Bricard Linkage Research Articles

Intrinsically Multi-Stable Spatial Linkages.

Multi-stable structures can be reconfigured with fewer, lightweight, and less accurate actuators. This is because the attraction domain in the multi-stable energy landscape provides both reconfiguration guidance and shape accuracy. Additionally, such structures can generate impulsive motion due to structural instability. Most multi-stable units are planar structures, while spatial linkages can generate complex 3D motion and hold a more promising potential for applications. This study proposes a generalized approach to design a type of intrinsically multi-stable spatial (IMSS) linkages with multiple prescriptible configurations, which are structurally compatible, and naturally stable at these states. It reveals that all over-constrained mechanisms can be transformed into multi-stable structures with the same design method. Single-loop bi-stable 4R and quadra-stable 6R spatial linkages modules with intrinsic non-symmetric stable states, which are transformed from fundamental kinematic linkage mechanisms unit such as Bennett and Bricard linkages, are designed to illustrate the basic idea and the superiority over the ordinary methods. Multi-loop assembly by these IMSS linkage modules shows potential for practical applications that are required for the deployability and impulsivity of reconfiguration. Two preliminary design cases of a deployable tube and an impulsive gripper are experimentally presented to validate this applicability. Further promisingly, this design method of IMSS linkages paves the way for morphing platforms with lightweight actuation, high shape accuracy, high stiffness, and prescribed impulsive 3Dmotion.

Design and kinematics analysis of a parabolic cylinder deployment mechanism with modular over-constrained triangular pyramid

Space missions require novel mechanisms that can have compact form in complex space environments. This study proposes a modular triangular pyramid parabolic cylinder deployment mechanism. The modular deployable unit contains an over-constrained triangular pyramid deployable mechanism and a symmetrical trapezoidal Bricard linkage that drives the longitudinal and transverse motions of the mechanism. A cable net is added between two symmetrical linkages to form a parabolic cylindrical reflector and is analyzed by the force density method. The Denavit-Hartenberg (D-H) coordinate method is used to analyze the kinematic characteristics. Based on the analytical solution for the angular displacement, the degrees of freedom of the mechanism are derived from screw theory. Subsequently, the angular velocity and acceleration of the joint points are obtained. Finally, kinematic models of the modular triangular pyramid parabolic cylinder deployable mechanism are established, and the accuracy of the theoretical model is verified using a numerical method. This novel parabolic cylinder deployable mechanism will have a significant influence in aerospace domain. Crucially, the work has value to design deployment mechanism for parabolic cylinder antenna.

Synthesis and Analysis of Plane–Space Switching Mechanisms Based on a Plane-Symmetric Eight-Bar Linkage

Abstract This paper proposes a novel synthesis method for constructing plane–space switching mechanisms based on the symmetric plane of the regular prism. First, the structure equation and motion characteristic of plane-symmetric eight-bar linkage are presented. Then, the plane-symmetric seven-bar linkage and rhombic Bricard linkage are obtained by locking the joint of the eight-bar linkage. Four types of plane–space switching mechanisms are constructed based on the synthesis method and switching linkage units. These switching mechanisms can expand completely into planar configurations and fold completely into spatial configurations. Subsequently, the kinematics of the coupled branch chain is analyzed, through which the folded and contractive characteristics of the mechanism are revealed. Then, the concept of the distributed circle of joints is proposed, and the enveloping performance of the mechanism is approximately analyzed. This paper provides a new idea and synthesis method for designing new deployable mechanisms.

On the kinematics of general plane-symmetric Bricard mechanisms

The spatial 6R single-loop overconstrained Bricard mechanisms have been attracting continuous attention owing to their remarkable mobility. However, kinematic analysis and characterization of diversified Bricard mechanisms with different structures remain challenging due to their geometric versatility. This paper presents a comprehensive method with an intuitive geometric interpretation for the kinematic modeling and analysis of general plane-symmetric Bricard mechanisms. The concept of ‘half Bricard linkage’ is first introduced to simplify the kinematics of the studied plane symmetric originals with the idea of kinematic equivalence. From a geometric perspective, the forward kinematics of half Bricard linkage can be transformed into the intersection of two coplanar ellipses and resolved straightforwardly. Using the proposed method, the kinematic properties of a general plane-symmetric Bricard mechanism, such as dead center configurations, motion branches, and their connectivity, can be characterized intuitively according to the intersection situations of two spatial surfaces generated from the planar ellipses. Several well-studied examples are elaborated as case studies to demonstrate the effectiveness and advantages of the novel geometric method for the kinematic analysis of general plane-symmetric Bricard mechanisms.

