restraining objects with curved effectors and its...

Restraining Objects with Curved Effectorsand Its Application to Whole-Arm Grasping

Jungwon Seo, Mark Yim, and Vijay Kumar

Abstract This paper develops the theory and algorithms for immobilizing/cagingpolyhedral objects using curved (for example, planar, cylindrical, or spherical) ef-fectors, in contrast to customary point effectors. We show that it is possible to im-mobilize all polyhedral objects with three effectors with possibly nonzero curvature,with finite extent. We further discuss how to cage the objects and obtain a stablegrasp from such a cage. The theory can also be applied to immobilize/cage polygo-nal objects on the plane. As one application of the theory, we address the problemof whole-arm grasping with robot arms.

1 Introduction

Our main interest is in immobilizing and caging objects. If an object is immobilized,it can neither translate nor rotate. Caging seeks to establish obstacles around an ob-ject such that it cannot escape arbitrarily far away. In contrast to previous work,we do not limit ourselves to point effectors (fingertip contacts): it has been custom-ary to consider point contacts and induced wrenches related to the contact normalsin studying the mobility of a grasped object [8, 6]. In addition, a mobility theorybased on the relative curvature of contacting bodies was established in [12, 13].Parallel-jaw grasping [2] is one way to obtain stable grasps, although the grasps donot immobilize objects. Recently, it has been shown that grasping can be facilitatedby forming cages first [11, 14]. There have also been efforts to compute cages [16].

In this work, we constructively show that all polyhedral objects can be immo-bilized with three curved (for example, planar, cylindrical, or spherical) effectorsproviding only frictionless, rigid, unilateral contacts. In [12], the authors proposeda conjecture that asks if general n-dimensional objects can be immobilized by nfrictionless, suitably concave effectors; our result thus confirms the conjecture for

J. Seo, M. Yim, and V. KumarUniversity of Pennsylvania, Philadelphia, PA, USA, e-mail: [email protected]

1

[email protected]

2 Jungwon Seo, Mark Yim, and Vijay Kumar

polyhedral objects. We also discuss how to establish cages exploiting the concavityof the curved effectors and obtain a stable grasp from the cages by simple motionplanning. We thus propose to integrate effector design into the synthesis of graspsand cages. Although not as simple as point effectors, the curved effectors are sim-ple enough to be easily manufactured, for example, by 3D printing, or emulated inmany ways, for example, by cupping the fingers and palm of a multi-fingered hand.

As one application of our theory, we address whole-arm grasping, where robotarms grasp an object not only using their end-effectors but also possibly exploit-ing other contacts with the arms and torso. Even contact planning for whole-armgrasping becomes intractable as the number of point contacts to be established in-creases. Whole-arm grasping has been addressed by data-driven approaches [4]. Incontrast, we shall introduce a model- and rule-based approach employing energy-based linkage reconfiguration [3, 5], which can be verified analytically. This workbuilds on our previous paper [15] that addressed the synthesis of spatial grasping;in this paper, we incorporate more complete algorithms and analysis for immobilityand caging conditions.

The paper is organized as follows. We begin by reviewing some preliminary con-cepts in Sec. 2. We discuss our theory and algorithms for immobilizing/caging poly-hedral/polygonal objects with curved effectors in Sec. 3. We then address the prob-lem of whole-arm grasping in Sec. 4 as one application of the theory. We concludein Sec. 5 with suggestions for future work.

2 Preliminaries on Grasping and Caging

We are concerned with an object in contact with effectors restricting its motion. Agrasp refers to such a configuration with additional information on contact wrenches[8, 6], force/moment pairs exerted at the involved contacts. We only consider thecontact wrenches of frictionless, rigid, unilateral contacts, which can be repre-sented as the positive linear combination of unit wrenches along the contact normals(Fig. 1a, b). A grasp can be in equilibrium if the resultant wrench can be made zeroin such a way that not all contact wrenches are equal to zero.

(a) (b) (c)

Fig. 1 (a) Immobilizing the regular triangle with the three point effectors located at the centerof each edge. (b) Clamping the tetrahedron with the two plane effectors contacting the vertex-face pair. (c) Caging the triangle with the three point effectors. The red arrows are involved unitwrenches in (a), (b), and all upcoming figures.

Restraining Objects with Curved Effectors and Its Application to Whole-Arm Grasping 3

If there is no object twist consistent with the contact wrenches of an equilibriumgrasp, the object is said to be immobilized to the first order (form-closure). Evenif such a twist exists, any finite motion may be restricted by considering curvatureeffects. For example, in Fig. 1a, the object can instantaneously rotate about its cen-troid (so a first-order kinematic analysis does not predict immobility), but any finiterotation will result in effector penetration. This idea is formalized using the conceptof second-order immobility [12, 13]. Seven (four) point effectors are required toimmobilize general three-dimensional objects to the first (second) order. Such im-mobility conditions are purely geometric; information on contact geometry is thussufficient to investigate first- or second-order immobility.

