Experimental Study of the Methodology for the Modelling...

International Journal of Advanced Robotic Systems Experimental Study of the Methodology for the Modelling and Simulation of Mobile Manipulators

Regular Paper

Luis Adrian Zuñiga Aviles1, 2, *, Jesus Carlos Pedraza Ortega1, 2 and Efren Gorrostieta Hurtado1, 2

1 Secretariat of National Defense and CIDESI, México 2 CIDIT-Facultad de Informática, Universidad Autónoma de Querétaro, México * Corresponding author E-mail: adrian_dgim@prodigy. net. mx Received 7 May 2012; Accepted 27 Jul 2012 DOI: 10. 5772/51867I © 2012 Aviles et al. ; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/3. 0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract This paper describes an experimental study of a novel methodology for the positioning of a multi‐articulated wheeled mobile manipulator with 12 degrees of freedom used for handling tasks with explosive devices. The approach is based on an extension of a homogenous transformation graph (HTG), which is adapted to be used in the kinematic modelling of manipulators as well as mobile manipulators. The positioning of a mobile manipulator is desirable when: (1) the manipulation task requires the orientation of the whole system towards the objective; (2) the tracking trajectories are performed upon approaching the explosive device’s location on the horizontal and inclined planes; (3) the application requires the manipulation of the explosive device; (4) the system requires the extension of its vertical scope; and (5) the system is required to climb stairs using its front arms. All of the aforementioned desirable features are analysed using the HTG, which establishes the appropriate transformations and interaction parameters of the coupled system. The methodology is tested with simulations and real experiments of the system where the error RMS average of the positioning task is 7. 91 mm, which is an acceptable parameter for performance of the mobile manipulator.

Keywords Methodology, Modelling, Simulation, Experimental study, Wheeled mobile manipulator, Positioning

1. Introduction

Mobile manipulators exhibit wide versatility in the handling of dangerous devices, such as the explosive ordinance devices (EOD) used by the world’s armies. Among the most important phases in the mechatronic design of mobile manipulators we have modelling and simulation, because upon them depends the proposed control for its correct operation and the fact that its behaviour will be verified by the simulation of several instances in order to obtain the parameters and ranks from the mobile manipulator that is being developed. In relation to the modelling of mobile manipulators, the work of Yamamoto, Alicja Mazur and Bayle is well known. Yamamoto is the most cited, with his paper on the locomotion coordination of mobile manipulators, published in 2004. Alicja Mazur and Bayle are the authors that have written most in relation to this topic. Since 1976,

1Luis Adrian Zuñiga Aviles, Jesus Carlos Pedraza Ortega and Efren Gorrostieta Hurtado: Experimental Study of the Methodology for the Modelling and Simulation of Mobile Manipulators

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ARTICLE

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there are recoteleoperated [2] and 1996 in the operaproposed; hplatform weBesides thispublished manipulatorsoperational kinematic manipulabili In order to oused in the transport of ethe developmmanipulatorsmulti‐articulaRobot MMRpresented in the mechatrkinematic mostudy of thepositioning throughout t The rest of tdescribes thethe full systeof operation.the basis of thmodelling asimulation anthe ideal padiscussed arespectively.

2. Modelling

This sectionwheeled modevelopmentfollows: prelposition andgraph, kinekinematic inconstraints, dynamic equsimulation.

2. 1 Prelimina

The robot Mmove on a suthe robot anmounted onbetween its

ords of worksmobile mani[3‐4], the dyn

ational case ohowever, the ere consideres, from 1994 on mobile s but withoutcases; they wredundancy, ity [5, 18, 26, 2

obtain the motasks for poexplosive deviment of a mods by applyingated wheeledR12‐EOD. Unlprevious woronic design, odelling in ree methodologin real operhis paper.

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presents thebile manipulat and experimliminary anald orientation,ematic schemnteractions, fodynamic paruation, real ex

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delling of moositioning in tices, the presedelling methog the mechatd mobile mlike the otherks, it considethe coupled

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ure 1. Robot MM

average veloc it is requiredctor. These fe the kind of e in order to fo

Master map

master map ihe mathematisystems and bile manipula internal consich are relatedterrain is conrdinates withrnal constrainonomy of theme with the nsystem) is cla

MR is the msystem, like the where the erating coopert of the princinsformations ordinate “0” rrdinate “C”. Tsubsystems forelation to thenipulation; in ded subsystemat which are

), at which it ms assumed thanisms, that tt all the axes ge surface of tsliding with thfficiently smathe guided aiggest problemocation with mobile platfo4‐DOF and

