Reconfigurable and Deployable Platonic Mechanisms with a Variable Revolute Joint

Abstract

This chapter presents for the first time a variable revolute joint and a group of reconfigurable and deployable Platonic mechanisms. Structure of the variable revolute joint is presented and demonstrated by its application to the construction of a reconfigurable generic 4R linkage which is capable of converting itself to a planar parallelogram 4R linkage, a spherical 4R linkage and a Bennett linkage. Then, with a two-phase variable revolute joint, a group of reconfigurable and deployable Platonic mechanisms are constructed and mobility of the proposed reconfigurable Platonic mechanisms is investigated by formulating their corresponding constraint matrices. Finally, kinematic characteristics of the proposed mechanisms are illustrated.

1 Introduction

Deployable polyhedral mechanisms (DPMs) have been raising interest from kinematicians and mathematicians since the pioneering work of Bricard on flexible polyhedrons [1] and of Verheyen [11] on the expandable polyhedral structures coined as “Jitterbug transformers”. Wohlhart proposed different synthesis methods leading to the generation of various deployable polyhedral mechanisms including the regular polyhedral linkages [15] which are directly related to the mechanisms proposed in this chapter. Kiper et al. [9], and Wei and Dai [14] revisited Wohlhart’s work and synthesized the polyhedral mechanisms with new approaches, and Röschel [10] investigated polyhedral mechanisms from the geometric point of view giving insight into the intrinsic geometric properties of the deployable polyhedral mechanisms. Reconfigurable mechanical systems including metamorphic mechanisms [3] satisfy the ever-growing market demands of adapting for various stipulations by changing topological configurations. In order to construct multifunctional mechanisms that are capable of changing mobility, mechanism topology and function through the metamorphic process to adapt themselves for different tasks and working environment without disassembling the mechanisms, Dai and Rees Jones [4] presented the concept of mechanism metamorphosis, Yan and Kuo [16] investigated variable topology mechanisms and kinematics pairs, Gan et al. [7] invented a reconfigurable Hooke (rT) joint, and Zhang et al. [17] proposed a variable axis (vA) joint.

In this chapter, a variable revolute (vR) joint is for the first time proposed which leads to the construction of reconfigurable mechanisms including a group of reconfigurable and deployable Platonic mechanisms.

2 A Variable Revolute Joint and a Reconfigurable Generic 4R Linkage

2.1 Structure of a Variable Revolute (vR) Joint

In the traditional mechanism design, a revolute joint is frequently employed to connect two links providing one degree of freedom relative motion and once a revolute joint is installed, direction of the axis of rotation as well as the corresponding structure parameters (e.g. D-H parameters [6]) between the two links are determined. However, in order to use revolute joints to construct reconfigurable/metamorphic mechanisms [3] having capability of changing topology structures and functions in different working stages, it is expected that the directions of their joint axes can be altered. Thus, inspired by the development of the reconfigurable Hooke joint (rT) [7] and the variable-axis (vA) joint [17], in this chapter, a variable revolute joint is invented and designed as illustrated in Fig. 1. It is denoted as vR joint, where R stands for a revolute joint and v stands for variable indicating that the orientation of the revolute joint is changeable. The joint consists of a reconfigurable connector (rC) which is rigidly attached to link i and contains a groove to accommodate a axis-variable revolute joint. As shown in Fig. 1a, the capacity of changing orientation of the joint axis is realized by adjusting the axis-variable revolute joint about the groove and connecting link j such that the relative geometric configuration between link i and link j can be consequently altered. After changing the direction of the revolute joint axis to a desired orientation, the joint is fixed by bolting it to the reconfigurable connector. Figure 1a illustrates one type of vR joint with the revolute joint embedded inside the groove, and Fig. 1b shows its variant in which the revolute joint is placed outside of the groove. Further, Fig. 1c gives the topological schematic diagram of the vR joint.

Fig. 1

Variable revolute (vR) joint and its variants

Connecting two links by a variable revolute (vR) joint, the two links can have various relative geometric configurations and thus variable relative structure parameters. Therefore, if a vR joint is used to replace the traditional R joint in a linkage, the linkage readily becomes a reconfigurable linkage that can change itself from one mechanism type to the other having different motion properties and functions.

2.2 A Reconfigurable Generic 4R Linkage with a Three-Phase Variable Revolute Joint

Figure 2a gives a conventional spherical 4R linkage with its four revolute joints arranged at the four corners of a square and their joint axes intersecting at a common point V. As shown in Fig. 2b, by replacing all R joints of the original spherical 4R linkage with vR joints, a reconfigurable spherical 4R mechanism can be obtained.

