kalman filtering

State Estimation from Radar/Seeker Measurements Using Kalman Filtering Aided With Multi Sensor Data Fusion

Department of Instrumentation and Control Engg National Institute of Technology, Tiruchirappalli – 620 015

Dr. A K SarkarScientist

Directorate of SystemsDRDL, Hyderabad-58

First Workshop of AUTOMATIC CONTROL & DYNAMIC OPTIMIZATION SOCIETY (ACDOS)

On Fascinating and Challenging Applications of Estimation

10th & 11th November 2011

•Introduction and Historical Perspectives of Estimation Theory

•Kalman Filtering

•Multi Sensor Data Fusion

•Case Study of Radar Data Fusion (Offline Study)

•Case Study of a Missile Tracking Problem Using Radar and Seeker (Realtime Study)

•Conclusions

History of Estimation Theory and its Application to Aerospace Problems

• First Application of ET to Track Planet Ceres again By Piazzi (Italian Astronomer) at the end of the year which he missed in January after observing for continuous 41 days (1802), He could tarce back the planet based on estimate given by Gauss who Invented Least Square Technique .

• Fisher Invented MMLE Estimator (Weighted Least Square for Batch Processing) Using Byes Rule and Gaussian Distribution (1922)

• Wiener First Time Designed Statistically Optimal Filter (Recursive Least Square ) In Frequency Domain To Remove Noise From Electrical Signal (1947)

• Kalman and Bucy Designed Time Domain Estimator Based on Wiener’s Research on State and Measurement Model (1958)

• Luenbarger’s Invention of Observer Theory (1964) Became Invaluable Tool To Apply Kalman Filter For Controlling A Plant Based on Noisy Measurements

• Research in MIT Lincoln Laboratory To apply Kalman Filter in Apollo Space Mission of NASA (1966-71)

• Application on Different Aerospace Problems (1971 Onwards)

Historical Evolution of Estimation Theory ( NASA TND 7647)

Johann Carl Friedrich Gauss

Born:1777 Brunswick, Germany

Died: February 23, 1855, Göttingen, Germany

By the age of eight during arithmetic class he astonished his teachers by being able to instantly find

the sum of the first hundred integers.

Curtsey:Dr. M R AnanthasayanamRetired Professor, IISc, Bangalore 12

Facts about Gauss

• Attended Brunswick College in 1792, where he discovered many important theorems before even reaching them in his studies

• Found a square root in two different ways to fifty decimal places by ingenious expansions and interpolations

• Constructed a regular 17 sided polygon, the first advance in this matter in two millennia. He was only 18 when he made the discovery

Ideas of Gauss

• Gauss was a mathematical scientist with interests in so many areas as a young man including theory of numbers, to algebra, analysis, geometry, probability, and the theory of errors.

• His interests grew, including observational astronomy, celestial mechanics, surveying, geodesy, capillarity, geomagnetism, electromagnetism, mechanism optics, and actuarial science.

Intellectual Personality and Controversy

• Those who knew Gauss best found him to be cold and uncommunicative.

• He only published half of his ideas and found no one to share his most valued thoughts.

• In 1805 Adrien-Marie Legendre published a paper on the method of least squares. His treatment, however, lacked a ‘formal consideration of probability and it’s relationship to least squares’, making it impossible to determine the accuracy of the method when applied to real observations.

• Gauss claimed that he had written colleagues concerning the use of least squares dating back to 1795

Curtsey:Dr. A.K. GhoshProfessor & Faculty InchargeFlight Lab.IIT, Kanpur – 208016Ph. (O) = +91-512-2597716 (Fax) = +91-512-2597716

ESTIMATION THEORY / SYSTEM IDENTIFICATION

Classification

• Classical problem (Simulation):given u and f, find x and Y

• Control Problem:given Y and f, find u

• Estimation Problem:given u and Y, find f and X

Problems in Kinematics and Dynamics

State EquationsInput

U

Output

Y ,,UXfX

What is Estimation Theory?

Aim:

To determine unknown model parameters and states X such that the model response Y matches well with the measured system response Z.

