Mechanism and Machine Theorypercro1.sssup.it/~antony/papers/gabardi2019.pdf214 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 The manipulator

Mechanism and Machine Theory 133 (2019) 211–228

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Mechanism and Machine Theory

journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

An optimization procedure based on kinematics analysis for

the design parameters of a 4-UPU parallel manipulator

M. Gabardi ∗, M. Solazzi , A. Frisoli

Scuola Superiore Sant’Anna, TeCIP Institute, PERCRO Laboratory, Italy

a r t i c l e i n f o

Article history:

Received 6 August 2018

Revised 29 October 2018

Accepted 21 November 2018

Keywords:

Parallel kinematics

Singularity analysis

Kinematics optimization

Screw theory

a b s t r a c t

The paper presents the kinematics analysis of a 4-UPU fully parallel manipulator and a

numerical procedure for its optimization. The 4-UPU kinematics is featured by four UPU

legs, moreover the prismatic joint of each leg is supposed to be actuated. The end-effector

has 4 degrees of freedom (Dofs), the three translations and the rotation along the direc-

tion perpendicular to the base platform. Singularity configurations have been analytically

determined and the analysis of both actuation and constraint Jacobian by screw theory has

been performed. Finally, a numerical procedure for the optimization of the design param-

eters of the manipulator is presented. The optimization procedure has been carried out

using adimensional parameters in order to generalize the obtained results. The presented

optimization procedure allows to define the ratios between the geometrical features of the

manipulator which maximize a generic performance parameter in a designed workspace

by avoiding all the possible singularity conditions. The obtained results are useful for the

design of all the manipulators based on the 4-UPU kinematics. In particular, the case study

of the design of a wearable fingertip haptic device is presented and discussed.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Thanks to their good characteristics of dynamics, precise positioning and stiffness, parallel manipulators play a fun-

damental role in many practical applications. Also in the scientific environment parallel manipulators have been widely

investigated, in particular, lower mobility manipulators having three degrees of freedom (3-DoF) fully-translational or fully-

rotational are very common. A particular class of 3-DoF reconfigurable parallel kinematics machines able to perform alter-

natively different types of motions, in particular pure rotational motion and pure translational motion, are described in [1] .

Moreover, in [2] , the reconfigurations analysis of a 3-DoF parallel mechanism is performed revealing its transition config-

urations, which corresponds with singular configurations. Furthermore, concerning the synthesis of fully-rotational manip-

ulators, both in [3] and [4] the analysis of non-overconstrained 3-DoF orientational manipulators is performed, whereas in

[5] a novel family of spherical parallel manipulators is presented, definition of such manipulators is based on the analysis

of the infinitesimal motion of 3-RRPRR kinematic architecture.

The number of studies on 4-DoF manipulators is relatively small instead. Possible lower-mobility parallel manipulators

having four Dofs have been synthesized in [6] by using of the special Plücker coordinates for describing the displacement of

the output link of a limb. In particular, structure synthesis of 4-DoF parallel manipulators with full-cycle mobility has been

investigated in [7] , by using the constraint-synthesis method as systematic method for the synthesis of the manipulators.

∗ Corresponding author.

E-mail addresses: [email protected] (M. Gabardi), [email protected] (M. Solazzi), [email protected] (A. Frisoli).

https://doi.org/10.1016/j.mechmachtheory.2018.11.015

0094-114X/© 2018 Elsevier Ltd. All rights reserved.

212 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

In [8] , the classes of 4-DoF and 5-DoF parallel manipulators with identical serial limbs are investigated, moreover, in [9] ,

[10] and [11] other studies on 4-Dofs fully parallel manipulators have been performed.

A particular class of 4-DoF mechanisms are Schoenflies-type manipulator. Schoenflies displacement (or motion) is a rigid

body motion consisting of linear motion in three dimensional space plus one orientation around an axis with fixed direction.

The possible general architectures of mechanical generators of Schoenflies motion are defined in [12] , whereas fully isotropic

parallel mechanisms, able to perform Schoenflies motion, have been investigated by Carricato et al. in [13] . Two different

Schoenflies manipulators based on parallel kinematics have been presented in [14] and [15] . In [16] the type synthesis of

leg structures leading to parallel manipulators able to perform 3T1R (3 translations, 1 rotation) motion with variable axis is

presented. 4-RUU parallel manipulator’s class has been analyzed in [17] revealing, by reconfiguration analysis, two Schoen-

flies modes (4-DoF) and one lower dimension operation mode (2-DoF). Furthermore, a 3T1R kinematic architecture called

Pantopteron is described in [18] . In the Pantopteron kinematics, three pantograph linkages are used to achieve amplifica-

tion of the movements between the actuators and the platform displacements, thus making the mechanism suitable for

fast positioning applications, such as pick and place applications. Concerning precision of robots based on 3T1R fully paral-

lel kinematics, in [19] , a simple method for error analysis is proposed, where the problem of error amplification is deeply

investigated.

Among Schoenflies-type manipulators the 4-UPU fully parallel kinematics presents a simple architecture with four iden-

tical legs and a good workspace-to-encumbrance ratio. The mentioned features make it suitable for several applications,

ranging from automatic machining applications up to the realization of haptic interfaces. A 4-UPU manipulator has been

described in [20] , where both the kinematic analysis and the singularity analysis of the Jacobian have been performed.

In the study of parallel manipulators the kinematics analysis can be very challenging, and tools like the screw theory can

be helpful for performing kinematic studies in a more efficient way [21] . A usage example of screw theory can be found

in [22] , where a 4-DoF parallel kinematics has been analyzed by screw theory and its singularity loci have been deter-

mined. Moreover, in parallel kinematics leg singularity has to be often considered and separately investigated, thus, it is a

kind of singularity non detectable by Jacobian analysis. An accurate analysis of the leg singularities in linear-actuated sym-

metrical spherical parallel manipulators is described in [23] . Examples of analytical parallel kinematics solution via screw

theory, where also mechanism design parameters optimization in a maximum singularity-free workspace is considered, are

described both in [24] and [25] . The first is applied to a 3-rRPS parallel manipulator whereas the second describes a 4-DoFs

2T2R parallel manipulator having different kinematics for each leg. Moreover, in [25] homogeneity of the jacobian matrix

used in the optimization procedure is considered.

Within this paper we present the complete kinematic analysis of the 4-UPU parallel manipulator by screw theory. Both

the Jacobian singularities and the singularity configurations not identifiable by the Jacobian analysis have been determined.

Moreover, a numerical procedure for the optimization of the geometrical parameter of the manipulator in a singularity-free

workspace is proposed. In the proposed treatise, the choice of the performance parameter used to perform the optimization

of the 4-UPU is decided taking into account the design specifications of a wearable fingertip haptic interface.

A wearable fingertip haptic interface is a device mounted on the users finger which is able to display cutaneous cues

on his fingertip while he is experiencing virtual objects manipulation or performing the exploration of surfaces in a virtual

environment.

