Compared with spatial mechanisms, planar mechanisms possess simple structural layouts that simplify theoretical derivation and numerical calculation, and they have a broad range of engineering application prospects. Similarly, spatial rotational Rubik's cube mechanisms adopt intricate spatial axis layouts and intersecting contact surfaces of revolute pairs (Fan et al., 2022). In contrast, all motions of planar Rubik's cube mechanisms are confined to a single plane, which brings about the merit of easy actuation. Meanwhile, planar Rubik's cube mechanisms also embody fascinating operational mechanisms. For instance, the eight-digit puzzle is a numerical game in which researchers rearrange eight numbered blocks in a 3 × 3 grid through moving operations. Scholars have carried out systematic research on its permutation rules and topological transformation laws. The most classic 3 × 3 Rubik's cube was invented by Ernő Rubik in 1974. Benefiting from its compact configuration and capacity to generate massive combinatorial configurations through basic movements, it has drawn extensive attention from both academia and the general public. Although the Rubik's cube was invented by Ernő Rubik, the core inspirations and concepts similar to it had long inspired the working people of ancient China. For example, Luo Shu is an ancient Chinese symbolic numerical pattern that inherently features the concept of transformation and rearrangement, a principle later reflected in Rubik's cubes. Luo Shu was employed to study weather, agriculture, solar terms, and other fields, ultimately promoting the development of production and daily life. After long-term inheritance and evolution, multiple derivative patterns derived from Luo Shu have been developed (Zeng et al., 2018). Later, mathematical games such as Magic Squares and the Backgammon appeared. The Backgammon consists of a 4 × 4 grid and 15 numbered tiles. The numbers 14 and 15 are swapped and misplaced in the initial state, and users can recover the standard arrangement by translating the number tiles.

The Rubik's cube offers entertainment and playability due to its vast number of combinatorial states, indicating that it inherently involves numerous important mathematical problems. Zassenhaus (1982) investigated group-theory-related issues in the Rubik's cube, such as subgroups, generators, and orders. Subsequently, Joyner (2008) extended the group-theoretic analysis of the Rubik's cube. Krutský and Šír (2025) introduced a novel symmetry-invariant neural architecture based on the rich symmetries and combinatorial properties of the Rubik's cube group. Mahindra Roshan et al. (2024) developed a deep-reinforcement learning approach for the combinatorial puzzle of the Rubik's cube, which learns to solve increasingly difficult states in reverse from the goal state.

Taking advantage of its property of massive combinatorial configurations, the Rubik's cube can also find application in the field of image encryption. The Rubik's cube, with an astronomical number of possible configurations, serves as an ideal encryption tool, and the combinatorial complexity of Rubik's cube, as well as other n-dimensional hypercubes, further opens up promising applications in cryptography, boasting significant potential for defense and other fields (Balasubramanian, 2024). Helmy et al. (2023) put forward a hybrid encryption architecture based on Rubik's cube theory to realize efficient encryption of facial, iris, and fingerprint biometric templates. Lai et al. (2020) leveraged the compact structural feature of the 3 × 3 Rubik's cube to design a modular Rubik's-cube-based microfluidic system, providing a promising solution to the problems of high cost and long processing time in the fabrication of microfluidic systems. Peng (2024) presented a Rubik's-cube-inspired educational tool in order to improve engineering ethics education and enhance lesson planning and assessment. Zhang et al. (2026) proposed a novel Rubik's-cube-inspired origami structure addressing spacecraft spatial needs, integrating electromagnetic and fluid functions. As a special type of reconfigurable mechanism, the planar prismatic Rubik's cube mechanism (PPRCM) realizes reconfiguration, topological variation, and structural deformation in a two-dimensional plane. It greatly simplifies the cognitive and analytical difficulty of complex reconfigurable mechanisms and lays a fundamental foundation for the subsequent extension to three-dimensional spatial mechanisms. Such variable and reconfigurable characteristics can be integrated with various engineering fields to produce cross-disciplinary research achievements. Over the last 5 decades, abundant research outcomes on Rubik's cube mechanisms have fully verified their research significance in modern multidisciplinary systems. Existing studies have proven that Rubik's cube mechanisms can be combined well with origami structures (Sharma et al., 2026), spatially deployable structures (Liu et al., 2022), and modular robots (Duan et al., 2026). In fact, Rubik's-cube-inspired design principles are applicable to almost all variable working scenarios, which can fully tap the application potential of Rubik's cube mechanisms in the field of deformable and reconfigurable equipment. Diversified novel Rubik's cube mechanisms can provide significant guidance for the innovative development of various disciplines. Accordingly, it is of great necessity to investigate the type synthesis method of the PPRCM.

