SINGULAR VALUE DECOMPOSITION AND LEAST SQUARES ORBIT DETERMINATION

Transcription

1 SINGULAR VALUE DECOMPOSITION AND LEAST SQUARES ORBIT DETERMINATION Zakhary N. Khutorovsky & Vladimir Boikov Vympel Corp. Kyle T. Alfriend Texas A&M University

2 Outline Background Batch Least Squares Nonlinear effects Singular Value Decomposition Modified least squares with nonlinear effects Algorithm Results Conclusions

3 Background Batch least squares (BLS) is the standard orbit determination method in the US and Russia for catalog maintenance. BLS performance is generally satisfactory for catalog maintenance because it is a relatively linear problem. BLS has convergence problems when the nonlinear effects become significant, which are: Poor initial or a priori state. Long time between tracks, sparse measurements. Poor or abnormal measurements Difficulty of obtaining a good Hessian approximation in regions that are remote from the solution. Problem occurs mostly with IOD and UCT correlation Propose using SVD to resolve these problems by choosing the dimension of the minimization subspace at each step.

4 Least Squares Least squares is concerned with the minimization of Fx 1 2 m k i i1 j1 ŝ ij s ij ij x 2 gradf x gx 0 Define F ŝ ij ŝij ij, s ij Standard BLS reduces to x s ij x ij s s 11,..., s 1k1, s 21,..., s 2k2,..., s m1,..., s mkm T ˆ ˆ T x s s x s s x b b, b sˆ s x T T T T b b s s g x F x b,, A x x x x x x A T Ax GN A T b T T

5 Revisit Least Squares Development Expand b in a Taylor series about a reference trajectory. b bx b x 1 2 b x xx 2 xt x.. x 2 xx b x b 2 b x A 2 b x x xx x 2 x 2 xx xx The least squares process reduces to A T A Bx A T b M B 2 M b x i b 2 i 2 s x i b x 2 i x x i1 i1 We propose an optimal strategy, which will initially move along the directions for which B is not significant, until it reaches the area of small residuals, where B can then be disregarded.

6 Singular Value Decomposition A USV T U T U V T V I S diag s 1, s 2,..., s n, s 1 s 2... s n T T g U b, y V x Sy g y g / s j j j Consider the probe vector y x k y, y,..., y,0,...,0 1 2 k k j Vy j1 Square of the normalized residual norm is m 2 k 2 k b Ax g j jk1 k k y v j T Need to find an index k such that the norm of the probe vector and the norm of the residual for this probe solution are small enough.

7 Proposed Procedure 1. Develop the matrix of trial vectors x k k j1 V j g j w j 2. For each trial vector compute the expected decrease of the least squares error function n k 2 g j 2 jk1 3. Check acceptability of each of the trial vectors. If not satisfied by all vectors normalize x j k by d min k x j c j If not satisfied, d j f c j k x,d min min j j f d j 4. Check relative decrease of the SVD method as we go to the next trial vector. If the inequality is satisfied then the trial vector x (k) is taken as the next iteration. 2 k1 2 k1 k 2 C 5. If the inequality in Step 4 is not satisfied then the previous trial vector is used. 6. After computing the least squares function with x (k) determine if the SVD method is converging sufficiently. Determine if F k1 F k If this inequality is satisfied go to the next step. If it is not satisfied then the modified least squares method is used because the Hessian is degenerate and the residuals need to be considered. F k1 C F

8 Residual Comparison

9 Residual Comparison

10 Residual Comparison

11 Results Geosynchronous satellite Angles only obs Short arc, 25 hour max ICs obtained using Laplace s method Theory is the Russian semi-analytic theory Table 1 O bservation Data Example # of obs Ob Time span (hr:min) 25:22 5:06 25:39 4:21 Table 2 Initial Conditions For the Orbit Determination Example Period Inclination Arg. of Per. Ascend Node Eccentricity (min) (deg) (deg) (deg)

12 Example 1 Fast Convergence Iter SS

13 Example 2 Intermediate Convergence Iter SS

14 Example 3 Slow Convergence Iter SS

15 Example 4 Slow Convergence Iter SS

16 Conclusions New method of orbit determination using singular value decomposition with least squares developed. A strategy using SVD and modified least squares that incorporates some nonlinear effects presented. Modified method used when Hessian is degenerate. SVD method applied to short arc, angles only 24-hour satellites shows significant improvement over standard least squares. As we look to the future of a catalog of more than 100,000 objects the US should look at Russian approaches to determine if they would help improve our catalog development and maintenance.