TOPIC 7: LEVEL 1 ESTIMATION

David L. Hall

TOPIC OBJECTIVES

Continue the introduction to Level 1 Fusion Focus on estimation – the combination of

data to obtain the best estimate of the position, velocity and characteristics of an entity

NOTE ABOUT LEVEL-1 ESTIMATION

The student is reminded of two things:

1) As in the previous lecture we will present Level 1 fusion (focused on estimation) as it pertains to tracking a physical target. However, the state estimation problem and techniques are very general and pertain to many problems in which we seek to determine an unknown state vector based on observations;

2) This lecture will contain slides with mathematical expressions – for those who do not have a mathematical background do not be alarmed; the in each instance we will describe the essence of the mathematical process. It is not necessary to “read” or understand the actual mathematical equations (they are included for the mathematically inclined).

LEVEL 1 ESTIMATION

ESTIMATION: THE NATURAL WAY Natural parameter estimation

Learned behavior (e.g., catching a ball)

Use of implicit processing for prediction, estimation & feedback

Computational estimation Use of explicit (mathematical

models) for equations of motion, observation prediction, state vector correction and update)

Models serve as an approximation to reality

INTRODUCTION TO PARAMETRIC FUSION

Problem StatementGiven a redundant set of N observations, Z(t), from one or more sensors, find the value of a state vector, x(t), that provides a best fit to the observational data

Aspects of the Estimation Problem Definition of the state vector, x(t) Observation model(s) Treatment/assumptions about observation noise Dynamic model for the state vector Definition of best fit Method of solution Fusion architecture Approach to handling data

CONCEPTUAL PROCESSING FLOW FOR LEVEL 1 FUSION

BulkGating

DataAssociation

& Correlation

Position/Kinematic/Attribute

Estimation

IdentityEstimation

• Observation File• Track File• Sensor Information

Sensor#1 Preprocessing Data

Alignment

Sensor#2 Preprocessing Data

Alignment

SensorN Preprocessing Data

Alignment

•

•

•

•

•

•

•

•

•

OBJECT POSITIONAL AND KINEMATIC ATTRIBUTE ESTIMATION

Object: Positional and Kinematic

Attribute Estimation

• Observation Equations• Equations of Motion• Dynamic Maneuver Model• State Vector Definition• Implementation - Data Editing - Coordinate Systems

• Sequential Processing - Kalman Filter - - Filter• Batch Processing• Covariance Error Formulation

CATE

GO

RY

FUN

CTIO

NTE

CHN

IQU

E

System Models

Optimization Criteria

Optimization Approach

Processing Approach

• Least Squares (LS)• Weighted LS• Mean Square Error• Maximum Likelihood• Constrained (Bayesian)

• Direct Methods - Non-Derivative Methods - Downhill Simplex - Direction Set - Derivative Methods - Conjugate Gradient - Variable Metric (Quasi-Newton)• Indirect Methods - Newton-Raphson Methods

DESIGN OPTIONS FOR MULTI-TARGET TRACKING

Assignment of observations to tracks Hard (unique) assignment Soft (non-unique) assignment

Allowable explanations for observations Single hypothesis Multiple hypothesis

When to make a final decision about observations After each observation (single scan) After N observations (multiple scan) After all observations are received (batch processing)

Processing approach Sequential estimation Batch estimation Covariance error analysis

INTRODUCTION TO PARAMETRIC FUSION

Problem StatementGiven a redundant set of N observations, Z(t),

from one or more sensors, find the value of a state vector, x(t), that provides a best fit to the observational data

Aspects of the Estimation Problem Definition of the state vector, x(t) Observation model(s) Treatment/assumptions about observation

noise Dynamic model for the state vector Definition of best fit Method of solution Fusion architecture Approach to handling data

In the late 1700s, Carl Frederick Gauss invented the method of least squares to compute the orbits of asteroids; addressing many of these identical issues!

THE IDEA BEHIND ESTIMATION

We observe a parameter, z, which varies as a function of t

We suspect that there is a relationship between z and t of some form (perhaps given by the equation z = a + bt2, where a and b are constants)

We don’t know the values of a and b. If we did know them, we could predict the value of z for any value of t.

