Circumnavigation From Distance Measurements Under Slow Drift Soura Dasgupta, U of Iowa With: Iman Shames, Baris Fidan, Brian Anderson

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

Problem ANU July 31, of 27

Problem ANU July 31, of 27

Problem ANU July 31, of 27

Problem ANU July 31, of 27

Problem ANU July 31, of 27

Problem ANU July 31, of 27

Problem Sufficiently rich orbit Sufficiently rich perspective Slow unknown drift in target 2 and 3 dimensions ANU July 31, of 27

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, of 27

If target stationary Measure distances from three noncollinear agent positions In 3d 4 non-coplanar positions Localizes target ANU July 31, of 27

If target stationary Move towards target Suppose target drifts Then moving toward phantom position ANU July 31, 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, 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, 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 ANU July 31, 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, of 27

Broad approach Stationary target From D(t) and y(t) localize agent  Force y(t) to circumnavigate as if it were x ANU July 31, of 27

ANU July 31, of 27

ANU July 31, 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 ANU July 31, 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, of 27

Rules on PE R(t) p.e. and f(t) in L 2  R(t)+f(t) p.e. –L 2 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, of 27

A basic principle Suppose x(t) is stationary and We can generate Then: If z(t) p.e. ANU July 31, of 27

Localization Dandach et. al. (2008) If x(t) stationary Algorithm below converges under p.e. Need explicit differentiation ANU July 31, of 27

Localization without differentiation If V(t) p.e. x stationary   ANU July 31, of 27

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, 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, of 27

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, of 27

The role of A(t) A(t) skew symmetric Φ(t,t 0 ) Orthogonal ||z(t)||=||z(t 0 ) || z(t) rotates ANU July 31, of 27

Control Law Features Will force Forces Rotation Overall still have p.e. Regardless of whether x drifts ANU July 31, of 27

Closed Loop ANU July 31, 2009 Nonlinear Periodic 30 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, of 27

The State Space ANU July 31, of 27

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, of 27

Regardless of drift ANU July 31, 2009 p.e. y(t) circumnavigates Stationary case: Need to show Drifting case: Need to show 34 of 27 Globally

Stationary Analysis p(t)=η(t)-m(t)+V T (t)x(t)  V(t) p.e.  ANU July 31, 2009 p.e.  p.e. 35 of 27

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, of 27

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, of 27

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, of 27

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, of 27

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, of 27

Selecting A(t) A(t): –Skew symmetric –Periodic –Derivative of z p.e. –P.E. parameters depend on d ANU July 31, of 27

2-Dimension A(t): –Skew symmetric –Periodic –Derivative of z p.e. ANU July 31, 2009 Constant 42 of 27

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, of 27

A(t) in 3-D Switch periodically between A 1 and A 2 Differentiable switch To preclude impulsive force on y(t) ANU July 31, 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, of 27

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

47 Circumnavigation Via Distance Measurements ANU July 31, 2009

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

49 Circumnavigation Via Distance Measurements ANU July 31, 2009

The Knee Initially this dominates –Zooms rapidly toward estimated location Fairly quickly rotation dominates ANU July 31, of 27

Conclusions Circumnavigation Distance measurements only Rich Orbit Slow but potentially unbounded drift Future work –Designing fancier orbits –Positioning at a distance from multiple objects –Noise analysis ANU July 31, of 27