In Fig. 1 an example of fingertip haptic interface is shown [26] . The device is based on a 3-RSR fully parallel kinematics

having 3 DoF. As it is possible to understand from Fig. 1 , a fingertip haptic interface needs to be small and lightweight,

moreover, high wearability level is required. Good results in terms of wearability can be achieved by adopting kinematics

that surrounds the fingertip in order to equally distribute the encumbrance around the finger. Since the 4-UPU allows to fit

the user’s finger in between the legs and uniformly surround the user’s fingertip, from the encumbrance distribution point

of view, such a kinematics results suitable for the development of a 4 DoF wearable fingertip haptic interface. Furthermore,

being a parallel manipulator, the 4-UPU is able for both precision positioning and good stiffness characteristic, which are

useful qualities to apply precise stimuli on the user’s fingertip.

In general, since the particular application of the device, small dimensions for both the actuation system and the struc-

ture are required. To this purpose, a numerical optimization procedure is presented. The aim of the optimization procedure is

to maximize a generic functional, describing the kinetostatic performance of the manipulator, as function of the fixed dimen-

sions of the manipulator. In this study the mean volume of the force manipulability ellipsoid (FME) in a given workspace

has been chosen as illustrative optimization objective. The FME is assumed as local performance parameter, wheres the

mean of the FME is used for a global evaluation of the manipulator in the given workspace. Such an optimization allows

to determine, among the evaluated geometries, the manipulator’s geometry that on average requires lower forces at the

actuation system to generate a generic force at the end effector. Thus, it allows to minimize the overall dimensions of the

actuation system, that usually represents the main part of the total weight of a wearable haptic interface.

In the following sections both the kinematic analysis of the 4-UPU and the proposed numerical optimization procedure

are presented and discussed. In detail, the solution of the kinematics is described in Section 2 , whereas the singularity anal-

yses of the constraint wrenches, the legs configurations and the Jacobian are respectively performed in Sections 3.1 , 3.2 , and

3.3 . In Section 4 , the optimization of the geometrical dimensions of the manipulator is carried out. The optimization process

has been performed using adimensional design parameters to identify the ratios between the geometrical dimensions of the

manipulator and a characteristic length. In this way, the obtained results hold for every possible 4-UPU manipulator, and

they are useful for the design process of devices based on such a kinematics. Finally, the results obtained from the opti-

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 213

Fig. 1. A 3-Dof fingertip haptic device realized by using a 3-RSR fully parallel kinematics [26] .

Fig. 2. Schematic representations of the 4-UPU kinematics, (a) single leg and (b) full kinematics. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

mization of the kinematics on a designed workspace suitable for the realization of a wearable fingertip haptic interface are

presented.

2. Kinematic modeling of the 4-UPU parallel manipulator

Let’s consider a general configuration of the 4-UPU parallel manipulator ( Fig. 2 ). Every leg of the manipulator is composed

by a prismatic joint placed between two universal joints ( Fig. 2 (a)). The four prismatic joints are the only actuated joints of

the manipulator.

214 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

The manipulator is provided with four base joints, two of them aligned along the y axis and placed at a distance D 1 from

origin O , whereas, the two others are aligned to the x axis at a distance D 2 from O . Each of the four legs is connected trough

its last joint to a different corner of the rectangular coupling part. The last joint rotation axis of each leg is perpendicularly

connected to the coupler surface. The coupler is a l x h rectangle, with h ≤ l . Referring to Fig. 2 (b), the projection ( E ) of the

center of the rectangular coupler on the plane passing through the centers of the universal joints at the coupler side, is

taken as reference for the position of the coupler in the x-y plane.

The axis of the base joint of each leg is aligned to the z axis, moreover, for each leg, parallelism between the rotation

axes of the second and fourth joints is always verified. Since the last joint ( z 5 i placed at O 5 i ) of each leg is attached to

the coupler in order to keep the joint axis always parallel to the z axis, each leg constraints a coupler rotation in the x-y

plane. Thus, the four legs together overconstrain the coupler in the x-y plane and only the coupler rotation parallel to the z

axis is allowed. Whereas, concerning the translations, all the three translations in the space are allowed, therefor the whole

manipulator has four degrees of freedom.

At this point, it is clear that the 4-UPU kinematics belongs to the category of the overconstrained parallel mechanisms. As

described in [27] mobility of overconstrained parallel mechanisms cannot be analytically evaluated by means of the classical

formulation of the Grübler-Kutzbach criterion. Therefor, in order to evaluate the mobility of the 4-UPU it is necessary to use

the Modified Grübler–Kutzbach criterion. Applicability and generality of such a criterion is well described in [28] . According

to the Modified Grübler–Kutzbach criterion, mobility (m) of the manipulator can be evaluated as:

m = d(n − g − 1) +

g ∑

i =1

f i + ν − ξ (1)

where, d = 6 is the dimension of the motion space of all the motion members of the mechanism, n = 10 is the number of the

links in the mechanism, g = 20 is the total number of the joints in the mechanism, f i is the number of DoF corresponding to

the i th kinematic pair (f = 1 for prismatic joints and f = 2 for universal joints), ν = 2 is the number of redundant constraints,

and ξ = 0 is the number of degrees of partial freedom or half partial freedom of links. By substituting the numerical values

to variables in Eq. (1) , m = 4 is obtained, which is the mobility previously determined for the studied 4-UPU mechanism.

2.1. Inverse kinematics solution

Coordinates of point E ( OE = [ x y z] T ) are used to define the position of the end-effector with respect to the base

reference system O xyz , whereas angle θ is used to define the coupler rotation around z axis.

The positions of points O 5 i in reference system O xyz have been evaluated as OO 5 i = OE + EO 5 i , where i denotes the num-

ber of the leg. EO 5 i depend by the lengths of the coupler sides, h and l , as shown in the group of Eq. (2) .

EO 51 =

[l

2

sin (θ ) +

h

2

cos (θ ) h

2

sin (θ ) − l

2

cos (θ ) 0

],

EO 52 =

[h

2

cos (θ ) − l

2

sin (θ ) l

2

cos (θ ) +

h

2

sin (θ ) 0

],

EO 53 =

[− l

2

sin (θ ) +

h

2

cos (θ ) − h

2

sin (θ ) +

l

2

cos (θ ) 0

],

EO 54 =

[l

2

sin (θ ) h

h

2

cos (θ ) h

2

sin (θ ) − l

2

cos (θ ) 0

]. (2)

The positions OO 1 i of joints O 1 i are determined by the geometrical parameters D 1 and D 2 of the kinematics. Furthermore,

both directions and lengths of the legs can be determined as functions of the pose of the end-effector as follows: O 1 i O 5 i =OO 5 i − OO 1 i .

Finally, for each leg it is possible to find out a plane that passes through the leg and that is also perpendicular to the x-y

plane. Let’s call the defined plane leg plane . According to Fig. 2 (b), each leg plane is defined by versors j 1 i and k 1 i .

Whereas, versor i 1 i = j 1 i × k 1 i defines the perpendicular direction to the leg plane . By means of the previously mentioned

relations it is now possible to determine the position and orientation of each joint of the kinematics as function of the pose

of the end-effector and the geometric dimensions l, h, D 1 , and D 2 .

