Mechanisms For Path Generation Research Articles

Structural–Parametric Synthesis of Path-Generating Mechanisms and Manipulators

This paper presents a structural–parametric synthesis of the four-link and Stephenson I, Stephenson II, and Stephenson III six-link path-generating mechanisms. The four-link path-generating mechanism is formed by connecting the output point and the base using an active closing kinematic chain (CKC) with two DOFs and a negative CKC of the type RR. The six-link path-generating mechanisms are formed by connecting the output point and the base by active, passive and negative CKCs. Active CKC has active kinematic pair, passive CKC has zero DOF, and negative CKC has a negative DOF. Active and negative CKCs impose geometrical constraints on the movement of the output point, and the geometric parameters of their links are determined by least-square approximation. Geometric parameters of the passive CKC are varied to satisfy the geometrical constraints of the active and negative CKCs. The CKCs of the active, passive and negative types, connecting the output point and the base, are the structural modules from which the different types of the path-generating mechanisms are synthesized. Numerical examples of the parametric synthesis of the four-link and six-link path-generating mechanisms are presented.

Optimal synthesis of a spatial RRSS mechanism for path generation

Optimal synthesis of a spatial RRSS mechanism for path generation

Dimensional synthesis of four-bar mechanisms using Rao algorithms and their variants

This paper presents the dimensional synthesis of four-bar mechanisms for path generation. The synthesis of the four-bar mechanisms is proposed using nine advanced optimization algorithms, namely the Rao-1 algorithm, Rao-2 algorithm, Rao-3 algorithm, self-adaptive multi-population (SAMP) Rao-1 algorithm, SAMP Rao-2 algorithm, SAMP Rao-3 algorithm, quasi-oppositional-based (QO) Rao-1 algorithm, QO Rao-2 algorithm, and QO Rao-3 algorithm. Six different cases of synthesis of a four-bar mechanism are considered. The brief literature review based on the synthesis of mechanisms is also presented. The designs of mechanisms obtained using Rao algorithms and their variants are compared with the designs acquired using other advanced optimization algorithms, such as genetic algorithm (GA), particle swarm optimization (PSO) algorithm, differential evolution (DE), imperialist competitive algorithm (ICA), hybrid GA-DE algorithm, simulated annealing (SA), Malaga university mechanism synthesis algorithm (MUMSA), IMMa optimization algorithm (IOA), hybrid Taguchi random coordinate search algorithm (HTRCA), hybrid GA and fuzzy logic (GA–FL) algorithm, ant-gradient method (AGM), teaching–learning-based optimization algorithm (TLBO), and hybrid Lagrange interpolation differential evolution algorithm (HLIDE). The comparison of results reveals the efficacy of Rao algorithms and their variants for synthesizing the four-bar mechanism.

Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints

Topology optimization is a powerful numerical tool in the synthesis of lightweight structures and compliant mechanisms. Compliant mechanisms present challenges for topology optimization, as they typically exhibit large displacements and rotations. Path-generation mechanisms are a class of mechanisms that are designed to follow an exact path. The characteristics of compliant mechanisms therefore exclude the validity of linear finite-element analysis to ensure the proper modeling of deformation and stresses. As stresses can exceed the limit when neglected, stress constraints are needed in the synthesis of compliant mechanisms. Both nonlinear finite-element analysis as well as the consideration of stress constraints significantly increase computational cost of topology optimization. Multiresolution topology optimization, which employs different levels of discretization for the finite-element analysis and the representation of the material distribution, allows an important reduction of computational effort. A multiresolution topology optimization methodology is proposed integrating stress constraints based on nonlinear finite-element analysis for path-generation mechanisms. Two objective formulations are used to motivate and validate this methodology: maximum-displacement mechanisms and path-generation mechanisms. The formulation of the stress constraints and their sensitivities within nonlinear finite-element analysis and multiresolution topology optimization are explained. We introduce two academic benchmark examples to demonstrate the results for each of the objective formulations. To show the practical, large-scale application of this method, results for the compliant mechanism structure of a droop-nose morphing wing concept are shown.

