We start with analyzing the degree of freedom (DOF) of a single-layer Yoshimura-ori structure. With rigid-folding assumption, the whole structure can be equivalent to a space truss frame. Hence, the DOF of a single-layer Yoshimura-ori structure can be determined by (Saito et al., 2015): N DOF = 3 ⋅ N vertex − N crease − 6 , (2) where N vertex is the total number of vertex, N crease is the total number of creases. For a single layer Yoshimura-ori structure, N vertex = 12 and N _crease = 30; hence, the DOF of the Yoshimura-ori structure is zero, which indicates that it cannot be transformed from the open configuration to the closed configuration by rigid folding. To achieve such transformation, facet bending is a necessity.

To understand the folding kinematics of the Yoshimura-ori structure, a cartesian coordinate system O-xyz is set, with the origin locating at the centroid of the equilateral triangle C 1 C 2 C 3 ¯ (its position and orientation remain fixed during folding). Considering the C 3 rotational symmetry (Figure 2B) and the top-down reflectional symmetry (i.e., symmetric with respect to the plane B 1 B 2 B 3 ¯ ) of a single layer Yoshimura-ori structure, examining a rectangle “unit” A 1 A 2 P 2 P 1 ¯ is sufficient to determine the folding motion. The positions of each vertex are not independent, instead, they are constrained by the geometric lengths of the creases “ A 1 A 2 ¯ ,” “ A 1 P 1 ¯ ,” “ A 2 P 2 ¯ ,” “ A 1 B 1 ¯ ,” and “ A 2 B 1 ¯ ,” which give rise to the following constraint equations: l A 1 A 2 ¯ = L , l A 1 P 1 ¯ = l A 2 P 2 ¯ = 3 L 6 , l A 1 B 1 ¯ = l A 2 B 1 ¯ = 3 L 2 . (3)

As mentioned early, folding of the Yoshimura-ori structure calls for elastic deformations of certain facets or creases. To understand the folding behavior, the lengths of the crease P 1 B 1 ¯ and P 2 B 1 ¯ (i.e., ‖ l P 1 B 1 ¯ ‖ and ‖ l P 2 B 1 ‖ ) are assumed to be mathematically variable during the whole folding process. Due to the reflection symmetry of the structure about the plane A 1 C 1 B 3 ¯ , ‖ l P 1 B 1 ¯ ‖ = ‖ l P 2 B 1 ¯ ‖ . This provides the structure with an additional DOF to transform between the closed and the open configurations. With such a conceptual simplification, the deformation of the structure can be quantified via a single parameter. Here, we choose the height of a single layer Yoshimura-ori structure (Figure 2A), denoted by h, as the independent variable, because it is related to the actuation in the robot prototype (to be discussed in Section 5). Hence, the position of each vertex can be derived: A 1 = [ 3 L / 3 , 0 , h ] , A 2 = [ − 3 L / 6 , L / 2 , h ] , B 1 = [ 1 2 ( 3 L 6 + L 2 − 3 h 2 12 ) , 3 2 ( 3 L 6 + L 2 − 3 h 2 12 ) , h 2 ] , P 1 = [ ( 3 L 3 − L 2 − 3 h 2 12 ) , 0 , h 2 ] , P 2 = [ − 1 2 ( 3 L 3 − L 2 − 3 h 2 12 ) , 3 2 ( 3 L 3 − L 2 − 3 h 2 12 ) , h 2 ] . (4)

Based on this, the distance between P 1 and B 1 can be calculated, and accordingly, the theoretical strain of the crease P 1 B 1 ¯ can be obtained. Figure 2C displays the evolution of the strain with respect to the height h . When h = 0  mm and h = 34.64  mm , the strain is zero, indicating that the structure has no elastic deformation at either the open ( h = 34.64  mm ) or the closed ( h = 0  mm ) configuration. With the increase of h from 0 to the maximum, the strain is non-zero and remains negative, suggesting that the crease has to be shortened to achieve the transformation from the closed state to the open state.

While the strain could effectively illustrate the non-rigid-folding of the structure, shortening of the free length P 1 B 1 ¯ does not agree with the actual deformation of a single-layer Yoshimura-ori prototype. To better describe the folding behavior, facet bending is taken into account. This is because the primary deformation mode of a facet is bending, rather than stretching. Specifically, to account for the bending behavior, a “virtual crease” is introduced into the bending facet, which represents the “hidden” DOF related to the out-of-plane bending deformation. According to experimental observations, two virtual creases, A 1 D 1 ¯ and A 2 D 2 ¯ , are placed, as shown in Figure 3A. Hence, facet A 1 B 1 P 1 ¯ is divided into two rigid triangular panels A 1 P 1 D 1 ¯ and A 1 D 1 B 1 ¯ ; A 2 B 1 P 2 ¯ is also divided into two rigid triangular planes A 2 D 2 B 1 ¯ and A 2 P 2 D 2 ¯ . Noting that panels A 2 D 2 B 1 ¯ and A 2 P 2 D 2 ¯ are mirror reflections of panels A 1 P 1 D 1 ¯ and A 1 D 1 B 1 ¯ , hence, in what follows, only the virtual crease A 1 D 1 ¯ and the associated panels A 1 P 1 D 1 ¯ and A 1 D 1 B 1 ¯ are considered.

