THE MODELING AND SIMULATION OF A ROBOTIC ARM

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1 Scetfc Bullet of the Electrcal Egeerg Faculty o. / 009 THE ODELING AND SILATION OF A ROBOTIC AR Paul Cpra PATIC ad Gabrel GORGHI Valaha versty of Târgovşte, 8-4 r Av E-al: ppatc@valaha.ro, ggorghu@yahoo.co Abstract A coplex apulator that cludes robot "ars" ca pck up ad apulate everythg fro delcate obects to large payloads. Every echas the apulator ust be defed as a obect, wth forato descrbg all the forces assocated wth the obect's oto or tasks. Ths cludes rotato ad posto atrces, ceter of ass, erta, exteral torque ad forces actg o t, otor torque, elastc torque etc. A operator ust costruct a graphcal odel, usg a atlab-sulk progra to descrbe the robot syste's topology. The operator drags ad drops obect blocks fro a lbrary, lks the together. Keywords: Robot, apulator, Ar, Sulato odel, atlab-sulk.. INTRODCTION Robotc ar are cooly used dustres. I ay feld applcatos where techcal support s requred, a-hadlg s ether dagerous or s ot possble. I such stuatos three or ore ar apulators are cooly used. Soe robots are used to spect dagerous areas or/ad to reove ad to destroy explosve devces. These robots ca be used to ake soe corrdors through ed battle felds, apulato ad eutralzato of the tact auto, specto of the vehcles, tras, arplaes ad buldgs. For these robots a good fuctoal actvty s to deterate the desos of the work space ad keatcs of the robotc ar.. THE STANDARD NERICAL SILATION ODEL About the uercal sulato odel t ca say wthout fear of error, there are a varety of such odels. The proble ourselves ow s to fd a odel or a class of sulato odels, whch we call the stadard odels the sese that all others are reducble to the. Experece has show that the ost coo sulato odels are dfferetal odel for cotuous systes, ad fte dfferece odel (dscrete te, for dscrete systes. All other ethods ay be reduced by approprate chages the two odels. The stadard uercal sulato odel (cotuous systes Ths class of systes s well represeted by dfferetal odel as a set of olear dfferetal equatos wth gve tal codtos ad sub-optalty tervals for state varables. Each of these equatos s actually a vector equato descrbg the evoluto status sub-vector x (,,..., fro state-space. Ths odel s: x (t A x (t + B u (t + f ( x, a + v (; x (0 x 0,,,,; g ( x v (, +, where: A ad B s the atrx of state ad cotrol respectvely, of approprate sze; f vector fuctos descrbg o-learty subsyste I; v the teractos betwee subsyste I ad other - subsystes; x (0 tal codto; α paraeters odeled sub-process. Addg soe tervals of sub-optalty oe obta: x x -Δx x x + Δx x ( x - beg the average aout of patet rege ad Δx s the ucertaty over the state x The cotrol varables u are those who are uder restrctos: u u u,,,,. The lear odel case (trasfer fucto. If the subsyste s lear, the state equato s wrtte as: x (t A x (t + B u (t + x (0 x 0., A x ( t ; 85

