Microsoft Word - Article 3 - 82-698-1-GA.docx ACTA IMEKO December 2013, Volume 2, Number 2, 3 – 12 www.imeko.org ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 3 Algorithm for a high precision contactless measurement system Tran Trung Nguyen, Arvid Amthor, Christoph Ament System Analysis Group,Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany Section: RESEARCH PAPER Keywords: optical measurement, multi laser tracker system, self-calibration Citation: Tran Trung Nguyen, Arvid Amthor, Christoph Ament, Algorithm for a high precision contactless measurement system, Acta IMEKO, vol. 2, no. 2, article 3, December 2013, identifier: IMEKO-ACTA-02 (2013)-02-03 Editor: Paolo Carbone, University of Perugia Received February 19th, 2013; In final form September 9th, 2013; Published December 2013 Copyright: © 2013 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unre- stricted use, distribution, and reproduction in any medium, provided the original author and source are credited Funding: This work was done in the framework of the project “Kompetenzdreieck Optische Mikrosysteme – Spitzenforschung und Innovation in den neuen Ländern”, which is supported by the German Ministry of Education and Research (FKZ: 16SV5473). Corresponding author: Tran Trung Nguyen, e-mail: tran-trung.nguyen@tu-ilmenau.de 1. INTRODUCTION The accurate position of the Tool Centre Point (TCP) during the production process is a crucial issue for efficient machine processing of complex tasks with smaller and smaller compo- nents and structures of steadily decreasing size. Capacitive and inductive position sensors achieve a limited measurement reso- lution and/or a limited working range and need to be integrated in the process machine. As the sensors measure the TCP posi- tion indirectly subsequent measurement errors are induced. Also all alignment errors of a measured serial kinematics deteri- orate the measured position of the TCP. In contrast, optical sensors offer a high precision and a large working range. Apart from that, optical sensors measure the TCP position directly and do not need to be integrated in the process machine. The preferred optical measurement device especially in high preci- sion applications is the laser tracker. A laser tracker is an optical instrument which permits a con- tactless 3D measurement of a moving target. The device con- sists of a very precise laser interferometer and a beam deflection system. The development of laser trackers began in the 1980’s when Lau et al. [7][8] used the laser tracking technique to determine the performance of a robot. This tracking system was imple- mented in a real-time system to identify the three-dimensional static as well as dynamic positioning accuracy of a robot end effector. The measuring volume of the described tracking sys- tem was approximately 3×3×3 m and a maximum speed of 300 mm/s was reached. In order to enhance the tracking speed Parker and Mayer developed an optical laser tracking system using a 2-axis rotational galvanometer scanner as beam deflec- tion unit [14][15][16]. Their system was used to measure the absolute position of a moving optical target which was mount- ed on a robot’s manipulator. The first who utilized a laser inter- ferometer for a precise distance measurement were Takatsuji et al. They presented an interferometer based laser tracking system called distance-only-measurement (DOM). This system deter- mined the position of a tracked target using the method of trilateration and consists of four single laser tracking units. Using four systems detecting one target offers the advantage of a redundant measurement. Thus, the position of the tracker units and the initial position of the target can be calculated from ABSTRACT This paper presents a self-calibration method for the multi laser tracker system (MLTS) which can track a retroreflector mounted on a kinematical system (e.g. positioning stage, robot manipulator etc.). Four laser trackers build up the MLTS. In the first part of the study the required algorithms enabling the MLTS to measure the position of the retroreflector are presented. The algorithms include the localization of the retroreflector, the communication between the laser trackers and the tracking controller as well as the calculation of the Tool Centre Point (TCP) position. In the second part of this study a detailed analysis of the high precision self-calibration algo- rithm is carried out. A sensitivity analysis shows that the quality of the parameter estimation highly depends on the optimal arrange- ment of the MTLS. ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 4 multiple position measurements. The resulting measurement error was about 40 microns with a target distance of one meter [18]. Due to the high resolution of the interferometer the DOM method is state of the art in non-contact high precision metrol- ogy. Beside the achievable precision an additional advantage of the proposed method is the fact that the three-dimensional position of the reflector is detected directly and hence Abbe’s principle is not violated. The crucial question concerning the trilateration is the determination of the absolute coordinates of the reflector and the position of each laser tracker given the initial measurement lengths. These parameters are calculated by self-calibration methods without the usage of a reference cali- bration device. In order to apply the self-calibration algorithm the number of free measurement points must be greater than the number of unknown parameters. This leads to an over- constrained system of equations, which can be solved using powerful numerical techniques. The authors of this contribution found that the quality of the self-calibration algorithm depends on various metrological arrangements of the measuring system e.g. the number of measurements, the working range, the distance of the laser tracker among each other and the measurement point. In [19] an optimal arrangement of four laser trackers is clarified. Nev- ertheless, there is no mathematical description for the various arrangements of measurement points and the laser trackers available up to date and hence an optimal configuration for the self-calibration is not given. Numerical simulations can (or will) be used to close this gap. The present contribution is organized as follows. Within the framework of the “Kompetenzdreieck Optische Mikrosystem (OPTIMI)”, which is supported by the Ministry of Education and Research (BMBF), the Systems Analysis Group of Ilmenau University of Technology has developed a MLTS for tracking a kinematical system [1][2][3][4][5][6] (see Figure 1). The hard- and software components of this experimental set-up is presented in the following section. Section 3 describes the required tracking control algorithm of one laser tracking unit and Section 4 presents the calculation of the TCP position using the trilateration method. Finally, selected results of the parameter estimation are shown in order to increase the accura- cy of the calibration method presented. The results are further concluded and a short outlook is given. 2. EXPERIMENTAL SET-UP The set-up of a single laser tracker module is decomposed of the opto-electronic components, the supply electronic unit and the opto-electronic detection unit (Figure 2). The opto-electronic components include a Michelson inter- ferometer and a galvanometer scanner. Both components are fixed on a base panel. The homodyne interferometer uses a stabilized He-Ne laser as beam source and realizes a position resolution on sub nanometer scale [9][10]. Beside the Michelson interferometer a four quadrant diode (4QD) unit is integrated. In this study the utilized 4QD has an active area of 5×5 mm. The four quadrants are separated by a gap, which amounts to 30 µm. This kind of diode is able to detect a misalignment between the interferometer and the retro-reflector. A corner cube mirror is used as a retro-reflector in this work [22]. The described opto-electronic components are arranged in such a way that the emitted beam hits the galvanometer scanner. The galvanometer scanner consists of two beam deflection units and the laser beam is reflected by both of them [1]. Each deflection unit consists of a mirror, a torque transducer, a high precision angle sensor and an analogue servo closed loop control. The servo is used to control the angles and the current required for the actuators. The developed algorithms (localization, communication, tracking control and determination of the TCP position) are implemented in Matlab/Simulink® using C-code-s-functions. The Simulink Coder® is used to automatically generate C code, which is carried out by a modular Rapid Control Prototyping System of dSpace®. All algorithms work with a sample rate of 10 kHz. Figure 3 depicts the functionality of one laser tracker module in detail. The laser beam is transported to the interferometer via an optical fibre. Inside the interferometer the laser beam hits the first polarizing beam splitter B1 and dividing the beam into a reference beam and a measuring beam. The reference beam then hits the fixed reference mirror and returns to the first beam splitter B1. Simultaneously, the measuring beam passes the second beam splitter B2 and leaves the interferometer. Starting from the interferometer the measurement beam hits the first mirror of the galvanometer scanner, which is rotating around the x-axis. The x-galvanometer scanner (x-scanner) deflects the measurement beam to the second mirror, which is rotating around the y-axis. The y-galvanometer scanner (y- scanner) deflects the measurement beam again to the retro- Figure 1. The developed MLTS for tracking kinematics. Figure 2. The developed tracker module. ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 5 reflector. The retro-reflector is fixed on the TCP of the kine- matics tracked. It reflects the measurement beam back to the galvanometer scanner. The reflected beam returns to the inter- ferometer via the galvanometer scanner and hits the second beam splitter B2. Part of the reflected beam reaches the 4QD. This signal of the 4QD is used to track the target. The other part of the reflected beam passes through the se- cond beam splitter B2 and is superimposed on the reference beam in the beam splitter B1. Furthermore, the superimposed beam hits the opto-electronic detection unit. A resolution of 0.1 nm is achieved for the position detection [10]. 3. TRACKING ALGORITHM 3.1. Localization of the retro-reflector The proposed localization method is based on an Archime- dean spiral which is derived in polar coordinates. The spiral is defined by three parameters. The radius r0 is used to describe the maximum rotational angle of the galvanometer scanner in polar coordinates. The duration t0 defines the rotation of the laser beam from the origin of the spiral to the predefined radius r0. The number of rotations is given by the parameter ω0. To describe the Archimedean spiral showed in Figure 4 the follow- ing equations in polar coordinates are given for the x-axis and for the y-axis:     0 0 0 0 0 0 0 0 cos 2 sin 2 r x t t t t t r y t t t t t                   (1) Due to the fact that the behaviour of the galvanometer scanner is specified in angle coordinates, the designed Archi- medean spiral has to be transformed from polar coordinates into angle coordinates. The rotation angles of the galvanometer scanner are calculated as follows:         arctan arctan x t t l y t t l                   (2) In Equation (2) the parameter l is the distance between the laser tracker and the localization plane. The distance is predefined by 1 m. The angle θ(t) describes the rotation of the galvanometer scanner about the x-axis and the angle φ(t) describes the rota- tion of the galvanometer scanner about the y-axis. 3.2. Communication of the MLTS The proposed localization algorithm is used for a single laser tracker module. Due to the fact that four tracker modules are used in an MLTS, the developed algorithm is expanded by a communication channel between the four tracker modules. Hence, the tracker modules are able to share the angle infor- mation and the retroreflector position with each other and this significantly accelerates the localization (all trackers have found the TCP) of the whole MLTS. To realize an effective accelera- tion of the localization phase, the approximate (low precision) TCP position is required in a global coordinate system (see Section 4.2). Initially, all tracker modules are searching for the TCP. If at least two trackers find the TCP the triangulation method will be used to determine the TCP position. The TCP position is subsequently transferred to the untraced tracker modules and therefore they find the TCP immediately. It has to be mentioned that during localization the approximate position coordinates of all tracker modules are required. For this pur- pose a low precision calibration process based on the angles and distances estimates the coordinates of the four tracker modules. For detailed information on the TCP localization process the reader is referred to [14]. Figure 3. The principle of the tracker module. Figure 4. The designed spiral in Cartesian coordinates. Figure 5. An example of the communication protocol. x-scanner photodiode reference mirror 4QD B1 B2 laser beam y-scanner retroreflector optical fibre 0 0.5 1 -0.1 0 0.1 -0.1 -0.05 0 0.05 0.1 z-axis in mx-axis in m y- ax is in m T1 T3 T2 Communication Determination of the radial distance Determination of the Tool Center Point Coordinate Transformation ReceiverTransceiver Position-Angle- Transformation ( , )   1 3,   ' '2 2,  ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 6 Figure 5 shows an example of a communication sequence. Initially, the tracker modules T3 as well as T1 have found the retroreflector. Based on the radial distance between the two tracker modules and the retro-reflector as well as the angle information the position of the retro-reflector is computed in the spherical coordinates of e.g. T1. The position of the retro- reflector is then transformed to the global coordinate system of MLTS. In a final step, the global TCP position is transformed to the polar coordinates of the residual (still searching) tracker modules, in our example T2. The local position is now shared with T2 and T2 finds the TCP immediately. To combine the localization algorithm with the communica- tion of the MLTS a state machine showed in Figure 6 was de- signed. This finite-state machine is used to model the switch behaviour of every tracker module in operation. The algorithm has three binary input variables (a, n, p) and two output varia- bles (θx, φy). The output variables are the rotation angles of the tracker module and the input variables are:  a: communication mode active  n: tracker does not hit the retroreflector  p: localization mode active Four finite-states (Z00, Z01, Z10 and Z11) are used during the localization:  Z00: The localization mode is deactivated. The tracker module has no information about the position of the retroreflector.  