When the Yoshimura-ori structure is axially contracted or extended, the position of vertex A 1 is still determined by Equation 4, while the newly introduced vertex D 1 is free to move, providing that the following geometric constraints are satisfied: l P 1 D 1 ¯ = constant ,   l D 1 B 1 ¯ = constant ,   l P 1 D 1 ¯ + l D 1 B 1 ¯ = L 2 , (5a) l A 1 D 1 ¯ = l P 1 D 1 ¯ 2 + l A 1 P 1 ¯ 2 = constant . (5b)

Equation 5a indicates that the crease lengths corresponding to P 1 D 1 ¯ and D 1 B 1 ¯ remain constant during folding, and their sum equals the initial distance between vertices P 1 and B 1 (i.e., L / 2 ). Here, the lengths of creases P 1 D 1 ¯ and D 1 B 1 ¯ are set based on observations. Equation 5b indicates that the length of the virtual crease A 1 D 1 ¯ remains constant during folding, which is determined from the open or closed configuration.

Combining Equations 4, 5, the coordinates of all vertices can be determined at each step of folding via numerical methods. Further, the dihedral angles between adjacent facets can be calculated based on the law of cosines, in which the normal vectors of the associated facets are used. For example, the dihedral angle at the crease A 1 B 1 ¯ , i.e., θ A 1 B 1 ¯ , can be calculated via V 1 → = D 1 B 1 → × B 1 A 1 → , (6a) V 2 → = B 1 A 1 → × A 1 A 2 → , (6b) θ A 1 B 1 ¯ = cos − 1 ( V 1 → ⋅ V 2 → ‖ V 1 → ‖ ⋅ ‖ V 2 → ‖ ) , (6c) Where V 1 → and V 2 → are normal vectors of panels A 1 D 1 B 1 ¯ and A 1 D 1 B 1 ¯ , respectively, ‖ • ‖ denotes the Euclidean length. Note that there are four dihedral angles that are related to the studied panels A 1 P 1 D 1 ¯ and A 1 D 1 B 1 ¯ , they are: θ A 1 B 1 ¯ , θ A 1 D 1 ¯ , θ A 1 P 1 ¯ , and θ A 1 A 2 ¯ (defined as the dihedral angle between the panel A 1 A 2 B 1 ¯ and the plane A 1 A 2 A 3 ¯ ). Figure 3B displays the four dihedral angles as functions of the height h . It reveals that when h = 0  mm and h = 34.64  mm , the virtual fold angle θ A 1 D 1 ¯ is 180 ∘ because the facet A 1 B 1 P 1 ¯ remains flat at the open and closed configurations. With the virtual crease, folding of the single-layer Yoshimura-ori structure can be treated as the rotation of rigid facets with respect to the elastic hinge-like creases. The obtained dihedral angles will be used to evaluate the mechanics of the Yoshimura-ori structure.

Bending motion is another important deformation pattern of the Yoshimura-ori structure. In this subsection, the bending motion of a single-layer Yoshimura-ori structure is examined, which is a prerequisite for evaluating the reachable workspace. Bending of a single-layer Yoshimura-ori structure can be achieved by reducing the distance between two vertices (e.g., A 1 and C 1 ). During the bending process, the fixed bottom plane C 1 C 2 C 3 ¯ and the top plane A 1 A 2 A 3 ¯ of the structure are assumed to be rigid. With the same O-xyz coordinate system (Figure 4A), the bending motion of a single-layer Yoshimura-ori structure can be described by the intersecting angle α between the planes A 1 A 2 A 3 ¯ and C 1 C 2 C 3 ¯ (Figure 4B) or the z coordinate of vertex A 1 (denoted by h 1 ). Through geometric analysis, the coordinates of A 1 , A 2 , and A 3 are given by: A 1 = [ 3 L / 3 + h 1 ⁡ tan ( α / 2 ) , 0 , h 1 ] , (7a) A 2 = [ 3 L 3 + h 1 ⁡ tan ( α 2 ) − 3 L 2 cos ⁡ α , L 2 , h 1 + 3 L ⁡ cos ⁡ α 2 ] , (7b) A 3 = [ 3 L 3 + h 1 ⁡ tan ( α 2 ) − 3 L 2 cos ⁡ α , − L 2 , h 1 + 3 L ⁡ cos ⁡ α 2 ] . (7c)

For bending motion, the structure is only reflectional symmetric about the plane A 1 C 1 B 2 ¯ , while the C 3 rotational symmetry and the top-down reflectional symmetry no longer hold. Hence, for bending motion, half of the single-layer Yoshimura-ori structure (denoted by shade in Figure 4B) needs to be taken out for investigation.

