RECENT ADVANCES in SIGNAL PROCESSING, ROBOTICS and AUTOMATION

Kinematic Analysis of a Two-Link Object for Whole Arm Manipulation
ZAKARYA ZYADA, YOSHIKAZU HAYAKAWA and SHIGEYUKI HOSOSE
RIKEN-TRI Collaboration Center for Human-Interactive Robot Research,
Nagoya Science Park, 2271-130 Anagahora, Shimoshidami, Moriyama-ku, Nagoya, Aichi 463-0003,
JAPAN{zzyada, hayakawa and hosoe}@nagoya.riken.jp, http://rtc.nagoya.riken.jp/index-e.html
Abstract: - This paper presents kinematic analysis of free-ended two-rigid-links object manipulated by two arms as
an initial step simplification towards the analysis of holding and manipulating a human body by a humanoid robot.
Rolling as well as sliding at the points of contact of the two-links object constrained by the holding two-arms is
explained. Position, velocity and acceleration analysis of the manipulated object, position, velocity andacceleration constraints as well as expressions for sliding displacement, sliding velocity and sliding acceleration
are presented. Simulation results for links sliding acceleration, the object joint acceleration and object
configurations for different input rolling accelerations are presented. The presented position, velocity andacceleration analysis proves that: 1- through manipulating two-links object by two-arms, rolling about one arm orboth arms at the points of contact is associated with sliding at both points of contact. There is no rolling without
sliding and there is no sliding without rolling. 2- It is possible to define the object links velocities or accelerations
as well as sliding accelerations from the contact rolling velocities and accelerations estimations or measurements.
3- It is possible to define the configuration of the object from contact points and hence contact angle
measurements, applying tactile sensors.
Key-Words: - Kinematics analysis, whole arm manipulation, dynamic manipulation, cooperative manipulation,
rigid body, position analysis and velocity analysis.

1 Introduction
Nursing robotics have received a considerable
attention by different research groups seeking theirpromising human-friendly assist and cooperation.
Among these groups is RIKEN-TRI robotics researchgroup who introduced RI-MAN for holding andtransferring a patient, [1], [2]. RI-MAN is designed tograsp, hold and transfer a human applying its whole
two-arms, Fig. 1. Whole arm manipulation is anapproach to manipulation that employs all theavailable manipulation surfaces of the robot to act
upon and sense the environment, [3]. Restraining large
size objects, lifting heavy loads or assembling matingparts are examples of suitable tasks that require wholearm manipulation. Whole arm grasps are formed bywrapping the arms (or fingers) around the objects, [4].
Manipulating a human, (the target is a patient), as
shown in Fig. 1, lies in the taxonomy of dynamic
manipulation in which task dynamics is significant foranalysis and planning.
Kinematic, static andquasi-static manipulation analysis are important steps
for dynamic manipulation. Dynamic manipulation has
many benefits such as increasing the repertoire ofactions to manipulators, increasing the speed ofmanipulation and saving the complexity and the

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Fig. 1. RI-MAN holding a dummy patient, [2]mass of the robot, [5]. With dynamic manipulation,some of the complexity of the robot system istransferred from the hardware (joints and actuators) toplanning and control, [6]. Many research works fordifferent dynamic manipulation tasks can be found in
the literature. Motion planning of a 1-DOF dynamic
pitching robot throwing a ball in a horizontal plane ispresented in [7]. Dynamic manipulation of an object
over a plate inspired by the handling of a pizza peel ispresented in [8]. Motion planning and controllability
for non-prehensile (without a form- or force-closure)
manipulation for throwing and catching a disc using
two planar manipulators are presented in [9].

