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A Novel Reconfigurable 3-URU
Parallel Platform
Luca Carbonari(B) , David Corinaldi, Matteo Palpacelli, Giacomo Palmieri,
and Massimo Callegari
Polytechnic University of Marche, 60131 Ancona, AN, Italy
Abstract. A novel parallel robot stemming from the 3-SRU (sphericalrevolute-universal) under - actuated joints topology is presented in this
paper. The conceptual design here proposed takes advantage of a reconfigurable universal joint obtained by locking, one at a time, different
rotations of a spherical pair by means of an automatized system. Such
local reconfiguration causes a slight modification of the robot legs mobility which is enough to provide the end-effector with different kinds of
motion. The first part of the paper is dedicated at formally demonstrating the motion capabilities offered by the 3-URU kinematics; in the second part, a mechanical solution for the realization of the reconfigurable
joint is shown.
Keywords: Parallel kinematics machine · Algebraic geometry · Study’s
parameters · Lockable joint
Due to the increasing demand of flexibility in industrial manufacturing, the
research in the field of robotics is focuest on versatile machines, which are able
to readily adapt themselves to new functions. A possible answer to such quest
of flexibility is certainly given by the definition of smart workstations based on
conventional industrial machines equipped with an on-board intelligence and
with some degrees of adaptive capacity; a more recent answer is addressed by
the development of novel mechanical solutions, thus the realization of reconfigurable manipulators. A reconfigurable or metamorphic manipulator is a machine
that is able to change its end-effector mobility according to a local change of
kinematics. Kinematotropic linkages can be also mentioned when the change of
mobility is originated by a gain of an infinitesimal mobility [1–3], namely when
a singular configuration is reached. Modular manipulators with serial kinematics are reconfigurable when they are assembled from elemental building blocks,
which can be combined to produce the desired sequence of joints [4,5].
Reconfigurable joints may have a great impact on parallel kinematics
machines (PKMs). For this family of robots, in fact, a little modification of
the legs kinematics may lead to substantial modifications of their mobility.
c Springer International Publishing AG 2018
C. Ferraresi and G. Quaglia (eds.), Advances in Service and Industrial Robotics,
Mechanisms and Machine Science 49, DOI 10.1007/978-3-319-61276-8 8
L. Carbonari et al.
This fact may help to overcome some of the drawbacks that are intrinsic in
PKMs, such as their small workspace. Moreover, the reconfiguration can be
often performed by acting some specific joints without disassembly or assembly
operations [6–9]: planar, pure translational, pure rotational or more complex
motions result from a manual intervention. Some examples of lockable joints,
instead, are provided in the literature [10,11]: the key idea in this kind of joints
is that of realizing higher kinematics pair from the composition of elementary
kinematic pairs and then locking them alternately. Nonetheless, the change of
configuration of the joints is often managed manually, without thinking about
the behaviour of the manipulator during the transition.
The researchers at the Robotics Lab of the Polytechnic University of Marche,
in Ancona, studied the possibility to realize a reconfigurable parallel robot
exploiting the peculiarities of the Cylindrical-Prismatic-Universal leg joints
topology [12]. The researches yield the design of a reconfigurable tripod, able
to perform motion of pure rotations and pure translations through the reconfiguration of the universal joints performed by means of a special lockable device
[13]. Notwithstanding the metamorphic capabilities shown, the 3-CPU architecture also owns some intrinsic drawbacks which pushed the research towards
different topology. In particular, the main issue that affects the 3-CPU is that of
needing a reconfiguration of the passive joints which connect the end-effector to
the limbs structure: such characteristic implies that the lockable device must be
part of the leg structure, with consequent complications in terms of mechanical
design (for limitation of weight and size of the device), and in terms of actual
payload reduction.
Such drawbacks led authors to investigate different topologies [14] capable
of performance similar to those of the 3-CPU architecture. In particular this
manuscript presents a novel concept for the design of a reconfigurable sphericaltranslational platform which exploits a 3-URU (universal - revolute - universal)
joints architecture, stemming from the under-actuated 3-SRU (spherical - revolute - universal) topology. The key concept of reconfigurability is presented in
the first part of the paper. Later, the kinematics of the two proposed joints configurations is analysed with the aim of both demonstrating the actual mobility
of the platform, and deriving the equations necessary for the solution of the
position kinematics problems. The approach is that of the algebraic geometry
which has proved to be successful in the case of the 3-CPU PKM.