During bending, constraint (3) still holds. To understand the bending behavior, the rigid-folding condition has to be broken by assuming that the lengths of the creases P 1 B 1 ¯ , P 2 B 1 ¯ , P 2 B 2 ¯ , P 3 B 2 ¯ , P 3 B 3 ¯ , and P 1 B 3 ¯ are mathematically variable. Considering the reflection symmetry, the following constraints are applied l P 2 B 1 ¯ = l P 2 B 2 ¯ = l P 3 B 2 ¯ = l P 3 B 3 ¯ ,   l P 1 B 1 ¯ = l P 1 B 3 ¯ . (8)

Solving Equations 3, 8 through numerical methods, the coordinates of vertices B 1 , B 2 , B 3 , P 1 , P 2 , and P 3 at each step of folding can be expressed by α and h 1 . Theoretically, the variables α and h 1 ( α > 0 ,   h 1 > 0 ) can be independent during bending motion. However, experimental observation indicates that they are partially interrelated. Hence, to reduce the numbers of variables, α and h ₁ are assumed to obey the following relation α ≈ α max − h 1 h 1 max α max , (9) Where α max is the maximum bending angle when h 1 reaches zero, and h 1 max is a value of h 1 corresponding to zero bending angle. In this research, α max and h 1 max can be obtained through geometric measurement of the prototype. Consequently, α is adopted as the independent variable to describe the bending motion, and the coordinates of all vertices can be expressed as functions of α . Particularly, if α max is assigned to be 0, the bending deformation model will degenerate into the axial deformation model discussed in Section 3.1.

Similar to the axial kinematics, assuming variable lengths of the creases does not agree with the structure’s actual deformation. To evaluate the evolution of the dihedral angles during bending, virtual creases are added again. As shown in Figure 5, six virtual creases are introduced into the facets A 1 B 1 P 1 ¯ , A 2 B 1 P 2 ¯ , A 2 B 2 P 2 ¯ , B 1 C 1 P 1 ¯ , B 1 C 2 P 2 ¯ , and B 2 C 2 P 2 ¯ , respectively; each virtual crease divide the corresponding facet into two rigid panels. Referring to Equations 5, 8 and considering the symmetry, the positions of the virtual vertices D i ( i = 1,2...6 ) are constrained by: l P 1 D 1 ¯ = l P 2 D 2 ¯ = l P 2 D 3 ¯ = l P 1 D 4 ¯ = l P 2 D 5 ¯ = l P 2 D 6 ¯ = constant , (10a) l D 1 B 1 ¯ = l D 2 B 1 ¯ = l D 3 B 2 ¯ = l D 4 B 1 ¯ = l D 5 B 1 ¯ = l D 6 B 2 ¯ = constant , (10b) l P 1 D 1 ¯ + l D 1 B 1 ¯ = l P 1 D 4 ¯ + l D 4 B 1 ¯ = l P 2 D 2 ¯ + l D 2 B 1 ¯ = l P 2 D 5 ¯ + l D 5 B 1 ¯ = l P 2 D 3 ¯ + l D 3 B 2 ¯ = l P 2 D 6 ¯ + l D 6 B 2 ¯ = L 2 , (10c) l A 1 D 1 ¯ = l A 2 D 2 ¯ = l A 2 D 3 ¯ = l C 1 D 4 ¯ = l C 2 D 5 ¯ = l C 2 D 6 ¯ = constant . (10d)

Similarly, Equations 10a, 10b indicate that during folding, the distances from vertices P 1 , B 1 , P 2 , B 2 to the newly introduced vertices D i   ( i = 1 , ... , 6 ) remain constant, Equation 10c implies that the corresponding sum equals to the initial distance between vertices P 1 and B 1 (or P 2 and B 2 , equal to L / 2 ), and Equation 10d suggests that the lengths of the virtual creases keep constant, which is determined from the open or closed configuration.

Combining Equations 7–10, the coordinates of all vertices can be determined at each step of bending deformation. Based on this, the dihedral angles between adjacent facets can also be calculated based on the law of cosines (exemplified in Equation 6). The obtained dihedral angles are preconditions for evaluating the mechanics of bending motion in Section 4.2.

