Spherical Linkages Research Articles

Design, Analysis and Experiment of a Modular Deployable Continuum Robot

Continuum robots, possessing great flexibility, can accomplish tasks in complex work scenes, regarded as an important direction in robotics. However, the current continuum robots are not satisfying enough in terms of fabrication and maintenance, and their workspace is limited by structure and other aspects. In this paper, to address the above problems, a modular deployable robot, which adopts an origami structure instead of a flexible hinge, is proposed. A fabrication method is innovated, the Spherical Linkage Parallel Mechanism (SLPM) unit is optimized, and the installation and fabrication process of the robot is simplified through modularization. The forward kinematics and inverse kinematics of the robot and its workspace are analyzed by using the screw theory. The prototype of the robot is constructed, and its folding performance, bending performance, and motion accuracy are tested, and the error analysis and compensation optimization are carried out. After the optimization, the position error of the robot is reduced by about 65%, and the standard deviation is greatly lowered, which effectively improves the motion accuracy and stability of the robot.

Morphable thick-panel origami

Beyond being a traditional art form, origami can be applied to create complex deployable systems. Origami structures with one degree of freedom (DOF) are easy to actuate and have high stiffness but lack the ability to achieve different tasks. Previous multiple DOF origami structures are the opposite, having good task generalizability but low stiffness. In this paper, we design morphable origami structures with multiple one-DOF motion branches by using reconfigurable linkages. However, the kinematic models of previous thick-panel techniques are spherical linkages and overconstrained linkages, which limit the ability of structure morphing. This paper first proposes an approach to create reconfigurable linkages by connecting two one-DOF linkages and removing common parts. The new linkages have two motion branches corresponding to the two original linkages, and each motion branch is one-DOF. Then, we use the relationships between these reconfigurable linkages and thick-panel origami structures to construct morphable thick-panel origami structures. Furthermore, we also analyze their kinematics and motion branches. The concept of this paper, using reconfigurable linkages to design thick-panel origami structures, will improve the potential of origami techniques for different tasks.

Kinematic behaviour of origami combining two four-crease vertices

The simplest rigidly foldable single-vertex origami structure is a four-crease origami (single 4C). It is a 1 DOF spherical linkage which possesses geometric properties that allow its copies to be combined into more complex rigidly foldable origami such as the Miura-ori. Merging two identical flat-foldable single 4Cs together produces a new construction, called the double 4C. We discover that this composite structure has some unique kinematic properties. Apart from being rigidly foldable and flat-foldable with 1 DOF, both single 4Cs in the double 4C are always in identical folded states. Moreover, it actually couples some panels together to form scissor-like mechanisms without physically crossing the panels. The double 4C can also have its kinematically equivalent thick-panel counterpart. By dissecting the double 4C and its thick panel counterpart, we find the constituent mechanisms forming them. We uncover how the constituent mechanisms evolve. Since the double 4C are fundamental units in many stackable tessellation origami structures, our kinematic analysis gives insight into the kinematic behaviour of these structures. Not only can the findings be used to guide the creation of thick panel counterparts of the stackable origami, but it may also enable the creation of new origami structures with single DOF by mimicking the vast number of deployable structures that contain scissor-like elements.

A flat-foldable equiangular spiral folding pattern inspired by sunflowers for deployable structures

Flasher origami pattern has been widely utilized to improve the stowage efficiency of deployable structures. Nevertheless, flasher origami cannot be folded fully flat, and they still have great potential for optimization in terms of storage volume and folding creases. In this paper, a flat foldable equiangular spiral folding pattern inspired by the sunflower disk is introduced. Then, a parametric design method for this equiangular spiral crease diagram is introduced in detail. Subsequently, a kinematic model of the equiangular spiral folding pattern is established based on the kinematic equivalence between rigid origami and spherical linkages. A simulation of the developed model demonstrates that the equiangular spiral folding pattern can be folded flat. Using the folded ratio as an evaluation index, the calculated results and experiments show that the equiangular spiral crease pattern can yield fewer creases and improve stowage efficiency in comparison to flasher origami pattern. Equiangular spiral folding pattern can save a considerable amount of space and provide a new approach to spatially deployable structures.

