Exact Constraint Design Research Articles

Improved dynamic performance in flexure mechanisms by overconstraining using viscoelastic material

Flexure mechanisms are commonly designed to be exactly constrained to favor determinism, though at the expense of limitations on the maximum parasitic natural frequencies and support stiffness. This paper presents the use of viscoelastic material for providing additional support stiffness in a certain frequency range without the indeterminism commonly associated with overconstraining. This design principle of dynamically stiffened exact-constraint design is exemplified by a parallelogram flexure mechanism. Experiments demonstrate that a custom synthesized elastomer compound can compensate for unintended misalignments without significant internal stress buildup, while improving the dynamic performance in terms of a higher first parasitic natural frequency. An analytical investigation clarifies the relationship between misalignment, internal load, stiffness and natural frequency. Using the buckling modes of the system, the nonlinear geometric stiffness is modeled accurately up to the bifurcation. The measurements and analytical model are corroborated by a nonlinear flexible multibody analysis.

How to consider Over-constrained Assemblies with Gaps in Tolerance-Cost Optimization?

Abstract Exact Constraint Design is well established in design theory to ensure a robust and predictable system behavior. In practice, however, it is often inevitable to deviate from this basic concept so that parts are constrained several times for rigidity or resilience. In doing so, gaps between mating parts have to be added to ensure assemblability and functionality. Despite the essential impact of gaps on the total product functionality, occurring uncertainties in part positioning are mostly neglected in tolerance analysis. In the context of tolerance-cost optimization, they have not been studied in detail so far. To overcome this drawback, this article addresses the research question how over-constrained systems with gaps can be considered in tolerance-cost optimization ensuring the identification of both reliable and cost-optimal tolerance values. Based on an initial discussion on the complexity of tolerance analysis of over-constrained systems with gaps, their impact on the results and the efficiency of tolerance-cost optimization are discussed. With the aim to ensure a sufficient model accuracy in time-consuming applications, a framework for tolerance-cost optimization considering assemblies with multiple gap configurations based on surrogate models is presented and applied to a case study of industrial complexity. In doing so, a novel framework for a proper and efficient modelling of systems with gaps in tolerance-cost optimization supporting both researchers and practitioners is presented.

Misalignments in an overconstrained flexure mechanism: A cross-hinge stiffness investigation

The exact-constraint design principle is commonly applied to flexure mechanisms to ensure deterministic behavior, but at the cost of reduced robustness, support stiffness, load capacity and usually an increased complexity of design. To explore the potential benefit of overconstrained design in flexure mechanisms, this paper investigates an elementary two-flexure cross-hinge with a single overconstraint as a case study. The stiffness effects of inadvertent stress due to misalignment are investigated experimentally, numerically and analytically.A measurement set-up with controllable misalignment has been designed. Measurements show that the first natural frequency of the cross-hinge decreases strongly with misalignment, suggesting that the actuation stiffness decreases due to the misalignment stress, and ultimately vanishes due to bifurcation buckling at a critical misalignment of the order of 0.1 mm for the mechanism at hand.Simulations with a detailed numerical model support the measurements and expose some additional factors, such as warping and shuttle compliance, which influence the system behavior. Importantly, they also show that the compliance in the support directions of the mechanism increases strongly at the critical misalignment, demonstrating that the mechanism no longer functions at the critical misalignment.An extensive analytical buckling analysis shows how the stress due to misalignment poses a functional operation limit on the overconstrained mechanism in terms of bifurcation buckling. The analysis serves to corroborate the numerical predictions. An expression is derived for the critical misalignment force and displacement as a function of the geometry and material of general cross-hinge mechanism designs.

Shifted axis angular positioning mechanisms – unconventional use of exact constraint design

Paper presents some remarks about designing angular positioners having significant angular displacement and shifted axis of rotation. Those driving mechanism are useful in specific applications where mechanism wraps around other object or there is a need of rotation around virtual axis. Examples of those applications like orthotic robots or gimbals for panoramic motions where developed at Faculty of Mechatronics WUT. Usage of exact constraint method was described including its influence to a kinematic structure of designed mechanism.

Exact Constraint Design and its Potential for Robust Embodiment

The design of exact, also referred to as minimal, constraints means applying just enough constraints between the various components of a mechanical assembly, in order to unambiguously define their positions in six degrees of freedom (3 translations, 3 rotations), their desired motions respectively. To ensure a predictable and reliable product performance, a systematic design of the corresponding elementary mechanical interfaces between components is of utmost importance. Over constraints, i. e. part-to-part connections with redundant interfaces which constrain one single degree of freedom, are largely susceptible to variation and therefore result in design solutions which frequently experience production/ assembly issues, reduced performance, excessive and non-predictable wear-rates, etc.Being a basic rule of embodiment design, literature provides various well-know and widely applied approaches for Exact Constraint Design. Examples are the calculation of a mechanisms’ mobility using the Grübler-Kutzbach criterion, the analysis of statically determinate assemblies by means of the screw theory or so called Schlussartenmatrizen, as well as the analysis of engaging surfaces in terms of location schemes or interface ambiguity. However, despite the various existing approaches, workshops with practitioners and academics have shown that the systematic design of optimal constraints appears to be cumbersome for many engineers. Based on an overview of the most relevant approaches for Exact Constraint design, this contribution therefore reviews the challenges experienced by the workshop participants, discusses the necessity of kinematically correct constraints for robustness, and derives an initial prescriptive procedure for a coherent design of constraints throughout the embodiment design phase, which, despite a variety of available approaches, seems to be still missing.

