Mechanics for Vibratory Manipulation

Transcription

1 Mechanics or Vibratory Manipulation Wesley H. Huang Matthew T. Mason Robotics Institute Carnegie Mellon Uniersity Abstract Vibratory manipulation is any mode o manipulation inoling repeated impacts due to a striker which ollows some periodic motion. In this paper, we study ibratory manipulation in the context o tapping planar objects which slide on a ixed support surace. We are interested in the behaiors an object exhibits under such excitation. There are two distinct types o tapping that can result: continuous tapping, in which the object is always in motion, and intermittent tapping, in which the object comes to rest between taps. We irst examine ibratory manipulation in one dimension, adapting results rom related work to ind conditions or stable periodic motion. The general two dimensional case is closely related to our preious work in impulsie manipulation which examined the mechanics o a sliding rotating object. In act, ibratory manipulation is an approximation to the iting cases o impulsie manipulation. We deelop the iting cases or intermittent and continuous tapping or rotationally symmetric objects and conclude with some examples. 1 Introduction Our preious work has ocused on impulsie manipulation, i.e. manipulating objects through impulsie orces. In this paper, we explore a closely related topic: ibratory manipulation. Vibratory manipulation broadly reers to any manipulation process inoling repeated impacts, generally at a high requency, due to a striker which ollows some regular or seroed periodic motion. In this paper, we consider ibratory manipulation in the context o manipulating planar parts that slide on a ixed support surace in between taps. Our interest is in stable periodic behaiors o an object that result rom such excitation. There are many dierent scenarios or this sort ibratory manipulation that we can consider; these include: Open-loop control o a pusher with superimposed ibration. A pusher with a ibrating tip ollows a predetermined trajectory while a striker tip executes some periodic motion. The actual object excitation is a high requency, low amplitude sequence o collisions between the striker and the object. Feedback control o a pusher with superimposed ibration. Same as aboe except that the gross motion o the pusher is determined by some eedback control law. Feedback control o impulses. The object is sensed beore eery impact, and a control law determines an appropriate impulse to drie the object towards some goal coniguration. Open-loop parts transer. A sequence o impacts is planned to moe an object rom start to goal, assuming ideal behaior o the slider and perect knowledge o all releant mechanical parameters. The shape o the striker and the details o the striking motion can be designed so that small errors gie rise to restoring impulses. There are a number o possible applications or such manipulation relating to micropositioning and to parts eeding. We ind that there are two types o tapping that result under ibratory manipulation: intermittent tapping, where the object comes to rest in between taps, and continuous tapping, where the object is always tapped again beore it comes to rest. Vibratory manipulation in one dimension that results in continous tapping is closely related to bouncing a ball and to robotic juggling. We reiew and adapt results rom work in these areas to determine the conditions or which the object displays stable periodic motion. Vibratory manipulation in two dimensions makes use o the same mechanics (o a sliding rotating planar object) as our preious work in impulsie manipulation. In act, ibratory manipulation (which concerns object behaior rom repeated impacts) approximates the iting cases o impulsie manipulation (which concerns the mechanics o a single impact). In our eorts to ormulate an understanding o the general case o ibratory manipulation, we hae deeloped the iting cases o impulsie manipulation or both intermittent and continuous tapping. We present detailed analysis o the mechanics o these iting cases or rotationally symmetric objects. Finally, we apply the results o this analysis to determine the range o possible motions under these iting cases or two dierent objects. Related Work Vibratory manipulation in which the support surace ibrates (and the object becomes airborne) is currently used

