Bennett Linkage Research Articles

A direct route to the reciprocal Bennett linkage

The notion of reciprocity between screws is extended by Baker (2000) to put forward the concept of ‘reciprocal linkages’, those governed by screw systems that are mutually reciprocal. In that article, the 13 four-bars of Delassus, including the Bennett ‘isogram’, are so categorized. Presented here is an alternative means of defining the Bennett loop's reciprocal system and linkage, one based upon the chain's own geometrical properties rather than its principal screw system.

On the skew network corresponding to Bricard's doubly collapsible octahedron

A prompt contributor to discussion of Bricard's marvellous revelation of deformable octahedra was Bennett, who related the findings to planar, spherical, and skew assemblages, the last-named consisting of a network formed by his own remarkable four-bar linkage. Since that time, many investigations have been directed to each of Bricard's and Bennett's linkages, but rarely to the notion of a connection between them. The present article draws upon recent discoveries of six-bar linkages synthesized from Bennett isograms to establish a surprising integration of three different families of six-bars and the skew network engendered by the doubly collapsible octahedron.

An Extended Myard Linkage and its Derived 6R Linkage

In this paper, a 6R linkage suitable as a building block for the construction of large deployable structures is presented. First, we report the possibility of construct an extended 5R Myard linkage by combining two complimentary Bennett linkages. Unlike the original 5R Myard linkage (also called Myard’s “number 1” linkage), the angle of twists in the Bennett linkages is not necessary to be π∕2. Then we show that a 6R linkage can be produced by merging two extended Myard linkages together and removing the common links. The closure equations for the 6R linkage are derived and its motion characteristics are discussed. Moreover, we demonstrate that a number of such 6R linkages can be assembled together to form a large-scale deployable structure, which opens to a flat profile.

On mobile assemblies of Bennett linkages

This paper deals with deployable structures formed by interconnected Bennett linkages. A total of eight cases that allow mobile assemblies of Bennett linkages being built have been found by considering the links that may contain sections with negative length. Among the eight assemblies, four are distinct ones, including the one that we reported previously, and the remaining can be obtained by modifying the four cases. All these assemblies consist of a grid of nested Bennett linkages. The layout of the assemblies can be repeated to form a large deployable structure. When the Bennett linkages are equilateral, the assemblies expand to form arches. For a non-equilateral case, the assemblies deploy into a helical shape with a cylindrical profile. They are geometrically overconstrained with a single degree of mobility. The newly found assemblies provide more choices in the design of deployable structures.

Deployable membrane structure based on the Bennett linkage

Deplayable structures based on the alternative forms of the Bennett linkage have been proposed and investigated by some researchers, which can be fully folded to a bundle and deployed to a flat surface. For aerospace applications, a deployable membrane structure based on the alternative form of the Bennett linkage is proposed and developed in the current paper. When fully deployed, four bars of the structure are not coplanar and the membrane tensioned by them can form a hyperboloidal surface. The compatibility between the structure and the membrane during the process of deployment is investigated, and preferred ranges of main design parameters are proposed to ensure the compatibility. The method to construct the assembly of the structure is also considered. In addition, three physical models were manufactured to verify the analytical investigations of the current paper.

Square deployable frames for space applications. Part 2: Realization

In a sister paper, the authors have proved mathematically that the three-dimensional 4 R overconstrained Bennett linkage has an alternative form allowing it to be folded up completely. In this paper, the ways of building such structures are shown. For design purposes, a set of design parameters is introduced that could be used for rapid prototyping because they are closely linked with the final expanded configuration. The relationships between the design and geometrical parameters, commonly used to describe the Bennett linkages by kinematicians, have been derived. Moreover, the concept of a single compact folding Bennett linkage has been extended to a grid in which the Bennett linkages are nested within each other. The new assembly retains all the common features of the single linkage. A series of physical models were made, which successfully demonstrate the concept.

