16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 9 Last time: Linearized error propagation 1s eSe= Integrate the errors at deployment to find the error at the surface. 11 1 T sss TT T Eee See S SE S = = = Or Φ can be integrated from: , where (0) () FI xfx df F dx Φ= Φ Φ = = = & & where F is the linearized system matrix. But this requires the full Φ (same number of equations as finite differencing). n t =time when the nominal trajectory impacts. 1 21 () () () nn rn r et t e et e e =Φ ==Φ where r Φ is the upper 3 rows of () n tΦ . Covariance matrix: 21 T rr EE=Φ Φ 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 () () () () () () () () () () () () () () T TT TTT T eFe Et etet Et etet etet Fe t e t e t e t F FE t E t F = = =+ =+ & & && You can integrate this differential equation to t n from 1 (0)EE= . This requires the full 66× E matrix. 2 () upper left 3 3 partition of ( ) TT rr rv n TT vr vv n ee ee Et ee ee EEt ?? =?? ?? =× For small times around t n , () () ()( ) () ( () ()( ) () ()( ) nnn nnnvn n nnn n et et vt t t et v t e t t t et v t t t =+ ? =+ + ? =+ ? 2 2 1()1 1 ()( )0 1 () 1 TTT vvvnn T v in T vn et e v t t t e tt v =+ ?= ?=? 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 3 2 2 2 position error at impact 1 1 1 "projection matrix" 1 T v n T vn T nv T vn e e ev v v Ie v = =? ?? =? ?? ?? [] 32 123 [] cos ... ... ... ... ... ... ... ... cos 111 ij ij ij eRe R R R θ θ ′ = ?? ?? = ?? = = 1 j = unit vectors along the j th axis of the 2 frame expressed in the coordinates of the 3 frame. 32 32 T eRe ERER ′= ′ = 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 3 3 3 () cos ( ) () () sin ( ) RRn n TT H Hn n et e v tt et e et e v tt γ γ =+ ? = =? ? Impact: 3 3 33 33 3 () sin( ) 0 1 () sin cos () sin cot () Hi H n i n in H n n Ri R H n RH Ti T et e v t t tt e v v et e e v ee et e γ γ γ γ γ =? ?= ?= =+ =+ = The transformation which relates R,H,T errors at the nominal end time to R and T errors when H=0 is: 33 3 4 33 () () cot 10cot 01 0 Ri Ti RH T et e et ee e ePe γ γ ?? = ?? ?? + ?? = ?? ?? ?? ′′=≡ ?? ?? If the s e defined earlier, based on integration of perturbed trajectories, is measured in R,T coordinates, then the sensitivity matrix defined at that point is equivalent to 1s r eSe SPR = =Φ 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 43 2 22 2 T RRT R RT RT T R T T EPEP RRT TR T σ μσρσ μσ ρσσ σ ′= ?? ??? ? == =?? ??? ? ??? ? ?? If all the original error sources are assumed normal, R and T will have a joint binormal distribution since they are derived from the error sources by linear operations only. This joint probability density function is () () 22 2 2 21 2 1 , 21 RRTT rrtt RT frt e ρ σσσσ ρ πσ σ ρ ?? ?? ?????? ???+ ?? ?????? ?? ?????? ? ? ?? = ? where , RT σ σ and ρ can be identified from 4 E . Recall that we are considering unbiased errors. Contour of constant probability density function is 22 2 2 RRTT rrtt cρ σσσσ ?? ?????? ?+= ?? ?????? ?? ?????? cos sin sin cos rx y tx y θ θ θ θ =? =+ Get: { 222 0 () () ()x xy y cθθθ = ++= Coefficient of ,x y equals zero for principal axes. 22 22 tan 2 RT RT RT RT ρσσ μ θ σ σσσ == ?? Use a 4 quadrant tan -1 function. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 8 Once θ is found, can plug into pdf expression, get x σ and y σ . 222 2 222 222 ()(2) 1 () 2 1 () 2 RT RT xRT yRT h h h σσ ρσσ σσσ σσσ =?+ =++ =+? cos sin ixi iyi xc yc σ φ σ φ = = May want to choose c to achieve a certain probability of lying in that contour. In principal coordinates, the probability of a point inside a “cσ ” ellipse is 2 2 1 c Pe ? =? People often choose c to find what is called the circular probable error (CPE). 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 1 () 2 xy σ σσ=+ Choosing P=0.5, c=1.177 0.588( ) xy CPE σ σ=+ This approximation is good to an ellipticity of around 3. Random Processes A random process is an ensemble of functions of time which occur at random. In most instances we have to imagine a non-countable infinity of possible functions in the ensemble. There is also a probability law which determines the chances of selecting the different members of the ensemble. We generally characterize random processes only partially. One important descriptor – the first order distribution. This is the classical description of random processes. We will also give the state space description later. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 8 1 ()x t is a random variable. [ ]( , ) ( ) , where ( ) is the name of a process and is the value taken (,) (,) Fxt Pxt x xt x dF x t fxt dx =≤ =