Experimental Study of the Methodology for the

ARTICLE 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. www.intechopen.com

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,

Int J Adv Robotic 2012, Vol. 9, 192:2012 Luis Adrian Zuñiga Aviles, Jesus Carlos Pedraza Ortega and Sy, Efren Gorrostieta Hurtado: Experimental Study of the Methodology for the Modelling and Simulation of Mobile Manipulators

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there are reco ords of workss related to thee programmin ng of teleoperated mobile manipulators [1]; moreover, m in 1988 [2] and 1996 [3‐4], the dyn namics of mo obile manipulaators in the operaational case of o the openin ng of a door was proposed; however, h the manipulator and the mo obile platform weere considereed and anaalysed separaately. Besides thiss, from 1994 to 2008, so ome works were w published on mobile manipulatorrs with pllanar manipulatorss but withoutt considering the load or other o operational cases; they were w focused only on con ntrol, kinematic redundancy, tracking trajectories and manipulabiliity [5, 18, 26, 29]. ators o the modelling of mo obile manipula In order to obtain used in the tasks for po ositioning in the t handling and transport of eexplosive deviices, the preseent work consiiders obile the developm ment of a mod delling methodology for mo manipulatorss by applying g the mechattronic design of a multi‐articulaated wheeled d mobile manipulator caalled Robot MMR R12‐EOD. Unllike the otheers methodolo ogies presented in previous worrks, it consideers the problem of the mechatronic design, the coupled full system and kinematic mo odelling in reeal operations. An experimeental study of thee methodolog gy capable off performing this positioning in real operational scenaarios is descrribed throughout this paper. The rest of the t paper is organized o as follows. Sectiion 2 describes thee methodology y for the kinem matic modellin ng of the full systeem and the ex xperiments in n several scenarios of operation. Later, section n 3 presents th he results in w which the basis of th he reference v values that weere obtained in n the modelling are a presented d with the errors from the simulation an nd the real exp periment, com mparing them with the ideal paarameters an nd both errorrs. This worrk is discussed and a is concluded in Secctions 4 and d 5, respectively. on Methodolo ogy 2. Modelling and Simulatio y for This section presents thee modelling methodology wheeled mobile manipulaators, describ bing the plann ning, developmentt and experim mentation. Itt is organized d as follows: prelliminary anallysis, master map, diagram m of position and d orientation,, homogenou us transformaation graph, kineematic schem mes of opeeration scenaarios, kinematic in nteractions, fo orward and inverse kinem matic, constraints, dynamic parrameters, con nsolidation off the dynamic equ uation, real ex xperimentation n and positio oning simulation. 2. 1 Preliminaary analysis The robot MMR12‐EOD M i considered by being able to is move on a su urface by the action of its w wheels mounteed in nd touching the surface. The wheelss are the robot an mounted on n devices th hat allow reelative movem ment between its attached poiint (point of guidance) an nd a 2

Int J Adv Robotic Sy, 2012, Vol. 9, 192:2012

must have a u unique rolling g surfface (the floor), at which it m conttact [6‐7]. It is assumed th hat the robot is being builtt with h rigid mechaanisms, that tthere is an lin nk guided by y everry wheel, thatt all the axes g guided are perrpendicular to o the floor, that the surface of tthe motion is a plane, and d thatt there is no sliding s with th he rolling surrface and thatt the friction is sufficiently smaall to allow th he rotation off any wheel about the guided aaxis [8‐10]. Slid ding with thee surfface is the biiggest problem m during thee moment off esta ablishing a lo ocation with high accura acy. Figure 1 1 show ws that the mobile platfo orm has 4‐D DOF, that thee man nipulator has 4‐DOF and that the addeed subsystem m does too.

