6R Linkage Research Articles

A Novel Ortho-Triplex Tensegrity Derived by the Linkage-Truss Transformation With Prestress-Stability Analysis Using Screw Theory

AbstractThis paper presents a novel tensegrity structure derived from the tensegrity triplex (also called the simplex or regular triangular prism) by using the linkage-truss transformation. In this paper, the tensegrity triplex is first transformed into a 6R linkage with vertical members as revolute joints and is coined the triplex linkage. With this, a novel 6R linkage was derived, whose joint axes are perpendicular to the joint axes of the triplex linkage and is coined the ortho-triplex linkage. Rigidity analysis based on screw theory demonstrates that both obtained linkages with infinitesimal mobility are prestress stable. Finally, transforming the ortho-triplex linkage to a truss, by using cables for tensional members and struts for compressional members, leads to a novel tensegrity that is coined ortho-triplex tensegrity. A non-dimensional quadratic form is further provided to analyze the sensitivity of prestress-stability in terms of the structural parameters. The process of derivation of this novel tensegrity presents a new way of designing tensegrity structures with prestress-stability analysis based on screw theory.

A double spherical 6R linkage with spatial crank-rocker characteristics inspired by kirigami

Grashof linkage was well studied in planar and spherical 4R linkage. In this paper, this characteristic is associated with a spatial linkage evolved from an kirigami loop. Both the overconstrained geometric conditions and the explicit closure equations are derived with the D-H method. It is found out that the 6R overconstrained linkage can work as a crank-rocker or a double crank linkage. Its bifurcation behaviours under two motion modes are analysed. We further transfer this 6R linkage back into the kirigami form and discuss its kinematic behaviours such as spatial 8-shaped trajectories. This work sets up the bridge between the spatial linkages and rigid kirigami forms to develop emerging kinematic behaviours.

Rigid foldability and mountain-valley crease assignments of square-twist origami pattern

Rigid foldability allows an origami pattern to fold about crease lines without twisting or stretching component panels. It enables folding of rigid materials, facilitating the design of foldable structures. Recent study shows that rigid foldability is affected by the mountain-valley crease (M-V) assignment of an origami pattern. In this paper, we investigate the rigid foldability of the square-twist origami pattern with diverse M-V assignments by a kinematic method based on the motion transmission path. Four types of square-twist origami patterns are analyzed, among which two are found rigidly foldable, while the other two are not. The explicit kinematic equations of the rigid cases are derived based on the kinematic equivalence between the rigid origami pattern and the closed-loop network of spherical 4R linkages. We also convert a non-rigid pattern into a rigid one by introducing an extra crease. The kinematic analysis of the modified pattern reveals an interesting bifurcation behaviour. This work not only helps to deepen our understanding on the rigid foldability of origami patterns and its relationship with the M-V assignments, but also provides us an effective way to create more rigidly foldable origami patterns from non-rigid ones.

Mobile assemblies of four-spherical-4R-integrated linkages and the associated four-crease-integrated rigid origami patterns

Rigid origami, which can be regarded as assemblies of spherical linkages, inspires a new paradigm of design for mechanical metamaterials and deployable structural systems with large deployable ratio. In this paper, the kinematic properties of assemblies of spherical 4R linkages are studied, and new rigid origami patterns are obtained. Kinematics of a single loop spherical 4R linkage is firstly presented, and by assigning four specified values for the four twist angles, and defining four pairs of specified input and output angles, 16 maps between the input and output angles are obtained resulting in 256 types of special spherical 4R linkages. By merging four identical spherical 4R linkages into a single loop, three basic mobile assemblies are constructed and the one-degree-of-freedom kinematic compatibility condition is formulated. Through alteration of the four vertex linkages in the three basic assemblies, variations of the basic assemblies are generated. Consequently, by adding further geometric conditions, novel rigid origami patterns are obtained leading to the tessellation of the spherical-4R-linkage-integrated assemblies. Hence, this paper provides a novel approach for generating new rigid origami patterns which can lead to the development of foldable structures and tessellations with potential applications in robotics, smart architectures, mechanical metamaterials and space exploration.