Optimum Synthesis of Four-Bar Mechanism by Using Relative Angle Method: A Comparative Performance Study

In this paper, the dimensional synthesis of the four-bar mechanism for path generation is formulated using the relative angle motion analysis and the link geometry parameterization with Cartesian coordinates. The Optimum Dimensional Synthesis using Relative Angles and the Cartesian space link Parameterization (ODSRA+CP) is stated as an optimization problem, and the solution is given by the differential evolution variant DE/best/1/bin. This study investigates the behavior and performance of such formulation and performs a comparative empirical study with the well-known synthesis method based on vector-loop equation motion analysis where different modifications in the metaheuristic algorithms are established in the literature to improve the obtained solution. Five study cases of dimensional synthesis for path generation with and without prescribed timing are solved and analyzed. The empirical results show that the way of stating the optimization problem in the ODSRA+CP significantly improves the search process for finding promising solutions in the optimizer without requiring algorithm modifications. Therefore, it is confirmed that the optimizer search process in the optimal synthesis of mechanisms is not the only way of improving the obtained solutions, but also the optimization problem formulation has a significant influence on the search for better solutions.

Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity

This paper proposes and investigates two formulations to topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. The first formulation extends the maximum output displacement robust approach with stress constraints to incorporate the effects of geometric nonlinear behavior during the optimization process. The second formulation relies on the concept of path-generating mechanisms, where not only the final configuration is important, but also the load–displacement equilibrium path. A novel path-generating formulation is thus proposed, not only to achieve the prescribed equilibrium path, but also to take stress constraints and manufacturing uncertainty into account during the optimization process. Although both formulations have different goals, the same main techniques are employed: density approach to topology optimization, augmented Lagrangian method to handle the large number of stress constraints, three-field robust approach to handle the manufacturing uncertainty, and the energy interpolation scheme to handle convergence issues due to large deformation in void regions. Several numerical examples are addressed to demonstrate applicability of the proposed approaches. The optimized results are post-processed with body-fitted finite element meshes. Obtained results demonstrate that: (1) the proposed nonlinear analysis based maximum output displacement approach is able to provide solutions with good performance in situations of large displacements, with stress and manufacturing requirements satisfied; (2) the linear analysis based maximum output displacement approach provides optimized topologies that show large stress constraint violations and rapidly varying stress behavior under uniform boundary variation, when these are post-processed with full nonlinear analysis; (3) the proposed path-generating formulation is able to provide solutions that follow the prescribed control points, including stress robustness.

A loop-by-loop defect rectification procedure for optimal synthesis of Stephenson III path generators

This paper presents a formulation of constraints for the synthesis of Stephenson III mechanism and a loop-by-loop defect-rectification procedure. The procedure divides the Stephenson III mechanism into two loops, namely, Loop I, i.e., four-bar, and Loop II, i.e., five-bar mechanisms. Then, the defects are identified using the established methodology to formulate the defect-specific constraints in the simplified form. Based on the constraints, an optimization problem is formulated to synthesize a Stephenson III mechanism for path generation. A well-established meta-heuristic algorithm is used to solve the constrained optimization problem. For demonstrating the effectiveness of the formulated constraints, two numerical examples are considered, in which Stephenson III path generator mechanisms are synthesized. It is found that the mechanisms synthesized using the proposed procedure are defect-free when constraints are imposed, which is verified using the stick-diagram.

A Stackelberg game theoretic multi-objective synthesis of four-bar mechanisms

In this article, a Stackelberg game theoretic approach is used for multi-objective optimal synthesis of four-bar mechanisms for path generation. Two objective functions, namely, tracking error (TE)...

Topology optimization of planar linkage mechanisms for path generation without prescribed timing

The synthesis of planar linkage mechanisms by a topology optimization method has recently received much attention because both their numbers and dimensions can be simultaneously determined without baseline layouts. To synthesize a mechanism to produce a desired path at its end-effector, a desired path can be defined with or without prescribed timing. While earlier topology optimizations of mechanisms were all concerned with paths with prescribed timing, no study to deal with the topology optimization of mechanisms for path generation without prescribed timing is carried out in spite of its importance. The aim of this study is to propose and set up a gradient-based topology optimization formulation to synthesize planar linkage mechanisms that generate desired paths without prescribed timing. To this end, the desired path of the end-effector is expressed by the centroid distance function which is then represented by its Fourier descriptors. Then a topology optimization formulation using the Fourier descriptors is developed and the sensitivity analysis based on the Fourier descriptors is derived. Several numerical case studies are considered to verify the effectiveness of the proposed formulation. Some numerical issues appearing with the use of the Fourier descriptors are also investigated.

