Screw-System-Variation Enabled Reconfiguration of

Ketao Zhang Centre for Robotics Research, Faculty of Natural and Mathematical Sciences, King’s College London, University of London, Strand, London WC2R 2LS, UK e-mail: [email protected]

Jian S. Dai Centre for Robotics Research, Faculty of Natural and Mathematical Sciences, King’s College London, University of London, Strand, London WC2R 2LS, UK e-mail: [email protected]

Screw-System-Variation Enabled Reconfiguration of the Bennett Plano-Spherical Hybrid Linkage and Its Evolved Parallel Mechanism This paper presents the Bennett plano-spherical hybrid linkage and proposes a novel metamorphic parallel mechanism consisting of this plano-spherical linkage as part of limbs. In light of geometrical modeling of the Bennett plano-spherical linkage, and with the investigation of the motion-screw system, the paper reveals for the first time the reconfigurability property of this plano-spherical linkage and identifies the design parameters that lead to change of constraint equations, and subsequently to variation of the order of the motion-screw system. Arranging this linkage as part of limbs, the paper further investigates the reconfiguration property of the plano-spherical linkage evolved parallel mechanism. The analysis reveals that the platform constraint-screw system varies following both bifurcation and trifurcation with motion branch variation in the 6R linkage integrated limb structure. Consequently, this variation of the platform constraintscrew system leads to reconfiguration of the proposed metamorphic parallel mechanism. The paper presents a way of analyzing reconfigurability of kinematic structures based on the screw-system approach. [DOI: 10.1115/1.4030015] Keywords: Bennett plano-spherical hybrid linkage, overconstrained 6R linkage, reconfiguration, constraint analysis, motion-screw system, constraint-screw system

1

Introduction

Parallel mechanisms and manipulators [1–4] with an endeffector connected to a base by multi-loop kinematic chains generally have higher accuracy [5], rigidity [6], and load/mass ratio [7] in comparison with serial mechanisms and manipulators. In the structural design of parallel mechanisms, closed loop subchains such as the 2-UU chains [8], planar four-bar parallelogram [9–13], and spherical six-bar linkages [14] have been employed in kinematic limbs that connect a platform to a base. This way of design led to a number of hybrid limb structures for constructing parallel mechanisms with limited degrees-of-freedom (DOFs) [15], including the parallel manipulators widely used in food production lines [16] and other packaging industries [17] as well as miniaturized medical devices [18]. Though parallel mechanisms and manipulators have been extensively investigated, the dedicated work on this type of mechanisms has mostly focused on kinematic structures with a constant topological configuration, unchanging motion behaviors, and a unitary function [19]. These features of conventional parallel mechanisms have limitations to accommodate tasks where variable functions are required. It is envisaged that mechanisms with an ability to reconfigure their structure, topological configuration and functionality are in demand for variable tasks and changing environment [20,21]. Zlatanov et al. [22] presented a typical reconfigurable parallel mechanism, the 3-URU DYMO, which has five types of platform motions corresponding to multiple threedimensional regions of its configuration space. Kong et al. [23] presented 3DOF parallel mechanisms with both spherical and translational modes and parallel mechanisms with more than one Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 9, 2014; final manuscript received March 6, 2015; published online April 15, 2015. Assoc. Editor: Oscar Altuzarra.

Journal of Mechanical Design

operation mode were subsequently synthesized [24]. Gan et al. [25] presented a design of the rT joint by integrating an extra rotational DOF in the traditional Hooke’s joint and developed two types of metamorphic parallel mechanisms. Further, Carbonari et al. developed a class of parallel robots with reconfigurability resorting to a locking system [26]. In comparison with the aforementioned reconfigurable parallel mechanisms in which the platform is connected to a base by serial kinematic chains, parallel mechanisms with closed loop subchains and capability of reconfiguration are a focus. Gogu [27] addressed the geometrical constraint singularity induced transitory phase [28] of bifurcated platform motion in parallel mechanisms and the mobility change. Zhang et al. [29] presented a novel metamorphic parallel mechanism which consists of two 8R linkages as closed loop subchains and reconfigures its structure and subsequently the motion characteristics of the platform. Bifurcated motion branches characterized by the distinct functionality were further revealed by analyzing the platform constraints in terms of screw theory [30]. Zeng et al. [31] developed a 4DOF kinematotropic hybrid parallel mechanism with a capability of reconfiguring the platform mobility without changing its topological structure. Ye et al. [32] synthesized a family of reconfigurable parallel mechanisms by integrating a diamond parallelogram four-bar linkage in the kinematic chains. With all of these developments, this paper is to study the course of reconfiguration based on the screw-system approach [33] and to investigate screw-system-variation enabled reconfiguration of the Bennett plano-spherical hybrid linkage and its evolved novel metamorphic parallel mechanism with three hybrid limbs. The geometry of the plane-symmetric Bennett plano-spherical linkage is described in detail. Following the geometry of the plano-spherical linkage, dimension of the mechanism motion-screw system and its variation in relation to inequality of design parameters of the linkage are revealed. This leads to unraveling of variation of the mechanism motion-screw system and constraint singularity

C 2015 by ASME Copyright V

JUNE 2015, Vol. 137 / 062303-1

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

enabled bifurcation and trifurcation [34] of the 6R linkage in accordance with the proposed inequality conditions of parametric design. The limb constraint-screw systems of the evolved metamorphic parallel mechanism are then revealed according to assemblage of the hybrid limb consisting of the plane-symmetric plano-spherical linkage and an RR chain. The motion characteristics and reconfiguration of the platform are hence analyzed in accordance with distinct motion branches of the 6R linkage integrated limb structure.

2 Geometry of the Plane-Symmetric Bennett Plano-Spherical Hybrid Linkage The Bennett plano-spherical hybrid linkage refers to the overconstrained mobile 6R linkage presented by Bennett [35] in his work on motion of the Sarrus linkage and related mechanisms [36]. This overconstrained 6R linkage is a typical form of double spherical 6R linkages [37–39] where one of the two spherical centers is at infinity leading to three parallel hinges. Accordingly, six hinges of this typical 6R linkage are grouped in two sets with three concurrent hinges having a common point and the remaining three hinges having parallel axes. A kinematic model and the corresponding geometric model of the plane-symmetric Bennett plano-spherical hybrid linkage are illustrated in Figs. 1(a) and 1(b), respectively. With the kinematic model illustrated in Fig. 1(a), axes of revolute joints R1, R2, and R6 have common point A and axes of revolute joints R3, R4, and R5 are parallel. The axes of two revolute joints at the distal end of each link are coplanar. Further, the Bennett plano-spherical hybrid linkage is symmetric with respect to plane P1 determined by axes of joints R1 and R4. It implies that axes of joints R6 and R5 are reflections of joints R2 and R3 with respect to symmetric plane P1, respectively. Plane P2 is perpendicular to axes of revolute joints R3, R4, and R5 which are parallel. This means axes of revolute joints R3 and R5 are parallel to symmetric plane P1. The axes of coplanar joints R2 and R3 have common point C and axes of coplanar joints R5 and R6 have common point D on the other side of plane P1. In a general configuration, two coplanar axes of joints R1 and R4 have instantaneous common point E. As illustrated in the geometric model of the hybrid linkage in Fig. 1(b), CC0 , EE0 and DD0 are axes of revolute joints R3, R4, and R5, where points C0 , D0 , and E0 are projections on plane P2 of common points C, D, and E. Point A0 is the projection on plane P2 of point A. Points B and B0 are the projections on EE0 and AA0