Although not immobilizing, clamping [2], also known as parallel-jaw grasping,is one way to realize a stable equilibrium grasp with two planar “jaws”. Moreover,the grasp can be force-closed [6] by considering friction. Even if not, the clampedobject can only move on the plane of the jaws. Consider the antipodal pair of aconvex, polyhedral object, i.e., the intersection of the object with a pair of parallelsupport planes, which can be a vertex-vertex, vertex-edge, vertex-face, edge-edge,edge-face, or face-face pair. According to [2], the last four types of antipodal paircan provide a clamp as shown in Fig. 1b. Note that they can determine the width ofthe object, the minimum distance between two parallel supporting planes.

Without contacting an object, effectors may just surround the object such thatit cannot escape from their cage (Fig. 1c). In [14], the concept of an F-cage wasformalized. Let F be a scalar function defined on effector configurations. Then anF-cage is a configuration of the effectors that cages an object even if they havefreedom to move while maintaining the value of F . An F-cage is an F-squeezing(stretching) cage if it still cages the object even if the effectors have freedom tomove while decreasing (increasing) the value of F . For two point effectors, F cansimply be the distance between them, and the prefix “F-” can be ignored as in [16].

3 Restraining Objects with Curved Effectors

In this section, we discuss how to immobilize and cage all polyhedral objects (or justpolyhedra for short) with at most three curved effectors providing frictionless, rigid,unilateral contacts. It is sufficient to consider only planar, cylindrical, and sphericaleffectors, shown in Fig. 2, which will be used only for a planar, a line, and a pointcontact at a face, a convex edge, and a pointed vertex of a polyhedron, respectively,as can be previewed in Fig. 3.


`

w `r

d ω

r

d

ω

Fig. 2 Examples of a planar, a cylindrical, and a spherical effector. The planar effector has arectangular shape. The latter two are cut parallel to the axis of revolution of a cylinder and asphere, respectively. Their dimensions are shown as w (width), r (radius), ` (length), d (depth), andω (aperture).

3.1 Immobilizing Polyhedral Objects with Curved Effectors

We first discuss how to synthesize immobilizing grasps using at most three curvedeffectors shown in Fig. 2. We also show that our grasps are complete, that is, theycan immobilize all polyhedra.

None of the effectors can individually immobilize polyhedra. However, two orthree effectors can provide immobility by exploiting the antipodal pairs of the con-vex hull of a given polyhedron, as will be explained below. The nonzero curvatureof a cylindrical or spherical effector and the multiple unit wrenches of a cylindricalor planar effector contact play an important role in the immobility. Note that if a vir-tual edge or face [10], that is, an edge or face belonging to the convex hull but notto the original polyhedron, needs to be contacted by a cylindrical or planar effector,w and ` (Fig. 2) should be large enough to entirely cover the virtual element.

Immobility using a vertex-vertex antipodal pair: Let (P,Q) denote the pair;see Fig. 3a. Suppose that the two support planes can be made perpendicular to PQand contact only P and Q, respectively. We can have two spherical effectors con-tact P and Q, respectively, such that their contact normals are collinear to ξ (theline collinear to PQ) because spherical effectors are locally flat and the vertices arepointed. Now the polyhedron can only rotate about ξ as long as the effectors havea radius less than 1

2 d(P,Q), where d(·, ·) is the Euclidean distance between the twoelements, because any finite displacement of PQ will result in penetrating the ef-fectors. An additional contact by a spherical effector at R can further restrict anyfinite rotation about ξ if the radius of the effector is less than d(R,ξ ) and its contactnormal intersects ξ .

Immobility using a vertex-face antipodal pair: Suppose that the pair deter-mines the width. Consider a planar effector contacting the face and a spherical ef-fector, whose radius can be arbitrary, contacting the vertex P such that its contactnormal, collinear to ξ , orthogonally intersects the planar effector; see Fig. 3b. Nowthe object can only rotate about ξ : the object is at least clamped by the two effectors(only planar motion on the planar effector possible); moreover, any finite translationon the plane results in penetrating the spherical effector. An additional contact bya cylindrical effector at one of the edges incident to P can further restrict any finiterotation about ξ if its contact normal intersects ξ : the edge with the least slope with


(a)

P

Q

R

ξ

(b)

P

ξ

b

(c)

ξ

Fig. 3 Immobilizing grasps using (a) a vertex-vertex antipodal pair, (b) a vertex-face antipodalpair, and (c) an edge-edge antipodal pair. The inscribed red line segment, right circular cone, andtetrahedron respectively in (a), (b), and (c) will be used in Fig. 6.

respect to the face satisfies the condition; the radius of the cylindrical effector shouldbe less than b as shown in the figure.

