27. Impact Mechanics of Manipulation

Transcription

1 27. Impact Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 27. Mechanics of Manipulation p.1

2 Lecture 27. Impact 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 27. Mechanics of Manipulation p.2

3 Outline. Impulse-momentum equations Impact of a particle Plastic and elastic impact Newtonian and Poisson restitution Impulse space Impact of a planar rigid body Solution of the sliding rod problem Lecture 27. Mechanics of Manipulation p.3

4 Frictional impact When rigid bodies collide: Discontinuity in velocity, Infinite forces, Zero time, Even for simple cases there is no agreement on laws, For multiple contact indeterminacies abound. We will adopt Routh s analysis of planar impact, which uses Poisson restitution. Lecture 27. Mechanics of Manipulation p.4

5 Impulse momentum equations Recall: Integrating: F = dp dt N = dl dt P = L = t1 t 0 t1 t 0 F dt N dt where P = P 1 P 0, L = L 1 L 0, F dt is the impulse, and N dt is the impulsive moment. Lecture 27. Mechanics of Manipulation p.5

6 Determining impulse. Particle in plane. We need impulse and impulsive moment as a function of state and physical parameters. Start with a frictionless particle on plane. Velocity changes discontinuously from v 0 to v 1 : v = 1 m P = 1 m I where v = v 1 v 0 The tangential impulse I t is zero, so v 1t = v 0t v 1n = v 0n + 1 m I n Lecture 27. Mechanics of Manipulation p.6

7 Plastic and elastic How to get the normal impulse I n? Two special cases: plastic impact and elastic impact I n = mv 0n v 1n = 0 I n = 2mv 0n v 1n = v 0n Lecture 27. Mechanics of Manipulation p.7

8 Newtonian restitution Newton hypothesized a continuum from plastic to elastic, defined by a coefficient of restitution e = v 1n v 0n (1) so that plastic impact corresponds to e = 0, and elastic impact corresponds to e = 1. Lecture 27. Mechanics of Manipulation p.8

9 Poisson restitution A competing definition of the coefficient of restitution was given by Poisson. Divide the collision into two stages. v n < 0 v n = 0 v n > 0 compression restitution Let I c be impulse accumulated during compression, and let I r be impulse accumulated during restitution. Poisson s hypothesis is that the ratio of these two parts is governed by material properties: e = I r I c (2) For simple cases no advantage over Newton, but for frictional rigid bodies Poisson restitution is superior. Lecture 27. Mechanics of Manipulation p.9

10 Applying Poisson restitution for frictionless parti Given m, v 0, and e, first find I c : I c = mv 0n then restitution impulse I r then total impulse I I r = emv 0n and resulting normal velocity: I = I c + I r = (1 + e)mv 0n v 1n = ev 0n The tangential velocity is unchanged. Lecture 27. Mechanics of Manipulation p.10

11 Frictional particle Consider frictional particle with initial leftward velocity. There should be some frictional impulse acting toward the right. Suppose we apply Coulomb s law, by looking at initial contact mode, and multiplying I n by µ. The result is absurd, and violates conservation of energy! Lecture 27. Mechanics of Manipulation p.11

12 A better model I fixed Lecture 27. Mechanics of Manipulation p.12

13 The left-drifting particle I n s-line 1 I c-line c-line: s-line: 0 Solution: Lecture 27. Mechanics of Manipulation p.13

14 Rigid body. Kinematics. First, we observe some kinematic relations c = x + r ċ = ẋ + ṙ = ẋ + ω r ċ = ẋ + ω r r c x Lecture 27. Mechanics of Manipulation p.14

15 Impulse momentum laws Now we can write the impulse-momentum laws: Substituting into the expression for ċ yields: m ẋ = I (3) ρ 2 m ω = r I (4) ċ = 1 m I + 1 (r ρ 2 I) r m (5) = 1 m I 1 r ρ 2 (r I) m (6) = 1 ( ρ 2 ρ 2 I 3 R 2) I m (7) where I 3 is the three by three identity matrix, and R is the cross-product matrix. Lecture 27. Mechanics of Manipulation p.15

16 Compression line and restitution line Substituting above and expanding, we obtain ( ċ t ċ n ) = 1 ρ 2 m ( (ρ 2 + rn) 2 r t r n r t r n (ρ 2 + rt 2 ) ) ( I t I n ) (8) The sticking line is defined by the equation ċ t = 0, ċ 0t + ρ2 + r 2 n ρ 2 m I t r tr n ρ 2 m I n = 0 (9) and the compression line is defined the equation ċ n = 0, ċ 0n r tr n ρ 2 m I t + ρ2 + r 2 t ρ 2 m I n = 0 (10) Lecture 27. Mechanics of Manipulation p.16

17 Rigid body example Lecture 27. Mechanics of Manipulation p.17

18 Analysis of rigid body example I 0 restitution compression left-sliding right-sliding Lecture 27. Mechanics of Manipulation p.18

19 Sliding rod problem Lecture 27. Mechanics of Manipulation p.19

20 Sliding rod solution 2 s-line I n c-line (1 + e)i nc I nc I t Lecture 27. Mechanics of Manipulation p.20

21 Next: Impact. 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 27. Mechanics of Manipulation p.21