of common points C and D. Hence, four points B, C, B0 , and D are located on a plane which is parallel to plane P2. CB0 is the common normal of AA0 and CC0 while DB0 is the common normal of AA0 and DD0 . As axes AC and AD of joints R2 and R6 are symmetric with respect to plane P1, two common normals CB0 and DB0 have equal length. Further, BC and BD, which are perpendicular to EE0 , have equal length denoted by r. Parameters aj (j ¼ 1, 2, 5, and 6) denote the angles between each pair of adjacent axes. Considering the symmetry with respect to plane P1 of the 6R linkage, a1 ¼ a6 and a2 ¼ a5. Further, the projections of points C and D on AE are coincide. Taking point G as the projection, it derives that AE is a normal of the plane determined by points C, D, and G. This further reveals AE is perpendicular to GM, where M is the midpoint of CD. The distance measured from point A to point G is denoted by h. Taking M0 as the projection on Z-axis of point M, it derives that the two right triangles DAFM0 and DMFG are similar, meaning /FMG ¼ /FAM0 ¼ b. Since AA0 and CC0 are parallel, angle /CAB0 ¼ a2. Symmetrically, angle /DAB0 ¼ a5. A Cartesian coordinate frame O-XYZ with the origin attached at common point A is set as global frame of the 6R linkage in Fig. 1. The Z-axis located in symmetric plane P1 is perpendicular to the axis of R4 and pointing upward, X-axis is aligned with AA0 and parallel to axis of R4 and Y-axis is set following the right-handed rule. Following the Denavit–Hartenberg convention [40], the parametric constraints [41,42] of the plane-symmetric 6R linkage in Fig. 1 are derived as a1 ¼ a2 ¼ a5 ¼ a6 ¼ 0; a1 ¼ a6 ¼ a16 ;

a3 ¼ a4 ¼ r

a2 ¼ a5 ¼ a25 ;

d1 ¼ d3 ¼ d4 ¼ d5 ¼ 0;

(1)

a3 ¼ a4 ¼ 0

(2)

d2 ¼ d6 ¼ h= cos a16

(3)

in which r, h, a16, and a25 are the design parameters of the linkage, a16僆[0, p/2] denotes two equal angles a1 and a6 while a25 denotes two equal angles a2 and a5. The angular displacement of each joint is denoted by hj (j ¼ 1, …, 6). Hence, the position vectors of points B, B0 , C, D, and G expressed in global frame O-XYZ are 2

hca25 =ca16

3

6 7 0 7 rb ¼ 6 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ðhsa25 =ca16 Þ  ðhta16 sh10 Þ þ rch40

(4)

Fig. 1 The plane-symmetric Bennett plano-spherical hybrid linkage and its geometrical model (a) kinematic model and (b) geometric model

062303-2 / Vol. 137, JUNE 2015

Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

2

hca25 =ca16

3

0

7 5

6 rb 0 ¼ 4

2

(5)

0 2 6 rc ¼ 6 4

hca25 =ca16

(16) (6)

hca25 =ca16 6 7 hta16 sh10 7 rd ¼ 6 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ðhsa25 =ca16 Þ  ðhta16 sh10 Þ

(7)

l0mm ¼ lmg cb þ l0gg

(9)

where lmg and l0gg denote the length of line segments MG and GG0 , respectively. Substituting Eqs. (4)–(8) into Eq. (9), it derives sa16 ch10 cb þ ca16 sb  ca25 ¼ 0

(10)

Solving the Eq. (10), expression of cb and sb are obtained and given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca25 ch10 sa16 6ca16 c2 a16  c2 a25 þ c2 h10 s2 a16 (11) cb ¼ c2 a16 þ c2 h10 s2 a16 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca16 ca25 ch10 sa16 c2 a16  c2 a25 þ c2 h10 s2 a16 sb ¼ (12) c2 a16 þ c2 h10 s2 a16

3 Variation of the Motion-Screw System and the Induced Bifurcation and Trifurcation 3.1 Motion-Screw System of the Linkage. In the 6R linkage, the joints Rj are described by motion screws Sj (j ¼ 1,…, 6). With Eqs. (11) and (12) for cb and sb and the position vectors in Eqs. (4)–(8), vectors sj ¼ [lj, mj, nj]T pointing in the direction of screws Sj are derived as 2 3 2 3 l1 sb s1 ¼ 4 m1 5 ¼ 4 0 5 (13) cb n1

6 7 6 s2 ¼ 4 m2 5 ¼ 6 4 n2

hca25 =ca16

¼ ½ l1

0

¼ ½ l2

m2

0 0 0 T

n1 n2

0 0 0 T

¼ ½ 1 0 0 0 n2

m2 T

¼ ½ 1 0 0 0 n2 þ rch40 ¼ ½ 1 0 0 0 n2 ¼ ½ l2

m2

n2

0 T

(17)

m2  T

0 0 0 T

The constraint screws are reciprocal to the motion screws [43–46] and the spanning constraint-screw system is derived as

in which h10 ¼ (p  h1)/2, h40 ¼ (h4  p)/2, “c” is a shorthand notation for cos(*), “s” for sin(*) and “t” for tan(*). With two similar right triangles DAFM0 and DMFG on symmetric plane P1, distance l0mm measured from point M to the Z-axis is given by

2

8 S1 > > > > > > > > S2 > > S4 > > > > > > S5 > > > : S6

(8)

hcb

3

Sm1

3

6 7 rg ¼ 4 0 5

l2

Accordingly, motion screws describing six hinges of the 6R linkage in global coordinate frame O-XYZ are given by

3

hsb

n2

3

7 hta16 sh10 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ðhsa25 =ca16 Þ  ðhta16 sh10 Þ

2

3

hca25 =ca16

ðhsa25 =ca16 Þ2  ðhta16 sh10 Þ2

n6

2

2

3

2

2 3 l2 7 6 6 7 6 7 hta16 sh10 7 s6 ¼ 4 m6 5 ¼ 6 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ¼ 4 m2 5 l6

