Synthesis and Construction of a Family of One-DoF Highly Overconstrained Deployable Polyhedral Mechanisms (DPMs)

Transcription

1 Synthesis and Construction of a Family of One-DoF Highly Overconstrained Deployable Polyhedral Mechanisms (DPMs) Guowu Wei Postdoctoral Research Associate School of Natural & Mathematical Sciences King s College London, University of London Strand, London, WC2R 2LS, UK Jian S Dai 1 Chair Professor of Mechanisms and Robotics School of Natural & Mathematical Sciences King s College London, University of London Strand, London, WC2R 2LS, UK Abstract This paper presents a family of one-dof highly overconstrained regular and semi-regular deployable polyhedral mechanisms (DPMs) that perform radially reciprocating motion. Based on two fundamental kinematic chains with radially reciprocating motion, i.e. the PRRP chain and a novel plane/semi-plane-symmetric spatial eight-bar linkage, two methods, i.e. the virtual-axis-based (VAB) method and the virtual-centre-based (VCB) method are proposed for the synthesis of the family of regular and semi-regular DPMs. Procedure and principle for synthesizing the mechanisms are presented and selected DPMs are constructed based on the five regular Platonic polyhedrons and the semi-regular Archimedean polyhedrons, Prism polyhedrons and Johnson polyhedrons. Mobility of the mechanisms is then analysed and verified using screw-loop equation method and degree of overconstraint of the mechanisms are investigated by combing the Euler s formula for polyhedrons and the Grübler-Kutzbach formula for mobility analysis of linkages. Singular configurations of the mechanisms are revealed and multifurcation of the deployable polyhedral mechanisms is identified. 1 Introduction Polyhedral mechanisms are the mechanisms developed by implanting elementary kinematic chains into faces, edges and vertices of the polyhedrons. Most of them belong to the symmetric and regular overconstrained mechanisms. Research interest in polyhedral mechanisms started from the pioneering work of Verheyen [1, 2] on the expandable polyhedral structures named Jitterbug transformers by following Fuller s [3] introduction of a geometrical structure Jitterbug. The polyhedral structures proposed by Verheyen [2] contain rigid pairs of polygonal elements arranged in double-layer polygons. However, probably it was the showpiece of a mobile octahedron, which was named Heureka-polyhedron [4] showing at the Heureka Exposition in Zurich that first aroused more interest for research of polyhedron mechanisms from kinematicians and structure researchers [5]. After the Heureka-polyhedron, quite a number of works on polyhedral mechanisms has been carried out and most notable contributions were brought out by Wohlhart [6-10] in which different kind of synthesis methods were proposed leading to the 1 Corresponding author: jian.dai@kcl.ac.uk and URL: 1

2 generation of the Turning Tower, the Breathing Ball, and the Star-cube, etc. During the same period, a popular toy Hoberman TM Switch Pitch appeared in the market which was developed based on the geared expanding structure [11]. The switchpitch ball is also called colour-changing ball, tossing it to the air, the ball flips and colour of the ball magically changes. It arouses interest of people in all levels including physical science classrooms and brings attention to the field of mechanisms and robotics with its geometry and kinematics in general and mobility in particular. Agrawal et al. [12] proposed a simple approach for constructing expanding polyhedrons based on prismatic joints which preserve their shape because of the rigidity of the vertices. Gosselin and Gagnon-Lachance [13] developed a family of expandable polyhedral mechanisms based on Platonic solids, pentagonal prism solids and gyrobifastigium solids by integrating 1-DOF regular polygon-shaped planar linkages into the faces of the solids and assembling them with spherical joints at the vertices of the solids. Laliberté and Gosselin [14] then proposed the concept of PAFs (polyhedrons with articulated faces) and constructed a series of polyhedral linkages/structures from Platonic solids, Archimedean solids, Johnson solids and Rhombic solids. Kiper et al. [15, 16] presented analytical methods of constructing regular polyhedral linkages based on Bennett loops, and Fulleroid-like linkages which belong to the special cases of the Röschel s unilaterally closed mechanisms [17-19]. Wei et al. [20] presented a method of constructing polyhedral mechanisms with the basic deployable cubic units that are generated by four branches of parallelogram linkages whose links are all of the same length. In order to emulate swelling of viruses, Kovács et al. [21] developed a class of expendable polyhedral structures consisting of prismatic faces connected by hinged plates along the edges of the polyhedrons. In their research, symmetry of the structures was investigated via group theory providing an effective method for analysing mobility and symmetry of the deployable polyhedral mechanisms/structures. The general principle of constructing polyhedral linkages was introduced by Goldberg [22], he pointed out that the polyhedral bases are not necessarily convex and tried to verify his statement with an example called Bricard deformable octahedron, which exists theoretically but cannot be obtained in practice due to the interferences between links of the mechanism. However, there are no deployable concave polyhedral mechanisms according to the existing literature. This paper presents two fundamental kinematic chains, i.e., a PRRP chain and a plane/semi-plane-symmetric eight-bar linkage with exact straight-line motion [23], and proposes two intuitive approaches for synthesis of deployable polyhedral mechanisms. Utilizing the proposed method, a family of highly overconstrained regular and semi-regular deployable polyhedral mechanisms (DPMs) are synthesized and constructed. Mobility associated with degree of overconstraint of the proposed mechanisms is then analysed based on constraint matrix and Euler formula for polyhedron, and multifurcation of the mechanisms is subsequently revealed. 2 Virtual-Axis-Based (VAB) Syntheses of Deployable Polyhedral Mechanisms (DPMs) In the previous research [24], we found that by arranging the PRRP chains into the faces, edges and vertices of the polyhedrons, DPMs that transfer rotation into radially reciprocating motion can be synthesized. The synthesis method is herein referred to as 2

