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(  ) a a=atan2(3,2) 0  -a  +a3  /22  -a aaaa = == 0  -a  +a3  /22  -a

Recovering the Plan Given  plans for j between 1 ~ A plan

 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 symmetry Compare plans  j ’ vs  i where 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(n 2 log n) and finds plans of length O(n 2 )  Compute squeeze function in O(n)  Step 2 takes O(n) time  Squeeze function defines O(n 2 ) s-intervals  Traverse list once  I is O(n 2 )  Sorting  O(n 2 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