3

Sc1 ¼ Sc11 ¼ ½ 1 0 0 0 0 0 T

(18)

where superscript “c” denotes the constraint screws and their spanned constraint-screw system. The above constraint screw represents a common constraint of the hybrid linkage and the dimension of the constraint-screw system, dim(Sc1 ), equals to 1. Reciprocally, the dimension of the mechanism motion-screw system is 5. In accordance with geometry of the 6R linkage and the mechanism constraint-screw system, link L2 is rotating about a virtual axis aligned with AA0 in relative to link L5. The virtual axis is aligned with the intersection of two planes determined by DADD0 and DACC0 . Further, the axis of joint R1 is rotating about Y-axis with respect to the virtual axis aligned with AA0 . 3.2 Degeneration of Motion-Screw System with Inequalities for Parameters. According to geometry of the Bennett plano-spherical hybrid linkage in Fig. 1 and Grassmann varieties [47–49] of ranks 1, 2, 3, and 4, the corresponding motion screws in Eq. (17) form a five-system in a general configuration and the spanning mechanism motion-screw system degenerates only when the link parameters satisfy the inequalities and equalities corresponding to following three cases: (1) r > ldg ¼ hta16, the axes of revolute joints R1, R2, R3, R5, and R6 are coplanar, except for the axis of joint R4; (2) r < ldg ¼ hta16, the axes of revolute joints R2, R3, R4, R5, and R6 are coplanar, except for the axis of joint R1; (3) r ¼ ldg ¼ hta16, the axes of all six joints are coplanar. where ldg denotes the distance between points D and G. For case (i), the linkage reaches the configuration in Fig. 2(a) with the axes of revolute joints R1, R2, R3, R5, and R6 located in a single plane only when the coordinates of screws in Eq. (17) satisfy the constraint equations l1 ¼ 1, n1 ¼ 0, and n2 ¼ 0 simultaneously. Substituting the vectors s1 and s2 expressed in Eqs. (13) and (14) into above constraints, the constraints can be rewritten as

7 hta16 sh10 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 ðhsa25 =ca16 Þ  ðhta16 sh10 Þ

(14)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca16 ca25 ch10 sa16 c2 a16  c2 a25 þ c2 h10 s2 a16 ¼ 1 (19) sb ¼ c2 a16 þ c2 h10 s2 a16

2 3 1 s3 ¼ s4 ¼ s5 ¼ 4 0 5 0

(15)

cb ¼

Journal of Mechanical Design

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca25 ch10 sa16 6ca16 c2 a16  c2 a25 þ c2 h10 s2 a16 ¼ 0 (20) c2 a16 þ c2 h10 s2 a16 JUNE 2015, Vol. 137 / 062303-3

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

Fig. 2 The singular configurations of the Bennett plano-spherical hybrid linkage (a) case (i): r > ldg, (b) case (ii): r < ldg, and (c) case (iii): r 5 ldg

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðhsa25 =ca16 Þ2  ðhta16 sh10 Þ2 ¼ 0

(21)

Solving Eqs. (19)–(21), design parameters a16 and a25 and joint variable h10 are derived as 8 p > < a16 ¼ a25 ¼ 4 (22) > : h10 ¼ p 2 Under such a condition, the motion screws in Eq. (17) of the 6R linkage change to

Sm2

8 S1 ¼ ½ 1 > > > > > > > > S2 ¼ ½ l2 > > > < S3 ¼ ½ 1 ¼ > > S4 ¼ ½ 1 > > > > > > S5 ¼ ½ 1 > > > : S6 ¼ ½ l2

0 0 0 0 0 T m2

0 0 0 0 T

0 0 0 0 m2 T 0 0 0 rch40

0 T

(23)

0 0 0 0 m2  T m2

0 0 0 0 T

The constraint screws reciprocal to the motion screws in Eq. (23) are yielded as

Sc2

¼

8 < Sc21 ¼ ½ 1 0

0 0 0 0 T

: Sc ¼ ½ 0 0 22

0 0 0 1 T

0 n 0 0 0 T

¼ ½ l2

m2

0 0 0 0 T

¼ ½ 1 0 0 0 0 m2 T ¼ ½ 1 0 0 0 0 0 T

(26)

¼ ½ 1 0 0 0 0 m2  T ¼ ½ l2

m2

0 0 0 0 T

The constraint screws reciprocal to the motion screws in Eq. (26) are derived as ( c S31 ¼ ½ 1 0 0 0 0 0 T c (27) S3 ¼ Sc32 ¼ ½ 0 0 0 0 1 0 T The above mechanism constraint-screw system is a two-system. It implies that the mechanism motion-screw system in Eq. (26) is a four-system degenerated from the five-system in Eq. (17). For case (iii), the linkage has capability to reach its configuration in Fig. 2(c) with all axes of the six revolute joints located in a single plane and axes of joint R1 and R4 aligned with each other. This case happens only when the coordinates of screws in Eq. (17) satisfy the constraints for both the cases (i) and (ii) simultaneously. Hence, design parameters r, h, a16 and a25, which lead to motion-screw system degeneration, are derived as ( p a16 ¼ a25 ¼ (28) 4 r¼h Under such a condition, the mechanism motion-screw system in Eq. (17) changes to

(25)

In this case, the motion screws in Eq. (17) of the plane-symmetric linkage change to 062303-4 / Vol. 137, JUNE 2015

¼ ½ l1

(24)

The above equation shows that the mechanism constraint-screw system of the 6R linkage becomes a two-system. It further implies that the mechanism motion-screw system corresponding to the configuration in Fig. 2(a) degenerates to a four-system. For case (ii), the linkage is able to reach the configuration in Fig. 2(b) with axes of revolute joints R2, R3, R4, R5, and R6 located in a single plane only when the coordinates of screws in Eq. (17) satisfy constraint equations n2 ¼ 0 and n2 þ rch40 ¼ 0 simultaneously. Considering link length r does not equal to zero and substituting n2 in Eq. (14) into the above constraints, the following relationship of design parameters a16 and a25 and joint variables h10 and h40 are obtained: 8 < sa25 ¼ sa16 sh10 p : h40 ¼ 2

Sm3

8 S1 > > > > > > S2 > > > > > S 4 > > > > > S5 > > > : S6

Sm4

8 S1 > > > > > > > > S2 > > S4 > > > > > > S5 > > > : S6

¼ ½ 1 0 0 0 0 0 T ¼ ½ l2

0 0 0 0 T

m2

¼ ½ 1 0 0 0 0 m2 T ¼ ½ 1 0 0 0 0 0 T

(29)

¼ ½ 1 0 0 0 0 m2  T ¼ ½ l2

m2

0 0 0 0 T

The constraint screws reciprocal to the motion screws in Eq. (29) are derived as 8 r S ¼ ½1 0 > > < 41 c S4 ¼ Sr42 ¼ ½ 0 0 > > : r S43 ¼ ½ 0 0

0 0 0 0 T 0 0 1 0 T 0 0 0 1

(30)

T

Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

The above equation shows that the mechanism constraint-screw system is a three-system. It implies that the mechanism motionscrew system in Eq. (29) degenerates and changes to a threesystem. Apart from the singularity configurations which are transitory positions in Fig. 2, the linkages are in singular configurations when axes of joints R2 and R3 align to axes of joints R6 and R5, respectively. In these singular configurations, the linkage is able to switch to an open kinematic chain. However, this singularity [50] does not allow a transitory position where the linkage switches between different motion branches with 1DOF. 3.3 Motion-Screw System Variation Induced Bifurcation and Trifurcation. According to geometry of the Bennett planospherical hybrid linkage and the mechanism motion-screw system at each singular configuration, the linkage can move to other motion branches by passing this singular position as a transitory position [34]. Bifurcation and trifurcation of the plan-symmetric Bennett plano-spherical hybrid linkage corresponding to the inequalities and equalities for parameters are summarized in Table. 1. The joints R2 and R6 are geometrically locked when the linkage moves to its planar parallelogram 4R motion branch while the joints R3 and R5 are geometrically locked in the spherical 4R motion branch.