Investigation of a Cylindroid Engendered by the Bennett Linkage

Most studies of the Bennett linkage over the past century have been concerned with the characteristics of its instantaneous motion, revealing a large number of features and relationships. Latterly, work has been carried out on properties of the finite motion, an evidently separate area of consideration. However, there is one item of commonality between the two zones of activity, namely, the instantaneous screw axis of the loop's coupler. This fact is directed towards the employment of results from the former area of study to the benefit of the latter.

Kinematic Investigation of the Deployable Bennett Loop

Despite the many studies devoted to it and its value as a learning tool, the Bennett linkage has never been employed as a working mechanism. It has recently found favor, however, among structural analysts as a possible unit in deployable networks owing to the potential for true spatial displacement without flexure. Although the loop can be analyzed in this application by means of purely geometrical methods, a wealth of kinematic examinations is available for more efficient treatment. The particular form that the chain must adopt as a deployable object and the special case of the linkage demanded by the purpose constitute the subject of the present exposition, which takes full advantage of prior analyses of the chain’s kinematic characteristics.

Square Deployable Frames for Space Applications. Part 1: Theory

The Bennett linkage is a three-dimensional 4R overconstrained mechanism consisting of four rigid links connected by revolute hinges whose axes of rotation are neither parallel nor concurrent. In general, it cannot be folded up compactly. This paper investigates the existence of alternative forms of the linkage in order to achieve the most compact folding and maximum expansion so that the linkage can be used as basic building blocks of large deployable structures for aerospace applications. This study has resulted in the creation of an effective deployable element based on the Bennett linkage. A simple method to build the Bennett linkage in its alternative form has been introduced and verified. The corresponding networks have been obtained following a layout similar to that the original Bennett linkage, which the authors discovered previously. This approach can also be extended to other types of overconstrained linkages so that they can be used for building large deployable assemblies as well.

On Certain Surfaces and Curves Associated with the Bennett Linkage

Although the subject of many more investigations than any other skew linkage, the Bennett loop does not yield up its properties easily. Recent work has revealed the axodes, which putatively determine its higher-order kinematics, and some of the derivable geometric features, but the study remains incomplete. The present paper uncovers previously hidden characteristics of significance, namely, the operative regulus in the fixed axode's central tangent surface and the consequent parametric equation of its curve of striction. The approach is extended to an examination of the hyperboloid which contains the loop's joint axes.

Using a Bennett linkage as a connector between other Bennett loops

The remarkable Bennett linkage has been used almost from the time of its announcement in 1903 as a fecund source of other kinematic loops and networks. Various techniques have been applied to yield an impressive number of products. Put forward here is a procedure by which the whole loop functions as an articulation between members of two other chains, resulting in a complex spatial network of a type having possible value as a deployable structure. An algebraic analysis also provides the determination of displacement-closure equations for two different forms of six-bar loop incorporated in the assemblage, one of them new to the literature.

Mobile assemblies based on the Bennett linkage

This paper presents a method of building large mobile assemblies using the Bennett linkage. The method is based on a basic single-layer layout consisting of overlapping 4 R loops, each of which is a Bennett linkage. The assemblies created have a single degree of freedom, and are overconstrained and scaleable, allowing unlimited extension by repetition. In general, they deploy into a circular or non-circular cylindrical profile. The joints of the assemblies move spirally on the surface during deployment. Under some particular geometrical conditions, the profiles of the assemblies can become arch-like or flat. Moreover, the single-layer assemblies can be extended to form multi-layer mechanisms, even mobile masts. The paper shows the great versatility of the Bennett linkage and demonstrates that the century-old invention can play an important role in the construction of deployable structures.

The Curve of Striction of the Bennett Loop’s Fixed Axode

In previous work, the algebraic representation of a fixed axode of the Bennett linkage has been revealed as extraordinarily cumbersome. In this sequel we use properties of the ruled surface to determine the central point of a typical generator of the axode and hence its curve of striction as the intersection of two comparatively simple surfaces. Because of its special significance in this case, we also obtain the equation to the central tangent surface. A feature of the investigation is the direct employment of screw vectors in dual format rather than unit line vectors.