Computer-aided synthesis of spherical and planar 4R linkages for four specified orientations

Abstract. According to the burmester theory, an infinite number of spherical or planar 4R linkages for a specific four-orientation task can be synthesized, but most of the linkage solutions calculated by this method are invalid because of motion defect, poor performance and others. In order to improve the synthesis efficiency, a program package based on Matlab is developed to find a satisfied linkage solution automatically and quickly. Firstly, the calculation on circle points and the center points based on the burmester theory in spherical problems is introduced. Secondly, the calculation methods of linkage defect discrimination, linkage type classification, linkage performance evaluation and solutions visualization based on the theory of spherical trigonometry are presented respectively. Thirdly, the synthesis calculation of program package is extended to the planar 4R linkage based on the theory of planar analytic geometry. Finally, the examples of the spherical synthesis problem and the planar synthesis problem based on solutions map are introduced to test the program package, the result proves this program package is effective and flexible.

Vertex-Splitting on a Diamond Origami Pattern

A diamond origami pattern is a well-known origami pattern consisting of identical six-crease vertices. As each vertex can be modeled as a spherical 6R linkage with three degrees of freedom (DOF), the tessellated pattern with multiple vertices is a multi-DOF system, which makes it difficult to fully control the motion in the desired symmetric manner. Here, two splitting schemes on the diamond vertex are proposed to generate three types of unit patterns to reduce the DOF. This vertex-splitting technique is applied to the multivertex diamond origami pattern to produce several one-DOF basic assemblies, which form a number of one-DOF origami patterns. Two of the one-DOF origami patterns are discussed: one of which is a flat-foldable origami pattern mixed with four- and six-crease vertices and the other is a nonflat-foldable one mixed with four-, five-, and six-crease vertices. In the one-DOF patterns, the symmetrically kinematic property of the original diamond origami pattern is well kept. Such property would significantly facilitate engineering applications comparing to the multi-DOF origami patterns. It also paves a new road to construct one-DOF origami patterns.

A Network of Type III Bricard Linkages

Among Bricard's overconstrained 6R linkages, the third type has two collapsed configurations, where all joint axes are coplanar. This paper presents a one-degree-of-freedom network of such linkages. Using the two coplanar states of the constituent Bricard units, the network is able to cover a large surface with a specific outline when deployed and can be folded compactly into a stack of much smaller planar shapes. Five geometric parameters describing each type III Bricard mechanism are introduced. Their influence on the outline of one collapsed configuration is discussed and inverse calculation to obtain the parameter values yielding a desired planar shape is performed. The network is built by linking the units, either using scissor linkage elements, if the thickness of the panels can be ignored, or with hinged parallelograms, for a thicker material. Two case studies, in which the Bricard network deploys as a rectangle and a regular hexagon, respectively, are presented, validating the analysis and design methods.

Solution Region Synthesis Methodology of RCCC Linkages for Four Poses

Abstract. This paper presents a synthesis methodology of RCCC linkages based on the solution region methodology, R denoting a revolute joint and C denoting a cylindrical joint. The RCCC linkage is usually synthesized via its two defining dyads, RC and CC. For the four poses problem, there are double infinite solutions of the CC dyad, but there is no solution for the RC dyad. However, if a condition is imposed that leads to a coupling of the two dyads, a maximum of four poses can be visited with the RCCC linkage. Unfortunately, until now, there is no methodology to synthesize the RCCC linkage for four given poses besides optimization method. According to the coupling condition above, infinite exact solutions of RCCC linkages can be obtained. For displaying these RCCC linkages, we first build a spherical 4R linkage solution region. Then solutions with circuit and branch defects can be eliminated on this solution region, so that the feasible solution region is obtained. An RCCC linkage can be obtained by using the prescribed spatial positions and selected a value on the feasible solution region. We take values on the feasible solution region by a certain step length and many exact solutions for RCCC linkages can be obtained. Finally we display these solutions on a map, this map is the solution region for RCCC linkages.

Novel spherical-planar and Bennett-spherical 6R metamorphic linkages with reconfigurable motion branches

A metamorphic linkage is capable of changing its motion branches and can be used as mechanisms for reconfigurable robots for various tasks. This paper presents two novel metamorphic linkages as the spherical-planar 6R metamorphic linkage and the Bennett-spherical 6R metamorphic linkage both of which have three various distinguished motion branches. Having established the close-loop equation of the spherical-planar 6R metamorphic linkage, the paper reveals the conditions of various motion branches and a set of transformations for switching motion branches. The paper further uses to reveal the inherent properties of this over-constrained metamorphic 6R linkage that is able to perform both spherical and planar motion with mobility one. Because of geometrical constraints at bifurcation points, the linkage is able to reconfigure to the deployed spherical motion branch, the planar motion branch and the folded spherical motion branch. The two spherical motion branches could be seen on both a large sphere that presents the deployed spherical motion and a small sphere that presents the folded spherical motion. This leads to the revelation of the novel Bennett-spherical 6R metamorphic linkage that has the transition from one deployed Bennett configuration branch to a spherical configuration branch and then to another folded Bennett configuration branch. Given the geometric parameters of both metamorphic linkages, it reveals that these linkages are special cases of Bricard line-symmetric 6R linkage.