Robot Description
The main objective of this paper is to investigate the possibility to realize
a reconfigurable multi-purpose parallel manipulator starting from the 3-SRU
under-actuated topology. To this aim, a design that takes advantage of a reconfigurable actuated joint used to connect each leg to the chassis of the robot is
proposed. Such joint is realized as a combination of revolute pairs; a locking
system allows to alternatively lock one of the two revolute joints, giving the
machine different 3-URU kinematic configurations which correspond to different
types of mobility.
A Novel Reconfigurable 3-URU Parallel Platform
Fig. 1. Reconfigurable 3-URU kinematic architecture.
The kinematic architecture of each leg is composed of a Spherical-RevoluteUniversal joints chain. The spherical pair, which must be reduced to a universal
joint in order to guarantee the functionality of the machine, connects the first
body of each leg to the chassis. The second link is connected to the first one by
a revolute joint. At last, a universal joint closes the kinematics chain connecting
the end-effector to the leg. The mutual arrangement of the joints, which is of
paramount importance for determination of the mobility of the moving platform,
is shown in Fig. 1. As visible, the three spherical pairs are located so that their
centres lie on the axes of the fixed reference frame {0} at the same distance, called
a, from the origin. The three rotations that compose each pair is realized through
three perpendicular revolute joints, whose axes concur in a point (namely the
centre of the spherical couple). The three axes are disposed as follows:
– the first revolute is coaxial to the respective axis of the reference frame {0}. In
the i-th leg local frame {l1,i } the unit vector describing such axis is {l1,i } r1,i =
100 ;
– the second revolute is perpendicular to the first one and it is directed along
the first body of the limb. In the frame {l2,i } it is {l2,i } r2,i = 0 0 1 ;
– the third and last rotation is perpendicular to both the first and the second
at home configuration. With respect to frame {l3,i } it is {l3,i } r3,i = 0 1 0 .
It is considered here that at home configuration, frames {l1,i }, {l2,i }, and {l3,i } are
coincident and for leg 1 they have the same orientation of the absolute frame {0}.
The locking mechanism is used for this application to produce two different
configurations, here called Joints Configuration A and Joints Configuration B (see
Fig. 1). In configuration A, the first rotation of the spherical pair is allowed and, in
L. Carbonari et al.
particular, it represents the actuated degree of freedom of the i-th leg. The second
rotation is locked, while the third one is free. In configuration B the first rotation
is locked, the second one is actuated and the third one is free. As demonstrated in
the following, such choice allows generating motions of pure rotations with Joint
Configuration A and motions of pure translations with Joint configuration B without further modification to the arrangement of the passive joints.
The remainder of the kinematic chain of each leg is composed by a revolute
joint, parallel to the last revolute of the reconfigurable spherical joint at a distance b, and by a universal joint that connects the second link of the leg to the
end-effector. As demonstrated by Palpacelli et al. in [14], the arrangement of the
last joint of the chain is also crucial for definition of the end-effector mobility.
In particular, for the reconfigurable 3-URU parallel robot the terminal universal
joint is disposed so that the first rotation is parallel to previous revolute joint at
a distance c, thus it is perpendicular the the leg plane πi (identified by point Ai ,
Bi , and Ci ). The second rotation lies on πi and is perpendicular to the previous
one; on the end effector, such axes are mutually perpendicular and concurrent
at the origin of the moving reference frame {1}. Moreover, such axes coincide
with the axes of {1}. With respect to such reference frame, the three attachment
points Ci are identified by vectors {1} C1 = 0 0 −d , {1} C2 = −d 0 0 , and
C3 = 0 −d 0 .
The locking mechanism described in the last part of this manuscript is dedicated at alternatively allowing the motion of the first and the second revolute
joint. A similar solution was already proposed by authors in [12]. Nonetheless,
the mechanical solution proposed for the 3-CPU robot was non negligibly different since it was used to modify the mobility of a passive universal joint. In the
present case, instead, the reconfiguration implies that the actuated joints of the
robots are to be changed. The mechanical design of the reconfigurable joint is
obviously affected by that. On the one hand, the need of connecting an actuator
to different joints relatively complicates the problem. On the other hand, the fact
that the lockable device lies on the chassis and it is not moving together with
the end effector relaxes the design parameters in terms of weight and dimension,
and consequently in terms of costs.
Mobility Analysis
In order to fully represent the attitude of the mobile platform, the Study’s parameters are chosen for the parametrization of its kinematics. This well known
notation [15,16] was introduced by Study who used a superabundant set of 8
parameters in the projective space of dimension 7 (P7 ) to describe the Euclidean
transformations. The 8 parameters [x0 : x1 : x2 : x3 : y0 : y1 : y2 : y3 ] are related
through a quadric polynomial equation (namely the Study quadric) and through
a metric equation which ensures that the transformations actually represent rigid
body motions.
