Pulling by Pushing, Slip with Infinite Friction, and Perfectly Rough Surfaces

Transcription

1 1993 IEEE International Conerence on Robotics and Automation Pulling by Pushing, Slip with Ininite Friction, and Perectly Rough Suraces Kevin M. Lynch Matthew T. Mason The Robotics Institute and School o Computer Science Carnegie Mellon University Pittsburgh, PA Abstract When one rigid object (the pusher) pushes another (the slider) across a horizontal support plane, Coulomb s law admits some surprising phenomena. First, it is possible to move the slider by moving the pusher away rom the slider: pulling. Second, it is possible to obtain slip between the two objects even with an ininite coeicient o riction. Thus the common conception that ininite riction prevents slip is in error. This paper shows examples o the phenomena with both quasi-static and dynamic analysis. It also addresses implications or the concept o perectly rough suraces. 1 Introduction Consider the problem o two rigid objects in rictional contact, supported by a horizontal planar surace, with gravity acting along the vertical. The motion o one object, the pusher, is given. The motion o the other object, the slider, is subject to Newton s laws. Frictional orces are governed by Coulomb s law. This paper constructs examples exhibiting two counterintuitive phenomena: 1. Pulling. The slider can maintain contact even when the pusher moves away rom the slider. 2. Slipping with ininite riction. Even with an ininite coeicient o riction, slip can occur between the slider and pusher. This second phenomenon has implications or the concept o perectly rough contact. Usually a perectly rough contact is deined as a contact that does not admit slip. Some treatments assume, erroneously, that this is equivalent to ininite riction. For example, Rutherord [11, page 41] states In the second case in which = 1 no sliding is possible and the suraces are said to be perectly rough. Intuitively this seems correct, but we now see that the two concepts are not equivalent. The paper begins with a review o the general method or solving rictional contact problems. Then we construct examples exhibiting the two phenomena. First we apply a quasi-static analysis, where orces o acceleration are assumed negligible, and then we ollow with a ull dynamic analysis. Finally we discuss alternative deinitions o ininite riction and the relation to perectly rough suraces. 2 Determining the motion o a pushed object The solution o rictional rigid-body contact problems presents some unusual diiculties. Under Coulomb s law, the orce elt by a point contacting a surace must satisy j t j n where t is the tangential rictional component, n is the normal component, and is the coeicient o riction. During sliding contact, j t j = n and the rictional orce is directed opposite the direction o sliding. Thus Coulomb s law does not directly speciy contact orces. Rather, it imposes constraints that vary depending on the contact mode: whether the contact is being maintained, and whether the contact is sliding and in which direction. Frictional contact problems are solved by case analysis. For each contact mode, we determine whether orces and accelerations exist satisying the simultaneous constraints o Coulomb s law, Newton s second law, and whatever kinematic constraints may be present. Each consistent set o orces and accelerations is deemed a solution to the problem. Problems with multiple solutions are ambiguous. Problems with no solutions are

2 v v 1 p s θ s l θ p 45 o Figure 1: Example pushing problem. v p center inconsistent. More detailed descriptions are given by Lötstedt [8], Erdmann [5], Rajan et al. [10], Brost and Mason [3], Bara [2], Dupont [4], and Wang et al. [12]. 3 Quasi-static pushing and pulling All o the examples in this paper are o the orm illustrated in Figure 1. The pusher is a straight ence. The slider is a ring o radius 1 and uniorm mass. The support orce is distributed evenly about the ring, and the coeicient o support riction is uniorm. Attached to the ring is a massless rod o length l which does not contact the support surace. The slider is pushed at the ree end o the rod. The rod makes an angle o 45 degrees with the ence. The coeicient o riction at the contact is. The orce applied by the pusher passes through the contact at an angle. The pusher is denoted v p, at an angle o p. The slider at the rod endpoint is, at an angle o s. All angles are measured with respect to the ence. We will say that a vector is directed into the slider i it has a nonnegative component along the outward-pointing normal o the ence, that is, i the angle is in the closed interval [0 180 ]. Otherwise we will say that it is directed away rom the slider. Force vectors are drawn with illed arrowheads and and acceleration vectors are drawn