Any movable system with definite motion can be regarded to be a mechanism. Therefore, such movable systems can be investigated from the perspective of mechanism theory. The PPRCM falls within the category of planar mechanisms, and existing theories on the type synthesis of planar mechanisms have become increasingly sophisticated. Historically, the synthesis of planar mechanisms originated from designers' experience and intuition. Subsequently, the synthesis of planar linkage mechanisms has evolved significantly, giving rise to a wealth of methods and theories. Davies and Crossley (1966) introduced a symbolic notation to represent plane pin-jointed kinematic chains and found all possible permutations of articulation for the purposes of censuses of 7-, 9-, 10-, and 11-bar chains. Tuttle et al. (1989) described a technique for enumerating all basic kinematic chains possessing N links and F degrees of freedom based on the theory of finite groups. Mruthyunjaya (1979) proposed a modified method of structural synthesis of linkage systems with positive, zero, or negative degrees of freedom by using the transformation of binary chains. Cao and Chu (1992) further developed the synthesis method based on the Assur group and synthesized all of the structural types of bar groups with multiple hinges up to eight bars with multiple hinges. Li et al. (2015) developed a new structural synthesis method by integrating Assur groups as elements in the newly developed group-based adjacency matrix. Slaboch and Voglewede (2011) represented the topological characteristics of reconfigurable mechanisms through the mechanism state matrices. Chan (2024) proposed a criterion for analyzing and synthesizing planar overconstrained mechanisms and obtained a systematic approach for synthesizing novel function cognates. Kong (2022) presented a method for the motion–structure mode analysis of n-RR planar parallelogram mechanisms. Santiago-Valentín et al. (2019) introduced the Santiago–Portilla method for representing any type of planar mechanisms independently of the number of elements or the link type. Huang et al. (2022) presented a computer-aided method to synthesize all feasible non-isomorphic mechanisms with one multiple-joint from the corresponding kinematic chains with one multiple-joint. Kang and Kim (2018) proposed a gradient-based topology optimization method to determine simultaneously the topology and joint type of planar linkage mechanisms. Deng and Purwar (2024) introduced a novel matrix-based approach for the simultaneous type and dimensional synthesis of planar four-bar linkage mechanisms. Vinícius Morlin et al. (2024) investigated the type synthesis of non-redundant planar kinematic chains applied to parallel mechanisms. Yim et al. (2021) proposed a neural-network-based big-data approach to simultaneously determine the mechanism topology and its end-effector location. The planar one-rotational one-translational and one-rotational two-translational parallel mechanisms that evolved from the crank-and-rocker mechanism are presented by means of Lie group theory (Jin et al., 2018). Ye et al. (2014) studied the characteristics of planar five-bar metamorphic linkages, and four kinds of planar five-bar metamorphic linkages were enumerated. Zhang et al. (2021) proposed a limb-constraint-based synthesis method that matches structural chains with the prescribed output motion of the moving platform. Liu et al. (2023) obtained a planar rhombus 4R mechanism by adding two or four revolute joints to a planar rhombus 4R mechanism, and then reconfigurable single-loop mechanisms were synthesized. Although mature type synthesis theories for planar linkages have been well established, due to the unique characteristics of Rubik's cube mechanisms, these methods are hardly applicable to the type synthesis design of the PPRCM.

