ADJUSTABLE GEOMETRIC CONSTRAINTS

1. ADJUSTABLE GEOMETRIC CONSTRAINTS

2. Adjusted Kinematic Coupling Kinematic Coupling Accuracy Repeatability Accuracy & Repeatability Non-Kinematic Coupling Desired Position Mated Position Why adjust kinematic couplings? • KC Repeatability is orders of magnitude better than accuracy • Accuracy = f( manufacture and assemble )

3. Serial and parallel kinematic machines/mechanisms • SERIAL MECHANISMS • Structure takes form of open loop • I.e. Most mills, lathes, “stacked” axis robots • Kinematics analysis typically easy • 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 Rotary 3 Linear 1 Rotary 2 Rotary 1 Base/ground 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 Prismatic Joint Mobile Mobile Base/ground Base/ground

5. Mobile Mobile platform Parallel mechanism: Variation • 6 DOF mechanism/machine by changing position of joints • Can have a combination of position and length changes 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. l n Initial Position of Ball’s Far Field Point dr dl Initial Contact Point dz dn Final Position of Ball’s Far Field Point Ball Center Initial Position of Groove’s Far Field Point z “Joint” motion r Final True Groove’s Far Field Point Final Contact Point Groove far field point Model of kinematic coupling mechanism

8. Ball’s center of curvature Symmetry intersect Contact Cone Groove’s center of curvature 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 • Groove center of curvature location • Groove orientation • Groove depth • Groove radii

9. Adjustment Ball’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) • Strain kinematic elements to correct inaccuracy (element size)

10. Position control in x, y, qz: Rotation axis offset from the center of the ball A A 180o e B B Example: Adjusting planar motion Eccentric left Eccentric right Patent Pending, Culpepper

11. y x ARKC demo animation 1 1 3 2 3 2 Input: Actuate Balls 2 & 3Output: Dy Input: Actuate Ball 1 Output: Dx and Dqz

12. y x z x Planar kinematic model • Equipping each joint provides control of 3 degrees of freedom View of kinematic coupling with balls in grooves (top platform removed) q1A Joint 1 Axis of rotation Offset, e Center of sphere qz q3A q2A Joint 3 Joint 2

13. r1a + r1b + r1c + r1d=rD r2a + r2b + r2c + r2d=rD r3a + r3b + r3c + r3d=rD Vector model for planar adjustment of KC A1 r1c M1’ r1d r1b M1 y’ x’ r1a qz rD r2d r3d y A3 x r3a A2 M3 r2c r3c r2a r3b r2b M2 M3’ M2’

14. r1a + r1b + r1c + r1d=rD r2a + r2b + r2c + r2d=rD r3a + r3b + r3c + r3d=rD Analytic position control equations • Vector equations: • System of 6 equations: • For standard coupling: 120o; offset E; coupling radius R; sin(qz) ~ qz:

15. q1C y y’ x x’ q q 1C Lower Limit Upper Limit Half Range % Error [ Degree ] [ Degree ] [ Degree ] 1 75 105 +/- 15 2 70 110 +/- 20 5 60 120 +/- 30 10 47 133 +/- 43 ARKC resolution analysis For E = 125 microns R90 = -1.5 micron/deg R0 = “0” micron/deg Limits on Linear Resolution Assumptions

16. Forward and reverse kinematic solutions

17. e Low-cost adjustment (10 mm) • 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 • Friction (of press fit) must be minimized… z Top Platform r q = 180o 2E Bottom Platform

18. Bottom Platform Moderate-cost adjustment (3 micron) • Shaft B positions z height of shaft A [ z, qx, qy ] • Shaft A positions as before [ x, y,qz ] • Force source preload I.e. magnets, cams, etc.. z Shaft A input r Magnet A Magnet B Teflon sheets (x 4) Shaft B input Top Platform qB = -360o e Ball l Bearing support

19. “Premium” adjustment (sub-micron) z r • Run-out is a major cause of error • Air bushings for lower run-out • Seals keep contaminants out • Dual motion actuators provide: • Linear motion • Rotary motion Actuator 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 }