2.2. Jacobian matrix definition

As previously mentioned screw theory can be a useful tool for parallel manipulators kinematic analysis. Basis of screw

theory, operations among screws and application of screw theory to parallel kinematics analysis can be found in [29] . By

means of screw theory and using Plücker coordinates, for each leg i , it is possible to arrange the twist $ ji associated to each

joint j of the leg in a system of twists:

$ i =

($ 1 i $ 2 i $ 3 i $ 4 i $ 5 i

)(3)

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 215

where each twist is written considering the origin of the base reference system O as the pole for the screw representation.

It is then possible to evaluate the constraint wrench W ci associated to the i − th leg as the reciprocal of system expressed

in (3) . The reciprocal to the system of screw is evaluated so that:

W ci ◦ $ ji = 0 , for j = 1 . . . 5 , (4)

where ◦ is the reciprocity product among screws. In the same way, the actuation wrench W ai associated to the i − th leg

is evaluated as the reciprocal wrench to the twists system composed by the twists associated to the leg joints except the

actuated one. Eq. (5) shows how the actuation wrench W ai is obtained by assuming the actuation at the prismatic joint

(third joint) of each leg. {W ai ◦ $ ji = 0 f or j = 1 . . . 5 , j � = 3

W ai ◦ $ ji � = 0 f or j = 3

(5)

Concerning a 4-UPU kinematics having the four prismatic joints actuated, the actuation wrench associated to the i th leg

represents a force applied to the coupler. Moreover, the direction determined by the W a 1 i , W a 2 i , and W a 3 i components of

the actuation wrench W ai associated to the i th leg (out of leg singularity condition) always represent a vector parallel to the

i th leg direction. Furthermore, the constraint applied to the coupler by the i th leg corresponds to a torque in the x-y plane.

The direction determined by the W c 4 i , W c 5 i , and W c 6 i components of the constraint wrench W ci associated to the i th leg

always represent a vector parallel to the direction of the projection of the i th leg on the x-y plane. For the i th leg, direction

defined by the mentioned components for both the actuation wrench W ai and the constraint wrench W ci are represented in

Fig. 2 (a) respectively in red and in green.

3. Analysis of singularities

The singularity analysis is conduced by means of the constraint wrenches and the actuation wrenches defined in

Section 2.2 . In detail, the constraint wrenches allow to find out the singularity conditions where the manipulator looses

a constraint and it is not isostatic. Moreover, by studying the actuation wrenches generated by the prismatic joints, it is

possible to discover the configurations where a particular load at the end effector, that cannot be balanced by any forces at

the actuators, exists. Finally, it is reported a kind of singularity which is not visible by the analysis of both the constraint and

actuation wrenches. It is the leg singularity. In these conditions the manipulator earns one ore more constraints depending

on the pose of its legs. In the following sections these three kinds of singularity are analyzed and discussed finding out the

singularity locus for each kind of singularity condition.

3.1. Global singularities

In order to find out the global singularities of the 4-UPU kinematics rank analysis of the constraint jacobian matrix J c is

performed. Let’s define the constraint Jacobian ( J c , Eq. (6) ) as the matrix realized by collecting the constraint wrenches of

∞ pitch W ci .

Since for rank analysis of the J c matrix only the directions of the four constraint wrenches are necessary, for each leg,

J c 4 i , J c 5 i , and J c 6 i components of the W ci wrench have been easily evaluated as the projection of the i th leg on the x-y plane.

Indeed, the i th leg projection coincide with the constraint torque direction applied on the coupler by the i th leg. Moreover,

since the constraint wrenches are pure torques, W c 1 i , W c 2 i , and W c 3 i components of each W ci wrench are equal to zero.

J c =

⎡

⎢ ⎢ ⎢ ⎢ ⎣

0 0 0 0

0 0 0 0

0 0 0 0

J c41 J c42 J c43 J c44

J c51 J c52 J c53 J c54

0 0 0 0

⎤

⎥ ⎥ ⎥ ⎥ ⎦

(6)

where:

J c41 = x +

l

2

s θ +

h

2

c θ ; J c42 = x − D 2 − l

2

s θ +

h

2

c θ ;

J c43 = x − l

2

s θ − h

2

c θ ; J c44 = x + D 2 +

l

2

s θ − h

2

c θ ;

J c51 = y + D 1 − l

2

c θ +

h

2

s θ ; J c52 = y +

l

2

c θ +

h

2

s θ ;

J c53 = y − D 1 +

l

2

c θ − h

2

s θ ; J c54 = y − l

2

c θ − h

2

s θ ;

and s θ = sin θ and c θ = cos θ .

216 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 3. Singular configuration of the kinematics caused by the parallelism among all the constraint wrenches.

Referring to Fig. 2 (b)), D 1 is the distance from the base center of the first joint of both leg 1 and leg 3 whereas D 2 is the

distance from the base center of the first joint of both leg 2 and leg 4. D 1 and D 2 are measured respectively along the y and

x axis. As said before, the ∞ pitch constraint wrench W ci associated to each leg lays on the x-y plane, therefor the matrix

J c cannot have a rank higher than two. When a constraint jacobian singularity condition occurs rank of matrix J c decreases

to one such as in the case represented in Fig. 3 , where all the constraint wrenches are parallel.

Let’s call J t ci

(with the apex t ) the vector composed by the last three components of the i th column of the constraint

jacobian matrix (i.e. J c 4 i , J c 5 i , and J c 6 i ), thus the components related to the directions of the constraint torques acting on

the i th leg. Singularity occurs when all the J t ci

are parallel, such a condition can be expressed by the following system of

equations:

J t c1 × J t c2 = c θ ( 2 lx − D 2 l − D 1 h + hl c θ ) + s θ ( 2 ly + D 1 l + D 2 h + hl s θ ) + 2 ( D 2 y − D 1 x + D 1 D 2 ) = 0 (7)

J t c1 × J t c3 = cos θ (lx + hy ) + sin θ (ly − hx ) − 2 xD 1 = 0 (8)

J t c2 × J t c4 = cos θ (hy − lx ) − sin θ (ly + hx ) − 2 yD 2 = 0 (9)

In order to solve the system of equations, it is possible to compute the terms xD 1 and yD 2 from Eqs. (8) and (9) . By

substituting the expressions obtained for xD 1 and yD 2 into (7) singularity condition reported in Eq. (10) is obtained.

(2 D 1 D 2 + hl) − (D 2 l + D 1 h ) cos θ + (D 1 l + D 2 h ) sin θ = 0 (10)

Eq. (10) has a solution if and only if the condition expressed in (11) is satisfied.

4 D

2 1 D

2 2 + h

2 l 2

(D

2 1

+ D

2 2 )(h

2 + l 2 ) ≤ 1 (11)

Since relation (11) depends only by geometrical features of the manipulator it is possible to avoid constraint wrenches

singularity by properly designing the kinematics.

On the other hand, when condition (11) is satisfied, it is possible to find out two suitable angles θ1 and θ2 ( Eq. (12) ) as

function of D 1 , D 2 , l , and h where constraint singularity occurs.

θ1 = arcsin

(−2 D 1 D 2 + hl

χ

)+ arctan

(D 2 l + D 1 h

D 1 l + D 2 h

),

θ2 = − arcsin

(−2 D 1 D 2 + hl

χ

)+ arctan

(D 2 l + D 1 h

D 1 l + D 2 h

)+ pi,

where χ =

√

(D 1 l + D 2 h ) 2 + (D 2 l + D 1 h ) 2 .