Modal Measurements and Model Corrections of A Large Stroke Compliant Mechanism

Abstract In modelling flexure based mechanisms, generally flexures are modelled perfectly aligned and nominal values are assumed for the dimensions. To test the validity of these assumptions for a two Degrees Of Freedom (DOF) large stroke compliant mechanism, eigenfrequency and mode shape measurements are compared to results obtained with a flexible multibody model. The mechanism consists of eleven cross flexures and seven interconnecting bodies. From the measurements 30% lower eigenfrequencies are observed than those obtained with the model. With a simplified model, it is demonstrated that these differences can be attributed to wrongly assumed leaf spring thickness and misalignment of the leaf springs in the cross flexures. These manufacturing tolerances thus significantly affect the behaviour of the two DOF mechanism, even though it was designed using the exact constraint design principle. This design principle avoids overconstraints to limit internal stresses due to manufacturing tolerances, yet this paper shows clearly that manufacturing imperfections can still result in significantly different dynamic behaviour.

Exact Constraint Design of a Two-Degree of Freedom Flexure-Based Mechanism1

We present the exact constraint design of a two degrees of freedom cross-flexure-based stage that combines a large workspace to footprint ratio with high vibration mode frequencies. To maximize unwanted vibration mode frequencies the mechanism is an assembly of optimized parts. To ensure a deterministic behavior the assembled mechanism is made exactly constrained. We analyze the kinematics of the mechanism using three methods; Grüblers criterion, opening the kinematic loops, and with a multibody singular value decomposition method. Nine release-flexures are implemented to obtain an exact constraint design. Measurements of the actuation force and natural frequency show no bifurcation, and load stiffening is minimized, even though there are various errors causing nonlinearity. Misalignment of the exact constraint designs does not lead to large stress, it does however decrease the support stiffness significantly. We conclude that designing an assembled mechanism in an exactly constrained manner leads to predictable stiffnesses and modal frequencies.

Flexible multibody modelling for exact constraint design of compliant mechanisms

In high precision equipment, the use of compliant mechanisms is favourable as elastic joints offer the advantages of low friction and no backlash. If the constraints in a compliant mechanism are not carefully dealt with, even small misalignments can lead to changes in natural frequencies and stiffnesses. Such unwanted behaviour can be avoided by applying exact constraint design, which implies that the mechanism should have exactly the required degrees of freedom and non-redundant constraints so that the system is kinematically and statically determinate. For this purpose, we propose a kinematic analysis using a finite element based multibody modelling approach. In compliant mechanisms, the system’s degrees of freedom are presented clearly from the analysis of a system in which the deformation modes with a low stiffness are free to deform while the deformation modes with a high stiffness are considered rigid. If the Jacobian matrix associated with the dependent coordinates is not full column or row rank, the system is under-constrained or over-constrained. The rank of this matrix is calculated from a singular value decomposition. For an under-constrained system, any motion in the mechanism that is not accounted for by the current set of degrees of freedom is visualised using data from the right singular matrix. For an over-constrained system, a statically indeterminate stress distribution is derived from the left singular matrix and is used to visualise the over-constraints. The analysis is exemplified for the design of a straight guiding mechanism, where under-constrained and over-constrained conditions are visualised clearly.

Design of an Instrument Guide for MRI-Guided Percutaneous Interventions

This paper describes the design of an instrument guidance device for percutaneous interventions in closed bore magnetic resonance (MR) imaging systems. The device consists of a curved arm piece that travels around a circular base and an additional needle holder that travels along the curved arm, thus providing two angular degrees of freedom that enable an ablation probe to pivot about a remote center of motion located at the skin entry point. The device is intended to be mounted onto a custom built MR coil that rests on the patient while they are imaged. Exact constraint design principles were used to incorporate translational bearings into the plastic parts. Thumbscrews were used for preload and locking so that the probe guide could be fixed along a specific trajectory. The device was prototyped via stereolithography as a proof of concept and demonstrated that a probe could be angled about a remote pivot point.

