An investigation on mobility and stiffness of a 3-DOF

translational parallel manipulator via screw theory

Qingsong Xu, Yangmin Li ∗

Department of Electromechanical Engineering, Faculty of Science and Technology,

University of Macau, Av. Padre Tomas Pereira S.J., Taipa, Macao SAR, P. R. China

Abstract

This paper analyzes the mobility and stiffness of a three-prismatic-revolute-cylindrical (3-

PRC) translational parallel manipulator (TPM). Firstly, the original 3-PRC TPM is con-

verted into a non-overconstrained manipulator since there exist some practical problems

for the overconstrained mechanism. By resorting to the screw theory, it is demonstrated

that the conversion brings no influences to the mobility and kinematics of the manipulator.

Secondly, the stiffness matrix is derived intuitively via an alternative approach based upon

screw theory with the consideration of actuations and constraints, and the compliances

subject to both actuators and legs are taken into account to establish the stiffness model.

Furthermore, the stiffness performance of the manipulator is evaluated by utilizing the ex-

tremum stiffness values over the usable workspace, and the influences of design parameters

on stiffness properties are presented, which will be helpful for the architecture design of

the TPM.

Key words: Parallel manipulators; Mobility; Stiffness; Analysis; Screw theory;

Mechanism design

∗ Corresponding author. Tel.: +853 3974464; Fax: +853 838314.

Email address: ymli@umac.mo (Yangmin Li).

Preprint submitted to Elsevier Science 12 February 2007

1 Introduction

A parallel manipulator typically consists of a mobile platform that is connected to

a fixed base by several limbs or legs in parallel. Generally, if carefully designed,

parallel manipulators can provide several attractive advantages over their serial

counterparts in terms of high stiffness, high accuracy, and low inertia [1], which

enable them become challenging alternatives for wide applications such as in as-

sembly lines, flight simulators, machine tools, ultra-precision instruments, medical

devices, and so on. Recently, the progress on the development of parallel manip-

ulators with less than six degree-of-freedom (DOF) has been accelerated because

these limited-DOF parallel manipulators own several other advantages including

the total cost reduction in manufacturing and operations in addition to the inherent

merits of parallel mechanisms. As a result, limited-DOF parallel manipulators have

been investigated and applied more and more extensively.

Among the limited-DOF manipulators, those translational parallel manipulators

(TPM) possessing three spatial pure translational DOF have drawn particular in-

terests from numerous researchers, since they satisfy the requirements of many

specific applications. Various TPM architectures have been proposed in the liter-

atures [2, 3], and the type syntheses of a 3-DOF TPM have been carried out by

investigations based on the screw theory [4,5], group theoretic approach [6,7], and

several other methods [8–11].

In our previous research [12], a 3-PRC (three-prismatic-revolute-cylindrical) TPM

with three P joints intersecting at a common point was presented and designed so

as to eliminate all the singularities from the workspace. It should be noted that this

3-PRC TPM possessed the same mechanism with the one that was proposed in [13]

with three P joints parallel to one another, and the used PRC linkage was equivalent

to the PRRP (letters R denote the revolute joints with parallel axes) translational

parallel kinematic chain that was enumerated by the type synthesis in [5].

2

As an overconstrained mechanism, the 3-PRC TPM has a very simple structure.

However, in practice, the problems of variable frictions in passive joints and large

reaction moment have to be considered so as to assure the mobility of the mo-

bile platform for a 3-PRC TPM. Otherwise, the mobile platform may not move

or the manipulator cannot work if there are some kinematic errors. Unfortunately,

due to the unavoidable errors arise from the manufacturing tolerance and imperfect

assembly, there will always exist kinematic errors. Considering this point, the orig-

inal 3-PRC TPM will be converted into a non-overconstrained TPM in this paper

in order to solve the movability problem by resorting to the screw theory [14–17].

Additionally, since high stiffness is one of the advantages of parallel manipulators,

it is necessary to investigate the stiffness issue of the 3-PRC TPM in more details.

Actually, the stiffness of a manipulator has direct impact on its position accuracy.

Hence, in the early design stage, it is desired to perform the stiffness modeling

and evaluation of a parallel manipulator for the precise manipulation purpose. The

second objective of this paper is to accomplish the stiffness analysis of the 3-PRC

TPM since there are no efforts made toward the stiffness characterization of this

type of manipulator yet.

In the remainder of this paper, after a brief review of the screw and reciprocal screw

systems in Section 2, and a short description of the 3-PRC TPM in Section 3, the

mobility of the manipulator is analyzed in Section 4, where the overconstrained

conditions of the mechanism are eliminated without any influences on its mobility

and kinematics. Then in Section 5, the stiffness model is identified with the consid-

eration of compliances subject to both actuators and legs, and the overall stiffness

matrix is established. Afterwards, the stiffness assessment is carried out in Sec-

tion 6 along with the derivation for the influences of design parameters on stiffness

characteristics. Finally, some concluding remarks are summarized in Section 7.

3

2 Overview of screw and reciprocal screw systems

In screw theory, a unit (normalized) screw is defined by a pair of vectors:

$ =

s

r × s + hs

, (1)

where s is a unit vector directing along the screw axis, r denotes the position vector

pointing from an arbitrary point on the screw axis to the origin of the reference

frame, the vector r × s defines the moment of the screw axis with respect to the

origin of the reference frame, and h represents the pitch of the screw. If the pitch

equals to zero, the screw becomes:

$ =

s

r × s

. (2)

While in case of infinite pitch, the screw reduces to:

$ =

0

s

. (3)

A screw can be used to represent a twist or a wrench. With $F and $L respectively

denoting the vectors of the first and last three components of a screw $, then $F and

$L respectively represent the angular and linear velocities when $ refers to a twist,

and the force and couple vectors when $ refers to a wrench.

