Research on the Type Synthesis of a Regular Hexagonal Prism Rubik’s Cube Mechanism

Abstract

The Rubik’s Cube mechanism (RCM) is a kind of reconfigurable mechanism with multiple characteristics such as multiple configurations, variable topology, strong coupling, and reconfigurability. Crossover research on the RCM with mathematics, chemistry, cryptography, and other disciplines has led to important breakthroughs and progress. It is obvious that the invention and creation of a new RCM can provide important ideological inspiration and theoretical guidance for the accelerated iterative updating of Rubik’s Cube products and the expansion of their applications. This paper investigates the type synthesis method for a regular hexagonal prism (RHP) RCM (RHPRCM). Through analysis of the reconfigurable movement process of the RCM, two mechanism factors are abstracted, a type synthesis process for the RHPRCM is proposed, a symmetry layout method for the RCM’s revolute axis based on the RHP space polyhedron is proposed, and an analysis method for the intersection of the revolute pair contact surfaces (RPCSs) based on the adjacency matrix is proposed. Taking a revolute axis passing through the center of an RHP and having only one RPCS for each revolute axis as an example, an RHPRCM with different topological structures is synthesized. The relevant research in this paper can provide methodological guidance for the synthesis of other spatial RCMs.

1. Introduction

The Rubik’s Cube, a complex mechanism characterized by strong coupling and variable topology, was invented by Professor Ernő Rubik in 1974. This three-dimensional puzzle has garnered widespread public attention due to its compact structure and its ability to achieve a vast number of combinations through simple rotational operations. As understanding of the Rubik’s Cube has deepened over time, researchers have discovered additional characteristics, such as continuous rotation and spatial transformation, rearrangement and reorganization, and rotation and circulation. Subsequently, various products leveraging these unique features of the Rubik’s Cube have been invented, including the Rubik’s Cube Satellite [1], Rubik’s Cube Lock [2], Rubik’s Cube Microfluid [3], AIE (aggregation-induced emission) Rubik’s Cube Hydroge [4], among others. This diverse array of inventions underscores the wide-ranging applications and significant impact of the Rubik’s Cube.

In the realm of mathematics, researchers have shown significant interest in the inherent symmetry of the Rubik’s Cube. Zassenhaus [5] conducted an analysis focusing on group theory aspects, such as subgroups, generators, and orders within the Rubik’s Cube. Joyner [6] delved into mathematical intricacies involving group theory and the symmetry inherent in the Rubik’s Cube. Capitalizing on the cube’s vast number of potential states, it has found applications in image encryption as well. Vidhya [7] introduced a chaotic image encryption method leveraging the Rubik’s Cube’s structural permutations and simple XOR diffusion. This method involves transforming an input image into a scrambled image based on the arrangement of the Rubik’s Cube, achieving efficient scrambling through rotational transformations in both row and column directions. Zhao et al. [8] innovatively combined quantum random walk with Rubik’s Cube transformation to generate a random sequence via quantum walk. This random sequence was then utilized to control Rubik’s Cube transformations, achieving the goal of image scrambling and completing the encryption process for digital images.

In the field of robotics, Ding et al. [9] conducted a study on the reconstruction theory for a snake-like Rubik’s Cube robot, comprising 27 cube modules in series. They analyzed the mathematical model and motion interference of its constituent units. Liu et al. [10] introduced a reconfigurable modular robot inspired by the Rubik’s Snake toy. This robot exhibits a robust capacity for configuration transformation, and the researchers explored geometric characteristics, kinematic modeling, isomorphic recognition, and motion sequence analysis, among other aspects. Kuo et al. [11] investigated the identification of degrees of freedom (DOF) and configuration reconstruction for a reconfigurable cubic mechanism. They proposed a method that involves both directional isomorphism and displacement isomorphism identification. They presented an algebraic method for the configuration transformation process for this reconfigurable cubic mechanism and obtained all non-isomorphic configuration representations [12]. Hou et al. [13] put forth a novel mathematical method for describing snake-shaped Rubik’s Cube blocks. They delved into the mathematical intricacies related to snake-shaped Rubik’s Cubes and designed additional Rubik’s Cube configurations with innovative structures. Li et al. [14] drew inspiration from the feature that the three internal axes of the third-order magic cube are perpendicular to each other. They designed a kinematic pair (rA) with six different modes by altering the angle between the internal axes of the magic cube, subsequently creating a new reconfigurable parallel mechanism utilizing this kinematic pair. Zeng et al. [15] established the mapping relationship between the topological structure and feature size of the RCM. They studied the relationship between the configuration transformation law of the RCM and its topological structure using graph theory and an adjacency matrix. The complex loop structure in the RCM generates overconstraints, resulting in a strong coupling relationship in kinematics. Based on screw theory, the researchers analyzed multi-loop coupling and hidden overconstraints in the RCM, proposing a new method for studying the RCM DOF [16].

The interdisciplinary exploration of the Rubik’s Cube in various scientific fields has yielded significant breakthroughs and results. Lee et al. [17] drew inspiration from the protein folding problem and introduced a Rubik’s Cube model. They delved into thermodynamic and kinematic behavior, asserting an intriguing similarity between the Rubik’s Cube and the dynamics of protein folding. Ji et al. [4] ingeniously combined the combinatory features of the Rubik’s Cube with AIE hydrogel materials, creating an exchangeable pattern array system. This innovative approach offers a fresh perspective on utilizing dynamic covalent hydrogel characteristics to construct a macro structure that is easily operable. Taking cues from the chemical molecules of normal alkanes, Feng et al. [18] pioneered a chain-reconfigurable modular molecular robot. They proposed a fully interconnected geometric feature algorithm for assessing non-isomorphic configurations. The third-order Rubik’s Cube, known for its compact structural form and ability to achieve a target configuration through up to 20 mechanical rotations, served as the basis for Lai et al.’s [3] design of a modular Rubik’s Cube microfluidic system. This system provides an effective solution to address the challenges of high cost and prolonged time consumption in the manufacturing process for microfluidic systems.