Immobility using an edge-edge antipodal pair: Suppose that the pair deter-mines the width. Consider two cylindrical effectors, whose radius can be arbitrary,respectively contacting the two edges such that their contact normals are all paral-lel to ξ , the common perpendicular of the two edges; see Fig. 3c. If ξ intersectsboth effectors, the object is immobilized: it is at least clamped by the two effectors;moreover, one of the cylindrical effectors only allows the object to translate alongits axis of revolution, but such translation is not allowed by the other effector.

The elements that the effectors contact, except for the edge in Fig. 3b, can easilybe reached because they belong to the convex hull; the minimum required apertureof a cylindrical or spherical effector can be determined by the object geometry.The following theorems verify the completeness of the grasps: all polyhedra can beimmobilized by applying the grasps. We first verify that even only the first type ofgrasp can be sufficient.

Theorem 1. Every polyhedron can be immobilized with three spherical effectors ofappropriately chosen dimensions.

Proof. We first show that every polyhedron has a vertex-vertex antipodal pair (P,Q)that allows two support planes perpendicular to PQ and contacting only P and Q,respectively. Consider the collection of the vertices of the polyhedron. Let (P,Q)be a pair of vertices determining the maximum distance between two vertices ofthe collection. Consider two planes ΠP and ΠQ perpendicular to PQ and contactingthe polyhedron respectively at P and Q. No other vertex of the polyhedron can belocated on ΠP and ΠQ because (P,Q) determines the maximum distance. Therefore,ΠP (ΠQ) is supporting the polyhedron only at P (Q); (P,Q) is the desired vertex pair.

Next, we show that an additional vertex, denoted as R, can be found such that acontact normal at R by a spherical effector intersects the line of PQ. Let R be thevertex that is the most distant from the line of PQ. Consider a plane ΠR parallel to


PQ, perpendicular to the plane of 4PQR, and contacting the polyhedron at R. Noother vertex of the polyhedron can be located on ΠR except on ξR, a line passingthrough R and parallel to PQ on ΠR, because R determines the maximum distance.However, R is not a mid-edge vertex. Therefore, we can make the contact normal ofa spherical effector intersect the line of PQ; R is the desired vertex.

We finally get immobility with three spherical effectors respectively contactingP, Q, and R as explained in immobility using a vertex-vertex antipodal pair. ut

The next theorem shows completeness in terms of general polyhedra, i.e., poly-hedra that do not have parallel elements that can be edges or faces.

Theorem 2. Every general polyhedron can be immobilized with either (1) two cylin-drical effectors or (2) a planar, a cylindrical, and a spherical effector of appropri-ately chosen dimensions.

Proof. Every general polyhedron can be clamped with either a vertex-face or anedge-edge antipodal pair. The polyhedron can then be immobilized using the antipo-dal pair by applying immobility using a vertex-face antipodal pair or immobilityusing an edge-edge antipodal pair. ut

Our approach can be generalized in a number of ways. On the one hand, moretypes of grasps can be considered by employing other types of antipodal pairs. Forexample, Fig. 4 shows an immobilizing grasp with two spherical effectors and oneplanar effector on an edge-face antipodal pair; it can be proved that if the involvedcontact wrenches can be in equilibrium, then we actually get immobility. On theother hand, the effector shapes can also be generalized because we only need thecurvature of contact points.

We now proceed to designing an algorithm to obtain the grasps. First, it takesO(n logn) expected time to compute the convex hull of a given polyhedron, where nis the number of the vertices. Then all of the antipodal pairs of the convex hull can befound in O(n2) time by applying a technique introduced in [2]. For a vertex-vertexor vertex-face antipodal pair, it additionally takes O(n) time to find the third contactlocation. Note that, however, the overall complexity will also depend on collisiondetection algorithms if the global geometry of the effectors is to be checked. Fig. 5shows some example grasps found by the algorithm; more than 100 grasps could befound in less than 5 seconds with a simple C++ implementation.

Fig. 4 An immobilizing graspusing the edge-face antipodalpair that determines the width.

P

Q


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

Fig. 5 The two or three curved effectors colored in green are grasping the rock model with 1,000faces (courtesy: Malcolm Lambert, Intresto Pty Ltd.). The rock is immobilized using (a) a vertex-vertex pair, (b) a vertex-face pair (a cylindrical effector is not shown), (c) an edge-edge pair, and(d) an edge-face pair.

3.2 Caging Polyhedral Objects with Curved Effectors

We now discuss the synthesis of cages using the global geometry of the effectormodels in Fig. 2. The cages are based on the immobilizing grasps in Fig. 3; it issufficient to use only the two effectors at the antipodal pairs. The inscribed shapesin Fig. 3, reproduced in Fig. 6, allow us to establish sufficient conditions for caging.