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Carrying a polygonal shape object applying two
mobile robots with non-prehensile manipulators has
been presented in [10]. So far the presented work isconcerned with one-link object to be manipulated by
one or two manipulators. However an application likeholding and/or transferring a human, (patient for
instance), is an example of multi-link object which is
not tackled in research literature, (up to the authors’knowledge), in robotics manipulation. To define,apply and assure stable holding and manipulating aswell as control, it is important to analyze andunderstand kinematics and dynamics of themanipulated multi-link object. As an initial step
towards analysis of holding and manipulating a humanbody object, this paper introduces two-dimensional
position, velocity and acceleration analysis of free
ended two-link rigid body object hold by two arms,
Fig. 3. Kinematic relations as well as kinematicconstraints are to be introduced. Through analysis andsimulation, it is aimed to answer the following
questions:
Given the acceleration of rolling around the points ofcontact, (defined by θ1 and θ2 in Fig. 3), for a
two-link object hold by two-arms, at any instant: a) Is
it possible to get information about object-joint
acceleration, velocity and position?, b) Is it possible toget information about the sliding accelerations,velocities and amount of sliding of links over arms?, c)
How are links sliding motions (acceleration, velocity
and position) associated with rolling motions(acceleration, velocity and position) ?, d) Is it possible
to get information about the object configuration?
This paper is organized as follows: rolling and slidingof one link object over one arm in a plane is presented
in section II. Position analysis of two-link object
constrained by two-arms with rolling and sliding
motions, position constraints, sliding displacement
expressions and position simulation results are
introduced in section III. Velocity analysis of two-link
object constrained by two-arms, velocity constraints,
sliding velocity expressions are introduced in section
IV.
Acceleration analysis of two-link object
constrained by two-arms, acceleration constraints,
sliding acceleration expressions and the acceleration
simulation results for different rolling accelerations
are introduced in section V. Conclusion and directions
for future work are to be presented in section VI.

In this section, rolling and/or sliding of a slim
rigid-link object over a rigid-link arm system, Fig. 2, is
explained. Contact type is assumed to be hard contact.
The coordinate system is introduced first followed byexplanation of the different possible rolling and/or
sliding motions.

2.1 Coordinate system
The coordinate system is defined as shown in Fig. 2. A
reference coordinate frame 0, 𝑂0 − 𝑥0 𝑦0 is defined at
the center of the circular cross section of the arm; twoframes 𝑆1𝑙 and 𝑆1𝑐 both initially having the origin at
the point of contact between the link and the arm, 𝑠1 ,are also defined. Frame 𝑆1𝑙 , (𝑂1𝑙 − 𝑥1𝑙 𝑦1𝑙 ), is fixed to
the link while frame 𝑆1𝑐 , (𝑂1𝑐 − 𝑥1𝑐 𝑦1𝑐 ), is at the
point of contact between the link and the arm; 𝑥1𝑐 is in
the dirction of line from 𝑂0 to 𝑠1 which is also normal
to the plane including the point of contact as shown in
Fig. 2. Frame 𝑒, 𝑂𝑒 − 𝑥𝑒 𝑦𝑒 is attached to the right endof the arm. The angle 𝜃1 , denoted as the angle ofcontact, is measured from 𝑥0 to 𝑥1𝑐 and being positive
counter clockwise; the length 𝑙1 expresses linksegment length from the origin of frame 𝑆1𝑐 to theorigin of frame 𝑒. The total length of the object link is
𝑎1 ; Clearly 𝑙1 is a variable while 𝑎1 is a constant.
The rolling and/or sliding motion between the link
object and the arm can be clarified in terms of therelative position of the two reference frames 𝑆1𝑙 and𝑆1𝑐 before and after the motion. After a motion ofrolling and/or sliding, frames 𝑆1𝑙 and 𝑆1𝑐 are denoted′′
respectively. Knowing the initial angle
by 𝑆1𝑙and 𝑆1𝑐of contact 𝜃1 0 and the initial length 𝑙1 (0) anddefining the motion, (rolling and/or sliding), thechange in angle of contact ∆𝜃1 and the change inlength ∆𝑙1 can be defined. To explain the motions,
three cases are considered: 1) sliding of the link over
the arm (without rolling); 2) rolling of the link over
the arm (without sliding); and 3) both sliding androlling of the link over the arm.

2.2 Sliding (without rolling)
Assuming that the link has the possibility to slide(without rolling) over arm, then the contact point after′
sliding will remain at its location before sliding, (𝑆1𝑐will be identical to 𝑆1𝑐 ). Hence the angle of contact
after sliding, 𝜃1 , will be equal to the initial angle ofcontact, 𝜃1 0 , then∆𝜃1 = 𝜃1 − 𝜃1 0 = 0(1)
Moreover the frame on the link side 𝑆1𝑙 as well as thewhole link will move according to the amount ofsliding displacement, 𝑑1 . 𝑆1𝑙 will become in a new

