9. Representing constraint

Transcription

1 9. Representing constraint Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 9. Mechanics of Manipulation p.1

2 Lecture 9. Representing constraint. 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 9. Mechanics of Manipulation p.2

3 Outline. Constraint using contact screw and reciprocal product. Repelling, reciprocal, contrary. Relation to Reuleaux. Examples. Lecture 9. Mechanics of Manipulation p.3

4 Remember Reuleaux s method? Perpendicular to constraint divides plane into positive IC s, negative IC s, and IC s of either sign. Doesn t extend to three dimensions. Great for humans, bad for computers. Sometimes equations are better than pictures. Lecture 9. Mechanics of Manipulation p.4

5 First order model of constraint Let û be contact normal, inward pointing Let p be contact point in the constrained body Let v p be the velocity of the point p. Then we write the bilateral velocity constraint as û v p = 0 and unilateral velocity constraint as û v p 0 p u ˆ Lecture 9. Mechanics of Manipulation p.5

6 Constraint using screw coordinates Let (ω, v 0 ) be screw coordinates of body velocity Then velocity of p is v p = v 0 + ω p We write the kinematic constraint û (v 0 + ω p) 0 Distribute dot product, play with triple product... û v 0 + (p û) ω 0 Reciprocal product! Lecture 9. Mechanics of Manipulation p.6

7 Contact screw Define contact screw to be Plücker coordinates of the contact normal (c, c 0 ) = (u, p û) Write the kinematic constraint as (c, c 0 ) (ω, v 0 ) 0 Lecture 9. Mechanics of Manipulation p.7

8 Reciprocal, contrary, repelling Definition 3.3: A pair of screws is reciprocal, contrary, or repelling, if their reciprocal product is zero, negative, or positive, respectively. Bilateral constraint: velocity screw (ω, v 0 ) and contact screw (c, c 0 ) must be reciprocal: (c, c 0 ) (ω, v 0 ) = 0 Unilateral constraint: velocity screw (ω, v 0 ) and contact screw (c, c 0 ) must be reciprocal or repelling: (c, c 0 ) (ω, v 0 ) 0 Lecture 9. Mechanics of Manipulation p.8

9 Connection to Reuleaux s method The contact screw (c, c 0 ) is a zeropitch screw the Plücker coordinates of the contact normal. For a planar motion of the ˆx-ŷ plane, the body velocity twist (ω, v 0 ) is also a zero-pitch screw a pure rotation. (ω, v 0 ) is the Plücker coordinates of the rotation axis perpendicular to the ˆx-ŷ plane. Contact normal and directed rotation axis must be reciprocal or repelling. Directed rotation axis must point down ( ) if it s to the right of the contact normal, and up (+) if it s to the left of the contact normal. Copy the figure from the board Lecture 9. Mechanics of Manipulation p.9

10 Ex 1: Screw coordinates of planar motions. Choose three bilateral constraints aligned with the ẑ axis. The screw coordinates for the constraints are: (s 1, s 01 ) = (s 2, s 02 ) = s 2,s s 02 3,s 03 (s 3, s 03 ) = s 1,s 01 y x Lecture 9. Mechanics of Manipulation p.10

11 Ex 1: Screw coordinates of planar motions. Choose three bilateral constraints aligned with the ẑ axis. The screw coordinates for the constraints are: (s 1, s 01 ) =(0, 0, 1, 0, 0, 0) (s 2, s 02 ) = s 2,s s 02 3,s 03 (s 3, s 03 ) = s 1,s 01 y x Lecture 9. Mechanics of Manipulation p.10

12 Ex 1: Screw coordinates of planar motions. Choose three bilateral constraints aligned with the ẑ axis. The screw coordinates for the constraints are: (s 1, s 01 ) =(0, 0, 1, 0, 0, 0) (s 2, s 02 ) =(0, 0, 1, 0, 1, 0) s 2,s s 02 3,s 03 (s 3, s 03 ) = s 1,s 01 y x Lecture 9. Mechanics of Manipulation p.10

13 Ex 1: Screw coordinates of planar motions. Choose three bilateral constraints aligned with the ẑ axis. The screw coordinates for the constraints are: (s 1, s 01 ) =(0, 0, 1, 0, 0, 0) (s 2, s 02 ) =(0, 0, 1, 0, 1, 0) s 2,s s 02 3,s 03 (s 3, s 03 ) =(0, 0, 1, 1, 0, 0) s 1,s 01 y x Lecture 9. Mechanics of Manipulation p.10

14 Ex 1: Screw coordinates of planar motions. Choose three bilateral constraints aligned with the ẑ axis. The screw coordinates for the constraints are: (s 1, s 01 ) =(0, 0, 1, 0, 0, 0) (s 2, s 02 ) =(0, 0, 1, 0, 1, 0) (s 3, s 03 ) =(0, 0, 1, 1, 0, 0) y s 3,s 03 x s 1,s 01 s 2,s 02 Let the twist be given by (t, t 0 ) = (t 1, t 2, t 3, t 4, t 5, t 6 ) Lecture 9. Mechanics of Manipulation p.10

