KINEMATIC AND SINGULARITY ANALYSIS OF THE

HYDRAULIC SHOULDER

A 3-DOF Redundant Parallel Manipulator

H. Sadjadian and H. D. Taghirad

Advanced Robotics and Automated Systems (ARAS),

Electrical Engineering Department,

K.N. Toosi University of Technology

P.O. Box 16315-1355, Tehran, Iran

Keywords: Parallel Manipulator, Kinematic Modeling, Forward Kinematics, Jacobian, Singularity Analysis.

Abstract: In this paper, kinematic modeling and singularity analysis of a three DOF redundant parallel manipulator

has been elaborated in detail. It is known, that on the contrary to series manipulators, the forward kinematic

map of parallel manipulators involves highly coupled nonlinear equations, whose closed-form solution

derivation is a real challenge. This issue is of great importance noting that the forward kinematics solution is

a key element in closed loop position control of parallel manipulators. Using the novel idea of kinematic

chains recently developed for parallel manipulators, both inverse and forward kinematics of our parallel

manipulator are fully developed, and a closed-form solution for the forward kinematic map of the parallel

manipulator is derived. The closed form solution is also obtained in detail for the Jacobian of the

mechanism and singularity analysis of the manipulator is performed based on the computed Jacobian.

1 INTRODUCTION

Over the last two decades, parallel manipulators

have been among the most considerable research

topics in the field of robotics. A parallel manipulator

typically consists of a moving platform that is

connected to a fixed base by several legs. The

number of legs is at least equal to the number of

degrees of freedom (DOF) of the moving platform

so that each leg is driven by no more than one

actuator, and all actuators can be mounted on or near

the fixed base. These robots are now used in real-life

applications such as force sensing robots, fine

positioning devices, and medical applications

(Merlet, 2002).

In the literature, mostly 6 DOF parallel mechanisms

based on the Stewart-Gough platform are analyzed

(Joshi et al., 2003). However, parallel manipulators

with 3 DOF have been also implemented for

applications where 6 DOF are not required, such as

high-speed machine tools. Recently, 3 DOF parallel

manipulators with more than three legs have been

investigated, in which the additional legs separate

the function of actuation from that of constraints at

the cost of increased mechanical complexity (Joshi

et al., 2003). Complete kinematic modeling and

Jacobian analysis of such mechanisms have not

received much attention so far and is still regarded

as an interesting problem in parallel robotics

research. It is known that unlike serial manipulators,

inverse position kinematics for parallel robots is

usually simple and straight-forward. In most cases

joint variables may be computed independently

using the given pose of the moving platform. The

solution to this problem is in most cases uniquely

determined. But forward kinematics of parallel

manipulators is generally very complicated. Its

solution usually involves systems of nonlinear

equations which are highly coupled and in general

have no closed form and unique solution. Different

approaches are provided in literature to solve this

problem either in general or in special cases. There

are also several cases in which the solution to this

problem is obtained for a special or novel

architecture (Baron et al., 2000, Merlet, 96, Song et

al., 2001, Bonev et al., 2001). Two such special 3

DOF constrained mechanisms have been studied in

(Siciliano, 99, Fattah et al., 2000), where kinematics,

Jacobian and dynamics have been considered for

125

Sadjadian H. and D. Taghirad H. (2005).

KINEMATIC AND SINGULARITY ANALYSIS OF THE HYDRAULIC SHOULDER - A 3-DOF Redundant Parallel Manipulator.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 125-131

DOI: 10.5220/0001185301250131

Copyright

c

SciTePress

such manipulators. Joshi and Tsai (Joshi et al., 2003)

performed a detailed comparison between a 3-UPU

and the so called Tricept manipulator regarding the

kinematic, workspace and stiffness properties of the

mechanisms. In general, different solutions to the

forward kinematics problem of parallel manipulators

can be found using numerical or analytical

approaches, or closed form solution for special

architectures (Didrit et al., 98, Dasgupta et al.,

2000).

In this paper, complete kinematic modeling has been

performed and a closed-form forward kinematics

solution is obtained for a three DOF actuator

redundant hydraulic parallel manipulator. The

mechanism is designed by Hayward (Hayward, 94),

borrowing design ideas from biological manipulators

particularly the biological shoulder. The interesting

features of this mechanism and its similarity to

human shoulder have made its design unique, which

can serve as a basis for a good experimental setup

for parallel robot research.