(12)

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 217

Fig. 4. Examples of singularities of the legs.

When Eq. (12) holds, Eqs. (8) and (9) describe two planes through the z axis. The intersection between the two planes is

the singularity locus for the founded condition of singularity.

3.2. Legs singularities

In this section the singularity conditions of the legs are discussed. It is not possible to derive these manipulator config-

urations from the Jacobian matrix because when leg singularity occurs the kinematics of the manipulator changes.

Single leg singularity.

When a leg aligns to the z axis ( Fig. 4 (a)), singularity occurs for that leg and a constraint wrench is added:

W b =

[a b 0 0 0 c

]T a, b, c ∈ R (13)

Rank of the matrix J c becomes then three, and one degree of freedom is lost by the manipulator.

In this configuration the leg can arbitrarily rotate around its axis constraining a translation in the x-y plane. Since it is

not possible to determine the leg plane, also the terms a, b and c cannot be univocally evaluated. End effector rotations

around the leg axis keep the singularity configuration.

Two legs singularity. Two legs are in singularity in two cases. The first case holds when one of the diagonals of the

rectangular coupler superimposes one of the diagonals of the rhombus determined by the first joint of each leg of the

manipulator ( 2 D 1 =

√

h 2 + l 2 ∨ 2 D 2 =

√

h 2 + l 2 ). The second case takes place with the superimposition of one of the coupler

sides with one side of the base rhombus ( l =

√

D

2 1

+ D

2 2

∨ h =

√

D

2 1

+ D

2 2 ).

When two legs present the singularity condition, constraint wrenches W bi of the same type of (13) are associated to each

leg in singularity.

Even in this case it is not possible to determine the planes of the legs in singularity and, concerning the manipulator’s

mobility, the following three different possibilities can occur.

First possibility holds when the two legs in singularity don’t have the same orientation. In this case the manipulator

looses the two translations in the x-y plane. Moreover, referring to Fig. 4 (b), the manipulator can rotate only around the a

axis, which is parallel to the z axis and passing through the intersection of the two additional wrenches.

Second possibility occurs when the legs in singularity are oriented in the same way but they have a different orientation

with respect to the other legs. In this case one translation in the x-y plane and the rotation parallel to the z axis are

constrained

Finally, the third possibility takes place when the legs in singularity are oriented in the same way and they have the

same orientation with respect to the other legs. In this case the manipulator looses the same DoF of case before but it gains

the rotation around the same direction of the constrained translation because of the singularity of the constraint wrenches.

Three and four legs singularity.

Because of the manipulators kinematics, the three legs singularity condition entails also that the fourth leg is in singu-

larity. This particular case can occur only if D 1 = D 2 =

l √

2 2 , θ = −π

4 and x = y = 0 . The four leg singularity condition causes

the indetermination of the directions of all the four additional wrenches. In the particular case all the added wrenches W

218 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 5. Example of manipulators singularity configuration which leads to Det (J −T

)= 0 .

intersect in the same point, the rotation of the end effector around the axis parallel to z axis and passing through that point

is allowed. Whereas, if all the additional wrenches directions are parallel, both the translation in the W bi direction and the

rotational DoF are constrained. But the manipulator gains the rotation around W bi direction instead. In the most general

case, when none of the two mentioned cases occurs, the manipulator can only translate in the z direction because all the

other DoF are constrained.

3.3. Actuation singularity

Within the performed analysis it is assumed that the actuation is placed in the prismatic joints of each leg. The actuation

wrench associated to each actuated joint is represented by a force vectors applied to the end-effector in the contact point

between each leg and the end-effector itself. The obtained four actuation wrenches W ai can be grouped in the actuation

Jacobian matrix J a which defines the relation between the actuation forces τ i and the resulting wrench at the end effector

W e according to equation W e = J −T a τ .

J −T a = ( W a 1 W a 2 W a 3 W a 4 ) τ = [ τ1 τ2 τ3 τ4 ]

T (14)

Row four and row five of the Jacobian J −T a refer to the torques in the x-y plane, thus the torques applied to the structure.

Since both row four and five are not essential for the actuation analysis, by neglecting the mentioned two rows of J −T a a 4

by 4 actuation Jacobian ( J −T ) is obtained. Equation (15) shows the determinant evaluated for the obtained square matrix.

Det (J −T ) = −z ( 2 D 1 D 2 + hl − ( D 1 h + D 2 l ) c θ + ( D 2 h + D 1 l ) s θ ) ( ( D 1 h − D 2 l ) c θ + ( D 1 l − D 2 h ) s θ ) (15)

By solving Det (J −T ) = 0 for theta and z the singularity loci can be found. This kind of singularity means that a non null

set of actuation forces τ i produce a self balanced system of forces applied to the end effector, i.e. it exists a load at the end

effector that actuation wrenches cannot balance.

Moreover, for any pose of the manipulator where no leg is in singularity, since z, D 1 , D 2 , h , and l cannot assume infinite

values, Det (J −T ) is limited too, therefor it always holds Det (J T ) � = 0 . It means that the manipulator always keeps its mobility

in the workspace, i.e. no loads W e at the end-effector can be entirely balanced by the structure reactions without a force

contribution by the motors.

When actuation singularity occurs ( J −T has not full rank), in general, means that the actuation wrenches are placed on

an hyperboloid regulus ( Fig. 5 ).

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 219

O

O

O

Actuation singularity occurs when:

• z = 0 ;

• θ = arctan

(D 2 l−D 1 h

D 1 l−D 2 h

)+ kπ where k ∈ N ;

• h = l ∧ D 1 = D 2 ; • 2 D 1 D 2 + hl − (D 1 h + D 2 l) cos (θ ) + (D 2 h + D 1 l) sin (θ ) = 0 .

The last condition, is the same singularity condition obtained for the constraint Jacobian ( J c ) in Section 3.1 , therefor it

is avoidable by properly designing the geometry of the manipulator. Moreover, that means that all the singularity of J care included within the singularity of the actuation Jacobian J . Finally, since first and second singularity conditions always

hold for any values of the geometrical parameters D 1 , D 2 h and l , it is not possible to completely avoid all the manipulator

singularities by design.

4. Performance optimization

In the following sections, a detailed procedure for the numerical optimization of the geometry of the 4-UPU manipulator

is proposed. The described procedure is based on the results shown in the first part of the manuscript. The goal of the

optimization is to define a systematic procedure to highlight the relation between the geometrical parameters, that are nec-

essary to define the kinematics, and the performance of the manipulator. Performance of the manipulator can be defined by

a generic performance parameter. Since the procedure does not depend by the chosen performance parameter, it is possible

to assume the procedure as a general way to optimize a 4-UPU manipulator. The results obtained within the previously

performed kinematic analysis are used in this second part to systematically exclude the singularity configurations during

the optimization procedure.

Fig. 2 (b) shows a minimum set of geometrical parameters for the mathematical definition of the 4-UPU kinemtics. Indeed,

in Fig. 2 (b), the geometry of the 4-UPU manipulator is completely defined by the four lengths D 1 , D 2 , h , and l . These lengths

determine both the coupler and the base shapes.

