Function Generation Synthesis Research Articles

Fourier-Based Function Generation of Four-Bar Linkages With an Improved Sampling Points Adjustment and Sylvester's Dialytic Elimination Method

Abstract Four-bar linkages are critical fundamental elements of many mechanical systems, and their design synthesis is often mathematically complicated with iterative numerical solutions. Analytical methods based on Fourier coefficients can circumvent these difficulties but have difficulties with sampling points adjustment and solutions of the design equations in previous studies. In this paper, an improved Fourier-based analytical synthesis method is presented, which transforms the function generation synthesis of planar four-bar linkages into a problem of solving design equations. Calculation of the Fourier coefficients is discussed, including the discretization of the prescribed function and an improved sampling points adjustment method. It is shown that the Fourier coefficients can be computed efficiently and accurately by discretizing the prescribed function with a small number of sampling points. The proposed sampling adjustment method overcomes the difficulty of easily resulting in non-Grashof solutions by considering the complete period of the prescribed function. An improved Sylvester's dialytic elimination method is presented to solve design equations. The method reduces the computation time and avoids cumbersome procedures without generating additional invalid solutions. Several examples are presented to demonstrate the advantages of the proposed synthesis method, which is easy-understanding and efficient, and yields more accurate solutions than available synthesis methods.

Complete Solutions for the Approximate Synthesis of Spherical Four-Bar Function Generators

Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves.

Multi-modal continuous approximate synthesis of planar four-bar function generators

Multi-modal continuous approximate synthesis of planar four-bar function generators

Utilization of Function Generation Synthesis on Biomimetics: A Case Study on Moray Eel Double Jaw Design.

Throughout history, humans have observed living or non-living things in nature and then imitated them in relation to these observations. This is due to the fact that the energy found in nature is generally consumed at an optimal level in order for it to endure. Biomimetic inspiration in many designs and applications is widely displayed, including within the field of engineering. In this paper, we were inspired by the double set of jaws found in the moray eel, which gives this fish a huge advantage while hunting, with a mobile pharyngeal jaw that works together with its oral jaw in order to overcome ineffective suction capabilities. A procedure that mimics the hunting motion of the moray eel was utilized by considering the overall movement as a single degree of freedom with multiple outputs on account of the repeating motion that is required during hunting. This procedure includes structural and dimensioning synthesis, wherein the latter was utilized with analytic kinematic synthesis for each linkage transfer. The flexibilities in parameters were taken into account with a novel multiple iterative kinematic synthesis algorithm that resulted in various mechanisms with the same purpose. Among the excessive number of resultant mechanisms, the optimization was carried out by considering the highest torque transmission ratio at critical timings that were specified as bio-constraints. In the end, the kinematic movement validation was utilized.

Multi-modal continuous approximate synthesis of planar four-bar function generators

Multi-modal continuous approximate synthesis of planar four-bar function generators

Integration of the function generation synthesis into the design of a multi-leaf collimator

Throughout recent history, medical robots have gained a lot of popularity in robotic assisted radiotherapy. Integration of robot manipulators into the field of radiotherapy has improved the chances of patient recovery by increasing system precision, operation reliability and treatment efficiency. In order to prevent healthy tissues from radiation exposures during treatments, linear accelerators of radiotherapy robots have been enhanced by the implementation of active collimator designs. Addition of collimator mechanisms to these systems has also increased possibilities of treatments by allowing adjustable beam geometries on the targets. However, as the collimator leaves are additional obstacles on the path of the beam, beam scattering has become one of the most crucial issues waiting to be improved for these systems. In order to propose a solution for this problem, current paper tries to implement function generation synthesis into a multi-leaf active collimator design, where the individual collimator blocks are stacked vertically. This study demonstrates the proposed approach by utilizing two single degree of freedom planar systems positioned side by side for a point beam source. Throughout the study, kinematic synthesis approach was introduced to design these planar systems to adjust beam contour geometries on a single axis. At the end of the study, proposed procedure was verified by a numerical example and the results were used to design four degrees of freedom collimator that is able to adjust beam contour geometries on the plane. The prototype of the system was also modelled in order to ensure the realization of the collimator design for contour adjustments.

A novel analytical method for function generation synthesis of planar four-bar linkages

In this paper, a novel analytical approach is presented to solve function generation synthesis problems for planar four-bar linkages with no limitations on the number of precision points. Firstly, the output rotation-angle function is formulated according to the Fourier series components of the input angle, and is substituted into a vector loop equation. Then, the relationship between the design parameters of the linkage and the Fourier coefficients of the output function is obtained. On the basis of this, the new design equations are established. Finally, a cubic polynomial equation is obtained by dialytic elimination. Three non-trivial solutions for the new design equations are ultimately obtained analytically. Three examples are provided to verify the effectiveness of the proposed technique. As a result, a general formula is derived for calculation of the design parameters of a planar four-bar linkage using the Fourier coefficients of the prescribed function.