Design and Fabrication of a Planar Three-DOFs MEMS-Based Manipulator

Abstract-This paper presents the design, modeling, and fabrication of a planar three-degrees-of-freedom parallel kinematic manipulator, fabricated with a simple two-mask process in conventional highly doped single-crystalline silicon (SCS) wafers (100). The manipulator's purpose is to provide accurate and stable positioning of a small sample (10 × 20 × 0.2 μm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), e.g., within a transmission electron microscope. The manipulator design is based on the principles of exact constraint design, resulting in a high actuation-compliance combined with a relatively high suspension stiffness. A modal analysis shows that the fourth vibration mode frequency is at least a factor 11 higher than the first three actuation-related mode frequencies. The comb-drive actuators are modeled in combination with the shuttle suspensions gaining insight into the side and rotational pull-in stability conditions. The two-mask fabrication process enables high-aspect-ratio structures, combined with electrical trench insulation. Trench insulation allows structures in conventional wafers to be mechanically connected while being electrically insulated from each other. Device characterization shows high linearity of displacement wrt voltage squared over ±10 μm stroke in the xand y-directions and ±2° rotation at a maximum of 50 V driving voltage. Out-of-plane displacement crosstalk due to in-plane actuation in resonance is measured to be less than 20 pm. The hysteresis in SCS, measured using white light interferometry, is shown to be extremely small.

Elastic Averaging in Flexure Mechanisms: A Three-Beam Parallelogram Flexure Case Study

Abstract Redundant constraints are generally avoided in mechanism design because they can lead to binding or loss in expected mobility. However, in certain distributed-compliance flexure mechanism geometries, this problem is mitigated by the phenomenon of elastic averaging. Elastic averaging is a design paradigm that, in contrast with exact constraint design principles, makes deliberate and effective use of redundant constraints to improve performance and robustness. The principle of elastic averaging and its advantages are illustrated in this paper by means of a three-beam parallelogram flexure mechanism, which represents an overconstrained geometry. In a lumped-compliance configuration, this mechanism is prone to binding in the presence of nominal manufacturing and assembly errors. However, with an increasing degree of distributed-compliance, the mechanism is shown to become more tolerant to such geometric imperfections. The nonlinear elastokinematic effect in the constituent beams is shown to play an important role in analytically predicting the consequences of overconstraint and provides a mathematical basis for elastic averaging. A generalized beam constraint model is used for these predictions so that varying degrees of distributed compliance are captured using a single geometric parameter. The closed-form analytical results are validated against finite element analysis, as well as experimental measurements.

Design and modeling of a six DOFs MEMS-based precision manipulator

In this paper a design is presented for a precision MEMS-based six degrees-of-freedom (DOFs) manipulator. The purpose of the manipulator is to position a small sample (10 μm × 20 μm × 0.2 μm) in a transmission electron microscope. A parallel kinematic mechanism with slanted leaf-springs is used to convert the motion of six in-plane electrostatic comb-drives into six DOFs at the end-effector. The manipulator design is based on the principles of exact constraint design, resulting in a high actuation compliance (flexibility) combined with a relatively high suspension stiffness. However, due to fabrication limitations overconstrained design has been applied to increase the stiffness in the out-of-plane direction. The result is a relatively large manipulator stroke of 20 μm in all directions combined with a high first vibration mode frequency of 3.8 kHz in relation to the used area of 4.9 mm × 5.2 mm. The motion of the manipulator is guided by elastic elements to avoid backlash, friction, hysteresis and wear, resulting in nanometer resolution position control. The fabrication of the slanted leaf-springs is based on the deposition of silicon nitride (SixNy) on a silicon pyramid, which in turn is obtained by selective crystal plane etching by potassium hydroxide (KOH). The design has been analyzed and optimized with a multibody program using flexible beam theory. A previously developed flexible beam element has been used for modeling the typical relatively large deflections and the resulting position-dependent behavior of compliant mechanisms in MEMS. The multibody modeling has been verified by FEM modeling. Presently only parts of the manipulator have been fabricated. Therefore, a scaled-up version of the manipulator has been fabricated to obtain experimental data and to verify the design and modeling.

MEMS-based clamp with a passive hold function for precision position retaining of micro manipulators

In this paper the design, modeling and fabrication of a precision MEMS-based clamp with a relatively large clamping force are presented. The purpose of the clamp is to mechanically fix a six-degree-of-freedom (DOF) MEMS-based sample manipulator (Brouwer et al J. Int. Soc. Precis. Eng. Nanotechnol. submitted) once the sample has been positioned in all DOFs. The clamping force is generated by a rotational electrostatic comb-drive actuator and can be latched passively by a parallel plate type electrostatically driven locking device. The clamp design is based on the principles of exact constraint design, resulting in a high actuation compliance (flexibility) combined with a high suspension stiffness. Therefore, a relatively large blocking force of 1.4 mN in relation to the used area of 1.8 mm2 is obtained. The fabrication is based on silicon bulk micromachining technology and combines a high-aspect-ratio deep reactive ion etching (DRIE), conformal deposition of low-pressure chemical vapor deposition (LPCVD) silicon nitride and an anisotropic potassium hydroxide (KOH) backside etching technology. Special attention is given to void reduction of SixNy trench isolation and reduction of heating phenomena during front-side release etching. Guidelines are given for the applied process. Measurements showed that the clamp was able to fix, hold and release a test actuator. The dynamic behavior was in good agreement with the modal analysis.