2. Rolling and Sliding of One Link
Object

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RECENT ADVANCES in SIGNAL PROCESSING, ROBOTICS and AUTOMATION′
position 𝑆1𝑙. The length 𝑙1 after sliding, will dependon its initial length, 𝑙1 0 , and amount of sliding 𝑑1 .
∆𝑙1 = 𝑙1 − 𝑙1 0 = 𝑑1(2)

after sliding
a1

x1c

y1c

s
y 1c

d

0

2.3 Rolling (without sliding)

s1l

1

Oe

x1c , x1l

y1c , y1lAssuming that the link has the possibility to roll
without sliding, then both 𝑆1𝑐 and 𝑆1𝑙 will be identicaland move on the circle of radius 𝑟, (the peripheral of
the arm), with the amount of rolling distance, 𝑟∆𝜃1 , to′′another position denoted by 𝑆1𝑐and 𝑆1𝑙. The change
in the angle of contact ∆𝜃1 is a function of the angleof contact after rolling, 𝜃1 , and the initial angle ofcontact, 𝜃1 0 ,∆𝜃1 = 𝜃1 − 𝜃1 0
(3)
The change in length because of rolling can be definedas:∆𝑙1 = 𝑙1 − 𝑙1 0 = −𝑟∆𝜃1
(4)

ye

x0

o

r

l1

x1l

y1l

s1l

xe

l1 (0)

y0 s1c1 (0)

link

ye

x0

o0

Oe

arm

xe

Initial state(a)after rolling

y0
1

a1

x1c , x1ly1c , y1l

2.4 Sliding and rollings1c , s1lAssuming that the link has the possibility to slide andto roll, then both 𝑆1𝑐 and 𝑆1𝑙 will move to different
new positions according to rolling angle and sliding′
according todisplacement. 𝑆1𝑐 will become 𝑆1𝑐′amount of roll, 𝑟∆𝜃1 , while 𝑆1𝑙 become 𝑆1𝑙
accordingto the amount of sliding, 𝑑1 . The new length 𝑙1 will
depend on 𝑟∆𝜃1 as well as 𝑑1 and the initial length
l1 0 . Expressions for change in the angle of contact
and the change in length will be the summation of boththe sliding case and the rolling case;∆𝜃1 = 𝜃1 − 𝜃1 0
(5)
The change in length because of both sliding androlling can be defined as:∆𝑙1 = 𝑙1 − 𝑙1 0 = 𝑑1 − 𝑟∆𝜃1
(6)

l1

x0

o

ye

x1c , x1l

y1c , y1l

Oes1l
y0 s1clink
1 (0)
r
x0
o

xe
ye

Oe

arm

xe

Initial state(b)after rolling & sliding

x1c

y1c

d1

y0
1

s1c ys1l1lo
y1c , y1l
y0

r

a1

x1l

l1

3. Position Analysis of Constrained
Two-Link Object

x0

x1c , x1l

s1l
s1c

o

ye
Oe

1 (0)

link

x0

arm
Initial state

Understanding the kinematics and dynamics of thehuman body as an object manipulated by the wholearms of a humanoid robot is important for stable
manipulation and control.
As an initial step
simplification towards this difficult problem, theauthor considers a two-dimensional system of two
slender-rigid-link object constrained by two-arms asshown in Fig. 3. In this section position analysis ispresented.
Sliding and/or rolling motion of
two-rigid-link object constrained by two arms is to bepresented.
The object is a two free-ended rigid-links, (link 1 andlink 2), with total lengths 𝑎1 and 𝑎2 connected with a

xe

ye
Oe

xe

(c)
Fig. 2. Graphical representation of rolling and/or
sliding motion of a link over an arm, (lower graph:
before motion; upper graph: after motion); (a) sliding
motion only, (b) rolling motion only, (c) both rollingand sliding motion.

