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Review Article

A Comprehensive Survey on Microgrippers Design: Mechanical Structure OPEN ACCESS

[+] Author and Article Information
Matteo Verotti

Department of Mechanical and
Aerospace Engineering,
Sapienza University of Rome,
Rome 00184, Italy
e-mail: matteo.verotti@uniroma1.it

Alden Dochshanov

Department of Mechanical and
Aerospace Engineering,
Sapienza University of Rome,
Rome 00184, Italy
e-mail: alden.dochshanov@uniroma1.it

Nicola P. Belfiore

Department of Mechanical and
Aerospace Engineering,
Sapienza University of Rome,
Rome 00184, Italy
e-mail: belfiore@dima.uniroma1.it

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 28, 2016; final manuscript received March 14, 2017; published online May 3, 2017. Assoc. Editor: Massimo Callegari.

J. Mech. Des 139(6), 060801 (May 03, 2017) (26 pages) Paper No: MD-16-1535; doi: 10.1115/1.4036351 History: Received July 28, 2016; Revised March 14, 2017

An atlas of 98 microgrippers that recently appeared in Literature is herein presented by using four different forms: (a) a restyled layout of the original mechanical structure, (b) its corresponding pseudorigid body model (PRBM), (c) its kinematic chain, and finally, (d) its related graph. Homogeneity in functional sketching (a) is assumed to be greatly helpful to understand how these grippers work and what are the most significant differences between them. Therefore, a unified and systematic set of aesthetics and proportionality criteria have been adopted. Analogously, unified criteria for obtaining pseudorigid (b), kinematic (c), and graph (d) representations have been also used, which made the atlas easy to be read and inspected. The distinction among lumped and distributed compliance has been also accepted to develop the structure of the atlas. A companion paper has been prepared to present a survey on the variety of operational strategies that are used in these microgrippers.

High-precision manipulation of micro and nano objects is a critical issue for a large class of applications, such as MEMS development, optical fiber alignment, electronic packaging of microcomponents, and biomedical engineering [13]. In particular, micromanipulation devices are becoming fundamental tools in the microbiology and in the microassembly fields.

In microbiology, new technological advances at the microscale allow scientists to characterize microorganisms from a new perspective [4]. Furthermore, the manipulation of a single cell represents the essential tasks to understand individual cell behaviors and interactions, since heterogeneity in cell populations has been ascertained [5]. Related activities in cellular microsurgery allow researchers also to modify the cell structure and to understand cell mechanics. These techniques played an important role in the development of assisted reproductive technology, expanding the repertoire of clinical methodology, and options [6]. As well, new operations, such as intracellular injections, can be performed when conventional methods, such as concentration gradients or electroporation, are not applicable or fail [79]. Finally, it was found that cells are sensible to mechanical stimulation and give a response through cytoskeletal reorganization and force generation [10,11]. In fact, the study of cell mechanical behaviors has been related to morphological changes and diseases [12].

Micro devices assembly (MDA) becomes necessary when a microsystem cannot be constructed by means of MEMS-based technologies, because of its geometry, required materials, or, more in general, limitations due to its technological process. Many investigations were focused on the development of tools for handling micro-objects [1316], and on the microassembly of 3D structures [17,18]. Generally, MDA operations, such as manipulation, insertion, and fixturing are difficult, costly, time-consuming, and require systems that are able to perform gripping, releasing, precise positioning, and joining. For these reasons, computer-based automated microdevices assembly (AMDA) could increase efficiency, reliability, and reduce costs [19,20]. However, both accurate positioning of micro-objects and automated microassembly remain challenging tasks [21].