Z01: The localization mode is activated and the com- munication is deactivated. The tracker module search- es the retroreflector using the proposed localization algorithm.  Z10: A minimum number of two tracker modules hit the target center of the retroreflector. The communi- cation mode is activated. Using this additional infor- mation the other tracker modules are looking for the retroreflector in a defined small area around the com- municated TCP position.  Z11: All tracker modules have found the retro- reflector. 3.3. Tracking control In the case that the laser beam has found the retroreflector, the beam has to track it. For this purpose a feedback control algorithm has to be implemented. A model based control ap- proach is one possibility to minimize the tracking error of the laser beam. The requirements in this case are stability, large bandwidth and robustness against external disturbances. The 4QD is used as a feedback sensor and to provide the displacement of the reflected beam as a x4QD - and y4QD – signal, respectively. If the beam hits the target center, then the laser beam is found in the origin of the 4QD coordinates. Therefore the x4QD-signal only affects the x-scanner and the y4QD-signal only affects the y-mirror. Figure 7 shows the structure of the closed loop tracking control scheme for one axis. The motion of the tracking kinematics xRetro in x-direction generates a 4QD signal (x4QD), which is handled as a dynamic disturbance. The error e between the set point w4QD and the disturbance signal x4QD is compensated by a PID controller, which generates an output, φfeedback. The output φdis is computed by the disturbance compensation unit GD in order to suppress the influence of the moving stage. The compensation unit GD reflects the inverse system dynamics of the galvanometer scanner GM and the beam path GB. Thus, the disturbance compensation unit acts as a feed forward controller and reduces the tracking error of the laser beam, which tracks the positioning stage. The regulating varia- ble φref is a combination of outputs φfeedback and φdis, and drives the motors of the beam deflection unit. First, the system model is introduced. The beam path de- scribes the relationship between the laser beam from the mirror to the retroreflector and the rotation angle of the galvanometer scanner. Moreover, the beam path model is a function of the absolute distance between the mirror and the retro-reflector. The beam path model is given as follows: Beam B rad x G 2     (3) The dynamics of the galvanometer scanner can be approximat- ed by a second order differential equation. The model parame- ters have to be identified using experimental data. The model can be expressed in Laplace space as follows: Figure 6. The state machine describing the performance of the localization algorithm. Figure 7. The control scheme. p nap 01Z 00Z 10Z 11Z nap nap nap nap p nap 0x  0y  x xF  y yF  x xS xC    y yS yC    x xS  y yS  np nap np np nap np p p 4QDw Re trox 4QDxe  Beamxfeedback dis ref BG DG CG PID MG   Scanner Mirror Beam path Disturbance Compensation ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 7 0 M 2 ref 0 1 2 b G a a s a s       (4) For detailed information regarding the system model the reader is referred to [1]. In the feedback path a digital PID controller is used to sup- press the error e. The PID parameters are determined using the pole placement method. The PID controller for single-input and single-output sys- tems can be described in Laplace space as follows: 2 I P D C K K s K s G s    (5) The controller affects on the error e, which is the difference between the control variable x4QD and the set point value w4QD. The set point of the feedback controller is w4QD=0. In the case that the motion of the tracking kinematics is known, the path planning signals of the kinematics can be used to compensate the disturbance xRetro. Hence, the disturbance compensator GD (see Figure 7) has to be taken into account. Its dynamical behaviour can be approximated by a second order system and is expressed in Laplace space: 2dis D 0 1 2 Re tro G k k s k s x      (6) To determine the parameters of the PID controller a trans- fer function of the whole closed loop shown in Figure 7 has to be found. The output of the transfer function is the control variable x4QD and the input is the motion of the retroreflector xRetro. Thus the transfer function can be expressed as follows: 4 QD D B M W Re tro C B M x 1 2G G G G x 1 2G G G     (7) Inserting Equations (3), (4), (5) and (6) in equation (7) yields:            2 2 2 1 1 0 0 2 W 1 D 0 P3 2 I 2 2 2 s s a k s a k a ka G a K a K K s s s a a a                  (8) with: 0 rad4b   (9) In Equation (8) the polynomial A(s) represents the numera- tor and B(s) describes the denominator of the transfer function. The transfer function decreases to zero for t  , consequent- ly s  0:    Ws 0 A s lim G 0 B s   (10) Equation (10) shows clearly that the proposed model-based controller is able to track the target with a tracking error of zero. The dynamical behaviour of equation (10) depends on the parameters of the denominator B(s) as well as the controller parameters KP, KD and KI. To calculate the parameters of the denominator B(s) the following desired polynomial P(s) is used: 3 2 2 1 0P( s ) s p s p s p    (11) Then the desired polynomial P(s) is equated with the de- nominator B(s) of equation (11) and the parameters of the PID controller are calculated as follows: 0 2 0 2 0 0 2 1 I P D 0 rad 0 rad 0 rad p a p a a p a a K , K , K 4b 4b 4b         (12) Equation (12) shows that the parameters KI, KP and KD are not constant and change as a function of the absolute distance between the laser tracker and the retroreflector. Due to the fact that the interferometers can only measure a relative distance, there is a need to determine the absolute distance. To solve this problem we have developed a triangulation method, which opens the possibility to determine the absolute distance with the precision needed by using at least two tracker modules [2] [3] [4]. The proposed control algorithm was experimentally validat- ed by a kinematical system, which moved along one axis. Figure 8a shows the sinusoidal motion of the tracked kinematics and Figure 8b shows the resulting tracking error of the laser beam. The dynamical set-points of the kinematics were generated by an analytic fourth order path planning algorithm, which is able to plan motions with arbitrary initial and final velocities [10]. The kinematics carried out a sinusoidal motion with amplitude of 100 mm and a frequency of 0.4 Hz. Without disturbance compensation the tracking error of the laser beam is approximately 1 V at peak and a clear phase shift is observable. Using the proposed model-based disturbance compensation the tracking error can be reduced by a factor of 2. Moreover, the phase shift is not noticeable anymore and this shows the effectiveness of the presented model based control approach. 4. CALCULATION OF THE TCP 4.1. Calculation of the TCP for one tracker module After the implementation of the required control algorithms one tracker module is able to measure the position of the retroreflector, viz. the TCP position. In the case of one tracker module the determination of the TCP position is carried out using vector analysis. This operation requires a precise geomet- rical model of the beam path. Figure 9 shows the geometric relation between laser beam and beam deflection system. The laser beam starts from point P0 and hits the x-scanner mirror in the point P1. The x-scanner mirror reflects the laser beam to the y-scanner mirror at point P2 and then the laser beam is reflected to the retroreflector at the TCP. To calculate the TCP the points P0, P1, P2 have to be known in a predefined coordi- nate system. The proposed coordinate system uses the rotation axes of the beam deflection system as orthogonal axes: T T 1 1 x 1 y 1 z 2 2 x 2 y 2 zv v ,v ,v ,v v ,v ,v          (13) First, the rotation axis of the y-scanner is used as the x-axis of the coordinate system. Second, the z-axis is defined by the cross product of the rotation axes, which is perpendicular to the Figure 8. The sinusoidal motion f = 0.4Hz of the tracked kinematics a) and the tracking error b). 0 5 10 15 20 -200 0 200 time in s x R e tr o in m m 0 5 10 15 20 -2 0 2 time in s tr ac ki ng e rr o r in V PID controller PID controller with disturbance compensation a) b) ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 8 x-axis. Finally, the y-axis is the cross product of x- and z-axis. All axes are given as follows:      1 1 2 x v z v v y x z        (14) The laser beam is considered to be a straight line represented by a position and a direction vector. The direction vector after an arbitrary deflection can be computed as follows:  1 0 0r r 2 r n n       (15) The vector r0 is the direction vector of the laser beam before the deflection and the vector n is the normal vector of the uti- lized mirror. In the case that the vector rP0 is known, the direc- tion vectors rP1, rP2 can be calculated using equation (15) (see Figure 9). Furthermore, the direction vectors are used to de- termine the reflection points P1, P2. The rotating mirror is treated as a plane described by a point on the surface and the according normal vector. The intersections P1 and P2 can be calculated as follows:  1 1 1 u P0 n P1 P0 rP0 rP0 n            (16)  2 2 2 u P1 n P2 P1 rP1 rP1 n            (17) The vectors n1, n2 are the normal vectors of the mirrors. The vectors u1 and u2 represent an arbitrary point on every mirror surface. A detailed description of the presented geomet- ric model is shown in [14], [15]. In the case of a moving TCP the points P1 and P2 are variant due to the rotation of the mir- rors. A normal vector of the mirror has to be updated online using the rotation