MR12‐EOD

city of the robd to support atures, as welsystem (non‐orm the maste

is a diagram thical model, inhomogenou

tor. Describedstraints, beingd to the operatnsidered as thh respect to nts are determhe mobile pnomenclature ssified, where

mobile to whhe lifting armadditional suratively and lopal manipulaof this subsystrather than bThe master maor the mobile e tasks of orithis case, the

m formed by twconsidered th

must have a uhat the robot there is an linguided are perthe motion is he rolling surall to allow thaxis [8‐10]. Slidm during thehigh accura

orm has 4‐Dthat the adde

bot MMR12‐EOa load of 10 ll as the critic‐holonomic) aer map.

hat shows the n which is repus transformad first are bothg those externting environmhe localizatioglobal coor

mined accordinplatform. AftM‐MMR (adde M is the manhich is addems or any subsubsystem is aocated in the sator), then the tem will relatebeing related ap has the stamanipulator entation, appmobile manipwo coupled lihe lifting and c

unique rollingis being builtnk guided byrpendicular toa plane, and

rface and thathe rotation ofding with thee moment ofacy. Figure 1DOF, that theed subsystem

OD is 0. 5 m/skg at its endal parametersare used as a

developmentpresented theations of theh the externalnal constraintsment. As such,on of its localrdinates. Theng to the non‐erwards, theding the othernipulator anded the othersystem. In thea manipulatorsame origin ashomogenouse to the originto the local

ages related toMMR12‐EOD

proaching andpulator has anfting arms (B.climbing tasks

g t y o d t f e f 1 e m

s d s a

t e e l s , l e ‐e r d r e r s s n l o D d n . s

2 Int J Adv Robotic Sy, 2012, Vol. 9, 192:2012 www.intechopen.com

[11‐14]. In order to model the robot MMR12‐EOD, seven stages in five real operation scenarios are considered [7], at which the assessed behaviour of the full system with and without load is considered. For the robot in the present work, the stages are: a) The additional subsystem is considered in the neutral

position and in operation with respect to the local coordinate “C”. This subsystem is called “A”. This stage is omitted if the manipulator has no additional subsystem.

b) The manipulator uses just its degrees of freedom to perform tracking trajectories tests in relation to the coordinate “0”, which is the base of the manipulator.

c) The mobile platform is moved forward with the manipulator mounted and performing a tracking trajectory test from the coordinate “C” to the global coordinate “G4”.

d) Considering the coupled system without allowing the advance of the mobile platform, the manipulator realizes the test of the tracking trajectory, assuming that the mobile platform rotates about its axis with respect to the local coordinate “C”.

e) The mobile platform is moved forward with the manipulator coupled while performing a tracking trajectory test from the coordinate “C” with respect to the global coordinate “G4”.

f) The coupled system is moved in a straight line, moving its manipulator in relation to the end effector with respect to “G4”.

g) The coupled system is subjected to the tests for tracking the trajectory in relation to the end effector with respect to the coordinate “G4”.

2. 3 Diagram of the position and orientation

The general description for the orientation and position of the system is described by the fourth transformation of the global coordinates with respect to GG, as shown in figure 2.

Figure 2. Fourth Transformation of the global coordinates with respect to the global coordinates

2. 4 Homogeneous transformation graph (HTG)

The third step of the proposed methodology is the homogenous transformation graph that, in a modular way, allows the organization of the subsystems that

constitute the mobile manipulator plus the added subsystem. This graph is an extension of the graph for obtaining the kinematics of the manipulators proposed by Robert Pauli, in 1983 [15], and used in several works by Tadeusz Skodny [16, 30] in order to perform the same applications. The extension of Pauli’s graph allows us to get the homogenous transformation graph and the kinematic modelling of wheeled mobile manipulators with any additional subsystem. These transformations are the central part of the methodology because they provide the relations with the coordinates “0” of manipulator and “C” of the mobile platform, as well as the transformations of the wheels and the lifting arms in relation to “C” and “C” with respect to “G4”, as shown in figure 3. Afterwards, coordinates systems to the selected points that allow us to obtain the coupled kinematic modelling of full system are assigned.

Figure 3. HTG for mobile manipulators

2. 5 Schemes for the kinematics of the operation scenarios

In order to realize the schemes for the kinematics of five real operational scenarios of the MMR12‐EOD robot, it is necessary to consider its design parameters.

2. 6 Interaction kinematics

Table 2 presents the parameters of the interaction kinematics obtained from the coupling of the subsystems of the robot MMR12‐EOD.