Two screws, namely, $r and $, are said to be reciprocal if they satisfy the following

condition [15].

$r ◦ $ = [∆$r]T $ = 0, (4)

4

where “◦” represents the reciprocal product operator, and the matrix ∆, which is

used to interchange the first and last three components of a screw ($r), is defined

by:

∆ ≡

0 I

I 0

, (5)

where 0 and I denote a zero matrix and an identity matrix in 3×3, respectively.

The physical meaning of reciprocal screws is that the wrench $r produces no work

along the twist of $.

Concerning a n-DOF spatial serial kinematic chain with n 1-DOF joints (n ≤ 6),

the joint screws (twists) associated with all the joints form an n-order twist system

or n-system if the n joint screws are linearly independent. The instantaneous twists

of the end-effector can be described as follows.

$ =n

∑

i=1

qi$i, (6)

where qi is the intensity and $i is the unit screw associated with the i-th joint.

The reciprocal screw system of the twist system consists of 6-n linearly indepen-

dent reciprocal screws (wrenches) and is called a (6-n)-order wrench system or

(6-n)-system. In what follows, the relevant results of screw theory are utilized for

both the mobility and stiffness investigations of a 3-DOF translational parallel ma-

nipulator.

3 Architecture description of the manipulator

The schematic diagram of a 3-PRC TPM is shown in Fig. 1. It consists of a mobile

platform, a fixed base, and three limbs with identical kinematic structure. Each limb

5

connects the fixed base to the mobile platform by a P (prismatic) joint, a R (revo-

lute) joint, and a C (cylindrical) joint in sequence, where the P joint is driven by a

linear actuator assembled on the fixed base. Thus, the mobile platform is attached

to the base by three identical PRC linkages.

To facilitate the analysis, as shown in Figs. 1 and 2, we assign a fixed Cartesian

frame O{x, y, z} at the centered point O of the fixed base, and a moving Cartesian

frame P{u, v, w} on the triangle mobile platform at the centered point P , along

with the z- and w-axes perpendicular to the platform, and the x- and y-axes parallel

to the u- and v-axes, respectively.

In addition, the i-th limb CiBi (i = 1, 2, 3) with the length of l is connected

to the mobile platform at Bi which is a point on the axis of the i-th C joint. B′

i

denotes the point on the mobile platform that is coincident with the initial position

of Bi, and the three points B′

i, for i = 1, 2, and 3, lie on a circle of radius b.

The three rails MiNi intersect one another at point D and intersect the x-y plane

at points A1, A2, and A3 that lie on a circle of radius a. The sliders of P joints Ci

are restricted to move along the rails between Mi and Ni. Moreover, the axis of P

joint is perpendicular to the axes of R and C joints within the i-th limb. Angle α

is measured from the fixed base to rails MiNi and is defined as the layout angle

of actuators. In order to obtain a compact architecture, the value of α is designed

within the range of [0◦, 90◦]. Angle ϕi is defined from the x-axis to−−→OAi in the

fixed frame, and also from the u-axis to−−→PB′

i in the moving frame. Without loss

of generality, let the x-axis point along−−→OA1, and the u-axis along

−−→PB′

1. Then, we

have ϕ1 = 0◦. Additionally, let dmax and smax denote the maximum stroke of linear

actuators and C joints, respectively, i.e.,

−dmax

2≤ di ≤

dmax

2, (7)

−smax

2≤ si ≤

smax

2, (8)

6

for i = 1, 2, and 3.

Furthermore, in order to achieve a symmetric workspace of the manipulator, both

∆A1A2A3 and ∆B1B2B3 are assigned to be equilateral triangles. The following

mobility analysis shows that in order to keep the mobile platform from changing

its orientation, it is sufficient for the joint axes within the same limb to satisfy

some certain geometric conditions. That is, the R joint axis (s2,i) and C joint axis

(s3,i or s4,i) within the i-th limb are parallel to the same unit vector si0, which is

perpendicular to the leg direction−−→CiBi, for i = 1, 2, and 3.

4 Mobility analysis and elimination of overconstrained conditions

4.1 Mobility determination of a 3-PRC TPM

The mobility determination, i.e., the DOF identification, is the first and foremost

issue in designing a parallel manipulator. The general Grubler-Kutzbach criterion is

useful in mobility analysis for many parallel manipulators, however it is difficult to

directly apply this criterion directly to mobility analysis of some kinds of limited-

DOF parallel manipulators. For example, the number of DOF of a 3-PRC TPM

given by the general Grubler-Kutzbach criterion is

F = λ(n − j − 1) +j

∑

i=1

fi = 6 × (8 − 9 − 1) + 12 = 0, (9)

where λ represents the dimension of task space, n is the number of links, j is the

number of joints, and fi denotes the degrees of freedom of joint i.

The zero number of DOF of a 3-PRC TPM given by the general Grubler-Kutzbach

criterion reveals that the 3-PRC TPM is an overconstrained parallel manipulator.

Another drawback of the general Grubler-Kutzbach criterion is that it can only

derive the number of DOF of some mechanisms but can not obtain the properties

7

of the DOF, i.e., whether they are translational or rotational DOF.

On the contrary, we can effectively determine the mobility of a 3-PRC TPM by

resorting to the screw theory. For a limited-DOF parallel manipulator, the motion

of each limb that can be treated as a twist system is guaranteed under some exerted

structural constraints which are termed as a wrench system. The wrench system

is a reciprocal screw system of the twist system for the limb. The mobility of the

manipulator is then determined by the effect of linear combination of the wrench

systems for all limbs.

