Mathematical Engineering Mathematical Engineering in in

Avionics ApplicationsAvionics Applications

Dr. SK ChaudhuriDr. SK Chaudhuri Sc. ‘H’Sc. ‘H’

Associate Director, RCIAssociate Director, RCI

9th June 2007, IISc Bangalore

FUNCTIONAL BLOCK DIAGRAM OF MAJOR MISSILE SUBSYSTEMSFUNCTIONAL BLOCK DIAGRAM OF MAJOR MISSILE SUBSYSTEMS

Reference Generation

System

KnowledgeGathering

system

DecisionProcess

ActionProcess

Airframe &Propulsion

Kinematics

~

Vx~

X~

M/T

Guidance Command

Target Trajectory

Acceleration Rates

~

Missile Trajectory

KNOWLEDGE GATHERING SYSTEM:

Navigation process for position, velocity and attitude etc.

DECISION PROCESS:

Missile guidance system based on available knowledge and stored guidance (if required)

ACTION PROCESS:

Flight control system with sensors, actuators

Reference Generation

system

NavigationComputer

GuidanceSystem

AutopilotActuation

SystemAirframe &Propulsion

Sensors

Kinematics

InertialSensors

RF/IR

Sensors

MISSILE CONFIGURATIONMISSILE CONFIGURATION

Rates & Acceleration

Missile Trajectory

Target Trajectory

~

Engine BayControl Surface

Wing

Propellant Tank

Guidance &

Control System

Nose ConeElectronic Bay

Radome

Warhead

Dynamic Eqn.s with Newton’s laws of motion Fluid dynamics Nonlinear Time varying differential Eqn.s Numerical Integration (Euler & RK4) Interpolation Flexibility dynamics in terms of generalized coordinates

Laplace Transforms Z-Transforms State space Methods Optimization Tech. Robust Design

Estimation TheoryRandom & Stochastic Process State space Methods Matrix algebra Iteration Techniques InterpolationOptimization Tech.

Quaternion algebra Matrix algebra Integration techniques Solid geometry with Geodetic, Geocentric and 3D representation

NavigationComputer

GuidanceSystem

AutopilotActuation

SystemAirframe &Propulsion

Sensors

Kinematics

InertialSensors

RF/IR

Sensors

Rates & Acceleration

Missile Trajectory

Target Trajectory

~

Fast Fourier Transforms Signal Processing Filtering techniques.

Curve Fitting Filtering techniques.

Kinematic Equations Linear and Matrix Algebra Integrations techniques

MATHEMATICAL ENGINEERING INVOLVED MISSILE SUBSYSTEMSMATHEMATICAL ENGINEERING INVOLVED MISSILE SUBSYSTEMS

Mathematical Modelling And SimulationMathematical Modelling And Simulation

ACTUAL SYSTEM

MATH MODEL COMPUTER SIMULATION

VALIDATION

COMPARISON

COMPARISON

VERIFICATION

System, Model & Simulation Correlation

BASIC TECH. COMPONENTS :1. Requirements which final Simulation must satisfy.

2. Equations for representing actual system.

3. Program Equations for Simulation.4. Compare Simulation Program to the Model and modify the mistakes.

5. Compare Simulation result with actual results.

VERIFICATION :Process to determine that a program causes computer to operate as intended by the software designer (i.e. Equations are programmed correctly).

VALIDATION :Process to determine that computer simulation behaves like actual system in all pertinent respects.

=tan-1(W/U)

=tan-1(V/U)

VM=(U2+V2+W2 )

axs = (Tx-Dx)/ Mays = Y /M+Yy /M +crazs = Z /M+Zp /M -cq

p = Lpp/Ixx+LRR /Ixx+ClQs/Ixx

q = M /Iyy+Mp /Iyy r = N /Izz+Ny /Izz

axs

ays

azs

p

q

r

INCREMENTAL ANGLES AND VELOCITIES

p

q

r

NAV.FUNC

U

A/D &

INERTIAL

PARAMET

ERS

ROTATIONAL AND TRANSLATIONAL LOOP JOB ROTATIONAL AND TRANSLATIONAL LOOP JOB ALLOCATION IN REAL TIME MISSILE 6DOFALLOCATION IN REAL TIME MISSILE 6DOF

