Matthew T. Mason angle, Mechanics of Manipulation
Today s outline angle, angle,
How do you move things around? angle, Kinematics, kinematic constraint. Force. Force of constraint; Gravity; ; Momentum.
How do you know where things are? You put them somewhere, or you look at them. And then, they stay put. Theorem (Liouville s theorem) In a Hamiltonian system (including an energy-conserving passive mechanical system) the uncertainty, measured as a probability distribution in phase space, remains constant as the system evolves. angle, Energy loss is essential. and plastic impact are essential.
An alternative reality What would it be like to live in a world without friction? Grabbing anything would be like grabbing a bar of soap. No tables. Bowls? Tables with edges? Walking wouldn t work. Drill footholds everywhere? (You re stranded in the middle of a frozen pond, perfectly flat and frictionless. Q: How do you get to shore? A: Throw one of your shoes to the opposite shore.) What would feet and hands look like? With no gravity? Where do you look for lost items? On the space shuttle, they find lost articles at the AC intake grille. angle,
Overview is complex and difficult to model. We will focus on simple approximations: generally, Coulomb s law of sliding friction, with known uniform coefficient of friction. Applied with common sense, this model is good enough to explore problems, and perform useful manipulation. But, know the limits. angle,
An experiment, in the style of Coulomb Clean surfaces, but not too clean dry, unlubricated. Pull on string with force f a, ramping up from 0. force f f will balance f a, up to a point. Max f f when not moving: µ s mg. f f µ smg µ dmg f f mg f a angle, Max f f when moving: µ d mg. From now on we will assume µ s = µ d = µ. f a
Coulomb s observations Coulomb conducted hundreds of experiments, and over a broad range of conditions he observed: al force is approximately independent of contact area. al force is approximately independent of velocity magnitude. Coefficient of friction depends on pairs of materials. angle, Materials µ metal on metal 0.15 0.6 rubber on concrete 0.6 0.9? 1.0 2.0? plastic wrap on lettuce Leonardo s number 0.25 (Do not believe these numbers!)
Apply Coulomb s law with care angle, It holds over a broad range, but not nearly everywhere. It is approximate. Coefficients of friction tables are terrible. How can you use something so unreliable? But, how can you not use it?
Contact modes We can write Coulomb s law: ẋ ẍ < 0 f t = µf n left sliding f n > 0 f t = µf n right sliding x = 0 < 0 f t = µf n left sliding = 0 > 0 f t = µf n right sliding = 0 = 0 f t µf n rest and define contact mode to be the right column left sliding, etc. f t angle,
angle Block at rest on plane with angle α: f n = mg cos α angle, f t = mg sin α At rest f t µf n. Maximum α: f t = µf n f n f t Substituting, mg sin α = µmg cos α α = tan 1 µ mg Sometimes called the friction angle or the angle of repose.
cone Definition Define the friction cone to be the set of all wrenches satisfying Coulomb s law for an object at rest, i.e. satisfying f t µ f n 2tan 1 µ f n f t angle, The friction cone is a polyhedral convex cone in wrench space. We can restate Coulomb s law using contact modes Left sliding: f n + f t right edge of friction cone Right sliding: f n + f t left edge of friction cone Rest: f n + f t friction cone
friction cone angle, cone is positive linear span of left edge unit vector and right edge unit vector. Moment labeling
Given a mobile body, several frictional contacts, and an applied force, is equilibrium possible? Approach: use moment labelling to identify all feasible contact wrenches; test whether one of them balances the applied force. Caveat A: Presence of balancing wrench in the cone does not imply nature will select that wrench. See the wedged plank example (static indeterminacy). Caveat B: Even if equilibrium is attained, stability may not be. Stability requires more careful analysis, depending on hypothesized disturbances. angle,
Pipe clamp design problem Why does pipe clamp work? Let diameter be 2 cm. Let length be 2 cm. Assume µ of 0.25. Find min moment arm. Extend to woodpecker toy? angle, fc1 f1 fa f2 fc2
Block on table angle,
Wedged plank and piranha angle,
Triangle and three fingers. angle,
tricks Measuring the coefficient of friction. Driving Coulomb s law doesn t work so well for rubber on concrete; Theory suggests you don t spin your wheels. Almost always spinning is bad. But sometimes, in snow or mud, spinning is a good idea. You can push a car sideways, if the wheels are spinning, turning sliding friction into viscous friction. Reducing imprecision due to friction: dithering. Reducing imprecision due to friction: impact. angle, In theory, there is no difference between theory and practice. In practice, there is. Anonymous