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passive revolute joint. The two-link object isconstrained by two arms, (right and left arms). Link 1
has a point contact with the cylindrical right-arm at 𝑠1
while link 2 has a point contact with the cylindricalleft-arm at 𝑠2 . For the purpose of analysis, the basecoordinate frame, (frame b), 𝑜𝑏 − 𝑥𝑏 𝑦𝑏 , the right arm
coordinate frame, (frame 𝑟 ), 𝑜𝑟 − 𝑥𝑟 𝑦𝑟 , left-arm
coordinate frame, (frame 𝑙), 𝑜𝑙 − 𝑥𝑙 𝑦𝑙 , contact points
coordinate frames, (frame 𝑠1 ), 𝑜𝑠1 − 𝑥𝑠1 𝑦𝑠1 and (frame𝑠2 ), 𝑜𝑠2 − 𝑥𝑠2 𝑦𝑠2 , (frame 3), 𝑜3 − 𝑥3 𝑦3 are defined asshown in Fig. 3.
The angles of contact, 𝜃1 is theangle from the 𝑥-axis of frame 𝑏 to the 𝑥-axis of frame𝑠1 while the angle of contact 𝜃2 is the angle from the𝑥-axis of frame 𝑏 to the 𝑥-axis of frame 𝑠2 , Fig. 3,
both being positive in a counter clockwise sense. 𝑙1
and 𝑙2 define link segment lengths from origins offrames 𝑠1 and 𝑠2 at the points of contact to the passive
joint, origin of frame 3, respectively. It is aimed to get
for a two-link manipulated object, constrained asshown in Fig. 3, the constraining relation between
𝑙1 , 𝑙2 and 𝜃1 , 𝜃2 .

xs1

y s1
yr rr ,s1

os1

rs1 ,3

1

ra

or

xr

y3

l1

In a similar way, the position of the passive joint,
frame 3, through the left-arm branch, can be describedas:𝑠𝑏𝑝3𝑏 = 𝑝𝑙𝑏 + 𝑟𝑙,𝑠
+ 𝑅𝑠𝑏2 𝑟𝑠22,3(8)2𝑏where 𝑝𝑙 is the position vector from the origin offrame b to the origin of frame l expressed in frame b;𝑏𝑟𝑙,𝑠is the position vector from origin of frame 𝑙 to the2𝑠origin of frame 𝑠2 expressed in frame b; 𝑟𝑠22,3 is the
position vector from the origin of frame 𝑠2 to theorigin of frame 3 expressed in frame 𝑠2 ; 𝑅𝑠𝑏2 is therotation matrix from frame 𝑠2 to frame 𝑏. The relative
position of the two arms can be expressed as:𝑟
𝑝𝑙𝑏 = 𝑝𝑟𝑏 + 𝑅𝑟𝑏 𝑟𝑟,𝑙(9)𝑟where 𝑟𝑟,𝑙is the position vector from the origin offrame 𝑟 to
the origin of frame 𝑙 expressed in frame 𝑟; and 𝑅𝑟𝑏 is
the rotation matrix from frame 𝑟 to frame 𝑏 .
Substituting (9) into (8) and equalizing (8) and (7),
then
𝑠
𝑠
𝑟
𝑏
𝑏
(10)
+ 𝑟𝑙,𝑠+ 𝑅𝑠𝑏2 𝑟𝑠22,3+ 𝑅𝑠𝑏1 𝑟𝑠11,3 = 𝑅𝑟𝑏 𝑟𝑟,𝑙𝑟𝑟,𝑠1
2from which, the following constraint equation can bederived,𝑥𝑟,𝑙 − 𝑟𝑎 𝑐𝜃1 − 𝑐𝜃2𝑠𝜃1𝑠𝜃2 𝑙10=;+−𝑐𝜃1 −𝑐𝜃2 𝑙2
𝑦𝑟,𝑙 − 𝑟𝑎 𝑠𝜃1 − 𝑠𝜃20(11)where 𝑠𝜃𝑖 = 𝑠𝑖𝑛(𝜃𝑖 ); 𝑐𝜃𝑖 = 𝑐𝑜𝑠(𝜃𝑖 ); 𝑥𝑟,𝑙 and 𝑦𝑟,𝑙 are𝑟
the components of position vector 𝑟𝑟,𝑙in 𝑥 and 𝑦
directions respectively; 𝑟𝑎 is the radius of bothright-arm and left-arm cross section.

y s2

a2

xs2

l2

rs2 ,3os 2 ylrl ,s3
2

rr ,l

ra ol

a1

3.2 Constrained sliding and rolling:xl

Link 2

Link 1o3

Left-arm

3

x3

Right-arm

pl

pr
yb

ob

3.3 Configuration determination from tactile
sensors measurements

xb
Base-frameFig. 3. Coordinate systems of free-ended two-link
object by constrained by two-arms