Since the performance of the manipulator does not depend by its size, the four geometrical parameters have been used

to define three adimensional quantities. The adimensional parameters allow to reduce the optimization variables from four

to three and to obtain results which hold independently from the overall dimensions of the manipulator. At this purpose the

four lengths have been divided by the biggest dimension of the rectangular coupler l . In this way the following adimensional

quantities have been defined: a 1 =

D 1 l

, a 2 =

D 2 l

, b =

h l . In the following paragraphs we refer to these quantities as design

parameters .

Usually, for a manipulator, a generic Performance Parameter (PP) depends by both its pose and the design parameters

that define its geometry. For a 4-UPU manipulator holds:

P P = f (x, y, z, θ, a 1 , a 2 , b) ; (16)

where, x, y, z and θ , define the pose of the 4-UPU manipulator; a1, a2 and b are the design parameters previously defined;

and f is an unknown function. This definition of PP is used to compare different 4-UPU geometries in a defined analysis

workspace. About the definition of the analysis workspace thorough discussions are made in Section 4.2 .

Once defined the analysis workspace, according to the generic assumption for the PP, the mean of the PP in the analysis

workspace is assumed as Performance Variable (PV). The PV depends only by the 4-UPU geometry.

P V = P P = g(a 1 , a 2 , b) ; (17)

where g is the mean of the function f in the analysis workspace.

In the following sections the guidelines for the definitions of both the analysis workspace and the search space for the

design parameters are defined. Both these definitions are carefully made in order to achieve the following objectives:

bj 1 to avoid the condition of existence of singularities expressed in (11) by making a proper choice for the search space

of the design parameters (as described in Section 4.1 );

bj 2 to avoid the actuation Jacobian singularities expressed in Section 3.3 by defining the desired analysis workspace of

the manipulator.

- Within this point it is possible to define the maximum range for the rotational DoF θ where the kinematics can

move avoiding singularities. The range for the rotational DoF is called rotational workspace ( Section 4.2 , step 2);

bj 3 to define the position of the analysis workspace along the z axis in order to maximize the performance of the manip-

ulator ( Section 4.2 , step 3).

In the following subsections both the procedures concerning the definition of the design parameters search space and

the design of the analysis workspace are explained and discussed. In Section 5 , a complete illustrative optimization case is

proposed, whereas, in Section 5.1 , the obtained numerical results are discussed. Results are presented as the mean of the PP

in an illustrative analysis workspace, assuming as PP the volume of the manipulability ellipsoid.

Finally, in Section 6 a practical application case study is presented. It is based on the workspace requirements necessary

for the realization of a wearable fingertip haptic interface.

220 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 6. In red the limits of the solution of equation (18) with b = 0 . 7 is plotted; the dashed area represents the solution of inequality (19b) . (For interpre-

tation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.1. Definition of the design parameters search space

As shown in the previously performed kinematic analysis of the 4-UPU manipulator, a proper choice of the design param-

eters allows to avoid the condition of singularity expressed in (10) . The condition of existence of this singularity condition

( Eq. (11) ) depends only by the geometrical features of the manipulator. As previously mentioned, the geometrical features

of the manipulator (except the scale) can be completely described by the three adimensional design parameters previously

defined: a 1 , a 2 , and b .

The b parameter can vary in the range between 0 and 1 because it represents the ratio between the sides of the rectan-

gular coupler ( b =

h l ), where l is the length of the biggest side of the coupler.

Once defined the range for the b parameter, the limits for the a 1 and a 2 parameters can be varied, depending on the b

value, in order to avoid the condition of singularity for both the constraint and actuation jacobians expressed in (10) . Using

the adimensional design parameters the condition of existence of this singularity condition can be rewritten as:

4 a 2 1 a 2 2 + b 2

(a 2 1

+ a 2 2 )(b 2 + 1)

≤ 1 . (18)

Equation (18) means that to avoid the singularity condition expressed in (10) , the following inequalities must be satis-

fied:

a 2 1 <

b 2 − a 2 2 (1 + b 2 )

1 + b 2 − 4 a 2 2

for a 2 2 <

1 + b 2

4

; (19a)

a 2 1 >

b 2 − a 2 2 (1 + b 2 )

1 + b 2 − 4 a 2 2

for a 2 2 >

1 + b 2

4

; (19b)

‖ b‖ > 1 for a 2 2 =

1 + b 2

4

. (19c)

For each value of b , the relations (19) define two different areas in the a 1 − a 2 plane. In Fig. 6 the areas obtained assuming

b = 0.7 are represented and colored in yellow. Both the yellow areas allow to avoid the singularity condition expressed in (18) .

In the general case, the choice of the suitable search space area in the a 1 − a 2 plane depends on the requirements of the

practical application the manipulator will be optimized for.

If the design of the manipulator admits or requires configurations placed in the white areas, the presented procedure can

be used to investigate manipulator’s performance also for those combinations of the design parameters. It is worth noting

that for combinations placed in the white areas singularity condition expressed in (18) holds. In order to evaluate the

manipulator in singularity-free workspace when singularity condition expressed in (18) holds, the rotational DoF workspace

has to be divided two times, according to the singularity loci solutions reported in Eq. (12) . Such divisions increase the

number of singularity-free possible rotational workspace by reducing the maximum range available for the rotational DoF.

Moreover, combinations of the design parameters placed in the white areas lead to configurations of the manipulator

having a diagonal of the base rhombus longer than the couplerâs major dimension l.

Since the majority of the parallel manipulators developed for practical applications are designed with the base larger

than the end effector, in the illustrative procedure the quarter comprising the mentioned geometrical condition is used to

perform the illustrative analysis. Indeed the quarter defined by the inequality (19b) is the adopted search space for the

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 221

a 1 and a 2 variables. Referring to Fig. 6 , the dashed area represents the search space assumed for a 1 and a 2 when b = 0 . 7 .

Different values of b leads to different search spaces having the same shape.

From the computational point of view, in order to simplify the analysis procedure, the search space in the a 1 − a 2 plane

is extended to all the combinations of a 1 and a 2 that satisfy the conditions: √

1 + b 2

4

≤ a 1 ≤√

1 + b 2

4

+ R (20a)

√

1 + b 2

4

≤ a 2 ≤√

1 + b 2

4

+ R (20b)

The two lines defined by a 1 =

√

1+ b 2 4 and a 2 =

√

1+ b 2 4 are two asymptotes for the relations (19) in the plane a 1 - a 2

and the range of R is arbitrarily decided in order to set an upper limit to the search space. Since such a definition for the

search space area includes some combinations of a 1 and a 2 near the asymptotes where the condition of existence (11) of the

singularity expressed in (10) is satisfied, in the results analysis those combinations won’t be considered as possible results.

4.2. Analysis workspace definition

Once defined the design parameters search space, it is possible to define the workspace where the performance of the

manipulator is evaluated. This part of the procedure is strictly related to the application the manipulator is going to be

optimized for. The analysis workspace definition is divided in three main steps:

• step 1 definition of the workspace shape and the position of its projection on the x − y plane. • step 2 definition of the range for the rotational degree of freedom. This definition is made in order to avoid the singu-

larity condition expressed by: ˜ θ = arctan

(a 2 −a 1 b

a 1 −a 2 b

).

