Type synthesis of freedom and constraint elements for design of ...

Mech. Sci., 4, 263–277, 2013 www.mech-sci.net/4/263/2013/ doi:10.5194/ms-4-263-2013 © Author(s) 2013. CC Attribution 3.0 License.

Mechanical

Sciences Open Access

Type synthesis of freedom and constraint elements for design of flexure mechanisms H.-J. Su1 and C. Yue2 1

Department of Mechanical and Aerospace Engineering, Ohio State University, Columbus, Ohio, 43210, USA 2 Department of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, Maryland, 21250, USA Correspondence to: H.-J. Su ([email protected]) Received: 3 March 2013 – Accepted: 19 April 2013 – Published: 17 July 2013

Abstract. In this paper, we present the type synthesis of freedom and constraint elements for design of gen-

eral flexure mechanisms. As an important step in the conceptual design stage, the goal of type synthesis is to qualitatively determine the topology or connectivity of flexure elements and rigid bodies in a mechanism. The synthesis procedure presented here is based on a recently emerging screw theory based approach for flexure mechanisms. We first categorize a list of commonly used atomic flexure primitives including blades, wires, notches and bellow springs etc. We then derive their twist and wrench matrices that mathematically represent their freedom and constraint spaces. The synthesis procedure rigorously follows screw algebra. Freedom elements including R-joints and P-joints are defined as basic motion elements that allow a single rotation or a single translation. By using parallel structures of these flexure primitives, eleven designs of R-joints and eight designs of P-joints are systematically synthesized. As a duality, constraint elements including P-constraints and R-constraints remove a single translation or rotation. In contract to freedom elements, we synthesized serial chains of flexure primitives and obtained six designs of P-constraints and three designs of R-constraints. These freedom and constraint elements form a catalogue of basic building blocks for designing more complex flexure mechanisms. At last we utilize four design examples to demonstrate how to synthesize hybrid structures with serial and parallel combination of these elements.

1

Introduction

Compliant mechanisms (Howell, 2001) or flexure mechanisms (Smith, 2000; Smith and Chetwynd, 1992), formed by a set of rigid bodies connected with compliant elements, produce a defined motion through elastic deformation of their compliant elements. They are widely used in various precision instruments and machines such as nano-manipulators Culpepper and Anderson (2004), nano-positioners (Chen and Culpepper, 2006; Brouwer et al., 2010; Yao et al., 2008; Dong et al., 2008) and precision manufacturing machines (Varadarajan and Culpepper, 2007a,b). The design of flexure mechanisms has been an ad hoc process that heavily relies on designers’ experience and intuition that are typically built up over years of training. One approach often used in precision engineering community is Published by Copernicus Publications.

called the constraint-based approach (Blanding, 1999; Hale, 1999; Awtar and Slocum, 2007). In a recent conference tutorial by Henein (2011), the author enumerated a list of flexure bearing designs based on their degree of freedom (DOF), Grubler mobility and degree of hyperstaticity (DOH). Many authors have attempted to systemize the constraint based approach. Recently this approach has been further formalized into the Freedom and Constraint Topology (FACT) framework (Hopkins, 2007a; Hopkins and Culpepper, 2010a,b). Essentially the constraint-based design approach and the FACT approach are mathematically equivalent to screw theory (Ball, 1998; Hunt, 1978; Phillips, 1984, 1990; Davidson and Hunt, 2004) that has been widely used in kinematics community for various problems such as type synthesis of parallel mechanisms (Kong and Gosselin, 2010) and mobility analysis of rigid body mechanisms (Huang et al., 2008).

264

H.-J. Su and C. Yue: Type synthesis

In recognizing this intrinsic connection between the constraint based approach and screw theory, a series of work (Su et al., 2009; Su and Tari, 2010, 2011; Hopkins and Culpepper, 2010c; Yu et al., 2010; Su, 2011) on screw theory based approach for type synthesis and analysis of flexure mechanisms have been done. This approach is completely based on screw algebra (Dai and Jones, 2001, 2003), which can be easily implemented in computer programs for truly automating the design process of flexure mechanisms, especially the conceptual design stage. However currently these work have been focusing on mechanisms with relatively simple topologies or mobility analysis (rather than synthesis). The type synthesis of general flexure mechanisms for any specified mobility is still yet to be done. In this paper, we present a systematic methodology based on screw theory for the type synthesis of general flexure mechanisms with serial, parallel or hybrid (combination of serial and parallel) topologies. First, a list of commonly used flexure primitives is studied. Then they are used to build basic freedom and constraint elements. At last these freedom and constraint elements are used for constructing more complex flexure mechanisms. 2

Screw theory overview

In this section, we first review basic concepts of screw theory as a background preparation for the following sections. 2.1

Twists and wrenches

A flexure mechanism subject to a general load represented by ˆ undergoes an instantaneous motion represented a wrench W by a twist Tˆ . From the freedom and constraint point of view, twists represent allowable motions while wrenches represent ˆ are 6 by 1 forbidden motions. Both twist Tˆ and wrench W column vectors, written as ( ) ( ) Ω Ω ˆ T= = , (1) V c × Ω + pΩ ( ) ( ) F ˆ = F = W , (2) M c × F + qF where p and q are called pitches of twist and wrenches. And ˆ satisfy the so called reciprocal condition: Tˆ and W ˆ = Ω · M + V · F = 0. Tˆ ◦ W

(3)

A general rotational or translational freedom respectively corresponds to a twist with zero or infinite pitch, written as ( ) ( ) Ω 0 ˆ ˆ TR = , TP = . (4) c×Ω V Similarly a general rotational or translation constraint removes a rotation or translation along a particular direction. They respectively correspond to a wrench with infinite or Mech. Sci., 4, 263–277, 2013

zero pitch, written as ( ) ( ) 0 F ˆ ˆ WR = , WP = M c×F

(5)

For convenience, we define six principal twists as the rotation and translations about the three coordinate axes,  T Rˆ x = 1 0 0 0 0 0  T Rˆ y = 0 1 0 0 0 0  T Rˆ z = 0 0 1 0 0 0  T Pˆ x = 0 0 0 1 0 0  T Pˆ y = 0 0 0 0 1 0  T (6) Pˆ z = 0 0 0 0 0 1 Similarly, we also define six principal wrenches as the rotational and translational constraint about the three coordinate axes  T Fˆ x = 1 0 0 0 0 0  T Fˆ y = 0 1 0 0 0 0  T Fˆ z = 0 0 1 0 0 0  T ˆx= 0 0 0 1 0 0 M  T ˆy = 0 0 0 0 1 0 M  T ˆz = 0 0 0 0 0 1 (7) M 2.2

Coordinate transformation of twists and wrenches

The coordinate transformation of a twist or wrench is calculated as 0 Tˆ = [Ad]Tˆ ,

ˆ 0 = [Ad]W, ˆ W

(8)

ˆ and Tˆ 0 , W ˆ 0 correspond to the twist and the where Tˆ , W wrench before and after the transformation. And [Ad] is the so-called 6 × 6 adjoint matrix, written as " # R 0 (9) [Ad] = DR R where [R] is a 3 by 3 rotation matrix and [D] is the 3 by 3 skew-symmetric matrix defined by the translational vector d = (d x , dy , dz )T . They have the form h [R] = x

y

i z ,

  0  [D] =  dz  −dy

−dz 0 dx

 dy   −d x   0

When the coordinate transformation is applied to a principal screw (twist or wrench), the resultant screw will along www.mech-sci.net/4/263/2013/

H.-J. Su and C. Yue: Type synthesis

265

the axes of the new coordinate system, e.g. ( ) x 0 Rˆ x = [Ad]Rˆ x = d×x ( ) 0 0 Pˆ x = [Ad]Pˆ x = x 3

(10) (11)

Flexure primitives

A flexure primitive is defined as an “atomic” flexure mechanism that consists of only one flexure element and zero intermediate body. They cannot be further divided into substructures. Recently Hopkins (2012) presented three commonly used flexures and their freedom and constraint spaces using FACT approach. In this section, we first categorize a more comprehensive set of commonly used flexure primitives and derive their freedom and constraint spaces. Then we will discuss a general synthesis methodology for constructing serial and parallel kinematic chains of these flexure primitives. 3.1

y, caused by beam bending. From the constraint point of view, a blade removes one rotation and two translations. Their twist and wrench matrices are h i h i ˆy [T b ] = Rˆ x Rˆ z Pˆ y , [Wb ] = Fˆ x Fˆ z M (14) 4. A long wire/rod flexure, denoted by “W”, removes the translation along its axis and allows the other five motions. A corner blade (a folded sheet) also provides a single constraint along its fold line. Their freedom space is equivalent to three R-joints and two P-joints, i.e. W=3R-2P. Mathematically its corresponding twist and wrench matrices are h i h i [T w ] = Rˆ x Rˆ y Rˆ z Pˆ y Pˆ z , [Ww ] = Fˆ x (15) 5. And lastly, a bellow spring, denoted by “Bs ”, removes a single rotation along its axis with a freedom space denoted by Bs =2R-3P. Its twist and wrench matrices are

