Orienting Micro-Scale Parts with Squeeze and Roll Primitives

Transcription

1 Orienting Micro-Scale Part with Squeeze and Roll Primitive Mark Moll Michael A. Erdmann Ron Fearing Ken Goldberg Department of Computer Science, Carnegie Mellon Univerity, Pittburgh, PA IEOR and EECS Department, UC Berkeley, Berkeley, CA Abtract Orienting part that meaure only a few micrometer in diameter introduce everal challenge that need not be conidered at the macro-cale. Firt, there are everal kind of ticking effect due to Van der Waal force and tatic electricity which complicate hand-off motion and releae of a part. Second, the degree of freedom of micro-manipulator are limited. Thi paper preent a complete algorithm that addree thee challenge. We will how that a equence of imple manipulation operation can uniquely orient an aymmetric part without ening. Thi allow u to apply the ame plan to many (identical) part imultaneouly. For aymmetric part we can find a plan of length O(n) in O(n) time that orient the part, where n i the number of vertice. 1 Introduction Increaed miniaturization of ma-produced conumer and indutrial product uch a dik drive, camera, diplay, and enor will require fundamental innovation in part handling. Conventional pick-and-place technique do not work well at the micro-cale where ticking affect dominate. We will ue the term ticking effect to decribe the combined effect of Van der Waal force, electrotatic urface charge and other attractive force that occur at the micro-level. Due to thee ticking effect part can tick to a manipulator without being graped. The attractive force drop rapidly in magnitude a the ditance between the part and manipulator increae. An additional complication for handling micro-part i that the ticking effect vary wildly, even if all the material propertie and geometrie of the object involved are known. In order to manipulate micro-part, we propoe manipulation trategy that conit of applying imple operation requiring no more than two degree of freedom. During an operation one degree of freedom will be active and the other will be compliant in order to maintain contact with the part. We maintain complete control over the part orientation. Figure 1 illutrate a plan that orient a part by rolling and Thi work wa upported in part by the NSF under grant??? Figure 1: Example of a queeze-roll plan orienting a polygonal part. After the initial operation the part i in one of ix orientation. After two more enorle operation the part i oriented up to 180 ymmetry. queezing it between two horizontal micro-cale jaw. Note that for thi type of manipulation we only need to conider the convex hull of the part. Initially the part can be in any orientation, but after execution of the plan the part will be in one of two orientation. The tate tranition after each operation are indicated by the dahed line. 2 Related Work 2.1 Micromanipulation When part are maller than one millimeter the effect of adheive force become ignificant. Fearing (1995) give an overview of all the different adheive force that occur at thi cale. Many reearcher have worked on reliable pick and

2 queeze roll queeze Figure 2: A equence of micro-manipulation operation, from left to right. We automatically follow each roll by a queeze. The mall arrow indicate in which direction the jaw have moved during each operation. The black arrow indicate a controlled motion, the gray arrow indicate a compliant motion. Note that part alway remain in contact π 2π π/2 π 3π/2 2π 3π/2 π π/2 0 0 π/2 π 3π/2 2π 3π/2 π π/2 0 0 π/2 π 3π/2 2π (a) A rectangular part (b) It diameter function (c) It queeze function (d) It roll function with upper jaw tranlating 1 unit to the left Figure 3: The diameter, queeze and roll function. Note that the queeze and roll function are monotone. place operation of micro-cale object. Miyazaki and Sato (1996) and Saito et al. (1999) decribe a ytem where a needle i ued to aemble 3D tructure compoed of particle ranging from 10nm to 1µm in ize. The ticking effect enable the needle to pick up a particle. By tranlating along a o-called hearing trajectory the needle releae the particle. A human operator control the manipulator, while a canning electron microcope (SEM) provide the viual feedback. Koyano and Sato (1996) decribe a imilar ytem for pick-and-place operation. They ue two different ized needle and define different part releae motion. One motion conit of clamping the part onto a urface with a thin needle, while the large