Whether it is numerical puzzle games such as Magic Squares, the eight-digit puzzle, and the Backgammon or casual games like Klotski game, all of them achieve the final target state by moving blocks within a plane. Their essence lies in planar mechanisms with prismatic pairs. To obtain more PPRCMs with diverse topological structures, expand the types of PPRCM, and enrich products centered on PPRCM, this paper investigates the reconfigurable motion of PPRCM and proposes a type synthesis method for the PPRCM. Section 2 thoroughly illustrates the internal structure and reconfigurable motion process of the PPRCM. Section 3 provides a rigorous definition and analysis of the prismatic-pair contact surface(PPCS). Section 4 derives the constraint criteria for realizing PPRCM reconfiguration motion. Section 5 puts forward the type synthesis method for the PPRCM. Section 6 presents specific type synthesis cases and obtains PPRCMs with specific topological structures. Finally, the conclusions are drawn in Sect. 7.

Next, taking the PPRCM prototype illustrated in Fig. 1 as a research case, we elaborate upon the structural composition. Figure 2 presents the corresponding 3D printed model. This PPRCM mainly consists of two parts, including a base and moving modules. The moving modules include moving module M1, moving module M2, and moving module M3. The base features uniformly distributed grooves along the X0 and Y0 directions, which can dynamically connect the moving modules, allowing them to achieve translational movement along their respective grooves, as illustrated in Fig. 3. This design draws inspiration from the classic ancient Chinese tenon-and-mortise joint structure. The moving grooves serve as the moving paths. The moving modules can perform translational movements on the base along the X0 and Y0 directions, denoted as moving directions 0K2 and 0K1, respectively. Six paths are evenly distributed along the X0 direction, including 0Tj10K2, 0Tj20K2, 0Tj30K2, 0Tj40K2, 0Tj50K2, and 0Tj60K2. Similarly, six paths are evenly distributed along the Y0 direction, including 0Tj10K1, 0Tj20K1, 0Tj30K1, 0Tj40K1, 0Tj50K1, and 0Tj60K1.

Three distinct motion modes are available for the PPRCM presented in Fig. 1. Firstly, the moving module translates along the Y0 direction; secondly, the moving module translates along the X0 direction; thirdly, the moving block translates along the X0 direction on the moving-block side piece (MBSP). The moving block consists of a moving block and MBSPs. The MBSPs are equipped with clamping pins, which ensure that the MBSPs are engaged in the grooves of the base when moving along the X0 and Y0 directions. Meanwhile, the MBSPs are provided with grooves that allow the moving block to be locked onto them during movement along the X0 direction, as shown in Fig. 4. The contact region between the MBSPs and the base is square-shaped, enabling the moving module to move within the range of the PPRCM base along the X0 and Y0 directions (i.e., the first and second movement mode). In practice, the moving block is “attached” to the MBSPs during these scenarios, meaning the movement of the moving module is essentially the movement of the MBSPs. Each MBSP occupies a specific “position” on the base, and a vacant unoccupied slot is defined as an “empty position”. For the third movement mode, the contact interface between the moving block and the MBSP serves as the prismatic-pair contact surface, and the intersection of the plane perpendicular to the movement direction with this contact surface forms the profile of the prismatic-pair contact surface for that direction. Since the prismatic-pair contact surfaces are identical for the first two scenarios but differ in the third, this provides more configuration possibilities for PPRCM design. Accordingly, the third motion mode is chosen as the core research object for the subsequent theoretical derivation and analysis.

To intuitively and clearly visualize the prismatic-pair contact surfaces and their corresponding profiles, the PPRCM is simplified as illustrated in Fig. 1. Subsequently, the internal structure of all PPRCMs discussed hereafter can be referenced to the PPRCM shown in Fig. 3.

Since the PPRCM only involves translational motion, at least one empty position is required to realize its reconfiguration capability. Furthermore, this paper mainly targets the third motion mode; it is necessary that only one MBSP MC1 occupies a single position, as shown in Fig. 1. In the third scenario, since the moving module translates along the X0 direction on the MBSP, there must be adjacent MBSP in the direction of the moving block's movement. As depicted in Fig. 1, the moving blocks Mm2-1 and Mm2-2 cannot move along the X0 direction because there are no adjacent MBSPs in their X0 direction. In contrast, the moving blocks Mm1-1 and Mm1-2 can move along the X0 direction as these two modules have an adjacent MBSP MC1 in the X0 direction.