Determination of the configuration of the manipulated
object is important in dynamic manipulation planningand control. Human do that applying his vision as wellas his skin force/position sensors. According to theconstraint equation (11), it is possible to define theconfiguration from tactile sensor measurements.
Constraint equation (11) defines the lengths 𝑙1 , 𝑙2 interms of the contact angles 𝜃1 , 𝜃2 and the relative
position of the two-manipulating arms. For given arm
𝑟size, 𝑟𝑎 , relative position, 𝑟𝑟,𝑙, of the two-arms; if it
possible to measure (or estimate) the angles of contact
𝜃1 and 𝜃2 , it is possible to determine the lengths 𝑙1 , 𝑙2
which completely define the configuration of theconstrained two-links object. Constraint equation (11)clarifies that if roll happens, the configuration will

3.1 Position of passive joint 3
The position of the passive joint, frame 3, through theright-arm branch, can be described as:𝑠𝑏𝑝3𝑏 = 𝑝𝑟𝑏 + 𝑟𝑟,𝑠
+ 𝑅𝑠𝑏1 𝑟𝑠11,3(7)1𝑏where 𝑝𝑟 is the position vector from the origin offrame b to the origin of frame r expressed in frame b;𝑏𝑟𝑟,𝑠is the position vector from origin of frame 𝑟 to1𝑠
origin of frame 𝑠1 expressed in frame b; 𝑟𝑠11,3 is the
position vector from origin of frame 𝑠1 to the origin offrame 3 expressed in frame 𝑠1 ; and 𝑅𝑠𝑏1 is the rotation
matrix from frame 𝑠1 to frame 𝑏.

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change. Also, in another way, if the configuration of
the object is changed, the angles of contact will bechanged. Measurements of the angles of contact canbe obtained through the tactile sensors proposed byanother research team in our institute for the humanoid
robot RI-MAN, [11].

4.2 Velocity constraint equation
The velocity constraint equation can be obtained bydifferentiating equation (11) as:𝑠𝜃1𝑠𝜃2 𝑙1+−𝑐𝜃1 −𝑐𝜃2 𝑙2
𝑟𝑎 𝑠𝜃1 + 𝑙1 𝑐𝜃1 −𝑟𝑎 𝑠𝜃2 + 𝑙2 𝑐𝜃2 𝜃1+−𝑟𝑎 𝑐𝜃1 + 𝑙1 𝑠𝜃1 𝑟𝑎 𝑐𝜃2 + 𝑙2 𝑠𝜃2 𝜃2
𝑥𝑟,𝑙
0
=
(15)
𝑦𝑟,𝑙
0

3.4 Rolling-sliding relation
It is important to define the sliding-rolling relation for
future study of planning and control for the current
problem of dynamic manipulation. As shown insection II, (equation 6), sliding, 𝑑𝑖 , is a function of thechange in angle of contact, ∆𝜃𝑖 and the change inlength ∆𝑙𝑖 for any two system states. The amount ofsliding, 𝑑𝑖 (referred to the frame 𝑠𝑖 ), can be expressedas:𝑑𝑖 = ∆𝑙𝑖 + 𝑟∆𝜃𝑖 ; 𝑖 = 1(12)
𝑑𝑖 = ∆𝑙𝑖 − 𝑟∆𝜃𝑖 ; 𝑖 = 2
Given the angles of contact 𝜃𝑖 , (𝑖 = 1,2), (and hence
the lengths 𝑙𝑖 , (𝑖 = 1,2), from equation 11), for any
two different instants, the change in length ∆𝑙𝑖 , and thechange in angle of contact ∆𝜃𝑖 can be obtained, fromwhich the amount of sliding 𝑑𝑖 , (𝑖 = 1,2) , can be
estimated at both points of contact. Equation (12)assures that for manipulating two-links by two-arms
both rolling and sliding are associated with each other.
If roll happened over one or the two arms, sliding at
the contact points exist. If sliding at the contact points
happened, roll also exists at one or two of contact
points. So, if the amount of roll is defined, (as it can be
estimated through tactile sensor measurements), thesliding can be defined by (12).

