^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A technique for intrinsic and extrinsic calibration of a laser triangulation sensor (LTS) integrated in an articulated arm coordinate measuring machine (AACMM) is presented in this paper. After applying a novel approach to the AACMM kinematic parameter identification problem, by means of a single calibration gauge object, a one-step calibration method to obtain both intrinsic—laser plane, CCD sensor and camera geometry—and extrinsic parameters related to the AACMM main frame has been developed. This allows the integration of LTS and AACMM mathematical models without the need of additional optimization methods after the prior sensor calibration, usually done in a coordinate measuring machine (CMM) before the assembly of the sensor in the arm. The experimental tests results for accuracy and repeatability show the suitable performance of this technique, resulting in a reliable, quick and friendly calibration method for the AACMM final user. The presented method is also valid for sensor integration in robot arms and CMMs.

The progressive spread of reverse engineering and digitalization in metrology and quality control tasks has increased sensor integration needs in instruments traditionally used for dimensional metrology. The latest improvements in equipment accuracy have resulted in metrology instruments capable of obtaining quick and accurate measurements approaching those of conventional coordinate measuring machines under certain circumstances. LTSs, able to obtain 3D coordinates from the projection of a laser line onto the surface to be measured, are based on the triangulation principle and are mainly composed of a camera (CCD or APS and lens) and a laser diode with a cylindrical lens capable of projecting a plane. This way, it is possible to reconstruct

The rapid integration of this type of 3D sensor in metrology equipment over recent years has been accompanied by a lack of standardization regarding their calibration procedures. For this reason different manufacturers have developed their own calibration procedures. However, these procedures do not reliably guarantee the accuracy of structured light optical measurement systems because they do not establish general evaluation procedures for the complete systems, due to the large number of parameters influencing the final system error. In particular, LTSs are nowadays the most commonly used non-contact sensors in traditional dimensional metrology equipment such as CMMs or AACMMs. This is due to their versatility and the fact that they are one of the most accurate structured light contactless measurement sensors, providing suitable accuracy values for most reverse engineering applications although, in general, these are not sufficient for metrological inspection tasks. AACMM applications with integrated laser sensors are, nowadays, mainly focused on the automotive, aeronautics and moulds sectors, and applications related to heritage conservation and general measurements of industrial components [

The difficulty of mathematically characterizing the influence of these parameters on the error in a general way for any LTS has traditionally prevented the development of calibration and later correction methods. Previous works [

Another conclusion reached by several authors [

Several possible assembly configurations using LTSs can be found nowadays in industry. Some use a CMM, a robot or an AACMM and others are assembled on specific high precision positioning systems or static structures under which the geometries to be digitized are displaced. The way to determine the relationship between the fixed frame (LTS or support) and the moving one will be of great influence on the final accuracy of the system.

Sensor manufacturers usually carry out the extrinsic calibration using a reference system on the sensor itself, making it necessary to subsequently transfer this extrinsic calibration to the LTS support. Several works have tried to solve this problem by scanning reference spheres in multiple spatial positions or by scanning the same sphere from three different sensor scanning paths in the same axis with a predefined offset between scans. Recent studies [

When the sensor is mounted on a manually operated AACMM, it is very difficult to apply these procedures without the aid of expensive instruments, because it is not possible to move the sensor only on a single axis of its reference system during the digitalization of points to obtain the conjugated pairs. Hence, it is necessary to find an alternative method of extrinsic calibration. This is the reason why LTS manufacturers carry out LTS or LTS-contact probe sets intrinsic and extrinsic calibration prior to mounting the sensor on a CMM. After that, if the set is calibrated, the extrinsic calibration is reduced to obtain, by the above-mentioned procedures, the geometric relationship between the LTS reference system and the contact probe reference system. Later, the complete set is mounted in an AACMM, integrating the sensor in its mathematical model through its well-known relation with the contact probe. If only the LTS has been calibrated, the usual technique for further integration of LTSs in AACMMs consists of mounting the sensor and digitizing a reference geometric primitive, usually a plane, from some spatial orientations. Later, a contact measurement of the same primitive is carried out with the AACMM (

