KINEMATICS FEATURE ANALYSIS OF A 3-DOF COMPLIANT MECHANISM FOR MICRO MANIPULATION

Kinematics and its related issues of a 3-DOF in-parallel compliant mechanismare focused on. The micro manipulation application that requires high accuracy is developed. Designof the developed micromanipulator is based on the modified Delta mechanism. The main advantages ofthis manipulator are the use of only revolute flexure hinges and the capability to produce puretranslation theoretically. The aim is to develop an efficient kinematic model used for positioningcontrol. For this purpose, the Jacobian matrix relating the end effector position with the actuatordisplacements is obtained by both theoretical derivation and experiment. Aiming at the abnormalityin the motion capabilities of the micromanipulator found in calibration experiment, the mobility ofthe compliant mechanism on a theoretical level is analyzed by using the matrix method and screwtheory. Both the experimental and theoretical results have verified that the compliant mechanismdoes have rotational motion.

Pseudo-rigid-body Model for Corrugated Cantilever Beam Used in Compliant Mechanisms

Common compliant joints generally have limited range of motion, reduced fatigue life and high stress concentration. To overcome these shortcomings, periodically corrugated cantilever beam is applied to design compliant joints. Basic corrugated beam unit is modeled by using pseudo-rigid-body method. The trajectory and deformation behavior of periodically corrugated cantilever beam are estimated by the transformation of coordinate and superposition of the deformation of corrugated beam units. Finite element analysis（FEA） is carried out on corrugated cantilever beam to estimate the accuracy of the pseudo-rigid-body model. Results show that the kinetostatic behaviors obtained by this method, which has a relative error less than 6%, has good applicability and corrugated cantilever beam has the characteristics of a large range of motion and high mechanical strength. The corrugated cantilever beam is then applied to design a flexible rotational joint to obtain a larger angle output. The paper proposes a pseudo-rigid-body model for corrugated cantilever beam and designed a flexible rotational joint with large angle output.

Topology Optimization of Compliant Mechanisms with Geometrical Nonlinearities Using the Ground Structure Approach

The majority of topology optimization of compliant mechanisms uses linear finite element models to find the structure responses.Because the displacements of compliant mechanisms are intrinsically large,the topological design can not provide quantitatively accurate result.Thus,topological design of these mechanisms considering geometrical nonlinearities is essential.A new methodology for geometrical nonlinear topology optimization of compliant mechanisms under displacement loading is presented.Frame elements are chosen to represent the design domain because they are capable of capturing the bending modes.Geometrically nonlinear structural response is obtained by using the co-rotational total Lagrange finite element formulation,and the equilibrium is solved by using the incremental scheme combined with Newton-Raphson iteration.The multi-objective function is developed by the minimum strain energy and maximum geometric advantage to design the mechanism which meets both stiffness and flexibility requirements, respectively.The adjoint method and the direct differentiation method are applied to obtain the sensitivities of the objective functions. The method of moving asymptotes（MMA） is employed as optimizer.The numerical example is simulated to show that the optimal mechanism based on geometrically nonlinear formulation not only has more flexibility and stiffness than that based on linear formulation,but also has better stress distribution than the one.It is necessary to design compliant mechanisms using geometrically nonlinear topology optimization.Compared with linear formulation,the formulation for geometrically nonlinear topology optimization of compliant mechanisms can give the compliant mechanism that has better mechanical performance.A new method is provided for topological design of large displacement compliant mechanisms.

