Rigidly foldable origami tubes with open ends have been reported in the past. Here, using a mechanism construction process, we show that these tubes can be used as building blocks to form new tubes that are rigidly foldable with a single degree-of-freedom (SDOF). A combination process is introduced, together with a possibility of inserting new facets into an existing tube. The approach can be applied to both single and multilayered tubes with a straight or curved profile. Our work provides designers great flexibility in the design of tubular structures that require large shape change. The results can be readily utilized to build new structures for engineering applications ranging from deployable structures, meta-materials to origami robots.

## Introduction

Origami tubes have been used for various applications ranging from medical devices [1] to worm robots [2]. There have been considerable efforts to effectively fold these tubular structures without distorting their surfaces. Guest and Pellegrino [3] proposed a method where the cylindrical surface of a tube was dissected into a set of triangular facets to enable packaging. However, they proved that such tubes could only be folded if facets were allowed to deform. In other words, the tube was not rigidly foldable. A large number of patterns for both tubes and cones were also devised by Nojima [4,5] who focused on whether the folding patterns could be formed from a flat piece of paper, and the tube could be folded flat eventually. It was found later that none of the tubes and cones could be rigidly folded longitudinally. In fact, it has been proven that it is impossible to fold a tube with closed ends rigidly without distortion of its facets [6]. The effort was therefore redirected to tubes with open ends. Using a geometrical method, Tachi [7,8] devised a set of tubes with parallelogram facets that are rigidly foldable and they can be extended longitudinally to form multilayered tubes by repeating the same foldable unit. Subsequently, Liu et al. [9] showed that this could be done with a kinematic approach. The cross sections of these straight and curvy tubes, defined by a loop of lateral crease lines, are commonly of even-sided plane- or line-symmetric polygons, such as a kite or parallelogram. In addition, it was also found that a set of rigidly foldable tubes with parallelogram cross sections could be placed side by side, forming the Tachi-Miura polyhedron bellows [10,11]. Separately, Schenk and Guest [12] investigated the geometry of metamaterial based on a stack of the Miura-ori patterns, which could be considered as a special case of the polyhedron bellows. Recently, Filipov et al. [13,14] presented a type of tubes with reconfigurable parallelogram cross sections. In the formation of all the compound tubes outlined above, the individual tubes are simply placed next to each other without removing or adjusting the common sides among the adjacent tubes.

Kinematically, the rigidly foldable tubes are linkages where rigid links are connected by revolute joints. In the past, Goldberg created a number of 5R and 6R linkages by combining a number of 4R Bennett linkages [15,16]. This prompted us to think whether the same approach could be adopted to create novel rigidly foldable tubes. Our endeavor leads to this article.

This paper focuses on the rigid foldable origami tubes formed by combining several tubes with parallelogram or kite cross sections. The layout of the paper is as follows. First, in Sec. 2, we conjoin several existing tubes by merging common sides or corners, resulting in a family of tubes with asymmetric polygonal cross sections. Then, in Sec. 3, we add transition parts into an existing tube, leading to the second set of origami tubes in which the crease lines between neighboring layers form nonplanar polygons. The formation of multilayered and curved tubes based on the above-mentioned tubes is discussed in Sec. 4. Finally, conclusions are drawn in Sec. 5, which ends the paper.

## Tubes Formed by Combination

Goldberg 5R and 6R linkages are obtained by merging two or more Bennett linkages through a summation or subtraction process depending upon the relative positions of adjoined Bennett linkages. Here, new tubes are also formed by applying a similar approach to the sections of tubes.

### Summation of Two Tubes.

Figures 1(a) and 1(b) show two one degree-of-freedom (DOF) rigidly foldable tubes: Tubes 1 and 2. The facets of both tubes are parallelograms. The Tube 1 with kite cross section in Fig. 1(a) is formed by two pieces with facets in different lengths. Both the top and bottom pieces are flat developable with $αT1+γT1=π$, $βT1+δT1=π$, and $α1T1+γ1T1=π$, $β1T1+δ1T1=π$. To achieve the flat foldability, $αT1=βT1$ and $α1T1=β1T1$. In order to connect them into a one-DOF tube, $a cos α1T1=b cos αT1$ [9]. For the Tube 2 with parallelogram cross section in Fig. 1(b), the left and right two pieces are flat developable and with identical geometry [7]. The cross sections of Tubes 1 and 2, ABCD, are plane-symmetric and line-symmetric, respectively. The tube with a parallelogram cross section has only one flat folding state, while the one with a kite cross section has two flat folding states. If the two tubes are placed side by side such that they share a common side, they can be joined together via the common side, forming the compound tube shown in Fig. 1(c), with the conditions that