4 A Bennett Plano-Spherical Hybrid Linkage Evolved Metamorphic Parallel Mechanism A novel parallel mechanism composed of three Bennett planospherical hybrid linkages with case (iii) singularity as closed loop subchains is shown in Fig. 3, where the base is labeled b and the tetrahedral platform is labeled p. Face DP1P2P3 of the tetrahedral platform is an equilateral triangle and the other three isosceles triangular faces meet at vertex O. Op is the center of face DP1P2P3 and three vertices Pi (i ¼ 1, 2, and 3) are symmetrically distributed with 120 deg intervals on the face, in which subscript “i” denotes sequence number of the limbs. Vertices Bi on the base plain of the parallel mechanism form an equilateral triangle and the center of the triangle is denoted by Ob. The platform is connected to the base with three identical limbs. Each limb consists of a closed loop subchain, the Bennett plano-sphrical hybrid linkage with trifurcation, and an open loop RR chain jointed to the closed loop chain successively by revolute joint Ri7. The axis of joint Ri7 is collinear with the axis of joint Ri1 of the closed loop suchchain. It implies that joints Ri1 and Ri7 form a compound revolute joint and each limb is a hybrid kinematic chain. The hybrid limb is connected to the base by joint Ri4 of the closed loop subchain and to the platform by joint Ri8 of the open loop chain. The axes of revolute joints Ri8 attached to the platform are aligned to three edges OP1, OP2 and OP3 of the tetrahedron. Hence, the parallel mechanism employs only revolute joints. As illustrated in Fig. 3, the axis of joint Ri8 passes common point Ai of the 6R linkage within the ith limb, meaning the axes of revolute joints Ri1, Ri2, Ri6, and Ri8 have common point Ai in the ith limb. The common points Ai of these three limbs coincide with vertex O of the platform simultaneously. The axis of joint Ri4 of each limb is aligned to ObBi. With such an assembly, the axes of Table 1

Cases

Fig. 3 The metamorphic parallel mechanism composed of Bennett plano-spherical hybrid linkages

revolute joints Ri4 are coplanar and intersect at center Ob of the base. The global reference frame Ob-XYZ is attached to center Ob and both X- and Y-axis are in the base plane defined by DB1B2B3. The X-axis is collinear with axis of R14, and Y-axis is in perpendicular. The Z-axis is perpendicular to the base plane and point upward. Another coordinate frame Op-uvw is attached to center Op of the platform and both u- and v-axis are located in triangular face DP1P2P3. The u-axis is aligned to P1Op, v-axis is parallel to P2P3 and w-axis is normal of triangular face DP1P2P3 and pointing upward. It means w-axis is collinear with OOp.

5 Constraint-Screw System Variation in a Limb of the Metamorphic Parallel Mechanism The kinematic model of a hybrid limb of the parallel mechanism is illustrated in Fig. 4. Taking account of geometry of the Bennett plano-sphrical hybrid linkage in Sec. 2, origin of local frame oi-xiyizi for the ith limb is located at common point A(O), the zi-axis is aligned with OOb, xi-axis is parallel to axis of revolute joint Ri4 and the yi-axis completes a right-handed coordinate frame. It implies that the xioizi plane is coplanar with symmetric plane P1 of the 6R linkage. As illustrated in Fig. 4(a), the closed loop subchain is working in its overconstrained 6R linkage motion branch, meaning all six revolute joints are active. As aforementioned in Sec. 3 in regarding to the motion characteristics of the Bennett plano-spherical hybrid linkage, axis of the revolute joint Ri1 implements rotational motion about the yi-axis while the common point Ai implements rectilinear motion along the zi-axis in Fig. 4. Hence, the 6R linkage is equivalent to a serial RvRvRvR chain in which axis of the last revolute joint R is restricted in the symmetric plane P1 and “Rv” stands for revolute joint of the equivalent serial chain. Subsequently, the equivalent serial kinematic chain of the hybrid limb in Fig. 4(a) is derived as RvRvRvRR chain in Fig. 4(b).

Summary of reconfiguration of the Bennett plano-spherical hybrid linkage

Inequality/equality

Constraint equations

Order of motion-screw system at singular position

Bifurcated motion branch

Trifurcation —

(i)

r > ldg ¼ hta16

a16 ¼ a26 ¼ p/4; h10 ¼ p/2

4

Planar 4R linkage

(ii)

r < ldg ¼ hta16

sa25 ¼ sa16sh10; h40 ¼ p/2

4

Spherical 4R linkage

—

(iii)

r ¼ ldg ¼ hta16

a16 ¼ a26 ¼ p/4; r ¼ h

3

—

Spherical 4R linkage; planar 4R linkage

Journal of Mechanical Design

JUNE 2015, Vol. 137 / 062303-5

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

Fig. 4 The hybrid limb with closed loop subchain working in overconstrained 6R linkage motion branch (a) kinematic model of the hybrid limb, (b) the equivalent RvRvRvRR kinematic chain

Since the axis of revolute joint Ri8 passes common point Ai, motion-screw system Smi of the equivalent serial kinematic chain expressed in local frame oi-xiyizi in Fig. 4 is given by

Smi

8 Sv1 ¼ ½ 1 0 0 0 2rch10 0 T > > > > T > > > < Sv2 ¼ ½ 1 0 0 0 rch10 rsh10  ¼ Sv3 ¼ ½ 1 sh10 ch10 0 0 0 T > > > > Si7 ¼ ½ li7 0 ni7 0 0 0 T > > > : Si8 ¼ ½ li8 mi8 ni8 0 0 0 T

(31)

where si7 ¼ [li7, 0, ni7]T and si8 ¼ [li8, mi8, ni8]T are the vectors pointing in the direction of screws Si7 and Si8, in which li7 ¼ s2h10/(1 þ c2h10) and ni7 ¼ 2c2h10/(1 þ c2h10). The constraint screws are reciprocal to motion screws and can be calculated from Eq. (31) based on the reciprocity of screw systems. The constraint-screw system of the equivalent serial kinematic chain is derived as ( Sci

¼

Sci1 ¼ ½ 1 0 0 0 0 0 T Sci2 ¼ ½ 0 th10

1 2rsh10

p q T

(32)