Based on the model of a single Yoshimura-ori layer, we then investigate the reachable workspace of the Yoshimura-ori structure without considering the effect of gravity. The reachable workspace is affected by the number of layers and the number of segments. Here, as an example, we first examine the reachable workspace of a six-layer Yoshimura-ori segment (Figure 5A). For describing purposes, the vertices of the structure are named as C i , j k in this part (Figure 5A), where the subscript i indicates the layer number from the bottom up ( j = 0 , ... , 6 ), the subscript j denotes the vertex number ( i = 1,2,3 ) in each layer, and the superscript k denotes the segment number ( k = I , II , ... ). The established coordinate system remains the same as that in Figures 2–4, 6. Hence, for layer #1, a one-to-one correspondence between the coordinates of the vertices in Figure 4A and those in Figure 5A can be obtained ( x C 0 , j k y C 0 , j k z C 0 , j k ) | j = 1,2,3 = ( x C j y C j z C j ) | j = 1,2,3 , (11a) ( x C 1 , j k y C 1 , j k z C 1 , j k ) | j = 1,2,3 = ( x A j y A j z A j ) | j = 1,2,3 . (11b)

In this research, the deformation of each Yoshimura-ori layer is characterized via the position of the endpoint, which locates at the centroid of the top equilateral triangle. During axial and bending motions, both the top and bottom equilateral triangles remain unchanged. Specifically, the coordinates of the centroid O i k of the plate C i , 1 k C i , 2 k C i , 3 k ¯ yields x O i k = x C i , 1 k + x C i , 2 k + x C i , 3 k 3 , y O i k = y C i , 1 k + y C i , 2 k + y C i , 3 k 3 , z O i k = z C i , 1 k + z C i , 2 k + z C i , 3 k 3 . (12)

Particularly, the coordinates of the endpoint O 6 I is employed to characterize the reachable workspace of the six-layer Yoshimura-ori structure as a whole. Based on the assumption that the constituent layers’ bending angles are uniform, the coordinates of O 6 I can be obtained via coordinate transformation [ x O i k , y O i k , z O i k ] = [ x O i − 1 k , y O i − 1 k , z O i − 1 k ] T + [ x O i − 1 k , y O i − 1 k , z O i − 1 k ] , i = 2 , ...6 , (13) Where T is the coordinate transformation matrix T = ( cos ⁡ α 0 − sin ⁡ α 0 1 0 sin ⁡ α 0 cos ⁡ α ) . (14)

To make a quantitative evaluation, the value of the variable α and the height h 1 max corresponding to zero bending angle are determined based on the observation of a six-layer Yoshimura-ori structure prototype (Figure 5B). Specifically, the angle α of a single layer is allowed to vary between 0 ∘ and 20 ∘ ( 119.6 ∘ / 6 ≈ 20 ∘ ), and h 1 max is set to be 16.6 mm ( 99.8 mm / 6 ≈ 16.6 mm ). Figure 5C shows the theoretical workspace of a six-layer Yoshimura-ori structure. The blue dots denote the reachable positions achieved by axial deformation, which ranges from 0 to 100 mm. The red dots represent the reachable positions achieved by a point-to-point bending motion. Considering that the orthogonal projection of the Yoshimura-ori structure is a hexagon, point-to-point bending can be achieved along six directions, representing by the six red curved tracks in Figure 5C. The lowest point of each curved track is attained when the Yoshimura-ori structure is fully bent, with α = 20 ∘ for each layer.

If involving more segments into the structure, the reachable workspace can be expanded significantly. As an example, we analyze the reachable workspace of a Yoshimura-ori structure with two segments (each segment consists of six layers). The connection between segments is rigid so that the deformation of each segment remains independent. Figure 5D displays a configuration in which segment I is under axial deformation, and segment II is under bending deformation along vertices C 0,2 II to C 6,2 II . With two segments, the furthest reachable distance of the endpoint O 6 II will expand twice accordingly. Moreover, if the two segments deform differently, the combination of their deformations would let the endpoint O 6 II reach new locations, thus, increasing the density of reachable positions in the workspace. Particularly, the endpoint could also reach the space with negative z values (gray dots). If the number of segments is increased to three, the furthest reachable distance of the endpoint O 6 III will raise three times, and the reachable positions in the workspace would become denser.

With the virtual creases, folding of the Yoshimura-ori structure can be considered as rotations of rigid facets with respect to the elastic hinge-like creases. Hence, the potential energy of the structure with respect to folding can be evaluated. By assigning k L as the torsional stiffness per unit length of the creases, and k B as the torsional stiffness per unit length of the virtual creases (bending stiffness), the elastic potential energy of the structure can be determined, which is the sum of the potential energies originated from the creases and the virtual creases Π crease = k L 2 [ 4 3 m n L 3 ( θ A 1 B 1 ¯ − θ A 1 B 1 ¯ 0 ) 2 + 4 L n ( m − 1 ) ( θ A 1 A 2 ¯ − θ A 1 A 2 ¯ 0 ) 2 + 3 n m L 3 ( θ A 1 P ¯ 1 − θ A 1 P 1 ¯ 0 ) 2 ] , (15a) Π virtual = 2 n m k B l A 1 D 1 ¯ ( θ A 1 D 1 ¯ − θ A 1 D 1 ¯ 0 ) 2 , (15b) Π total = Π crease + Π virtual . (15c)

