This paper presents an algebraic strategy for formulating the configuration
transformation of a special class of reconfigurable cube mechanism (RCM) made
by 2

There is a special class of reconfigurable cube mechanism (RCM) equivalent to an eight-bar closed-loop spatial linkage with cyclic reconfiguration, Fig. 1. The mechanism can demonstrate eight different operation configurations, i.e., configuration A to H in Fig. 1, where nine different figures are exposed on the outer surface of the RCM. An earlier study (Kuo and Su, 2017) to this special mechanism has shown that the manipulation of this elegant artifact can be interpreted by using the mechanism theories in terms of variable mobility and isomorphism identification.

When investigating the topological properties of reconfigurable mechanisms, configuration transformation is one interesting topic to be explored. In literature, a couple of studies have made significant contributions to the configuration transformation analysis for some specific reconfigurable mechanisms. For example, Wohlhart (1996) introduced a special linkage, namely “kinematotropic linkage,” that could permanently change its mobility by reconfigure the linkage into different working configurations. Dai and Rees Jones (1999a, b, 2005) studied a special foldable/erectable mechanism and developed an EU-elementary matrix operation for formulating its topology transformation (Dai and Rees Jones, 2005). Zhang and Dai (2008, 2009) proposed an evolutionary reconfiguration algorithm of general spatial metamorphic mechanisms. Yan and Kang (2009a) studied the configuration transformation of variable topology mechanisms based on the concept of mapping function, leading to a general methodology for configuration synthesis of variable topology mechanisms (Yan and Kang, 2009b). Yan and Kuo (2007, 2009) put forward a systematic approach for configuration analysis and synthesis for general variable topology mechanisms where the topological reconfiguration can be described by graph (Yan and Kuo, 2006a), finite state machine (Yan and Kuo, 2006b), and screw matrix (Kuo and Yan, 2007). Ghrist and Peterson (2007) realized the reconfiguration of reconfigurable systems in robotics and biology by using state complex technique.

In addition to the general theories of reconfiguration, some tailor-made approaches for formulating the configuration transformation were available for several specific applications. For instance, Liu and Dai (2002) investigated the folding process of packaging cartons, leading to a reconfiguration methodology and algorithm for the reconfigurable carton folding. Ding and Yang (2012) evaluated the geometry and reconfiguration principles of a special Mandala-type artifact and discussed its application for aerospace engineering. Wei et al. (2010, 2011) analyzed the configuration singularity and reconfiguration properties of a Hoberman switch-pitch ball. Ding et al. (2013) designed a deployable polyhedral linkage for which the changeable configuration was presented by joint screws. Ding and Lu (2013) analyzed the motion sequence and isomorphism of a chain-type cube mechanism. Gan et al. (2009, 2010, 2013a, b) proposed a metamorphic parallel mechanism whose configuration was changed via a special reconfigurable joint. Zhang et al. (2010a, 2012) presented other fantastic metamorphic parallel mechanisms via the concepts of variable-axis joints, origami folding (Zhang et al., 2010b), and kirigami (Zhang and Dai, 2014).

The reconfigurable cube mechanism (RCM).

On the other hand, topology synthesis of reconfigurable mechanisms is also an interesting and challenging problem. In the past decades, topology synthesis of reconfigurable mechanisms has been attempted for several special mechanisms, e.g., the kinematotropic linkages (Galletti and Fanghella, 2001), metamorphic mechanisms (Zhang et al., 2008; Zhang and Dai, 2009), variable topology mechanisms (Yan and Kuo, 2009; Kuo and Chang, 2014; Shieh et al., 2011), and reconfigurable mechanisms (Kuo et al., 2009; Huang et al., 2010). However, those studies were mostly focused on the deployable or folding mechanisms. For the reconfigurable mechanisms formed by connected sub-cubes that we study here, its topology synthesis and enumeration tasks are still an open problem.

Therefore, the aim of this work is to develop an algebraic formulation for
describing the configuration transformation of the RCM made by 2

By observing the RCM in Fig. 1, the topological and reconfiguration characteristics of the RCM can be concluded as follows.

The topological characteristics of the RCM include:

The RCM is always a single-loop closed kinematic chain during reconfiguration.

All the sub-cubes are topologically similar class. In the RCM, the links are all binary links, the joints are all revolute joints, and the linkage is a single-loop chain. Therefore, each link is topologically identical to each other, i.e., the links are similar class (Harary, 1964).

The orientation of each joint is not changed after the joint is
reconfigured. Referring Fig. 4 in (Kuo and Su, 2017), for example, when
the blue sub-cube is grounded, the RCM can verify a series of configuration
changes as shown from Fig. 4a to h. Then, it can be verified that the
orientation of each joint will remain the same in all configurations, even if
the joint axis is displaced from some configuration to another one. For
example, joint

Each configuration must have joints pointing at the

Since all the joints are incident to the edges of the sub-cubes, all the joints form an orthogonal pattern in each configuration, i.e., they are either parallel or orthogonal to one another in each configuration.

In each configuration, there must have exactly two, two, and four joints pointing at the three axial directions, respectively.
For example, there may have two joints pointing at the

The reconfiguration characteristics of the RCM include:

When any two joints are being coaxial, they will become a pair of workable joints, i.e., the degree of freedom of motion of the joint is not restrained by the configurations or the link shape.

The workable joints must appear in pairwise—there is no single workable joint on an axis.

