Transcript Lecture 31
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 31: State Noise Compensation University of Colorado Boulder Homework 9 due Friday Lecture Quiz due Friday at 5pm ◦ It will be posted tonight Exam 2 ◦ Returned to students and discussion on Friday 11/13 ◦ Lecture 33 will not be posted to D2L until Monday 11/16 University of Colorado Boulder 2 Last homework due 12/4 (in-class students) I need to turn grades in on Friday December 18 Exam 3 will be take home ◦ ◦ ◦ ◦ Will be posted on Friday Dec. 4 In-class: Due by 5pm, Friday Dec. 11 CAETE students: Due by 11:59pm (Mountain), Sunday Dec. 13 -10 pts for each 24 hours late (includes weekend days!) Final Projects “Freebies” not applicable for exam or project We are happy to accept your exam and project earlier! ◦ In-class & CAETE students: Due by noon, Monday Dec. 14 ◦ -10 pts for each 24 hours late University of Colorado Boulder 3 Process Noise University of Colorado Boulder 4 What happened to u (modeling error) ? ◦ This is true process noise… Can we ignore it? How do we account for it? University of Colorado Boulder 5 University of Colorado Boulder 6 Random process u maps to the state through the matrix B ◦ Consider it a random process for our purposes Usually (for OD), we consider random accelerations: University of Colorado Boulder 7 For the sake of our discussion, assume: University of Colorado Boulder 8 For the sake of our discussion, assume: In other words, Gaussian with zero mean and uncorrelated in time ◦ Dubbed State Noise Compensation (SNC) University of Colorado Boulder 9 This is a non-homogenous differential equation The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: University of Colorado Boulder 10 If we want to instead map between two discrete times: University of Colorado Boulder 11 For the case of a noise process with zero mean: The zero-mean noise process does not change the mapping of the mean state University of Colorado Boulder 12 What about the covariance matrix? The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: University of Colorado Boulder 13 University of Colorado Boulder 14 The previous discussion considered the case where the noise process is continuous, i.e, Things may be simplified if we instead consider a discrete process: University of Colorado Boulder 15 Using the discrete noise process, we instead get (for zero mean process): University of Colorado Boulder 16 This defines, mathematically, how we can select the minimum covariance to prevent saturation ◦ Saturation is mostly a problem when we have dynamic model error ◦ With a stochastic (probabilistic) description of the modeling error, we have our minimum University of Colorado Boulder 17 University of Colorado Boulder 18 The addition of a noise process is better suited for a sequential filter ◦ Would include the process noise transition matrix in the Batch formulation ◦ Changes the mapping of the state (deviation) back to the epoch time, which requires alterations to the H matrix definition ◦ Tapley, Schutz, and Born (p. 229) argue that this is cumbersome and impractical for real-world application Advantage: Kalman, EKF, Potter, and others University of Colorado Boulder 19 Let’s derive the process noise model for a simple case ◦ Noise process defined in the acceleration ◦ Time between measurement small enough to treat velocity as constant What is the process noise transition matrix (PNTM)? University of Colorado Boulder 20 Recall that: University of Colorado Boulder 21 University of Colorado Boulder 22 Recall that we are assuming a small time between observations, i.e., velocity is constant University of Colorado Boulder 23 If velocity is constant, change in position is linear in time: University of Colorado Boulder 24 University of Colorado Boulder 25 Derived under the assumptions that: ◦ Noise process defined only in the acceleration ◦ Time between measurements small enough to treat velocity as constant University of Colorado Boulder 26 University of Colorado Boulder 27 This definition of Q is in the inertial Cartesian frame ◦ Is that always a good idea? ◦ What are some of the things you should consider? University of Colorado Boulder 28 We can define Q in any frame, and rotate the matrix: University of Colorado Boulder 29 The previous derivation assumed Q was known ◦ How do we select Q ? ◦ Ideally, Q describes the magnitude of the uncertainty of the acceleration acting on the satellite ◦ What if we don’t know the magnitude? (after all, we are trying to account for an unknown acceleration) Often determined by trial and error ◦ You will do this in Homework 11 University of Colorado Boulder 30 Can we estimate the Q matrix or other parameters of the process noise? ◦ Gauss Markov Process (Dynamic Model Compensation) ◦ Multiple Model Adaptive Estimation (MMAE) and Heirarchical Mixture of Experts (HME) ◦ Others in the literature University of Colorado Boulder 31