ENGINETHRUST V

W

QUATUPDATE

VB=([DCM] )RS

TVR

Qm

VX

DCM

VB1-3

1-3

t Vm Z

Undue Roll oscillations due to low damping introduced by gimballed engines, thrust frame and hardware actuator compliance

TWD EFFECTS IN 6-DOF MODELTWD EFFECTS IN 6-DOF MODEL

MISSILE AUTOPILOT WITH FLEXIBILITYMISSILE AUTOPILOT WITH FLEXIBILITY

Unstable Autopilot Response Modified Stable Autopilot Response

0 100 200 300 400 500 600 700-200

-150

-100

-50

0

50

Psi E

rror(a

rc mi

n)

PII-06 Launch T.A Results (Psi Error, Del Vn Plots)

blue = Optically measured psi error

red = AKF estimated psi error

0 100 200 300 400 500 600 700-0.1

-0.05

0

0.05

0.1

Del V

n(m/s)

PII-06 Launch TA Results

Time (sec)

Demonstrated 7-state AKF based TA for SSMs launched from Moving Platform.

Fdbk gains are selected using Linear Quadratic Gaussian Regulator and offline Matrix Riccati equation solution.

Integrated the above with EKF based GPS-INS data fusion for Dhanush extended range missions.

Validated through Van, Aircraft, Ship & Flight trials.

TRANSFER ALIGNMENT (TA) SCHEME FOR SHIP LAUNCHED MISSILETRANSFER ALIGNMENT (TA) SCHEME FOR SHIP LAUNCHED MISSILE

update

AKF(Adaptive Kalman Filter)

Feed backController

Slave INS(SDINS)

S : System

Missile q, r 1.2 0/s

-

+

s^

GPS/DGPS LOG

Master Vel, Lat, Long

Conversion to

Error quaternion Alignment

corrections

S curve 0.15 m/s2

Master INS

Slave Accn

Slave Vel, Lat, Long

Fdbks

Meas & Noise

Process NoiseStates

Ship 100/s SS

red = AKF estimated syi error

Blue = Optically measured syi error

Demonstrated 17-state Extended Kalman Filter (EKF) based GPS-INS Data Fusion in OBC for extended range Prithvi missions.

GPS-INS DATA FUSION SCHEME FOR EXT. RANGE PRITHVI MISSION GPS-INS DATA FUSION SCHEME FOR EXT. RANGE PRITHVI MISSION

0 500 1000 1500 2000 2500 3000 3500 4000 4500-40

-30

-20

-10

0

10

20

30

time (secs)

Pos

ition

cor

r (m

)

Position corrections

X corrY corrZ corr

FUSEDNAVIGATION

GUIDANCEMODULE

TVC

CONTROL ACTUATIONSYSTEM

GPSKF

MODULE

IMU

LC

Guidance Commands

Defln

Pos, Vel

Nominal Trajectory

Rates

Accln

Corrections

Pos, Vel, DCM

PURENAVIGATION

Quat, Pos, Vel

GPS Data

ADC

Defln

Quaternion

1985 2000 2010

EMBEDDED ONBOARD PROCESSORS

PR

OC

ESS

OR

CL

ASS

Year

2005

8086

80486

•Pentium Class •Power PC•COTS•Multi Protocol Connectivity

System On Chip

GUIDANCE SYSTEM ENGINEERING

1990 2010

1 Km

40 mCE

P of

Pri

thvi With Strapdown Inertial Implicit Guidance (CEP < 1 Km)

1 m

2003

With TA & GPS-INS data fusion (CEP < 40 m)

Inertial & Seeker Guided PGMs (CEP < 1 m)

Year2006

Inertial, Radar & Seeker fused Guidance (CEP < 10 m)

NUMBER OF FLIGHT TRIALS OF PRITHVINUMBER OF FLIGHT TRIALS OF PRITHVI

1984 1988 20040

64

38

Year

No. of FlightTrials

Prithvi

12

(Planned)

(Planned)

(Actual)

Requirement of number of flight trials is reduced because of HILS.

1996