2. 7 Forward and inverse kinematics

2. 7. 1 Forward kinematics

Based on the homogenous transformation graph and the table of the kinematic interactions, we develop the forward kinematics, starting with the coordinate “C”, and from here describe the transformations of the wheels, the


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lift arms aneffector “E” w6 [17, 28‐30].

Figure 4. Later

a) Mobile pla The homogeis given byparameters ∆λC, zC� � λC�var. The incrrelationship:

∆λC � �fo

Then the tranand the tranand the leobtained[29‐3 b) Lifting arm The transformrespect to thelifting arm ofvar, lCS� � coarm in relatiGTCS� � ACAsecond, third c) Manipulat The homogerespect to the“C” are transformatiodepends upoconst � 0, l� �A� � �x�y�z�parameters: transformatio

nd the maniwith the load

ral view of the M

atform

nous transfory AC � �xG

��yare: AC: xC� �� const � 0,rement of λC

lCS�SC� � λCor lCS�SC� � λC

0 for anothensformation onsformations feft side, G30].

ms

mations of the e local coordinf the right fronnst � 0. Thereion to the gloC�ACS�. In a s and fourth lif

tor

enous transfoe coordinate “now pres

on from “C” on the follow� const � 0 α�� x�y�z��, d

θ� � var,on A� � �x�y�

ipulator, cons“Eμ”, as show

MMR12‐EOD in

rmation of theyG

��zG�� xCy

� var, yC� � va, θC� � var, lCS�C depends up

C� � rC� � r � 0er case

, r �

f the local coofor the wheelsTR � ACARand

lifting arms anate “C”. The nt side are giveefore, the tranobal coordinateimilar way, thfting arms are o

ormations of “0” in relationsented. Theto “0”, A� � �wing parame� � �0°. Thedepends upoλ� � 0, l� � 0 z� � x�y�z��,

sidering the wn in figures 4

n the lifting posi

e mobile platf

CzC�, where r, zC � var � λ� � const � 0,pon the follow

� constant � 0

ordinate, GTCs of the right d GTL � ACA

are developed parameters foen by: ACS�: θCnsformation ofes is expressehe relations foobtained.

its 4‐DOFsn to the coordie homogen�xCyCzC � x�yters: θ� � 0°,e transformaon the followα� � �0°. depends upon

end 4 and

ition

form the

λC� �, � �wing

(1)

� AC side

ALare

with or the CS� �f first ed as or the

with inate neous y�z��,

λ� �ation wing The

n the

folloTheupocons�x�ypara ThefourE �paraIn owithrelaconsandof tconsjoin Themanthe simpSen�trancoorthe Eµ_Ptranrela[19]

whe � �CCS�SCC�

2. 7.

HercoorparaeffeCartdevequinve

owing parame transformaon the followst � 0, α� � �0y�z� � x�y�z��ameters:: θ� �

transformatiorth degree of f�x�y�z� � x�yameters areorder to obtainh the end tionship Eµ �stituted by Eµ the rotationathe prismatic st � 0, l� � 0 αt are as follow

n, the transnipulator are m

manipulatoplification �A � �� CASnsformation frdinates givenend effector Eµ_R. Finallnsformation frtion to the glo.

�A � � xC� � � yCC � � zC0 0 0

ere:

S�SC � CCC��C��, � � SCS��, ��S� � C�CC, � 2 Inverse kinem

re, the equrdinates of thameters are ctor from thetesian coordieloped using ations 3 to 11erse kinematic

eters: θ� � 0°tion A� � �xwing parame0° and th� depends

� var, λ� � 0, l�

ons of the endfreedom of they�z��, where e E: θ� � 0°, λ�n transformateffector, thesEµ_PEµ_R, wh

µ_P y Eµ_R, whal load, respecjoint are as

α� � 0°. The pws: Eµ_R: θ� � 0

sformations multiplied so r is T�, u

Cos�A � �� SB � SACB � SAfrom the man by GT� � ACwith the loay, we obrom the load obal coordinat

XM �GT�

� CC�S�λ� � l�C�� SC�S�λ� � l�C�

� � ∆λC � λ� � λ�

C�, � � SCC��C� � �C��, � �� S��S�.

matics

uations that he task spacdescribed in e mobile maninates. The the homogene1 from the schcs, which is rep

°, λ� � var, l� �x�y�z� � x�y�zeters: θ� � vahe transform

upon the� � 0 α� � 0°.

