circumnavigation from distance measurements under slow drift soura dasgupta, u of iowa with: iman...
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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
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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Problem
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Problem
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Problem
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Problem
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Problem
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Problem
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Problem
Sufficiently rich orbitSufficiently rich perspective
Slow unknown drift in target
2 and 3 dimensions
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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
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If target stationary
Move towards target
Suppose target drifts
Then moving toward phantom position
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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
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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
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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
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Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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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
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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
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Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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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
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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.
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Localization• Dandach et. al. (2008)• If x(t) stationary• Algorithm below converges under p.e.• Need explicit differentiation
))()(()()(2
1
))(()')(()(2
xtytytDtD
xtyxtytD
T
))(ˆ)(())(ˆ)()(()()(2
1xtxtytxtytytDtD TT
)))(ˆ)()(()()(2
1)(()(ˆ txtytytDtDtytx T
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Localization without differentiation
)(2
1)()()(
||)(||2
1)()()(
)(2
1)()()(
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
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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.
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Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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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
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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
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Closed Loop
)(2
1)()()(
||)(||2
1)()()(
)(2
1)()()(
33
222
211
tytztztV
tytztztm
tDtztzt
))(ˆ)()()()(()(ˆ txtVtmttVtx T
))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty
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Nonlinear Periodic
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Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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The State Space
)(2
1)()()(
||)(||2
1)()()(
)(2
1)()()(
33
222
211
tytztztV
tytztztm
tDtztzt
))(ˆ)()()()(()(ˆ txtVtmttVtx T
))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty
)(
)(ˆ)(~)(
)(
)(
3
2
1
ty
xtxtx
tz
tz
tz
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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
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Regardless of drift
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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
tx
xtx
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Globally
Stationary Analysis
• p(t)=η(t)-m(t)+VT(t)x(t)
V(t) p.e.
ANU July 31, 2009
2ˆ
)(
)(~
0
)()()(
)(
)(~
Lx
tp
txtVtVtV
tp
tx T
0)(
)(~
tp
tx
)(ˆ)( txty p.e. p.e.)(ty
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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
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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
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)()(ˆ)()(
)()(ˆ)()(
txtxtqty
txtxtytq
)()(ˆ)()(1 txtxtqtq
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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.
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)(~)()(
],~,[
txtwtw
pxw TT
)())(),(()( tttrFt
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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
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x
Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection – Selecting A(t)
• Simulation• Conclusion
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Selecting A(t)
• A(t):– Skew symmetric
– Periodic
– Derivative of z p.e.
– P.E. parameters depend on d
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)(ˆ)( txty ))(ˆ)()(( txtytA
)()()( tztAtz 2
2
2
1 )0()()()0(01
zdsszszz TTt
t
))(ˆ)()()(ˆ( 22 txtydtD
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2-Dimension
• A(t):– Skew symmetric
– Periodic
– Derivative of z p.e.
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)()()( tztAtz
01
10)( ctA
Tctcttztz )sin()cos()()( 0
Constant
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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
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)()()( tztAtz
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A(t) in 3-D
• Switch periodically between A1 and A2
• Differentiable switch • To preclude impulsive force on y(t)
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000
001
010
11 aA
010
100
000
22 aA
))(ˆ)()(())(ˆ)()()(ˆ()(ˆ)( 22 txtytAtxtydtDtxty
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Outline• The Problem
– Motivation– Precise Formulation
• Broad Approach• Localization• Control Law• Analysis
– Stationary target– Drifting target
• Rotation selection• Simulation• Conclusion
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47
Circumnavigation Via Distance Measurements
Distance Measurements
Target Position Error
Trajectories
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Circumnavigation Via Distance Measurements
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Circumnavigation Via Distance Measurements
Distance Measurements
Target Position Error
Trajectories
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50
Circumnavigation Via Distance Measurements
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The Knee
• Initially this dominates– Zooms rapidly toward estimated location
• Fairly quickly rotation dominates
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)(ˆ)( txty ))(ˆ)()()(ˆ( 22 txtydtD ))(ˆ)()(( txtytA
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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
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