$αT1=αT2, βT1=βT2$
(1a)

$θT1+θT2=2π$
(1b)

where superscripts T1 and T2 represent Tubes 1 and 2, respectively. In addition, the widths of the facets of two tubes should match. Should these conditions be met, the combination does not alter the motion of each tube, and the combined tube has only one DOF, i.e., the compound tube is also rigidly foldable. Now we can remove the common side of the two tubes, resulting in a new origami tube that is rigidly foldable with one DOF, see Fig. 1(d). It can be found that the cross section of the resultant tube is neither in line- nor plane-symmetry. In fact, it can be an arbitrary polygon.

When vertices A of Tubes 1 and 2 are positioned at the same point, and
$γT1+γT2=π, δT1+δT2=π$
(2)

two adjacent facets from each tube can be welded together into one piece so that a new tube with a cross section consisting of an odd number of sides can be formed. The schematic diagrams in Figs. 2(a) and 2(b) summarize the summation approach in which a kite tube and a parallelogram one are combined, together with physical models demonstrating the folding of the resultant tubes. It should be pointed out that when the two tubes are joined via one of their longer sides, the folding on the short side of kite tube is not disturbed, and the combined tube still has two folding states as Fig. 2(b).

### Subtraction of Two Tubes.

Figure 3 illustrates the subtraction process in which the tube with smaller cross section, Tube 2, is nested inside the larger one, Tube 1. When geometric conditions

$γT1=γT2, δT1=δT2$
(3a)

$θT1=θT2$
(3b)
are satisfied and the widths of the facets of two tubes match each other, the subtraction can be done along one common side, see Fig. 3(c). Or if geometric conditions
$αT1=αT2, βT1=βT2$
(4a)

$γT1=γT2, δT1=δT2$
(4b)

$θT1=θT2$
(4c)

are met and the widths of the facets of two tubes match each other, the subtraction can be done at the common corner of the two tubes, Fig. 3(e), leading to two common sides. After removing the common parts, new tubes with combined and nonsymmetric cross sections can be obtained, see Figs. 3(d) and 3(f). The resultant tubes from either of the subtraction process retain the rigid foldability with one DOF. Figures 4 and 5 provide schematic diagrams of the subtraction process involving a common side and a common corner, respectively, accompanied by the folding sequences of physical models obtained through this process.

### Combination of More Tubes.

Not only can a pair of rigidly foldable tubes be combined to create new rigidly foldable tubes, but more tubes can be united using both the summation and subtraction approaches. An example is given in Fig. 6, which involves three tubes. First, Tube 1 and Tube 2 are summed up together, and then Tube 3 is subtracted from it, which results in a new tube with seven-side polygonal cross section (see Fig. 6(a)). Similar to Goldberg's method, we can also take away the summation of the Tubes 1 and 2 from Tube 3 to produce a clipped tube. The process is shown in Fig. 6(b). All of the new tubes retain rigid foldability, which is demonstrated by physical models.

## Tubes Formed by Adding Transition Parts

A rigidly foldable origami tube can also have transition parts added to produce a new rigidly foldable tube known as a shifted tube.

### Geometry and Kinematics of a Shifted Tube.

The origami tubes can be seen as an assembly of spherical 4R linkages [9], and the coordinate frames on the links and joints in a spherical 4R linkage can be set up as that shown in Fig. 7 following the D-H notations [17]. Here, $Zi$ (i = 1, 2, 3 and 4) is the coordinate axis along the revolute axis of joint i; $Xi$ is the coordinate axis normal to both $Zi−1$ and $Zi$ ($Xi=Zi−1×Zi$, when i − 1 = 0, it is replaced by 4); $θi$ is the revolute variable of joint i, which is the rotation angle from $Xi$ to $Xi+1$, positively about $Zi$; and $αi(i+1)$ is known as the twist angle of link i(i + 1) (when i + 1 > 4, it is replaced by 1), which is the rotation angle from $Zi$ to $Zi+1$ positively about axis $Xi+1$. The closure equation of this spherical linkage is