Commonly used flexure primitives

h [T bs ] = Rˆ x

According to the mobility or the rank of their twist system, we can categorize the most commonly used flexure primitives as shown in Table 1. 1. A notch hinge, denoted by symbol “R”, allows a rotation about the centerline and constraints other motions. The shape of the cross section may be circular, elliptical, hyperbolic etc. Meanwhile short beams, living hinges that have one dimension significantly smaller than others can function as a notch hinge. A split tube which is a tube sliced along its longitudinal direction also allows a single rotation about its axis. If we define the axis of the R-joint to be the z-axis, its twist and wrench matrices can be written as h i h i ˆx M ˆy [T r ] = Rˆ z , [Wr ] = Fˆ x Fˆ y Fˆ z M (12) 2. A spherical notch or short wires/rod, denoted by “S”, allows three rotations and constrains three translations. Kinematically it is equivalent to a serial chain of three R-joints, i.e. S=3R. Their twist and wrench matrices are h [T s ] = Rˆ x

Rˆ y

i Rˆ z ,

h [W s ] = Fˆ x

Fˆ y

Fˆ z

i

(13)

3. Blade/sheet flexures also called leaf springs, denoted by “B”, allow two rotations and one translation, i.e. B = 2R− P. The rotational symmetric cylinders and disc rings also have the same freedom and constraint spaces. As shown in Table 1, we define the normal of the blade as the y-axis and the longitudinal direction being the x-axis. The two rotations are about two in-plane axes (x, z) due to beam torsion and beam bending respectively. And the translation is along the normal direction www.mech-sci.net/4/263/2013/

Rˆ y

Pˆ x

Pˆ y

i Pˆ z ,

h i ˆz [Wbs ] = M

(16)

Table 1 summarizes the aforementioned flexure primitives and their freedom space and twist and wrench matrices. These primitives are basic building blocks for constructing more complex flexure systems. 3.2

Serial chains of flexure primitives

The freedom space of a rigid body represents all of its allowable motion in space. For any given flexure element or building block, its freedom space is a f system that is represented by a twist matrix [T] formed by f independent twists, written as h i [T] = Tˆ1 Tˆ2 · · · Tˆf (17) When a mechanism is formed by m flexure elements or building blocks that are connected in serial, its twist matrix (freedom space) can be obtained by combining the twist matrix of each building block column-wise, mathematically written as [T] = [Ad1 T 1

Ad2 T 2

···

Adm T m ]

(18)

where [Adj ] represent the coordinate transformation from the j-th building block to the functional stage, defined in (9). This formulation has been previously presented by Su (2011). Also see similar work by Hopkins (2007b); Hopkins and Culpepper (2011). By applying a column-wise reduction, we can easily obtain a set of independent r = rank(T ) twists that forms the basis of the freedom space. r is also called the mobility or degree-of-freedom of the mechanism. Any set of r independent twists called basis can span the entire freedom space. Mech. Sci., 4, 263–277, 2013

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H.-J. Su and C. Yue: Type synthesis

Table 1. The motion and constraint spaces of commonly used flexure primitives.

Flexure y

Freedom Symbol x

h i Rˆ z

R

z

h

Fˆ x

Fˆ y

Fˆ z

ˆx M

Fˆ x

Fˆ y

Fˆ z

Fˆ x

Fˆ z

ˆy M

ˆy M

i

z Split Tube

y

y

x

x z

z Spherical Notch

h Rˆ x

Rˆ y

Rˆ z

i

h

B=2R-P

h Rˆ x

Rˆ z

Pˆ y

i

h

i

y

x

z

S=3R

Short Wire/Rod

x

z y

Rotational Symmetric Cylinder x

i

x

z Blade/Sheet/ Leaf Spring

[W]

y

x

Notch/Living Hinge Short Beam

y

[T]

Disc Coupling

z y

W=3R-2P

h Rˆ x

Rˆ y

Rˆ z

Pˆ y

Pˆ z

i

h

Bs =2R-3P

h

Rˆ y

Pˆ x

Pˆ y

Pˆ z

i

h

Fˆ x

i

x Long Wire/Rod

Corner Blade

y x z

Rˆ x

ˆz M

i

Bellow Spring

The transformation from one basis to another is done by a linear operation to the twist matrix. A freedom space can be spanned by different sets of basis twists. This can be very useful in design practices. For instance, consider a freedom space formed by rotations about two parallel axes. The twist matrix of these two rotations is h

[T] = Tˆ R1

Tˆ R2

i

# Ω Ω = c1 × Ω c2 × Ω "

(19)

lations. They are formulated in screws as # h i " Ω Ω Ω ˆ ˆ ˆ [T] = T R1 T R2 T R3 = (21) c1 × Ω c2 × Ω c3 × Ω " # Ω 0 0  (22) , c1 × Ω (c2 − c1 ) × Ω (c3 − c1 ) × Ω where the second and third columns represent translations. See the bottom part of Fig. 1 for illustration of this case. 3.3

Subtracting the second column from the first one yields " [T] 

Ω 0 c1 × Ω (c2 − c1 ) × Ω

# (20)

where  represents a column-wise linear operation. Note the second column of the above matrix represents a translation along the direction normal to both the rotation axis and the line c1 c2 . Basically this means that a serial chain of two rotations is equivalent to a serial chain of a rotation and a translation. See the top part of Fig. 1. Similarly, a serial chain of three (non-coplanar) parallel rotations is equivalent to a rotation plus two orthogonal transMech. Sci., 4, 263–277, 2013

Parallel chains of flexure primitives

The constraint space of a rigid body represents all the forbidden motions of the body subject to a constraint arrangement. In screw theory, a constraint space can be represented by a wrench matrix [W] combined by c independent wrenches, written as h i ˆ1 W ˆ 2 ··· W ˆc [W] = W (23) The constraint space of a flexure mechanism formed by m building blocks connected in parallel is given by a wrench matrix that is obtained by assembling the wrench matrix of each building block, written as [W] = [Ad1 W1

Ad2 W2

···

Adm Wm ]

(24)

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131 132

1 [ ] F 0 constraints removes0 two rotations and where the second and third columns represent transla- 143 translational (28) c1 × F (cThis × F (c3 −with c1 ) ×the F following m 2 − cis 1 ) illustrated c1 the bottom c2 part of Fig. 1 for illustration of 144 translation. tions. See 145 ematical formulation 267 this H.-J. case.Su and C. Yue: Type synthesis R2 [ R1 ] [ F F ˆP W ˆP = ˆP W [W ] = W P 3 1 2 c c 1 × F c2 × F R2P1 R1 2 [ ] c1 ˆ 0 F 0 W P2  P2 c1 × F (c2 − c1 ) × F (c3 − c1 ) × F c1 c2

R3

c3

ˆ W P1

R2

R1

ure 1. Equivalent freedom spaces. (Top) A serial chain of c2 P1 parallel rotations produces c1 a translation. (Bottom) A sechain of three (non-coplanar) parallel rotations produce P2 translation.

functional body

reference body

ˆ W P3

ˆ W P2

R3

ˆ W P2

c3

Parallel Chains Flexure Primitives Figure 1. of Equivalent freedom spaces. (Top) A serial two of Figure 1. Equivalent freedom spaces. (Top) A chain serialof chain