Multi-objective Topology Optimization of Thermo-mechanical Compliant Mechanisms

The material characteristics of a structure will change with temperature variation,and will induce stress within the structure.Currently,the optimal design for the topology of compliant mechanisms is mainly performed in single physical field.However,when compliant mechanisms work in high temperature environments,their displacement outputs are generated not only by mechanical load,but also by the temperature variation which may become the prominent factor.Therefore,the influence of temperature must be considered in the design.In this paper,a novel optimization method for multi-objective topology of thermo-mechanical compliant mechanisms is presented.First,the thermal field is analyzed with finite-element method,where the thermal strain is taken into account in the constitutive relation,and the equivalent nodal thermal load is derived with the principle of virtual work.Then the thermal load is converted into physical loads in elastic field,and the control equation of the thermo-mechanical compliant mechanism is obtained.Second,the mathematical model of the multi-objective topology optimization is built by incorporating both the flexibility and stiffness.Meanwhile,the coupling sensitivity function and the sensitivity analysis equations of thermal steady-state response are derived.Finally,optimality criteria algorithm is employed to obtain numerical solution of the multi-objective topology optimization.Numerical examples show that the compliant mechanisms have better performance and are more applicable if the temperature effect is taken into account in the design process.The presented modeling and analysis methods provide a new idea and an effective approach to topology optimization of compliant mechanisms in electrothermic coupling field and multiphysics fields.

CONTINUUM TOPOLOGY OPTIMIZATION FOR MONOLITHIC COMPLIANT MECHANISMS OF MICRO-ACTUATORS

A multi-objective scheme for structural topology optimization of distributed compliant mechanisms of micro-actuators in MEMS condition is presented in this work, in which mechanical flexibility and structural stiffness are both considered as objective functions. The compliant micro-mechanism developed in this way can not only provide sufficient output work but also have sufficient rigidity to resist reaction forces and maintain its shape when holding the work-piece. A density filtering approach is also proposed to eliminate numerical instabilities such as checkerboards, mesh-dependency and one-node connected hinges occurring in resulting mechanisms. SIMP is used as the interpolation scheme to indicate the dependence of material modulus on element-regularized densities. The sequential convex programming method, such as the method of moving asymptotes （MMA）, is used to solve the optimization problem. The validation of the presented methodologies is demonstrated by a typical numerical example.

Analytical Compliance Modeling of Serial Flexure-Based Compliant Mechanism Under Arbitrary Applied Load

Analytical compliance model is vital to the flexure- based compliant mechanism in its mechanical design and motion control. The matrix is a common and effective approach in the compliance modeling while it is not well developed for the closed-loop serial and parallel compliant mechanisms and is not applicable to the situation when the external loads are applied on the flexure mem- bers. Concise and explicit analytical compliance models of the serial flexure-based compliant mechanisms under arbitrary loads are derived by using the matrix method. An equivalent method is proposed to deal with the situation when the external loads are applied on the flexure mem- bers. The external loads are transformed to concentrated forces applied on the rigid links, which satisfy the equa- tions of static equilibrium and also guarantee that the deformations at the displacement output point remain unchanged. Then the matrix method can be still adopted for the compliance analysis of the compliant mechanism. Finally, several specific examples and an experimental testare given to verify the effectiveness of the compliance models and the force equivalent method. The research enriches the matrix method and provides concise analytical compliance models for the serial compliant mechanism.

Stiffness Analysis of Corrugated Flexure Beam Used in Compliant Mechanisms

Conventional flexible joints generally have limited range of motion and high stress concentration. To overcome these shortcomings, corrugated flexure beam（CF beam） is designed because of its large flexibility obtained from longer overall length on the same span. The successful design of compliant mechanisms using CF beam requires manipulation of the stiffnesses as the design variables. Empirical equations of the CF beam stiffness components, except of the torsional stiffness, are obtained by curve-fitting method. The application ranges of all the parameters in each empirical equation are also discussed. The ratio of off-axis to axial stiffness is considered as a key characteristic of an effective compliant joint. And parameter study shows that the radius of semi-circular segment and the length of straight segment contribute most to the ratio. At last, CF beam is used to design translational and rotational flexible joints, which also verifies the validity of the empirical equations. CF beam with large flexibility is presented, and empirical equations of its stiffness are proposed to facilitate the design of flexible joint with large range of motion.

Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the crosssection(i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio Independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design Insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.

