Vibrtions in Physicl Systems 018, 9, 01803 (1 of 9) Dynmics of n Inertilly Driven Robot Pweł FRITZKOWSKI Institute of Applied Mechnics, Poznn University of Technology, ul. Jn Pwł II 4, 60-965 Poznn, pwel.itzkowski@put.poznn.pl Romn STAROSTA Institute of Applied Mechnics, Poznn University of Technology, ul. Jn Pwł II 4, 60-965 Poznn, romn.strost@put.poznn.pl Abstrct An inertilly-driven system tht slides on rough horizontl plne due to periodic oscilltions of two ernl bodies is studied. Three-phse ernl motions with piecewise-constnt reltive ccelertions re considered. It is shown tht n lwys forwrd motion of the robot cn be chieved. The effect of the model prmeters on the displcement nd verge velocity of the system is nlyzed. Keywords: motion control, movble ernl mss, vibrtion-driven robot 1. Introduction Mobile robots without ny externl moving prts (like legs, wheels, trcks etc.) hve drwn ttention of mny reserchers in recent yers. Within this ctegory, body tht cn slide on rough surfce due to oscilltory ernl motions is perhps the most curious conception. Such robot cn be imgined s box contining one or severl msses which perform oscilltions nd erct with the primry body. The ernl (reltive) motions, rectiliner or rottionl, cn be controlled to push the entire system in the desired direction. More precisely, the ide is to produce certin inerti forces in order to control the norml force of the robot on the ground (nd hence the iction force) s well s the force tht tends to displce the box. Thus, systems of this type re clled vibrtiondriven or inertilly driven robots. The bsence of outer movers mkes the systems reltively simple in design. This feture cretes promising possibilities for use in prcticl pplictions. For exmple, one cn more esily produce robot of very smll size (micro-robot), or provide complete enclosure (wterproof, lek-protected etc.) for the robot. Most ppers devoted to vibrtion-driven systems re focused on optimiztion of the robot motion (e.g. its verge velocity) with respect to prmeters of the ernl vibrtions. Systems moving long horizontl stright line, hving single ernl body, were investigted by Chernous ko [, 3]. Fng nd Xu nlyzed the sme robot immersed in resistive medium [4], nd system of two identicl, coupled modules [5]. Figurin [6] considered the optimiztion problem for the rectiliner motion of robot up n inclined plne. Dynmics of system contining two ernl msses, moving horizontlly or verticlly, ws nlyzed by Bolotnik et. l [1]. Experimentl studies of systems contining unblnced rotors were presented by Sobolev nd Sorokin [7].
Vibrtions in Physicl Systems 018, 9, 01803 ( of 9) This pper is devoted to n inertilly-driven system tht slides on rough horizontl plne due to periodic oscilltions of two ernl bodies: one of them moves horizontlly, the other verticlly. Three-phse ernl motions re considered, where the reltive ccelertions re piecewise-constnt. The objective of the studies is to nlyze the effect of the model prmeters on the displcement nd velocity of the robot, nd to ccess efficiency of the proposed, reltively simple conception in comprison with the ides provided in literture.. Mthemticl model Consider system tht consists of crrying body (box, primry body) of mss M nd two ernl prticles of mss m 1 nd m (see Fig. 1). The former one lies on horizontl plne nd cn move long the globl x xis. The erction between the box nd the ground is chrcterized by the dry iction coefficient, µ. The ernl bodies cn move horizontlly or verticlly under the ction of driving control forces F 1 nd F. The prticles erct with the box, e.g. they slide long guide brs rigidly ttched to the crrying body. The reltive motions re described using the locl reference mes fixed to the box: x 1y 1 nd x y. Figure 1. The crrying body with movble ernl prticles Let x denote the displcement of the whole mechnicl system. For the ernl bodies, equtions of the horizontl nd verticl motion in the inertil me tke the form: m1 ( x x 1) F1, m y F. (1) The bsolute motion of the system, in turn, is described by ( M m x F F, () ) where F is the dry (Coulomb) iction force. Moreover, one cn write the following equilibrium eqution relted to the y direction: 1 ( M m g F N, (3) 1 ) where N denotes the norml component of the ground rection. Obviously, this norml force is involved in the iction lw: F F F F sgn x, if x 0, if x 0 nd F F sgn F, if x 0 nd F F (4)
Vibrtions in Physicl Systems 018, 9, 01803 (3 of 9) In the bove formul, F denotes the kinetic iction force while F is the resultnt of ll ernl (inertil) forces: F F N, (5) m ( x x ) m x ], (6) [ 1 1 Assume tht the reltive ccelertions of the prticles re known functions of time:,. Using Eqs. (1), one cn write eqution of motion () s x 1 1 ( t ) y ( t) m s x m, (7) 1 1 F where m s = M + m 1 + m is the totl mss of the system. Substituting Eqs. (1) nd (3) o reltion (5) leds to F ( msg m ). (8) When it comes to force F, tking o ccount the discussion presented in Ref. [1], formul (6) cn be replced with the simpler one: 1 1 F m. (9) Thus, the robot motion is described by Eq. (7), where the iction force is determined by (4) together with (8) nd (9). Let us roduce the dimensionless time nd displcements: t t T e, x X, T g e u x1, T g x e u y, (10) T g y e where T e is chrcteristic time (some ervl length) of the excittion function 1(t). Consequently, one obtins the non-dimensionl form of the eqution of motion: where f f f f X, (11) 1 x f sgn X, if X 0, if X 0 nd f f sgn f, if X 0 nd f f (1) nd f 1 ), f 1 x, (13) ( y 1 m 1 / m s, m / ms, x u x 1 / g, y u y / g. (14) Obviously, now n overdot denotes the derivtive with respect to the dimensionless time t * (the sterisk will be omitted in further considertions).
