ADJUSTABLE GEOMETRIC CONSTRAINTS 2001 MIT PSDAM AND PERG LABS
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1 ADJUSTABLE GEOMETRIC CONSTRAINTS
2 Why adjust kinematic couplings? KC Repeatability is orders of magnitude better than accuracy Accuracy = f ( manufacture and assemble ) Kinematic Coupling Accuracy Adjusted Kinematic Coupling Repeatability Accuracy & Repeatability Non-Kinematic Coupling Mated Position Desired Position
3 Serial and parallel kinematic machines/mechanisms SERIAL MECHANISMS Rotary 3 Structure takes form of open loop I.e. Most mills, lathes, stacked axis robots Rotary 2 Linear 1 Kinematics analysis typically easy Rotary 1 Base/ground PARALLEL MECHANISMS Structure of closed loop chain(s) I.e. Stuart platforms & hexapods Kinematics analysis usually difficult 6 DOF mechanism/machine Multiple variations on this theme Photo from Physik Instruments web page
4 Parallel mechanism: Stewart-Gough platform 6 DOF mechanism/machine Multiple variations on this theme with different joints: Spherical joints: 3 Cs Permits 3 rotary DOF Ball Joint Prismatic joints: 5 Cs Permits one linear DOF Sliding piston Planar: 3 Cs Permits two linear, one rotary DOF Roller on plane E.g. Changing length of legs Spherical Joint Mobile Prismatic Joint Mobile Base/ground Base/ground
5 Parallel mechanism: Variation 6 DOF mechanism/machine by changing position of joints Can have a combination of position and length changes Mobile Mobile platform Base Base
6 Kinematic couplings as mechanisms Ideally, kinematic couplings are static parallel mechanisms IRL, deflection(s) = mobile parallel kinematic mechanisms How are they mobile? Hertz normal distance of approach ~ length change of leg Far field points in bottom platform moves as ball center moves ~ joint motions Top Platform Bottom Platform Far-field points modeled as joints Hertz Compliance
7 Model of kinematic coupling mechanism l Initial Position of Ball s Far Field Point δ l δ r n Final Position of Ball s Far Field Point z δ n r δ z Final Contact Point Initial Contact Point Initial Position of Groove s Far Field Point Joint motion Final True Groove s Far Field Point Ball Center Groove far field point
8 Accuracy of kinematic mechanisms Since location of platform depends on length of legs and position of base and platform joints, accuracy is a function of mfg and assembly Parameters affecting coupling centroid (platform) location: Ball center of curvature location Ball orientation (i.e. canoe ball) Ball centerline intersect position (joint) Ball radii Ball s center of curvature Groove center of curvature location Groove orientation Groove depth Groove radii Symmetry intersect Contact Cone Groove s center of curvature
9 Utilizing the parallel nature of kinematic couplings Add components that adjust or change link position/size, i.e.: Place adjustment between kinematic elements and platforms (joint position) Adjustment Strain kinematic elements to correct inaccuracy (element size) Ball s center of curvature
10 Example: Adjusting planar motion Position control in x, y, θ z : Rotation axis offset from the center of the ball A A e B 18 o B Eccentric left Eccentric right Patent Pending, Culpepper
11 ARKC demo animation y x Input: Actuate Balls 2 & 3 Output: y Input: Actuate Ball 1 Output: x and θ z
12 Planar kinematic model Equipping each joint provides control of 3 degrees of freedom View of kinematic coupling with balls in grooves (top platform removed) Joint 1 θ 1A Axis of rotation Offset, e y x θ z Center of sphere θ 3A θ 2A z Joint 3 Joint 2 x
13 Vector model for planar adjustment of KC r 1c A 1 M 1 r 1b r 1d M 1 y x r 1a r r 3d r 2d θ z y x A 3 A 2 M 3 r 3a r 2c r 3c r 3b r 2a r 2b M 3 M 2 M 2 r 1a + r 1b + r 1c + r 1d = r r 2a + r 2b + r 2c + r 2d = r r 3a + r 3b + r 3c + r 3d = r
