Circumnavigation From Distance Measurements Under Slow Drift

1. Circumnavigation From Distance Measurements Under Slow Drift SouraDasgupta, U of Iowa With: Iman Shames, Baris Fidan, Brian Anderson

2. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

3. Problem ANU July 31, 2009

4. Problem ANU July 31, 2009

5. Problem ANU July 31, 2009

6. Problem ANU July 31, 2009

7. Problem ANU July 31, 2009

8. Problem ANU July 31, 2009

9. Problem Slow unknown drift in target Sufficiently rich orbit Sufficiently rich perspective 2 and 3 dimensions ANU July 31, 2009

10. Motivation • Surveillance • Monitoring from a distance • Require a rich enough perspective • May only be able to measure distance • Target emitting EM signal • Agent can measure its intensity  Distance • Past work • Position measurements • Local results • Circumnavigation not dealt with • Potential drift complicates ANU July 31, 2009

11. If target stationary Measure distances from three noncollinear agent positions In 3d 4 non-coplanar positions Localizes target ANU July 31, 2009

12. If target stationary Move towards target Suppose target drifts Then moving toward phantom position ANU July 31, 2009

13. Coping With Drift • Target position must be continuously estimated • Agent must execute sufficiently rich trajectory • Noncollinear enough: 2d • Noncoplanar enough: 3d • Compatible with goal of circumnavigation for rich perspective ANU July 31, 2009

14. Precise formulation • Agent at location y(t) • Measures D(t)=||x(t)-y(t)|| • Must rotate at a distance d from target • On a sufficiently rich orbit • When target drifts sufficiently slowly • Retain richness • Distance error proportional to drift velocity • Permit unbounded but slow drift ANU July 31, 2009

15. Quantifying Richness • Persistent Excitation (p.e.) • The ai are the p.e. parameters • Derivative of y(t) persistently spanning • y(t) avoids the same line (plane) persistently • Provides richness of perspective • Aids estimation ANU July 31, 2009

16. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

17. Broadapproach • Stationary target • From D(t) and y(t) localize agent • Force y(t) to circumnavigate as if it were x ANU July 31, 2009

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20. Coping With drifting Target • Suppose exponential convergence in stationary case • Show objective approximately met when target velocity is small • x(t) can be unbounded • Inverse Lyapunov arguments • Wish to avoid partial stability arguments ANU July 31, 2009

21. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

22. Rules on PE • R(t) p.e. and f(t) in L2  R(t)+f(t) p.e. • L2 rule • R(t) p.e. and f(t) small enough  R(t)+f(t) p.e. • Small perturbation rule • R(t) p.e. and H(s) stable minimum phase  H(s){R(t)} p.e. • Filtering rule ANU July 31, 2009

23. A basic principle • Suppose x(t) is stationary and • We can generate • Then: If z(t) p.e. ANU July 31, 2009

24. Localization • Dandach et. al. (2008) • If x(t) stationary • Algorithm below converges under p.e. • Need explicit differentiation ANU July 31, 2009

25. Localization without differentiation x stationary  If V(t) p.e. ANU July 31, 2009

26. Summary of localization • Achieved through signals generated • From D(t) and y(t) • No explicit differentiation • Exponential convergence when derivative of y(t) p.e. • x stationary • Implies p.e. of V(t) • Exponential convergence  robustness to time variations • As long as derivative of y is p.e. ANU July 31, 2009

27. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

28. Control Law • How to move y(t)? • Achieve circumnavigation objective around • A(t) • skew symmetric for all t • A(t+T)=A(t) • Forces derivative of z(t) to be p.e. ANU July 31, 2009

29. The role of A(t) • A(t) skewsymmetric • Φ(t,t0) Orthogonal • ||z(t)||=||z(t0) || • z(t)rotates ANU July 31, 2009

30. Control Law Features • Will force • Forces Rotation • Overall still have • p.e. • Regardless of whether x drifts ANU July 31, 2009

31. Closed Loop Nonlinear Periodic ANU July 31, 2009

32. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

33. The State Space ANU July 31, 2009

34. Looking ahead to drift • When x is constant • Part of the state converges exponentially to a point • Part (y(t)) goes to an orbit • Partially known • Distance from x • P.E. derivative • Standard inverse Lyapunov Theory inadequate • Partial Stability? • Reformulate the state space ANU July 31, 2009

35. Regardless of drift p.e. y(t) circumnavigates Stationary case: Need to show Drifting case: Need to show Globally ANU July 31, 2009

36. Stationary Analysis • p(t)=η(t)-m(t)+VT(t)x(t) •  V(t) p.e. •  p.e. p.e. ANU July 31, 2009

37. Nonstationary Case • Under slow drift need to show that derivative of y(t)remains p.e • Tough to show using inverse Lyapunov or partial stability approach • Alternative approach: Formulate reduced state space • If state vector converges exponentially then objective met exponentially • If state vector small then objective met to within a small error • y(t) appears as a time varying parameter with proven characteristics ANU July 31, 2009

38. Key device to handle drift • q(t) p.e. under small drift • Reformulate state space by replacing derivative of y(t) by • q(t) is p.e. under slow enough target velocity • Partial characterization of “slow enough drift” • Determined solely by A(t), and d ANU July 31, 2009

39. Reduced State Space • q(t) p.e. under small drift • r(t)=1/(s+α){q(t)} p.e. • Reduced state vector: • Stationary dynamics: • eas when r(t) p.e. ANU July 31, 2009

40. Reduced State Space • q(t) p.e. under small drift • r(t)=1/(s+α){q(t)} p.e. • Reduced state vector: • Nonstationary dynamics • G and H linear in • Meet objective for slow enough drift ANU July 31, 2009

41. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Selecting A(t) • Simulation • Conclusion ANU July 31, 2009

42. Selecting A(t) • A(t): • Skew symmetric • Periodic • Derivative of z p.e. • P.E. parameters depend on d ANU July 31, 2009

43. 2-Dimension • A(t): • Skew symmetric • Periodic • Derivative of z p.e. Constant ANU July 31, 2009

44. 3-Dimension • A(t): • Skew symmetric • Periodic • Derivative of z p.e. • Will constant A do? • No! • A singular Φ(t) has eigenvalue at 1 ANU July 31, 2009

45. A(t) in 3-D • Switch periodically between A1 and A2 • Differentiable switch • To preclude impulsive force on y(t) ANU July 31, 2009

46. Outline • The Problem • Motivation • Precise Formulation • Broad Approach • Localization • Control Law • Analysis • Stationary target • Drifting target • Rotation selection • Simulation • Conclusion ANU July 31, 2009

47. Circumnavigation Via Distance Measurements Distance Measurements Trajectories Target Position Error ANU July 31, 2009

48. Circumnavigation Via Distance Measurements ANU July 31, 2009

49. Circumnavigation Via Distance Measurements Distance Measurements Trajectories Target Position Error ANU July 31, 2009

50. Circumnavigation Via Distance Measurements ANU July 31, 2009