Positive linear span n For now, use n-dimensional vector space R . Later, wrench space and velocity twist space. n Let v be any non-zero vector in R . Then the set of vectors fkv j k ≥ 0g (1) describes a ray. Let v1, v2 be non-zero and non-parallel. Then the set of positively scaled sums fk1v1 + k2v2 j k1; k2 ≥ 0g (2) is a planar cone—sector of a plane. Generalize by defining the positive linear span of a set of vectors fvig: pos(fvig) = fP kivi j ki ≥ 0g (3) (The positive linear span of the empty set is the origin.) Lecture 13. Mechanics of Manipulation – p.4 Relatives of positive linear span The linear span lin(fvig) = fP kivi j ki 2 Rg (4) The convex hull conv(fvig) = fP kivi j ki ≥ 0; P ki = 1g (5) Lecture 13. Mechanics of Manipulation – p.5 Varieties of cones in three space 1 edge a. ray 2 edges b. line c. planar cone 3 edges d. solid cone e. half plane f. plane 4 edges g. wedge h. half space i. whole space Lecture 13. Mechanics of Manipulation – p.6 n Spanning all of R n Theorem: A set of vectors fvig positively spans the entire space R if and only if the origin is in the interior of the convex hull: n pos(fvig) = R $ 0 2 int(conv(fvig)) (6) n Theorem: It takes at least n + 1 vectors to positively span R . Lecture 13. Mechanics of Manipulation – p.7 Two contacts Given two frictionless contacts w1 and w2, total wrench is the sum of possible positive scalings of w1 and w2: k1w1 + k2w2; k1; k2 ≥ 0 (12) i.e. the positive linear span pos(fw1; w2g). Generalizing: Theorem: If a set of frictionless contacts on a rigid body is described by the contact normals wi = (ci; c0i) then the set of all possible wrenches is given by the positive linear span pos(fwig). Lecture 13. Mechanics of Manipulation – p.12 Force closure Definition: Force closure means that the set of possible wrenches exhausts all of wrench space. It follows from theorem ? that a frictionless force closure requires at least 7 contacts. Or, since planar wrench space is only three-dimensional, frictionless force closure in the plane requires at least 4 contacts. Lecture 13. Mechanics of Manipulation – p.13 Example wrench cone Construct unit magnitude 3 w 3 force at each contact. 4 nz y Write screw coords of w 1 fx wrenches. x fy 2 w 2 Take positive linear span. 1 w 4 Exhausts wrench space? Lecture 13. Mechanics of Manipulation – p.14.