In Equation 15, θ denotes the dihedral angles between adjacent facets, which have been derived in Section 3 and are functions of the height h ; n and m represent the number of basal rectangles and the number of layers, respectively (Figure 1A). The stress-free configuration is defined as the open configuration (Figure 1B), and its corresponding dihedral angles are indicated by the superscript “0.” The potential energy profile can then be used to calculate the constitutive force-displacement relation by taking the derivative with respect to the displacement h , i.e., F = d Π total d h . (16)

We then experimentally examine the mechanical behavior of the Yoshimura-ori structure during axial deformation. With the fabrication method introduced in Section 2, four 3 × 6 Yoshimura-ori structure prototypes are fabricated with PETE films of different thicknesses: 0.025, 0.05, 0.075, and 0.1 mm. For each prototype, three quasi-static compression and extension tests are performed (see the insets of Figure 7). The original length of the structure is 100 mm, we first compress it to 20 mm and then stretch it to 140 mm, with a loading speed of 0.5 mm/s. By averaging the three test results for each prototype, the corresponding force-displacement curve can be obtained (Figure 7). It shows that the constitutive force-displacement profile can be effectively tailored by adjusting the film thickness. With thicker film, the restoring force at the maximum-length configuration is much larger. For example, the restoring force grows more than 5.5 times from 0.67 to 3.72 N as the film thickness increases from 0.05 to 0.1 mm.

Similar to the scenario of axial deformation, the potential energy corresponding to the bending motion can also be derived via dihedral angles, lengths of creases, and torsional stiffness Π crease = k L 2 { 2 3 m L 3 [ ( θ A 1 B 1 ¯ − θ A 1 B 1 ¯ 0 ) 2 + ( θ A 2 B 1 ¯ − θ A 2 B 1 ¯ 0 ) 2 + ( θ A 2 B 2 ¯ − θ A 2 B 2 ¯ 0 ) 2 + ... ( θ C 1 B 1 ¯ − θ C 1 B 1 ¯ 0 ) 2 + ( θ C 2 B 1 ¯ − θ C 2 B 1 ¯ 0 ) 2 + ( θ C 2 B 2 ¯ − θ C 2 B 2 ¯ 0 ) 2 ] + 3 m L 3 [ ( θ A 1 P 1 ¯ − θ A 1 P 1 ¯ 0 ) 2 + ... ( θ C 1 P 1 ¯ − θ C 1 P 1 ¯ 0 ) 2 + 2 ( θ A 2 P 2 ¯ − θ A 2 P 2 ¯ 0 ) 2 + 2 ( θ C 2 P 2 ¯ − θ C 2 P 2 ¯ 0 ) 2 ] + 2 L ( m − 1 ) [ 2 ( θ A 1 A 2 ¯ − θ A 1 A 2 ¯ 0 ) 2 + ...2 ( θ C 1 C 2 ¯ − θ C 1 C 2 ¯ 0 ) 2 + ( θ A 2 A 3 ¯ − θ A 2 A 3 ¯ 0 ) 2 + ( θ C 2 C 3 ¯ − θ C 2 C 3 ¯ 0 ) 2 ] } , (17a) Π virtual = m k B l A 1 D 1 ¯ [ ( θ A 1 D ¯ 1 − θ A 1 D 1 ¯ 0 ) 2 + ( θ A 2 D 2 ¯ − θ A 2 D 2 ¯ 0 ) 2 + ( θ A 2 D 3 ¯ − θ A 2 D 3 ¯ 0 ) 2 + ... ( θ C 1 D 4 ¯ − θ C 1 D 4 ¯ 0 ) 2 + ( θ C 2 D 5 ¯ − θ C 2 D 5 ¯ 0 ) 2 + ( θ C 2 D 3 ¯ − θ C 2 D 6 ¯ 0 ) 2 ] , (17b) Π total = Π crease + Π virtual , (17c) Where the variables m , L , k B , k L has been defined in Section 4.1. Note that during bending, the bending angle α is the independent variable. Therefore, all the dihedral angles, and hence, the total potential energy of the structure, can also be expressed as functions of α . However, it is worth pointing out that during quasi-static tests, bending of the 3 × 6 Yoshimura-ori structure is achieved by adjusting the distance d between vertices C 6 , j k and C 0 , j k , which can also be expressed as a function of α (Figure 5A). Thus, the force for bending the structure can be obtained by: T = d Π total d α ⋅ d α d d . (18)