A considerable amount of work has been devoted to satisfy the increasing demand of high-efficiency, high-precision, and reliable microgrippers and some state-of-the-art surveys have been already completed to analyze micromanipulation systems under different points of view. Cecil et al. considered gripping and manipulation techniques for microassembly applications [19,20], whereas Fantoni and Porta focused on the releasing strategies [22]. Manipulation of micro and nano objects in electron microscopes was investigated by Denisyuk et al. [23]. Wei and Xu analyzed working principles, detection accuracies, advantages, and disadvantages of microforce sensing methods [24], whereas an overview on gripping force measurement (using two-fingered microrobotic systems) was presented by Boudaoud and Regnier [25]. A review of conceptual designs of nanoscale manipulators was presented by Mekid et al. [26], who described the characteristics of 40 recent patents. Several MEMS microgripper actuators and sensors were compared by Jia and Xu, who suggested some guidelines for different scenarios [27]. Nikoobin and Niaki compared several types of microgrippers, and then, they were able to derive effective design and performance parameters, such as displacement amplification factor, gripping range and stroke, jaw motion characteristic, ideal shape of tips, number of degree-of-freedom, and microactuator specifications [28].

The present contribution is exclusively dedicated to the electro mechanical microgrippers, because these structures are widely used in research and applications and also because their large number makes it interesting to compare them with one another. Therefore, some peculiar classes of tweezers, such as ultrasonic, molecular, fluorescence resonance energy transfer, optical, laser, acoustic, dielectrophoretic, or freezer tweezers, were not investigated. A selection and analysis process was necessary to build the group of structures, and a new atlas was created by gathering the different elements in three classes. These categories were established on the basis of classical mechanical characteristics. Compliance played a fundamental role in classification because the mechanical component of any MEMS consists conceptually in a compliant mechanism.

The first two classes are composed of micromechanisms which embody either lumped (class I) or distributed (class II) compliance, whereas those microgrippers which do not obey to the two previous criteria have been gathered into the third class, as special structures (class III).

An effort has been made in order to extract not only the functional representation of the microgrippers but also their most popular corresponding simplified representations, namely, the pseudorigid body models (PRBMs) [2934]. Such model is obtained by reducing the original compliant mechanism to an ordinary one (i.e., composed of only rigid bodies and kinematic pairs). Both the functional and the PRBM representations cannot be uniquely defined. However, while the functional sketch is quite simple and straightforward, a more complex procedure must be used to obtain the PRBM.

Once the PRBM has been defined, the resulting ordinary mechanism gives rise to its corresponding kinematic chain and graph, which can be allocated to a family and a group, according to a classic method of mechanisms categorization [35]. As a consequence, the atlas becomes an effective and systematic tool, which is well integrated to number and type synthesis, and which allows designers to appreciate the microgrippers topological characteristics.

The mechanical structure of a microgripper is strictly related to the adopted actuation strategy, because the value of the actuating force or torque depends on the mechanical constraints which characterize the corresponding kinematic chain. Both the elastic joints configuration and the mechanism posture have a great effect on actuation. For this reason, a comparative analysis of the possible actuation strategies has been developed and proposed in a companion paper [36]. This analysis, which includes force feedback, sensing, and releasing strategy, represents a complementary part of the present paper. A wide range of different actuator types, specially designed for microgrippers, is therein explored and some survey tables are introduced. At the same time, a comparison of the basic performance characteristics of three major actuator types, namely, electrostatic, thermal, and piezoelectric, is proposed.

Since decades, the systematic development of mechanisms atlases (see, for example, Refs. [3740]) has been supported by type and number synthesis. Graph theory played an important role in separating the concepts of structure and function, as recently remarked [41], and therefore, it has been widely adopted as the main and preferred resource in generating atlases of kinematic structures of mechanisms. Provided that a systematic approach is adopted for enumeration (for example, algorithms based on graph theory), the atlases are, generally, both exhaustive (i.e., all the structures with the selected characteristics are considered) and not redundant (i.e., there are no idle copies of the same structure in the group). Graph-based algorithms perform best during the generation of large classes of kinematic structures with specified topological characteristics (i.e., up to a given number l links, with specified F degrees-of-freedom, LIND loops, and so on). Taking l = 4, F = 1, and LIND = 1, an atlas of grippers (at the macroscale) has been presented in 1997 [40]. This contribution offered a full atlas of grippers obtained by combining two elementary four bar linkages in symmetric and asymmetric arrangements, by attaching the jaws to the couplers. This atlas has been used as a source for generating new microgrippers. For example, in 2005, Tsai et al. [42] applied the operation of kinematic joint transformation to the grippers reported in atlas [40] and generated 28 compliant microgrippers. The compliant mechanisms and their corresponding pseudorigid body models have been related by means of equivalent joint transformations.