Motion and path planning of a novel multi-mode mobile parallel robot based on chessboard-shaped grid division

PurposeThe purpose of this paper is to put forward a rolling assistant robot with two rolling modes, and the multi-mode rolling motion strategy with path planning algorithm, which is suitable to this multi-mode mobile robot, is proposed based on chessboard-shaped grid division (CGD).Design/methodology/approachBased on the kinematic analysis and motion properties of the mobile parallel robot, the motion strategy based on CGD path planning algorithm of a mobile robot with two rolling modes moving to a target position is divided into two parts, which are local self-motion planning and global path planning. In the first part, the mobile parallel robot can move by switching and combining the two rolling modes; and in the second part, the specific algorithm of the global path planning is proposed according to the CGD of the moving ground.FindingsThe assistant robot, which is a novel 4-RSR mobile parallel robot (where R denotes a revolute joint and S denotes a spherical joint) integrating operation and rolling locomotion (Watt linkage rolling mode and 6R linkage rolling mode), can work as a moving spotlight or worktable. A series of simulation and prototype experiment results are presented to verify the CGD path planning strategy of the robot, and the performance of the path planning experiments in simulations and practices shows the validation of the path planning analysis.Originality/valueThe work presented in this paper is a further exploration to apply parallel mechanisms with two rolling modes to the field of assistant rolling robots by proposing the CGD path planning strategy. It is also a new attempt to use the specific path planning algorithm in the field of mobile robots for operating tasks.

Motion Cycle and Configuration Torus With Their Relationship to Furcation During Reconfiguration

In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an 8π range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a 4π range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.

Rigid Foldability of Generalized Triangle Twist Origami Pattern and Its Derived 6R Linkages

Rigid origami is a restrictive form of origami that permits continuous motion between folded and unfolded states along the predetermined creases without stretching or bending of the facets. It has great potential in engineering applications, such as foldable structures that consist of rigid materials. The rigid foldability is an important characteristic of an origami pattern, which is determined by both the geometrical parameters and the mountain-valley crease (M-V) assignments. In this paper, we present a systematic method to analyze the rigid foldability and motion of the generalized triangle twist origami pattern using the kinematic equivalence between the rigid origami and the spherical linkages. All schemes of M-V assignment are derived based on the flat-foldable conditions among which rigidly foldable ones are identified. Moreover, a new type of overconstrained 6R linkage and a variation of doubly collapsible octahedral Bricard are developed by applying kirigami technique to the rigidly foldable pattern without changing its degree-of-freedom. The proposed method opens up a new way to generate spatial overconstrained linkages from the network of spherical linkages. It can be readily extended to other types of origami patterns.

Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case

This paper for the first time reveals a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids, and presents a complete theory of the branch reconfiguration of the Bricard plane-symmetric linkages. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities, which lead to design of the plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities, it is found that the linkage can be reconfigured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special configuration connecting the branches. The study of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection, which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero finite degrees-of-freedom (DOF) but with higher instantaneous mobility. This paper is the second part of a paper published in parallel by the authors in which the method is applied to the line-symmetric case.

Rigid Body Guidance of Planar 4R Linkages Based on Gray-coded Type Map

In order to improve the calculation efficiency of 4-position guidance of 4R linkages based on burmester theory, and to solve the problem of blindness in center point choice, a computer aided method of gray-coded type map was proposed in this paper. Firstly, center curve and circle curve in burmester theory were calculated respectively by the methods of projective geometry and vector subtraction; then, all the combinations of center points were mapped into a two-dimensional set that was gray-coded according to the transmission performance of linkages; finally, 8 subsets of gray-coded map were redrew according to linkage type, on this basis, solutions satisfied the 4-position task were easily found by this map. The method proposed by this paper could improve the calculation efficiency of rigid body guidance.