3 virtual-axis-based (VAB) method and it contains two approaches, i.e. Approach I by integrating PRRP chains into vertices and edges of the polyhedrons and Approach II by integrating PRRP chains into faces, vertices and edges of the polyhedrons. This synthesis method has been discussed in detail in [24], the steps are summarized in a diagram shown in Fig. 1 and herein an additional example is Fig. 1 Procedure of the virtual-axis-based synthesis of DPMs given to illustrate the construction procedure via Approach I of the VAB synthesis method. Since Approach II of the VAB method is equivalent to the VCB (virtual-centre-based) synthesis method, it will be discussed in Section 3 together with the VCB method. Figure 2 gives a hexagonal prism with two identical hexagonal faces connected by six identical rectangular sides. The prism has twelve vertices and eighteen edges of the same length. Using Approach I of the virtual-axis-based synthesis method, a PRRP chain is Fig. 2 A hexagonal prism and a PRRP chain 3

4 Fig. 3 A 18-PRRP-integrated framework in a hexagonal prism implanted into the prism in such a way that axes of the two prismatic joints are placed at vertices A and A' and aligned with diagonals AD' and A'D, the two sliders are denoted as V A and V A' and they are connected by a links along edge AA'. Subsequently, by carrying out the similar operation and integrating seventeen more PRRP chains into the basis with all the P joints located at the vertices and axes of the P joints collinear with the corresponding diagonals as illustrated in Figure 3, an 18-PRRP-integrated framework is generated. It should be pointed out that there are two groups of PRRP chains in the mechanism, the PRRP chains along edges AA', BB', CC', DD', EE' and FF' are one identical group, and the PRRP chains along AB, BC, CD, DE, EF, FA, A'B', B'C', C'D', D'E', E'F' and F'A' are another identical group. A frame is formed by the guides of the prismatic joints and all the axes of the prismatic joints intersect at the centre point O. Removing the frame as shown in Fig. 3, a highly overconstrained mechanism is readily obtained in Fig. 4. In the mechanism, the original joint axes of the prismatic joints become virtual axes of the vertices and the centre point O becomes the virtual centre of the mechanism. Like the motion possessed by the whole family of the deployable polyhedral mechanisms to be developed in this paper, (a) (b) (c) Fig. 4 A deployable hexagonal prismatic mechanism and its typical phases 4

5 the mechanism can perform radially reciprocating motion in such a manner that when V-plates (vertices) V A, V C, V E, V B', V D' and V F' move radially towards virtual centre O, V-plates (vertices) V B, V D, V E, V A', V C', and V E' move radially away from virtual centre O and vice versa. It should be noted that the links placed at the vertices and the faces of the base polyhedron are respectively referred to as V- plates and F-plates. One can see that although the guides of the prismatic joints for the original 18-PRRP-integrated framework shown in Fig. 3 are removed, in the novel deployable hexagonal prismatic mechanism, the V-plates can still perform radially reciprocating motions along the axes of the original guides. Therefore, this synthesis method is named virtual-axis-based (VAB) method. Thus, using Approach I of the virtual-axis-based synthesis method, a deployable hexagonal prismatic mechanism is developed. As to be discussed in Section 3, this mechanism can also be constructed based on the virtual-centre-based method by implanting six eightbar linkages into the six sides and combining them together. 3 Virtual-Centre-Based (VCB) Syntheses of Deployable Polyhedral Mechanisms (DPMS) The virtual-centre-based (VCB) method can be used to synthesize both regular and semi-regular deployable polyhedral mechanisms (DPMs), this method is based on the fundamental plane-symmetric eight-bar linkage and semi-plane-symmetric eight-bar linkage with exact straight-line motion. Therefore, in this section the two eight-bar linkages that perform exact straight-line motion are introduced and syntheses of regular and semi-regular deployable polyhedral mechanisms are subsequently presented. 3.1 A Plane-symmetric Eight-bar Linkage with Exact Straight-line Motion A plane-symmetric eight-bar linkage with exact straight-line motion has been proposed by the authors [23] as illustrated in Fig. 5. The linkage contains four links 1, 3, 5 and 7 with two parallel joint axes at each end and four vertexes V 0, V 2, V 4 and V 6 in isosceles triangle shape with two revolute joint axes being spread at both ends of the vertexes by φ 1 and φ 2. Vertices V 0 and V 4 are identical and so are vertices V 2 and V 6. The eight-bar linkage is a linkage of mobility two, thus in order to define the configuration of the linkage, two inputs are required. Regarding the linkage as a parallel mechanism by picking two of the vertices, i.e. V 0 and V 4, one (V 0 ) as base and the other (V 4 ) as platform. The base and the platform are connected by two identical limbs each of which consists of two links, one isosceles triangular vertex and four joints. It is found that in the case when the lengths of links 1, 3, 5 and 7 are the same and the dimensions of vertexes V 0 and V 4, and V 2 and V 6 are respectively identical, the linkage becomes a plane-symmetric linkage with its two limbs symmetric to a plane π which passes the centre points of V 0 and V 4 and is perpendicular to vertices V 0 and V 4. If the two joints connected to the base are assigned to be actuated joints, with the symmetric inputs, i.e. θ 11 = θ 21 (see Fig. 5), vertex V 4, as the platform of the linkage, performs exact straight-line motion with respect to the base. According to the detailed analysis in [23], the 5

6 Fig. 5 A plane-symmetric eight-bar linkage angle β between this straight-line traced by the trajectories of point P of V 4 and the base (a plane which is coplanar with vertex V 0 ) only dependents on the value of angles φ 1 and φ 2 as, arctan 1 cot sin 2 1 cos 2 1 csc tan 2 2, (1) with γ being the angle between two adjacent ormal (see n 0, n 2, n 4 and n 6 as shown in Fig. 4) of the vertices. As indicated in [24], the orientations of the ormal n 0, n 2, n 4 and n 6 maintain constant and the four normal intersect at a common centre O which is referred to as virtual centre of the eight-bar linkage as shown in Fig. 1, thus in every configuration the angle γ can be represented by the structure parameters as tan 2 1 cos 1 2 arccos sin2. (2) It should be pointed out that since all the four links 1, 3, 5 and 7 are of the same length and vertices V 0 and V 4, and V 2 and V 6 are identical, the linkage is also symmetric to another plane that passes through the centre points of V 2 and V 6 and is perpendicular to vertices V 2 and V 6. Therefore this eight-bar linkage is referred to as spatial plane-symmetric eight-bar linkage. sections. Based on the eight-bar linkage, a family of regular deployable polyhedral mechanisms can be synthesized in the following 6

