A class of reconfigurable deployable platonic mechanisms

Mechanism and Machine Theory 105 (2016) 409–427

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

A class of reconfigurable deployable platonic mechanisms Ruiming Li a, Yan-an Yao a,⁎, Xianwen Kong b,⁎ a b

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK

a r t i c l e

i n f o

Article history: Received 3 December 2015 Received in revised form 21 July 2016 Accepted 22 July 2016 Available online xxxx Keywords: Reconfigurable mechanism Deployable mechanism Angulated element Hoberman sphere Radially reciprocating motion

a b s t r a c t This paper deals with the design of a new class of reconfigurable deployable Platonic mechanisms (RDPMs). At first, reconfigurable angulated element (RAE) with four half-platforms is designed and analyzed based on the reconfigurable property of angulated element. Using RAE as module, a method of synthesizing RDPMs is then proposed and three RDPMs are constructed. Starting from Type I generalised angulated element (GAE), a reconfigurable generalised angulated element (RGAE) comprised of two GAEs is investigated with its reconfigurable conditions identified. n pairs of straight elements (SE) and n pairs of angulated elements (AE) are further inserted into 2-RGAEs to construct n-SEs-2-RGAEs and n-AEs-2-RGAEs. The fully-retracted state and fully-expanded state of n-SEs-2-RGAEs and n-AEs-2-RGAEs are also detailed. Using 1-AE-2-RGAEs, three RDPMs with large magnification ratio are constructed in 3D software and fabricated using aluminum to verify their feasibility. This work provides a systematic approach to the design of RDPMs which can switch between two kinds of conventional deployable polyhedral mechanisms, including deployable polyhedral mechanisms with radially reciprocating motion and Hoberman spheres. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Deployable polyhedral mechanisms are generally one degree-of-freedom (DOF) multi-loop overconstrained mechanisms that adopt polyhedral layout and preserve their global shape during deployment. Deployable polyhedral mechanisms and their synthesis methods have been studied by numerous researchers. Verheyen [1] presented a complete set of Jitterbug transformers, which are able to perform a symmetrical expansion and impansion motion. Kiper [2] synthesized dipyramidal and stellated Fulleroid-like linkages using symmetrical spatial 8R closed chains. Wohlhart [3] constructed zig-zag linkages by inserting multiple scissors into edges of Platonic polyhedra. Kovács et al. [4] proposed double-link expandohedra to imitate the reversible expansion of viruses. Wei et al. [5] created a class of expandable structures for spatial objects which are constructed by connecting a series of parallelogram mechanisms. Gosselin and Gagnon-Lachance [6] constructed a family of expandable mechanisms using regular planar polygonal linkages, and the expansion ratio can be increased by inserting more additional parallelograms into the legs of the polygonal linkages. Based on the principle of Cardan motion, Kiper et al. [7] proposed a synthesis method for designing linkages with slider assemblies for scaling polyhedral shapes. Hoberman [8,9] constructed reversibly expandable truss structures based on angulated elements, which later were generalised by You and Pellegrino [10]. Hoberman spheres, as three-dimensional reversibly expandable truss structures, can be constructed by inserting angulated elements into the edges of polyhedra with a circumscribed sphere. Hoberman spheres always maintain their curved geometry during deployment. Deployable polyhedral mechanisms with radially reciprocating motion (RRM) are especially fascinating due to their radial deployment. The synthesis of such deployable polyhedral mechanisms has also been approached from different perspectives based on several modular linkages, such as CRRC kinematic chains [11], PRRP kinematic chains [12] and symmetric 8-bar linkage [13].

http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.019 0094-114X/© 2016 Elsevier Ltd. All rights reserved.

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R. Li et al. / Mechanism and Machine Theory 105 (2016) 409–427

(a)

(b)

Fig. 1. Equivalent mechanism of Hoberman's angulated element: (a) A pair of PRRP mechanisms, (b) Transition configuration.

Here, C, R and P represent cylindrical, revolute and prismatic joints, respectively. In addition, Cardan motion [14] along radial axes has also been used to synthesize such deployable polyhedral mechanisms. The deployable polyhedral mechanisms based on Platonic, semiregular and Johnson solids have been synthesized and constructed using the above mentioned approaches. Meanwhile, reconfigurable mechanisms have been one of emerging areas in mechanism science and robotics in past decade, known as metamorphic mechanism [15], or multiple operation modes mechanism [16]. Recently, some reconfigurable mechanisms are also used as deployable mechanisms to increase flexibility and adaptability, such as deployable canopy [17]. Wei and Dai [18] even constructed a group of reconfigurable deployable Platonic mechanisms (RDPM), which can transform themselves from Fulleroid-like linkage type to star-transformer linkage type without reassembly. However, the reconfigurable method of Wei and Dai's RDPMs is based on locking and unlocking joints, which increases the complexity of mechanisms. Recently, the authors of this paper proposed a method of constructing RDPMs [19] based on reconfigurable angulated elements. A tetrahedral RDPM derived from switch-pitch ball mechanism was constructed therein. The tetrahedral RDPM, which didn't need to lock and unlock joints during reconfiguration, is more practical and versatile to meet the emerging requirements for reconfigurable deployable mechanisms that can deploy in different modes. As a systematic extension, this paper determined the parameters of reconfigurable generalised angulated elements and identified their fully-retracted states and fully-expanded states by considering the interference between revolute joints. The aim of this paper is to synthesize and construct a class of RDPMs, which have more compact structures and can transform themselves among Hoberman sphere mode and RRM mode. The term “reconfigurable mechanism” in this paper refers to a mechanism that can change assembly modes without disassembly and reassembly. Hoberman sphere mode and RRM mode are two kinds of assembly modes, since RDPMs in these two modes have different effective numbers of links and joints. The layout of this paper is as follows. The concept of reconfigurable angulated element (RAE) is proposed in Section 2, and the mechanical design and transition configuration analysis of RAE are also detailed in this section. In Section 3, a method of constructing RDPMs is proposed. In Section 4, reconfigurable generalised angulated element is introduced and extended to n-AE-2RGAEs and n-SE-2-RGAEs cases. In Section 5, RDPMs with large magnification ratio are constructed based on 1-AE-2-RGAEs modules. Three prototypes are fabricated to test and verify the feasibility of the proposed construction method in Section 6. Conclusions and discussions are given in Section 7. 2. Reconfigurable angulated element In the early 1990s, Hoberman invented angulated element (AE), which consists of two identical angulated rods, each rod has a middle and two end pivots and is connected to the other rod at the middle pivot. Since four end pivots are constrained to move