In the present investigation, an inverse process, namely, from the compliant mechanism to a corresponding PRBM, has been used to generate the PRBMs that appear in the new atlas. In fact, the adopted inclusion criterion consisted in considering only those mechanisms which have been really fabricated or, at least, simulated and, then, presented in Literature. Although the topological approach has not been adopted for enumeration purposes, it has been used as a fundamental tool for classifying the structures. Another interesting difference between the review approach adopted in this paper and the typical enumeration algorithms consists in the contents of the atlas itself. In fact, whereas enumeration is tailored for generating classes of kinematic structures with a great number of links (i.e., l8), the actual microgrippers used in applications have rarely more than eight links.

In this atlas, micromechanisms are classified considering three levels. The first level focuses on the compliant structures and defines a class of mechanisms. The second level assesses a classification of compliant structures according to the topological nature of the corresponding PRBMs, whereas the third one considers the family and the group.

Functional Classification.

A compliant mechanism modifies its neutral configuration either through the elastic deformations of its flexible parts (flexures) or through a distributed deformation along the whole mechanism.

The advantages of flexure-based mechanisms (e.g., high accuracy, no backlash, no need of lubrication) provide great opportunities for high-precision microgrippers establishment [43]. For example, beams with uniform cross section in straight configuration can serve as flexures in leaf-spring guidance systems [44] or as amplification beams in MEMS devices [45,46]. Constant-curvature beams, with uniform cross section, have also been considered as primitive flexures for compliant mechanism [47,48] or employed to realize complex flexures [49] and micromechanisms [5053].

Beams with variable cross section have also been considered as flexures. Notch hinges, for example, are characterized by the geometric configuration (circular, corner-filleted, parabolic, hyperbolic, elliptical, inverse parabolic, and secant designs) [5457] and by axial symmetry [58,59]. Figure 1 shows a circular, an elliptical, and a corner-filleted notch hinge.

Whereas notch hinges are peculiar elements of the compliant mechanisms that are characterized by lumped compliance, uniform cross section beams, in straight or curved configuration, may introduce both lumped and distributed compliance, depending on the mechanism configuration.

To improve the performance of compliant mechanisms in terms of stress concentration, off-axis-to-axial stiffness ratio, range of motion, and precision of rotation [60], complex flexures have been developed combining more flexible elements or involving contact systems [49,6163]. Large deflections, within the elastic range, and precision of motion are among the most important features of microgrippers, and such characteristics are often in contrast to each other.

The goal of recent investigations [64,65] has been the realization of advanced flexures, with capability of achieving wide range of motion and high accuracy in rotations.

Some microgrippers embed both long beams and notch hinges within their compliant structure. In fact, these structures combine both lumped and distributed compliance to guarantee motion. All these mechanisms have been grouped in the above-mentioned third group, namely, the special structures group. The mechanical structure has an effect on its compliance response to externally applied load. The importance of such response has been recently pointed out by Rabenorosoa et al. [21], and compliance optimization, well-established in rigid-body mechanics (for example, by means of active stiffness regulation [6669]) has been recently considered in the design of a microgripper [70].

Topological Classification.

A kinematic chain is characterized by its degrees-of-freedom (F), number of links (), number of joints (m), and number of independent loops (LIND). These parameters are related by Grübler's and Euler's equations Display Formula

(1)F=3(1)2mLIND=m+1

and so only two of these parameters are independent. The pair F- defines a family of kinematic chains. The kinematic chains of the same family have the same number of links , but, generally, different numbers of binary, ternary, and jnary links (j). Then, within the same family, it is possible to define groups of linkages having the same numbers of binary, ternary, jnary links.

In Secs. 57, microgrippers presentation has been ordered according to the following criteria:

  1. (1)special topological cases as first, including those mechanisms which are not classifiable as members of a family (e.g., open chain with LIND = 0);
  2. (2)increasing degrees-of-freedom;
  3. (3)increasing family number;
  4. (4)increasing group, starting from the minimum number of the lowest j-nary links lj.