Dynamic SystemInput

U

Output

Z

Mathematical ModelInput

U

Output

Y

),(),()(

),(),()(

tUtXgty

tUtXftX

StateEquations

MeasurementEquations

Process noise(turbulence)

Inputs

Sensors

states

Sensor model(calibration factor, bias error)

Measurement Noise

Outputs

Block Schematic of Stochastic System Model

Estimation Theory Applications ( NASA TND 7647)

Kalman Filtering (Recursive Estimation Algorithm)

http://academic.csuohi/edu/simond for Some matlab codes on Kalman Filtering

(Dan Simon, Clievland University)

DIFFERENT ESTIMATION ALGORITHMS DEVELOPED (Since 20th Century)

•MATHEMATICAL TOOLS DEVELOPED PRACTICAL PROBLEMS SOLVED

Recursive Techniques (Online realtime application) Extended Kalman Filter (EKF) Radar/EOTS/Accelerometer/Gyro Data processing Adaptive EKF (AEKF) EKF Filter Tuning

Batch Processing (Offline application)

Maximum Likelihood Estimation (MLE) Flight Data Compatibility Check Multiple Linear Regression (MLR) Aerodynamic Moment Coefficients Estimation

Genetic Algorithms (GA)

Artificial Neural Network Nonlinear Least Square (NLS) Post Flight Trajectory Reconstruction

Orthogonal Least Square (OLS) Roll Moment Coefficient, Roll Derivative Estimation

• OFFLINE APPLICATIONS FOR POST FLIGHT ANALYSIS

• Estimation of Sensor Bias and optimal trajectory from EOTS and Radar Data

• Aerodynamic Coefficient Estimation From Flight Data

• ONLINE APPLICATION FOR GUIDANCE AND NAVIGATION

• State Estimation from Tracking Radar measurements for Guidance • State estimation from Sonar data for underwater application• State Estimation from Seeker measurements for Guidance

• GPS/INS Integration, INS Alignment Problems • Multi Sensor Data Fusion / Fuzzy Logic, Neural Network / Intelligent

Control

Application of Kalman Filtering in a Missile System

ASTRA MISSION SPECIFICATION

TARGET

ASTRA

TERMINALHOMING

= MIDCOURSE INERTIAL

PLATFORM = Mirage2000, Mig29, SU 30 and LCA

TARGET MANEUVER = 9g AT SEA LEVELLAUNCH ALTITUDE = SL TO 20 KMLAUNCH SPEED = 0.6 – 2.2MINTERCEPT RANGE > 80 kmGUIDANCE

TERMINAL HOMING

LAUNCHFROM LCA

L C A

DATA LINK

PRELIMINARY SYSTEM DESIGN: •LAUNCH BOUNDARY AGAINST HIGH MANEUVERING TARGET •PN GUIDANCE IN BOTH MIDCOURSE AND TERMINAL PHASE

MISSILE

AIRBORN RADAR

AIRCRAFT

(R,A,E) AS RADAR MEASUREMENT

Missile Kinematics

Autopilot & Actuator

Guidance Law

Achieved Acceleration

+

+

Predicted State & Update

Radar Measurements

PredictedMeasurements

2

1

1

1

s

s

Random Acceleration

EKF Estimator

Radar Noise +

+

Commanded Acceleration

+

–

x̂

PURSUER EVADER ENGAGEMENT SIMULATION IN CLOSE LOOP

Why Estimator is used?

Multi Sensor Data Fusion (MSDF)

Measurements• Sensors System (Radar, Sonar, EOTS, Seeker ) as external measurements• Imaging sensors • Pitot Static Probe• GPS, DGPS as external measurements (inertial)• Onboard measurements (SDINS)

MSDF Activities

Information from multiple sources A multilevel process dealing with detection, association, correlation, estimation Data availability at different time tags Combination of all sets of data in a statistical sense

Applications Robotics Military applications (coast guard, air traffic control, remote sensing) Medical diagnosis (information fusion of sensors such as X-ray, magnetic

resonance, tomography images)

Aerospace applications Improved target detection and tracking with less tracking error Fire control system employing multiple sensors for acquisition, tracking and

command guidance

Desirable features of MSDF architecture Modularity Parallelism Distributed structure (Military C3 system) Robustness/ Survivability

Fusion algorithms

• State Vector Fusion (bank of Kalman Filters)• Measurement Fusion (One Kalman Filter )

Review paper

Text book

Case Studies

Two Case Studies to be discussed

• Available External Measurements (Cass 1) (Post flight analysis)

Track data containing position information of Strategic Flight Vehicle from Tracking Radar located at three different tracking stations .