2 in bowl eeders (see Boothroyd et al. [4] or descriptions and Boothroyd [3], Caine [6], and Christiansen et al. [7] or recent work on analysis and automated design or bowl eeders) and in the SONY APOS system (Hitakawa [10]). Böhringer et al. [] oriented planar parts by taking adantage o the ibrational modes o the support surace. The ibratory manipulation which we describe in this paper (inoling a sliding phase o the object in between taps) is closely related to the problem o bouncing a ball on a ibrating table and to robotic juggling. The bouncing ball problem has been analyzed by Holmes [11] and Bapat et al. [1]; Tuillaro and Albano [16] and Wieseneld and Tuillaro [17] hae perormed experiments with this system. Schaal and Atkeson [15] hae studied almost the exact same system in the context o robotic juggling. Bühler and Koditschek [5] demonstrated a system that juggles a puck on an inclined plane using a bat. In the area o impulsie manipulation, Higuchi [9] demonstrated linear micropositioning using an electromagnetic coil to delier an impulsie orce. More recently, Yamagata and Higuchi [18] hae deeloped a piezoelectric deice to delier impulsie orces to do micropositioning or precision optical components. Mirtich and Canny [14] introduced the idea o impulsebased dynamic simulation, in which all contact between objects is modeled as a series o microcollisions. The iting cases o impulsie manipulation deeloped in this paper build upon the results o Huang et al. [1] which presented an analytical ormulation and solution o the mechanics o tapping a rotationally symmetric object. These iting cases examine the mechanics as the impulse deliered by a tap approaches zero. Many o the results o this paper are coered in greater detail in Huang and Mason [13]. The present work also makes use o the eigendirections o Goyal et al. [8] in their analysis o the dynamics o sliding objects. 3 Vibratory Manipulation in 1D We consider in one dimension the problem o determining the object behaior when we delier a sequence o impacts. We note that i the object is kept in continuous motion (continuous tapping) then it is subject to a constant (Coulomb) rictional orce. Suppose the striker moes with some periodic ibration superimposed on a constant elocity. When iewed in a rame moing at that constant elocity (see Figure 1), this system is equialent to a ball bouncing on a ibrating table between impacts, the object is subject to a constant acceleration produced by riction whereas the ball is subject to acceleration produced by graity. This system has been studied in the context o chaotic systems and in the context o robotic juggling. We can apply the results o these analyses to ibratory manipulation in order to relate parameters o the striker s Figure 1: An illustration o regular periodic bouncing. The table moes with a sinusoidal ibration, and the object ollows a parabolic trajectory while in light. This same illustration applies to continuously tapping an object using a striker with a sinusoidal ibration superimposed on a constant elocity (when iewed in the rame o that constant elocity). motion to possible object behaiors. In this section, we describe some o the main results. 3.1 The Bouncing Ball Problem The problem o a ball bouncing on a sinusoidally ibrating table has been studied in some detail by Holmes [11] and Bapat et al. [1]. This system can exhibit bounded chaotic behaior, though we are primarily interested in regular periodic behaiors. One key result rom Holmes is the relation between striker requency and amplitude or stable periodic bouncing. We assume the motion o the table is X(t) =A sin!t. Holmes ormed the return map which maps the elocity o the ball (just beore an impact) and the phase o the impact (relatie to the table oscillation) rom one impact to the next and showed that this return map has a number o ixed points which correspond to dierent modes o periodic bouncing. Stability analysis o these ixed points showed that in order to maintain stable periodic juggling, the amplitude o the table ibration must satisy: s ng (1, e)! (1 + e) <A<g n! (1, e) +1 (1) (1 + e) where the object motion has period one and strikes the table eery n table cycles; e is the coeicient o restitution between the object and the striker, and the coeicient o riction between the object and the surace. This relationship is shown graphically in Figure. 