This method is common in combined AACMM-LTS commercial systems in which the manufacturer of the LTS performs the intrinsic calibration of the sensor and defines, by means of local extrinsic calibration, its reference system. The later integration of the sensor in the AACMM is carried out by the previously described approximate determination procedure of the above-mentioned matrix. During this procedure, the capture of points of the digitized primitive requires manual displacements of the measurement arm that gather the influence of errors due to kinematic parameters and of dynamic errors, which are generally dependant on the position of the joints at the moment of digitalization. These errors are later reduced by the optimization procedure to obtain the final extrinsic parameters that will transform coordinates in the sensor coordinate system to the global arm coordinate system for any arm position and orientation. Therefore, a transformation matrix is obtained that is highly dependant on the digitized zone of the gauge primitive and only suitable for digitalization trajectories similar to those used during the data capture.

In this work, the mathematical modelling of a commercial LTS is presented first, followed by the complete procedure of sensor intrinsic and extrinsic calibration with the LTS already mounted in the AACMM. This method circumvents the use of approximated methods to determine extrinsic parameters, such as the digitalization of gauge objects prior to extrinsic calibration of the sensor, thus also avoiding the introduction of possible digitalization errors during scanning paths for extrinsic calibration. Moreover, the transformation matrix is obtained analytically in a single step. This is necessary in order to obtain 3D coordinates in the AACMM global coordinate system from the laser line points image coordinates in any capture position. Therefore, the time and the cost necessary to calibrate the whole equipment with current methods are reduced.

The kinematic model of the arm, the parameters considered, and the identification process used [

In the present work, the AACMM used is a six degrees-of-freedom (dof) Sterling series FARO arm with a typical 2-2-2 configuration and a-b-c-d-e-f deg rotation, in accordance with ASME B89.4.22-2004. Each of the six joints is characterized by the four geometrical parameters (distances _{i}_{i}_{i}_{i}

In _{i}_{0i}

By means of successive transformations of the coordinates, by pre-multiplying successively the transformation matrix for a given position between a frame and the previous one, it is possible to obtain the global transformation matrix of the arm, which gives the coordinates of the centre of the probe sphere with regards to the base of the AACMM.

In this manner, considering 0 as the global fixed reference system of the base and 6 as the reference system moving with the rotation of the last joint (

All the calibration procedures, both for robotic arms and AACMMs, are based on the establishment of a system which materialises coordinates or nominal distances in the workspace, in order to capture points which allow the error to be evaluated and minimized. The number of identification and data capture methods for robotic arms contrasts with the scant bibliographical resources regarding capture methods for parameter identification in measurement arms, currently identical to those used in robot parameter identification techniques. The different nature of robot arms and AACMMs requires the development of strategies to obtain the results desired in each case. A continuous data capture method has been implemented [

Therefore, apart from characterising and optimising the behaviour of the arm with regard to error in distances, its capacity to repeat measurements of a same point is also tested and subsequently optimized. Hence, automatic arm position capture software has been developed, probing each considered sphere of the gauge and replicating the arm behaviour in the ASME B89.4.22-2004 single-point articulation performance test, but in this case including the positions captured in the optimization from the point of view of this repeatability. The objective function used in the Levenberg-Marquardt [_{ijk}_{0jk} the nominal distance materialized by the gauge and _{Xij}

LTS modelling must establish the geometric relations necessary to obtain 3D coordinates, in the global coordinate system, of the points from the 2D CCD image corresponding to the line formed by the intersection of the laser plane and the surface to be digitized. The parameters to consider and to calibrate subsequently by means of the implemented method include intrinsic and extrinsic parameters of the camera and the laser plane equation.