Upon reviewing developments over nearly 50 years, it becomes evident that research on the Rubik’s Cube and its associated fields has yielded fruitful results. This underscores the profound significance of the Rubik’s Cube within the contemporary disciplinary framework. Simultaneously, sustained interest and investment from researchers across diverse disciplines have fostered the integration and intersection of the Rubik’s Cube with various fields, thereby advancing its applications and associated concepts. This trend suggests that the exploration of new types of RCM can offer valuable inspiration for the further development of diverse disciplines. However, the systematic exploration of RCM design is essential. Any movable system with a defined motion can be considered a mechanism, and, thus, movable systems like the Rubik’s Cube can be examined from a mechanistic perspective. As the core and framework of machinery, the innovative design of mechanisms holds paramount importance in the development of machine equipment. For a complex mechanism such as the RCM, empirical design alone is no longer adequate to meet fundamental requirements. There is an imperative need to explore theoretical perspectives in studying the design methods of the RCM.

A mechanism, as an assembly of components with defined motion for transferring motion and force, derives its topological structure from the number of components it contains and the connection relationships (kinematic pairs) between these components. This topological structure plays a pivotal role in determining the movements the mechanism can execute and the functions it can achieve. Mechanism type synthesis aims to obtain the topological structure of the mechanism [19]. The topology directly reflects the DOF of the mechanism’s motion and can convey characteristics, such as motion decoupling, Jacobian matrix, redundancy, mechanism configuration transformation modes, and more. Clearly, mechanism type synthesis involves a highly organized and systematic derivation process, necessitating a comprehensive mathematical depiction of the topological information within the mechanism and the formulation of rigorous logical operation rules. Commonly employed mathematical tools include graph theory, screw algebra, group theory, G_F set, set theory, position and orientation set, etc. [20]. Mechanisms can be categorized into various types, including series mechanisms, parallel mechanisms, multi-loop mechanisms, single-loop mechanisms, foldable mechanisms, overconstrained mechanisms, etc. Each type of mechanism has its own applicable methods for type synthesis.

In recent years, significant advancements have been made in the type synthesis theory of mechanisms, particularly those represented by parallel mechanisms and reconfigurable mechanisms. Huang and Li [21] introduced a constraint synthesis method based on screw theory for synthesizing low-DOF parallel mechanisms. This groundbreaking work, for the first time, resulted in the synthesis of symmetric four-DOF and five-DOF parallel mechanisms [22], marking the inception of lower-DOF parallel mechanism synthesis. Given the instantaneous characteristic of screw theory, determining the continuity of the mechanism’s DOF is crucial. To obtain mechanisms with non-instantaneous motion, Hervé [23] initially utilized displacement manifolds to describe the topological information of the mechanism. Through continuous refinement and development, Huang, Hervé, and Li [24] introduced the displacement manifold synthesis theory for any lower-DOF parallel mechanism in 2004. Ye et al. [25] presented the type synthesis of a 4-DOF non-overconstrained parallel mechanism with symmetrical structures based on screw theory, obtaining three classes of hybrid limbs composed of a special topology structure. In the evolution of type synthesis theory, the mechanism’s DOF is often combined with other characteristics of the mechanism to carry out type synthesis. Examples include the integration of the mechanism’s DOF with the Jacobian matrix [26,27], of the mechanism’s DOF with the decoupling situation [28], of the mechanism’s DOF with the redundancy situation [29,30], and more.

As a representative of spatial and multi-loop mechanisms, the type synthesis of parallel mechanisms has elevated type synthesis theory overall, providing a robust theoretical foundation for reconfigurable mechanism synthesis. Unlike traditional mechanisms, reconfigurable mechanisms exhibit characteristics such as changing topology, DOF, and working modes. The versatility of reconfigurable mechanisms, with their ability to change and reconstruct their kinematic geometric constraints during movement, makes them more complex in mathematical modeling, including in topology description and synthesis, compared with traditional mechanisms. The type synthesis method for reconfigurable mechanisms facilitates the alteration and reconstruction of kinematic geometric constraints during the mechanism’s movement by employing reconfigurable kinematic pairs. This enables the realization of reconfigurable movements and functions within the mechanism. Kong et al. [31] introduced a parallel-mechanism type synthesis method suitable for multiple working modes, systematically generating a three-DOF parallel mechanism with spherical and translational motion modes. Locking specific kinematic pairs in the mechanism can alter its configuration, resulting in mechanisms with different motion modes. Kong et al. [32] proposed a type synthesis method for a 3-DOF translational/spherical reconfigurable parallel mechanism with lockable joints and established a type synthesis method for multi-mode mechanism systems. Ye et al. [33] designed several closed-loop mechanisms with singular characteristics and applied them to reconfigurable parallel configuration synthesis. Utilizing the 4R planar diamond mechanism as a base, different reconfigurable parallel mechanisms were derived. Liu et al. [34] explored the type synthesis of reconfigurable single-loop mechanisms based on screw theory, introducing a new method of inserting two or four revolute pairs into a planar diamond 4R mechanism to synthesize reconfigurable single-loop mechanisms. The resulting reconfigurable single-loop mechanism can achieve a planar 4R mode, Bricard mode, and Bennett mode. Li et al. [35] introduced a method for constructing reconfigurable and expandable polyhedral mechanisms based on the polyhedron and mechanism layering method. Leveraging the reconfigurable characteristics of the bending rod group, a coupled-motion relationship between two bending rod groups was established. Whether through the introduction of a limb chain with constraint singularity [36], the utilization of a reconfigurable kinematic pair [37], the locking of a kinematic pair [38], etc., these approaches can be considered indirect methods for conducting the type synthesis of reconfigurable mechanisms.

