Geometric construction of 4-finger force-closure grasps for polyhedral objects

Geometric construction of 4-finger force-closure grasps for polyhedral objects Ricardo Prado Gardini, Raúl Suárez Feijoo IOC-DT-P-2005-23 Novembre 2005 1 Geometric Construction of 4-Finger Force-Closure Grasps for Polyhedral Objects Ricardo Prado Gardini Raúl Suárez Feijóo Institute of Industrial and Control Engineering (IOC) Polytechnic University of Catalonia (UPC) Diagonal 647, planta 11. 08028 Barcelona; España {prado, suarez}@upc.edu} Index 1. Introduction..........................................................................................2 2. Assumptions and basic nomenclature................................................3 3. Force-closure grasp..............................................................................3 4. Selection of the sets of four faces that allow FCG.............................4 4.1 Selection of faces according to their orientations.......................................4 4.2 Selection of faces according to their positions...........................................7 5. Quality of the sets of faces ...................................................................8 6. Determination of the contact points ...................................................9 Case of concurrent grasps ..............................................................................9 Case of flat-pencil grasps .............................................................................10 Case of regulus grasps..................................................................................10 7. Examples .............................................................................................11 8. Conclusion...........................................................................................12 References ..................................................................................................13 This work was partially supported by the CICYT projects DPI-2004-03104 and DPI-2002-03540. 2 1. Introduction A force-closure grasp has the property to reject external forces applied on the grasped object by means of the forces applied by the fingers. The theory regarding force-closure grasps has been deeply studied, and different techniques have been proposed for different cases [1-10]. The force-closure grasps with four non-planar contact forces are classified into three categories [8]: concurrent, flat-pencil and regulus. In a concurrent grasp the lines of action of the four contact forces intersect in a point (Figure 1a). In a flat-pencil grasp the lines of action of two contact forces intersect in a point and those of the other two forces intersect in another point, with these two points laying on the intersection of the planes defined by each pair of lines of action (Figure 1b). In a regulus grasp two lines of action are on different sides of a plane parallel to them and at a distance d to this plane, in the same way the other two lines of action are on different sides of a plane parallel to them and at a distance d to this plane, and the projections of each pair of lines of actions on their corresponding parallel plane must form a concurrent or a flat-pencil (regulus grasp, Figure 1c). Ponce et al. [8] developed a technique to build concurrent grasps, they determine the sets of four faces whose relative orientations satisfy a sufficient condition and their relative positions allow this type of grasp, and then the set of faces that optimizes an objective function is selected to be contacted by the fingers. However, the method does not work for flat-pencil and regulus grasps. Sundang and Ponce [9] proposed a method for the construction of the three types of non-planar grasps over four faces whose relative orientations satisfy a sufficient condition but assuming that the relative positions of the faces to be contacted by the fingers allow flat-pencil and regulus grasps. Yoshikawa [12] developed a model to determine the internal forces for the three types of non-planar grasps considering that the contact points are known. In this work, a method to build the three types of non-planar grasps is proposed. First, the sets of four object faces whose relative orientations satisfy a necessary and sufficient condition and whose relative positions allow the existence of three (concurrent, flatpencil and regulus) or two (flat-pencil and regulus) types of non-coplanar grasps are selected (the determination of sets of four faces that only allow regulus grasp is not considered in this paper). Second, from these sets the one that maximizes a quality function is selected and, finally, on the selected faces four contact points assuring a force-closure grasp are determined according to the possible types of non-planar grasps. 