3. Robotic Juggling Schaal and Atkeson [15] hae implemented paddle jugging using a trampoline-like racket and a ping-pong ball. They obsered that negatie acceleration o the paddle trajectory at impact seems to proide stability. They calculated the basin o attraction or periodic juggling to be 0.57 the area o the iable space o initial conditions. Howeer, or a number o reasons (including mechanical ariability and air resistance) the basin o attraction was signiicantly larger. In act, they were unable to aoid getting a periodic (mostly period one) juggling pattern. Bühler and Koditschek [5] demonstrated a system that juggles a puck on an inclined plane using a bat. In order to achiee stable juggling, they deeloped a mirror control law which seros the bat to nonlinearly relect the position

3 Amplitude (mm) requency (Hz) Figure : The relationship between the requency and amplitude or a sinusoidally ibrating striker in order to achiee period one, n =1 bouncing or ibratory manipulation. Here we hae assumed e = 0:8 and = 0:5. Increasing the amplitude beyond the range shown here will result in period doubling and will eentually lead to chaotic behaior. Using an amplitude below this range will likely result in the object coming to rest against the striker and being pushed along. o the puck. Using this sort o simple control law or ibratory manipulation is appealing but sensing the object would be diicult or high requency, low amplitude motions. 4 The Limiting Cases o Impulsie Manipulation Vibratory manipulation in two dimensions is much more complicated than in one dimension. There is strong coupling between translational and rotational elocities, and the rictional load on the object is generally a unction o the object elocities and the object orientation. It will require greater understanding o the dynamics o a sliding object in order to understand two dimensional ibratory manipulation in general. We hae started by ormulating the iting cases o impulsie manipulation or rotationally symmetric objects. Suppose we manipulate an object by deliering a series o identical taps at some requency. We take the iting case to be the iting behaior o the object as we let the number o taps approach ininity and the impulse o each tap approach zero. Here, we will ocus on regular period one motions, i.e. the object will repeatedly ollow the same path in state space ater each impact. Thereore, we can ocus on the resultant object motion rom a single tap as the impulse deliered by that tap approaches zero. In this section, we deelop the iting cases or intermittent tapping and continuous tapping. First, howeer, we coer a ew preinaries or sliding rotationallysymmetric objects in two dimensions. 4.1 Preinaries or D The Physics o a Sliding Rotating Object Rotationally symmetric objects are, or our purposes, objects whose pressure distribution is a unction o radius only. These objects hae the special property that they slide in a straight line. (Thus we need only consider a two dimensional state space.) As discussed in Huang et al. [1], the net orce and torque on a rotationally symmetric object are unctions o only the ratio!. The equations o motion are:,f (! ) = m _ (),T (! ) = I _! (3) where F and T are the orce and torque loads on the object. The initial elocities are 0 and! 0. During the motion, the object will translate a distance x and rotate an angle x = = Z t 0 Z t 0 dt (4)!dt (5) where t is the time that the object slides Trajectory Scaling Through straightorward use o the chain rule, we can show that i (t) and!(t) are solutions to the equations o motion, then so are k( t k ) and k!( t k ). I the irst pair o trajectories coer a distance x and an angle, then the second coer a distance k x and an angle k Centers o Rotation (CORs) We can describe any single planar motion o a rigid object by a center o rotation (COR). The COR is the point about which the net moement o the object appears to be a pure rotation. Pure translation corresponds to a COR at ininity. Gien the displacement ~x and rotation, thecor is located at ^z~x. We can also use CORs to describe the instantaneous motion o an object; gien the linear elocity ~ and rotational elocity!, thecor is located at ^z~!. We can characterize a mode o manipulation by the locus o CORs which it can eect upon an object. This enables us to compare dierent modes o manipulation, and it also suices as the inormation required in order to create plans or that mode o manipulation. In this paper, we construct the locus o CORs with respect to the initial translational elocity o the object and assume that rotations are positie. Negatie rotations result in a dierent locus o CORs which is simply the mirror image with respect to the initial elocity. 