Parasitic Motion Research Articles

Analytical compliance analysis and finite element verification of spherical flexure hinges for spatial compliant mechanisms

This paper introduces and investigates a novel Spherical Flexure (SF), specifically conceived for application on spherical compliant mechanisms. The flexure features an arc of a circle as a centroidal axis and an annulus sector as cross-section, circle and annulus having a common center coinciding to that of the desired spherical motion. In this context, each element of the SF spatial compliance matrix is analytically computed as a function of both flexure dimensions and employed material. The theoretical model is then validated by relating analytical data with the results obtained through three-dimensional Finite Element Analysis. Finally, SFs are compared to Circularly Curved-Beam Flexures (CCBFs) in terms of parasitic motions.

Design, analysis and simulation of a novel 3-DOF translational micromanipulator based on the PRB model

A novel 3-DOF translational compliant mechanism based on flexure hinges is designed, analyzed and simulated in this paper. The mechanism is completely decoupled in XY directions, while there is a very small parasitic motion in Z direction when the XY axes are driven due to the lack of symmetry in Z direction. The pseudo-rigid-body model (PRBM) of the mechanism is established to analyze the relationship between the stiffness of the beam and the input displacement. Considering the flexible characteristics and displacement loss of mechanism, the displacement transformation efficiency from the input-end to the output-end is calculated firstly in the flexure-based micromanipulator field. To obtain the natural frequencies of the mechanism, the Hamilton's principle is successfully used to establish the dynamic model. The analytical results are validated by the finite element model, which indicate that the calculations and designs for the compliant mechanism are feasible. The maximal coupling error is only 1.82% when input displacement is maximal. The dynamic simulation using the backstepping dynamic surface control (BDSC) method is successfully applied in the compliant mechanism to obtain a good tracking performance, the tracking errors gradually decrease along with the input frequency reduction.

Kinematic analysis and testing of a 6-RRRPRR parallel manipulator

The Gough-Stewart platform has been successfully used in a wide variety of fields ranging from medical to automotive applications. This paper proposes a 6-RR RPRR parallel manipulator with orthogonal non-intersecting RR-joint configurations and ball screw actuators without guide mechanisms. A novel methodology is developed to define the dependent RR-joint variables and a numerical algorithm is employed to calculate the joint variables. The parasitic motion caused by the helical motion of the ball screw can be expressed and solved with vector method. The inverse kinematics of this manipulator can be solved. To verify the effectiveness of the proposed approach, simulations are performed with software package ADAMS. A prototype of this manipulator is manufactured. Its resolution, accuracy, and repeatability are measured. It is shown that the presented method is effective for this parallel manipulator.

A New XYZ Compliant Parallel Mechanism for Micro-/Nano-Manipulation: Design and Analysis

Based on the constraint and position identification (CPI) approach for synthesizing XYZ compliant parallel mechanisms (CPMs) and configuration modifications, this paper proposes a new fully-symmetrical XYZ CPM with desired motion characteristics such as reduced cross-axis coupling, minimized lost motion, and relatively small parasitic motion. The good motion characteristics arise from not only its symmetric configuration, but also the rigid linkages between non-adjacent rigid stages. Comprehensive kinematic analysis is carried out based on a series of finite element simulations over a motion range per axis less than ±5% of the beam length, which reveals that the maximum cross-axis coupling rate is less than 0.86%, the maximum lost motion rate is less than 1.20%, the parasitic rotations of the motion stage (MS) are in the order of 10−5 rad, and the parasitic translations of the three actuated stages (ASs) are in the order of 10−4 of the beam length (less than 0.3% of the motion range), where the beam slenderness ratio is larger than 20. Furthermore, the nonlinear analytical models of the primary translations of the XYZ CPM, including the primary translations of the MS and the ASs, are derived and validated to provide a quick design synthesis. Moreover, two practical design schemes of the proposed XYZ CPM are discussed with consideration of the manufacturability. The practical designs enable the XYZ CPM to be employed in many applications such as micro-/nano-positioning, micro-/nano-manufacturing and micro-/nano-assembly. Finally, a spatial high-precision translational system is presented based on the practical design schemes, taking the actuator and sensor integration into account.

Symmetrical 2R1T Parallel Mechanism without Parasitic Motion

Symmetrical 2R1T Parallel Mechanism without Parasitic Motion

Kinematic Analysis and Workspace of a New Four Legged 3-Dof Parallel Manipulator with No Parasitic Motions

Kinematic Analysis and Workspace of a New Four Legged 3-Dof Parallel Manipulator with No Parasitic Motions

Design and analysis of a single-flexure parallelogram mechanism-based X-Y nanopositioning stage

This paper presents the design and analysis of a flexure-based X-Y direction translational nanopositioning stage. Mechanism used in this present design has a circular flexure only on the drive end of the mechanism unlike the conventional compliance mechanism, providing advantage of both minimum parasitic motion and also higher bandwidth. Theoretical calculations are carried out to evaluate range and stiffness of the mechanism and further the designed flexure mechanism is analysed for dynamic and static behaviour of mechanism using FEA. Experiments conducted demonstrate that the design has a low cross-coupling of 0.2% of total motion, and a travelling range of 114 × 114 µm. However the mechanism is capable of moving upto 657 µm, which will be tested in our future work.