The direct type synthesis of reconfigurable mechanisms not only involves describing the topology information within the mechanism but also researching the methods of switching between multiple topologies, DOF, and motion modes, which presents certain theoretical challenges. Tian et al. [39] synthesized the basic kinematic chain based on the functional requirements of each working stage. They achieved the transformation between two adjacent working stages by modifying the constraint of the kinematic joint, proposing a configuration synthesis method based on the source metamorphic mechanism. Wei et al. investigated each configuration space in the movement process for reconfigurable mechanisms using Lie groups and differentiable manifold theory. They provided the mathematical preconditions for each branch movement switch, revealing the motion mechanisms for reconfigurable parallel mechanisms. They synthesized reconfigurable parallel mechanisms with 1R2T and 2R1T [40] and reconfigurable parallel mechanisms with planar and spherical motions [41]. Kang et al. [42] explored a dual-loop metamorphic mechanism with kinematic pairs, capable of achieving the functions of special mechanisms under different motion branches. This research offers insights into the design of dual-loop metamorphic mechanisms with kinematic pairs. Tian et al. [43] categorized the type synthesis of reconfigurable mechanisms into closed-loop linkage mechanisms and reconfigurable platforms to synthesize reconfigurable generalized parallel mechanisms. Overvelde et al. [44] applied origami technology to construct prism geometric structures with reconfigurable and foldable characteristics. They utilized the adjustable ability of these structures to design three-dimensional reconfigurable building materials composed of rigid components and flexible hinges.

In summary, it is evident that existing mechanism type synthesis methods are not universally applicable to all mechanisms. Whether considering the completeness and conciseness of mechanism topology description or the interpretability of physical meaning, it is essential to analyze and synthesize the essence of mechanism topology for specific mechanisms. Moreover, type synthesis necessitates a mathematical description of key topological information, such as the types and quantities of components and kinematic pairs in the mechanism, along with the formulation of logical operation rules corresponding to the motion of the mechanism. For the RCM, several challenges arise. On the one hand, compared with existing mechanisms, the RCM boasts a large number of components and a compact structure. For instance, a third-order RCM comprises 27 components, while a 6-SPS with a six-DOF parallel mechanism has 14 components. This abundance of components naturally results in more kinematic pairs in the mechanism, all presented in the form of combined kinematic pairs. On the other hand, existing reconfigurable mechanisms exhibit relatively limited reconfigurable configurations and configuration transformations. In contrast, the RCM’s tens of millions of combined states necessitate numerous configuration transformation methods. Given the kinematic characteristics of the RCM and the constraints for achieving reconfigurable motion, current mechanism type synthesis theory, whether the increasingly mature traditional mechanism type synthesis theory or early-stage research on reconfigurable mechanism type synthesis, faces challenges in its application to the type synthesis of the RCM.

Ref. [45] conducted pertinent research on the kinematics of the RCM from the perspective of reconfigurable motion. The study proposed a piece matrix capable of describing the topological information of the RCM segment. Within a reconfigurable motion analysis based on the introduced piece matrix, the revolute axis and RCM’s RPCSs emerged as two crucial factors. Consequently, in this paper, these two factors, as reflected in the piece matrix, are referred to as the mechanism factors of the RCM. The paper introduces a symmetry layout method for the revolute axis based on the RHP and an analysis method for the intersection topology of the RPCSs based on the adjacency matrix. Section 2 outlines a type synthesis method for RHPRCM. Section 3 and Section 4, respectively, offer a detailed analysis of the revolute axis symmetry and the RPCS symmetry of the RCM. Section 5 establishes an intersection topology model for the RPCSs. In Section 6, using the example of a revolute axis passing through the center of an RHP and having only one RPCS on each revolute axis for the type synthesis of an RCM, a segment of the RHPRCM is chosen for prototype production to validate the correctness and rationality of the proposed synthesis method.

2. The Synthesis Method for RCM Based on RHP

On the one hand, the reconfigurable characteristics and symmetry of the topological structure in an RCM are closely related, and the symmetry is reflected in the two factors of the revolute axis and the RPCSs. On the other hand, because the pieces of the RCM are all obtained by intersecting the RPCSs, the intersection between the RPCSs also determines the topological structure of the RCM. For example, the overall shape of the third-order RCM shown in Figure 1 is a regular hexahedron. A coordinate system ⁰O—⁰X⁰Y⁰Z with the RCM center as the origin is established. Six revolute axes ⁰L₁, ⁰L₂, ⁰L₃, ⁰L₄, ⁰L₅, ⁰L₆ passing through the center intersect at the origin ⁰O, and each revolute axis has an RPCS ⁰Π₁, ⁰Π₂, ⁰Π₃, ⁰Π₄, ⁰Π₅, ⁰Π₆. The RCM pieces are formed by the intersection of these RPCSs. For example, the corner piece ⁰B₁ shown in Figure 2 is formed by the intersection of three RPCSs: ⁰Π₁, ⁰Π₃ and ⁰Π₅. When the intersection between the RPCSs is different, the generated pieces are also different, corresponding to different rotational characteristics.

In summary, research on the topological structure of the RCM mainly includes revolute axis symmetry, RPCS symmetry, and RPCS intersection. The overall process of the synthesis method for the RHPRCM is as follows:

(1)

Symmetrical arrangement of the revolute axis of the RCM based on RHP;

(2)

Symmetrical arrangement of the RPCSs;

(3)

Based on the adjacency matrix, the intersection of the RPCSs is analyzed;

(4)

An RCM is obtained with a determined topological structure.

3. Analysis and Arrangement of Revolute Axis Symmetry

Firstly, the reconfigurable motion process of the RCM must meet the symmetry of the revolute axis. The axis of the RCM can be equivalent to a line in space. In line geometry, the determination and representation of any line in space require six parameters, where three parameters represent the direction vector of the line and three parameters are related to the coordinates of any point on the line. There are many revolute axes in the RCM, and using the six parameters in linear geometry to represent them not only appears complicated but also cannot conveniently and intuitively represent the symmetry of the revolute axes of the RCM, requiring additional constraints. Therefore, adopting linear geometry to describe the symmetry of the revolute axis not only represents complexity but also increases the difficulty of the synthesis process of the RCM. Based on the above analysis, this paper proposes a symmetric arrangement method for the revolute axis of the RCM based on RHP. A coordinate system with the center of the RHP as the origin is established, and the revolute axis starts from the origin while passing through the symmetrical parts of the RHP, such as vertices, face centers, midpoints of edges, etc., which can also be a combination of these situations, as shown in Figure 3.

