14. the oriented plane
TRANSCRIPT
14. The Oriented PlaneMechanics of Manipulation
Matt [email protected]
http://www.cs.cmu.edu/~mason
Carnegie Mellon
Lecture 14. Mechanics of Manipulation – p.1
Exhortations“Hands are levers of influence on the world that made intelligenceworth having. Precision hands and precision intelligence coevolvedin the human lineage, and the fossil record shows that hands led theway.”—Steven Pinker, in How the Mind Works
Lecture 14. Mechanics of Manipulation – p.2
Lecture 14. The Oriented Plane.
Chapter 1 Manipulation 11.1 Case 1: Manipulation by a human 11.2 Case 2: An automated assembly system 31.3 Issues in manipulation 51.4 A taxonomy of manipulation techniques 71.5 Bibliographic notes 8
Exercises 8
Chapter 2 Kinematics 112.1 Preliminaries 112.2 Planar kinematics 152.3 Spherical kinematics 202.4 Spatial kinematics 222.5 Kinematic constraint 252.6 Kinematic mechanisms 342.7 Bibliographic notes 36
Exercises 37
Chapter 3 Kinematic Representation 413.1 Representation of spatial rotations 413.2 Representation of spatial displacements 583.3 Kinematic constraints 683.4 Bibliographic notes 72
Exercises 72
Chapter 4 Kinematic Manipulation 774.1 Path planning 774.2 Path planning for nonholonomic systems 844.3 Kinematic models of contact 864.4 Bibliographic notes 88
Exercises 88
Chapter 5 Rigid Body Statics 935.1 Forces acting on rigid bodies 935.2 Polyhedral convex cones 995.3 Contact wrenches and wrench cones 1025.4 Cones in velocity twist space 1045.5 The oriented plane 1055.6 Instantaneous centers and Reuleaux' s method 1095.7 Line of force; moment labeling 1105.8 Force dual 1125.9 Summary 1175.10 Bibliographic notes 117
Exercises 118
Chapter 6 Friction 1216.1 Coulomb' s Law 1216.2 Single degree-of-freedom problems 1236.3 Planar single contact problems 1266.4 Graphical representation of friction cones 1276.5 Static equilibrium problems 1286.6 Planar sliding 1306.7 Bibliographic notes 139
Exercises 139
Chapter 7 Quasistatic Manipulation 1437.1 Grasping and fixturing 1437.2 Pushing 1477.3 Stable pushing 1537.4 Parts orienting 1627.5 Assembly 1687.6 Bibliographic notes 173
Exercises 175
Chapter 8 Dynamics 1818.1 Newton' s laws 1818.2 A particle in three dimensions 1818.3 Moment of force; moment of momentum 1838.4 Dynamics of a system of particles 1848.5 Rigid body dynamics 1868.6 The angular inertia matrix 1898.7 Motion of a freely rotating body 1958.8 Planar single contact problems 1978.9 Graphical methods for the plane 2038.10 Planar multiple-contact problems 2058.11 Bibliographic notes 207
Exercises 208
Chapter 9 Impact 2119.1 A particle 2119.2 Rigid body impact 2179.3 Bibliographic notes 223
Exercises 223
Chapter 10 Dynamic Manipulation 22510.1 Quasidynamic manipulation 22510.2 Briefly dynamic manipulation 22910.3 Continuously dynamic manipulation 23010.4 Bibliographic notes 232
Exercises 235
Appendix A Infinity 237
Lecture 14. Mechanics of Manipulation – p.3
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
Reuleaux’s spaceReuleaux represents PCC’s in planardiff’l twist space.
3D twist space is 6D.
planar twist space is 3D.
Reuleaux does it in 2D!
What space do signed rotation cen-ters live in?
Plane with + and/or − label,
and points at infinity?
Stolfi and Guibas: the oriented plane.
Lecture 14. Mechanics of Manipulation – p.5
Formal definitionConsider homogeneous coordinates (x, y, w) 6= (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
Central projectionProject each point(x, y, w) to w = 1 plane.
Attach appropriate sign.
Ideal points miss theplane but hit the equator.
w
xy
positive andnegative planessuperimposed
line at in finity
Lecture 14. Mechanics of Manipulation – p.7
Relation to projective planeMaybe we should call it the OrientedProjective Plane.
Projective plane is set of lines throughthe origin of E
3.
Oriented plane is set of directed linesthrough the origin of E
3.
The projective plane is the sphereS(2) with antipodes identified.
The oriented plane is the sphere S(2).
Northern hemisphere is thepositive plane,
Southern hemisphere is thenegative plane,
Equator is the ideal line.
w
xy
positive andnegative planessuperimposed
line at in finity
(x,y,w)
(x/w,y/w,1)
(x,y,w)p
x2 + y2 + w 2
Lecture 14. Mechanics of Manipulation – p.8
Why do we need it?Because, e.g. Reuleaux’s method is easier than analyzingpolyhedral convex cones in planar differential twist space.
