Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36115-494 Cognitive Robotics04/04/07Manipulation with Friction15-494 Cognitive Robotics David S. Touretzky & Ethan Tira-Thompson Carnegie Mellon Spring 2007215-494 Cognitive Robotics04/04/07Introduction1.Friction 2.Jacobians3.Dynamics4.Control (P, PD, PID)15-494 Cognitive Robotics04/04/073Friction: Coulomb’s LawFigure 6.1 - Mason, Mechanics Of Robotic ManipulationPulling forceStatic friction force balances pulling force, up to maximum specified by static friction coefficientFriction ForceOnce object begins moving, frictional force drops to constant value, called sliding friction or kinetic friction415-494 Cognitive Robotics04/04/07Friction: Coulomb’s Law•For common tasks, independent of velocity and surface area•With extreme pressures, coefficient rises•With extreme velocities, coefficient drops•Coefficients of friction are different for every pair of surfaces — table lookup•also differ for every change in temperature, humidity, dust/dirt, vibration, celestial alignment, etc. — not terribly accurate515-494 Cognitive Robotics04/04/07Friction within Joints•Static friction is a headache for fine motor control•motor has to ramp up power to overcome static friction within gears, but as soon as it succeeds in doing so, it’s now providing too much power and will “jump” to life.•this is the fundamental reason you see the Aibo’s joints twitch from time to time•the higher the gear ratio, the bigger the problem615-494 Cognitive Robotics04/04/07Computing with Forces•Forces are defined by a line through space, and a magnitude•usually represented by a vector and a point•but the point is not unique — any point along the vector is equally valid (“line of action”)Figure 5.1 - Mason,Mechanics Of Robotic Manipulation715-494 Cognitive Robotics04/04/07Friction with Objects•Now we can define a friction cone:•Edges of the cone define maximum angle allowed for forces without slippage•If you break applied force into normal force fn and tangental force ft, friction cone is defined as |ft| ≤ µ|fn|, with interior angle 2 tan-1µlLlRftftfnfn815-494 Cognitive Robotics04/04/07Friction with Objects•Remember Reuleaux’s Method?•Works with friction cones as well•Now we’re analyzing forces, not displacements, a different interpretation!(be careful about trying to mix them...)•Only forces which agree with the all of the contacts’ constraints can be applied by the contact(s):lLlR++––++––915-494 Cognitive Robotics04/04/07Combining Forces•Adding multiple contacts allows you to apply any force in the linear span of their friction cones•Remember that forces act along a line through space•slide forces along line of action to intersection•Resultant force is the vector sum of the two forces, acting through common intersectionf1f2f1 + f215-494 Cognitive Robotics04/04/0710Friction with Objects:Examples–NONONOYESYESYESNO+YESYESDon’t actually care aboutthe object itself once contactshave been analyzed+–For reference:1115-494 Cognitive Robotics04/04/07Center of Friction•Similar to center of mass, center of friction is the integrated pressure over the support region•Allows you to treat the interaction as a single contact•Hard to model — with a rigid body, small variances completely throw off pressure distribution•Ever play Jenga?1215-494 Cognitive Robotics04/04/07Applying Friction & Forces•Use weight to flip brick•Use wall to direct ball (extra arm)•Get ball away from wall•Use wall to align/direct brick•Stand bone upright•Insert objects without jamming or wedging1315-494 Cognitive Robotics04/04/07The Jacobian Matrix•One of the most important tools in analyzing and controlling robot motion!•Provides the instantaneous velocity in each of the 6 freedoms (translation and orientation along/around each of x, y, and z) as a function of each of the robot’s links= Jacobian (6×n) — a function of current joint positions (q)= joint velocity vector (length n)= workspace velocity vector (length 6)1415-494 Cognitive Robotics04/04/07The Jacobian Matrix:Usage•Find current workspace velocity/force•Determine contribution ofindividual joints•Analyze rank to detect singularities (for better or worse)•a singularity occurs when joints become aligned, causing a loss in effector mobility (but increased strength along that axis!)•under-actuated robots always haveincomplete rankFull (planar) mobilitySingularity:cannot move along y axis, but also don’t have to do any work to resist forces along it1515-494 Cognitive Robotics04/04/07The Jacobian Matrix:Usage•Things to watch out for at/near singularities:•Small workspace movements/forces may require instantaneous joint motion (infinite motor torque!)•Usually occur at workspace limits•May have infinite inverse kinematic solutions•Test for configuration “quality”:Swap J(q) and JT(q) if under-actuated (i.e. J(q) is less than full rank)M(q) becomes zero at singularities15-494 Cognitive Robotics04/04/0716The Jacobian Matrix:Composition= Position component (sometimes )= Orientation component= Linear velocity vector of end effector= Angular velocity vector of end effectorJacobian is split into two components:15-494 Cognitive Robotics04/04/0717The Jacobian Matrix:Position ComponentIf Joint i is prismatic:If Joint i is revolute:Where:zi = z axis of joint i p = position of the end effectorpi = position of joint i’s origin(all relative to base frame)Remember that a joint’s z axis is always defined to point along its axis of motion15-494 Cognitive Robotics04/04/0718The Jacobian Matrix:Orientation ComponentIf Joint i is prismatic:If Joint i is revolute:15-494 Cognitive Robotics04/04/0719The Jacobian Matrix:ExampleA planar RRR armThese are all given to you by the forward kinematics: each joint’s transformation matrix holds the current z vector in the 3rd column and the current position in the 4th column15-494 Cognitive Robotics04/04/0720The Jacobian Matrix:ExampleA planar RRR armNotation:s1 = sin(θ1)c123 = cos(θ1+θ2+θ3)15-494 Cognitive Robotics04/04/0721The Jacobian Matrix:ExampleA planar RRR armNotation:s1 = sin(θ1)c123 = cos(θ1+θ2+θ3)2215-494 Cognitive Robotics04/04/07Dynamics•How