with open arrowheads. Velocities will sometimes be represented as. The center is the point in the plane about which the motion is a pure rotation. Similarly, initial accelerations will sometimes be represented as acceleration. Now suppose that the slider motion is slow enough that inertial orces are negligible compared with contact orces. Then a solution obtains whenever the support rictional orces balance the pusher contact orce. Consider Figure 2, or example. For the center shown, the rictional support orces acting on the slider are precisely balanced by the pusher orce passing through the contact. For a pusher v p equal to the solution slider, and a large enough coeicient o riction, we have a solution. There is nothing unusual about this solution. Since the pusher v p is directed into the slider, we have pushing, not pulling. But consider the example o Figure 3, where the applied Figure 2: Quasi-static pushing. v p center Figure 3: Quasi-static pulling. orce passes closely on the other side o the ring center. Here a solution is obtained or v p equal to but directed away rom the slider: pulling. Note that a second solution is that the slider simply breaks contact and does not move. This problem is ambiguous, as are all o the pulling examples that we will construct. To show that the solutions shown in Figures 2 and 3 really are solutions, we integrate the support rictional orces around the ring and show that the total rictional orce the slider applies to the support equals. We use two coordinate systems (Figure 4), one system x-y aligned with the pusher, and a primed system x 0 -y 0 aligned with the applied orce but centered on the slider. With some modiications, Goyal [6] gives the ollowing expressions or rictional orce and torque applied by the ring to the support surace in the primed system: Z 2 x 0 = 0 (1) W 1 +! cos v y 0 = sgn(v) d (2) 2 q1 +(!v )2 + 2!v cos 0 0 = sgn(v) W 2 0 Z 2 0! + cos v d (3) q1 +(!v )2 + 2!v cos

3 y y x y θ s φ φ Figure 4: Coordinate conventions. x x where W is the slider weight multiplied by the support riction coeicient,! is the slider angular, and v is the o the slider center along the y 0 axis. (The o the slider center along the x 0 axis is zero. The center will all a directed distance ;v=! along the x 0 axis.) The integrals are elliptic integrals which could be reduced to normal orm, as in [6]. The rictional orce and torque measured in the unprimed system is: x = y 0 cos (4) y = y 0 sin (5) 0 = l y 0 sin( ; 45 ) (6) For quasi-static balance the torque about the contact point must be zero: 0 = 0 (7) Equation 7 implicitly deines a center as a unction o the orce angle. In principle, we could solve equation 7 or!=v as a unction o. The quantity ;v=! gives the center as a directed distance along the x 0 -axis and the sign o ; 0 0 gives the rotation sense. In practice, the relation o the pusher orce to the slider motion can be obtained numerically. We can also employ the limit surace o Goyal et al. [7] to represent this relation. The limit surace is the locus o ( x y ) arising rom support riction during all possible slider motions. It is a closed, convex, two-dimensional surace enclosing the origin. Goyal showed that i a slider motion (v x v y!) causes the slider to apply a orce ( x y ) to the support, then (v x v y!) is parallel to the outward-pointing normal o the limit surace at ( x y ). For the present case, the limit surace can be reduced to a curve in the plane. The limit surace is constructed with orces measured in the x-y coordinate system and torques measured about the pushing contact. But we are interested only in those motions producing no torque about this point. Figure 5: Limit curves or dierent rod lengths. Thereore, we intersect the limit surace with the plane 0 = 0 to obtain a limit curve representing the locus o rictional orces corresponding to possible solutions o the slider motion. Now choose to be one such orce on the limit curve, and let be the corresponding o the slider contact point. Then is normal to the limit curve at. Figure 5 shows limit curves or several dierent values o l. (The limit curves may appear to be elliptical, but they are not.) As l approaches zero, the limit curve approaches a circle. As l approaches ininity, the limit curve approaches a line segment aligned with the rod. The orce and the are parallel only at local extrema o the limit curve. The orce which the support riction can resist is maximized at equal to 45 degrees and 225 degrees, which correspond to pure lengthwise translations o the slider, and minimized at equal to 135 degrees and 315 degrees. At these our orce angles, s is equal to. 3.1 Friction s and s Coulomb s law implies a simple constraint on the total orce acting at a pointcontact: the orce must act on a line through the contact and inside a riction deined by the directed lines which make an angle o tan ;1 with the contact normal. Similarly, we can deine the [9] to be a constraint on the possible directions o the slider contact point. Each orce inside the contact riction is mapped by the limit curve to a corresponding contact point angle s. These vector directions sweep