d effector in re manipulator

E � Trans�0,� const � 0, ltions that conse are obtaihere Eµ is thhich are the pctively [18]. Thfollows: Eµ_P: parameter of 0°, λ� � 0, l� �

from the jo that the tranusing the � CACB � SASBABis obtaineanipulator to

CCT�. The tranad is determitain the h“X” in the entes described

E Eµ=

� � l�� CCλ�S�� l�� SCλ�S��

�C�� C�λ� � l�S�1

C� � CCS�, � �CCC��S� � C�

connect the in relationorder to ca

nipulator to thinverse kineeous transformheme for forwpresented by f

� 0 α� � ��0°.z�� dependsar, λ� � 0, l� �mation A� �e following

relation to ther are given by0, λ�� and its� � 0 α� � 0°.nnect the loadined by thehe total load,prismatic loadhe parameters

θ� � 0°, λ� �the rotational0 α� � �0°.

oints of thensformation oftrigonometric

B � CAB, .ed by theo the globalnsformation ofined by Eµ �homogeneousnd effector inby equation 2

� CCλ�S�� SCλ�S��

�� λ�C�� (2)

S��C�, � �SC, � �

he Cartesiann to the jointarry the endhe mentionedematics weremations of theward and thefigure 5.

. s

��g

e y s . d e , d s

�l

e f c . e l f

�s n 2

)

n t d d e e e


Figure 5. Schem

� �

� �

d

2. 8 Constrain

The differentbased on thenot have an DOFs than tfigure 6. In ofrom the syangles in rel(yG, φG) andorientation oreference is and θ� C� � 0,equations [22

q � �xC� yC�

J � ��s�nφ�cosφ

0

It is known ththen: J��

me for the inver

φ

θ� � cos��

λ� �

tan��

� ; K

1�0 � K � q�;

d � cos��

q�� s�n��

x� � d��C��λy� � d��S��λnts

tial kinematicse non‐holonomintegrable so

the controllaborder to obtaiystem, we colation to the gd (zG, γG), assof the local refconstant in

θ� CS� � 0. Tabl2‐25].

� φC θ� θ� z

cosφ 0 0φ �s�nφ �b r

0 �1 ��

hat J�q�q� � 0 � � �

|J�| �adj�J��

rse kinematics

� tan��

� ��

�Z� �� C�� C�

� 1�0 � �0 �

; r � �h� � d��

�� ; q��

��Z��C� � ; q�

λ�S� � �� C�

λ�S� � �� C

s in relation tomy constraintolution and itble DOFs [20‐in the relationonsidered thaglobal coordinsuming that ference in relathe movem

le 1 shows the

zC� θC� θCS�

0 0 0 0r 0 0 0��

�� 0 0

and that J�q� i�T, �adj�J��T

��

��

� S�

� �1; h � � ��SE

� ;

�� 2hd�C�� ;

1�0 � � � d1;

� q�� q��

�� λ�S��

�� λ�S��)

o ZG��is develo

t ‐ namely, it t represents m‐21] in relations of the velocat the orientanates are, (xGthe position ation to the glent of the re constraints o

θ� d� θ� θ��T

0 0 0 0 00 0 0 0 00 0 0 0 0

�

is a square mabecause

(3)

(4)

(5)

; (6)

; (7)

(8)

(9)

(10)

(11)

oped does more on to cities ation

G, δG), and lobal robot of the

T (15)

(16)

atrix, the

detemanmobJ��q Thin symgrothemaof ��

��

Ththedriwhdisleft

Tabl

2. 9

The comtranthe the itranOn Lis obis odevethe obtathe d

2. 10

TheA�q�comDerit in

M

eterminant onipulator has bile platformq� � �J��q�

e mobile platforthe direction

mmetry with ound, where ��e coordinates oass and � is the the platform m

��[26]. e other two ce rolling constraiving wheels dhere θ�, θ� are splacements of t wheels, respec

le 1. Constraint

Dynamic param

dynamic mmputational alnsformations anletter L to depinteractions tabnsformations inL3is developedbtained the pseobtained the meloped the termCoriolis and ained the matrdynamic equat

M�q�q� �0 Consolidation

two column� and are line

mbination of iving q � we han equation 14 g

M�q� � �d11 …

� …d121 …

��S�C

0q� � M�q��

S�q� � �

f J� is|J�| �good manip

m [27] ‐ J�qJ��q�.

rm must move ofits axis of respect tothe

��, ��, �� are of its centre of heading angle

measured from

constraints are aints ‐ i. e., the do not slip ‐ the angular the right and ctively [26].

equations to th

meters

model was lgorithm, basnd the interactpict the steps. Lble. On L2 is en the G. T. H. ad the matrix Ueudo‐inertia mmatrix of inerms h��, and inthe centrifugerix of gravity tion 17 on L10.