A study on the flexible Bricard mechanisms with plane-symmetric structure

The six-bar single-loop overconstrained Bricard linkages have been attracting continuous attention due to their special geometry and remarkable mobility. During the past century, a great number of papers have been published in various aspects, however, almost all of them focus on those only consisting of rigid links and joints. In this paper, the concept of ‘Flexible Bricard Mechanisms’ is introduced to extend these classical overconstrained linkages from the category of rigid-body systems to the scope of flexible ones. In these novel designs, slender elastic beams are used to replace the rigid links and joints, such that the mobility of flexible Bricard mechanisms is achieved not only via the relative motion from rigid joints, but also the large-scale deformations of elastic links. The plane-symmetric case is studied to demonstrate the realization, analysis methodology, and potential applications of this new type of Bricard mechanism. A candidate with a specialized structure is elaborated as a case study, about which a proof-of-concept prototype is developed. Preliminary experiments are conducted to verify the feasibility of the proposed idea.

Three-Dimensional Mobile Assemblies Based on Threefold-Symmetric Bricard Linkages

Abstract Multiple mobile assemblies have been created based on two-dimensional tessellations of linkages for deployable structures. However, few three-dimensional tessellations of linkages have been created, especially mobile assemblies with bifurcation. Here, we proposed four types of mobile assemblies of kaleidocycles, a special type of threefold-symmetric Bricard linkages, based on cubic cellulation and symmetry. Kinematic analysis of them is carried out based on the matrix method and numerical method. Two assemblies have bifurcations with two motion paths following cuboid symmetry and tetrahedral symmetry, respectively. Meanwhile, the other two have one motion path with a single degree-of-freedom. The designing process facilitates the creation of new mobile assemblies under symmetry, and the four assemblies have the potential application for designing metamaterials.

A metric description of flexible octahedra

A new description of flexible Bricard octahedra is obtained using conditions in terms of edge lengths. It is suitable for the study of a number of problems in the metric geometry of octahedra and, in particular, for searching for a proof of the conjecture of Sabitov on the vanishing of all but the leading coefficients of the polynomial for the volume of a type $3$ octahedron. Bibliography: 17 titles.

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.

The Bricard Linkage Network Constructed by Bennett Linkage: Design Method and Kinematic Analysis

The paper focuses on a Bricard Linkage network constructed by Bennett Linkage and discusses the possibility of connecting Bricard linkage using Bennett linkage as the connection. To simplify the problem, accessible geometric parameters are set without much loss of generality. This new design can also evolve into some new mechanisms by connecting the linkages in different ways. Based on the designed connection idea, the corresponding three-dimensional model is created by Onshape. Using Onshape, the movement and deployment of each mechanism could be observed. Then the kinematic analysis is carried out with Adams software, to find out how the kinematic angles vary as time increases. A physical model is made to observe the deployment process more clearly, indicating a potential application of a retractable table. Since the problem has been simplified by setting accessible parameters, the mechanism may show some further performance worthy of being explored if the parameters are more complicated.

A one-DOF compliant gripper mechanism with four identical twofold-symmetric Bricard linkages

AbstractGripper is one of the most important parts of robot because of contacting with workpieces directly and has attracted lots of research interests. However, the existing grippers are either simple in function or complex in structure. In this paper, we will propose a one-DOF gripper based on a compliant mechanism with four identical twofold-symmetric Bricard linkages. A mobile network with four identical twofold-symmetric Bricard linkages with particular design parameters is constructed at first. Kinematics, such as mobility, singularity, and folding performance, is then analyzed to show the potential of realizing the function of grasping. The result is demonstrated with a physical prototype. To simplify the fabricating process, a compliant mechanism of the network is designed and fabricated with a single polypropylene board, and the grasping function is realized by a cable-driven scheme. Some grasping experiments are carried out on different types of objects which shows that the proposed and fabricated four-figure gripper is simple in structure and has a great grasping function. The work provides a new idea for the design of grippers with low cost, simple structure, and rich functions.

Acoustic Wave Focusing From Reconfigurable Acoustic Arrays Based on a Bricard-Miura Synthesis

Abstract Recent studies have shown that reconfigurable acoustic arrays inspired from rigid origami structures can be used to radiate and focus acoustic waves. Yet, there is a need for exploration of single-degree-of-freedom deployment to be integrated with such arrays for sake of tailoring wave focusing. This research explores a reconfigurable acoustic array inspired from a regular Miura-ori unit cell and threefold-symmetric Bricard linkage. The system focuses on acoustic waves and has single-degree-of-freedom motion when incorporated with a modified threefold-symmetric Bricard linkage. Three configurations of the array are analyzed where array facets that converge towards the center axis are considered to vibrate like baffled pistons and generate acoustic waves into the surrounding fluid. An analytical model is constructed to explore the near-field acoustic focusing behavior of the proposed acoustic array. The wave focusing capabilities of the array are verified through proof-of-principle experiments. The results show that the wave focusing of the array is influenced by the geometric parameters of the facets and the relative distance of facets to the center axis, in agreement with simplified ray acoustics estimates. These findings underscore the fundamental relationship between focusing sound radiators and geometric acoustics principles. The results encourage broader exploration of acoustic array designs inspired from integrated single-degree-of-freedom linkages and origami structures for sake of straightforward array deployment and reconfiguration.