It worth noting that, for every combination of the design parameters, different limits for the maximum range of the

rotational workspace have to be defined. • step 3 Optimization of the z coordinate of the workspace center ( ̃ z ) as function of the three design parameters.

In this step, a lower limit for ( ̃ z ) is introduced in order to avoid points of the analysis workspace where z ≤ 0.

This step allows to discover, for each set of design parameters, the distance ( ̃ z ) between the workspace center and the

x − y plane that allows the best manipulator’s performance.

Step 1 depends only by the requirements of the application the manipulator will be used for, whereas the other two

steps are strictly related to the kinematic behavior of the manipulator.

In Step 2 the limits defining the maximal range for the rotation of the coupler avoiding singularity conditions are defined.

As said before, the rotational workspace is defined in order to avoid the singularity condition expressed by: ˜ θ =arctan

(a 2 −a 1 b

a 1 −a 2 b

). This equation specifies two opposite points on the trigonometric circle that it is necessary to avoid. There-

fore, referring to the mentioned singularity condition, it is possible to find out two singularity - free maximal ranges of πradians for the rotational degree of freedom ( θ ). The limit angles of the ranges change for each of the design parameters

combination evaluated. Each range defines a possible rotational workspace. Therefor, in order to perform a complete analysis

of the performance of the manipulator, all the following evaluations will be carried out in parallel by using both the defined

workspace.

In the next sections the two possible workspace will be called Upper rotational workspace (URW) (21a) , and Lower rota-

tional workspace(LRW) (21b) .

˜ θ < θ <

˜ θ + π, (21a)

˜ θ − π < θ <

˜ θ, (21b)

In Fig. 7 (a) and (b) the minimum and the maximum limits of the θ variable for the two identified workspaces respec-

tively are represented. The limits are functions of the design parameters only.

In the last step, step 3 , the z coordinate of the workspace center ( ̃ z ) is defined. ˜ z is a function of a 1 , a 2 , b , and takes into

account the limits of the other DoF of the workspace defined in step 1 and 2, and the PP used for the entire optimization

process. Indeed, for each combination of a 1 , a 2 , and b , ˜ z is evaluated as the mean of z in the projected workspace x-y- θ(defined in step 1 and step 2) and z is defined as the value of the z coordinate which maximizes the PP in the generic pose

of the manipulator x − y − θ .

The definition of ˜ z as function of a 1 , a 2 , b allows to evaluate the performance of every manipulator’s geometry, defined

by a combination of design parameters, in a workspace whose center is optimized along the z direction for the geometry

itself. In other words, step 3 allows to compare the performance that correspond to different design solutions of the manip-

ulator, by evaluating the Performance Variable (PV) in a workspace positioned at a suitable height for every design solution

analyzed.

222 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 7. Minimum and maximum values of the rotational degree of freedom in the two possible workspace for different combinations of the design param-

eters a 1 , a 2 and b .

4.3. Numerical evaluation of the optimization results

Once the performance functional, the parameters search space, and the analysis workspace are defined as described in

the previous sections, it is possible to perform the numerical evaluation of the results.

For each combination of the design parameters ( a 1 , a 2 , and b ) included in the search space, the Performance Variable

(PV) is evaluated as the mean of the Performance Parameter (PP) in the defined analysis workspace. The obtained numerical

results allow to find out the design parameters that allow the best manipulator’s performance associated to the chosen

performance functional, the parameters search space, and the analysis workspace. As output of the procedure, 3D plots can

be used to visualize the obtained numerical results in order to better understand the behavior of the analyzed kinematics.

In the next section an illustrative example of the optimization procedure is carried out by following the general steps

described before.

5. Illustrative performance optimization procedure

Minimization of the required actuation forces is a common target for manipulator’s optimizations. At this purpose, ac-

cording with the previous definitions, to optimize the 4-UPU kinematics the volume of the force manipulability ellipsoid

(FME) is chosen as performance parameter (PP), and the mean volume of the FME (Mean V) in the defined workspace as

performance variable (PV).

The volume of the FME is defined as:

V =

4

3

π‖ Det (MJ −T

)‖; (22)

where M is a metric matrix used to uniform the unit of measure between the elements of J −T . M is defined as:

M =

⎡

⎢ ⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0

2 l

⎤

⎥ ⎦

. (23)

where the 2 l

coefficient defines the metrics used to compare torques with forces. In addition to being responsible for ho-

mogeneity of the quantities considered in the performance parameter, the metric matrix coefficients also weights both the

forces and torque contributions in the performance parameter. For the mentioned reason the optimization results can be

affected by the chosen metric matrix coefficients. In the performed analysis the metric matrix coefficient 2 l

has been arbi-

trarily defined with the same order of magnitude of the distance between the application points of the forces on the coupler

and the coupler center. Aim of this choice is to keep torque contributions comparable to the force contributions in the force

manipulability ellipsoid without neither favor nor penalize any of the force/torque contributions. It is worth noting that the

presented optimization procedure does not depend by the adopted metric coefficient.

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 223

Fig. 8. Mean values on the workspace x, y and θ ( URW 8 (a), LRW 8 (b)) of the z coordinate that entails the maximum value for the PV as function of the

design parameters a 1 , a 2 and b .

According to the procedure previously described, after the definition of the performance parameter, both the design

parameters search space and the analysis workspace have to be defined. In this illustrative procedure a 3 mm range is used

for the design parameters search space, thus equations (24) define the limits of the adopted search space.

√

1 + b 2

4

≤ a 1 ≤√

1 + b 2

4

+ 3 (24a)

√

1 + b 2

4

≤ a 2 ≤√

1 + b 2

4

+ 3 (24b)

Concerning the workspace definition, by following step 1 ( Section 4.2 ), a cube of side l with the corners aligned with x,

y and z axis of the base reference system is assumed as workspace shape. Moreover, since both the base and the coupler

shapes are symmetric with respect to the base reference system, it seems reasonable to place the workspace center on the

z axis. For this reason, within the illustrative procedure, the workspace is assumed to be centered in x = 0 , y = 0 and the

performance of the manipulator close to the origin of the x-y plane have been investigated. Finally, as rotational workspace

(step 2), for both the LRW and the URW the maximal range possible without singularities is assumed (Eq. (21)).

By using the mentioned definitions for the PP, the PV, the design parameters search space and the workspace, the fol-

lowing results have been obtained:

Fig. 8 shows the results evaluated for the optimal values of ˜ z l

as function of the three design parameters. In detail,

Fig. 8 (a) shows the results obtained for the Upper rotational workspace , whereas Fig. 8 (b) represents the results obtained

for the Lower rotational workspace . Since the defined analysis workspace shape is a cube of side l , for each combination of

the adimensional parameters, the manipulator is considered to translate along the z axis between the values ˜ z /l − 0 . 5 and

˜ z /l + 0 . 5 for every value of x and y . Moreover, since z = 0 is a singularity condition unavoidable by design, the values of ˜ z /l,

have been saturated in order to obtain min ( ̃ z /l) > 0 . 5 and avoid all the possible singularities inside the analysis workspace.