(a)

(b)

Fig. 2. Diagram of RAE: (a) Assembly diagram of RAE, (b) Transition configuration.

R. Li et al. / Mechanism and Machine Theory 105 (2016) 409–427

(a)

411

(b)

Fig. 3. Single PRRP mechanism mode of RAE: (a) CP-ab moves toward center O, (b) CP-cd moves toward center O.

along two intersecting lines, we can interpret the angulated element as a pair of PRRP mechanisms [20] connected at the middle pivot. As shown in Fig. 1(a), two angulated rods AQD and BQC are connected at middle pivot point Q, four terminal pivot points A, B, C, and D are jointed to two intersecting lines AB and CD, respectively. To ensure this mechanism has 1 DOF, the following conditions should be satisfied ∘

CQ ¼ DQ; AQ ¼ BQ; ∠AQD ¼ ∠BQC ¼ 180 −ϕ

ð1Þ

where ϕ denotes the angle between AB and CD. Although the angulated element has been studied by many authors and used in numerous cases such as retractable roofs and expandable toys, its singularity is rarely mentioned, probably due to mechanical design issues. It is apparent that singularity happens when links AQD and BQC coincide (Fig. 1(b)). In the transition configuration, AQ/BQ and CQ/DQ are perpendicular to the directions of two prismatic (P) joints, respectively. Two revolute (R) joints passing A/C and B/D coincide, and two PRRP mechanisms merge into one PRRP mechanism. In this configuration, the mechanism may enter two motion modes: angulated element (two PRRPs) motion mode or single PRRP motion mode. The above analysis shows that the angulated element is reconfigurable. Such a characteristic has been ignored in the literature either intentionally or unintentionally. Now let us consider the practical structure of a reconfigurable angulated element (RAE). A slider is cut into two identical halfplatforms (HP), and two HPs on the same side need to be assembled face to face. Therefore an RAE is consisted of six components: two jointed angulated links and four half-platforms. Fig. 2(a) shows the assembly diagram of RAE. Four HPs are represented by HP-a, HP-b, HP-c and HP-d, respectively. The axes of the two prismatic joints pass through HP-a, HP-b, HP-c, HP-d, respectively, at centers A, B, C and D. A1, B1, C1, D1 and Q are the centers of five revolute joints. AA1, BB1, CC1 and DD1 are the offsets of HP-a, HP-b, HP-c, HP-d off two prismatic joints, and AA1 ⊥ OA, BB1 ⊥OB, CC1 ⊥ OC, DD1 ⊥ OD. To make two HPs on the same side to merge into a compound-platform (CP), the offsets on the same side should be equal to each other AA1 ¼ BB1 ; CC 1 ¼ DD1

ð2Þ

Fig. 2(b) shows the transition configuration of RAE, in which configuration links AQD and BQC are coincident, HP-a and HP-b merge into CP-ab, HP-c and HP-d merge into CP-cd, the axes of revolute joints A1 and B1 are collinear, and the axes of revolute

Fig. 4. Diagram of symmetric RAE: (a) Transition configuration, (b) Angulated element motion mode.

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Fig. 5. The diagram of tetrahedron and RAE-A1B1QC1D1: (a) Tetrahedron, (b) RAE-A1B1QC1D1.

joints C1 and D1 are also collinear. The following conditions hold AQ⊥OA; BQ ⊥OB; CQ⊥OC; DQ ⊥OD

ð3Þ

The key to design and utilize RAE is to identify its transition configuration. Since links AQD and BQC coincide, we choose the front link BQC, HP-b and HP-c as the single PRRP mechanism to determine the parameters (OB and OC) of transition configuration. For a given ϕ, BB1, CC1, B1Q, C1Q, we can obtain a set of positive solutions of OB and OC by solving Eqs. (4) and (5). The circumcircle radius of RDPM constructed by RAE in transition configuration can also be measured by the solutions of OB and OC. 2