2 Scetfc Bullet of the Electrcal Egeerg Faculty o. / 008 Applyg to the equato of Laplace trasforato oe obta: s[y (s-x (0A Y (s + B F (s + ad fally: Y (s (si-a - [sx (0I + B F (s +,, A Y ( s A Y ( s, where: Y (s L{x (t}; Y (s L{x (t}; F (s L{u (t}. The stadard uercal sulato odel (dscrete systes I the case of dscrete systes, the class of uercal sulato odels s best represeted by a set of equatos wth fte dffereces, olear, wth tal gve codtos ad wth sub-optalty tervals for x (,,, varables. x (k+ A x (k + B u (k + f (a,k + v (; x (0 x 0,,,,;, + g ( x Assocatg the sub-optalty tervals: x x x ad the restrctos for the cotrol varables are: u u u Note. The uercal stadard odel of dscrete sulato ca be obtaed by drect odelg of the process or dscrete syste, but such that kd of odel ca be obtaed by eshg of the cotuous odel. The lear odel case (trasfer fucto. I the case of dscrete lear subsystes, the odel (state equato s wrtte as: x (t+ A x (t + B u (t +, A x ( t Applyg Z trasforato to the equato oe have: z[ (z x (0 A (z + B Ψ (z + ad fally: where: (z(zi A - [zx (0I + B Ψ (z +,, A ( t A ( z z Z{x (t} ; z Z{x (t} ; ( ( ψ ( z Z{u (t}. The dscrete evet sulato odel The hybrd odel s cluded dscrete evet sulato odel, whose stadard for s a set of Boolea dfferetal equatos. A dscrete evet syste ca be odeled as a sequetal autoatc ad ca be descrbed wth: G {,, Y, δ, λ, T}. ( k Y ( k ( k Y ( k... ( k T { ( k, ( k,..., ( k}... Yl ( k The sgfcato of the sybols s wrtte below: - - s the lot of the status syste wth dscrete evets; - - the lot of put; - Y - the lot of output; - δ : - trasto fucto; - λ : Y - output fucto; - T the lot of the oets of dscrete te. If,, Y Y takes values fro Boolea lot {0, }, oe dfferetal Boolea odel of the dscrete evets syste ca be buld as follows: If T k, k, k,..., k,..., k } s the lot of oets { 0 f of dscrete te, the the odel whch represet the evoluto of the dscrete evets syste ca be a Boolea dfferetal odel: ( k + F ( ( k, ( k,..., ( k, ( k, ( k,..., ( k Y k + G ( ( k, ( k,..., ( k, ( k, ( k,..., ( ( k whch : F s the status fucto of the dscrete evets syste; G the output fucto of the syste fro above; F ad G Boolea fuctos who takes values fro the terval {0,}. If x (t ad x (k are the status of the aalyzed syste oe ca presue that, the case of the cotuous process, the t varable s eshed the sae aer as T. 3. APPLICATION. THE SILATION OF THE ANIPLATOR/ROBOTIC AR The apulator ar s represeted the followg schee: 86

3 Scetfc Bullet of the Electrcal Egeerg Faculty o. / 009 atr where:, asses of the two ars; I ertal oet of the secod ar; Robot apulator cossts of two segets artculate wth each other. The frst oe ca oly realze a traslatoal oto of the Ox axs drecto uder the force F. The secod ca oly execute a rotatg oto by Oz axs (perpedcular to the plae xoy, uder the acto of torque. I the fgure are deoted: a ceter of gravty posto of the frst seget; x ceter of gravty coordate of the frst seget (y, 0 b ceter of gravty posto of the secod seget; x, y ceter of gravty coordate of the secod seget; - agle betwee the secod seget ad the vertcal. Syste, wth two degrees of freedo, ca be characterzed by two depedet varables; are cosdered as state varables x ad. All other varables ca be expressed ters of these as: y 0; x 0; x + bs y bcos Dfferetal equato characterzg the evoluto of the syste uder the acto of exteral stul s: + a; [ ( x[ x + [ H ( x[ x + [ g( [ G( x[ u whch: x [ x - status varables vector; [x,[x frst ad secod order dervatves, of the state varables vector; + bcos - ertal bcos I + b s [ ( 0 H b s 0 0 [ ( - atrx of pulse; 0 [ g ( - the forces ad couples gbs vector due the gravty; F [ G ( x[ u - exteral stul vector. Sulk odel, whch descrbes the behavor of the equato, ust be brought the for of equatos of state, respectvely: x C( u A( x B( g [ [ [ [ [ ultplyg at the left the equato wth [( - ad detfyg wth the ters of the equato result: whch [ ( wth [ A( [ ( x[ H ( [ B( g [ ( x[ g( [ C( u [ ( x[ G( x[ u I + b cos b s b cos +, ( + (I + b s ( bcos Dog atrx calculatos are obtaed: ( I + b s bs [ A( x[ ( b s cos [ B( g ( I + b F b s ( cos [ C( u ( b cos g ( b s cos g ( + bs F + ( + Result, the fal, expressos the for of state equatos of two varables:, 87