4.3 Sliding velocity estimation
Differentiating equation (12), sliding velocity 𝑑𝑖 canbe obtained as a function of velocity of ith link, 𝑙𝑖 andith rolling velocity, 𝜃𝑖 as follows:
𝑑𝑖 = 𝑙𝑖 + 𝑟𝜃𝑖 ; 𝑖 = 1(16)
𝑑𝑖 = 𝑙𝑖 − 𝑟𝜃𝑖 ; 𝑖 = 2
Estimating the velocities of rolling 𝜃1 , 𝜃2 , thevelocities of links, 𝑙1 , 𝑙2 , and sliding velocities 𝑑1 , 𝑑2 ,can be estimated.

5. Acceleration Analysis of Constrained
Two-Link Object
In this section, the analysis is extended to include
object links’ accelerations. The acceleration of thepassive joint, the acceleration constraint equation andthe sliding acceleration expressions as well as
simulation results are introduced.

5.1 Acceleration of passive joint 3
Differentiating the velocity equation (14), theacceleration of the passive joint can be obtained as,

𝑏
𝑏
𝑝3𝑥
𝑝𝑟𝑥
−𝑟𝑎 𝑠𝜃1 − 𝑙1 𝑐𝜃1 −𝑠𝜃1 𝜃1=𝑏𝑏 + 𝑟 𝑐𝜃 − 𝑙 𝑠𝜃𝑐𝜃1 𝑙1
𝑝𝑟𝑦
𝑝3𝑦
𝑎
1
1 1
−𝑟𝑎 𝑐𝜃1 + 𝑙1 𝑠𝜃1 −2𝑐𝜃1 𝜃12+−𝑟𝑎 𝑠𝜃1 − 𝑙1 𝑐𝜃1 −2𝑠𝜃1 𝜃1 𝑙1

4. Velocity Analysis of Constrained
Two-Link Object
In this section, the analysis of two-link object
constrained by two-arms is extended to include object
links’ velocities. The velocity of the passive joint, thevelocity constraint equation and the sliding velocity
expressions are introduced.

(17)

5.2 Acceleration constraint equation
In a similar way, differentiating the velocity constraint
equation (15), will lead to the acceleration constraint
equation as,

4.1 Velocity of passive joint 3
Differentiating the position equation (7), the velocityof the passive joint can be obtained,𝑠𝑠𝑏𝑝3𝑏 = 𝑝𝑟𝑏 + 𝑟𝑟,𝑠
+ 𝑅𝑠𝑏1 𝑟𝑠11,3 + 𝑅𝑠𝑏1 𝑟𝑠11,3(13)
1
Substituting with the individual components, then
𝑏
𝑏
𝑝3𝑥
𝑝𝑟𝑥
−𝑟𝑎 𝑠𝜃1 − 𝑙1 𝑐𝜃1 −𝑠𝜃1 𝜃1
= 𝑏 +
(14)
𝑏
𝑟𝑎 𝑐𝜃1 − 𝑙1 𝑠𝜃1𝑐𝜃1 𝑙1𝑝𝑝3𝑦𝑟𝑦

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𝑠𝜃2 𝑙1𝑐𝜃1 𝑐𝜃2 𝜃1 𝑙1++2−𝑐𝜃2 𝑙2𝑠𝜃1 𝑠𝜃2 𝜃2 𝑙2
𝑟𝑎 𝑠𝜃1 + 𝑙1 𝑐𝜃1 −𝑟𝑎 𝑠𝜃2 + 𝑙2 𝑐𝜃2 𝜃1+−𝑟𝑎 𝑐𝜃1 + 𝑙1 𝑠𝜃1 𝑟𝑎 𝑐𝜃2 + 𝑙2 𝑠𝜃2 𝜃2

𝑠𝜃1
−𝑐𝜃1

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RECENT ADVANCES in SIGNAL PROCESSING, ROBOTICS and AUTOMATION

