Structure of Parallel Mechanism Combined with Waterbomb-Base-Inspired Origami †

: Structure from the geometry and analysis of the three-spherical kinematic chain-base parallel mechanism have been studied. The parallel mechanism evolved from an origami fold as chain legs with three spherical kinematic chains becoming rigid bodies. The parallel mechanism with a three 6R kinematic chain as three chain legs is complicated. The reconﬁguration of the parallel mechanism with full tilt–circle movement, kinematic, and workspace are investigated, too. This parallel mechanism can be applied in speciﬁc applications with certain treatments.


Introduction
The parallel mechanism is the system that converts the motions of several bodies into constrained motions of other bodies [1]. The parallel mechanism has a closed-loop as a type of mechanism [1], and that is made of an end-effector (mobile platform) [2] and a fixed platform, linked together by independent kinematic chains [1]. Furthermore, structure, workspace considerations, singularities, and link interference need to be considered in the design [2]. The parallel mechanism has much potential in several fields, including industrial, space, medical science, and miscellaneous applications [2].
Origami is Japanese cultural art, which is the art of folding paper. In general, origami starts with flat paper (2D object), then folds become a 3D object with various shapes and forms without stretching, cutting, or gluing [3]. Moreover, origami has become an inspiration for engineers in various fields. Especially waterbomb-base origami has become an inspiration in engineering for several applications. Fonseca and Savi [3] presented an investigation of the origami waterbomb-base pattern from its unit cell and explored the different formulations for origami structure. Liang, Gao, Huang, and Li [4] presented the design of a pneumatic rigid-flexible coupling origami gripper from a waterbombbase origami pattern. Salerno, Zhang, Menciassi, and Dai [5] proposed the concept of a miniaturized surgical tool grasper as a 3-DOF parallel module inspired by the waterbombbase origami.
It is common to see the creases of origami mechanisms as the compliant mechanism joint with flexible material. However, the origami mechanism becomes rigid if the creases are replaced with non-flexible material. One of the rigid kinematic origami models describes the creases with the revolute joint mechanism [6].

Geometry of Parallel Mechanism
A parallel mechanism consists of a base, a platform, and legs to sustain the platform. The leg structure is obtained from waterbomb-base origami, which involves the spherical kinematic chain with a close loop structure. The parallel mechanism consists of three legs and one actuator in the middle of the mechanism. kinematic chain with a close loop structure. The parallel mechanism consists of three legs and one actuator in the middle of the mechanism.

Waterbomb-Base Origami for Spherical Mechanisms
As shown in Figure 1a,b, origami starts with flat paper and then folds to become creases. The creases consist of two types. The first type has a convex shape called Mountain creases, denoted by M, and the second type has a concave shape called Valley creases, represented by V [7,8]. Waterbomb-base origami consists of six planes and six creases. Of the six creases, Ac and Af are the valley-crease types, and Ab, Ad, Ae, and Ag are the mountain-crease types. As shown in Figure 1c, the creases and the planes on the waterbomb-base structure are represented for the revolute pairs and the links, respectively [7]. Thus, the waterbombbase origami form contains six revolute pairs and six links. Barreto et al. [8] made an analogy between the spherical mechanism and the origami vertex as a concept of mechanism design.
Maekawa's-Justin's Theorem [9] is as follows. Let M be the number of mountain creases and V be the number of valley creases adjacent to a vertex in a flat origami crease pattern. Then, M − V = ± 2. Moreover, Kawasaki-Justin's Theorem [9] lets v be a vertex of degree 2n is an origami crease pattern and … , be the consecutive angles between the creases. Then, the creases adjacent to v (locally) fold flat only if Waterbomb-base origami consists of six planes and six creases. Of the six creases, Ac and Af are the valley-crease types, and Ab, Ad, Ae, and Ag are the mountain-crease types. As shown in Figure 1c, the creases and the planes on the waterbomb-base structure are represented for the revolute pairs and the links, respectively [7]. Thus, the waterbomb-base origami form contains six revolute pairs and six links. Barreto et al. [8] made an analogy between the spherical mechanism and the origami vertex as a concept of mechanism design. Maekawa's-Justin's Theorem [9] is as follows. Let M be the number of mountain creases and V be the number of valley creases adjacent to a vertex in a flat origami crease pattern. Then, M − V = ±2. Moreover, Kawasaki-Justin's Theorem [9] lets v be a vertex of degree 2n is an origami crease pattern and α 1 . . . , α 2n be the consecutive angles between the creases. Then, the creases adjacent to v (locally) fold flat only if That is, the two theorems correspond with waterbomb-base origami form. As shown in Figure 1d,e, the figure mapping and the spherical mechanism from the origami evolve. In the waterbomb base origami, Point A is the intersection of the whole of the connected Eng. Proc. 2023, 38, 51 3 of 10 creases, while in the spherical mechanism, point A is the virtual point intersection of the axis from the whole revolute pair in the spherical kinematic chain. The revolute pair and links are denoted with S and R, respectively. The link R i (i = 1, 2, . . . , 6) is deformed by revolute joints S i−1 and S i . Due to this close loop structure, link R 6 forms by revolute joint S 6 and S 1 . R 3 and R 6 have a revolute joint for connecting mobile and base platforms. Link R i (i = 1, 2, . . . , 6) has the angle α i (i = 1, 2, . . . , 6). The relationship between the angles in waterbomb-base origami is α 1 = α 2 = α 4 = α 5 ; meanwhile, α 3 = α 6 .
The initial shape of waterbomb-base origami can be a square (bd = de) or a rectangular (bd < de), depending on the intended use, application, and other parameters. Determining it requires further discussion.