Figu ure 1. Robot MM MR12‐EOD

The average veloccity of the rob bot MMR12‐EO OD is 0. 5 m/ss d to support a load of 10 kg at its end d and it is required effector. These features, as welll as the critical parameterss a used as a a and the kind of system (non‐‐holonomic) are orm the masteer map. basee in order to fo 2. 2 Master map The master map iis a diagram th hat shows the developmentt of th he mathematiical model, in n which is rep presented thee subssystems and homogenou us transforma ations of thee mob bile manipulator. Described d first are both h the externall and internal consstraints, being g those extern nal constraintss whiich are related d to the operatting environm ment. As such,, nsidered as th he localizatio on of its locall the terrain is con h respect to global coorrdinates. Thee coorrdinates with internal constrain nts are determ mined accordin ng to the non‐‐ holo onomy of th he mobile p platform. Afterwards, thee scheeme with the n nomenclature M‐MMR (add ding the otherr subssystem) is classified, wheree M is the man nipulator and d MM MR is the mobile m to wh hich is addeed the otherr subssystem, like th he lifting arm ms or any subssystem. In thee casee where the additional su ubsystem is a a manipulatorr (opeerating cooperratively and lo ocated in the ssame origin ass thatt of the principal manipulaator), then the homogenouss tran nsformations o of this subsysttem will relatee to the origin n coorrdinate “0” rather r than b being related to the locall coorrdinate “C”. T The master maap has the sta ages related to o the subsystems fo or the mobile manipulator MMR12‐EOD D in relation r to thee tasks of orientation, app proaching and d man nipulation; in this case, the mobile manip pulator has an n add ded subsystem m formed by tw wo coupled lifting arms (B.. A. ) at which are considered th he lifting and cclimbing taskss 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”.

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.

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 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

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 www.intechopen.com

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

Luis 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 an nd the maniipulator, conssidering the end effector “E” w with the load “Eμ”, as show wn in figures 44 and 6 [17, 28‐30].

Figure 4. Laterral view of the M MMR12‐EOD in n the lifting posiition

a) Mobile plaatform The homogenous transforrmation of thee mobile platfform ��

is given by y AC � �xG yG zG � xC yC zC �, where the

parameters are: AC : xC� � var, yC� � var, zC � var � λC� � ∆λC , zC� � λC�� const � 0,, θC� � var, lCS�� const � 0,, � � var. The incrrement of λC depends up pon the follow wing relationship:

lCS� SC� � λC� C �r ∆λC � �fo or lCS� SC� � λC� C � r � 0 , r � constant � 0 (1) 0 for anotheer case

Then the tran nsformation of the local coo ordinate, GTC � AC and the tran nsformations ffor the wheelss of the right side d GTL � AC AL are and the leeft side, GTR � AC AR and obtained[29‐330]. ms b) Lifting arm mations of the lifting arms aare developed with The transform respect to thee local coordin nate “C”. The parameters fo or the lifting arm off the right fron nt side are giveen by: ACS� : θCS� � C var, lCS� � const � 0. Thereefore, the tran nsformation off first obal coordinatees is expresseed as arm in relatiion to the glo GTCS� � AC AC� ACS� . In a similar way, th he relations fo or the obtained. second, third and fourth liffting arms are o c) Manipulattor ormations of its 4‐DOFs with The homogeenous transfo respect to thee coordinate ““0” in relation n to the coordiinate “C” are now pressented. Thee homogen neous on from “C” to “0”, A� � �x � C yC zC � x� y� z� �, transformatio on the follow depends upo wing parameters: θ� � 0°, λ� � const � 0, l� � const � 0 α� � �0°. Thee transformaation A� � �x� y� z� � x� y� z� �, depends d upo on the follow wing The parameters: θ� � var, λ� � 0, l� � 0 α� � �0°. on A� � �x� y� z� � x� y� z� �, depends upon n the transformatio