3 2. Assumptions and basic nomenclature The following assumptions are considered in this work: • The objects are polyhedrons. • The grasp is done using four fingers and each finger contacts with a different face of the object. • Only the fingertips will contact with the object surface and the contact is a point (then for stability reasons, the contact points cannot be on an object edge). • The friction coefficient μ is constant. The following basic nomenclature will be used: Pi: contact point on the object surface (i =1,2,3,4). Ai: contacted face of the object (i =1,2,3,4). ni: unitary vector with object inward direction normal to Ai. α=tg-1 μ : half-angle of the friction cone (α<π/2). Cfi: friction cone with half-angle α, axis parallel to ni and vertex at Pi. Ci: friction cone with half-angle α, axis parallel to ni and vertex at the origin of the reference system (representation of Cfi in the force space). fi: contact force applied at contact Pi (fi⊂Cfi). cm: object center of mass. 3. Force-closure grasp A force-closure grasp (FCG) must satisfy [3]: 1 n iex i= = ∑ f F and 1 n ii ex i= ×= ∑r f M (1) where n is the number of contact points, ri is the vector from the object center of mass to the contact point Pi, and Fex and Mex are, respectively, any external arbitrary force and torque applied on the object. Fig. 1. Types of grasps: a) Concurrent; b) flat-pencil; c) regulus. c) A2 P4 a) A1 A3 A4 f2 f1 f4 f3 P1 P2 P3 Π1 Π2 Π1∩Π2 A3 b) d Π 1 ∩ Π 2 A1 P1 f1f2 P2 A2 f3 f4 A4 P4 Π 2 Π 1 Π1∩Π2 A2 P2 f2 Π2 P4 f4 A3 f3 A1 P1 f1 d d d A4 4 A necessary condition for the existence of a FCG is that equations (1) must be satisfied for Fex=0 and Mex=0 [3]-[8]. In the case of a FCG with four fi without any three of them acting in the same plane, the following two conditions must be satisfied when Fex=0 and Mex=0 [8][9][10]. C1: f1, f2, f3 and f4 span ℜ3. C2: The lines of action of the applied forces form a concurrent, flat-pencil, or regulus grasp. Ponce et al. (1997) demonstrated that if fi∈Cfi, i=1,2,3,4, satisfy the conditions C1 and C2, then the contact points allow a FCG. Also, if a set of four faces allows a concurrent grasp with the contact points in the interior of the face (i.e. the contacts do not belong to the face boundary) then it is always possible to determine flat-pencil and regulus grasps on the same set of faces; moreover, the different types of grasp can be reached using the same directions of force by changing only the contact points. In the same way, if a set of four faces allows a flat-pencil grasp (but not necessarily a concurrent grasp) then it is always possible to determine a regulus grasp on the same set of faces. The approach presented in this report determines non-coplanar grasps on sets of faces that allow at least two types of non-coplanar grasps, i.e. the sets of four faces that allow only regulus grasp are not considered in this work. 4. Selection of the sets of four faces that allow FCG The selection of the sets of four object faces that allow at least two types of non-planar grasps is done in two phases: 1. Selection of faces according to their orientations. 2. Selection of faces according to their positions (from those passing the first phase). Each phase is described in the following subsections. 