7 3.2 A Semi-plane-symmetric Eight-bar Linkage with Exact Straight-line Motion Using the plane-symmetric eight-bar linkage, regular deployable polyhedral mechanisms based on the five Platonic polyhedrons can be synthesized. However, in order to construct semi-regular deployable polyhedral mechanisms, the plane-symmetric eight-bar linkage itself is not enough. Herein, a semi-plan-symmetric eight-bar linkage is introduced and analysed leading to the synthesis and construction of a family of semi-regular deployable polyhedral mechanisms. Figure 6 shows a semi-plane-symmetric eight-bar linkage with the similar structure to the plane-symmetric eight-bar linkage investigated in Section 3.1. Comparing to the plane-symmetric eight-bar linkage, vertices V 0 and V 4 in the eight-bar linkage presented here are identical but not necessary in isosceles triangle shapes, vertices V 2 and V 6 are in two different isosceles triangle shapes. Links 1 and 3 are in the same length as a 11 = a 13 = a 1, and links 5 and 7 have the same length as a 21 = a 23 = a 2, but a 1 a 2. This is still an eight-bar linkage of mobility two and in this study, if the linkage is still considered to be a two-limbed parallel mechanism, one can find that the platform V 4 of this linkage can still perform exact straight-line motion if the two inputs satisfy certain constraints. The straight line produced by the trajectory of V 4 lines in a plane which is formed by the normal of V 0 passing through the orthocentre of the vertex triangle and the normal of V 4 passing through the orthocentre of the vertex triangle. Further, four axes, i.e. O 0 O, O 2 O, O 4 O and O 6 O are the normals of vertices V 0, V 2, V 4 and V 6 that pass through the orthocentres of vertices, they are referred to as vertex axes. The angle between vertex axes O 0 O and O 2 O equals the angle between vertex axes O 2 O and O 4 O, and the angle between vertex axes O 0 O and O 6 O equals the angle between vertex axes O 6 O and O 4 O. Further, comparing to the plane-symmetric eight-bar linkage Fig. 6 A semi-plane-symmetric eight-bar linkage 7

8 in Section 3.1 which is symmetric to two planes, the eight-bar linkage herein is only symmetric to a plane forms by the normals of V 2 and V 6 passing through points O 2 and O 6 of plates V 2 and V 6, therefore the eight-bar linkage herein is referred to as semi-planesymmetric eight-bar linkage. Similar to the study of the plane-symmetric eight-bar linkage in [23], the characteristic of the semi-plane-symmetric eight-bar linkage can be investigated. It is found that the platform V 4 of this linkage can perform exact straight-line motion in the plane if the two inputs, i.e. θ 11 and θ 21 comply with A sin B cos C A sin B cos C, (3) Where, 2 sin 1 cos, B a sin cos 1 cos A a and C1 b1sin 1sin2 2h1sin 1cos1, and sin 1 cos, B a sin cos 1 cos A a and C2 b2sin 2sin6 2h2sin 2cos Imagining that if point O is the fixed frame of the eight-bar linkage, and if the two inputs θ 11 and θ 21 satisfy Eq. (3), the four vertices V 0, V 2, V 4 and V 6 should perform radially reciprocating motion in such a way that vertices V 0 and V 4 move towards point O, vertices V 2 and V 6 move away from point O, and vice versa. Therefore, point O in the eight-bar linkage is referred to as virtual centre of the linkage as shown in Fig.9. Based on this semi-plane-symmetric eight-bar linkage and the previous plane-symmetric eight-bar linkage, a family of semiregular deployable polyhedral mechanisms can be synthesized using virtual-centre-based (VCB) method as follows. 3.3 Synthesis of the Regular Deployable Polyhedral Mechanisms (DPMs) It is well known that there are only five regular convex polyhedrons named Platonic polyhedrons. Based on the five Platonic polyhedrons and the proposed plane-symmetric eight-bar linkage, a family of deployable polyhedral mechanisms (DPMs) can be synthesized and constructed. In this section, the synthesis of the regular deployable polyhedral mechanisms (DPMs) starts from the synthesis of a deployable tetrahedral mechanism by implanting a group of eight-bar linkages into a regular tetrahedron base. The method used for synthesizing the deployable tetrahedral mechanism can then be applied to the synthesis of the whole family of regular DPMs. In Fig. 7, a regular tetrahedron ABCD is given with its six edges denoted by e 1 to e 6. O 1, O 2, O 3 and O 4 are the centres of the four equilateral triangular faces such that AO 1, BO 2, CO 3 and DO 4 are all perpendicular to the faces they pass through and therefore O is the centroid of the tetrahedron. AO 1, BO 2, CO 3 and DO 4 are heights of the tetrahedron. The central angle is denoted as α. 8

9 Fig. 7 A tetrahedron and its geometry In the tetrahedron, as shown in Fig. 8a, a proposed eight-bar linkage is implanted along edge e 3 in such a way that two identical vertices V A and V C of the eight-bar linkage are perpendicular to AO 1 and CO 3 such that AO 1 and CO 3 become vertex axes [23] of vertices V A and V C. The other two identical vertices of the eight-bar linkage, i.e. vertices V 2 and V 4 are placed on faces 2 and 4 in such an arrangement that BO 2 passes through the centre of V 2 and is perpendicular to V 2, and DO 4 passes through centre of V 4 and is perpendicular to V 4. The revolute joints of the two chains in the eight-bar linkage, i.e. joints A 1, B 1 C 1 and D 1, and joints A 2, B 2, C 2 and D 2 are arranged in such a configuration that joints A 1 and B 1 are parallel to edge e 4, joints C 1 and D 1 are parallel to edge e 5, joints (a) Fig. 8 Synthesis of a deployable tetrahedral mechanism (b) A 2 and B 2 are parallel to edge e 1, and joints C 2 and D 2 are parallel to edge e 2. The lengths of the two link groups, i.e. links A 1 B 1 and C 1 D 1, and A 2 B 2 and C 2 D 2 are arbitrary allocated with the condition that A 1 B 1 = C 1 D 1 = A 2 B 2 = C 2 D 2. Arranged in this way, it can be 9

