Circumnavigation From Distance Measurements Under Slow Drift

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

– Stationary target– Drifting target

• Rotation selection• Simulation• Conclusion

ANU July 31, 2009 2 of 27

Problem

ANU July 31, 2009 3 of 27

Problem

ANU July 31, 2009 4 of 27

Problem

ANU July 31, 2009 5 of 27

Problem

ANU July 31, 2009 6 of 27

Problem

ANU July 31, 2009 7 of 27

Problem

ANU July 31, 2009 8 of 27

Problem

Sufficiently rich orbitSufficiently rich perspective

Slow unknown drift in target

2 and 3 dimensionsANU July 31, 2009 9 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, 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

t21 )(')(0

1

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

• R(t) p.e. and H(s) stable minimum phase H(s){R(t)} p.e.– Filtering rule

ANU July 31, 2009 22 of 27

A basic principle• Suppose x(t) is stationary and• We can generate

• Then:

))(ˆ)(()( xtxtztv T

)()()(ˆ tvtztx

xtx )(ˆ If z(t) p.e.

ANU July 31, 2009 23 of 27

Localization• Dandach et. al. (2008)• If x(t) stationary• Algorithm below converges under p.e.• Need explicit differentiation

))()(()()(21

))(()')(()(2

xtytytDtD

xtyxtytD

T

))(ˆ)(())(ˆ)()(()()(21 xtxtytxtytytDtD TT

)))(ˆ)()(()()(21)(()(ˆ txtytytDtDtytx T

ANU July 31, 2009 24 of 27

Localization without differentiation

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)((

)(ˆ)()()(

xtxtV

txtVtmtT

T

))(ˆ)()()()(()(ˆ txtVtmttVtx T

xtx )(ˆ If V(t) p.e. p.e. )(ty

x stationary

ANU July 31, 2009 25 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, 2009 26 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 27 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.

)(ˆ tx

)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

||)(ˆ)(||)(ˆ txtytD

)()()( tztAtz ANU July 31, 2009 28 of 27

The role of A(t)

• A(t) skew symmetric

• Φ(t,t0) Orthogonal

• ||z(t)||=||z(t0) ||

• z(t) rotates

)(),()(

)()()(

00 tztttz

tztAtz

ANU July 31, 2009 29 of 27

Control Law Features• Will force

• Forces Rotation• Overall still have

• p.e. • Regardless of whether x drifts

)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

)()()( tztAtz

dtD )(ˆ

dtD )(ˆ

)(ˆ)( txty

ANU July 31, 2009 30 of 27

Closed Loop

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)()()()(()(ˆ txtVtmttVtx T

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

ANU July 31, 2009

Nonlinear Periodic

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, 2009 32 of 27

The State Space

)(21)()()(

||)(||21)()()(

)(21)()()(

33

222

211

tytztztV

tytztztm

tDtztzt

))(ˆ)()()()(()(ˆ txtVtmttVtx T

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

)(

)(ˆ)(~)()()(

3

2

1

tyxtxtx

tztztz

ANU July 31, 2009 33 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, 2009 34 of 27

Regardless of drift

ANU July 31, 2009

dtxty ||)(ˆ)(||

)(ˆ)( txty p.e.

y(t) circumnavigates )(ˆ tx

Stationary case: Need to showDrifting case: Need to show

xtx )(ˆ

small is )( toclose gets )(ˆ

txxtx

35 of 27

Globally

Stationary Analysis

• p(t)=η(t)-m(t)+VT(t)x(t) V(t) p.e.

ANU July 31, 2009

2ˆ

)()(~

0)()()(

)()(~

Lx

tptxtVtVtV

tptx T

0)()(~

tptx

)(ˆ)( txty p.e. p.e.)(ty

36 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 characteristicsANU July 31, 2009 37 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, 2009

)()(ˆ)()(

)()(ˆ)()(

txtxtqty

txtxtytq

)()(ˆ)()(1 txtxtqtq

38 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, 2009

)(~)()(

],~,[

txtwtw

pxw TT

)())(),(()( tttrFt

39 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, 2009

)(~)()(

],~,[

txtwtw

pxw TT

)()()]())(),(([)( xHtxGttrFt

40 of 27

x

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 41 of 27

Selecting A(t)

• A(t):– Skew symmetric– Periodic– Derivative of z p.e.– P.E. parameters depend on d

ANU July 31, 2009

)(ˆ)( txty ))(ˆ)()(( txtytA

)()()( tztAtz 2

22

1 )0()()()0(01

zdsszszz TTt

t

))(ˆ)()()(ˆ( 22 txtydtD

42 of 27

2-Dimension

• A(t):– Skew symmetric– Periodic– Derivative of z p.e.

ANU July 31, 2009

)()()( tztAtz

0110

)( ctA

Tctcttztz )sin()cos()()( 0

Constant

43 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, 2009

)()()( tztAtz

44 of 27

A(t) in 3-D• Switch periodically between A1 and A2

• Differentiable switch • To preclude impulsive force on y(t)

ANU July 31, 2009

000001010

11 aA

010100000

22 aA

))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty

45 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 46 of 27

47

Circumnavigation Via Distance Measurements

Distance Measurements

Target Position Error

Trajectories

ANU July 31, 2009

48

Circumnavigation Via Distance Measurements

ANU July 31, 2009

49

Circumnavigation Via Distance Measurements

Distance Measurements

Target Position Error

Trajectories

ANU July 31, 2009

50

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, 2009

)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA

51 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, 2009 52 of 27