axis vi = [vix viy viz] as well as the rotation angle : i in n , i=1,2  R (18) with       2 ix ix iy iz ix iz iy 2 ix iy iz iy iy iz ix 2 ix iz iy iy iz ix iz i i i v C v v v S v v v S v v v S v C v v v S , v v v S v v v S v C 1 cos , C cos , S sin , i 1,2                                   R (19) After the determination of the deflection points P1 and P2 the TCP can be calculated by using the straight line with an initial position at point P2 and a direction vector rP2. The TCP is given as follows:   TCP P2 rP2     (20) The parameter ℓ is the radial distance between the y-scanner mirror and the TCP and ℓ is the sum of the unknown initial length ℓ0 and the measured relative distance Δℓ. The proposed calculation method of the TCP requires the determination of the unknown system parameters as well as the unknown initial length. After the calibration process the pre- sented geometric model is used to correct the length measure- ment of the interferometer [14][15]. 4.2. High precision determination of the TCP position using the MLTS The TCP position can be calculated by the relative length measurements of all interferometers. Figure 10 shows the four laser trackers (T1, T2, T3 and T4) and the position of the retro- reflector PTCP = [X, Y, Z] T in the global Cartesian coordinate system. Every tracker module has three location parameters Ti = [xi, yi, zi] T and hence the number of the location parameters lead to a total of twelve location parameters describing the whole multi-laser tracking system. To reduce the number of the unknown location parameters from twelve to six and simplify the calculation of the TCP position, the geometric configuration in Figure 10 is chosen. For detailed information regarding the choice of the global coordinate system the reader is referred to [17] [18]. The tracker T1 is located in the origin of the coordinate system and has position [0, 0, 0] T. The tracker T2 is located in the x-axis, the tracker T3 is located on the x-y-plane and the position of the tracker T4 is freely selectable. The parameter ℓi describes the radial distance of the four laser trackers and is defined as fol- lows: i 0i i ,i 1 4      (21) The parameter ℓ0i represents the offset length (dead zone) after the initialization phase and the interferometer detects the relative radial distance Δℓi. If the positions of all trackers and all Figure 9. The geometric representation of the beam deflection system. 1T 2T 3T 4T y x z   2 4 3 1 TCPP Figure 10. The position configuration of the MLTS. P0 P1 P2 TCP rP0 rP1 rP2 x-scanner mirror 1 y-scanner mirror 2 2v 2k 2u 1u 2n 1k 1n 1v ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 9 distances ℓ0i are known, the TCP position can be determined by an approach called multilateration. The four spherical equations in the Euclidian space are given by:  22 2 2 01 1X Y Z      (22)    2 22 22 02 2X x Y Z       (23)      2 2 223 3 03 3X x Y y Z        (24)        2 2 2 24 4 4 04 4X x Y y Z z         (25) After the insertion of Equation (22) in Equation (23), Equa- tion (24), Equation (25) and using of the relation ℓi = ℓ0i + Δℓi, the following linear system of equations can be defined [17]: 2 2 2 2 1 2 2 2 2 2 2 3 3 1 3 3 3 2 2 2 2 2 4 4 4 1 4 4 4 4 M x 0 0 X x 2 x y 0 Y x y x y z Z x y z                                         (26) It is possible to calculate the TCP if the matrix M can be in- verted: 2 2 2 1 2 2 1 2 2 2 2 1 3 3 3 2 2 2 2 2 1 4 4 4 4 X x 1 Y x y 2 Z x y z                         M       (27) The inverted matrix M is given by: 2 1 3 2 3 3 4 3 4 4 2 4 2 3 4 3 4 4 1 0 0 x x 1 0 x y y x x y y 1 x z x y z y z z                        M (28) 4.3. Estimation algorithm of the system parameters Equation (25) can be used to prove the validity of the calcu- lated TCP position. Therefore, the location parameters x4, y4 and z4 as well as ℓ04 are needed. The offset length ℓ04 is calculat- ed by substituting equation (27) in equation (25). The relative distances can be set to zero in order to determine ℓ04 which is a function of the location and the remaining offset length param- eters:   with 0404 2 3 3 4 4 4 01 02 03 1 2 3 f x , x , y , x , y , z , , , 0              (29) To achieve a simplification the location parameters and the independent offset length parameters can be denoted as fol- lows:     l 2 3 3 4 4 4 o 01 02 03 p x , x , y , x , y , z p , ,      (30) In order to calculate the TCP position very precisely the ob- jective is the determination of the optimal parameter values. As the location parameters are static they can be estimated offline. For this purpose a pattern-search algorithm is used [23]. That method is rather time-consuming but has to be accomplished only once. In the case that there is no reference position of the kinematics the offset length parameters change with every shift of the initial position of the TCP. The strategy is to identify the location parameters in a previous step with a pattern-search algorithm. Then, the offset length parameters are