4. The Limiting Case or Intermittent Tapping For intermittent tapping, we delier a sequence o identical impacts, letting the object come to rest beore the next impact. Here, the object repeatedly ollows a path in state space starting at the initial elocities and ending at the origin (zero elocities). We take the iting case to be the

4 x / θ x Net COR x tan (θ /) Figure 3: Construction or inding the net COR or a tap where the object moes a distance x and turns an angle. resultant behaior as the impulse approaches zero but the ratio o the initial elocities remains the same. We irst determine the net COR or a motion in the it and then ind the locus o CORs possible or this iting case Net COR in the Limit Gien the distance traeled and the angle turned, we can determine the net COR or the motion with a geometric construction as illustrated in Figure 3. The location o the net COR is: ~x + ^z ~x (6) tan Since we let the impulse o a tap approach zero by scaling the initial elocities, we can apply the trajectory scaling o Section 4.1. (scaling the elocities by k). In the it, the COR is then k ~x k!0 + ^z k ~x tan k θ = k!0 ^z k ~x tan k Applying l Hôpital s rule (dierentiating both numerator and denominator with respect to k), we get k!0 ^z ~x sec k (7) = ^z ~x (8) Thus, as the impulse deliered by a tap approaches zero, the distance rom the center o mass (COM) to the net COR approaches x. 4.. Locus o CORs In order to characterize this iting case, we construct the locus o CORs. This represents all possible motions o the object or a gien initial elocity direction. The trajectory scaling o Section 4.1. shows that each alue o!0 0 corresponds to a single alue o x, no matter what the magnitude o the initial elocities or the resulting displacements. In other words x is a unction o!0 0. Huang and Mason [13] urther showed that x is a continuous monotonic decreasing unction o!0 0. Thereore the minimum and maximum alues o!0 0 that we can achiee ia an impact correspond to the maximum and minimum alues o x or the resulting displacements. These alues o x in turn determine the closest and urthest CORs possible or this iting case. Huang et al. [1] ormulated the impact cone which represents the set o elocities that can be achieed by striking an object at rest. We can restate this cone in terms o the ratios o initial elocities possible! 0 p rrm (9) 0 I 1+ r The minimum alue o!0 0 is 0, corresponding to pure translational elocity (which results in a pure translation o the x object: = 1); the maximum alue is gien by the equality condition o Equation 9. Let c min = x or the maximum alue o!0 0 that satisies Equation 9. The locus o CORs possible or the iting case o intermittent tapping is then c^z ^ 0 j c [c min ; 1]g (10) 4.3 The Limiting Case o Continuous Tapping in D For continuous tapping, we delier a sequence o regularly spaced impacts, always striking the object beore it comes to rest. Here, the object ollows a path in state space starting at its initial elocities and continues until the object is tapped again, kicking the state back to the original initial elocities. We take the iting case to be the resulting behaior o the object as the amount o time beore the next impact approaches zero. We take an analogous approach (to that or the iting case o intermittent tapping) to determine the location o the net COR in the it. We start with the same expression (Equation 6) or the net COR gien the displacement and rotation o the object: ~x + ^z ~x (11) tan As we decrease the time that the object slides (t ), ~x and both approach zero: t!0 ~x + ^z ~x tan = t!0 ^z ~x (1) tan Assume or the moment that this it exists. Applying l Hôpital s rule (dierentiating numerator and denominator with respect to t ), we get t!0 ^z d ~x dt sec d dt = t!0 R ^z d t dt 0 R ~dt d t dt 0!dt = ^z ~(t ) = ^z ~ 0 (13) t!0!(t )! 0 Thus, the COR in the it is simply the initial (instantaneous) COR. The remainder o this subsection examines the conditions under which this it exists and then inds the locus o CORs possible under this iting case.