Furthermore, the symmetry of the revolute axis is elaborated on by selecting the revolute axis passing through the face center of the RHP. For the RHP shown in Figure 4, its center is selected as the intersection point, and the other points of the eight revolute axes pass through the face center. According to the geometric characteristics of the RHP, six revolute axes passing through the side face centers can rotate and coincide around the revolute axes passing through the upper and lower face centers. Among the divisors of 6, 3, 2, and 6 are greater than 1 but less than or equal to 6. Correspondingly, the six revolute axes are divided into two groups, three groups, and one group. When the six revolute axes are divided into two groups, namely L_1−s1, L_1−s3, L_1−s5 and L_1−s2, L_1−s4, L_1−s6, the revolute axis can rotate 360°/3 = 120° by an integer multiple and then overlap. This is Case One, as shown in Figure 4a. When the six revolute axes are divided into three groups, they are L_2−s1 and L_2−s4, L_2−s2 and L_2−s5, and L_2−s3 and L_2−s6, so that the revolute axes can rotate 360°/2 = 180° by an integer multiple and then overlap. This is Case Two, as shown in Figure 4b. When divided into one group, they are L_3−s1, L_3−s2, L_3−s3, L_3−s4, L_3−s5, L_3−s6, so that the axis can rotate 360°/6 = 60° by an integer multiple and then overlap. This is Case Three, as shown in Figure 4c. Among these, L_p−sq represents the q−th revolute axis under the symmetry of the revolute axis. In this paper, p is taken as 1~3, and q is taken as 1~6. The above method can be used to study other situations where the revolute axis passes through the symmetrical part of the RHP.

4. Analysis and Arrangement of the RPCSs

For the classic third-order RCM, there is an RPCS on each revolute axis, and the rotation in the RCM is combined rotation, so the kinematic pair is also a combined kinematic pair. On the other hand, the ability of reconfigurable mechanisms to achieve reconfigurable motion is due to the reconstruction of internal constraints. For the RCM, the revolute pair is an intuitive form of constraint. The revolute joint constraints in the RCM are divided into many sub-revolute joint constraints, namely sub-RPCS. Reconstructing and combining revolute joint constraints through the combination of revolute joint constraints would ultimately achieve RCM reconfigurability. As can be seen from the above, in an RCM with multiple revolute axes, the number of RPCSs naturally increases. Therefore, it is necessary to provide a definite description of the RPCSs in order to determine the similarities and differences between RPCSs.

Based on screw theory, the contact geometry of a kinematic pair was studied in [46], where the revolute pair element was defined as a spatial rotating surface formed around the revolute axis. This spatial rotating surface is obtained by rotating the generatrix around the revolute axis, as in Figure 5. The RCM has multiple revolute axes; there can be multiple RPCSs on the same revolute axis in a higher-order RCM. Therefore, it is necessary to determine a description of the RPCSs. Here, the intersection point of the RPCS and the revolute axis is selected as the origin. The direction of the revolute axis in a line pointing towards the movable linkage is the ordinate, and the direction perpendicular to the revolute axis and passing through the origin is the abscissa. The curve formed by the intersection of the plane formed by the abscissa and longitudinal coordinates with the RPCS is the generatrix of the RPCS. The RPCS can be uniquely represented using the equation of the generatrix in the coordinate system. For example, when the revolute axis is taken as the ordinate, any direction perpendicular to the revolute axis that is passing through the origin is the abscissa. As shown in Figure 5, X₁ or X₂ can be selected as the abscissa axis, and the plane formed by the abscissa axis and the ordinate axis is Π₁ or Π₂. The part formed by the intersection of the plane composed of two coordinate axes and the RPCS is G₁ or G₂, respectively, which is the corresponding generatrix of the corresponding RPCS under the corresponding coordinate system. It can be found that no matter which abscissa axis is selected, the representation of the corresponding generatrix in the corresponding coordinate system is the same.

The symmetry arrangement of the RPCS is based on the symmetry of the revolute axis. In Section 3, the symmetry arrangement of the revolute axis was shown passing through the face centers of the RHP. At the time, there were various different situations of symmetry for the revolute axis. Here, we only analyze the symmetry that can be achieved by the revolute axis passing through the upper and lower face centers, and the RPCSs on the revolute axis passing through the upper and lower face centers are the same, as shown in Figure 4. When the revolute axes are divided into one group, with six revolute axes in each group, they need to be rotated by an integer multiple of 60° to coincide. Therefore, when arranging the RPCSs symmetrically, the RPCSs on the six revolute axes must be identical, and the RPCSs must have the same equation representation in the coordinate system with the revolute axis as the ordinate. At this point, the RPCSs on these six revolute axes will intersect with the RPCS on the revolute axis passing through the upper and lower face center simultaneously. This situation is relatively simple, and the following analysis will mainly focus on the symmetry of the revolute axes in the other two cases.