needle releae the part. The ticking effect are not large enough to make the part tick to the thin needle. Another part releae motion conit of rotating the needle around the part in order to minimize the contact area. Zech and Fearing (1998) explore how one can orient part by puhing them with an AFM cantilever equipped with a force enor. The force enor i ued to detect obtacle and changing contact condition. In (Shimada et al., 2000) a ytem i decribed coniting of two orthogonal one degree-of-freedom tweezer. The tweezer are equipped with force enor. Shimada et al. decribe two different way to orient the part with thee tweezer: one way i to roll the part between them, another i to pivot the part around a fixture. (Shimada et al., 2000) contain a lot more good micro-manipulation reference 2.2 Part Orienting The problem of how to bring part into a deired orientation ha been well tudied at the macro-cale. It i not neceary to grap an object in order to orient it. Maon (1982, 1985) puhing howed how to orient part by puhing them. Thi can be ued to deign a equence of fence over a conveyor belt (Pehkin and Sanderon, 1988; Wiegley et al., 1996; Berretty et al., 1998). Akella et al. (1997) howed that intead of a equence of fence one can alo ue one fence with one rotational degree of freedom. Goldberg (1993) howed that it i poible to orient po- queezing lygonal part with a frictionle parallel-jaw gripper without enor. Goldberg conjectured and Chen and Ierardi (1995) proved that for every n-ided polygonal part, a equence of queeze can be computed in O(n 2 ) time that will orient it up to ymmetry. The length of uch a equence i O(n). Akella et al. (2000) that with partial enor information the length of thi equence can be reduced to O(m), where m i the maximum number of tate with the ame enor value. Bicchi and colleague howed that by imply rolling an rolling object between the two hand of a parallel-jaw gripper it i poible to orient and poition polyhedral part (Ceccarelli et al., 2000; Marigo et al., 1997) and mooth 3D part (Marigo and Bicchi, 2000). The jaw are equipped with tactile enor, which allow the ytem to recontruct the hape of unknown mooth object (Bicchi et al., 1999). In (Rao et al., 1995) an algorithm i decribed to orient pivoting polyhedral part uing o-called pivot grap. A part i

3 graped with two hard finger contact and i then free to rotate around the axi formed by the contact. tilting Erdmann and Maon (1988) developed a tray-tilting en- orle manipulator that can orient planar part in the preence of friction. If it in t poible to bring a part into a unique orientation, the planner would try to minimize the number of final orientation. In (Erdmann et al., 1993) it i hown how (with ome implifying aumption) three-dimenional part can be oriented uing a tray-tilting manipulator. In particular, for polyhedral part with n face a equence of tilt of length O(n) can be found in O(n 3 ) time. Zumel (1997) ued a variation of the tray tilting idea to orient planar part. Zumel ued two actuated arm connected at a hinge to tilt part from one arm to the other. vector In recent year a lot of work ha been done on pro- field grammable force field to orient part (Böhringer et al., 1997, 1999; Kavraki, 1997; Reznik et al., 1999) The idea i that ome kind of force field (implemented uing e.g. MEMS actuator array) can be ued to puh the part in a certain orientation. Kavraki (1997) preented a vector field that induced two table configuration for mot part. Böhringer et al. ued Goldberg algorithm (1993) to define a equence of queeze field to orient a part. They alo gave an example how programmable vector field can be ued to imultaneouly ort different part and orient them. 3 Two Micro-Manipulation Primitive We manipulate part with a pair of parallel jaw gripper. We aume there i no lip between the jaw and the part. We can realize thi aumption, for intance, by making the part and jaw out of a hydrophobic material and embedding them in a fluid. We alo aume both jaw will alway be in contact with the part and that there are no udden change in the poe of the part due to ticking effect. One jaw can tranlate in the horizontal direction, the other jaw can tranlate in the vertical direction. With each pair of gripper we can perform the following two operation: Squeeze We cloe the jaw and, imultaneouly, allow the jaw that can tranlate in the horizontal direction to move compliantly until a table grap i reached. Thi i equivalent to a frictionle