V�q, q� �q� � G�n of the dynami

ns from S�q� arly independtwo colum

ave q� � S�q�ν�gives as the re

… d112… �… d1212

� q� �

SC CC 0CC �SC �b0 0 b

��q�τ � AT�q

��SC CC�CC �SC �

0 0

�1, and sopulability accoq� � �J��q�, J��

y� C�cosφ � x�

x� C�cosφ � y�

φ� � �

he robot MMR12

done usingsed on the tions table [28L1 is describedstablished the and the forwa

U��, L4 the matrmatrix for each lrtia M�q� � �dn L8 is obtainede V�q, q� � � �hG�q� � �c��T. A.

�q� � ��q�τ � Ac equation.

are the nuldent. It presenmns from Sν� �t�+S� �q�ν�t� asult of equatio

�h1�

h12� q� � �

c1�

c12�

0 0 … 0r 0 … 00 r … 0

�

q�λ � V�q, q� �q� �0 0 0 …

�b r 0 …b 0 r …

the mobileording to the�q��. Finally,

C�s�nφ � 0 (12)

y� C�s�nφ � φ� b �θ� �r (13)

�� θ� � � θ� ��(14)

2‐EOD

g Lagrange´shomogeneous

8]. We will used every link inhomogeneous

ard kinematics.rix U��. On L5link and on L6

d��. On L7 isd the matrix of

h��T. On L9 isAs, we obtain

AT�q�λ (17)

l space fromnts q� as linearS�q�, q� � S�q�νand raplacingons 21 and 22:

�

��0 0

� �1 00 1� �0 0�

��

�τ��τ�

� �

(18)

� G�q�� (19)000

� (20)

e e ,

s s e n s . 5 6 s f s n

)

m r ν g :

�

)

)


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ST�q� �M�q�S�q�ν� �t� � M�q�S� �q�ν�t� � ��q, q� �q� � ��q�� E�q�� AT�q�λ�ST�q� (21)

ST�q��M�q�S�q�ν� �t� � M�q�S� �q�ν�t� � ��q, q� �q� � ��q�� (22) Is verified that M�q� complies with n=m (square), MT�q�= M�q� (symmetric). It is not singular (det �M�q�� 8. 0��e � 021 � 0 � ��M�q�T� � 8. 0��e � 021 �0; d�� 0. 21� � 0), ) and therefore is inverse. M��q� �; M��q� � �M��q�� 0. It is also positive definite, in that M��q� � 0 for all q and its determinant�M�q�� 0. 3. Real experiment and positioning simulation

The real experiment and the simulation of positioning from the mobile manipulator is done according to the 5 real operation tasks. In order to realize these activities, the control strategy published by Yamamoto was used [29] in the paper dedicated to the control and coordination of movement from mobile manipulators in

2004. Using the vector in the state space TT Tx q ,

we can represent the constraints and the motion’s equations from the mobile manipulator in the state space:

12

0

( )

TS

xf S MS

(23)

where 12 ( ) ( ) T T Tf S MS S MS S V . This equation is

simplified in order to apply the following linear feedback:

2( ) TS MS u f . Then, it could assume a new input u , which will linearize equation 23. Considering the linearization, it could be used in the desired trajectory dy in order to feedback the error de y y :

( ) ( ) d d dd py v y K y y K y y (24)

From equation 24, v is obtained and then u and, therefore, x . Finally, by integration, x can be obtained. a) Orientation (stage 2, task 1). The driving wheels are located on the left side at the front and right side rear, the alternate movement v� � �v� allows the rotation about axis z in the reference ratio of ø 1130 mm. This is using the transformation GTC � AC and qM � �φC θ�θ� �T

. Therefore, it is assumed that the movement is in the horizontal plane [13], that the contact wheel is at one point, that the wheels are not deformable, that there is pure rotation at the contact point v = 0, that there is no slippage, and that the axis direction is orthogonal to the surface. The wheels are connected by a rigid body (chassis). b) Approximation / trajectories tracking (stage 4, task 2). The system executes the tracking of the desired trajectory,

based in the transformation GTC � AC. The mobile platform is moved according to this equation, qMR ��xyzφC θ�θ� �T

. c) Manipulation (stage 4, task 3). In the rank x=2. 5, y=2. 5 and z=1. 5 m, the kinematics is given by XM �GT�E Eµ by the equation qM � �φC θ�d�θ�θ� �T