10 found that the four angles φ A, φ B, φ 2 and φ 4 of the isosceles triangle vertices are identical and they all equal 60. It should be noted that according to the previous definition, herein, V A and V C can be referred to as V-plate and V 2 and V 4 can be referred to as F-plate. Thus, it is a plane-symmetric eight-bar linkage that is integrated into the tetrahedron and the four perpendiculars AO 1, BO 2, CO 3 and DO 4 become vertex axes of the four vertices (plates) V A, V 2, V C and V 4. This should lead to the result that γ 1 = γ 2 = α. Substituting φ A = φ B = φ 2 = φ 4 = 60 into Eq.(2), it has γ 1 = γ 2 = 70.53, and according to the geometric property of the tetrahedron, it has α = 70.53, thus γ 1 = γ 2 = α holds. According to Section 2, regarding vertices V C as a base and V A as a platform, given two symmetric inputs at joints A 1 and A 2, vertex V A can perform exact straight-line motion in a plane passing through points A and C, and perpendicular to edge e 6. Similarly, treating vertices V 2 as a base and V 4 as a platform, given two symmetric inputs at joints B 1 and C 1, plate V 4 can perform exact straight-line motion in a plane passing through points B and D, and perpendicular to edge e 3. Then, imagining that the linkage is fixed at point O, given symmetric inputs at any pair of joints in any of the four vertices, the four vertices perform radially reciprocating motions along their vertex axes and the centre point O of the tetrahedron become the virtual centre of the eight-bar linkage. Taking the same procedure, and integrating another eight-bar linkage into the tetrahedron base along edge e 2 as illustrated in Fig. 8b, it can be seen that based on the first eight-bar linkage, in order to form a second eight-bar linkage in the tetrahedron base, only two new vertices V B and V 1, and three new links A 3 B 3, C 3 D 3, C 4 D 4 are needed to be added and connected by six new revolute joints A 3, B 3, C 3, D 3, C 4 and D 4. Vertex V 1 is placed on face 1 of the tetrahedron, and vertex V B is placed at vertex B and is perpendicular to BO 2, the new revolute joints A 3 and B 3 are parallel to edge e 6, C 3 and D 3 are parallel to edge e 5 and C 4 and D 4 are parallel to edge e 3. Arranged in this way, it can be found that the second eight-bar linkage is also a plane-symmetric eight-bar linkage and the four vertices are all in isosceles triangle shape. This secures that the angles between the adjacent vertex axes satisfy γ 4 = γ 3 = γ 2 = γ 1 = α, and the virtual centre of the second eight-bar linkage coincides with the first eight-bar linkage. Further, taking the same procedure, integrating three more plane-symmetric eight-bar linkages into the tetrahedron base along edges e 1, e 4 and e 5, and carrying out a detailed structure design, a deployable tetrahedral mechanism can be generated as shown in Fig. 9. Once the deployable tetrahedron mechanism is constructed, all the virtual centres of the five eight-bar linkages are coincident at one common point, i.e. the virtual centre of the mechanism. Therefore, this synthesis method is referred to as virtual-centre-based (VCB) synthesis method. 10

11 (a) (b) (c) Fig. 9 A deployable tetrahedral mechanisms and its typical phases Figure 9 shows a deployable tetrahedral mechanism that is synthesized based on plane-symmetric eight-bar linkages; the mechanism is a highly overconstrained mechanism of mobility one that can perform radially reciprocating motion in such a manner that vertices V A, V B, V C and V D move radially towards the virtual centre O, vertices V 1, V 2, V 3 and V 4 move radially away from the virtual centre O, and vice versa. It should be pointed out that, in every work configuration, the four vertices V 1, V 2, V 3 and V 4 locate on the faces of a virtual tetrahedron. Subsequently, based on the same principle, the whole family of regular PDMs can be synthesized and constructed based on the five regular Platonic polyhedrons shown in Fig. 10. They are all overconstrained mechanisms of mobility one. In all the mechanisms, each edge of the virtual polyhedrons contains a plane-symmetric eight-bar linkage presented in Section 3.1 with various vertex angles φ 1 and φ 2. Further, the mobility of the mechanisms can be verified in Section 4 through the constraint matrix using the screw-loop method proposed in [25]. (a) (b) (c) (d) Fig. 10 The regular deployable polyhedral mechanisms 11

12 3.4 Synthesis of the Semi-Regular DPMs Section 3.3 presents the principle of virtual-centre-based synthesis of regular DPMs. In this section, the method is used for the synthesis of semi-regular DPMs using the eight-bar linkages based on the semi-regular polyhedrons, i.e. Archimedean polyhedrons, Prism polyhedrons, and Johnson polyhedrons. The principle and procedure of this synthesis is interpreted based on the synthesis of a deployable rectangular prismatic mechanism. A rectangular prism belongs to the Prism solid which is defined to be a solid with two ends that are parallel and of the same size and shape, and with sides whose opposite edges are equal and parallel. A Prism is not a fully regular polyhedron because it does not contain identical faces. Thus, to synthesize the deployable polyhedral mechanisms based on Prismatic solids, both the planesymmetric eight-bar linkage and semi-plane-symmetric eight-bar linkage must be employed. A rectangular prism is indicated in Fig. 11, which contains six faces each of which is a rectangle, twelve edges from e 1 to e 12 and eight vertices A, B, C, D, E, F, G and H. The rectangular faces can be divided into three pairs as pair f 1 and f 3, pair f 2 and f 4, and pair f 5 and f 6, and each pair are identical. O 1 to O 6 denote the centres of the rectangular faces and O denotes centroid of the rectangular prism. The diagonals of the rectangular prism and O 1 O 3, O 2 O 4, O 5 O 6 all pass through O. The central angle α 1 denotes the angle between O 1 O 3 and diagonal AF, α 2 denotes angle between O 2 O 4 and diagonal BE, and α 3 denotes angle between O 5 O 6 and diagonal AF. Fig. 11 A rectangular prism and its geometric parameters In the rectangular prism, as indicated in Fig. 12, an eight-bar linkage is placed along edge e 7 in such a way that two V-plates V C and V F of the eight-bar linkage are perpendicular to diagonals CH and AF such that CH and AF become vertex axes of plates V C and V F. The other two F-plates V 1 and V 4 are placed on faces f 2 and f 4 with orthocentres of V 1 and V 4 coincident with O 1 and O 4 so that vertex axes of V 1 and V 4 are collinear with O 1 O and O 4 O. The revolute joints of the two chains in the eight-bar linkage, i.e. joints A 1, B 1 C 1 and D 1, and joints A 2, B 2, C 2 and D 2 are arranged in such a configuration that joints A 1 and B 1 are perpendicular to the diagonal DF of face f 1, joints C 1 and D 1 are perpendicular to the diagonal CE of face f 1, joints A 2 and B 2, and joints C 2 and D 2 are perpendicular 12

