1. Texas A&M University, Department of Aerospace Engineering AN EMBEDDED FUNCTION TOOL FOR MODELING AND SIMULATING ESTIMATION PROBLEMS IN AEROSPACE ENGINEERING D. Todd Griffith, James D. Turner, and John L. Junkins Texas A&M University Department of Aerospace Engineering College Station, TX 77840

2. Texas A&M University, Department of Aerospace Engineering Presentation Outline Introduction Overview of automatic differentiation by OCEA Development of higher-order GLSDC algorithms Examples

3. Texas A&M University, Department of Aerospace Engineering Introduction (1) Estimation of dynamical systems broadly addresses the following challenge: “Given your best model of a dynamical system, including sensors, and given the erroneous measured output of sensors, now what?!” (*) (*) Junkins First Sermon on Estimation of Dynamical Systems Dynamics model: Measurement model: Typically, first order differential correction method utilized:

4. Texas A&M University, Department of Aerospace Engineering Introduction (2) In this work, we develop higher-order (first through fourth-order) generalizations of the Gaussian Least Squares Differential Correction (GLSDC) algorithm. These generalizations are implemented by using a Reversion of Series Solution:, and so on……..

5. Texas A&M University, Department of Aerospace Engineering Presentation Outline Introduction Overview of automatic differentiation by OCEA Development of higher-order GLSDC algorithms Examples

6. Texas A&M University, Department of Aerospace Engineering Overview of automatic differentiation by OCEA (1) OCEA (Object Oriented Coordinate Embedding Method) –Extension for FORTRAN90 (F90) written by James D. Turner –Automatic Differentiation (AD) equation manipulation package in which new data types are created in order to define independent and dependent variables (functions) OCEA description –Automatic differentiation enabled by coding rules for differentiation - Chain rule of calculus –Can compute first through fourth-order partial derivatives of scalar, vector, matrix, and higher order tensors –Standard mathematical library functions (e.g. sin, cos, exp) are overloaded –Scalars are replaced by differential n-tuple, for second-order:

7. Texas A&M University, Department of Aerospace Engineering Overview of automatic differentiation by OCEA (2) OCEA description (cont’d) –Intrinsic operators such as ( +, -, *, /, = ) are also overloaded to enable, for example, addition and multiplication of OCEA variables: –Partial derivative computation is hidden to the user - takes place in the background without user intervention. –Access to partial derivatives (e.g. Jacobian, Hessian, etc.) made simple by overloading of the “ = “ sign.

8. Texas A&M University, Department of Aerospace Engineering Overview of automatic differentiation by OCEA (3) The need for computation of partial derivatives is found in numerous applications Previous applications include –Root solving and Optimization –Automatic generation and integration of equations of motion (AAS 04-242) In this paper, we consider higher-order GLSDC algorithms………………….

9. Texas A&M University, Department of Aerospace Engineering Presentation Outline Introduction Overview of automatic differentiation by OCEA Development of higher-order GLSDC algorithms Examples

10. Texas A&M University, Department of Aerospace Engineering Higher-order GLSDC Algorithms (1) Given the generic nonlinear measurement model... First, we look at computing the required sensitivities for the Series Reversion Solution. … the task is to compute partial derivatives of this model w.r.t. the unknown parameters ---> Sensitivities 1st order: 2nd order: Need to compute first and higher-order state transition matrices……..

11. Texas A&M University, Department of Aerospace Engineering Higher-order GLSDC Algorithms (2) Notation for higher-order state transition matrices 1st order: 2nd order: 3rd order: 4th order:

12. Texas A&M University, Department of Aerospace Engineering Higher-order GLSDC Algorithms (3) Higher-order state transition matrix differential equations Repeated differentiation of the integral form of the equations of motion, followed by time differentiation, leads to first and higher-order state transition matrix differential equation: Fourth-order equations too long to show here, but are computed by continuing from above………………….

13. Texas A&M University, Department of Aerospace Engineering Higher-order GLSDC Algorithms (4) Summary The automatic differentiation capability allows the analyst to produce a general estimation tool because the dynamical model and the measurement model can be simply specified and differentiated. The work involved in computing and validating partial derivatives by hand is not required. Dynamics model partials Measurement model partials

14. Texas A&M University, Department of Aerospace Engineering Presentation Outline Introduction and previous work Overview of automatic differentiation by OCEA Development of higher-order GLSDC algorithms Examples

15. Texas A&M University, Department of Aerospace Engineering Ballistic Projectile Identification Problem (1) Here, we estimate model parameters for pitch,  and yaw,  angle models for an aerodynamically and inertially symmetric projectile. Models for the angles given by: 14 unknown parameters include:

16. Texas A&M University, Department of Aerospace Engineering Ballistic Projectile Identification Problem (2) SUBROUTINE NONLINEAR_FX( T, EB_VAR, EB_FCTN ) USE EB_HANDLING IMPLICIT NONE ! ARGUMENT LIST VARIABLES REAL(DP)::T TYPE(EB), DIMENSION(NV), INTENT(IN ):: EB_VAR TYPE(EB), DIMENSION(NF), INTENT(INOUT):: EB_FCTN ! DEFINE LOCAL + EMBEDDED VARIABLES TYPE(EB):: K1, K2, K3, K4, K5, LAM1, LAM2, LAM3, OMEG1, OMEG2, OMEG3, DEL1, DEL2, DEL3 ! ASSIGN LOCAL VARIABLES K1=EB_VAR(1);K2=EB_VAR(2);K3=EB_VAR(3);K4=EB_VAR(4);K5=EB_VAR(5) LAM1=EB_VAR(6);LAM2=EB_VAR(7);LAM3=EB_VAR(8) OMEG1=EB_VAR(9);OMEG2=EB_VAR(10);OMEG3=EB_VAR(11) DEL1=EB_VAR(12);DEL2=EB_VAR(13);DEL3=EB_VAR(14) ! COMPUTE NONLINEAR FUNCTION USING EMBEDDED ALGEBRA EB_FCTN(1) = K1*EXP(LAM1*T)*COS(OMEG1*T+DEL1) + K2*EXP(LAM2*T)*& COS(OMEG2*T+DEL2) + K3*EXP(LAM3*T)*COS(OMEG3*T+DEL3) + K4 EB_FCTN(2) = K1*EXP(LAM1*T)*SIN(OMEG1*T+DEL1) + K2*EXP(LAM2*T)*& SIN(OMEG2*T+DEL2) + K3*EXP(LAM3*T)*SIN(OMEG3*T+DEL3) + K5 END SUBROUTINE NONLINEAR_FX

17. Texas A&M University, Department of Aerospace Engineering Ballistic Projectile Identification Problem (3)

18. Texas A&M University, Department of Aerospace Engineering Planar orbit example (1) Drag model: EQM: Measurements: Truth: Observed for 20 seconds at 1 second intervals

19. Texas A&M University, Department of Aerospace Engineering Planar orbit example (2) Case I: Zero drag

20. Texas A&M University, Department of Aerospace Engineering Planar orbit example (3) Case II: With drag

21. Texas A&M University, Department of Aerospace Engineering Another look at the Reversion of Series Solution First-order Second-order And, so on…..

22. Texas A&M University, Department of Aerospace Engineering Conclusions Introduced an automatic differentiation (AD) tool OCEA Presented higher-order Gaussian Least Squares Differentiatal Correction (GLSDC) algorithms Simulated two examples using first and second-order algorithms Future work includes: –Simulation of third and fourth-order algorithms –Extensions in propagation of uncertainty