Geometry of Transformable Metamaterials Inspired by Modular Origami

Modular origami is a type of origami where multiple pieces of paper are folded into modules, and these modules are then interlocked with each other forming an assembly. Some of them turn out to be capable of large-scale shape transformation, making them ideal to create metamaterials with tuned mechanical properties. In this paper, we carry out a fundamental research on two-dimensional (2D) transformable assemblies inspired by modular origami. Using mathematical tiling and patterns and mechanism analysis, we are able to develop various structures consisting of interconnected quadrilateral modules. Due to the existence of 4R linkages within the assemblies, they become transformable, and can be compactly packaged. Moreover, by the introduction of paired modules, we are able to adjust the expansion ratio of the pattern. Moreover, we also show that transformable patterns with higher mobility exist for other polygonal modules. The design flexibility among these structures makes them ideal to be used for creation of truly programmable metamaterials.

Geometric constraints and motion branch variations for reconfiguration of single-loop linkages with mobility one

This paper explores essence of geometric constraints induced reconfiguration of single-loop kinematic chains with mobility one. Reconfigurable kinematic chains are firstly classified into four categories in light of variations of parameters in the Chebychev–Grübler–Kutzbach mobility criterion. Within these four categories, single-loop kinematic chains with mobility one but distinct motion branches are further classified in accordance with degeneration of degree-of-freedom of certain revolute joints. With the essence of reconfiguration, the interrelationship of motion-branch changes through constraint singularity induced transitory positions of reconfigurable single-loop linkages is revealed in the context of reciprocity of screws. Four basic geometric constraints leading to transitory positions are explored by analysing the Bennett plano-spherical linkage, a kinematic embodiment of Grassmann varieties. Geometric constraints induced screw-system variation and motion branch changes of a novel asymmetric 7R linkage and the line- and plane-symmetric Bricard 6R linkage with capability of reconfiguring their motion branches are subsequently analysed for interpreting the fundamentals explored.

Historical Review of Studies of Spatial nR Linkages

This paper presents a historical review of spatial nR linkages studies by Russian researches, shows history of the Kazan School of mechanics and points out some of the World researches of overconstrained nR linkages. For a long time Russian scientists investigated the overconstrained spatial nR linkages in isolation from the rest of the World. This led to the formation of unique scientific schools, developing own methods of research. There are an interesting historical paradox: Bennett obtained the new one-degree-of-freedom 4R linkage in 1907. But in 1925 Verkhovskiy, regardless of Bennett research, also obtained this mechanism. Later, in 1947 Verkhovskiy obtained a family of 6R linkages. Some of these linkages was previously obtained by Bricard and Goldberg, but Verkhovskiy used completely new synthesis methods and get the same results as foreign scientists. This paper shows the main achievements of Russian researches, determines the place of Russian studies in the world's researches of spatial nR linkages.

Deployable Prismatic Structures With Rigid Origami Patterns

Rigid origami inspires new design technology in deployable structures with large deployable ratio due to the property of flat foldability. In this paper, we present a general kinematic model of rigid origami pattern and obtain a family of deployable prismatic structures. Basically, a four-crease vertex rigid origami pattern can be presented as a spherical 4R linkage, and the multivertex patterns are the assemblies of spherical linkages. Thus, this prismatic origami structure is modeled as a closed loop of spherical 4R linkages, which includes all the possible prismatic deployable structures consisting of quadrilateral facets and four-crease vertices. By solving the compatibility of the kinematic model, a new group of 2n-sided deployable prismatic structures with plane symmetric intersections is derived with multilayer, straight and curvy variations. The general design method for the 2n-sided multilayer deployable prismatic structures is proposed. All the deployable structures constructed with this method have single degree-of-freedom (DOF), can be deployed and folded without stretching or twisting the facets, and have the compactly flat-folded configuration, which makes it to have great potential in engineering applications.

空間6R機構によるOrigami Springの挙動解析

空間6R機構によるOrigami Springの挙動解析

A technique for deriving equational conditions on the Denavit–Hartenberg parameters of 6R linkages that are necessary for movability

A closed 6R linkage is generically rigid. Special cases may be mobile. Many families of mobile 6R linkages have been characterized in terms of the invariant Denavit–Hartenberg parameters of the linkage. In other words, many sufficient conditions for mobility are known. In this paper we give, for the first time, equational conditions on the invariant Denavit–Hartenberg parameters that are necessary for mobility. The method is based on the theory of bonds. We illustrate the method by deriving the equational conditions for various well-known linkages (Bricard's line symmetric linkage, Hooke's linkage, Dietmaier's linkage, and recent a generalization of Bricard's orthogonal linkage), starting from their bond diagrams; and by deriving the equations for another bond diagram, thereby discovering a new mobile 6R linkage.