A Novel Reconfigurable 3-URU Parallel Platform
Thus, the key idea is to map the Euclidean space through a one to one
representation x in the projective space P7 :
x : 0 T1 ∈ SE (3) → x ∈ P7
If the homogeneous notation is used, the transformation 0 T1 describing the
pose of the robot platform looks like:
T1 =
0 x20 + x21 + x22 + x23
⎡ 2
x0 + x21 − x22 − x23 2 (x1 x2 − x0 x3 ) 2 (x1 x3 + x0 x2 )
R1 = ⎣ 2 (x1 x2 + x0 x3 ) x20 − x21 + x22 − x23 2 (x2 x3 − x0 x1 ) ⎦
2 (x1 x3 − x0 x2 ) 2 (x2 x3 + x0 x1 ) x20 − x21 − x22 + x23
2 (−x0 y1 + x1 y0 − x2 y3 + x3 y2 )
p1 = ⎣2 (−x0 y2 + x1 y3 + x2 y0 − x3 y1 )⎦
2 (−x0 y3 − x1 y2 + x2 y1 + x3 y0 )
The 8 parameters [x0 : x1 : x2 : x3 : y0 : y1 : y2 : y3 ] in (2)–(4) are called Study’s
parameters; a set of such parameters corresponds to only one transformation in
the Euclidean space and allows to uniquely define the configuration of a rigid
body, e.g. the mobile platform, when the Study’s quadric equations are satisfied:
x0 y0 + x1 y1 + x2 y2 + x3 y3 = 0
x20 + x21 + x22 + x23 = 0
If the inequality in (5) is assumed equal to an arbitrary constant, the expression
becomes a normalizing equation that ensures that matrix (2) is effectively a non
singular transformation matrix; typically it is assumed x20 + x21 + x22 + x23 = 1.
Thus, Eq. (5) yields the two polynomials:
σ1 : x0 y0 + x1 y1 + x2 y2 + x3 y3 , σ1 = 0
σ2 = 0
σ2 : x20 + x21 + x22 + x23 − 1,
The vanishing sets of polynomials σ1 and σ2 relate the 8 Study parameters and
represent an intrinsic characteristic that the parameters set must fulfil. It is then
necessary to provide a complete algebraic description of the kinematics of the
parallel platform taking advantage of the constraints imposed by the architecture
of each joints arrangement. As shown in the following, a distinction can be made
among the constraint equations that compose the kinematic model of the moving
platform based on their dependency on the robot actuation parameters. Some
equations, in fact, can be directly gathered by legs kinematic architecture. Other
ones, instead, take into consideration the effect of the motion of actuated joints
on the whole robot. Subsequently, such sets of equations are deducted for both
the translational and rotational 3-URU manipulators.
L. Carbonari et al.
Fig. 2. Kinematics of the i-th leg for (a) spherical and (b) translational 3-URU architectures.
The Spherical 3-URU Robot
In order to obtain a complete description of robot kinematics, Study quadric
Eq. (5) must be put together to the algebraic relations which are characteristic
of the specific platform. Firstly, the relations which are peculiar of the legs architecture are introduced. This procedure is also useful to formally demonstrate the
mobility of the platform within its workspace. It is not trivial, in fact, to figure
out what kind, or kinds, of motions a parallel manipulator might be capable of.