To experimentally examine the relation between the tension force and the bending motion of the Yoshimura-ori structure, a special setup is designed (Figure 8A). Specifically, the Yoshimura-ori structure is clamped between two acrylic plates; the right plate connected with the support is fixed with the bottom platform of the Instron universal testing machine. To measure the force required for bending deformation, an inextensible string is employed to bend the structure by reducing the distance between vertices C 0 , j 1 and C 6 , j 1 . A pulley is fixedly installed on the support, and the string, which is threaded through the pulley, is connected with the left plate at the top vertex, and the other end of the string is connected with the top movable platform of the testing machine. The pulley could effectively eliminate the effect of friction. Hence, when the top platform is lifted, the distance between vertices C 6 , j I and C 0 , j I will be reduced, which equivalently bends the Yoshimura-ori structure (see insets in Figure 8A), and the required force for bending can be recorded accordingly. Bending tests are performed on four Yoshimura-ori structure prototypes with thicknesses 0.025, 0.05, 0.075, and 0.1 mm, and each prototype is tested three times. The original length of the prototype (i.e., the initial distance d between C 6 , j I vertices C 0 , j I and) is 100 mm, and we will reduce it to 0 mm, with a loading speed of 0.5 mm/s. Averaging the three test results of each prototype, the corresponding force-displacement curve can be obtained. Figure 8B indicates that the maximum force is reached when the structure is fully bent ( d = 0 ). Moreover, the flexural rigidity (i.e., the slope of the curve) can be tailored by changing the film thickness. As the film thickness grows from 0.025 to 0.1 mm, the restoring force will increase more than 20 times from 0.08 to 1.64 N.

Study on the mechanics of Yoshimura-ori structure could provide useful guidelines for robot design and actuator selection. For robot development purposes, folding of the Yoshimura-ori structure has to be robust and consistent, and the restoring forces at the maximum-length and the maximum-angle configurations cannot exceed the actuation limit. These two requirements are sometimes contradictory. With thicker PETE film, the folding process will be more robust and consistent, but meanwhile, it needs a larger actuation force to deform axially or to bend, which will thus restrict the deformability. Hence, compromising is always necessary during robot design. In this research, the PETE film with thickness 0.05 mm is selected to fabricate the robot segment, which corresponds to reasonable magnitudes of the restoring force at the maximum-length and the maximum-angle configurations (0.69 and 0.45 N, respectively), and the obtained origami structure could exhibit robust and consistent folding deformations.

To acquire active 3D deformability, two types of actuators are incorporated into the robot segment: a pneumatic balloon for axial deformation and four SMA springs for bending deformation. Considering that the Yoshimura-ori structure cannot exhibit radial deformation during folding, additional bristle-like structures made of electromagnets are added to achieve anchoring effects. Figure 9A shows the CAD design of a single segment.

In detail, for each robot segment, a 3 × 6 Yoshimura-ori structure (Figure 1A) is clamped between two acrylic plates. The actuators, including the pneumatic balloon, the SMA springs, and the electromagnets, are also attached to the acrylic plates (Figure 9A). Details of these actuators are listed in Table A1 at the end of the paper. The balloon actuator locates at the axial axis inside the Yoshimura-ori structure and is controlled by pressure with high accuracy via the E/P pressure regulator. By increasing the pressure, the balloon will expand along the axial direction and extend the Yoshimura-ori structure, switching the robot segment into the axially-extended configuration; when the pressure is released, the balloon could provide sufficient force to pull the robot segment back to the axially-contracted configuration. Four SMA springs are set at the top, bottom, left, and right sides of the robot segment, and each SMA spring can be actuated independently. When applying electric current to an SMA spring, it will shorten quickly and thus bend the robot segment in the associated direction; when the current is removed, the SMA spring will be cooled and gradually return to its original length. By combining the deformations of the balloon and the SMA springs, the bending deformation can be achieved more effectively. Hence, the pneumatic balloon actuator is needed in both axial and bending deformations, while the SMA springs are only used in bending. Moreover, to facilitate robotic locomotion, electromagnets are installed at the bottom of the acrylic plates, which could anchor with magnetic media when it is energized.

Note that due to the introduction of actuation components, the kinematical properties of the robot segment will be different from the Yoshimura-ori structure segment, including the length of the prototype at the deformation-free state, the maximum bending angle, etc.

Figure 9A also displays the photo of a robot segment prototype. It weighs only 29.8 g. The mechanical properties of the Yoshimura-ori structure have been comprehensively studied in Section 4. Here, the effectiveness of the actuators is demonstrated.

First, axial actuation of the robot segment through the pneumatic balloon is tested. The air pressure P of the balloon can be controlled precisely via the E/P Pressure Regulators (ITV ITV0030). In this research, the air pressure is set to be 0.162 MPa, which could effectively inflate the balloon. Figure 9B shows two configurations of the robot segment, one at the axially-contracted configuration with the internal balloon uninflated, and the other at the axially-extended configuration with the internal balloon inflated. Through inflation, the segment is extended from 43.4 to 79.7 mm, with a net deformation of 36.3 mm. The diameter of the internal balloon increases from 4 to 33.2 mm, which, however, does not affect the radial dimension of the robot segment. Note that the reachable maximum length is small than the original length of the Yoshimura-ori structure, this is due to the constraints from the SMA spring and to prevent the burst of the balloon.

Second, the mechanical properties of the SMA spring are evaluated. The SMA spring is fixed on the Instron universal testing machine and its initial length is set to be 40 mm. At t = 6 s , a current of 1.5 A is applied to heat the SMA spring and lasts for 24 s, during which, the contraction force is recorded. The test is performed three times, and the actuation force profile can be obtained by averaging the three test results (Figure 9C). It shows that the actuation force profile of the SMA spring is nonlinear in the overall trend. When there is no current (from 0 to 6 s), the actuation force remains zero; with the current input, the actuation force keeps increasing and reaches the maximum (4.39 N) at 30 s, which is larger than the maximum restoring force (0.46 N) for the bending motion of a single Yoshimura-ori segment. Hence, the SMA spring can provide sufficient force to help the Yoshimura-ori structure achieve bending.

To vividly demonstrate the 3D deformability, a prototype with three robot segments is tested. To test the deformability, one end of the prototype is fixed on the support; we use a camera to record the deformation process and measure the geometry. Figures 10A–D illustrates the axial contraction, axial extension, up-contraction, and left-contraction of the three-segment Yoshimura-ori structure prototype. At the axially-contracted and the axially-extended configurations, the axial length of the prototype is 160.9 and 273.9 mm, respectively. Hence, the net axial deformation of the three-segment prototype is 113.0 mm, which is 60.23% of its original length (160.9 mm). Such excellent axial deformability is desirable in earthworm-like robot development for achieving fast locomotion. Note that theoretically, the structure can be fully contracted to a very small height; while practically, its reachable axial deformation is restricted by the actuators. At the up-contracted and left-contracted configurations, the bending angle can be up to 42.4° and 54.2°, respectively, which is also favorable to robot development for achieving effective turning and rising motions. Note that the bending angle of the up-contracted configuration is smaller than that of the left-contracted configuration, which is induced by the effect of gravity.

Based on the designed robot segments a five-segment origami-based earthworm-like robot is prototyped by connecting five robot segments in series (Figure 11A). A camera is set at the head of the robot to capture the video of the working environment. The entire robot is 410 mm in length and 331.6 g in weight, and the weight-length ratio is only 0.81 g/mm. Comparing with other earthworm-like robots (e.g., 76.0 g/mm for the continuous meshworm robot (Boxerbaum et al., 2012), 1.39 g/mm for the earthworm-like robot with a spring-steel-belt body (Fang et al., 2014), and 1.27 g/mm for the underground earthworm-like explorer (Omori et al., 2009)), the robot has a very low weight-length ratio, which is beneficial for 3D locomotion.

Particularly, note that the weight of the robot forepart (consisting of a robot segment and the camera) is about 1.76 N, which is lower than the maimum actuation force of the SMA spring (4.39 N), indicating the feasibility for lifting the robot head. Moreover, the nominal attraction force of a single electromagnet is 10 N; with multiple embedded electromagnets and with a proper locomotion gait, the robot is possible to hold its weight (3.25 N), anchor with the magnetic media, and climb a wall, providing that the contact surface is not too smooth. However, further investigation on the locomotion performance by incorporating the robot weight, the electromagnet attraction force, and the friction coefficient is necessary, which is also an important part of our ongoing research.

Learning from the earthworm’s locomotion mechanism, i.e., the retrograde peristalsis wave, periodic gaits are proposed for the robotic locomotion. Fundamentally, a locomotion gait is defined as a sequence of the robot segment states with respect to the time, with the initial state being identical to the final state (Fang et al., 2014). Specifically, seven states of the robot segment are defined to construct the locomotion gait. Table 1 lists the seven states and the corresponding driving patterns.

For the developed Yoshimura-ori based earthworm-like robot, seven parameters are needed to construct the locomotion gait, namely, the total number of the segments N total , the number of the axially-extended segments N extend , the number of the upward-contracted segments N up , the number of the right-contracted segments N right , the number of the downward-contracted segments N down , the number of the left-contracted segments N left , the number of actuated electromagnets N emagnet (anchored), and the number of propagating segments N prop . These seven parameters are not independent, rather, they should satisfy the following constraints: N extend ≥ 0 , N up ≥ 0 , N right ≥ 0 , N down ≥ 0 , N left ≥ 0 , N emagnet > 0 , N prop > 0 , N extend + N up + N right + N down + N left ≥ 1 , N total > 2 ( N extend + N up + N right + N down + N left ) − N prop , (19)

The segments with inflatable balloons ( N extend + N up + N right + N down + N left ) are assumed to be gathered together, and they stay at the forepart of the robot at the initial state. To mimic the retrograde peristalsis wave of the earthworm, the actuation states (including the axially-extended, up/down/right/left contracted) are asked to propagate backward by N prop segments in each transition. In addition, there should always exist an anchoring point in the robot so that the other segments can lean against them and deform to achieve displacement. Hence, when the segments start to extend axially or to bend, at least one electromagnet in the rear of the actuated segments should be energized to achieve anchoring; when the actuated segments’ balloons start to deflate, the posterior electromagnets should be de-energized, while the anterior electromagnet of the front-most actuated segment should be energized so that the posterior segments can be pulled forward by the contraction of the balloons.

Figure 11B display the architecture of the robot gait controller, which consists of a computer (with Arduino Software IDE), Arduino UNO hardware, power supply, SMA controllers, and E/P Pressure Regulators. Inputting the seven gait parameters to the controller, admissible gaits of different locomotion can be generated. Gait signal will then be transmitted to voltage to the E/P Pressure Regulators, electromagnet, and SMA controllers, and finally actuate the SMA springs and balloon. However, the locomotion control of this robot is different from the traditional earthworm-like robot.

Based on the robot prototype shown in Figure 11A and the gait controller provided in Figure 11B, locomotion tests are carried out on a horizontal iron plane. The robot’s locomotion is recorded by cameras, from which the trajectories of the robot head can be extracted and analyzed. To demonstrate the predicted 3D locomotion capability of the robot, Figure 12 shows the video snapshots of four locomotion tests, which include three types of locomotion, namely, rectilinear locomotion, turning locomotion, and rising locomotion.

Figure 12A (Supplementary Video S1) displays the snapshots for a rectilinear locomotion test and the corresponding gait (demonstrated as the state sequence of the robot segments). For the gait used in this test, each period of the robotic locomotion contains 10 steps, and only one segment is extended (i.e., the balloon is actuated) in every other step. Specifically, in the first step, the balloon in the head segment (segment #1) is pneumatized in 0.8 s, and the rear electromagnet of segment #1 is activated, which can push the robot head forward. In the second step, the front electromagnet of segment #1 is actuated, and the balloon of segment #1 is deflated in 1.1 s, which would pull the rest of the robot forward. In the following steps, actuation of the balloon and the electromagnet as a whole will propagate backward, generating a retrograde peristalsis wave, similar to the earthworm’s wave. With such a gait, the robot moves 327.1 mm in a period of 20.33 s, with an average speed of 16.09 mm/s. Figure 11A shows the snapshot at 40.65 s, where a 654.1 mm displacement is achieved by the robot. With five segments, the metameric robot could be equipped with more than one rectilinear locomotion gaits. Figure 12B (Supplementary Video S2) depicts another rectilinear locomotion with a different gait. The main difference is that two balloons are pneumatized in every other step. Hence, the robot needs only eight steps to finish a period, and the robot achieves faster locomotion with an average speed of 22.24 mm/s (i.e., move 339.2 mm in a period of 15.25 s).

In the third test, we demonstrate the planar locomotion, and the corresponding gait control is shown in Figure 12C (Supplementary Video S3). In this case, as an example, the three anterior segments are asked to perform right-contraction for turning purposes, and the two posterior segments are asked to perform axial extension for propelling purposes. In the first step, the right SMA spring of segment #1 is heated and contract in 4 s, and its rear electromagnet is energized so that the robot head turns to the right. In the second step, the rear electromagnet of segment #1 keeps energized, and the balloon of segment #1 is inflated to push the robot head forward right. In the third step, the anterior electromagnet of segment #1 is activated, and the balloon of segment #1 is deflated to pull the rest of the robot forward right. In the following steps, actuation of the balloon, the SMA spring, and the electromagnet as a whole will propagate backward. With such a gait, the robot can move 445.6 mm in two periods (50.48 s), with an average speed of 8.83 mm/s, and the robot head turns an angle of 64.1°.

The last test is the rising locomotion test, which is significantly distinct from the rectilinear and planar locomotion. For the gait used in this test, each period of the robotic locomotion contains five steps (Figure 12D (Supplementary Video S4)). In the first step, the last three electromagnets at the rear of the robot are energized to firmly anchor with the ground so that the robot can remain stable when the head is raised. In the second step, the top SMA springs of segments #1, #2, and #3 are heated and contract in 13 s, so that the robot head would be lifted. In the third step, the balloons in segments #1, #2, and #3 are inflated, which pushes the robot head upward. In the next step, the balloons are deflated and the robot head is lowered. Finally, all actuators are deactivated and the robot returns to its initial state. With such a gait, the robot can raise its head to 179.6 mm, and the rising angle could reach 109.9°.