Cage using two spherical effectors: Given a vertex-vertex antipodal pair, (P,Q),we consider how to cage PQ with two spherical effectors; see Fig. 6a. Suppose thatthe effectors are only allowed to move such that their axes of revolution are alwayscollinear; the axis is denoted as η . Then PQ is caged with the effectors if δ is smallenough to guarantee (1) P and Q are respectively in the “pockets”, i.e., the interiorof the convex hull, of the effectors and (2) c < d(P,Q), where c is the maximumopening between the two effectors.

Cage using a spherical and a planar effector: Given a vertex-face antipodalpair determining the width, we consider how to cage the cone in Fig. 6b with aspherical and a planar effector. First suppose that two infinitely large planar effec-tors are clamping the cone at the apex and base. We now allow the effectors to movein such a way that they remain parallel to each other. If their distance is less thana, the side length of the cone, the clamp can stably be recovered by just makingthe effectors approach each other; moreover, the distance between the apex and theeffector at the base is always larger than h, the height of the cone. Instead of the pla-nar effector at the apex, now consider a spherical effector only allowed to relativelytranslate along its axis η perpendicular to the other planar effector (Fig. 6b). Thenthe cone is caged with the effectors if δ is small enough to guarantee (1) the apexis in the pocket of the spherical effector, (2) δ + d < a, and (3) δ < h: (2) and (3)guarantee that the apex cannot escape from the pocket by the analysis above. Notethat the planar effector at the base only has to be large enough to contain a disk ofradius r+ rc centered at O, where η intersects the planar effector.

Cage using two cylindrical effectors: Given an edge-edge antipodal pair de-termining the width, we consider how to cage the tetrahedron in Fig. 6c with twocylindrical effectors. Consider again the two infinitely large planar effectors clamp-ing the tetrahedron at the edge pair (AB,CD). If their distance is less than a, the


(a)

P

Q

cδ

η

(b)

η

a

r

rc

δ +d

O

δh

(c)

Aη

B

CD

d1

δ

d2

Fig. 6 Cages using two curved effectors. The red inscribed shapes are reproduced from Fig. 3.(a) A cage using two spherical effectors. c = c(δ ) denotes the maximum distance between a pointon the rim of one effector and a point on the surface of the other effector; thus PQ (and thusthe polyhedron) cannot escape from the effectors if c < d(P,Q). (b) A cage using a sphericaland a planar effector. If δ + d < a, the vertex of the cone cannot be located below (closer to theplanar effector than) its current position. Therefore if additionally δ < h, the cone (and thus thepolyhedron) cannot escape from the two effectors. (c) A cage using two closed-ended cylindricaleffectors. Note that δ denotes the vertical distance between the two effectors in all the three cases.

smallest value among d(←→AB,C), d(

←→AB,D), d(

←→CD,A), and d(

←→CD,B), where

←→AB is the

line of AB, and so on, the clamp can stably be recovered by just making the effectorsapproach each other. Moreover, the lowest (closest to the bottom effector) possiblepositions of A and B can be found by rotating the tetrahedron about CD lying on thebottom effector. The highest possible positions of C and D can also be found sim-ilarly. Instead of the planar effectors, now consider two cylindrical effectors facingeach other and only allowed to relatively translate on their common perpendicularη . To simplify analysis, assume that the sides of the effectors are closed as can beseen in Fig. 6c. Then the tetrahedron is caged with the effectors if δ is small enoughto guarantee (1) AB and CD are respectively in the pockets of the cylindrical ef-fectors, (2) δ + d1 + d2 < a, and (3) d1 (d2) is large enough to contain the lowest(highest) positions of A and B (C and D): (2) and (3) guarantee that the edges cannotescape from the pocket by the analysis above.

As a corollary of Theorems 1 and 2, the three types of cages are complete. More-over, other effector geometry can also be considered; using cylindrical or sphericalsurfaces is just one way to satisfy the caging conditions.

The following theorem states what happens if the two caging effectors get closer.

Theorem 3. For the three types of cages discussed above, an equilibrium grasp iseventually obtained if the two effectors are controlled such that the relative velocityis along η and δ monotonically decreases.

Proof. We first show that δ is a grasping function [14] for the cages. In each type,the configuration space of the two effectors can be represented as M = SE(3)×SE(3); δ is a semi-algebraic scalar function δ : M → R invariant with respect tothe rigid transformations of the effectors as a whole in that it is the distance betweenthe effectors (Fig. 6). Furthermore, the preimages of δ do not cage the object below


(above) a certain value m (M) such that m < M, for example, m = 0 and M = h inFig. 6b. Then δ is a grasping function according to [14].