The basic camera model, based on the perspective projection principle, obtains ^{T}^{T}_{int}_{ext}^{T} a vector with the intrinsic and extrinsic camera parameters. Although there are diverse approaches to the consideration of these parameters, the basic intrinsic parameters make reference firstly to the geometry and optics of the camera, involving: (1) focal length _{0} and _{0} coordinates of the principal point in pixels, (3) _{i}_{u}_{v}

From

From

Expressing

Only two of the three equations obtained are linearly independent. Thus, operating with these equations, it is possible to extract the linearly independent equations from (10)–(12)

_{ij}

The aim of the LTS is to obtain the coordinates expressed in the 3D global coordinate system of the points identified in an image belonging to the laser plane, through its projection onto the piece to be digitized. Therefore, a point

The laser plane contributes with the additional information necessary to complete the equation of the straight line of the camera model and to achieve a system of three equations with three variables for each identified point, so that their 3D global coordinates can be extracted from their 2D screen coordinates

AACMM-LTS integration demands the determination of the geometric relationships between the LTS frame and the AACMM last joint frame or, in other words, the extrinsic parameters of the sensor once integrated in the arm. Thus, the extrinsic calibration procedure of a LTS mounted on an AACMM consists of determining the sensor frame origin coordinates and its direction related to the AACMM last joint frame, linking both mathematical models. Once these extrinsic parameters are determined, the laser line point coordinates are obtained in the LTS frame and, therefore, also with respect to the AACMM global frame in any arm pose.

Traditional integration methods are based on digitalization of gauge geometric primitives, generally planes or spheres. These methods start with the sensor already calibrated. Then, they perform multiple scanning paths over the gauge primitive without knowing the geometric relationship between the LTS frame and the last joint frame of the AACMM. By comparing the contact measurement of the gauge primitive, taken as nominal, and the least-squares one reconstructed from digitized points, it is possible to define a measurement error. The matrix that links the mathematical model of the sensor with the mathematical model of the AACMM is then obtained by an optimization procedure that minimizes the error mentioned changing the terms of such unknown matrix starting from an approximate initial value. Thus, the obtained matrix allows to subsequently expressing the coordinates of the digitized points in the AACMM global frame. The optimization methods used in these techniques are commonly based on the gradient method, so the success of the optimization procedure and its speed of convergence depend on the initial value considered for the matrix terms.

The calibration method presented in this section performs the intrinsic and extrinsic calibration of the sensor in a single step, so it is not necessary to have the LTS previously calibrated. Furthermore, it is based on the capture of an image of a gauge object in a single AACMM position, so the error influence of the arm due to the error made during the scan paths is avoided, absorbing only the measurement error in the contact measurement procedure of the gauge object and the error in the AACMM capture position of the image for calibration. Finally we obtain the transformation matrix between the LTS reference system and the last joint frame of the arm following an analytical scheme, thus avoiding optimization procedures. The result of these optimization procedures depends on the type and number of scanning paths because it adjusts the matrix terms to minimize the error with the captured data in each case. With the proposed method explained in this section, the digitization of a geometric primitive is also avoided.

On the other hand, there are many influences over the final accuracy of a LTS. The digitization of a geometric primitive using a manually operated instrument like an AACMM implies that it is not possible to maintain constant neither the distance from the sensor to the scanned surface nor the perpendicular orientation of the laser to the surface of the gauge primitive, affecting also the manual operation to the digitalization conditions such as scanning speed. The use of scanning paths in the traditional methods implies that the digitized points will be affected by these error sources. Thus, these errors will be subsequently absorbed by the least squares based calculation of the gauge geometric primitive with the captured points, and by the optimization procedure of the link matrix, affecting the final accuracy depending on the data and scans considered.

This section presents the required steps to perform the calibration method presented, which avoids data-dependent optimization procedures and the consideration of an initial value for the matrix terms, and also influences of the mentioned error mechanisms.

The target object used in this work is shown in

The first step of the implemented integration procedure consists of the alignment of the AACMM reference system with the gauge object coordinate system. Once placed in a position accessible to the arm, the gauge object is measured by contact with the AACMM to align a reference frame attached to the calibration object and calculate a transformation matrix ^{AACMM}M_{CAL}

The ^{AACMM}M_{CAL}

In the current integration procedures based on error optimization over digitalized data, once the LTS intrinsic calibration has been done on a CMM, due to the point reconstruction process nature used in the LTS model, only points belonging to the captured laser line are known in the LTS frame when the LTS is linked to the arm. Thus, the coordinates of these points in the AACMM global frame cannot be obtained in this situation. In order to avoid approximate optimization procedures so as to determine the sensor position and orientation in AACMM coordinate system, it is necessary to do the LTS intrinsic calibration once it is already mounted onto the AACMM, when the camera gauge object point coordinates can be known in the LTS frame. LTS calibration implies the determination of the intrinsic and extrinsic parameters of the camera, and therefore the terms of the perspective transformation matrix of

