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
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