2

2

2

2

OB þ BQ ¼ OC þ CQ

2

ð4Þ 2

2

OB þ OC −2OB  OC  cosϕ ¼ BQ þ CQ −2BQ  OC  cos∠BQC

ð5Þ

where BQ = BB1 + B1Q, CQ = CC1 + C1Q, ∠BQC= 180∘ − ϕ. In single PRRP mechanism mode, the RAE undergoes RRM. When CP-ab moves radially toward center O, CP-cd moves radially away from center O and vice versa, as shown in Fig. 3. Note that the offsets BB1(AA1), CC1(DD1) and the link sizes B1Q(A1Q), C1Q(D1Q) can be chosen arbitrarily. If BB1(AA1) equals to CC1(DD1) and B1Q(A1Q) equals to C1Q(D1Q), the RAE turns to a symmetric mechanism, whose symmetry axis is line OQ, as shown in Fig. 4(a). Line OQ bisects angle between line OB and line OC. In angulated element motion mode (Fig. 4(b)), HP-a and HP-c moves by the same amount away from center O, HP-b and HP-d moves by the same amount toward center O. In Section 3, we will use symmetric RAE to synthesize RDPM.

Fig. 6. The construction steps of RDTM: (a) Insert one RAE, (b) Insert three RAEs, (c) The resulted RDTM.

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Fig. 7. Hoberman sphere mode of RDTM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

3. Reconfigurable deployable platonic mechanism The PRRP mechanism has been used to construct the deployable polyhedral mechanisms [12,14] that perform RRM. RAE in its transition configuration can be treated as a PRRP mechanism. So, we can use RAE to construct the RDPMs in their transition configurations. The construction steps are as follows: Step Step Step Step

1: 2: 3: 4:

Pick a Platonic polyhedron and find out all necessary axes. Design an RAE module and identify its transition configuration. Insert RAEs into the polyhedron faces along corresponding axes. Merge the overlapping CPs.

Here we take the simplest Platonic polyhedron (tetrahedron) as an example to illustrate the construction steps. The tetrahedron is shown in Fig. 5(a), it consists of four vertices V1, V2, V3 and V4. Three among four vertices determine a face, for example face F-123 is determined by V1, V2 and V3. The center of tetrahedron is denoted by O. The centers of four faces are denoted by O123, O124, O134 and O234, respectively. We can obtain four vertex axes passing center O and four corresponding vertices, also four face axes passing center O and four face centers. Since a Platonic polyhedron has both an inscribed sphere and a circumscribed sphere, four vertex axes are coincident with four face axes. Since each face of a tetrahedron is a triangle and three adjacent faces intersect with each other at one vertex, the RAE module of tetrahedron consists of four HPs and each HP is designed to be triangle shape possessing three symmetrical revolute joints. The RAE module RAE-A1B1QC1D1 is shown in Fig. 5(b). The included angles satisfy ∠A1QD1 = ∠B1QC1 = 109.47°. The link lengths comply with AA1 = BB1 = CC1 = DD1 and A1Q = B1Q = C1Q = D1Q as the special case illustrated in Fig. 4. The transition configuration occurs when four HPs merge into two CPs. We insert the RAE-A1B1QC1D1 in its transition configuration into the face F-123 in such way that axis-O-V1 passes the center of CP-cd normal to the platform plane and axis-O-O123 passes the center of CP-ab normal to the platform plane, as shown in Fig. 6(a). Similarly, we can also insert RAE-A1B1QG1H1 and RAE-A1B1QI1J1 into face F-123, as shown in Fig. 6(b). It is worth noting that there are three overlapping CP-ab, we should merge duplicate CPs to make sure the CP-ab is connected to CP-cd, CP-gh and CP-ij via six coplanar revolute joints. Following the same procedure, we can insert nine more RAEs into their corresponding faces. An RDPM derived from a tetrahedron can be finally obtained by merging duplicate CPs on each vertex. It is named as

Fig. 8. RRM mode of RDTM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

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Fig. 9. RAE modules: (a) RAE module for RDHOM, (b) RAE module for RDDIM.