$Q12Q23Q34Q41=I$
(5)
where I is a 3 × 3 unit matrix and
$Qi(i+1)=[ cos θi sin θi0−cos αi(i+1) sin θi cos αi(i+1) cos θi sin αi(i+1) sin αi(i+1) sin θi−sin αi(i+1) cos θi cos αi(i+1)], i=1,2,3,4$
(6)

Now we shall show how a pair of transition parts can be added into a rigidly foldable tube to produce a new tube termed as a shifted tube. Figure 8(a) presents a rigidly foldable origami tube with a kite cross section. It is subsequently separated into two parts: the blue P1 and yellow P2. A pair of identical transition parts shown in Fig. 8(b), T1, are to be added between them. All of the facets in T1 have parallelogram shapes. The new tube with added transition parts is illustrated in Fig. 8(c). Next, we shall identify the conditions under which rigid foldability of the resultant tube can be achieved.

Since a tube can be seen as the assembly of spherical 4R linkages at each vertex, prior to adding the transition parts, at the vertex A surrounded by twist angles $α12$, $α23$, $α34$ and $α41$, there is
$cos α12 cos α41−sin α12 sin α41 cos θ1=cos α23 cos α34−sin α23 sin α34 cos θ3$
(7)

This is obtained by substituting Eq. (6) to Eq. (5).

Once a pair of transition parts T1 are added, vertex A becomes two: $A′$ with twist angles $α12$, $γ1$, $γ2$ and $α41$, and $A″$ with twist angles $π−γ1$, $α23$, $α34$, $π−γ2$, respectively. Applying closure Eqs. (5) and (6) to both vertices, we have
$cos α12 cos α41−sin α12 sin α41 cos θ1=cos γ1 cos γ2−sin γ1 sin γ2 cos θγ$
(8a)
and
$cos(π−γ1)cos(π−γ2)−sin(π−γ1)sin(π−γ2)cos θγ=cos α23 cos α34−sin α23 sin α34 cos θ3$
(8b)

Merging Eqs. (8a) and (8b) gives Eq. (7). The same conclusion could be drawn for the two vertices on the back of Fig. 8(b) where the other T1 is added. This demonstrates that adding a pair of identical transition parts formed by parallelogram facets to an existing tube does not change the relationship among the angles of the original tube. Hence, we can conclude that the new shifted tube is still rigidly foldable with one-DOF.

The addition of a transition pair may alter the foldability of the tube. In general, if $γ1≠γ2$, the resultant cross sections of the shifted tube are no longer planer, and thus, the shifted tube is not flat-foldable even when the original tube is flat foldable. Figure 9(a) shows such an example. On the other hand, if the original tube is flat foldable, and the added parts have $γ1=γ2$, then the shifted tube remains flat-foldable with a planar cross section. One such an example is shown in Fig. 9(b). The tubes with reconfigurable polygonal cross sections given in Ref. [14] belong to this category.

### Further Discussions of Shifted Tubes.

Using the method outlined above, more than a pair of transition parts can be added to the original tube without changing its DOF. Each pair must contain two identical parts with parallelogram facets. Figure 10 shows the rigid folding sequence of a physical model in which two transition pairs, labeled as T1 and T2, are added to a kite tube. Moreover, a pair of transition parts can be added to different places of the same original tube. We use two physical models given in Figs. 11(a) and 11(b), respectively, to illustrate this. In Fig. 11(a), the original tube with a six-sided polygonal cross section is divided into a four-sided left part and a two-sided right part before inserting a pair of transition parts, T1, to form a shifted tube. In Fig. 11(b), the same six-sided polygonal cross section of the original tube is partitioned into a three-sided left part and a three-sided right part instead, and then the transition pair T1 is added. The resultant tubes in Figs. 11(a) and 11(b) are obviously different, but both are rigidly foldable.

## Multilayered Straight and Curved Tubes

Multilayered tubes can be obtained by stacking single layered tube discussed in Sec. 3 with parallel cross sections to form a straight tube or nonparallel cross sections to form a curved one. In the latter case, some of the parallelogram facets are altered into trapezoid ones.

