IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018 1

Robotic Edge-Rolling Manipulation: A GraspPlanning Approach

Andrew Specian1, Caio Mucchiani1, Mark Yim1, and Jungwon Seo2

Abstract—This paper presents a novel robotic manipulationtechnique that we call robotic edge-rolling. It refers to trans-porting a cylindrical object by rolling on its circular edge, ashuman workers might maneuver a gas cylinder on the ground.Our robotic edge-rolling is achieved by controlling the objectto both roll on its bottom edge in contact with the ground andslide on the surface of the robot’s end-effector. It can thus beregarded as a form of robotic dexterous, in-hand manipulationwith nonprehensile grasps. We address the problem of graspplanning for edge-rolling by studying how to design appropriatelyshaped end-effectors with zero internal mobility and how to findfeasible grasps for stably rolling the object quasidynamicallywith our simple end-effectors. An extensive set of experimentsis performed with a conventional manipulator arm on not onlyflat surfaces but also a U-shaped half-pipe track. Long-rangeedge-rolling is demonstrated with a modular mobile manipulatorwhich is capable of active steering control.

Index Terms—Grasping; Grippers and Other End-Effectors;Dexterous Manipulation

I. INTRODUCTION

WE introduce robotic edge-rolling, a novel robotic ma-nipulation technique for moving cylindrical objects by

rolling them on the circular edge of the bottom face. Thismethod, with possible applications to object transportation,part reorientation, and the like, is an alternative to traditionalgrasp-lift-and-carry manipulation, which might not be possiblewhen it comes to handling large, heavy objects beyond thecarrying capacity of the robot. Fig. 1 depicts our robots edge-rolling the cylindrical object, along with human edge-rolling,as commonly performed with the gas tank.

As successfully demonstrated in [1], [2], cylindrical objectscan be transported by rolling on their curved face. However,the potentially large footprints of the configuration can make ithard to navigate through a cluttered, unstructured environment.

Manuscript received: February 24, 2018; Revised April, 15, 2018; AcceptedJune, 10, 2018.

This paper was recommended for publication by Editor Han Ding uponevaluation of the Associate Editor and Reviewers’ comments.

1Andrew Specian, Caio Mucchiani, and Mark Yim are with the De-partment of Mechanical Engineering and Applied Mechanics, Universityof Pennsylvania, Philadelphia, PA, USA. aspecian@seas.upenn.edu,caio@seas.upenn.edu, yim@seas.upenn.edu

2Jungwon Seo is with the Departments of Mechanical and Aerospace/-Electronic and Computer Engineering, The Hong Kong University of Scienceand Technology, Clear Water Bay, Hong Kong junseo@ust.hk

This letter has supplemental downloadable multimedia material availableat http://ieeexplore.ieee.org, provided by the authors. This includes onemultimedia MP4 format movie clip, showing detailed edge-rolling maneuversand successful results for a conventional manipulator and mobile base. Thismaterial is 7 MB in size. References to video are noted within paper as ”seethe video attachment”.

Digital Object Identifier (DOI): see top of this page.

(a)

object↓

(b)

object↓

(c)

Fig. 1. (a) Robotic edge-rolling with the mobile robot (the object rolls onthe ground). (b) Robotic edge-rolling with the conventional manipulator (theobject rolls on the U-shaped vertical half-pipe). (c) Human edge-rolling.

It may take a great deal of effort commensurate with the inertiaof the object to steer the heading of the object rolling on itsface. In contrast, edge-rolling requires very small footprints inthe sense that the object stands on a point contact in principle.

This paper presents grasp planning for robotic edge-rollingmanipulation with implementation on real robotic systems: anindustrial robot arm and a modular mobile robot (Fig. 1(a-b)).Our technique is realized with very simple end-effectors withno internal mobility which can be fabricated as a single rigidbody (Fig. 1(a-b)). It can be considered as a form of roboticin-hand manipulation in a nonprehensile manner since theobject is controlled to slide on the surface of the zero-mobilityend-effector while rolling on the ground surface and is, thus,closely related to the important issue of robotic dexterity.Utilizing our previous work [3], we also show that it is possibleto acquire the resulting target grasps for edge-rolling in anautonomous manner.

It is also possible to edge-roll objects using multi-fingered,dexterous hands, as in human edge-rolling (Fig. 1(c)). Forexample, first the object may be firmly grasped by the multi-fingered gripper, and then the object-gripper system may berotated together by means of the mobility of the rest ofthe robot. This prehensile approach has advantages includingstraightforward grasp planning. However, errors in controllingthe object or the robot may result in large internal forces,which can get the system damaged and make edge-rollingimpossible. We believe our approach using nonprehensilegrasps with simple end-effectors can be more effective fornegotiating large internal forces. Likewise, we believe that theapproach will allow errors and uncertainties in perception andcontrol to be handled in a more energy-efficient manner likemany other underactuated, passive manipulation scenarios. Inaddition, our approach can be more practical in that our end-effectors use zero actuators.

After outlining relevant literature (Sec. II) and presenting

2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018

our problem statement (Sec. III), in Sec. IV we presentour approach to effector design. In Sec. V we investigatehow to find feasible grasps for stable edge-rolling. Sec. VIdemonstrates edge-rolling with real robotic systems in somedifferent environments.

II. RELATED WORK

The challenge task, edge-rolling, studied in the paper can bemodeled as a quasidynamic manipulation process to addressour target scenario in which a mobile robotic platform (forexample, an industrial robot arm or a wheeled mobile base)manipulates the object with negligible momentum. Accordingto [4], quasidynamic manipulation is intermediate betweenstatic and dynamic manipulation as if excessive damping pre-vents accelerations from integrating into significant velocities.Examples in the literature include the well-known peg-in-hole task, tray tilting for object reorientation [5], pushingprimitives of a parallel-jaw gripper for regrasping [2], and,most recently, the planning of a stable pushing strategy forin-hand manipulation [6].

The main scope of work here is the problem of graspplanning for stable edge-rolling manipulation. Grasp planningconcerns designing end-effectors and choosing grasps on ob-jects of interest [4]. The problem has been addressed mainlyin the context of robotic grasping and fixturing, which canbe modeled as a quasistatic process. Previously, we addressedgrasp planning for whole-arm grasping [7]. Important notionsof the closure properties of robotic grasps include force-closure, form-closure, and immobilization [8]. In addition,partial closure properties (for example, partial force-/form-closure) have also been investigated [9], [10]. The dynamicstability of a grasp depends on a variety of factors such as thegeometry around contact, contact models, and material proper-ties [11], [12]. Both model-based and data-driven algorithmsfor constructing stable robotic grasps are an active researcharea [13], [14].

Beyond grasping and fixturing, robotic pushing [15] is an-other practically important quasistatic manipulation technique.Other examples in the literature include object handling withflat contact surfaces [16], knocking an object over with a singlecontact [17], pivoting operation [18], push-grasping [19], andour previous work on stable picking and placing via two rigidlyconnected contacts [3]. Although dynamic manipulation is arelatively new area, some early examples can be seen in roboticjuggling [20] and dynamic nonprehensile manipulation [21].

Also relevant to this work is robotic in-hand manipulation.It refers to the capability to reconfigure an object within arobotic hand. Various in-hand manipulation techniques havebeen realized in the forms of robotics regrasping, dexterousmanipulation, and finger gaiting; see [22]–[26] and referencestherein.

III. PROBLEM DESCRIPTION

The problem we address in the paper is concerned witha novel robotic manipulation capability that we call roboticedge-rolling (recall Fig. 1). It refers to rolling a cylinder-shaped object on the circular edge of its bottom face. We will

investigate grasp planning for edge-rolling: designing the end-effector and choosing grasps that enable stable edge-rolling.We will model edge-rolling as a quasidynamic process inthat dynamic loads are necessary to counterbalance the non-jamming, applied forces, but the process is highly overdampedso that accelerations of the object do not yield significantmomentum.

(a)

A

end-effector

B

G

(b)

A

B

G

time t = t0

t = t1 > t0

t = t2 > t1

Fig. 2. (a) The cylindrical object can be in force-closure with two pointcontacts with friction at A, B produced by the end-effector and anotherfrictional contact at G on the ground surface. (b) Rolling the object on Gwhile sliding on A, B is not straightforward. The object can easily fall.

In our approach, objects of interest are thus modeled as rightcircular cylinders. We are interested in developing a robotictechnique for edge-rolling, particularly using very simple end-effectors—in essence, single rigid body end-effectors with nointernal degrees of freedom (DOF), which can also be seenin our previous work on object caging, fixturing, picking-and-placing [3], [7]. The lack of mobility in those simple end-effectors can render robotic edge-rolling very practical yetchallenging at the same time.

When simply holding the object, it is sufficient to have anend-effector that can make two point contacts with friction[27] and acquire force-closure [4] grasps along with anotherfrictional contact on the ground surface. In [28], it is exam-ined how to obtain force-closure grasps with three frictionalcontacts. This idea can be realized using, for example, an end-effector with two parallel fingers as illustrated in Fig. 2(a).Note that the relative configuration of the fingers can befixed (no internal mobility) and that all the contact forcesare unilateral. We will consider how to adapt the effectorshapes and the grasps shown in Fig. 2(a) to successfullyrealize the desired contact mode for edge-rolling: rolling theobject on the ground surface and, in the meantime, slidingthe object on the contacts with the end-effector such thatthe relative configuration between the end-effector and itsrobot body can remain fixed. In taking advantage of suchnonprehensile grasps, our resulting edge-rolling technique canthus be considered as a form of in-hand manipulation, whichstill remains as a great challenge.

IV. END-EFFECTOR SHAPE DESIGN FOR EDGE-ROLLING

This section presents our end-effector design approach torealizing stable edge-rolling.

SPECIAN et al.: ROBOTIC EDGE-ROLLING MANIPULATION: A GRASP PLANNING APPROACH 3

A. Local and Global Geometry

When it comes to transitioning from holding (Fig. 2(a))to edge-rolling, the three frictional point contacts that workwell for holding the object are not effective in that it canbe hard to maintain force equilibrium in the desired contactmode (Fig. 2(b)). For example, if one supposes that the weightof the object is negligible, then the three contact forces canbe in static equilibrium only in the case they are necessarilycoplanar, according to screw theory [29]. This is almostimpossible here in the sense that the direction of the frictionalcontact wrenches at the end-effector cannot be controlled whilethe object is sliding on the end-effector.

(a)

A1 A2

B2 B1

G

end-effector

(b)

AB

G

Fig. 3. (a) Four point contacts A1, A2, B1, and B2 made by the polyhedralend-effector. (b) Two point contacts A and B made by the end-effector withthe curved contact surfaces. In the panels, contact A(·) is antipodal to B(·) asseen in the direction of the axis of the object.

Our approach to making the end-effector (that is, the twopoint contacts {A,B} that are antipodal as seen in the directionof the axis of the object) more suitable for stable edge-rollingis to consider how to impart non-degenerate contact wrenchcones even when the object is sliding on the end-effector.This can be achieved by fixing the local geometry aroundthe two contacts by incorporating (1) additional contacts and(2) surface curvature effects. For example, Fig. 3 is illus-trating scenarios in which two additional contacts are added(Fig. 3(a)) and the object is in contact with the concavity ofthe curved effector surfaces (Fig. 3(b)), as seen in the directionof the axis of the object.

Additional contacts can produce contact wrenches based onthe first-order effects of the additional contact normals. Forexample, in Fig. 3(a) there are now four contact normals atA1, A2, B1, and B2 that can induce contact wrenches regardlessof contact modes (in Fig. 2, only two contact normals atA and B). Due to the concavity at the contacts, the objectin Fig. 3(b) cannot move tangentially at all into the curvedsurface. Therefore, the effects of the concavity of a smoothcontact surface (in other words, the second-order geometric,surface curvature effects [30]) can also be understood usingmultiple contact wrenches in the direction of, and orthogonalto, the contact normal, regardless of contact modes. Note thatin practice, elastic deformation contact models are needed toexplain the additional tangential contact wrenches induced bythe second-order effects [30]. Therefore, the object in contactwith the curved effector may need to be dislocated from thenominal contact position to induce sufficiently large contactwrenches.

By virtue of these extra design features, the contactwrenches incorporating the additional contact normals or thecurvature effects can span (1) all the forces perpendicular tothe object’s axis (force-direction closure [31] except for thedirection of the object’s axis) and (2) all the pure torques(torque-closure [31]) along with the frictional contact at G,with neither friction effects nor dependence on contact modesbetween the end-effector and the object. We shall explain howto address the incomplete force-direction closure using activecontrol in Sec. VI-B.

In terms of global geometry, we intend to fabricate theend-effector as a single rigid body modeled as a closed,two-dimensional manifold with a genus such that objects indifferent sizes can be addressed in the hole. Our end-effector,having no DOF, will then be operated in a passive manner,interfacing the wrist of a manipulator and the object. Thisis a conservative approach, compared with the potential roleof dexterous in-hand manipulation by the human operator(Fig. 1), but it has the potential to be more practical. Figs. 7and 10 show our two- and four-contact end-effectors realizingthe design approach here. They all can accommodate objectsin various sizes.

B. Modeling for Grasp Planning

For planning purposes, our end-effector, modeled as a singlerigid body with no internal DOF, will further be abstractedagain as two rigidly connected point contacts that are antipodalas seen in the direction of the axis of the object. Whethersliding occurs or not, each of the two contacts is able toexert a planar cone of wrenches by the effects of the designfeatures (the additional contact normals and the curvatureeffects) discussed in Sec. IV-A. See Fig. 4(a).

Our four-contact end-effector (Fig. 3(a)) actually makes thetwo contact pairs (A1,A2) and (B1,B2), and there is a questionof whether this is equivalent to such an antipodal contact pair.The resultants are assumed to act at A (B) on arc A1A2 (B1B2);for example, A can be A1, A2, or the midway point betweenA1 and A2 on the perimeter. The approximation improves asthe distance between A1 and A2 (B1 and B2) gets smaller.For an object that is a rigid line segment, any modelingerrors will vanish. Therefore, for relatively slender objects (forwhich edge-rolling can be an effective way of handling), themodeling errors can be insignificant. The planar wrench coneat A (B) is spanned by the unit wrenches parallel to the contactnormals at A1 and A2 (B1 and B2). The aperture of the wrenchcone, denoted ψA (ψB) in Fig. 4(a), is thus equal to the anglebetween the contact normals at A1 and A2 (B1 and B2).

Our ellipse-shaped, two-contact end-effector (Fig. 3(b)) canindeed make such an antipodal contact pair. The geometry ofthe planar wrench cone can be determined by linearizing theshape of the effector as a polygon and following the sameprocedure as in the previous paragraph. To be more rigorous,elastic deformation has to be taken into account as discussed inSec. IV-A, but we assume that it is negligible. More thoroughdynamics analysis is left as future work.

4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018

(a)

A

ψA

B

ψB

G

(b)

φ

h

dA

B

G

B

AG

θ

Fig. 4. (a) Our edge-rolling scenario with two planar wrench cones at contactsA and B that are antipodal as seen in the direction of the object axis. Theplane of the wrench cone at A (B) is perpendicular to the object’s axis. (b)Grasp parameters h, θ , and φ (d remains fixed) used for both of our two- andfour-contact end-effectors.

V. GRASP SYNTHESIS FOR STABLE EDGE-ROLLING

According to our model presented in Sec. IV-B, a grasp isdefined as a triple (h,θ ,φ) in which h is the distance fromthe bottom face to the end-effector, θ is the angle betweenthe ground surface and the axis of the object, and φ is theangular position of the contacts on the circular base of theobject (Fig. 4(b)). We here discuss how to find grasps that aresecure and capable of edge-rolling—grasps that are able torealize the desired contact mode (sliding between the objectand the effector, and rolling between the object and the groundsurface) without losing the object.

A. Grasps That Are Secure

In order to find grasps that can securely hold the objectduring edge-rolling, we search for grasps that can indeedfulfill the promise of the design features: grasps in force-direction closure, except for the direction of the object’s axis,and torque-closure with the three contacts at A, B, and G(Fig. 4(a)). The contact wrenches at G form a friction cone.The contact wrenches at A and B form planar wrench conesinduced by the design features as explained in Sec. IV-B; theydo not depend on friction such that the closure properties canbe independent of friction and contact modes at A and B.The closure properties can be formulated as an optimizationproblem (linear programming) as in [32].

To visualize the set of the secure grasps, we performednumerical simulations. Data points were randomly sampledin hθφ -space and the closure properties were checked bysolving the linear programming problem for each data point.Fig. 5 shows the point cloud of the secure grasps on acylindrical object. It can be seen that the absolute value ofφ is upper bounded. As the grasp parameter h increases, theset of allowable φ and θ expands. It can also be seen thatthe larger the friction at G, the higher the upper bound on theabsolute value of φ .

B. Grasps That Are Capable of Edge-Rolling

Now we examine how to find grasps capable of edge-rolling, in the set of the secure grasps. The resulting grasps

(a)

(b)

(c)

h (m)

h (m)

θ(◦

)φ

(◦)

h (m)θ ( ◦)

φ(◦

)

Fig. 5. (a) The set of the secure grasps represented as a point cloud, computedfor a cylindrical object whose radius is 9.5mm. d (recall Fig. 4) was set tobe 0.14m and µG = 0.7 (the coefficients of friction at G). Contacts A and Bare assumed to exert wrench cones whose aperture is 0.2rad. (b) The pointcloud projected onto hθ -plane. (c) The point cloud projected onto hφ -plane.

will realize the desired contact mode (sliding contacts betweenthe object and the end-effector, and a rolling contact betweenthe object and the ground surface) in a quasidynamic manner.The application of a quasidynamic analysis here is justified byconsidering our edge-rolling technique can be thought of as ahighly overdamped dynamical process. It is not fully dynamicin that accelerations do not integrate into significant velocities,but quasistatic analysis might not be expressive enough in thatdynamic loads (or inertial forces) are necessary to counterbal-ance the non-jamming, applied forces. This situation is similarto the well-known peg-in-hole insertion task, an exemplaryquasidynamic process.

The closure properties of the secure grasps allow us toperform a planar analysis on the plane of the bottom face ofthe object in which only φ does matter in terms of configuringgrasps. Fig. 6 explains how to generate the accelerationthat can roll the object to the right (clockwise) in dynamicequilibrium. The figure features frictional contact forces at Aand B in order to understand how to overcome the friction andmake edge-rolling possible by changing φ . Thus, contacts Aand B are able to impart only the wrenches (wA and wB) lyingon the edge of the friction cone consistent with the directionof motion. It is a conservative approach in that the contactwrenches induced by the design features are not taken intoaccount.

The dynamic load f due to rolling without slipping in thex-direction can be obtained by finding the dual transform[4] of the acceleration center [4], G. f is parallel to theground surface and tangent to the object if we select theunit of length to be the object’s radius. According to theformalism of moment labeling [4], the “−” labeled gray setsrepresent the composite wrench cone of the dynamic load f andthe negated contact wrenches −wA and −wB. Then the setsinclude all wrenches at G that would lead to the desired contactmode (sliding at A, B and rolling at G). The diagram can beinterpreted as follows. When φ is too small (Fig. 6(a)), thereis no external wrench wext at G belonging to the compositewrench cone (that is, not intersecting the gray set, and makinga negative (−) moment about each point in the “−” labeled

SPECIAN et al.: ROBOTIC EDGE-ROLLING MANIPULATION: A GRASP PLANNING APPROACH 5

(a)

B

A Groundsurface

G

−x

y

wext

−wA

−wB

f

(b)

B

A

G

−x

y

wext

−wA

−wB f

Fig. 6. The moment labeling diagrams of the object rolling clockwise seenin the direction of its axis. The panels show the positions of the contacts A,B (on the end-effector), G (on the ground surface), the friction cone edgesat A, B, and the wrenches acting on the object (the arrows) in case φ is (a)small and (b) large. Note that at A and B, the wrenches are negated (−wAand −wB). wext leads to the desired contact mode in (b), but not in (a).

set). When φ is sufficiently large (Fig. 6(b)), it is possible tofind such wext in the positive linear span of the wrenches:

wext ∈ pos({f,−wA,−wB})

Or equivalently, the contact wrenches wext , wA, and wB canbe combined to produce the dynamic load f.

Therefore it can be seen that φ should be lower boundedin order to maintain quasidynamic equilibrium in the desiredcontact mode. In Fig. 6, this happens in case −wB can makea positive moment about G. It can also be seen that the lowerbound depends on the friction at A and B. For example, in casethe friction coefficient at B is µB = 0.2, the lower bound on φ

is computed to be 22.61◦. The larger the friction cone at A andB, the larger the lower bound on φ . The conservative analysishere clearly shows the advantage of adjusting φ , particularlyin the case of using end-effectors taking advantage of thesecond-order curvature effects (such as the ellipse-shaped onein Figs. 7 and 10) whose mechanical behavior can be difficultto model.

C. Conclusion: How to Grasp and Regrasp

In order for stable edge-rolling to happen with the right con-tact mode, φ ’s lower bound, that will enable rolling (discussedin Sec. V-B), should indeed be less than its upper bound forthe closure properties (discussed in Sec. V-A). As discussedabove, low friction at A, B and high friction at G will facilitatethis; in contrast, the combination of a “sticky” end-effectorand “slippery” ground can render edge-rolling impossible. Wetake the intersection of the conditions to get the collectionof feasible (that is, both secure and capable of edge-rolling)grasps. One viable approach to choosing a grasp is elaboratedbelow:Strategy for Choosing a Grasp: First, φlb (the lower boundon φ ) can be found by the quasidynamic analysis in Fig. 6.Second, h is chosen such that the upper bound on φ (see Fig.5(c)), denoted φub, is greater than φlb. Third, φ is chosen suchthat φlb ≤ φ ≤ φub. Fourth, θ is chosen such that it is less thanits upper bound given h (see Fig. 5(b)).

It is possible to change the grasp parameters h, θ , and φ

(in other words, “regrasping”) in a continuous manner withoutlosing stability because the set of the secure grasps can berepresented as a connected space with a single component(recall Fig. 5). Useful regrasping operations include changingthe sign of φ for switching the direction of edge-rolling and θ

for tilting the object. θ and φ can be changed by making use ofthe mobility of the arm to which our passive end-effectors areattached since the end-effector and the object move togetherin unison (see Figs. 8(a) and (b)). Although our end-effectorlacks any controllable degrees of freedom to directly changeh, it can be changed by appropriately steering the object (seeFigs. 11 and 12).

One way to further facilitate edge-rolling, given φ , is toincrease θ and decrease h. As θ is increased, it is possible tobalance the force of gravity with smaller end-effector forcesbecause the moment of gravity about G would decrease. Simi-larly, it takes less effort to balance ground reaction wrenches ash is decreased. These all contribute to reducing internal forcesbetween the object and the end-effector. However, there is atrade-off between ease of edge-rolling and grasp stability: asthe grasp parameter h (θ ) decreases (increases), the set of thesecure grasps shrinks as can be seen in Fig. 5. If h (θ ) is toosmall (large), the moments of all the contact forces can beunilateral and thus the object might not be in torque-closure.

One side effect of our approach to edge-rolling with thesecure grasps is that, as long as the grasp is maintained, theother contact mode with a sliding contact on the ground androlling (in essence, fixed) contacts on the end-effector canalso happen. Whether or not the object will actually edge-roll depends on the magnitude of the internal forces. Thiscan be formulated as a complementarity problem and solvedfor each of the contact modes. An example can be seenin [2]. However, the question cannot be fully addressed bysuch a rigid body mechanics approach, which can result ininconsistency or indeterminacy. A compliant contact modelcan make the analysis more complete.

VI. IMPLEMENTATION AND EXPERIMENTATION

In this section, we demonstrate the soundness of our graspplanning for stable edge-rolling using an industrial robot armand a wheeled mobile robot equipped with our zero-mobilityend-effectors.

(a) (b) (c) (d)

Fig. 7. (a) The Sawyer arm with the four-contact end-effector holding theobject on the treadmill. (b) The four-contact end-effector and the object seenin the direction of the object axis. (c) The Sawyer arm with the two-contactend-effector holding the object on the treadmill. (d) The two-contact end-effector and the object seen in the direction of the object axis.

6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018

(a) (b) (c) (d)

Fig. 8. (a) Object tilting and adjusting φ on the way to the target grasp from the initial configuration (left to right). (b) Before (left) and after (right) regraspingin φ . (c-d) Edge-rolling on the vertical half-pipe track performed with the four- and two-contact end-effectors, respectively (left to right).

A. Edge-Rolling with a Conventional Manipulator

Using a seven-DOF Rethink Robotics’ Sawyer robot arm,we demonstrate that stable edge-rolling can be realized withour end-effector design approach and grasp planning. SeeFig. 7. Two types of end-effectors following the design ap-proach presented in Sec. IV-A were 3D printed: four- andtwo-contact end-effectors. When viewed in the direction ofthe axis of the object, the four-contact effector looks like arhombus, making contact with the object on each of its edges.The ellipse-shaped, two-contact end-effector is supposed tomake two contacts with an object at the end points of its semi-major axes. The curvature around the contact points providesthe second-order effects. The Sawyer is controlled to edge-rollobjects on a small treadmill because of the limited size of itsworkspace.

Initially, a cylindrical object is in the upright configuration,and the Sawyer arm is away from the object. If coefficients offriction1 are assumed to be µA = µB = 0.2 (between plasticand metal at contacts A and B), the lower bound on φ iscomputed to be 22.61◦ as discussed in Sec. V-B. If µG = 0.7(between metal and concrete at contact G) and the aperture ofthe wrench cones at A, B is 100◦ (this is the angle betweenthe two adjacent contact normals that our four-contact end-effector can make on an object whose diameter is 1.9cm), then(h,θ ,φ) = (0.13m,45◦,45◦) can be a legitimate target graspbecause it lies in the space of the secure grasps (as in Fig. 5)and φ > 22.61◦ (the lower bound). The arm is then positioncontrolled to bring the object into the target grasp (Fig. 8(a)):

• Phase I Object tilting: The robot is first controlled tocage the object in the hole of the end-effector. h ischosen at this point. Then the robot tilts the object tothe desired θ value by following a position trajectoryin which the path maintains quasistatic stability throughin-hand manipulation even without gripper mobility, aspresented in our previous work [3].

• Phase II Adjusting φ : The robot then rotates its end-effector about the axis of the object up to the desiredφ value. Due to the closure properties of the grasp, theobject may also rotate with the end-effector (it may thusroll on the ground surface). The arm is also controlled tocompensate for the relocation of the object.

1These values are from sources on engineering practice so the values areapproximate.

Success Rate

5/5 (100%)

4/5 (80%)

3/5 (60%)

2/5 (40%)

1/5 (20%)

0/5 (0%)

(a)0.05 0.13 0.18304578.75

φ = 10

φ = 22.5

φ = 45

φ = 67.8

h(m)θ (◦)

φ (◦)

(b)0.05 0.13 0.18304578.75

φ = 10

φ = 22.5

h(m)θ (◦)

φ (◦)

Fig. 9. 3D histogram summarizing the result of our 200+ experiments. Thecoordinates of each block represents a target grasp. Edge-rolling was testedfive times for each target grasp. The color of the block represents the successrate. The experiments were performed with (a) the four- and (b) the two-contact end-effector.

Now edge-rolling is attempted by rolling the treadmill. Ifthe object can make five revolutions, the trial is considereda success. Fig. 9 summarizes the result of our experimentsperformed with the two types of end-effector. See the videoattachment too. As discussed in Sec. V-C, as h gets smaller,the success rates increase (easier to edge-roll). It can also beseen that as θ increases, the success rates first increase thendecrease eventually, for example, when h = 0.18 and φ = 45◦

with the four-contact end-effector. This may be explained bythe trade-off between ease of edge-rolling and grasp stabil-ity discussed in Sec. V-C. Indeed, with relatively large θ

(θ = 78.75◦), edge-rolling fails as the object slips laterallyor escapes from the end-effector. When h is relatively large(h=0.18m) and θ relatively small (θ = 30◦), sliding at G is themajor failure mode, which reconfirms that these grasps can bewith large internal forces. If we put together the grasps withh = 0.05,0.13m, θ = 30,45◦, and φ = 22.5,45◦, the overall

SPECIAN et al.: ROBOTIC EDGE-ROLLING MANIPULATION: A GRASP PLANNING APPROACH 7

success rate is 76/82= 92.7%. The complement of this set hassuccess rate 37/125 = 29.6%. Note that, however, if the anglebetween the two adjacent contact normals decreases (recallthat this angle was 100◦ in our four-contact end-effector), thesuccess rates can drop significantly in particular for the graspswith large φ . The effectiveness of the second-order curvatureeffects can be assessed by comparing the outcomes of thetwo- and four-contact end-effectors: our result shows that thetwo-contact fails 25% more out of 60 trials.

Furthermore, our robot is also able to switch the directionof edge-rolling through regrasping. In Fig. 8(b), the robotwas controlled to regrasp from (h,θ ,φ) = (0.13m,45◦,45◦)to (h,θ ,φ) = (0.13m,45◦,−45◦). The treadmill was thenadvanced in the opposite direction, and the object was ableto roll counterclockwise (clockwise before regrasping); see thevideo attachment. To further verify the robustness of our edge-rolling technique, the robot was controlled to perform edge-rolling along a 36cm tall vertical half-pipe track (Fig. 8(c)).Here the arm was position-controlled in a feedforward mannerwith no feedback to follow a trajectory on which the relativeconfiguration between the end-effector and the tangent planeto the half-pipe track at the contact G remained constant. Inother words, nominally (h,θ ,φ) = (0.13m,45◦,45◦) on thetrajectory; the grasp was chosen for the four-contact effectorby following the strategy specified in Sec. V-C. It was possibleto successfully (10 successes / 10 trials) edge-roll the objectfrom the bottom of the track, where the normal at G isantiparallel to gravity, to the top, where the normal is parallelto gravity (see the video attachment).

