Kinematic Analysis of Lower Mobility Cooperative Arms by Screw

Theory

Philip Long, Wisama Khalil and St´ephane Caro

Institut de Recherche en Communications et Cybern´etique de Nantes, UMR CNRS n

◦

6597

Ecole Centrale de Nantes, 1 Rue de la No¨e, 44321 Nantes, France

Keywords:

Cooperative Manipulators, Humanoid Robots, Screw Theory.

Abstract:

This paper studies the kinematic modeling and analysis of a system with two cooperative manipulators, work-

ing together on a common task. The task is deﬁned as the transportation of an object in space. The cooperative

system is the dual armed humanoid Nao robot, where the serial architecture of each arm has ﬁve degrees of

freedom. The mobility of the closed loop system is analyzed and the nature of the possible motion explored.

The serial singular conﬁgurations of the system are considered. Furthermore the parallel singularities of the

closed loop system associated with each actuation scheme are analyzed.

1 INTRODUCTION

The capability of dual independent arms when pro-

cessing parts reduces the need for custom ﬁxtures and

permits the use of a simpler end effector. The system

can then execute sophisticated tasks that may be difﬁ-

cult for a single arm system. For example, rather than

using a large serial robot a cooperative system dis-

tributes a heavy load among several smaller robots.

Similarly if the object is of an unwieldy, non-rigid

or awkward composition, the single arm robot may

struggle to manipulate it.

By using a cooperative system of two or more ma-

nipulators, both the location and the internal forces

of the object can be controlled. The two principal

approaches that avail of force sensors on the robot

are: hybrid position/force control (Uchiyama and

Dauchez, 1988) and impedance control (Sadati and

Ghaffarkhah, 2008), (Caccavale et al., 2008).

Another approach is to formulate kinematic rela-

tions that create a task space describing the multi-arm

system while grasping an object. The main meth-

ods are known as Symmetric formulation (Uchiyama

and Dauchez, 1988) and Task orientated formulation

(Chiacchio et al., 1996; Caccavale et al., 2000). Both

create a cooperative task space of velocity variables,

describing the object motion in space and the relative

motion between the end effectors (i.e forces applied

on object). On the other hand, the system can also be

viewed as a redundantly actuated parallel manipula-

tor. In this case kinematic constraint equations are de-

rived that establish a relationship between the chosen

independent and dependent joint variables (Yeo et al.,

1999; Liu et al., 1999; Cheng et al., 2003;

Ozkan and

Ozg¨oren, 2001). The dependent joint variables adopt

values that ensure loop closure at each instant.

Most of the preceding work has been carried out

with dual arm systems, where both arms are either

of 3-DOF or 6-DOF spatial composition. Thus away

from singularities, the grasped object has a mobility

of 3 or 6 respectively. On the other hand the study

of lower mobility cooperative manipulators has been

limited. In (Yeo et al., 1999) the cooperation be-

tween a 5-DOF and 4-DOF robot is used in conjunc-

tion with a passive joint in order to execute a 4-DOF

position/force task. In (Zielinski and Szynkiewicz,

1996) admissible path planning for two 5-DOF robots

is explored. In (Bicchi et al., 1995) a generalized

method based on the Jacobian matrix of each arm,

and their constraint relations with the object, is for-

mulated. Analysis of these matrices permits the cal-

culation of the mobility and possible ﬁrst order differ-

ential motions of general multiple limb robots.

Lower mobility systems suffer from three types of

singularities, limb (serial) singularities, actuation and

constraint (parallel) singularities (Amine et al., 2011).

In (Liu et al., 1999) the presence of parallel singu-

larities of a cooperative system with passive joints is

explored. The issue of a valid selection of actuators

is addressed in (

Ozkan and

Ozg¨oren, 2001). In both

cases an analysis of the Jacobian matrix is carried out.

Conversely screw theory can be used to locate and un-

280

Long P., Khalil W. and Caro S..

Kinematic Analysis of Lower Mobility Cooperative Arms by Screw Theory.

DOI: 10.5220/0004039202800285

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 280-285

ISBN: 978-989-8565-22-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

derstand parallel singularities in closed chain mecha-

nisms (Zlatanov et al., 2002).

In this paper the cooperative system, deﬁned by

the two arms of Aldebaran NAO T14 humanoid robot

and a grasped object, is examined. The system has

been modeled as a closed chain mechanism (Sec-

tions 2 and 3). The originality of this paper lies in the

use of screw theorytechniques to analyze the system’s

mobility, singularities and motion type. The beneﬁt of

this approach is that special conﬁgurations such as the

loss of stiffness, loss of DOF etc., can be determined

without the complex derivation of the Jacobian matri-

ces (or their inverses) (Sections 4 and 5).