estimated in real time. As an optimization method for the offset length parameters the Gauss-Newton approach is applied. The Gauss-Newton approach is a specific modification of the Newton method and convenient for nonlinear problems [20]. The calculation of the Hessian is very complex and requires a lot of computing re- sources. Hence, a quasi-Newton approach is applied that does not require the calculation of the Hessian. The objective function is based on equation (25) and is solved for Δℓ4:  4c4 l o 1 2 3f p , p , , ,        (31) Equation (31) is a function dependent on the offset length as well as the location parameters. The different distances can be interpreted as input signals. Hence, the objective function re- sults in the difference of the calculated and measured distance of tracker T4. To apply the estimation algorithm a definite number of measurements is needed leading to the residual function is defined as follows: with c i 4 i 4 iR i 1 N        (32) where N is the index of the measurement data. Since the objec- tive function is clarified the next step is the application of the Gauss-Newton algorithm. Subsequently, the algorithm shall be outlined. The goal is to minimize the sum of squares of Ri: min N 2 i i 1 R   (33) The Gauss-Newton approach is an iterative process and the current estimated parameters are modified by a correction term c:     with with k k 1 T T T 1 2 N p p c c r r R , R , , R        D D D  (34) The Jacobian matrix D is determined by the derivatives of the residual function Equation (32), i.e. the offset length pa- rameters. Every row correlates with a measurement value. The Jacobian matrix reads as follows: 1 1 1 01,1 02,1 03,1 2 2 2 01,2 02,2 03,2 01, 01, 01, n n n n n n R R R R R R R R R                                                      D (35) Due to the fact that the residuum changes continuously with the adjustment of the parameters the Jacobian D has to be updated in every iteration step. To determine the end of the iteration a criterion is necessary. Here, the norm of the differ- ence of the modified (pk) and the current (pk-1) parameter vector is used. The calculation will be terminated as soon as the norm gets below a defined threshold, is e: 1k kp p e  (36) The calibration process runs online and automatically. It shall be depicted in the following. ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 10 After the measurement system is activated the control unit handles the initial procedures that include enabling the actuator and sensor units. If the target is localized by all laser trackers, the system is prepared for the calibration procedure. Here, a cube dimensioned motion with 100×100×100 mm³ is executed. This space is marked with 27 grid points and at every point the positioning stage stops for a short period of time. The trajecto- ry is quasi-static because at every grid point several measure- ment values are taken to generate an average value. After the cube trajectory is finished the measurement data is used to estimate the offset length parameters online. All 27 measure- ment points compose the residual function R and the Jacobian D. When the termination criterion is reached the parameters are identified. Then, they are written as fixed values in the control system and the multi laser tracker system is ready to measure the motion of the target. 4.4. Influence of the initial tracker arrangement on the parameter estimation After the self-calibration, a precise estimation of all nine sys- tem parameters is available. The real values within the simula- tion are well known. To evaluate the quality of the estimation, the maximum difference between the known and the estimated parameters is calculated: realQ max | p p |     First investigations have shown that the optimization algo- rithm is able to converge to the designated minimum. To ana- lyse the influence of noise and local minima, the starting point for optimization is set into an area of ±50 mm around the real parameters. The measurement noise of the interferometers has a Gaussian distribution with standard deviation of 1 µm. These assumptions match with the experimental setup. The choice of the boundary values has significant influence on the accuracy of the self-calibration algorithm. Then, the number of measurement points (aG), the distance between the different trackers among each other (aTT), the size of the work- ing range for the reference points (abTCP), the distance be- tween the laser trackers and the measurement area (aTTCP) are investigated. In the simulation these configuration values are varied within the following intervals:  aG: [27, 64, 125, 256]  aTT in [m]: [0.2, 0.3, 0.5, 0.8]  abTCP in [m]: [0.3, 0.4, 0.5, 0.6, 0.7, 0.8]  aTTCP in [m]: [0.5,0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9] For aG and aTT there exist four variations of boundary val- ues, six for abTCP and eight for aTTCP. The total number of calibration set-ups NC is the product of the configuration varia- tions providing the different maximum combination options. NC is calculated as follows: CN aG aTT abTCP aTTCP 4 4 6 8 768         (37) For every combination the optimization is carried out and the quality Q is determined. The current boundary and the optimization results are stored. For the evaluation the maxi- mum aberration between the estimated parameters and the real parameters is compared with a threshold value of 50 µm. The combination is valid if the maximum aberration is below the threshold value. In the case that all combinations are valid, a maximum number of valid calibrations NMAX can be defined as the quotient between the total number of the calibrations NC and the number of the used intervals for every boundary:  aG → NMAX = 192  aTT → NMAX = 192  abTCP → NMAX = 128  aTTCP → NMAX = 96 The results of a simulation without measurement noise are presented in Figure 11. The number of valid calibrations is minimal below the maximum number of valid calibrations NMAX. It can be stated that the utilized estimation algorithm is appropriate for the selected boundary values. Figure 12 depicts the simulation results with measurement noise. It can clearly be seen that the choice of the boundaries influences the quality of the parameter estimation. Only 508 out of 768 calibrations are below the threshold value. The boundary aTT shows the strongest sensitivity regard- ing a valid calibration. Furthermore, a large distance between the trackers among each other significantly increases the num- ber of the valid calibrations. A step-by-step rise of the aTTCP decreases the number of valid calibrations. In contrast, an in- creasing aG as well as an increasing abTCP rises the number of the valid calibrations only slightly. To show the effect of the boundary values, the threshold Figure 11. Variation of four boundary values and number of valid calibra- tions without measurement noise, threshold value = 50 µm. Figure 12. Variation of four boundary values and number of valid calibra- tions with measurement noise, threshold value = 50 µm. ACTA IMEKO | www.imeko.org December 2013 | Volume 2 | Number 2 | 11 value is changed in the simulation from 50 µm to 10 µm. In Figure 13 the variation of the boundary values and the related valid calibrations is depicted. Only 240 of 768 calibrations are below the threshold value. It can be seen that the number of successful optimizations varies extremely with the choice of the boundaries. The number of valid calibrations increases with the number of measurement points aG as well as with an increasing distance between the laser trackers aTT. The results further improve with a larger area of reference points. The best performance for aTTCP is reached for a minimal distance between the trackers and the measurement interval. Further simulations showed that unequally spaced reference points have no negative impact on the estimation error as long as every direction in space is used for measurement. If points only lie in one or two spatial directions, one cannot act on the assumption that all effects of the parameters are gathered. In Table 1 the estimated parameter values for a non-uniform dis- tribution are shown, following the rules mentioned above. 5. CONCLUSIONS In the first part of this contribution we present the devel- oped multi laser tracker system and the required control algo- rithms allowing to track a highly dynamic moving target. The proposed control algorithms include the model based tracking control, the localization routines for finding the retro-reflector and the communication between the four laser trackers. In the second part of this work the authors present a meth- od to determine the position of the TCP based on a multilatera- tion algorithm. The key point in the calculation of the TCP is the determination of nine system parameters. The estimation of these parameters is highly dependent on the initial arrangement of the MTLS. In order to evaluate the sensitivity of the parame- ter estimation various arrangements of the MLTS were investi- gated. The presented simulation shows that the choice of the boundary values affects the estimation quality significantly. A sufficiently large distance between laser trackers among each other is required to achieve good results. Furthermore, the reference points should be distributed in a large area and a minimal distance between the laser trackers and the measure- ment area is recommended. Finally, a large number of meas- urement points ensures the convergence of the parameter esti- mation. ACKNOWLEDGEMENT This work was done within the framework of the project “Kompetenzdreieck Optische Mikrosysteme – Spitzen- forschung und Innovation in den neuen Ländern”, which is supported by the German Ministry of Education and Research (FKZ: 16SV5473). The authors would also like to thank all colleagues, who offered help with the work presented. REFERENCES [1] T. T. Nguyen, Q. T. Nguyen, A. Amthor, C. 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