15 Ex 1: Form reciprocal products. The twist must be reciprocal to (s 1, s 01 ): t 6 = 0... to (s 2, s 02 ):... and to (s 3, s 03 ): t 6 t 2 = 0 t 6 + t 1 = 0 Thus the twist must be of the form (t, t 0 ) = (0, 0, t 3, t 4, t 5, 0) y s 3,s 03 x s 1,s 01 s 2,s 02 Lecture 9. Mechanics of Manipulation p.11

16 Ex 1: Interpreting the answer The twist must be of the form To get the pitch: (t, t 0 ) = (0, 0, t 3, t 4, t 5, 0) p = t t 0 t t = 0 The direction vector (0, 0, t 3 ) is parallel to ẑ The point closest to the origin is t t 0 t t = ( t 5 t 3, t 4 t 3, 0)/t 2 3 = ( t 5 /t 3, t 4 /t 3 ) So the twist represents the rotation center using homogeneous coordinates. As a special case, when t 3 = 0, we obtain a pure translational velocity (t 4, t 5, 0). Lecture 9. Mechanics of Manipulation p.12

17 Ex 2: Squeezing the corners of a cube We will consider the simpler bilateral problem s 4,s 04 (s 1, s 01 ) = (s 2, s 02 ) = (s 3, s 03 ) = (s 4, s 04 ) = s 5,s 05 s 6,s 06 x z O s 1,s 01 y s 3,s 03 s 2,s 02 (s 5, s 05 ) = (s 6, s 06 ) = Lecture 9. Mechanics of Manipulation p.13

18 Ex 2: Squeezing the corners of a cube We will consider the simpler bilateral problem s 4,s 04 (s 1, s 01 ) = (1, 0, 0, 0, 1, 0) (s 2, s 02 ) = (0, 1, 0, 1, 0, 0) (s 3, s 03 ) = (0, 0, 1, 0, 0, 0) (s 4, s 04 ) = ( 1, 0, 0, 0, 0, 1) s 5,s 05 s 6,s 06 x z O s 1,s 01 y s 3,s 03 s 2,s 02 (s 5, s 05 ) = (0, 1, 0, 0, 0, 1) (s 6, s 06 ) = (0, 0, 1, 1, 1, 0) Let (t, t 0 ) be a differential twist. Reciprocal with respect to (s 1, s 01 ) t 4 + t 2 = 0 Lecture 9. Mechanics of Manipulation p.13

19 Ex 2: Solving the constraint equations Reciprocal to all 6 contact screws: t 4 +t 2 = 0 s 5,s 05 s 4,s 04 t 5 +t 1 = 0 t 6 = 0 t 4 t 3 = 0 t 5 +t 3 = 0 s 6,s 06 x z O s 1,s 01 y s 3,s 03 s 2,s 02 t 6 t 1 t 2 = 0 The solutions are of the form (t, t 0 ) = k(1, 1, 1, 1, 1, 0) Lecture 9. Mechanics of Manipulation p.14

20 Ex 2: Interpreting the solution The solutions are of the form (t, t 0 ) = k(1, 1, 1, 1, 1, 0) s 5,s 05 s 4,s 04 Pitch: t t 0 /t t = 0. Point on line closest to origin: t t 0 /t t = Direction vector: t = (1, 1, 1). s 6,s 06 x z O s 1,s 01 y s 3,s 03 s 2,s 02 I.e., as expected, the diagonal of the cube. Lecture 9. Mechanics of Manipulation p.15

21 Next: Cspace transform and motion planning. 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 93 Chapter 8 Dynamics Forces acting on rigid bodies Newton s laws Polyhedral convex cones A particle in three dimensions Contact wrenches and wrench cones Moment of force; moment of momentum Cones in velocity twist space Dynamics of a system of particles The oriented plane Rigid body dynamics Instantaneous centers and Reuleaux s method The angular inertia matrix Line of force; moment labeling Motion of a freely rotating body Force dual Planar single contact problems Summary Graphical methods for the plane Bibliographic notes Planar multiple-contact problems 205 Exercises Bibliographic notes 207 Exercises 208 Chapter 6 Friction Coulomb s Law 121 Chapter 9 Impact Single degree-of-freedom problems A particle Planar single contact problems Rigid body impact Graphical representation of friction cones Bibliographic notes Static equilibrium problems 128 Exercises Planar sliding Bibliographic notes 139 Chapter 10 Dynamic Manipulation 225 Exercises Quasidynamic manipulation Brie y dynamic manipulation 229 Chapter 7 Quasistatic Manipulation Continuously dynamic manipulation Grasping and fixturing Bibliographic notes Pushing 147 Exercises Stable pushing Parts orienting 162 Appendix A Infinity Assembly Bibliographic notes 173 Exercises 175 Lecture 9. Mechanics of Manipulation p.16