ˆ W P1

parallel rotations produces a translation. (Bottom) A serial chain of ˆ two parallel rotations produces arepresents translation. (Bottom) A seW functional body reference body P1 e constraint of a rigid the threespace (non-coplanar) parallelbody rotations produce two all translation. rial chain of three (non-coplanar) parallel rotations produce bidden motions of the body subject to a constraint ˆ W P3 two translation. angement. In screw theory, a constraint space can functional body reference body 133 Again, matrices [Ad j ] represent the coordinate transformarepresented by a wrench matrix [W ] combined by c ˆ W tion from j-th building block to the functional stage. P2 Figure 2. Equivalent constraint spaces. (Top) A parallel chain of Figure 2. Equivalent constraint spaces. (Top) A parallel ependent wrenches, written as of flexures, the constraint space Similar to serial chains two parallel translational constraints exerts a translational plus a ro134 3.3 Parallel Chains of Flexure Primitives of two parallel translational constraints exerts a trans] [ parallel ˆ translational constraints assembled in parallelchaintational constraint to the functional body. (Bottom) A parallel chain ˆ 1 ofW ˆtwo (23) ]= W · · · W ˆ 2 c lational plus(non-coplanar) a rotational constraint to the functional body. W 2) is represented by two force body wrenches with identical parallel translational constraints P1 exerts a The (Fig. constraint space of a rigid represents all theof three (Bottom) A parallel chain of three (non-coplanar) parallel direction F of aofflexure forbidden motions the body subject formed to a constrainttranslation plus two rotational constraints. The arrows represent the The constraint space mechanism translational constraints exerts a translation plus two rotaremoved motions. " # h i arrangement. In screw theory, a constraint can constraints. Thefunctional m building blocks connected in parallel is given byspace tional F F body arrows body represent thereference removed moˆ P1 W ˆ [W] = W (25) represented byP2obtained a=wrench matrix by c c1 × F by c2 ×assembling F [W ] combined wrenchbematrix that is the tions. 4.1 Synthesis Tˆ R constraint spaces. (Top) A par Figure 2.of R-joints Equivalent independent writtenwritten as nch matrix of eachwrenches, building block, as Subtracting the second column from the first one yields chain of twoofparallel translational ] [ The freedom space an R-joint is given by constraints Tˆ R shown in exerts a tr ˆ "12 WW ˆ2 2 · ···· · Ad ˆmc# Wm ] (23)(4). Without [W ]= ] = [Ad Ad (24) W 1W 1 W lational plus a rotational constraint the functional b loss of generality, we assume the axisto of R-joint F 0 T 146 of Freedom Elements [W] = (26)4 Synthesis (Bottom) A parallel chain of three (non-coplanar) par F (cspace ×ofF coordinate The constraint a flexure mechanism 2 − c1 )the ain, matrices [Adc1j ]×represent transfor- formedthrough the origin, i.e. c = (0, 0, 0) . This gives us constraints exerts a translation plus two r (translational ) tion from building blockconnected to the functional stage. by mjth building blocks in parallel is given by Ω 147 In this section, we use flexure primitives listed(29) in Tational constraints. The arrows represent the removed Note the second column of the above matrix represents a roTˆ R = imilara to serial chains of flexures, the constraint wrench matrix that is obtained by assembling the 0 tions. 148 ble 1 to design flexure joints that have only one DOF, tational constraint along the direction perpendicular to both ce of two parallel translational assembled wrench matrix of each written as the constraint axis F and building theconstraints line c1 c2block, . Basically this says 149thatrotational (R) or translational (P). To design freedom Our goal is to remove the other five unwanted motions: parallel (Fig. 2) ischain represented by two force wrenches a parallel of two translational constraints removes150oneelements, we use parallel chains of primitives to remove two rotations and three translations. Since we consider paral[W ] = [Ad W Ad W · · · Ad W ] (24) 1and1one 2 2 m m h identicalrotation direction F translation. 151 unwanted lel structures of at least two and each limb must apply 146 4 freedoms. Synthesis of limbs Freedom Elements [ chain of three (non-coplanar) ] And a parallel parallel trans1–5 constraints to the functional body, primitive “R” (notch ] 135 [ Again, matrices [Ad ] represent the coordinate transforj F removesFtwo rotations and one translaˆ P lational ˆ P constraints ]= = (25) W W hinge) cannot be used in the design as it is indeed a R-joint. 1 2 136 mation to themathematical functionalforstage. 147 In this section, we c1building × F with c2block ×F tion. from This isjth illustrated the following use flexure with primitives listed in 152 4.1 This Synthesis R-Joints TˆRcombinations leads us of total 11 possible the other Similar mulationto serial chains of flexures, the constraint 148 ble 1 to design flexure joints that have only one D four primitives B, S, W and Bs . Among them, cases 1-5 use btracting theofsecond column "from the first one # yields ˆR shown space two constraints assembled The no freedom space and of is at given by h parallelitranslational 149 bellow rotational (R)an orR-joint translational design freed springs cases 6–11 use least(P). oneTTo bellow FFF [ ] ˆ ˆ ˆ W Pis = (27)in (4). [W] = W P1 W inFparallel (Fig. represented two force wrenches 3 losswe of use generality, assume the axis ofto rem spring. 0 P22) c1 × Fc2 × Fcby 150 Without elements, parallelwe chains of primitives 3×F ] = with identical" direction F (26) # R-joint through thefreedoms. origin, i.e. c = (0, 0, 0)T . This gives c1 × F (c2 − c1 ) × F 151 unwanted F00 [ ]  (28) Use flexure primitives B and W only [ ] − c ) × F(c − c ) × F c1 × F(c ˆP W ˆ P 2= 1 F 3 1 F [W ] = W (25) 1 2 c1 × F c2 × F The first cases concerns designs with and w.jn.net Mechanical 152 4.1threeSynthesis of R-Joints TˆRprimitives BSciences

W only: 2B, B-2W, 5W. These are the most commonly used

Subtracting the of second column from the first one yieldsdesigns. 4 Synthesis freedom elements The freedom space of an R-joint is given by TˆR sh [ ] in (4). Without loss of generality, we assume the ax F 0 [W ] In = this section, we use flexure primitives listed in Table 1 (26)4.1.1 R-joint Case 1: through 2B the origin, i.e. c = (0, 0, 0)T . This g c1 ×flexure F (cjoints F only one DOF, rotational to design 2 − cthat 1 ) ×have (R) or translational (P). To design freedom elements, we use parallel chains of primitives to remove unwanted freedoms.

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The synthesis with two parallel blades (2B) is described as the following. As shown in Fig. 3, let us denote the

Mechanical Scien

Mech. Sci., 4, 263–277, 2013

167 168 169

where [Wb ] is the wrench matrix of blade written in (14). 186 And vectors xi , yi , zi , di denote the orientation and po- 187 188 268 of ith blade. sition 189 190

y2

191

x2

192

This design evolves from the case 2B. Since a blade allows two rotations and one translation, we just need to remove one rotation and wires. H.-J.one Su translation and C. Yue:with Typetwo synthesis We have shown in (27) that two parallel constraints remove translation andoften one used rotation. rein Fig. one 4a. This is the most flexureTherefore, hinge design. placing one blade T of the 2B design with two parallel If x1 = z2 = (1, 0, 0) , we have a design with blade 1 subject wires yieldsand a B-2W in Fig.4(d). to twisting blade 2design, subjectshown to bending, Fig. 4b. If x = 1

y1 blade 2

z2

193

x1 blade 1

194

In order to obtain a hinge design with five wires, we

195

4.1.2 2: B-2W simply Case replace the blade of the B-2W design with three

196

d2

z1

197 198

d1

y

199

x functional body

z

x2 = (1, 0, 0)T , we have a design with both blades subject to 4.1.3 Case 4c.5W twisting, Fig. 3:

200

R-joints withwith a parallel connection of two 201 Figure 3. 3. Synthesis Synthesisof of R-joints a parallel connection blades. two blades are rigidly to the functionaltobody. of two The blades. The two bladesconnected are rigidly connected the 202 functional body. 203

co-planar wires. This is due to the fact that a blade This design evolves from the case 2B. Since a blade allows is equivalent to three co-planar wires. See Fig.4(e). An two rotations and one translation, we just need to remove one alternative synthesis procedure using screw algebra for rotation and one translation with two wires. We have shown the case five wires can be found in (Su and Tari, 2010). in (27) that two parallel constraints remove one translation and one rotation. Therefore, replacing one blade of the 2B Use flexure primitives B, Wyields a B-2W design, shown design with two parallelS,wires 4d.consider designs using primitives S, B, W with in Fig.we Now at least one S-joint. There are three possible combina4.1.3 Case 3: 5W tions: 2S, S-2W, B-S.

coordinate transformation of two blades relative to the funcMechanical Sciences tional body by [Ad1 ] and [Ad2 ] which are the six by six adjoint matrices with the form of (9). By applying the coordinate transformation to both blades, their wrench matrices are calculated by (10,11) as " # xi zi 0 [Wi ] = [Adi ][Wb ] = , i = 1, 2, (30) di × xi di × zi yi

In order to obtain a hinge design with five wires, we simply www.jn.net replace the blade of the B-2W design with three co-planar wires. This is due to the fact that a blade is equivalent to three co-planar wires. See Fig. 4e. An alternative synthesis procedure using screw algebra for the case five wires can be found in Su and Tari (2010).

where [Wb ] is the wrench matrix of blade written in (14). And vectors xi , yi , zi , di denote the orientation and position of i-th blade. Requiring the reciprocity of the [Wi ] with Tˆ R yields the necessary condonations regarding to the orientation and position of the blades

Now we consider designs using primitives S, B, W with at least one S-joint. There are three possible combinations: 2S, S-2W, B-S. However the case B-S is not qualified as a R-joint for the following reasons. Without loss of generality, we let the coordinate system align with the local coordinate system of the blade shown in Table 1. And we denote the position of Sjoint by d = (d x , dy , dz )T . Therefore the wrench matrix of the blade and the S-joint are " # h i i j k [Wb ] = Rˆ x Rˆ z Pˆ y , [W0s ] = (36) d×i d× j d× k

Ω · (di × xi ) = 0, Ω · (di × zi ) = 0, Ω · yi = 0.

=⇒ =⇒

xi · (di × Ω) = 0 zi · (di × Ω) = 0

(31) (32) (33)

Simplifying the first two conditions yields di · yi = 0,

Ω · yi

= 0.