Once the calibration object point coordinates in the AACMM global reference frame are known by means of

From the captured image it is possible to determine the image coordinates

The perspective transformation matrix being homogenous, the solution is modified by a scale factor, reason why the condition _{34}_{z}_{z}

Knowing the coordinate pairs _{W}_{W}_{W}_{34}

LTS calibration, in addition to giving the camera intrinsic parameters, defines the position and orientation of the sensor global coordinate system. In this way, by means of this calibration, the sensor global coordinate system is defined coincident with the gauge object local coordinate system (

On the other hand it is necessary to identify the screen coordinates of the laser line points in order to determine the equation that defines the laser plane in the coordinate system considered. The captured laser line has a greater width than a single pixel in the image, which is the reason why the identification of the laser line point is carried out by means of a gray level centroid estimation algorithm for each cross section of the line [

Once the laser line points screen coordinates have been detected, and considering that the _{W}_{W}, Y_{W} coordinates of the identified points using

Finally, in the subsequent LTS operation, the information provided by the laser plane complements

It is necessary to note that the sensor calibration has been shown without considering distortion effects on the reconstructed points. These effects are very low in the modelled sensor because the capture distance to the surface only allows the capture of points in the range of ±5 mm around the central line of the captured image, where the distortion effects are minimum. In order to verify distortion effects on the reconstructed points, the calculation of the screen coordinates corresponding to the gauge object points after the calibration has been made, obtaining mean values of 0.224 pixels in maximum error for the

Once the sensor calibration from the captured image has been done, not only the laser line points but also the calibration object points coordinates are known in the LTS global frame. This calibration defines the sensor global reference system that matches the gauge object local coordinate system for the position of image capture. In this way, the matrix that relates the sensor global coordinate system to the AACMM global coordinate system for the AACMM capture image position is the transformation matrix obtained by contact measurement of the gauge object in ^{AACMM}M_{W_LTS}^{AACMM}M_{CAL}

With ^{6_AACMM}_{0_AACMM} that will coincide with the inverse matrix of the product of matrices _{1}_{6}

With this, it is possible to define the desired matrix by means of

In _{LTS_Probe}

In order to analyze the accuracy and repeatability of the developed calibration procedure, several calibration tests have been carried out using the FARO AACMM already described. A commercial LTS (DATAPIXEL Optiscan H-1040-L) was linked to the arm. The nominal working characteristics of this sensor are frame rate, 60 fps; working distance, 100 mm; measurement range, 40 mm; field of view, 40 mm; triangulation angle 30°, as well as accuracy, according to the manufacturer, of ±0.010 mm. It is equipped with a 1/3 CCD sensor. This LTS is able to obtain 30,000 pts/sec with nominal repeatability of 10 μm. Previous studies on this sensor mounted in a CMM [

Ten different calibrations have been carried out giving 10 _{LTS_Probe}

As a repeatability analysis of the calibration procedure, a gauge plane has been digitized obtaining 10 different point clouds for the same plane. For each one of these clouds, the

A plane has been chosen as a gauge geometric primitive in the first test. The nominal value for the plane equation was obtained as the average result from 10 AACMM contact measurements of the plane,

Once the point clouds were reconstructed, the digitized plane equation for each one was determined by a least squares estimation algorithm that includes segmentation and point filtration techniques based on the standard deviation of point distances to the calculated plane. Two error parameters were chosen. Firstly, the angle between the nominal and the calculated plane normal vectors, and finally the difference in the Z coordinate of the central point of the cloud projected on the nominal and the calculated plane.