• Available External Measurements (Cass 2) (Real time guidance)

Seeker data and radar data fusion fusion for guidance purpose in real time for Air Defence (exo-atmosphere application)

Measurement Fusion Algorithm

Similar Sensors (Radar #1, Radar #2, Radar #3)

Based on different sets of measurements, get fused measurement covariance

One Kalman Filter Used with fused measurements

State Vector Fusion Algorithm

Dissimilar Sensors (Radar, EOTS, Seeker, GPS)

Estimate the state variables from each set of measurements

Bank of Kalman Filters Used

Use both state and covariance informations to obtained fused estimates and covariance

A Typical Case Study (Case study #1)

Agni AE02 Radar Data Processing Using EKF/MBFS and MSDF

(Post Flight Analysis)

Sarkar A K: Flight Data Compatibility Check Using BFGS Under Limited Measurements With Multi Sensor Data Fusion, Paper No AIAA-99-4176, (1999).

Typical SFV Tracking Situation

Radar data Precessing using three sets of radar data

Use CJ model with 12 states and three measurements

Use EKF in forward pass and MBFS in backward pass

Run three filter/smoothers (EKF/MBFS) in parallel

Combine estimates from three sources using state vector fusion technique of MSDF.

Comparison of estimated position from three sets of Radar Data (MSDF using Algorithm #2)

1 sigma estimation error of position, velocity components (MSDF (Algorithm #2))

Flight Data Compatibility Check

Matching System Output With External Radar Measurements

Estimation Of Bias And Scale Factor From All Telemeter Data

Flight Estimated (Cm_alpha, Cm_delta) variation with respect to Mach No (SFV)

Introspections based on present radar data fusion

Estimates are more accurate using Algorithm #2 due to inclusion of cross-covariance among different sensors

Using the fused position data flight data compatibility check was carried out for aerodynamic parameter estimation

In a nutshell information available from all radar data was used

In a practical situation state vector fusion was used

Measurement fusion not used (lack of data from all sensors at a time)

A Typical Case Study (Case study #2)

Seeker and Radar data processing Using EKF and MSDF(Realtime application)

Ananthasayanam M R, Sarkar A K, Bhattacharya A, Tiwari P and Vorha P: Nonlinear Observer State EstimationFrom Seeker Measurements and Seeker-Radar Measurements Fusion , Paper No AIAA-2005-6066-CP (2005).

EKF Formulation for Seeker Measurements

EKF Formulation for Radar Measurements

MSDF (Algorithm #1)

MSDF (Algorithm #2)

Results

Conclusion

Seeker EKF Formulation

Schematic Diagram of Pursuer and Evader Engagement

Different Axes System For Seeker

•Pursuer Body Frame

•Pursuer Fin Frame

•Pursuer Seeker Gimbal Frame

•Local Vertical Frame

•Inertial Frame

Axis System for Kinematic Modeling of State Equations

PROBLEM DEFINITION

• Estimation of Relative Position, Relative velocity and Target acceleration from noisy (Range rate, Gimbal angles and Gimbal angle rates ) Seeker Measurements

CONSTRAINTS • Measurement Data at 25 milliseconds interval • Measurement noise is non gaussian due to effect of eclipsing, glint, thermal noise and RCS fluctuations•There is an aperiodic data loss in LOS rates due to eclipsing, RCS fluctuations

Measurements Eclipsing Zone Non Eclipsing Zone

Range Rate A A

Gimbal Angle (yaw) A A

Gimbal Angle (pitch) A A

LOS rate (yaw) NA A

LOS rate (pitch) NA A

NA = Not Available A = Available

Total 5 and 3 measurements during non eclipsing and eclipsing time zones respectively

Different Seeker Measurements

KALMAN FILTER FORMULATION (CP Frame)

Radar EKF Formulation

Tracking Radar at launch point (schematic)

Derivation of measurement equations from radar measurements

Filter Tuning Elements (P_0, Q)

Time History of measured range and range rate estimation error averaged over 25 MC (Filter #1, Filter #2)

Measured yaw, pitch gimbal angle estimation error hostory averaged over 25 MC (Filter #1, Filter #2)

Time History of LOS rates (yaw + pitch) estimation error averaged over 25 MC (Filter #1, Filter #2)

Time History of Delta x estimation error averaged over 25 MC (Filter #1, Filter #2)

Time History of Delta V_ x estimation error averaged over 25 MC (Filter #1, Filter #2)

Seeker LOS rate (yaw) measurement, estimated and estimation error

Pursuer evader trajectory

RECAPITULATION AND CONCLUSION

•Historical Evolution of Estimation Theory

•Kalman Filtering and its Applications

•Brief Introduction to MSDF and its utility in real world

•Case study of radar data processing from different tracking stations (Case study #1)

.Seeker and Radar Data processing (Case Study #2)

MSDF Performance is very bad if the sensors have deterministic bias

THANK YOU FOR THE PATIENT HEARING !

Other ContributorsDr. M R Ananathasayanam

Dr. S Vathsal