4.1 Selection of faces according to their orientations In this phase the sets of four faces whose relative orientations allow the application of forces fi∈Ci i=1,2,3,4 that span ℜ3 are selected. Then, for each of these sets of faces, subsets *C of the friction cones Ci are determined such that, if fi∈*Ci i=1,2,3,4, the four fi span ℜ3 with independence of the contact point. A sufficient condition for the existence of fi∈Ci, i=1,2,3,4, that span ℜ3 is that 0∈ConvexHull(n1,n2,n3,n4) [8]. Therefore the sets of faces that satisfy this condition are selected as candidates for a FCG. However, due to friction, there exist sets of faces with 0∉ConvexHull(ni) that also allow applied forces fi∈Ci, i=1,2,3,4, that span ℜ3. These sets of faces satisfy the conditions in Proposition 1 below and are also considered as candidates for a FCG. Let: Πl be the closest plane to the origin that contains a face of ConvexHull(n1,n2,n3,n4), i=1,2,3,4 (Figure 2). ϕ be the angle between Πl and any of the three ni that determines Πl (note that ϕ is the same for any i). 'Πl be the plane parallel to Πl that contains the origin. 5 vil and vir be the two unitary vectors that indicate the two boundary directions of 'Πl∩Ci, i=1,2,3,4, respectively (vil and vir are not defined when 'Πl∩Ci=∅ and vil=vir when Ci is tangent to 'Πl). Proposition 1. Given four faces such that 0∉ConvexHull(n1,n2,n3,n4), ∃ fi∈Ci, i=1,2,3,4 spanning ℜ3 iff: 1. ϕ < α. 2. 0∈ConvexHull(v1l, v1r, v2l, v2r, v3l, v3r,v4l, v4r) ■ Proof. If ϕ ≥ α then all four cones Ci lie in one of the half-spaces of ℜ3 defined by 'Πl and therefore the vectors in the other half-space can not be obtained as a linear combination of any four vectors from the cones Ci (note that vil and vir do not exist for ϕ>α). Then ϕ < α must be satisfied. If ϕ < α and 0∉ConvexHull(v1l, v1r, v2l, v2r, v3l, v3r,v4l, v4r) then the plane 'Πl can not be spanned by a linear combination of the components on 'Πl of any four vectors from the cones Ci, and therefore some vectors of ℜ3 cannot be obtained. If ϕ < α and 0∈ConvexHull(v1l, v1r, v2l, v2r, v3l, v3r,v4l, v4r) then the plane 'Πl can be spanned by a linear combination of the components on 'Πl of four vectors from the friction cones Ci and, at the same time, there are force components in the two halfspaces of ℜ3 defined by 'Πl, as a consequence any vector of ℜ3 can be obtained as a linear combination of four vectors, one from each Ci. ■ After selecting the set of faces that allow FCG, the subsets *Ci of the friction cones directions Ci that assure the FCG have to be determined. It is done as follows. Let: +S and –S be the two half-spaces defined by 'Πl (Figure 3a), with +S containing the ni that does not define Πl. +Ci and –Ci be the two largest cones contained in Ci ∩+S and Ci∩–S, respectively (if Ci ∩+S=∅ then +Ci does not exist and –Ci=Ci, and vice versa if Ci ∩–S=∅). +ni and –ni be unitary vectors along the axis of +Ci and –Ci respectively (if +Ci=Ci ⇒+ni= ni and if –Ci=Ci ⇒–ni= ni). Now, four unitary non-coplanar vector *ni∈Ci, i=1,2,3,4, are computed as: o C1 v1l v1r ' Π l∩C1 o n 1 n2 n3 n4 ϕ Π l 'Πl Fig. 2. Selection of faces according to their orientations for a set of faces with 0∉ConvexHull(n1,n2,n3,n4). convexhull ( n 1 , n 2 , n 3 , n 4 ) 6 • If the extremes of ni i=1,2,3,4 are non-coplanar (Figure 3b), then three *ni are equal to the –ni corresponding to the three ni that define Πl, and the fourth *ni is equal to the +ni of the remaining ni (i.e. the one that does not defines Πl). • If the extremes of ni i=1,2,3,4 are coplanar (Figure 3c), then the convex hulls defined by the sets of four vectors –ni or +ni i=1,2,3,4 are computed. Then, *ni i=1,2,3,4 are equal to the corresponding four vectors (either –ni or +ni) that determine the convex hull with largest volume. Let now: Πjkr be the plane parallel to the triangle defined by the extremes of *nj, *nk and *nr for j,k,r∈{1,2,3,4} with j≠k≠r, and passing through the origin (since the extremes of *ni i=1,2,3,4, are not coplanar there exist four different Πjkr, Figure 4a). +Si and -Si be the two half-spaces defined by Πjkr, where *ni∈+Si, i,j,k,r∈{1,2,3,4} with i≠j≠k≠r. The independent subset *Ci of each friction cone Ci is determined such that *Ci⊂+Si and *Cj, *Ck and *Cr are included in -Si, therefore *Ci⊂+Si∩-Sj∩-Sk∩-Sr. +Si∩-Sj∩-Sk∩-Sr determines a polyhedral convex cone Ti of three faces, each one on a plane Πijk, i=1,…,4 and j,k∈{1,2,3,4} with i≠j≠k. For instance, the faces of T4 (Figure 4a) are on the planes Π124, Π134 and Π234, respectively, and their edges are on the intersections of each pair of these planes. Π123 doest not determine