𝑟𝑎 𝑐𝜃1 − 𝑙1 𝑠𝜃1𝑟𝑎 𝑠𝜃1 + 𝑙1 𝑐𝜃1
𝑥𝑟,𝑙
0
=
𝑦𝑟,𝑙
0

−𝑟𝑎 𝑐𝜃2 − 𝑙2 𝑠𝜃2 𝜃12+−𝑟𝑎 𝑠𝜃2 + 𝑙2 𝑐𝜃2 𝜃22

to estimate the joint relative accelerations (velocities).
It is possible to estimate the rolling velocities (and/or
accelerations) using tactile sensor measurements; 3- asa result it is possible to estimate the amount of sliding
and of the amount of link motion. Also, equations (11,12) have the same interpretation. Fig. 4c presents theacceleration of the passive joint 3 while Fig. 4dpresents some configurations of the object. The
presented configurations are the results of positions
analysis simulations from equation (11). If the pointsof contact are measurable, (applying tactile sensor),and hence the angles of contact, then the lengths fromthe points of contact to the passive joint are definable
which completely define the configuration. Fig. 4c
and Fig. 4d show that not only object configuration isdetectable but also the object joints’ accelerations(velocities) which are important for future dynamic
manipulation planning and control.

(18)

5.3 Sliding acceleration estimation
Differentiating equation (16), sliding acceleration 𝑑𝑖can be obtained as a function of the acceleration 𝑙𝑖 androlling acceleration, 𝜃𝑖 as follows:

𝑑𝑖 = 𝑙𝑖 + 𝑟𝜃𝑖 ; 𝑖 = 1
𝑑𝑖 = 𝑙𝑖 − 𝑟𝜃𝑖 ; 𝑖 = 2(19)

5.4 Acceleration Simulation Results
The presented governing equations for position,
velocity and acceleration analysis as well as positionand velocity constraints are simulated through an
implemented MATLAB program. Because of thelimited space, it is presented here the simulation
results of acceleration and acceleration constraints
only. The independent input variables are the rolling
accelerations, (𝜃1 and 𝜃2 ), and initial velocities (𝜃1 (0)and 𝜃2 (0)) and initial positions, (𝜃1 (0) and 𝜃2 (0)).
The rolling accelerations are assumed as a nonlinear
function of time; the rolling velocities 𝜃1 , 𝜃2 and therolling angles 𝜃1 and 𝜃2 are obtained through simple
integration. The sampling time is assumed to be 0.01 s.
the relative velocity of the arms is assumed zero. In
real time implementation, the angles of contact can beobtained from tactile sensor measurements and therolling velocity and accelerations, can be estimated.
The simulation results are shown in Fig. 4. The
relative acceleration, 𝑙1 , sliding velocity, 𝑑1 , (of link1
over the right-arm at the point of contact, s1 ), and thelinear velocity, 𝑟𝑎 𝜃1 , (due to a rolling acceleration,
𝜃1 , of link 1 over the right-arm), are shown in Fig. 4a.
Similarly, the relative acceleration, 𝑙2 , sliding
acceleration, 𝑑2 , (of link 2 over the left-arm at the
point of contact, 𝑠2 ), and the linear acceleration, 𝑟𝑎 𝜃2 ,(due to a rolling acceleration, 𝜃2 , of link 2 over theleft-arm), are shown in Fig. 4b. These two figures, 4aand 4b assure that: 1- if there is rolling then there isassociated sliding; it can be said for a two free ended
object manipulated by two whole arms; there no
rolling without sliding and there is no sliding withoutrolling; both are associated with each other; 2- if it ispossible to estimate the acceleration (or velocity) ofrolling, it is possible to estimate the object joint
relative accelerations (or velocities) as well as thesliding accelerations (velocities); also if it is possible

ISSN: 1790-5117

6. Conclusion
Kinematic analysis and acceleration simulation resultsof free-ended two-link object motion hold by two arms
are presented. The main conclusions derived from thepresented research work can be summarized as: 1- fordynamic manipulation of two-link object by two arms,
rolling of a link about an arm is associated with slidingof the link over the arm at the point of contact. Thereis no rolling without sliding and there is no sliding
without rolling; 2- through the measurements ofcontact points and hence the angles of contact,
applying a tactile sensor, the configuration of themanipulated two-link object can be determined; 3through estimating the rolling accelerations(velocities), joint relative accelerations (velocities)
and hence the sliding accelerations (velocities) at both
points of contact can be estimated.