The above constraint screws show that each limb of the parallel mechanism provides two constraints including one pure force and one general constraint. The constraint force is collinear with the xi-axis. When the closed loop subchain, Bennett plano-spherical hybrid linkage, changes to the spherical 4R motion branch, the hybrid kinematic limb of the parallel mechanism changes to a distinct motion branch with axes of all active joints Ri1, Ri2, Ri4, Ri6, Ri7, and Ri8 passing common point Ai in Fig. 5. In this motion branch, origin oi of local frame oi-xiyizi coincides with common point Ai of the closed loop subchain and vertex O of the tetrahedral platform. Resultantly, the common points Ai, vertex O of platform and geometric center Ob of the base are coincide simultaneously. Considering the geometry of the intersecting revolute joints, any link connected to joint Ri1 of the evolved spherical 4R linkage is implementing spherical motion and the closed loop subchain can be taken as a serial spherical kinematic chain with three intersecting revolute joints, meaning RvRvRv chain. Since the axes of joints Ri1 and Ri7 are collinear, the equivalent serial kinematic chain of the hybrid limb in Fig. 5(a) is then derived as RvRvRvR chain in Fig. 5(b). The motion-screw system, Smi , of the equivalent serial kinematic chain in Fig. 5(b) expressed in local frame oi-xiyizi becomes

in which p ¼ rsh10(ni8th10  2li8)/mi8 and q ¼  rsh10t2h10. Smi

8 Sv1 ¼ ½ 1 0 0 0 0 0 T > > > > < S ¼ ½ 1 sh ch10 0 0 0 T v2 10 ¼ > > Si7 ¼ ½ li7 0 ni7 0 0 0 T > > : Si8 ¼ ½ li8 mi8 ni8 0 0 0 T

(33)

where si7 ¼ [li7, 0, ni7]T and si8 ¼ [li8, mi8, ni8]T are the vectors pointing in the direction of screws Si7 and Si8, in which li7 ¼ s2h10/(1 þ c2h10) and ni7 ¼ 2c2h10/(1 þ c2h10). The constraint-screw system of the hybrid limb is reciprocal to motion-screw system Sci of the equivalent serial chain and can be calculated from Eq. (33), that is 8 c S ¼ ½ 1 0 0 0 0 0 T > > < i1 Sci ¼ Sci1 ¼ ½ 0 1 0 0 0 0 T (34) > > : c Si2 ¼ ½ 0 0 1 0 0 0 T

Fig. 5 The hybrid limb with closed loop subchain in spherical 4R motion branch (a) kinematic model of the hybrid limb, (b) the equivalent RvRvRR kinematic chain

062303-6 / Vol. 137, JUNE 2015

The above constraint screws show that each limb provides three pure forces passing common point Ai. When the Bennett plano-spherical hybrid linkage in the hybrid limb moves to the planar 4R motion branch, the hybrid limb of Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

Fig. 6 The hybrid limb with closed loop subchain in planar 4R motion branch (a) kinematic model of the hybrid limb and (b) the equivalent RvPvRR kinematic chain

the parallel mechanism can change to the third motion branch in Fig. 6. Resultantly, axes of all active joints of the closed loop subchain are parallel while the axis of joint Ri8 passes common point O in the ith limb. In this motion branch in Fig. 6, origin oi of local frame oi-xiyizi is coincident with common point Ob on the base. The axis of joint Ri1 passing common point O is implementing pure translation motion in symmetric plane P1. Considering the motion characteristics of the planar 4R linkage, the closed loop subchain can be taken as an RvPvRv chain. Hence, the equivalent serial kinematic chain of the hybrid limb in Fig. 6(a) becomes an RvPvRR chain in Fig. 6(b) since the axes of joint Rv and joint Ri7 are collinear. The motion-screw system of the equivalent serial kinematic chain in Fig. 6(b) expressed in local frame oi-xiyizi is

Smi

8 Sv1 ¼ ½ 1 0 0 0 0 0 T > > > > < S ¼ ½ 0 0 0 0 0 1 T v2 ¼ > > Si7 ¼ ½ 1 0 0 0 2rch10 0 T > > : Si8 ¼ ½ li8 mi8 ni8 2rmi8 ch10 2rli8 ch10

0 T (35)

where si8 ¼ [li8, mi8, ni8]T is the vector pointing in direction of screw Si8. The constraint-screw system of the hybrid limb in this motion branch is reciprocal to motion-screw system Smi in Eq. (35) and can be derived as ( r Si1 ¼ ½ 1 0 0 0 2rch10 0 T (36) Sci ¼ Sri2 ¼ ½ 0 0 0 0 ni2 mi2 T The above constraint screws show that each limb exerts one constraint force and one constraint couple on the platform. The constraint force passes vertex O of the platform and parallel to xi-axis.

6 Reconfiguration Analysis of the Metamorphic Parallel Mechanism As the hybrid limbs of the parallel mechanism in Fig. 3 are capable of reconfiguring their structure and changing the constraints when the Bennett plano-spherical hybrid linkage varies between three distinct motion branches, the tetrahedral platform which is jointed to the three hybrid limbs is able to reconfigure its full-cycle motion characteristics in regarding to the variation of constraints provided by three limbs. In this section, the distinct motion branches of the metamorphic parallel mechanism with all three limbs in identical motion branch are presented. Journal of Mechanical Design

6.1 Motion Branch With the Helical Motion. When all three hybrid limbs of the parallel mechanism are in the same motion branch with closed loop subchain working as overconstrained 6R linkage, the parallel mechanism is in the motion branch-I illustrated in Fig. 3. In this motion branch, all revolute joints are active and each limb provides one constraint force and one general constraint expressed in local frame oi-xiyizi. In order to analyze the motion characteristics of the platform, all constraints of three limbs are transformed to global frame Ob-XYZ which is attached to the base. Considering distribution of joints Ri4 connected to the base and the arrangement of local frames oi-xiyizi and global frame Ob-XYZ, the constraint-screws of each limb can be transformed to the global frame by shifting along the Z-axis and subsequently rotating about the Z-axis   0 Rðui Þ o c ðSsik ÞT Sik ¼ ðTi ÞðScik ÞT ¼ (37) Ai Rðui Þ Rðui Þ where the presuperscript “o” denotes the global coordinate frame, Ti is the transformation matrix [33,51,52] form the local frame of ith limb to global frame, R(ui) is the rotational matrix with rotational angle ui ¼ 0, 2p/3, and 2p/3. The skew-symmetric matrix Ai representing the shifting is given by 2 3 0 zoi yoi 0 xoi 5 Ai ¼ 4 zoi yoi xoi 0 in which [xoi, yoi, zoi]T is the position vector of common point Ai in the global frame. Hence, the constraints exerted on the platform in motion branch-I by the three limbs expressed in frame Ob-XYZ span the platform constraint-screw system, that is

Scb1

8 c o > S11 > > > > o c > S12 > > > > > > o > Sc > > > 21 > > < ¼ o Sc 22 > > > > > > > o c > > > S31 > > > > > > > > : o Sc32