4 y y riction v y v y x v x x v x Figure 6: Mapping a riction ( = 2) to a through the limit curve or. out the. Note that the convexity o the limit curve implies that s changes monotonically with. Now it is rather easy to see how pulling is possible. Coulomb s law is satisied only by orces applied into the slider, but there is no similar requirement on velocities. For a large coeicient o riction, the can include velocities away rom the slider. Figure 6, or example, shows the or and = 2. Note, however, that the quantity must be nonnegative; that is, the pusher must perorm nonnegative work in order to move the slider. Thereore, the will always lie within 90 degrees o the riction. 3.2 Slipping with ininite riction An ininite coeicient o riction results in a riction spanning 180 degrees, including exactly those orces directed into the slider. Due to symmetry in the limit curve, the also spans 180 degrees. But unless this is aligned with the riction, slipping may occur at the pusher-slider contact. Slipping occurs or any pusher such that (1) v p has a positive component in the contact normal (v py > 0), and (2) v p is outside the. The irst constraint ensures that the slider must move in response to the push: the pushing constraint is active. The second constraint ensures that the slider is not equal to the pusher v p : the contact is slipping. For sticking contact to occur the ence would have to apply a orce away rom the slider, which is impossible without adhesion. Figure 7 shows the s or ininite and several choices o l. The pusher-slider contact will slip or p just less than 180 degrees. Figure 8 demonstrates slipping contact or. The rod endpoint cannot equal the pusher v p because v p lies to the let o the. Thereore, the contact is slipping Figure 7: Ininite riction s. to the right and the applied orce lies on the let edge o the riction. This gives rise to a rod endpoint on the let edge o the. The pusher, slider, and slipping velocities are illustrated in Figure 8. Sticking contact will always occur or ininite, regardless o the slider s support distribution, i the pusher moves exactly in the direction o the contact normal. The ininite riction must lie within 90 degrees o the riction, and thereore the must always include the contact normal. Slip is impossible or a normal push with ininite because it implies that the pusher perorms negative work, i.e., is negative. 4 Dynamic pushing and pulling In this section we irst analyze the dynamic motion o a slider on a rictionless support, and then we incorporate support rictional orces. In the case o pure dynamic pushing, support rictional orces are negligible compared to inertial orces. It is a simple matter to show that the two eects demonstrated above or the quasi-static case also apply to the pure dynamic case. The example used here is analogous to that or the quasi-static case. The slider has unit mass uniormly distributed over the ring, giving it a unit radius o gyration. The linear acceleration o the pusher is a p and the acceleration o the rod endpoint is a s. We will consider the initial motion o the slider starting rom rest. We will employ an acceleration, which is analogous to the or quasi-static pushing. The orces which may be applied by the pusher, represented by the riction, map to a set o possible rod endpoint accelerations, represented by the acceleration. The acceleration possesses properties similar to those o the. In particular, the angle o a s changes monotonically with, and a s must always be positive, i.e., the acceleration always lies within 90 degrees o

5 + accel. (α > 0) lip v p riction accel. - accel. (α < 0) (a) Figure 8: Slipping with ininite riction or. the riction. This is another way o saying that the mass matrix o the slider must be positive deinite. For a slider o unit mass, unit radius o gyration, and the general rod angle, we ind the generalized acceleration o the slider center o mass (a cx a cy c ): a cx = x (8) a cy = y (9) c = ( x l sin ; y l cos ) (10) The rod endpoint acceleration a s is then a sx = a cx + c l sin = x (1 + l 2 sin 2 ) ; y l 2 sin cos (11) a sy = a cy ; c l cos = y (1 + l 2 cos 2 ) ; x l 2 sin cos (12) Pulling is possible when the rod endpoint acceleration is away rom the slider (a sy < 0), i.e., when y (1 + l 2 cos 2 ) ; x l 2 sin cos <0 (13) Substituting y or x and choosing = 45 and, this condition becomes 3 y ; 2 y < 0, which is satisied or >1:5. Figure 9 shows the possible initial acceleration (ound