Spherical 4R Linkages Research Articles

Tapered Origami Tubes With Non-Planar Cross Sections

Abstract Rigidly foldable origami tubes are widely used in origami-inspired engineering designs. Here, using a mechanism construction process, we show that these tubes can be combined with tapered adding parts to form new tubes with different-sized cross sections that are rigidly foldable. A tapered tube is proposed, whose geometries are provided based on the kinematics of spherical 4R linkages. Several variations of the tapered tubes are presented, and the flat-foldability of these tubes is studied, leading to the right-angled and non-right-angled tubes which can be folded along their radial direction. The approach can be applied to both single and multilayered tubes. Moreover, the thick-panel form of the right-angled tubes is developed. Our work provides designers great flexibility in the design of tubular structures that require large shape changes. The results can be readily utilized to build new structures for engineering applications ranging from deployable structures, meta-materials to origami robots.

Deployable origami polyhedrons with one-DOF radial motion

Deployable polyhedrons, especially transformable polyhedrons, are mathematically interesting yet kinematically challenging. This paper presents a synthesis method for constructing a group of origami polyhedrons with synchronized radial motion and deployable transformability. First, an origami-synchronized mechanism with a threefold-symmetric motion feature is proposed by integrating a spatial 9R linkage and three pairs of spherical 4R linkages. Subsequently, by embedding the proposed origami-synchronized mechanism cells into the surface of Archimedean polyhedrons, three one-DOF transformable polyhedral mechanisms with distinct symmetries are constructed, the corresponding prototypes are fabricated to verify their kinematic properties. Further, referring to the dimensional shortening operations, structural variations of origami polyhedrons with mechanism topology isomorphism are demonstrated to realise various polyhedral transformations with different volumetric deployable ratios. The newly found kinematic strategy and construction method can be readily extended to other polyhedron groups with potential applications in the fields such as deployable structures for aerospace exploration and architecture, as well as unit cells for mechanical metamaterials.

A Zigzag-Based Thickness-Accommodating Foldable Prismatic Structure With a Single-Degree-of-Freedom

Abstract Origami inspires various designs of foldable prismatic structures with zero-thickness facets. However, there is a difficulty in directly applying them to thick materials. This paper presents a novel design of the foldable prismatic structure inspired by the zigzag pattern, which can accommodate the thickness of materials easily by placing hinges on the top or bottom surfaces of panels. The foldable prismatic structures are constructed by connecting multiple zigzag strips, in which each strip is made up of uniform-thickness hexagonal panels. By identifying the relationship between the foldable structure and spatial linkages, we analyze the mobility of the assembly based on the matrix method with DH notations. The result reveals that the foldable prismatic structure is equivalent to a network of spherical 4R linkages and Bennett linkages, and its motion has a single-degree-of-freedom. Based on the proposed foldable prismatic structure, a foldable manipulator is developed to demonstrate its potential engineering applications. The actuation strategy is designed by employing a motor-cable-driven system and torsional spring hinges. The physical prototype of the foldable manipulator is fabricated, and experimental results prove that our designs are feasible and effective.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

Thick-panel origami-based parabolic cylindrical antenna

This paper proposes a novel parabolic cylindrical antenna with great folding efficiency based on two symmetrically arranged Miura-ori thick-panel origami patterns and the removal of partial material from each panel. A design rule for the transition from the given parabolic equation to the geometric parameters of the thick-panel origami is defined. To ensure that the deployable structure has one degree of freedom (DOF), specific connectors between the two thick-panel origami patterns are designed from spherical 4R linkages and Bennett linkages, which are verified by kinematic analysis based on the assembly of linkages corresponding to the thick-panel origami pattern. Two joint-removing techniques based on the kinematic equivalence of linkages are constructed to obtain the continuous and smooth parabolic cylindrical surface and maintain the one-DOF property. The deployed/folded ratio influencing factors are fully studied, finding that a small panel thickness and a large origami pattern sector angle are necessary to obtain a high deployed/folded ratio. Finally, a prototype of the deployable structure is manufactured to verify and demonstrate its excellent performance.

Thick-Panel Origami Tubes With Hexagonal Cross-Sections

Abstract Rigidly foldable origami tubes can be kinematically regarded as assemblies of spherical linkages. They have exhibited excellent properties for deployable structures. Yet, for the engineering applications, the corresponding thick-panel forms have to be designed. In this paper, the spherical 4R linkages in tubes with hexagonal cross-sections are partially replaced by spatial linkages, leading to a method to construct the thick-panel tubes, which can reproduce kinematic motions equivalent to those realized using zero-thickness origami. Based on the D–H matrix method, the rotational symmetric and symmetric tubes are introduced, together with their four types of vertexes, where the specific spherical 4R linkages are replaced by Bennett and Bricard linkages to obtain the thick-panel foldable tubes. The approach can be applied to multilayered tubes with a straight or curved profile, whose manufacture can be simplified by removing extra links. The results can be readily utilized to the design of deployable tubular structures whose thickness cannot be disregarded.