Design and development of a SLPM-based deployable robot

AbstractWith the progress of industry, people are facing more and more complicated tasks, which cannot be completed by conventional rigid robot. For this, a deployable robot based on spherical linkage parallel mechanism was proposed to satisfy relevant requirements for the degrees of freedom in this study. Based on the design of robot model and its control box, a mathematical model for the robot was established, and the relationship between motion space and drive space was deduced accordingly. Subsequently, a control system consisting of the upper and lower computers was introduced. Two control modes, that is, Joystick control and remote control, were developed. The upper computer control interface for the robot was completed by MATLAB construction. At last, the two control modes as well as autonomous detection were demonstrated by motion test. This achievement will further advance the applications of deployable robot in job aid and intelligent exploration.

Dimensional synthesis of a spherical linkage crank slider mechanism for motion generation using an optimization algorithm

Abstract. In the present study, Fourier theory is applied to establish the expression of rigid-body poses of a spherical four-bar crank slider rigid-body guidance mechanism. According to an analysis of the harmonic components of the trajectory curve and rigid-body rotation angle, it has a certain relationship with the geometric parameters of the mechanism. On this basis, the rigid-body poses are normalized by preprocessing. Then, the rotation angle of the curve around the y axis and z axis is determined, respectively. The theoretical formulas used for calculating the real sizes and the installation position parameters of the desired spherical four-bar crank slider rigid-body guidance mechanism are established. Besides this, a genetic optimization algorithm and theoretical formulas are applied to solve the dimensional synthesis of motion generation for the spherical four-bar crank slider mechanism. The effectiveness of the proposed method is illustrated by an example. The maximum Euclidean distance error of the rigid-body position of the results with the highest similarity is 0.0086, and the average Euclidean distance error is 0.0044. The maximum error of the rigid-body orientation is 0.0179, and the average error is 0.0065.

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.

Kinematic and mechanical error analysis of serial spherical linkages using a modular approach

Although spherical mechanisms have been widely implemented in medical and industrial fields, there still lacks a simple and systematic approach for analyzing their kinematics and mechanical errors. Therefore, the purpose of this paper is to provide a modular approach for kinematic and mechanical error analysis of serial spherical mechanisms that can be decomposed into predefined modules. Firstly, the forward and inverse kinematics, described by rotation matrices, of the spherical serial manipulator are derived. Besides, the error propagations of the spherical serial manipulator are formulated based on the derived kinematic equations. The derived algorithms are programed into subroutines to analyze the kinematics and mechanical error of a serial spherical linkage by analyzing each of its substructures as the illustrative examples have shown. Until now, the presented module in this paper is applicable to a variety of serial spherical mechanisms, and the feasibility of this method adopting high-class modules will be explored in the future.

Algebraic Method for Exact Synthesis of One-Degree-of-Freedom Linkages With Arbitrarily Prescribed Constant Velocity Ratios

Abstract This paper addresses the synthesis of one degree-of-freedom (1DOF) linkages that can exactly transmit angular motion between coplanar axes (i.e., parallel axes or intersectant axes) with arbitrarily prescribed constant velocity ratios. According to motion polynomials over dual quaternions and pure rolling models between two circles, an algebraic approach is presented to precisely synthesize new 1DOF linkages with arbitrarily prescribed constant velocity ratios. The approach includes four steps: (a) formulate a characteristic curve occurred by the pure rolling, (b) compute the motion polynomial of the minimal degree that can generate the curve, (c) deal with the factorization of the motion polynomial to construct an open chain, and (d) convert the open chain to a 1DOF linkage. Using this approach, several 1DOF planar, spherical, and spatial linkages for angular motion transmission between parallel axes or intersectant ones are constructed by designating various velocity ratios. Taking the planar and spherical linkages with a constant 1:2 velocity ratio as examples, kinematics analysis is implemented to prove their motion characteristics. The result shows that the generated linkages indeed can transmit angular motion between two coplanar axes with constant velocity ratios. Meanwhile, three-dimensional (3D)-printed prototypes of these linkages also demonstrate such a conclusion. This work provides a framework for synthesizing linkages that have great application potential to transmit motion in robotic systems that require low inertia to achieve reciprocating motion with high speed and accuracy.

