Manipulation with Friction 15 494 Cognitive Robotics David S Touretzky Ethan Tira Thompson Carnegie Mellon Spring 2008 04 02 08 15 494 Cognitive Robotics 1 Introduction 1 Friction 2 Jacobians 3 Dynamics 4 Control P PD PID 04 02 08 15 494 Cognitive Robotics 2 Friction Coulomb s Law ff mg Once object begins moving frictional force drops to constant value called sliding friction or kinetic friction ff s mg d mg Friction Force Static friction force balances pulling force up to maximum specified by static friction coefficient fa Pulling force fa Figure 6 1 Mason Mechanics Of Robotic Manipulation 04 02 08 15 494 Cognitive Robotics 3 Friction 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 04 02 08 also differ for every change in temperature humidity dust dirt vibration celestial alignment etc not terribly accurate 15 494 Cognitive Robotics 4 Friction within Joints Static friction is a headache for fine motor control 04 02 08 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 problem 15 494 Cognitive Robotics 5 Computing 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 x1 f x2 f F N line of action Figure 5 1 Mason Mechanics Of Robotic Manipulation 04 02 08 15 494 Cognitive Robotics 6 Friction with Objects Now we can define a friction cone lL lR 2 tan 1 fn ft 04 02 08 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 fn with interior angle 2 tan 1 15 494 Cognitive Robotics 7 Friction with Objects Remember Reuleaux s Method Works with friction cones as well Now we re analyzing forces not lL lR 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 04 02 08 15 494 Cognitive Robotics 8 Combining Forces Adding multiple contacts allows you to apply any force in the linear span of their friction cones f1 Remember that forces act along a line through space 04 02 08 f2 slide forces along line of action to intersection f1 f2 Resultant force is the vector sum of the two forces acting through common intersection 15 494 Cognitive Robotics 9 Friction with Objects Examples YES NO NO Don t actually care about the object itself once contacts have been analyzed YES NO YES YES NO YES For reference 04 02 08 15 494 Cognitive Robotics 10 Center 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 Different surfaces provide different friction coefficients thus center of friction is a weighted average of the mass over its contacts Uniform bar Example Center of Friction Chewing gum 100 Center of Mass Table 5 Applied Force 04 02 08 15 494 Cognitive Robotics 11 Center of Friction Hard to model with a rigid body small variances completely throw off pressure distribution e g spinning dinner plates Ever play Jenga Photographer Derek Mawhinney http en wikipedia org wiki Jenga 04 02 08 15 494 Cognitive Robotics 12 Applying 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 wedging 04 02 08 15 494 Cognitive Robotics 13 The 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 the joint angles J q Jacobian 6 n a function of J q q x 04 02 08 current joint angles q joint velocity vector length n q x workspace velocity vector length 6 15 494 Cognitive Robotics 14 The Jacobian Matrix Usage Find current workspace velocity force Determine contribution of individual joints Analyze rank to detect singularities for better or worse 04 02 08 a singularity occurs when joints become aligned causing a loss in effector mobility but increased strength along that axis under actuated robots always have incomplete rank 15 494 Cognitive Robotics y x Full planar mobility y x Singularity cannot move along y axis but also don t have to do any work to resist forces along it 15 The 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 quality Test for configuration M q becomes zero at singularities 04 02 08 M q det J q JT q 15 494 Cognitive Robotics Swap J q and JT q if under actuated i e J q is less than full rank 16 The Jacobian Matrix Composition J q q x Jacobian is split into two components Jp q Position component sometimes Jv q Jo q Orientation component Jp q Jo q q p p Linear velocity vector of end effector Angular velocity vector of end effector 04 02 08 15 494 Cognitive Robotics 17 The Jacobian Matrix Position Component Jp q px q1 px q2 py q1 py q2 pz q1 Jp q 1 px qn py qn pz q2 pz qn Jp q 2 Jp q n Remember that a joint s z axis is always defined to point along its axis of motion If Joint i is prismatic Jp q i zi If Joint i is revolute Jp q i zi p pi Where zi z axis of joint i p position of the end effector 04 02 08 pi position of joint i s origin all relative to base frame 15 494 Cognitive Robotics 18 The Jacobian Matrix Orientation Component Jo q x q1 x q2 x qn y q1 y q2 y qn z q1 z q2 z qn Jo q 1 Jo q 2 Jo q n If Joint i is prismatic Jo q i 0 If Joint i is revolute Jo q i zi 04 02 08 15 494 Cognitive Robotics 19 The Jacobian Matrix Example y y A planar RRR arm a1 y 1 x 2 a2 a3 y 3 x x J q 04 02 08 z0 pe p0 z0 x These are all given to you by the forward kinematics each joint s transformation matrix holds the current z vector in the 3rd column and …