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owing parameeters: θ� � 0°°, λ� � var, l� � 0 α� � ��0°.. follo The transformation A� � �xx� y� z� � x� y� z� � dependss upo on the follow wing parameeters: θ� � vaar, λ� � 0, l� � consst � 0, α� � �0 0° and th he transform mation A� � �x� y� z� � x� y� z� � � depends upon thee following g para ameters:: θ� � var, λ� � 0, l� � 0 α� � 0°. The transformatio ons of the end d effector in rrelation to thee fourrth degree of ffreedom of thee manipulatorr are given by y E � �x� y� z� � x� y� z� �, where E � Trans�0, 0, λ� � and itss para ameters aree E: θ� � 0°, λ� � const � 0, l� � 0 α� � 0°.. In order o to obtain n transformattions that con nnect the load d with h the end effector, thesse are obtaiined by thee relationship Eµ � Eµ_P Eµ_R , wh here Eµ is th he total load,, consstituted by Eµ_P hich are the prismatic p load d µ y Eµ_R , wh and the rotationaal load, respecctively [18]. Th he parameterss t prismatic joint are as follows: Eµ_P : θ� � 0°, λ� � of the consst � 0, l� � 0 α� � 0°. The p parameter of the rotationall joint are as follow ws: Eµ_R : θ� � 0 0°, λ� � 0, l� � 0 α� � �0°. oints of thee Then, the transsformations from the jo nipulator are multiplied m nsformation off so that the tran man the manipulator is T� , u using the trigonometricc plification Cos�A � �� CA CB � SA SB � CAB , . . simp Sen�A � �� CA SB � SA CB � SAAB is obtaineed by thee tran nsformation from f the maanipulator to o the globall coorrdinates given n by GT� � AC CT� . The tran nsformation off the end effector with the loaad is determiined by Eµ � Eµ_P Eµ_R . Finally, we obtain the homogeneous h s tran nsformation frrom the load “X” in the en nd effector in n relation to the glo obal coordinattes described by equation 22 [19].

�

A � C 0

� � � 0

� � � 0

XM �GT� E Eµ=

xC � CC �S� λ� � l� C�� l� � � CC λ� S�� CC λ� S�� yC � SC �S� λ� � l� C�� l� � � SC λ� S�� SC λ� S�� zC� � ∆λC � λ� � λ� C�� C� λ� � l� S�� λ� C�� 1

(2))

wheere: � � S� SC � CC C�� C� , � � SC C�� C� � CC S� , � � S�� C� , � � CC S�� , � � SC S�� , � � �C�� , � � CC C�� S� � C� SC , � � SC C�� S� � C� CC , � � S�� S� . matics 2. 7. 2 Inverse kinem

uations that connect th he Cartesian n Herre, the equ coorrdinates of th he task space in relation n to the jointt para ameters are described in order to ca arry the end d effector from thee mobile man nipulator to th he mentioned d Carttesian coordiinates. The inverse kineematics weree developed using the homogeneeous transform mations of thee heme for forw ward and thee equations 3 to 111 from the sch inveerse kinematiccs, which is rep presented by ffigure 5.

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deteeterminant of J� is|J� | � �1, and so the mobilee man nipulator has good manip pulability acco ording to thee mob bile platform m [27] ‐ J�q� � �J� �q�, J� �q��. � Finally,, �� J�� q q� � �J� �q�J� �q�. The mobile platforrm must move in the direction ofits axis of sym mmetry with respect tothe gro ound, where �� , �� , �� are thee coordinates o of its centre of ma ass and � is the heading angle of the platform measured m from �� [26]. The other two cconstraints are thee rolling constraaints ‐ i. e., the driiving wheels do d not slip ‐ wh here θ� , θ� are the angular dissplacements of the right and leftt wheels, respecctively [26].

Figure 5. Schem me for the inverrse kinematics ��

φ � tan�� (3)

θ� � cos ��

�

��

�Z � �� C� � �� S�

λ� �

C�

� (4)

(5) �

; � � tan�� ; K � 1�0 � �0 � �1; h � �S� � ; (6) �

E

� � 1�0 � K � q� ; r � �h� � d�� 2hd� C�� ; (7) ; d � cos ��

��

�� q�� s�n

� ; q�� 1�0 � � � d1; (8)

�Z �� C �

� ; q� � q�� q�� (9)

x� � d� �C� �λλ� S� � �� C�� λ� S�� (10)

y� � d� �S� �λλ� S� � �� C�� λ� S�� ) (11)