Parallel Mechanism
The parallel structure mechanism consists of a mobile platform, a fixed platform, three legs, and a truss in the middle of the mechanism. Meanwhile, the legs are represented by the spherical kinematic chain involved in waterbomb-base origami, and then, the structure has three spherical kinematic chains for the legs [7]. The truss consists of a prismatic connector hinge and two offset universal hinges to connect to the mobile platform and base platform. The truss component has supported the movement of the mobile platform concerning the base platform. Figure 2 shows the revolute pair S ij and the link R ij , where i (i = 1, 2, 3) is the number of legs and j (i = 1, 2, . . . , 6) is the number of a revolute pair. P and B are the revolute joints paired with the mobile and base platforms.
That is, the two theorems correspond with waterbomb-base origami form. As shown in Figure 1d,e, the figure mapping and the spherical mechanism from the origami evolve. In the waterbomb base origami, Point A is the intersection of the whole of the connected creases, while in the spherical mechanism, point A is the virtual point intersection of the axis from the whole revolute pair in the spherical kinematic chain. The revolute pair and links are denoted with S and R, respectively. The link = 1, 2, . . , 6 is deformed by revolute joints and . Due to this close loop structure, link forms by revolute joint and . and have a revolute joint for connecting mobile and base platforms. Link = 1, 2, . . , 6 has the angle = 1, 2, . . , 6 . The relationship between the angles in waterbomb-base origami is = = = ; meanwhile, = . The initial shape of waterbomb-base origami can be a square ( = ) or a rectangular ( ), depending on the intended use, application, and other parameters. Determining it requires further discussion.

Parallel Mechanism
The parallel structure mechanism consists of a mobile platform, a fixed platform, three legs, and a truss in the middle of the mechanism. Meanwhile, the legs are represented by the spherical kinematic chain involved in waterbomb-base origami, and then, the structure has three spherical kinematic chains for the legs [7]. The truss consists of a prismatic connector hinge and two offset universal hinges to connect to the mobile platform and base platform. The truss component has supported the movement of the mobile platform concerning the base platform. Figure 2 shows the revolute pair and the link , where = 1, 2, 3 is the number of legs and = 1, 2, . . , 6 is the number of a revolute pair. P and B are the revolute joints paired with the mobile and base platforms.

Kinematic of Structure Mechanism
In this section, the displacement analysis of the parallel mechanism is presented. This parallel mechanism establishes global reference and local reference. Point Ob-XYZ is established as the global reference frame at the base platform. The X-axis and the Y-axis are perpendicular and parallel to the axis of the revolute hinge with the center point , respectively. At the same time, the Z-axis is normal to the base following the right-hand rule [7]. In addition, the mobile platform has a point Op-XYZ as the midpoint, and the legs and the truss component have the local reference frame. Table 1 explains the procedure of the DH Convention for knowing the forward kinematics [10].

Kinematic of Structure Mechanism
In this section, the displacement analysis of the parallel mechanism is presented. This parallel mechanism establishes global reference and local reference. Point Ob-XYZ is established as the global reference frame at the base platform. The X-axis and the Y-axis are perpendicular and parallel to the axis of the revolute hinge with the center point B 1 , respectively. At the same time, the Z-axis is normal to the base following the right-hand rule [7]. In addition, the mobile platform has a point Op-XYZ as the midpoint, and the legs and the truss component have the local reference frame. Table 1 explains the procedure of the DH Convention for knowing the forward kinematics [10].
The table presents the DH parameter determined by four transformation parameters [10] from the link and joint parameters [11]. Link parameters are the length of the link (a i ) and the angle of a twist of a link (α i ); meanwhile, joint parameters as the offset of link (d i ) and joint angle (θ i ) represent the relative positions of the following links [11]. Step 1 Determining and labeling of the joint axis Z 0 , . . . , Z n−1 .
Step 2 Determine the base frame. Set the origin on Z 0 -axis. X 0 -axis and Y 0 -axis are chosen conveniently, according to the right-hand rule frame. For i = 1, . . . , n − 1 Step 3 The origin is O i , where the common is normal to Z i and intersects between Z i−1 and Z i . If Z i intersects with Z i−1 , then the O i is located at that intersection, or if Z i and Z i−1 are parallel, the origin O i is located in any convenient position along Z i .
Step 4 Determine the X i from the origin O i along the common normal between Z i−1 and Z i , or if Z i−1 and Z i intersects, the direction normal to the Z i−1 − Z i plane.
Step 5 Determine Y i to complete the frame according to the right-hand rule.
Step 6 Determine the end-effector frame O n − X n Y n Z n . Define the origin O n along Z n , preferably at the center of the end-effector, and set X n and Y n following the right-hand rule.
Step 7 Create a table of DH parameters.
Step 8 Substituting the DH parameters into the equation to obtain the form of the homogeneous transformation matrices.
Step 9 The product result from step 8 is then given the position and orientation of the end-effector with respect base coordinate frame.
In Ref. [10], • a i = distance from the intersection of the x i , and z i−1 -axis to point O i are measured along x i ;

Leg
In Ref. [3], the waterbomb chain is related to a close chain, and the last revolute joint pairs with the first revolute joint. Therefore, waterbomb chain does not have an end effector. In this regard, evaluating the close loop equation is necessary, rather than considering the end-effector [3].

Truss Parallel Mechanism
The truss component section consists of a prismatic hinge connector, and two offset universal hinges connect to the mobile platform and the base platform. Figure 2 [12]. l represents the prismatic hinge length, the distance between the center point O 3 and point O 4 . Each hinge axe variable has the offset distance between the center of the local coordinate hinge axe. The distance between point Ob and point Op represents the height of the mechanism (h). Table 1 shows the procedure of the DH convention. As shown in Figure 3, the axis of each joint is already available, which corresponds from step 1 to step 7. Table 2 shows the Denavit-Hartenberg (DH) parameters [13] of the truss component for each link. According to step 8 in Table 1, substituting Equation (3) obtains the form of the homogeneous transformation matrices.
Equation (5) gives the position and orientation of the mobile platform with respect base platform, as shown in Figure 4.  According to step 8 in Table 1, substituting Equation (3) obtains the form of the homogeneous transformation matrices.

Motion Characteristics of a Structure Mechanism
In this structure, the truss component is active, and the legs are passive. Therefore, the active component moves as the structure drives, while the passive component limits the driven movement. Figure 5 shows the rotation around the X-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the -axis. The rotation of the -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.

Motion Characteristics of a Structure Mechanism
In this structure, the truss component is active, and the legs are passive. Therefore, the active component moves as the structure drives, while the passive component limits the driven movement. Figure 5 shows the rotation around the X-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the Z 1 -axis. The rotation of the Z 6 -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.

