1. 1 MONALISA Compact Straightness Monitor Simulation and Calibration Week 2 Report By Patrick Gloster

2. 2 A recap of the simulation Generate a random ‘true’ position Generate measurements to this position from observer coordinates which will have an error Calculate an approximation to the true position using these measurements and compare with the true position

3. 3 Refinements and additions Most of the last week has been spent refining and adapting the program We can now input a separate error to be associated with each measurement, choose our observer coordinates and even jump to a particular simulation

4. 4 User is prompted to input information by the program Just enter simulation(N) in Matlab, where N is the number of samples you want, and you will be led through the rest of the program.

5. 5 Results You can also examine the results in more detail Instead of just jumping to the final value, you can plot each successive calculated value of the position. If this is taking too long you can jump to the final answer instead.

6. 6 Successive iterations

7. 7 Ironing out some problems Last week there was an occasional complex number in the result This was because sometimes in calculating the approximate position I was solving a problem with no real solutions: This was solved by taking our approximation as the value exactly between the two centres of the circles When adding an error to the measurements this sometimes reduces both to the extent that the circles don’t overlap

8. 8 The distribution of the results Having written the simulation we wanted to be able to examine the spread of the results This was done by plotting a histogram showing the spread of the difference between true and calculated positions

9. 9 The new problem Now we are working on the reverse; we have n locations with measurements taken from the observer coordinates We want to work out where the observer coordinates are We know that the observer coordinates must be fixed with respect to each other

10. 10 The plan Where before we had we now also have So we use the variational principle with two Lagrange multipliers (k1 and k2) and find a new (more complicated) equation for