The made definition of the variable workspace allows to avoid actuation singularities in the whole workspace as long

as it holds the relation between the geometrical dimensions D 1 , D 2 , h, l expressed in (11) . Furthermore, all the singularity

conditions can be prevented taking into account the conditions of existence for the leg singularities expressed in paragraph

3.2 .

By representing the obtained numerical results of ˜ z /l by means of 3D plots, surfaces depict in Fig. 8 (a) and (b) are

obtained. From both the Fig. 8 (a) and (b) it is possible to notice that the value for the ˜ z l

factor presents a significant variation

in the a 1 − a 2 plane but it remains quite constant with variations of the b parameter.

Finally, as results of the performed optimization, 3D plots in Fig. 9 show the surfaces representing the behavior of the

evaluated mean volume of the FME (Mean V) assumed as performance variable (PV) as function of the design parameters.

The two Fig. 9 (a) and (b) refer to the URW and the LRW respectively.

224 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 9. Mean volume of the manipulability ellipsoid in the two possible workspaces for different combinations of the design parameters a 1 , a 2 and b .

5.1. Results discussion

Referring to Fig. 9 , both the analyzed workspace allow better performance when the design parameter b tends to 0, but,

the maximum value of Mean V is obtained within the Lower Rotational Workspace ( Fig. 9 (b)).

From the shape of the resulting surfaces it is possible to notice that the manipulator displays better performances when

the shape of the base tends to be a square and the shorter side of the coupler tends to 0. In fact, as shown in the contour

plots of Fig. 10 , when the a 1 = a 2 condition holds, the value of Mean V reaches its maximum for b = 0 . Whereas, increasing

b the maximum of Mean V moves to combinations of a 1 and a 2 farther from the bisector a 1 = a 2 .

As result of the optimization analysis, the best combination of adimensional design parameters is composed by b = 0 and

a 1 = a 2 with a 1 and a 2 placed at the limits of the analyzed search space. This follows directly by the asymptotic behavior of

the b = 0 surface in the search space. Obtained results hold only for a 4 − U P U manipulator, working on a cubical workspace

of size l with the center placed in x = 0 , y = 0 and z = ˜ z with ˜ z function of the design parameters.

6. Case study: Optimization of the 4-UPU parallel kinematics for the realization of a fingertip haptic interface

In order to optimize the kinematics for the realization of a 4 Dof wearable fingertip haptic interface, the requirements for

the workspace have to be defined. Similarly to the configuration of the 3-DoF haptic interface shown in Fig. 1 , end-effector’s

displacements in the plane tangential to the fingertip skin in the contact point and the end-effector rotation are used to

transmit the skin-stretch stimuli to the user. Accordin to the geometrical definition of the 4-UPU kinematics previously

made, the plane tangential to fingertip skin is assumed parallel to the x − y plane. Whereas the z direction is the normal

direction used to generate the contact.

A previous study conduced by Provancher at al. [30] assessed that, a fingertips skin stretch of at least 1 mm is necessary

for the detection of the direction of a slow tangential stimulus. In order to fulfill the requirements given by the psychophys-

ical background, the following workspace has been chosen for the kinematic optimization:

−2 mm ≤x ≤ 2 mm

−2 mm ≤y ≤ 2 mm

˜ z − 5 mm ≤z ≤ ˜ z + 5 mm

where ˜ z will be find out within the optimization procedure.

By assuming the mentioned DoF ranges, with respect to the workspace center, the kinematics has the capability to exert

tangential stimuli on the fingertip skin ensuring displacements up to 2 mm in every direction of the x-y plane, which is the

double of the minimum displacement required. Concerning the rotational DoF, since the limits evaluated in Section 4.2 are

independent with respect to the designed workspace, by using the same search space defined in Section 4.1 it is possible to

assume the same values adopted for the previous analysis.

Furthermore, according to the application requirement of reduced encumbrance, 10 mm length is assumed for the biggest

side of the coupler ( l ).

Finally, by using the same search space defined in (20a) , the obtained PV numerical results visualized in 3D graphs are

shown in Fig. 11 . In detail, the graphs in Fig. 11 (a) and (b) show the results obtained for the URW and the LRW respectively.

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 225

Fig. 10. Contour plots of the mean volume of the manipulability ellipsoid in the Lower rotational workspace for each value of b evaluated.

226 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

Fig. 11. Mean volume of the manipulability ellipsoid in the two possible workspaces for different combinations of the design parameters a 1 , a 2 and b ,

obtained within the realization of the haptic interface example.

Fig. 12. Contour plot of the ˜ z /l optimal values obtained for b = 0 evaluated in the lower rotational workspace.

It is possible to notice that the performance of the kinematics lightly increased for this small workspace with respect to

the illustrative case of Section 5 , whereas the shape of the surfaces remained quite the same. Moreover, also for the case

study, the LRW allows better performances. Referring to Fig. 11 (b), it is possible to choose as best values for the design

parameters b = 0 and a 1 = a 2 = 3 . 5 . It’s worth noting that also for the performed case study the found solution is placed at

the boundary of the search space. Fig. 12 shows the contour plot of the ˜ z /l optimal values in the a 1 − a 2 plane, evaluated

for the lower rotational workspace, and corresponding to b = 0 . It is possible to notice that the optimal value of the center

of the analysis workspace ˜ z /l is equal to 1.9.

7. Conclusion

As previously mentioned, the 4-UPU fully parallel manipulator is of particular interest because of its simple structure,

good dynamic and stiffness qualities (common qualities in parallel manipulators). Moreover, such a kinematics is able to

independently control the four schoenflies-motion DoF.

M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228 227

Within the paper, both kinematic analysis and the optimization of the 4-UPU kinematics have been carried out. General

results, useful for the design of every manipulator based on 4-UPU kinematics have been obtained. Moreover, as shown in

the case study, the proposed optimization procedure can be easily applied to practical applications.

In detail, in the first part of the manuscript, the 4-UPU kinematics has been solved by screw theory. All the possible

singularities and the associated singularity loci have been analytically described. Also the singularity configurations due

to the kinematics of the legs and not visible by the Jacobian analysis, have been found out and thoroughly discussed. As

relevant result, it has been found out, that the singular configuration for the constraint Jacobian J c are included in the

singular configurations for the actuation Jacobian J .

In the second part of the manuscript, a numerical procedure for the optimization of the design parameters of the manip-

ulator has been presented. It uses the mean volume of the FME in a given workspace as performance variable. Furthermore,

design parameters combinations are chosen in order to avoid singularity conditions depending by the manipulator’s geome-

try. The numerical procedure starts from the definition of the analysis workspace and the optimization of it’s center distance

from the manipulator’s base. Indeed, the first phase of the presented procedure aims to find out the optimal position of the

workspace along the z axis of the base frame, as function of the adimensional design parameters of the manipulator. More-

over, also the limits of the rotational DoF are varied as functions of the design parameters in order to avoid singularity

conditions in all the analysis workspace.

The generalization of the algorithm using adimensional design parameters makes the results independent from the size

of the manipulator.

As results the behavior of the performance variable as function of the design parameters has been presented and dis-

cussed for different starting workspace. One of the studied cases refers to the design of a wearable fingertip haptic interface

with 4 DoF. From the optimization analysis useful guidelines for the design process of 4-UPU manipulators have been found.