In addition, the cages are δ -squeezing cages [14] in that the object remains cagedeven if δ decreases, and then there exists a path in M that leads the effectors into aconfiguration that can realize an equilibrium grasp. Furthermore, in terms of the one-dimensional set representing the relative configuration space of the two effectors, δ

can be considered as a convex, i.e., linear, function. Then we get to a configuration torealize an equilibrium grasp only by moving the effectors such that δ monotonicallydecreases by the result of [14]. ut

A translation that monotonically decreases δ will be referred to as a squeezingmotion in the remaining discussion. Note that, however, the resultant grasp might notbe the one we have expected; for example, the circular rim of a spherical effectormay unexpectedly contact the object. Still, it is guaranteed to be a configuration torealize equilibrium. We may then add more contacts to further secure the grasp.

3.3 Immobilizing/Caging Polygonal Objects with Curved Effectors

In this subsection, the results of Sec. 3.1 and 3.2 are applied to polygonal objects(or just polygons for short) with curved effectors on the plane.

Corollary 1. Every polygon can be immobilized with two circular effectors of ap-propriately chosen dimensions.

Corollary 2. Every general polygon, whose convex hull does not have paralleledges, can be immobilized with a linear and a circular effector of appropriatelychosen dimensions.

Here is a sketch of proof. For Corollary 1, consider a pair of vertices determiningthe diameter. Two circular effectors contacting the vertices can then provide planarimmobility. For Corollary 2, we use the fact that a general polygon allows clampingonly at an edge-vertex pair. Then a linear and a circular effector at the pair can pro-vide planar immobility. To illustrate, see Fig. 7a, b and imagine the circular (linear)effectors contacting the vertices (edge).

(a)

η

r

d

(b)

η

`(c)

η

Fig. 7 The polygon can be caged by (a) the two circular effectors, (b) the linear and circulareffectors, and (c) the two point effectors. Effector dimensions are shown as r (radius), ` (length),and d (depth).


We can also construct planar cages of two curved effectors inspired by the ge-ometry of the grasps; see Fig. 7a, b. As implied by the red line segment and cone inthe figures, the caging conditions can be established similarly to Fig. 6a, b. We canalso add more grasp/cage pairs. As shown in Fig. 7c, two point effectors suffice toimmobilize/cage some concave polygons; the related caging condition is discussedin [16]. Finally, the following corollary can be proved similarly to Theorem 3.

Corollary 3. For the cages shown in Fig. 7, an equilibrium grasp is eventually ob-tained if the two effectors are controlled such that the relative velocity is along η

and the distance monotonically decreases.

4 An Application to Whole-Arm Grasping

Our approach in Sec. 3 seeks to minimize the number of restraining effectors with-out losing stability. We show how this approach can be applied to whole-arm grasp-ing with robot arms, which can be effective for grasping large, bulky objects withrelatively small end-effectors and the armchain between them.

4.1 Approach

Whole-arm grasping is naturally related to an open kinematic chain. Here, we partic-ularly consider a planar open kinematic chain where revolute joints are connectingrigid links moving on the plane; the platform will be just referred to as a manipula-tor. The planar architecture suffices to realize even our spatial grasps in that the threeeffective contact wrenches respectively from the three contacts should be from a pla-nar pencil and thus coplanar. We again assume that the manipulator only providesfrictionless, rigid, unilateral contacts. But, our strategy based on such conservativeassumptions can also be effective for physical environments with nonzero friction

(a) (b) (c)

d

r

Fig. 8 (a) A manipulator for whole-arm grasping. The base and the two end-effectors of the planaropen kinematic chain provide a planar and spherical surfaces, respectively. We can also consider acollection of exchangeable effectors with various sizes and shapes. (b) An example of an immobi-lizing whole-arm grasp on the rock model with the manipulator. (c) The flexion of the PR2’s elbowcan emulate a circular (or cylindrical) effector shown in the figure.


and compliance. At least two of the manipulator links should be shaped to providethe curved effectors in order to apply our theory; we call them end-effectors. Theymay be made exchangeable to suit the size and shape of an arbitrary object (Fig. 8a,b). Without making dedicated end-effectors, the curved shapes may be emulated insome ways, e.g., the flexion of a joint for a circular effector (Fig. 8c) for planargrasping (Sec. 3.3).

Our approach to whole-arm grasping is composed of two phases: preshaping andsqueezing. In the preshaping phase, we move the manipulator such that its two end-effectors can cage the object. The fact that we can aim at any of our cages, whosecollection is not a set of measure zero, can facilitate involved motion planning. In thesqueezing phase, the two end-effectors perform a squeezing motion. Meanwhile, wecan have other links contact the object without losing stability, which can be justifiedby the following corollary:

Corollary 4. Suppose that the end-effectors of a manipulator are caging an objectas shown in Fig. 6, 7. An equilibrium grasp is eventually obtained if the manipulatormoves such that the end-effectors are performing a squeezing motion.

