using the resolver-to-digital conversion. The original approach takes sample at positive peak values of excitation signal which increases the system's complexity. This paper applies an approach which is to get information by taking sample at other positions in an excitation period. This approach yields an experimental result with the higher sample frequency and more real-time. Meanwhile, in order to solve the problem of complex computations or large data space for checking tables in the process of directly using arctangent to get angle degrees, it adopts the approach, which converts the tangent value into sine and cosine values. Finally, it uses the LabVIEW simulation to test to meet the requirement better.

Stability of Trigonometric Functional Equations in Generalized Functions

Abstract: We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

A non-standard approach to introduce simple harmonic motion

Abstract: We'll be presenting an approach to solve the equation of simple harmonic mo- tion (SHM) which is non-standard as compared with the usual way of solution presented in textbooks. In addition to help students avoid the unnecessary memorization of formulas to solve physics problems, this approach could help instructors to present the subject in a teaching framework which integrates conceptual and mathematical reasoning, in a systemic way of thinking that will help students to reinforce their quantitative reasoning skills by using mathematical knowledge already familiar to students in a first calculus-based introductory physics course, such as the chain rule for derivatives, inverse trigonometric functions, and integration methods.

Nonlinear Vector Filtering for Impulsive Noise Removal from Color Images

Abstract: In this paper, a comprehensive survey of 48 filters for impulsive noise removal from color images is presented. The filters are formulated using a uniform notation and categorized into 8 families. The performance of these filters is compared on a large set of images that cover a variety of domains using three effectiveness and one efficiency criteria. In order to ensure a fair efficiency comparison, a fast and accurate approximation for the inverse cosine function is introduced. In addition, commonly used distance measures (Minkowski, angular, and directional-distance) are analyzed and evaluated. Finally, suggestions are provided on how to choose a filter given certain requirements.

A fully integrated CMOS inverse sine circuit for computational systems

Abstract: An inverse trigonometric function generator using CMOS technology is presented and implemented. The development and synthesis of inverse trigonometric functional circuits based on the simple approximation equations are also introduced. The proposed inverse sine function generator has the infinite input range and can be used in many measurement and instrumentation systems. The nonlinearity of less than 2.8% for the entire input range of 0.5 Vp-p with a small-signal bandwidth of 3.2 MHz is achieved. The chip implemented in 0.25 μm CMOS process operates from a single 1.8 V supply. The measured power consumption and the active chip area of the inverse sine function circuit are 350 μW and 0.15 mm2, respectively.

Multiplier-less and Table-less Linear Approximation for Square-Related Functions

Abstract: Square-related functions such as square, inverse square, square-root and inverse square-root operations are widely used in digital signal processing and digital communication algorithms, and their efficient realizations are commonly required to reduce the hardware complexity. In the implementation point of view, approximate realizations are often desired if they do not degrade performance significantly. In this paper, we propose new linear approximations for the square-related functions. The traditional linear approximations need multipliers to calculate slope offsets and tables to store initial offset values and slope values, whereas the proposed approximations exploit the inherent properties of square-related functions to linearly interpolate with only simple operations, such as shift, concatenation and addition, which are usually supported in modern VLSI systems. Regardless of the bit-width of the number system, more importantly, the maximum relative errors of the proposed approximations are bounded to 6.25% and 3.13% for square and square-root functions, respectively. For inverse square and inverse square-root functions, the maximum relative errors are bounded to 12.5% and 6.25% if the input operands are represented in 20bits, respectively.

Efficient evaluation methods of elementary functions suitable for SIMD computation

Abstract: Data-parallel architectures like SIMD (Single Instruction Multiple Data) or SIMT (Single Instruction Multiple Thread) have been adopted in many recent CPU and GPU architectures. Although some SIMD and SIMT instruction sets include double-precision arithmetic and bitwise operations, there are no instructions dedicated to evaluating elementary functions like trigonometric functions in double precision. Thus, these functions have to be evaluated one by one using an FPU or using a software library. However, traditional algorithms for evaluating these elementary functions involve heavy use of conditional branches and/or table look-ups, which are not suitable for SIMD computation. In this paper, efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches. We implemented these methods using the Intel SSE2 instruction set to evaluate their accuracy and speed. The results showed that the average error was less than 0.67 ulp, and the maximum error was 6 ulps. The computation speed was faster than the FPUs on Intel Core 2 and Core i7 processors.

A bi-orthogonal system of trigonometric functions

Abstract: In this paper, we construct a system of bi-orthogonal trigonometric functions and we connect this system to a family of orthogonal functions on the unit circle that can be considered like generalized polynomials in the variable z 1/2. Some properties such as recurrence relations, kernel representations and a Favard-type theorem are studied.

Summation of multiple trigonometric series by generalized methods formulated using δ -like finite functions

Abstract: We consider the generalized sums of multiple trigonometric series. We investigate the sufficient conditions of convergence of the series obtained by termwise differentiation of the series for Lebesgue integrable functions as well as the errors of approximation of functions by sequences of generalized partial sums of series.

Complex inclusion functions in the CoStLy C++ class library

Abstract: The C++ class library CoStLy for the rigorous computation of complex function values or ranges is presented. Rectangular complex interval arithmetic is used for the computations. In the CoStLy procedures, all truncation and roundoff errors are calculated during the course of the floating-point computation and enclosed into the result. The library contains procedures for root and power functions, the exponential, trigonometric and hyperbolic functions, their inverse functions, and some auxiliary functions, such as the absolute value or the argument function.