Future WorkFor dynamic analysis and control, it important toanalyze forces including gravity forces and friction
forces at the points of contact. It is also important todefine stable manipulation and to control object
sliding. So, force analysis as well as sliding control
for stable manipulation will be of future interest.
References[1] T. Odashima, M. Onishi, K. Tahara, T. Mukai, S.
Hirano, Z. Luo and S. Hosoe, Development and
Evaluation of a Human-interactive Robot Platform

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RECENT ADVANCES in SIGNAL PROCESSING, ROBOTICS and AUTOMATION“RI-MAN”, Journal of Robotics Society of Japan,
vol. 25, No. 4, 2007, pp. 554–565.
[2] M. Onishi, Z. Luo, T. Odashima, S. Hirano, K.
Tahara and T. Mukai, Generation of Human Care
Behaviors
by
Human-Interactive
Robot“RI-MAN”, IEEE International Conference on
Robotics and Automation, 2007, pp. 3128–3129.
[3] K. Salisbury, W. Townsend, B. Ebrman and D.
DiPietro, Preliminary Design of a Whole-Arm
Manipulation
System(WAMS),IEEE
International Conference on Robotics andAutomation, 1988, pp. 254–260.
[4] P. Song, M. Yashima and V. Kumar, Dynamics
and Control of Whole Arm Grasps, IEEE
International Conference on Robotics andAutomation, 2001, pp. 2229–2234.
[5] M. Mason and K. Lynch, Dynamic Manipulation,
Proc. Of the 1993 IEEE/RSJ International
Conference on Intelligent Robots and Systems,
1993, pp. 152–159.
[6] K. Lynch and M. Mason, Dynamic UnderactuatedNonprehensile Manipulation, Proceedings of the2006 IEEE International Conference on Intelligent
Robots and Systems, 2006, pp. 889-896.
[7] W. Mori, J. Ueda and T. Ogasawara, 1-DOF
Dynamic Pitching Robot that Independently
Controls Velocity, Angular velocity, and Directionof a Ball: Contact Models and Motion Planning,
IEEE International Conference on Robotics andAutomation, 2009, pp. 1655–1661.
[8] M. Higashimori, K. Utsumi, Y. Omoto and M.
Kaneko, Dynamic Manipulation Inspired by the
Handling of a Pizza Peel, IEEE Transactions on
Robotics, Vol. 25, No.4, 2009, pp. 829-838.
[9] B. Beigzadeh, M. Ahmadabadi and A. Meghdari,
Two Dimensional Dynamic Manipulation of a
Disc Using Two Manipulators, Proceedings of the2006 IEEE International Conference on
Mechatronics and Automation, 2006, pp.
1191-1196.
[10]A. Gupta and W. Huang, A Carrying Task for
Nonprehensile Mobile Manipulators, Proceedingsof the 2003 IEEE/RSJ International Conference onIntelligent Robots and Systems, 2003, pp.
2896-2901.
[11]T. Mukai, M. Onishi, T. Odashima, S. Hirano, and
Z. Luo, Development of the Tactile Sensor Systemof a Human-Interactive Robot “RI-MAN”, IEEE
Transactions on Robotics, Vol. 24, No.2, 2008, pp.
505-512.

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0

-0.1

accelerations, [m/s 2]

-0.2

-0.3

-0.4

-0.5

r1
l1

-0.6

d1
-0.7

0

0.1

0.2

0.3

0.4

0.5
0.6
Time, [s]

0.7

0.8

0.9

1

a) roll linear acceleration, joint relative accelerationand sliding acceleration of link 1 over right-arm.
0.2
0.1
0

accelerations, [m/s 2]

-0.1
-0.2
-0.3
-0.4
-0.5
r2

-0.6

l2

-0.7
-0.8

d2
0

0.1

0.2

0.3

0.4

0.5
0.6
Time, [s]

0.7

0.8

0.9

1

b) roll linear acceleration, joint relative accelerationand sliding acceleration of link 2 over left-arm.
0.4

Joint 3 acceleration, [m/s2]

0.2

0

-0.2

-0.4

-0.6
d2P3x /dt 2-0.8

d2P3y/dt 2
-1

0

0.1

0.2

0.3

0.4

0.5
0.6
Time, [s]

0.7

0.8

0.9

1

c) acceleration components of the passive joint 3
0.6
0.5
0.4

Location, [m]

0.3
0.2
0.1
0
-0.1
-0.2
-0.3

right arm

-0.4
-0.2

0

left arm
0.2
0.4
Location, [m]

0.6

0.8

d) configurations changes
Fig. 4. Simulation results of free-ended two-link
object constrained by two-arms for rolling around therespective arms at both s1 and s2.

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ISBN: 978-960-474-157-1