A Rotation Joint for Shape Morphing Space Truss Structures

8/17/2019 A Rotation Joint for Shape Morphing Space Truss Structures

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) 1277–1284 doi:10.1088/0964-1726/16/4/040

A rotational joint for shape morphing

space truss structuresA Y N Sofla, D M Elzey and H N G Wadley

Materials Science and Engineering Department, University of Virginia, Charlottesville,

VA 22904, USA

E-mail: [email protected] (A Y N Sofla)

Received 28 November 2006, in final form 10 June 2007Published 5 July 2007Online at stacks.iop.org/SMS/16/1277

AbstractA rotational joint is introduced for use in shape morphing or deployablespace truss structures. The joint is a chain mechanism comprising up to sixpivoted linkages that provide a compact mechanism for the connection of upto six structures at a node. The total number of degrees of freedom for aconstrained joint mechanism with six links is found. A closed chain model isthen used to determine the angle between the adjacent links as the structure ischanged during shape morphing. The limiting pivot rotation angle isestablished by physical interference and is determined for a model problem.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Space truss structures consist of truss members (struts) joined

at nodes [1]. The nodes restrict axial movement of the truss and

impede truss rotation at the nodes. This rotational resistance

could be a significant limitation for actuated truss structures

in which some trusses are replaced by linear displacement

actuators. To illustrate, consider a simple, tetrahedral three-

dimensional (3D) truss structure supported on one of its

triangular sides by a foundation, figure 1(a). The number of

inextensional mechanisms, M , of such a pin-jointed structure

is given by Maxwell’s stability criterion, which relates the

number of non-foundation joints ( j ) and non-foundation trussmembers (b) to M [2]. The three-dimensional form of the

criterion applicable to the tetrahedral shown in figure 1 can be

written:

M = 3 j − b. (1)For the pin-jointed truss structure shown in figure 1(a), b = 3and j = 1 (one of the triangles consisting of three nodes andthree bars is the foundation). In this case, M = 0, whichdefines a statically determinate structure. If an external load

is therefore applied to the structure, the force in every strut can

be determined from the equations of mechanical equilibrium

at the nodes. The structure is also kinematically determinate

since thelocationof the joints can also be uniquely determined.

If M has a positive value, the structure is kinematicallyindeterminate and the location of one or more nodes can then

no longer be uniquely determined by the length of the trusses.

Removing one of the non-foundation trusses of the

statically determinate structure, figure 1(b), results in a

structure that behaves as a mechanism with one degree of

freedom ( M = 1). Replacing the removed truss with anextensional actuator can restore the static determinacy of the

tetrahedral truss structure and allows it to then exhibit shape

changing behavior provided the nodes marked 1 and 2 in

figure 1(b) have a rotational degree of freedom. Such structures

have attracted recent interest for shape morphing applications

since they do not develop states of self stress when shape

changes are caused to occur by longitudinal deformations of

the linear actuator [3]. They are therefore candidates for high

authority, shape morphing systems.The very simple statically determinate system shown in

figure 1(a) is of limited utility for smart structures. However,

Hutchinson et al [4] have identified a 3D kagome ‘plate like’

truss structure in which the trusses are connected by pin joints

with no rotational resistance, figure 2. This structure has been

converted to a shape morphing truss plate by replacing some of

the trusses with linear actuators.

Unfortunately the trusses in such space truss structures

are connected at nodes by either welding or by mechanical

fastening (for example by screwing the trusses to a spherical

node) and they are unable to rotate about the node.

Deformation of the truss (induced by the actuators) then results

in either elastic or/and plastic deformation near these joints.The storage of strain energy at the joints limits the shape

morphing capability of the plate and repeated cycling of the

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A Y N Sofla et al

Figure 1. A tetrahedral space truss unit attached on one side to arigid foundation. (a) A pin-jointed statically determinate truss.(b) Elimination of one truss member changes the structure to amechanism provided nodes 1 and 2 permit rotational degrees of freedom.

Figure 2. Example of a 3D kagome plate for shape morphing platestructure [4]. The welded node construction causes stresses todevelop in the trusses during actuation. This limits actuationauthority and increases the susceptibility of the structure to failure byfatigue.

structure’s shape could lead to premature failure by fatigue.

Shape morphing structures of this type could therefore be

improved if the trusses were connected using nodes that had

a low rotational resistance.

