ADJUSTABLE GEOMETRIC CONSTRAINTS 2001 MIT PSDAM AND PERG

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

Serial and parallel kinematic machines/mechanisms SERIAL MECHANISMS ¤ 2 ry ¤ Structure takes form of open loop I. e. Most mills, lathes, “stacked” axis robots Kinematics analysis typically easy Ro ta ¤ Rotary 3 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 © 2001 MIT PSDAM AND PERG LABS Photo from Physik Instruments web page Linear 1

Parallel mechanism: Stewart-Gough platform 6 DOF mechanism/machine Multiple variations on this theme with different joints: ¤ ¤ ¤ Spherical joints: Prismatic joints: Planar: 3 Cs 5 Cs 3 Cs Permits 3 rotary DOF Permits one linear DOF Permits two linear, one rotary DOF Ball Joint Sliding piston Roller on plane E. g. Changing length of “legs” Spherical Joint Prismatic Joint ile Mobile Base/ground © 2001 MIT PSDAM AND PERG LABS Base/ground

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

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

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 Ball’s center of curvature Symmetry intersect Contact Cone Groove’s center of curvature © 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) Adjustment ¤ Strain kinematic elements to correct inaccuracy (element size) Ball’s center of curvature © 2001 MIT PSDAM AND PERG LABS

Example: Adjusting planar motion Position control in x, y, qz: Ø Rotation axis offset from the center of the ball A A e 180 o B B Eccentric left Eccentric right Patent Pending, Culpepper © 2001 MIT PSDAM AND PERG LABS

ARKC demo animation y x 1 3 1 2 Input: Output: Actuate Balls 2

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 q 1 A Axis of rotation Offset, e Center of sphere y x qz q 3 A Joint 3 © 2001 MIT PSDAM AND PERG LABS q 2 A Joint 2 z x

Vector model for planar adjustment of KC A 1 r 1 c M 1 ’ M 1 r 1 d r 1 b y’ x’ r 1 a r. D y M 3 r 3 b r 3 c r 2 c A 2 r 2 a M 2 M 3 ’ r 3 d qz A 3 x r 3 a r 2 d r 2 b M 2 ’ r 1 a + r 1 b + r 1 c + r 1 d = r. D r 2 a + r 2 b + r 2 c + r 2 d = r. D r 3 a + r 3 b + r 3 c + r 3 d = r. D © 2001 MIT PSDAM AND PERG LABS

Analytic position control equations Vector equations: r 1 a + r 1 b + r 1 c + r 1 d = r. D r 2 a + r 2 b + r 2 c + r 2 d = r. D r 3 a + r 3 b + r 3 c + r 3 d = r. D System of 6 equations: For standard coupling: 120 o; offset E; coupling radius R; © 2001 MIT PSDAM AND PERG LABS sin(qz) ~ qz:

ARKC resolution analysis For E = 125 microns R 90 = -1. 5 micron/deg R 0 = “ 0” micron/deg q 1 C y’ y x’ q x Limits on Linear Resolution Assumptions q 1 C © 2001 MIT PSDAM AND PERG LABS % Error Lower Limit [ Degree ] Upper Limit [ Degree ] Half Range [ Degree ] 1 75 105 +/- 15 2 70 110 +/- 20 5 60 120 +/- 30 10 47 133 +/- 43

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

Low-cost adjustment (10 mm) Peg shank and convex crown are offset Light press between peg and bore in plate Adjustment with allen wrench e Epoxy or spreading to set in place Friction (of press fit) must be minimized… z r Top Platform q = 180 o 2 E Bottom Platform © 2001 MIT PSDAM AND PERG LABS

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

“Premium” adjustment (sub-micron) z r Run-out is a major cause of error Seals keep contaminants out Dual motion actuators provide: ¤ ¤ Actuator Air bushings for lower run-out Flexible Coupling Linear motion Rotary motion Seal Air Bushings Seal Shaft Ball Groove Flat © 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 } © 2001 MIT PSDAM AND PERG LABS