2 SYSTEM DESCRIPTION

The system is described by the Modiﬁed Denavit-

Hartenberg (MDH) notation (Khalil and Kleinﬁnger,

1986),in Table 1. The right arm consists of joints 1-5

and the left arm consists of joints 6-10. The transfor-

mation matrix

a( j)

, from frame a( j), the antecedent

of j, to frame j is the 4× 4 matrix given by:

a( j)

= rot

(γ

) · trans

) · rot

(α

) ·

trans

) · rot

(θ

) · trans

) (1)

where rot

(θ) indicates a rotation of θ radians about

the ith axis and trans

(l) a translation of l meters

along the ith axis. It should be noted that γ

= b

= 0

when x

(a( j))

is perpendicular to z

. Once the object is

grasped a closed loop is formed. As shown in Fig. 1,

link 5 of the closed chain is composed of link 5, link

10 of the open chain and the object. Frame 10 is ﬁxed

on link 5. We introduce frame 11, which is equiva-

lent to frame 10, but its antecedent is frame 5. The

parameters of frame 11 are deﬁned once the robot has

grasped the object. The system has, in this case, only

nine bodies. Hence joint 10 is denoted as the cut joint.

Chain 1

Chain 2

LINK 5

Figure 1: Closed Loop Formulation.

Table 1: MDH Parameters of the closed loop chain.

j a( j) γ b d α θ r

1 0 0 b

0 −

-r

2 1 0 0 0

3 2 0 0 -d

4 3 0 0 0

−π

5 4 0 0 0

6 0 0 b

−π

7 6 0 0 0

8 7 0 0 d

9 8 0 0 0

−π

10 9 0 0 0

11 5 γ

3 KINEMATIC CONSTRAINTS

The location and velocity of the cut joint frame must

be equivalent when calculated via either chain. This

ensures a constant object grasp throughout the trajec-

tory. In the closed loop formulation, joints are des-

ignated either as actuated or passive. q

contains the

joint variables that are actuated, q

contains the pas-

sive joint variables and q

contains the joints that are

cut. The passive and cut joint variables can be ob-

tained in terms of the active joint variables using the

following geometric constraint equations:

(2)

Similarly, the kinematic constraints are given by:









(3)

By substituting in the joint velocity vectors:

(4)

where

and

contain the joint velocities of the

right arm and the left arm, respectively.

is the lin-

ear velocity and

the angular velocity of frame j in

frame 0,

is the 6× 5 kinematic Jacobian matrix of

frame j w.r.t. frame 0. By rearranging the rows and

columns of (4), a relationship is obtained between the

passive joint velocities and the actuated joint veloci-

ties













= 0 (5)

From (5), we obtain:

= −G

−1

(6)

= −G

−1



− G

−1



(7)

KinematicAnalysisofLowerMobilityCooperativeArmsbyScrewTheory

281

The mobility from section 4 is found to be equal

to 4, hence the dimension of G

is 5× 4 , G

is 5× 5

is a scalar (4, 5, 1 being the number of active,

passive and cut joints respectively). G

degenerates at

conﬁgurations corresponding to parallel singularities

as seen in section 5.2.

4 MOBILITY ANALYSIS

The degree of freedom (DOF) of the system, is equal

to the number of independent coordinates required to

deﬁne the pose of the end-effector. The DOF can be

obtained by several methods for example Chebychev-

Gr¨ubler-Kutzbach, or Gogu’s Method (Gogu, 2007).

In order to elucidate the motion type, screw theory is

used. In summary, each serial arm has 5-DOF how-

ever once the object is ﬁrmly grasped by each arm, a

closed chain is formed and the object DOF becomes

four.

4.1 Screw Theory

4.1.1 Mobility Analysis based on Screw Theory

Screw theory can be used to analyze the instantaneous

motions of complex mechanisms (Hunt, 1978; Ball,

1900). A screw of pitch λ is deﬁned as:



s× r+ λs



(8)

s is the unit vector of the rotational axis of the screw.

r is the vector from any point on the axis to the ori-

gin. A zero-pitch screw and an inﬁnite-pitch screw

are given respectively by:



s× r



, $

∞



3×1



(9)

For every screw system, consisting of n linearly inde-

pendentscrews, there exists a reciprocal screw system

of dimension 6− n. Two screws $

and $

are recip-

rocal if their instantaneous power is zero, namely,



3×3





= 0 (10)

The following reciprocity conditions are deﬁned

from (10):