In general, connecting more than two moving trusses or

linkages to a single joint complicates the mechanical design

and associated fabrication processes. For instance, two rods

may be connected by a spherical joint with three degrees of

freedom. Such ball joints are relatively easy to manufacture

and designs are available as spherical bearings. However

adding only one more link to a two-rod ball joint in such

a way that the new member possesses at least one degree

of freedom makes the design considerably more complicated.The earliest multilink joint design appears to be that used in

variable geometry truss (VGT) structures [5]. A VGT structure

is a statically determinate truss with tetrahedral, octahedral or

other simple geometric unit cells. The truss structure’s shape

can be varied by changing the length of some of the struts.

Two of the trusses in the VGT structure’s joint are directly

hinged to a main hub while four others are hinged to two

pivots which can rotate with respect to the hub [6]. Such a

joint is an example of an open chain mechanism [7]. The truss

rotation angle is limited by the physical interference between

truss members at the joint. The trusses connections are usually

offset at thejoint and do notthereforeintersectat a single point,

making analysis of the joint quite complicated.A few other approaches for multi-truss rotational connec-

tion have been proposed. Stewart platform type mechanisms

Figure 3. A pivot pin passing through the center of two concentricspherical shells enables free rotation of the two shells about the pin.The outer radius of the inner shell is assumed equal to the innerradius of the outer shell.

utilize a different multi-truss articulating joint [8]. In this ap-

proach six links are joined together between a platform and abase using a parallel mechanism design. The joint has a sig-

nificant offset because the links are connected using a series of

hinges. Scissors mechanisms have been exploited by Hamlin

and Sanderson [9] to connect three trusses. The links in their

concentric joint intersect at a point so there is no offset, but the

joint is difficult to fabricate. Bosscher and Ebbert-Uphoff [10]

suggested two spherical joint mechanisms to implement mul-

tiple co-located spherical joints. Their ‘scalable’ design is an

improvement over Hamlin’s mechanism in that it reduces the

number of intermediate links, but the design then loses strength

and becomes impracticalfor many applications. Song etal [11]

proposed a spherical node that is capable of connecting three

or more trusses. In this approach each truss passes through ahole in an outer spherical shell and is joined to an inner sphere.

However, the joint only offers a very small range of rotations

for the trusses because of the small clearance of the truss mem-

bers inside the holes.

Here we introduce a compact joint design suitable for

constructing shape morphing or deployable truss structures. It

allows the connection of up to six truss structures (or more

generally any six rigid bodies (linkages)) using low rotation

resistance pivots. This hexa-pivotal (H-P) joint has been

fabricated at the millimeter size scale and appears to offer

adequate strength for structural morphing applications over

a wide range of size scales. Interrelationships between the

rotational angles of the joint mechanism to determine unknownangles between the joint links are derived. Examples are

provided to illustrate the application of the analysis.

2. Hexa-pivotal joint design

Consider the rotation of two concentric spheres with respect

to each other about a pivot pin that connects two spheres by

penetration through their common center, figure 3.

The pin itself is free to rotate about its axis. A hexa-pivotal

joint consists of spherical shell elements (here called links)

which can similarly rotate with respect to each other without

interference. Figure 4(a) illustrates a pair of overlapped links

cut from a pair of spherical shells and connected by a pivot pinthat permits rotation of the links about the pin axis. The pivot

pin passes through the common center of curvature of the two

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A rotational joint for shape morphing space truss structures

Figure 4. (a) Two concentric spherical links can be cut fromspherical shells. Both links can freely rotate about the common pivot

pin though the axial movement of the links is restricted (viamechanical retainers). (b) A hexa-pivotal joint consisting of sixspherical links ( L1 − L6) held in position using six pivoting pins(P1 − P6). Two trusses (not shown) can be attached to each link. The joint is shown with all the pivot pins lying in a common plane.

links. To ensure that the links remain in sliding contact, they

are fabricated from spherical shells, where the outer radius of

the smaller sphere is equal to the inner radius of the larger one.

The spherical shell links are free to rotate about the pin but are

restricted from moving axially along the pin by axial retainers

(e.g. a cotter or ring and grooved pin).

A spatial closed chain mechanism (the links form a closed

loop) can be formed by sequentially connecting six sphericalshell links, figure 4(b). This closed mechanism is referred

to as a hexa-pivotal joint. Although a closed chain H-P joint

is considered here, the joint can be used as an open chain

mechanism by disengaging any of the pivots or by using a

similar mechanism with fewer than six links. An open chain

joint has more total degrees of freedom but loses strength.