With ω = [ωx ωy ωz]T and υ = [υx υy υz]

T respectively denoting the vectors for

the angular and linear velocities, the twist of the mobile platform can be defined as

$p = [ωT υT ]T . Considering that a C joint is equivalent to the combination of a

P joint with a coaxial R joint, the connectivity of each limb for a 3-PRC TPM is

equal to 4 since each limb consists of four 1-DOF joints. Hence, the instantaneous

twist $p of the mobile platform can be expressed as a linear combination of the four

instantaneous twists, i.e.,

$p = di$1,i + θ2,i$2,i + θ3,i$3,i + si$4,i, (10)

for i = 1, 2, 3, where θj,i is the intensity and $j,i denotes a unit screw associated

with the j-th joint of the i-th limb with respect to the instantaneous reference frame

P , and

$1,i =

0

s1,i

, (11)

$2,i =

s2,i

ci × s2,i

, (12)

8

$3,i =

s3,i

bi × s3,i

, (13)

$4,i =

0

s4,i

, (14)

can be identified, where sj,i represents a unit vector along the j-th joint axis of the

i-th limb, 0 denotes a 3×1 zero vector, bi =−−→PB, ci =

−→PC = bi − l li0, and

s2,i = s3,i = s4,i = si0 since the R and C joint axes are parallel to each other.

The screws that are reciprocal to all the joint screws of one limb of a 3-PRC TPM

form a 2-system. Hence, two reciprocal screws of the i-th limb can be identified as

two infinite-pitch wrench screws as follows.

$r,1,i =

0

h1,i

, (15)

$r,2,i =

0

h2,i

, (16)

where h1,i and h2,i are two different arbitrary vectors perpendicular to si0 of the i-th

limb. $r,1,i and $r,2,i denote two unit couples of constraints imposed by the joints of

the i-th limb, and are exerted on the mobile platform.

For simplicity, let h1,i lie in the u-v plane and h2,i be vertical to the u-v plane,

respectively, i.e.,

h1,1 = [1 0 0]T ,

h1,2 = [−1

2

√3

20]T ,

9

h1,3 = [−1

2−

√3

20]T ,

h2,1 = h2,2 = h2,3 = [0 0 1]T .

It is observed that the six wrench screws are linearly dependent and form a screw

system of order 3, namely a 3-order wrench system of the mobile platform. Since

the directions of each R and C joint axes satisfy the conditions described earlier,

i.e., they are invariable, the wrench system restricts three rotations of the mobile

platform with respect to the x-, y- and z-axes of the fixed frame at any instant. Thus

leads to a TPM with three translational DOF along the x-, y- and z-axes of the fixed

frame.

It should be noted that the mobility of a 3-PRC TPM can also be determined by

adopting other methods, such as a recent theory of degrees of freedom for complex

spatial mechanisms proposed in [18], or a group-theoretic approach recommended

in [19], etc.

4.2 Elimination of overconstraints of a 3-PRC TPM

As an overconstrained mechanism, the problems of variable friction in passive

joints and large reaction moment in the 3-PRC TPM have to be considered to assure

the mobility of the mobile platform. Otherwise, the mobile platform may not move

or the manipulator cannot work if there are some kinematic errors. In this paper, a

R joint with its axis along the unit vector s5,i is added in each limb of the 3-PRC

TPM to eliminate the overconstrained conditions. It will be demonstrated that the

added R joints are all inactive and do not affect the mobility and kinematics of the

manipulator at all.

For the convenience of assembly, the R joint is added with the common axis of the

P joint in the i-th limb, i.e., s5,i = s1,i and di = ci, which allows the generation

of a 3-CRC parallel manipulator indeed. The DOF number of the 3-CRC parallel

10

manipulator can be given by the general Grubler-Kutzbach criterion as:

F = λ(n − j − 1) +j

∑

i=1

fi = 6 × (8 − 9 − 1) + 15 = 3, (17)

which indicates that it is not an overconstrained manipulator.

In addition, the joint screw of the added R joint can be expressed as:

$5,i =

s5,i

di × s5,i

, (18)

where di is the position vector of the center of the added R joint with respect to the

reference frame P .

Then, the instantaneous twist $p of the mobile platform can be expressed as a linear

combination of the 5 instantaneous twists, i.e.,

$p = di$1,i + θ2,i$2,i + θ3,i$3,i + si$4,i + θ5,i$5,i, (19)

for i = 1, 2, and 3.

The screws that are reciprocal to all the joint screws of a limb of the manipulator

form a 1-system. Hence, one reciprocal screw of the i-th limb can be identified as

an infinite-pitch wrench screw as follows.

$c,i =

0

ki

, (20)

where ki is a unit vector defined by

ki ≡s2,i × s5,i

||s2,i × s5,i||. (21)

It is observed that $c,i in Eq. (20) represents a unit couple of constraints imposed by

11

the joints of the i-th limb, and the couple is exerted on the mobile platform around

the direction of ki.

Taking the reciprocal product of both sides of Eq. (19) with $c,i, yields

[∆$c,i]T $p = 0, (22)

which can be rewritten into the matrix form:

Jc$p = 0, (23)

where

Jc =

kT1 0T

kT2 0T

kT3 0T

3×6

(24)

is called the Jacobian of constraints [20]. Each row in Jc denotes a unit wrench of

constraints imposed by the joints of a limb, the combination of which constrains

the mobile platform to a 3-DOF motion. Hence, if ki ( i = 1, 2 and 3) are linearly

independent, the unique solution to Eq. (23) is

ω = 0, (25)

which exhibits that the 3-CRC manipulator has three translational DOF.

Substituting Eq. (25) into Eq. (19), results in

θ2,is2,i + θ3,is3,i + θ5,is5,i = 0. (26)

In view of the condition that s2,i = s3,i 6= s5, the following two equations can be

derived from Eq. (26):

θ2,i + θ3,i = 0, (27)

12

θ5,i = 0. (28)

Equation (28) reveals that the added R joints in a 3-CRC TPM do not rotate around

the C joints at all and do not influence the mobility and kinematics of the original

3-PRC TPM, but they are introduced to eliminate the internal constraints in a 3-

PRC TPM and lead to a non-overconstrained parallel manipulator as indicated by

Eq. (17).