A restatement of the optimum synthesis of function generators with planar four-bar and slider-crank mechanisms examples

The problem of optimum synthesising of a mechanism to best approximate a function, while simultaneously ensuring good motion transmission characteristics is discussed. Distinction is made between the design variables that determine the shape of the input-output (I/O) function of the mechanism, and the design variables that affect the degree of overlap between this I/O function and the function to be mechanised, through scaling, mirroring and rotations in 90o increments. Examples are given of designing the planar four-bar and slider-crank linkages of a logarithmic scale, and that of a tangent-function generator. These are performed on modified mechanisms with an added degree-of-freedom, which substantially simplify the synthesis problem.

A Function Generation Synthesis Methodology for All Defect-Free Slider-Crank Solutions for Four Precision Points

The well-established methodology for slider-crank function generation states that five precision points can be achieved without structural error. The resulting designs, however, do not necessarily satisfy all of the kinematic requirements for designing a slider-crank linkage used in common machine applications such as driving the ram of a mechanical press. First, linkage solutions to the five precision point synthesis problem may need to change circuits to reach the precision points. Second, there is no guarantee that the input crank is fully rotatable. This paper presents a modification to the function generation synthesis methodology that reveals a continuum of defect-free, slider-crank solutions for four precision points. Additionally, the methodology allows the specification of velocity or acceleration at the precision points. Although smaller accelerations at a point of zero slide velocity are associated with longer dwell, a point having zero velocity and acceleration is shown not to be possible. Examples are included to illustrate this kinematic synthesis methodology.

Function generation synthesis of spherical 5R mechanism with regional spacing and Chebyshev approximation

The Chebyshev approximation is well known to be applicable for the approximation of single input–single output functions by means of a function generator mechanism. The approximation method may be also applied to multi-input functions, although until recently, it was not used for function generation with multi-degrees-of-freedom mechanisms. In a recent study, the authors applied the approximation method to a two-degrees-of-freedom mechanism for the first time, however the selection and iteration of the design points at which the errors were minimized were not satisfactory. In this study, an alternative method of selection and iteration for these design points is introduced and the corresponding spacing is called the “regional spacing”. As a case study for the application of the approximation of multi-input functions, a spherical 5R mechanism is used to generate a two input-single-output function. The input joints of the mechanism are selected as one of the fixed joints and the moving mid-joint, whereas the remaining fixed joint represents the output. The synthesis problem is analytically formulated and presented in polynomial form for five and six unknown parameters. The synthesis problem for five unknown parameters is illustrated as a numerical example. Regional spacing is used for the selection and iteration of design points for the synthesis. The Chebyshev approximation along with the Remez algorithm is utilized to find the unknown construction parameters and the error of the function. The design points and the coefficients of the approximation polynomial are determined by numerical iteration using six moving points. At each iteration step, the design points are relocated at the extremum error points in their respective regions. Iterations are repeated until the magnitudes of the extremum point errors are approximately equal. Finally, the construction parameters of the mechanism are determined and the variation of the percentage error between the desired and generated function values is obtained.

Function generation with the RRRCR spatial linkage

This paper presents a systematic approach to perform the function generation synthesis of the spatial RRRCR linkage for six precision points. The mathematical model associated with the synthesis process is a system composed of six multivariate polynomial equations and two quadratic equations, which must be solved for the design parameters of the linkage. Due to the highly nonlinear and extremely complex nature of the design equations, there was used a computer package (BERTINI©) in order to find all numerical solutions. The applicability of the proposed approach is illustrated by the generation of three nonlinear functions.

High-Precision Digital Audio Waveform Synthesis Using a Multi-Frequency Interpolation Technique

A description is given of a digital, real-time, multi-frequency function generator for audio bandwidth applications. A novel feature of the system is the embedded algorithm, which employs a bank of parallel interpolators, rather than a look-up table, to synthesise complex functions with high precision and flexibility. The complete instrumentation comprises a high-level software design tool operating on a conventional desktop or notebook PC and a separate hardware unit incorporating a powerful real-time digital signal processor unit, operating at 275 million multiplication-accumulations per second (MMACs) and a dual channel 24-bit codec with a maximum output rate of 200 kilo-samples per second. Using total harmonic distortion (THD) analysis, the purity of the waveforms has been compared with those generated by traditional analogue instruments; this analysis has confirmed that the new instrument has a consistently superior performance over such devices. Testing established that for a 2 kHz sine wave, for example, there was an order of magnitude improvement in THD performance over a commercially available analogue function generator. The algorithm and associated hardware offer the combined advantages of function flexibility, ease of implementation, accuracy and the unique ability to alter, in real-time, the frequencies, magnitudes and phases of individual harmonics within any given arbitrary waveform. Digital function generation and waveform synthesis represent key components of any DSP course; at the University of Manchester it is taught as a formal discipline through lectures, and from a practical context during our laboratory classes, in which students gain an understanding of digital methods and how they excel over traditional analogue circuits designed for the same purpose.

Optimal synthesis of function generation in steering linkages

This paper presents an approach to optimize planar mechanisms for function generation synthesis. It is applied to the design of different steering linkages used in road vehicles where the Ackermann condition is formulated as the objective function. Four different mechanisms are kinematically defined and synthesized in terms of the proposed method. Limitations of the size of the elements are also considered by means of inequality constraints. Solutions are presented as the difference between desired behaviour, established by Ackermann, and actual generated performance. The results show that the accuracy achieved by the proposed procedure is satisfactory for working conditions in this type of linkages.