reconfigurable deployable tetrahedral mechanism (for short RDTM). As shown in Fig. 6(c), the obtained RDTM contains 8 identical CPs (16 HPs) and 24 identical angulated links. The eight CPs are denoted by CP-ab, CP-cd, CP-ef, CP-gh, CP-ij, CP-kl, CP-mn and CP-op. Fig. 7 illustrated the RDTM in Hoberman sphere mode. This mode starts from the transition configuration (also its fullyexpanded state), as shown in Fig. 7(a). Among sixteen HPs, the conjugate HPs move away from each other. As shown in Fig. 7(b), HP-b, HP-d, HP-f, HP-h, HP-j, HP-l, HP-n and HP-p move by the same amount toward the virtual center, while their conjugate HP-a, HP-c, HP-e, HP-g, HP-i, HP-k, HP-m and HP-o move by the same amount away from the virtual center. The minimum state occurs when the eight inner HPs interfere with each other, as shown in Fig. 7(c). RDTM in RRM mode has many multifurcation configurations, which has been discussed in [13] and don't belong to the scope of this paper. In this paper, we pick two typical configurations (represented by RRM configuration I and RRM configuration II) to illustrate the RDTM in RRM mode. In RRM configuration I, CP-ab, CP-ef, CP-kl, CP-op move toward the virtual center and CP-cd, CP-gh, CP-ij, CP-mn move away from the virtual center, as shown in Fig. 8(b). In RRM configuration II, CP-ab, CP-ef, CP-kl, CP-op move away from the virtual center, CP-cd, CP-gh, CP-ij, CP-mn move toward the virtual center, as shown in Fig. 8(c). Using the same method, we can synthesize and construct other four RDPMs with their corresponding RAE modules. As the hexahedron and octahedron are dual to each other and so are the dodecahedron and icosahedron, the reconfigurable deployable hexahedral/dodecahedral mechanism and the reconfigurable deployable octahedral/icosahedral mechanism are isomorphic. We shall call these two kinds of mechanisms as reconfigurable deployable hex-octahedral mechanism (for short RDHOM) and reconfigurable deployable dodec-icosahedral mechanism (for short RDDIM). Since hexahedron contains 8 vertices and 6 quadrilateral faces, while octahedron contains 6 vertices and 8 triangle faces, 16 triangle HPs and 12 quadrilateral HPs are needed for constructing RDHOM. Since dodecahedron contains 20 vertices and 12 pentagon faces, while icosahedron contains 12 vertices and 20 triangle faces, 40 triangle HPs and 24 pentagon HPs are needed for constructing RDDIM. The RAE modules for RDHOM and RDDIM are shown in Fig. 9. The RAE module of RDHOM consists of two angulated links, two triangle HPs and two quadrilateral HPs, the included angles of its angulated links satisfy ∠A1QD1 = ∠B1QC1 = 125.26°., the link lengths comply with AA1 = BB1 = CC1 = DD1 and A1Q = B1Q = C1Q = D1Q. The RAE module of RDDIM consists of two angulated links, two triangle HPs and two pentagon HPs, the included angles of its angulated links satisfy ∠ A1QD1 = ∠ B1QC1 = 142.62°., the link lengths also comply with AA1 = BB1 = CC1 = DD1 and A1Q = B1Q = C1Q = D1Q. The transition configuration of RAE modules occurs when two conjugate half-platforms merge into one compound-

Fig. 10. Hoberman sphere mode of RDHOM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

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Fig. 11. RRM mode of RDHOM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

platform. We insert these two kinds of RAE modules into the faces of the hexahedron/octahedron and dodecahedron/icosahedron to construct RDHOM and RDDIM. The resulted RDHOM consists of 28 HPs and 24 pairs of angulated elements, the HPs are connected to angulated links via revolute joints to form a multi-loop mechanism. In its transition configuration, an RDHOM can switch to Hoberman sphere mode, as shown in Fig. 10, also RRM mode, as shown in Fig. 11. In RRM configuration I, 8 triangle CPs move away from the virtual center and 6 quadrilateral CPs move toward the virtual center. While 6 quadrilateral CPs move away from the virtual center and 8 triangle CPs move toward the virtual center in RRM configuration II. The RDDIM consists of 64 HPs and 60 pairs of angulated elements. Starting from the transition configuration, RDDIM can switch to Hoberman sphere mode, as shown in Fig. 12, also RRM mode, as shown in Fig. 13. In RRM configuration I, twenty triangle CPs move away from the virtual center and 12 pentagon CPs move toward the virtual center. While 12 pentagon CPs move away from the virtual center and 20 triangle CPs move toward the virtual center in RRM configuration II. Link numbers of the RDPMs and included angles of angulated links are listed in Table 1.

4. Reconfigurable generalised angulated element Using the above method, we can construct a class of RDPMs based on RAE modules. However, the deployable scale of these RDPMs in the Hoberman sphere mode or RRM mode is relatively small. To enlarge the magnification ratio, the intuitive method is to increase the number of angulated elements in RAE, say m-RAEs. It is noted that at the singular configuration, the problem arises as that an m-RAEs degenerates into (m + 1) R (R represents the revolute joint) serial chain and the serial chain has undesirable self-motion. One way to solve this problem is to use the Type I generalised angulated element [10] to construct reconfigurable modules. For convenience, we call the generalised angulated element with reconfigurable function as reconfigurable generalised angulated element (for short RGAE). Taking the number of GAEs into account, the RGAE based on m GAEs is named as m-RGAEs. As we know, the general type I GAE with two or more parallelograms can be used as 1-RGAE. Only the simplest GAE is accounted in this paper by considering the combination of GAEs and other scissor elements. Fig. 14(a) shows the simplest m-RGAEs, 2-RGAEs, which consists of two simplest GAEs: GAE-A1B1Q1RS and GAE-C1D1Q2RS. They are jointed at points R and S. GAE-A1B1Q1RS is connected to HP-a and HP-b at points A1 and B1 via revolute joints, respectively.