Additionally, Eq. (27) reveals that the intermediate two R joints always rotate in

the inverse directions at the same angular velocity. It follows that the two R joints

rotate in the same motion range, which provide some guidelines for the selection

of mechanical joints to construct a 3-PRC or a 3-CRC TPM prototype. The CAD

model of a 3-CRC TPM is shown in Fig. 3, where the P joint within the first C joint

is driven by a lead screw linear actuator, that is easy to implement in practice.

It should be noted that the overconstraints of a 3-PRC TPM can be eliminated

by other approaches as well. For example, one R joint can be added between the

original P and R joints to construct a 3-PUC parallel manipulator, which is also

a non-overconstrained manipulator. And it can be shown easily via screw theory

that the 3-PUC parallel manipulator has the same mobility and kinematics with the

original 3-PRC TPM.

5 Stiffness model identification

Concerning a rigid body elastically suspended by elastic devices, if only small dis-

placements from its unpreloaded equilibrium position are considered, the overall

spatial force-deflection relation of the mechanism is linear and described by a 6×6

symmetric positive semidefinite matrix [21], i.e., the stiffness matrix. Generally,

the stiffness characteristics of a parallel manipulator can be described by the 6×6

13

stiffness matrix, which relates the the vector of compliant deformations of the end-

effector to an external static wrench applied on the manipulator [22]. By taking into

account the flexibilities of every compliant elements, the stiffness model of 6-DOF

parallel manipulators with six legs can be established straightforwardly via a proper

investigation [23, 24]. While for those limited-DOF parallel manipulators, it is not

easy to derive their overall stiffness matrices since the rank of conventional Jaco-

bian matrix for a limited-DOF parallel manipulator is less than six. There are also

various examples illustrating how to perform the stiffness modeling in the litera-

tures. For instance, the stiffness of a tripod-based parallel manipulator is modeled

in [25] by decomposing the whole machine structure into two separated substruc-

tures, and formulating the stiffness model of each substructure by means of virtual

work principle. A stiffness model of the 3-DOF CaPaMan parallel manipulator is

established in [26] by taking into account the kinematic and static features of the

three legs in view of the motions of every joint and link. In addition, the stiffness

modeling of four tripod-based parallel manipulators is carried out in [27] with the

consideration of actuator flexibility, and the leg bending and axial deformations,

where the stiffness matrix is derived using the force and infinitesimal motion re-

lationships in each serial limb, while the processes for the derivation of Jacobian

matrices are not given in details.

An observation of the commonly stiffness modeling of limited-DOF parallel ma-

nipulators reveals that it is not obvious what is the best way since these existing ap-

proaches are not intuitive enough. Recently, the screw theoretic approach is adopted

in [28] to derive the stiffness model of a limited-DOF parallel manipulator with the

determination of spring constants of six connecting springs, which is then applied

for the stiffness modeling of the Tricept parallel manipulator with variable-length

legs. In this paper, we will derive the stiffness model of a 3-PRC TPM based upon

the screw theory as well. However, it will be illustrated that the stiffness model of a

limited-DOF parallel manipulator with fixed-length legs is different from that with

14

variable-length legs, since there are nine springs instead of only six ones connecting

the mobile platform to the fixed base.

5.1 Derivation of the Jacobian of actuations

The overall Jacobian of a 3-CRC TPM can be generated by the combination of

the Jacobian of actuations with the Jacobian of constraints, where the former one

has been derived as expressed in Eq. (24), and the latter one can be determined as

follows.

Let the P joint in each limb be locked, then the reciprocal screws of each limb form

a 2-system which includes one screw $r,1,i identified earlier. The other basis screw

being reciprocal to all the passive joint screws of the i-th limb can be identified as

a zero-pitch screw along the direction of the i-th leg, i.e.,

$a,i =

li0

bi × li0

, (29)

where li0 is a unit vector along−−→CiBi.

Taking the reciprocal product of both sides of Eq. (19) with $a,i, results in

[∆$a,i]T $p = dil

Ti0s1,i, (30)

which can be rewritten into the following matrix form:

Ja$p = q, (31)

15

where q = [d1 d2 d3]T denotes the actuated joint rates and

Ja =

(b1×l10)T

lT10

s1,1

lT10

lT10

s1,1

(b2×l20)T

lT20

s1,2

lT20

lT20

s1,2

(b3×l30)T

lT30

s1,3

lT30

lT30

s1,3

3×6

(32)

is called the Jacobian of actuations, which transforms the angular and linear ve-

locities of the mobile platform to the input joint rates. Here, it is assumed that all

singularities have been eliminated from the workspace by the mechanism design

rules proposed in [12].

5.2 Stiffness modeling

Referring to Eqs. (20) and (29), it can be observed that the wrench system, which is

the infinite-pitch reciprocal screw system of constraints, exerts three constraint cou-

ples to the mobile platform with the direction of ki, and the wrench system which is

the zero-pitch reciprocal screw system of actuations imposes three constraint forces

to the mobile platform with the directions just along the legs. This means that each

leg is subject to a force and a couple. With the assumption that the rigidities of the

passive R and C joints and the mobile platform are infinite, the compliances subject

to actuators and legs can be derived as follows.

5.2.1 Compliance subject to actuators

Let us consider the compliance of actuators in their actuation directions. Referring

to Fig. 3, for the lead screw actuation system, the torque τi of the i-th actuator can

be expressed as:

τi = Kα,i∆αi, (33)

16

where Kα,i denotes the torsional stiffness of the actuator and ∆αi represents the

angular deformation.