. d) The lifting of the mobile manipulator in the rank from 0. 15 up to 0. 52 m based on the transformation GTC �AC, z is given by zC � var � λC� � ∆λC, zC� � λC� �constant � 0, θC� � var�a�le, lCS� � constant � 0, � ��, where the increment ∆λC of z, depends upon the lifting arms. e) The lifting arms climb in the rank from 0. 15 up to 0. 47 m. The mobile manipulator with a mass m is moved on the sloped surface at an angle ψ, with respect to the horizontal, given by the transformation CTCS�, and the equation F�� g�senψ � µcosψ�, where g is gravity when θCS � �0�, λC � ��C � lCS� � senθCS�, in order to realize the climbing task. λCis variable and depends upon θCS.

4. Results

From the forward kinematics, the reference values for every task were taken. The parameters of the coordinate local “C” which is the base for obtaining the reference values are as follows: AC: xC� � 2, yC� � 2, zC � λC� � ∆λC, zC� � λC� �0. 15, θC� � �

� , lCS� � 0. ��, φ � �� , φ � 20�.

∆λC � var�a�le � �lCS�SC� � λC� � r

for lCS�SC� � λC� � r � 00 for another case

, r � 0. 08 (25)

a) Equation 26 is the transformation of the mobile platform rotating about its own axis:

GTC � �1 2. ��3e � 015 0 0

�2. ��3e � 015 1 0 00 0 1 0. 150 0 0 1

� (26)

b) Equation 27 is the transformation of the platform according to these parameters: xC � 2, yC � 2, λC� �0. 15, φC � �

� , θR � 2�, θL � �2�, � � 0. �, ∆λC � 0.

GTC � AC � �0. 7071 �0. 7071 0 20. 7071 0. 7071 0 2

0 0 1 0. 150 0 0 1

� (27)

c) Equation 28 is the transformation of the manipulator with the end effector: xC � 2, yC � 2, λC� � 0. 15, φC �� , λ� � 0. 1�, l� � 0. 33, θ� � ��

� , λ� � 1. �, θ� � � �� , l� �

0. 15, θ� � �� , θC� � �

� , lCS� � 0. ��, r � 0. 08, λ� �0. 38, , λ� � 0. 25:


XM �GT�E

d) Equation platform is li

GTC � A

e) Equation climbing: xC�� , θCS� � � �

�

GTCS

Taking the requation 30 fout, where obtained, as s

Trackintrajectory aits geome

centre.

Trackintrajectory osinusoidal c

Table 2. Tasks

Mean erroin the man

Mean e

Table 3. Tasks

E Eµ �� 0. 7071

� 0. 7071 00

29 is the tranifting: xC � 0, y

AC � �0. 70710. 7071

00

30 is the tra� 0, yC � 0, �C, � � 0. 35, �CS

S� � �0. 70710. 7071

00

reference valufor the real opthe errors ashown in table

ng about etric .

Samplreferencoordi

Mean e

ng of the curve.

�� 2these c

��

Mean e

for orientation

r of 0. 050 mmnipulation task

error of 0. 040

for manipulatio

0. 5000 0. 500. 5000 0. 50

�0. 7071 0. 700 0

nsformation atyC � 0, θC� � �

�

�0. 7071 00. 7071 0

0 10 0

ansformation C� � 0. 15, �C �

� 0. 33, �CS� �0 �0. 7070 0. 707

�1 00 0

ues, several serational scenand performaes 2 and 3.

ing time: �nce C is lonates:

� ��

error of 40 mm.200 �. The referecurves: � �� 30 �� 40 �� error of 2. 37 mm

and the trackin

m k.

Mean errin the

mm in the cli

on, lifting and c

000 2. 8567000 2. 8567071 1. 33050 1 ��

��

t the time tha��:

0 00 01 0. 52000 1

�

correspondin� �

� , θC� �� 0. 44, � � 0. 171 01 0

0. 471

�

simulations unarios were carance index w

�� 200 �. Tocated on the

� � 0 � � 0 . ence is located

�� 0. 25 �� 0. 25 �� 2. 5m.

ng trajectory

ror of 0. 020 me lifting task.

imbing task.

climbing

(28)

at the

(29)

ng to

10.

(30)

using rried were

The ese

on

�

mm

In tmanmin 301.

2.

1.

2.

3.

Tablclim

In fexpein th

Figu

5. D

The laboregigrouand its simuare pfor e In tasimutaskwithexpeerro

able 4, the renipulator are nimum errors a

0 samples . In the test fororientation therrors were t. Approaching(trajectories tracking). . Manipulation(positioning)

. Lifting.

