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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

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

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( X 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[ 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

.

.

.

.

.

.

.

.

.

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

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

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]

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 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

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

GREEN=T, RED=D, BLUE=UNRESOLVED

ONLY ONE FRAME OF CLUTTER: YELLOW

(TRANSFORMED FROM POLAR

CO-ORDS)

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

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

with inverse Gaussian cost function

Distribution of predicted miss

with uj = 0 for j>=k,

i.e. for zero pursuer effort

Guidance via particle filter

with inverse Gaussian cost function

with quadratic cost function

Distribution of predicted miss

with uj = 0 for j>=k,

i.e. for zero pursuer effort

Guidance via particle filter

with quadratic cost function

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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