1. A method of measuring mirror-tilt error in laser trackers
Reporter: Zhang Zili
Academy of Opto-electronics, Chinese Academy of Sciences
October 14th, 2014

2. Content 1 2 3 4 5 6 7 Introduction Laser Tracker Measuring Principle
Laser Tracker Error Analysis 3
Mirror Tilt Error Analysis 4
Mirror Tilt Error Calibration 5
Experimental Results 6
Summary 7

3. Application
Laser tracker is in widespread use for 3D data acquisition in equipment manufacturing，civil engineering architecture, cultural heritage and industrial metrology such as the aerospace, automobile, ship building, mechanical and nuclear industry with the characteristics of large measuring scale and high accuracy.

4. Laser Tracker Measuring Principle
The expression of target position
The laser tracker is a large-scale measurement system which tracks the movement of a target and calculates its position in the form of spatial coordinates. The laser beam originated from the interferometer (IFM) or absolute distance meter (ADM) of the laser tracker is firstly reflected by the reflected mirror fixed on the transit axis, and then reflected by the target and returned to the data processing device in the system to get the spatial information of the target, as is shown in figure 1. The X, Y and Z axes form a mutually orthogonal fixed Cartesian system with its origin at O. The system measures the distance (D) from the system origin (O) to the target as well as the horizontal and vertical angles (H, V).Then the position of the target can be stated in the following formulation.
Laser tracker measuring principle

5. Prototype
In our laboratory, a laser tracking system is developed to detect the spatial information of the object. The main configuration of hardware system can be divided into two parts, the tracking head and the control box. The tracking head includes both the distance and angle measuring devices. The distance measuring devices include the relative distance measuring device(IFM) and the absolute distance measuring device(ADM). The main component of the angle measuring device, the precision rotating stage, is comprised of the precision axis and encoders. Except for the elements in the tracking head, the data processing unit, tracking and control unit and power supply unit are all set in the control box. Apart from the hardware mentioned above, accessories such as retroreflectors, laser tracker stands, retroreflector supports(adapters) are also needed for various measurement tasks.

6. Interferometer Design of IFM unit The construction of IFM unit
The interferometer accomplishes measuring tasks using the traditional Michelson interferometer which can determine relative distances with accuracies on the nanometer level when the reflector is tracked continuously. However, measurement in this mode has its limit due to the working principle. An IFM is unable to determine an absolute position without having a known starting point first. When the laser beam is interrupted, the reflector has to move back to ‘HOME’ to reset the initial distance which is quite inconvenient and time-consuming for the users to realize the
measuring process.
Design of IFM unit
The construction of IFM unit

7. Absolute Distance Meter
The ADM enables the trackers to locate the target without keeping continuous tracking. When the laser beam is broken, users can easily move the reflector back to its previous location where the laser is interrupted. Then the ADM would reestablish the broken beam and immediately start measuring process, providing absolute distance. But the accuracy of ADM is far lower than that of IFM.
Design of ADM unit
Realization of ADM unit

8. Angle Measurement Device
The angle measuring device is capable of measuring horizontal and vertical angles from the instrument to the target. The Renishaw circular gratings are used to measure the angles by detecting the signals. To eliminate un-concentric errors, two scanning heads are utilized to provide more accurate results. An error model is established which can eliminates the second harmonic components.
Angle measurement device

9. Tracking and Control Unit
The servo control and electronic control unit is one of the most important parts in the system which has functions of data
collection and communication, control, synchro-clock output and multitask management. The unit collects measuring signals of the distances (IFM and ADM), the angles and PSD positions, state parameters of motor working current, controller supply voltage and limit switch, as well as the environmental parameters of temperature, humidity, pressure and tilt sensors. Also the unit is responsible for sending controlling commands such as system self-checking, resetting, ‘back to HOME position’, tracking, safety protection, status indication and so on. In the meantime, the unit sends synchro-clock signal of ADM, IFM, encoders, PSD, data management unit and servo drivers. To enhance working efficiency, the unit utilizes the ARM processor to realize multitask management of communication and control. The whole unit is comprised of ARM, FPGA and peripheral A/D and D/A chips. Also this module takes charge of communication with the upper computer software.
Control box
Tracking experiment

10. Software Geometric element analysis unit Error compensation unit
The software is mainly in charge of data communication, command control, data analysis, calibration and error compensation. The software configuration is categorized into six parts: (1) data communication and command control module for data transmission, parameter setting, hardware monitoring and working state control; (2) calibration and error compensation module for error modeling, error parameter calibration and compensation; (3) data analysis module for geometrical element calculation and fitting and geometric elements relationship calculation; (4) database management module for database manipulation; (5) 3D display module for geometrical element display; (6)man-machine interface for
interactive manipulation of the tracker.
Error compensation unit
3D display unit

11. Error Analysis and Compensation
Distance
Measuring
Error
Geometric
Error
The errors of the laser tracker can be categorized into three parts, the distance measuring error, the angle measuring error and the geometric error.
Angle
Measuring
Error

12. Error Analysis and Compensation
Distance
Measuring
Error
Angle
Measuring
Error
Geometric
Error
IFM measuring error
ADM measuring error
Home distance error
Horizontal encoder
compensation
Vertical encoder
Mirror tilt
Mirror offset
Laser beam tilt
Laser beam offset
Transit tilt
Transit offset
The distance measuring error includes the IFM, ADM measuring error and the home distance error. The angle measuring error comprises of the horizontal and vertical angle encoder error. The geometric error includes the transit tilt, transit offset, laser beam tilt, laser beam offset, mirror tilt and mirror offset. The transit tilt error analysis will be emphatically described in this report.

