Gps Algorithm
Lab Report for Computer Assignment #2 GPS Algorithm
Introduction In order to guide each object correctly on the earth, the GPS satellites need to locate them very accurately. The goal of the assignment is to use some optimization methods to locate an object with given pseudo-range data. The signals which are sent from satellites to the target are not accurate because the systematic errors caused by inaccurate receiver clock and some other random noise. The un-accurate range is known as pseudo-range comparing with true range. Since the pseudo-range data, which is calculated, is given, we need to first build a pseudo-range equation, which includes true range, clock bias error and noise error. The equation is like: yl=RlS+b*e+vl ; e = [1 1 1 1]T
The random noise term is i.i.d with p.d.f. N(0, σ2). The clock bias error b is caused by an inaccurate clock in the GPS receiver. Knowing the satellite locations, Sl, l = 1, … , 4, we have m pseudo-range, yl, to every satellite, so we can estimate the receiver location, S, and the clock bias b, using Gradient Descent and Gauss-Newton methods for solving this non-linear least squares problems.
Procedure
1. Linearization
Linearize the pseudo-range equation, yl=RlS+b*e+vl ;
Let X=Sb; X=Sb; ylXl=hXl+vl ;
The pseudo-range equation is a non-linear equation since Rl (S) is nonlinear. To linearize a non-linear equation, Taylor series expansion is used, yX+∆X=yX+∂∂XyX∆X+h.o.t; ∂∂XyX=∂∂XhX=rlT1=(S-Sl)TRl1=HX; yX+∆X- yX=HX∆X+h.o.t.; ∆y=HX∆X+h.o.t.;
Since rlT= (S-Sl)TRl, if the true range increases, rlTbecomes smaller. Then, ∆S=(S-Sl), gets smaller. We can conclude that the larger the true range is, the better linearized approximation will be.
2. Algorithm Development
In order to find the XMLE where the object interested is most likely to be located in space, we need to minimize the Loss function about X. minXlX=12y-h(X)2 To find the minimum Loss function lX, we will find
Introduction In order to guide each object correctly on the earth, the GPS satellites need to locate them very accurately. The goal of the assignment is to use some optimization methods to locate an object with given pseudo-range data. The signals which are sent from satellites to the target are not accurate because the systematic errors caused by inaccurate receiver clock and some other random noise. The un-accurate range is known as pseudo-range comparing with true range. Since the pseudo-range data, which is calculated, is given, we need to first build a pseudo-range equation, which includes true range, clock bias error and noise error. The equation is like: yl=RlS+b*e+vl ; e = [1 1 1 1]T
The random noise term is i.i.d with p.d.f. N(0, σ2). The clock bias error b is caused by an inaccurate clock in the GPS receiver. Knowing the satellite locations, Sl, l = 1, … , 4, we have m pseudo-range, yl, to every satellite, so we can estimate the receiver location, S, and the clock bias b, using Gradient Descent and Gauss-Newton methods for solving this non-linear least squares problems.
Procedure
1. Linearization
Linearize the pseudo-range equation, yl=RlS+b*e+vl ;
Let X=Sb; X=Sb; ylXl=hXl+vl ;
The pseudo-range equation is a non-linear equation since Rl (S) is nonlinear. To linearize a non-linear equation, Taylor series expansion is used, yX+∆X=yX+∂∂XyX∆X+h.o.t; ∂∂XyX=∂∂XhX=rlT1=(S-Sl)TRl1=HX; yX+∆X- yX=HX∆X+h.o.t.; ∆y=HX∆X+h.o.t.;
Since rlT= (S-Sl)TRl, if the true range increases, rlTbecomes smaller. Then, ∆S=(S-Sl), gets smaller. We can conclude that the larger the true range is, the better linearized approximation will be.
2. Algorithm Development
In order to find the XMLE where the object interested is most likely to be located in space, we need to minimize the Loss function about X. minXlX=12y-h(X)2 To find the minimum Loss function lX, we will find