Transcript Lecture 16
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 16: Statistical Least Squares and Minimum Norm Estimator University of Colorado Boulder Exam 1 – Friday, October 9 ◦ More details on Friday Homework 5 Due Friday Lecture Quiz Due Friday by 5pm University of Colorado Boulder 2 Statistical Least Squares Solution for Nonlinear System University of Colorado Boulder 3 p. 196-197 of textbook (includes corrections) University of Colorado Boulder 4 University of Colorado Boulder 5 The batch filter depends on the assumptions of linearity ◦ Violations of this assumption may lead to filter divergence ◦ If the reference trajectory is near the truth, this holds just fine The batch processor must be iterated 2-3 times to get the best estimate ◦ The iteration reduces the linearization error in the approximation Continue the process until we “converge” ◦ Definition of convergence is an element of filter design University of Colorado Boulder 6 University of Colorado Boulder 7 If we know the observation error, why “fit to the noise”? University of Colorado Boulder 8 No improvement in observation RMS Magnitude of the state deviation vector Maximum number of iterations University of Colorado Boulder 9 Instantaneous observation data is taken from three Earth-fixed tracking stations ◦ Why is instantaneous important in this context? Filter estimates the spacecraft’s position/velocity and the ground station locations University of Colorado Boulder 10 University of Colorado Boulder 11 Range Residuals (m) Range Rate Residuals (m/s) 20 0 0 Pass 1 2000 -2000 -20 0 100 200 300 400 0 100 observation number 200 300 400 observation number RMS Values (Range σ=0.01 m, Range-Rate σ = 0.001 m/s) -3 Pass 2 x 10 1 5 0 0 -1 Pass 1 -5 0 100 200 300 400 0 100 observation number 200 300 400 observation number -3 Pass 3 x 10 0.05 2 0 0 -0.05 Pass 2 Pass 3 Range (m) 732.748 0.319 0.010 Range Rate (m/s) 2.9002 0.0012 0.0010 -2 0 100 200 300 400 observation number University of Colorado Boulder 0 100 200 300 400 observation number 12 University of Colorado Boulder 13 Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009. FLIR – Forward-looking infrared (FLIR) imaging sensor University of Colorado Boulder 14 University of Colorado Boulder 15 University of Colorado Boulder 16 Inverting a potentially poorly scaled matrix Solutions: Numeric Issues ◦ Matrix Decomposition (e.g., Singular Value Decomposition) ◦ Orthogonal Transformations ◦ Square-root free Algorithms ◦ Resulting covariance matrix not symmetric ◦ Becomes non-positive definite (bad!) University of Colorado Boulder 17 Minimum Norm Estimator University of Colorado Boulder 18 For the simple least squares filter estimate to exist, m≥n and H must be full rank. Consider a case with m<n and rank(H)=m<n ◦ There are more unknowns than linearly independent observations! University of Colorado Boulder 19 Option 1: Reduce the number of estimated variables to yield m=n ◦ Yields multiple solutions Option 2: Use the minimum norm criterion to uniquely determine an estimate state ◦ Constrained optimization problem University of Colorado Boulder 20 University of Colorado Boulder 21 University of Colorado Boulder 22 University of Colorado Boulder 23 Propagation of Estimate and Covariance Matrix University of Colorado Boulder 24 Well, we’ve kind of covered this one before: Note: Yesterdays estimate can become today’s a priori… ◦ Not used much for the batch, but will be used for sequential processing University of Colorado Boulder 25 How do we map our uncertainty forward in time? X* University of Colorado Boulder 26 University of Colorado Boulder 27 Monte Carlo University of Colorado Boulder Linear Map Unscente Transform 28