The Rigid Origami Patterns for Flat Surface

Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA

DETC2013-12947

THE RIGID ORIGAMI PATTERNS FOR FLAT SURFACE Sicong LIU Nanyang Technological University 50 Nanyang Ave, Singapore 639798 Email: [email protected]

Yan CHEN* Tianjin University 92 Weijin Road, Nankai District, Tianjin, China 300072 Email: [email protected]

Guoxing LU Nanyang Technological University 50 Nanyang Ave, Singapore 639798 Email: [email protected]

covers all the types with explicit, necessary and sufficient geometric conditions of the rigid origami structures. From the viewpoint of mechanism, when four fold creases intersect at a single vertex, the creases and facets around this vertex can be represented by a spherical 4R linkage [2, 11], which has mobile one [12]. As the simplest spherical linkage, the spherical 4R linkage has been used as basic element to construct other mechanisms, such as Hooke’s linkage and double Hooke’s linkage [13]. Recently, Wang and Chen used the assembly of spherical 4R linkages to study the rigid origami patterns [14, 15]. They have successfully derived the general geometric condition of the identical linkage type pattern. In this paper, the focus is drawn on the kinematic study of the rigid origami patterns for flat surfaces. The layout of the paper is as follows. Firstly, the kinematics of a single spherical 4R linkage will be introduced in Section 2. The kinematics of the assembly of spherical 4R linkages and its compatibility conditions will be derived in section 3. Section 4 gives the solutions of one-DOF mobile assembly of four spherical 4R linkages and their corresponding rigid origami patterns. The conclusion and discussions in Section 5 end the paper.

ABSTRACT Because of the internal mobility, rigid origami structures have great potential in engineering applications. In this research, a kinematic model of the rigid origami pattern is proposed based on the assembly of spherical 4R linkages. To ensure the rigid origami pattern with mobility one, the kinematic and geometric compatibility conditions of the kinematic model are derived. Four types of flat rigid origami patterns are obtained, including three existing types as well as a novel one called the supplementary type. To testify and display the mobile processes of the patterns, their simulation models are built accordingly. KEYWORDS: Rigid origami, mobility one, spherical 4R linkage INTRODUCTION Origami is to fold paper into sculpture as a traditional Japanese art with hundred-years history [1]. For decades origami has caught great attentions in the mathematical field to explore the geometrical laws [2]. Rigid origami is referred to the case that each surface surrounded with crease lines is not stretching or bending during folding [3]. When its facets and crease lines are replaced by rigid panels and hinges, rigid origami can be realized by mechanisms [4]. The rigid origami inspired structures with wide engineering applications in deployable structures, such as the solar panels of the satellites, radar antennas and building boards. Through various approaches in previous studies, five types of one-DOF rigid origami patterns for flat surfaces have been discovered: the identical linkage type [5], planar-symmetric type [5, 6], isogonal type [5, 7], orthogonal type [8], and Miuraori type [9, 10]. However, there is no general approach which

THE KINEMATICS OF A SINGLE SPHERICAL 4R LINKAGE In Fig. 1, the coordinate frame i on the link i(i+1) and the joint i is set up in such a way that Zi is the axis of revolute joint i and X i is the axis commonly normal to Zi and Zi1 , ( Xi  Zi  Zi1 ). Then, the geometric parameters of the links are defined as follows. The length of link i(i+1), ai (i 1) , is the distance between axes Zi and Zi1 , positively along axis

X i ; the twist of link i(i+1), i (i 1) , is the rotation angle from

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axes Zi to Zi1 , positively about axis X i ; and the offset of joint i, d i , is the distance from link (i-1)i to link i(i+1), positively along axis Zi . The revolute variable of the linkage,

the four axes of the linkage have a coplanar configuration, i.e., the linkage is capable of being fully deployed into a flat status.

 i , is the rotation angle from Xi1 to Xi , positively about Zi .

Figure 2. THE DEFINITHION OF A SINGLE SPHERICAL 4R LINKAGE. Figure 1. THE DEFINITION OF THE PARAMETERS IN THE LINKS (i-1)i AND i(i+1).