Fig. 12. Hoberman sphere mode of RDDIM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

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Fig. 13. RRM mode of RDDIM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

GAE-C1D1Q2RS is connected to HP-c and HP-d at points C1 and D1 via revolute joints, respectively. The axes of all revolute joints are parallel to each other. Axes of two prismatic joints pass through HP-a, HP-b, HP-c, HP-d, respectively, at centers A, B, C, D. To ensure the 2-RGAEs mechanism has 1 DOF, the following conditions on link lengths should be satisfied AA1 ¼ BB1 ; CC 1 ¼ DD1 ; A1 Q 1 ¼ B1 Q 1 ; C 1 Q 2 ¼ D1 Q 2 ; Q 1 R ¼ Q 1 S; Q 2 R ¼ Q 2 S

ð6Þ

The angles of four angulated links should satisfy the following equation ∘

ϕ ¼ 2  180 −

∠A1 Q 1 S þ ∠B1 Q 1 R þ ∠C 1 Q 2 S þ ∠D1 Q 2 R 2

ð7Þ

To reach transition configuration, HP-a, HP-b, HP-c, HP-d have to merge into CP-ab and CP-cd simultaneously. Normally the bearings will be installed in the revolute joints, the diameter of the revolute joint, say w, is wider than rest parts of angulated link, so the interference happens between two adjacent revolute joints, RS must be greater than or equal to w. Because HP-a and HP-c are located above HP-b and HP-d, ∠ B1Q1R should be larger than ∠A1Q1S and ∠D1Q2R should be larger than∠C1Q2S to guarantee deployment. As shown in Fig. 14(b), in transition configuration, the angle between Q1R and Q1S equals to ∠B1Q1R minus ∠A1Q1S, the angle between Q2R and Q2S equals to∠D1Q2R minus∠C1Q2S. The mechanism can achieve its reconfiguration as long as Q1S, Q2S,∠B1Q1R, ∠A1Q1S, ∠D1Q2R and∠C1Q2S comply with Eqs. (8) and (9), in addition to Eqs. (6) and (7). Q 1 S sin

∠B1 Q 1 R‐∠A1 Q 1 S ∠D1 Q 2 R−∠C 1 Q 2 S ¼ Q 2 S sin 2 2

2Q 1 S sin

ð8Þ

∠B1 Q 1 R‐∠A1 Q 1 S ≥w 2

ð9Þ

where w denotes the diameter of revolute joints. 4.1. n-SEs-2-RGAEs Theoretically, one may add any pairs of straight elements (SE) into 2-RGAEs in order to increase the magnification ratio, as done by Wohlhart [3]. For the simplicity, 2-RGAEs with n pairs of straight elements is named as n-SEs-2-RGAEs. Apparently, the combination of 2 GAEs and n SEs are countless regarding the link lengths and arrangement. According to the practical experience, the more exotic the mechanisms become, the less useful the obtained mechanisms are. So in this paper, two GAEs are identical and so do n SEs. Lengths from middle pivot point to either end pivot point for two GAEs are the same and denoted as lg, Lengths from middle pivot point to either end pivot point for all SEs are also the same and denoted as ls.

Table 1 Link numbers and angle parameters of the RDPMs. Reconfigurable deployable Platonic mechanisms

CP number

HP number

AE number

Included angle

Reconfigurable deployable tetrahedral mechanism Reconfigurable deployable hexa-octahedral mechanism Reconfigurable deployable dodeca-icosahedral mechanism

8 14 32

16 28 64

12 24 60

109.47° 125.26° 142.62°

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Fig. 14. Diagram of 2-RGAEs: (a) Assembly diagram of 2-RGAEs, (b) Transition configuration.

The diagram of n-SEs-2-RGAEs is shown in Fig. 15, n-SEs-2-RGAEs has a retracted state, in which state two GAEs interfere simultaneously, as shown in the lower part of Fig. 15. n-SEs-2-RGAEs also has an expanded state (transition configuration), in which state, four HPs merge into two CPs at the same time, as shown in the upper part of Fig. 15. α denotes the larger included angle of GAEs, β denotes the smaller included angle of GAEs, Here we still assume that the interference occurs between cylindrical surfaces of two adjacent revolute joints, whose diameters are identical and represented by w. α, β and lg satisfy the following conditions αþβ ϕ ∘ ¼ 180 − 2 2

α−β ≥2 arcsin

w 2lg

ð10Þ ! ð11Þ

Substituting Eq. (10) into Eq. (11), yields

α ≥ arcsin

w 2lg

! ∘

þ 180 −

ϕ 2

ð12Þ

Fig. 15. The diagram of n-SEs-2-RGAEs in Hoberman sphere mode.

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Fig. 16. Two special cases of 3-SEs-2-RGAEs: (a) α=180∘, lg = ls, (b)α= arcsin(w/2l)+180∘ −ϕ/2, lg = ls.

Since α is less than or equal to 180∘, we can obtain the range of values for angle α. ∘

180 ≥α ≥ arcsin

w 2lg

! ∘

þ 180 −

ϕ 2

ð13Þ

We can obtain various n-SEs-2-RGAEs by changing the value of angle α. Especially, when lg equals to ls and α reaches its upper limit (α= 180∘), n-SEs-2-RGAEs can achieve the fully-retracted state, in which state, two side GAEs interfere and all SEs interfere with each other simultaneously, as shown in Fig. 16(a). When lg is equal to ls and α reaches its lower limit (α = arcsin (w/2l∘ g) + 180 − ϕ/2), n-SEs-2-RGAEs can achieve its fully-expanded state, in which state, four HPs merge into two CPs and all SEs interfere with each other at the same time, as shown in Fig. 16(b). 4.2. n-AEs-2-RGAEs Similarly, one may insert Hoberman's simple angulated elements (AE) into 2-RGAEs to construct n-AEs-2-RGAEs. To maintain the symmetry and reduce the complexity, n-AEs-2-RGAEs consists of two identical GAEs and n identical pairs of AEs. The diagram of n-AEs-2-RGAEs is shown in Fig. 17. The link lengths satisfy the following conditions AA1 ¼ BB1 ¼ CC 1 ¼ DD1 ¼ l

ð14Þ

A1 Q 1 ¼ B1 Q 1 ¼ C 1 Q 2 ¼ D1 Q 2 ¼ Q 1 R1 ¼ Q 1 S1 ¼ Q 2 R2 ¼ Q 2 S2 ¼ lg

ð15Þ

Fig. 17. The diagram of n-AEs-2-RGAEs in Hoberman sphere mode.

