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Lecture 14 Polyhedral Convex Cones Lecture 14 Polyhedral Convex Cones Motivation, context Positive linear span Lecture 14 Types of cones Edge and face Polyhedral Convex Cones representation Supplementary cones; polar Representing Matthew T. Mason frictionless contact Cones in wrench space Force closure Cones in velocity Mechanics of Manipulation twist space Conclusion Lecture 14 Today’s outline Polyhedral Convex Cones Motivation, context Motivation, context Positive linear Positive linear span span Types of cones Types of cones Edge and face representation Edge and face representation Supplementary cones; polar Representing Supplementary cones; polar frictionless contact Cones in wrench Representing frictionless contact space Force closure Cones in velocity Cones in wrench space twist space Force closure Conclusion Cones in velocity twist space Conclusion Lecture 14 What use is a cone? I: velocity twists. Polyhedral Convex Cones Motivation, context Positive linear I Reuleaux’s analysis of unilateral constraint yields a span set of nominally feasible motions. Types of cones Edge and face I What does the corresponding set of IC’s look like, in representation the plane? Supplementary A triangle of signed rotation centers. cones; polar I What does the corresponding set of velocity twists Representing look like, in three space? frictionless contact Cones in wrench A batch of vertical, zero pitch screws hitting the space horizontal plane in a triangle. Force closure Cones in velocity I What does the corresponding set of velocity twists twist space look like, in velocity twist space (!z ; v0x ; v0y )? Conclusion A polyhedral convex cone. Lecture 14 What use is a cone? I: velocity twists. Polyhedral Convex Cones Motivation, context Positive linear I Reuleaux’s analysis of unilateral constraint yields a span set of nominally feasible motions. Types of cones Edge and face I What does the corresponding set of IC’s look like, in representation the plane? Supplementary A triangle of signed rotation centers. cones; polar I What does the corresponding set of velocity twists Representing look like, in three space? frictionless contact Cones in wrench A batch of vertical, zero pitch screws hitting the space horizontal plane in a triangle. Force closure Cones in velocity I What does the corresponding set of velocity twists twist space look like, in velocity twist space (!z ; v0x ; v0y )? Conclusion A polyhedral convex cone. Lecture 14 What use is a cone? I: velocity twists. Polyhedral Convex Cones Motivation, context Positive linear I Reuleaux’s analysis of unilateral constraint yields a span set of nominally feasible motions. Types of cones Edge and face I What does the corresponding set of IC’s look like, in representation the plane? Supplementary A triangle of signed rotation centers. cones; polar I What does the corresponding set of velocity twists Representing look like, in three space? frictionless contact Cones in wrench A batch of vertical, zero pitch screws hitting the space horizontal plane in a triangle. Force closure Cones in velocity I What does the corresponding set of velocity twists twist space look like, in velocity twist space (!z ; v0x ; v0y )? Conclusion A polyhedral convex cone. Lecture 14 What use is a cone? I: velocity twists. Polyhedral Convex Cones Motivation, context Positive linear I Reuleaux’s analysis of unilateral constraint yields a span set of nominally feasible motions. Types of cones Edge and face I What does the corresponding set of IC’s look like, in representation the plane? Supplementary A triangle of signed rotation centers. cones; polar I What does the corresponding set of velocity twists Representing look like, in three space? frictionless contact Cones in wrench A batch of vertical, zero pitch screws hitting the space horizontal plane in a triangle. Force closure Cones in velocity I What does the corresponding set of velocity twists twist space look like, in velocity twist space (!z ; v0x ; v0y )? Conclusion A polyhedral convex cone. Lecture 14 What use is a cone? II: wrenches. Polyhedral Convex Cones Motivation, context Positive linear span Types of cones Edge and face I What is the set of wrenches that can be applied by representation frictionless contacts on a rigid body? Supplementary cones; polar Polyhedral convex cones. Representing frictionless contact I What if we add friction? Cones in wrench Polyhedral convex cones. space Force closure Cones in velocity twist space Conclusion Lecture 14 Cones, the agenda. Polyhedral Convex Cones Motivation, context Positive linear span I Today: introduction to polyhedral convex cones; Types of cones Edge and face I Next time: the oriented plane. (The plane of signed representation points, i.e. Reuleaux’s plane. Convex polygons in the Supplementary cones; polar oriented plane are a way of representing cones.) Representing frictionless contact I Following: graphical methods, using cones and the Cones in wrench oriented plane, to work with wrenches and velocity space