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Target tracking and guidance using particles

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Target tracking and guidance using particles

David Salmond

QinetiQ Farnborough UK

collaborators: Nick Everett, Neil Gordon (DSTO Australia),

Kevin Gilholm, Malcolm Rollason

Target tracking is usually a means to an end:

e.g. to generate a guidance demand

Contents:

1 The guidance / control problem

2 An example scenario

3 Illustrative results

Structure of estimator / controller

Cost function

- a function of current

and future state vectors:

X k+ = { x k , x k+1 , x k+2 , … , x N }

+ control effort

Sensors

Control

law

Estimator

Measurements

Zk

Demand

uk

“Effectors”

Available information for estimator / controller design

General problemGuidance problem

System dynamics models- Model of pursuer dynamics

as a function of control demand

- Dynamics models of target

and other scenario objects

- Model of scenario development

birth / death of objects

Measurement models - Model relating pursuer’s sensor

(measurements as a functionmeasurements to target, other

of system state)scenario objects and clutter

Cost function- Interception requirement in

terms of miss distance

For guidance problems usually force problem into a

Linear Quadratic Gaussian (LQG) formulation, i.e. assume

1 All models (dynamics and measurement) are linear

2 All disturbances and errors are Gaussian

3 The cost function is quadratic

In this case the Certainty Equivalence principle holds.

Control depends

only on expected

value of x k

Optimal filter/controller:

z k = H kx k + vk

x k

z k

u k

Control

law

Estimator

Measurements

Demand

Kalman filter:

Linear state regulator:

x k = x k + K k (z k - H kxk )

u k = Gk x k

- In practice, for many (most) guidance problems, none of the LQG assumptions are valid. For example,
- - for a Cartesian state vector, the measurement model is non-linear (sensors provide polar measurements)
- - in stressing scenarios with multiple objects and clutter, a quadratic cost function is not appropriate
- - again, due to measurement association uncertainty, the measurement information is far more complex than a simple Gaussian perturbation.
- Certainty Equivalence does not hold for such problems. (Extended) Kalman filter - linear state regulator combination is often markedly sub-optimal.

A more general structure from a Bayesian point of view LQG assumptions are valid. For example,

Information state

- current state of

system knowledge

Measurement

likelihood

p(z k | xk)

p(x k | Zk)

u k

z k

Control

law

Estimator

Measurements

Demand

Bayes rule

etc

Select demand u k to minimise

expected value of cost function

Particle filter implementation LQG assumptions are valid. For example,

Information state

- current state of

system knowledge

Measurement

likelihood

p(z k | xk)

Bayes rule

etc

p(x k | Zk)

u k

z k

Control

law

Estimator

Measurements

Demand

Sample set

Sk = { x k*(i) : i=1,…,NS}

Particle

filter

u k =uk(Sk)

IN GENERAL, CONTROL SHOULD DEPEND ON FULL SAMPLE SET - NOT JUST THE MEAN

- CERTAINTY EQUIVALENCE IS A POOR USE OF THE AVAILABLE INFORMATION

Stochastic control problem: minimise current and future costs

At time step k, define:

Sequence of future statesX k+ = { x k , x k+1 , x k+2 , … , x N }

Sequence of future controlsU k+ = { u k , u k+1 , u k+2 , … , u N-1 }

Available measurementsZ k = { z 1 , z 2 , z 3 , … , z k }

Previous controls (known)U k-1 = { u 1 , u 2 , u 3 , … , u k-1 }

Find the sequence of future controls U k+ that minimises the cost:

J [Z k , U k-1] = min {E [ g( X k+ , U k+ ) | Z k , U k-1 ] }

U k+

Available information

Expectation over all uncertainty:

current state, future dynamics,

and future measurements

Specified

future cost

g( costsX k+ , U k+ ) g†( x k , u k )

Approximations to make the problem tractable

1 Ignore the information that future measurements will become available - Open Loop Optimal Feedback (OLOF) principle -

- so expectation over future measurements is ignored (no possibility of dual effect)

2 For guidance problem, assume particular forms for cost function

(predicted miss) and future controls to reduce dimensionality:

So,

J [Z k , U k-1] = min {E[ g( X k+ , U k+ ) | Z k , U k-1]}

U k+

= min {E[ g†( x k , u k )| Z k , U k-1]}

u k

Expectation over uncertainty

in current state only

E costs[ g†( x k , u k )| Z k , U k-1]

=g†( x k , u k ) p( x k| Z k , U k-1 ) dx k

g†( x k*(i), u k )

NS

i=1

Evaluation of expected cost using particles (for given u k)

Samples from particle filter,

approximately distributed as p( x k| Z k , U k-1 )

Hence optimisation problem reduced to:

NS

min { g†( x k*(i), u k ) }

u k

i=1

. costs

.