13 Fig. 12 Synthesis of a rectangular prismatic mechanism to diagonals BF and CG of face f 4. The lengths of two link groups, i.e. links A 1 B 1 and C 1 D 1, and A 2 B 2 and C 2 D 2 are arbitrary allocated with the condition that A 1 B 1 = C 1 D 1 and A 2 B 2 = C 2 D 2. Arranged in this way, it can be found that the four angles φ C, φ F, φ 1 and φ 4 have the relation φ C = φ F φ 1 φ 4 leading to γ 1 γ 2. Thus, it is a semi-plane-symmetric eight-bar linkage that is integrated to the rectangular prism. If the input condition in Eq. (3) is satisfied accompanying with the conditions that the vertex angles equal the central angles as γ 1 = α 1 and γ 2 = α 2, treating V F as a base and V C a platform, given two inputs at joints A 1 and A 2 that comply with Eq. (3), plate V C can perform exact straight-line motion in a plane formed by diagonals AF and CH. Then imagining that the linkage is fixed at point O, given two inputs at joints A 1 and A 2 that satisfied Eq. (3), the four plates perform radially reciprocating motions along their vertex axes and the centroid point O of the rectangular prism become the virtual centre of the eight-bar linkage. Carrying out the same procedure, and integrating another eight-bar linkage into the tetrahedron along edge e 3 as shown in Fig. 13, it can be seen that based on the first eight-bar linkage, in order to form a second eight-bar linkage in the tetrahedron, only two new Fig. 13 Integrate a second eight-bar linkage into a rectangular prism 13

14 plates, i.e. V-plate V D and F-plate V 5, and three new links A 3 B 3, A 4 B 4, C 4 D 4 are needed to be added and they are connected by six new revolute joints A 3, B 3, A 4, B 4, C 4 and D 4. Plate V 5 is placed on face f 5 of the rectangular prism and vertex axis is aligned with O 5 O, plate V D is placed at vertex D and is perpendicular to diagonal DG, the new revolute joints A 3 and B 3 are perpendicular to diagonal DF of face f 1, A 4 and B 4, and C 4 and D 4 are respectively perpendicular to diagonals BD and AC of face f 5. Arranged in this way, it can be found that the second eight-bar linkage is also a semi-plane-symmetric eight-bar linkage such that φ C = φ D φ' 1 φ 4 leading to γ' 2 γ 3. Therefore, in order to make the second eight-bar linkage work is such a way that when virtual centre O is imaged to be fixed, the four plates V 1, V 5, V C and V D perform radially reciprocating motion along their vertex axes, input condition in Eq. (3) must be satisfied. Further, to ensure that the virtual centre of the second eight-bar linkage coincident with the first eight-bar linkage, the common links and joints, i.e. plates V C and V 1, links C 3 D 3 (C 1 D 1 ), and joints C 3 (C 1 ) and D 3 (D 1 ) must have the same structure parameters so that it has γ' 2 = γ 2 = α 2 and in addition γ 3 = α 3. Further, taking the same procedure, integrating more eight-bar linkages into the tetrahedron along edges e 1, e 2, e 4 to e 6, and e 8 to e 11, ensuring that structure parameters of links and joints shared by different eight-bar linkages are identical and carrying out a detailed structure design, a deployable rectangular prismatic mechanism can be generated in Fig. 14. It should be noted that only eleven semiplane-symmetric eight-bar linkages are needed to form a deployable rectangular prism mechanism because the mechanism contains only eleven independent loops according to the Euler s formula of independent loops. (a) (b) (c) Fig. 14 A deployable rectangular prismatic mechanism and its typical phases As it is expected in the beginning, the deployable rectangular prismatic mechanism is a highly overconstrained mechanism of mobility one. The mechanism can perform a radially reciprocating motion in such a manner that when plates V A, V B, V C, V D, V E, V F, V G, and V H move radially towards the virtual centre O, plates V 1, V 2, V 3, V 4, V 5 and V 6 move radially away from the virtual centre, and vice versa. It is interested to find that it is the eight V-plates V A, V B, V B, V D, V E and V F rather than F-plates V 1, V 2, V 3, V 4, V 5 and V 6 that locate on a virtual rectangular prism that magnifies and shrinks proportionally with respect to the original rectangular prismatic base. 14

15 Further, based on the synthesis of the deployable rectangular prismatic mechanism, using the same principle and method, integrating the semi-plane-symmetric eight-bar linkages or/and the plane-symmetric eight-bar linkages into the semi-regular polyhedrons, i.e. Archimedean polyhedrons, Prism polyhedrons, and Johnson polyhedrons, a family of highly overconstrained semiregular DPMs of mobility one can be synthesized and constructed. Figures give the deployable cuboctahedral mechanism, the deployable rhombicuboctahedral mechanism, the deployable triangular prismatic mechanism, and the deployable square pyramid mechanism. Fig. 15 Two of the typical phases of a deployable cuboctahedral mechanism Fig. 16 A deployable rhombicuboctahedral mechanism and two of the typical phases Fig. 17 Two of the typical phases of a deployable triangular prismatic mechanism 15

