11. Kinematic models of contact Mechanics of Manipulation

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1 11. Kinematic models of contact Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 11. Mechanics of Manipulation p.1

2 Lecture 11. Kinematic models of contact. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system 1. Issues in manipulation 1.4 A taxonomy of manipulation techniques 7 1. Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics 1 2. Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 6 Exercises 7 Chapter Kinematic Representation 41.1 Representation of spatial rotations 41.2 Representation of spatial displacements 8. Kinematic constraints 68.4 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 Rigid Body Statics 9.1 Forces acting on rigid bodies 9.2 Polyhedral convex cones 99. Contact wrenches and wrench cones Cones in velocity twist space 104. The oriented plane 10.6 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 19 Exercises 19 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 17 Exercises 17 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 22 Exercises 22 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Brie y dynamic manipulation Continuously dynamic manipulation Bibliographic notes 22 Exercises 2 Appendix A Infinity 27 Lecture 11. Mechanics of Manipulation p.2

3 Outline. Grübler Review of kinematic mechanisms Mobility and connectivity Grübler s formula Salisbury Taxonomy of contacts Mobility and connectivity of grasp Lecture 11. Mechanics of Manipulation p.

4 Kinematic mechanisms Link: a rigid body; Joint: imposes one or more constraints on the relative motion of two links; Kinematic mechanism: a bunch of links joined by joints; lower pairs joints involving positive contact area. Planar freedoms Cylindrical 2 freedoms Prismatic 1 freedom Spherical freedoms Revolute 1 freedom Helical 1 freedom Lecture 11. Mechanics of Manipulation p.4

5 Mobility and connectivity mobility of a mechanism: DOFs with one link fixed. connectivity DOFs of one link relative to another. What is the mobility of the five bar linkage at right? L2 L L1 L4 L What is the connectivity of Link 1 relative to link two? Link relative to link 1? Link relative to link 4? Lecture 11. Mechanics of Manipulation p.

6 Mobility and connectivity mobility of a mechanism: DOFs with one link fixed. connectivity DOFs of one link relative to another. What is the mobility of the five bar linkage at right? Two. L2 L L1 L4 L What is the connectivity of Link 1 relative to link two? Link relative to link 1? Link relative to link 4? Lecture 11. Mechanics of Manipulation p.

7 Mobility and connectivity mobility of a mechanism: DOFs with one link fixed. connectivity DOFs of one link relative to another. What is the mobility of the five bar linkage at right? Two. L2 L L1 L4 L What is the connectivity of Link 1 relative to link two? One. Link relative to link 1? Link relative to link 4? Lecture 11. Mechanics of Manipulation p.

8 Mobility and connectivity mobility of a mechanism: DOFs with one link fixed. connectivity DOFs of one link relative to another. What is the mobility of the five bar linkage at right? Two. L2 L L1 L4 L What is the connectivity of Link 1 relative to link two? One. Link relative to link 1? Two. Link relative to link 4? Lecture 11. Mechanics of Manipulation p.

9 Mobility and connectivity mobility of a mechanism: DOFs with one link fixed. connectivity DOFs of one link relative to another. What is the mobility of the five bar linkage at right? Two. L2 L L1 L4 L What is the connectivity of Link 1 relative to link two? One. Link relative to link 1? Two. Link relative to link 4? One. Lecture 11. Mechanics of Manipulation p.

10 Grübler s formula Given n links joined by g joints, with u i constraints and f i freedoms at joint i. (Note that u i + f i = 6.) Assume one link is fixed and constraints are all independent. The mobility M is M = 6(n 1) u i = 6(n 1) (6 f i ) = 6(n g 1) + f i Or, for a planar mechanism: M = (n 1) u i = (n g 1) + f i Lecture 11. Mechanics of Manipulation p.6

11 Grübler: special case for loops The previous formula works (sort of) for all mechanisms. For loops there is a variant. One loop: n = g, so M = f i + 6( 1) Two loops: make a second loop by adding k links and k + 1 joints: M = f i + 6( 2) Every loop increases excess of joints over links by 1. For l loops: for a spatial linkage, and M = f i 6l M = f i l Lecture 11. Mechanics of Manipulation p.7

12 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 10 M = (n g 1) + f i = M = f i l = Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = M = 6(n g 1) + f i = M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

13 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = M = f i l = Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = M = 6(n g 1) + f i = M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

14 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = 1 M = f i l = Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = M = 6(n g 1) + f i = M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

15 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = 1 M = f i l = 1 Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = M = 6(n g 1) + f i = M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

16 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = 1 M = f i l = 1 Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = 2 M = 6(n g 1) + f i = M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

17 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = 1 M = f i l = 1 Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = 2 M = 6(n g 1) + f i = 2 M = f i 6l = Lecture 11. Mechanics of Manipulation p.8

18 Common sense Example: what is the mobility of Watt s linkage? Planar Grübler s formula: M = (n 1) u i = 1 10 M = (n g 1) + f i = 1 M = f i l = 1 Spatial Grübler s formula: Independent constraints is a very strong assumption. M = 6(n 1) u i = 2 M = 6(n g 1) + f i = 2 M = f i 6l = 2 Lecture 11. Mechanics of Manipulation p.8

19 Kinematic models of contact A grasp is like a kinematic mechanism. Assume fingers do not lift or slip. Model each contact as a joint Apply Grübler s formula! Lecture 11. Mechanics of Manipulation p.9

20 Taxonomy of contact types In previous slide, contact was modeled as spherical joint. Are there other possibilities? Salisbury s PhD thesis, 1982, included a taxonomy. No contact 6 freedoms Line contact without friction 4 freedoms Point contact without friction freedoms Point contact with friction freedoms Planar contact without friction freedoms Soft fi nger 2 freedoms Line contact with friction 1 freedom Planar contact with friction 0 freedoms Lecture 11. Mechanics of Manipulation p.10

21 Mobility and connectivity of grasp Salisbury suggests four measures: M Mobility of the entire system with the finger joints free. M Mobility of the entire system, with the finger joints locked. C Connectivity of the object relative to a fixed palm, with the finger joints free. C Connectivity of the object relative to a fixed palm, with the finger joints locked. If C = 6 then object can make general motions. If C 0 then hand can immobilize object. Lecture 11. Mechanics of Manipulation p.11

22 Example: the Salisbury hand What is C? What is C? Lecture 11. Mechanics of Manipulation p.12

23 Next: Foundations of statics. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system 1. Issues in manipulation 1.4 A taxonomy of manipulation techniques 7 1. Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics 1 2. Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 6 Exercises 7 Chapter Kinematic Representation 41.1 Representation of spatial rotations 41.2 Representation of spatial displacements 8. Kinematic constraints 68.4 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 Rigid Body Statics 9.1 Forces acting on rigid bodies 9.2 Polyhedral convex cones 99. Contact wrenches and wrench cones Cones in velocity twist space 104. The oriented plane 10.6 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 19 Exercises 19 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 17 Exercises 17 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 22 Exercises 22 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Brie y dynamic manipulation Continuously dynamic manipulation Bibliographic notes 22 Exercises 2 Appendix A Infinity 27 Lecture 11. Mechanics of Manipulation p.1