Anyone who wishes to achieve a complete picture of the actual State-of-the-Art of microgripper design encounters many difficulties because each contribution is depicted through an individual sketching style. The variety of applications leads to a rather inhomogeneous set, where the representations are neither standard nor uniform. As a consequence, comparing different microgrippers can be a time consuming and inaccurate task. For example, two different layouts may seem rather similar, while two similar structures may be treated as very different microgrippers. The quality of the present atlas, according to the Authors experience [3740,71], depends strongly on how much systematic is the representation method and on the efficacy of the adopted esthetics. The representation method is based on four main features:

  • an appropriate and effective way of representing the geometrical and structural characteristics of the microsystems,

  • a systematic way of representing the functional characteristics of the microstructures (PRBM),

  • a standard way of representing the kinematic chain, and

  • a uniform style for displaying the corresponding graph.

Furthermore, the following general criteria to represent mechanism geometry, PRBM, kinematic chains, and graphs have been adopted:

  • drawings have been positioned inside widows which have all an equal size and rectangular shape;

  • pin joints have been all represented by circles with equal size, whereas a generic prismatic pair has been represented as the usual box sliding inside a slot;

  • sensible points, such as the centers of the revolute joints or the polygons vertices, have been positioned on a base grid;

  • graph nodes have been represented by characterizing the nature of their corresponding link: actuator, frame, or output vertices have been represented by black, double, or checkered circles, respectively.

Moreover, the following general criteria have been adopted to represent mechanisms geometry:

  • geometrical remapping of the original design (in order to locate the jaws always on the right hand side);

  • horizontal orientation of the symmetry (when applicable) or gripping axis;

  • symmetry (when applicable);

  • introduction of modular and standard graphical elements to represent the jaws or the fixed revolute joints;

  • introduction of a minimum size for the compliant parts of the compliant mechanism;

  • introduction of a maximum size for the pseudorigid parts of the compliant mechanism;

  • introduction of a maximum-to-minimum size ratio for the parts of the compliant mechanism.

  • simplification of truss structures and possible substitution of truss regions with T-labeled block.

Finally, the following actions have been applied to determine the PRBM corresponding to the analyzed structures:

  • notch hinges have been replaced by revolute (pin) joints located in correspondence of the minimum cross-sectional area of the flexure;

  • flexible beams have been also replaced by revolute joints, placing the pin joint in the center of the beam elastic weights;

  • revolute joints which are incident to the frame link have been represented with the same shape and size in all the drawings;

  • multiple revolute joints have been represented by two concentric circles;

  • actuator producing a linear displacement (such as piezoelectric or chevron) have been replaced by a prismatic joint;

  • sensing hinges have been replaced by torsional springs;

  • actuation forces or torques, or their effects on specific parts of the structures, have been schematically represented with bold black arrows;

  • sensing elements, or their effects on specific parts of the structures, have been schematically represented with white-colored black-bordered arrows.

The class of microgrippers based on lumped compliance shows a large variety of structures. In this investigation, this group counts 35 different mechanisms, which have been illustrated in Figs. 27.

The common feature consists in the presence of some zones, with restricted width, which operates as flexures. The conceptual stage of design, which most includes designer's creativity, can be based on the invention of a PRBM which can be easily transformed into a compliant mechanism. There are not canonical transformation rules, although the Authors of the paper think that the criterion based on the selection of the center of elastic weights can be adopted with great accuracy to detect the center of relative rotations between two rigid parts. In fact, as demonstrated in Ref. [47], in case of uniform flexures, the position of the center of rotation can be found analytically. The center lies on the axis of symmetry of the flexure and tends to its center of mass for loads approaching zero.

In the devices belonging to class I, motion relies on the deflections of flexures that are considerably smaller than the rest of the structure. These regions are exposed to stress levels that could compromise the device, if not adequately considered in the design steps. This condition could be worsened by possible imperfections due to the fabrication processes that give rise to stress concentrations.