(34)

The first equality is interpreted as that the translation di must in the blade plane xi zi . The second condition means that yi , the normal direction of the blade, must be perpendicular to the axis of the R-joint (Ω). If Ω = (1, 0, 0)T , yi must be in the plane yz of the functional body. Picking any two independent directions in the yz plane leads us a solution, e.g. y1 = (0, 1, 0)T ,

y2 = (0, 0, 1)T

(35)

Once yi is determined. The axes of xi and zi are chosen arbitrarily as long as the two blades are not co-planar (redundant). Depending on how xi and zi are chosen for each blade, we can obtain three possible designs. If z1 = z2 = (1, 0, 0)T , we have a design with both blades subject to bending, shown Mech. Sci., 4, 263–277, 2013

Use flexure primitives S, B, W

where vectors i, j, k are unit vectors along three coordinate axes. And [W0s ] is the wrench matrix with an appropriate coordinate transformation, i.e. [W0s ] = [Ad][W s ]. We write the wrench matrix of the parallel structure B-S and apply a linear operation to obtain   0 0  1 0 0 1  0 0 0 0 1 0   h i 0 1 0 0 0 1  (37) [Wbs ] = Wb W0s =   0 0 0 0 −dz dy  0 0 1 d  0 −d x  z   0 0 0 −dy d x 0

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H.-J. Su and C. Yue: Type synthesis

269

(a)

(b)

(c)

(g)

(h)

(i)

(d)

(e)

(j)

(k)

(f)

(l)

(m)

Figure 4. Various designs of R-joints with flexure primitives: B, W, S and B . (a–c) case 1 with two blades (BB). (d) case 2 with one blade

s Various designs of (B-2W). R-joints with primitives: B,twoW, S and Bs(2S). . (a-c) two blades (BB) and two wires (e) case 3 withflexure five wires (5W). (f) case 4 with spherical notches (g) casecase 5 with1onewith spherical notch with wires (S-2W).(B-2W). (h) case 6 with bellow3spring, bladewires and one (5W). spherical notch (Bs -B-S). case 7two with one bellow spring, one ade and two two wires (e)onecase withonefive (f) case 4 (i)with spherical notches (2S) blade and one wire (Bs -B-W). (j) case 8 with one bellow spring, one spherical notch and one wire (Bs -S-W). (k) case 9 with one bellow herical notch with two wires (S-2W). (h) case 6 with one bellow spring, one blade and one spherical notc spring and four wires (Bs -4W). (l) case 10 with two bellow springs and one spherical notch (2Bs -S). (m) case 11 with two bellow springs three wires (2Bs -3W). The double arcs one represent the rotation allowed by (j) flexure R-joints. The box one represents the functional body. one sphe with one and bellow spring, one bladearrow and wire (Bs -B-W). case 8 with bellow spring, re (Bs -S-W). (k) case 9 with one bellow spring and four wires (Bs -4W). (l) case 10 with two bellow s al notch (2Bs -S). (m) case 11 with two bellow springs and three wires (2Bs -3W). The double arrow arc     n allowed by flexure R-joints. The box represents the functional body.  0 0  1 0 0

e second ).

0 0  1 0 0 0  0 0 0 0 1 0   0 1 0 0 0 0     0 −dz dy  0 0 0This  rotation. results 0 0 1 0 0 0    0 0 0 −dy d x 0

(38)

a design shown 281 282 283

From Eq. (38), we can draw two conclusions. (1) If dy , 0, [Wbs ] has a full rank which means that the body is fully con-284 e 9: Bs -4W: strained by this structure. (2) If dy = 0, i.e. the S-joint is on285 blade we plane,first [Wbs ]use has athree rank of 2wires since the 4th and 6th size this the case, aligning column would be zeros. As a matter of fact, its reciprocal286 coordinate axes tois remove translations. freedom space formed by three the two rotations allowed by287 the blade flexure. Therefore, parallel structures of cannot s us three rotations. We then remove twoB-Smore 288 form a R-joint design.

by two more wires. By Eq.(26), we know that 289 el constraints remove 4.1.4 Case 4: 2S one additional rotation. 290 we let aThe fourth wire be parallel to one of the 2S case is trivial. When two S-joints are connected in291 s aforementioned. This wireabout removes a 292 parallel, this results in a fourth single rotation the line connecting to their joint centers, Fig. 4f. bout the normal line to the plan formed by arallel wires. At last, a bellow spring is used 4.1.5 Case 5: S-2W another rotation. This results a hinge design 293 The case S-2W is synthesized as the following. The S-joint Fig. 4(k).itself removes three translations. We just need to remove two294

ases 10:

additional rotations with two wire flexures. By using a wire 295 parallel to y-axis with a non-zero offset d = (d, 0, 0)T , the ro2B s -S tation Rˆ z is removed. Similarly, a wire parallel to z-axis with a non-zero offset d = (d, 0, 0)T removes the rotation Rˆ y . Mathematically, the wrench matrix of S-joint andS-joint the two wires are296 2B is simple. The res -S case

esis of ee translations and two bellow springs remove 297 www.mech-sci.net/4/263/2013/ three rotations, Fig. 4(l). 298 299

    0 1 0 1 0    0 1  0 0 1  [W s ] =   , [W2w ] =  0 0  0 0 0 where V represent 0 −d  0 vector    0 0 d 0 0 0 0 P-joint.

(39)

the translational d

We arespace interested in simple parallel struc The constraint of S-2W is obtained by combining [W s ] and [W ], to which we apply a column-wise reduction and 2w at least two limbs that remove three rotation obtain translations. And each limb must apply at leas  1 0 0 0 0  straint and allow least one translational m 0 at 1 0 0 0   h i body. the functional primitives “R 0 0 1Therefore 0 0  (40) [W s−2w ] = W s W2w =   0 0 0 0 0  no translation. T are not qualified0as 0they allow 0 0 −d   three possible primitives: 0 0 0 d B, 0 W and Bs from

can have eight possible cases. Three of them u least o

Obviously its reciprocal twist matrix is Rˆ x which corresponds springs andthethe other fiveis shown use at tolow a single rotation about x-axis. This design in 4g. Fig. spring. Use flexure primitives Bs , S, B, W

Designs with flexures B, W At last, we synthesize the design cases withonly at least one bellow spring. If one Bs flexure is used, there are four possible design cases: Bs -B-S,only Bs -B-W, Bs -S-W, B Bs -4W. Considering flexure andAnd W,if two we bellow springs are used, there are two additional combina2B, B-2W and 5W. tions: 2Bs -S and 2Bs -3W. Note 2Bs -B is not possible as B removes one rotation and 2Bs remove another two rotation which results in a freedom space with no rotation. Let us dis4.2.1 Case 1:the2B cuss these six cases in following.

have t

For designs with two blades, the reciprocity Mech. Sci., 4, 263–277, 2013 of TˆP in (41) with the wrench matrices [W yields

270

H.-J. Su and C. Yue: Type synthesis

4.1.6

Case 6: Bs -B-S:

From Eq. (38), we have concluded that a B-S parallel structure may result in a freedom space of two rotations if the center of S-joint is on the blade plane. Now we just need to use a bellow spring Bs to remove one rotation in order to yield a freedom space of only one rotation, hence a hinge design in Fig. 4h. 4.1.7

Case 7: Bs -B-W:

As we know one blade has a freedom space of 2R-P. To obtain a single rotation, we can use a bellow spring to remove one rotation and a wire to remove the translation. See this design in Fig. 4i. 4.1.8

Case 8: Bs -S-W:

The S-joint has a freedom space of 3R. To obtain a single rotation, we can use a bellow spring to remove one rotation then use a wire offset to the center of the S-joint to remove the second rotation. This results a design shown in Fig. 4j. 4.1.9

Case 9: Bs -4W:

To synthesize this case, we first use three wires aligning the three coordinate axes to remove three translations. This leaves us three rotations. We then remove two more rotations by two more wires. By Eq. (26), we know that two parallel constraints remove one additional rotation. Therefore we let a fourth wire be parallel to one of the three wires aforementioned. This fourth wire removes a rotation about the normal line to the plan formed by the two parallel wires. At last, a bellow spring is used to remove another rotation. This results a hinge design shown in Fig. 4k. 4.1.10

Cases 10: 2Bs -S

The synthesis of 2Bs -S case is simple. The S-joint removes three translations and two bellow springs remove two of the three rotations, Fig. 4l.

where vector V represent the translational direction of P-joint. We are interested in simple parallel structures with at least two limbs that remove three rotations and two translations. And each limb must apply at least one constraint and allow at least one translational motion for the functional body. Therefore primitives “R” and “S” are not qualified as they allow no translation. This leaves three possible primitives: B, W and Bs from which we can have eight possible cases. Three of them use no bellow springs and the other five use at least one bellow spring.

Designs with flexures B, W only

Considering only flexure B and W, we have three cases: 2B, B-2W and 5W.