Motion Characteristics of a Structure Mechanism
In this structure, the truss component is active, and the legs are passive. Therefore, the active component moves as the structure drives, while the passive component limits the driven movement. Figure 5 shows the rotation around the X-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the -axis. The rotation of the -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.   Figure 6 shows the rotation around the Y-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the Z 2 -axis. The rotation of the Z 5 -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.  Figure 6 shows the rotation around the Y-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the -axis. The rotation of the -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.  Figure 7 shows the translation along the X-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotations occur on the -axis and the -axis. The translational moving away depends on the stroke of a prismatic hinge. Furthermore, the angle degree of the -axis is the same as with the -axis in the opposite direction. Thus, the rotation on the revolute hinge can rotate clockwise or counter-clockwise.   Figure 7 shows the translation along the X-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotations occur on the Z 2 -axis and the Z 5 -axis. The translational moving away depends on the stroke of a prismatic hinge. Furthermore, the angle degree of the Z 2 -axis is the same as with the Z 5 -axis in the opposite direction. Thus, the rotation on the revolute hinge can rotate clockwise or counter-clockwise.  Figure 6 shows the rotation around the Y-axis of the global coordinate frame in the base platform. The rotation happens due to the rotation of the -axis. The rotation of the -axis determines the orientation of the end effector. The revolute hinge rotates clockwise or counterclockwise. Activating the prismatic hinge adjusts the radius of the structure's rotation. The minimum and maximum radii are the prismatic hinge's minimum and maximum strokes, respectively.  Figure 7 shows the translation along the X-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotations occur on the -axis and the -axis. The translational moving away depends on the stroke of a prismatic hinge. Furthermore, the angle degree of the -axis is the same as with the -axis in the opposite direction. Thus, the rotation on the revolute hinge can rotate clockwise or counter-clockwise.   Figure 8 shows the translation along the Y-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotation happens on the Z 1 -axis and the Z 6 -axis. The translational moving away depends on the stroke of a prismatic hinge. Moreover, the angle degree of the Z 1 -axis is the same as with the Z 6 -axis in the opposite direction. Thus, the revolute hinge rotates clockwise or counterclockwise.  Figure 8 shows the translation along the Y-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotation happens on the -axis and the -axis. The translational moving away depends on the stroke of a prismatic hinge. Moreover, the angle degree of the -axis is the same as with the -axis in the opposite direction. Thus, the revolute hinge rotates clockwise or counterclockwise.  Figure 9 shows the translation along the Z-axis of the global coordinate frame in the base platform. The translation movement only activates a prismatic hinge. The high of the structure's parallel mechanism depends on the stroke of the prismatic joint.   Figure 9 shows the translation along the Z-axis of the global coordinate frame in the base platform. The translation movement only activates a prismatic hinge. The high of the structure's parallel mechanism depends on the stroke of the prismatic joint.  Figure 8 shows the translation along the Y-axis of the global coordinate frame in the base platform. The translation movement activates the combination of two rotations and a prismatic hinge. Meanwhile, the rotation happens on the -axis and the -axis. The translational moving away depends on the stroke of a prismatic hinge. Moreover, the angle degree of the -axis is the same as with the -axis in the opposite direction. Thus, the revolute hinge rotates clockwise or counterclockwise.  Figure 9 shows the translation along the Z-axis of the global coordinate frame in the base platform. The translation movement only activates a prismatic hinge. The high of the structure's parallel mechanism depends on the stroke of the prismatic joint.

Rotation with Tilt Movement
As shown in Figure 10, there is a combination of the structure movement parallel mechanism. The mechanism shows a tilt with moving around. All motors can be active and need control, depending on the requirements.

Rotation with Tilt Movement
As shown in Figure 10, there is a combination of the structure movement parallel mechanism. The mechanism shows a tilt with moving around. All motors can be active and need control, depending on the requirements.

Conclusions
The parallel structure mechanism consists of a mobile platform, a fixed platform, three legs, and a truss in the middle of the mechanism. Point Ob-XYZ is established as the global reference frame at the base platform. Movement of the structure to the Ob-XYZ is observed in the base platform. There are six motion characteristics of the structure mechanism.

Conclusions
The parallel structure mechanism consists of a mobile platform, a fixed platform, three legs, and a truss in the middle of the mechanism. Point Ob-XYZ is established as the global reference frame at the base platform. Movement of the structure to the Ob-XYZ is observed in the base platform. There are six motion characteristics of the structure mechanism.