Through the torque transmission, the force acting on the i-th nut and its linear

displacement can be respectively expressed by [29]:

fi =2τi

dp tan (λ + ψ), (34a)

∆ti = L ∆αi, (34b)

where dp and L respectively denote the pitch diameter and lead of the lead screw,

and λ = tan−1( Lπdp

) and ψ = tan−1(

µc

cos (αt)

)

respectively denote the lead angle and

friction angle, with αt and µc representing the radial angle of thread and coefficient

of friction between screw and the nut, respectively.

Combining Eq. (33) with Eq. (34), allows the derivation of

∆ti = Cifi, (35)

where

Ci =Ldp tan (λ + ψ)

2Kα,i

(36)

denotes the compliance of the i-th linear driving device.

The compliance of the i-th actuator due to the constraint force in the i-th leg’s

direction can be derived as

Cs,i = lTi0s1,iCi. (37)

In view of Eqs. (36) and (37), one can get that

Cs,i =Ldp tan (λ + ψ) lTi0s1,i

2Kα,i

. (38)

17

5.2.2 Compliance subject to legs

The i-th leg CiBi bears a constraint force (F ci ) along the direction of li0 and a

constraint couple (M ci ) around the direction of ki exerted at the position of point

Bi, where the force causes a linear elastic deformation (∆li) of the leg along its axis

direction. As illustrated in Fig. 2, the constraint couple can be decomposed into two

couples with the directions of k1,i and k2,i, which are along and perpendicular to

li0, respectively. Therefore, the i-th constraint couple causes a torsional deformation

∆θi of the i-th leg around the k1,i direction and a bending deformation ∆βi about

the direction of k2,i.

Let Cl,i, Cθ,i, and Cβ,i be the longitudinal, torsional, and bending compliances of

the i-th leg, respectively. Then, the elastic deformations of the i-th leg caused by

the constraint force and couple can be expressed as:

∆li = Cl,iFci =

l

EAF c

i , (39)

∆θi = Cθ,iMci ||k1,i|| =

l

GIp

||k1,i||M ci , (40)

∆βi = Cβ,iMci ||k2,i|| =

l

EIk

||k2,i||M ci , (41)

where k1,i + k2,i = ki, ||k1,i|| = lTi0ki, ||k2,i|| =√

1 − ||k1,i||2, l and A denote the

length and cross section area of each leg, E and G are the Young’s modulus and

shear modulus of elasticity, Ip represents the polar moment of inertia, and Ik is the

moment inertia around an axis parallel to k2,i, respectively.

Then, from Eqs. (39), (40), and (41), one can obtain

Cl,i =l

EA, (42)

Cθ,i =l

GIp

||k1,i||, (43)

Cβ,i =l

EIk

||k2,i||. (44)

18

5.2.3 Stiffness model

From the above discussions, it is observed that the constraint couple of the i-th

limb is equivalent to two decomposed couples along the directions of k1,i and k2,i,

respectively. In view of the inverse relationship between stiffness and compliance,

the stiffness of actuations and constraints can be respectively obtained as

Ka,i = C−1a,i = (Cs,i + Cl,i)

−1, (45a)

Kc,1,i = C−1c,1,i = (Cθ,i)

−1, (45b)

Kc,2,i = C−1c,2,i = (Cβ,i)

−1, (45c)

for i = 1, 2, and 3.

Consequently, the stiffness model of a 3-CRC TPM can be established by consid-

ering that the mobile platform is connected to the fixed base by 3 compliant limbs,

each of which consists of one linear spring and two rotational springs as shown in

Fig. 4.

5.3 Derivation of the overall Jacobian matrix

Since the constraint couple of each limb is equivalently decomposed into two sepa-

rate couples for the convenience of compliant analysis, Eq. (20) can be divided into

$c,i = $c,1,i + $c,2,i, where

$c,1,i =

0

k1,i

, (46)

$c,2,i =

0

k2,i

. (47)

19

Taking the reciprocal product of both sides of Eq. (19) with $c,k,i, leads to

[∆$c,k,i]T $p = 0, (48)

for k = 1 and 2, and i = 1, 2 and 3, which can be rewritten into the matrix form:

Jcr$p = 06×1, (49)

where

Jcr =

kT1,1 0T

kT2,1 0T

kT1,2 0T

kT2,2 0T

kT1,3 0T

kT2,3 0T

6×6

(50)

is the extended Jacobian of constraints.

Then, complementing Eq. (31) with Eq. (49) allows the generation of

qo = J $p, (51)

where qo = [d1 d2 d3 0 0 0 0 0 0]T , and

J =

Ja

Jcr

9×6

(52)

is called the overall Jacobian of a 3-CRC TPM, which includes the effects of both

actuations and constraints.

20

5.4 Stiffness matrix determination

In view of Plucker’s conventions, a twist is expressed in the axis coordinate and

a wrench is described in the ray coordinate. While in above discussions, the twist

$p = [ωT υT ]T is expressed in the Plucker ray coordinate. The conventional twist

can be derived as T = ∆$p = [υT ωT ]T . Assuming that there are no preload

effects on the manipulator, the stiffness model of the TPM is established based on

the following discussions.

Let f = [fx fy fz]T and m = [mx my mz]

T respectively denote an external

force and torque exerted on the mobile platform. The wrench can be expressed as

w = [fT mT ]T in the Plucker ray coordinate. Additionally, let τ a = [τa,1 τa,2 τa,3]T

and τ c = [τc,1,1 τc,2,1 τc,1,2 τc,2,2 τc,1,3 τc,2,3]T represent the reaction forces/torques

of actuations and constraints, respectively. In the absence of gravity, the external

wrench is balanced by the reaction forces/torques exerted by the actuations and

constraints, i.e.,

w = [Ja∆]T τ a + [Jcr∆]T τ c, (53)

where the reaction forces/torques can be expressed as

τ a = χa∆qa, (54a)

τ c = χc∆qc, (54b)

with ∆qa and ∆qc denoting the displacements of actuations and constraints, re-

spectively. And the diagonal matrices are

χa = diag[Ka,1, Ka,2, Ka,3], (55a)

χc = diag[Kc,1,1, Kc,2,1, Kc,1,2, Kc,2,2, Kc,1,3, Kc,2,3 ]. (55b)

Moreover, let ∆x = [∆x ∆y ∆z]T and ∆θ = [∆θx ∆θy ∆θz]T be the infinites-

imal displacements of translation and rotation of the mobile platform with respect

21

to three axes of the reference frame. Then, applying the principle of virtual work

by neglecting the gravitational effect, allows the generation of

wT ∆X = τ Ta ∆qa + τ T

c ∆qc, (56)

where ∆X = [∆xT ∆θT ]T denotes the mobile platform’s twist deformation in the

axis coordinate.