OPTIMAL SYNTHESIS OF EIGHT BAR MECHANISM FOR SPECIFIED PATH GENERATION WITH VARIABLE TOPOLOGY

This paper proposes an analytical synthesis method for optimal synthesis of planar eight bar mechanism for path generation with variable topology. The eight bar planar mechanism has been considered to be made of three phases where each of the phases is a four bar mechanism of RRRR type. The method of kinematic synthesis involves the use of Cheybeychev’s spacing for locating precision points and Freudenstein’s equations for determining link lengths. For any given function, link lengths of the mechanism are determined by assuming the range of input and output angular rotations. Optimized length of rocker link and output angle of the first phase is used as input values for the second phase and same procedure is continued to find the dimensions of the third phase. The combined objective function is established for optimization with transmission angle and link length ratios as design variables. The optimized values of lengths of links of the resulting mechanism satisfy Grashof’s condition and have controlled transmission angle. The coupler connected to third phase generates the coupler curve for obtained optimized link lengths. The method developed herein is illustrated with a numerical example.

Dimensional synthesis of six-bar Stephenson III mechanism for 12 precision points path generation

The technological advancements require sophisticated manufacturing procedures wherein requirement to design mechanisms that can precisely follow a specified path through a given number of precision points are to be fulfilled. With these objectives, a six-bar Stephenson III linkage mechanism, having one degree of freedom has been dimensionally synthesised for 12 precision points. The complex number dyadic and triadic loop closure equations have been used to synthesise the mechanism for path generation. In this approach, loop closure equations are solved simultaneously for 12 displacement positions of coupler tracing point and 12 orientation positions of various links for which the output link oscillates. The prescribed parameters are displacement vector (δj) and coupler link motion and the designer is not constrained to restrict the remaining orientation angles of links. Finally, a code has been developed in MATLAB to solve these loop closure equations for determination of the dimensional length of each link.

Optimum Synthesis of Planar Mechanisms for Path Generation Based on a Combined Discrete Fourier Descriptor

A two-phase synthesis method is described, which is capable of solving quite challenging path generation problems. A combined discrete Fourier descriptor (FD) is proposed for shape optimization, and a geometric-based approach is used for the scale–rotation–translation synthesis. The combined discrete FD comprises three shape signatures, i.e., complex coordinates (CCs), centroid distance (CD), and triangular centroid area (TCA), which can capture greater similarity of shape. The genetic algorithm–differential evolution (GA–DE) optimization method is used to solve the optimization problem. The proposed two-phase synthesis method, based on the combined discrete FD, successfully solves the challenging path generation problems with a relatively small number of function evaluations. A more accurate path shape can be obtained using the combined FD than the one-phase synthesis method. The obtained coupler curves approximate the desired paths quite well.

Synthesis of Rigid-Body Guidance Mechanism with Path and Pose Generation Independently

This paper proposes a practical method for the synthesis of rigid-body guidance mechanism by dividing the guidance task into path and pose generation independently. As a first step, based on the mechanism combination principle, a basic four-bar mechanism for path generation is synthesized and a binary-link is combined to the mechanism for achieving the execute link’s poses, then the potential kinematic chain with ability to realize the desired coupler poses is selected from all of the combination possibilities. And then a strategy by fixing a Cam to the basic four-bar mechanism to control the linkage poses is also proposed. Finally, the Cam’s profile is synthesized and the combined cam-link mechanism is able to guide the coupler through all desired linkage positions. One example is presented to validate the proposed guidance mechanism synthesis method.

Synthesis of spherical 4R mechanism for path generation using Differential Evolution

The problem of path generation for the spherical 4R mechanism is solved using the Differential Evolution (DE) algorithm. Formulas for the spherical geodesics are employed in order to obtain the parametric equation for the generated trajectory. Direct optimization of the objective function gives the solution to the path generation task without prescribed timing. Therefore, there is no need to separate this task into two stages to perform the optimization. Moreover, the order defect problem can be solved without difficulty by means of manipulations of the individuals in the DE algorithm. Two examples of optimum synthesis showing the simplicity and effectiveness of this approach are included.