Proof. Recall Theorem 3. Here, the configuration space M of the manipulator canbe thought of as SE(3)×Sm, where the first (second) term addresses the configura-tion of the base (m joints), but the same argument can also be applied by regardingδ , the distance between the two effectors, as a grasping function again. ut

Although other links, besides the end-effectors, contact the object, the corollaryshows that the grasp can still be in equilibrium as long as the end-effectors aresqueezing. The squeezing phase can be performed in a blind manner without directfeedback of the object pose. In fact, the approach is also popular in multi-fingeredgrasping [7], where data-driven approaches have mainly been used for preshapingand squeezing, i.e., closing the hand. In contrast, we use a model-based approachpossibly for a hyper-redundant arm that does not have an obvious closing motion.

4.2 Planning and Executing Whole-Arm Grasping

4.2.1 Planning Whole-Arm Grasping

The central algorithm is Algorithm 1; the key idea is as follows. Let C de-note the configuration space of a given object-manipulator system, isomorphic toSE(3)× SE(3)× Sm for two independent rigid bodies (the object and the base ofthe manipulator) and m manipulator joints. It is assumed that the kinematic modelof the system is known. The algorithm takes as input an initial configuration of thesystem ci ∈ C , a 6+6+m-dimensional vector; it returns a reference trajectory forthe manipulator, γ(s) : [0,1]→ SE(3)×Sm, where s is a non-dimensional parameterincreasing with time. The following paragraphs elaborate each line of the algorithm.


Algorithm 1 Motion planning for whole-arm graspingInput: An initial configuration of an object-manipulator system, ci ∈ COutput: A reference trajectory for whole-arm grasping1: Construct two configurations: cp ∈ C for preshaping, cs ∈ C for squeezing.2: Plan a trajectory γip for the preshaping: from ci to cp.3: Plan a trajectory γps for the squeezing: from cp to cs (possibly in parallel with Line 2).4: Concatenate γip and γps into γips, the resultant trajectory from ci to cs via cp.

(a)

cp

cs

(b)

p j p j+1

Fig. 9 (a) The configuration cp (in grey) results in a caged configuration with the two sphericalend-effectors. The wireframe shows the configurations of the two end-effectors at cs after a squeez-ing motion (see the arrows), where the longest link is also intersecting the object as shown. If weadditionally consider a virtual link connecting the end-effectors, the manipulator configurationscan be described as simple polygons. (b) The level set of d(·, ·) appeared in Eq. (2) allows us to ad-dress the planar shape of the link p jp j+1 connecting the joints p j and p j+1 for collision avoidance.Some of the level sets are shown as solid boundaries.

Line 1: We first construct cp,cs ∈ C that are supposed to describe configurationsat which preshaping and squeezing should aim, respectively (Fig. 9a). They canthus be interpreted as desirable waypoints. cp is constructed such that the two end-effectors cage the object in a kinematically feasible manner. cs is constructed suchthat the manipulator deliberately intersects the object; one simple strategy is to makejust the two end-effectors approach and intersect the object, but other links can alsobe considered as shown in Fig. 9a. Any collision-aware inverse kinematics algorithmcan be employed here. At cp and cs, the manipulator should be described as simplepolygons (Fig. 9a) to facilitate the squeezing as will be explained; one way to do thisis to consider a virtual link connecting the end-effectors for collision avoidance. Inplanar grasping for polygons, cp may be constructed simply by enclosing the objectwith the manipulator such that the opening is less than the width of the object.

Line 2: We plan for a trajectory from ci to cp. During the motion, we do not wantthe manipulator to interact with the object; thus any collisions should be avoided.This can be considered as an ordinary motion planning problem where some off-the-shelf algorithms can be applied. Ultimately, some manipulations such as toppling ortilting may be needed to reach cp; but it is outside the scope of this paper.

Line 3: We now plan for a trajectory from cp to cs while ignoring the object ge-ometry. In order to realize a squeezing motion, the manipulator should be regardedas a closed kinematic chain. Furthermore, the length of the virtual link between thetwo squeezing end-effectors must monotonically decrease. This is a hard problemin general, but can be efficiently solved by Iben et al.’s algorithm [5], for interpo-lating between two planar, simple polygons without any self-intersections. In the


algorithm, link lengths can be fixed or monotonically changed, which allows us toimplement squeezing. The resultant motion basically reconfigures the two polygons“towards each other” according to a metric defined between a pair of polygons, e.g.,the `2-norm on the vector of vertex coordinates. Whenever the direct reconfigurationincreases the value of an energy function such as

E = ∑1

d(pi,p jp j+1)2 , each term is for joint pi and link p jp j+1 (i 6= j, j+1) (1)

we follow the downward gradient of E to avoid self-collisions.