### Multilayered Tubes.

Stacking the same single-layered tubes outlined in Sec. 3 yields a long multilayered tube, which has straight profile in general. Several examples of such tubes are given in Fig. 12. The single-layered tubes used can be a combined or shifted tube.

### Curved Tubes With Combined Cross Sections.

A straight tube with a planar cross section can be transformed into a curved one by plane slicing away a portion of the tube. Figure 13 shows two single layered curved tubes created by this method. For the first tube, shown on the left in Fig 13(a), the top slicing plane has an inclination angle $εT1$, whereas that of the bottom slicing plane is $νT1$. As a result, the front and back parallelogram facets become trapezoid ones. If the same is applied to the second tube, Tube 2, with slicing plane angles $εT2$ and $νT2$, respectively, These two tubes can be combined in the same way as those straight tubes illustrated in Sec. 2. However, there are additional conditions, which are

$αT1=αT2, βT1=βT2$
(9a)

$θT1+θT2=2π$
(9b)

$εT1=εT2 and νT1=νT2$
(9c)
and the side lengths of the facets on commonly shared side must match. After removing the shared facets, a combined curved tube with six sides is obtained (see the left side of Fig. 13(b)). Moreover, if
$γT1 +γT2=π and δT1 +δT2=π$
(10)

the common creases of the combined tube can be removed, and the two facets on either sides of them can be bonded into a single facet. Now the resultant tube has five sides (see the right side of Fig. 13(b)).

Placing a number of such curved tubes one on top of another could lead to a multilayered curved tube. Figure 13(c) shows the folding sequence of a curved combined tube.

### Curved Shifted Tubes.

It is also possible to add transition parts into a curved tube while retaining its rigid foldability. Figure 14(a) is a portion of a rigidly foldable curved tube prior to any transition part is added. Denote the dihedral angles of adjacent facets between two neighboring layers by $ηL$ and $ηR$,respectively; there is

$cos α12 cos α41+sin α12 sin α41 cos ηL=cos δL=cos α23 cos α34+sin α23 sin α34 cos ηR$
(11)
where $δ$ is the angle between the ridgelines on the left side of the tube. Now add a pair of transition parts made with rectangular facets to the tube. Only the front part of the transition pair is shown in Fig. 14(b), and $γ1=γ2=(π/2)$. Geometrically, for the left side of the transition part, we have
$cos α12 cos α41+sin α12 sin α41 cos ηL=cos δL=cos γ1 cos γ2+sin γ1 sin γ2 cos δT=cos δT$
(12a)
For the right side of the transition part, there is
$cos δT=cos α23 cos α34+sin α23 sin α34 cos ηR$
(12b)

Comparing Eqs. (12a) with (12b) yields the same equation as Eq. (11). Hence, the relationship between two dihedral angles $ηL$ and $ηR$ is maintained despite that a transition part made from rectangular facets has been inserted.

The above derivation can also be applied to other facets in the transition part. As a result, the tube with multiple layers becomes rigidly foldable when a pair of transition parts consisting of only rectangular facets are appended. The reason why a pair is required is because doing so, the P1 and P2 are only translated.

We have yet to prove whether it is possible to produce shifted curved tubes with an additional transition pair with general parallelogram or trapezoid facets.

Figure 14(c) shows a curved tube constructed by this way. The original tube prior to adding the transition parts is flat-foldable, and it remains so after adding a pair of transition parts made of rectangular facets.

## Conclusions

In this paper, two methods to construct origami tubes using known rigid origami tubes are presented. The resultant tubes, known as combined and shifted tubes, remain rigidly foldable with a SDOF. They may have an asymmetric planar or nonplanar cross section. We also extended our approach to build multilayered straight and curved tubes while maintaining rigid foldability. Some of these tubes can be arranged side by side to produce compound tubes. The approach presented offers much great flexibility to designers when they are to fabricate rigidly foldable tubes to create meta-materials, origami robots, and other devices that require large shape change. The rigid foldability of these tubes ensures no facet distortion during such a shape change.

## Acknowledgment

Y. Chen acknowledges the support of the National Natural Science Foundation of China (Projects 51290293 and 51422506) and the Ministry of Science and Technology of China (Project 2014DFA70710). Z. You wishes to acknowledge the support of Air Force Office of Scientific Research (FA9550-16-1-0339). He was a visiting professor at Tianjin University while this research was carried out.

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