4.2.1

For designs with two blades, the reciprocity conditions of Tˆ P in (41) with the wrench matrices [Wi ] in (30) yields V · xi = 0,

i = 1, 2

(42)

4.2.2

Case 2: B-2W

The design with one blade and two wires can be easily obtained by replacing one blade with two wires that are parallel to the blade plane. Figure 5b.

Case 3: 5W

Cases 11: 2Bs -3W

This case is also trivial. First the three wires aligning the three coordinate axes remove three translations. And we then use two bellow springs remove two rotations. This design is shown in Fig. 4m. 4.2

V · zi = 0,

from which we conclude that V must be parallel to both yi (normal of the blade plane). This means that two blades must be parallel with the normal of the blade plane along the axis of P-joint. This is the well known parallel sheet design of P-joints shown in Fig. 5a.

4.2.3 4.1.11

Case 1: 2B

Synthesis of P-Joints Tˆ P

Now let us synthesize P-joints with flexure primitives listed in Table 1. The freedom space of an P-joint is given by a single twist ( ) 0 Tˆ P = (41) V Mech. Sci., 4, 263–277, 2013

This design is obtained from the B-2W design by replacing one blade with three co-planar wires and the other one with two wires. This case can also be computationally synthesized with screw theory. Assume the P-joint is along the x direction, i.e. V = (1, 0, 0)T . First of all, compute the reciprocal wrench matrix of Pˆ using linear algebra as  0 1  0 [W] =  0 0  0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

 0  0 0  0  0  1

(43)

www.mech-sci.net/4/263/2013/

10

H.-J. Su and C. Yue: Type synthesis

271 371

We then apply a column-wise linear operation to this wrench matrix to obtain   0 0 0 0 0 1 1 0 0 1   0 0 1 1 0 0 (44) [W ] =   0 0 0 0 1 0 0 0 1 0   0 1 0 0 0

372 373 374 375 376

(a)

(b)

[T1 ] =

(c)

By the twist m

0

Each column of [W ] is a force wrench which can be realized with a long wire flexure (W). The 5W design is shown in Fig. 5b. See the work by Su and Tari (2010) for an alternative synthesis procedure for this case.

[T ] = (d)

(e)

(f)

[T ′ ] =

There are five cases with at least one bellow spring: Bs -BW, Bs -4W, 2Bs -B, 2Bs -3W, 3Bs -2W. We discuss each in the following.

377 378

(g)

Case 4: Bs -B-W

A blade B has a freedom space of 2R-P. To obtain a P-joint design, we must remove the two rotations. First we use a wire flexure with axis being parallel and has a non-zero distance to the blade plane to remove one rotation. The second rotation is removed by a bellow spring flexure. See Fig. 5d. 4.2.5

Case 5: Bs -4W

This design can be obtained from the case Bs -B-W by replacing the blade with three co-planar wires. A computational way to synthesize this case is applying an alternative linear operation to the wrench matrix [W] in (43). This time 353 we would like to have four force wrenches and one couple 354 wrench. The new wrench matrix is written as 355 356   0 0 0 0 0 357 1 1 0 0 1   358 0 0 1 0 0 00 [W ] =  (45) 359  0 0 0 0 1 360 0 0 0 1 0   361 0 1 0 0 0 362 363

of which the fourth column is a couple wrench and the other 364 four are force wrenches. Realizing each force wrench with a 365 wire and the couple wrench with a bellow spring yields the 366 design of Bs -4W. See Fig. 5e. 367 4.2.6

Case 6: 2Bs -B

368

This case is trivial since the two bellow springs remove the 369 two rotations of the blade. This leaves a freedom space with 370 a single translation, hence a P-joint design shown in Fig. 5f. www.mech-sci.net/4/263/2013/

[

Substit column

Designs with flexures B, W and Bs

4.2.4

center proof w assume system x axis, blades

(h)

379

380 designs of of P-joints P-jointswith withparallel parallelstructures structures Figure 5. Various Various designs of of flexure primitives: B, BW, Bs .case (a) case 1 with two blades flexure primitives: B, W, 1 with two blades (2B). s . (a) (2B). case one 2 with and two wires(c)(B-2W). (c) 381 (b) case(b) 2 with bladeone andblade two wires (B-2W). case 3 with casewires. 3 with(d)five wires. (d)one casebellow 4 with one bellow spring, five case 4 with spring, one blade and one 382 blade(Band one (e) wire casespring 5 with wire case(B5s -B-W). with one(e) bellow andone fourbellow wires s -B-W). spring and case and 6 with (B (f) four case 6wires with (B two bellow(f) springs one two bladebellow (2Bs - 383 s -4W). s -4W). 384 springs and7 one (2Bs -B). (g) and casethree 7 with B). (g) case withblade two bellow springs wirestwo (2Bbellow s -3W). springs three (2B (h)two case 8 with (h) case 8and with threewires bellow springs and wires (3Bsthree -2W). belThe 385 s -3W). low springs two the wires (3Bs -2W). The arrowed lines inarrowed linesand indicate direction of translation. The box repredicate direction of translation. The box represents the sents thethe functional body. functional body.

4.2.7 Case 7: 2Bs -3W whose complementary freedom space consists of three This case evolves case 2Bs -B replacing the blade 386 rotations plus twofrom translations, i.e.by3R-2P. ˆ P , we 387 with co-planarthe wires. See Fig. 5g.constraint W Tothree synthesize translational 388 consider serial chains of at least two flexure primitives ˆ P . 389 to make up the complementary freedom space of W 4.2.8 Case 8: 3Bs -2W We do not include the long wire (W) flexure in the designthis as case, itselfthe is 3indeed translational Also For bellowasprings remove constraint. three translations bellow springs (B ) are not qualified either as they aland two wires remove s two translations. This design is shown low three translations. As a result, only three types of in Fig. 5h. primitives S, B and R are considered in design. Based on 390 the degree of freedom, there are six possible combina5tions: Synthesis constraint elements S-S, S-B,ofS-2R, B-B, B-2R and 5R. However these primitives must satisfy some restriction on their relative 391 As a duality and to the design of elements, this secorientation position. Infreedom what follows, we in derive the 392 ˆ ˆ tion, we design basic constraint elements W and W R P which 393 geometric conditions for each case.

which along y remove remove 5.1.2

Figure plying matric yields [Ts ]

=

[Tb ]

=

Sinc prescri rocal t S-joint

F · (d1

F · (d1 F · (d1 which

F × d1

This im along t And to

remove one rotation and one translation respectively. These 394 constraint elements are most often used in designing parallel 5.1.1 Case S-S F · (d2 structures for 1: removing a particular set of motions. The basis approach is to use a serial chain of flexure primitives listed F · (d2 By intuition, we know that a serial chain of two S-joints to construct the complementary spacethe of in Table 1one removes translation along the linefreedom connecting the constraint element to be synthesized. Mechanical Sciences

Mech. Sci., 4, 263–277, 2013

397 398 399 400

272

The first equality implies that the spherical notch must 425 be on the blade plane. And the second equality means 426 that the normal of the blade must be perpendicular to 427 the direction of F. The design is shown in Fig. 7(b). H.-J. Su and C. Yue: Type synthesis 428

435

where means dicular not be be red constra ented, shown tude d Fig. 7( of the

436

5.1.5

429

5.1

ˆP Synthesis of P-constraints W

A P-constraint element removes a translational freedom of the functional body while allows all other motions. The constraint space of the translational constraint is given by a pure force given in (5).Without loss of generality, we assume the axis of translational constraint through the origin of the coordinate system, i.e. c = (0, 0, 0)T . That is ( ) F ˆ WP = 0

430

y1

x1

431 432 433

spherical notch

y2

z1

434

x2 blade d1

z2

(46)

437 438

d2

y

439

x

whose complementary freedom space consists of three rotations plus two translations, i.e. 3R-2P. ˆ P , we consider To synthesize the translational constraint W serial chains of at least two flexure primitives to make up the ˆ P . We do not include the complementary freedom space of W long wire (W) flexure in the design as itself is indeed a translational constraint. Also bellow springs (Bs ) are not qualified either as they allow three translations. As a result, only three types of primitives S, B and R are considered in design. Based on the degree of freedom, there are six possible combinations: S-S, S-B, S-2R, B-B, B-2R and 5R. However these primitives must satisfy some restriction on their relative orientation and position. In what follows, we derive the geometric conditions for each case.

