1. N ONDETERMINISTIC U NCERTAINTY & S ENSORLESS P LANNING

2. Sensing error Partial observability Unpredictable dynamics Other agents

3. U NCERTAINTY MODELS This class : Nondeterministic uncertainty f(x,u) -> a set of possible successors Probabilistic uncertainty P(x’|x,u): a probability distribution over successors x’, given state x, control u

4. U NCERTAINTY IN MOTION : R EASONING WITH STATE SETS x’ = x + ,   [-a,a] t=0 t=1 -aa t=2 -2a2a Belief State: x(t)  [-ta,ta]

5. B ELIEF S TATE D YNAMICS F(x,u) = { x’ | x’=f(x,u) is a possible successor } e.g., F(x,u) = [x+u- , x+u+  ]

6. B ELIEF S TATE D YNAMICS

8. S ENSORLESS P LANNING

9. M ARBLE IN A TILTING MAZE EXAMPLE Use natural dynamics of compliance to ensure that uncertainty stays bounded / shrinks Command Uncertain outcome On contact, certainty is increased A guaranteed solution

10. Q UESTIONS Is a solution possible? How is uncertainty represented / transformed? Need for actions that reduce uncertainty

12. G OLDBERG (1998): O RIENTING P OLYGONAL P ARTS WITHOUT S ENSING A single parallel jaw gripper can orient parts that have a polygonal convex hull.

13. A SSUMPTIONS : All motions appear in the plane The gripper has two parallel jaws The gripper moves orthogonally to its jaws The object’s convex hull behaves rigidly The object is presented in isolation The object is initially between the jaws Both jaws make contact simultaneously with the object Once contact is made, the jaws stay in contact with the object throughout the grasping motion. There is no friction between the jaws and the object

14. S QUEEZE A CTIONS Let a squeeze action, α, be the combination of orienting the gripper at angle α w.r.t. a fixed world frame, closing the jaws as far as possible, and then opening the jaws. To avoid squeezing the object right at its endpoints, a squeeze action, α, can always be followed by closing the gripper at angle α + ε, rotating it by -ε, and then opening it.

15. O RIENTING A R ECTANGLE A 0 0 squeeze action, followed by a 45 0 squeeze action can orient a rectangular part up to symmetry.

16. D IAMETER F UNCTIONS The diameter function, d:S 1 ->R, describes how the distance between jaws varies, if they were to make contact with the object at angle θ. the diameter of the grasped object at direction θ is defined to be the maximum distance between two parallel supporting lines at angle θ.

17. S QUEEZE F UNCTIONS The squeeze function, s:S 1 ->S 1, is a transfer function, such that if θ is the initial orientation of the object w.r.t. the gripper, then s(θ) is the object’s orientation after the squeeze action is completed. Define an s- interval to be [ a,b ), s.t. a, b are points of discontinuity in the squeeze function. Define the s- image of a set, s(Θ), to be the smallest interval containing the following set: {s(θ)|θ is in Θ}.

18. O RIENTING A R ECTANGLE ( CONTINUED )

19. S QUEEZE F UNCTIONS ( CONTINUED ) Θ 1 – is an example of an s-interval s(Θ 1 ) – is the s-image of Θ 1

20. O RIENTING A R ECTANGLE ( CONTINUED ) Gripper Orientatio n Possible Orientations Before the Squeeze Action Possible Orientations After the Squeeze Action 1. 2. 3. 4.

21. T HE A LGORITHM Goal: 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.

22. T HE A LGORITHM 1) Compute the squeeze function; 2) Find the widest single step in the squeeze function and set Θ 1 equal to the corresponding s-interval; 3) While there exists an s-interval Θ s.t. |s(Θ)|<|Θ i |: Set Θ i+1 equal to the widest such s-interval Increment i ; 4) Return the list (Θ 1, Θ 2, …, Θ i ).

23. O RIENTING A R ECTANGLE ( CONTINUED ) The squeeze function plots illustrate steps 2 and 3 of the algorithm: The output list is: (Θ 1, Θ 2 )

24. R ECOVERING THE A CTION P LAN Goal: Given a list of i s-intervals (Θ 1, Θ 2, …, Θ i ) generated by the described algorithm, construct a plan, ρ i,consisting of i squeeze actions (a i, a i-1, …, a 1 ) that collapses all orientations in Θ i to the unique orientation s(Θ 1 ).