twists. Force closure Cones in velocity I Applications: grasping, manipulation. twist space Conclusion Lecture 14 Positive linear span Polyhedral Convex Cones n I For now, use n-dimensional vector space R . Later, Motivation, context Positive linear wrench space and velocity twist space. span n I Let v be any non-zero vector in R . Then the set of Types of cones vectors Edge and face representation fkv j k ≥ 0g Supplementary cones; polar describes a ray. Representing frictionless contact I Let v1, v2 be non-zero and non-parallel. Then the set Cones in wrench of positively scaled sums space Force closure Cones in velocity fk1v1 + k2v2 j k1; k2 ≥ 0g twist space Conclusion is a planar cone—sector of a plane. I We want to generalize to an arbitrary finite set of vectors ::: Lecture 14 Positive linear span Polyhedral Convex Cones Motivation, context Positive linear span Definition Types of cones Edge and face representation The positive linear span pos(·) of a set of vectors fvi g is Supplementary P cones; polar pos(fvi g) = f ki vi j ki ≥ 0g Representing frictionless contact Cones in wrench space I (The positive linear span of the empty set is the Force closure Cones in velocity origin.) twist space Conclusion Lecture 14 Relatives of positive linear span Polyhedral Convex Cones Motivation, context Positive linear Definition span Types of cones The linear span lin(·) is Edge and face representation P lin(fvi g) = f ki vi j ki 2 Rg Supplementary cones; polar Representing frictionless contact Definition Cones in wrench space The convex hull conv(·) is Force closure Cones in velocity twist space P P conv(fvi g) = f ki vi j ki ≥ 0; ki = 1g Conclusion Lecture 14 Varieties of cones in three space Polyhedral Convex Cones 1 edge a. ray Motivation, context Positive linear span 2edges b. line c. planar cone Types of cones Edge and face representation Supplementary cones; polar 3edges Representing d. solid cone e. half plane f. plane frictionless contact Cones in wrench space Force closure Cones in velocity twist space 4edges Conclusion g. wedge h. half space i. whole space I By the way, how many different ways are there of positively spanning a linear space? Start with Chandler Davis, Theory of positive linear dependence, American Journal of Mathematics, 76(4), Oct. 1954. n Lecture 14 Spanning all of R Polyhedral Convex Cones Motivation, context Positive linear Theorem span Types of cones n A set of vectors fvi g positively spans the entire space R Edge and face if and only if the origin is in the interior of the convex hull: representation Supplementary n cones; polar pos(fvi g) = R $ 0 2 int(conv(fvi g)) Representing frictionless contact Cones in wrench space (Review meaning of “interior”.) Force closure Cones in velocity Theorem twist space It takes at least n + 1 vectors to positively span Rn. Conclusion Lecture 14 Representing cones Polyhedral Convex Cones Motivation, context Positive linear span Types of cones Edge and face representation I Two ways to represent cones: edge representation Supplementary and face representation. cones; polar Representing I Edge representation uses positive linear span. Given frictionless contact a set of edges fei g, the cone is given by pos(fei g). Cones in wrench space Force closure Cones in velocity twist space Conclusion Lecture 14 Face representation of cones Polyhedral Convex Cones Motivation, context Positive linear I First represent planar half-space by inward pointing span normal vector n. Types of cones Edge and face representation half(n) = fv j n · v ≥ 0g Supplementary cones; polar (Here we use dot product, but when working with Representing twists and wrenches we will use reciprocal product.) frictionless contact Cones in wrench I Consider a cone with face normals fn g. Then the space i Force closure cone is the intersection of the half-spaces: Cones in velocity twist space Conclusion \fhalf(ni )g Lecture 14 Supplementary cone; polar Polyhedral Convex Cones Motivation, context Positive linear span Definition Types of cones Given a polyhedral convex cone Edge and face representation V , the supplementary cone Supplementary supp(V ) (also known as the polar) cones; polar Representing comprises the vectors that make frictionless contact non-negative dot products with all Cones in wrench space the vectors in V : Force closure Cones in velocity fu 2 Rn j u · v ≥ 0 8v 2 V g twist space Conclusion Lecture 14 Supplementary cone; polar; representation Polyhedral Convex Cones Motivation, context Positive linear span I The supplementary cone’s Types of cones edges are the original cone’s Edge and face representation face normals, and vice versa. Supplementary So if cones; polar Representing frictionless contact V = pos(fe g) = \fhalf(n )g i i Cones in wrench space then Force closure Cones in velocity twist space supp(V ) = pos(fni g) = \fhalf(eiConclusion)g Lecture 14 Frictionless contact Polyhedral Convex Cones Motivation, context Positive linear span Types of cones Characterize contact by set of possible wrenches.
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