Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 4, pp. 855-870, Warsaw 2009 APPLICATION OF MODAL ANALYSIS SUPPORTED BY 3D VISION-BASED MEASUREMENTS Piotr Kohut Piotr Kurowski AGH-University of Science and Technology, Deparetment of Robotics and Mechatronics, Cracow, Poland; e-mail: pko@agh.edu.pl; kurowski@agh.edu.pl In the paper, applications of the 3Dvision techniques to themodal ana- lysis method are presented. The goal of the project was to develop a methodology for vibration amplitude measurements and a software to- ol for modal analysis based on visual data. For this purpose, dedicated procedures and algorithms based on the vision technique methods were developed. 3D measurements of vibrations and structure geometry we- re obtained by application and developing passive 3D vision techniques. The amplitude of vibrations was calculated for selected points on the structure. Necessary vision data were received from the high-speed digi- tal camera ”X-StreamVision” in formof ”avi”files thatwere used as the input data for the developed software tool. The amplitude of vibration displacements obtained fromvision-basedmeasurementswere treated as the inputdata for operationalmodal analysis algorithms. In this domain, the Balanced Realization algorithm has been used. Key words: passive 3D vision techniques, structure from motion, vibra- tion measurements, modal analysis 1. Introduction The recent tendency in the realm of experimental modal analysis is reduction of time associated with preparing and carrying out modal tests. Non-contact techniques of signal registration have conformed to the new tendency in the construction design, and they meet all contemporary modal-analysis require- ments. The basics of modal-analysis requirements can be classified as: test- accuracy increasing, increasing of frequency-bandwidth andmeasuring-points number, reducing testing-preparation time and result-analysis time, facilita- ting analyses and tests. Vibration measurement tools are employed in reali- 856 P. Kohut, P. Kurowski sation of modal analysis. Non-contact optical techniques of displacement and vibration measurement are often encountered (Chen et al., 2003; Freymann et al., 1996; Mitchell, 2005; Moreno et al., 2005; Peeters et al., 2004; Schmidt et al., 2003; Synnergren and Sjödahl, 1999; Van der Auweraer et al., 2002a,b; Vanlanduit et al., 1998; Zisserman and Hartley, 2004). Among many vario- us methods based on 3D optical measurements, two categories of systems in the area of vision techniques prevail: active and passive. In the case of active methods for the purpose of depth measurement, supplementary devices (e.g. lasers, LCD projector) for generating suitably formed light (e.g. in form of a regular grille) are used (Moreno et al., 2005; Van derAuweraer et al., 2002a,b, polytec). Great advantages of active methods are their high accuracy (up to pm, polytec), the downside being their cost. They are however expensive, but getting cheaper. Moreover, these methods are completely insensitive to diver- sity of texture in the scene and they are not always feasible, especially for modelling distant or fast-moving objects. Passive methods (Kohut and Kurowski, 2005, 2006; Peeters et al., 2004; Schmidt et al., 2003) refer to the measurement of visible radiation already present in the scene. In the case of this methods, depth measurements are carried out on the basis of image sequences captured by one ormore cameras. Theyacquire images of a scene observed fromdifferentviewpoints andpossibly different illuminations. Based on these images, the scene shape and reflectan- ce at every surface point is computed. Many systems comprise combination of passive imaging devices (one or more cameras) and active devices (lasers/ LCDprojectors)mutually calibrated (Mitchell, 2005; SynnergrenandSjödahl, 1999; Van der Auweraer et al., 2002a,b). A three-dimensional scene structure is determined bymeans of geometrical triangulation. Amongactive techniques used to modal estimation of parameters, the most popular are methods ba- sed on lasers (Chen et al., 2003; Freymann et al., 1996; Van der Auweraer et al., 2002a,b; Vanlanduit et al., 1998). There are two basic approaches: older technology engagesDoppler vibrometers or scanningvibrometers.Another ba- sic type of laser measurement systems are interferometric systems, including ESPI (Electronic Speckle Pattern Interferometry). They present a new tech- nology which uses holography and coherent and monochromatic property of laser rays. Vision systems for two and three-dimensional measurements of geometry and object motion are employed in modal analysis (Kohut and Kurowski, 2005, 2006; Peeters et al., 2004; Schmidt et al., 2003). They basically relay on passivemethods. A great challenge for designers of vision systems is to create systems estimating three dimensional scenes based on obtained images and Application of modal analysis... 857 external illuminations only. In contrast to the active techniques, the passive methods are often more flexible, but computationally more expensive and dependent on the structure of the scene itself. Stereovision methods belong to the most natural ones, since they make use of two images of the same scene to inference about three-dimensional properties of the scene. Another approach to stereovision techniques involves, in general, replacing a pair of cameras by a single moving camera. In this case, the single camera records two images of a scene at two different locations and at two different time moments. The reconstruction procedure is identical to stereovision: the three- dimensional structure is determined by means of triangulation, based upon two obtained images. The advantage of this approach is its low cost (single camera) and ergonomic properties. The paper explains the implementation of the method of ”structure from motion” belonging to the group of passive techniques. Vibration amplitude characteristics for selected structural points will be presented as well as 3D geometry of the structure, found on themanufactured test stand bymeans of devices for realisation of modal analysis and estimation of quantities charac- terising dynamic properties of the structure. 