In a former study by the authors, different numerical

approaches have been used to solve the forward

kinematic map of this manipulator (Sadjadian et al.,

2004). The numerical approaches are an alternative

to estimate the forward kinematic solution, in case

such solutions cannot be obtained in closed form. In

this paper, however, the novel idea of kinematic

chains developed for parallel manipulators structures

(Siciliano, 99, Fattah et al., 2000), is applied for our

manipulator, and it is observed that in a systematic

manner the closed form solution for this manipulator

can also be obtained in detail.

The paper is organized as following: Section 2

contains the mechanism description. Kinematic

modeling of the manipulator is discussed in section

3, where inverse and forward kinematics is studied

and the need for appropriate method to solve the

forward kinematics is justified. In section 4, The

Jacobian matrix of the manipulator is derived

through a complete velocity analysis of the

mechanism, and finally, in section 5, singularity

analysis is performed using the configuration-

dependent Jacobian.

2 MECHANISM ESCRIPTION

A schematic of the mechanism, which is currently

under experimental studies in ARAS Robotics Lab,

is shown in Fig. 1. The mobile platform is

constrained to spherical motions. Four high

performance hydraulic piston actuators are used to

give three degrees of freedom in the mobile

platform. Each actuator includes a position sensor of

LVDT type and an embedded Hall Effect force

sensor. The four limbs share an identical kinematic

structure. A passive leg connects the fixed base to

the moving platform by a spherical joint, which

suppresses the pure translations of the moving

platform. Simple elements like spherical and

universal joints are used in the structure. A complete

analysis of such a careful design will provide us with

required characteristics regarding the structure itself,

its performance, and the control algorithms.

From the structural point of view, the shoulder

mechanism which, from now on, we call it "the

Hydraulic Shoulder" falls into an important class of

robotic mechanisms called parallel robots. In these

robots, the end effector is connected to the base

through several closed kinematic chains. The

motivation behind using these types of robot

manipulators was to compensate for the

shortcomings of the conventional serial manipulators

such as low precision, stiffness and load carrying

capability. However, they have their own

disadvantages, which are mainly smaller workspace

and many singular configurations. The hydraulic

shoulder, having a parallel structure, has the general

features of these structures. It can be considered as a

shoulder for a light weighed seven DOF robotic arm,

which can carry loads several times, its own weight.

Simple elements, used in this design, add to its

lightness and simplicity. The workspace of such a

mechanism can be considered as part of a spherical

surface. The orientation angles are limited to vary

between -π/6 and π/6. No sensors are available for

measuring the orientation angles of the moving

platform which justifies the importance of the

forward kinematic map as a key element in feedback

position control of the shoulder with the LVDT

position sensors used as the output of such a control

scheme.

3 KINEMATICS

Fig. 2 depicts a geometric model for the hydraulic

shoulder manipulator which will be used for its

kinematics derivation.

The parameters used in kinematics can be defined

as:

ib

CAl = , CPl

p

=

4

y

id

PPl =

4

z

ik

PPl =

:α The angle between

4

CA and

0

y

:C Center of the reference frame

:P Center of the moving plate

:

i

l Actuator lengths i=1, 2, 3, 4

:

P

i

Moving endpoints of the actuators

:

A

i

Fixed endpoints of the actuators

ICINCO 2005 - ROBOTICS AND AUTOMATION

126

1

l

2

l

3

l

4

l

C

1

A

2

A

3

A

4

A

1

P

2

P

d

l

k

l

0

X

4

X

α

0

Y

4

Y

0

Z

4

Z

b

l

p

l

P

}{A

}{B

Figure 1: The hydraulic shoulder in movement

Two coordinate frames are defined for the purpose

of analysis. The base coordinate frame {A}:

000

zyx is

attached to the fixed base at point C (rotation center)

with its

0

z

-axis perpendicular to the plane defined

by the actuator base points

4321

AAAA and an

0

x -axis

parallel to the bisector of angle ∠A

1

CA

4

. The

second coordinate frame {B}:

444

zyx

is attached to

the center of the moving platform P with its z-axis

perpendicular to the line defined by the actuators

moving end points (P

1

P

2

) along the passive leg. Note

that we have assumed that the actuator fixed

endpoints lie on the same plane as the rotation center

C. The position of the moving platform center P is

defined by:

T

zyx

A

pppp ],,[=

(1)

Also, a rotation matrix

B

A

R is used to define the

orientation of the moving platform with respect to

the base frame:

)(θ)R(θ)R(θR

xxyyzz

=

B

A

R

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−+

+−

=

xyxyy

xzxyzxzxyzyz

xzxyzxzxyzyz

ccscs

sccssccssscs

sscsccsssccc

θθθθθ

θθθθθθθθθθθθ

θθθθθθθθθθθθ

(2)

where

zyx

θ

θ

θ

,, are the orientation angles of the

moving platform denoting rotations of the moving

frame about the fixed

,, y

x

and

z

axes respectively.

Also

θ

c and

θ

s denote )cos(

θ

and )sin(

θ

respectively.

With the above definitions, the

44 × transformation

matrix

B

A

T is easily found to be:

⎥

⎦

⎤

⎢

⎣

⎡

=

10

pR

T

A

B

A

B

A

(3)

Hence, the position and orientation of the moving

platform are completely defined by six variables,

from which, only three orientation angles

zyx

θ

θ

θ

,, are

independently specified as the task space variables

of the hydraulic shoulder.

Figure 2: A geometric model for the hydraulic shoulder

manipulator

3.1 Inverse Kinematics

In modeling the inverse kinematics of the hydraulic

shoulder we must determine actuator lengths (

i

l ) as

the actuator space variables given the task space

variables

zyx

θ

θ

θ

,, as the orientation angles of the

moving platform. First, note that the passive leg

connecting the center of the rotation to the moving

platform can be viewed as a 3-DOF open-loop chain

by defining three joint variables

,,

21

θ

θ

and

3

θ

as the

joint space variables of the hydraulic shoulder.

Hence, applying the Denavit-Hartenberg (D-H)

convention, the transformation

B

A

T can also be

written as:

B

A

B

A

TTTTT

3

33

2

22

1

11

).().().(

θθθ

=

(4)

The D-H transformation matrices

j

i

T are computed

using the coordinate systems for the passive leg in

Fig. 3, according to the D-H convention. As shown

in Fig. 3, the

0

x axis of frame {A} points along the

first joint axis of the passive leg; the first link frame

is attached to the first moving link with its

1

x axis

pointing along the second joint axis of the passive

leg; the second link frame is attached to the second

moving link with its

2

x axis pointing along the third

joint axis of the passive leg; and the third link frame

is attached to the moving platform in accordance

with the D-H convention. Using the above frames,

the D-H parameters of the passive leg are found as

in Table (1).

Table 1: D-H Parameters for the passive supporting leg

i

1−i

α

1−i

a

i

d

i

θ

1

°

90 0 0

1

θ

2

°

90

0 0

2

θ

3

°

90 0 0

3

θ

B 0 0

p

l 0

KINEMATIC AND SINGULARITY ANALYSIS OF THE HYDRAULIC SHOULDER: A 3-DOF Redundant Parallel

Manipulator

127

1

l

2

l

3

l

4

l

C

1

A

2

A

3

A

4

A

1

P

2

P

d

l

k

l

0

X

4

X

α

0

Y

4

Y

0

Z

4

Z

b

l

p

l

P

}{A

}{B

3

X

2

Y

1

Z

1

X

2

Z

3

Y

1

Y

2

X

3

Z

Figure 3: D-H Frame attachments for the passive

supporting leg

Using the D-H parameters in Table (1), the D-H

transformation matrices in (4) can be found as:

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

=

1000

00

0100

00

11

11

1

θθ

θθ

cs

sc

T

A

,

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

=

1000

00

0100

00

22

22

2

1

θθ

θθ

cs

sc

T

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

=

1000

00

0100

00

33

33

3

2

θθ

θθ

cs

sc

T

,

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

1000

100

0010

0001

3

p

B

l

T

(5)

Substituting (5) into (4) yields:

⎥

⎦

⎤

⎢

⎣

⎡

=

10

pR

T

A

B

A

B

A

(6)

Where:

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−−−

−

+−+

=

213132131321

23232

213132131321

θθθθθθθθθθθθ

θθθθθ

θθθθθθθθθθθθ

ssccscsscccs

csscs

sccssccssccc

R

B

A

(7)

and:

[]

T

ppp

A

sslclsclp

21221

θθθθθ

=

(8)

For the inverse kinematics, the three independent

orientation angles in (2) are given. Hence, equating

(2) to (7) yields:

)(cos)(cos

1

)3,2(

1

2 xzxyzB

A

sccssR

θθθθθθ

−==

−−

(9)

Once

2

θ

is known, we can solve for

1

θ

and

3

θ

as:

),(2tan

2

)3,1(

2

)3,3(

1

θθ

θ

s

R

s

R

A

B

A

B

A

=

(10)

And

),(2tan

2

)1,2(

2

)2,2(

3

θθ

θ

s

R

s

R

A

B

A

B

A

−

=

(11)

Provided that 0

2

≠

θ

s .Having the joint space

variables

,,

21

θ

θ

and

3

θ

in hand, we can easily solve

for the position of the moving platform using (8).

Now, in order to find the actuator lengths, we write a

kinematic vector-loop equation for each actuated leg

as:

ii

B

B

AA

iii

apRpslL −+== .

(12)

where

i

l is the length of the

th

i actuated leg and

i

s is

a unit vector pointing along the direction of the

th

i actuated leg. Also, p

A

is the position vector of the

moving platform and

B

A

R is its rotation matrix.

Vectors

i

a and

i

B

p denote the fixed end points of

the actuators (A

i

) in the base frame and the moving

end points of the actuators respectively, written as:

(

)

T

A

Aa 0cosα

l

sinα

l

bb

11

−

== ,

(

)

T

A

Aa 0cosα

l

sinα

l

bb

22

−

−== ,

(

)

T

A

Aa 0cosα

l

sinα

l

bb

33

−== ,

(

)

T

A

Aa 0cosα

l

sinα

l

bb

44

== ,

and

(13)

(

)

T

B

p

ll

0

kd

1

−−

=

,

(

)

T

B

p

ll

0

kd

2

−

= ,

(14)

Hence, the actuator lengths

i

l can be easily

computed by dot-multiplying (12) with itself to

yield:

][][.

2

ii

B

B

AAT

ii

B

B

AA

ii

T

i

apRpapRplLL −+−+==

(15)

Writing (15) four times with the corresponding

parameters given in (7),(8),(13) and (14), and

simplifying the results yields:

325321314213221

2

1

)(

θθθθθθθθθθ

sskscccsksckckkl +−+++=

(16-a)

325321314213221

2

2

)(

θθθθθθθθθθ

sskscccsksckckkl +−−−+=

(16-b)

325321314213221

2

3

)(

θθθθθθθθθθ

sskscccsksckckkl +−+−−=

(16-c)

325321314213221

2

4

)(

θθθθθθθθθθ

sskscccsksckckkl +−−+−=

(16-d)

where:

222

1

)(

kpdb

llllk −++=

)cos()(2

2

α

kpb

lllk

−

=

)sin()(2

3

α

kpb

lllk

−

−

=

)sin(2

4

α

db

llk

=

)cos(2

5

α

db

llk

−

=

(17)

Finally, the actuator lengths are given by the square

roots of (16), yielding actuator space variables as the

unknowns of the inverse kinematics problem.

3.2 Forward Kinematics

Forward kinematics is undoubtedly a basic element

in modeling and control of the manipulator. In

forward kinematic analysis of the hydraulic

shoulder, we shall find all the possible orientations

of the moving platform for a given set of actuated

leg lengths. Equation (16) can also be used for the

ICINCO 2005 - ROBOTICS AND AUTOMATION

128

forward kinematics of the hydraulic shoulder but

with the actuator lengths as the input variables. In

fact, we have four nonlinear equations to solve for

three unknowns. First, we try to express the moving

platform position and orientation in terms of the

joint variables

,,

21

θ

θ

and

3

θ

using (7)-(8). As it is

obvious from (16), the only unknowns are the joint

variables

,,

21

θ

θ

and

3

θ

, since actuator lengths are

given and all other parameters are determined by the

geometry of the manipulator. Hence, we must solve

the equations for six unknowns from which only

three are independent. Summing (16-a) and (16-b)

we get:

325221

2

2

2

1

222

θθθ

sskckkll ++=+

(18)

Similarly adding (16-c) and (16-d) yields:

325221

2

4

2

3

222

θθθ

sskckkll +−=+

(19)

Subtracting (19) from (18), we can solve for

2

θ

c as:

2

2

4

2

3

2

2

2

1

2

4k

llll

c

−−+

=

θ

(20)

Substituting (20) into the trigonometric

identity

1

2

2

2

2

=+

θθ

cs , we get:

2

22

1

θθ

cs −±=

(21)

Having

2

θ

s and

2

θ

c in hand, we can solve for

3

θ

s

from (18) as:

25

221

2

2

2

1

3

2

22

θ

θ

θ

sk

ckkll

s

−−+

=

(22)

Similarly:

2

33

1

θθ

sc −±=

(23)

To solve for the remaining unknowns,

1

θ

c and

1

θ

s ,

we sum (16-b) and (16-c) to get:

3252131

2

3

2

2

222

θθθθ

ssksckkll +−=+

(24)

Having computed

2

θ

s and

3

θ

s , we obtain:

23

2

3

2

23251

1

2

22

θ

θθ

θ

sk

llsskk

c

−−+

=

(25)

And finally:

2

11

1

θθ

cs −±=

(26)

Hence, the joint space variables are given by:

),(2tan

111

θ

θ

θ

csa=

),(2tan

222

θ

θ

θ

csa=

(27)

),(2tan

333

θ

θ

θ

csa=

Also, the moving platform position p

A

and

orientation

B

A

R are found using (7)-(8). The final

step is to solve for the orientation angles θ

x,

θ

y

and θ

z

using (3) which completes the solution process to

the forward kinematics of the hydraulic shoulder. It

should be noted that there are some additional

erroneous solutions to the forward kinematics as

stated above due to several square roots involved in

the process. These solutions must be identified and

omitted. Another important assumption made in our

solution procedure was that all four actuator fixed

endpoints are coplanar, just as the actuator moving

endpoints.

4 JACOBIAN ANALYSIS

The Jacobian matrix of a 3-DOF parallel

manipulator relates the task space linear or angular

velocity to the vector of actuated joint rates in a way

that it corresponds to the inverse Jacobian of a serial

manipulator. In this section, we derive the Jacobian

for the Hydraulic shoulder as a key element in

singularity analysis and position control of this

manipulator. For the Jacobian analysis of the

Hydraulic shoulder, we must find a relationship

between the angular velocity of the moving

platform,

ω , and the vector of leg rates as the

actuator space variables,

[]

T

llll

4321

=l , so that:

ωl J=

(28)

From the above definition, it is easily observed that

the Jacobian for the Hydraulic shoulder will be a

34

×

rectangular matrix as expected, regarding the

mechanism as an actuator redundant manipulator.

Using the same idea of mapping between actuator,

joint and task space, we find that the Jacobian

depends on the actuated legs as well as the passive

supporting leg. Therefore, we first derive a

64

×

Jacobian,

l

J , relating the six-dimensional velocity

of the moving platform,

v , to the vector of actuated

leg rates,

l

. Then, we find the 36 × Jacobian of the

passive supporting leg,

p

J The Jacobian of the

Hydraulic shoulder will be finally derived as:

pl

JJJ

=

(29)

4.1 Jacobian of the actuated legs

The Jacobian of the actuated legs,

l

J , relates the six-

dimensional velocity of the moving platform,

v , to

the vector of actuated leg rates,

l

, such that:

vl

l

J=

(30)

We can write the six-dimensional moving platform

velocity as:

[]

T

zyxzyx

A

ppp

p

ωωω

=

⎥

⎦

⎤

⎢

⎣

⎡

=

ω

v

(31)

where p

A

is the velocity of the moving platform

center and

ω is the angular velocity of the moving

platform. Differentiating the kinematic vector-loop

equation (12) with respect to time we get:

i

B

B

AA

iiiii

pRplssl ×+=×+ ω

)(

ω

(32)

where

i

ω

is the angular velocity of the i

th

leg written

in the base frame. Dot multiplying (32) by

i

s

we

have:

ω

T

ii

B

B

AAT

ii

spRpsl )( ×+=

(33)

KINEMATIC AND SINGULARITY ANALYSIS OF THE HYDRAULIC SHOULDER: A 3-DOF Redundant Parallel

Manipulator

129

writing the above equation four times for each

actuated leg and comparing the result to (30) gives

the actuated legs Jacobian as:

64

424

313

222

111

)(

)(

)(

)(

×

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

×

×

×

×

=

TB

B

AT

TB

B

AT

TB

B

AT

TB

B

AT

l

spRs

spRs

spRs

spRs

J

(34)

4.2 Jacobian of the passive leg

In order to find the manipulator Jacobian, we need to

find a relationship between the six-dimensional

velocity vector of the moving platform,

v , and the

angular velocity of the moving platform,

ω . First,

by differentiating (8) with respect to time we get:

θ

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

=

0

00

0

2121

2

2121

θθθθ

θ

θθθθ

cslscl

sl

cclssl

p

pp

p

pp

A

(35)

where

[]

T

321

θθθ

=θ is the vector of passive joint

rates. The angular velocity of the moving platform

can also be expressed as:

1−

=

B

A

B

A

RR

ω

(36)

Substituting

B

A

R from (7) and computing (36), we

have:

θω

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−=

211

2

211

0

01

0

θθθ

θ

θθθ

ssc

c

scs

(37)

Solving (37) for θ

and substituting it in (35) yields:

ω

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

=

0

0

0

212

2121

221

θθθ

θθθθ

θθθ

slpclpc

slpcslps

lpcslps

p

A

(38)

Complementing the above equation with the identity

map

ωω

3

I= , we finally obtain:

ω

ω

v

p

A

J

p

=

⎥

⎦

⎤

⎢

⎣

⎡

=

(39)

where:

36

212

2121

221

100

010

001

0

0

0

×

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

−

=

θθθ

θθθθ

θθθ

sclcl

sclssl

clssl

J

pp

pp

pp

p

(40)

Having

l

J and

p

J in hand, the Hydraulic shoulder

Jacobian,

34×

J

will be easily found using (29).

5 SINGULARITY ANALYSIS

As shown in the previous section, the linear

velocities of the actuators

l

are related to the

angular velocity of the moving platform

ω

by (28),

in which

J was the 34

×

Jacobian matrix of the

hydraulic shoulder. Singularities will occur if the

Jacobian rank is lower than three, the number of

DOF of the moving platform, or equivalently if:

0)det( =JJ

T

(41)

Such a case occurs only if the determinants of all

33

×

minors of J are identically zero. These square

minors correspond to the Jacobian matrices of the

hydraulic shoulder with one of the actuating legs

removed. Therefore, the redundant manipulator will

be in a singular configuration only if all the non-

redundant structures resulted by suppressing one of

the actuating legs are in a singular configuration.

Such a case will not occur in the workspace of the

hydraulic shoulder owing to the specific and careful

design of the mechanism (Hayward, 94). In fact, one

of the remarkable features of adding the fourth

actuator is the elimination of the loci of singularities.

Fig.(5), shows the determinants of the four minor

Jacobian matrices, computed in the workspace of the

manipulator where DM1-DM4 denote the non-

redundant minor determinants. This can be of

interest in cases where one of the actuators

malfunctions. It can be seen that the possibility of

getting into a singular configuration is increased

when either of the redundant actuators are removed.

Figure 4: Minor Determinants for the non-redundant

structures

6 CONCLUSIONS

In this paper, kinematic modeling and singularity

analysis of a 3-DOF actuator redundant parallel

ICINCO 2005 - ROBOTICS AND AUTOMATION

130

manipulator has been studied in detail. The closed

form solution to the forward kinematic is obtained

using a vector approach by considering the

individual kinematic chains inherent in such parallel

mechanisms. It is proposed to consider suitable

mapping between actuator, joint and task spaces in

both kinematic and Jacobian modeling of the

manipulator. The proposed method paves the way

for the feedback position control of the manipulator,

using a closed-form solution to the forward

kinematics and leaving out the approximation errors

inherent in numerical identification methods. It is

also shown that the forward kinematics map

provides us with some extra solutions which should

be regarded properly. Singularity analysis was also

performed using the analytic Jacobian obtained for

the mechanism. The manipulator workspace was

shown to be free of singularities due to the

redundancy in actuation. Future work will consider

Dynamic analysis of the hydraulic shoulder

manipulator.