To form a closed loop, the product of the transformation matrices along a closed loop equals the unit matrix [16]. So for the general spherical 4R linkage, the closure condition is

Thus, the coordinate transformation from frame i to frame i+1, can be represented as

Fi Tz (di )Tx (ai (i 1) )R x (i (i 1) )Rz (i 1 )  Fi 1 ,

Rx ( )Rz (2 )Rx ( )Rz (3 )R x ( )Rz (4 )R x ( )Rz (1 )  I , (5)

(1)

which can be rearranged into two forms,

in which, homogeneous matrices of translation d i along axis

Zi

is

Tz (di ) , translation

ai (i 1)

along axis

Xi

-1 -1 -1 Rz (1 )R x ( )Rz (2 )R x ( )  R-1 x ( )R z (4 )R x ( )R z (3 ) , (6)

is

Tx (ai (i 1) ) , rotation i (i 1) about axis X i is R x (i (i 1) ) ,

-1 -1 -1 Rz (2 )R x ( )Rz (3 )R x ( )  R-1 x ( )R z (1 )R x ( )R z ( 4 ) . (7)

and rotation  i about axis Zi is R z (i ) [16]. 1 0 Tz (di )   0  0

0 1 0 0 cos  1 0 0  i (i 1) ; R x ( i (i 1) )   0 1 di  0 sin  i (i 1)   0 0 1 0 0 0 0

cos i  sin  i R z (i )    0   0

 sin i cos i 0 0

0 0 0 0  ; 1 0  0 1

1  0 Tx (ai (i 1) )   0  0

0  sin  i (i 1) cos  i (i 1) 0

0 0  ; 0  1 

Expand matrix equations (6) and (7). Corresponding elements of the matrix at the left and right sides of equation M L and M R must be equal. Considering elements L R L R L R , M2,3 and M3,2 in (6) and (7), M3,3  M3,3  M 2,3  M3,2

(2)

to eliminate cos 4 , cos 1 , cos 2 and cos 3 , the following relationships can be obtained.

0 0 ai (i 1)   1 0 0  . 0 1 0   0 0 1 

cos sin  sin  cos1  cos sin  sin  cos 2  cos sin  sin  cos1 cos 2  sin  sin  sin 1 sin  2  cos  cos  cos  cos   0;

For general spherical 4R linkage, all four revolute axes intersect at one vertex, as shown in Fig. 2. Hence, its geometric parameters are

 sin  sin  sin 2 sin 3  cos  cos  cos  cos   0;

12   ,  23   ,  34   ,  41   ; d1  d 2  d3  d 4  0;

(8)

cos sin  sin  cos 2  cos sin  sin  cos3  cos sin  sin  cos 2 cos3

(9)

cos sin  sin  cos3  cos sin  sin  cos 4  cos sin  sin  cos3 cos 4

(3)

 sin  sin  sin 3 sin  4  cos  cos  cos  cos   0;

a12  a23  a34  a41  0.

(10)

cos sin  sin  cos 4  cos sin  sin  cos1  cos sin  sin  cos 4 cos1

When

 sin  sin  sin  4 sin 1  cos  cos  cos  cos   0.

        2 ,

(4)

2

(11)


Equations (8-11) show the relationships between two adjacent rotation angles  i and i 1 ( i  1, 2, 3, 4 ), which can be represented by functions

2  f12 (1 )， 3  f23 (2 )， 4  f34 (3 ) and 1  f 41 (4 ) , (12) respectively. Thus, when geometric parameters i (i 1) of a spherical 4R linkage are assigned, with one input rotation angle  i , the other three rotation angles can be obtained. THE KINEMATICS OF THE ASSEMBLY OF SPHERICAL 4R LINKAGES AND ITS COMPATIBILITY CONDITIONS

Figure 4. THE CLOSED ADJACENT ASSEMBLY OF FOUR SPHERICAL 4R LINKAGES.

When two spherical 4R linkages are connected as shown in Fig. 3, the axes Zai of linkage A and Zbi of linkage B are aligned, but in opposite directions to ensure the axes of each spherical 4R linkages are always pointed to its own spherical center. Then the link i(i+1) of linkage A and link (i-1)i of linkage B , are rigidified into one link. So are the link (i-1)i of linkage A and link i(i+1) of linkage B. Therefore, the rotation angles  ai and  bi become synchronized during the

The kinematic compatibility condition of the closed adjacent assembly

movement of the assembly, i.e. ai  bi for the rigid origami patterns. Thus, joints ai and bi can be represented as one joint. In this study, only the flat panels are used in the origami patterns, ie. the axes Za(i1) and Zb(i1) are coplanar, so are

a)

axes Za(i1) and Zb(i1) , then ai  bi . Furthermore, the facets of the rigid origami patterns are set as quadrilaterals. Thus, only four linkages are assembled adjacently into a closed chain, see Fig. 4, which is considered as a unit model of the whole pattern.

b) Figure 5. THE INPUT AND OUTPUT RELATIONSHIP OF THE ROTATION ANGLES IN THE CLOSED ADJACENT ASSEMBLY: a) ONE CYRCLE; b) TWO HALF-CIRCLES.