jaw grap (Goldberg, 1993). Roll We tranlate one jaw in the horizontal direction by a given amount and allow the other jaw to move compliantly. To make ure that the part i alway in one of a finite number of orientation, we automatically follow each roll by a queeze. Thee two operation are illutrated in figure 2. Thee operation can be defined more formally a function that map orientation to orientation. Let S 1 be the et of orientation x=d(co β co( α+ β)) d α Figure 4: Relationhip between x, the tranlation of the upper jaw, and α, the change in orientation during a roll operation. in the plane. Conider the diameter function d : S 1 R, which, given an orientation, return the ditance between the jaw when they jut touch the part in thi orientation. We define the queeze function, : S 1 S 1, uch that if θ i the initial orientation, (θ) i the orientation after the queeze i completed. Note that for any θ (θ) i a local minimum of the diameter function. For the roll operation we can define a parametrized family of function, r x : S 1 S 1, uch that if θ i the initial orientation, r x (θ) i the orientation after the upper jaw ha been tranlated by x and the part ha been queezed. We define a local frame uch that if x i greater than 0, the jaw move to the left. A roll function correpond roughly to a hifted queeze function. Figure 3 how thee function for a rectangular part. Suppoe that during a roll the contact point do not change. For a given tranlation x of the upper jaw the change in orientation i then equal to α = co 1 (co β x/d) β, where d i the ditance between the contact point and β the angle between the x-axi and the line through the contact point. See figure 4. Note that x and α alway have the ame ign. If the contact point do change during a roll, we divide the roll into maller tep uch that during each tep the contact point do not change. The total change in orientation i imply the um of change during each tep. It i not hard to ee that for a given amount of tranlation the roll function i monotone in the orientation. We will ue thi property later. 4 Planning Algorithm Define a tate a an edge of the part being aligned with one of the jaw. Since there are 2 jaw, an n-ided polygon can be in 2n tate. The queeze and roll primitive are cloed under the et of part tate: any primitive will map from any tate to another tate in the et. For each tate we compute the minimum and maximum amount of tranlation needed to make a clockwie and counter-clockwie tranition to the next table edge. Thee minima and maxima correpond to critical point where the outcome of a roll operation will change. Now conider the orted lit of all critical point for all tate. To determine all poible outcome of a roll operation applied to a et of tate, it i ufficient to look d β

4 r(-3.05) 1,1 r(-1.51) 2,1 0,0 r(+4.25) r(+3.41) r(-5.67) r(+1.12) r(+5.59) r(-11.86) r(+2.19) r(-4.65) 4,0 r(-2.16) 3,1 r(+10.18) r(-2.16) 3,0 r(+10.18) r(-4.65) r(+2.19) (a) Tranition graph r(-4.65) r(+1.12) r(+4.25) r(+3.41) r(-11.86) r(+5.59) 4,1 (b) Tranition from tate [0, 0] to [3, 1] 0,1 r(-5.67) r(-1.51) 2,0 1,0 r(-3.05) Figure 5: Each node conit of a pair [e, j], where e = 0,..., n 1 i the edge index and j = 0, 1 i the jaw index. up the outcome at all the midpoint between conecutive critical point. In other word, even though we can roll a part by any continuou value, we only need to conider a finite et of O(n) roll function. The roll and queeze function induce a labeled graph on the tate, where each edge i labeled with the appropriate function. Figure 5 how an example of uch a graph for a given part. Baed on the roll and queeze function we can alo contruct a graph where the node are et of tate, which we will call hypertate. There exit an edge from node v to v if and only if applying a roll or queeze operation to each tate in v reult in the et v. Each edge i labeled with the correponding operation. The goal i now to find a path in thi graph from the et of all tate to a et with a few element a poible. The goal et will alway have at leat two element, becaue we can not ditinguih any two orientation that are 180 degree apart. We are intereted in finding the hortet path from the node repreenting all tate to node with a minimal number of tate. Thee path correpond to plan that orient a part with the mallet number of operation. Natarajan (1989) wa the firt to analyze the complexity of thi problem. He howed that given k function a plan can be found, if one exit, in time O(kn 4 ). Epptein (1990) preented an algorithm that given k monotone function find the hortet plan in O(kn 2 ) time. Goldberg (1993) and Chen and Ierardi (1995) improved on thi bound for the pecial cae where function correpond to queeze in a finite number of direction. In thi cae a plan of length O(n) can be found in time O(n 2 ). Recently, Berretty et al. (1998) analyzed another monotone function called the puh function. The puh function, p α : S 1 S 1, when given an orientation θ return the orientation of the part p α (θ) after it ha been puhed from direction α by a fence orthogonal to the puh direction. With the puh operation it i poible to uniquely orient a part. Berretty et al. preented an O(n 3 log n) algorithm to find the hortet plan. Since we have O(n) roll function and one queeze function, we can find a plan in O(n 3 ). Baed on the reult in the next ection we conjecture that for aymmetric part the final hypertate contain alway exactly two tate. It i poible that there exit many path of the ame length that lead to node with the ame number of tate. We can impoe additional contraint to find the bet path. For intance, we might prefer queeze operation over roll operation. Or we can minimize the total amount of tranlation required by a plan. Finally, we can elect the path that i the mot robut. It i poible to orient a polygonal by repeatedly applying the ame operation. Conider the et of minimal ditance uch that a counter-clockwie roll with thi ditance a parameter will caue a tranition to another tate. Let d be a ditance between the econd larget and larget element of that et. If we perform a roll operation with ditance d n 1 time, the part will be in the tate correponding to the larget element of the et. So we can compute a plan of length n 1 that will orient a part in linear time. 5 Random Polygonal Part Figure 1 how a plan that wa found uing the algorithm. The algorithm take the convex hull of the part a input. To get a ene for what kind of polygon take many operation to orient, we teted the algorithm on a et of random convex polygon. Random convex polygon are generated in the following way. We can regard a convex n-ided polygon a a et of n vector ubject to the contraint that the vector add up to the zero vector. We pick the x-coordinate (and y-coordinate) of the vector a follow: we pick a uniformly random point inide a n 1 dimenional hyperphere and rotate the reulting point (padded with a zero to make it

5 1 0.8 Cumulative probability Number of vertice Figure 7: Plot of number of operation v. number of node. Cumulative ditribution function of number of operation required to orient a part with 2, 3,, 12 node. primatic joint r(-4.49) r(+3.50) r(-1.30) Figure 6: For part with many table grap many operation are needed to orient it. jaw jaw part primatic joint length n) to lie in the hyperplane defined by n i=1 x i = 0. The vector can be computed in O(n) time. To create a polygon we need to ort the vector by angle, cauing the overall run time to be O(n log n). Thi algorithm i due to Lambert (1994). In figure 6 an example i hown of a 5-ided polygon that require 4 operation to orient it. The polygon i queezed firt, followed by three roll tranition. In general, the more table grap a part ha, the more operation will be needed to orient it. If all tate correpond to table grap, queezing the part ha no effect and only roll operation can be ued to orient it. Under the probability ditribution function induced by the algorithm decribed above, we found that almot all random polygonal part hape can be oriented in le than four operation. In figure 7 i hown the cumulative ditribution function of the number of operation required to orient a random polygon. Thi function i computed by ampling 5000 polygon for a given number of node. From thi figure it can be concluded that the expected number of operation needed i mall and increae lowly with the number of node. Figure 8: The kinematic of a queeze-roll micro-gripper 6 Dicuion We have preented a complete algorithm that compute a plan to orient a polygonal part at the micro-cale. The manipulation primitive take into account the ticking effect that occur at the micro-cale a well a the limited degree of freedom that manipulator typically have at that cale. The part orienting trategy i completely enorle. Reference Akella, S., Huang, W. H., Lynch, K. M., and Maon, M. T. (1997). Senorle part orienting with a one-joint manipulator. In Proc IEEE Intl. Conf. on Robotic and Automation. Akella, S., Huang, W. H., Lynch, K. M., and Maon, M. T. (2000). Part feeding on a conveyor with a one joint robot. Algorithmica, 26: Berretty, R.-P., Goldberg, K., Overmar, M. H., and van der