Vibrtions in Physicl Systems 018, 9, 01803 (4 of 9) 3. Internl three-phse motions We restrict our ttention to one of the simplest scenrios for the reltive motions of the prticles, i.e. periodic three-phse oscilltions. Let T denote the oscilltion period, while v,. First, consider the horizontl motion. For prcticl resons, we impose x u x v y u y the conditions: u x(0) = u x(t) = 0 nd v x(0) = v x(t) = 0. Moreover, the period includes three ervls of constnt reltive ccelertion. Denoting the ervl durtion by t i (i = 1,, 3) nd the chrcteristic time pos by t i = t i-1 + t i (in prticulr t 0 = 0, t 3 = T), we ssume the excittion function in the following form: x1 for 0 t t1 x ( t) x for t1 t t (15) for t t t x3 where xi re rel constnts. Consequently, the reltive velocity nd displcement re piecewise-liner nd piecewise-prbolic, respectively (see Fig. ). 3 Figure. Reltive ccelertion, velocity nd displcement of the ernl mss over one period Consider the system t rest: X X 0. According to Eq. (1), motion of the crrying body cn be excited by the first prticle if f f, i.e. x 1 y x min (16) 1
Vibrtions in Physicl Systems 018, 9, 01803 (5 of 9) Now, the ccelertion vlues for ech subervl re defined s c, (17) x1 x1 x min, x cxx min, x3 cx3x min where c xi re positive coefficients. In the initil phse c x1 > 1, which leds to motion of the box to the right (ctive phse). Next, to void its bckwrd motion, it is necessry to tke c x < 1. With this ssumption, it is impossible to obtin both zero velocity nd displcement. Velocity v x reches zero for cx1 t0 t0 t1 t1, (18) c hence the second phse durtion time is t = c tt 0, where c t > 1. The third indispensble stge is usully lso rest phse of the box, i.e. c x3 < 1, nd the following reltion holds: t 1 < t < t 3. When it comes to the verticl ernl motion, the reltive ccelertion is constrined vi the condition of the constnt contct between the box nd the ground (N 0). Tking o ccount (5) nd (13), one cn obtin y y min x 1 (19) Now, the ccelertion vlues for ech subervl t i re given by, (0) y1 y0, y cyx, y3 cyx3 where y0 > y min is the bse vlue, nd c y = y1/ x1. Thus, the verticl nd horizontl ccelertions re proportionl, nd the dependency y(t) is similr to the one shown in Fig.. The conditions of zero reltive velocity nd zero reltive displcement t the end of period led to x t t t 0 (1) 1 1 x x3 3 t1 t t3 1 3 0 () x x x This system of equtions llows one to determine two prmeters, while the others re ssumed to be constnt. 4. Numericl results Let us strt with numericl solutions of Eq. (11) for selected vlues of the model prmeters. The bsic dt set is s follows: 0.1, 1 0. 15, c x1, c x 0. 9, y0 0. 1, t 1 1, c t 1. 8. (3)
Vibrtions in Physicl Systems 018, 9, 01803 (6 of 9) Hence x min 0.66 nd y min -6.67. The ccelertion nd durtion time of the third phse hve been determined s the solution of system (1)-(): x3-0.05 (which mens c x3 0.038), t 3 10.8. For such vlues of the model prmeters, the horizontl reltive displcement nd velocity over two periods s well s the resulting displcement nd velocity of the whole system re shown in Fig. 3. As cn be seen, there is reltively short ervl of the box forwrd motion (ctive phse), which is followed by much longer rest phse. A closer look t the grphs indictes tht the pek-shpe velocity of the system reches zero t the beginning of the second phse of the ernl motions. Moreover, u x is round ten times lrger thn X. The effect of prticulr prmeters on the crrying body motion cn be ssessed vi the totl displcement nd the verge velocity. Assume tht the motion occurs without sticking. Direct egrtion of Eq. (11) over t 1 nd t llows one to