Figure 3. AN ASSEMBLY OF TWO SPHERICAL 4R LINKAGES.

In this model, the twists i (i 1) are constant geometric

As shown in Fig. 5, the closed assembly can be considered as an open chain closed by connecting its head and tail, while relationships c4  d4 , a1  d1 and a2  b2 are already

parameters, while rotation angles  i are kinematic variables, which should satisfy following relationships.

a2  b2 , b3  c3 , c4  d4 , d1  a1 .

satisfied. In this open chain,  c3 can be considered as input and  b3 as the output as shown in Fig. 5a. In order to achieve the closed mobile assembly, the input and output should be equal, i.e. b3  c3 . Considering Eqn. (12), we have

(13)

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b3  fb23 ( fa12 ( fd41 ( fc34 (c3 ))))  c3 .

(14)

However, Eqn. (14) is too complicated. Alternatively, as shown in Fig. 5b,  a1 is set as input, through linkages A and B

to derive the output  b3 . Similarly,  d1 is set as input and through linkages D and C to derive the output  c3 . Because

relationship a1  d1 already exists, the closure condition is still b3  c3 , i.e., -1 -1 b3  f b23 ( fa12 (a1 ))  fc34 ( fd41 (d1 ))  c3 ,

(15) Figure 6. THE TRANSFORMATIONS FROM COORDINATE FRAME Xd1  Zd1 TO Xb3  Zb3 .

-1 -1 in which f d41 and f c34 are the inverse function of f d41 and f c34 . Substituting Eqns. (8-11, 13) into (15) gives

cos a cos d  ; tan a2 tan d4 cos a sin  a sin a cos a2  cos d sin  d sin  d cos  d4 ;

Because both paths have the same start and end coordinate frames, and due to the Eqns. in (13), the following transformation can be derived.

(16)

R x ( )Tz ( DA)R x ( a )R x ( )Tz ( AB)R x (  b )

cosa sin  a sin  a  cos d sin  d sin  d ;

 R x1 ( d )Tz1 (CD)R x1 ( )R x1 ( c )Tz1 ( BC )R x1 ( )

cos a  cos  a cos  a cos  a  cos d  cos  d cos  d cos  d ,

cos b cos c  ; tan  b2 tan c4 cos b sin  b sin  b cos  b2  cosc sin  c sin  c cos c4 ;

After expanding (18) and considering elements R L R L R L R , M3,4 , M2,2 and M2,3 ,  M2,4  M3,4  M 2,2  M2,3

L M2,4

(17)

we have

 AB sin  a  BC sin( d   c )  CD sin  d ;

cos b sin  b sin  b  cos c sin c sin  c ; cos b  cos  b cos  b cos  b  os c  cos c cos  c cos  c .

AB cos  a  DA  BC cos( d   c )  CD cos  d ; cos( a   b )  cos( d   c );

To eliminate  i , Eqns. in (16) are divided by corresponding Eqns. in (17), which derives the kinematic compatibility conditions for the closed adjacent assembly as follows.

cos a sin  a sin  a cos d sin  d sin  d  ; cos b sin  b sin  b cos c sin  c sin  c

(20)

sin( a   b )   sin( d   c ),

which are the geometric compatibility conditions. Equations in (18) and (20) form the compatibility conditions to ensure the closed adjacent assembly with mobility one.

cos a cos d ;  cos b cos c cos a sin  a sin  a cos d sin  d sin  d  ; cos b sin  b sin  b cos c sin  c sin  c

(19)

THE ONE-DOF MOBILE ASSEMBLIES AND RIGID ORIGAMI PATTERNS Solving Eqns. in (18) and (20) gives four types of one-ODF mobile assemblies.

(18)

cos a  cos  a cos  a cos  a cos d  cos  d cos  d cos  d .  cos b  cos  b cos  b cos  b cos c  cos  c cos  c cos  c

The planar-symmetric type To make the corresponding terms in the Eqns. in (18) equal, the relationships between two adjacent linkages can be derived

The geometric compatibility condition of the closed adjacent assembly As shown in Fig. 6, the transformations from coordinate frame Xd1  Zd1 to Xb3  Zb3 can go through two different paths. One path starts at linkage D, through linkage C reaches the coordinate frame in linkage B. The other starts and ends at the same coordinate frames, but via linkage A.

 a   d;  a   d;  a   d;  a   d;

   

b 

;c

b 

;c

 ;c b  . c

b

(21)

Substituting (21) into the geometric condition (20) gives

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AB  CD ; BC  2 CD cos  d  DA .