When there exists a pair of workable joints in the configuration,
this configuration is able to transform into the next one. For example, in
Fig. 4h of (Kuo and Su, 2017), joints (

The configuration of the RCM can be classified into “operation status” or “transition status.” When the RCM is at an operation status, it has two or more pairs of workable joints that are pointing at two different orientations. On the other hand, when the RCM is at a transition status, it has only one pair of workable joints. Therefore, referring Fig. 4 in (Kuo and Su, 2017), configurations A, C, D, E, G, and H are at operation status, whereas configurations B and F are at transition status.

The transformation from configuration

The operation of a working joint is either a forward or a backward
operation. A configuration is said under a forward operation when its working
joints are different from the ones in the previous two
transformations

The two coaxial working joints of a configuration
can point at either

A backward operation of the working joints will induce a repeated
transformation or reverse the direction of transformation of the
reconfiguration. For instance, in the previous example, the operation of
joints (

Each configuration has only one forward operation for forming a non-repeated, cyclic transformation. This property can be verified from Fig. 2.

The reconfiguration sequence of the RCM.

The configuration transformation of the RCM is an interesting problem In our previous study (Kuo and Su, 2017), we have shown that the configurations of the RCM can be represented by a matrix of joint screws. In what follows, a computational procedure is presented for deriving all the configurations of the RCM based on a given screw matrix of the initial configuration.

The flowchart of the computational procedure for formulating the configuration transformation process is given in Fig. 3. The detailed procedure is introduced as follows.

A computational procedure for determining all feasible configurations of the RCM.

The RCM at the initial configuration.

When an initial configuration of the RCM is given, a Cartesian coordinate system is attached to some link as a referencing coordinate system for the reconfiguration. For convenience, the origin of the system is set to some corner of the sub-cube, and the three coordinate axes are to point along the edges of the cube. For example, Fig. 4a is a given initial configuration of an RCM (same as Fig. 5a in Kuo and Su, 2017) and a coordinate system is attached onto link 2 as shown. Note that the arrangement of the coordinate system is independent of the derived results, i.e., it can be arbitrarily set to any links and any corner.

The movable and immovable link/joint groups.

Now, the joint-screw matrix

Since the RCM linkage has no specific ground link, one link should be grounded to be the reference of the relative motions of the other sub-cubes. Any link of the RCM can be selected as the ground. For example, for the RCM in Fig. 4a, link 2 is selected as the ground link. Accordingly, the RCM becomes a linkage mechanism as depicted in Fig. 4b.

As stated in Sect. 2, a workable joint is a joint whose degree of freedom of
motion is not restrained by the configuration or the link shapes. According
to the reconfiguration characteristics described in Sect. 2, when two joints
become coaxial, they will together form a pair of workable joints. So our
next step is to identify the group of the workable joints from the
joint-screw matrix. For example, it can be easily identified from Eq. (1)
that vectors (

Now one group of the workable joints is chosen as the group of working joints
for actuating the RCM at the configuration. In order to derive all follow-up
feasible configurations of the RCM, i.e., a cyclic reconfiguration without
repeated configurations, only the group of working joints that is at
forward operation as introduced in Sect. 2 can be chosen. To do this, the
selected working joint group should be compared with the working joint groups
in the previous two configurations. If the selected working joint group is
not as same as that in the previous two configurations, it will be a feasible
working joint group. This comparison should be continued until the working
joint group has been identified. Since each configuration has only one
forward operation, the selection result of the working joint group will be
unique. If there are no previous configurations to be compared, e.g., the
current configuration is the first or second configuration, this check can be
ignored. For example, for the configuration in Fig. 4, joint group (

After the working joint group is identified, all the remaining joints can be
divided into two kinds, the movable joints and immovable joints. A movable
joint means that its joint axis will be displaced as the working joints are
actuated. Reversely, the location of the joint axis of an immovable joint
will be unchanged when the working joints are functioning. The identification
of movable and immovable joints is illustrated via the topological graph in
Fig. 5. In this graph, the vertices represent the links with their numbering
and the edges are the joints with their labeling. The ground link, link 2, is
labeled with a concentric circle. In the previous step, joint group (

When a pair of joints (

The position vector of a joint in the unit RCM is measured from the origin of
the reference system to the middle point of the joint. For example, the
position vector of joint

Orientation and position vectors of the RCM at the new configuration.

After obtaining the new position vectors of all movable joints, the
joint-screw matrix of the new configuration can be constructed. First, the
joint screws of the immovable joints will be the same in the new
configuration. Second, for the moveable joints, the joint orientations will
be invariant in all configurations and the position vectors of the joints
have been derived in the previous step. So the joint-screw matrix

The RCM at the new configuration.

Initial configuration of the illustrative RCM.

Computation results for the configuration transformation of the illustrative RCM.

After the joint-screw matrix

Here the presented computational procedure is illustrated by taking Fig. 7 as an example. Figure 7 shows the initial configuration of an RCM to be manipulated. For illustration, the lower-right sub-cube is selected as the ground link in each following configuration. The computation results are summarized in Fig. 8.

Based on the configuration transformation algorithm proposed above, all
possible topological configurations of the RCMs with 2

A computational procedure was presented for formulating the configuration
transformation of the RCM with 2

The authors declare that they have no conflict of interest. Edited by: X. Ding Reviewed by: three anonymous referees