442

445

5.1.2 401

402 403 404 405

409 410 411 412 413 414 415 416 417 418 419

By the formulation for serial structures in Eq.(18), the twist matrix of the S-S chain is written as # h i "i j k i j k [T] = T1 T2 = (48) 0 0 0 d×i d× j d× k Substituting d = (d, 0, 0)T and subtracting the last three columns from the first three columns yield "

i [T ] = 0 0

j 0

k 0

0 0 0 0 d k −d j

# (49)

which indicates three rotations and two translation along yand z-axes, i.e. the translation along x-axis is removed. See Fig. 7a. The arrow line represents the removed translation. Mech. Sci., 4, 263–277, 2013

Figure 6 shows a serial chain of S and B flexures. Applying 5.1.3 Case 3: S-2R a general coordinate transformation to the twist matrices of 448 the the spherical in (13, 14) that yieldsan S-joint 449 Forblade this and design case, we notch first recognize " provides three rotations (3R). We# just need to make 450 x1 y1 z [T (50) 451 ups ]two= more translations with 1 two, notch hinges (R). d1 × x1 d1 × y1 d1 × z1 We have shown in Section 3.2# that two parallel rota- 452 " z2 0 three parallel rotations 453 tions producex2a translation and [T (51) b] = d2 ×translations. x2 d2 × z2 Therefore y2 produce two we can have two 454 possible designs. One design utilizes two perpendicular 455 Since both freedom spaces are complementary to the pre- 456 R-joints. Each R-joint is parallel one rotation axis of ˆ to scribed translational constraint, W in (46) is reciprocal to the S-joint. The direction of the Ptranslational constraint 457 both [T s ] and [Tb ]. The reciprocity condition of S-joint leads 458 is along the direction that is perpendicular to both Rto joint axes. See Fig. 7(c). two R-joints which F · The (d1 × other x1 ) = 0design =⇒ uses x1 · (F × dparallel (52) 459 1) = 0 together with one rotation of the S-joint produce two F · (d1 × y1 ) = 0 =⇒ y1 · (F × d1 ) = 0 (53) 460 translations. The S-joint and two R-joints axes are co461 F · (d1 × The z1 ) = direction 0 =⇒ ofz1 the · (F × d1 ) = 0 (54) planar. translational constraint is 462 parallel to the R-join axis. And the constraint line passes 463 which are reduced to through the center of S-joint. See Fig. 7(d). Note the along the direction(55) inFfunctional × d1 = 0, body =⇒ cannot d1 = dtranslate 1F dicated with its orientation fixed. This implies that the position of the S-joint must be along the direction of F. www.jn.net And the reciprocity requirement for the blade leads to 447

408

By intuition, we know that a serial chain of two S-joints removes one translation along the line connecting the center of the joints. Here let us give a mathematical proof with screw theory. Without loss of generality, we assume one S-joint being at the origin of the coordinate system and the other one being at a distance of d along x-axis, i.e. d = (d, 0, 0)T . The twist matrices of these two blades are " # " # i j k i j k [T1 ] = , [T2 ] = (47) 0 0 0 d×i d× j d× k

Case 2: S-B 446

407

Case 1: S-S

441

Figure 6. 6. Synthesis SynthesisofofP-constraint P-constraint a serial one Figure withwith a serial chainchain of oneofblade 443 blade one spherical (S).isThe blade with is con(B) and(B) one and spherical notch (S). notch The blade connected the 444 nected with spherical notchbody via(not an drawn) intermediate body spherical notchthe via an intermediate (not drawn)

406

5.1.1

440

functional body

z

420

F · (d2 × x2 ) = 0 F · (d2 × z2 ) = 0 F · y2 = 0

=⇒ =⇒

x2 · (F × d2 ) = 0 z2 · (F × d2 ) = 0

(56) (57) (58)

which are reduced to F × d2 = d2 y2 ,

F · y2 = 0

(59)

Substituting (55) into (59) yields d1 × d2 = d1 d2 y2 ,

F · y2 = 0

(60)

www.mech-sci.net/4/263/2013/

Since a can ma R-joint mal of rotatio joint. tional See Fig 5.1.6

Synthe tions w designs have tw R-joint vantag a trans transla And which the ax tional allel R

5.2 S

A R-co allows tationa (5), wh ˆR= W

H.-J. Su and C. Yue: Type synthesis

273

12

The first equality implies that the spherical notch must be on the blade plane. And the second equality means that the normal of the blade must be perpendicular to the direction of F. The design is shown in Fig. 7b.

They m the foll

M · xi =

(a)

which i 5.1.3

Case 3: S-2R

(f)

For this design case, we first recognize that an S-joint provides three rotations (3R). We just need to make up two more translations with two notch hinges (R). We have shown in Sect. 3.2 that two parallel rotations produce a translation and three parallel rotations produce two translations. Therefore we can have two possible designs. One design utilizes two perpendicular R-joints. Each R-joint is parallel to one rotation axis of the S-joint. The direction of the translational constraint is along the direction that is perpendicular to both R-joint axes. See Fig. 7c. The other design uses two parallel R-joints which together with one rotation of the S-joint produce two translations. The S-joint and two R-joints axes are co-planar. The direction of the translational constraint is parallel to the R-join axis. And the constraint line passes through the center of S-joint. See Fig. 7d. Note the functional body cannot translate along the direction indicated with its orientation fixed.

(b) (g) (c)

5.2.3

487 488

(i)

ˆ Pserial ˆ P with Figure 7.7.Various Various designs of P-constraint W with serial Figure designs of P-constraint W chains of 491 chains primitives of flexureS,primitives S, Bdouble and R. The double head 492 flexure B and R. The head arrows represent arrows represent direction of the constrained the direction of the the constrained translation. One end oftranslation. the chains is 493 One end of the chains is fixed. notches (a) case(2S). 1 with two 2spherical fixed. (a) case 1 with two spherical (b) case with one 494 notches notch (2S). and (b) one case 2 with one spherical notch one spherical blade (S-B). (c–d) case 3 with one and spherical bladeand (S-B). (c-d) hinges case 3(S-2R). with one notch two notch two notch (e–f)spherical case 4 with two and intersectnotch hinges (S-2R). (e-f) case one 4 with twoand intersecting ing blades (2B). (g) case 5 with blade two notch blades hinges (2B). (g) case 5 with one blade two (5R). notch hinges (B-2R). (B-2R). (h–i) case 6 with five notchand hinges (h-i) case 6 with five notch hinges (5R).

464 465 466 467 468 469 470 471

473

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490

484

486

Case 5: B-2R

Since a blade has two rotations and one translation, we can make up one rotation and one translation with two R-joints. Obviously one R-joint must be along the normal of the blade plane in order to make up the third rotation. The second Rjoint is parallel to the first R-joint. These two parallel R-joints produce one additional translation along the local z-axis of the blade. See Fig. 7g.

(h)

485

472

5.1.5

489

Tee des B flexu two Rthe two only so ing dire the tor separat Fig 8(b

483

(d)

(61) (62)

where we have applied a vector algebra operation. This means that the normal of both blades must be perpendicular to the direction F. However the two blades must not be parallel to each other as otherwise they would be redundant. The intersection line of the blades is the constraint line. Depending on how the blades are oriented, we can have two possible designs. In the design shown in Fig. 7e, the constraint line is along the longitude direction of the blades. And in the design shown in Fig. 7f, the constraint line is along the width direction of the blades.

5.2.2

478

482

The synthesis of B-B follows a similar procedure of the case S-B. For this case, both blades must satisfy the conditions in Eq. (59), i.e. F · y1 = 0 F · y2 = 0

480

477

481

Case 4: B-B

d1 = d1 y1 × F, d2 = d2 y2 × F,

479

Therefo to each two par tance in directio

475 476

(e) 5.1.4

y1 = y2

474

This ca That is other tw two pla

5.1.6 Case 6: 5R Our goal is to use serial chains of at least two flexure Synthesis case W isˆ simple. translations primitivesoftothis design is produce to make two up its compleR . ThatTo with a serial chain of 5R, we canconsists have twoofpossible designs mentary freedom space which two rotations by rotations. In(P). oneSince design, we have two R(R)using and parallel three translations a bellow spring is joints to oneconstraint, plane and another itself parallel a rotational we do two notR-joints considerparallel it in our design.plane. And This wiresdesign and spherical notchesofare exto a second takes advantage thealso fact that cluded in design as their freedom space already consists two parallel rotations produce a translation as proven previof three rotations. we only consider primitives ously. The directionTherefore of the translational constraint is along. B and See Fig.R. 7h.There are three potential designs B-B, B-2R and 5R.the other design uses three parallel R-joints which And

produce two translations in the plane normal to the axis of the R-joints. direction of the translational constraint is 5.2.1 Case The 1: B-B along the the axis of the three parallel R-joints. See Fig. 7i. To synthesize a serial chain of B-B for a rotational constraint, we first write the twist matrix of the blades as ˆR 5.2 Synthesis of R-Constraints W [ ] xi zi 0 [TiR-constraint ]= = 1, 2 rotation while (64) A element removes, a isingle aldi × xi di × zi yi lows all other motions. The constraint space of a rotational Mechanical Sciences

Mech. Sci., 4, 263–277, 2013

Figure with ser arcs rep One end

ˆ P with serial Figure 7. Various designs of P-constraint W 491 chains of flexure primitives S, B and R. The double head 492 arrows represent the direction of the constrained translation. 493 One end of the chains is fixed. (a) case 1 with two spherical 274 494 notches (2S). (b) case 2 with one spherical notch and one blade (S-B). (c-d) case 3 with one spherical notch and two ˆ R given constraint is pure couple W in (5), which we notch hinges (S-2R). (e-f)wrench case 4 with two intersecting blades copy for convenience (2B).here (g) case 5 with one blade and two notch hinges (B-2R). (h-i) case ( )6 with five notch hinges (5R).