estimte the durtion of the ctive phse. Next, the totl displcement per period is pproximted by X T ( 1x1 y1) X ( T) [ 1( x1 x) ( y1 y)], (4) ( ) 1 x y nd the verge velocity is V = X T /T. For dt set (3) one obtins X T 0.075, V 0.005. Filled contour plots of the totl displcement (scled by fctor of 10 ) on the prmeter plnes (c x1, c x) nd (c x1, y0) re presented in Fig. 4; the other prmeters tke the vlues (3). The effect of coefficient c x1 on X T is much greter thn tht of c x. Generlly, X T increses with incresing c x1 nd decresing y0. Obviously, the verge velocity is better mesure of the robot driving efficiency. By nlogy to Fig. 4, mps of V (scled by fctor of 10 3 ) re shown in Fig. 5, b. Unlike for the displcement, both c x1 nd c x ffect the velocity strongly; the vlues om the bsic dt set re relted to quite high velocity zone. When it comes to y0, its decrese produces n increse in V; it is even dvntgeous to chnge its sign (ntiphse ernl motions). Additionlly, the prmeter plnes (c x1, c t) nd (c t, y0) re tken o ccount in Fig. 5c, d. Compred to c x1, coefficient c t hs weker effect on V. Nevertheless, bsed on the grphs it is possible to select such vlues of the prmeters tht mximize the verge velocity. ) b) Figure 3. Results for the smple dt: ) reltive displcement (solid) nd velocity (dotted) of the ernl mss, b) displcement (solid) nd velocity (dotted) of the box
Vibrtions in Physicl Systems 018, 9, 01803 (7 of 9) ) b) Figure 4. The totl displcement X T 10 for vrying prmeters: ) c x1 nd c x; b) c x1 nd y0 ) b) c) d) Figure 5. The verge velocity V 10 3 for vrying prmeters: ) c x1 nd c x; b) c x1 nd y0; c) c x1 nd c t; d) c t nd y0
Vibrtions in Physicl Systems 018, 9, 01803 (8 of 9) Figure 6. Displcement of the crrying body: the bsic dt (grey solid); c x1 =.1, y0 = -0.5 (solid); c x1 =.3, c t =.4 (dshed); c t =.4, y0 = -0.5 (dotted) Figure 6 includes severl numericl solutions of the eqution of motion obtined for modified vlues of prticulr prmeters. The reference X(t) dependence (om Fig. 3b) is mrked in grey. The blck curves correspond to certin chnges mde on the bsis of the contour plots in Fig. 5b-d. In ll these cses, the jump in X(t) is greter thn for the reference dt set. It should be emphsized tht, even theoreticlly, prmeter y0 cnnot be decresed without bounds. Aprt om constr (19), condition (16) must be fulfilled only in the first phse of the ernl motion, ccording to the ssumptions. However, considerble negtive vlues of y0 mke the second phse to be ctive, i.e. the bckwrd motion of the box occurs, nd estimtion (4) is no longer vlid. 5. Conclusions The discussed conception of the inertilly-driven system is reltively simple. As result of rectiliner periodic oscilltions of two ernl prticles, n lwys forwrd motion of the crrying body cn be chieved. The ctive phse is rther short, but creful selection of the ernl motions prmeters llows one to mximize the verge velocity of the robot. Most of the vibrtion-driven systems nlyzed in literture involve unblnce vibrtion exciters. Actully, it is one of the simplest methods to excite hrmonic oscilltions. However, such n pproch leds to both forwrd nd bckwrd motion of the crrying body, nd vrious control strtegies must be used to minimize the effect [1, 4, 5]. From prcticl po of view, further studies should incorporte some constrs imposed on the reltive displcement, velocity or ccelertion of the ernl prticles, e.g. due to limited dimensions of the crrying body. Acknowledgments This work ws supported by grnts of the Ministry of Science nd Higher Eduction in Polnd: 0/1/DS-PB/3513/018.
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