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Fig. 18. The curves of OA and OB with respect to γg in Hoberman sphere mode.

R1 T 1 ¼ S1 T 1 ¼ R2 T n ¼ S2 T n ¼ T i U i ¼ T i V i ¼ U i T iþ1 ¼ V i T iþ1 ¼ la

for i ¼ 1; 2…n−1

ð16Þ

where n denotes the pairs of AEs. The included angles of AEs and GAEs satisfy the following conditions ∠A1 Q 1 S1 ¼ ∠C 1 Q 2 S2 ¼ β; ∠B1 Q 1 R1 ¼ ∠D1 Q 2 R2 ¼ α

ð17Þ

∠R1 T 1 V 1 ¼ ∠S1 T 1 U 1 ¼ ∠R2 T n V n‐1 ¼ ∠S2 T n U n‐1 ¼ θ

ð18Þ

∠U i T iþ1 V iþ1 ¼ ∠V i T iþ1 U iþ1 ¼ θ

ð19Þ

for i ¼ 1; 2…n−2

α, β and θ must satisfy the following equation ∘

ϕ ¼ ð2 þ nÞ  180 −ðα þ β þ nθÞ

Fig. 19. The kinematic diagram of equivalent PRRP mechanism.

ð20Þ

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Fig. 20. The curve of OA with respect to OD in RRM mode.

Let γg denote ∠B1Q1S1 and γa denote ∠S1T1V1, γg and γa vary during the deployment process. In parallelogram Q1R1T1S1, the following condition holds lg sin

α−γ g θ−γa ¼ la sin 2 2

ð21Þ

It is very interesting to note that n-AEs-2-RGAEs can reach its fully-retracted state and its fully-expanded state in one assembly as long as some special conditions are satisfied. In the following, we are going to find out these special conditions. In order to reach fully-retracted state, GAEs and AEs have to interfere at the same time. In other words, γg and γa need to reach their minimum simultaneously. Eq. (21) can be rewritten as lg sin

α−γ g min θ−γa min ¼ la sin 2 2

ð22Þ

where γg min ¼ 2arc sin 2lwg , γa min ¼ 2arc sin 2lwa . In order to reach fully-expanded state, the distance between points R1 and S1 should be equal to w and γg should increase to β, as shown in the upper part of Fig. 17. The following equation holds w ¼ 2lg sin

α‐β 2

ð23Þ

Solving the transcendental equations formed by Eqs. (20), (22) and (23), we can normally get a set of numerical solutions for α, β and θ, and especially get a set of analytical solutions for α, β and θ when lg = la. When lg = la = L, we can simplify Eq. (22) as follows α¼θ

ð24Þ

Fig. 21. Hoberman sphere mode of LR-RDTM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

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Fig. 22. RRM mode of LR-RDTM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

Substituting Eq. (24) into Eqs. (20) and (23), then solving Eqs. (20) and (23), α, β and θ can be expressed analytically as follows

α¼θ¼

β¼

2 arcsin

w ∘ þ ðn þ 2Þ  180 −ϕ 2L nþ2

ðn þ 2Þ  180∘ −ð2n þ 2Þ arcsin nþ2

w −ϕ 2L

ð25Þ

ð26Þ

The obtained α, β and θ can guarantee n-AEs-2-RGAEs to reach its fully-retracted state, as shown in the lower part of Fig. 17 and its fully-expanded state, as shown in the upper part of Fig. 17. This special property of n-AEs-2-RGAEs may have potential application value in actual deployable structures to maximize the magnification ratio. n-AEs-2-RGAEs and n-SEs-2-RGAEs have their own merits. The n-AEs-2-RGAEs can have its fully-retracted state and fullyexpanded state in one assembly to enlarge its magnification ratio, while the design and manufacturing of n-SEs-2-RGAEs are relatively easy because straight elements reduce the complicated constraints for included angles of angulated elements. In the following, the kinematics of n-AEs-2-RGAEs in Hoberman sphere mode and RRM mode will be presented. As shown in Fig. 17, in the Hoberman sphere mode, the distance between points B and D can be written as follows BD ¼ 2X g þ X a þ 2l cos

ϕ 2

ð27Þ

where Xg and Xa can be determined by X g ¼ 2lg sin

  γg ϕþβ π cos − 2 2 2

Fig. 23. Hoberman sphere mode of LR-RDHOM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

ð28Þ

422

R. Li et al. / Mechanism and Machine Theory 105 (2016) 409–427

Fig. 24. RRM mode of LR-RDHOM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

Xa ¼

2la sin

γa nπ−nθ sin 2 2 π−θ sin 2

ð29Þ

So OB and OA in Hoberman sphere mode can be written as follows

OB ¼

BD 2 sin

ϕ 2

¼

2X g þ X a þ 2l cos 2 sin

OA ¼ OB þ AB ¼

ϕ 2

ϕ 2

2X g þ X a þ 2l cos 2 sin

ϕ 2

ð30Þ

ϕ 2 þ 2l sin β−γg g 2

ð31Þ

Let l = 50 mm, lg = la = 60 mm, ϕ = 60°, w = 15 mm, n = 3, we get α = θ = 170.87°, β = 156.51°, the range of γg is [14.36°, 156.51°]. We can plot the curves of OA and OB with respect to γg during the deployment, as shown in Fig. 18. The maximum of OA in Hoberman sphere mode (represented by OAHmax), which is the same as the maximum of OB, equals to 659.24 mm when γg equals to 156.51°, the minimum of OA in Hoberman sphere mode (represented by OAHmin) equals to 273.23 mm when γg equals to 14.36°. As shown in Fig. 19, in RRM mode, the n-AEs-2-RGAEs can be treated as a PRRP mechanism. The kinematic equations can be derived as follows l cosϕ þ A1 D1 sinη þ l ¼ OA sinϕ

Fig. 25. Hoberman sphere mode of LR-RDDIM: (a) Transition configuration, (b) Middle state, (c) Minimum state.

ð32Þ

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423

Fig. 26. RRM mode of LR-RDDIM: (a) Transition configuration, (b) RRM configuration I, (c) RRM configuration II.

OA cosϕ þ l sinϕ þ A1 D1 cosη ¼ OD

ð33Þ

where ϕ represents the angle between OA and OD, η represents the sharp angle between A1D1 and OD, the length of A1D1 can be calculated as 2Xgmax + Xamax (refer to Eqs. (28) and (29)). Due to the symmetry of n-AEs-2-RGAEs, we only need to analyze one configuration of RRM mode, in which configuration, point D moves toward virtual center O. The minimum value of OA in RRM mode (represented by OARmin) appears when OD equals to zero. When η = 90°, the maximum value of OA in RRM mode (represented by OARmax) can be expressed as OAR max ¼

A1 D1 þ l l þ sinϕ tanϕ

ð34Þ

Let l = 50 mm, lg = la = 60 mm, ϕ = 60°, w = 15 mm, n = 3, we can plot the curve of OA with respect to OD, whose range is [0, ODRmax(OARmax)], as shown in Fig. 20. For each given OD, there are two solutions for OA. But for above-mentioned RRM mode, the smaller solution should be omitted. So OARmax equals to 747.83 mm when A1D1 is perpendicular to OD, OARmin equals to 611 mm when point D is coincident with center O. The obtained OAHmax, OAHmin, OARmax, OARmin will be used to measure the magnification ratio of RDPMs, which will be discussed later in Section 5. 5. RDPMs with large magnification ratio Referring to the construction steps of RDPMs, we can also construct RDPMs using RGAEs to enlarge magnification ratio. An RDPM with large magnification ratio is called LR-RDPM for simplicity reason. It is obvious that both n-AEs-2-RGAEs and n-SEs2-RGAEs can be used as modules to construct LR-RDPMs. Here we use symmetrical 1-AE-2-RGAEs, whose parameters satisfy Eqs. (14)–(19), (25) and (26), to construct LR-RDPMs (LR-RDTM, LR-RDHOM, LR-RDDIM). As shown in Fig. 21, the resulted LR-RDTM consists of 16 triangle HPs and 12 1-AE-2-RGAEs components. The Hoberman sphere mode of LR-RDTM is similar to RDTM. It starts from its fully-expanded state (transition configuration), as shown in Fig. 21(a), stops when 1-AE-2-RGAEs reaches its fully-retracted state, as shown in Fig. 21(c). Figs. 22(b) and (c) show RRM configuration I and RRM configuration II of LR-RDTM respectively. They also start from transition configuration (Fig. 22(a)). When four CPs move by the same amount toward the virtual center, other four CPs move by the same amount away from the virtual center, and vice versa. The resulted LR-RDHOM is shown in Figs. 23 and 24, it consists of 16 triangle HPs, 12 quadrilateral HPs and 24 1-AE-2-RGAEs components. The Hoberman sphere mode of the LR-RDHOM is shown in Fig. 23. RRM configuration I and RRM configuration II of the LR-RDHOM are shown in Fig. 24. Six quadrilateral CPs move toward the virtual center and 8 triangle CPs move away from the virtual center in RRM configuration I, and vice versa in RRM configuration II. Table 2 Link numbers and parameters of LR-RDPMs. LR-RDPM

HP number

AE number

n

l(mm)

la, lg(mm)

w(mm)

ϕ (deg)

α (deg)

β (deg)

θ (deg)

LR-RDTM LR-RDHOM LR-RDDIM

16 28 64

12 24 60

1 1 1

50 50 50

60 60 60

15 15 15

70.53 54.74 37.38

161.28 166.54 172.33

146.92 152.18 157.97

161.28 166.54 172.33

424

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Fig. 27. Four enveloping spheres of LR-RDDIM: (a) Smallest state in Hoberman sphere mode, (b) Biggest state in Hoberman sphere mode, (c) Smallest state in RRM mode, (d) Biggest state in RRM mode.