Although a joint can conceptually consist of two or more

spherical links, here we are interested in a joint with six links:

three inner and three outer links with a common radius of

curvature at their contacting surfaces. The H-P joint can be

used to connect 12 truss members, labeled T 1–T 12, figure 5.

The trusses can be fixed to the spherical shell links by either

mechanical attachment or by welding. The pivot pins can alsobe used as truss members although they must be permitted to

freely rotate about their longitudinal axis.

Figure 5. A hexa-pivotal joint with 12 truss members ( T 1–T 12)

attached. In practice the trusses can be rigidly fixed to the links bymechanical fasteners or by welding.

3. Kinematic analysis

A closed chain hexa-pivotal joint is a constrained mechanism

in which six links are connected by six revolute joints (pivots).

Moving any link in the mechanism, as during shape morphing,

can cause the other links to be reconfigured. The new shape

of the closed chain H-P joint can be uniquely defined from the

rotation angles of adjacent pairs of links about the common

revolute joint of each pair. Therefore, in the shape morphing

applications using the H-P joint, the shape of the structurecan be controlled by actuators which control the rotation angle

between the adjacent links.

The number of actuators acting on a single H-P joint

must be equal to the total degrees of freedom of the H-P

joint. Although there are six angular rotations between the

adjacentspherical linkages (because there is a total of six pivots

in the H-P joint), the number of degrees of freedom of the

closed chain mechanism needs to be determined. For a general

spherical linkage in which each of the links is constrained to

rotate about the same fixed point in space, the mobility, F , is

given by the number of links and revolute joints [7]:

F

=3(n

−1)

−2 p, (2)

where n is the number of links and p is the number of revolute

joints. For the H-P joint, n = 6 and p = 6, so the total degreesof freedom of the assembly is three. Therefore, controlling

three (out of six) rotation angles of the adjacent spherical

links can uniquely determine the H-P joint shape. The three

unknown rotation angles of the H-P joint need to be found to

fully determine the joint shape.

The six spherical links in H-P joint are labeled L1,

L 2, . . . , L 6, figure 4(b). An adjacent pair of links, e.g. L i−1and L i , are hinged by a pivot pin, Pi . A flat joint shape,

figure 4(b), results when the axis of all six pivot pins lie in

the plane. The relative rotation angle of a pair of adjacent

links, L i−1 and L i , about their common revolute joint, Pi , isdenoted θ i (note that L6 precedes L1 in the closed chain and

the angle is denoted θ 1; see figure 6). This angular rotation is

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Figure 6. The pivot rotation angle, θ 1, refers to the relative rotationof two adjacent links, L 6 and L 1, about their common pivot pin, P1,with respect to the flat shape. All pivot pin axes initially lay in aplane (shown by the hexagon) in the flat shape.

represented with respect to the flat shape where all the rotation

angles are defined to be zero. Figure 6 shows an example of

a reconfigured H-P joint. The rotation angle, θ 1, results from

rotation of links L 6 and L 1 about pivot P1 from the flat shape

(the hexagon in figure 6). The joint mechanism of figure 6

is the same as the one in figure 4(b). For clarity, the truss

members are not shown in figure 6. However, two holes can

be seen in each spherical linkage that can be used to connect

truss members to each link.

Denavit and Hartenberg have suggested a notation

for systematically representing a chained mechanism (by

sequentially labeling and assigning a coordinate system to the

links) in order to relate their kinematic motions [12]. The

system of coordinates W i ( x i , yi , zi ) attached to link L i is

illustrated in figure 7(a), in the original notation proposed by

Denavit and Hartenberg. For the system of coordinates, W i( x i , yi , z i ) in figure 7(a), z i is the direction along the axis of a

revolute joint, x i lies along the common normal from z i−1 to ziand axis y i completes the right-handed coordinate frame. The

geometric relationship between a pair of successive systems of

coordinates, W i−1 and W i , is defined by using general Denavit–Hartenberg (D–H) parameters (ai , si , αi , θ i ) [12], where aiis the distance between the origin of W i and W i−1 measuredalong x i ; si is the distance between x i−1 and x i measured along zi−1; αi is the angle between z i−1 and z i , measured in a right-hand sense about x i ; and θ i is the angle between x i−1 and x imeasured in a right-hand sense about z i−1; see figure 7(a).

The general D–H notation, figure 7(a), is applied to an H-

P joint in figure 7(b). The axis z i−1 is selected to be the axisof pivot Pi , which allows rotation at the pivot Pi be identified

as θ i . The rotation angle of links L1 and L 6 about the axis of

pivot P1 is θ 1, figure 7(b) (recall that the axis identified as z 6precedes the axis z1 in the closed chain).

