1. Mathematical Engineering in Avionics Applications Dr. SK Chaudhuri Sc. ‘H’ Sc. ‘H’ Associate Director, RCI 9 th June 2007, IISc Bangalore

2. FUNCTIONAL BLOCK DIAGRAM OF MAJOR MISSILE SUBSYSTEMS Reference Generation System Knowledge Gathering system Decision Process Action Process Airframe & Propulsion Kinematics  ~ VxVx ~ 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

3. Reference Generation system Navigation Computer Guidance System Autopilot Actuation System Airframe & Propulsion Sensors Kinematics Inertial Sensors RF/IR Sensors MISSILE CONFIGURATION Rates & Acceleration Missile Trajectory Target Trajectory  ~ Engine Bay Control Surface Wing Propellant Tank Guidance & Control System Nose Cone Electronic Bay Radome Warhead

4.  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 Theory  Random & Stochastic Process  State space Methods  Matrix algebra  Iteration Techniques  Interpolation  Optimization Tech.  Quaternion algebra  Matrix algebra  Integration techniques  Solid geometry with Geodetic, Geocentric and 3D representation Navigation Computer Guidance System Autopilot Actuation System Airframe & Propulsion Sensors Kinematics Inertial Sensors 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 SUBSYSTEMS

5. Mathematical Modelling And Simulation ACTUAL SYSTEM MATH MODELCOMPUTER SIMULATION VALIDATION 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.

6.  =tan -1 (W/U)  =tan -1 (V/U) V M =  (U 2 +V 2 +W 2 ) a xs = (T x -D x )/ M a ys = Y   /M+Y   y /M +cr a zs = Z   /M+Z   p /M -cq p = L p p/I xx +L  R  R /I xx +ClQs/I xx q = M   /I yy +M   p /I yy r = N   /I zz +N   y /I zz         a xs a ys a zs p q r INCREMENTAL ANGLES AND VELOCITIES p q r NAV. FUNC U   A/D & INERTIAL PARAMET ERS  ROTATIONAL AND TRANSLATIONAL LOOP JOB ALLOCATION IN REAL TIME MISSILE 6DOF   ENGINE THRUST V W QUAT UPDATE V B = ( [DCM] ) R S T VRVR QmQm VXVX DCM  V B 1-3  1-3 tVmZ

7.  Undue Roll oscillations due to low damping introduced by gimballed engines, thrust frame and hardware actuator compliance TWD EFFECTS IN 6-DOF MODEL

8. MISSILE AUTOPILOT WITH FLEXIBILITY MISSILE AUTOPILOT WITH FLEXIBILITY Unstable Autopilot ResponseModified Stable Autopilot Response

9. 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 MISSILE  update AKF (Adaptive Kalman Filter) Feed back Controller Slave INS (SDINS) S : System Missile q, r  1.2 0 /s - + s ^ GPS/DGPSLOG Master Vel, Lat, Long Conversion to Error quaternion Alignment corrections S curve  0.15 m/s 2 Master INS Slave Accn Slave Vel, Lat, Long Fdbks Meas & Noise Process Noise States Ship 10 0 /s SS red = AKF estimated syi error Blue = Optically measured syi error

10.  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 FUSED NAVIGATION GUIDANCE MODULE TVC CONTROL ACTUATION SYSTEM GPS KF MODULE IMU LC Guidance Commands Defln Pos, Vel Nominal Trajectory Rates Accln Corrections Pos, Vel, DCM PURE NAVIGATION Quat, Pos, Vel GPS Data ADC Defln Quaternion

12. NUMBER OF FLIGHT TRIALS OF PRITHVI 1984 1988 2004 0 64 38 Year No. of Flight Trials Prithvi 12 (Planned) (Actual)  Requirement of number of flight trials is reduced because of HILS. 1996