1. $

is reciprocal to $

∞

if their axes ⊥;

2. $

∞

is always reciprocal to another $

∞

;

3. Two $

are reciprocal if their axes are coplanar;

A zero-pitch twist ε

corresponds to a pure rotation

about its axis. An ∞-pitch twist ε

∞

corresponds to a

pure translation along its axis. A zero-pitch wrench

corresponds to a pure force along its axis. An ∞-

pitch wrench ζ

∞

corresponds to a pure moment about

its axis. The twist system T

and the wrench system W

of a serial kinematic chain i composed of f joints are

given by:

⊕

j=1

, W

j=1

(11)

The twist system T and the wrench system W of a par-

allel kinematic chain composed of m serial chains are

given by:

T =

i=1

, W =

⊕

i=1

(12)

A =

⊕

j=1

means A is spanned by vectors B

1··· f

while

A =

j=1

means A is intersection of vectors of B

1··· f

Table 1 shows that NAO T14 is composed of ﬁve

revolute joints. From (11), the twist systems T

and T

of its right and left arms are deﬁned as

= span(ε

, ε

) (13)

= span(ε

, ε

010

) (14)

The ﬁrst two joints of each arm intersect and consti-

tute a universal (U)-joint. The last three joints also

intersect and are equivalent to a spherical (S)-joint.

Since the twist system of each arm is a 5-system,

its reciprocal wrench system is a 1-system. By reci-

procity condition 3, it can be shown that each arm ap-

plies one pure force (a zero-pitch wrench ζ

, i = 1, 2)

to the object along an axis intersecting the U- and S-

joint axes. Thus, the wrench systems of the two arms

are deﬁned as:

= ζ

, W

= ζ

(15)

From (12), the constraint wrench system applied to

the object is a 2-system given by:

= W

⊕ W

= span(ζ

, ζ

) (16)

The object twist system is reciprocal to W

. Thus, it is

a 4-system and the object has four DOF.

4.1.2 Motion Analysis based on Screw Theory

When the constraint forces are parallel, i.e., ζ

k ζ

reciprocity condition 1 states that there are two inde-

pendent ∞-pitch twists, ε

∞1

and ε

∞2

, reciprocal (nor-

mal) to ζ

and ζ

. Reciprocity condition 3 states

r and l stand for right and left.

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

282

∞1

∞2

object

Figure 2: Reciprocal twists to parallel constraint forces

(2T2R motion mode).

∞

object

Figure 3: Reciprocal twists to intersecting constraint forces

(1T3R motion mode).

that there are also two independent zero-pitch twists,

and ε

, reciprocal (coplanar) to ζ

and ζ

shown in Fig. 2. Therefore, the corresponding motion

mode is 2T2R

When ζ

∦ ζ

, reciprocity condition 1 states

there is one ∞-pitch twist ε

∞1

reciprocal (normal) to

both ζ

and ζ

and condition 3 states there are three

independent zero-pitch twists, ε

, ε

and ε

, re-

ciprocal (coplanar) to both ζ

and ζ

as shown in

Fig. 3. Consequently, the correspondingmotion mode

is 1T3R.

5 SINGULARITY ANALYSIS

This section deals with the singularity analysis of the

NAO 14 when it ﬁrmly grasps an object. Limb sin-

gularities can be characterized by a loss of DOF of

the limb, while a gain of DOF or a lack of stiffness

is known as a parallel singular conﬁguration (Amine

et al., 2011; Amine et al., 2012).

T and R stand for Translation and Rotation, respectively.

5.1 Limb Singularities

A limb singularity is similar to the singularity of a

serial manipulator. It occurs for the dual-arm system

when the limb kinematic screw system (twist system)

degenerates. Consequently, the grasped object loses

one or more DOF in such a conﬁguration. From (9)

and (13) the kinematic Jacobian matrix of the right

arm can be written as:



× r



(17)

Let the wrist center be the origin of the frame where

vectors r

, i = 1, . . . , 5 are expressed. Equation (17)

becomes



× r



(18)

The right arm reaches a limb singularity when J

is rank deﬁcient. The relations leading to rank de-

ﬁciency can be examined. Since s

⊥ s

, s

⊥ s

and

⊥ s

these degeneracies can be eliminated. Further-

more, the architecture of the arm means the S-joint

cannot lie on s

, leaving two singular conﬁgurations.

Firstly when s

k s

obtained at q

= 0 ± π , secondly

when s

× r

= 0

obtained at q

= atan



−r



5.2 Parallel Singularities

The loss of stiffness due to a parallel singularity can

be characterizedby a degeneracyof screw system rep-

resenting the wrenches. Examples are given in Ta-

ble 2 (Hunt, 1978).