In the evolved reconfigurable spherical 4R linkage, the variable revolute (vR) joints are connected by reconfigurable connectors (rCs) which are rigidly attached to the links (Herein, for the sake of clarity, the detailed structure of the reconfigurable connectors is not illustrated in the figure, readers can refer to Fig. 1b for it). The reconfigurable connectors have the functions of releasing and locking the vR joints such that directions of the joint axes can be adjusted so as to reconfigure the types of linkages.

Fig. 2

A reconfigurable general 4R linkage. a A spherical 4R linkage. b Reconfiguration of vR joint from PI to PII and from PII to PIII. c A reconfigurable spherical 4R linkage. d A reconfigurable Bennett linkage (Note rC stands for reconfigurable connector. PI, PII and PIII stand respectively for phase I, phase II and phase III)

In the reconfigurable spherical 4R linkage, since four vR joints are distributed at four corners of a square formed by points A, B, C and D, and axes of the four joints intersect at a common point V, in the configuration shown in Fig. 2b, the angle between the axis of any joint and a line passing through point V and point O (which is the centre point of the square) equals $\gamma$. Axes of joints A and C lie in a same vertical plane, and axes of joint B and D forms another same plane $\pi$ (see Fig. 2b).

We keep the directions of the axes of joints A and C unchanged and as shown in Fig. 2c, release reconfigurable connectors at joints B and D, and adjust the axis directions of vR joints B and D by rotating them respectively about axes ${{\varvec{e}}}_\mathrm{{B}}$ and ${{\varvec{e}}}_\mathrm{{D}}$ by angle $\alpha$ (Where axes ${{\varvec{e}}}_\mathrm{{B}}$ and ${{\varvec{e}}}_\mathrm{{D}}$ are aligned with central axes of the grooves embedded in their corresponding rCs passing through points B and D respectively, and are perpendicular to plane $\pi$.), then fix the vR joints B and D when they reach phase II. In such a configuration as shown in Fig. 2d, the axes of joints A and C intersect at point V$_1$, and the axes of joints B and D intersect at point V$_2$, points V$_1$, V$_2$ and O are collinear, and the angle between axis of any joint and a line passing through V$_1$, V$_2$ and O remains $\gamma$. In such a case, the reconfigured linkage turns out to be a Bennett linkage [8].

Further, releasing all the vR joints and adjusting their axis directions to form a configuration that axes of all the four joints are parallel to each other, i.e. each joint reaches phase III as shown in Fig. 2c, then fastening the reconfigurable connectors (rCs) so as to fix the vR joints, a planar parallelogram 4R linkage can be obtained and in this case the angle $\gamma$ equals 0.

In the stage of Bennett linkage, mobility of the linkage can be calculated and verified by computing the dimension of nullity of the constraint matrix [12] of the linkages as

$$\begin{aligned} m=\mathrm{{dim}}\left( N\left( \mathbf{{M}}_c\right) \right) . \end{aligned}$$

(1)

Where the constraint matrix of the linkage according to its associated graph can be formulated as

$$\begin{aligned} \mathbf{{M}}_c=\left[ \begin{array}{cccc} {{\varvec{S}}}_A&{{\varvec{S}}}_B&{{\varvec{S}}}_C&{{\varvec{S}}}_D \end{array} \right] , \end{aligned}$$

(2)

with screws [5] for the joint axes ${{\varvec{S}}}_A$, ${{\varvec{S}}}_B$, ${{\varvec{S}}}_C$ and ${{\varvec{S}}}_D$ derived according to Fig. 2d as

$$\begin{aligned} \left\{ \begin{array}{llll} {{\varvec{S}}}_A=\left[ \frac{\sqrt{2}}{2}\cos \gamma \,\,\frac{\sqrt{2}}{2}\cos \gamma \,\,\sin \gamma \,\, -a\sin \gamma \,\, a\sin \gamma \,\, 0\right] ^T\\ {{\varvec{S}}}_B=\left[ \frac{\sqrt{2}}{2}\cos \gamma \,\, -\frac{\sqrt{2}}{2}\cos \gamma \,\, \sin \gamma \,\,-a\sin \gamma \,\, -a\sin \gamma \,\, 0\right] ^T\\ {{\varvec{S}}}_C=\left[ -\frac{\sqrt{2}}{2}\cos \gamma \,\, -\frac{\sqrt{2}}{2}\cos \gamma \,\, \sin \gamma \,\, a\sin \gamma \,\, -a\sin \gamma \,\, 0\right] ^T\\ {{\varvec{S}}}_D=\left[ -\frac{\sqrt{2}}{2}\cos \gamma \,\, \frac{\sqrt{2}}{2}\cos \gamma \,\, \sin \gamma \,\, a\sin \gamma \,\, a\sin \gamma \,\, 0\right] ^T \end{array}\right. . \end{aligned}$$

(3)

Where $a$ denotes the length of a link in the linkage.

Substituting Eq. (2) into Eq. (1) with joint screws provided in Eq. (3) gives $m=1$, which indicates that the linkage at Bennett linkage type has one mobility.

Therefore, this example shows that by integrating variable revolute (vR) joints into a conventional spherical 4R linkage, a reconfigurable generic 4R linkage can be generated which is capable of transforming itself into a planar 4R linkage, a spherical 4R linkage and a Bennett Linkage by adjusting the directions of the joint axes through the reconfigurable connectors (rCs). In order to precisely place axes of the joints in the correct directions, three slots can be fabricated in the reconfigurable connectors which clearly define three phases for each vR joint. In such a way, starting from the spherical 4R linkage configuration shown in Fig. 2b, keeping either pair of joints B and D, or joints A and C unchanged, and adjusting the other pair from phase I to phase II as shown in Fig. 2c, the linkage transforms from a spherical 4R linkage into a Bennett linkage, and vice versa. If all joints are locked at phase III, the linkage turns out to be a planar parallelogram 4R linkage.

As a result, this section demonstrates that the proposed vR joint can be used to replace the traditional R joint leading to reconfigurable linkages/mechanisms which can be converted into linkages/mechanisms of different types that perform diverse motions and functions. In Sect. 3, it shows that the vR joint can lead to the construction of a group of reconfigurable and deployable Platonic mechanisms.

3 Construction of Reconfigurable and Deployable Platonic Mechanisms with a Two-Phase vR Joint

In the previous work, based on a dual-plane-symmetric spatial eight-bar linkage [14] and an overconstrained [2] spatial eight-bar linkage [13], the authors of this chapter synthesized a group of deployable Platonic mechanisms with radially reciprocating motion as well as a group of Fulleroid-like deployable Platonic mechanisms. All the mechanisms are constructed by purely using R joints to connect the links. Figure 3 shows examples of the two types of Platonic mechanisms, i.e. a deployable dodecahedral mechanism with radially reciprocating motion and a Fulleroid-like deployable dodecahedral mechanism. As indicated in Fig. 3a, for the deployable dodecahedral mechanism with radially reciprocating motion, in each pentagonal facet all R joints are parallel to the facet, while, for the Fulleroid-like dodecahedral mechanism (see Fig. 3b), in each pentagonal facet all R joints are perpendicular to the facet.

Fig. 3

Two different types of deployable dodecahedral mechanisms. a A deployable dodecahedral mechanism with radially reciprocating motion. b A Fulleroid-like deployable dodecahedral mechanism

In this chapter, it is found that by replacing all R joints in either type of aforementioned deployable Platonic mechanisms with two-phase vR joints, a group of reconfigurable and deployable Platonic mechanisms can be constructed. The mechanisms are capable of transforming themselves from one type to the other executing motions and functions possessed by both types of aforementioned deployable Platonic mechanisms.

Fig. 4

A reconfigurable and deployable tetrahedral mechanism. a In Fulleroid-like linkage type. b vR joints in phase I configuration. c vR joints in phase II configuration. d In star-transformer linkage type

Figure 4 shows a reconfigurable and deployable tetrahedral mechanism which is constructed by using a two-phase vR joint. In the mechanism, both ends of each link are connected by vR joints which are mounted in two reconfigurable connectors (rCs), one rC is rigidly embedded in the facet component and the other rC is rigidly fixed in the vertex component. Figure 4a shows the mechanism in a Fulleroid-like deployable tetrahedral mechanism type, in which the vR joints are placed in their phase I configurations such that all joint axes are perpendicular to their corresponding facet as shown in Fig. 4b. Then, moving the mechanism to the configuration that the centre axes of the two rCs at both ends of each link are collinear (see Fig. 4b) and adjusting all the vR joints from phase I to phase II configurations such that all joint axes are parallel to their corresponding facet as shown in Fig. 4c, subsequently the mechanism turns out to be in another type which is capable of performing radially reciprocating motion.

Hence, by integrating the variable revolute (vR) joints into a deployable tetrahedral mechanism, the mechanism becomes a reconfigurable one which has the capacity of converting itself from one type to another and performing diverse functions. For the Fulleroid-like tetrahedral mechanism in Fig. 4a, all facet components perform screw motions around the axes that are perpendicular to their corresponding facets and pass centroid point O, and all vertex components execute radially reciprocating motions along the axes passing through their corresponding vertexes and point O. However, for the other type of tetrahedral mechanism shown in Fig. 4d, all components carry out radially reciprocating motions along the aforementioned axes towards/outwards the centroid point O forming star-like tetrahedron which is one of the so called polyhedral star-transformers [15].

Moreover, by applying the two-phase vR joints to the deployable Platonic mechanisms developed by Wohlhart [15] and Wei and Dai [13, 14], a group of reconfigurable and deployable Platonic mechanisms can be constructed as illustrated in Fig. 5. All the reconfigurable and deployable Platonic mechanisms can transform themselves without disassembly from a Fulleroid-like mechanism type (see left column in Fig. 5) to a star-transformer mechanism type (see right column in Fig. 5) or vice versa.

Fig. 5

A group of reconfigurable and deployable Platonic mechanisms. a A reconfigurable and deployable hexahedral mechanism and its two types. b A reconfigurable and deployable octahedral mechanism and its two types. c A reconfigurable and deployable dodecahedral mechanism and its two types. d A reconfigurable and deployable icosahedral mechanism and its two types

4 Mobility and Kinematics of the Reconfigurable Platonic Mechanisms

For the reconfigurable Platonic mechanisms proposed in this chapter, when the mechanisms are in the star-transformer type, mobility of the mechanisms was verified by the present authors in Ref. [12] and kinematics of the mechanisms was indicated in Ref. [14]. When the reconfigurable mechanisms are in the Fulleroid -like type, mobility and kinematics of the mechanisms can be similarly formulated and analysed.

By following the mobility analysis of star-transformer type mechanisms [12], referring to the coordinate systems provided in Fig. 6 and the joint screws in individual facet presented in Fig. 7a, constraint matrix $\mathbf M '_c$ of the Fulleriod-like tetrahedral mechanism can be formulated with the joint screws in the individual facet being modified as

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\varvec{S}}}_{i1}=\left[ 0\ 0\ 1\ b\ 0\ 0\right] ^T,\ {{\varvec{S}}}_{i2}=\left[ 0\ 0\ 1\ -b/2\ \sqrt{3}b/2\ 0\right] ^T\\ {{\varvec{S}}}_{i3}=\left[ 0\ 0\ 1\ -b/2\ -\sqrt{3}b/2\ 0\right] ^T,\ {{\varvec{S}}}'_{i1}=\left[ 0\ 0\ 1\ b+l\cos \beta \ -l\sin \beta \ 0\right] ^T,\\ {{\varvec{S}}}'_{i2}=\left[ 0\ 0\ 1\ -b/2-l(\cos \beta -\sqrt{3}\sin \beta )/2\ \sqrt{3}b/2+l(\sqrt{3}\cos \beta +\sin \beta )/2\ 0\right] ^T,\\ {{\varvec{S}}}'_{i3}=\left[ 0\ 0\ 1\ -b/2-l(\cos \beta +\sqrt{3}\sin \beta )/2\ -\sqrt{3}b/2-l(\sqrt{3}\cos \beta -\sin \beta )/2\ 0\right] ^T. \end{array}\right. } \end{aligned}$$

(4)

Mobility of the mechanism can then be obtained as $m=\mathrm{{dim}}(N(\mathbf{{M}}'_c))=1$.

Fig. 6

Geometry and coordinate systems of a Fulleroid-like tetrahedral mechanism. a Initial configuration and coordinate systems. b A random configuration and coordinate systems

Fig. 7

Joint screws in individual facet and kinematics of the mechanism. a Geometry and joint screws in individual facet. b Trajectories of the facet and vertex components

Further, using the coordinate systems established in Fig. 6, kinematic analysis of the Fulleroid-like deployable tetrahedral mechanism can be carried out leading to the illustration of kinematic characteristics of the mechanism as shown in Fig. 7b. Figure 7b indicates that the Fulleroid-like tetrahedral mechanism has such a property that the facet components execute screw motions around the axes that are perpendicular to their corresponding facets and pass centroid point O, and vertex components execute radially reciprocating motions along the axes passing through their corresponding vertexes and point O. Trajectories of the motions are illustrated in Fig. 7b.

The above mobility and kinematic analysis method can then be extended to the whole group of reconfigurable Platonic mechanisms when they work in the Fulleroid-like mechanism type. Integrating these analysis with those investigations of mobility and kinematics for the star-transformer type mechanisms in [12, 14], kinematic properties of all the reconfigurable and deployable Platonic mechanisms can be fully characterized.

5 Conclusions

In this chapter, a variable revolute (vR) joint was for the first time proposed and its structure was presented. Using a three-phase vR joint, a reconfigurable generic 4R linkage was developed verifying the application of the proposed vR joint. The generic 4R linkage can transform itself into a planar 4R linkage, a spherical 4R linkage and a Bennett linkage. Subsequently, with a two-phase variable revolute joint, a group of reconfigurable and deployable Platonic mechanisms were constructed which have the capacity of converting themselves from the Fulleroid-like linkage type to the star-transformer linkage type performing screw motion integrated with radially reciprocating motion or pure radially reciprocating motion. Mobility of the group of reconfigurable Platonic mechanisms was further investigated and kinematics properties of the mechanisms were characterized with numerical simulation.