Approximating cylinders with bundle-folding plane-symmetric Bricard linkages

This paper proposes a geometric method via isosceles trapezoids or isosceles trapezoidal prisms (ITP) to approximate a smooth planar curve or a cylinder. The method is then extended for designing double-layer cylinder-approximating deployable mechanisms (CDMs) via ITP mechanisms. A bundle-folding plane-symmetric Bricard linkage, which has an isosceles trapezoidal deployed configuration, is developed. The mappings between geometric parameters and the working configurations are discussed. Kinematics of the isosceles trapezoidal Bricard linkage are analyzed. By linking two sets of isosceles trapezoidal Bricard mechanisms and rectangular Bricard mechanisms under certain geometric conditions, the novel ITP mechanism is synthesized. Two types of ITP mechanisms, which have the same folded and deployed configurations but different intermediate configurations, are obtained. The geometric constraints for constructing general CDMs via ITP mechanisms are presented. The construction method is then validated by three cases of approximating a cylindrical cylinder, a parabolic cylinder, and a sinusoidal cylinder, respectively. The proposed method, which can be applied to all cylinders, provides the geometric guide to construct double-layer CDMs and facilitates engineering applications of the plane-symmetric Bricard linkages.

On New 6r Linkages Based on Generalized Type Iii Bricard Linkage and Bennett Linkage

On New 6r Linkages Based on Generalized Type Iii Bricard Linkage and Bennett Linkage

A 3D-Printable Robotic Gripper Based on Thick Panel Origami.

Origami has been a source of inspiration for the design of robots because it can be easily produced using 2D materials and its motions can be well quantified. However, most applications to date have utilised origami patterns for thin sheet materials with a negligible thickness. If the thickness of the material cannot be neglected, commonly known as the thick panel origami, the creases need to be redesigned. One approach is to place creases either on top or bottom surfaces of a sheet of finite thickness. As a result, spherical linkages in the zero-thickness origami are replaced by spatial linkages in the thick panel one, leading to a reduction in the overall degrees of freedom (DOFs). For instance, a waterbomb pattern for a zero-thickness sheet shows multiple DOFs while its thick panel counterpart has only one DOF, which significantly reduces the complexity of motion control. In this article, we present a robotic gripper derived from a unit that is based on the thick panel six-crease waterbomb origami. Four such units complete the gripper. Kinematically, each unit is a plane-symmetric Bricard linkage, and the gripper can be modelled as an assembly of Bricard linkages, giving it single mobility. A gripper prototype was made using 3D printing technology, and its motion was controlled by a set of tendons tied to a single motor. Detailed kinematic modelling was done, and experiments were carried out to characterise the gripper’s behaviours. The positions of the tips on the gripper, the actuation force on tendons, and the grasping force generated on objects were analysed and measured. The experimental results matched well with the analytical ones, and the repeated tests demonstrate that the concept is viable. Furthermore, we observed that the gripper was also capable of grasping non-symmetrical objects, and such performance is discussed in detail in the paper.

Nonlinear Dynamics Analysis of Flexible Deployable Linkage Mechanisms Using the Finite Particle Method

Deployable mechanisms have notable applications in mechanical engineering, civil engineering, and space technology. Although often ignored, deployable linkage mechanism exhibits additional flexibility beyond rigid folding owing to the deformation of nonrigid components. The actual behavior of flexible deployable linkage usually involves the dynamic effect and geometric nonlinearity. Using the finite particle method (FPM), this study investigated the nonlinear responses of flexible deployable linkage mechanisms. As a particle method, the FPM is displacement based, explicit, and can avoid iterations to solve nonlinear equilibrium equations. It can be used for nonlinear analyses of structures with rigid body motion and infinitesimal mechanisms, and to determine the internal force and structural deformation. To investigate the nonrigid Bennett linkage and Bricard linkage, formulations of a three-dimensional beam element and revolute hinge element were derived for FPM analysis. Agreement between analytical solutions and numerical simulations demonstrated the efficiency of the proposed approach for nonlinear motion analysis of nonrigid mechanisms. The FPM results revealed that mechanism flexibility can cause deviation from rigid compatibility paths, and the internal force and deformation of mechanisms should be considered when designing nonrigid mechanisms.

