1. © 2001 MIT PSDAM AND PERG LABS ADJUSTABLE GEOMETRIC CONSTRAINTS

2. © 2001 MIT PSDAM AND PERG LABS Why adjust kinematic couplings? KC Repeatability is orders of magnitude better than accuracy Accuracy = f ( manufacture and assemble ) Desired PositionMated Position Non-Kinematic Coupling Kinematic Coupling Adjusted Kinematic Coupling AccuracyRepeatability Accuracy & Repeatability

3. © 2001 MIT PSDAM AND PERG LABS 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 Base/ground Rotary 1 Rotary 2 Rotary 3 Linear 1 Photo from Physik Instruments web page

4. © 2001 MIT PSDAM AND PERG LABS Parallel mechanism: Stewart-Gough platform 6 DOF mechanism/machine Multiple variations on this theme with different joints:  Spherical joints:3 CsPermits 3 rotary DOFBall Joint  Prismatic joints:5 CsPermits one linear DOFSliding piston  Planar:3 CsPermits two linear, one rotary DOFRoller on plane E.g. Changing length of “legs” Spherical Joint Prismatic Joint Base/ground Mobile Base/ground Mobile

5. © 2001 MIT PSDAM AND PERG LABS Parallel mechanism: Variation 6 DOF mechanism/machine by changing position of joints Can have a combination of position and length changes Base Mobile Base Mobile platform

6. © 2001 MIT PSDAM AND PERG LABS Kinematic couplings as mechanisms Bottom Platform Top Platform Far-field points modeled as joints Hertz Compliance 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

7. © 2001 MIT PSDAM AND PERG LABS rr zz nn ll Initial Position of Ball’s Far Field Point Final Position of Ball’s Far Field Point r z l n Final True Groove’s Far Field Point Initial Position of Groove’s Far Field Point Initial Contact Point Final Contact Point “Joint” motion Model of kinematic coupling mechanism Ball Center Groove far field point

8. © 2001 MIT PSDAM AND PERG LABS 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 Accuracy of kinematic mechanisms Ball’s center of curvature Symmetry intersect Groove’s center of curvature Contact Cone

9. © 2001 MIT PSDAM AND PERG LABS 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) Adjustment Ball’s center of curvature

10. © 2001 MIT PSDAM AND PERG LABS Example: Adjusting planar motion Position control in x, y,  z :  Rotation axis offset from the center of the ball 180 o BB AA e Eccentric leftEccentric right Patent Pending, Culpepper

11. © 2001 MIT PSDAM AND PERG LABS ARKC demo animation Input: Actuate Ball 1 Output:  x and  z Input: Actuate Balls 2 & 3 Output:  y x y 1 23 1 23

12. © 2001 MIT PSDAM AND PERG LABS Planar kinematic model Equipping each joint provides control of 3 degrees of freedom View of kinematic coupling with balls in grooves (top platform removed) x y zz Joint 1 Axis of rotation  2A  3A 1A1A Joint 3 x z Joint 2 Center of sphere Offset, e

13. © 2001 MIT PSDAM AND PERG LABS Vector model for planar adjustment of KC x y rr r 1d r 1c r 1b r 1a r 3a r 3c r 3b r 2a r 2b r 2c r 2d r 3d zz M1M1 M3M3 M2M2 A3A3 A2A2 A1A1 x’ y’ M2’M2’M3’M3’ M1’M1’ 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. © 2001 MIT PSDAM AND PERG LABS Analytic position control equations Vector equations: System of 6 equations: For standard coupling: 120 o ; offset E; coupling radius R; sin(  z ) ~  z : 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 

15. © 2001 MIT PSDAM AND PERG LABS ARKC resolution analysis Limits on Linear Resolution Assumptions For E = 125 microns R 90 = -1.5 micron/deg R 0 = “0” micron/deg x y  1C1C x’ y’  1C Lower LimitUpper LimitHalf Range % Error [ Degree ] 175105+/- 15 270110+/- 20 560120+/- 30 1047133+/- 43

16. © 2001 MIT PSDAM AND PERG LABS Forward and reverse kinematic solutions

17. © 2001 MIT PSDAM AND PERG LABS e Low-cost adjustment (10  m) 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… Bottom Platform  = 180 o r z Top Platform 2E

18. © 2001 MIT PSDAM AND PERG LABS Moderate-cost adjustment (3 micron)  B = -360 o Bottom Platform Shaft A input Top Platform Magnet B r z Magnet A Shaft B input Teflon sheets (x 4) e Ball l Shaft B positions z height of shaft A[ z,  x,  y ] Shaft A positions as before[ x, y,  z ] Force source preload I.e. magnets, cams, etc.. Bearing support

19. © 2001 MIT PSDAM AND PERG LABS “Premium” adjustment (sub-micron) r z Shaft Ball Seal Groove Flat Air Bushings Flexible Coupling Actuator 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

20. © 2001 MIT PSDAM AND PERG LABS 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 }