When the revolute axes are divided into two groups, there are only three revolute axes in each group, which need to be rotated by an integer multiple of 120° before they overlap. This is Case One. For Case One, the RPCS Π_1−s1, Π_1−s3, Π_1−s5 on the revolute axis L_1−s1, L_1−s3, L_1−s5 is the same, and Π_1−s1, Π_1−s3, Π_1−s5 has the same parameter representation in the coordinate system with axis L_1−s1, L_1−s3, L_1−s5 as the ordinate; the RPCS Π_1−s2, Π_1−s4, Π_1−s6 on the revolute axis L_1−s2, L_1−s4, L_1−s6 is the same, and Π_1−s2, Π_1−s4, Π_1−s6 has the same parameter representation in the coordinate system with axis L_1−s2, L_1−s4, L_1−s6 as the ordinate. Among these, Π_1−sq represents the q−th RPCS under the symmetry distribution for Case One of the revolute axis, and a symmetry distribution diagram of the RPCS of the revolute pair under the symmetry distribution of the revolute axis is shown in Figure 6a. The above content is expressed as follows:

Π_1−s1 = Π_1−s3 = Π_1−s5, Π_1−s2 = Π_1−s4 = Π_1−s6

(1)

When the revolute axes are divided into three groups, there are only two revolute axes in each group, which need to be rotated by an integer multiple of 180° before they overlap. This is Case Two. For Case Two, the RPCS Π_2−s1, Π_2−s4 on the revolute axis L_2−s1, L_2−s4 is the same, and Π_2−s1 and Π_2−s4 are represented by the same parameters in the coordinate system with the axis L_2−s1 and L_2−s4 as the ordinate; the RPCS Π_2−s2, Π_2−s5 on the revolute axis L_2−s2, L_2−s5 is the same, and Π_2−s2 and Π_2−s5 are represented by the same parameters in the coordinate system with the axis L_2−s2 and L_2−s5 as the ordinate; the RPCS Π_2−s3, Π_2−s6 on the revolute axis L_2−s3, L_2−s6 is the same, and Π_2−s3 and Π_2−s6 are represented by the same parameters in the coordinate system with the axis L_2−s3 and L_2−s6 as the ordinate. Among these, Π_2−sq represents the q−th RPCS under the symmetry distribution for Case Two of the revolute axis, and a symmetry distribution diagram of the RPCS of the revolute pair under the symmetry distribution of the revolute axis is shown in Figure 6b. The above content is expressed as follows:

Π_1−s1 = Π_1−s4, Π_1−s2 = Π_1−s5, Π_1−s3 = Π_1−s6

(2)

There can be multiple different RPCSs on each revolute axis, and these different RPCSs can be symmetrically arranged according to different revolute axis arrangements. For example, it is better to select a revolute axis that passes through the face center of RHP where there is an RPCS on the revolute axis that passes through the upper and lower face center and two RPCSs on the revolute axis that passes through the side face center. At this time, analysis shows that the two different symmetrical arrangements of the RPCS in Figure 6a,b can be combined to obtain a symmetrical arrangement of the RPCSs, as shown in Figure 6c. Among these, the RPCSs Π_c−s1/1, Π_c−s2/1, Π_c−s3/1, Π_c−s4/1, Π_c−s5/1, Π_c−s6/1 and Π_1−s1, Π_1−s2, Π_1−s3, Π_1−s4, Π_1−s5, Π_1−s6 are the same, respectively, and the RPCSs Π_c−s1/2, Π_c−s2/2, Π_c−s3/2, Π_c−s4/2, Π_c−s5/2, Π_c−s6/2 and L_2−s1, L_2−s2, L_2−s3, L_2−s4, L_2−s5, L_2−s6 are the same, respectively.

Through analysis, it can be found that when selecting symmetry Case One for the revolute axis, the RPCSs Π_c−s1/1, Π_c−s3/1, Π_c−s5/1 and Π_c−s2/1, Π_c−s4/1, Π_c−s6/1 are the same, and the RPCS Π_c−s1/2, Π_c−s2/2, Π_c−s3/2, Π_c−s4/2, Π_c−s5/2, Π_c−s6/2 cannot overlap by rotating by an integer multiple of 120°; when selecting symmetry Case Two for the revolute axis, the RPCSs Π_c−s1/2, Π_c−s4/2; Π_c−s2/2, Π_c−s5/2; and Π_c−s3/2, Π_c−s6/2 are the same, and the RPCS Π_c−s1/1, Π_c−s2/1, Π_c−s3/1, Π_c−s4/1, Π_c−s5/1, Π_c−s6/1 cannot overlap by rotating by an integer multiple of 180°. Among these, Π_c−sq/1 and Π_c−sq/2 represent the first and second RPCSs of the corresponding revolute pair on the q−th revolute axis in Case One and Case Two of revolute axis symmetry, respectively, with q still taking values of 1~6. The symmetrical arrangement of the RPCSs shown in Figure 6c includes two different situations, so there can be different symmetrical arrangements of RPCSs in the same RCM. The above content is expressed as follows:

$$\begin{array}{l}{\Pi }_{c-s1/1}={\Pi }_{1-s1}={\Pi }_{c-s1/3}={\Pi }_{1-s3}={\Pi }_{c-s1/5}={\Pi }_{1-s5}\\ {\Pi }_{c-s2/1}={\Pi }_{1-s2}={\Pi }_{c-s4/1}={\Pi }_{1-s4}={\Pi }_{c-s6/1}={\Pi }_{1-s6}\\ {\Pi }_{c-s1/2}={\Pi }_{2-s1}={\Pi }_{c-s4/2}={\Pi }_{2-s4}\\ {\Pi }_{c-s2/2}={\Pi }_{2-s2}={\Pi }_{c-s5/2}={\Pi }_{2-s5}\\ {\Pi }_{c-s3/2}={\Pi }_{2-s3}={\Pi }_{c-s6/2}={\Pi }_{2-s6}\end{array}$$

(3)

5. Analysis of the Intersection of the RPCSs

The pieces in the RCM are formed by the intersection of RPCSs, so the intersection between RPCSs directly affects the motion and topology of the RCM. On the other hand, different pieces are generated by the intersection of the RPCSs at the outermost part of the RCM, thereby exhibiting different topological structures in the RCM. Therefore, the intersection between the RPCSs of the RCM mentioned in this paper occurs on the outer surface of the RCM. The intersection of RPCSs in the RCM is also regular. This paper uses the adjacency matrix in graph theory to analyze the intersection of RPCSs.