Design of Single Degree-of-Freedom Triangular Resch Patterns with Thick-panel Origami

As a series of tessellation origami patterns consisting of more than one type of polygons, Resch patterns are generally rigid foldable but with a large number of degrees of freedom (DOFs). The patterns are not flat foldable, instead the fully folded configuration is a sandwich style in flat profile. In order to achieve one-DOF forms of triangular Resch pattern units, the thick-panel technique is employed at vertices to replace spherical linkages with spatial linkages. The compatibility among all the vertices is studied by a systematic kinematic analysis of the thick-panel Resch pattern. Finally, two design schemes are obtained to form a one-DOF origami structure between the initial planar structure and the fully folded 3D sandwich structure with dome-curved intermediate folded configurations. The one-DOF Resch pattern structure can be used to generate reliable shape transformation.

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.

Clearance-Induced Orientation Uncertainty of Spherical Linkages

Abstract This paper proposes a kinematic model to evaluate the orientation uncertainty range of spherical linkages caused by the joint clearances. Based on the concepts of imaginary clearance link, spherical N-bar rotatability laws, and the invariant link rotatability, the uncertainty of the output angle can be treated as a mobility problem. The uncertainty region of the end-effector is treated as a workspace problem for the remodeled linkage. This paper highlights the orientation error by isolating the kinematic effects of joint clearance from other error factors. The discussion is carried out through spherical four-bar linkages and five-bar linkages. Numeric examples are presented to demonstrate the uncertainty range of the output angle and the uncertainty region of the end-effector. The result shows that, in the worst case, the error of each joint clearance will be magnified in a closed-loop structure compared with linearly adding all the clearance error. This implies that from a kinematics point of view, closed-loop spherical linkages or parallel manipulators will lead to a greater deviation on the end-effector than their open-loop counterparts. Using more passive joints in the manipulator may result in bigger error possibilities.

A Miniature Flapping Mechanism Using an Origami-Based Spherical Six-Bar Pattern

In this paper, we suggest a novel transmission for the DC motor-based flapping-wing micro aerial vehicles (FWMAVs). Most DC motor-based FWMAVs employ linkage structures, such as a crank-rocker or a crank-slider, which are designed to transmit the motor’s rotating motion to the wing’s flapping motion. These transmitting linkages have shown successful performance; however, they entail the possibility of mechanical wear originating from the friction between relative moving components and require an onerous assembly process owing to several tiny components. To reduce the assembly process and wear problems, we present a geometrically constrained and origami-based spherical six-bar linkage. The origami-based fabrication method reduces the number of the relative moving components by replacing rigid links and pin joints with facets and folding joints, which shortens the assembly process and reduces friction between components. The constrained spherical six-bar linkage enables us to change the motor’s rotating motion to the linear reciprocating motion. Due to the property that every axis passes through a single central point, the motor’s rotating motion is filtered at the spherical linkage and does not transfer to the flapping wing. Only linear motion, therefore, is passed to the flapping wing. To show the feasibility of the idea, a prototype is fabricated and analyzed by measuring the flapping angle, the wing rotation angle and the thrust.