DA  AB cos a  CD cos  d  BC cos( d   c ) . (22)

Thus, all four linkages in this assembly are identical and connected each other in a rotational fashion. The planar quadrilateral in this assembly can be general convex quadrilateral, see Fig. 7c. This type was first established in [5].

In this case, the planar quadrilateral ABCD must be an equilateral trapezoid with AD || BC. The relationships in (21) make linkages A and B the mirror images of linkages D and C, respectively. This type is named the planar-symmetric type in [6]. A schematic model of this type is built in Fig. 7a.

The orthogonal type The orthogonal type is studied in [6] with geometric conditions. Apart from the relationships introduced in theorem 4 of [6], in order to build the assembly with complete geometric parameters, following additional relationships are necessary,

The supplementary type Considering the supplementary angles in the trigonometric functions, a new set of solution is derived

a  d a   d  a  d a  d

 d   c   ; a   b   .

  ; b  c   ;

(28)

Thus, the complete and explicit geometric conditions of the orthogonal type are as

  ; b   c   ;   ;  b  c   ;

(27)

(24)

 a   d  0;  b   c  0;  d  c   ; cos   cos 

  ;  b  c   ;

AB  DA cos  a  BC cos  b  CD . in which the two corresponding angles from two linkages A and D or B and C are supplementary. So this type is named the supplementary type. The planar quadrilateral ABCD can be a general trapezoid with its edges AB || DC, see Fig. 7b. The supplementary type is a novel type. It includes the isogonal type because based on the Eqns. in (24), at each vertex of the isogonal type, opposite angles are congruent [6]. The Miura-ori type is another special case of the supplementary type, which is constructed by identical parallelograms.

 a   d  0;  b   c  0; a   b   ; cos  ; cos 

(29)

AB  CD ; BC  2 CD cos  d  DA . The quadrilateral ABCD must be an equilateral trapezoid with AD || BC, as shown in Fig. 7d.

The identical linkage type As previously studied, if condition (14) is satisfied, the rotation angles of this assembly are compatible with each other. For example, linkage C should satisfy

c3  fc23 ( fc12 ( fc41 ( fc34 (c3 )))) . Compare

(14)

with

(25),

when

(25)

functions fc41  fd41 ,

a)

fc12  fa12 and fc23  f b23 , condition (14) will be satisfied. thus,

a  b  c  d ; a   b  c  d ; a  b  c  d; a  b  c  d ,

(26)

which is one solution of Eqn. (18). Considering Eqn. (20), we have b)

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e)

c)

f) Figure 8. THE FLAT DEPLOYING PROCESSES: a) THE PLANARSYMMETRIC TYPE; b) THE SUPPLYMENTARY TYPE: GENERAL CASE; c) THE SUPPLYMENTARY TYPE: MIURA-ORI; d) THE IDENTICAL LINKAGE TYPE; e) THE ORTHOGONAL TYPE.

CONCLUSION AND DISCUSSION In this paper, a kinematic model of the rigid origami pattern has been proposed based on the closed adjacent assembly of four spherical 4R linkages. To make the assembly with mobility one, the kinematic and geometric compatibility conditions of the kinematic model have been studied systematically. Given the flat surface condition, the solutions to the compatibility conditions include three existing types and a novel supplementary type. Totally, four distinct types of flat rigid origami patterns for flat surfaces are obtained. However, due to the complicity of both the kinematic and geometric compatibility conditions of the assembly unit, we have yet to find the complete sets of the solutions for all possible mobile assemblies and the corresponding rigid origami patterns, which will be the further study of this topic.

d) Figure 7. a) THE PLANAR-SYMMETRIC TYPE; b) THE SUPPLEMENTARY TYPE; c) THE IDENTICAL LINKAGE TYPE; d) THE ORTHOGONAL TYPE.

The rigid origami patterns Following the square tiling, the assemblies previously introduced can be extended to networks of unlimited numbers of linkages, which can be considered as the rigid origami patterns. Their modeling is shown in Fig. 8, in which the wellknown Miura-ori pattern is considered as a special case of the flat supplementary type.

REFERENCES This work is financially supported by the Natural Science Foundation of China (Projects No. 51275334 and No. 51290293). a)

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