Usually, class I is characterized by a fair correspondence between the original compliant mechanism and its corresponding PRBM, because it is quite easy to identify, with a good approximation, the relative rotation center between two adjacent pseudorigid links within a restricted region around the flexure. This allows designers to adopt more easily the classical algorithms from kinematic synthesis. However, an accurate detection of the above-mentioned center is still quite challenging because its position actually depends on the loads applied on the pseudorigid links.

Table 1 reports the microgrippers with lumped compliance classified in terms of families and groups.

The class of microgrippers that are based on distributed compliance shows also a similarly large variety of structures. This group counts 32 different mechanisms, which have been illustrated in Figs. 813.

Embedded in these mechanisms, there may be also some thin beams with higher flexibility. However, in such case, these regions are as large as the pseudorigid ones, and therefore, it is not easy to predict where the centers of relative rotations between the two adjacent pseudorigid parts are, for a given static load on the structure. As a consequence, the conceptual stage of design relies on the designer's mastery of statics, which must include the capability of deciphering the compliance behavior of the whole complex structure, rather than of its simplified PRBM. It is still possible to find a PRBM corresponding to a given microgripper with distributed compliance. However, for the class of distributed compliance, the equivalence among the original compliant mechanism and its corresponding PRBM will be rather weaker than for the class of lumped compliance.

With respect to lumped compliance structures, the devices belonging to class II are characterized by flexible elements whose dimensions are comparable to the dimensions of the whole structure. Stress is then distributed along extended regions and not concentrated to small parts of the device. For this reason, distributed compliance structures could be less vulnerable to low-yield fabrication processes and stress concentrations.

The PRBM reported in Figs. 810 have been still developed by using the criterion based on the selection of elastic weights center of the most flexible parts of the structures. However, the variation of the position of the relative rotation center could imply a loss of significance for the PRBM, determining also a loss of precision in case designers apply kinematic synthesis to the above-mentioned PRBM.

Table 2 reports the microgrippers with distributed compliance classified in terms of family and group.

As in any human activity, creativity is something which can be hardly classified. In fact, together with structures for which a lumped or distributed compliance is self-evident, some other structures did not showed neither the former nor the latter as clearly as for the first two groups.

This particular class of microgrippers counts 31 very different mechanisms, which have been illustrated in Figs. 1419 and consist of

  • structures with one or more parallel substructures embedded in the mechanical structure (see, for example, Fig. 17(a));

  • structures for which motion and operation are due to a particular symmetry (as in Fig. 18(b));

  • structures for which sliding is allowed (as in Fig. 16(f));

  • elementary structures (as in Fig. 16(d));

  • complex structures (as in Fig. 18(e));

  • truss structures, as, for example, Figs. 14(d), 17(a), 17(c), 17(e), and 18(b));

  • distributed wedge simple mechanisms (Fig. 16(f)),

  • ratchet structures (Fig. 15(b)),

  • distributed compliance simple cantilever structures (Figs. 14(b), 14(c), 15(c)15(f), 16(a), 16(e), 17(b), and 17(d)),

  • out-of-plane simple cantilever structures with two (Fig. 15(f)), four (Fig. 14(c)), or six (Fig. 15(e)) fingers,

  • distributed compliance parallel cantilevers structures (Figs. 17(f), 18(a), 18(c)18(e), and 19(a)).

As for the case of distributed compliance, it is not easy to predict where the relative rotations centers (between any two adjacent pseudorigid parts) are, and therefore, the conceptual stage of design relies only on designer's individual skills in understanding and interpreting the compliance of the whole structure due to its peculiar geometrical characteristics. Furthermore, similarly to the case of distributed compliance, the equivalence among the original compliant mechanism and its corresponding PRBM will be rather weak. However, the specification of a PRBM is still of great importance in order to understand how the original system operates.

The PRBM reported in Figs. 1416 have been developed by using the criterion based on the selection of the rotation centers as coincident to the center of elastic weights of the most flexible parts of the structures. Furthermore, prismatic joints also appear, as floating dyads, to represent relative translations.

Table 3 lists the microgrippers with distributed compliance classified in terms of families and groups.