The quantity x(a,b) is the unknown “state vector”

z = a + bt2 is the observation equation For simplicity, we assume a and b do not

vary in time.

z (t)

t

How can we find a and b, given observations of z?

Note: the observations, denoted by blue circles do not fit exactly on the dashed line – this is due to observation noise – our inability to observe z exactly

THE LEAST SQUARES BATCH ESTIMATION SOLUTION

Observe a number of values of z (e.g., z (t1), z (t2), …)

Guess at a value of a and b (say a0, and b0 – this corresponds to an initial guess at the state vector, x0)

Given these guessed values of a and b, compute predicted values of z at times t1, t2, …

Compare the values of the predicted values of z

predicted(t1), etc, with the actual observed values (these are called observation residuals)

Compute the sum of the squares of the residuals Systematically vary the guessed value of (a,b) until

the sum is a minimum (i.e., we have found the value of (a and b) that “best” fits the observed values; this is our state estimate.

z (t)

t

How can we find a and b, given observations of z?

Predicted value of z at time ti given an assumed or guessed value of the quantities a and b

ti

Observation residual

LEAST SQUARES ESTIMATION CONTINUED

We showed the concept of “batch” processing – viz, we waited until all of the observations have been received to estimate the value of a,b. Note there are numerous methods for batch estimation including; The least squares method The weighted least squares method Mean square error method Maximum likelihood method Many others (the choice depends upon our assumptions about the

models)

What happens if we want to estimate a and b, before all of the observations are obtained? This is the case of sequential estimation. The next charts show a common approach known as Kalman filtering.

DEFINING THE OBSERVATION AND STATE MODELS

THE STATE MODEL:THE STATE MODEL:STATE VECTOR: A set of independent parameters, x(t), which, if known, would allow complete specification of the current and future state of a system (e.g., position, velocity, and attributes)

_

STATE EQUATIONS: A set of equations that relate the state vector, x(t0), at an epoch in time to the state vector at any other time. Typically, the state equations are a set of simultaneous nonlinear differential equations.

_

THE OBSERVATION MODEL:THE OBSERVATION MODEL:

x(t) = (t, t0) x(t0)_ _

OBSERVATION MODEL: The observation model relates the state vector to predicted observations,

z(t) = H x (t) + obs. noise_ _

This equation enables us to predict an observation that would be observed at time, t, if the state vector were known.

Interpretation: The state at time, t, equals a transition function times the state at time t 0)

Interpretation: predicted observation at time, t, equals the estimated state vector at time t, times a function H

RECURSIVE ESTIMATION

zi = x + vi ( i = 1, 2, . . .k)

• Suppose we are estimating a scalar deterministic constant based on k noisy measurements:

• The unbiased minimum-variance estimate is:

x̂K = 1/K Zi

k

add new measurement ZK+1 and

X̂K+1 = 1/(K+1) Zi

it can be shown that (new est) = (old est) + (weighted measurement residual)

X̂K+1 = XK + [1/(K+1)][ZK+1 - XK] ^ ^

Interpretation: the new estimate of the state vector, x, (based on k+1 observations) is equal to the old or previous estimate of x (based on k observations) plus a constant times the difference between the new observation at k+1 minus the predicted observation at k+1, given the previous estimate of x.

SUMMARY OF DISCRETE KALMAN FILTER EQUATIONS

System Model xk = k-1 xk-1 + wk ~N(0, Qk)

Measurement zk = Hkxk + vk, vk ~N(0, Rk)

Initial Conclusions E[x(0)] = x0, E[(x(0) - x0)(x(0) - x0)T = P0

Other Assumptions E[ wk vjT ] = 0 for all j, k

State Estimate Extrapolation xk (-) = k-1 xk-1 (+)

Error Covariance Extrapolation Pk(-) = k-1 Pk-1(+) Tk-1

+ Qk-1

State Estimate Update (Kalman Equation) xk (+) = xk (-) + Kk [zk - Hk xk (-)]