2. 8 Constrain nts

��

The differenttial kinematicss in relation to o ZG is develo oped based on thee non‐holonom my constraintt ‐ namely, it does not have an integrable so olution and itt represents more m DOFs than the t controllab ble DOFs [20‐‐21] in relatio on to figure 6. In order o to obtaiin the relation ns of the veloccities from the sy ystem, we co onsidered thaat the orientaation angles in rellation to the global g coordin nates are, (xG , δG ), (yG , φG ) and d (zG , γG ), asssuming that the position and of the local refference in relaation to the gllobal orientation o reference is constant in the movement of the robot r of the and θ�C� � 0, θ�CS� � 0. Tablle 1 shows thee constraints o equations [222‐25].

q � �xC�

yC��

φC

�s�nφ cosφ φ �s�nφ J � ��cosφ 0 0

θ�

θ�

0 �b �1

zC�

0 r �

��

θC�

0 0 � �

��

θCS�

0 0 0 0 0 0

θ�

d�

θ�

0 0 0 0 0 0 0 0 0 0 0 0

θ� �T (15) 0 0� 0

(16)

It is known th hat J�q�q� � 0 and that J�q� iis a square maatrix, � �� the then: J� � |J | �adj�J� ��T , �adj�J� ��T because �

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y� C� cosφ � x� C� s�nφ � 0 (12)

x� C� cosφ � y� C� s�nφ � φ�b � θ�� r (13) φ� �

�

��

�θ�� θ�� (14)

Tablle 1. Constraint equations to th he robot MMR12 2‐EOD

2. 9 Dynamic param meters m was done using g Lagrange´ss The dynamic model com mputational allgorithm, bassed on the homogeneouss tran nsformations an nd the interacttions table [28 8]. We will usee the letter L to dep pict the steps. L L1 is described d every link in n the iinteractions tab ble. On L2 is established the homogeneouss tran nsformations in n the G. T. H. aand the forwa ard kinematics.. On L L3is developed d the matrix U U�� , L4 the matrrix U�� . On L55 is ob btained the pseeudo‐inertia m matrix for each llink and on L66 is obtained o the matrix m of inerrtia M�q� � �d d�� . On L7 iss deveeloped the term ms h�� , and in n L8 is obtained d the matrix off the Coriolis and the centrifugee V�q, q� � � �h h� �T . On L9 iss ained the matrrix of gravity G�q� � �c� �T . As, A we obtain n obta the d dynamic equattion 17 on L10.. M�q�q� � V�q, q� �q� � G�q� � ��q�τ � AT �q�λ (17))

2. 10 0 Consolidation n of the dynamic equation.

The two column ns from S�q� are the null space from m dent. It presen nts q� as linearr A�q� and are linearly independ com mbination of two colum mns from S�q�, q� � S�q�ν ν Deriving q� we haave q� � S�q�ν� �t�+S��q�ν�t� and a raplacing g n equation 14 g gives as the result of equatio ons 21 and 22:: it in

d11 �� M�q� M d121

… … …

�SSC ��C CC 0

0 0 �� c1 � � τ� h1 d112 1 0� � � �� q� � � � � q� � � � � � � 0 1 � � τ� c12 d1212 h12 �� 0 0� CC 0 0 0 … 0 �SC �b r 0 … 0� (18) 0 b 0 r … 0

q� � M�q�� q�τ � AT �q�λ � V�q, q� �q� � G�q�� (19)) �SC S�q� � ��CC 0

CC �SC 0

0 0 0 … 0 � �b r 0 … 0� (20)) b 0 r … 0


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S T �q� �M�q�S�q�ν� �t� � M�q�S��q�ν�t� � ��q, q� �q� � ��q��

�E�q�� AT �q�λ�S T �q� (21)

S q��M�q�S�q�ν� �t� � M�q�S��q�ν�t� � ��q, q� �q� � ��q�� T�

� � (22)

Is verified that M�q� complies with n=m (square), M T �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 T

2004. Using the vector in the state space x  qT  T  ,   we can represent the constraints and the motion’s equations from the mobile manipulator in the state space:

 0 S   x      T  (23) 1  f S MS ( )  2   

where f2  (ST MS)1( ST MS  ST V ) . This equation is simplified in order to apply the following linear feedback:   ST MS(u  f2 ) . Then, it could assume a new input u , which will linearize equation 23. Considering the linearization, it could be used in the desired trajectory yd in order to feedback the error e  y d  y :

d

d

d

 y  v   y  Kd ( y  y )  K p ( 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 T transformation GTC � AC and qM � �φC θ� θ� � . 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, 6


