14. The Oriented Plane Mechanics of Manipulation

Transcription

1 14. The Oriented Plane Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 14. Mechanics of Manipulation p.1

2 Exhortations Hands are levers of influence on the world that made intelligence worth having. Precision hands and precision intelligence coevolved in the human lineage, and the fossil record shows that hands led the way. Steven Pinker, in How the Mind Works Lecture 14. Mechanics of Manipulation p.2

3 Lecture 14. 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 14. Mechanics of Manipulation p.3

4 Outline. Reuleaux s space...what is it? Formal definitions. Central projection. Convexity in the oriented plane. Relation to polyhedral convex cones. Rotation centers and the oriented plane. Lecture 14. Mechanics of Manipulation p.4

5 Reuleaux s space Reuleaux represents PCC s in planar diff l twist space. 3D twist space is 6D. planar twist space is 3D. Reuleaux does it in 2D! What space do signed rotation centers live in? Plane with + and/or label, and points at infinity? Stolfi and Guibas: the oriented plane. Lecture 14. Mechanics of Manipulation p.5

6 Formal definition Consider homogeneous coordinates (x, y, w) (0, 0, 0). A point in the oriented plane is a ray of triples: {(kx, ky, kw) k > 0} (1) Three cases: w > 0 : Signed Euclidean point (x/w, y/w, ). w < 0 : Signed Euclidean point (x/w, y/w, ). w = 0 : Ideal point: point at infinity. Lecture 14. Mechanics of Manipulation p.6

7 Central projection Project each point (x, y, w) to w = 1 plane. Attach appropriate sign. Ideal points miss the plane but hit the equator. x w y positiveand negativeplanes superimposed lineatin finity Lecture 14. Mechanics of Manipulation p.7

8 Relation to projective plane Maybe we should call it the Oriented Projective Plane. Projective plane is set of lines through the origin of E 3. Oriented plane is set of directed lines through the origin of E 3. x w y positiveand negativeplanes superimposed lineatin finity The projective plane is the sphere S(2) with antipodes identified. The oriented plane is the sphere S(2). Northern hemisphere is the positive plane, Southern hemisphere is the negative plane, Equator is the ideal line. Lecture 14. Mechanics of Manipulation p.8

9 Why do we need it? Because, e.g. Reuleaux s method is easier than analyzing polyhedral convex cones in planar differential twist space. (or rather, Reuleaux s method is an easy way to analyze PCCs in planar differential twist space.) In practice, we work directly in the oriented plane, as Reuleaux did, not worrying too much about the projection. But to answer deep questions, refer back to the projection. Lecture 14. Mechanics of Manipulation p.9

10 Geometry An example question. We know what points are in the oriented plane. Are there lines? Lecture 14. Mechanics of Manipulation p.10

11 Geometry An example question. We know what points are in the oriented plane. Are there lines? Transform to a question about spheres. In spherical geometry, are there lines? How are they defined? Lecture 14. Mechanics of Manipulation p.10

12 Geometry An example question. We know what points are in the oriented plane. Are there lines? Transform to a question about spheres. In spherical geometry, are there lines? How are they defined? So... Two non-antipodal points determine a line. (The ideal line, or a Euclidean line labeled ±.) Every pair of lines intersects in two antipodal points. Lecture 14. Mechanics of Manipulation p.10

13 Convexity Another deep question: How does convexity work in the oriented plane? Lecture 14. Mechanics of Manipulation p.11

14 Convexity Another deep question: How does convexity work in the oriented plane? How is convexity defined for Euclidean space? Lecture 14. Mechanics of Manipulation p.11

15 Convexity Another deep question: How does convexity work in the oriented plane? How is convexity defined for Euclidean space? If a and b are in the figure, then so is ab. Lecture 14. Mechanics of Manipulation p.11

16 Convexity Another deep question: How does convexity work in the oriented plane? How is convexity defined for Euclidean space? If a and b are in the figure, then so is ab. What is ab in the oriented plane? What is conv({a, b})? Lecture 14. Mechanics of Manipulation p.11

17 Convexity: plus and plus Consider the polyhedral convex cone determined by the two rays. Project it to the plane. So, if a and b have the same sign then conv({a, b}) is the line segment between them with the same sign. Lecture 14. Mechanics of Manipulation p.12

18 Convexity: plus and minus Consider the PCC... Define the external line segment. Lecture 14. Mechanics of Manipulation p.13

19 Problem Set 4 Let s do PS4! How do you construct the convex hull of some set of points? 1. Pick two points, and construct their convex hull. 2. Repeat step 1, ad nauseum. Lecture 14. Mechanics of Manipulation p.14

20 Reuleaux s method Given a differential velocity twist (v 0x, v 0y, ω z ), what is the rotation center? Velocity at a point p = (x, y) is v 0 + ω p ( v 0x ωy = v 0y + ωx ) Set equal to (0, 0) and solve for (x, y)... Rotation center is at ( ) v 0y /ω v 0x /ω Central projection, and rotation of coordinates. Lecture 14. Mechanics of Manipulation p.15

21 Next: Moment labeling. 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 14. Mechanics of Manipulation p.16