. Climbing.

le 4. Real experimbing tasks.

figure 6, we eriment approhe target.

ure 6. Mobile ma

Discussion

real experimeoratory, whichstered parameund were not instrumentatioparameters. Tulations and thpresented in taeach task, whic

able 6 is showulation and thks. The figuresh the resultseriment in redor.

esults of real shown, focusaccording to t

r he next aken:

Er4

g Er0

n ).

Er0

Er

Er4

iments for the m

see the moboaching and p

anipulator in a r

ent was carrieh helped in eters; howevetaken due to on for the corrThe comparishe real experimable 5. As manch validates the

wn the differehe real expers are describeds of the simd, including t

experiments ising on the mhe reference v

Deviatio

rror min. 40 mm

E

rror min. 0. 5 mm

E

rror min. 0. 5 mm

E

rror min. 4 mm

E

rror min. 4. 6 mm

E

manipulation, lif

bile manipulapositioning the

real experiment

ed out on the the measure

er, measuremea lack of adeqresponding meson of the ements of the 5 eny as 30 sample kinematic mo

ences in errorsriments of thed in the Gaussmulation in bthe mean and

in the mobilemaximum andvalues.

ons

Error max. 80 mm.

Error max. 8 mm.

Error max. 8 mm.

Error max. 6 mm.

Error max. 8 mm.

fting and

ator in a reale end effector

t

floor inside aement of theents in roughquate methodseasurements oferrors of theevaluated tasksles were takenodelling.

s between thee 5 evaluatedsian bell form,blue and thed the R. M. S

e d

l r

a e h s f e s n

e d , e S


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Task

Orientation.

Approachin

Manipulatio

Lifting.

Climbing.

Dynamic analysis with software CAE

According toconsidered wit

Table 5. Simul

Comparis

1. Orientati

2. Approach

3. Handling

4. Lifting.

5. Climbing.

Table 6. Errors

Simulation (��

40 and 40. 1

Error due

g. 2. 37 and 2.

The 2 te

on. 0. 050 and 0

This is d

0. 020 and 002144

Due to adju

0. 040 and 0040743

This is due tand its adjus

8. 497

With lo

The mass varies in � 6

The g

The stress dload of 14 k

The deforofλ�is 1. 055

It was validthe normal

o the manual of eth a maxim um ha

lation and real e

son of the simu

Tasks

on.

hing.

g.

�� 6.

RMS 6.

s between the si

Performance [

�� y RMS) Real

161 60 an

to the wheel slip

376 6. 5 a

ests were perform

0. 051 4. 25

ue to slip of whe

0. 5 an

ustment and weig

0. 6. 3 a

to the deformationstment, in addition

8. 5299 16

oad

centre 60.

The in �

greatest variation

due to a kpa.

The load

rmation 5.

The is 0.

dated to the fullesstate of the prism

explosives, deactiandling error of 10

experiments for

lation and the r

e�RM

e�RM

e�RM

e�RM

. 26 mm

. 28 mm

imulation and th

[mm]

l system (�� y RM

nd 60. 0922

pping on the floor

and 6. 5236

med at 0. 5 m /s.

5 mm and 4. 2500

eels on the floor.

nd 5. 037658

ght of system.

and 6. 30918

n of the rear when to its weight.

6. 41 16. 44

Without load

mass centre vari40.

n occurs in Z.

stress due to d of 7 kpa.

deformation of 03.

st extension and matic joint.

ivation [24] must mm.

the tasks.

real experiment

Difference

e� 20 mmMS 19. 94 m

e� 4. 12 mmMS 4. 147 me� 4. 2 mmMS 4. 19 mme� 4. 98 mmMS 5. 01 mm

he real experim

S)

r.

02

els

47

ies

a

λ�

in

be

t

mm

mmmmmmm

ment.