Structures representations have been built to convey to the reader some reasonable interpretations of the topological characteristics of the microgrippers. Of course, it will be the readers' tolerance and understanding that the actual mechanical features of any discussed microgripper are those described in the original paper (always referenced in figures captions). In fact, while the atlas has been built by taking into account esthetics, the original layouts have been developed for the sake of functionality.

Actuation, sensing and operation will be discussed systematically in the above-mentioned companion paper. However, before addressing the reader to that work [36], it is worth noting that the morphology has some direct implication on some characteristics, such as the mechanical advantage and output motion.

For example, the jaws motion, usually rather limited in amplitude, can be enhanced involving different kinds of displacement amplification, such as Scott–Russel mechanism [72] (see Fig. 7(b)), leverage mechanism [73] (see Fig. 5(f)), the combination of flexure hinge and flexure beam [43] (see Fig. 5(e)), the integration of lever, and parallelogram mechanisms [74] (see Fig. 7(a)), differential mechanism [75] (see Fig. 7(e)).

Furthermore, most of the existing microgrippers are designed with parallelogram flexures [7678] (see Figs. 6(a), 4(d), and 11(b)) to generate a pure translational motion of gripper tips [79] (see Fig. 12(e)) that makes the architecture of the gripper mechanism rather complex. Another side-effect of the displacement amplification is the reduction of the actuation force [80] (see Fig. 11(a)). The last, in turn, may require high input voltage [43,76,81] (see Figs. 5(e), 6(a), and 4(d)). Sometime, piezoelectric bimorph [8284] or linear actuators employing a lever mechanism [85] can be conveniently used. The designs adopted in this kind of microgrippers are mainly asymmetric [75] (see Fig. 7(e)) and improves gripping accuracy as well [74] (see Fig. 7(a)). Moreover, grasping can be performed asymmetrically: the left jaw accomplishes grasping, while the right one provides rotations [75].

Finally, other interesting morphology–to–function relations appear, for example, in multifinger grasping and rotating operations [86] (see Fig. 5(d)), piezoelectric stack actuator (PSA)-driven mechanisms [79] (see Fig. 12(e)), and roberts mechanisms combined in parallel to provide both grasping and rotating capability [86].

Optimal criteria and graphical esthetics have been used to generate an atlas of microgrippers composed of a large selection of structures from the recent literature. Each mechanism has been represented by means of a functional form, which is quite similar to the original layout, a pseudorigid body equivalent mechanism, together with the corresponding kinematic chain and graph. The collected structures have not been obtained by means of type or number synthesis, because the purpose of the present investigation consists in offering a survey of those microgrippers that have been actually fabricated or, at least, simulated. All the sketches that have been included in the atlas appear homogeneous, easy to inspect, and quite useful to understand the basic ideas standing behind the reviewed microgrippers. This feature does not appear in any other review dedicated to microgripping. The strategies employed to operate all the collected devices have been also reviewed, but, for the sake of brevity, such additional survey has been presented in another contribution, namely, the second part of the present paper. The Authors hope that their new catalog could help designers to develop their own new microgrippers.

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Figures

Grahic Jump Location
Fig. 1

Circular (a), elliptical (b), and corner-filleted (c) notch hinge

Grahic Jump Location
Fig. 2

Geometry of class I: structures with lumped compliance (1–6): (a) see Ref. [53], (b) see Ref. [87], (c) see Ref. [88], (d) see Ref. [89], (e) see Refs. [90] and [91], and (f) see Ref. [92]

Grahic Jump Location
Fig. 3

Geometry of class I: structures with lumped compliance (7–12): (a) see Refs. [9395], (b) see Ref. [96], (c) see Ref. [97], (d) see Ref. [98], (e) see Ref. [99], and (f) see Ref. [81]

Grahic Jump Location
Fig. 4

Geometry of class I: structures with lumped compliance (13–18): (a) see Refs. [9] and [100], (b) see Refs. [101] and [102], (c) see Ref. [103], (d) see Ref. [77], (e) see Ref. [104], and (f) see Ref. [105]

