THE MATHEMATICAL MODEL OF A WALKING ROBOT WITH THREE FREE JOINTS

Transcription

1 THE MATHEMATICAL MODEL OF A WALKING ROBOT WITH THREE FREE JOINTS Constantn MARIN Faclty of Atomaton and Compters, Unversty of Craova, Romana cmarn@atomaton.cv.ro Anca PETRISOR Unversty of Craova, Faclty of Electromecancs, Romana apetrsor@em.cv.ro Abstract: Ths paper presents the mathematcal model of a planar robot where the actve par of leg has three free ont. The mathematcal model of the robot s determned consderng all the ponts n the x-plane as beng complex nmbers. Takng nto accont a possble symmetrcal strctre, only the vertcal x-plane evolton s consdered. The reslts can be extended to three-dmensonal space. The approach s best on varable dynamc systems, consderng that each leg of the par can change the state dependng of the casalty order. The approach s also best on the mechancal strctre of a planar robot presented n (Petr'or, Marn 3). It s consdered that the frst leg has both onts free and the second leg has the second ont free. A systemc approach of the Varable Casalty Dynamc Systems (VCDS) s presented wth reslts whch allow an easy nmercal mplementaton. The model s mplemented n MATLAB envronment and some evoltons examples are presented. Key words: walkng robots, casal orderng, control algorthms. 1. INTRODUCTION Behavor of walkng robots s charactered by a specfc type of movement called legged locomoton, (Thron, et al., 1), (Cbero, 1). Legged locomoton combnes contnos tme dfferental systems and logcal systems concepts, whch allow descrbng the both fndamental aspects: leg movements and leg coordnaton. As a reslt, dfferent types of legged movements are possble as walkng gates and clmbng movements. Many control algorthms mplemented on the exstng walkng robots, (WMC 3), (CWR 3), are based on "state of the art" technologes to control the movements of artclated lmbs and ont actators. Ths paper presents the mathematcal model of a planar robot where the actve par of leg has three free ont. The mathematcal model of the robot s determned consderng all the ponts n the x-plane as beng complex nmbers. Takng nto accont a possble symmetrcal strctre, only the vertcal x-plane evolton s consdered. The reslts can be extended to three-dmensonal space. It s consdered that the frst leg has both onts free and the second leg has the second ont free.. GEOMETRICAL STRUCTURE Let s consder a planar walkng robot (PWR) strctre as depcted n Fg.1., havng three normal legs p L,L,L and a head eqvalent to another leg L contanng the robot centre of gravty G placed n ts foot. The robot body RB s charactersed by two poston vectors O, 1 O and the leg onng ponts (hps) denoted R, R p, R. The onng pont of the head L s the central pont O, R = O, so the robot body RB s nvocally charactered by the set, 1 p RB = {O,O,,,, }, (1) where =. The robot has a rgd body f the three scalars (, k, ) are constant n tme. The geometrcal strctre of the PWR s defned by 1 O O = e () R = O + e (3) R = O + e (4) p p R = O + e (5) R = O + e = O (6) from whch, (7) p p (8) p p (9) G R 1,, R, O 1, G =G R, 1,p O 1 G R p 1, Fg.1. The robot geometrcal strctre.,p G p

