ADJUSTABLE GEOMETRIC CONSTRAINTS
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
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
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
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
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
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
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
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
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
ARKC demo animation y x 1 1 3 2 3 2 Input: Actuate Balls 2 & 3 Output: y Input: Actuate Ball 1 Output: x and θ z
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
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
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 ) 1 2 3 y 3E 3 ( sin( θ θ ) ( θ θ )) 2C 2 A 3C 3A 1 z ( z1 + z2 + z3 ) 3 θ θ Y Z 1 3 3 3 E R ( z z ) 2 1 ( sin( θ θ ) + sin( θ θ ) + ( θ θ )) 1C 1A 2C 2 A 3C 3A
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 5 3 6 9 12 15 18-5 -1 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 ] 1 75 15 +/- 15 2 7 11 +/- 2 5 6 12 +/- 3 1 47 133 +/- 43
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 1.571 radians.1969 inches θ 2A 33 degrees 5.76 radians z 2 5 microns θ3 A 21 degrees.19685 inches 3.665 radians z 3 2 microns θ 1C 87.773 degrees.7874 inches 1.531 radians θ 2C 331.1455 degrees 5.78 radians θ3 C 211.1467 degrees 3.685 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 -.424 radians.6 µradians -424.4889 µradians. degrees -.23586 degrees x 5.5 microns θ Y -.331 radians.197 inches -33.74 µradians -.173647 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 1-2.2927 degrees x 5. -.1 microns θ 2 1.1455 degrees θ3 1.1467 degrees y..1 microns ERROR SUM 2698.1 <- 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.
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
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
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
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