Iben et al.’s algorithm employed in Line 3 is basically for line segment linkswithout joint limits, i.e., θi ∈ (−π,π) where θi is the angle of joint i (θi = 0 whenthe two links are collinear). We further discuss how to adapt the algorithm so asto address link shapes that are not line segments and joint ranges narrower than(−π,π), i.e., θi ∈ [−ì,ui] where 0 < ì,ui < π . First, in order to address nonzerolink volume, we propose to use (d(pi,p jp j+1)− δ j)

2 as the denominator of eachterm of E where δ j is determined for each link p jp j+1 such that the collectionof x’s on the level set d(x,p jp j+1)− δ j = 0 can address the collision hull of thelink (Fig. 9b). Note that two adjacent links overlap each other. Second, If (1) anyθi is close to its limit, i.e., θi ∈ [−ì,−ì + ε] or θi ∈ [ui− ε,ui] for some ε > 0and (2) the manipulator is described as a concave polygon, we propose to apply anexpansive motion [3] to straighten all joints such that θi can return to the “safe” rangebeyond the margin of ε . During an expansive motion for a closed chain, every joint ismonotonically unfolded until the chain is convexified. Such a motion always existsas long as the chain is described as a simple, concave polygon and is computed byconvex optimization [3]. In case the manipulator is described as a convex polygon,there also exists such an angle-monotone motion to address the joint limit issues [1].It can be shown that the adaptations do not affect the convergence of the algorithm.

The performance of the inverse kinematics and motion planning in Lines 1 and 2depends on off-the-shelf algorithms. Iben et al’s algorithm was shown to terminatein a finite number of steps, computing the integral curve of a vector field [5].

4.2.2 Executing Whole-Arm Grasping

We make the manipulator follow the resultant trajectory in a quasi-static mannerbecause we are basically concerned with the relative configuration of the object andmanipulator. In fact, the motion will necessarily be interrupted on the way because itis designed to collide with the object. If there is neither friction nor compliance, theconfiguration where the manipulator stops moving can realize a caged, equilibriumgrasp by Corollary 4; the configuration can thus be an acceptable grasp.

In many cases where we cannot ignore friction, the manipulator might get stuckon the way. In other words, it might not be able to reach the configuration of the idealcase because nonzero friction can cause jamming and wedging [6]. However, bothphenomena imply force-closure [6], which in turn implies involved wrenches are in


equilibrium. Thus the jammed or wedged configuration can also be an acceptablegrasp; we then have more candidates for acceptable grasps due to nonzero friction.

Even if we cannot ignore compliance, the local stability of the resultant grasp isguaranteed. If the manipulator happens to immobilize the object, the object remainslocally dynamically stable under a common stiffness model [12]; nonzero frictioncan further enhance the stability. Even if not immobilizing, the force-closure graspby the caging effectors can be made stable [9]. The guaranteed stability allows us toexert internal forces, further securing the resultant grasp, only by position control,i.e., simply by letting the manipulator “move” as planned. A terminating condi-tion can then be stated as follows: terminate the motion of the manipulator after itstops moving and its joint torque values exceed appropriate threshold values. Thethreshold values should take the safety of the system into account. More refinedterminating conditions can be considered if visual or tactile sensing is available.

4.3 Implementing Whole-Arm Grasping

We implemented whole-arm grasping with CKbot1, our chain style modular robotsystem. In terms of kinematics, each module can be used as an 1-d.o.f. swivel orelbow joint (Fig. 10a). Fig. 10b shows the subassemblies of two 3-d.o.f. arms anda 3-d.o.f. spine. The arms are planar chains where the energy-based squeezing canbe applied; the spine provides all the three rotational degrees of freedom by realiz-ing z-y-z Euler angles. Fig. 10c shows 3D printed curved end-effectors compatiblewith CKbot; the arms in Fig. 10b actually have the two spherical end-effectors. InFig. 10d, the finished two-armed modular manipulator is shown.

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

Fig. 10 (a) CKbot modules providing one rotational degree of freedom. (b) Two 3-d.o.f. arms anda 3-d.o.f. spine between them. (c) One planar effector, two spherical effectors, and two cylindricaleffectors (left to right). (d) A two-armed modular manipulator.