The H-P joint is designed so that all pivot pins always passthrough the center of curvature of the links. Because all the

links are concentric, the axes of pivot pins, z1 to z6, always

intersect ensuring that the D–H parameters ai and si are zero:

ai = 0, si = 0 for i = 1, 2, . . . , 6. (3)

In addition, the in-plane (in the flat shape plane) angles

between any pair of adjacent pivot pins, αi , are always 60◦.The interrelationships of the six rotation angles play animportant role in using the H-P joint for the design of shape

morphing or adapting structures. The interrelationships of the

angles at six pivots and, consequently, the joint shape can be

determined when one or more of the angles are changed using

the D–H approach [12]. Using the D–H notation [12], the

[4 × 1] vector of coordinates is defined as

v = [ x , y, z, 1]T. (4)

The [4 × 4] transformation matrix between W i−1 and W i is

Ai =

cos θ i

−sin θ i cos αi sin θ i sin αi ai cos θ i

sin θ i cos θ i cos αi − cos θ i sin αi ai sin θ i0 sin αi cos αi si0 0 0 1

(5)

such that,

vi−1 = Ai vi . (6)The D–H parameters for the H-P joint are

ai = 0, si = 0 and αi = 60◦.

The transformation matrix for the H-P joint then becomes

Ai =

cos θ i

−12

sin θ i√

32

sin θ i 0

sin θ i 12 cos θ i −√ 32 cos θ i 00 1

2

√ 3

2 0

0 0 0 1

. (7)

Because all the fourth row and column’s elements, except

for the element (4, 4) which is 1, have become zero the

analysis can be simplified by defining a new equivalent [3× 3]transformation matrix, Bi :

Bi =

cos θ i − 12 sin θ i√

32

sin θ i

sin θ i12

cos θ i −√

32

cos θ i

0 12

√ 3

2

. (8)

The new vector of coordinates, u , and the transformation rule,equation (10), are then written as

u = [ x , y, z]T (9)

ui−1 = Bi ui . (10)The transformation rule, equation (10), can be used to relate

the coordinate of the neighboring links such that

u1 = B2u2, u2 = B3u3, . . . , u6 = B7u7. (11)

A closed chain mechanism requires the last link in the chain to

be the same as the first one; therefore B7

= B1 and u 7

= u1.

By using equation (11) we will have

u1 = B2 B3 B4 B5 B6 B1u1. (12)

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Figure 7. (a) The Denavit–Hartenberg (D–H) notation is used to label and assign coordinates to chain mechanisms [ 12]. (b) The D–Hcoordinates and parameters corresponding to the H-P joint cross section.

Figure 8. Rotation of the linkages for (a) the 123-configuration with(0, 0, γ ), (b) the 123-configuration with (0, β, 0), and (c) the 134configuration with (0, 0, γ ).

Taking the symmetry of the chain into account and by using

equation (12),

B1 B2 B3 B4 B5 B6 =

1 0 0

0 1 0

0 0 1

. (13)

The dependent system of equation (13) defines the interrela-

tionship between the H-P joint parameters and can be used to

determine the three unknown rotation angles. The examples

later in the paper further clarify this method.Although the total number of degrees of freedom of the

joint is three, several configurations are possible for a joint

having three known (input) angles. For instance, an H-P

joint with three known pivot angles (α,β,γ ) is called a 123-

configuration if (θ 1 = α, θ 2 = β and θ 3 = γ ) while a joint with three known angles (α,β,γ ) is referred to as a

234-configuration joint if (θ 2 = α, θ 3 = β and θ 4 = γ ).The second joint is identical to the first but rotated 60◦ aboutthe x axis (the x axis in figure 7(b) is perpendicular to the

paper). However, a joint in which two of three known angles

(α , β , γ ) are non-adjacent (e.g. a 124-configuration, (θ 1 = α,θ 2 = β and θ 4 = γ )) has different shape from a 123-configuration with the same known angles. Expectedly, a joint

with three known angles (α , β , γ ) with no known adjacent

angles, such as the 135-configuration (θ 1 = α, θ 3 = β andθ 5 = γ ), is totally different from the 123-configuration or 124-configuration with the same known angles. This leads to three

families of configurations for any set of three known angles

(α , β , γ ), as summarized in table 1. As shown in table 1,

20 unique arrangements of pivots may accept a set of threeknown rotation angles (α , β , γ ). The shape and orientation

of the H-P joint, therefore, can be identified by three known

rotation angles and the corresponding pivots (123, 124, . . .,

etc) amongst the 20 possible arrangements represented in

table 1.