2 The robot poston n the vertcal plane s defned by 1 the par of the poston vectors O, O 1 where O O = 1, or by the vector O and the scalar, the anglar drecton of the robot body. Each of the for robot legs L,L,L p,l, s charactersed by an so called Exstence Relaton ER(L) dependng on specfc varables as we presented n (Petr'or, Marn 3). The mathematcal model of ths obect s a Varable Casalty Dynamc Systems VCDS (Marn, 3) and wll be analysed from ths pont of vew. A par of legs { L,L } constttes the so called Actve Par of Legs (APL) f the robot body poston s the same rrespectve of the feet poston of all the other legs dfferent of L and L. A label s assgned to each possble APL. The APL label s expressed by a varable q called Index of Actvty (IA) whch can take N a vales, nmbers or strngs of characters. For example the strng of characters, q = '' ponts ot that the par { L,L } s an APL. Instead of strngs of characters the IA can take nmercal vales as for example, q = 1 q = '' q = 3 q = 'p' (1) q = 31 q = 'p' All the other legs that at a tme nstant do not belong to APL are called Passve Legs (PL). The leg n APL, havng a free onng pont (FJP) s called slave leg the opposte of the motor (or master) leg whose both onng ponts are external controlled (EC). 3. CAUSALITY ORDERING OF AN APL WITH THREE FREE JOINTS Let be APL={ L,L } that means q = '' q = 1. Changng the ndces, the below relatons are avalable for any APL. The knematcs strctre of ths APL can be followed n Fg.1. In ths strctre only one angle s external controlled (EC) so three onts are free. Consderng also the par L, L as APL we denote ths by p q = '', s = motor, motor1, s (11) or p q = '', s = motor, motor, s. (1) In ths paper t s consderng only the casalty orderng: s=[motor, motor 1, s p ] In ths casalty orderng the angle 1, s EC and, the angle s free. The block dagram of ths casalty strctre, 1, consderng to be EC, s represented as a connecton of orented sbsystems, s llstrated n Fg.. R 1, R 1, G Fg.. Block dagram of a casalty strctre wth three free angles. In ths strctre, the poston vectors R, G, G can be external controlled. The vector R depends on R, G and satsfes R R = r. Now, the G vector G does not affect R. The otpt varables are, 1, R = H R,G, (13) 1, 1 1,, 1,, 1, = F R,G,G, (14) = F R,G,G, (15) = F R,G,G, (16) It can be observed that the and depend on and G only. 1, = F (R,G,,s ) (17) f R,G f R,G R, G G R R s=[motor, motor1, s p ] R R = r R = (18) = (19) bt the command angles 1,,, are referred wth respect to the robot body so they depend on the poston angle so they depend on G and 1,. R R = r 1,, R, 1,, R,

3 The relatons (17), (18) represent the mathematcal model solton of an solate leg wth two free onts. So, for the leg L we have ˆ [,,s ] = f (R,G,s,a,b ) () The otpt varable R,,, are calclated from the exstence relatons R = G e AB (1), 1, AB = e [b + a e ] () of the leg L and the knematc restrcton (3) where = r =constant (4) The npt-state-otpt relatons of the leg L n casalty strctre motor1 are llstrated n Fg3. Fg.3. The npt-state-otpt relatons of the leg L. For example, the same par of vales ( R, G ) can be realed n two ways: [ ],, (5) and, (, ] ŝ R, S ŝ = 'down' = 'p' (6) G R, [,] ŝ = 'down', (, ] ŝ = 'p' Fg.4. The achevement of the vales ( R, G ). The eqatons (1), (3) represent two complex eqatons, so 4 scalars eqatons from whch are, calclated the otpts, R, that means two scalars and a complex nmber. Replacng for R R e = (7) from (3), n (1) are obtaned ( ) G + AB R 1,, R = = f(,,r,g ) (8) + AB from whch on calclate G R R R = ( ) (9) + AB s ŝ 1, R 1,,, S G By replacng (9) n (7) s obtane, for G R e = (3) + AB for whch G R e = (31) + AB ths, s obtaned the knematc restrcton condton moved to the npt, + AB = G R. (3) The relaton (3), n explcty form becomes,, 1, + e [b + a e ] = G R (33) The angle vale s obtaned from (33),, 1, = = F (,R,G,s ˆ ) (34) The state ŝ s ntrodced to restore the npt-otpt nvocty. We denote, 1, W = b + a e (35), where the sbscrpt meanng that angle (here ) s free, so by dvson wth W,(33) becomes, G R + e = W W (36) We denote, x 1, Q = = q + q = Q ( ) W (37) G R 1, g = = g (,R,G ). (38) W It s obtaned a polynomal eqaton, wth nknown, scalar, presented n the form as, Q + e = g (39) It s obtaned x,, q + q + cos( ) + sn( ) = g (4) x x,, (q ) + (q ) + 1+ q cos( ) + q (sn ) = g a lnear eqaton n sn and cos. (41) We denote, x 1, a = q = a ( ) (4) 1, b = q = b ( ) (43) x c = (q ) (q ) 1+ g = Q 1+ g (44) 1, c = c (,R,G ) From (39), wth (4) (44) s obtaned a trgonometrc eqaton,, acos( ) + bsn( ) = c (45) wth the followng soltons of the t crrent moment:,, 1, ˆ = (t) = F (,R,G,s ) (46)