¼ ½1

0 0 0 2rch10

0 T T

¼ ½0

th10 1 0 p q  pffiffiffi pffiffiffi 1 T 3 ¼ ½ 0 3rch10 rch10 0  2 2  pffiffiffi T pffiffiffi 1 1 3 3 ¼ th10 1  p q th10 p 2 2 2 2 pffiffiffi pffiffiffi 1 3 T ¼ ½  0  3rch10 rch10 0  2 2  pffiffiffi T pffiffiffi 3 1 ¼  3 th10 1 th10 1 p  p q 2 2 2 2 (38)

where subscript “b1” indicates motion branch-I. The possible motion screws are reciprocal to Scb1 and the platform motion-screw system can be calculated from Eq. (38), that is Sb1 ¼ S ¼ ½ 0 0 1 0 0 q T

(39)

The motion screw in Eq. (41) shows that the platform has only 1DOF and implements helical motion along the Z-axis. 6.2 Motion Branch With the 3DOF Spherical Motion. When all three hybrid limbs of the parallel mechanism are in the same motion branch with the closed loop subchain working as spherical 4R linkage, the parallel mechanism changes to motion branch-II illustrated in Fig. 7. In this motion branch, the axes of all active revolute joints pass vertex O which is coincident with center Ob of the base. As derived in Sec. 3, each limb provides three constraint forces passing the origin of local frame oi-xiyizi. It implies that all constraints forces pass center Ob of the base since origin of oi of the local frame is coincident with center Ob. JUNE 2015, Vol. 137 / 062303-7

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

Fig. 7 The motion branch of the parallel mechanism implementing 3DOF spherical motion

Considering distribution of revolute joints Ri4 connected to the base and arrangement of local frames oi-xiyizi with respect to the global frame, the constraint screws of each limb can be transformed to the global frame by rotating about Z-axis following Eq. (37). Since all the constraints provided by three limbs are pure forces and passing the vertex O of the platform, only three of them are independent and span a three-system. Under such constraints, the platform motion-screw system spanned by a set of bases is given by 8 T > > < S1 ¼ ½ 1 0 0 0 0 0  (40) Sb2 ¼ S2 ¼ ½ 0 1 0 0 0 0 T > > : T S3 ¼ ½ 0 0 1 0 0 0  The motion-screw system in Eq. (40) shows the platform in the motion branch-II has 3DOFs and implements spherical motion with the motion center at point Ob. 6.3 Motion Branch With the 1DOF Pure Translational Motion. When all three limbs of the parallel mechanism change to the common motion branch with the closed loop subchain

Fig. 8 The motion branch of the parallel mechanism implementing 1DOF pure translation

062303-8 / Vol. 137, JUNE 2015

acting as planar 4R linkage, the parallel mechanism changes to motion branch-III in Fig. 8. The axes of four active joints of the closed loop subchain are parallel to the base plane defined by DB1B2B3. The vertex O, common point of axes of joints Ri8, moves along the vertical Z-axis. Considering geometry of the parallel mechanism in Fig. 8 and the setting of local frame oi-xiyizi for each limb in Fig. 5, the constraint screws of each limb can be transformed to global frame Ob-XYZ by rotating about the Z-axis in terms of the transformation expressed in Eq. (37). The constraint-screw system of the platform in motion branch-III is derived as 8 c o > S ¼ ½ 1 0 0 0 2rch10 0 T > > 11 > > o c > S ¼ ½ 0 0 0 0 n12 m12 T > > > 12  T > pffiffiffi > > pffiffiffi 1 3 > o c > S ¼ > 0  3rch10 rch10 0  21 > > 2 2 > >  T < pffiffiffi 3 Scb3 ¼ o Sc ¼ 0 0 0 1 n12 2 n12 m12 22 > > 2 > >  T > pffiffiffi > > pffiffiffi > 1 3 o c > S ¼ >  0 3 rch rch 0  31 10 10 > > 2 2 > > >  T pffiffiffi > > > > : o Sc32 ¼ 0 0 0  3 n12 1 n12 m12 2 2 (41) The possible motion screws are reciprocal to Scb3 in Eq. (41) and the platform motion-screw system changes to Sb3 ¼ S ¼ ½ 0 0 1 0 0 0 T

(42)

The motion screw in Eq. (42) shows that the platform has only 1DOF and implements pure translation along the Z-axis. 6.4 Application of the Parallel Mechanism With Redundant Actuation. Since reconfiguration of the proposed parallel mechanism is induced by the trifurcation of the Bennett planospherical hybrid linkage, the switch of motion branches of the parallel mechanism is realized by changing motion branches of the Bennett plano-spherical hybrid linkages in three limbs. Considering the motion-screw system degeneration at the transitory position, redundant actuation [53–55] can be employed for generating the drive torque to facilitate the Bennett plano-spherical hybrid linkage to get out from the transitory position and switch between the three motion branches. The two revolute joints R1 and R4 are selected as two joints to be actuated and the actuation scheme is characterized by the domain of two angles h10 and h40 in Eqs. (4)–(8) and the domains corresponding to distinct motion branches are listed in Table 2. Mechanisms with capability of reconfiguring their structure are desirable in tasks such as search and rescue in disastrous environments and space exploration, where mechanisms go through unexpected situations and perform multitasks that are difficult for devices with traditional kinematic structures [56–58]. The new mechanisms including metamorphic mechanisms and kinematotropic mechanisms and other mechanisms with reconfigurability are required for mechanical devices to adapt to various conditions and meet the demands. The presented new parallel mechanism, with the Bennett planospherical hybrid linkage as closed loop subchains, has a potential to be used in the production where the mobility change helps reconfigure the mechanism in the required motion range and change to other required motion ranges. Further, in light of the helical motion, 3DOF spherical motion and 1DOF translation allowed by three distinct motion branches, the parallel mechanism is able to serve as an integrated vehicle testing-repairing platform enabling a big lifting force and a convenient reconfigurable platform for drive-on and drive-off. The Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

Table 2 Actuation scheme for reconfiguration of the plano-spherical linkage Inequality/equality

Constraint equations

Domain of h10 and h40

Motion branches

r > ldg ¼ hta16

a16 ¼ a26 ¼ p/4; h10 ¼ p/2

6R linkage Planar 4R linkage

h10僆[0, p/2), h40僆 [0, p/2) h10僆[0, p/2), h40僆 (p/2, p]

r < ldg ¼ hta16

sa25 ¼ sa16sh10; h40 ¼ p/2

r ¼ ldg ¼ hta16

a16 ¼ a26 ¼ p/4; r ¼ h

6R linkage Spherical 4R linkage 6R linkage Planar 4R linkage Spherical 4R linkage

h10僆[0, p/2), h40僆[0, p/2) h10僆(p/2, p], h40僆[0, p/2) h10僆[0, p/2), h40僆[0, p/2) h10僆[0, p/2), h40僆 (p/2, p] h10僆(p/2, p], h40僆[0, p/2)

parallel mechanism has a reconfiguration in three motion branches is only composed of revolute joints which make it simple for construction.

7

Conclusions

This paper investigated screw-system-variation enabled reconfiguration of the plane-symmetric Bennett plano-spherical hybrid linkage and its evolved novel metamorphic parallel mechanism. The geometry and parametric constraints of the plane-symmetric Bennett plano-spherical hybrid linkage were analyzed. The order of the mechanism motion-screw system of the plano-spherical linkage in a general configuration and its variation in relation to the inequalities conditions for design parameters were further analyzed in terms of reciprocity of screw systems and of line geometry. This revealed that the degeneration of the mechanism motionscrew system at the singular position enables the typical 6R linkage to reconfigure its motion branch by passing transitory positions. Following the study of the reconfigurable 6R linkage, the paper further presented the novel metamorphic parallel mechanism employing the 6R linkage as a closed loop subchain in each kinematic limb. The limb constraint-screw systems corresponding to three distinct motion branches of the plano-spherical linkage were analyzed in terms of equivalent serial kinematic chains, leading to identification of distinct constraints exerted to the platform of the parallel mechanism. The constraints analysis further revealed that variation of the platform constraint-screw system of the multiloop parallel mechanism allows three distinct motion branches of the platform as 1DOF helical motion, 3DOF spherical motion, and 1DOF pure translation. Hence, the paper presented a new way of exploring reconfiguration of closed loop mechanisms using the screw system approach.

Acknowledgment The authors acknowledge the European Commission for the support in the human–robot interaction project SQUIRREL in the name of Clearing Clutter Bit by Bit under Grant No. 610532 and the National Natural Science Foundation of China under Grant Nos. 51205016 and 51175366.

References [1] Merlet, J.-P., 2001, Parallel Robots, Kluwer Academic Publishers, Dordrecht, Netherlands. [2] Gosselin, C., and Angeles, J., 1989, “The Optimum Kinematic Design of a Spherical Three-Degree-of-Freedom Parallel Manipulator,” ASME J. Mech. Des., 111(2), pp. 202–207. [3] Dasgupta, B., and Mruthyunjaya, T. S., 2000, “The Stewart Platform Manipulator: A Review,” Mech. Mach. Theory, 35(1), pp. 15–40. [4] Gogu, G., 2008, Structural Synthesis of Parallel Robots, Springer, Dordrecht, Netherlands. [5] Briot, S., and Bonev, I. A., 2007, “Are Parallel Robots More Accurate Than Serial Robots,” Trans. Can. Soc. Mech. Eng., 31(4), pp. 445–455. [6] Lee, K. M., and Shah, D. K., 1988, “Dynamic Analysis of a Three-Degrees-OfFreedom In-Parallel Actuated Manipulator,” IEEE J. Rob. Autom., 4(3), pp. 361–367. [7] Pashkevich, A., Chablat, D., and Wenger, P., 2009, “Stiffness Analysis of Overconstrained Parallel Manipulators,” Mech. Mach. Theory, 44(5), pp. 966–982.

Journal of Mechanical Design

[8] Claver, R., 1988, “Delta, a Fast Robot With Parallel Geometry,” Proceedings of the International Symposium on Industrial Robot, Switzerland, pp. 91–100. [9] Tsai, L. W., Walsh, G. C., and Stamper, R. E., 1996, “Kinematics of a Novel Three DOF Translational Platform,” Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, pp. 3446–3451. [10] Ceccarelli, M., 1997, “A New 3 D.O.F Spatial Parallel Mechanism,” Mech. Mach. Theory, 32(8), pp. 895–902. [11] Pierrot, F., and Company, O., 1999, “H4: A New Family of 4-DOF Parallel Robots,” Proceedings of the 1999 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, GA, pp. 508–513. [12] Gao, F., Li, W., Zhao, X., Jin, Z., and Zhao, H., 2002, “New Kinematic Structures for 2-, 3-, 4-, and 5-DOF Parallel Manipulator Designs,” Mech. Mach. Theory, 37(8), pp. 1395–1411. [13] Liu, X., and Wang, J., 2003, “Some New Parallel Mechanisms Containing the Planar Four-Bar Parallelogram,” Int. J. Rob. Res., 22(9), pp. 717–732. [14] Zhang, K., Fang, Y., Fang, H., and Dai, J. S., 2010, “Geometry and Constraint Analysis of the Three-Spherical Kinematic Chain Based Parallel Mechanism,” ASME J. Mech. Rob., 2(3), p. 031014. [15] Zhao, T. S., Dai, J. S., and Huang, Z., 2002, “Geometric Synthesis of Spatial Parallel Manipulators With Fewer Than Six Degrees of Freedom,” Proc. IMechE, 216(12), pp. 1175–1185. [16] Dai, J. S., and Caldwell, D. G., 2012, “Robotics and Automation for Packaging in the Confectionery Industry,” Robotics and Automation in the Food Industry: Current and Future Technologies, Woodhead Publishing, Cambridge, pp. 401–419. [17] Wurdemann, H., Aminzadeh, V., Dai, J. S., Reed, J., and Purnell, G., 2011, “Category-Based Food Ordering Processes,” Trends Food Sci. Technol., 22(1), pp. 14–20. [18] Salerno, M., Zhang, K., Menciassi, A., and Dai, J. S., 2014, “A Novel 4-DOFs Origami Enabled, SMA Actuated, Robotic End-Effector for Minimally Invasive Surgery,” Proceedings of the IEEE International Conference on Robotics and Automation (ICRA2014), Hong Kong, pp. 2844–2849. [19] Kuo, C. H., and Dai, J. S., 2012, “Kinematics of a Fully-Decoupled Remote Center-Of-Motion Parallel Manipulator for Minimally Invasive Surgery,” ASME J. Med. Devices, 6(2), p. 021008. [20] Dai, J. S., and Rees Jones, J., 1999, “Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds,” ASME J. Mech. Des., 121(3), pp. 375–382. [21] Coppola, G., Zhang, D., and Liu, K., 2014, “A 6-DOF Reconfigurable Hybrid Parallel Manipulator,” Rob. Comput.-Integr. Manuf., 30(2), pp. 99–106. [22] Zlatanov, D., Bonev, I. A., and Gosselin, C. M., 2002, “Constraint Singularities as C-Space Singularities,” Advances in Robot Kinematics, Kluwer, Dordrecht, Netherlands, pp. 183–192. [23] Kong, X., Gosselin, C. M., and Richard, P. L., 2007, “Type Synthesis of Parallel Mechanisms With Multiple Operation Modes,” ASME J. Mech. Des., 129(7), pp. 595–601. [24] Kong, X., 2013, “Type Synthesis of 3-DOF Parallel Manipulators With Both a Planar Operation Mode and a Spatial Translational Operation Mode,” ASME J. Mech. Rob., 5(4), p. 041015. [25] Gan, D. M., Dai, J. S., and Liao, Q. Z., 2009, “Mobility Change in Two Types of Metamorphic Parallel Mechanisms,” ASME J. Mech. Rob., 1(4), p. 041007. [26] Carbonari, L., Callegari, M., Palmieri, G., and Palpacelli, M.-C., 2014, “A New Class of Reconfigurable Parallel Kinematic Machines,” Mech. Mach. Theory, 79, pp. 173–183. [27] Gogu, G., 2009, “Branching Singularities in Kinematotropic Parallel Mechanisms,” Proceedings of the 5th International Workshop on Computational Kinematics, Duisburg, Germany, pp. 341–348. [28] Wohlhart, K., 1996, “Kinematotropic Linkages,” Recent Advances in Robot Kinematics, Kluwer Academic, Dordrecht, The Netherlands, pp. 359–368. [29] Zhang, K. T., Dai, J. S., and Fang, Y. F., 2009, “A New Metamorphic Mechanism With Ability for Platform Orientation Switch and Mobility Change,” Proceedings of the ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots, London, pp. 626–632. [30] Zhang, K. T., Dai, J. S., and Fang, Y. F., 2010, “Topology and Constraint Analysis of Phase Change in the Metamorphic Chain and Its Evolved Mechanism,” ASME J. Mech. Des., 132(12), p. 121001. [31] Zeng, Q., Fang, Y., and Ehmann, K. F., 2011, “Design of a Novel 4-DOF Kinematotropic Hybrid Parallel Manipulator,” ASME J. Mech. Des., 133(12), p. 121006. [32] Ye, W., Fang, Y., Zhang, K., and Guo, S., 2014, “A New Family of Reconfigurable Parallel Mechanisms With Diamond Kinematotropic Chain,” Mech. Mach. Theory, 74, pp. 1–9.

