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 (ωz , v0x , v0y )? Conclusion A polyhedral convex cone. Lecture 14 What use is a cone? I: velocity twists. 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 {kv | k ≥ 0} 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 {k1v1 + k2v2 | k1, k2 ≥ 0} 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 {vi } is Supplementary P cones; polar pos({vi }) = { ki vi | ki ≥ 0} 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({vi }) = { ki vi | ki ∈ R} 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({vi }) = { ki vi | ki ≥ 0, ki = 1} 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 {vi } 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({vi }) = R ↔ 0 ∈ int(conv({vi })) 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 {ei }, the cone is given by pos({ei }). 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) = {v | n · v ≥ 0} 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 {n }. Then the space i Force closure

cone is the intersection of the half-spaces: Cones in velocity twist space

Conclusion ∩{half(ni )} 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 {u ∈ Rn | u · v ≥ 0 ∀v ∈ V } 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({e }) = ∩{half(n )} i i Cones in wrench space then Force closure Cones in velocity twist space supp(V ) = pos({ni }) = ∩{half(eiConclusion)} Lecture 14 Frictionless contact Polyhedral Convex Cones

Motivation, context

Positive linear span

Types of cones Characterize contact by set of possible wrenches. Edge and face representation

Assume uniquely determined contact normal. Supplementary cones; polar

Assume frictionless contact can give arbitrary magnitude Representing force along inward-pointing normal. frictionless contact Cones in wrench Then a frictionless contact gives a ray in wrench space, space Force closure

pos(w), where w = (c, c0) is the contact screw. Cones in velocity twist space

Conclusion Lecture 14 Two contacts Polyhedral Convex Cones

Motivation, context Given two frictionless contacts w1 and w2, total wrench is Positive linear span the sum of possible positive scalings of w1 and w2: Types of cones

Edge and face k1w1 + k2w2; k1, k2 ≥ 0 representation

Supplementary i.e. the positive linear span pos({w1, w2}). cones; polar Representing Generalizing: frictionless contact Cones in wrench space Theorem Force closure If a set of frictionless contacts on a rigid body is Cones in velocity twist space

described by the contact normals wi = (ci , c0i ) then the Conclusion set of all possible wrenches is given by the positive linear span pos({wi }). Lecture 14 Force closure Polyhedral Convex Cones

Motivation, context

Positive linear span Deﬁnition Types of cones Edge and face Force closure means that the set of possible wrenches representation exhausts all of wrench space. Supplementary cones; polar

Representing It follows from a previous theorem that a frictionless force frictionless contact Cones in wrench closure requires at least 7 contacts. Or, since planar space wrench space is only three-dimensional, frictionless force Force closure Cones in velocity closure in the plane requires at least 4 contacts. twist space Conclusion Lecture 14 Example wrench cone Polyhedral Convex Cones

Motivation, context

Positive linear span Construct unit Types of cones magnitude force at Edge and face representation each contact. w 3 Supplementary nz cones; polar y Write screw coords of w 1 Representing wrenches. frictionless contact fx x fy Cones in wrench

w 2 Take positive linear space Force closure w 4 span. Cones in velocity Exhausts wrench twist space space? Conclusion Lecture 14 Cones in velocity twist space Polyhedral Convex Cones

Motivation, context

Positive linear span Cannot use ﬁnite displacement twists. They are not Types of cones Edge and face vectors. representation Supplementary Velocity twists are vectors! cones; polar

Representing Let {wi } be a set of contact normals. frictionless contact Let W = pos({w }) be set of possible wrenches. Cones in wrench i space First order analysis: velocity twists T must be reciprocal Force closure Cones in velocity or repelling to contact wrenches: T = supp(W ). twist space Conclusion Lecture 14 “Reciprocal or repelling” ≡ supp(·) Polyhedral Convex Cones

Motivation, context

Positive linear span

Types of cones

I The PCC of possible wrenches, and the PCC of Edge and face feasible velocity twists, are supplementary cones! representation Supplementary I This raises some interesting questions. Most cones; polar Representing speciﬁcally, we have a keen two-dimensional way of frictionless contact

representing the feasible velocity twists: Reuleaux’s Cones in wrench space method. Can we do the same thing for wrenches? Force closure

More later! Cones in velocity twist space

Conclusion Lecture 14 Conclusion Polyhedral Convex Cones

Motivation, context

Positive linear span

I You can represent contact constraints as PCC’s in Types of cones

wrench space. Edge and face representation

I You can represent feasible velocities as PCC’s in Supplementary velocity twist space. cones; polar Representing I The twist cone will be supplementary to the wrench frictionless contact cone. Cones in wrench space I So ... what about those defective cases, where Force closure Cones in velocity Reuleaux gives false positives? Example: triangle twist space

with a force focus. Conclusion