T4, but determines T1, T2 and T3, respectively and defines the half-spaces +S4 and -S4 such that T4⊂+S4 and T1, T2 and T3 are included in -S4. a) n4 n3 n2 n1 n1 o n2 n3 n4 n2 n4 n 3 n1 o n2 n4 n3 n1 b) c) Fig. 3. a) Determination of * n1; b) two cases where the extremes of ni are not coplanar; c) two cases where the extremes of ni are coplanar. α -C1 ' Π l ∩ C1 +C1 C1 +S -S n1 n2n3 n4 ' Π l +n1 ( α + ϕ )/2 *n1= -n1 ϕ 7 Since the four convex cones Ti (i=1,…,4) are determined by the four planes Πjkr (j,k,r∈{1,2,3,4} with j≠k≠r), then it is always satisfied that the three edges of the negated of Ti, represented as -Ti (Figure 4b), are respectively an edge of Tj, Tk and Tr. This implies that any vector that belongs to -Ti can be obtained as a linear combination of three vectors, one from Tj, Tk and Tr, respectively. As a consequence any four vectors, one from each Ti, i=1,2,3,4, always span ℜ3. Note that by construction Ti≠∅, i=1,2,3,4, even if 0∉ConvexHull(*n1,*n2,*n3,*n4). The conditions satisfied by the orientations of the faces to allow a FCG assures that Ti∩Ci≠∅, then *Ci is determined as *Ci=Ti∩Ci and it is non null. Finally, each set of directions *Ci is approximated by the largest cone included in Ti∩Ci (Figure 4c). 4.2 Selection of faces according to their positions From previous section, it must be satisfied fi∈*Ci⊂Ci, i=1,2,3,4, in order to allow the FCG with independence of the particular contact points on the object faces. In order to have the largest range of variation of the directions of fi to keep a FCG when Fex≠0 and Mex≠0, it is desirable the direction of fi to be aligned with the axis, nfi, of *Ci when Fex=0 and Mex=0, and so is considered as a constraint in this selection of faces. Now, given a set of four faces Ai, i=1,2,3,4, the procedure to test if it is valid to produce a FCG is the following: 1. Determine the volumen, Di, swept by Ai when Ai is displaced in the direction of nfi, i=1,2,3,4, (Figure 5). 2. Compute Dj∩Dk, for any j∈{1,2,3,4} and ∀k∈{1,2,3,4} with j≠k. If ∀k Dj∩Dk=∅ then return (Invalid). *n4 n4 Π 123 Π124 Π234 Π 134 C4 T4 ∩ C4 n4 a) c) Fig. 4. Determination of: a) the plane Π jkr; b) cone T4; c) *Cf4. *Cf4 *n3 *n2 *n1 *n2 *n3 -T4 T3 b) Π123 T4 +S4 -S4 T4=+S4 ∩ -S1 ∩ -S2 ∩ -S3 *n1 Π234∩Π124 Π134∩Π124 Π234 ∩ Π 134 T2 T1 8 In this case it is not possible for the line of action of fj, with direction of nfj, to intersect with the line of action of any fk, with the direction of nfk. 3. Determine two pairs (Dj∩Dk) and (Dr∩Dh), where at least Dj∩Dk≠∅, with {j,k,r,h}={1,2,3,4}. Consider the plane define by nfj and nfk and the plane defined by nfr and nfh, and let nI be the vector parallel to intersection of these two planes. 4. Determine the volumes, D' and D'', swept by Dj∩Dk and Dr∩Dh with {j,k,r,h}={1,2,3,4}, respectively, when they are displaced in the direction of nI. • If D'∩D'' = ∅ then return (Invalid). In this case, the intersection point of the lines of action of fj and fk and the intersection point of the lines of action of fr and fh do not lay on the intersection of the planes defined by each pair of lines of action (note that this intersection is always parallel to nI). • If D'∩D'' ≠ ∅ then return (Valid). In this case, the projections on Ai with directions of nfi of any two points contained in a straight line parallel to nI and belonging to D'∩D''∩Dj∩Dk and D'∩D''∩Dr∩Dh respectively, always determine a FCG. If a valid set of faces satisfies D1∩D2∩D3∩D4 ≠ ∅ (note that D1∩D2∩D3∩D4⊂D'∩D'') then it allows concurrent, flat pencil and regulus grasps; otherwise it only allows flat pencil and regulus grasps. This is a conservative approach because it selects sets of four faces that allow FCG and also must allow to apply fi with direction of nfi that allow at least two types of noncoplanar grasps, therefore there may be sets of faces that actually allow FCG that are not considered as valid in this procedure. 5. Quality of the sets of faces In order to select the set of faces to be contacted by the fingers, all valid sets of faces are evaluated according to a quality measure that considers: D1 D2 D1∩D2 Fig. 5. Selection of faces according their relative positions. nf2 nf1 D4 D3 D3∩D4 nf3 nf4 D3∩D4 n I D1 ∩ D2 CD D' D'' D'' ∩ D'