Kirigami-inspired foldable 3D cellular structures with a single degree of freedom

It is common to fold a hollow 3D Archimedean solid flat using paper folding. However, little has been done to create 3D cellular assemblies of polyhedrons consisting of rigid facets that can be folded flat with a single degree of freedom. Inspired by both origami and kirigami, we present two groups of foldable truncated octahedrons composed of rigidly thin and thick facets, respectively. Starting by identifying seven possible connection types between a pair of foldable truncated octahedrons, we use these connection types to construct various foldable 3D cellular arrays. Kinematically, these arrays contain spherical 4R linkages or a mixture of spherical 4R linkages and Bennett 4R linkages if facets of uniform and finite thickness are used. Through a thorough kinematic analysis, we are able to obtain the relationships amongst all dihedral angles along each folding crease. We demonstrate that the foldable cellular assembly has a single degree of freedom. Moreover, the arrays made from thin and thick facets are kinematically equivalent. Physical models were made, which have successfully validated our findings. The proposed foldable cellular structures can be used to construct space habitats or create robot arms.

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.

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.

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.

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.

Reconfiguration of the plane-symmetric double-spherical 6R linkage with bifurcation and trifurcation

This study presents a double-spherical 6R overconstrained linkage which is a variant of the Sarrus linkage and investigates its constraint-induced bifurcation and trifurcation. In light of the unique geometry of the double-spherical 6R linkage, which is also a typical plane-symmetric Bricard 6R loop, its parametric constraints are explored. Based on constraint analysis in screw system theory, both design and motion parameters that lead to constraint singularities are revealed and the transitory positions for bifurcation and trifurcation are identified among geometric constraints induced singular positions. The analysis reveals that the presented 6R overconstrained linkage is able to reconfigure its configurations by passing transitory positions and to evolve to distinct motion branches including spherical 4R linkages and serial kinematic chains.

Rolling 4R linkages

In this paper, a family of mobile 4R linkages is proposed to complete the rolling gait. They are derived from the general 4R four-bar linkage. The first subgroup of this family consists of three planar straight rolling 4R linkages: (1) the linkage driven by four active revolute joints, (2) the linkage driven by an arbitrary active revolute joint, and (3) the overconstrained linkage driven by a motor mounted at the intersection of two additional links. The spherical 4R linkage which rolls along a circle belongs to the second subgroup. The third subgroup includes the spatial Bennett mechanism which rolls along a polygonal line. These rolling 4R linkages have alterable fixed links and are proved to roll successfully by simulations and experiments after the analyses of dynamics, stability and the strategy of motor acts. They can be used to develop new spatial four-bar or multi-bar linkages which could move straight and take a turn.

A Simple Synthesis Method for Spherical 4R Linkages Reaching Four Positions

For spherical 4R linkage synthesis reaching four specified task positions, we introduce a simple derivation method of spherical Burmester curve equation by employing a displacement matrix method. Then we presented a method to calculate the coordinates of circle and center points, so the spherical Burmester curves can be drawn by the software developed.

Dynamic Balancing of Spherical 4R Linkages

A mechanism, which, for any motion, does not apply reaction forces on the base, is said to be statically or force balanced. It is moment balanced or dynamically balanced if, moreover, it does not apply torques on the base. In this paper, an approach to determine the complete set of statically balanced spherical four-bar linkages is presented. Furthermore, it is shown that for all possible design parameters, it is not possible to dynamically balance such linkages.

Optimal selection of precision points for function synthesis of spherical 4R linkage

In order to reduce synthesis error, a new approach is reported to select precision points for function synthesis of spherical 4R linkages. Based on a closed-form symbolic solution of the set of function synthesis equations of spherical 4R linkages, the structure error equation has been derived by introducing a scaling factor. After this, an optimization model was constructed to search for the precision points. The initial solution of the optimization model was chosen using a numerical interpolation method. The global optimal solution was obtained using the fractal algorithm. To illustrate this, an example is presented and discussed before finally deriving conclusions.

Some improvements on the exact kinematic synthesis of spherical 4R function generators

Abstract It is shown that a careful definition of the design coefficients may improve the kinematic synthesis of spherical 4R linkages – intended for function generation – for three and four precision points. As a result, the design process is based on a simple system of linear equations whose solution is obtained in closed form. Moreover, the synthesis process does not lead to algorithmic problems during the determination of the link lengths. Furthermore, useful discussions for obtaining positive link lengths are also provided. Finally, several application examples are presented to prove the feasibility and the validity of the proposed method.