4 [ (I + b s b s [ g ( b s cos } x {(I + b s F ( b cos respectve, [ ( b cos F + ( + ( b s cos g ( + b s odelg the apulator s ar atlab-sulk evroet wll be acheved by tegratg the equatos of state, twce. Note that the expressos deped, besdes the echacal paraeters of the syste, by the state varable ad ts frst order dervatve, whch eas cosderg the as tegrato reacto Sulk schee of the two varables. To obta a geeral odel that ca be used regardless of the values of echacal paraeters of the syste, Sulk odel wll be developed wth foral paraeters. But, before started the sulato, the uercal values of echacal paraeters (,, I, b, g 0 wll be talzed atlab evroet. Is doe Sulk the followg schee: obtag values of state varables x ad. Note takg, respose, the teredate varable [, usg [ to calculate the ters fro the produce A( x. Also, the varable s used all blocks, beg take as respose The fuctos perfored by each block are: x_a: - (I + *b^*(s(u[^* *b*s(u[*(u[ ^; x_b: -( *b^*s(u[*cos(u[*g 0 ; x_c: (I + *b^*(s(u[3^*u[- ( *b*cos(u[3*u[; t_a: ( *b^*s(u[*cos(u[*(u[^; t_b: ( + * *b*s(u[*g 0 ; t_c: -( *b*cos(u[3*u[+( + *u[; /D, /D: u[/ (( + *(I + *b^*(s(u[^- ( *b*cos(u[^ All tegrators wll have zero tal codtos, except Fgure. Sulk Schee regardg the ar odelato. The localzato of the blocks Sulk sub-lbrares s: x_a, x_b, x_c, /D, t_a, t_b, t_c, /D blocks Fc Fuctos&Tables; F, blocks Costat Sources; Su, Su blocks Su ath; u ux blocks ux Sgals&Systes; Itegrator, Itegrator,. blocks Itegrator Cotuous; Rad-grd block Ga ath; F, Teta blocks Scope Sks The blocks /D ad /D realzed the dvso wth the atrx deterat [(, obtaed the dervatves of order II of the two state varables. It tegrates two tes, to obta posto that tegrates agular velocty (tegrator 3, whch wll be posed as tal codto the value of p / 4. The block rad-grd beg Ga type, wth value 80/p, trasfor the posto fro radas to degrees, ust for vewg. As a ethod of tegrato, wll choose the ethod wth varable step ode45, axu tegrato step s posg wth [s ad fal te (stop te 0 [s. The axu step was chose to track the fluece of chagg values the sulato stul o the evoluto of the syste. After the odel realzato t s creates a ATLAB fle by. type whch t s talzes the values of echacal paraeters: ; ; I0.0; b0.; g09.8; brat; Whe typg atlab wdow, the ae of ths fle wll be loaded atlab space the echacs paraeter values, the last le of the fle causg the opeg odel. It wll a the sulato, the evoluto of the syste to chages exteral stul.

5 The results exeplfed correspod to a force applcato F [N at te of t sec. Fgure 4. The evoluto of the syste chagg exteral stul. 4. CONCLSION Fgure. The evoluto of the syste chagg exteral stul. The followg coclusos ca be forulated accordg to the cosderatos preseted: The a keatc echacal copoet of the oe kd of robot s the robotc ar. The desos of the robotc ar are essetal for local work space of the robot. The keatc odels are very portat for to fd the posto of the characterstc pot of the robotc ar. A fuctoal sulato usg a 3D odel s very portat for obtag a optu verso of a robotc ar ad for a robot. 5. REFERENCES Fgure 3. The evoluto of the syste chagg exteral stul. [ Hakldr,., Tasdele, I., odelg ad sulato of a atropoorphc robot ar by usg Dyola. 5th Iteratoal Syposu o Itellget aufacturg Systes IS 006, Sakarya, Turkey, 9 3 ay, 006. [ Hrzeger,., Fscher,., Advaces robotcs. Iteratoal Joural of Robotcs Research, 999. [3 Scavcco, L., Sclao, B., odellg ad cotrol of robot apulators. Sprger, 000. [4 Spog,., Vdyasagar,., Robot dyacs ad cotrol. Joh Wley ad Sos, 989. [5 Staculescu Flor, odelarea ssteelor de are coplextate, Ed. Tehcă, 003. [6 Stareţu I., Ioescu., Keatcs ad Fuctoal Sulato of a Robotc Ar fro a Pyrotechc Robot, [7 SSE_Lab.htl

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