by the orce-dual method o [3]) and the acceleration s or and two dierent values o the coeicient o riction: (a) = 2; and (b) = 1. The pusher and slider are initially at rest. In Figure 9(a), pulling is a consistent solution i the pusher acceleration a p is inside the acceleration and away rom the slider (a py < 0). In Figure 9(b), slipping contact is the only solution i the pusher acceleration a p is outside the acceleration with a py > 0. Continuing the analogy with the quasi-static case, we note that the mapping between the applied orce angle and the angle o a s is governed by an ellipsoidal surace analogous to the limit surace o the quasi-static case. An accel. riction + accel. (b) (α > 0) + or - - accel. (α < 0) Figure 9: Acceleration s or. (a) = 2. (b) = 1. applied orce ( x y ) maps to an acceleration (a x a y ) normal to this ellipsoid at ( x y ). The 0 = 0 plane o this ellipsoid is an ellipse with a major axis o hal-length c aligned with p the rod and a perpendicular minor axis o hal-length c= 1 +(l=) 2, where is the (unit) radius o gyration and c is a scaling actor depending on the orce magnitude. Figure 10 shows the ellipses corresponding to several rod lengths and the mapping o a riction to an acceleration or the case. These ellipses clearly indicate the possibility o pulling and slipping with ininite riction. We note that the dynamic pulling example is equivalent to a well known example o rictional ambiguity: a rod touching a nearly vertical wall in a gravity ield. (See [8, 5, 10].) As ar as we know, no example o slip with ininite riction has previously been published. Finally we address a case where neither riction nor dynamic orces can be neglected. In such a case neither the nor the acceleration is directly applicable, but it is still easy to construct examples where the initial motion involves pulling or slip with ininite riction. Again we consider the initial motion o an object starting rom rest. The initial acceleration is represented by an acceleration center. This acceleration center is also the

6 y initially at rest, each line o orce is perpendicular to the line through the acceleration center and the slider center o mass. riction x a y acceleration a x 2. Both lines o orce are directed so as to oppose the acceleration. 3. Thereore, the two lines o orce are parallel to each other, and to their resultant orce. 4. Both lines o orce lie on the opposite side o the ring center rom the acceleration center. 5. I the acceleration center is outside the ring, each line o orce passes through the ring. Figure 10: Ellipses governing the mapping rom to a s or the pure dynamic case and our dierent rod lengths. Also shown is the mapping rom a riction to an acceleration or. + accel. center a p, a s Figure 11: A pulling example, including support riction and inertial orces.. impending center, rom which we can derive the support rictional orces. Figure 11 shows a possible pulling solution or. Consider an initial acceleration center on the ring as shown. For this acceleration center, both the dynamic orce and the rictional orce act along a line through the contact point and tangent to the ring. Whatever their relative magnitudes, the resultant must act along the same line. A dynamic balance results rom an applied orce acting along the same line in the opposite direction, as shown. Notice that this acceleration center gives an initial rod endpoint acceleration away rom the slider: pulling. Demonstrating slip with ininite riction is a little more involved. We note the ollowing geometrical relations that apply to the lines o action o both the support rictional orce and the dynamic orce: 1. Given an initial acceleration center with the slider 6. I the acceleration center is on the ring, each line o orce is tangent to the ring. 7. I the acceleration center is inside the ring, each line o orce passes outside the ring. Now we consider several cases, depending on the direction o the applied orce. First, suppose the applied orce acts within the riction, but along a line wholly to the right o the ring. Then both the support riction and the dynamic orce must also pass the ring on the same side. Hence the acceleration center is inside the ring, on a line perpendicular to the applied orce, and on the opposite side o the ring center. As we sweep the applied orce through this range, the possible acceleration must all within a sector o the ring interior. Second, suppose the applied orce is tangent to the ring. Both the support riction and the dynamic orce must act along the same line. The acceleration center must lie on the opposite side o the ring. Third, suppose the applied orce passes to the right o the ring center, but through the ring. The support riction and the dynamic orce must act on the same side o the ring center and through the