As visible in Fig. 2, the configuration A is built so as to constrain the axis
of the last revolute joint on the plane πi . Equivalently, one can say that the
line li , identified by the axis of such joint, has an intersection point with all the
lines that lie on the plane πi , for example the axis of the first revolute of the
reconfigurable universal joint, here called mi . The two lines can be parametrized
through their direction plus a passage point, as:
li : αi 0 R1 {1} Ci + 0 p1 mi : βi {l1i } r1,i
with αi , βi ∈ R. It must be remarked that, in joints configuration A, the direction
of the unit vector {l1i } r1,i is fixed. For the leg constraint to be fulfilled it might
exist a set of αi and βi ∈ R so that:
li = mi → αi 0 R1 {1} Ci + 0 p1 = βi {l1i } r1,i
or, in matrix form:
R1 {1} Ci
{l1 } 0 − i r1,i
p1 αi βi 1 = 0
A non trivial solution of this last homogeneous equation can be found only letting
the determinant of the respective matrix vanish. Substituting (3) and (4) for 0 R1
A Novel Reconfigurable 3-URU Parallel Platform
and 0 p1 in (9), three polynomials in the Study parameters can be found, one for
each leg. After simplification [17], the three constraints can be expressed by the
varieties of the following polynomials:
g1 : x0 y2 + x1 y3 + x2 y0 + x3 y1 g1 = 0
g2 : x0 y3 + x1 y2 + x2 y1 + x3 y0 g2 = 0
g3 : x0 y1 + x1 y0 + x2 y3 + x3 y2 g3 = 0
The vanishing set of polynomials g1 , g2 , and g3 , together with Study quadric
Eq. (6) are still not sufficient to fully describe the kinematics of the 3-URU platform with joints configuration A, since a set of actuation dependent equations
are also needed. Nonetheless, such relations provide all the necessary information
about the mobility of the end-effector [16,18]. With the aim of performing a complete kinematics analysis of the robot it is possible to decompose the polynomial
ideal in its primary components. Each primary components, then, is peculiar of a
particular branch of robot mobility (the interested reader can approach authors’
previous work [12]). In any case, the main target of this section is to demonstrate
the ability of the manipulator to perform motions of pure rotations and, to this
aim, it is sufficient to ensure that the conditions provided by such mobility are
compatible with the constraints provided by the three legs. In particular, it can
be easily verified that the general transformation matrix 0 T1 describes a pure
rotation when the four Study parameters y0 , y1 , y2 , and y3 are null. In this case
the translation vector 0 p1 vanishes, denoting a pure rotational behaviour of the
end-effector. Also, it is as much easy to verify that such condition fulfils both
the constraints expressed by Eq. (10), and the Study quadric σ1 = 0.
Being demonstrated that the pure rotational behaviour is allowed by the
Joints Configuration A, the homogeneous transformation matrix can be simplified consequently as:
T1 =
01×3 x20 + x21 + x22 + x23
A set of simple equations dependent on the actuated variables can now be
derived. The line li can be used again: given that li lies on plane πi , it follows
that such line is perpendicular to the unit vector {l3,i } r3,i , whose expression in
the fixed reference frame is only function of the angle qi of rotation of the first
revolute joint. For the three legs it is:
sin q2
cos q3
r3,1 = ⎣cos q1 ⎦ r3,2 = ⎣ 0 ⎦ r3,3 = ⎣ sin q3 ⎦
sin q1
cos q2
The perpendicularity of vectors (12) with the lines li is respected when:
rT3,i 0 Ri {1} Ci = 0
L. Carbonari et al.
Substitution of the simplified homogeneous transformation matrix (11) in (13)
yields the three polynomials:
g4 : −2 cos q1 (x0 x1 − x2 x3 ) + sin q1 x20 − x21 − x22 + x23 d g4 = 0
g5 : −2 cos q2 (x0 x2 − x1 x3 ) + sin q2 x20 + x21 − x22 − x23 d g5 = 0
g6 : −2 cos q3 (x0 x3 − x1 x2 ) + sin q3 x20 − x21 + x22 − x23 d g6 = 0
Now the kinematic problem of the 3-URU robot for motions of pure rotations,
namely the joints configuration A, is fully posed as the variety of the polynomial
ideal HA = g4, g5, g6, σ2 . It is worth remarking that the solutions of such
system of equations intrinsically belong to the Study projective space being
verified the quadric σ1 .
The Translational 3-URU Robot
At this point, the kinematic problem of the 3-URU spherical platform is completely defined, and its mobility is verified. Similar steps should now be made
for Joints Configuration B so as to demonstrate its ability to perform motions
of pure translation. To this aim, two sets of equations similar to (13) and (14)
are to be found also for this joint arrangement.
Even though they are very similar, joints configurations A and B offer different sets of polynomials because of the different mobility that each one of the legs
provide the manipulator with. Even in this case, equations that do not depend
on actuated joints variables can be expressed through the co-planarity of two
lines. As well visible in Fig. 2.b, lines li and ni both lie on the leg plane πi . In
this case, the ni line equation is ni : γi l2i r2,i + Ai . As done for configuration A,
it is possible to state that the coplanarity constraint
lines li and
ni is ful
filled when the determinant of the matrix 0 R1 {1} Ci −{l2i } r2,i 0 p1 − Ai
vanishes. The three polynomials, one for each leg, coming from the determinant
of such matrix can be simplified as done for (10). The result is the following:
h1 : a (x0 x1 − x2 x3 ) + 2 (x1 y1 + x2 y2 ) h1 = 0
h2 : a (x0 x2 − x1 x3 ) + 2 (x2 y2 + x3 y3 ) h2 = 0
h3 : a (x0 x3 − x1 x2 ) + 2 (x1 y1 + x3 y3 ) h3 = 0
For the general transformation (2) to describe motions of pure translations, the
Study parameters representing the rotation should assume the values x0 = 1,
x1 = 0, x2 = 0 and x3 = 0. Under these conditions, the rotation matrix 0 R1
is an identity matrix, and the translation vector becomes 0 p1 = −2 y1 y2 y3 .