Substituting Eq. (54) into Eq. (56), results in

wT ∆X = ∆qTa χa∆qa + ∆qT

c χc∆qc. (57)

In addition, substituting Eq. (53) into Eq. (56), leads to

τ Ta [Ja∆]∆X + τ T

c [Jcr∆]∆X = τ Ta ∆qa + τ T

c ∆qc, (58)

which holds for any values of τ a and τ c. Consequently, the virtual displacements

of actuations and constraints can be derived as

∆qa = [Ja∆]∆X, (59a)

∆qc = [Jcr∆]∆X. (59b)

Then, substituting Eq. (59) into Eq. (57), one can obtain

wT ∆X = ∆XT [Ja∆]T χa[Ja∆]∆X + ∆XT [Jcr∆]T χc[Jcr∆]∆X. (60)

Since Eq. (60) holds for any arbitrary virtual displacements ∆X, we conclude that

w = K ∆X, (61)

where

K = [Ja∆]T χa[Ja∆] + [Jcr∆]T χc[Jcr∆] (62)

is defined as the 6×6 stiffness matrix of a 3-CRC TPM including the effects of

22

actuations and constraints, which can be further written into the form:

K = [J∆]T χ [J∆], (63)

where the 9×9 diagonal matrix χ = diag[χa χc].

An observation of the units of matrix Ja in Eq. (32) reveals that the last three

columns are dimensionless while the first three ones are related to the unit of length

which is introduced by the position vector bi. It is necessary to homogenize the

units of the Jacobian matrices so as to generate a stiffness matrix and performance

index invariant of the length unit adopted. The dimensionally homogeneous Jaco-

bian of actuations can be achieved by

Jah = JaD, (64)

with

D = diag[

1, 1, 1,1

b,

1

b,

1

b

]

(65)

where the mobile platform radius b is chosen as the characteristic length [30] to

homogenize the dimension of the Jacobian matrix.

In consequence, the overall Jacobian in Eq. (52) becomes

J =

Jah

Jcr

9×6

, (66)

which is homogenous in terms of units.

6 Stiffness evaluation of the TPM

For a given design of a parallel manipulator, the stiffness varies with the variation of

the manipulator configurations within its workspace as well as the direction of the

23

applied wrenches. Once the stiffness model is obtained, it is desired to predict its

stiffness characteristics over the workspace in order to assess whether the design

is satisfied with the stiffness requirements or even further to perform an optimal

design with the stiffness considered especially in the design stage.

As far as the approaches for the stiffness evaluation are concerned, several dif-

ferent performance indices have been proposed and utilized in the literatures. A

simple way to predict the stiffness is to use stiffness factors, i.e., the terms of the

stiffness matrix [22, 25]. Besides, the stiffness can be evaluated using the eigen-

value of stiffness matrix which is experienced in the direction of the corresponding

eigenvector [22, 31]. It has been shown that the stiffness is bounded by the mini-

mum and maximum eigenvalues of the stiffness matrix [32]. Based on this concept,

the stiffness has been predicted by the minimum, maximum, and average eigen-

values, even magnitude of the ratio of the maximum and minimum eigenvalues of

the stiffness matrix [32]. Additionally, the determinant of stiffness matrix, that is

the product of its eigenvalues, has been adopted to assess the stiffness of parallel

manipulators [26, 33]. And the trace of the compliance matrix (the inverse of the

stiffness matrix), which is the sum of its eigenvalues, has been used to measure

the kinetostatic property of tripod-based parallel manipulators [27]. Furthermore,

similar to the condition number of Jacobian matrix, the condition number of the

stiffness matrix has been introduced, based on which, a global stiffness index de-

fined as the the inverse of the condition number of stiffness matrix integrated over

the reachable workspace and divided by the workspace volume is presented to as-

sess the stiffness of a 3-DOF spherical parallel manipulator [34] and the hexaslide

machine tools [24].

Among these usually used stiffness performance indices, the stiffness factors are

preferred to be applied to evaluate the stiffness matrix only with a diagonal form.

Because for a stiffness matrix with the generic form, the off-diagonal terms cou-

ple the forces/torques applied in the corresponding directions, and the individual

24

stiffness factors can not totally reflect the stiffness property in any directions. Con-

cerning the determinant or trace of the stiffness matrix, it can not distinguish the

situations in which the manipulator has a very low stiffness in one direction while

a very high stiffness in another one, that leads to a high value of the determinant or

trace although the low stiffness prohibits the use of the manipulator for some situ-

ations such as machine tool applications. In consequence, neither the determinant

nor the trace of the stiffness matrix is a good choice for the stiffness evaluation of

parallel manipulators. And the same problem arises for the average of the eigenval-

ues of the stiffness matrix. Additionally, although the condition number of stiffness

matrix can indicate the ill-conditioning of the stiffness matrix, it does not provide

enough information of the stiffness values. For a precise manipulation, the mini-

mum stiffness over the workspace should be larger than a specified value in order

to ensure the accuracy of the manipulation everywhere in the workspace. Hence,

the minimum and maximum values of stiffness and their variances appear to be the

most direct and reasonable indices for the stiffness evaluation, which are adopted

as stiffness performance indices in this paper.