(or rather, Reuleaux’s method is an easy way to analyze PCCsin 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
GeometryAn example question. We know what points are in the orientedplane. Are there lines?
Transform to a question about spheres. In spherical geometry, arethere 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
GeometryAn example question. We know what points are in the orientedplane. Are there lines?
Transform to a question about spheres. In spherical geometry, arethere 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
GeometryAn example question. We know what points are in the orientedplane. Are there lines?
Transform to a question about spheres. In spherical geometry, arethere 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
ConvexityAnother “deep” question: How does convexity work in the orientedplane?
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
ConvexityAnother “deep” question: How does convexity work in the orientedplane?
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
ConvexityAnother “deep” question: How does convexity work in the orientedplane?
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
ConvexityAnother “deep” question: How does convexity work in the orientedplane?
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
Convexity: plus and plusConsider the polyhedral convex conedetermined by the two rays.
Project it to the plane.
So, if a and b have the same signthen conv({a, b}) is the line segmentbetween them with the same sign.
Lecture 14. Mechanics of Manipulation – p.12
Convexity: plus and minusConsider the PCC . . .
Define the external line segment.
Lecture 14. Mechanics of Manipulation – p.13
Problem Set 4Let’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
Reuleaux’s methodGiven a differential velocity twist (v0x, v0y, ωz), what is the rotationcenter?
Velocity at a point p = (x, y) is
v0 + ω × p
=
(
v0x − ωy
v0y + ωx
)
Set equal to (0, 0) and solve for (x, y) . . .
Rotation center is at(
−v0y/ω
v0x/ω
)
Central projection, and rotation of coordinates.
Lecture 14. Mechanics of Manipulation – p.15
Next: Moment labeling.
Chapter 1 Manipulation 11.1 Case 1: Manipulation by a human 11.2 Case 2: An automated assembly system 31.3 Issues in manipulation 51.4 A taxonomy of manipulation techniques 71.5 Bibliographic notes 8
Exercises 8
Chapter 2 Kinematics 112.1 Preliminaries 112.2 Planar kinematics 152.3 Spherical kinematics 202.4 Spatial kinematics 222.5 Kinematic constraint 252.6 Kinematic mechanisms 342.7 Bibliographic notes 36
Exercises 37
Chapter 3 Kinematic Representation 413.1 Representation of spatial rotations 413.2 Representation of spatial displacements 583.3 Kinematic constraints 683.4 Bibliographic notes 72
Exercises 72
Chapter 4 Kinematic Manipulation 774.1 Path planning 774.2 Path planning for nonholonomic systems 844.3 Kinematic models of contact 864.4 Bibliographic notes 88
Exercises 88
Chapter 5 Rigid Body Statics 935.1 Forces acting on rigid bodies 935.2 Polyhedral convex cones 995.3 Contact wrenches and wrench cones 1025.4 Cones in velocity twist space 1045.5 The oriented plane 1055.6 Instantaneous centers and Reuleaux' s method 1095.7 Line of force; moment labeling 1105.8 Force dual 1125.9 Summary 1175.10 Bibliographic notes 117
Exercises 118
Chapter 6 Friction 1216.1 Coulomb' s Law 1216.2 Single degree-of-freedom problems 1236.3 Planar single contact problems 1266.4 Graphical representation of friction cones 1276.5 Static equilibrium problems 1286.6 Planar sliding 1306.7 Bibliographic notes 139
Exercises 139
Chapter 7 Quasistatic Manipulation 1437.1 Grasping and fixturing 1437.2 Pushing 1477.3 Stable pushing 1537.4 Parts orienting 1627.5 Assembly 1687.6 Bibliographic notes 173
Exercises 175
Chapter 8 Dynamics 1818.1 Newton' s laws 1818.2 A particle in three dimensions 1818.3 Moment of force; moment of momentum 1838.4 Dynamics of a system of particles 1848.5 Rigid body dynamics 1868.6 The angular inertia matrix 1898.7 Motion of a freely rotating body 1958.8 Planar single contact problems 1978.9 Graphical methods for the plane 2038.10 Planar multiple-contact problems 2058.11 Bibliographic notes 207
Exercises 208
Chapter 9 Impact 2119.1 A particle 2119.2 Rigid body impact 2179.3 Bibliographic notes 223
Exercises 223
Chapter 10 Dynamic Manipulation 22510.1 Quasidynamic manipulation 22510.2 Briefly dynamic manipulation 22910.3 Continuously dynamic manipulation 23010.4 Bibliographic notes 232
Exercises 235
Appendix A Infinity 237
Lecture 14. Mechanics of Manipulation – p.16