TOPOLOGY SYNTHESIS OF GEOMETRICALLY NONLINEAR COMPLIANT MECHANISMS USING MESHLESS METHODS

This paper presents a new method for topology optimization of geometrical nonlinear compliant mechanisms using the element-free Galerkin method （EFGM）. The EFGM is employed as an alternative scheme to numerically solve the state equations by fully taking advantage of its capability in dealing with large displacement problems. In the meshless method, the imposition of essential boundary conditions is also addressed. The popularly studied solid isotropic material with the penalization （SIMP） scheme is used to represent the nonlinear dependence between material properties and regularized discrete densities. The output displacement is regarded as the objective function and the adjoint method is applied to finding the sensitivity of the design functions. As a result, the optimization of compliant mechanisms is mathematically established as a nonlinear programming problem, to which the method of moving asymptotes （MMA） belonging to the sequential convex programming can be applied. The availability of the present method is finally demonstrated with several widely investigated numerical examples.

Compliant Mechanism Synthesis by Using Elastic Similitude

Compliant mechanisms have several advantages, especially smaller number of elements and therefore less movable joints. The flexural members furthermore allow an integration of special functions like balancing or locking. Especially fiber reinforced materials exhibit a wide range of function integration considering their compliance in passive as well active applications. To take advantage of compliant elements in applications a robust synthesis tool is needed. The synthesis based on topology optimization method or the pseudo rigid body approach leads to complex structures. Considering the use of fiber reinforced material a synthesis approach which leads to less complex structures is more suitable. For building up simple structures, with only one cantilever beam as compliant element（B） a graphical approach using the elastic similitude is the most efficient method. A step-by-step synthesis procedure is presented to synthesize compliant mechanisms with rotatory joints（R） and prismatic joints（P） to develop RRB/PRB- and RPB-linkages. Using the elastic similitude to implement these results into a graphical synthesis algorithm is the innovation part of this paper. It can be shown that this approach leads to a comfortable handling of beam elements during the synthesis, where the two free parameters can be directly coupled to scale and fix the orientation of the beam element. This advantage inherently shortens the development process. In giving an example the focus lies of the experimental approach, which also shows that the simple BERNOULLI beam model is valid and so the synthesis by using the elastic similitude. The method is presented and discussed by using an application for a cup holder mechanism made of fiber reinforced material.

TOPOLOGY OPTIMIZATION OF MULTIPLE INPUTS AND MULTIPLE OUTPUTS COMPLIANT MECHANISMS

An optimal topology design method for multiple inputs and multiple outputs compliant micro-manipulation system is presented. Firstly, the topology design problem is posed in terms of a multiple inputs load and several specified output deflections. The compliance and stiffness of the system are expressed by the mutual potential energy and strain energy, respectively, which can be controlled by a multi-critaria objective function. Secondly, based on the optimality criteria method, a model solution algorithm is presented. Finally, a numerical example is presented to show the validity oftbe presented technique. The optimal topology ofa 4 inputs and 4 outputs compliant mechanism is obtained by using the method, and the corresponding micro-positioning stage system is further designed.

Analysis of Frequency Characteristics and Sensitivity of Compliant Mechanisms

Based on a modified pseudo-rigid-body model,the frequency characteristics and sensitivity of the large-deformation compliant mechanism are studied.Firstly,the pseudo-rigid-body model under the static and kinetic conditions is modified to enable the modified pseudo-rigid-body model to be more suitable for the dynamic analysis of the compliant mechanism.Subsequently,based on the modified pseudo-rigid-body model,the dynamic equations of the ordinary compliant four-bar mechanism are established using the analytical mechanics.Finally,in combination with the finite element analysis software ANSYS,the frequency characteristics and sensitivity of the compliant mechanism are analyzed by taking the compliant parallel-guiding mechanism and the compliant bistable mechanism as examples.From the simulation results,the dynamic characteristics of compliant mechanism are relatively sensitive to the structure size,section parameter,and characteristic parameter of material on mechanisms.The results could provide great theoretical significance and application values for the structural optimization of compliant mechanisms,the improvement of their dynamic properties and the expansion of their application range.