How to Choose From a Synthesized Set of Path-Generating Mechanisms

Partially compliant mechanisms inherit the attributes of fully compliant and rigid-body linkages and offer simpler, compact design alternatives to accomplish complex kinematic tasks such as tracing large nonsmooth paths. This paper describes qualitative and quantitative criteria that can be employed to select the linkage configuration. The proposed criteria are categorized as general or specific. General criteria pertain to often-used kinematic attributes whereas specific criteria address the application at hand. The veracity and viability of each mechanism are evaluated with respect to compactness, design simplicity, static and dynamic failure, number of rigid-body joints, relative ease of fabrication, and other relevant criteria. Three decision-making techniques, namely, Pugh decision matrix, analytic hierarchy process, and a variant of the Pugh decision matrix are used to perform the evaluation. An example of a displacement-delimited gripper with a prescribed large nonsmooth path is used to illustrate linkage selection.

A methodology to learn designing optimal mechanisms for path generation

AbstractThe study of four‐bar linkages to trace a desired path is an important part of teaching in mechanical design. When the number of precision points exceeds a certain number, most recent approaches utilize intelligent optimization methods based on too complex computer science theories to be implemented by an engineering student. In this article we develop and implement new mechanism design results, reducing simultaneously the design space to facilitate finding the optimal mechanism. Finally, we apply global optimization methods that do not require analytical expression of the objective function and are freely available for educational use with Matlab. The proposed computerized methodology focuses student motivation on the mechanical aspects of the problem. Design examples presented illustrate the effectiveness of the approach that provides a solution quality comparable to that of the recently proposed intelligent optimization methods with simplicity of implementation and fast convergence. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18: 87–92, 2010; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20181

Forward and inverse analyses of smart compliant mechanisms for path generation

In this paper, an open loop compliant mechanism consisting of two elastic links actuated through piezoelectric actuators has been analyzed. The links may be joined through rigid or elastic hinge connection. The effect of piezoelectric actuators placed on an elastic beam is considered as two concentrated self-balancing moments acting on the beam at the edges of the actuator. Considering the large deflection of the two links of the mechanism under these self-balancing moments as well as end loads, the forward and inverse analyses of the mechanism are carried out. Two numerical schemes, namely, non-linear shooting (NLS) and Adomian decomposition methods (ADM) have been used for solving these problems. Numerical results are included to demonstrate the potential of the proposed methods.

Pareto optimal synthesis of four-bar mechanisms for path generation

Abstract In this paper, hybrid multi-objective genetic algorithms (GAs) are used for Pareto optimum synthesis of four-bar linkages considering the minimization of two objective functions simultaneously. The important conflicting objective functions that have been considered in this work are tracking error (TE) and transmission angle’s deviation from 90° (TA). The obtained Pareto fronts demonstrate that trade-offs between these two objectives can be recognized so that a designer can optimally compromise for the selection of a desired four-bar linkage. Moreover, the obtained optimum points are also compared with those given in the literature in terms of both tracking error (TE) and transmission angle error (TA).

Synthesis of adjustable spherical six-bar mechanisms for path generation

An optimal synthesis method of adjustable spherical StephensonⅢsix-bar mechanisms for multi-path generation is put forward. Based on kinematic synthesis of spherical Ste- phensonⅢsix-bar mechanisms, a set of nonlinear equations of multi-path kinematic synthesis for adjustable spherical six-bar mechanisms is established. The relationship of the number of tasks, the number of coupler positions, and the number of arbitrary choice of parameter are analyzed. In order to make the crank be continuous rotation one-way and the mechanisms have good transmission quality, the condition of crank existing of the spherical StephensonⅢsix-bar mecha- nisms is deduced, and the formula of transmission angle is de- rived. The mathematical model for optimization is presented, and a virus evolution genetic algorithm is utilized to obtain the global optimal solution. The virus evolution genetic algorithm has good convergence speed and convergence stability, which combines dynamically main group's overall evolution with virus group's local evolution. It can be ensureed to obtain the global optimal solution of the mechanism synthesis. A numeri- cal example is given to illustrate the effectiveness of the pro- posed method.

Optimal Synthesis of Mechanisms for Path Generation Using Refined Numerical Representation Based Model and AIS Based Searching Method

This paper presents an optimal synthesis method for path generation of planar mechanisms, in which a new path-description method named refined numerical representation is proposed to define the object function of the optimization model for path generation, and then the artificial immune network searching method is introduced to search candidate solutions. As a result, desired mechanisms can be generated independent of the scale, rotation, and translation transformation as well as sampling uniformity of initial sampling points. Experiment results demonstrate the effectiveness of the approach.