25. R ECOVERING THE A CTION P LAN Observe that for the list (Θ 1, Θ 2, …, Θ i,.. Θ n ): |s(Θ i+1 )|<|Θ i |, therefore we could collapse the interval s(Θ i+1 ) to a point with a single squeeze action at angle 0 if s(Θ i+1 ) could be aligned with Θ i. This can be done by rotating the gripper by: s(θ i+1 ) - θ i = a i degrees, where θ i and s(θ i+1 ) can be taken as the start of the s- interval Θ i and the start of the s-image of s(Θ i+1 ), respectively.

26. O RIENTING A R ECTANGLE ( CONTINUED ) One possible way of aligning s(Θ 2 ) with Θ 1 for the rectangular part example is to rotate the gripper by 45 0, which results in shifting s(Θ 2 ) by -45 0 = -π/4.

27. A LGORITHM A NALYSIS Correctness: For any plan ρ, ρ (θ + T) = ρ (θ) + T, where T is the smallest period in the object’s squeeze function. Thus, the plan ρ orients the part up to symmetry. It also does it in the minimal number of steps (see the paper). Completeness: Given a starting s-interval, we can either find a larger s- interval that has a smaller s-image, or the starting s- interval is a period of symmetry in the squeeze function. Therefore, we can generate a sequence of s-intervals (Θ 1, Θ 2, …, Θ i ) of increasing measure. Complexity: Runs in time O ( n 2 log n ) and finds plans of length O ( n 2 ) (Note: there are newer derivations of lower complexity)

28. E XTENSION : P USH -G RASP A CTIONS Assuming simultaneous contact is unreliable Let a push-grasp action, α, be the combination of orienting the gripper at angle α w.r.t. a fixed world frame, translating the gripper in direction α + π/2 for a fixed distance, closing the jaws as far as possible, and then opening the jaws. Can derive a push-grasp squeeze function as before and apply the same algorithm

29. N ONDETERMINISM IN S ENSING Achieve goal for every possible sensor result in belief state Move Sense 12 34 Minimizing # of sensor results reduces branching factor Outcomes

30. I NTRUDER F INDING P ROBLEM  A moving intruder is hiding in a 2-D workspace  The robot must “sweep” the workspace to find the intruder  Both the robot and the intruder are points robot’s visibility region hiding region 1 cleared region 2 3 4 5 6 robot

31. D OES A SOLUTION ALWAYS EXIST ? Easy to test: “Hole” in the workspace Hard to test: No “hole” in the workspace No !

32. I NFORMATION S TATE  Example of an information state = (x,y,a=1,b=1,c=0)  An initial state is of the form (x,y,1, 1,..., 1)  A goal state is any state of the form (x,y,0,0,..., 0) (x,y) a = 0 or 1 c = 0 or 1 b = 0 or 1 0  cleared region 1  hidding region

33. C RITICAL L INE a=0 b=1 a=0 b=1 Information state is unchanged a=0 b=0 Critical line

34. A BCD E C RITICALITY -B ASED D ISCRETIZATION Each of the regions A, B, C, D, and E consists of “equivalent” positions of the robot, so it’s sufficient to consider a single position per region

35. C RITICALITY -B ASED D ISCRETIZATION A B CD E (C, 1, 1) (D, 1)(B, 1)

36. C RITICALITY -B ASED D ISCRETIZATION A BC D E (C, 1, 1) (D, 1)(B, 1)(E, 1)(C, 1, 0)

37. C RITICALITY -B ASED D ISCRETIZATION A B CD E (C, 1, 1) (D, 1)(B, 1)(E, 1)(C, 1, 0) (B, 0)(D, 1)

38. C RITICALITY -B ASED D ISCRETIZATION A CD E (C, 1, 1) (D, 1)(B, 1)(E, 1)(C, 1, 0) (B, 0)(D, 1) Discretization chosen explicitly to get small search trees B

39. R EMARKS Planners that use nondeterministic uncertainty models optimize for the worst case In general, devising good discretizations / information state representations takes a bit of ingenuity

40. N EXT TIME Probabilistic uncertainty