The RPCSs in the RCM are selected as the research object, and the adjacency matrix corresponding to the intersection of the RPCSs is:

$$AM\left(G\right)={\left[\right(am{)}_{i_3{j}_3}]}_{n\times n}$$

(4)

where $AM\left(G\right)$ represents the intersection relationship between the RPCSs of the RCM, G, and the elements in the matrix are determined by Equation (5).

$${\left(am\right)}_{i_3{j}_3}=\{\begin{array}{l}1,\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{R}\mathrm{P}\mathrm{C}\mathrm{S}{i}_3\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{R}\mathrm{P}\mathrm{C}\mathrm{S}{j}_3\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}.\\ 0,\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{R}\mathrm{P}\mathrm{C}\mathrm{S}{i}_3\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{R}\mathrm{P}\mathrm{C}\mathrm{S}{j}_3\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}.\end{array}$$

(5)

According to the symmetry and reconfigurable motion characteristics of the RCM, the corresponding adjacency matrix has the following characteristics:

(1) The adjacency matrix is a real symmetric matrix, in which the elements only contain 1 and 0. The diagonal element is 0, indicating that there is no intersecting relationship between the same RPCSs.

(2) In the adjacency matrix, the RPCS coincides with the revolute axis. The coincidence of the revolute axis is determined by taking the sub-revolute axis as the unit. The RPCS on the revolute axis around which the sub-revolute axis moves should intersect or not intersect with the RPCS on the sub-revolute axis at the same time. At this time, the element in the corresponding adjacency matrix is 1 or 0 at the same time.

Here, distribution Case Two of the revolute axis in Figure 4b is selected as an example to explain. The RPCSs on each group of revolute axes are the same, i.e., Π_1−s1 = Π_1−s3 = Π_1−s5 ≠ Π_1−s2 = Π_1−s4 = Π_1−s6, and, at this point, the RPCSs on the revolute axes L_1−s7, L_1−s8 are the same, i.e., Π_1−s7 = Π_1−s8, as shown in Figure 7. Therefore, the RPCSs Π_1−s7 and Π_1−s8 will intersect or not intersect with the RPCSs of a different group of revolute axes simultaneously. A state in which the RPCS Π_1−s7, Π_1−s8 and the RPCS Π_1−s1, Π_1−s3, Π_1−s5 do not intersect is selected as the initial state, as shown in Figure 7a; the RPCS Π_1−s7, Π_1−s8 and the RPCS Π_1−s1, Π_1−s3, Π_1−s5 intersect simultaneously, and the RPCS Π_1−s7, Π_1−s8 and the RPCS Π_1−s2, Π_1−s4, Π_1−s6 do not intersect simultaneously, as shown in Figure 7b; the RPCS Π_1−s7, Π_1−s8 and the RPCS Π_1−s1, Π_1−s3, Π_1−s5 intersect simultaneously, and the RPCS Π_1−s7, Π_1−s8 and the RPCS Π_1−s2, Π_1−s4, Π_1−s6 intersect simultaneously, as shown in Figure 7c.

(3) The RPCSs on revolute axes with a collinear reverse direction will not intersect, and the element in the corresponding adjacency matrix is 0; different RPCSs on the same revolute axes will not intersect, and the element in the corresponding adjacency matrix is 0. For example, in the symmetrical arrangement of the RPCSs, as shown in Figure 6c, the RPCSs Π_c−s7/1 and Π_c−s7/2 on the collinear opposite axes L_c−s7 and L_c−s7 do not intersect. The RPCSs Π_c−s1/1 and Π_c−s1/2 on the revolute axis L_c−s1 do not intersect.

(4) When the elements corresponding to the row or column where any RPCS is located are all 0, this RPCS will not intersect with any RPCS, so it will not form interchangeable pieces and cannot enable the RCM to complete reconfigurable motion. Therefore, such an RPCS does not exist.

6. Type Synthesis of the RHPRCM

There is an RPCS on each revolute axis. The symmetry arrangement of the RPCSs and the intersection between the RPCSs under the two cases of revolute axis symmetry distribution are analyzed. The adjacency matrix of the intersection of the RPCSs is given, and an RHPRCM with a definite topological structure where the revolute axis passes through the face center of the RHP is synthesized.

(1) The symmetrical arrangement of the revolute axis in Case One when the revolute axis passes through the face center of the RHP.

There are eight revolute axes of the RHP passing through the face center, and each revolute axis is equipped with an RPCS, resulting in a total of eight RPCSs. Based on the geometric characteristics and the characteristics of the adjacency matrix of Case One of the symmetrical layout of the revolute axis, the adjacency matrix of the RPCS intersection in Case One is given:

$$AM\left(O_{1s}\right)=\begin{array}{ccccccccc}\begin{array}{c}\\ \begin{array}{c}{\Pi }_{1-s1}\\ {\Pi }_{1-s2}\\ {\Pi }_{1-s3}\\ {\Pi }_{1-s4}\\ {\Pi }_{1-s5}\\ {\Pi }_{1-s6}\\ {\Pi }_{1-s7}\\ {\Pi }_{1-s8}\end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s1}\\ [\begin{array}{c}0\\ \\ \\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s2}\\ \begin{array}{c}{a}_{1-s1}\\ 0\\ \\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s3}\\ \begin{array}{c}{a}_{1-s2}\\ a_{1-s1}\\ 0\\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s4}\\ \begin{array}{c}0\\ a_{1-s3}\\ a_{1-s1}\\ 0\\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s5}\\ \begin{array}{c}{a}_{1-s2}\\ 0\\ a_{1-s2}\\ a_{1-s1}\\ 0\\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s6}\\ \begin{array}{c}{a}_{1-s1}\\ a_{1-s3}\\ 0\\ a_{1-s3}\\ a_{1-s1}\\ 0\\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s7}\\ \begin{array}{c}{a}_{1-s4}\\ a_{1-s5}\\ a_{1-s4}\\ a_{1-s5}\\ a_{1-s4}\\ a_{1-s5}\\ 0\\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{1-s8}\\ \begin{array}{c}{a}_{1-s4}\\ a_{1-s5}\\ a_{1-s4}\\ a_{1-s5}\\ a_{1-s4}\\ a_{1-s5}\\ 0\\ 0\end{array}]\end{array}\end{array}$$

(6)

where ${\Pi }_{1-s1}$, …, ${\Pi }_{1-s8}$ represents the first, …, eighth RPCS, and $a_{1-s1}$, …, $a_{1-s5}$ represents the five parameters of the RPCS intersection.

