The Analysis of Coriolis Effect on a Robot Manipulator

Transcription

1 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) he Analyss of Corols Effect on a Robot Manpulator Pratap P homas Assstant Professor Department of Mechancal Engneerng K G Reddy college of Engneerng and echnology, Monabad, elangana Abstract: hs paper attempts to desgn a logc controller for controllng the poston of the arms of a two-lnk manpulators takng nto account the dsturbances caused by nertal loadng, couplng reacton forces between jonts (Corols and centrfugal), and gravty loadng effects. he desgn for the poston control ncorporates a torque computaton block that compensates for the dsturbances caused by these effects by a logc controller. he smulaton results for the poston control of a two-lnk robotc manpulator usng the compensated logc model were then compared wth the results obtaned usng an optmzed PID controller (proportonal-ntegral-dervatve controller). he results clearly ndcated that the performance of the compensated logc controller was much better than the optmzed PID. I. INRODUCION Robotcs s a relatvely young feld of modern technology that crosses tradtonal engneerng boundares. Understandng the complexty of robots and ther applcatons requres knowledge of electrcal engneerng, mechancal engneerng, systems and ndustral engneerng, computer scence, economcs, and mathematcs. New dscplnes of engneerng, such as manufacturng engneerng, applcatons engneerng, and knowledge engneerng have emerged to deal wth the complexty of the feld of robotcs and factory automaton. In ths paper, a new spatal 3 DOF parallel manpulator wth three non-dentcal chans s developed. he movable platform has three degrees of freedom, whch are two degrees of translatonal freedom and one degree of rotatonal freedom, wth respect to the base plate. he knematcs problems and velocty equaton of the new parallel manpulator are gven. hree knds of sngulartes are presented. hs paper s concerned wth the effects of actuaton schemes on three measures of knematc performance whch depend upon a manpulators Jacobean Matrx, the manpulablty (yoshkawa,1985) and condton number (Salsbury and crag, 1982; angeles, 1992; krcansk, 1994; angeles,1995). We begn by presentng a smple framework on how to ncorporate actuator locaton n the knematc model. For each measure we drve propertes relatng ts jonts space to ts actuator space descrpton. Fnally we employ these concepts n the desgn of a spatal parallel manpulator. II. LIERAURE SURVEY Davd E. Orn and Wllam W. Schrader dscussed the paper on sx dfferent methods for computng the Jacoban matrx for a general n degrees of freedom manpulator. In ths, they compared several approaches that have been proposed for computng Jacoban matrx, ncludng the new approach ntroduced. he result s found to be effcent for Jacoban matrx, when t s based on end effector coordnates. B. Roth frst nvestgated the relatonshp between knetc geometry and manpulator workspace. J. Denavt and R.S Hartenberg gave a descrpton of manpulator n terms of lnk lengths, lnk twsts and jont angles. Accordng to G.Strang, the condton number of a matrx s used n numercal analyss to estmate the errors generated n the soluton of lnear equatons by error on the data. he concept was attracted the attenton of the several researchers. hs concept was utlzed by Salsbury J.K. and Crag J.J [14] for research n knematcs and control ssues n contest of an artculated mult-fnger mechancal hand.he work s related to analyzng and optmzng the knematc structure of manpulators wth out regard to the form of actuaton of the jonts. Robot sngulartes cause many problems n moton synthess, both for non redundant and for redundant robots. Although redundant robots may be able to avod sngulartes nsde the workspace boundares, non-redundant robots have to fnd the method to pass through the sngular confguratons, f moton specfed n the task coordnates s to be acheved nsde the whole robot workspace Corols Effect : It s a deflecton occurred n movng objects when they are vewed n a rotatng reference frame. In a reference frame wth clockwse rotaton, the deflecton s to the left of the moton of the object; n Specal Issue NCREEFOSS ISSN:

2 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) another wth counter-clockwse rotaton, the deflecton s to the rght. Corols Effect n case of a robot s the deflecton or the error occurred n the target acheved when one arm of a robot s rotatng above the another rotatng arm he desgn for the poston control ncorporates a torque computaton block that compensates for the dsturbances caused by these effects by a logc controller. he smulaton results for the poston control of a two-lnk robotc manpulator usng the compensated logc model were then compared wth the results obtaned usng an optmzed PID controller (proportonal-ntegral-dervatve controller). he results clearly ndcated that the performance of the compensated logc controller was much better than the optmzed PID. 2.1 Robot Dynamcs: Robot dynamcs bascally deals wth mathematcal formulatons of the equatons of robot arm moton. he dynamc equatons of moton of a manpulator are a set of mathematcal equatons descrbng the dynamc behavor of the manpulator. Such equatons of moton are useful for computer smulaton of the robot arm moton, desgn of sutable control equatons of a robot arm and evaluaton of knematc desgn and structure of a robot arm. he actual dynamc model can be obtaned from known physcal laws such as the laws of Newtonan and Lagrangan mechancs. 2.2 L-E Formulaton :he general moton equatons of a manpulator can convenently be expressed through the drect applcaton of the Lagrange Euler (L-E) formulaton to non - conservatve systems. he L-E equaton s gven as follows d / dt ( ) - = 1,2,3, n where L = Lagrangan functon = knetc energy K- potental energy P K = total knetc energy of the robot arm P = total potental energy of the robot arm q = generalzed coordnates of the robot arm q = frst tme dervatve of the generalzed coordnate, q From the above Lagrange-Euler equaton, one s requred to properly choose a set of generalzed coordnates to descrbe the system. Generalzed coordnates are used as a convenent set of coordnates that completely descrbe the locaton (poston and orentaton) of a system wth respect to a reference coordnate frame. For a smple manpulator wth rotary-prsmatc jonts, varous sets of generalzed postons of the jonts are readly avalable because they can be measured by potentometers or encoders or other sensng devces, they provde a natural correspondence wth the generalzed coordnates. hus, n the case of a rotary jont, q, the jont angle span of the jont. 2.3 Moton Equatons of a manpulator: he dervaton of equatons of moton of a manpulator requres the computaton of the total knetc energy and the total potental energy of the manpulator. 2.4 Knetc Energy of the manpulator: If dk s the knetc energy of a partcle wth dfferental mass dm n lnk ; then dk = ½(x + y + z ) dm = ½ trace (v v ) dm =½ r (v v ) dm where v U j q j ) r 0 j-1 U j = A j-1 Q j A j vares from 1 to for j <= 0 for j > Q j s a 4 x 4 matrx, gven as Specal Issue NCREEFOSS ISSN:

3 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) for revolute jont...(1) r = fxed pont at rest n a lnk w.r.t homogeneous coordnates of the th lnk coordnate A j = homogeneous coordnate transformaton matrx whch relates j th coordnate frame to the th coordnate frame. he total knetc energy of all the lnks can be found out by ntegratng dk over all the lnks. he total knetc energy K of a robot arm after ntegraton wll be found as K = p r U p ( J ) U r qp q r ) ] p, r vares from 1 to (2) where J s the nerta of all the ponts on lnk gven by J = r r dm 2.5 Potental Energy of the Manpulator Let P be the th lnk s potental energy. he P s gven by P = -m g ( 0 A ) = 1, 2,.. n where 0 A s the coordnate transformaton matrx whch relates the th coordnate frame to the base coordnaton frame s the a fxed pont n the th lnk expressed n homogenous coordnates wth respect to the th lnk coordnate frame. he total potental energy P of the robot manpulator s found out by ntegratng P over all the lnks. P -m g ( 0 A ) vares from 1 to n (3) Where g = (0, 0, - g, 0) and g s the gravtatonal constant. (g = m/sec 2 ) 2.6 L-E formulaton for the two-lnk manpulator From equatons (2) and (3), the Lagrangan functon L = K-P s gven by L = j k [ r (U j J U k ) qj q k m g ( 0 A ) vares from 1 to n, j,k vares from 1 to Applyng the Lagrange-Euler formulaton to the Lagrangan functon, the generalzed torque for jont actuator to drve the th lnk of the manpulator, k r (U jk U j ) qk k m r (U jkm U j ) qk q m - j m j g U j j j for = 1, 2, 3 n where j vares from to n; k and m vares from 1 to j Specal Issue NCREEFOSS ISSN:

4 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) he above equaton can be expressed n a much smpler matrx form as 1 2 n (t)) q(t) = an n x 1 vector of the jont varables of the robot arm and can be expressed as q(t) = (q 1 (t), q 2 (t),., q n (t)) q(t) = an n x 1 vector of the jont velocty of the robot arm and can be expressed as q(t) = (q 1 (t), q 2 (t),., q n (t) ) q(t) = an n x 1 vector of the jont acceleraton of the jont varables q(t) and can be expressed as q(t) = (q 1 (t), q 2 (t),., q n (t) ) (q) = an n x n nertal acceleraton-related symmetrc matrx whose elements are D k r (U jk U j ), k = 1,2..n j vares from max(,k) to n h(q, q ) = an 1x 1 nonlnear Corols and centrfugal force vector whose elements are h(q,q ) = (h 1, h 2,..h n ) Where h m q k q m =1,2,.n. (4) and h km r (U jkm U j ), k, m = 1,2..n j vares from max(,k,m) to n c(q) = an n x 1 gravty loadng force vector whose elements are c(q) = (c 1, c 2, c n ) Where c ( -m j g U j j j ) = 1,2,.n....(5) j vares from to n. he computatonal complexty of the L-E formulaton ncreases wth the 4 th power of the number of degrees of freedom of the robot arm. Hence for the smulaton purpose a specfc smple case of a two-lnk manpulator wth revolute jonts s taken. We assume the followng n our dervaton of the generalzed torque equaton for the two- 1, 2,mass of the lnks = m 1, m 2, 1 2 = 0; d 1 = d 2 = 0; a 1 = a 2 = l. Specal Issue NCREEFOSS ISSN:

5 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) he homogenous coordnate transformaton matrces -1 A ( = 1,2) are obtaned as 0 A1 = [(C 1, -S 1, 0, lc 1 ), (S 1, C 1, 0, ls 1 ), (0, 0,1, 0), (0, 0, 0, 1)]..(6) 1 A2 = [(C -S, 0, lc ), (S, C, 0, l, ), (0, 0,1, 0), (0, 0, 0, 1)]..(7) A2 = 0 1 A A2 1 = [(C -S, 0, l(c + C ), (S, C, 0, l(s + S 12, ), (0, 0,1, 0), (0, 0, 0, 1)] where C ; S ; C j ); fs j ) From the eqns 1, 6, 7, 8 we can derve the followng results...(8) 2.7 Inerta Effects: D 11 = 1/3 m 1 l 2 + 4/3 m 2 l 2 + m 2 l 2. D 12 = D 21 = 1/3 m 2 l 2 + ½ m 2 l 2..(9) (10) D 22 = 1/3 m 2 l 2...(11) Where D 11, D 12, D 21, D 22 are nerta elements. 2.8 Corols Effects: he velocty related coeffcents n the Corols and Centrfugal terms could be obtaned from equatons 4 and 5 as h 1 = - ½ m 2 l m 2 2 h 2 = - ½ m 2 2.(12)..(13) 2.9 Gravty Effects: he gravty loadng force vector elements can be obtaned from equaton 5 as c 1 = ½ m 1 + ½ m m 2..(14) c 2 = ½ m (15) Fnally, the L-E equatons of moton for the two-lnk manpulator are found to be 1 = 1/3 m 1 l 2 + 4/3 m 2 l 2 + m 2 l 2 + 1/3 m 2 l 2 + ½ m 2 l 2 ½m 2 l m2 2 + ½ m 1 +½ m m 2 (16) 2 = 1/3 m 2 l 2 + ½ m 2 l 2 + 1/3 m 2 l 2 ½m ½ m2 2.(17) he nerta, gravty and Corols Effect have been dentfed as the major contrbutors for the naccuracy n the postonng of the robot arms. Hence an accurate controller nvarably has to overcome these adverse effects by ncludng some knd of compensaton logc n the desgn. o vew robotcs as an applcaton of the prncples of motons, together wth motors to provde motons and sensors to provde locaton and velocty may mss the nherent complexty of the dscplne. A real robot does face potental errors due to a number of reasons, ncludng: ncorrect parameters (for example mass, drecton, dstance) values, frctonal forces,terran estmatons, play at the lnk jonts, calbraton Specal Issue NCREEFOSS ISSN:

6 Internatonal Journal of Innovatons n Engneerng and echnology (IJIE) errors n sensors, error n the values read from the sensors. Among such errors centrpetal and Corols effects are of great mportance when robot manpulator s movng at hgh speeds. III. CONCLUSION AND FUURE SCOPE hs superor performance of the controller s complmented by the reduced oscllatons of the control varable about the set pont.he magnfed porton of the smulaton graph to demonstrate the ampltude of oscllatons experenced by an arm about the set pont n the case of optmzed PID controllers wth accurate poston control s requred for a senstve ndustral or medcal applcaton, ths knd of oscllatons about the set pont s hghly undesrable. Hence the proposed desgn of controller has the followng advantages over the conventonal controllers: 1. More accurate trackng of the set pont varatons. 2. Reduced oscllatons about the set pont 3. Settlng tme s comparatvely smaller. REFERENCES [1] Angeles s, J., 1992, he Desgn of Isotropc Manpulator Archtectures n the Presence of Redundances. he Internatonal Journal of Robotc Research, Vol.11;No.3, pp [2] Angeles s, J.,1995, Knematc Isotropy n Humans and machnes, processedngs of the 9 th World Congress of the heory of Machnes and Mechansms, Vol. I,Mlano, Italy [3] Horn,R..A.,. and Johnson,C.R..1991, opcs n Matrx Analyss, Cambrdge unversty press [4] Krcansk, M.V., 1994, Robotc Isotropc and Optmal Robot Desgn of Planar Manpulators,.Proc.IEEE Internatonal Conference on Robotcs And Automaton, San Dego, USA, pp [5] Klen,C.A.,and Blaho, B,E.,1987, Dexterty Measures for the Desgn and Control of Knematcally Redundant Manpulators, he Internatonal Journal of Robotc Research, Vol.6, No.2,pp,72-83, [6] Matone.R, 1997, he Effects of Actuaton Schemes on the Performance of Manpulators, PhD hess, Department of Mechancal Engneerng, Standard Unversty Stanford, CA. [7] Legos, A., 1977, Automatc Supervsory Control for the Confguraton and Behavor of MultbodyMechansms, IEEE rans. of System.Man,Cybernatcs, SMC-7 (12),pp [8] Salsbury.J.K., and Crag,J,J.,1982, Artculated Hands:Force Control and Knematc Issues, he Internatonal Journal of Robotc Research, Vol.,I.,No.1, pp Specal Issue NCREEFOSS ISSN: