Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Satellite Tracking Example of SNC and DMC ASEN 5070 LECTURE 32 12/01/08
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Colorado Center for Astrodynamics Research The University of Colorado 7 Again the off diagonal terms are small and the diagonal terms are Aproximately=1. We assume that the velocity is constant over the interval so
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Colorado Center for Astrodynamics Research The University of Colorado The J 3 Example
Colorado Center for Astrodynamics Research The University of Colorado Tracking Configuration
Copyright The J 3 Problem Gravitational Potential Function for the J 3 Model where
Copyright J 3 Equations of Motion Taking the partials of the Gravitational Potential to get x,y,z accelerations
Copyright Conventional Kalman Filter (CKF)
Copyright Conventional Kalman Filter (CKF) Position RSS = m Velocity RSS = x m/s RSS errors are based on X T – (X * + ) at each stage RSS=4.18 m/s
Copyright Extended Kalman Filter (EKF) No State Noise Added EKF implemented after 154 minutes Note that pre-fit residuals for the CKF are much larger that the EKF but post-fit residuals are identical indicating that the reference trajectory for the CKF is in the linear range
Copyright Extended Kalman Filter (EKF) Position RSS = m Velocity RSS = x m/s RSS position and velocity errors for CKF and EKF are nearly identical
Copyright Band Diagonal State Noise Matrix See web page handouts or previous slides for derivation
Copyright Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value: 2 =
Copyright Conventional Kalman Filter with SNC Position RSS = 8.43 m Velocity RSS = 1.62 x m/s Band Diagonal State Noise Matrix results for optimum value: 2 =
Copyright Conventional Kalman Filter with SNC Band Diagonal State Noise Matrix results for optimum value: 2 = This is what happens if an input error inject a huge amount of process noise i.e. 2 = instead of Kalman gain is so large that data is fit almost perfectly – well below noise levels of 1cm and 1mm/s
Copyright Conventional Kalman Filter with input error Position RSS = m Velocity RSS = 1.22 x m/s Band Diagonal State Noise Matrix results for optimum value: 2 = Small residuals do not a good orbit Make – RSS errors are larger than Previous results 2 standard deviations of estimation error covariance are red lines
Copyright Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix
Copyright Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix Looked for optimum values of 2 by analyzing residuals and errors Optimum value to minimize position errors: 2 =
Copyright Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix results for optimum value: 2 =
Copyright Extended Kalman Filter with State Noise (SNC) Band Diagonal State Noise Matrix results for optimum value: 2 = Position RSS = m Velocity RSS = x m/s Position RSS = m The CKF and EKF results are Nearly identical indicating that Initial conditions are in the linear range
Copyright Extended Kalman Filter with Fading Adds a fading term to the time update – this downweights earlier data by Keeping the Kalman gain elevated where t = time between measurements = 20 seconds = age-weighting time constant >1
Copyright Extended Kalman Filter with Fading Looked for optimum value of s to minimize the residuals and errors Optimum value of s ~ This corresponds to a of 10 minutes
Copyright Extended Kalman Filter with Fading Optimum values of s to minimize position and velocity errors : s =
Copyright Extended Kalman Filter with Fading Optimum values of s to minimize position and velocity errors : s = Position RSS = m Velocity RSS = x m/s Position RSS=8.688 m
Copyright Kalman Filter Results Comparison Conclusions: The CKF and EKF produce comparable results indicating that the reference orbit for the CKF is in the linear range for this example. Fading produces comparable results to SNC
Copyright Dynamic Model Compensation (DMC) DMC accounts for unmodeled or inaccurately modeled accelerations acting on the spacecraft - J 3 in this problem. The state vector was augmented to the following: where x, y, and z are the accelerations A Gauss-Markov process is used to account for these accelerations: where u(t) is white Gaussian noise with and Where is a time constant
Copyright Dynamic Model Compensation and were optimized to give the lowest position and velocity errors: 2 =
Copyright Dynamic Model Compensation Optimized Results Position RSS = m Velocity RSS = x m/s
Copyright Dynamic Model Compensation Optimized Results Errors in acceleration estimates in x,y, And z directions Note: the DMC did a poor job of recovering accelerations. More work is needed on optimizing and We should do a better job of recovering accelerations and the state while reducing tracking residuals. Actual vs estimated accelerations
Copyright Filter Comparisons All filters achieved comparable results; however we should do better with the DMC
Copyright Added deviations to the initial conditions so that they are outside linear range to show Convergence for EKF with SNC Divergence for CKF with SNC Original Initial Conditions [ , , , , , ] m and m/s Perturbed Initial Conditions Deviation [8, 5, 5, 8, 5, 5] m and m/s [ , , , , , ] m and m/s
Copyright Conventional Kalman Filter Band Diagonal State Noise Matrix results for optimum value: 2 = Deviated Initial Conditions
Copyright Conventional Kalman Filter Position RSS = m Velocity RSS = 2.88 m/s Band Diagonal State Noise Matrix results for optimum value: 2 = Deviated Initial Conditions
Copyright Extended Kalman Filter Band Diagonal State Noise Matrix results for optimum value: 2 = Deviated Initial Conditions
Copyright Extended Kalman Filter Position RSS = m Velocity RSS = 2.53 x m/s Band Diagonal State Noise Matrix results for optimum value: 2 = Deviated Initial Conditions
Copyright Filter Comparisons While the CKF does not diverge, its solution is significantly In error
Copyright Added deviations to the initial conditions to show Convergence for EKF with SNC Divergence for the Batch Processor Original Initial Conditions [ , , , , , ] m, m/s Perturbed Initial Conditions Deviation [1000, 1000, 1000, 500, 500, 500] m,m/s [ , , , , , ] m, m/s
Copyright Batch Processor Pass 1 Range RMS = km Range rate RMS = km/s Pass 3 Range RMS = km Range rate RMS = km/s Note that the batch processor is not converging and successive Iterations show divergence
Copyright Extended Kalman Filter Band Diagonal State Noise Matrix results for various values of 2 Deviated Initial Conditions Trajectory updated after 30 minutes Note: Values of 2 from to were tested. However, the residuals and errors were orders of magnitude higher for value of 2 between to Therefore, those values are not shown on the plots. Note that the optimal value for 2 is the same as for small initial condition errors.
Copyright Extended Kalman Filter Band Diagonal State Noise Matrix results for 2 = Deviated Initial Conditions Trajectory updated after 30 minutes
Copyright Extended Kalman Filter Band Diagonal State Noise Matrix results for 2 = Deviated Initial Conditions Position Errors Position Errors after 250 minutes Position RSS = m
Copyright Extended Kalman Filter Band Diagonal State Noise Matrix results for 2 = Deviated Initial Conditions Velocity Errors Velocity Errors after 250 minutes Velocity RSS = x m/s Note that position and velocity RSS are comparable to those on slide 40. If IC errors are large it may be Advantageous to use the EKF to Obtain ICs for the batch
Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 50
Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 51
Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 52
Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 53
Copyright 2005 Transformation of State and Covariance Matrix to Alternate Frames 54
Copyright 2005 Assume we wish to transform Φ(t i, t j ) from the A frame to the B frame Let Recall that and Transformation of State Transition Matrix to Alternate Frames 55
Copyright 2005 But Hence, where Transformation of State Transition Matrix to Alternate Frames 56