After the repeatability was analyzed, an estimation of the complete system accuracy was made by means of a reference ceramic sphere digitized five times and reconstructing the clouds of points with the calibration close to the average values of error obtained in the angle between the normals and Z. To emphasize the influence of the points capture strategy, the sphere was digitized five times orienting the laser plane perpendicular to the surface, and another 5 times with an orientation of the LTS similar to that used for its calibration. The results of

The best results are obtained, for orientation of the laser perpendicular to the sphere surface, showing accuracies around of 50 μm both in radius error (R_{NOMINAL}-R_{MEASURED}) and in distance between centres error. In

This paper presents an intrinsic and extrinsic LTS-AACMM calibration method, the calibration procedure being performed in a single step with the LTS already mounted in the AACMM, with no need to previously characterize the LTS-Contact probe set geometry by means of calibration methods on CMM. The developed method also avoids the use of approximated techniques to optimize the LTS position and orientation subsequent to the assembly of the sensor in the arm; techniques that are based on contact measurement and digitalization error of gauge primitives in several trajectories. These approximated techniques use estimated initial values of sensor position and orientation and are common practice in almost all commercial AACMM-LTS systems. This achieves a simple and cheap calibration method for the final user, required for any portable measurement equipment. By means of the use of a gauge object that materializes points in different planes with respect to a local reference frame, it is possible to obtain the equation of the sensor laser plane, its perspective transformation matrix and the necessary conjugated pairs of points in the LTS frame and the AACMM frame for the extrinsic calibration in a single operation in any AACMM image capture posture. The experimental results show the repeatability of the calibration process by means of digitalization of gauge primitives, with suitable accuracies for AACMM-LTS digitalization systems.

A procedure of kinematic calibration for AACMMs has also been presented. This method is based on the continuous capture of arm positions by directly probing the centre of the spheres of a gauge ball bar by way of a self-centring kinematic coupling probe. Oppositely, current methods are based on the capture of identification data probing surface points of geometrical primitives of different gauge objects. Parameter identification relies on a Levenberg-Marquardt scheme with an objective function including terms of error in distances and terms of standard deviation which allow to consider the influence of arm repeatability, given its capacity to probe the same point from different orientations.

Contact and non contact measurement of a gauge plane to obtain an estimation of position and orientation of LTS coordinate system in AACMM last frame by optimization.

Model definition posture of FARO AACMM with D–H convention [

Balls measured and distances between sphere centres calculated [

Perspective projection of pin-hole camera model without distortion [

The global coordinates of a point

Gauge object with calibration points [

Gauge object contact measurement. Alignment of AACMM and calibration object coordinate systems.

Image capture for LTS intrinsic calibration in AACMM calibration position.

Coordinate systems and transformations in calibration pose.

Gauge plane: (a) Contact measurement for nominal data on AACMM global frame. (b) Digitalization.

System repeatability and calibration process influence: (a) Angle between normals. (b) Projected central point error.

Calibration influence on system repeatability: (a) Angle between normals mean repeatability. (b) Projected central point error mean repeatability.

Accuracy estimation of the whole system. Radius and centre distance errors digitizing a reference sphere.

Identified values for the model parameters by L-M algorithm [

_{i} |
_{i}^{o}) |
_{i} |
_{0i} ^{o}) | |
---|---|---|---|---|

1 | 0.036962 | −90.052249 | −0.000002 | −0.126434 |

2 | 0.102485 | 90.044751 | 47.891183 | 14.942165 |

3 | 0.097868 | −90.020699 | 645.780523 | −88.99688 |

4 | −0.133079 | 90.068899 | 54.240741 | −3.636896 |

5 | 0.057606 | 90.011014 | 615.242600 | 89.770488 |

6 | 0.367275 | −0.522698 | 0.150712 | −0.878373 |

| ||||

_{probe} |
_{probe} |
_{probe} |
||

0.367276 | 139.450887 | 54.657060 |

Quality indicators for the identified set of model parameters over seven ball bar locations (10780 AACMM positions) [

Causing Pos. | POS2 | Causing Pos. | POS 1 |

Causing Dist. | D1 | Causing Sph. | B1 |

Min. | 0.005550 | Causing Coord. | Z |

Causing Pos. | POS1 | Min. | 0.035286 |

Causing Dist. | D2 | Causing Pos. | POS4 |

Causing Sph. | B6 | ||

Causing Coord. | Y | ||