The PPRCM only contains prismatic kinematic pairs with diversified structural forms. To facilitate the analysis and design of PPRCM, a parametric representation of the PPCS is first established. A prismatic pair refers to a movable connection between two components with relative linear motion, serving as a constraint that provides only 1 translational degree of freedom. As illustrated in Fig. 5a, the moving component translates relative to the fixed component along the moving direction z1, and the direct contact area between the two components forms the PPCS. A plane that is perpendicular to the moving direction and capable of sectioning both the fixed component and the moving component simultaneously is defined as the cross-sectional plane. The intersection line formed by this cross-sectional plane and the PPCS is the profile l1 of the PPCS.

Next, a parametric description of the prismatic pair is performed. A Cartesian coordinate system o1-x1y1z1 is constructed, as illustrated in Fig. 5, where the cross-sectional plane serves as the o1-x1y1 plane and where the translational direction of the movable component is taken as the z1 axis. Since the PPCS is a curved surface formed by extruding the profile l1 along the moving direction, it can be uniquely determined under the coordinate system o1-x1y1z1 through the profile l1. Furthermore, as the profile of the PPCS lies entirely inside the o1-x1y1 plane of the coordinate system, it is expressed by the equation f1(x1,y1)=0. Additionally, when multiple PPCSs of different types exist in a single coordinate system, the parametric representations of all PPCSs can be obtained using the above-mentioned modeling approach.

In addition, owing to diversified reconfiguration motions, PPCSs can take different forms, as illustrated by the two distinct types shown in Fig. 5. A PPCS is classified as symmetric if there exists a plane parallel to the moving direction, with respect to which the PPCS is symmetric. Conversely, a PPCS is deemed to be asymmetric if no such plane parallel to the moving direction can be found to satisfy the symmetry condition. Specifically, as illustrated in Fig. 5a, there exists a plane N1 parallel to the moving direction, with respect to which the corresponding PPCS is symmetric; hence, this PPCS is categorized as symmetric. As shown in Fig. 5b, however, no plane N2 parallel to the moving direction can be identified to achieve symmetry for the relevant PPCS. Therefore, this PPCS should be defined as asymmetric. For simplicity, only symmetric PPCSs are adopted for all subsequent theoretical analyses in this paper.

For the components in the PPRCM to interchange positions and achieve reconfigurable motion, the following conditions must be satisfied.

Identical movement slopes.

The movement slope refers to the moving direction. In the same PPRCM, only components sharing identical translational axes are mutually interchangeable. Component interchange occurs between parts located on different moving paths but with the same moving direction, thereby enabling reconfigurable motion. When moving directions are inconsistent, components do not exhibit identical motion behaviors.

Based on the above analysis of the conditions for realizing reconfigurable motion of the PPRCM, a type synthesis method for PPRCM is proposed as follows:

Determine the moving direction number Ki in the PPRCM, namely the number of movement slope Ki (i=1, 2, …, n). Since the movement is linear within a plane, the moving directions of the PPRCM can be regarded as straight lines in the plane coordinate system. Different moving paths under the same moving direction are parallel straight lines, and different moving directions are straight lines with different slopes. Theoretically, here, the movement slopes can be arbitrarily assigned. In the PPRCM illustrated in Fig. 1, the number of moving directions is 2 along the Y0 direction and X0 direction, denoted as 0K2 and 0K1, respectively.

The procedural steps of the above-mentioned type synthesis method are summarized in Fig. 9.

Through the analysis of the reconfigurable motion of the PPRCM, the factors affecting the topological structure of the PPRCM can be obtained, including the number Ki of movement slopes (i=1, 2, …, n), the number j of moving paths in the moving direction Ki, and the types of PPCSs in the moving direction Ki. Next, different topological structures of the PPRCM can be obtained by specifying each of these factors.

This paper conducts a comprehensive analysis of the structural composition, topological configuration, and reconfigurable motion process of the PPRCM. Three factors that dominate the topological characteristics of the PPRCM are summarized, and the essential constraint criteria for achieving its reconfigurable motion are derived. A type synthesis method for the PPRCM is proposed, with specific synthesis steps given. By assigning values to the topological structure-influencing factors, PPRCMs with different topological structures are synthesized, and corresponding 3D prototype models are established, verifying the correctness of the proposed type synthesis method. The type synthesis method for the PPRCM proposed in this work can offer valuable theoretical references for relevant research on planar mechanisms and reconfigurable mechanisms.