16 Fig. 18 A deployable square pyramid mechanism and two of the typical phases Thus, in this section, the virtual-centre-based method, which is equivalent to the Approach II of the virtual-axis-based synthesis method, is used to synthesize and construct deployable regular and semi-regular polyhedral mechanisms based on the Platonic polyhedrons, Archimedean polyhedrons, Prism polyhedrons and Johnson polyhedrons. From the above construction, it can be found that for the deployable regular DPMs only the plane-symmetric eight-bar linkages are involved; while, for the construction of the semi-regular DPMs, both the plane-symmetric and semi-plane-symmetric eight-bar linkages are involved, the plane-symmetric eightbar linkages are implanted into the polyhedral bases along the edges formed by two identical regular faces and the semi-planesymmetric eight-bar linkages are integrated into the polyhedral bases along the edges containing two different types of regular faces. The steps of using VCB method for synthesizing DPMs is summarized in Fig. 19. Fig. 19 Procedure of the VCB synthesis of DPMs 16

17 It should be pointed out that all the deployable mechanisms synthesized in this section are of mobility one with radially reciprocating motion. Further, since the virtual-centre-based method is equivalent to the Approach II of the virtual-axis-based method, the mechanisms developed in this section can equivalently be constructed with Approach II of the virtual-axis-based method. 4 CONSTRAINT MATRIX AND MOBILITY OF THE REGULAR AND SEMI-REGULAR DEPLOYABLE POLYHEDRAL MECHANISMS (DPMS) Mobility of the family of DPMs can be analysed through the screw-loop equation which is evolved from the mechanical network stemmed from Kirchhoff s circulation law. This is illustrated by taking the deployable tetrahedral mechanism as an example, as (a) (b) Fig. 20 Geometry of a deployable tetrahedral mechanism and its constraint graph illustrated in Fig. 20a, a general configuration of the deployable tetrahedral mechanism is picked and a reference coordinate system is established with its origin locating at the virtual centre of the mechanism and x-axis passing through the middle points of edges AD and BC, y-axis passing through the middle points of edges AB and CD, and z-axis passing through the middle points of edges AC and BD. In vertices V 1, V 2, V 3 and V 4, local coordinate system O i -x i y i z i are established with origin O i locating at the centre of the ith vertex (i = 1, 2, 3, 4) as shown in Fig. 19. In the local coordinate system, z i -axis is collinear with OO i, and x i -axis is parallel to one of the sides of the equilateral triangle vertex. The lengths of the binary links are all l, the distance between the centre of the vertex and the joint axis is b, and the distances between the virtual centre of the deployable tetrahedral mechanism and the centres of vertices V 1, V 2, 17

18 Fig. 21 Screws in an individual vertex V 3 and V 4 are all d. From Fig. 21, the joint screws in every individual vertex can be obtained in the corresponding local coordinate system as T Si b, Si b, T T Si b, Si lsin b lcos, T S i lsin 2 lsin 2 b lcos, T S i lsin 2 lsin 2 b lcos T (4) In the above equation, the first subscript i = 1, 2, 3 and 4 indicates the number of vertex. The joint screws in the local coordinate system of each individual vertex can be transformed to the reference coordinate system through a screw transformation matrix Ri 0 Ti p iri Ri, with R i being the rotation transformation matrix and p i being a skew-symmetric matrix derived from p i presenting the displacement of point O i in the reference coordinate system. From Fig. 18a and according to the geometry of a tetrahedron, R i and p i (i = 1, 2, 3, 4) can be obtained as R , R , , (5) R , R and T p1d , 2 d p, T p3d , 4d p T T. (6) 18

19 Thus, through the screw transformation matrix T i, all the joint screws in the mechanism can be obtained in the reference coordinate system. According to Euler s formula for independent loop of mechanical graph, the mechanism contains five independent loops such that the constraint graph of the mechanism can be obtained in Fig. 18b. Based on the constraint graph, the constraint matrix [25] of the deployable tetrahedral mechanism can be obtained as M C é S S 0 0 S S S S S S ù S S S13 -S 0 0 S S S S = S S S22 -S 0 0 S S 0 0 S S S31 -S 31 S32 S S41 S S43 -S S21 -S ê S23 -S S41 -S 41 -S42 -S ë úû.(7) This is a matrix with 0 = [ ] T and through computation, the mobility of the mechanism can be given [25] as m nc rank M C , (8) where m denotes the mobility of the mechanism and n c is the number of joints. Above analysis proves that the mobility of the deployable tetrahedral mechanism is actually one and it is an overconstrained mechanism. The mobility of the whole proposed family of DPMs proposed in this paper can be verified with the same approach. 5 Degree of Overconstraint of the Deployable Polyhedral Mechanisms (DPMS) The deployable polyhedral mechanisms proposed in this paper are composed of a significantly large number of links and joints forming complex multi-looped kinematic chains. As aforementioned, these mechanisms are highly overconstrained of mobility one, thus it is interesting to compute the degree of overconstraint c of these mechanisms based on the Euler formula for polyhedrons and the Grübler-Kutzbach formula for mechanisms. Based on the Grübler-Kutzbach formula, the mobility of a mechanism is g m d ng 1 f, (9) i1 i where m is the mobility of the mechanism as predicted by the above formula, d is the degree of freedom of an unconstrained rigid body, n the number of rigid bodies, g the number of joints and f i is the number of degree of freedom of the ith joint. This formula is 19

