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 structures
A 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 2007
Published 5 July 2007
Online at stacks.iop.org/SMS/16/1277
Abstract
A rotational joint is introduced for use in shape morphing or deployable
space truss structures. The joint is a chain mechanism comprising up to six
pivoted linkages that provide a compact mechanism for the connection of up
to six structures at a node. The total number of degrees of freedom for a
constrained joint mechanism with six links is found. A closed chain model is
then used to determine the angle between the adjacent links as the structure is
changed during shape morphing. The limiting pivot rotation angle is
established 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 threedimensional (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 truss
members (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 = 3
and j = 1 (one of the triangles consisting of three nodes and
three bars is the foundation). In this case, M = 0, which
defines 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 the location of the joints can also be uniquely determined.
If M has a positive value, the structure is kinematically
indeterminate and the location of one or more nodes can then
no longer be uniquely determined by the length of the trusses.
0964-1726/07/041277+08$30.00
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 an
extensional 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
© 2007 IOP Publishing Ltd Printed in the UK
1277
A Y N Sofla et al
Figure 1. A tetrahedral space truss unit attached on one side to a
rigid foundation. (a) A pin-jointed statically determinate truss.
(b) Elimination of one truss member changes the structure to a
mechanism provided nodes 1 and 2 permit rotational degrees of
freedom.
Figure 2. Example of a 3D kagome plate for shape morphing plate
structure [4]. The welded node construction causes stresses to
develop in the trusses during actuation. This limits actuation
authority and increases the susceptibility of the structure to failure by
fatigue.
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 the joint and do not therefore intersect at a single point,
making analysis of the joint quite complicated.
A few other approaches for multi-truss rotational connection have been proposed. Stewart platform type mechanisms
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Figure 3. A pivot pin passing through the center of two concentric
spherical shells enables free rotation of the two shells about the pin.
The outer radius of the inner shell is assumed equal to the inner
radius of the outer shell.
utilize a different multi-truss articulating joint [8]. In this approach six links are joined together between a platform and a
base using a parallel mechanism design. The joint has a significant 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 multiple 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 impractical for many applications. Song et al [11]
proposed a spherical node that is capable of connecting three
or more trusses. In this approach each truss passes through a
hole 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 members 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 unknown
angles 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 pin
that permits rotation of the links about the pin axis. The pivot
pin passes through the common center of curvature of the two
A rotational joint for shape morphing space truss structures
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 by
mechanical fasteners or by welding.
3. Kinematic analysis
Figure 4. (a) Two concentric spherical links can be cut from
spherical shells. Both links can freely rotate about the common pivot
pin though the axial movement of the links is restricted (via
mechanical retainers). (b) A hexa-pivotal joint consisting of six
spherical links ( L 1 − L 6 ) 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 spherical
shell 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 also
be used as truss members although they must be permitted to
freely rotate about their longitudinal axis.
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 structure
can 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
adjacent spherical 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 degrees
of 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 L 1 ,
L 2 , . . . , L 6 , figure 4(b). An adjacent pair of links, e.g. L i−1
and 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 , is
denoted θi (note that L 6 precedes L 1 in the closed chain and
the angle is denoted θ1 ; see figure 6). This angular rotation is
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A Y N Sofla et al
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 an
important 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 Wi−1 and Wi is
⎡
Figure 6. The pivot rotation angle, θ1 , refers to the relative rotation
of 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 a
plane (shown by the hexagon) in the flat shape.
cos θi
⎢ sin θi
Ai = ⎣
0
0
− sin θi cos αi
cos θi cos αi
sin αi
0
sin θi sin αi
− cos θi sin αi
cos αi
0
⎤
ai cos θi
ai sin θi ⎥
⎦
si
1
(5)
such that,
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, θ1 , 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 Wi (xi , yi , z i ) 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, Wi
(xi , yi , z i ) in figure 7(a), z i is the direction along the axis of a
revolute joint, xi lies along the common normal from z i−1 to z i
and axis yi completes the right-handed coordinate frame. The
geometric relationship between a pair of successive systems of
coordinates, Wi−1 and Wi , is defined by using general Denavit–
Hartenberg (D–H) parameters (ai , si , αi , θi ) [12], where ai
is the distance between the origin of Wi and Wi−1 measured
along xi ; si is the distance between xi−1 and xi measured along
z i−1 ; αi is the angle between z i−1 and z i , measured in a righthand sense about xi ; and θi is the angle between xi−1 and xi
measured 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 HP joint in figure 7(b). The axis z i−1 is selected to be the axis
of pivot Pi , which allows rotation at the pivot Pi be identified
as θi . The rotation angle of links L 1 and L 6 about the axis of
pivot P1 is θ1 , figure 7(b) (recall that the axis identified as z 6
precedes the axis z 1 in the closed chain).
The H-P joint is designed so that all pivot pins always pass
through the center of curvature of the links. Because all the
links are concentric, the axes of pivot pins, z 1 to z 6 , always
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v i−1 = Ai v i .
(6)
The D–H parameters for the H-P joint are
ai = 0,
si = 0
αi = 60◦ .
and
The transformation matrix for the H-P joint then becomes
⎡
cos θi
⎢ sin θi
Ai = ⎢
⎣
0
0
− 12 sin θi
1
2
√
1
2
3
sin θi
2√
− 23 cos θi
√
3
2
0
0
cos θi
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 :
√
⎡
⎤
3
cos θi − 12 sin θi
sin θi
2√
⎢
⎥
1
Bi = ⎣ sin θi
(8)
cos θi − 23 cos θi ⎦ .