Inverse proportionality between the b design parameter and the FME performance variable has been highlighted. Further-

more, within the limits of the analyzed search space, direct proportionality between the values of a 1 , a 2 and the FME

performance variable, when b tends to 0, has been observed.

It is worth noting that the shown results have been obtained without consider any constraint due to the real encum-

brance of the legs. In the prototype design phase, when the overall encumbrance of the manipulators legs are known, the

described optimization procedure can be performed in a more focused way considering also realistic environmental con-

straints in the form of limits in the passive joints which affect the possible choices for the design parameters.

References

[1] L. Carbonari , M. Callegari , G. Palmieri , M.-C. Palpacelli , A new class of reconfigurable parallel kinematic machines, Mech. Mach. Theory 79 (2014)173–183 .

[2] X. Kong , Reconfiguration analysis of a 3-dof parallel mechanism using euler parameter quaternions and algebraic geometry method, Mech. Mach.Theory 74 (2014) 188–201 .

[3] M. Karouia , J.M. Hervé, Non-overconstrained 3-dof spherical parallel manipulators of type: 3-rcc, 3-ccr, 3-crc, Robotica 24 (1) (2006) 85–94 .

[4] M. Karouia , J.M. Hervé, A Family of Novel Orientational 3-Dof Parallel Robots, in: Romansy 14, Springer, 2002, pp. 359–368 . [5] R. Di Gregorio , A new family of spherical parallel manipulators, Robotica 20 (4) (2002) 353–358 .

[6] F. Gao , W. Li , X. Zhao , Z. Jin , H. Zhao , New kinematic structures for 2-, 3-, 4-, and 5-dof parallel manipulator designs, Mech. Mach. Theory 37 (11)(2002) 1395–1411 .

[7] Y. Fang , L.-W. Tsai , Structure synthesis of a class of 4-dof and 5-dof parallel manipulators with identical limb structures, Int. J. Rob. Res. 21 (9) (2002)799–810 .

[8] Q. Li , Z. Huang , Type synthesis of 4-dof parallel manipulators, in: Proceedings of ICRA 2003 IEEE International Conference on Robotics and Automation.,

1, IEEE, 2003, pp. 755–760 . [9] M. Zoppi , D. Zlatanov , C.M. Gosselin , Analytical kinematics models and special geometries of a class of 4-dof parallel mechanisms, IEEE Trans. Robot.,

21 (6) (2005) 1046–1055 . [10] F. Pierrot , et al. , H4: a new family of 4-dof parallel robots, in: Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatron-

ics., IEEE, 1999, pp. 508–513 . [11] C. Wen-Jia , Z. Ming-Yang , C. Shu-Hong , W. Hong-Guang , X. Zhi-Gang , F. Li-Jin , A novel 4-dof parallel manipulator and its kinematic modelling, in:

Proceedings of ICRA 2001 IEEE International Conference on Robotics and Automation, 4, IEEE, 2001, pp. 3350–3355 .

[12] C.-C. Lee , J.M. Hervé, Type synthesis of primitive Schoenflies-motion generators, Mech. Mach. Theory 44 (10) (2009) 1980–1997 . [13] M. Carricato , Fully isotropic four-degrees-of-freedom parallel mechanisms for Schoenflies motion, Int. J. Rob. Res. 24 (5) (2005) 397–414 .

[14] P.-C. Lee , J.-J. Lee , On the kinematics of a new parallel mechanism with schoenflies motion, Robotica (2016) 1–15 . [15] J. Angeles , S. Caro , W. Khan , A. Morozov , Kinetostatic design of an innovative Schönflies-motion generator, Proc. Instit. Mech.Eng. Part C 220 (7) (2006)

935–943 . [16] S. Yang , T. Sun , T. Huang , Type synthesis of parallel mechanisms having 3t1r motion with variable rotational axis, Mech. Mach. Theory 109 (2017)

220–230 .

[17] L. Nurahmi , S. Caro , P. Wenger , J. Schadlbauer , M. Husty , Reconfiguration analysis of a 4-ruu parallel manipulator, Mech. Mach. Theory 96 (2016)269–289 .

[18] S. Briot , I.A. Bonev , Pantopteron-4: a new 3t1r decoupled parallel manipulator for pick-and-place applications, Mech. Mach. Theory 45 (5) (2010)707–721 .

[19] S. Briot , I.A. Bonev , Accuracy analysis of 3t1r fully-parallel robots, Mech. Mach. Theory 45 (5) (2010) 695–706 . [20] J.-S. Zhao , Y.-Z. Fu , K. Zhou , Z.-J. Feng , Mobility properties of a schoenflies-type parallel manipulator, Robot Comput. Integr. Manuf. 22 (2) (2006)

124–133 .

[21] J. Gallardo , J. Rico , A. Frisoli , D. Checcacci , M. Bergamasco , Dynamics of parallel manipulators by means of screw theory, Mech. Mach. Theory 38 (11)(2003) 1113–1131 .

[22] J. Gallardo-Alvarado , J.M. Rico-Martínez , G. Alici , Kinematics and singularity analyses of a 4-dof parallel manipulator using screw theory, Mech. Mach.Theory 41 (9) (2006) 1048–1061 .

[23] D. Gan , J.S. Dai , J. Dias , L. Seneviratne , Forward kinematics solution distribution and analytic singularity-free workspace of linear-actuated symmetricalspherical parallel manipulators, J. Mech. Robot. 7 (4) (2015) 041007 .

228 M. Gabardi, M. Solazzi and A. Frisoli / Mechanism and Machine Theory 133 (2019) 211–228

[24] D. Gan , J. Dias , L. Seneviratne , Unified kinematics and optimal design of a 3rrps metamorphic parallel mechanism with a reconfigurable revolute joint,Mech. Mach. Theory 96 (2016) 239–254 .

[25] D. Gan , J.S. Dai , J. Dias , R. Umer , L. Seneviratne , Singularity-free workspace aimed optimal design of a 2t2r parallel mechanism for automated fiberplacement, J. Mech. Robot 7 (4) (2015) 041022 .

[26] D. Leonardis , M. Solazzi , I. Bortone , A. Frisoli , A wearable fingertip haptic device with 3 dof asymmetric 3-rsr kinematics, in: World Haptics Conference(WHC), 2015 IEEE, IEEE, 2015, pp. 388–393 .

[27] J.S. Dai , Z. Huang , H. Lipkin , Mobility of overconstrained parallel mechanisms, J. Mech. Des. 128 (1) (2006) 220–229 .

[28] Y. Li , L. Wang , J. Liu , Z. Huang , Applicability and generality of the modified Grübler-Kutzbach criterion, Chin. J. Mech. Eng. 26 (2) (2013) 257–263 . [29] Z. Huang , Q. Li , H. Ding , Theory of Parallel Mechanisms, 6, Springer Science & Business Media, 2012 .

[30] B.T. Gleeson , S.K. Horschel , W.R. Provancher , Design of a fingertip-mounted tactile display with tangential skin displacement feedback, Haptics, IEEETransactions on 3 (4) (2010) 297–301 .