16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde E4= PE3 ITR T grr」LpoR0r0 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 f(r,)= where o,, Or and p can be identified from E4. Recall that we are co ng unbiased errors Contour of constant probability density function is r=xcos 8- yin 6 ne Get (6)x2+()xy+()y2=c Coefficient of x, y equals zero for principal axes tan 20=LpGROT Use a 4 quadrant tan.function. ge 5 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 4 3 2 2 2 2 2 2 T R RT R R T RT T R T T E PE P R RT TR T σ µ σ ρσ σ µ σ ρσ σ 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 ( ) ( ) 2 2 2 2 2 1 2 1 , 2 1 R RT T r rt t R T f rt e ρ σ σσ σ ρ πσ σ ρ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ − + ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ − − ⎣ ⎦ = − where , σ R T σ and ρ can be identified from E4 . Recall that we are considering unbiased errors. Contour of constant probability density function is 2 2 2 2 R RT T r rt t ρ c σ σσ σ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ − += ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ cos sin sin cos rx y tx y θ θ θ θ = − = + Get: { 2 22 0 () () () θθ θ x xy y c = ++= Coefficient of x, y equals zero for principal axes. 22 22 2 2 tan 2 R T RT RT RT ρσσ µ θ σ σ σσ = = − − Use a 4 quadrant tan-1 function