According to the characteristics of the adjacency matrix in Section 3, when the row or column element of a certain RPCS is 0, that RPCS and any other RPCSs do not intersect. Therefore, the RPCS does not play any role in the reconfigurable movement of the RCM. It can be considered that the RPCS does not exist, and the corresponding revolute axis can also be considered as non-existent. In order to ensure that each revolute axis has an effective RPCS, it is necessary to ensure that the row or column of each RPCS in the adjacency matrix is not all zeros. By combining the values of the parameters in the adjacency matrix with 0 or 1, RCMs with different topological structures can be obtained. Some 3D models of RCMs are shown in Figure 8. In Figure 8, the revolute axis $L_{1-s1}$, $L_{1-s2}$…$L_{1-s8}$ represents the first, second, and eighth revolute axes. The numbers in the upper-left corner of each parameter represent the situation. The corresponding adjacency matrix also adds numbers in the upper-left corners of each element to represent the intersections of RPCSs under different conditions.

In Figure 8a, the RCM can rotate around any of the eight revolute axes after rotating 120° around the axes ¹L_1−s7, ¹L_1−s8. Due to the fact that the RPCSs on each revolute axis passing through the side face center are the same, the RCM can also rotate by an integer multiple of 60° around the revolute axis passing through the upper and lower face centers and continue to rotate around any of the eight revolute axes. Of course, you can also rotate around the axis passing through the side face center by an integer multiple of 180° and then continue to rotate around any of the eight revolute axes. Generally, the RPCSs on the two groups of revolute axes on the side are different, and the topological structure is shown in Figure 8b. The RCM can only rotate around any of the eight revolute axes after being rotated by an integer multiple of 120° around the revolute axis ¹L_1−s7, ¹L_1−s8 that passes through the upper and lower face centers. In the above two types of RCM, there is no intersection between the same RPCSs on the same groups of revolute axes. When the RPCSs on one group of revolute axes intersect, the topology structure is shown in Figure 8c; when the RPCSs on two groups of revolute axes intersect, the topological structure is shown in Figure 8d. Obviously, the more intersections that occur between the RPCSs, the more complex the topological structure of the RCM formed. Based on the RCM in Figure 8d, when the RPCS intersection parameters on the two groups of revolute axes are completely the same, the RCM in Figure 8e is synthesized. The RCM shown in Figure 8e has better symmetry; therefore, it has more combination states and stronger reconfigurability.

The RPCS on the revolute axis passing through the upper and lower face center can also intersect with the RPCSs in the two groups of revolute axes passing through the side face center; the RCMs in Figure 8f–j are synthesized.

Among these, the RPCSs in the same group of revolute axes passing through the side face center in Figure 8f do not intersect; the group of RPCSs in the same group of revolute axes passing through the side face center in Figure 8g or in Figure 8h intersect. Therefore, the topological structure of the RCM in Figure 8f is relatively simple. Under the condition of ensuring that the RPCSs on the revolute axis passing through the upper and lower face center intersect with the RPCSs on the two sets of revolute axes passing through the side face center, the RPCSs on different groups of revolute axes passing through the side face center cannot intersect, as shown in Figure 8i,j. Compared with the previous RCMs, the topology of the two RCMs this time is relatively simple, and there are significantly fewer pieces that can be exchanged. When the intersection parameters of the RPCSs on the two groups of revolute axes are completely the same, the RCM in Figure 8i is obtained. Compared with the RCM in Figure 8j, the RCM in Figure 8i has higher symmetry, resulting in more final combined states.

(2) The symmetrical arrangement of the revolute axis in Case Two when the revolute axis passes through the face center of the RHP.

In Case Two, the six revolute axes passing through the side face center are coplanar and uniformly distributed, and the RPCSs on any three adjacent revolute axes are different. In order to complete the reconfigurable motion of the RCM, the RPCS on the revolute axis passing through the upper and lower face center must intersect with the six revolute axes passing through the side face center simultaneously. The specific explanation is as follows.

In Case One, the RPCS passing through the upper and lower face center can only intersect with a portion of the RPCSs passing through the side face center, such as the RCM in Figure 8f. This is because, in Case One, without considering the RPCS of the revolute axis passing through the upper and lower face center, the six revolute axes passing through the side face center are divided into two groups; two RPCSs adjacent to one RPCS are the same and can continue to rotate around the two adjacent RPCSs when the RPCS is rotated by an integer multiple of 180°. Even if the RPCS passing through the upper and lower face center does not intersect with a portion of the RPCSs, this portion of the non-intersecting RPCSs can still achieve reconfigurable motion through rotation with the adjacent and intersecting RPCSs. For example, for the RHPRCM shown in Figure 8f, the RPCSs ${}^6\Pi _{1-s7}$ and ${}^6\Pi _{1-s8}$ on the revolute axes ${}^{6}L_{1-s7}$ and ${}^{6}L_{1-s8}$ intersect with the RPCS ${}^6\Pi _{1-s2}$, ${}^6\Pi _{1-s4}$, ${}^6\Pi _{1-s6}$ on the revolute axis ${}^{6}L_{1-s2}$, ${}^{6}L_{1-s4}$, ${}^{6}L_{1-s6}$ and do not intersect with the RPCS ${}^6\Pi _{1-s1}$, ${}^6\Pi _{1-s3}$, ${}^6\Pi _{1-s5}$ on the revolute axis ${}^{6}L_{1-s1}$, ${}^{6}L_{1-s3}$, ${}^{6}L_{1-s5}$. Although the RPCS ${}^6\Pi _{1-s7}$, ${}^6\Pi _{1-s8}$ does not intersect with the RPCS ${}^6\Pi _{1-s1}$, ${}^6\Pi _{1-s3}$, ${}^6\Pi _{1-s5}$, the RPCSs on the adjacent revolute axis passing through the side face center intersect. Therefore, when the RPCS ${}^6\Pi _{1-s1}$ rotates by an integer multiple of 180°, it can continue to rotate with the adjacent RPCSs ${}^6\Pi _{1-s2}$ and ${}^6\Pi _{1-s6}$, thereby completing the reconfigurable motion of the RCM.