Nonlinear Analytical Modeling and Characteristic Analysis of a Class of Compound Multibeam Parallelogram Mechanisms

This paper deals with nonlinear analytical models of a class of compound multibeam parallelogram mechanisms (CMPMs) along with the static characteristic analysis. The CMPM is composed of multiple compound basic parallelogram mechanisms (CBPMs) in an embedded parallel arrangement. First, nonlinear analytical models for the CBPM are derived using the free-body diagram method through appropriate approximation strategies. The nonlinear analytical models of the CMPM are then derived based on the modeling results of the CBPM. Nonlinear finite element analysis (FEA) comparisons, experimental testing, and detailed stiffness analysis for the CBPM are finally carried out. It is shown that the analytical primary motion model agrees with both the FEA model and the testing result very well but the analytical parasitic motion model deviates from the FEA model over the large primary motion/force. It is also shown from the analytical characteristic analysis that the primary translational stiffness increases with the primary motion but the parasitic motion stiffness decreases with the primary motion, and the stiffness ratio of the parasitic motion stiffness to the primary translation stiffness also decreases with the primary motion. It is found that the larger the beam slenderness ratio is, the larger the stiffness or stiffness ratio is, and the more apparent the change of the stiffness or stiffness ratio is. The varied stiffness ratio indicates the mobility change of the CBPM.

Design and analysis of a new compliant XY micropositioning stage based on Roberts mechanism

The demand for a compliant micropositioning stage with a large workspace and high precision is increasing. This paper presents a new design of XY micropositioning stage based on Roberts mechanism, which is a traditional linear guiding mechanism. By extending the traditional rigid-body Roberts mechanism, two kinds of new compliant guiding mechanisms are designed and a novel parallel-kinematic XY compliant guiding stage is synthesized. Pseudo-rigid-body model (PRBM) is developed to build the quantitative models of the compliant Roberts mechanisms. In addition, finite-element analysis (FEA) is carried out to validate its performance. A prototype is fabricated and tested through experimental studies. Results show that the XY stage delivers a work range larger than 12mm in each axis. Moreover, the parasitic motion of the stage in the non-working direction is less than 1.7% of the motion stroke. It indicates that the designed stage meets the requirements of a large stroke and high precision linear guiding mechanism, which demonstrates the feasibility of the proposed design ideas using the Roberts mechanism.

Continuous motion of a novel 3-CRC symmetrical parallel mechanism

Parallel mechanisms (PMs) with three degrees of freedom (DOFs) have been studied extensively, especially the PMs with two rotational and one translational DOFs (2R1T PMs). One major problem of the 2R1T PMs is the inherent parasitic motion. In this paper, a novel 2R1T symmetrical parallel mechanism with no parasitic motion is proposed and studied. The moving platform and the base of this mechanism are mirror symmetric with respect to a mid-plane. This moving platform can realize continuous rotation about any axis or any point on the mid-plane and can have continuous translation along the normal line of the mid-plane. The constraint and motion characteristics of this mechanism are analyzed. The kinematics solutions and the Jacobian matrix are derived. The singularities of this PM are discussed. In the end, several numerical examples are given to show the continuous rotations and continuous translations of this PM. This kind of PMs has outstanding advantages of easy path planning and controlling.

On the correlation between the structure and one stepping characteristic of a piezo-driven rotary actuator

The parasitic motion principle was recently proposed to design piezo-driven actuators. Based on this principle, linear and rotary actuators with features of compact structure, large motion range, high velocity and high positioning resolution have been reported. However, some shortcomings such as the backward motion and the nonlinearity still exist and their causes are not clarified. To find potential reasons leading to these shortcomings, one stepping characteristic of a piezo-driven rotary actuator designed by the parasitic motion principle was investigated in this paper, and the correlation between the structure and one stepping characteristic was attempted to be established by theoretical analysis and simulations. Analysis results indicated that the nonlinearity in one stepping characteristic of the rotary actuator was correlated to the small gap between the inner and outer rings of the bearing as well as the self-deformation of the flexible gripper after the contact stiffness between the flexible gripper and the rotor reaches to the maximum value. The backward motion could be mainly resulted from the non-ideal driving wave. These analysis results enhance the understanding of the parasitic motion principle, which are meaningful for design and performance improvement of the parasitic motion principle linear or rotary actuators.