The architectural parameters of a 3-CRC TPM are shown in Table 1. By adopting

a numerical searching method described in [35], the reachable workspace of the

manipulator is generated as illustrated in Fig. 5. In addition, the physical parameters

of the designed TPM are elaborated in Table 2.

According to Eqs. (38) and (42)—(44), the compliance parameters for the TPM

can be calculated. Let the home position of the mobile platform be in the case of

mid-stroke of linear actuators, i.e, di = 0 (i = 1, 2, 3), in which the stiffness matrix

25

is calculated as follows.

K0 =

0.3653 0 0 0 −0.4871 0

0 0.3653 0 0.4871 0 0

0 0 1.2990 0 0 0

0 0.4871 0 0.7277 0 0

−0.4871 0 0 0 0.7277 0

0 0 0 0 0 0.1132

× 105, (67)

where the units of terms are N/m for {K011, K0

22, K033}, N/rad for {K0

15, K024},

N·m/m for {K051, K0

42}, and N·m/rad for {K044, K0

55, K066}.

In order to ensure the accuracy of the TPM manipulation anywhere within the

workspace, the minimum stiffness over the workspace should be larger than a spec-

ified value. It follows that the minimum stiffness is the most important index for the

3-CRC TPM. In this subsection, both the minimum and maximum eigenvalues ob-

tained through the conventional eigenvalue decomposition of the stiffness matrix

are used as stiffness indices to have a global view of the stiffness values over the

workspace.

The distributions for the minimum (Kmin) and maximum (Kmax) stiffness in the

planes of z = −0.30 m, z = z0 = −0.40 m (home position height) and z = −0.50 m

are illustrated in Fig. 6. The first observation from Fig. 6 is that, similar to the

reachable workspace, the distribution of stiffness in a x-y plane is 120 degree-

symmetrical about the axial directions of three actuated P joints. Additionally, in

a particular height plane, the lowest value of minimum stiffness occurs around the

boundary of the workspace, so does the highest value of the maximum stiffness,

which arise from the reason that the manipulator approaches singular when it comes

26

near the workspace boundary. Moreover, both the lowest and the highest stiffness

values occur on the lowest height plane in the workspace.

Since around the boundary of the reachable workspace, the TPM takes on a bad

stiffness property, it is reasonable to restrict the TPM to operate in a subworkspace

located within the reachable workspace. According to the manipulator tasks and

performances, there are several ways to define this subworkspace. Here, for the

main purpose of examining the influences of the kinematic parameters on stiffness

values, the subworkspace is assigned as a cylinder shape usable workspace with the

radius of 0.08 m and height of 0.16 m, whose geometry center lies at the home po-

sition point of the mobile platform. In view of the stiffness distributions illustrated

in Fig. 6, it can be deduced that both the lowest value of minimum stiffness and the

highest value of maximum stiffness arise on the boundary of the lowest plane of

the usable workspace, i.e., a circle of radius 0.08 m at the height of (z0 − 0.08) m.

This deduction will greatly simplify the procedure for the stiffness assessment.

Following the aforementioned process, the analytical expression of stiffness value

for the TPM can be easily obtained via a symbolic calculation tool such as Math-

ematica or Matlab, etc. However, it is a time-consuming work to derive the an-

alytical solution, and the final symbolic expression is too long. For example, an

attempt to derive the analytical solution of stiffness values via Matlab running on a

personal computer (Intel Pentium 4 CPU 3.00GHz, 512MB RAM) with Microsoft

Windows XP operating system exhibits that it needs more than 10 hours to obtain

solely the analytical solutions for the stiffness values. On the other hand, the nu-

merical assessment of stiffness is quite straightforward and simple especially due to

the aforementioned deduction. For instance, by equally sampling one circle (360◦)

into 60 segments, and the range of actuators layout angles (90◦) into 90 segments,

the calculation of the maximum and minimum stiffness values (Fig. 7(d)) requires

only less than one minute. The high computational efficiency of numerical method

is very obvious. Hence, in order to have a quick view of the TPM stiffness, the

27

numerical evaluation approach is adopted in this paper.

Along with the varying of architectural parameters of the 3-CRC TPM, the ten-

dency of variation on the minimum and maximum stiffness over the usable workspace

is depicted in Figs. 7(a)—7(d), from which it is clear to see the impacts of design

parameters (a, b, l, and α) on the stiffness property of the manipulator. It can be

observed from Figs. 7(a) and 7(b) that the increasing of the fixed platform size

has almost the same effect as the decreasing of the mobile platform size on the

stiffness properties as anticipated. In view of Fig. 7(c), one can see that as the in-

creasing of the leg length, the minimum stiffness values decrease monotonously,

while the maximum stiffness does not vary uniformly which attains the lowest

value at l = 0.335 m. Additionally, as the increasing of the actuators layout an-

gle, the maximum stiffness decreases and reaches at a lowest value around 60◦,

while the variation of the minimum stiffness is not monotonously as elaborated in

Fig. 7(d). It appears that the minimum stiffness arises to the highest value around

28◦ of actuators layout angle. It is noticeable that the lowest and height values

of the extremum stiffness can be generated alternatively by differentiating analyt-

ical formulas of stiffness values with respect to the design parameters once the

analytical-form solutions are derived. However, it is quite a time-consuming work,

which is not preferred in this paper.

Besides, these figures can also be used to design the manipulator in order to assure

that the lowest stiffness is larger than a predefined value. For example, referring

to Fig. 7(c), if the minimum stiffness is designed as Kmin > 1000 N/m, then the

length of the legs should be designed as l < 0.56 m accordingly.