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

�xyzφC θ� θ� � . 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 T

the equation qM � �φC θ� d� θ� θ� � . 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. λC is 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�. lCS� SC� � λC� � r for l ∆λC � var�a�le � � CS� SC� � λC� � r � 0 , r � 0. 08 (25) 0 for another case

a) Equation 26 is the transformation of the mobile platform rotating about its own axis: 1 �2. ��3e � 015 GTC � � 0 0

2. ��3e � 015 1 0 0

0 0 0 0 � (26) 1 0. 15 0 1

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.

0. 7071 0. 7071 GTC � AC � � 0 0

�0. 7071 0. 7071 0 0

0 2 0 2 � (27) 1 0. 15 0 1

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:

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� 0. 7071 � XM �GT� E Eµ � � � 0. 7071 0 � � 0

0. 5000 0. 50 000 2. 8567� 0. 5000 0. 50 000 2. 8567� � �0. 7071 0. 70 071 1. 3305� 0 0 1 �

(28)

d) Equation 29 is the tran nsformation att the time thaat the � platform is liifting: xC � 0, yC � 0, θC� � : �

0. 7071 0. 7071 GTC � AC � � 0 0

�0. 7071 0. 7071 0 0

0 0 0 0 � (29) 1 0. 5200 0 1

e) Equation 30 is the traansformation correspondin ng to � climbing: xC � 0, yC � 0, �C� � 0. 15, � � , θ � C C C� � � �

�

, θCS� � � � , � � 0. 35, �CS � 0. 33, �CS� � 0. 44, � � 0. 10. 1 GTCSS�

0. 7071 0. 7071 �� 0 0

0 0 �1 0

�0. 707 71 0 0. 7071 0 � (30) 0 0. 47 0 1

Taking the reference r valu ues, several simulations s u using equation 30 ffor the real operational scen narios were carrried out, where the errors and a performaance index were w obtained, as sshown in tablees 2 and 3. Trackin ng trajectory aabout its geomeetric centre..

Trackin ng trajectory o of the sinusoidal ccurve.

In table 4, the reesults of real experiments in i the mobilee nipulator are shown, focussing on the maximum m and d man min nimum errors aaccording to the reference v values. 30 0 samples Deviatio ons 1.. In the test forr E Errror min. Error max. orientation th he next 440 mm 80 mm. errors were taken: 2.. Approaching g Errror min. Error max. E (trajectories 00. 5 mm 8 mm. tracking). 1.. Manipulation n E Errror min. Error max. 8 mm. (positioning)). 00. 5 mm Errror min. Error max. E 2.. Lifting. 4 mm 6 mm. Errror min. Error max. E 3.. Climbing. 44. 6 mm 8 mm. Tablle 4. Real experiiments for the m manipulation, liffting and clim mbing tasks.

In figure f 6, we see the mob bile manipula ator in a reall expeeriment appro oaching and p positioning thee end effectorr in th he target.

T Sampling time: �� 200 �. The referen nce C is lo ocated on theese coordinates: � �� 0 � �� 0 Mean eerror of 40 mm.. �� 200 2 �. The refereence is located on these ccurves: � �� 30 �� 0. 25 �� 40 �� 0. � 25 �� 2. 5 �

Mean eerror of 2. 37 mm m.

Table 2. Tasks for orientation and the trackin ng trajectory

Figu ure 6. Mobile maanipulator in a rreal experimentt

5. D Discussion

Mean error of 0. 050 mm m in the man nipulation task k.

Mean errror of 0. 020 m mm in thee lifting task.