5 stable eigendirection negatie impact cone ω ω ω negatie impact cone (a) (b) (c) Figure 4: Illustrations o the state space or a rotating sliding rotationally symmetric object; the stable eigendirection is shown as the dashed-dotted line. Figure (a) shows a typical ector ield, which is a unction o only!. Figure (b) illustrates the case where the initial slope o the path in state space aligns with the negatie impact cone aboe the stable eigendirection. In this case, the trajectory cures in to the cone. Figure(c) illustrates the case where this alignment occurs below the eigendirection; in this case, the trajectory cures away rom the cone.! Disc COR! Ring COR Intermittent LC Continuous LC Impact Cone Eigendirection Radius Table 1: Motion constraints or a disc and a ring or both iting cases (LCs). The distance to the COR gien or the impact cone, iting cases, and pushing is the distance to the closest COR. The distance to the COR corresponding to the stable eigendirection and the radius o the object are gien or comparison. Quasistatic pushing has the same motion constraints as the iting case o continuous tapping. These results were numerically generated, and assume that = r =0: Impact Constraint or Moing Objects For the iting case o intermittent tapping, we reormulated the impact cone (the cone in state space that represents the elocities that can be achieed by tapping an object at rest) in terms o the achieable ratios o!0 0. For a gien point in the impact cone, we can compute the striker elocity and contact point required to achiee that change in elocity. Not surprisingly, so long as we recreate the same relatie elocities between the striker and the object, we can achiee the same change in elocity. This implies that we can restate Equation 9 in terms o change in elocities j!j p rrm (14) I 1+ r The result is that we can moe the impact cone to any point in state space. We will primarily make use o the negatie impact cone, which we deine as the set o states rom which we can reach the current state with a single impact. Since the impact cone is inariant to its position in state space, we orm the negatie impact cone by simply relecting the impact cone through the current state Existence o the Limit & Locus o CORs The it taken in Equations 1 and 13 exists i and only i the path o the object in state space passes through the negatie impact cone (at the initial elocities) or a nonzero period o time. In order to determine which initial conditions satisy this property, we must examine properties o paths in state space. The eigendirections o Goyal et al. [8] sere to guide our analysis. There are a number o properties o paths in state space or a sliding rotationally symmetric object. We state them here in terms o the ector ield (_; _!) =(,F=M;,T=I) oer the state space. as! increases, the orientation o (_; _!) monotonically increases rom to 3. (_; _!) points towards the origin only at alues o! corresponding to eigendirections; most objects hae a single stable eigendirection that is neither pure translation nor rotation. at all points below the stable eigendirection, (_; _!) points aboe the origin (and ice ersa), drawing the system towards the stable eigendirection. These properties are illustrated in Figure 4a. I we place the negatie impact cone on the line corresponding to! = 0(a COR at ininity), the path in state space will lie entirely inside the cone. As we increase the alue o!, the ector (_; _!) will rotate counterclockwise. The path o the object in state space will continue to pass through the cone until this ector aligns with the bottom edge o the cone. At that point, we hae F (! ) T (! ) = R q (15) 1+ 1 r Let 1=c min be the alue o!0 0 which satisies this equation. I this alignment occurs aboe the stable eigendirection then the path will cure into the cone (see Figure 4b), so the set o possible CORs is c^z ^ 0 j c [c min ; 1]g (16) I this alignment occurs below the stable eigendirection then the path will cure away rom the cone (see Figure 4c), so the set o possible CORsis c^z ^ 0 j c (c min ; 1]g (17) Interestingly, the set o possible CORs or pushing an object is identical to Equation 16. Pushing can achiee any COR so long as! corresponds to a orce/torque load due to an applied orce in the riction cone (in orce space ) at the contact point; the iting case o continuous tapping can achiee any COR so long as!0 0 results in a orce/torque within the negatie impact cone, which is simply a transormation o the riction cone (in impulse space) at the contact point. The mapping between orce/torque due to riction and the object elocities is the same or both modes o manipulation.