From the above analysis, it can be seen that the next step is only to analyze the intersection between the RPCSs on the revolute axis passing through the side face center. Therefore, the adjacency matrix of the RPCS intersection in this case is given:

$$AM\left(O_{2s}\right)=\begin{array}{ccccccccc}\begin{array}{c}\\ \begin{array}{c}{\Pi }_{2-s1}\\ {\Pi }_{2-s2}\\ {\Pi }_{2-s3}\\ {\Pi }_{2-s4}\\ {\Pi }_{2-s5}\\ {\Pi }_{2-s6}\\ {\Pi }_{2-s7}\\ {\Pi }_{2-s8}\end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s1}\\ [\begin{array}{c}0\\ \\ \\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s2}\\ \begin{array}{c}{a}_{2-s1}\\ 0\\ \\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s3}\\ \begin{array}{c}{a}_{2-s4}\\ a_{2-s2}\\ 0\\ \\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s4}\\ \begin{array}{c}0\\ a_{2-s5}\\ a_{2-s3}\\ 0\\ \\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s5}\\ \begin{array}{c}{a}_{2-s5}\\ 0\\ a_{2-s6}\\ a_{2-s1}\\ 0\\ \\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s6}\\ \begin{array}{c}{a}_{2-s3}\\ a_{2-s6}\\ 0\\ a_{2-s4}\\ a_{2-s2}\\ 0\\ \\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s7}\\ \begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 0\\ \end{array}\end{array}& \begin{array}{c}{\Pi }_{2-s8}\\ \begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 0\\ 0\end{array}]\end{array}\end{array}$$

(7)

By combining the parameters in the adjacency matrix shown in Equation (7) with 0 or 1, the RPCS intersections on the revolute axis passing through the side face center can be determined. Figure 9 provides the RPCS intersections and the final topology of the RCM. For the RCM shown in Figure 9, the RPCSs on the adjacent revolute axes intersect. At this point, one operation step for the RCM to complete reconfigurable motion is as follows: first, rotate 180° around the RPCS on the two collinear revolute axes that pass through the side face center; then, rotate 180° around the revolute axes that pass through the upper and lower face center; and, finally, rotate 180° around the RPCS on the first revolute axis.

Taking the RCM shown in Figure 9a as an example, first rotate 180° around the RPCS ${}^1\Pi _{2-s2}$ on the revolute axis ${}^{1}L_{2-s2}$ and the RPCS ${}^1\Pi _{2-s5}$ on the revolute axis ${}^{1}L_{2-s5}$; then, rotate 180° around the RPCS ${}^1\Pi _{2-s7}$ on the revolute axis ${}^{1}L_{2-s7}$ or the RPCS ${}^1\Pi _{2-s8}$ on the revolute axis ${}^{1}L_{2-s8}$. Finally, rotate 180° around the RPCS ${}^1\Pi _{2-s2}$ on the revolute axis ${}^{1}L_{2-s2}$ and the RPCS ${}^1\Pi _{2-s5}$ on the revolute axis ${}^{1}L_{2-s5}$, respectively. The reconfigurable motion of the RCM can be achieved by selecting different revolute axes that pass through the side face center and are collinear and repeating the above rotation operations in sequence. When the RPCSs on each group of revolute axes are the same, the RCM in Figure 9b is synthesized, and it clearly has better symmetry. The rotation operation of the RCM can not only meet the rotation operation requirements of the RCM in Figure 9a, it can also continue to rotate around the revolute axis, passing through the upper and lower face centers after being rotated by an integer multiple of 60° around the revolute axis passing through the side face centers. Obviously, the RCM in Figure 9b has more reconfigurable motion states.

RPCSs on the same group of revolute axes, that is, RPCSs on revolute axes separated by one revolute axis, can also intersect. The RCM is obtained in Figure 9c and has more a complex topological structure compared with the RCM in Figure 9a. When all elements of the parameters in the adjacency matrix are 1, the RPCS intersections and the RCM of the final topological structure are shown in Figure 9d. Obviously, this RCM has more complex topological structures and more exchangeable pieces than the two RCMs in Figure 9c,d.

The RPCSs passing through the side face center can also not intersect, and, by only intersecting the RPCS on the upper and lower face centers, an RCM that completes reconfigurable motion can also be obtained, as shown in Figure 9e. The RPCSs passing through the side face centers can also partially intersect, and by intersecting with the RPCS passing through the upper and lower face centers, an RCM that completes reconfigurable motion can be obtained, as shown in Figure 9f.

7. Conclusions

This paper investigates the type synthesis of RHPRCMs. Firstly, in the process of reconfigurable motion analysis of the RCM, two mechanism elements—the revolute axis and the RPCS—were obtained, and then the type synthesis process for the RHPRCM was presented. Based on the geometric characteristics of RHPs, the symmetrical arrangement of the revolute axis was obtained, and then the symmetrical arrangement of the RPCSs was given. The intersection between the RPCSs was determined by using the adjacency matrix. The revolute axis was selected to pass through the face center of the RHP, and three different situations of revolute axis symmetry were obtained through symmetry analysis of the revolute axis. The first two situations were selected, and only one RPCS was arranged on each revolute axis as an example. The adjacency matrix corresponding to the intersection of the RPCSs was given. By giving the parameter values in the adjacency matrix, the RHPRCM that determines the topology was obtained. Taking two types of RCM with a revolute axis in Case One as an example, a model prototype was made to verify the rationality, feasibility, and correctness of the proposed synthesis method, laying the foundation for the type synthesis of other spatial RCMs. The research in this paper will help to continuously supplement and improve the type synthesis theory of reconfigurable mechanisms, and it provides more inspiration for the application of the RCM and interdisciplinary research on the RCM.