Characterization of Spatial Parasitic Motions of Compliant Mechanisms Induced by Manufacturing Errors

This paper proposes a theoretical model for characterizing manufacturing error induced spatial parasitic motions (MESPM) of compliant mechanisms (CM), and investigates the inherent statistic features of MESPM using Monte Carlo simulation. It also applies and extends a novel finite beam based matrix modeling (FBMM) method to theoretically derive the elastic deformation behavior of an imperfect flexural linkage (IFL), which is a basic element of a wide spectrum of compliant mechanisms. A case study of a well-known double parallelogram compliant mechanism (DPCM) is also conducted, and the practical parasitic motions of a prototype DPCM are characterized by laser interferometer based measurements.

Positioning control effectiveness of optical knife edge displacement sensor-embedded monolithic precision stage

This paper presents positioning control effectiveness of optical knife edge sensors (OKES) for compact nanopositioning stage applications. The OKES performance was characterized in terms of precision, dynamic characteristics, working range and control effectiveness. The OKES consists of the laser diode, collimation lens, beam splitter, mirror, double-side sharp knife edge, and two lensed photodiodes. The double compound notch type flexure stage was fabricated by additive manufacturing process, stereolithography, and the OKES was successfully embedded into the flexure stage to control the positioning in a compact and efficient configuration. The voice coil motor was aligned along the moving axis of the stage without offset to mitigate Abbe error, and the OKES was setup at the center perpendicular to the moving axis. The positioning of flexure stage was proportional-integral-derivative feedback controlled. As a result, the AM flexure mechanism was successfully position controlled to a margin of 50nm within a 500μm range. It was also found that the OKES is sensitive to torsional motion and it can be used for angular motion measurement such as parasitic motions with high sensitivity. Therefore, the OKES and voice coil actuator can be a good alternative to the capacitive sensors and piezoelectric transducers in terms of performance: measuring and actuation range, positioning accuracy, resolution, bandwidth and control effectiveness.

Dynamic analysis and experiment of a novel ultra-precision compliant linear-motion mechanism

The parasitic motion of double parallel four-bar mechanism (PFBM) is diminished at the cost of introducing a non-controllable motion. A family of ultra-precision compliant linear-motion mechanism (CLMM) is therefore proposed. In this paper, the interaction between inertia and stiffness, which is determined by the structure dimensions, is considered. Then the dynamic model is developed using Lagrange equation. Furthermore, the influence of material properties, shape parameters, and geometric parameters on natural frequency is evaluated to provide quantitative insight for designers. The analysis shows that the proposed CLMM has prominent advantage in its frequency characteristics. Finally, the analytic model is verified by finite element analysis (FEA) and experiment. The analytical results are within 6.1% error compared with FEA and within 10.8% error compared with the experimental results.

2D shape optimization of leaf-type crossed flexure pivot springs for minimum stress

Leaf-spring crossed flexure pivots are used in an increasingly large number of applications, many of which require the fatigue life and/or the rotational capability of the pivots to be maximized. Since these two qualities are determined by the endurance limit or the yield stress of the leaf-springs’ material, maximizing them will necessarily require reducing the stress levels arising in the aforementioned springs during operation. Partly for the sake of simplicity, partly due to manufacturing constraints, constant thickness leaf-springs have traditionally been used in crossed flexure pivots. However, minimizing stress simply by increasing the length or by decreasing the thickness of the leaf-springs is often not viable, either due to spatial constraints or to the attendant degradation in spring performance. This paper investigates the scope for stress reduction through shape optimization of the leaf-springs. To this end, a procedure combining a linear strain energy formulation, a parametric thickness profile definition and a series of optimization algorithms is employed. The resulting optimized thickness profiles are proven to be not only independent of the angular rotation at which the pivot operates, but also linearly scalable to leaf-springs of any length, minimum thickness and width. Validated using non-linear finite element analysis, the results show very significant reductions in relative maximum stress, of up to as much as 24% in the case of some pivot configurations. The optimized profiles and their corresponding constant thickness counterparts are also compared in terms of stiffness, strain energy and parasitic motion characteristics. It is concluded that shape optimization offers great potential for extending the fatigue life and/or rotational range of crossed flexure pivots.