2
0
1
2
√
3
2
The new vector of coordinates, u , and the transformation rule,
equation (10), are then written as
u = [x, y, z]T
(9)
u i−1 = Bi u i .
(10)
The transformation rule, equation (10), can be used to relate
the coordinate of the neighboring links such that
u 1 = B2 u 2 , u 2 = B3 u 3 , . . . , u 6 = B7 u 7 .
(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 = u 1 .
By using equation (11) we will have
u 1 = B2 B3 B4 B5 B6 B1 u 1 .
(12)
A rotational joint for shape morphing space truss structures
Figure 7. (a) The Denavit–Hartenberg (D–H) notation is used to label and assign coordinates to chain mechanisms [12]. (b) The D–H
coordinates 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 134
configuration with (0, 0, γ ).
Taking the symmetry of the chain into account and by using
equation (12),
1 0 0
B1 B2 B3 B4 B5 B6 = 0 1 0 .
(13)
0 0 1
The dependent system of equation (13) defines the interrelationship 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 123configuration if (θ1 = α , θ2 = β and θ3 = γ ) while a
joint with three known angles (α, β, γ ) is referred to as a
234-configuration joint if (θ2 = α , θ3 = β and θ4 = γ ).
The second joint is identical to the first but rotated 60◦ about
the 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, (θ1 = α ,
θ2 = β and θ4 = γ )) has different shape from a 123configuration 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 (θ1 = α , θ3 = β and
θ5 = γ ), is totally different from the 123-configuration or 124configuration 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 three
known 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 rotation
angle of the motor is set at 30◦ ; therefore, the three known
angles are (0, 0, 30), which means α = 0◦ , β = 0◦ , γ = 30◦ .
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A Y N Sofla et al
Figure 10. Three symmetric arrangement of a hexa-pivotal joint. A
flat shape with all the pins in a plane (a), hinge arrangement where
two pins rotate relative to two others (b) and tripod shape where
odd-number pivots have equal but different angle than the
even-number pivots.
Table 1. Three families of configurations with known pivot rotation
angles (α, β, γ ). Any three-digit number represents those pivots with
the known rotation angles. For example, 123 means θ1 = α , θ2 = β
and θ3 = γ while 124 corresponds to θ1 = α , θ2 = β and θ4 = γ .
Description of
the family
Figure 9. Unknown pivot rotation for (a) the 123-configuration, with
fixed θ1 = θ2 = 30◦ and variable θ3 , (b) the 124-configuration with
θ1 = θ2 = 30◦ versus θ4 and (c) the 135-configuration with fixed
θ1 = θ3 = 30◦ and variable θ5 .
If these known angles belong to the pivots # 1, #2, and #3 (123configuration) then we have
θ 1 = 0◦ ,
θ 2 = 0◦ ,
θ3 = 30◦ .
(14)
By substituting (14) into the system of equation (13), the
unknown rotation angles of pivots in the joint are found to
be θ4 = 0◦ , θ5 = 0◦ , θ6 = 30◦ . The motion of this joint
is simulated in figure 8(a), where two of the rotation angles
are set to zero and a motor rotates its adjacent links with
respect to each other. Zero angle for a pivot means that the
relative rotation of the two adjacent links connected by the pin
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Family-1: the three
pivots with known
angles are neighbors
Family-2: only two
of the pivots with
known angles are
neighbors
Family-3: none of
the pivots with
known angles are
neighbors
Configuration of
the input angles
of the joint
Number of
possible
configurations
in the family
123, 234, 345,
456, 561, 612
6
124, 125, 235,
236, 346, 341,
451, 452, 562,
563, 613, 614
135, 246
12
2
The total number of 20
possible pivot
arrangement
for a set of three
known rotation
angles
(α, β, γ )
is restricted such that the pin and two neighboring pins always
lie in a plane. Figure 8(b) illustrates motion of the joint for
A rotational joint for shape morphing space truss structures
Figure 11. (a) The range of rotation is limited by geometrical interference. In this design the interference of the external spherical linkage and
the 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 rotation
angle at pivot P2 at 30◦ while the rotation angle of links at
pivot P3 can vary between 0 and 90◦ (known). This joint is
therefore referred to as a 123-configuration with known angles,
(30◦ , 30◦ , γ ). To determine the joint shape, three unknown
angles (θ4 , θ5 and θ6 ) are found by solving equation (13). The
angles (θ4 , θ5 and θ6 ) are plotted in figure 9(a) as a function of
rotation angle at pivot P3 , θ3 .
Now, consider a 124-configuration joint with θ1 = 30◦
and θ2 = 30◦ (similar to the first joint), while for this case the
rotation angle at pivot P4 varies between 0 and 90◦ . Figure 9(b)
plots θ3 , θ5 and θ6 as a function of θ4 when θ1 = θ2 = 30◦ .
Finally, for a 135-configuration joint with θ1 = θ 3 = 30◦
and variable θ5 , unknown rotation angles θ2 , θ4 and θ6 are
plotted 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 aligned
pins. 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 evennumber ones. As an example, for a 135 arrangement with input
angles (30◦ , 30◦ , 30◦ ), the other three angles are determined by
means of figure 9(c) as θ2 = θ4 = θ6 = −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 θL can be calculated using the
geometrical relationship, equation (15), for the design shown
in the figure 11(b).
sin(m)
sin(n)
−1
θL = sin
+ sin−1 √
(h/a)2 + sin2 (n)
(h/a)2 + sin2 (m)
√
(15)
( π6
h2,
rt
)
a
( π6
rp +d
).
a
−
m=
−
and n =
−
As
where a =
illustrated in figure 10(b), rt is the radius of the truss member,
rp 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.
R2
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 rotation
angles 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 for
precise positioning applications.
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