13. Polyhedral Convex Cones

Transcription

1 13. Polyhedral Convex Cones Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 13. Mechanics of Manipulation p.1

2 Lecture 13. Cones. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system Issues in manipulation A taxonomy of manipulation techniques Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Exercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Exercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic systems Kinematic models of contact Bibliographic notes 88 Exercises 88 Chapter 5 Rigid Body Statics Forces acting on rigid bodies Polyhedral convex cones Contact wrenches and wrench cones Cones in velocity twist space The oriented plane Instantaneous centers and Reuleaux s method Line of force; moment labeling Force dual Summary Bibliographic notes 117 Exercises 118 Chapter 6 Friction Coulomb s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Exercises 139 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 173 Exercises 175 Chapter 8 Dynamics Newton s laws A particle in three dimensions Moment of force; moment of momentum Dynamics of a system of particles Rigid body dynamics The angular inertia matrix Motion of a freely rotating body Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Exercises 208 Chapter 9 Impact A particle Rigid body impact Bibliographic notes 223 Exercises 223 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Brie y dynamic manipulation Continuously dynamic manipulation Bibliographic notes 232 Exercises 235 Appendix A Infinity 237 Lecture 13. Mechanics of Manipulation p.2

3 Outline. 1. Positive linear span 2. Types of cones 3. Edge and face representation 4. Supplementary cones; polar 5. Representing frictionless contact 6. Cones in wrench space force closure 7. Cones in velocity twist space Lecture 13. Mechanics of Manipulation p.3

4 Positive linear span For now, use n-dimensional vector space R n. Later, wrench space and velocity twist space. Let v be any non-zero vector in R n. Then the set of vectors describes a ray. {kv k 0} (1) Let v 1, v 2 be non-zero and non-parallel. Then the set of positively scaled sums {k 1 v 1 + k 2 v 2 k 1, k 2 0} (2) is a planar cone sector of a plane. Generalize by defining the positive linear span of a set of vectors {v i }: pos({v i }) = { k i v i k i 0} (3) (The positive linear span of the empty set is the origin.) Lecture 13. Mechanics of Manipulation p.4

5 Relatives of positive linear span The linear span The convex hull lin({v i }) = { k i v i k i R} (4) conv({v i }) = { k i v i k i 0, k i = 1} (5) Lecture 13. Mechanics of Manipulation p.5

6 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

7 Spanning all of R n Theorem: A set of vectors {v i } positively spans the entire space R n if and only if the origin is in the interior of the convex hull: pos({v i }) = R n 0 int(conv({v i })) (6) Theorem: It takes at least n + 1 vectors to positively span R n. Lecture 13. Mechanics of Manipulation p.7

8 Representing cones Two ways to represent cones: edge representation and face representation. Edge representation uses positive linear span. Given a set of edges {e i }, the cone is given by pos({e i }). Lecture 13. Mechanics of Manipulation p.8

9 Face representation of cones First represent planar half-space by inward pointing normal vector n. half(n) = {v n v 0} (7) (Here we use dot product, but when working with twists and wrenches we will use reciprocal product.) Consider a cone with face normals {n i }. Then the cone is the intersection of the half-spaces: {half(n i )} (8) Lecture 13. Mechanics of Manipulation p.9

10 Supplementary cone; polar Supplementary cone supp(v ) (also known as polar) comprises the vectors that make non-negative dot products with vectors in V : {u R n u v 0 v V } (9) The supplementary cone s edges are the original cone s face normals, and vice versa. So if V = pos({e i }) = {half(n i )} (10) then supp(v ) = pos({n i }) = {half(e i )} (11) Lecture 13. Mechanics of Manipulation p.10

11 Frictionless contact Characterize contact by set of possible wrenches. Assume uniquely determined contact normal. Assume frictionless contact can give arbitrary magnitude force along inward-pointing normal. Then a frictionless contact gives a ray in wrench space, pos(w), where w = (c, c 0 ) is the contact screw. Lecture 13. Mechanics of Manipulation p.11

12 Two contacts Given two frictionless contacts w 1 and w 2, total wrench is the sum of possible positive scalings of w 1 and w 2 : i.e. the positive linear span pos({w 1, w 2 }). k 1 w 1 + k 2 w 2 ; k 1, k 2 0 (12) Generalizing: Theorem: If a set of frictionless contacts on a rigid body is described by the contact normals w i = (c i, c 0i ) then the set of all possible wrenches is given by the positive linear span pos({w i }). Lecture 13. Mechanics of Manipulation p.12

13 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

14 Example wrench cone w n w 1 Construct unit magnitude force at each contact. Write screw coords of wrenches. w 2 Take positive linear span. w Exhausts wrench space? Lecture 13. Mechanics of Manipulation p.14

15 Cones in velocity twist space Cannot use finite displacement twists. They are not vectors. Velocity twists are vectors! Let {w i } be a set of contact normals. Let W = pos({w i }) be set of possible wrenches. First order analysis: velocity twists T must be reciprocal or repelling to contact wrenches: T = supp(w ). Lecture 13. Mechanics of Manipulation p.15

16 Next: The Oriented Plane. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system Issues in manipulation A taxonomy of manipulation techniques Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Exercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Exercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic systems Kinematic models of contact Bibliographic notes 88 Exercises 88 Chapter 5 Rigid Body Statics Forces acting on rigid bodies Polyhedral convex cones Contact wrenches and wrench cones Cones in velocity twist space The oriented plane Instantaneous centers and Reuleaux' s method Line of force; moment labeling Force dual Summary Bibliographic notes 117 Exercises 118 Chapter 6 Friction Coulomb' s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Exercises 139 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 173 Exercises 175 Chapter 8 Dynamics Newton' s laws A particle in three dimensions Moment of force; moment of momentum Dynamics of a system of particles Rigid body dynamics The angular inertia matrix Motion of a freely rotating body Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Exercises 208 Chapter 9 Impact A particle Rigid body impact Bibliographic notes 223 Exercises 223 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Briefly dynamic manipulation Continuously dynamic manipulation Bibliographic notes 232 Exercises 235 Appendix A Infinity 237 Lecture 13. Mechanics of Manipulation p.16