Error Covariance Update Pk (+) = [I-Kk Hk] Pk (-)

Kalman Gain Matrix Kk = Pk (-) HkT [Hk Pk (-) Hk

T + Rk]-1

^^^

^^

^^^

DISCRETE KALMAN FILTER TIMING DIAGRAM

Hk-1, Rk-1 Hk, Rk

xk-1^ xk-1

^(-) (+)

Pk-1 Pk-1

(-) (+)

tk-1 tk

xk^ xk

^(-) (+)

Pk Pk

(-) (+)

k-1, Qk-1 k, Qk

KALMAN FILTERING PROCESSING STEPS

1

1)(1111

)(11

T

kkkkT

kkk HPHRHPK

PredictForward

Compute Kalman Gain

ReceiveMeasurement

kkkk wxΦx

)()(1 ˆˆ

kTkkkk QΦPΦP

)()(

1

UpdatePrediction

)(1111

)(1

)(1 ˆˆˆ

kkkkkk xHzKxx

)(111

)(1

kkkk PHKIP

1)(111 ˆ

kkkk vxHz

THE KALMAN FILTER IN WORDS

To find a new estimate of the state vector, x at time ti+1 we do the following Predict forward in time

Update the old state vector from the time of the last observation, t i, to the time of the current observation at time, t i+1 (using the equations of motion)Update the estimate of the uncertainty of the state vector to time, ti+1 (using an “equation of motion” for the error growth)

Compute the Kalman gain- Compute the Kalman gain – this is a factor used to change our estimate of the state vector and associated uncertainty based on several factors; our confidence in the accuracy of our current estimate of the state vector (how well do we think we know the answer) and our confidence in the accuracy of the observations (are we receiving “good” observations or relatively “bad” observations?)

Receive a new observationObtain a new observation at time, ti+

Update the estimates of the state vector & its uncertaintyUpdate the estimate of the state vector at time ti+1 using the Kalman equation (which says in essence, “new state vector equals olds state vector times a constant multiplied by the difference between the actual observation and the predicted observation”.Similarly update the estimate of the uncertainty of the state vector.

WHERE DOES THE FUSION OCCUR?

Notice that: The only places in the Kalman equations that depend upon the

type of sensor are the variables; z (the observation), R (the observation noise) and H (the quantity that predicts the observation based upon the current estimate of the state vector)

The dimension of the state vector and the observation are unrelated

Measurements from different types of sensors can be interleaved in the sequential estimation process

Where does the fusion occur?

MEASUREMENT FUSION WITH SYNCHRONOUS DATA: SCALAR APPROACH

• Given I observations at time, tk+1: zk+1, i

• The Kalman Filter update equations may be solved sequentially for each observation component. For zk+1,1; zk+1,2 . . . :

STATE:

GAIN:

COVARIANCE:

Kk+1,i = Pk+1/k+1,i-1HTk+1,i (Hk+1,i Pk+1/k+1,i-1 HT

k+1,i + Rk+1,i)-1

Pk+1/k+1,i = Pk+1/k+1,i-1 -Kk+1,i Hk+1,i Pk+1/k+1,i-1

xk+1/k+1,i = xk+1/k+1,i-1 + kk+1,i (zk+1/i -h(xk+1/k+1,i-1))^^ ^

+ Kk+1, i (zk+1, i - hi (xk+1/k+1,i-1))I

i=1x̂k+1/k+1 = xk+1/k ^ ^

MEASUREMENT FUSION WITH SYNCHRONOUS DATA: VECTOR (PARALLEL)

FORMULATION• Given I observations at time, tk+1: zk+1, i

• The Kalman Filter update equations for the state vector, the Kalman Gain, and the state covariance can be formulated in a vector (or parallel) approach:

STATE:

GAIN:

COVARIANCE:

^+ Kk+1, i (zk+1, i - hi (xk+1/k))I

i=1

xk+1/k+1 = xk+1/k^ ^^

Kk+1, i = Pk+1/k+1 HTk+1, i R-1

k+1, i

P-1k+1/k+1 = P-1

k+1/k + (Hk+1, i R-1k+1, i Hk+1, i)

I

i=1

TRACK FILE FUSION:PROBLEM CONCEPT

xTGT

SENSOR 1

SIGNAL PROCESSOR #1

H1 +v1

TRACKPROCESSOR

#1

SENSOR 2

SIGNAL PROCESSOR #2

H2 +v2

TRACKPROCESSOR

#2

XFUSED

_?