Unified synthesis of a planar four-bar mechanism for function generation using a spring-connected arbitrarily sized block model

A unified mechanism synthesis method for simultaneously determining the type and dimension of a planar four-bar mechanism is proposed for function generation synthesis. A spring-connected arbitrarily sized rectangular block model is proposed, and the unified synthesis process is formulated as an optimization problem using this block model. By solving the optimization problem, the type and dimension of a planar four-bar mechanism can be obtained simultaneously for a function generation synthesis. Several function generation synthesis problems are successfully solved with the proposed method. An index to define the level of difficulty of a function generation synthesis problem is proposed based on the standard deviation of curvature and the length of the curve of the desired function.

Synthesis of function generating spherical four bar mechanism for the six independent parameters

Abstract This study deals with the function generation synthesis of spherical four bar mechanism for six independent construction parameters φ 0 , ψ 0 , α 1 , α 2 , α 3 , and α 4 by giving six or more design points with respect to the methods that are used in synthesis procedure. Quaternion algebra is used to derive the objective function of spherical four bar mechanism by following some rotational sequences. Three different methods as interpolation approximation, least squares approximation and Chebyshev approximation are used during synthesis procedure. During the consecutive trials in Chebyshev approximation, a new approach is taken to renew the design points φ i by plotting the graph of the objective functions derivative and taking the roots of it as new design points with two boundary points. Separate examples are given for each section and the results are tabulated. Discussions about the study and comparisons between the used methods are given at the end of the study.

Defect-Free Synthesis of Stephenson-II Function Generators

This article presents a novel method for the dimensional synthesis of Stephenson-II function generators without order, circuit, and branch defects. The explicit equations relating the generated output and specified variables of Stephenson-II function generators are derived by eliminating and transforming angular variables on two five-bar loops and by using the specified values of input and output variables. Then, the deviation between the generated and required angular positions of the output link of the synthesized mechanism is obtained for the objective function. Using the characteristics of order, circuits, branches, and the geometric features of dead-center configurations for Stephenson-II function generators, the constraint equations for avoiding these defects are proposed for the optimal synthesis of Stephenson-II function generators. The synthesized mechanisms are shown to be defect-free and satisfy all functional requirements. An example is given to demonstrate the feasibility of the proposed method.

Optimal synthesis of function generator of four-bar linkages based on distribution of precision points

This article describes a method for optimizing the synthesis of a four-bar linkage mechanism to generate a definite mathematic function. In this research, the objective function was defined based on the least squares of error between the generated function and the desired function. Because of non-linearity of the objective function and the constraint, SQP method has been used to find the optimal mechanism. This method of optimization is a gradient-based method and its result is more trustable than heuristic methods. In this study, five precision points have been used to synthesize four-bar linkages; therefore, by using this amount of precision points, a set of non-linear equations was obtained for mechanism synthesis. Unlike the previous researches that dimensional parameters are used as optimization variables, precision points distribution was used in current research. The main innovation of this paper is presentation of some directions to have estimative prediction for distribution of precision points in optimal mechanism which maybe useful for designers. These directions were obtained based on the method that presented in current work and regard to shape of desired function and its first and second derivative.

Optimal synthesis of planar mechanisms via an extensible-link approach

This paper presents a novel approach to optimize the design of planar mechanisms with revolute joints for function generation or path synthesis. The proposed method is based on the use of an extensible-link mechanism model whose strain energy is minimized to find the optimal rigid design. This enables us to get rid of assembly constraints and the use of natural coordinates makes the objective function simpler. Two optimization strategies are developed and then discussed. The first one relies on alternate optimizations of design parameters and point coordinates. The second one uses multiple partial syntheses as starting point for a full synthesis process. The question of finding the global optimum is also addressed and developed. A simple algorithm is proposed to find several local optima among which the designer may choose the best one taking other criteria into account (e.g. stiffness, collision, size,...). Three applications are presented to illustrate the strategies while mentioning their limits.

Some improvements on the exact kinematic synthesis of spherical 4R function generators

Abstract It is shown that a careful definition of the design coefficients may improve the kinematic synthesis of spherical 4R linkages – intended for function generation – for three and four precision points. As a result, the design process is based on a simple system of linear equations whose solution is obtained in closed form. Moreover, the synthesis process does not lead to algorithmic problems during the determination of the link lengths. Furthermore, useful discussions for obtaining positive link lengths are also provided. Finally, several application examples are presented to prove the feasibility and the validity of the proposed method.

Optimum synthesis of the four-bar function generator in its symmetric embodiment: the Ackermann steering linkage

The problem of optimum synthesis of the planar four-bar function generator is investigated and the practical case of the Ackermann steering linkage considered as an example. The reduced number of design parameters of this symmetric four-bar linkage allowed inspecting the design space of various types of objective functions through 3D representations, and their properties suggestively highlighted. For practical purposes, the numerical results were summarized in a set of parametric design-charts useful to the automotive engineer in conceiving the steering linkage of a new vehicle.