Spring-Joint Method for Topology Optimization of Planar Passive Compliant Mechanisms

There is seldom approach developed for the initial topology design of flexure-based compliant mechanisms. The most commonly-used approaches, which start with an existing rigid-body mechanism, do not consider the performances between different topologies. Moreover, they rely heavily on the rigid-body topology, therefore limit the diversity of compliant mechanisms topology. To obtain the optimal initial topology of such mechanisms directly from problem specifications without referencing to the existing mechanism topologies, a spring-joint method is presented for a restricted class of the serial passive flexure-based compliant mechanisms, which are the building blocks of parallel compliant mechanisms. The topology of the compliant mechanisms is represented by a serial spring-joint mechanism（SSJM） that is a traditional rigid-body mechanism with a torsional spring acting on each joint, and is described by position vectors of the spring-joints. A simplified compliance matrix, determined by the position vectors, is used to characterize the tip of the SSJM kinematically, and is optimized to ensure the desired freedoms of the compliant mechanisms during optimization. The topology optimization problem is formulated as finding out the optimal position of the spring-joints in a blank design domain with an objective function derived from the simplified compliance matrix. In design examples, syntheses of the compliant mechanisms with both single freedom and two decoupled freedoms are presented to illustrate the proposed method. The proposed method provides a new way for the initial design of flexure-based compliant mechanisms.

Design and Analysis of a Novel Compliant Mechanism with RPR Degrees of Freedom

A novel compliant mechanism with RPR degrees of freedom(DOF)is proposed where R and P represent rotation and translation DOFs,respectively.The proposed compliant mechanism is obtained from dimension synthesizing a 2-RPU-UPR rigid parallel mechanism with the method of optimization of motion/force transfer characteristic.R,P and U represent rotation,translation and universal pairs,respectively.Firstly,inverse kinematics and Jacobian matrix are analyzed for the dimensional synthesis.Then,output transmission indexes of branches in the parallel mechanism are given.Dimensional synthesis is completed based on the normalized design parameter.And optimization of flexure joints based on constrained energy is carried out.Afterwards,the novel compliant mechanism is obtained by direct replacing method.Mechanical model of the compliant mechanism including static stiffness and input stiffness is built based on the pseudo-rigid body modeling method and virtual work principle.Finally,FEA simulation by Ansys Workbench is carried out to verify DOF,effectiveness of the dimension synthesis,and compliant model.Optimization of motion/force transfer characteristic is first applied for the design of compliant mechanisms to suppress drift of rotation axis in the paper.

Simulation optimal design of a compliant mechanism with circular notch flexure hinges based on the equivalent beam methodology

This paper presents an in-depth study of Equivalent beam model (EBM).Firstly three-dimensional (3D) finite element analysis (FEA) model for circular flexure hinge developed by Zettl et al.was verified by the comparison with Smith's experimental results and the 3D FEA model was feasible within 5.5% error.Then the accuracy of Timoshenko short-beam due to shear force was verified based on finite element method.The results showed that the EBM has good accuracy within 5% error for 1≤r/t≤3.Finally the EBM methodology was applied for the simulation optimal design of a bridge-type compliant mechanism.The results showed that the EBM methodology has very high numerical efficiency and satisfactory accuracy for simulation optimal design of planar compliant mechanism with flexure hinges.

Enabling the transfer matrix method to model serial-parallel compliant mechanisms including curved flexure beams

Compliant mechanisms with curved flexure hinges/beams have potential advantages of small spaces,low stress levels,and flexible design parameters,which have attracted considerable attention in precision engineering,metamaterials,robotics,and so forth.However,serial-parallel configurations with curved flexure hinges/beams often lead to a complicated parametric design.Here,the transfer matrix method is enabled for analysis of both the kinetostatics and dynamics of general serial-parallel compliant mechanisms without deriving laborious formulas or combining other modeling methods.Consequently,serial-parallel compliant mechanisms with curved flexure hinges/beams can be modeled in a straightforward manner based on a single transfer matrix of Timoshenko straight beams using a step-by-step procedure.Theoretical and numerical validations on two customized XY nanopositioners comprised of straight and corrugated flexure units confirm the concise modeling process and high prediction accuracy of the presented approach.In conclusion,the present study provides an enhanced transfer matrix modeling approach to streamline the kinetostatic and dynamic analyses of general serial-parallel compliant mechanisms and beam structures,including curved flexure hinges and irregular-shaped rigid bodies.

