# SKC IISc Performance Analysis Laboratory ECE IISc

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```					Mathematical Engineering
in
Avionics Applications
Dr. SK Chaudhuri
Sc. ‘H’
Associate Director, RCI

9th June 2007, IISc Bangalore
FUNCTIONAL BLOCK DIAGRAM OF MAJOR MISSILE SUBSYSTEMS

Reference
Generation
System
Guidance                            Acceleration
Command                             Rates
Target                                                           
Knowledge                                           ~
Trajectory
Decision               Action         Airframe &
Gathering                                                                   Kinematics
Process                Process        Propulsion
system
X
~
Vx
~

~   M/T
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
Engine Bay
Surface                                           Electronic Bay
Wing                                       Nose Cone

Guidance &
Propellant Tank
System

MISSILE CONFIGURATION

Reference
Generation
system
                            Rates &
~
Navigation       Guidance                      Actuation     Airframe & Acceleration
Autopilot
Computer          System                        System       Propulsion

Sensors

Inertial
Sensors
Target
Trajectory     RF/IR
Kinematics
Sensors         Missile
Trajectory
MATHEMATICAL ENGINEERING INVOLVED MISSILE SUBSYSTEMS

 Estimation Theory                                                         Dynamic Eqn.s with
Random & Stochastic Process                                                 Newton’s laws of motion
 State space Methods                                                       Fluid dynamics
 Matrix algebra                                                            Nonlinear Time varying
 Iteration Techniques                                                       differential Eqn.s
 Interpolation                                                             Numerical Integration
Optimization Tech.                                                          (Euler & RK4)
 Interpolation
 Laplace Transforms                  Flexibility dynamics
 Quaternion algebra                                                                            in terms of generalized
 Matrix algebra                                          Z-Transforms
 State space Methods                  coordinates
 Integration techniques
 Solid geometry with                                     Optimization Tech.
Geodetic, Geocentric                                     Robust Design
and 3D representation

Rates &
~             Airframe &   Acceleration
Autopilot
Computer      System                     System     Propulsion

 Fast Fourier Transforms                                                                                     Curve Fitting
 Signal Processing                                                   Sensors                                 Filtering techniques.
 Filtering techniques.
Inertial
Sensors

Target         RF/IR
Trajectory   Sensors      Missile        Kinematics
Trajectory                                               Kinematic Equations
 Linear and Matrix Algebra
 Integrations techniques
Mathematical Modelling And Simulation

ACTUAL SYSTEM

COMPARISON
MATH MODEL                                              COMPUTER SIMULATION
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.
ROTATIONAL AND TRANSLATIONAL LOOP JOB
ALLOCATION IN REAL TIME MISSILE 6DOF

axs
axs = (Tx-Dx)/ M                 ays

ays = Y /M+Yy /M +cr
         azs
   azs = Z /M+Zp /M -cq


                                                             p            p

p = Lpp/Ixx+LRR /Ixx+ClQs/Ixx q            q    INCREMENTAL
                                                 ANGLES AND
q = M /Iyy+Mp /Iyy                        r    VELOCITIES
                                
    r = N /Izz+Ny /Izz           r

A/D &                             U                      VX                    1-3
ENGINE                               =tan-1(W/U)
INERTIAL                                            R T
NAV. Qm QUAT
THRUST                              =tan-1(V/U)   V   VB=([DCM] S) VR          FUNC
PARAMET                                                       DCM
UPDATE
VM=(U2+V2+W2 ) W
ERS
 VB1-3
t   Vm      Z
TWD EFFECTS IN 6-DOF MODEL

Undue Roll oscillations due to low damping introduced by gimballed
engines, thrust frame and hardware actuator compliance
MISSILE AUTOPILOT WITH FLEXIBILITY

Unstable Autopilot Response   Modified Stable Autopilot Response
TRANSFER ALIGNMENT (TA) SCHEME FOR SHIP LAUNCHED MISSILE
GPS/DGPS     LOG

S : System
Slave Accn

Master Vel,
Master INS                Lat, Long                   F update

Meas & Noise
^           States
Ship 100/s                                 Process Noise
-
s
SS                                                           +                    AKF
S curve  0.15 m/s 2                                                                            Fdbks        (Adaptive
Kalman
Filter)
Missile                Slave INS
q, r 1.2 0/s           (SDINS)                 Slave Vel,
Lat, Long

Feed back
Conversion to
Alignment                                             Controller
corrections        Error quaternion

PII-06 Results (Psi Error, Del Vn
PII-06 Launch T.A Launch TA Results Plots)
50
 Demonstrated 7-state AKF based TA           for SSMs
launched from Moving Platform.
Psi Error(arc min)

0

-50           Blue = Optically measured
blue = Optically measured psi error syi error                  Fdbk gains are selected using Linear Quadratic
Gaussian Regulator and offline Matrix Riccati
-100       red = AKF estimated psi error error
red = AKF estimated syi
equation solution.
-150                                                                        Integrated the above with EKF based GPS-INS data
fusion for Dhanush extended range missions.
-200                                                                        Validated through Van, Aircraft, Ship & Flight trials.
0     100       200       300         400      500    600     700
Time (sec)
GPS-INS DATA FUSION SCHEME FOR EXT. RANGE PRITHVI MISSION

LC

Nominal
PURE         Quat, Pos, Vel                  Trajectory

Rates                           Pos, Vel                  GUIDANCE
IMU
Accln                                                      MODULE

Corrections
FUSED                          Quaternion
Guidance

Pos, Vel, DCM

Defln         Defln
GPS Data              KF
Position corrections                                     GPS
30                                                                                                           MODULE
X corr
Y corr
20                                                              Z corr

10

CONTROL ACTUATION
Position corr (m)

0
SYSTEM
-10

-20
 Demonstrated 17-state Extended Kalman Filter (EKF)
-30
based GPS-INS Data Fusion in OBC for extended
-40
range Prithvi missions.
0   500   1000   1500   2000 2500          3000   3500    4000     4500
time (secs)
EMBEDDED ONBOARD PROCESSORS                                  GUIDANCE SYSTEM ENGINEERING

1 Km
PROCESSOR CLASS

CEP of Prithvi
With Strapdown Inertial Implicit Guidance (CEP < 1 Km)
System On Chip

•Pentium Class
•Power PC
•COTS
•Multi Protocol
Connectivity                                   40 m            With TA & GPS-INS data fusion (CEP < 40 m)

80486                                                                   Inertial, Radar & Seeker fused Guidance (CEP < 10 m)
8086
1m                            Inertial & Seeker Guided PGMs (CEP < 1 m)

Year
1985     2000     2005         2010                                1990 2003 2006           2010
Year
NUMBER OF FLIGHT TRIALS OF PRITHVI

64       Prithvi
(Planned)

No. of
Flight
Trials

38
(Planned)

12
(Actual)

0
1984 1988    1996     2004
Year

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

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