Lecture 20. Planar Sliding

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1 Matthew T. Mason Mechanics of Manipulation

2 Today s outline

3 Planar sliding: s Planar sliding is surprisingly common in manipulation. Shakey used planar sliding to move things around. To locate and orient part accurately in grasping. Object slip in parallel jaw gripper, for compliant motion. For precision motion of object on a workbench, clamp it down and tap it. Moving a couch or a refrigerator. Dealing cards, gathering chips,... Even if you want not to move something, you need to understand planar sliding: Holding an object without slip. Workholding on a workbench. Avoiding slip of tires or feet.

4 Planar sliding: Features You don t need form or force closure. You can do it with one contact. You don t need a hand that conforms to object shape. You don t need to lift. You don t need to know where the object is!

5 Shakey Pushing, the simplest kind of manipulation:

6 Stanford Programmable Assembly demo

7 CMU PRL, using physics models

8 History and disclaimer Disclaimer Pushing is a central interest of mine since PhD, so I m prejudiced. How I got interested. I was trying to find a good thesis topic. Try to prove correctness of a program to put block A on block B, including uncertainty and error in position sensing. You cannot grasp block A without moving it. How to predict the motion of the object prior to the grasp? With that on my mind, I saw the Stanford hinge-picking demo.

9 What would we like to know? Fundamental mechanics: Relation of motion to reaction forces. Prediction, analysis: Given applied force or kinematic constraints or initial motion, predict motion. s: Will tire slip? How far will card glide? Planning, design: Given desired motion, find required applied forces, constraints, initial velocity. s: push the fridge into the corner.

10 Pressure distribution is key Frictional forces are distributed over support surface. Coulomb s law gives each frictional force a weight proportional to pressure. Pressure is generally underdetermined. Classify problems by pressure assumptions: Determined: Tactile sensor; One, two, or three contact points with known resultant (e.g. known weight wrench); Linear (elastic layer between two flat rigid bodies) with known resultant Underdetermined: Known support region; Known resultant; We will assume known pressure distribution, then address the more general case.

11 of planar sliding Let R be support region. Let v(r) be vel of some point r R. Let p(r) be the pressure at r, Let da be element of area at r, Then normal force at r is given by p(r) da Assume µ uniform over R. Coulomb s law at r: for v(r) 0. µ v(r) p(r) da (1) v(r)

12 Integrating... To obtain total force and moment we integrate over R: v(r) f f = µ p(r) da R v(r) n f = µ r v(r) p(r) da v(r) R For known pressure distribution we can evaluate the integrals. Generally we cannot.

13 for translation Although the integrals cannot be evaluated, they can be simplified for translational sliding. Assume translation: v(r) = v. We can pull velocity out of integral: f f = µ v p(r) da v R n f = µ r p(r) da v v R

14 Let f 0 be the total normal force: f 0 = p(r) da Let r 0 be the centroid of the pressure distribution. r 0 = 1 r p(r) da f 0 R R Substituting above: f f = µ v v f 0 n f = r 0 f f This is Coulomb s law for a point contact!

15 Definition Define the center of friction to be the centroid of the pressure distribution r 0. In translation, frictional force acts through center of friction. Generalization to non-uniform coefficient of friction µ is straightforward. Generalization to rotation... nope. (But see the related construction Effective center of friction due to Lynch.)

16 Let r IC be the instantaneous center of rotation Velocity at r is: Direction of motion at r is: v(r) = ω (r r IC ) = θ ˆk (r r IC ) v(r) v(r) = sgn( θ) ˆk r r IC r r IC Substituting f f = µ sgn( θ) ˆk r r IC p(r) da R r r IC n fz = µ sgn( θ) r r IC r p(r) da r r IC R

17 There is a variational, expressed as the maximum power inequality: Given a slider motion, the frictional wrench maximizes the power dissipated. First, we need to define frictional load, and consider the complexities of the motion-to-force mapping.