Dynamic Modeling and Experimental Verification of an RPR Type Compliant Paralle Mechanism with Low Orders

Efficiency of calculating a dynamic response is an important point of the compliant mechanism for posture adjustment.Dynamic modeling with low orders of a 2R1T compliant parallel mechanism is studied in the paper.The mechanism with two out-of-plane rotational and one lifting degrees of freedom(DoFs)plays an important role in posture adjustment.Based on elastic beam theory,the stiffness matrix and mass matrix of the beam element are established where the moment of inertia is considered.To improve solving efficiency,a dynamic model with low orders of the mechanism is established based on a modified modal synthesis method.Firstly,each branch of the RPR type mechanism is divided into a substructure.Subsequently,a set of hypothetical modes of each substructure is obtained based on the C-B method.Finally,dynamic equation of the whole mechanism is established by the substructure assembly.A dynamic experiment is conducted to verify the dynamic characteristics of the compliant mechanism.

Compliant mechanisms design based on pairs of curves

The success of compliant mechanism design by structural topology optimization approach depends,to a large extent,on its structural geometry representation scheme.In this work,a novel representation scheme based on pairs of curves is presented.In the representation,the structure is characterized by a set of input/output(I/O) regions.While it is still unknown how the rest of the design space will be occupied by the structure,the I/O regions must exist somewhere because any structure must have parts which interact with its surroundings by way of at least one loading region,one support region,and one output region.For a valid structural design,pairs of Bezier curves are used to connect I/O regions in order to form one single connected load-bearing structure.The boundary is explicitly described,so the need for smoothening of the blurred and jagged edges can be avoided by developing such a representation scheme to directly generate smooth boundary structures.With the scheme,shape and topology can be optimized simultaneously,and the obtained topology solutions have no check-board phenomena nor intermediate zones.A multi-objective genetic algorithm is then applied to couple with the representation scheme for defining and encoding the structural geometry in the form of graph.The solution framework is integrated with a nonlinear fixed grid finite element method(FG-FEM) code for large-displacement analyses of the compliant structures.Simulation results from a displacement inverter indicated that the proposed representation scheme is appropriate.

The Honeybee＇s Protrusible Glossa is a Compliant Mechanism

Many biological structures can perform highly-dexterous actions by using dynamic surfaces. To deal with the contradictive demands of high feeding efficiency and low energy expenditure during nectar feeding, the glossal surface of a honeybee un- dergoes shape changes, in which glossal hairs erect together with segment elongation in a drinking cycle. In this paper, we extracted a transmission link embedded in the glossa from postmortem examination and found that the compliance of the in- tersegmental membranes provides more possibilities for this highly kinematic synchronicity. According to the morphing phe- nomena of honeybee＇s glossa, we proposed a compliant mechanism model to predict the deformation behavior of honeybee considering elastic properties of the glossal intersegmental membranes. The increase of membrane stiffness may improve the capacity of elastic potential energy transfer, but will still result in the increase of mass. An index is introduced to evaluate the contradiction for optimizing structural parameters. This work may arouse new prospects for conceptual design of mi- cro-mechanical systems equipped with bio-inspired compliant mechanisms.

A brief review on nonlinear modeling methods and applications of compliant mechanisms

Compliant mechanisms （CMs） have become one of the most popular research themes in mechanisms and robotics because of their merits. This paper aims to provide a brief systematic review on the advances of nonlinear static modeling approaches and the applications of CMs to promote interdisciplinaly/multidisciplinary development for associated theories and other new applications. It also predicts likely fh^tre directions of applications and theory development.