6 eigendirection continuous LC 3 intermittent LC 4 impact cone cm eigendirection continuous LC 3 intermittent LC 4 impact cone Figure 5: Graphic illustration o the motion constraints on a disc and a ring due to tapping. Points 4 are the closest COR achieable ia the respectie mode o manipulation. The ull set o CORs include all the points ertically aboe, going out to ininity. Point 1 represents the stable eigendirection; the dotted lines show the path the object would take i it continued to moe about that COR. 5 Examples or the Limiting Cases Table 1 shows the results o analyzing the possible motions o a uniorm disc and a ring under the two iting cases. These results are graphically illustrated in Figure 5. Note that the alignment or the iting case o continuous tapping occurs below the stable eigendirection (i.e. the alue o! (slope in state space) or the iting case is less than that or the eigendirection). 6 Summary In this paper, we deeloped the idea o ibratory manipulation in the context o objects sliding on a planar surace. We ound two distinct modes o ibratory manipulation: intermittent tapping, where the object comes to rest between impacts, and continuous tapping, where the object is always tapped again beore it comes to rest. We irst examined one dimensional continuous tapping; this mode o manipulation is closely related to bouncing a ball on a ibrating table. We applied work on the bouncing ball problem to determine the relationship between striker requency and amplitude in order to achiee regular periodic bouncing under continuous tapping. Research in robotic juggling suggests that the basin o attraction or stable periodic bouncing is larger than theory indicates. The two dimensional case o ibratory manipulation is more complex. Vibratory manipulation (which concerns object behaior rom repeated impacts) is an approximation to the iting cases o impulsie manipulation (which addresses the mechanics o a single impact). We deeloped the iting cases or rotationally symmetric objects and ound that the possible motions or the iting case o continuous tapping are (nearly) identical to those or quasistatic pushing. Finally, we applied our analysis to determine the possible motions or a disc and a ring under both iting cases. Reerences [1] C. N. Bapat, S. Sankar, and N. Popplewell. Repeated impacts on a sinusoidally ibrating table reappraised. Journal o Sound and Vibration, 108(1):99 115, [] K. Böhringer, V. Bhatt, and K. Goldberg. Sensorless manipulation using transerse ibrations o a plate. In IEEE International Conerence on Robotics and Automation, olume, pages , [3] G. Boothroyd. Assembly Automation and Product Design. Marcel Dekker, 199. [4] G. Boothroyd, C. Poli, and L. E. Murch. Automatic Assembly. Marcel Dekker, 198. [5] M. Bühler and D. E. Koditschek. From stable to chaotic juggling: Theory, simulation, and experiments. In IEEE International Conerence on Robotics and Automation, pages , Cincinnati, OH, [6] M. Caine. The design o shape interactions using motion constraints. In IEEE International Conerence on Robotics and Automation, [7] A. Christiansen, A. Edwards, and C. Coello. Automated design o part eeders using a genetic algorithm. In IEEE International Conerence on Robotics and Automation, olume 1, pages , [8] S. Goyal, A. Ruina, and J. Papadopoulos. Planar sliding with dry riction. Part 1. Limit surace and moment unction. Wear, 143: , [9] T. Higuchi. Application o electromagnetic impulsie orce to precise positioning tools in robot systems. In International Symposium on Robotics Research, pages Cambridge, Mass: MIT Press, [10] H. Hitakawa. Adanced parts orientation system has wide application. Assembly Automation, 8(3): , [11] P. J. Holmes. The dynamics o repeated impacts with a sinusoidally ibrating table. Journal o Sound and Vibration, 84(): , 198. [1] W. Huang, E. P. Krotko, and M. T. Mason. Impulsie manipulation. In IEEE International Conerence on Robotics and Automation, [13] W. H. Huang and M. T. Mason. Limiting cases o impulsie manipulation. Technical Report CMU RI TR 96 4, The Robotics Institute, Carnegie Mellon Uniersity, September [14] B. Mirtich and J. Canny. Impulse-based dynamic simulation. In K. Goldberg, D. Halperin, J.-C. Latombe, and R. Wilson, editors, Algorithmic Foundations o Robotics, pages A. K. Peters, Boston, MA, [15] S. Schaal and C. G. Atkeson. Open loop stable control strategies or robot juggling. In IEEE International Conerence on Robotics and Automation, pages 3: , Atlanta, GA, [16] N. B. Tuillaro and A. M. Albano. Chaotic dynamics o a bouncing ball. American Journal o Physics, 54(10): , October [17] K. Wieseneld and N. B. Tuillaro. Suppression o period doubling in the dynamics o a bouncing ball. Physica D, 6(1 3):31 335, May-June [18] Y. Yamagata and T. Higuchi. A micropositioning deice or precision automatic assembly using impact orce o piezoelectric elements. In IEEE International Conerence on Robotics and Automation, 1995.