Grasp Planning: How to Choose a Suitable Task Wrench Space

Transcription

1 Grasp Planning: How to Choose a Suitable Task Wrenh Spae Ch. Borst, M. Fisher and G. Hirzinger German Aerospae Center - DLR Institute for Robotis and Mehatronis 8223 Wessling, Germany [Christoph.Borst, Max.Fisher]@dlr.de Abstrat For the evaluation of grasp quality, different measures have been proposed that are based on wrenh spaes. Almost all of them have drawbaks that derive from the nonuniformity of the wrenh spae, omposed of fore and torque dimensions. Moreover, many of these approahes are omputationally expensive. In this paper, we address the problem of hoosing a proper task wrenh spae to overome the problems of the non-uniform wrenh spae and show how to integrate it in a well-known, high preision and extremely fast omputable grasp quality measure. I. INTRODUCTION With the development of flexible and highly integrated dexterous gripping devies (e.g. the DLR Hands I and II, the researh on grasp and manipulation analysis and planning an be applied to the real world. So the need for effiient and implementable methods to perform these analysis and planning tasks inreases. Fig. 1. The DLR Hand II exeuting a speial grasping task One main topi in this field is the stati analysis of fores and torques that an be applied to an objet through grasp ontats. The evaluation of the effiieny of a grasp to ounterat disturbanes in order to keep the objet firmly fixed in the gripper is the next important researh topi. A basi quality riterion for a grasp is the fore losure property, first proposed for grasping appliations by Lakshminarayana [5]. Effiient tests for this property have been developed using different models, like the grasp matrix or the grasp wrenh spaes [1], [3], [8]. Also, many approahes deal with the onstrution of a fore losure grasp. The fore losure property, however, is only a minimal quality requirement for a grasp. It is more relevant, how effiiently a grasp an ompensate for arbitrary disturbanes or balane a speial set of disturbanes that is expeted when exeuting a desired task. To quantify this effiieny of a grasp, the onept of wrenh spaes an be used. The set of all wrenhes that an be applied to the objet through the grasp ontats is alled the Grasp Wrenh Spae (. A ommonly used and effiient way to approximate the is to alulate the onvex hull over the disretized frition ones [9], [1], [11]. The problem with all these approahes is the disretization of the frition ones, where signifiant errors may be introdued when approximating the one with only a few vetors to ahieve fast omputation (e.g. 4 vetors lead to an error of 3 %, 8 vetors still 8 % [2]. Moreover, Teihmann and Mishra [13] showed that there are problems to be expeted with this method for large frition oeffiients. This approximation orresponds to the idea that the sum of all applied fores is onstrained to one, whih has only a weak physial interpretation for multifingered grippers. Ferrari and Canny [3] proposed a method for alulating a physially well interpretable approximation, where the fores applied in the ontat sum up to the number of ontats, by alulating the onvex hull over the Minkowski sum of the frition ones. The drawbak with diretly alulating this approximation is that it is omputationally expensive. To rate the quality of a grasp, task direted and task independent measures were introdued. Kirkpatrik et al. [4] use the largest wrenh sphere that just fits within the as a task independent quality measure of the grasp. The measure is not sale invariant and depends on the seletion of the torque origin (r in fig. 5. To ahieve invariane to the seletion of the referene point, Li and Sastry [6] propose to use the volume of an ellipsoid generated by the grasp matrix as a measure of grasp quality. They also suggest to model the task wrenh spae as a six dimensional ellipsoid and fit it in the. The problem with this approah is how to model the task ellipsoid for a given task, whih they state to be quite ompliated. Pollard [11] introdues the Objet Wrenh Spae (OWS whih inorporates the objet geometry into the grasp evaluation. With her approah, however, this OWS represents the best grasp that an be ahieved for an objet and gives no diret measure for an arbitrary grasp. To rate the quality of a grasp, the largest insribed sphere is used. To ahieve sale invariane, the torque omponent of the wrenhes is saled