The Composition of Bennett’s Hyperboloids From the Loop Itself

Despite the tidiness with which the Bennett linkage can be related geometrically to its associated quadric surfaces, the corresponding algebraic determinations have generally turned out to be cumbersome, and so practically noninvertible. In particular, although earlier work has delineated the linkage in terms of parameters peculiar to its quadric surfaces, the inverse dependences could not be given. We here use the loop strictly as a basis for derivation of the surfaces and employ cartesian co-ordinates throughout, thereby effectively completing the conversion process.

Dimensional Synthesis of Bennett Linkages

Three position synthesis of a spatial RR chain yields two solutions that can be assembled to form a Bennett linkage. In this paper, we show that the cylindroid associated with Bennett’s linkage can be generated directly from the three specified positions, and used to simplify the design equations. Analysis of these equations shows that there are always two solutions and that their symmetry yields a Bennett linkage. An example design is presented.

Bennett's linkage and the cylindroid

Bennett's linkage is a spatial 4R closed chain that can move with one degree of freedom. The set of relative displacement screws that form the one-dimensional workspace of this device defines a ruled surface known as a cylindroid. The cylindroid is generally obtained as a result of a real linear combination of two screws. Thus, the workspace of Bennett's linkage is directly related to a one-dimensional linear subspace of screws. In this paper, we examine in detail Bennett's linkage and its associated cylindroid, and introduce a reference pyramid which provides a convenient way to relate the two. These results are fundamental to efficient techniques for solving the synthesis equations for spatial RR chains.

Algebraic-geometric properties of the coupler curves of the RCCC spatial four-bar mechanism

This paper uses an analytical approach, based on (3×3) orthogonal transformation matrices with dual number elements and the principle of transference, to obtain three parametric equations for the coupler curves of the RCCC four-bar mechanism. The equations express the Cartesian coordinates of an arbitrary point fixed in the coupler link as a function of a single variable; namely, the input angle. From this systematic approach, the curve describing the path of the coupler point is shown to be 16th-order. The coupler curve equations for the spherical four-bar and the Bennett linkage are deduced as special cases. The paper then presents plots of coupler curves of a general geometry and a special geometry RCCC mechanism. The latter example has a branch extending to infinity which is directly related to the limiting positions of the mechanism. Finally, the paper includes plots of typical coupler curves of the spherical four-bar and the Bennett linkage.

Analytical and Graphical Solutions to the Forward Position Problem of Two Robots Manipulating a Spatial Linkage Payload

This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a spatial four-bar linkage payload. To simplify the kinematic analysis, the Bennett linkage, which is a special geometry spatial four-bar, will be regarded as the payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown joint displacements in the wrist subassembly of each robot. An important result is that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. Repeated roots of the polynomials are shown to correspond to the stationary configurations of the system. The paper emphasizes that an understanding of the kinematic geometry of the system is essential to verify the number of possible solutions to the forward position problem. Graphical methods are also presented to provide insight into the assembly and stationary configurations. A numerical example of the two robots manipulating the Bennett linkage is included to demonstrate the importance of the polynomial and closed-form solutions.

A comparative survey of the bennett-based, 6-revolute kinematic loops

Among the large number of mobile 6-revolute spatial loops so far isolated are several which have been synthesized from Bennett linkages alone. They are scattered in the literature, presented differently and two of them have not been previously discussed. A path through the maze of findings is made more tortuous by the extraneous solutions which have appeared among the valid ones. An attempt is made here to differentiate among the cases by providing comparable sets of closure equations for all. Consideration is also given to related topics which are of assistance in comprehending the range of results.

The Bennett linkage and its associated quadric surfaces

The 4-revolute Bennett linkage has been intensively studied by several researchers over many years, and yet there remain depths to be plumbed. In this study, we begin by considering separately the manner in which the Bennett linkage is related to the hyperboloid defined by its joint axes and the one defined by its links. We go on to algebraically inter-relate these and other aspects of the loop, including as associated sphere and an equivalent screw of the overall instantaneous motion.