14 Analytic position control equations Vector equations: r 1a + r 1b + r 1c + r 1d = r r 2a + r 2b + r 2c + r 2d = r r 3a + r 3b + r 3c + r 3d = r System of 6 equations: cos 1A 1 L 1D sin 1A L 1B L 1D L 1A cos 1A L 1C cos 1C sin 1A 1 L 1D cos 1A L 2B L 1D L 1A sin 1A L 1C sin 1C cos 2A 1 L 2D sin 2A L 3B L 2D L 2A cos 2A L 2C cos 2C sin 2A 1 L 2D cos 2A x L 2D L 2A sin 2A L 2C sin 2C cos 3A 1 L 3D sin 3A y L 3D L 3A cos 3A L 3C cos 3C sin 3A 1 L 3D cos 3A z L 3D L 3A sin 3A L 3C sin 3C For standard coupling: 12 o ; offset E; coupling radius R; sin(θ z ) ~ θ z : x E 3 ( 2sin( θ θ ) + sin( θ θ ) + ( θ θ )) 1C 1A 2C 2 A 3C 3A θ X 1 3 ( 2z z z ) y 3E 3 ( sin( θ θ ) ( θ θ )) 2C 2 A 3C 3A 1 z ( z1 + z2 + z3 ) 3 θ θ Y Z E R ( z z ) 2 1 ( sin( θ θ ) + sin( θ θ ) + ( θ θ )) 1C 1A 2C 2 A 3C 3A
15 ARKC resolution analysis For E = 125 microns R 9 = -1.5 micron/deg R = micron/deg θ 1C x [micron] Centroid X Position, 12 Degree Coupling 15 True Position Plot 1 Linearized Position Plot y x y x θ -15 θ 1c [degrees] Limits on Linear Resolution Assumptions θ 1C Lower Limit Upper Limit Half Range % Error [ Degree ] [ Degree ] [ Degree ] / / / /- 43
16 Forward and reverse kinematic solutions PART I: COUPLING CHARACTERISTICS Use this to specify groove angles and input ball rotations and heights used to calculate coupling position in part II θ 1A 9 degrees z 1 5 microns radians.1969 inches θ 2A 33 degrees 5.76 radians z 2 5 microns θ3 A 21 degrees inches radians z 3 2 microns θ 1C degrees.7874 inches radians θ 2C degrees 5.78 radians θ3 C degrees radians E 125 microns.49 inches R T 5715 microns Note: R T = L id 2.25 inches PART II: CALCULATED MOVEMENTS This takes input from part I to calculate the position of the top part of the coupling. θ Z. radians θ X radians.6 µradians µradians. degrees degrees x 5.5 microns θ Y radians.197 inches µradians degrees y -.14 microns z 235. microns. inches.9252 inches PART III: REVERSE SOLUTION Use this to input a desired position and goal seek to solve for the position of the balls DESIRED POSITION POSITION ERROR BALL SETTINGS θ Z... µrad See part I for modified ball angles, the following angles are the difference between groove and ball angles θ degrees x microns θ degrees θ degrees y..1 microns ERROR SUM <- x, y, θ z <- Use the solver on this value to set it to a value (ideally zero) that is small, but greater than or equal to.
17 Low-cost adjustment (1 µm) Peg shank and convex crown are offset Light press between peg and bore in plate Adjustment with allen wrench Epoxy or spreading to set in place e Friction (of press fit) must be minimized z r Top Platform θ = 18 o 2E Bottom Platform
18 Moderate-cost adjustment (3 micron) Shaft B positions z height of shaft A [ z, θ x, θ y ] Shaft A positions as before [ x, y, θ z ] Force source preload I.e. magnets, cams, etc.. z r Teflon sheets (x 4) Shaft A input Magnet A Magnet B Shaft B input Top Platform θ B = -36 o Bearing support Ball e l Bottom Platform
19 Premium adjustment (sub-micron) Run-out is a major cause of error z r Air bushings for lower run-out Seals keep contaminants out Actuator Dual motion actuators provide: Linear motion Rotary motion Flexible Coupling Seal Air Bushings Seal Shaft Ball Groove Flat
20 Mechanical interface wear management Wear and particle generation are unknowns. Must investigate: Coatings [minimize friction, maximize surface energy] Surface geometry, minimize contact forces Alternate means of force/constraint generation At present, must uncouple before actuation Sliding damage
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