20 based solely on the topology (and not on the geometry) of kinematic chains, it leads to erroneous results when applied to overconstrained mechanisms and thus it can be used to determine the degree of overconstraint of mechanisms. Further, since the mechanisms proposed in this paper are closely related to the numbers of faces, edges and vertices of the polyhedrons, thus, combing Eq. (9) with the Euler formula, the modified Grübler-Kutzbach formula for the DPMs can be presented and the degree of overconstraint can be obtained. For a polyhedron with v vertices, e edges and F faces, based on Euler formula [26], it has v ef 2. (10) for the deployable polyhedral mechanisms constructed based on Approach I of the virtual-axis-based method, based on the constructing principle, each mechanism consists of n = v + e, i.e. n = 2e F + 2 rigid bodies, and g = 2e revolute joints. Further, since all the mechanisms are purely connected by revolute joints, each joint has one degree of freedom. Substituting these into Eq. (9), the mobility m 1 of the mechanisms can be represented based on the Grübler-Kutzbach formula with respect to the number of faces and edges of the polyhedrons as m1 61F 2e. (11) Further, for the deployable polyhedral mechanisms constructed based on Approach II of the virtual-axis-based method or the virtual-centre-based method, based on the constructing principle, referring to Table 6.1, for each mechanism it has n v F s F, F F i.e. n e sj 2 rigid bodies with s j denotes the number of sides of the jth face, and g 2 s j revolute joints. Substituting these j into Eq. (9), the mobility m 2 of the mechanism based on the Grübler-Kutzbach formula can be obtained as j j j 2 F m 6 e1 4 s. (12) j 1 j As it is known that the real mobility of the deployable polyhedral mechanisms developed in this paper is one, the degree of overconstraint c i written as c i = 1 m i (i = 1, 2) can be obtained as follows. For the mechanisms developed by Approach I of the virtual-axis-based method it has, c 1m 6F 2e 5. (13) 1 1 And for the mechanisms generated by virtual-centre-based method or Approach II of the virtual-axis-based method, it has F c 1m 4 s 6e5 2 2 j. (14) j1 20

21 Based on Eqs. (13) and (14), the degrees of overconstraint of the deployable polyhedral mechanisms discussed in this paper can be calculated. For the deployable hexagonal prismatic mechanism developed in Section 2 based on Approach I of the virtual-axis method, the degrees of overconstraint can be obtained from Eq. (13) as c 1 = 7. For the five Platonic polyhedrons, the relation F s j 2e holds such that Eq. (14) can be reduced to c 2 2e 5. Therefore, the j 1 degrees of overconstraint of the deployable polyhedral mechanisms constructed by virtual-centre-based method based on the five Platonic solids can be obtained as: c 2 = 7 for the deployable tetrahedral mechanism, c 2 = 19 for the deployable hexahedral and octahedral mechanisms because they are dual to each other and have the same number of edges, and c 2 = 55 for the deployable dodecahedral and icosahedral mechanisms because they are dual to each other and have the same number of edges. For the Archimedean polyhedrons, since the types of the faces of the polyhedrons are variable, to calculate the degree of overconstraint of the deployable polyhedral mechanisms constructed using virtual-centre-based method based on the Archimedean solids, Eq. (14) has to be strictly complied. Following Eq.(14), degree of overconstraint of the deployable cuboctahedral mechanism is c 2 = 43, and that of the deployable rhombicuboctahedral mechanism is c 2 = 99. Further, for the Prism polyhedrons each of which contains two s-sided regular polygons and s square faces, thus with s being the number of sides of the end regular polygons, the number of edge can be obtained as e = 3s. Substituting these in the Eq. (14), it has c2 6s 5. Therefore, for the deployable rectangular prismatic mechanism, it has c 2 = 19, and for the deployable triangular prismatic mechanism, its degree of overconstraint is c 2 = 13. Finally, each of the Johnson polyhedrons contains certain numbers of different kind of regular faces so that the term F j 1 s j in Eq (14). is always variable. Thus, in order to calculate the degrees of overconstraint of the deployable mechanisms constructed by virtualcentre-based method based on the Johnson solids, Eq. (14) must be strictly followed. Based on Eq. (14), the degree of overconstraint of the deployable square pyramid mechanism is c 2 = Singularity and Multifurcation of the Deployable Polyhedral Mechanisms (DPMS) Equation 7 gives the instantaneous mobility of the deployable tetrahedral mechanism. Using numerical method, it is shown that in normal configurations, the rank of constraint matrix M C is 23 so that the degree of freedom of the mechanism is 1 corresponding to Eq. (8). However, it is found that in the cases that θ = 0, namely Case I or θ = 180 ψ, namely Case II, with ψ denotes the dihedral 21

22 angle of tetrahedron, the rank of the constraint matrix M C decreases to 20 and the mechanism reaches singular configurations. In Case I, the singular configuration shows that vertices V 1, V 2, V 3 and V 4 are respectively coplanar with the corresponding three links connected to them. And in Case II, each of vertices V A, V B, V C and V D is coplanar with its three associative links. In both of these two singular configurations, the mechanism gains three extra degrees of freedom such that for Case I, after passing the singular Fig. 22 Multifurcation of the deployable tetrahedral mechanism for Case I configuration, vertices V 1, V 2, V 3, and V 4 are possible to be on either side of their singular positions, and for Case II, after passing the singular configuration, vertices VA, VB, VC and VD are possible to be on either side of their singular positions. These lead to the multifurcation phenomenon [27] of the deployable polyhedral mechanisms. As shown in Fig. 22, after the deployable tetrahedral mechanism passing through singular configuration θ = 0, multifurcation occurs and it leads to the eight possible configurations. Figures 20(a) and (e) show the two uniform configurations that all vertices V 1, V 2, V 3, and V 4 more outward or toward the virtual centre O after passing through the singularity position. Figures 20(b) and (f) indicate the two cluttered configurations that vertices V 1 moves outward/towards the virtual centre O while vertices V 2, V 3 and V 4 move towards/outward the virtual centre. Figures 20(c) and (g) show the case that vertices V 2 and V 3 move towards/outward the virtual centre while vertices V 1 and V 4 move outward/towards the virtual centre. Figures 20(d) and (h) present the case that vertex V 4 moves outward/towards the virtual centre O but vertices V 1, V 2 and V 3 move towards/outward the virtual centre. Similarly, Fig. 23 illustrates the multifurcation of the deployable tetrahedral mechanism for Case II. From the analysis, it is found that the number of configurations for each case of multifurcation can be gives as 22