Moreover, the normalization equation σ2 = 0 is fulfilled, while the Study quadric
σ1 = 0 is verified when y0 = 0. It is easy to verify that such values also belong
to the vanishing sets of the constraint polynomials (15), demonstrating that the
pure translational motion is actually allowed by the 3-URU platform with Joints
Configuration B.
At last, three actuation dependent polynomial equations can be found as
done for Configuration A. To this aim, it is possible to observe that the vector
A Novel Reconfigurable 3-URU Parallel Platform
connecting the reconfigurable universal joint and the origin of the moving frame
{1}, namely the vector 0 p1 − Ai , lies on the leg plane πi . Therefore, it is also
perpendicular to the unit vector r3,i whose coordinates only depend on the
rotation of the i-th actuated joint. Being the second rotation of the reconfigurable
joint active while the first locked, the three r3,i vectors assume the values:
− sin q1
cos q3
r3,1 = ⎣ cos q1 ⎦ r3,2 = ⎣− sin q2 ⎦ r3,3 = ⎣ 0 ⎦
cos q2
− sin q3
Then, the three constraint conditions rT3,i 0 p1 − Ai = 0 yield the three polynomials:
h4 : sin q1 (2y1 + a) + 2y2 cos q1 h4 = 0
h5 : sin q2 (2y2 + a) + 2y3 cos q2 h5 = 0
h6 : sin q3 (2y3 + a) + 2y1 cos q3 h6 = 0
whose vanishing set provide a complete description of the 3-URU kinematics
when Joints Configuration B is used.
Reconfigurable Spherical Joint
The functional design that is proposed here is based on a bevel gear coupling, as
shown in Fig. 3. Such solution is similar to the one proposed by authors in [13].
Anyhow, in the present case the actuation of the locking mechanism has been
re-conceived in order to allow the actuation of the bevel gear D. The motion of
the sliding cursor C reconfigures the spherical joint alternatively locking one of
the rotation of the spherical joint as described in the following.
The cursor C is driven by an actuator in two different positions, which provide
the joint as many different working modes. The cursor has a cylindrical shape
with an external splined shaft and an internal splined hub at the top. In the
Configuration I (as in Fig. 3.a) the cursor C mechanically engages the splined
hub of the mechanical fork B. In this case B, C and D globally behave like a
rigid body. Due to the solid rotation of the fork B together with the bevel gear
D, the second fork E does not rotate with respect to B. The body F remains
free of rotating since it is not constrained anyhow. Considering the reference
frame represented in Fig. 3.a, such configuration turns to allow rotations about
axes x and y as required by the 3-URU Joints Configuration A.
If the cursor is moved downward to Configuration II (Fig. 3.b), the splined
shaft engages with both bodies A and B. Being the body A fixed to ground, the
rotation of B remains locked as well. The rotation of the actuated bevel gear
is transmitted instead to the second fork E which represents now the actuated
degree o freedom of the leg. Thus, configuration II allows an actuated rotation
about axis z and a free rotation around y, exactly as required by the robot Joints
Configuration B.
L. Carbonari et al.
Fig. 3. Functional scheme of the actuated reconfigurable joint for spherical motions
(a) and translational motions (b) of the 3-URU.
This manuscript introduced the mobility analysis of a novel reconfigurable parallel platform, based on the 3-URU kinematic architecture. The study has been
performed by means of tools deriving from algebraic geometry, which allowed the
demonstration of some interesting reconfigurability aspects of the robot. In particular, the paper shows that the 3-URU platform allows motions of both pure
rotations and pure translations through a little reconfiguration of the legs architecture performed by means of a spherical reconfigurable joint. To this aim, a set
of equations for each joints configuration was found exploiting the legs mobility.
Such equations, written in polynomial form thanks to a particular projective
approach, were simplified. Then, it was verified that the conditions necessary
for the manipulator to perform particular motions (namely spherical in the one
case, translational in the other) also fulfil the constraints imposed by legs mobility. The information obtained with such strategy allowed the simplification of a
general transformation matrix for the two cases of pure rotations and translations. This step brought to simple relations also for computation of the forward
and inverse kinematics problems through two further sets of equations. At last,
a possible solution for the mechanical realization of the spherical reconfigurable
joint was illustrated.
A Novel Reconfigurable 3-URU Parallel Platform
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