4 The case b = and a c = arccos [, ], a (47) a f a >! = " (48) f a > #, f (t $ ) > ŝ = 'p' () t = " # f, (t $ )% ŝ = 'down' (49) The case b = and a =, s possble only f c and we sppose,! =, =, (t) = (5) The case b On calclate: a! = arctg ( /, /) b (51) b c = arcsn [ /, / ] b (a ) + (b ) (5), 1, If (t $ )% s = 'down'! +, f! + % () t = " (53) #! + f! + >, 1, If (t $ ) > s = 'p'! +, + f! + % () t = " (54) #! + f! + > The solton (34) of the eqaton (33) eqvalent wth (5) s gven by one of the expresson (49), (5), (53) or (54). Each form, n ts valdty condtons, respects the knematc restrcton of rgd body (4). Fg.5 presents a block dagram whch llstrate the, sbordonaton of dfferent relatons n angle rotne comptaton ŝ a! G (49) R (4) (48) g b (5) (38) (43) (5) 1, W (53) Q c (35) (37) (44) (51) (54) a b Fg.5. The block dagram., whch s sch as (34), 1, = F (,R,G,s ˆ ) (55) concs llstrated dependence n Fg. 6. Fg.6. Concs block dagram.,, Knowng the angle vale =, on calclate the npt varables vector o the leg L ~,, 1, AB = e W = e [b + a e ] (56) whch assmed the knematc restrcton. In sch of condtons the angle and the vector R are calclated, by replacng (56) n (3) and (8), G R e = (57) ~ + AB whence & G R ' 1, = arg & F ~ ' = (, R,G,s ˆ ) (58) & AB + ' ~ ( ) G + AB R 1, = = ~ ˆ + AB R H (,R,G,s )(59) Also, the knowledge of the angle allows the calclaton of the free angles 1,,, of the leg L, abot whch are already known the angles, estmated by the fncton f ( ), (). The angles 1,,, are 1, = (6), = (61) A calclaton alternatve sppose frst assessment of R the vector by relaton (59) and then the angle s estmated from the vectors R ' R as n relaton (6) 1, R R H (,R,G,s ˆ ) R = arg = arg (6) nstead of relaton (58). 4. CONCLUSIONS G R,, 1, F ( ) The mathematcal model on VCDS allows the best mplementaton of the robot behavor. Based on relatons obtaned t was conceved and mplemented compter programs for smlaton and control. s =

5 5. EXAMPLES OF EVOLUTIONS USING MATLAB ENVIRONMENT Fg.8. Intal state of the robot. Fg.1. The state of the robot for 1, x = ct ; R = ct ; R =,818. Fg.9. The state of the robot for 1, x = ct ; R =,6; R = ct. Fg.1. The state of the robot for 1, x = ct ; R = ct ; R =, 978. Fg.1. The state of the robot for 1, x = ct ; R =,76; R = ct. Fg.1. The state of the robot for 1, =,18; R = ct ; = ct. x R

6 REFERENCES Fg.1. The state of the robot for 1, =,3; R = ct ; = ct. x R Fg.1. The state of the robot for 1, =,16; R = ct ; = ct. x R Beom H.R, Cho H.S. (1995). A sensor-based Navgaton for a Moble Robot Usng Fy Logc and Renforcement Learnng. IEEE Trans.on SMC, vol.5, No 3, Cbero S., (1). A 6-Legged Hybrd Walkng and Wheeled Vehcle. 7 -th Internatonal Conference on Mechatroncs and Machne Vson n Practce,USA. CWR (3). The Clmbng and Walkng Robots Home Page, Galt S., Lk L. (1999). Intellgent walkng gat generaton for legged robots. Proc. -th Internatonal Conference on Clmbng and Walkng Robots, Jang J.G (1999). Fy Modelng Control for Robotc Gat Synthess. Proc. Of 36 -ty IEEE CDC San Dego Ca, Dec.1999, Marn C. (3). On the Varable Casalty Dynamc Systems. SINTES Craova, 5 October, to be pblshed. Petrsor A., Marn C. (3). Control algorthm for walkng robots n ncertan envronments. Internatonal Conference CSCS14 Bcre4t, 4-6 Jne. Thron B., Thry L. (1). Concrent Programmng for the Control of Hexapod Walkng. 7 -th Internatonal Conference on Mechatroncs and Machne Vson n Practce,USA. WMC (3). The Walkng Machne Catalog nes_katalog.html.