1. Carnegie Mellon University TRAJECTORY MODIFICATION TECHNIQUES IN COVERAGE PLANNING By - Sanjiban Choudhury (Indian Institute of Technology, Kharagpur, India) Sanjiv Singh (Carnegie Mellon University, USA)

2. About the Institute…

3. … and the robots

4. Motivation

5. Problem At Hand NO – GO AREAS New outgrowths

6. Sample Field

7. Main aspects of work  Handling new obstacles without re-planning.  Ensuring all routes respect curvature constraints.  Optimal turnings without entering “no-go” areas.

8. Current Approaches  Exact Cellular Decomposition  - Trapezoidal Decomposition (F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, 1991)  - Boustrophedon Decomposition (H. Choset and P. Pignon, Coverage path planning: The boustrophedon decomposition, 1997)  - Split and Merge Method (T.Oksanen and A.Visala, Path Planning Algorithms for Agricultural Field Machines, 2007)

9. Split and Merge Method

17. Problems with the approach  Obstacles are not handled efficiently.  Requires extra pre-determined spaces for turning, known as headlands.  Headlands have to be provided externally by operator.

18. Problem with obstacles  Re-planning required  Distribution of cells completely changed  Trivial cases may have over-complicated solutions.

19. Headlands  They are pre-determined and taken to be a considerably large enough value to allow turning.  No optimization of headland width with respect to – Curvature Constraints – Obstacle profile

20. Modified Solution

21. ‘Bending’ Lines  Each track is represented as a B-spline  Tracks are subjected to forces by 1. Obstacles 2. Curvature of track 3. Interaction forces between tracks

22. Vector fields

24. Modification of curves  Splines are used because of their local modification property.  Knot intervals containing the obstacle are modified.

25. Algorithm  1. A track passing through the obstacle is taken.  2. Knots are inserted appropriately.  3. Energy associated with each knot interval is calculated.  4. Points corresponding to the knots are acted upon by forces given by the energy gradient.

26. Knot insertion This part frozen These points span over a distance of (π/2)*R to alter INITIAL heading These points span over a distance of (π/2)*R to alter FINAL heading These points carry the curve over obstacles

27. Knot insertion This part frozen These points span over a distance of (π/2)*R to alter INITIAL heading These points span over a distance of (π/2)*R to alter FINAL heading These points carry the curve over obstacles

28. Energy of tracks  Each track has an energy associated with it.

29. Movements of spline points

30. Head to head comparisons

32. MethodRe-PlanningBending Paths Distance Travelled 4689.49 m5014.65 m Moving Time 4689.49 s5014.65 s Turning Time 2594.73 s1569.66 s Total Time7284.22 s6584.31s Moving speed = 1m/s, Turning speed = 0.5 m/s, improvement = 9.6%

33. Headlands

34. Turning at headlands The vehicle turns at 1. Field edges which are un-restricted. 2. Field edges which are restricted, i.e, “No Go Areas” 3. Edges of obstacles

35. Solution for the problem  The problem is framed as an “Optimal Control Problem” (OCP) (Oksanen, T., & Visala, A. (2004, October). Optimal control of tractor-trailer system in headlands)

36. Vehicle Model Used x and y are the coordinates of the rear axle center θ is the heading angle α is the steering angle u is the steering rate v is the velocity L is the wheel base

37. The OCP x is a state vector f dynamics function t0 and tf the time limits J the cost functional F the cost function C the path constraint function E the boundary condition function

38. Case 1 : No restriction on headlands d (0,0) (0,d)

39. Case 2: “No Go” Area  For such cases, solutions have been derived only for given headland width.  The OCP problem is reframed to solve for optimal headland width and time. d (0,0) (0,d) No go

40. Case 2: “No Go” Area Initial Values : Path Constraints : d (0,0) (0,d) No go φ Cost Function :

41. Case 3: Obstacles  Obstacles are also “no-go” areas inside the field.  In a similar fashion, the optimal turn is calculated.

42. Fitting the turns  Two sets of turns, one for no restriction, and another for restriction is stored for 10< φ <170 degrees, where φ is the angle between adjacent tracks.  The angle between tracks is rounded of to nearest multiple of 10, and the corresponding turn route is applied.

43. Local adjustment of turns φ Turn in no go area

44. Local adjustment of turns φ Turn in no go area Trajectory is converted to a spline and “bent” to respect both curvature as well as experience repulsion by the restricted areas.

45. Optimisation of Cellular Traversal

46.  Currently all cellular traversal is boustrophedon. (back and forth)  However a circle takes more time back and forth than just spiraling down from its edges.  A rectangle would take more time spiraling than just covering it back and forth.

47. Algorithm  Choose the cell to optimise.  Offset the polygon inside by track width. Compute traversal time of the spiral and add it with traversal time of the remaining polygon.  Repeat till offsetting not possible.  Take the best solution.

48. Some optimal solution

49. Results  Guaranteed better solutions than Split and Merge method.  No significant increase in computation time.

50. Results Total Time = 4965Total Time = 4270 Improvement in time = 14%

51. Results Total time = 7060Total Time = 7951 Improvement in time = 11.1%

52. Some more results

53. THANK YOU