6 Stappen, A. F. (1998). Computing fence deign for orienting part. Computational Geometry: Theory and Application, 10: Bicchi, A., Marigo, A., and Prattichizzo, D. (1999). Dexterity through rolling: Manipulation of unknown object. In Proc IEEE Intl. Conf. on Robotic and Automation, page , Detroit, Michigan. Böhringer, K. F., Bhatt, V., Donald, B. R., and Goldberg, K. Y. (1997). Algorithm for enorle manipulation uing a vibrating urface. Algorithmica. Accepted for publication. Böhringer, K. F., Donald, B. R., and MacDonald, N. C. (1999). Programmable vector field for ditributed manipulation, with application to MEMS actuator array and vibratory part feeder. Intl. J. of Robotic Reearch, 18(2): Ceccarelli, M., Marigo, A., Piccinocchi, S., and Bicchi, A. (2000). Planning motion of polyhedral part by rolling. Algorithmica, 26(4): Chen, Y.-B. and Ierardi, D. (1995). The complexity of obliviou plan for orienting and ditinguihing polygonal part. Algorithmica, 14. Epptein, D. (1990). Reet equence for monotonic automata. SIAM J. Computing, 19(3): Erdmann, M. A. and Maon, M. T. (1988). An exploration of enorle manipulation. IEEE J. of Robotic and Automation, 4(4): Erdmann, M. A., Maon, M. T., and Vaněček, Jr., G. (1993). Mechanical part orienting: The cae of a polyhedron on a table. Algorithmica, 10: Fearing, R. S. (1995). Survey of ticking effect for micro part handling. In Proc IEEE/RSJ Intl. Conf. on Intelligent Robot and Sytem, volume 2, page , Pittburgh, PA. Goldberg, K. Y. (1993). Orienting polygonal part without enor. Algorithmica, 10(3): Kavraki, L. E. (1997). Part orientation with programmable vector field: Two table equilibria for mot part. In Proc IEEE Intl. Conf. on Robotic and Automation, page , Albuquerque, New Mexico. Koyano, K. and Sato, T. (1996). Micro object handling ytem with concentrated viual field and new handling kill. In Proc IEEE Intl. Conf. on Robotic and Automation, page , Minneapoli, MN. Lambert, T. (1994). Empty-Shape Triangulation Algorithm. PhD thei, Department of Computer Science, Univerity of Manitoba, Winnipeg, Manitoba. Marigo, A. and Bicchi, A. (2000). Rolling bodie with regular urface: Controllability theory and application. IEEE Tran. on Automatic Control, 45(9): Marigo, A., Chitour, Y., and Bicchi, A. (1997). Manipulation of polyhedral part by rolling. In Proc IEEE Intl. Conf. on Robotic and Automation, page Maon, M. T. (1982). Manipulator Graping and Puhing Operation. PhD thei, AI-TR-690, Artificial Intelligence Laboratory, MIT. Maon, M. T. (1985). The mechanic of manipulation. In Proc IEEE Intl. Conf. on Robotic and Automation, page , St. Loui. Miyazaki, H. and Sato, T. (1996). Fabrication of 3D quantum optical device by pick-and-place forming. In IEEE 9th Int l Workhop on MEMS, page , New York, NY. Natarajan, B. K. (1989). Some paradigm for the automated deign of part feeder. Intl. J. of Robotic Reearch, 8(6): Pehkin, M. A. and Sanderon, A. C. (1988). Planning robotic manipulation trategie for liding object. IEEE J. of Robotic and Automation, 4(5). Rao, A., Kriegman, D., and Goldberg, K. (1995). Complete algorithm for reorienting polyhedral part uing a pivoting gripper. In Proc IEEE Intl. Conf. on Robotic and Automation. Reznik, D., Mohkovich, E., and Canny, J. (1999). Building a univeral part manipulator. In Böhringer, K. and Choet, H., editor, Ditributed Manipulation. Kluwer. Saito, S., Miyazaki, H., and Sato, T. (1999). Pick and place operation of a micro object with high reliability and preciion baed on micro phyic under SEM. In Proc IEEE Intl. Conf. on Robotic and Automation, page , Detroit, MI. Shimada, E., Thompon, J., Yan, J., Wood, R., and Fearing, R. (2000). Prototyping millirobot uing dextrou microaembly and folding. In Sympoium on Microrobotic ASME Int. Mechanical Engineering Cong. and Exp., Orlando, FL. Wiegley, J., Goldberg, K., Pehkin, M., and Brokowki, M. (1996). A complete algorithm for deigning paive fence to orient part. In Proc IEEE Intl. Conf. on Robotic and Automation. Zech, W. and Fearing, R. S. (1998). Alignment of micropart uing force controlled puhing. In SPIE Conference on Microrobotic and Micromanipulation, Boton, MA. Zumel, N. B. (1997). A Nonprehenile Method for Reliable Part Orienting. PhD thei, Robotic Intitute, Carnegie Mellon Univerity, Pittburgh, PA.