Grahic Jump Location
Fig. 5

Geometry of class I: structures with lumped compliance (19–24): (a) see Ref. [106], (b) see Ref. [107], (c) see Refs. [108] and [109], (d) see Ref. [86], (e) see Ref. [43], and (f) see Ref. [73]

Grahic Jump Location
Fig. 6

Geometry of class I: structures with lumped compliance (25–30): (a) see Ref. [76], (b) see Ref. [110], (c) see Ref. [89], (d) see Ref. [111], (e) see Ref. [112], and (f) see Ref. [89]

Grahic Jump Location
Fig. 7

Geometry of class I: structures with lumped compliance (31–35): (a) see Ref. [74], (b) see Ref. [72], (c) see Ref. [113], (d) see Refs. [114] and [115], and (e) see Ref. [75]

Grahic Jump Location
Fig. 8

Geometry of class II: systems with distributed compliance (1–6): (a) see Ref. [116], (b) see Ref. [118], (c) see Ref. [119], (d) see Refs. [120] and [121], (e) see Ref. [122], and (f) see Ref. [123]

Grahic Jump Location
Fig. 9

Geometry of class II: systems with distributed compliance (7–12): (a) see Refs. [46], [124], and [125], (b) see Ref. [126], (c) see Ref. [127], (d) see Ref. [128], (e) see Ref. [129], and (f) see Ref. [130]

Grahic Jump Location
Fig. 10

Geometry of class II: systems with distributed compliance (13–18): (a) see Ref. [45], (b) see Ref. [131], (c) see Ref. [132], (d) see Refs. [133] and [134], (e) see Refs. [135137], and (f) see Ref. [138]

Grahic Jump Location
Fig. 11

Geometry of class II: systems with distributed compliance (19–24): (a) see Refs. [80], [139], and [140], (b) see Ref. [78], (c) see Ref. [141], (d) see Ref. [142], (e) see Refs. [143] and [144], and (f) see Ref. [145]

Grahic Jump Location
Fig. 12

Geometry of class II: systems with distributed compliance (25–30): (a) see Ref. [146], (b) see Ref. [147], (c) see Ref. [148], (d) see Refs. [149] and [150], (e) see Ref. [79], and (f) see Refs. [14] and [151]

Grahic Jump Location
Fig. 13

Geometry of class II: systems with distributed compliance (31–32): (a) see Ref. [152] and (b) see Ref. [153]

Grahic Jump Location
Fig. 14

Geometry of class III: special structures (1–6): (a) see Ref. [154], (b) see Ref. [155], (c) see Ref. [156], (d) see Ref. [157], (e) see Refs. [158160], and (f) see Ref. [161]

Grahic Jump Location
Fig. 15

Geometry of class III: special structures (7–12): (a) see Ref. [162], (b) see Ref. [163], (c) see Refs. [164] and [165], (d) see Ref. [166], (e) see Ref. [167], and (f) see Ref. [168]

Grahic Jump Location
Fig. 16

Geometry of class III: special structures (13–18): (a) see Ref. [169], (b) see Ref. [170], (c) see Ref. [171], (d) see Ref. [172], (e) see Ref. [173], and (f) see Ref. [174]

Grahic Jump Location
Fig. 17

Geometry of class III: special structures (19–24): (a) see Ref. [175], (b) see Refs. [176] and [177], (c) see Ref. [178], (d) see Ref. [179], (e) see Ref. [180], and (f) see Ref. [181]

Grahic Jump Location
Fig. 18

Geometry of class III: special structures (25–30): (a) see Ref. [182], (b) see Ref. [183], (c) see Ref. [184], (d) see Ref. [185], (e) see Ref. [186], and (f) see Ref. [17]

Grahic Jump Location
Fig. 19

Geometry of class III: special structures (31): (a) see Refs. [187] and [188]

Tables

Table Grahic Jump Location
Table 1 Classification of class I structures
Table Footer Notea1R open chain.
Table Footer NotebRR open chain.
Table Grahic Jump Location
Table 2 Classification of class II structures
Table Grahic Jump Location
Table 3 Classification of class III structures
Table Footer Notea1R open chain.
Table Footer Notebone higher pair, cam-follower.

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