Table 2: Linear Dependence of Screws.

No. ζ

Condition

≥ 2 ζ

collinear axes

≥ 2 ζ

∞

parallel axes

> 3 ζ

intersect the same point

≥ 6 ζ

intersect the same line

5.2.1 Constraint Singularities

A constraint singularity occurs when the constraint

wrench system (16) degenerates, i.e., when ζ

and

are linearly dependent (condition 1 in Table 2).

The closed loop system reaches such a conﬁguration

when the two S-joint centers lie on s

and s

, from the

geometric model:

= atan



−r



and q

= atan





(19)

KinematicAnalysisofLowerMobilityCooperativeArmsbyScrewTheory

283

From section 5.1 it is noted that when the closed

loop system reaches a constraint singularity, both

arms reach a limb singularity. Four wrenches form-

ing a 3-system as described in Fig. 4 are applied on

the object: the constraint wrenches ζ

and ζ

and

the wrenches due to the serial singularity of each arm

0s1

and ζ

0s2

. As a consequence, the object has 3-

DOF in this conﬁguration.

0s1

0s2

Figure 4: Constraint singularity of the dual arm system.

5.2.2 Actuation Singularities

In this section a selection criterion is given for suitable

actuated joints. Once an actuator is locked, it imposes

a wrench on the object. The actuator wrench for joint

i is denoted as ζ

λj

. This wrench is reciprocal to all

the twists of the arm except the actuator twist itself

and furthermore it should not lie in W

. The actuation

wrench system W

applied on the object is the span of

actuation wrenches from both arms, namely,

= span(W

, W

) (20)

The constraint wrenches and actuation wrenches of

both arms should form a 6-system. This system is

denoted as the global wrench system and is deﬁned

as:

W = W

⊕ W

(21)

An actuation singularity occurs when (21) degener-

ates while (16) does not. Since it is possible to ac-

tuate any four of the ten joints there are

10!

4!(10−4)!

= 210 possible actuation schemes. By excluding

non-symmetrical actuation schemes, 110 actuation

schemes remain. It should be noted that since all the

joints in the arm can be actuated, a redundant actua-

tion scheme can decrease the likelihood of actuation

singularities.

Example 1: q

= [q

]. Since it is generally

preferable to actuate joints close to the base, the case

when the base U-joints are actuated is examined. The

global wrench system is derived:

• A pure force ζ

k s

and intersecting s

, s

coplanar to all twists except that generated by q

• A pure force ζ

k s

and intersecting s

, s

coplanar to all twists except that generated by q

• A pure force ζ

k s

and intersecting s

, s

coplanar to all twists except that generated by q

• A pure force ζ

, k s

and intersecting s

, s

coplanar to all twists except that generated by q

The global wrench system as illustrated in Fig. 5,

is given by W = span(ζ

). A line

between point A and point B (the S-Joints centers of

either arm) can be drawn which is intersected by all

constraint forces regardless of the conﬁguration of the

robot. Thus this actuation scheme is not admissible,

for any conﬁguration the global wrench system de-

generates, i.e. rank(W) = 5, rank(W

) = 2, rank(W

) =

4, resulting in an uncontrollable closed chain mecha-

nism.

k s

Figure 5: Non-admissible actuation scheme.

Example 2: q

= [q

]. To ﬁnd the ac-

tuation wrench applied by q

, the planes Π

, Π

spanned by (s

, s

) and (s

, s

) respectively, are ex-

amined:

1. if Π

∦ Π

, there is a pure force ζ

acting along

the line formed by the two planes

2. if Π

k Π

, there is a pure moment ζ

∞3

acting

around the line normal to both two planes

The global wrench system for case 1 is shown in

Fig.6, W = span(ζ

). An actuation

singularity occurs when the line ζ

contains points A

or B. Other singularities can be obtained by observ-

ing when the wrench system formed by two or more

wrenches degenerates(Table 2), for instance when ζ

and ζ

become collinear.

6 CONCLUSIONS

This work has presented a study of the NAO robot’s

two arms engaged in a cooperative task. The DOF of

an object, simultaneously grasped by both arms, was

explored. The serial singularities are straightforward,

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

284

k s

Figure 6: Admissible actuation scheme.

more interesting is the presence of constraint singu-

larities. The issue of actuation singularities, present

due to dependent joint variables, was investigated.

By considering the wrenches exerted by the actuators,

both an admissible and inadmissible actuation scheme

were shown.

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