The following examples illustrate application of the

analysis in order to determine the shape (configuration) of the

joint when the rotation angles of three of six pivots are known

as input.

Example 1.

The rotation of two pairs of links about their common

pivots is restricted while a motor controls the rotation angle

of a third pair in a closed chain H-P joint. The output rotationangle of the motor is set at 30◦; therefore, the three knownangles are (0, 0, 30), which means α = 0◦, β = 0◦, γ = 30◦.

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Figure 9. Unknown pivot rotation for (a) the 123-configuration, with

fixed θ 1 = θ 2 = 30◦ and variable θ 3, (b) the 124-configuration withθ 1 = θ 2 = 30◦ versus θ 4 and (c) the 135-configuration with fixedθ 1 = θ 3 = 30◦ and variable θ 5.

If these known angles belong to the pivots # 1, #2, and #3 (123-

configuration) then we have

θ 1 = 0◦, θ 2 = 0◦, θ 3 = 30◦. (14)

By substituting (14) into the system of equation (13), the

unknown rotation angles of pivots in the joint are found to

be θ 4 = 0◦, θ 5 = 0◦, θ 6 = 30◦. The motion of this jointis simulated in figure 8(a), where two of the rotation angles

are set to zero and a motor rotates its adjacent links withrespect to each other. Zero angle for a pivot means that the

relative rotation of the two adjacent links connected by the pin

Figure 10. Three symmetric arrangement of a hexa-pivotal joint. Aflat shape with all the pins in a plane (a), hinge arrangement wheretwo pins rotate relative to two others (b) and tripod shape whereodd-number pivots have equal but different angle than the

even-number pivots.

Table 1. Three families of configurations with known pivot rotationangles (α , β , γ ). Any three-digit number represents those pivots withthe known rotation angles. For example, 123 means θ 1 = α, θ 2 = βand θ 3 = γ while 124 corresponds to θ 1 = α, θ 2 = β and θ 4 = γ .

Description of the family

Configuration of the input anglesof the joint

Number of possibleconfigurationsin the family

Family-1: the threepivots with knownangles are neighbors

123, 234, 345,456, 561, 612

6

Family-2: only two

of the pivots withknown angles areneighbors

124, 125, 235,

236, 346, 341,451, 452, 562,563, 613, 614

12

Family-3: none of the pivots withknown angles areneighbors

135, 246 2

The total number of possible pivotarrangementfor a set of threeknown rotationangles(α , β , γ )

20

is restricted such that the pin and two neighboring pins always

lie in a plane. Figure 8(b) illustrates motion of the joint for

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Figure 11. (a) The range of rotation is limited by geometrical interference. In this design the interference of the external spherical linkage andthe connected truss member of the adjacent linkage identifies the limiting rotation. (b) A spherical linkage design.

a 123-configuration with (0, β, 0). In this figure the motor

operates on the middle of three consecutive pivots. Figure 8(c)

shows a 134-configuration with angles (0, 0, γ ), a family-2

configuration.

Example 2.

To illustrate the effect of the H-P joint family type on the

joint shape, consider three H-P joints with 123-configuration,

124-configuration and 135-configuration in which the rotation

angles at two pivots are known and a motor operates at a

third pivot to change the rotation angle of the links, which are

connected by the third pivot.

For the H-P joint with 123-configuration, the rotation

angle at pivot P1 is assumed to be fixed at 30◦ and the rotationangle at pivot P2 at 30

◦ while the rotation angle of links atpivot P3 can vary between 0 and 90

◦ (known). This joint istherefore referred to as a 123-configuration with known angles,

(30◦, 30◦, γ ). To determine the joint shape, three unknownangles (θ 4, θ 5 and θ 6) are found by solving equation (13). The

angles (θ 4, θ 5 and θ 6) are plotted in figure 9(a) as a function of

rotation angle at pivot P3, θ 3.

Now, consider a 124-configuration joint with θ 1 = 30◦and θ 2 = 30◦ (similar to the first joint), while for this case therotation angle at pivot P4 varies between 0 and 90

◦. Figure 9(b)plots θ 3, θ 5 and θ 6 as a function of θ 4 when θ 1 = θ 2 = 30◦.