Adaptive pseudo-rigid-body model for generalized cross-spring pivots under combined loads

Generalized cross-spring pivots (CSPs) are widely used as revolute joints in precision machinery. However, pseudo-rigid-body (PRB) models cannot capture the parasitic motions of a generalized CSP exactly under combined loads; moreover, the characteristic parameters used in PRB methods must be recomputed using optimization techniques. In this study, we develop two simple and accurate PRB models for generalized CSPs. First, a PRB method for a beam is developed based on the beam constraint model and the instantaneous center model, where the beam is modeled as two rigid links joined at a pivot via a torsion spring. Subsequently, two PRB models of the generalized CSP, comprising a four-bar model for accuracy and a pin-joint model for stiffness, are constructed based on a kinematic analysis using the proposed PRB method. A deflection characteristic analysis is then conducted to determine the relationship between the proposed model and the existing models. Finally, the PRB models for the pivot under the action of combined loads are validated via finite element analysis. The error evaluation indicates that the proposed PRB models are more accurate than the results from existing methods. The PRB models proposed here can be used in parametric design of compliant mechanisms.

Design principle of high-precision flexure mechanisms based on parasitic-motion compensation

In design of flexure mechanism, diminishing the parasitic-motion is a key point to improve the accuracy. However, most of existing topics concentrate on improving the accuracy of linear-motion flexure mechanisms via compensating the parasitic error, but few research the multi-dimensional flexure mechanisms. A general design principle and method for high-precision flexure mechanisms based on the parasitic-motion compensation is presented, and the proposed method can compensate the parasitic rotation in company with translation, or the parasitic translation in company with rotation, or both. The crucial step for the method is that the parasitic motion of a flexure mechanism is formulated and evaluated in terms of its compliance. The overall compliance matrix of a general flexure mechanism is formulated by using screw theory firstly, then the criteria for the parasitic motions is introduced by analyzing the characteristics of the resultant compliance matrix as well as with aid of the concept of instantaneous rotation center. Subsequently, a compliance-based compensation approach for reducing parasitic-motion is addressed as the most important part. The design principles and procedure are further discussed to help with improving the accuracy of flexure mechanisms, and case studies are provided to illustrate this method. Finally, an analytical verification is provided to demonstrate that the symmetry design philosophy widely used in flexure design can effectively improve accuracy in terms of the proposed method. The proposed compensation method can be well used to diminish the parasitic-motion of multi-dimensional flexure mechanisms.

Generalized Two-Point Method for Straightness Profile Measurement - Error Propagation and Experimental Results

In the straightness profile measurement of a mechanical workpiece, hardware datums have been the traditional standard. However, when the straightness profile is measured using a scanning displacement sensor set on an X-stage as the hardware datums, output of a displacement sensor includes the signal of straightness profile and the sensor’s parasitic motion, i.e. straightness error motion. Then, error separation techniques of the straightness profile from parasitic motions have been developed. For example, two-point method uses two displacement sensors and separates the sensor’s straightness error motion from the straightness profile. However, the conventional two-point method cannot measure a large-scale workpiece because the large sampling number causes random error amplification. In this article, the influence of the random error of generalized two-point method is shown. As the result of the theoretical analysis and numerical analysis, random error propagation decrease when sampling number increase. Further, experimental results obtained by generalized two-point method with large sampling number are analyzed using Wavelet transform and influence of error of the generalized two-point method is discussed in the space-spatial frequency domain.

Design, modeling, control and experiment for a 2-DOF compliant micro-motion stage

A new XY micro-motion stage is proposed with the double four-bar prismatic joints to transfer linear motions and mechanical displacement amplifier. The compliance models of the amplifier, the prismatic joints, and the whole stage are established based on the flexibility matrix method. The simulation is made by using finite element analysis ANSYS software. The output cross-talk is about 2% and the parasitic motions of the input points in the other limbs are less than 1.8%, which shows a good decoupling property. The mechanical prototype is fabricated, the experimental results show that the input/output of the stage has a very good linearity, the ratio of output displacement to input displacement is 5.06 and the working range is 156 μm × 156 μm. To tackle for the serious hysteresis nonlinear problems of the stage, model reference adaptive PID controller is designed, the final resolution can reach ±0.2 μm.

Workspace-constrained optimal design of three-degrees-of-freedom parallel manipulators with minimum parasitic motions by integrating interval analysis, region mapping and differential evolution

Three-degrees-of-freedom (3-DOF) parallel manipulators have many advantages such as simple structure, fewer actuators, and lower maintenance cost. However, parasitic motions may degrade the positioning accuracy of the platforms. In order to design 3-DOF parallel manipulators which can fulfil specified workspace requirements and exhibit minimum parasitic motions, the design problem is formulated into a minimax problem with workspace constraints. Then, an interval-based method is exploited to determine the feasible solution set which is derived as a union of many scattered parameter intervals (boxes). Then, a new approach based on region mapping and a powerful optimizer (namely differential evolution) is proposed to solve the optimization problem over scattered search regions. Benchmark tests show the superiority of the proposed approach. Then, the approach and interval analysis are used to solve a real-world design problem involving a 3-DOF manipulator. Numerical results demonstrate the effectiveness and advantages of the proposed design method.