REFERENCES

J.P. Merlet, Still a long way to go on the road for parallel

mechanisms, ASME 2002 DETC Conference,

Montreal, Canada, 2002

J.P. Merlet, Parallel Robots: Open problems, In 9th Int'l.

Symp. of Robotics Research, Snowbird, 9-12 October

1999.

S. Joshi and L.W. Tsai, The kinematics of a class of 3-

DOF , 4-Legged parallel manipulators, Trans. Of the

ASME, vol. 125, March 2003, 52-60.

S. Joshi and L.W. Tsai, A comparison study of two 3-DOF

parallel manipulators: One with three and the other

with four supporting legs, IEEE Trans. On Robotics &

Automation, vol. 19, No. 2, April 2003, 200-209.

L.Baron and J. Angeles, The kinematic decoupling of

parallel manipulators using joint-sensor data, IEEE

Trans. On Robotics & Automation, vol. 16, No. 6, Dec

2000, 644-651.

J.P.Merlet, Closed-form resolution of the direct kinematics

of parallel manipulators using extra sensors data, IEEE

Intl. Conf. On Robotics & Automation, 1993, 200-

204.

J.P.Merlet, Direct kinematics of planar parallel

manipulators, Intl. Conf. On Robotics & Automation,

Minnesota, April 1996, 3744-3749.

S.K.Song and D.S.Kwon, Efficient formulation approach

for the forward kinematics of the 3-6 Stewart-Gough

platform, Intl. Conf. on Intelligent Robots and

Systems, Hawaii, USA, Oct. 2001, 1688-1693.

Bonev, I. A., Ryu, J., Kim, S.-G., and Lee, S.-K., A

closed-form solution to the direct kinematics of nearly

general parallel manipulators with optimally located

three linear extra sensors, IEEE Trans. On Robotics &

Automation, vol. 17, No. 2, April 2001, 148-156.

B. Siciliano, The tricept robot: inverse kinematics,

manipulability analysis and closed-loop direct

kinematics algorithm, Robotica, vol. 17, pp.437-445,

June 1999.

A. Fattah and G. Kasaei, Kinematics and dynamics of a

parallel manipulator with a new architecture,

Robotica, vol. 18, pp. 535-543, Sept. 2000.

O.Didrit, M.Petitot and E.Walter, Guaranteed solution of

direct kinematic problems for general configurations

of parallel manipulators, IEEE Trans. On Robotics &

Automation, April 1998, 259-266.

B. Dasgupta, T.S. Mruthyunjaya, The Stewart platform

manipulator: a review, Elsevier Science, Mechanism

& Machine theory,2000,15-40.

Hayward, V. and Kurtz, R.: Modeling of a parallel wrist

mechanism with actuator redundancy, Int'l. J.

Laboratory Robotics and Automation, VCH

Publishers, Vol. 4, No. 2.1992, 69-76.

Hayward, V.: “Design of a hydraulic robot shoulder based

on a combinatorial mechanism” Experimental

Robotics III: The 3rd Int'l Symposium, Japan Oct.

1994. Lecture Notes in Control & Information

Sciences, Springer-Verlag, 297-310.

Hayward, V.: “Borrowing some design ideas from

biological manipulators to design an artificial one” in

Robots and Biological System, NATO Series,

Springer-Verlag, 1993, 135-148.

H. Sadjadian and H.D. Taghirad, Numerical Methods for

Computing the Forward Kinematics of a Redundant

Parallel Manipulator, IEEE Conf. on Mechatronics

and Robotics, pp 557-562, Sept. 2004, Aachen,

Germany.

KINEMATIC AND SINGULARITY ANALYSIS OF THE HYDRAULIC SHOULDER: A 3-DOF Redundant Parallel

Manipulator

131