Thick-panel Origami Cube

This paper presents a new method for constructing a novel thick-panel origami cube. By replacing the equivalent 4R-spherical linkage in a four-crease zero-thickness origami vertex with a plane-symmetric Bricard linkage, the thick-panel form of a general plane-symmetric four-crease origami vertex is identified and constructed. The proposed thick-panel vertex preserves the kinematics of the original four-crease origami vertex. Then, by utilizing the proposed thick-panel vertex, a thick-panel cube is constructed based on the zero-thickness origami cube that was proposed in our previous work. Through mechanism decomposition, geometric constraints and kinematic properties of the corresponding integrated mechanism are investigated and formulated, which reveals the kinematic equivalence between the thick-panel and zero-thickness forms of the origami cube. In addition, a prototype of the proposed thick-panel origami cube is fabricated, verifying its kinematic properties. The proposed technique can be extended to the design and construction of thick-panel polyhedrons with potential applications in the fields such as aerospace exploration, robotics and architecture.

Double-Layer Deployable Mechanical Network Constructed of Threefold-Symmetric Bricard Linkages and Sarrus Linkages

Abstract Threefold-symmetric (TFS) Bricard linkages are known for their excellent deployment performance properties. This paper proposes a novel networking method of TFS Bricard linkages and a double-layer mechanical network. First, the angle relationship for parts of the TFS Bricard linkage is analyzed. Then, the angle relationship of two TFS Bricard linkages connected by a scissor mechanism is studied. The result suggests that when the twist angles of the two TFS Bricard linkages are equal, their corresponding planes are parallel, and the link lengths have no effect on the parallel relationship. A novel networking method of the TFS Bricard linkage is recommended according to these results. This mechanical network is constructed of two different sized units and can be plane deployed and be folded with a smaller height. We also propose a hybrid linkage constructed of the TFS Bricard linkage and Sarrus linkage. Two kinds of double-layer mechanical networks are suggested by applying the hybrid linkage to a smaller unit in the mechanical network and using the hybrid linkage as the interlayer pillar. The new networking method and the double-layer mechanical network provide convenience for the TFS Bricard linkage's engineering application.

Reconfigurability of the origami-inspired integrated 8R kinematotropic metamorphic mechanism and its evolved 6R and 4R mechanisms

Origami engineering as an interdisciplinary subject brings thriving progress of mechanisms innovation. In this paper, a construction approach from origami to multiple spherical-integrated mechanisms is proposed and subtly sets up the bridge between origami and Bennett linkage, Bricard linkage. Subsequently, the attention is drawn to a special origami-inspired integrated 8R kinematotropic metamorphic mechanism. The paper applies screw theory to illustrating the reconfigurability of singular configurations with distinct parametric constraints in the 8R mechanism. Two overconstrained linkages, a 4R linkage and a 6R linkage, are evolved under peculiar geometrical constraints respectively. Furthermore, the kaleidocycle and deployable properties of the 8R mechanism are revealed. The paper hence not only presents an interesting work extracted from origami but also sheds light on the promising investigation about the intersection of distinct types of reconfigurable mechanisms. Finally, by using the kinematotropic metamorphic 8R mechanism as the reconfigurable trunk, a novel quadruped robot is built and its characteristics will be investigated in future research.

A Variable-Dimension Overtube for Natural Orifice Transluminal Endoscopic Surgery

In natural orifice transluminal endoscopic surgery, the insertion of surgical instruments can cause patient discomfort, even bleeding and perforation. This paper proposes a design of an overtube which can vary its radial dimension. It can be inserted through the natural orifice in a small diameter configuration and then enlarge its size and form a hollow and rigid channel to allow passing instruments, while to reduce the friction between the instruments and the surrounding tissue. A Bricard linkage is used as the basic unit of the overtube and a mechanism is designed to drive multiple Bricard units to fold or unfold. Optimization of the design parameters has been studied and comparison experiments have been completed to verify the feasibility of the design on reducing the shear forces and normal forces exerted during surgical instrument insertion. The ex vivo experiment is also completed to prove the protection of the overtube on the esophagus during the insertion.

Mobility Analysis of the Threefold-Symmetric Bricard Linkage and Its Network

Abstract In this study, the mobility of the threefold-symmetric Bricard linkage and its network are analyzed using screw theory. First, the screw motion equation of the linkage is derived. By applying the modified Grübler–Kutzbach criterion, we deduce that the degree of freedom (DOF) of the linkage is equal to 1. Then, we analyze the mechanical network constructed of threefold-symmetric Bricard linkages and provide its topological constraint graph. Using graph theory and screw theory, the constraint matrix of the mechanical network is obtained. Then, we solve the matrix rank via linear column transformation. Results show that the DOF of the mechanical network is equal to 1. The mobility analysis method of the mechanical network proposed in this study facilitates the solution of the constraint matrix rank and can be used as a reference for other mechanical networks constructed of single-loop linkages.