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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.
the paper is oe methodologyem and the ex Later, sectionhe reference vare presentednd the real exparameters anand is concl
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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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face (the floortact [6‐7]. It ih rigid mechary wheel, thatfloor, that tht there is no sfriction is suwheel about
face is the biablishing a lows that the nipulator has s too.
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
3Luis 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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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
4 Int J Adv Robotic Sy, 2012, Vol. 9, 192:2012 www.intechopen.com
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 :
�
)
)
5Luis 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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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:
6 Int J Adv Robotic Sy, 2012, Vol. 9, 192:2012 www.intechopen.com
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
7Luis Adrian Zuñiga Aviles, Jesus Carlos Pedraza Ortega and Efren Gorrostieta Hurtado: Experimental Study of the Methodology for the Modelling and Simulation of Mobile Manipulators
www.intechopen.com
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
8 Int J Adv Robotic Sy, 2012, Vol. 9, 192:2012 www.intechopen.com
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.
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