Finally, for a 135-configuration joint with θ 1

= θ 3

= 30◦

and variable θ 5, unknown rotation angles θ 2, θ 4 and θ 6 areplotted in figure 9(c).

4. Experimental implementation

A prototype H-P joint is fabricated by connecting stainless

steel links by tubular pivots, figure 10. The links are made

by machining. Three interesting arrangements of the joint

are demonstrated using the prototype in figure 10. In the

flat shape, figure 10(a), all pivots lie in a plane whereas

in the hinge configuration, figure 10(b), two of the pins

(three corresponding linkages) are hinged with respect to two

pins (the other three linkages) about the remaining alignedpins. Examples for a hinge arrangement are example 1

and figure 8(a). Figure 10(c) shows another symmetric

arrangement called the tripod shape, where all odd-number

pivots have same rotation angle but different than the even-

number ones. As an example, for a 135 arrangement with input

angles (30◦, 30◦, 30◦), theother threeangles are determined bymeans of figure 9(c) as θ 2 = θ 4 = θ 6 = −7.5◦.

Rotation at each pivot is ultimately restricted by

interference of the joint parts. For example, for the design in

this paper which carries two truss members at each linkage in

addition to the extended pivot pins, the interference happens

between an external spherical linkage and the truss member of

the adjacent link, figure 11(a).

The limiting rotation angle θ L can be calculated using the

geometrical relationship, equation (15), for the design shownin the figure 11(b).

θ L= sin−1

sin(m) (h/a)2 + sin2(m)

+sin−1

sin(n)√

(h/a)2 + sin2(n)

(15)

where a =√

R2 − h2, m = ( π6 − r t

a ) and n = ( π

6 − r p+d

a ). As

illustrated in figure 10(b), r t is the radius of the truss member,

r p the radius of the pivot pin, d is the distance of the hole from

edge of the linkage, R the internal radius of external spherical

linkage and h is half of the vertical distance between the truss

connection center holes in the linkage.

5. Conclusion

Deployable and shape morphing 3D truss structures require

efficient joint designs which offer adequate degrees of

freedom and are easily manufactured. The hexa-pivotal joint

mechanism consists of six spherical linkages connected by

six pivots to form a closed chain mechanism. The spherical

linkages are designed to freely rotate about the common pivot.

The outer surface of internal linkages contacts the internal

surface of the neighboring outer linkages at common pivots

without rotational interference, providing a design for strength

and accuracy. The joint mechanism has a total of three degrees

of freedom leading to interrelationship of the six rotationangles at the pivots. By knowing three of the angles as

input, the other three rotation angles are determined. Twenty

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A Y N Sofla et al

different arrangements for the joint mechanism are possible

for any set of three known angles, making the joint capable

of having several different shapes. The hexa-pivotal joint is

easy to manufacture and can be fabricated to the desired size

and strength. Having no offset between the links is another

advantage of the joint which makes it easily programmable forprecise positioning applications.

References

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[2] Maxwell J C 1864 On the calculation of the equilibrium andstiffness of frames Phil. Mag. 27 294

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[4] Hutchinson R G, Wicks N, Evans A G, Fleck N A andHutchinson J W 2003 Kagome plate structures for actuation Int. J. Solids Struct. 40 6969–80

[5] Padmanabhan B, Arun V and Reinholtz C F 1992 Closed forminverse kinematic analysis of variable-geometry trussmanipulators ASME J. Mech. Des. 114 438–43

[6] Hornett H C 1994 Joint for a variable geometry truss andmethod of constructing same Patent ApplicationLAR-15136-1 Serial No. 08/325,723 (Nasa LangleyResearch Center)

[7] McCarthy J M 2000 Geometric Design of Linkages(New York: Springer)

[8] Zanganeh K E and Angeles J 1994 Instantaneous kinematicsand design of a novel redundant parallel manipulator Proc. IEEE Int. Conf. on Robotics and Automation (San Diego,

CA) pp 3043–8[9] Hamlin G J and Snaderson A C 1998 Tetrobot: A Modular

Approach to Reconfigurable Parallel Robotics

(Norwell, MA: Kluwer)[10] Bosscher P and Ebbert-Uphoff I 2003 A novel mechanism for

implementing multiple collocated spherical joints Proc. IEEE Int. Conf. on Robotics and Automation (Taipei,

Taiwan) pp 336–41[11] Song S, Kwon D and Kim W S 2001 Spherical joint for

coupling three or more links together at one point US Patent Application 20010002964

[12] Denavit J and Hartenberg R S 1955 A kinematic notation for

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