This case is trivial as there is only one possible design. That is, three R-joints are parallel to one plane and the other two R-joints parallel to a second plane. And these H.-J. Su to and C. Yue: Type two planes are perpendicular each other. See synthesis Fig 8(c).

(a)

473

0 (63) M Our goal is to use serial chains of at least two flexure Our goal is to use serial of at least two flexure primˆ Rchains primitives to design W . That is to make up its compleˆ R . That itives to design W is to make up its complementary mentary freedom space which consists of two rotations freedom space which consists of two rotations (R) and three (R) and three translations (P). Since a bellow spring is translations (P). Since a bellow spring is itself a rotational itself a rotational constraint, we do not consider it in constraint, we do not consider it in our design. And wires our design. And wires and spherical notches are also exand spherical notches are also excluded in design as their cluded in design as their freedom space already consists freedom space already consists of three rotations. Therefore of three rotations. Therefore we only consider primitives we only consider primitives B and R. There are three potenB and R. There are three potential designs B-B, B-2R tial designs B-B, B-2R and 5R. and 5R.

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(c)

ˆR= W 464 465 466 467 468 469 470 471 472

(b)

To a serial chain of B-B for afor rotational constraint, Tosynthesize synthesize a serial chain of B-B a rotational conwe first write the twist matrix of the blades as straint, we first write the twist matrix of the blades as " # [ xi ] zi 0 [T i ] = , 0 i = 1, 2 (64) x z [Ti ] = di × xii di × zi i yi , i = 1, 2 (64) di × xi di × zi yi

ˆR ˆ R with serial Figure 8.8.Various Various designs ofrotational the rotational constraint W Figure designs of the constraint W with serial chains of flexure primitives B and R. arrowed chains of flexure primitives B and R. The arrowed arcsThe represent the arcs represent the rotation constrained by the design. rotation constrained by the flexure design. One endflexure of the chains is One end of the chains is fixed. fixed.

ˆ R in (63). This leads us the folThey must be reciprocal to W Mechanical Sciences lowing necessary conditions

6

M · xi = 0,

M · zi = 0

(65)

which implies that y1 = y2 = M

(66)

Therefore, we conclude that two blades must be parallel to each other with their y-(normal) axis along M. The two parallel blades must be separated by a nonzero distance in order to produce sufficient translations in three directions. This design is shown in Fig. 8a. 5.2.2

Case 2: B-2R

Tee design of B-2R is synthesized as the following. The B flexure has a freedom space of 2R-P. We have to use two Rjoints to make up two extra translations. And the two translations must be in the blade plane. The only solution is to have one R-joint parallel to the bending direction of the blade and the other R-joint parallel the torsion direction of the blade. And they should be separated by nonzero distance. This design is shown in Fig 8b. 5.2.3

Case 3: 5R

This case is trivial as there is only one possible design. That is, three R-joints are parallel to one plane and the other two R-joints parallel to a second plane. And these two planes are perpendicular to each other. See Fig. 8c. Mech. Sci., 4, 263–277, 2013

Synthesis of hybrid structures

www.jn.net We can build more complex flexure mechanisms with hybrid structures of flexure primitives together with the freedom and constraint elements synthesized in the previous sections. Here a hybrid structure is a structure with both serial and parallel connections. As a matter of fact, many spatial flexure mechanisms in practice are in hybrid structures (Yao et al., 2008; Dong et al., 2008). In this section, we use four examples to demonstrate how to design hybrid structures. 6.1

Hybrid designs of joints and constraints

In the previous sections, we have presented synthesis of R-joints, P-joints, R-constraints, P-constraints with flexure primitives in Table 1. Although these procedures concern designs of notch hinges (R), blades (B), spherical notches (S), long wires (W) and bellow springs (Bs ), designers should be aware that there may be multiple choices for these primitives. For instance, a blade flexure of any of these designs can be replaced by a rotational symmetric cylinder or a disc coupling since they have identical freedom and constraint space. Similarly, a wire flexure can be substituted with a corner blade and a spherical notch with a short wire, a notch hinge with a short mean or a split tube. Moreover, a notch hinge of any design can be replaced with the R-joint designs listed in Fig. 4. These designs are by no means the complete list of the basic freedom and constraint elements as many flexure mechanisms are in hybrid structures which are combinations of serial and parallel chains. For instance, Fig. 9a shows the well known parallelogram 4-bar design with four notch hinges. This structure is considered as a hybrid structure that is www.mech-sci.net/4/263/2013/

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As This we discussed previously, we can use a errors serialand chain design is widely used to reduce parasitic 559 As another example, we would like to design a parof R-joints and P-joints tostroke. construct a freedom space 14allel structure with three translations. We 275 536 increase the displacement 560 just need to H.-J. Su and C. Yue: Type synthesis with any combination of DOF. For instance, a serial 561 use three rotational constraint elements to remove all for all chain of three P-joints can make up a freedom space562of rotations. If we choose the BB design in Fig 8(a) Acknowle ported by B 563 three rotational constraints, we obtain the design shown three translations. If we use the double parallel flexure CMMI-11 564 in Fig 12. The functional body A can translate inrecommen all distructure for the P-joint design, we obtain a serial PPP author(s) design with three decoupled translations. See Fig. 10.565Of rections while its rotations are constrained. tional Sci course, one can further increase the structural stiffness (a) (b) (c) 566 7 Conclusions with assembling multiple serial chains of PPP in parallel Referenc B Figure 9. Designs of P-joints with hybrid structures. Figure 9. Designs of P-joints with hybrid structures. (Awtar et al., 2011). 567 A list of commonly used flexure primitives is firstAwtar, cate-S. 535

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of Mech Awtar, S. matic F www.jn.net Mechanical594 Sciences 595 Degree A A 596 Ball, R. S B 597 Press, C 598 and rev 599 introdu Figure11. 11.AAparallel parallelkinematic kinematic chain three translational 600 Blanding, Figure chain of ofthree translational conB constraints in a design with rotations three rotations about theof 601 Kinema straints resultsresults in a design with three about the center of thebody functional A.are The body B are fixed. 602 Brouwer, thecenter functional A. The body body B fixed. 603 eling of 604 Precisio 605 Chen, S.-C 6.3 Synthesis of parallel chains with constraint 606 scale na elements 607 314 – 3 608 Culpeppe In duality, one can also synthesize parallel chains with the609 cost na constraint elements synthesized in the previous sections. For Figure 10. A serial chain of three P-joints connected with two in610 tial com B B Figure 10. A serial chain of three P-joints connected with two termediate bodies. Each P-joint is a parallel blade design shown in – 482, 2 instance, if we would like to design a parallel structure with611 intermediate bodies. Each parallel blade Fig. 5a. Body B is the fixed P-joint base. BodyisA a is the functional body. design three rotations, we just need to use three translational con-612 Dai, J. S. 613 system shown The in Fig. 5(a).represent Bodythree B istranslations the fixed base. A is the arrow lines along three Body coordinate straint elements to remove all A translations. If we choose the 614 cations axes. functional body. The arrow lines represent three translations BB design in Fig. 7e for all three translational constraints,615 2001. we obtain the designBshown in Fig. 11. The functional body616 Dai, J. S. along three coordinate axes. A can rotate about its center relative to the base body B while617 Obtain System itsFigure translations are constrained. formed by two identical limbs assembled in parallel. Each 12. A parallel kinematic chain of three rotational con- 618 619 Davidson, As another example, we would like to design a parallel limb is a serial chain of two R-joints. It is not hard to prove straints result in a design with three translations along the ory: Ap 6.3 Synthesis of Parallel with Elements directionwith indicated the figure. The B areto fixed. structure three intranslations. We body just need use The three620 that this hybrid structure Chains allows only oneConstraint translation indi621 Oxford body A isconstraint the functional body.to remove all rotations. If we rotational elements cated in the figure. If we replace one of the RR chain with 622 Dong, J., In duality, can chains choose the BB design in Fig. 8a for all three rotational con-623 a blade one flexure, we also obtainsynthesize another designparallel shown in Fig. 9b. with kinema straints, we obtain the design shown in Fig. 12. The func-624 And Fig. 9celements shows a double parallelogram design that is secthe constraint synthesized in the previous sitionin 569 tional freedom is synthesized with parallel bodyelements A can translate in all directions while itsconnecrotations625 ysis, Pr formed by a serial connection of two parallelogram 4-bar detions. For instance, if we would like to design a parallel of these primitives. In duality, translational and 626 Hale, L. C constrained. sign. This design is widely used to reduce parasitic errors and 570 aretions structure with rotations, 571 rotational constraint elements are synthesized with se- 627 sion M increase the three displacement stroke. we just need to use three 572 rial chains of flexure primitives. These freedom and 628 Henein, S translational constraint elements to remove all translaof flexu 573 7 constraints elements form a catalogue of basic build- 629 Conclusions tions. If we choose the BB design in Fig 7(e) for all three on Com 574 ing blocks for constructing more complex flexure mech- 630 631 delft, T 575 in We have also with examples the translational constraints, we obtain the design Aanisms. list of commonly useddemonstrate flexure primitives is first catego6.2 Synthesis of serial chains with freedom elementsshown 632 Hopkins, use of building withand design of hybridfreedom strucrized. A these complete list ofblocks rotational translational Fig 11. The functional body A can rotate about its 576 cen633 straint 577 tures forisany combinations of freedom. The importance elements synthesized with parallel connections of these634 As we discussed previously, we can use a serial chain of tional C ter relative to the base body B while its translations578areof this work lies in the fact that it formalizes the type primitives. In duality, translational and rotational constraint635 Hopkins, R-joints and P-joints to construct a freedom space with any 579 synthesis (type/toplogy selection) of flexure mechanisms constrained. and co combination of DOF. For instance, a serial chain of three Pelements are synthesized with serial chains of flexure prim-636 580 which has been an ad hoc process. It also paves the way Institut As another example, we would to designIfa parjoints can make up a freedom space of like three translations. itives. These freedom and constraints elements form a cat-637 581 towards to design automation and systematic invention 638 Hopkins, we use the double parallel translations. flexure structure for P-joint alogue basic building blocks for constructing more com-639 allel structure with three Wethejust need toof newofflexure 582 machinery. mechat 592 593

