Figure A.2: Basic principle of the classical autocollimator, according to the LDS-2000 manual.
on the back of the focal plane of the collimated lens. If the value of the focal length of the collimated lens is F, then the lateral displacement will be:
ΔY = F · tan(2 ·Δθ) (A.1) whereΔθ is the angular displacement of the mirror.
In the case of an electronic autocollimator this operation is performed automatically [59]. In fact, the measuring eyepiece (E in Figure A.2) is replaced by a position-sensitive device (PSD). Moreover, the source reticle is a laser diode. The PSD is an electronic device that delivers analog signals proportional to the displacement of the spot light position. These two analog signals are then converted into digital signals. Those values are theΔY of equation A.1. At this point it is possible to calculate the angular displacements.
In comparison with the interferometer, the autocollimator does not suffer as much from thermal drift. Its great disadvantage is the limited measuring course.
A.3 Compensation of Systematic Errors
The use of a 6 DOF measuring system is undoubtedly a more practical and effective solution than using a single device mounted in different configurations during different measuring sessions. The advantage of collecting the data as the same time is evident.
Moreover, by using a 6 DOF measuring system it is also possible to actively correct the interferometers systematic errors, namely the cosine error and the abbe error.
A.3.1 Cosine Error Compensation
The cosine error is caused by an angular misalignment between the measuring axis and the object to be measured (Figure A.3 on the next page). By using the reading of the autocolli- mators it is possible to compensate for it for each axis [84]:
Δl = l ·(1 − cos(α)) (A.2) Where l is the measure acquired with the interferometer,α is the misalignment value and Δl is the cosine error.
Measure Axis α
Δl l - Δl
Interferometer
Beam from Sensor Head l
Figure A.3: The cosine error.
By using the reading of the two autocollimators and applying A.2 we are able to compen- sate the cosine error in the measure of the three interferometers:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
xcor r ected = xsi os·cos
θy
cos (θz)
ycor r ected = ysi os·cos (θx) cos (θz)
zcor r ected = zsi os·cos (θx) cos
θy
(A.3)
Where xsi os, ysi os and zsi os are the distances measured with the interferometers andθx,
θyandθz are the angles measured with the autocollimators.
A.3.2 Abbe Error Compensation
The remaining systematic error coming from an angular error together with an offset be- tween the axis of measure and the object axis is called Abbe error.
A way to minimize this effect consists in mounting the interferometer in order to point the beam as near as possible to the mirror center (this method has been used in [27]). This method is not always reliable, because as the robot moves, the laser does not point any more to the mirror center, adding the abbe error.
Considering Figure A.4, if a reading of a andα is available, it is possible to compensate this effect in real-time using the following equation:
Δl = a · tan (α) (A.4) Where a is the distance between the laser and the mirror’s center of rotation andα is the tilt of the measuring device.
Again, by using Equation A.4, the reading of the autocollimators and the reading of the interferometers, we are able to compensate for this effect:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ xcor r ected = xsi os− ayt an θy − azt an (θz) ycor r ected = ysi os− bxt an (θx)− bzt an (θz) zcor r ected = zsi os− cxt an (θx)− cyt an θy (A.5)
Where xsi os, ysi os and zsi os are the distances measured with the interferometers,θx,θy
andθzare the angles measured with the autocollimators and the a, b, and c are the relative
displacements of the laser beam from the center of rotation along each axis. Each of those last coefficients is calculated in the following way, respectively for each axis:
A.3. COMPENSATION OF SYSTEMATIC ERRORS 135
α
Δl
Beam from Sensor Head
a Interferometer
Figure A.4: The abbe error.
ay= ay0+ ysi os (A.6)
The ay0 values are measured at the beginning of the measuring session, by taking a pic-
ture of the cube and using an image analysis software to determine with a precision of 1 mm the relative spot position in zero.
Finally, the following system of equations represents the correction of the cosine error together with the abbe error:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
xcor r ected = xsi os·cos
θy cos (θz)− ayt an θy − azt an (θz)
ycor r ected = ysi os·cos (θx) cos (θz)− bxt an (θx)− bzt an (θz)
zcor r ected = zsi os·cos (θx) cos
θy − cxt an (θx)− cyt an θy (A.7)
Appendix B
The Calibration System and the Software
The whole system is depicted in Figure B.1. Two computers are used to run the whole cali- bration process. A real-time computer (the Control Computer) is used to control the robot(s), while a second computer is used to collect the measurements and handle the thermal stabi- lization (the Measuring Computer). The two PCs are connected together through an Ethernet network and the measurement computer commands the real-time computer. The interfer- ometers, autocollimators, the A/D converter for the thermal sensors are all connected to the measuring computer. The measuring computer sends the command to move to the next po- sition to the control computer. Once the measures are stable those are recorded and a new position is sent to the control computer.
For the data processing we use a third personal computer on which Matlab is installed. The software to control the robot is written in Visual C++ and the software to perform the data acquisition is written with LabView 8. In the next sections we will show some details about the LabView software used for the data-acquisition software and for indentation.
B.1 Data-acquisition Software
We wrote a LabView 8 software to handle all the data acquisition process. In this section we will show the entire modules used to acquire the data.
In Figure B.2 we show the acquisition of the 13 thermal readings, plus the target temper- ature for the three interferometers bases (Consigne).
The thermal measures are used to stabilize the three interferometers bases. In Figure B.3 we see the graphs of the current given to the Peltiers, the PID parameters and the current limit for the Peltiers amplifiers.
Once the interferometers bases are stabilized, the measures are started. It is possible to monitor the quality of the interferometers reading in the part of software displayed in Figure B.4. The top graph displays the measure of the interferometer. The central graph displays the corresponding standard deviation. In the lower graph we display the reading of the interferometers in the zero. The evolution of the zero is useful to understand if there are major problems in the measures (as measure “jumps” or interferometer lost of signal).
Ethernet Network
Control Computer
Robot
Measuring Computer A/D Converter
Interferometers
Autocollimators
Amplifier
Anti Vibration Table Peltier Cell
Thermal Sensor
1 axis 2 axes
Figure B.1: The complete system.
In Figure B.5 we show the reading of the autocollimators and in Figure B.6 the software’s part used to choose the trajectory file and to launch the measures.