ring. The acceleration center must lie on a line perpendicular to the applied orce, on the opposite side o the ring center, and outside the ring. As the applied orce is swept through this region, the acceleration center is conined to a sector o the plane outside the ring. Fourth, suppose the applied orce passes through the ring center. The support riction and the dynamic orce cannot act on opposite sides o the center, so they must both act through the center. The acceleration center is at ininity, corresponding to a translation in the direction o the applied orce. There are three more cases, but they are similar to the irst three. Taking all seven cases together, we ind that the possible acceleration or are conined to the region shown in Figure 12. The corresponding initial accelerations o the rod endpoint are also shown. No initial acceleration outside that is possible. Slipping occurs

7 + accel. - accel. in which deinition 1 prevents sticking contact, deinition 2 also disallows sticking contact. Deinition 2, however, also disallows slipping contact; the slipping examples given above have no solution. Neither deinition implements the perectly rough surace o classical mechanics. a p Acknowledgments We thank John Maddocks or an interesting discussion on ininite riction and the members o the Manipulation Lab at CMU or their comments. This work was supported by NSF Grant IRI Reerences Figure 12: Slip with ininite riction, including support riction and inertial orces.. or an initial pusher acceleration a p at angles just less than 180, regardless o the magnitude o the acceleration. (Note that the bounds on the initial accelerations shown are a conservative approximation ound using the geometrical acts given. Tighter bounds may be obtained with an exact analysis, but these bounds serve to demonstrate the phenomenon.) 5 Remarks One o the diiculties in this work was that deinitions o ininite riction and perectly rough are generally imprecise and vary rom one author to the next. In our sampling it appears that most authors really intend perectly rough to mean a kinematic constraint that prevents slip, which is the deinition we adopted. This deinition does not need to be amended, but it is wise to bear in mind that it sometimes implies adhesive orces. With respect to the notion o ininite riction the situation is less clear. It should be noted that slipping is possible with ininite riction because zero normal orce is applied to the slider, thus allowing the tangential rictional orce to be inite during slipping. Alexander and Maddocks [1] adopt a dierent deinition o ininite riction which prevents a tangential orce in the absence o a normal orce. This deinition changes the range o orce angles which may be applied by the ence rom the closed interval [0 180 ] to the open interval (0 180 ). These deinitions o ininite riction correspond to (1) n = 0 ) t unrestricted (and during slip, the rictional orce is directed opposite the slipping ), and (2) n = 0 ) t = 0, respectively, where n is the normal orce and t is the tangential rictional orce. In our examples, any situation [1] J. C. Alexander and J. H. Maddocks. Bounds on the rictiondominated motion o a pushed object. International Journal o Robotics Research, to appear. [2] D. Bara. Coping with riction or non-penetrating rigid body simulation. Computer Graphics, 25(4): 31 40, [3] R. C. Brost and M. T. Mason. Graphical analysis o planar rigid-body dynamics with multiple rictional contacts. In International Symposium on Robotics Research, pages , Aug [4] P. E. Dupont. Theeect o Coulomb riction on the existence and uniqueness o the orward dynamics problem. In IEEE International Conerence on Robotics and Automation, pages , [5] M. A. Erdmann. On motion planning with uncertainty. Master s thesis, MassachusettsInstitute o Technology, Aug [6] S. Goyal. Planar Sliding o a Rigid Body With Dry Friction: Limit Suracesand Dynamics o Motion. PhD thesis, Cornell University, Dept. o Mechanical Engineering, [7] S. Goyal, A. Ruina, and J. Papadopoulos. Planar sliding with dry riction. Part 1. Limit surace and moment unction. Wear, 143: , [8] P. Lötstedt. Coulomb riction in two-dimensional rigid body systems. Zeitschrit ür Angewandte Mathematik und Mechanik, 61: , [9] M. T. Mason. Mechanics and planning o manipulator pushing operations. International Journal o Robotics Research, 5(3): 53 71, Fall [10] V. T. Rajan, R. Burridge, and J. T. Schwartz. Dynamics o a rigid body in rictional contact with rigid walls. In IEEE International Conerence on Robotics and Automation, pages , [11] D. E. Rutherord. Classical Mechanics. Oliver and Boyd, London, [12] Y.-T. Wang, V. Kumar, and J. Abel. Dynamics o rigid bodies undergoing multiple rictional contacts. In IEEE International Conerence on Robotics and Automation, pages , 1992.