Least squares adjustment/regression - two points known distance apart

[gge]

Hi All,

I'm struggling with finding a solution to an adjustment I'm working on. Thought someone else may have some thoughts?

I have a kinematic time series of X,Y positions for two points (X1,Y1,X2,Y2). I know that the two points were a distance D (e.g., 100 m) apart from each other (the constraint).

I'm attempting to find a solution that finds a best fit to X1,Y1 and to X2,Y2 based on the constraint that the distance between them is D. The path formed by the points is not linear -- I'm starting with a quadratic model.

Application: Picture two GPS receivers rigidly mounted on top of a car. They both have positions and both have noise in their position. I'd like to best fit to both of their positions, but with the constraint on that known distance between them.

In coming up with observation equations, my initial thought was something like:

X1(t) = At^2+Bt+C
Y1(t) = Dt^2+Et+F
X2(t) = Gt^2+Ht+I
Y2(t) = Jt^2+Kt+L

Then:

sqrt((X1-X2)^2+(Y1-Y2)^2) = 100

So this leaves me with 12 parameters. I have ample observations to solve this (observations are X1, Y1, X2, Y2).

Does it appear that I'm on the right track?

I'm not sure what the best method is to proceed from here (i.e. how to set up the LS problem). Any suggestions?

Thanks in advance for any help!

GGE

 

[mfb]

You could change your coordinates to include the constraint there. One example would be to track the central position and the angle of the line between both points.

If that is not useful, I think many fitting programs have some way to consider external constraints in some way.

I would expect that your uncertainties for the positions will be heavily correlated, as the atmosphere is the same for measurements close together in time and space. Differential GPS can get cm-accurary.

 

[Studiot]

You have two variables that could be in error - the x and y coordinates.

To set up the least squares line for this you calculate the perpendicular distance from each point to the assumed line and minimise the sum of the squares of these distances.

 

[hotvette]

The first part of the below link describes how to solve least squares problems with constraints.

http://kom.aau.dk/~borre/kalman/lecture2/p403.pdf

However, if the constraint applies to each time stamp I don't think you have a viable problem. Let's say you want to use 20 observations to solve for 12 variables in a least squares sense. That would work, but you also have 20 constraints. For a constrained least squares problem involving n variables, you can have at most n-1 constraints. Otherwise it is impossible to satisfy all of the constraints and have a meaningful least squares problem to solve.

 

[mfb]

However, if the constraint applies to each time stamp I don't think you have a viable problem. Let's say you want to use 20 observations to solve for 12 variables in a least squares sense. That would work, but you also have 20 constraints. For a constrained least squares problem involving n variables, you can have at most n-1 constraints. Otherwise it is impossible to satisfy all of the constraints and have a meaningful least squares problem to solve.

Good point. The quadratic model for X_i, Y_i will not satisfy the constraints for each step in time apart from some special conditions (like no rotation).
Alternative parameters solve this problem.