Mean eerror of 0. 040 mm in the cliimbing task. Table 3. Tasks for manipulatio on, lifting and cclimbing www.intechopen.com

The real experimeent was carrieed out on the floor inside a a oratory, which h helped in the measureement of thee labo registered parameeters; howeveer, measuremeents in rough h grou und were not taken due to a lack of adeq quate methodss and instrumentatio on for the corrresponding meeasurements off its parameters. The T comparisson of the errors e of thee simu ulations and th he real experim ments of the 5 eevaluated taskss are p presented in taable 5. As man ny as 30 samplles were taken n for eeach task, whicch validates thee kinematic mo odelling. able 6 is show wn the differeences in errorss between thee In ta simu ulation and th he real experriments of thee 5 evaluated d task ks. The figuress are described d in the Gausssian bell form,, with h the resultss of the sim mulation in blue b and thee expeeriment in red d, including tthe mean and d the R. M. S S erro or.


7

Task

Performance [[mm] Simulation (�� y RMS) 40 and 40. 1161

Orientation.

Reall system (�� y RMS) 60 an nd 60. 0922

Error due to the wheel slip pping on the floorr. 2. 37 and 2. 376

Approaching.

6. 5 aand 6. 5236

The orientation eerror is very laarge due to w wheel slippagee t ground, as a well as thee tracking of the trajectory y on the wheere the trajecttory is a poin nt referenced by the masss centtre of the mo obile manipu ulator; the afo orementioned d variiations are du ue to the comp pensation of the t control off the non‐holonom my of the system caused by the non‐‐ ons of the wheeels. integrable solutio

The 2 teests were perform med at 0. 5 m /s. 0. 050 and 00. 051

Manipulatio on.

4. 255 mm and 4. 250002

This is due to slip of wheeels on the floor. 0. 020 and 00. 02144

Lifting.

5 an nd 5. 037658

Due to adju ustment and weig ght of system. 0. 040 and 00. 040743

Climbing.

6. 3 aand 6. 30918

This is due tto the deformation n of the rear wheels and its adjusstment, in addition n to its weight. 8. 497

8. 5299

166. 41

With lo oad The mass centre varies in � 660.

16. 4447

The mass centre variies in � 40.

The g greatest variation n occurs in Z.

Dynamic analysis with software CAE

The stress d due to a load of 14 k kpa.

The stress due to a load d of 7 kpa.

The deforrmation ofλ� is 1. 0555.

The deformation of λ� is 0. 03.

It was valid dated to the fullesst extension and in the normal state of the prism matic joint. According too the manual of explosives, e deactiivation [24] must be considered witth a maxim um haandling error of 10 mm.

Table 5. Simullation and real eexperiments for the tasks. Comparisson of the simulation and the rreal experimentt Tasks

Difference

1. Orientation. 2. Approach hing.

RM MS

19. 94 mm

e�

4. 12 mm m

��

RMS

6.. 26 mm

20 mm m

RM MS

4. 147 mm m

RM MS

4. 19 mm m

RM MS

5. 01 mm m

e�

4. Lifting.

5. Climbing.

e�

e�

3. Handling g.

4. 2 mm m

4. 98 mm m

6.. 28 mm

Table 6. Errorss between the siimulation and th he real experim ment.

8


Without load

Figu ure 7. Error simu ulation against tthe error of the real expeeriment for the m manipulation taask