XK = K1 • X1 + K2 • X2

MULTIPLE SENSOR EXTENSIONS:TRACK FILE FUSION

• Assume autonomous (track) fusionAssume autonomous (track) fusion– Optimize combinations of state vector predictions

– Estimation error is given by

where C accounts for differences in observation spaces

• Choose a value of K to minimize estimation errorChoose a value of K to minimize estimation error• The solution givesThe solution gives

Є(k) = (I - KC) Є 1(kn) + K Є 2 (km)

K = [P11(k)CT + P12(k)][P22(k) - P21(k)CT

- CP12(k) + CP11(k)CT]-1

x(k) = K1x1 (kn) + K2x2(km)^ ^ ^

x(k) = x1(kn) [x2 (km) - C • x1 (kn)]^^^ ^

KALMAN FILTERING ASSUMPTIONS

Kalman filtering is a specific case of Bayesian tracking

Relies on two important assumptions:

− Gaussian noise distribution

− Linear system models

Computationally efficient and performs well if the assumptions

hold true

KALMAN FILTERING LIMITATIONS AND ALTERNATIVES

Disadvantages Usually the underlying noise distribution not known Practical applications usually involve non-linear

systems

Alternatives Extended Kalman Filter (EKF) Particle Filter (PF)

MANEUVER DETECTION Any maneuver involves a prolonged acceleration. Acceleration is not measured directly Acceleration is often caused by human volition An estimate of acceleration frequently is not carried as a result

of processing load. Estimation of acceleration is challenging (even if a model exists)

because of high degree of correlation between velocity and acceleration.

If all of the above were true, a lag problem would still exist; measurements would lag behind the actual onset of the maneuver.

What can be done? Try using the measurement strategy anyway Try using deviations from straight line behavior Try other creative, ad hoc techniques

MANEUVER PROCESSING

Four general approaches: Maintain filter form, modify parameters Augment filter state model with correlated accelerations Use multiple filters with different maneuver models, select best output Interacting multiple models (use multiple filters with different maneuver models

and integrate model outputs)

Design problems: Miscorrelations Noise False alarms triggering maneuver detector

EXAMPLE OF INTERACTING MULTIPLE MODELS

Figure 2 from a paper entitled, Comparison of Various Schema of Filter Adaptivity for the Tracking of Maneuvering Targets, Center of Mathematical Research, Montreal by Jouan

EVOLVING RESEARCH IN ESTIMATION

New types and generalizations of estimation filters Higher order filters Explicit consideration of observation cross correlation Monte-Carlo approach for estimating underlying probability

density functions Random set theoretic formulations Increasing sophistication of MHT New methods for association/correlation integrated with

estimation Etc

TOPIC 7 ASSIGNMENTS Preview the on-line topic 7 materials Read chapter 4 of Hall and McMullen pp 129 – 132 Writing assignment 6: Write a brief description of how level 1 estimation

applies to your selected application; what entities, objects or activities need to be tracked; what are the components of a state vector; what is the relationship between what can be observed and what must be tracked; what are the issues (if any) in predicting or modeling the evolution or trajectory of the entities to be tracked?

Discussion 3: Discuss the issue of association and correlation for clearly definable physical objects such as automobiles or individual humans versus non-physical objects or distributed objects such as human groups, vapor clouds, or other “objects” that cannot be clearly located in a small physical volume.

DATA FUSION TIP OF THE WEEK

There are numerous variations in estimation techniques, based on different levels of knowledge or assumptions about our understanding of the inherent models, error characteristics (e.g., of the sensor observations, growth of error in the state vector propagation, etc. The key is not which model is the most sophisticated, but rather which model is appropriate based on our understanding of the problem at hand. Not every problem is a nail!