The resulted LR-RDDIM is shown in Figs. 25 and 26, it consists of 40 triangle HPs, 24 pentagon HPs and 60 1-AE-2-RGAEs components. The Hoberman sphere mode of the LR-RDDIM is shown in Fig. 25. RRM configuration I and RRM configuration II of the LR-RDDIM are shown in Fig. 26. Twelve pentagon CPs move toward the virtual center and 20 triangle CPs move away from the virtual center in RRM configuration I, and vice versa in RRM configuration II. The link numbers and parameters of LR-RDPMs are listed in Table 2. The magnification ratio is an essential criterion for deployable mechanisms. Considering the reconfigurable property of LRRDPM, we need to get the smallest and biggest states for both two modes to calculate the overall magnification ratio. Since our reconfigurable deployable mechanisms have spherical shapes, we can envelope the smallest and biggest states of LRRDDIM in Hoberman sphere mode and RRM mode with four spheres of different radius, say RHmin, RHmax, RRmin and RRmax, which can be calculated as OAHmin, OAHmax, OARmin and OARmax (see Section 4) without considering the thickness of platforms and links. Fig. 27 illustrates the four spheres that envelope the smallest and biggest states of LR-RDDIM in Hoberman sphere mode and RRM mode. The magnification ratio of LR-RDPMs in Hoberman sphere mode (RHm) equals to RHmax divided by RHmin. The magnification ratio of LR-RDPMs in RRM mode (RRm) equals to RRmax divided by RRmin. The overall magnification ratio of LR-RDPMs (Rm) can be calculated as follow

Rm ¼

maxfRH max ; RR max g minfRH min ; RR min g

ð35Þ

Table 3 The magnification ratios of LR-RDPMs. LR-RDPM

RHmin/mm

RHmax/mm

RRmin/mm

RRmax/mm

RHm

RRm

Rm

LR-RDTM LR-RDHOM LR-RDDIM

218.17 256.60 331.61

361.07 470.73 696.09

360.26 374.68 369.96

423.83 516.73 726.43

1.66 1.83 2.10

1.17 1.38 1.96

1.94 2.01 2.19

R. Li et al. / Mechanism and Machine Theory 105 (2016) 409–427

Fig. 28. Prototype of LR-RDTM: (a) Transition configuration, (b) Hoberman sphere configuration, (c) RRM configuration I, (d) RRM configuration II.

Fig. 29. Prototype of LR-RDHOM: (a) Transition configuration, (b) Hoberman sphere configuration, (c) RRM configuration I, (d) RRM configuration II.

425

426

R. Li et al. / Mechanism and Machine Theory 105 (2016) 409–427

Fig. 30. Prototype of LR-RDDIM: (a) Transition configuration, (b) Hoberman sphere configuration, (c) RRM configuration I, (d) RRM configuration II.

6. Prototypes To test the feasibility of RDPMs, three aluminum prototypes (LR-RDTM, LR-RDHOM, LR-RDDIM) were fabricated according to the parameters in Table 2. The radii of enveloping spheres including RHmin, RHmax, RRmin and RRmax for each prototype are listed in Table 3. RHm, RRm, and Rm for each prototype also can be calculated correspondingly. The prototypes showed that LR-RDPMs can freely switch among Hoberman sphere mode and RRM mode in transition configuration and have one DOF in each mode. Figs. 28, 29 and 30 show the LR-RDTM, LR-RDDIM and LR-RDDIM, respectively, in transition configuration, Hoberman sphere mode, RRM configuration I and RRM configuration II. It is noted there are also some problems arising when manually operating the prototypes, mainly in RRM mode. In RRM mode, the multifurcation increases the DOFs of LR-RDPMs instantaneously, and it becomes difficult to manually switch between different modes especially when the number of CPs is high. One way to eliminate the multifurcation in RRM mode is to use synchronized gears, as done by Hoberman for designing geared expanding structures [21]. Another way is to use symmetrical redundant actuations, which will increase the difficulty of actuation and control. So the mechanical design, actuation and control of RDPMs aiming at eliminating the multifurcation in RRM mode will be our core work in the near future.

7. Conclusions This paper has revealed the reconfigurable property of angulated element, and designed RAE and RGAE with half-platforms and straight/angulated elements. A systematic approach has been proposed to the design of RDPMs. Several RDPMs with small and large magnification ratios have been constructed in 3D software and the feasibility of the proposed method has been verified using physical prototypes. This paper has also given a method to measure the magnification ratios of LR-RDPMs in Hoberman sphere mode and RRM mode, and the overall magnification ratio of LR-RDPMs has been derived correspondingly. Considering practical application, manipulators or other devices can be mounted on outer half-platforms of RDPMs, whose size (l in Section 4.2) is independent of the other link parameters and can be chosen according to the installation needs. The manipulators possess two kinds of motion modes. In the Hoberman sphere mode, they retract or expand together by the same amount. In RRM mode, they can be divided into two groups. When manipulators in one group retract, manipulators in the other group expand, and vice versa. It is worth noting that reconfigurable deployable polyhedral mechanisms based on semiregular polyhedra and Johnson solids can also be constructed by merging half-platforms and collinear joints. The corresponding parameters of RAEs and RGAEs will be discussed in our future work. The method of constructing RDPMs by merging half-platforms and collinear joints may open up a new field in the reconfigurable design of deployable mechanisms to increase their flexibility and adaptability.

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427

Acknowledgement This work was supported by the National “Climbing” Program of China (2015BAK04B02).

Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.019.

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