.

.

.

.

.

.

.

.

Cost functions for guidance problems

Cost is usually some function of the miss distance:

Pursuer’s prediction of

miss distance is imperfect

principally due to:

i) Uncertainty in current target state x k

ii) Uncertainty in future target behaviour

PURSUER

MISS

DISTANCE

TARGET

For significant measurement association uncertainty (i) will dominate

so assume miss = m( x k ,u k) >= 0 ,

- i.e. achieved miss depends only on current state and future controls

Cost function is of the form

g( X k+ , U k+ ) = g†( x k , u k ) = f( m(x k , u k) )

4.5 costs

4

3.5

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

Quadratic cost: cost rises as square of miss - unbounded

- always drives system towards mean of cost function

Inverse Gaussian cost: cost of missing essentially constant

when miss exceeds 3 normalised units

i.e. “ a large miss is as bad as a very large miss”

QUADRATIC

COST (UNBOUNDED)

COST f( m )

INVERSE GAUSSIAN

COST (BOUNDED)

MISS DISTANCE m

o costs

o

o

o

o

*

o

D

*

*

*

*

o

INITIALLY

UNRESOLVED

T

o

o

Example scenario: single target (T) in dense random clutter with

intermittent spurious object (D)

D is spawned in the vicinity of T and with a similar velocity

Sensor resolution is limited: T / D pair may initially be unresolved

The sensor takes measurements

of range and bearing and is

carried by the pursuer

Measurements are corrupted

by dense random clutter

A (very poor) classification

flag may be associated with each

measurement [but T and D cannot be

distinguished the following example]

Particle filter includes: costs

Second order dynamics for T and D (noise driven constant velocity)

Markov model to represent birth / death of D objects

Measurement association uncertainty via assignment hypotheses

Classification data within measurement likelihood

A possible assignment for Nk measurements received at time step k:

Type

Meas. number

1

Target

D

J unresolved

Clutter

2

3

4

.

.

.

N

k

{1,2,..., Nk } {T,D,J,C}

Unknown assignment

Pursuer model costs

Pursuer moves at a constant speed VM

Heading is controlled by a turn rate (guidance) demand uk

updated at every time step (no lag)

So heading: f k+1 = fk + ukDt

where | uk | < a MAX / VM .

Assume that choice of future controls U k+ is restricted to a

constant turn rate, so uj = uk for j>k

Guidance problem is to select a single number uk from

the range ( - a MAX / VM , a MAX / VM ) to minimise the expected

cost:

f(m(x k*(i), u k ) )

NS

i=1

Object paths costs

1.15

T path

D path

1.1

D deployed

at this point

1.05

y

1

Constant

turn rate

Constant

velocity

0.95

T and D indistinguishable

at split

0.9

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

ALL OBJECT MEASUREMENTS: costs

GREEN=T, RED=D, BLUE=UNRESOLVED

ONLY ONE FRAME OF CLUTTER: YELLOW

(TRANSFORMED FROM POLAR

CO-ORDS)

D present costs

and resolved

D present but

not resolved

D not

present

Prob.

from

filter

Particle filter’s assessment of scenario state

BIFURCATION OF PARTICLE SET ON DEPLOYMENT OF D costs

1.2

Some particles from time steps 40 50 60 70 80 90

1.15

1.1

T PATH

1.05

Y

1

0.95

D PATH

0.9

0.85

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

X

Guidance via particle filter costs

with inverse Gaussian cost function

TARGET TRACK AND “TRUE” MEASUREMENTS costs

Guidance via particle filter

with inverse Gaussian cost function

Distribution of predicted miss costs

with uj = 0 for j>=k,

i.e. for zero pursuer effort

Guidance via particle filter

with inverse Gaussian cost function

Guidance via particle filter costs

with quadratic cost function

TARGET TRACK AND “TRUE” MEASUREMENTS costs

Guidance via particle filter

with quadratic cost function

Distribution of predicted miss costs

with uj = 0 for j>=k,

i.e. for zero pursuer effort

Guidance via particle filter

with quadratic cost function

Conclusions costs

1 Have demonstrated a guidance law for exploiting

output of a particle filter

2 Guidance law is based on a bounded cost function of the predicted miss distance

3 A smooth transition from a hedging / learning strategy to a firm selection decision has been demonstrated

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