Theon twhecentvarithe inte

Figuexpe

6. C

A articdevandthe struhas for andthe is rparaschemodthe of th Thewithexpeare whiexplman Thewithcontmod

orientation ethe ground, aere the trajecttre of the moiations are dunon‐holonomgrable solutio

ure 7. Error simueriment for the m

Conclusion

methodology culated mobelopment of a assessed, whhomogeneou

ucture for the the followingmechatronic simulation ofestimation of required; (3) ameters of inteemes in a coudelling when analysis of thhe real operati

simulations hh regard to erimentationspermissible eich is specifielosives [30]. nipulator with

major advanh previous tributions of tdelling in a

error is very laas well as thetory is a poinobile manipuue to the compmy of the syons of the whe

ulation against tmanipulation ta

for the pobile manipula kinematical mhere the use ous transformaforward kine

g advantages: (design when f the mobile mthe homogenoit allows t

eraction whenupled way; (4any added suhe performancional scenario

have an averathe reference

s have an erroerrors in relatied in the task

The maxih a load of 10 k

ntages of thiwork are cthis paper: thecoupled wa

arge due to we tracking of nt referenced ulator; the afopensation of tstem caused eels.

the error of the ask

ositioning ofator by memodel is definof a novel scheations achievematics. The (1) it providesit develops t

manipulator; (2ous transformthe determinn it is necessar4) it is easy ubsystem is ince index basedos:

age error RMSes values, wor RMS of 7. 9ion to the errks for the deim deformatkg was 1 mm.

s proposal inconsidered, se developmentay for a wh

wheel slippagethe trajectoryby the massorementionedthe control ofby the non‐

real

f the multi‐eans of thened, explainedeme to devisees the givenmethodologys a big picturethe modelling2) it simplifiesmation when itation of thery to make thefor kinematicncorporated ind on the error

S of 3. 47 mmwhile the real91 mm. Theseror of 10 mm,eactivation oftion of the

n comparisonsuch as thet of kinematiceeled mobile

e y s d f ‐

‐e d e n y e g s t e e c n r

m l e , f e

n e c e


manipulator with 12‐DOF, the extension of homogeneous transformation graphs which organize in a modular way the transformations of every link so as to obtain the kinematics of the mobile manipulators and the proposed methodology formodelling the positioning in real operational scenarios. As to future work, by means of a visual feedback system, will be determined the position of the explosive device incorporating a vision system to detect the coordinates of targets, placing some sensors at the end effector for use in different kinds of manipulation based in the deviceʹs form, and finally implementing the proposed modelling methodology in the design of mobile systems with different manipulators.

7. Acknowledgements

The authors would like to acknowledge the Centro de Ingeniería y Desarrollo Industrial (CIDESI) [30] for funding this work.

8. References

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[10] Skodny T. Forward and Inverse kinematics of IRb‐6 Manipulator. Mach. Theory, Pergamon Press, London, 1995; 30(7): 1039‐1056.

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[13] Arai T. Robots with integrated locomotion and manipulation and their future. IROS, Osaka, Japan, November 1996: 541–545.

[14] Ghaffari A., Meghdari A., Naderi D., Eslami S. Stability enhancement of mobile manipulators via soft computing. International Journal of Advanced Robotic Systems. 2004; 3(3): 191‐198.

[15] Aviles J. A. Z., Pedraza J. C., Gorrostieta E., García‐V L. H. Development of a Mechatronic Unit applied to the Manipulation of Explosives. International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE, 2008; ISBN: 978‐1‐4244‐2499‐3.

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[17] Choset H., Lynch K. M., Hutchinson S., Kantor G., Burgard W., Kavraki L. E., Thrun S. Principles of Robot Motion: Theory, Algorithms, and Implementation. MIT Press, USA, 2005; ISBN: 1098765432.

[18] Bayle B., Fourquet J. Y., Renaud M. Manipulability of Wheeled Mobile Manipulators: Application to Motion Generation. International Journal of Robotics Research, 2003; 22(708): 565‐ 581.

[19] Velinsky S. A., Gardner J. F. Kinematics of mobile manipulators and implications for design. Journal of Robotics Systems, 2000: 309‐320.

[20] Tang C. P., Bhatt R. M., Abou‐Samah M., Krovi V. A Screw‐Theoretic Analysis Framework for Payload Transport by Mobile Manipulator Collectives. IEEE/ASME Transactions on Mechatronics, 2006: 169‐178.

[21] Tan J., Xi N., Wang Y. Integrated Task Planning and Control for Mobile Manipulators. Proc. International Journal of Robotics Research, 2003: 337‐354.

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[23] Goris K. Autonomous Mobile Robot Mechanical Design. VrijeUniversiteitBrussel, Engineering Degree Thesis, Brussels, Belgium, 2005.

[24] Dimitrios S. A. Analytical Configuration of Wheeled Robotic Locomotion. Carnegie Mellon University, Doctoral Thesis, U. S. A., 2001.

[25] Zuñiga L. A., PedrazaJ. C. , Gorrostieta E., García L., Ramos J. M., Herrera R. Modeling and Simulation of a Mechatronic Unit EOD/IEDD. International


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