6. C Conclusion A methodology for the po ositioning off the multi‐‐ articculated mob bile manipulator by meeans of thee development of aa kinematical m model is defin ned, explained d here the use o of a novel scheeme to devisee and assessed, wh us transformaations achieves the given n the homogeneou ucture for the forward kineematics. The methodology y stru has the following g advantages: ((1) it providess a big picturee t modelling g for mechatronic design when it develops the manipulator; (2 2) it simplifiess and simulation off the mobile m ous transform mation when itt the estimation of the homogeno r (3) it allows tthe determination of thee is required; para ameters of inteeraction when n it is necessarry to make thee scheemes in a cou upled way; (44) it is easy for kinematicc mod delling when any added su ubsystem is in ncorporated in n the analysis of th he performancce index based d on the errorr he real operatiional scenario os: of th have an averaage error RMS S of 3. 47 mm m The simulations h h regard to the referencees values, while w the reall with expeerimentationss have an erro or RMS of 7. 91 9 mm. Thesee are permissible errors e in relatiion to the errror of 10 mm,, whiich is specifieed in the task ks for the deeactivation off expllosives [30]. The maxiim deformattion of thee man nipulator with h a load of 10 k kg was 1 mm. ntages of this proposal in n comparison n The major advan h previous work are cconsidered, such s as thee with conttributions of tthis paper: thee developmentt of kinematicc mod delling in a coupled waay for a wheeled mobilee 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. 8. References [1] Zúñiga L. Diseño de un Effector Final para Manipulación de Artefactos Explosivos. Engeeniring Thesis, Engineers Military School from Mexican Army, 2003. [2] Krus P. Distributed Techniques for Modelling and Simulation of Engineering Systems. Technical Report, Department of Mechanical Engineering Linköping University, Linköping, Sweden, 2000. [3] Korayem M. H., Banirostam T. Design, Modeling and Experimental Analysis of Wheeled Mobile Robots. 3rd IFAC Symposium on Mechatronic Systems, 2/a. ed. Vol. 12, September 2004: 821‐228. [4] Li Li, Fei‐Yue W. Advanced Motion Control and Sensing for Intelligent Vehicles. Springer, University of Arizona, U. S. A., 2007; ISBN: 0‐387‐44407‐6. [5] BirtaG., Louis A. G. Modelling and Simulation Exploring Dynamic System Behaviour. Springer‐ Verlag, London, U. K., 2007; ISBN‐10: 1‐84628‐621‐2. [6] Josephs H., Huston R. Dynamics of Mechanical Systems. CRC Press LLC. U. S. A., 2004. [7] Bayle B., Fourquet J. Y., Renaud M. Nonholonomic Mobile Manipulators: Kinematics, Velocities and Redundancies. Journal of Intelligent and Robotic Systems, Vol. 36, January 2003. [8] Yamamoto Y., Yun X. Coordinating Locomotion and Manipulation of a Mobile Manipulator. IEEE Transactions on Automatic Control, 6th. Ed., Vol. 39, June 1994. [9] Paul R. P. Robot Manipulators: mathematics, programming and control. MIT Press, Massachusetts, 1981; ISBN: 0‐262‐16082. [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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[11] Pavlov V., Timofeyev A. Construction and stabilization of programmed movements of a mobile robot‐manipulator. Engineering Cybernetics, 1976; 14(6):70–79. [12] Dubowsky S., Tanner A. B. A study of the dynamics and control of mobile manipulators subjected to vehicle disturbances. In Robotics research: 4th International Symposium, 1988: 111–117. [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. [16] ZúñigaL. A., PedrazaJ. C., Gorrostieta E., Ramos J. M. Analysis of Dynamic Behavior of an EOD Mechatronic Unit. International Conference on Electronics, Robotics and Automotive Mechanics Conference, CERMA, 2009. [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. [22] Roth B. “Advances in Robot Kinematics”. Springer, Stanford University, U. S. A., 2006. [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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Conference on Methods and Models in Automation and Robotics, 2009. [26] Fruchard M., Morin P., Samson C. A Framework for the Control of Nonholonomic Mobile Manipulators. International Journal of Robotics Research, 2006; 25(8): 745‐780. [27] Mazur A., Szakiel D. On Path Following Control of Nonholonomic Mobile Manipulators. Int. J. Appl. Math. Comput. Sci. 2009; 19(4): 561‐574. [28] Kelly, R., Santibáñez, V. Control de Movimiento de Robots Manipuladores. Editorial Prentice Hall, edición 2003; ISBN: 84‐205‐3831‐0.

[29] Zúñiga L. A., PedrazaJ. C. , Gorrostieta E., Ramos J. M. New approach to Modeling and Simulation Methodology for the Mechatronic Design of IEDD‐ Unmanned Wheeled Mobile Manipulator. International Conference on Electronics, Robotics and Automotive Mechanics Conference, CERMA, México, D. F., 2010. [30] Zúñiga L. A., Skodny T., PedrazaJ. C. , Gorrostieta E. Systematic Analysis of an IEDD Unit based in a New Methodology for Modeling and Simulation. Journal of Advanced Robotics, INTECH, Austria, 2010; 7: 93‐ 100.

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