ADJUSTABLE
GEOMETRIC
CONSTRAINTS
© 2001 MIT PSDAM AND PERG LABS
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



Rotary 3
Structure takes form of open loop
I.e. Most mills, lathes, “stacked” axis
robots
Kinematics analysis typically easy
Linear 1
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
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
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
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
Model of kinematic coupling mechanism
Initial Position of Ball’s
Far Field Point
dr
dl
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
Groove
Groove
Groove
center of curvature location
orientation
depth
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:

Rotation axis offset from the center of the ball
A
A
e
180o
B
B
Eccentric left
Eccentric right
© 2001 MIT PSDAM AND PERG LABS
Patent Pending, Culpepper
ARKC demo animation
y
x
1
3
1
2
Input:
Output:
Actuate Balls 2 & 3
Dy
© 2001 MIT PSDAM AND PERG LABS
3
2
Input:
Output:
Actuate Ball 1
Dx and Dqz
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
q1
A
Axis of rotation
Offset, e
Center of sphere
y
x
qz
q3A
Joint 3
© 2001 MIT PSDAM AND PERG LABS
q2A
Joint 2
z
x
Vector model for planar adjustment of KC
A1
r1c
M 1’
M1
r1d
r1b
r1a
rD
r2d
r3d
qz
y
A3
x
M3
r3a
r2c
r3c
r3b
r2a
M2
M 3’
A2
r2b
M 2’
r1a + r1b + r1c + r1d = rD
r2a + r2b + r2c + r2d = rD
r3a + r3b + r3c + r3d = rD
© 2001 MIT PSDAM AND PERG LABS
Analytic position control equations
Vector equations:
r1a + r1b + r1c + r1d = rD
r2a + r2b + r2c + r2d = rD
r3a + r3b + r3c + r3d = rD
System of 6 equations:
0
0
 cos  q 1A

0
0
 sin  q 1A

0
cos  q 2A
0


0
sin  q 2A
0

0
0
cos  q 3A


0
0
sin  q 3A

1 0
0 1
1 0
0 1
1 0
0 1


  L 1B 

L 1D cos  q 1A   L 
 2B 

L 2D sin  q 2A
   L 3B 


L 2D cos  q 2A   x 

L 3D sin  q 3A   y 


L 3D cos  q 3A   q z 
L 1D sin q 1A
  L 1D  L 1A  cos  q 1A  L 1C cos  q 1C 


L

L

sin
q

L

sin
q






 1D
1A
1A
1C
1C 


  L 2D  L 2A  cos  q 2A  L 2C cos  q 2C 
  L 2D  L 2A  sin  q 2A  L 2C sin  q 2C 


L

L

cos
q

L

cos
q






3A
3A
3C
3C 
 3D
  L  L   sin  q   L  sin  q  
3A
3A
3C
3C 
 3D
For standard coupling: 120o; offset E; coupling radius R;
x
y
z
E
 2 sin( q1C  q1 A )  sin( q 2C  q 2 A )  (q 3C  q 3 A ) 
3
3E
sin( q 2C  q 2 A )  (q 3C  q 3 A ) 
3
1
z1  z 2  z 3 
3
© 2001 MIT PSDAM AND PERG LABS
qX  
1
2 z1  z 2  z 3 
3
qY  
3
 z 2  z1 
3
qZ 
sin(qz) ~ qz:
1 E
sin( q1C  q1A )  sin( q 2C  q 2 A )  (q 3C  q 3 A )
3 R
ARKC resolution analysis
For E
= 125 microns
Centroid X Position, 120 Degree Coupling
R90 = -1.5 micron/deg
150
True Position Plot
R0 = “0” micron/deg
100
Linearized Position Plot
C
50
x [micron]
q1
0
0
30
60
90
120
150
180
-50
-100
-150
q 1c [degrees]
y
x
q
Limits on Linear Resolution Assumptions
q1C
© 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
ARKC Kinematic Analysis Spread Sheet
Do not change cells in red, only change cells in blue
PART I: COUPLING CHARACTERISTICS
Use this to specify groove angles and input ball rotations and heights used to calculate coupling position in part II
q1A
q2A
90
1.571
330
degrees
radians
degrees
z1
5
microns
0.0001969 inches
z2
500
5.760
radians
qA
210
3.665
degrees
radians
q1C
87.7073
1.531
331.1455
5.780
211.1467
3.685
degrees
radians
degrees
radians
degrees
radians
E
125
0.0049
microns
inches
RT
57150
2.2500
microns
inches
z3
q2C
qC
microns
0.0196850 inches
200
microns
0.0078740 inches
Note: RT = LiD
PART II: CALCULATED MOVEMENTS
This takes input from part I to calculate the position of the top part of the coupling.
qZ
0.0000
0.0006
0.000000
radians
radians
degrees
qX
-0.004024 radians
-4024.4889 radians
-0.230586 degrees
x
5.0005
0.000197
microns
inches
qY
-0.003031 radians
-3030.7040 radians
-0.173647 degrees
y
-0.0014
0.000000
microns
inches
z
235.0000
0.009252
microns
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
qZ
0.000
0.00
x
5.000
-0.001
BALL SETTINGS
0.0
rad
microns
See part I for modified ball angles, the following angles are the difference between groove and ball angles
q1
-2.2927
degrees
q2
1.1455
1.1467
degrees
degrees
q
y
0.000
ERROR SUM
© 2001 MIT PSDAM AND PERG LABS
0.001
2698.1
microns
<- x, y, qz
<- Use the solver on this value to set it to a value (ideally zero) that is small, but greater than or equal to 0.
Low-cost adjustment (10 m)
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 = 180o
2E
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
qB = -360o
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