Our software implementing Algorithm 1 is organized as ROS2 packages. Theyprovide methods for grasp synthesis, preshaping, and squeezing. Fig. 11a shows asimulated manipulator with two 5-d.o.f. arms and a 3-d.o.f. spine is preshaping for acage. Fig. 11b shows two examples of the energy-based squeezing for the 10-d.o.f.armchain. Finally, in the top row of Fig. 12, the tetrahedral carton, with the texture

1 http://www.modlabupenn.org/ckbot2 http://www.ros.org

http://www.modlabupenn.org/ckbot

http://www.ros.org


(a)

(b)cp1

cp2

cs

cs

Fig. 11 (a) In the first two panels, the simulated, 13-d.o.f. modular manipulator is moving to thepreshape where the end-effectors are caging the rock. In the last two panels, the end-effectors werezoomed in. Uncertainty in sensing and control can be accommodated in the cages. (b) In eachrow, the real, 10-d.o.f. armchain is reconfiguring from the preshape, cpi, to the common squeezedconfiguration, cs (left to right).

of marble, is grasped by the chain of 4-d.o.f. and 2-d.o.f. arms in the first panel,the chain of two 3-d.o.f. arms in the second panel, and the chain of 2-d.o.f. and 4-d.o.f. arms in the last panel. The grasps are respectively derived from the cages ofone planar and one spherical, two cylindrical, and two spherical effectors as can benoticed by the end-effectors used in each grasp.

Fig. 12 Some example graspsby the modular manipulator(top) and the conventionalmanipulator, PR2 (bottom).The PR2 is applying planargrasping (Sec. 3.3) that canbe effective for the prismaticobjects.

5 Concluding Remarks

We have presented the theory and algorithms for immobilizing and caging objectsusing at most three simple, curved effectors with only frictionless, rigid, unilateralcontacts. Based on these results, we addressed the problem of whole-arm graspingfor grasping objects that are large compared to the size of end-effectors. Our futurework addresses optimizing the geometry of the curved effectors and incorporatingmore elaborate sensing capabilities.


Acknowledgements We gratefully acknowledge the support of NSF 1328805, 1138847, and ARLGrant W911NF-10-2-0016.

References

[1] Aichholzer, O., Demaine, E.D., Erickson, J., Hurtado, F., Overmars, M., Soss,M.A., Toussaint, G.T.: Reconfiguring convex polygons. Computational Ge-ometry: Theory and Applications 20(1–2) (2001)

[2] Bose, P., Bremner, D., Toussaint, G.T.: All convex polyhedra can be clampedwith parallel jaw grippers. Computational Geometry: Theory and Applications6, 291 – 302 (1996)

[3] Connelly, R., Demaine, E., Rote, G.: Straightening polygonal arcs and convex-ifying polygonal cycles. Discrete Comput Geom 30, 205–239 (2003)

[4] Hsiao, K., Lozano-Perez, T.: Imitation learning of whole-body grasps.IEEE/RSJ Int. Conference on Intelligent Robots and Systems (IROS) (2006)

[5] Iben, H., O’Brien, J., Demaine, E.: Refolding planar polygons. Discrete &Computational Geometry 41(3), 444–460 (2009)

[6] Mason, M.T.: Mechanics of robotic manipulation. MIT press (2001)[7] Miller, A.T., Knoop, S., Christensen, H.I., Allen, P.K.: Automatic grasp plan-

ning using shape primitives. In: Robotics and Automation, 2003. Proceedings.IEEE International Conference on, vol. 2, pp. 1824–1829 (2003)

[8] Murray, R.M., Li, Z., Sastry, S.S.: A mathematical introduction to robotic ma-nipulation. CRC (1994)

[9] Nguyen, V.D.: Constructing stable grasps. The International Journal ofRobotics Research 8(1), 26–37 (1989)

[10] Peshkin, M., Sanderson, A.: Reachable grasps on a polygon: the convex ropealgorithm. Robotics and Automation, IEEE Journal of 2(1), 53–58 (1986)

[11] Rimon, E., Blake, A.: Caging planar bodies by one-parameter two-fingeredgripping systems. The International Journal of Robotics Research 18(3), 299–318 (1999)

[12] Rimon, E., Burdick, J.W.: Mobility of bodies in contact - part I: A 2nd-ordermobility index for multiple-finger grasps. Robotics and Automation, IEEETransactions on 14, 696 – 708 (1998)

[13] Rimon, E., Burdick, J.W.: Mobility of bodies in contact - Part II: How forcesare generated by curvature effects. Robotics and Automation, IEEE Transac-tions on 14, 709 – 717 (1998)

[14] Rodriguez, A., Mason, M., Ferry, S.: From caging to grasping. The Interna-tional Journal of Robotics Research 31(7), 886–900 (2012)

[15] Seo, J., Kumar, V.: Spatial, bimanual, whole-arm grasping. In: IEEE/RSJ In-ternational Conference on Intelligent Robots and Systems (IROS) (2012)

[16] Vahedi, M., van der Stappen, A.: Caging polygons with two and three fingers.The International Journal of Robotics Research 27(11-12), 1308–1324 (2008)

restraining objects with curved effectors and its...

Documents