B. Long-Range Edge-Rolling Manipulation

We demonstrate long-range edge-rolling by a modular mo-bile manipulator whose workspace is not constrained by jointlimits like the Sawyer. The robot is assembled with CKBotV3 modules [33]. It has a differential-drive and a one-DOFarm mounted on top. The two- and four-contact end-effectorscan be fixed to the arm (Fig. 10(a)).

(a) (b)

arm

M1

M3

objectM2

end-effector −→x

y↑ γ

Fig. 10. (a) Our mobile robot with three servos (M1, M2, and M3) and aneffector-arm substructure on top of M3. M1 and M2 drive the robot in adifferential-drive manner. M3 is able to rotate the effector-arm substructureabout the axis perpendicular to the ground (see the video attachment). (b)γ denotes the difference between the headings of the object and the mobilebase. The object may escape from the end-effector as γ builds up.

The one-DOF mobility of the arm is used for steeringthe object about an axis perpendicular to the ground surface.This active steering capability is the key to edge-rolling longdistances in a robust manner. If the heading of the object is

not aligned perfectly with the direction of the motion of thebase during edge-rolling (that is, nonzero γ in Fig. 10(b)),the object may end up sliding down or up on the end-effector surface while rolling on the ground. This is becauseof the nonprehensile nature of our grasp in incomplete force-direction closure. Suppose that the mobile base in Fig. 10(b)moves in the positive y-direction. Positive (negative) γ resultsin the object sliding up (down). As seen from the end-effector,the object moves along a helical path, like the motion of ascrew. This can result in loss of the grasp if not addressed(Fig. 11).

Fig. 11. For a small (γ = 10◦) misalignment in object heading the grasp canlead to the object sliding out of the grasp and the track of the treadmill (downin photograph).

The robot is able to measure the relative translation ofthe object and the end-effector by receiving feedback froma motion capture system. An integral (I) controller was imple-mented to correct the error and adjust the relative orientationbetween the mobile base and the object-effector-arm system:

u(t) = Ki

∫ t

0(h(ξ )−h0)dξ

u(t) is an angular position command for the steering arm. It isproportional to the cumulative error in the grasp parameter h.If the value is positive (negative), the arm is controlled to turncounterclockwise (clockwise). The value of u(t) was limitedto the range [−17◦,17◦]. If the arm is allowed to steer moreaggressively, we empirically observe that it can jeopardize thecontrol of the object.

Fig. 12 illustrates a sample path. With active steering on, theamount of the unwanted translation was bounded and the robotwas able to edge-roll the object and travel around the square atleast twice, 15.9m or 265 times the perimeter of the object. Thenominal grasp here was also chosen by following the strategyspecified in Sec. V-C. With active steering off, the objecttranslates off the grasp, traveling only 0.3m. See the videoattachment. We believe the minor alignment errors (small γ)during the experiment allow for the simple, purely feedbackI-controller to be effective. A model predictive controller,taking the model of dynamics into account, may be worthinvestigating for more responsive active steering.

VII. CONCLUSION

This paper presents grasp planning for robotic edge-rollingwith zero mobility, passive end-effectors that can be fabricatedas a single rigid body. The generated set of feasible graspstakes into account the closure properties and quasidynamicequilibrium. Successful edge-rolling was demonstrated with aconventional manipulator arm and a modular mobile platform

8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2018

(a)(b)h(

t)−

h 0(c

m)

time (s)

Fig. 12. Time-lapse action sequence of edge-rolling around the 2.4m×1.55mworkspace. The mobile robot is with the two-contact end-effector. The imageof the robot darkens with time. The plot shows the translational error in theaxial direction vs. time when the active steering controller was (a) on and (b)off (it lasts only 10 seconds because at the moment the object falls out of theend-effector as the errors build up).

on a treadmill, the ground surface, and a U-shaped verticalhalf-pipe track.

This work establishes the first steps towards more practi-cal and complete robotic edge-rolling manipulation. Possibledirections for future work include incorporating more so-phisticated models of contact/friction, improving the steeringcontrol capability to better hold the object, implementing fullyautonomous roll-and-place manipulation in the unstructuredenvironment, and generalizing into a wider range of objectssuch as prismatic objects other than cylinders.

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