23 Fig. 23 Multifurcation of the deployable tetrahedral mechanism for Case II N m ìï 2 NVI for Case I = ï í, (15) ïï 2 NVII for Case II î where N m stands for the number of configurations, N VI denotes the number of vertices involved in singular Case I and N VII denotes number of vertices involved in Case II. Thus, numbers of configurations for the multifurcation of the deployable polyhedral mechanisms proposed in this paper are listed in Table. 1. Table 1 Number of configurations for the multifurcation of DPMs Mechanism Number of configurations Case I Case II Deployable hexagonal prismatic mechanism Deployable tetrahedral mechanism 8 8 Deployable hexahedral mechanism Deployable octahedral mechanism Deployable dodecahedral mechanism Deployable icosahedral mechanism Deployable rectangular prismatic mechanism Deployable cuboctahedral mechanism Deployable rhombicuboctahedral mechanism Deployable triangular prismatic mechanism Deployable square pyramid mechanism

24 7 CONCLUSIONS This paper presented intuitive and efficient approaches for the synthesis and construction of a family of deployable polyhedral mechanisms (DPMs) that perform radially reciprocating motion based on the regular and semi-regular polyhedrons using the virtualaxis-based method and the virtual-centre-based method. Based on the properties of the regular and semi-regular polyhedrons, constructions of the deployable polyhedral mechanisms (DPMs) based on different type of polyhedrons were demonstrated with selected examples. Mobility of the mechanisms has been verified utilizing the screw-loop equation method and degree of overconstraint of the mechanisms was presented combining the Euler formula of polyhedrons and the Grübler-Kutzbach formula of mobility. It was found in this paper that to construct deployable mechanisms based on the five regular polyhedrons, only the planesymmetric eight-bar linkages are involved and the mechanisms preserve the shapes of the basis solids; on the other hand, to construct the deployable mechanisms based on the semi-regular polyhedrons, both the plane-symmetric and semi-plane-symmetric eight-bar linkages are needed. The multifurcation of the deployable polyhedral mechanisms was investigated revealing various configurations the mechanisms may have after passing through their singular configurations. The mechanisms proposed in this paper have great potential application in the fields of architecture, manufacturing and space exploration. ACKNOWLEDGMENTS The authors gratefully acknowledge the supports from the EU 7th Framework Programme under grant No and No , and the supports from the National Natural Science Foundation of China under grant number and from Tianjin Technology Scheme project under grant number 12JCZDJC REFERENCES [1] Verheyen, H. F., 1984, Expandable polyhedral structures based on dipolygonids, in: Proc. 3rd Int. conf. Space Structures, Elsevier, London, UK. [2] Verheyen, H. F., 1989, The complete set of Jitterbug transformers and the analysis of their motion, Computers Math. Applic. 17(1-3), pp [3] Fuller, R. B., 1975, Synergetics: exploration in the geometry of thinking, Macmillan, New York. [4] Stachel, H., 1994, The Heureka-Polyhedron, In Fejes Tóth, G. (ed.): Intuitive Geometry. Colloq. Math. Soc. János Bolyai, 63, pp , North-Holland, Amsterdam. 24

25 [5] Guest, S. D., 1994, Deployable structures: concepts and analysis, PhD dissertation, Corpus Christi College, Cambridge University. [6] Wohlhart, K., 1993, Heureka octahedron and Brussels folding cube as special cases of the turning tower, in: Proc. the Sixth IFToMM International Symposium on Linkages and Computer Aided Design Methods, Bucharest, Romania. [7] Wohlhart, K., 1995, New overconstrained spheroidal linkages, in: Proc. the Ninth World Congress on the Theory of Machines and Mechanisms, Milano, Italy. [8] Wohlhart, K., 1998, Kinematics of Röschel polyhedra, in: Advances in Robot Kinematics: Analysis and Control, J. Lenarcic and M. Husty eds., Kluwer Akademie Publisher. [9] Wohlhart, K., 2008, Double-ring polyhedral linkages, in: Proc. Interdisciplinary Applications of Kinematics, Peru, Lima. [10] Wohlhart, K., 2008, New polyhedral star linkages, in: Proc. the 10th International Conference on the Theory of Machines and Mechanisms, Liberec, Czech Republic. [11] Hoberman, C., 2004, Geared expanding structures, US patent: US B2. [12] Agrawal, S. K., Kumar, S., and Yim, M., 2002, Polyhedral single degree-of freedom expanding structures: design and prototypes, ASME Journal of Mechanical Design, 124(9), pp [13] Gosselin, C. M., and Gagnon-Lachance, D., 2006, Expandable polyhedral mechanisms based on polygonal one-degree-offreedom faces, Proc. IMechE Part C: J. Mechanical Engineering Science, 220, pp [14] Laliberté, T., and Gosselin, C. M., 2007, Polyhedra with articulated faces, in: Proc. of the 12th IFToMM world congress, Besancon, France. [15] Kiper, G., Söylemez, E., and Kisisel, A. U. Ö., 2007, Polyhedral linkages synthesized using Cardan motion along radial axes, in: Proc. of the 12th IFToMM world congress, Besancon, France. [16] Kiper, G., Fulleroid-like linkages, 2009, in Proc. EUCOMES 08, M. Ceccarelli (ed.), Springer. [17] Röschel, O., 1995, Zwangläufig Bewegliche Polyedermodelle I, Math. Pannonica, 6, pp [18] Röschel, O., 1996, Zwangläufig Bewegliche Polyedermodelle II, Studia Sci. Math. Hung, 32, pp [19] Röschel, O., 2001, Zwangläufig Bewegliche Polyedermodelle III, Math. Pannonica, 12, pp [20] Wei, X., Yao, Y., Tian, Y., and Fang, R., 2006, A new method of creating expandable structure, Proc. IMechE Part C: J. Mechanical Engineering Science, 220, pp [21] Kovács, F. Tarnai, T., Fowler, P. W., and Guest S. D., 2004, A class of expandable polyhedral structures, International Journal of Solids and Structures, 41, pp