28

7 Conclusions

In this paper, the mobility determination and stiffness analysis for a 3-DOF TPM

are carried out based upon the screw theory. A 3-CRC TPM is constructed from

a previously presented overconstrained 3-PRC TPM by adding a revolute joint in

each limb. It has been demonstrated that the conversion does not bring any im-

pacts on the mobility and kinematics of original manipulator since the added joints

are idle indeed, but it can eliminate the overconstrained conditions of a 3-PRC

TPM. The stiffness matrix is derived based on an overall Jacobian via the theory

of reciprocal screws in view of the effects of actuations and constraints. And the

rigidities in both actuators and legs are taken into account to establish the stiffness

model of the manipulator. Through a survey of the commonly used stiffness perfor-

mance criteria, the minimum and maximum eigenvalues of the stiffness matrix over

a cylinder usable workspace are adopted to evaluate the stiffness of the TPM. Fur-

thermore, the variation tendencies of stiffness within the workspace are illustrated

and the impacts of variation for design parameters on the stiffness characteristics

are presented.

The main contribution of this paper is the conversion of an overconstrained 3-PRC

TPM into a non-overconstrained 3-CRC TPM without affecting the mobility and

kinematic properties, and the stiffness modeling of the 3-DOF TPM via an alter-

native approach, i.e., the screw theoretic approach, along with the quantitatively

evaluation of the TPM stiffness. This paper provides a basis for the architectural

design of a 3-CRC TPM with stiffness properties taken into account. Moreover,

the methodology presented here can be extended for the stiffness analysis of other

types of limited-DOF parallel manipulators (e.g., with architectures of 3-PUU, 3-

PRS, 3-RPS, etc.) as well.

29

Acknowledgments

The authors appreciate the fund support from the research committee of University

of Macau under grant no.: RG068/05-06S/07R/LYM/FST and Macao Science and

Technology Development Fund under grant no.: 069/2005/A.

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33

Figure Captions

Fig. 1. Schematic representation of a 3-PRC TPM.

Fig. 2. Representation of vectors.

Fig. 3. A 3-CRC TPM.

Fig. 4. Stiffness model of a 3-CRC TPM.

Fig. 5. Reachable workspace of a 3-CRC TPM.

Fig. 6. The distributions for minimum and maximum stiffness in the planes of (a)(b)

z = −0.30 m, (c)(d) z = −0.40 m, and (e)(f) z = −0.50 m, respectively.

Fig. 7. Stiffness indices versus design parameters of (a) the fixed platform size, (b)

the mobile platform size, (c) the length of legs, and (d) actuators layout angle.

34

1A

2A

3A 1C

3CD

O x

yz2ϕα

α

α

10l20l

30l

2B

P

wv

u 1 1( )B B'

3 3( )B B '

2( )B '

Base platform

Mobile platform

P joint

R joint

C joint

3ϕ

2C

Fig. 1. Schematic representation of a 3-PRC TPM.

zy

x

ia

α

iAiM

iN

iC

O

0il l

p

iB

iB'Pu

vw

2,is

1,5,(

)i

i

ss

3, 4,( )i i

s s

ik1,ik

2,iki

B

Fig. 2. Representation of vectors.

35

Fig. 3. A 3-CRC TPM.

2C

α1A

2A

3A

3C

O x

yz

α

α

2B

P 1B

3B

1C

,1aK

,1,1cK

,2,1cK

Fig. 4. Stiffness model of a 3-CRC TPM.

36

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

x (m)y (m)

z (

m)

Home position height

Fig. 5. Reachable workspace of a 3-CRC TPM.

37

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

1800

2000

2200

2400

2600

x (m)y (m)

Km

in

(a)

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

1

1.1

1.2

1.3

x 105

x (m)y (m)

Km

ax

(b)

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

1000

1500

2000

2500

x (m)y (m)

Km

in

(c)

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

1.25

1.3

1.35

1.4

1.45

x 105

x (m)y (m)

Km

ax

(d)

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

500

1000

1500

2000

x (m)y (m)

Km

in

(e)

−0.1−0.05

00.05

0.1

−0.1−0.050

0.050.1

1.72

1.74

1.76

1.78

1.8

x 105

x (m)y (m)

Km

ax

(f)

Fig. 6. The distributions for minimum and maximum stiffness in the planes of (a)(b)

z = −0.30 m, (c)(d) z = −0.40 m, and (e)(f) z = −0.50 m, respectively.

38

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80

1000

2000

3000

4000

5000

a (m)

Km

in

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.81

2

3

4

5

6x 10

5

Km

ax

Kmin

Kmax

(a)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2000

4000

b (m)

Km

in

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

2

4

6x 10

5

Km

ax

Kmin

Kmax

(b)

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

2000

4000

6000

8000

l (m)

Km

in

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.71

1.5

2

2.5x 10

5

Km

ax

Kmin

Kmax

0.56

(c)

0 10 20 30 40 50 60 70 80 90900

1000

1100

1200

1300

1400

1500

α (deg.)

Km

in

0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6x 10

6

Km

ax

Kmin

Kmax

(d)

Fig. 7. Stiffness indices versus design parameters of (a) the fixed platform size, (b) the

mobile platform size, (c) the length of legs, and (d) actuators layout angle.

39

Tables

Table 1

Architectural parameters of a 3-CRC TPM

Parameter Value Parameter Value

a 0.6 m α 45 ◦

b 0.3 m ϕ1 0 ◦

l 0.5 m ϕ2 120 ◦

dmax 0.4 m ϕ3 240 ◦

smax 0.2 m

Table 2

Physical parameters of a 3-CRC TPM

Parameter Value Parameter Value

E 2.03×1011 N/m2 Kα,i 1.45×10

6 N·m/rad

G 7.85×1010 N/m2 dp 31.75×10

−3 m

A 4.52×10−4 m2 L 6.35×10

−3 m

Ip 3.26×10−8 m4 αt 14.5 ◦

Iy 1.63×10−8 m4 µc 0.20

40