Orienting Polygonal Parts without Sensors

Kenneth Y. Goldberg

Presented by Alan Chen

Outline

Background Assumptions Algorithm: Grasper Push – Grasp Proof Discussion/Conclusion

Part Feeder

Hardware vs. Software Redesigned vs. Reprogrammed

No sensors

Previous Work

Exact motion planning Compliant motion

planning Open loop, sensorless Corner example Preimage backchaining

Mechanical part feeders Vibrations

Bowl feeder Bad for fragile parts

Fence Grasper/Pusher

Grasper

Squeeze action No sensors

Assumptions

All motions in plane and inertial forces are negligible Gripper consists of two parallel linear jaws Gripper motion orthogonal to jaws Convex hull treated as rigid planar polygon Part is isolated Part’s initial position is constrained within jaws * Jaws make contact simultaneously Once contact made, surfaces remain in contact Zero friction between part and jaws

Definitions

Diameter function

Squeeze function

s()

Symmetry

S(+T) = s()+T; T period T = 2/ r(1+(r mod 2))

r - rotational symmetry r = 1 T = : no symmetry r = 3 T = /3: equilateral triangle r = 4 T = /2: square

Parts Feeding Problem

“Given a list of n vertices describing the convex hull of a polygonal part, find the shortest sequence of squeeze actions guaranteed to orient the part up to symmetry.” (Goldberg, 11)

Algorithm

Compute the squeeze function Find widest step in the squeeze function and set

1 = corresponding s-interval

While there exists |s()| < |i| Set i+1 = widest s-interval

Increment i

Return the list (1,2…)

Continue until |i| = T

s()

01

1s 1s

2

2s1

3

2a

a=atan2(3,2)

0 -a +a 3/2 2-a

a aa a

aa 2,=

a2

=

2

=

2T

0 -a +a 3/2 2-a

Recovering the PlanGiven plans i ,...,, 21 ifor j between 1 ~A plan

i 11,...,, jjj

),1,1(

0

jjijfori

11

1)(2

1

jjjjj

jjj

s

s

21,02

44

422

2

1)(

2

1

21122

211

aas

aas

s()

Push-Grasp vs Grasp

Jaws do not contact simultaneously

Radius function

Push function

Correctness

Plan will orient the part up to symmetry 2 = (1,2)

No shorter plan orients up to symmetryCompare plans j’ vs i where j < i |j’| > |j|

Algorithm |’| < ||

|’| < || & |’| > ||Cannot happen so no shorter plan

Completeness

Theorem – For any polygonal part, we can always find a plan to orient the part up to symmetry

h – measure of some s-interval

Complexity

For a polygon of n sides, the algorithm runs in time O(n2 log n) and finds plans of length O(n2)Compute squeeze function in O(n)Step 2 takes O(n) timeSqueeze function defines O(n2) s-intervalsTraverse list once I is O(n2)Sorting O(n2 log n)

Discussions

Plan verified experimentally Carnegie Mellon University using a PUMA robot w/ electric

LORD Co. gripper USC using an IBM robot w/ pneumatic Robotics and Automation

Corp. gripper Occasionally failed: not enough pushing distance

Practicality Feed rate: i actions with 1 gripper vs. i grippers performing 1 action 1 D.O.F. not constrained Limited to flat 2-D polygons

Working on curved edges 3-D orient the part so its sitting on its most stable face before

grasping

Conclusions

Algorithm that rapidly analyzes part geometry

Sensorless and easily reprogrammable

Complete and correct Not practical for

industry