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design, we obtain a serial PPP design with three decoupled plex flexure mechanisms. We have also demonstrate with exuse three rotational constraint elements to remove all translations. See Fig. 10. Of course, one can further increase amples the use of these building blocks with design of hyrotations. If we stiffness choosewith theassembling BB design in serial Fig 8(a) Mechanical the structural multiple chains for all brid structuresSciences for any combinations of freedom. The imthree rotational constraints, obtain the design shown of PPP in parallel (Awtar et al.,we 2011). portance of this work lies in the fact that it formalizes the in Fig 12. The functional body A can translate in all diwww.mech-sci.net/4/263/2013/ Mech. Sci., 4, 263–277, 2013 rections while its rotations are constrained.

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Figure 11. A parallel kinematic chain of three translational 600 constraints results in a design with three rotations about the 601 center of the functional body A. The body B are fixed. 602

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Blanding, D. L.: Exact Constraint: Machine Design U Kinematic Processing, ASME Press, New York, NY, 1 Brouwer, D., de Jong, B., and Soemers, H.: Design and m 276 H.-J. Su and C. Yue: Type synthesis 603 eling of a six DOFs MEMS-based precision manipula 604 Precision Engineering, 34, 307 – 319, 2010. Culpepper, M. L. and Anderson, G.: Design of a low-cost nano605 Chen, S.-C. and Culpepper, M. L.: Design of a six-axis mi manipulator which utilizes a monolithic, spatial compliant mech606 scale nanopositioner-µHexFlex, Precision Engineering anism, Precis. Eng., 28, 469–482, 2004. 607 J. S. and 314Jones, – 324, 2006. Dai, J. R.: Interrelationship between screw systems and Culpepper, corresponding reciprocal and applications, 608 M. L. systems and Anderson, G.: Mech. Design of a Mach. Theory, 36, 633–651, 2001. 609 cost nano-manipulator which utilizes a monolithic, Dai, J. S. and Jones, J. R.: A Linear Algebraic Procedure in Ob610 tial compliant mechanism, Precision Engineering, 28, B B taining Reciprocal Screw Systems, J. Robot. Syst., 20, 401–412, 611 2003. – 482, 2004. 612 Dai, J.and S. Hunt, and K. Jones, J. R.: Davidson, J. K. H.: Robots andInterrelationship Screw Theory: Appli-between sc 613 and systems and ap cations systems of Kinematics andcorresponding Statics to Robotics,reciprocal Oxford University A Press, New York, NY, 2004. 614 cations, Mechanism and Machine Theory, 36, 633 – Dong, J., Yao, Q., and Ferreira, P. M.: A novel parallel-kinematics 615 2001. mechanism for integrated, multi-axis nanopositioning: Part 2: 616 Dai, J. S. and Jones, J. R.: A Linear Algebraic Procedur B Dynamics, control and performance analysis, Precis. Eng., 32, 617 Obtaining Reciprocal Screw Systems, Journal of Rob 20–33, 2008. 618 Systems, 20, Techniques 401–412,for 2003. Figure A parallel kinematic kinematic chain of three constraints con-Hale, L. C.: Principles and Designing Precision MaFigure 12. 12. A parallel chain of rotational three rotational result in a design with three translations along the direction indichines, Ph.D. thesis, MIT, Cambridge, MA, 619 Davidson, J. K. and Hunt, K.1999. H.: Robots and Screw T straints result in a design with three translations along the cated in the figure. The body B are fixed. The body A is the funcHenein, S.:ory: Flexures: simply subtle,oftutorial on the design of flexure620 Applications Kinematics and Statics to Robo direction indicated in the figure. The body B are fixed. The mechanisms, Second International Symposium on tional body. Compliant 621 Oxford University Press, New York, NY, 2004. body A is the functional body. Mechanisms, IFToMM/ASME CoMe 2011, delft, the Nether622 Dong, J., Yao, Q., and Ferreira, P. M.: A novel para lands, 2011. 623 kinematics mechanism forFlexure integrated, multi-axis nan Hopkins, J.: Modeling and Generating New Constraint Eltype synthesis (type/toplogy selection) of flexure mecha624 sitioning: Part 2: Dynamics, control and performance a ements, in: Proc. of the 12th EUSPEN International Conference, nisms which has been an ad hoc process. It also paves the freedom elements is synthesized with parallel connec- 625424–428, Stockholm, Sweden, 2012. ysis, Precision Engineering, 32, 20 – 33, 2008. way towards to design automation and systematic invention Hopkins, J. B.: Design of parallel flexure systems via freedom and tionsof of newthese flexureprimitives. machinery. In duality, translational and 626 Hale, L. C.: Principles and Techniques for Designing P topologies (FACT), MS, Massachusetts Institute of rotational constraint elements are synthesized with se- 627constraint sion Machines, PhD Thesis, MIT, Cambridge, MA, 1 Technology, Cambridge, MA, 2007a. 628 Henein, S.: Flexures: simply subtle, tutorial rial chains of flexure primitives. These freedom andHopkins, J. B.: Design of flexure-based motion stages for mecha- on the de of flexure-mechanisms, Second International Acknowledgements. material is based upon work supvia freedom, actuation and constraint topologies Sympos constraints elements This form a catalogue of basic build- 629tronic systems ported by the National Science Foundation under Grant No: Compliant Mechanisms, IFToMM/ASME thesis, Massachusetts Institute of Technology, CoMe 2 ing blocks for constructing more complex flexure mech- 630(FACT),onPh.D. CMMI-1161841. Any opinions, findings, and conclusions or Cambridge, MA, 2007b. 631 delft, The Netherlands, 2011. anisms. We have expressed also demonstrate with examples recommendations in this material are those of the theHopkins, J. B. and Culpepper, M. L.: Synthesis of multi-degree of Hopkins, J.: Modeling and Generating New Flexure C author(s) do not necessarily reflect the views of of the National use of theseand building blocks with design hybrid struc- 632freedom, parallel flexure system concepts via Freedom and Con633 straint Elements, in: Proc. of the 12th EUSPEN Inte Science Foundation. straint Topology (FACT) – Part I: Principles, Precis. Eng., 34, tures for any combinations of freedom. The importance 634 tional Conference, pp. 424–428, Stockholm, Sweden, 2 2010a. of this work lies in the fact that it formalizes the type 635259–270, Edited by: G. Hao Hopkins, J. B.: Design parallelofflexure systems via free Hopkins, J. B. and Culpepper, M. L.:ofSynthesis multi-degree of synthesis (type/toplogy selection) Reviewed by: two anonymous referees of flexure mechanisms 636freedom, parallel flexure system concepts via freedom and con- Massachus and constraint topologies (FACT), M.S., (FACT). Part II: Practice, Precis. Eng., 34, 271– 2007a. which has been an ad hoc process. It also paves the way 637straint topology Institute of Technology, Cambridge, M.A., 278, 2010b. towards to design automation and systematic invention 638 Hopkins, J. B.: Design of flexure-based motion stages References Hopkins, J. B. and Culpepper, M. L.: A screw theory basis for quanof new flexure machinery. 639 systems freedom, actuation and titative mechatronic and graphical design tools thatvia define layout of actuators Awtar, S. and Slocum, A. H.: Constraint-based design of parallel kinematic xy flexure mechanisms, J. Mech. Des.-T. ASME, 129, Mechanical 816–830,Sciences 2007. 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