2 with the length of the longest objet axis. Strandberg [12] proposes to evaluate grasps using disturbane fores in order to overome the problem of torque origin seletion and to take the geometry of the objet into aount. The quality evaluation method is very reliable, however, the omplexity of this approah is very high as the geometri information has to be evaluated for eah grasp andidate. In the next setion, we want to illustrate shortly some drawbaks of the so far proposed grasp measures. Then we give a desription of exat Grasp and Task Wrenh Spaes that are physially motivated and propose a very intuitive grasp measure using these wrenh spae definitions. The final part of the paper deals with a very effiient method for alulating this new grasp measure. II. A COMPARISON OF PROPOSED QUALITY MEASURES AND THEIR INVARIANCES The different methods mentioned above for rating the quality of a grasp in a task independent manner have different properties as to saling the grasped objet, hanging the referene point, and saling torque axes. Here we shortly present the invarianes of the different methods with some simple examples to illustrate in whih ase the quality measures are intuitive and physially orret and where they may fail. We ompare the largest insribed sphere method and the same method with uniform saled torques (as suggested by Pollard with the volume of the. The first example ompares the same geometri grasp onfiguration applied at different loations on the objet and the hange of referene point (fig. 2. In the seond example, the grasp and objet are saled by a fator two (fig. 3. In the last example, we apply the same grasp onfiguration to different objets (fig. 3. A B Fig. 2. The same grasp onfiguration applied at different loations on an objet (ase A and C and the hange of referene point with the same grasp on the same objet (ase A and B D E F G C method fig. 2 fig. 3 fig. 3 InsrBall A B D E F = G A C InsrBall A B D = E F G (torque saled A C volume of A = B D < E F = G A = C Human A = B D = E F > G Intuition A > C Fig. 4. Comparing the different grasp measures with human intuition in some simple ases reason why grasps on larger objet should be better than those on smaller. For the last ase, F is superior to G as the torque disturbanes that are likely to our are muh larger in G than in F, while the grasp abilities are the same. III. A PHYSICALLY MOTIVATED MODEL TO SPECIFY GRASPS AND GRASPING TASKS There are three main questions regarding the stati part of grasping: What are the fores/torques that an be applied to the objet by the grasp? Whih disturbanes are expeted to at on the objet? The third question is about the quality of the hosen grasp. A good quality measure for a grasp is a salar that desribes how well the grasp an resist the expeted disturbanes. For all three questions above various models and measures have been developed. From a physial or mehanial point of view, however, all an be modeled similarly and in a simple manner. There are only fores and torques ating on the objet, either as a disturbane anywhere on the objet or in the grasp ontats to ounterat the disturbanes. Both, the set of disturbane fores/torques and the set of possible grasp fores/torques, are usually represented in a vetor spae [3], [1], [11]. A. The Grasp Wrenh Spae Let us assume that the grasp onsists of k point ontats with frition. So in eah ontat a fore within the frition one an be applied to the objet (Fig. 5. Fig. 3. The same grasp and objet topology saled by a fator of two (D and E and the same grasp onfiguation applied to different objets (F and G. How we expet the different quality measures to ompare in the different grasp situations is listed in the table of fig. 4. For the results on human intuition in fig. 4, some remarks to explain the bakground of our deisions may be useful. For the first example in fig. 2, we would qualify situation A and B as equal grasps but C as inferior sine we move away from the enter of mass. The saled grasp and objet (fig. 3 situations lead to an equal qualifiation as there is no obvious Fig. 5. A single ontat point in 3D illustrating the frition one and a sample k-ontat grasp (k=3 on a planar objet. The length of the applied fore vetor is normalized to a unit fore as we assume that eah finger an apply the same magnitude of fore and only one ontat arises for eah finger (preision grasp. The diretion of the fore f i that an be applied at ontat point i is onstrained by the frition one speified by the

3 frition oeffiient µ, the ontat point i, and the ontat normal n i. The onstraint an be written as: f i (f i n i n i µ(f i n i. (1 Any fore ating at a ontat point on the objet also reates a torque relative to a referene point r that an be arbitrarily hosen. Often the enter of mass is used as that referene point to give it a physial meaning. The torque τ i orresponding to f i is then: τ i = ( i r f i For onveniene, these fore and torque vetors an be onatenated to a wrenh : ( f w i = i ( i r f i Next we speify the set of wrenhes that an be reated by frition one unit fores ating in one ontat. We all this set the Cone Wrenh Spae (CWS. It is used to larify the onstrution of the Grasp Wrenh Spae. CWS i = ( w i w i = f i ( i r f i f i (f i n i n i µ(f i n i f i 1} (2 The Grasp Wrenh Spae ( should ontain all wrenhes that a given grasp an ounterbalane by applying fores in its k ontats. This spae an be omposed from all k one wrenh spaes in the following manner: } k = w w = w i w i CWS i (3 i=1 Note that equation 3 is an exat desription of the. It orresponds to the idea that eah finger of the manipulator is apable to exert a unit magnitude of fore to the objet. The only drawbak of this kind of definition is that it is only desriptive but not onstrutive. To find the linear ombination of finger fores to ounterbalane a disturbane wrenh is diffiult. This problem is addressed in setion IV. B. The Task Wrenh Spae The wrenhes that are expeted to our for a given task an be speified as a so-alled Task Wrenh Spae (TWS. For the TWS two ases an be distinguished, either the task to be exeuted is known and a speifiation in the wrenh spae is given or the task is unknown and no speifiation exists. 1 Given Task Speifiation: If there is a detailed desription of the task given by a set of wrenhes that are applied to the objet during the manipulation one an use the onvex hull over these task disturbane wrenhes as a Task Wrenh Spae. Li and Sastry [6] propose to approximate the task wrenh spae by a task ellipsoid but they state that the data aquisition is diffiult. 2 Unknown Task Speifiation: If one knows nothing about a grasping task, one at least an assume that a grasp should hold an objet (1 against gravity, (2 against fores and torques arising from aelerating the objet (whih has the same effet for translational aelerations and (3 against fores that result from ontats of the objet with the environment. A ommonly used approah to model an unknown task wrenh spae is to use a unit sphere in the wrenh spae. With this approah it is assumed that the probability for every wrenh diretion to our as a disturbane is equal. However, this has no physial or mehanial interpretation. Torques are typially aused by fores ating on the boundary of the objet and therefore a general task wrenh spae is not uniform for most objets. A more natural way to desribe an unknown TWS that takes the objet geometry into aount is the Objet Wrenh Spae (OWS as introdued by Pollard [11]. In this work, we want to ombine the idea of the task ellipsoid [6] with the onept of the objet wrenh spae. This enables us to automatially obtain a task independent, yet physially motivated desription of the wrenh spae that takes all possible disturbanes into aount and thus is a kind of generalization over all task dependent wrenh spaes. C. The Objet Wrenh Spae The OWS should ontain any wrenh that an be reated by a distribution of n disturbane fores ating anywhere on the surfae of the objet. As we are interested in the effet of a normalized disturbane on the objet, the sum of the length of all n fores should be 1. By ontrast, the number of fores that at on the objet is unlimited (so n 1.. }; see fig. 6 for illustration. The OWS an again be omposed of the union of one wrenh spaes, in the following way: OWS = n w w = α i w i i=1 } n α i = 1 w i CWS i n 1.. } (4 i=1 This desription represents the resulting wrenhes of any possible disturbane of a ertain magnitude that at on the surfae. To add gravity, whih ats not on the surfae but in the enter of mass, one an merge this OWS with the wrenh spae that is produed by fores of any diretion ating in the enter of mass. If the referene point r is equal to the enter of mass, then one merges a sphere in the fore domain to the OWS, saled with the mass of the objet. For the general ase, the mass wrenh spae (MWS generated by gravity g ating in any diretion in the enter of mass (m (dependent on the objet rotation an be written as: ( MWS = w w = f (m r f } f m g Suh an OWS MWS desribes the general ase where nothing about the task is known very naturally, as any possible disturbane and also gravity is represented. The drawbak is again that there is no onstrutive desription to really alulate the set. (5

4 Fig. 6. Illustration of different fore distributions that produe the wrenh set of the OWS. Eah distribution ontributes one single wrenh to the OWS set. The length of all fore vetors sum to the unit length. D. The Physially Motivated Grasp Quality Measure QM BF As stated in the introdution, many different quality metris for grasps have been introdued but almost all of them have drawbaks that arise from the different units or saling in the fore/torque dimensions in the grasp wrenh spae or are even dependent on the seletion of the referene point. With the OWS defined above, we an propose a quality metri that overomes all these drawbaks and rates grasps in a physially interpretable and intuitive way. Of ourse, the non-uniformity of the wrenh spae remains. However, our onept generates a physially interpretable saling between fores and torques automatially. We take the ability of a grasp to ounterat the possible disturbanes on an objet as a measure of the grasp quality. The Grasp Wrenh Spae ( of a given grasp C k represents the apabilities of the grasp, while the Objet Wrenh Spae (OWS of a given objet o defines whih disturbanes may our. So the largest fator by whih we an sale the OWS to just fit in the gives us a measure of the grasp quality. Formally expressed, we get QMS BF (C k, o = k; k Max x OWSo : k x Ck }(6 With this measure, we are independent of the seletion of the referene point as we use it for the reation of both wrenh spaes. Moving the referene point an be expressed by a linear transformation that is norm onserving and so it takes no effet on the saling fator. It should be noted that the above desriptions are an exat view on the stati grasp situation whih is the base for the grasp planning problem that we want to address with our grasp quality measure. The question of how to ontrol the fores exerted by the manipulator to resist ertain disturbanes in a real dynami grasp situation is not addressed. See [7] for a review. Also, no other phenomena that may our when the grasped objet dynamially touhes the enviroment are onsidered. The problem with our desribed measure is to find a way to effiiently alulate the saling fator of the task wrenh spae. This problem is onsidered in the following setions. IV. A MODIFIED GRASP QUALIFICATION PROCEDURE The next problem to be solved is to integrate the OWS onept in a omputationally effiient manner into a grasp qualifiation proedure: Our grasp measure alulation uses the more omplex but physially more relevant Minkowskisum based grasp quality measure [2]. For ases where no task wrenh spae is given, we use the radius of the largest insribed sphere. Our grasp quality measure alulation is of extremely high performane. This stems from the fat that we alulate the not ompletely but iteratively only at its relevant, weakest regions, i.e. where the insribed sphere touhes it. For obtaining the weakest wrenh diretion during the iterative omputation, we projet it into the grasp ontat frition ones. This way we easily get the linear ombination of CWS wrenhes that sum up to the largest possible wrenh in this diretion. (BTW this also avoids the need for a frition one disretisation, whih is a major error ause, as mentioned above. As motivated in setion III-B.2 we now want to ompare the OWS (and no longer the largest insribed sphere with the of the grasp that is atually evaluated. That means we searh the largest saling fator for a given OWS to fit it into a. In order to keep this algorithm of the same omplexity, we annot use the sampled OWS diretly. Instead, we irumsribe the OWS with an ellipsoid and use the orresponding inverse linear mapping for the. Thus we redue the problem to the above mentioned sphere fitting problem with an additional linear mappping per vetor (see fig. 7. TWS onv(tws TWS TWS Fig. 7. Approximating the OWS with an ellipsoid: 1. The sampled OWS (exat spae, exat quality measure 2. Convex Hull over the sampled OWS (approx. spae, exat quality measure 3. Enlosing ellipsoid (approx. spae, approx. quality measure 4. Linear transformation of ellipsoid and (sphere algorithm appliable A. Calulating an OWS approximation ellipsoid We are looking now for the smallest ellipsoid (spanned by the quadrati form x T Qx 1, Q symmetri and positive definite that enloses the QWS, more formally, we look for Q that fulfills x OWS : x T Qx 1 Q Q : V (Q < V (Q V (Q = 1/ det(q being the volume of the ellipsoid spanned by Q. In the following, we outline a four step proedure to effiiently alulate a small (not neessarily the smallest OWS approximation ellipsoid: (1 Sample a disrete set of OWS wrenhes, (2 find a speial, analytially desribable hull HU LL(OWS for these, (3 irumsribe an ellipsoid to HU LL(OWS and (4 integrate this OWS representation in our grasp qualifiation algorithm. 1 OWS sampling: Given a polyhedral objet model, we get a set of wrenhes (OW S that sample and approximate the orresponding OWS by alulating the wrenhes generated from frition one approximation fores at the orners of all polygons of the objet. Of ourse, one might use this set

5 diretly to ompute the smallest enlosing ellipsoid. This, however, is omputationally expensive and doesn t allow for keeping the spherial struture of the OWS s fore dimensions. 2 An OWS hull: Let s start with a short disussion of the OWS struture: First, by definition, the fores generating the OWS have all unit length and an have any diretion (on most objets, thus the OWS projetion to the fore dimensions an be tightly enlosed by a unit sphere. To add the effets of gravity for the (most relevant ase that we use the enter of mass as our referene point, we add a saling fator = Max(f ontat, f gravity and get Hull(OWS fores = f 1 } f 2 = 1 Next, the form and size of the OWS projeted to its torque dimensions OW S torques is determined by the objet geometry. From examples with different test objets we an see that this projetion an be approximated by a 3 dimensional ellipsoid without introduing a large error (see fig. 8. For people like us who like some rough visual or lower dimensional interpretation of the things they deal with: In 3D this orresponds to a ylinder, but instead of being bounded by its height and a irle, this 6D objet is bounded by a sphere and a ellipsoid in its fore and torque dimensions, respetively. 3 The OWS approximation ellipsoid: The next thing to be done now is to find a small enlosing 6D ellipsoid for this ylinder : We hoose a oordinate system for the following in whih the torque ellipsoid is entered at the origin and its main axes are aligned with the oordinate system axes of the torque dimensions. Let s all this transformation M. In this oordinate system M the shape of W is W = MW = t 1 t 2 t 3 As there is no need for a transformation in the fore dimensions (a sphere, we need to look for a quadrati form k 1 k 2 x T k Qx 1, Q = 3 k 4 k 5 k 6 To determine the k 1..k 6 we again ome from the 3D intuition: If we want to alulate the smallest enlosing ellipsoid for a ylinder, we know that this ellipsoid needs to touh the ylinder in the ylinder s bounding irle, whih means, in partiular, that the projetion to the ylinders irle dimensions also looks like a irle. In 6D this requires that the ellipsoid should touh the bounding fore sphere, and projeted to the fore dimension it looks like a sphere: Fig. 8. The struture of the OWS projetions in fore and torque spae for two sample objets a ube and a hampagne glass. In fore spae (middle one an see that not every diretion an be generated by a single disturbane fore, due to the limited surfae normal diretions on a ube. For the glass the fore spae is almost a perfet sphere. The torque spae for the ube is symmetri in all oordinate axes and the enlosing ellipsoid would be a sphere in this speial ase. For the glass the torque spae is flat for torques round the symmetry axis of the glass. Assuming that we already have given the ellipsoid whih enloses OW S torques as a quadrati form with the symmetri matrix W (see set. IV-A.5, we get Hull(OWS torques = t t T Wt = 1 } with t = t t origin where t origin is the enter of the torque enlosing ellipsoid (again see set. IV-A.5. Finally, we an ombine all this information (independene of the fore and torque dimensions an be expressed as maximum norm with w = (f T, t T T Hull(OW S = w 1 } f 2, t T Wt = 1 k 1 = k 3 (7 k 2 = k 3 (8 The property of touhing the unit fore sphere means in 2D that the edge points of a retangle touh its smallest enlosing ellipse. Its 6D analog is. x T Qx = 1 t 4 x, expressed by the following equations t 5, t 6 2 k 3 + t 2 4k 4 = 1 (9 2 k 3 + t 2 5k 5 = 1 (1 2 k 3 + t 2 5k 5 = 1 (11 The equations 7 through 11 determine a family of ellipsoid

6 generating quadrati forms k 3 k 3 k 3 Q k3 = 1 2 k 3 t k 3 t k 3 t 2 6 To ompute k 3 orresponding to the smallest ellipsoid, we minimize the trae (a measure for the volume of an axis aligned ellipsoid of the matrix S that transforms a unit sphere to the ellipsoid orresponding to Q = (SS 1. As 1 tr(s =, we need to minimize tr(g 1 tr(s = k k k k 3 t 2 4 t 2 5 t 2 6 and get the (not surprising solution k 3 = 2 /2, whih results in the ellipsoid generating matrix Q = 2 2t t t 2 6. In fig. 1 through 13 we plotted several 2D projetions of OW S for a hampagne glass (fig. 9, Hull(OW S (the inner irle and the retangle, resp. and the final ellipsoid, resulting from Q (Note that the plot axes are saled so that the outer ellipsoid is mapped to a irle. Although the projetions look rather onservative, it should be noted that OW S touhes the ellipsoid and that the shape of OW S is approximated very well (espeially in the fore dimensions, Q adds no distortion at all!. 4 Integration in Our Grasp Measure Algorithm: The inorporation of Q in the previously mentioned grasp quality measure is straightforward: For the inremental alulation we map eah vetor with the inverse mapping whih maps a 6D unit sphere to our ellipsoid: Note that we determined Q in a rotated and translated oordinate system M (see above. M is derived from the mean µ T and the ovariane matrix Σ T of the torques of OW S (see se. IV-A.5. The translational part of M is µ T, the rotational part is the smallest rotation R of Σ T s main axes to the oordinate axes (note that there is no rotation and translation in the fore dimensions, only in the torque dimensions. Now we just need to map eah vetor w to ( w = R (w 1 µ T and an use our high performane, high preision, physially well motivated grasp quality measure not only for a spherial task wrenh spae but also for a physially motivated task wrenh spae with a negletible loss of performane! 5 Enlosing Torque Ellipsoid: The only thing that remains to be done now is to determine the smallest enlosing ellipsoid for the OWS projeted into torque spae. Although there may be better algorithms, we use first and seond order statistis here beause of high omputational performane and suffiiently good results: Given an OWS as a set of sampled wrenhes (OW S, see se. IV-A.1, we projet the wrenhes to a set of torques in torque spae T = OW S torque and alulate the mean µ T and the ovariane matrix Σ T. Σ 1 is already very similar to a quadrati form of an T enlosing ellipsoid, we just need to add a saling fator ( σ = max t T Σ 1 T t t T µ T and get Q T = (σ 2 Σ T 1 V. CONCLUSION In this paper we give a well defined and pysially motivated desription of a general task wrenh spae based on an OWS for ases where no exat task speifiation is known. From this TWS we derive a quality measure QM BF and show how it an be implemented very effiiently. The main improvement this measure adds to our grasp planner is for grasping long, thin objets, e.g. a srewdriver: While using a sphere as a general task wrenh spae speifiation as we did before results in good grasps without aounting for different torques in different objet dimensions, using our TWS approximation takes are of those now. It has also to be mentioned that the omputation time of the new measure is only slightly above the time for the old one. That is due to the preproessing of the grasp objet to alulate the ellipsoid approximation of the OWS (mainly the alulation of the mean and the ovariane matrix, whih takes some 1 ms on a 2 GHz Pentium IV and is done only one per objet. The online mapping of the inreases the omputational omplexity for alulating the grasp quality by 1 multipliations per iteration step (number of ontats 6 2, whih still leads to a small total omputation time of 1 2 ms for the grasp quality. REFERENCES [1] Antonio Bihi. On the losure properties of roboti grasping. The Int. Journal of Robotis Researh, 14(4, [2] Ch. Borst, M. Fisher, and G. Hirzinger. A Fast and Robust Grasp Planner for Arbitrary 3D Objets. In Proeedings of the IEEE Int. Conferene on Robotis and Automation, pages , Detroit, Mihigan, May [3] C. Ferrari and J. Canny. Planning Optimal Grasps. In Proeedings of the IEEE Int. Conferene on Robotis and Automation, pages , Nie, Frane, May [4] David Kirkpatrik, Bhubaneswar Mishra, and Chee-Keng Yap. Quantitative Steinitz s Theorems with Appliations to Multifingered Grasping. In Pro. of the 2th ACM Symp. on Theory of Computing, pages ACM, 199. [5] K. Lakshminarayana. Mehanis of form losure. Tehnial report, ASME Tehnial Report 78-DET-32, [6] Zexiang Li and Shankar S. Sastry. Task-Oriented Optimal Grasping by Multifingered Robot Hands. IEEE Journal of Robotis and Automation, 4(1:32 44, February 1988.

7 [7] G. F. Liu, J.J. Xu, and Z.X. Li. A Comparative Study of Geometri Algorithms for Real-time Grasping Fore Optimization. In Proeedings of the IEEE Int. Conferene on Robotis and Automation, pages , Taipei, Taiwan, September [8] Yun-Hui Liu. Qualitative Test and Fore Optimization of 3-D Fritional Form Closure Grasps Using Linear Programming. IEEE Transations on Robotis and Automation, 15(1: , February [9] Andrew T. Miller, Steffen Knoop, Henrik I. Christensen, and Peter K. Allen. Automati Grasp Planning Using Shape Primitives. In Proeedings of the IEEE Int. Conferene on Robotis and Automation, pages IEEE, September 23. [1] B. Mishra, J.T. Shwartz, and M. Sharir. On the existene and synthesis of multifinger positive grips. Algorithmia, Speial Issue: Robotis, 2(4: , November [11] Nany S. Pollard. Parallel methods for synthesizing whole-hand grasps from generalized prototypes. Tehnial Report AI-TR 1464, MIT Artifiial Intelligene Laboratory, [12] Morton Strandberg. A grasp evaluation proedure based on disturbane fores. In Proeedings of the IEEE/RSJ Int. Conferene on Intelligent Robots and Systems, pages , EPFL, Lausanne, Switzerland, IEEE. [13] Marek Teihmann and Bud Mishra. The Power of Frition: Quantifying the Goodness of Fritional Grasps. In A. K. Peters, editor, Algorithms for Roboti Motion and Manipulation, pages A. K. Peters, Wellesley, MA, USA, Fig. 11. Projetion to fore dimensions 2 and torque dimension 4. The box is the projetion of the Hull(OW S, the outer irle is the projetion of the enlosing ellipsoid, resulting from Q. (Note that the plot axes are saled so that the outer ellipsoid is mapped to a irle Fig. 9. The objet and its sampled OWS in fore and torque spae for whih the OWS projetions were alulated. Fig. 12. Projetion to fore dimensions 3 and torque dimension 4 (for explanation see fig. 11 Fig. 1. Projetion to fore dimensions 1 and 2. The inner irle is the projetion of the Hull(OW S, the outer irle is the projetion of the enlosing ellipsoid, resulting from Q. Fig. 13. Projetion to torque dimensions 4 and 6 (for explanation see fig. 1