13. Error Analysis and Compensation
IFM distance error
Angle measuring error calibration
In our job, the IFM distance error compensation, HOME distance error calibration, angle measuring error calibration and geometric error calibration are all carried out, as is shown in the figures. The tilt error calibration and compensation is analyzed in detail in this report.
HOME distance calibration
Geometric error calibration

14. Mirror Tilt Error Analysis
In the specific position when the reflected mirror is lying in the XOZ plane，the normal vector of the mirror can be stated as :
Whereas the actual normal vector is which rotate about the Z axis by c. It can be stated as:
The X axis that is defined as the transit axis and its normal in the XY plane (Y-axis) are both attached to the instrument head which rotate about the standing axis (Z-axis). In an ideal case, the transit axis (X-axis) is perpendicular to the standing axis (Z-axis) whereas it may be tilted in reality. The transit tilt is referred to as the tilt of transit axis relative to the standing axis. An error model is presented to describe its influence on the measurement results. The nominal horizontal and vertical angles are defined as H and V with transit tilt (θ) and the actual angles are defined as H' and V‘. Thus the relationship between the two vectors can be described in the formulation.
Mirror tilt error model

15. Mirror Tilt Error Analysis
When the vertical angle is V，the angle between the mirror and Z axis is Vm=V/2-45°
Then can be stated as:
In the situation mentioned above, Hz=90°. When Hz=H，The vector can be stated as:

16. Mirror Tilt Error Analysis
In the initial coordinate frame,
is the vector of the injected laser beam ,
is the vector of the emergent laser beam. The relationship between them is:
Thus can be calculated :
Set the parameters of to be px, py and pz. The actual horizontal and vertical angles is and Finally they can be expressed as:

17. Mirror Tilt Error Calibration
Measurement device
The transit tilt error calibration process can be categorized into two steps. Firstly, a test device is established to detect the contrail of laser beam with the same horizontal angles and diverse vertical angles in two-face mode. Then the simulation is manipulated which estimates the beam trail in different conditions. When the simulated contrail coincides with the actual one, the corresponding error paramter in the simulation is the same as the actual transit tilt.
Error parameter calibration process

18. Mirror Tilt Error Calibration
Calibration process
Keep the standing axis of the tracker still and motorize the tracker to make the transit axis rotate from 0°to 360°every 15°and record the readout of autocollimator B. The data can be summarized as (in the X direction):
In the formulation, Ax is the angle between the mirror plane and the plane perpendicular to the transit axis, Bx is the angle between the axis of the autocollimator and the transit axis. Фx is the initial phase.

19. Mirror Tilt Error Calibration
The formulation can be expressed as:
In the formulation:
Using the least square method to minimize the error:

20. Mirror Tilt Error Calibration
Thus the parameters ax, bx and Bx can be obtained. Also, Ax and Øx can be calculated in the following formulations:
Rotate the standing axis of the tracker by 90°. Adjust the rotary angle of the transit axis to make the tracker mirror perpendicular to the axis of autocollimator B in its Y direction. Then record the readout in the X direction B'.
Thus the angle between the tracker mirror and the transit axis, as well as the mirror tilt error, can be stated as:
c=Bx-B'

21. Calibration Setup Calibration Diagram
As is shown in the figure,a gradienter is used to make the standing axis of the laser tracker perpendicular to the horizontal plane. Then the laser beam of the tracker is projected onto a vertical plane set in a designated distance from the tracker with equal horizontal angles and diverse vertical angles (from -30°to 30°) in two-face mode. An autocollimator and a multi-mirror polygon are used to make sure that the horizontal angles are invariable. Also a total station is used to detect the contrail of laser beam in high precision.
Calibration Diagram

22. Software compensation
Experimental Result
Hardware adjustment
Using the proposed method, the mirror tilt error can be detected so that the researchers can adjust the position of the mirror to reduce the deviation. The mirror tilt can be adjusted from 80” to less than 5”.
Software compensation
Further software compensation have been implemented by using the formulations mentioned in the mirror error analysis chapter. The two-face experiments showed that the deviation between the front face and back face can be deduced from 1mm to less than 0.5mm in the distance of about 10m.
The simulation is manipulated estimating the beam trail in the same direction as the parameter identification test with different transit tilt errors. The coordinates of the designated points constituting the contrail can be easily calculated using the formulations mentioned in Section 3 with known distance, horizontal angles and vertical angles. Different tilt error parameters are tested to verify whether the simulated contrail coincide with the actual one. When the two contrails coincide with each other with least deviations, the corresponding tilt error in the simulation is considered to be the actual error. Thus the tilt error is obtained according to the comparison of actual result against the simulated one.

23. Summary Advantage Disadvantage Next Work
The introduced method can obtain and compensation the mirror tilt error parameter with convenience and efficiency.
Disadvantage
The data of the tracker mirror position after rotating 90°is quite limited in the measuring process which may bring in extra errors in the final measuring results.
The software compensation is not quite effective which has been proved in the experiments.
Next Work
Detailed analysis needs to be done to improve the precision of the method and the software compensation.

24. Acknowledgement Sponsor Cooperator
Ministry of Science and Technology of the People’s Republic of China
Chinese Academy of Sciences
National Natural Science Foundation of China
State key laboratory of precision measuring technology and instruments (Tianjin University)
Cooperator
Xi’an Institute of Optics and Precision Mechanics, CAS

25. Thank you for your attention！