Circumnavigation Circumnavigation From Distance From Distance Measurements Under Measurements Under Slow Drift Slow Drift Soura Dasgupta, U of Iowa Soura Dasgupta, U of Iowa With: Iman Shames, Baris With: Iman Shames, Baris Fidan, Brian Anderson Fidan, Brian Anderson
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Circumnavigation From Distance Measurements Under Slow Drift Soura Dasgupta, U of Iowa With: Iman Shames, Baris Fidan, Brian Anderson.
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Circumnavigation From Circumnavigation From Distance Measurements Distance Measurements Under Slow DriftUnder Slow Drift
Soura Dasgupta, U of IowaSoura Dasgupta, U of IowaWith: Iman Shames, Baris Fidan, Brian With: Iman Shames, Baris Fidan, Brian
AndersonAnderson
Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
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 complicatesANU July 31, 2009 10 of 27
If target stationary Measure distances from three noncollinear agent positions
In 3d 4 non-coplanar positions
Localizes target
ANU July 31, 2009 11 of 27
If target stationary
Move towards target
Suppose target drifts
Then moving toward phantom position
ANU July 31, 2009 12 of 27
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 13 of 27
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 14 of 27
Quantifying Richness
• Persistent Excitation (p.e.)
• The i 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
IdyyITt
t
21 )(')(01
ANU July 31, 2009 15 of 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 16 of 27
Broad approach• Stationary target
• From D(t) and y(t) localize agent
• Force y(t) to circumnavigate as if it were x
xtx )(ˆ
)(ˆ tx
ANU July 31, 2009 17 of 27
ANU July 31, 2009 18 of 27
ANU July 31, 2009 19 of 27
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
|)(|)()( txKdtxty
ANU July 31, 2009 20 of 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 21 of 27
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