Origami cubes with one-DOF rigid and flat foldability

Rigid origami is a branch of origami with great potential in engineering applications to deal with rigid-panel folding. One of the challenges is to compactly fold flat a polyhedron made from rigid facets with a single degree of freedom. In this paper, we present a new method to design origami cubes with three fundamental characteristics: rigid foldability, flat foldability and one degree of freedom (DOF). A total of four cases of crease patterns that enable origami cubes with distinct folding performances have been proposed, which covers all possible layouts of the diagonal creases on the square facets of origami cubes. The folding utilises spherical linkages formed by intersecting creases, and based on the kinematic equivalence between the rigid origami and the spherical linkages, the corresponding spherical linkage loops are introduced and analysed to reveal the motion properties of the origami cubes. The newly found method can be readily utilized to design deployable structures for various engineering applications including cube-shaped cartons, small satellites, containers, etc.

A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages

This paper presents a screw theory approach for the computation of the instantaneous rotation axes of indeterminate spherical linkages. Since the last part of the XIX century, the determination of the instantaneous rotation, or velocity, centers of planar mechanisms has been studied using the Aronhold-Kennedy theorem, extended to the spatial case until the second part of last century. In the beginning of the XX century, it was found that there were planar mechanisms for which the application of the Aronhold-Kennedy theorem was unable to find all the instantaneous rotation centers; these mechanisms where denominated complex or indeterminate. The beginning of this century saw a renewed interest on the complex or indeterminate mechanisms, both planar and spherical; and very soon, taken advantage of the results obtained for planar mechanisms, there were several methods for the determination of the indeterminate instantaneous rotation axes. The new screw theory approach, presented here, provides a more straightforward method for setting up the equations; furthermore, the algebraic equations to be solved are simpler than the ones published up to date. In addition, the method is part of a comprehensive method for the determination of the instantaneous screw axes of spatial mechanisms or their many special cases. The method is based on a systematic application of screw theory, which is isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).

Using a Point-Line-Plane Representation for Unified Simulation of Planar and Spherical Mechanisms

Abstract This paper presents a geometric constraints driven approach to unified kinematic simulation of n-bar planar and spherical linkage mechanisms consisting of both revolute and prismatic joints. Generalized constraint equations using point, line, and plane coordinates have been proposed which unify simulation of planar and spherical linkages and are demonstrably scalable to spatial mechanisms. As opposed to some of the existing approaches, which seek to derive loop-closure equations for each type of mechanism separately, we have shown that the simulation can be made simpler and more efficient by using unified version of the geometric constraints on joints and links. This is facilitated using homogeneous coordinates and constraints on geometric primitives, such as point, line, and plane. Furthermore, the approach enables simpler programming, real-time computation, and ability to handle any type of planar and spherical mechanism. This work facilitates creation of practical and intuitive design tools for mechanism designers.

Optimization of a Spherical Decoupled Mechanism for Neuro-Endoscopy Based on Experimental Kinematic Data

ABSTRACTThe neuro-endoscopy is a surgical technique that allows the neurosurgeon to maintain a visual contact while operating inside the brain of a patient. A special instrument called the neuro-endoscope is inserted in the brain until the neurosurgeon reaches his/her target. Its manipulation requires a high level of training for neurosurgeons. To enforce both quality and safety of neuro-endoscopy, we propose a robotic manipulator based on a Spherical Decoupled Mechanism. This mechanical architecture has been modified from a 5-Bar Spherical Linkages and adapted to this medical application. It is able to generate a Remote Center of Motion of 2 Degrees of Freedom. It merges the advantages of parallel mechanisms with the kinematic and control simplicity of decoupled mechanisms, while having a very simple architecture. Motion capture experiments using a brain simulation model have been performed with a team of neurosurgeons to obtain the kinematic data of the neuro-endoscope during brain exploration. Based on the identified workspace, the mechanism has been optimized using kinematic performance and architectural compactness as criteria. An optimum mechanism has been selected, showing better kinematic performances than the original 5-bar spherical linkage mechanism.

On rigid origami I: piecewise-planar paper with straight-line creases

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

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.

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.