JUNE 2015, Vol. 137 / 062303-9

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms

[33] Dai, J. S., Huang, Z., and Lipkin, H., 2006, “Mobility of Overconstrained Parallel Mechanisms,” ASME J. Mech. Des., 128(1), pp. 220–229. [34] Zhang, K., and Dai, J. S., 2014, “Trifurcation of the Evolved Sarrus-Motion Linkage Based on Parametric Constraints,” Advances in Robot Kinematics, Springer International Publishing, Switzerland, pp. 345–353. [35] Bennett, G. T., 1905, “The Parallel Motion of Sarrut and Some Allied Mechanisms,” Philosophy Magazine, 6th series, Vol. 9, Taylor & Francis, London, pp. 803–810. _ C., 2014, “Function [36] Alizade, R. I., Kiper, G., Ba gdadio glu, B., and Dede, M. I. Synthesis of Bennett 6R Mechanisms Using Chebyshev Approximation,” Mech. Mach. Theory, 81, pp. 62–78. [37] Chen, Y., 2003, Design of Structural Mechanisms, University of Oxford, Oxford. [38] Cui, L., and Dai, J. S., 2011, “Axis Constraint Analysis and Its Resultant 6R Double-Centered Overconstrained Mechanisms,” ASME J. Mech. Rob., 3(3), p. 031004. [39] Baker, J. E., 2012, “The Six-Revolute Hybrids of Four-Revolute Loops and Their Single Reciprocal Screws,” Mech. Mach. Theory, 53, pp. 128–144. [40] Denavit, J., and Hartenberg, R. S., 1955, “A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices,” ASME J. Appl. Mech., 77, pp. 215–221. [41] Baker, J. E., 1980, “An Analysis of Bricard Linkages,” Mech. Mach. Theory, 15, pp. 267–286. [42] Viquerat, A. D., Hutt, T., and Guest, S. D., 2013, “A Plane Symmetric 6R Foldable Ring,” Mech. Mach. Theory, 63, pp. 73–88. [43] Dai, J. S., and Rees Jones, J., 2001, “Interrelationship Between Screw Systems and Corresponding Reciprocal Systems and Applications,” Mech. Mach. Theory, 36(5), pp. 633–651. [44] Zhao, J. S., Feng, Z. J., Zhou, K., and Dong, J. X., 2005, “Analysis of the Singularity of Spatial Parallel Manipulator With Terminal Constraints,” Mech. Mach. Theory, 40(3), pp. 275–284. [45] Fang, Y., and Tsai, L. W., 2002, “Structure Synthesis of a Class of 4-Degree of Freedom and 5-Degree of Freedom Parallel Manipulators With Identical Limb Structures,” Int. J. Rob. Res., 21(9), pp. 799–810.

062303-10 / Vol. 137, JUNE 2015

[46] Huang, Z., and Li, Q. C., 2002, “General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators,” Int. J. Rob. Res., 21(9), pp. 131–145. [47] Merlet, J. P., 1989, “Singularity Configurations of Parallel Manipulators and Grassmann Geometry,” Int. J. Rob. Res., 8(9), pp. 45–56. [48] Monsarrat, B., and Gosselin, C. M., 2001, “Singularity Analysis of a Three-Leg Six-Degree-of-Freedom Parallel Platform Mechanism Based on Grassmann Line Geometry,” Int. J. Rob. Res., 20(4), pp. 312–328. [49] Amine, S., Masouleh, M. T., Caro, S., Wenger, P., and Gosselin, C. M., 2012, “Singularity Analysis of 3T2R Parallel Mechanisms Using Grassmann–Cayley Algebra and Grassmann Geometry,” Mech. Mach. Theory, 52, pp. 326–340 [50] Gosselin, C. M., and Angeles, J., 1990, “Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290. [51] McCarthy, J. M., and Soh, G. S., 2010, Geometric Design of Linkages, Springer Science & Business Media, New York. [52] Angeles, J., 1998, “The Application of Dual Algebra to Kinematic Analysis,” Computational Methods in Mechanical Systems, Springer, Berlin, Heidelberg, pp. 3–32. [53] Cheng, H., Liu, G. F., Yiu, Y. K., Xiong, Z. H., and Li, Z. X., 2001, “Advantages and Dynamics of Parallel Manipulators With Redundant Actuation,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Maui, HI, pp. 171–176. [54] Tosi, D., Legnani, G., Pedrocchi, N., Righettini, P., and Giberti, H., 2010, “Cheope: A New Reconfigurable Redundant Manipulator,” Mech. Mach. Theory, 45(4), pp. 611–626. [55] Moosavian, A., and Xi, F., 2014, “Design and Analysis of Reconfigurable Parallel Robots With Enhanced Stiffness,” Mech. Mach. Theory, 77, pp. 92–110. [56] Merlet, J. P., 2000, “Parallel Robots: Open Problems,” Robotics ResearchInternational Symposium, Vol. 9, pp. 27–32. [57] Tadokoro, S., and Kobayashi, S., 2002, “A Portable Parallel Motion Platform for Urban Search and Surveillance in Disasters,” Adv. Rob., 16(6), pp. 537–540. [58] Zhang, K., Dai, J. S., and Fang, Y., 2013, “Geometric Constraint and Mobility Variation of Two 3SvPSv Metamorphic Parallel Mechanisms,” ASME J. Mech. Des., 135(1), p. 011001.

Transactions of the ASME

Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 05/28/2015 Terms of Use: http://asme.org/terms