# Cooperative Localization using angular measures - PowerPoint PPT Presentation

Cooperative Localization using angular measures

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Cooperative Localization using angular measures

## Cooperative Localization using angular measures

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##### Presentation Transcript

1. Sorawish Dhanapanichkul Advisor : Dr. AttawithSudsang Cooperative Localization using angular measures

2. Our problem • Localization • Multi-robot localization

3. Our problem

4. Our problem

5. Our problem • Output • A Positional pattern • Input • Angular measurements A11,…,A14 A21,…,A24 (X1,Y1) (X2,Y2) Cooperative Localization Algorithm A31,…,A34 A41,…,A44 (X3,Y3) A51,…,A54 (X5,Y5) (X4,Y4)

6. Our problem Line Of Sight(LOS)

7. Introduction to problem • Unknown correspondence between measurements and robots’ ID

8. Introduction to problem • Matching Problem • Between measurements and robots’ name Naïve O(NN)

9. Proposed algorithms • Geometric based algorithm • Based on triangulation • Sensitive to noise • Convex optimization based algorithm • Transform the problem to convex optimization problem • More flexibility

10. Scope • 2D planar space • Fully visible • Include Uncertainty (measurement’s noise)

11. Geometric based algorithm • Use property of convex hull • Reduce matching complexity • Based on triangulation

12. Triangulation • Example • 3 robots • 2 known coordinatesand 1 LOS 3 1 2

13. Triangulation : Ghost node • > 3 robots (ex. 4 robots) 4 3 1 2

14. Ghost node elimination • 1 more known coordinate + 2 LOSs 4 3 coordinates 3 LOSs 3 1 2 5

15. Geometric based algorithm • Compute the position of 3 robots • Triangulation using angular measures of 3 robots with known position

16. Our matching algorithm • Convexity of LOS Graph 4 Boundary lines 3 1 2 5 Boundary points

17. Define set of boundary point • A boundary point • have 2 measurements which all the others reside between these two measurements • These 2 measurements are called leftmost(LM) and rightmost(RM) • RM  LM <= 180 degrees LM1 RM2 4 RM4 LM4 1 2 LM5 5 RM5 LM2 RM1

18. Find the boundary points which connected with that reference • Choose reference point • From set of boundary points • By using LM and RM • Set the distance between one of them to be 1 1 2 RM1 1 LM2 5 LM5 RM5 (0,0)

19. Find last LOS • Try to find the last LOS • To find the last coordinate 1 2 1 5 (0,0)

20. Find last LOS • These 3 robots are forming a triangle • Assuming that there are S robots inside this triangle Convex!!! S S+1 1 2 1 5 (0,0)

21. Find last LOS • Example: S = 3 • After sorting, compare 1st measurement of robot 1 and 2 3 1 2 1 5 (0,0)

22. Find last LOS • Example: S = 3 • After sorting, compare 2nd measurement of robot 1 and 2 3 1 2 1 5 (0,0)

23. Find last LOS • Example: S = 3 • After sorting, compare 3rd measurement of robot 1 and 2 3 1 2 1 5 (0,0)

24. Find last LOS • Example: S = 3 • After sorting, compare 4th measurement of robot 1 and 2 3 3+1 1 2 1 5 (0,0)

25. Algorithm summary • Use our matching algorithm to find 3 LOS • Calculate all intersection from 2 robots • Use 3rd robot’s measurements to eliminate ghost node Time complexity : O(N2) + O(N2) + O(N2 lg(N)) 1 2 5

26. Measurement noise • Due to our comparing method (opposite direction) • Change comparing method

27. Ghost returns • Special case Ex. 6 robots Ghost node

28. Experimental result Wrong rate Number of robot

29. Convex optimization based algorithm

30. Convex optimization based algorithm • Propose iterative method :: try to minimize error • Reduce problem to convex optimization problem

31. Iterative method - flow • Random the answer • Update new answer by step vector • Meet termination condition Yes No End

32. Iterative method - Example 0th step -Random Actual Answer B A C C A B

33. Iterative method - Example 1st step Actual Answer B A C C A B

34. Iterative method - Example 2nd step Actual Answer B A C C B A

35. Iterative method - Example 3rd step Actual Answer B A B C C A

36. Iterative method - Example Termination condition Actual Answer B B C C A A

37. Step vector • Difference between “Sum vector” of actual and answer • Ex. Sum vector A A 0th step Actual Answer B A Step vector A C C A B

38. Example • C# Answer Actual

39. Example • C# Answer Actual

40. Error Actual errorAB • Total error = Σerrorij • Mean angular error = total error / no. of input B B Result A A

41. Experimental result Mean angular error (Radian) Number of robot

42. Mathematical explanation • Compare iterative method with gradient descent • Proof of correctness

43. Update eqn <-> Gradient descent Optimization problem !!! • Update equation • Gradient descent