18 Frictional load Definition Define frictional load to be wrench applied by slider to planar support. This is a change of sign. The integrals we developed earlier give the frictional load wrench as a function of slider velocity twist. Consider the relation of frictional load to slider velocity for a finite number of support points. We will see that the motion-force mapping is neither one-to-many nor many-to-one.

19 First case: One point of support Let v be the velocity of the particle. Let f be the frictional load. Coulomb s law can be stated: slip: f v, and f = µf n, where µ is the coefficient of friction, and f n is the support force. stick: f µf n.

20 Limit Curve: maximum power inequality Consider the set of all possible frictional loads, with f n fixed. That is a disk centered at the origin of radius µf n. Definition Define the limit curve LC to be the circle of radius µf n at the origin of force space. Coulomb s law is equivalent to the maximum power inequality: f LC (f f ) v 0 I.e. motion v yields a load that is extremal in the v direction. When slip occurs f is on the limit curve, and v is normal to the limit curve at f. LC f y f f x v

21 More than one point Let r vary over the support region Construct a limit curve LC(r) at each point r, At each point r the maximum power inequality holds: f (r) LC(r) (f(r) f (r)) v(r) 0 How do we put them all together??? Wrenches and twists!

22 Frictional load wrench and velocity twist Let p be the total frictional load wrench f x p = f y = f x (r) f y (r) n 0z r r f(r) Let q be the velocity twist q = v 0x v 0y ω z

23 Power of Coulomb load versus arbitrary load Let f(r) be a distribution of frictional loads satisfying Coulomb s law Let f (r) be arbitrary, except that at each r, f (r) is in the corresponding limit curve: r f (r) LC(r) Let p and p be the total frictional load wrench for f(r) and f (r) respectively. The power dissipated by the Coulomb load can be written two ways p q = r f(r) v(r) Similarly for the arbitrary load can write p q = r f (r) v(r)

24 Power difference between Coulomb and arbitrary loads Taking the difference yields (p p ) q = r (f(r) f (r)) v(r) By maximum power inequality, every term in the sum on the right hand side is non-negative. So: (p p ) q 0 Another maximum power inequality! This time in wrench space!!! Summary: Consider all loads such that f (r) LC(r). The correct load maximizes the power dissipated.

25 Limit Surface No motion: q = 0, power dissipated is zero for any frictional load. So, form the set of all possible total frictional load wrenches p Define the limit surface to be the surface of this set. : the frictional load wrench yields maximum power over all wrenches in the limit surface. Equivalently: during slip the total frictional load wrench p lies on the limit surface, and the velocity twist q is normal to the limit surface at p.

26 : two point contact Two points of contact, support evenly divided. 1. LS a : all frictional loads at a. An elliptical disk in wrench space. 2. LS b : all frictional loads at b. Also an elliptical disk. 1 f x y f y 3. The desired limit surface LS is the Minkowski sum: 1 m x LS = {w a +w b w a LS a, w b LS b }

27 Barbell Limit Surface fl at elliptical facet Point vertex f x n 0z f Section on f y is a square Section on f x is a circle f y 0plane Section on n 0z is a circle 0plane 0plane vertex corresponding to pure moment n 0z f f x vertex corresponding to pure force f x 0 0

28 Barbell Limit Surface sweeping

29 Barbell Limit Surface sweeping

30 Barbell Limit Surface sweeping

31 LS properties The barbell LS illustrates some properties that hold generally: Closed, convex, enclosing the origin of wrench space. Symmetric when reflected through origin. Orthogonal projection onto the f x,f y plane is a circle of radius µf n. Each discrete point of support yields two antipodal flat facets. On each facet several loads map to one motion (rotation about the support point.) (No discrete points: LS is strictly convex and load-motion mapping is one-to-one.) Collinear discrete support is even weirder: vertices on LS where one load maps to several velocities (rotation about point collinear with support).

16. Friction Mechanics of Manipulation

16. Friction Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 16. Mechanics of Manipulation p.1 Lecture 16. Friction. Chapter 1 Manipulation