2. Experimental set-up In the experimental research, a test stand enabling automatic two-axis con- trol of a camera (Fig.1a) was designed and manufactured. A frame structure was built, in which the guiding-rail system enables straight-line motion of the camera. Additionally, a system realising rotational motion of the camera was built-in. In order to control the test stand, software making it possible to combine the hardware-software part of the standwith the software part of the vision system was created. The tool was developed for the purpose of modal analysis and estimation of the quantities characterising dynamic properties of the structure based on vision signals. The produced software comprises the set of functions and procedures written in theMatlab environment dedicated for the purpose of determining vibrations of visible areas of mechanical systems. System parameters: frequency range 0.1-150Hz measurement accuracy 0.08-1.00mm range of camera straight-line motion 1000mm camera rotation range 360◦ 858 P. Kohut, P. Kurowski Fig. 1. (a), (b) Developed and constructed vision system for 3Dmeasurements of geometry andmotion of analysed objects. (c) Developed andmanufactured tool for carrying outmodal analysis and estimation of quantities characterising dynamic properties of the structure based on vision signals. (d) Hardware specification (linear and revolutemotion based on twoMitsubishi servomotors) Application of modal analysis... 859 height of camera attachment to the frame base 350-1800mm resolution of measurement of angle location of the camera (1/1820)◦ resolution of measurement of camera location during straight-line motion 0.05mm VIOMAsoftware (Virtual InOperationModalAnalysis,Kurowski (2001)), which is available at theDepartment ofRobotics andMechatronics was exten- ded. VIOMA is a toolbox consisting of advanced numerical procedures imple- mented in theMatlab programming environment.Themain task ofVIOMA is to estimate parameters of the experimentalmodalmodel. Such estimation can be performed based on the classic experiment withmeasurement of the extor- tion force or based on exploitation analysis performed during regular work of the device examined. VIOMA has been developed at the Department of Robotics andMechatronics. It is commercially available and gained numerous positive references fromusers in educational and industrial sectors. Below, the general purpose algorithm of the created software is presented. The algorithm was designed in accordance with its stages that had been carried out during execution of the modal experiment. The first stage is connected with prelimi- nary analysis of the examined structure. In the second stage, a geometrical model of the examined object should be built. Third stage involves execution of measurements on the examined object. The purpose of the next stage is to prepare measurements for carrying out the further analysis. This stage is extremely important in the case of preserving data in form of time histories during the measurement. Data in such a form should be processed to a form acceptable by suitable modal analysis algorithms. The next two stages are: realisation of modal analysis and visualization of the results. In order to fulfil the abovementioned assumptions the software toolkit was introduced in the Matlab programming environment. In oreder to facilitate dealing with the whole set of tools, a Graphical User Interface – GUI was created (Fig.1c), integrating and consolidating all the tools in one coherent form. The GUI interface also allows integrating a tool with VIOMA toolbox. Basic properties of the tool are as follows: 1) Camera calibration based on the available tools (Camera Calibration Toolbox forMatlab byBouguet andPerona (1998); CalibrationToolbox forMatlab by Heikkilä (2000)). 2) Determination of vibration characteristics and amplitude, which is reali- sed in several stages according to the following algorithm: a) Areas are selected, in which vibration tracking will be executed. There should be at least a few characteristic points that can be 860 P. Kohut, P. Kurowski easily distinguished in each image frame. These points will be au- tomatically selected by the tracking algorithm and, as a result of this selection, vibrations will be determined in consecutive stages of the program (Fig.2a). b) Algorithm of vibration identification must be selected. Three diffe- rent algorithms arebeing currently implemented into the tool based on ’structure from motion’ techniques (with orthographic model, scaled and para-perspective one (Fig.2b). c) Carrying out vibration identification (Fig.2c). d) Visualization of vibrations and its registering (Fig.2d). 3) Modal analysis operation. Fig. 2. Example of developed software toolkit: (a) embedded objects ready for vibration-characteristic tracking, (b) selection of areas on objects subject to tracking procedure, (c) tracking result: white spots –markers subject to tracking, red spots – markers resulted from tracking, (d) time characteristic of vibration found after conducting tracking procedure In order to curry out modal analysis, a module of VIOMA software is applied. Data conversed by the module described above are ready to use by modal analysis modules. Application of modal analysis... 861 3. Methodology – structure from motion In order to obtain 3D geometry of an object and vibration measurements in selectedmeasurement points, algorithms based on the ’structure frommotion’ methodwere drawnup.The feature trackingmethodwas based on the tracker described byMa et al. (2004), Trucco and Verri (1998). The factorizationmethodbelongs to thegroupof sparsemethods: structure from motion. The key features of the factorization approach are as follows: the knowledge concerning the number of objects is not required, no initial segmentation is necessary and the measurement matrix is globally factorized into twomatrices (motionmatrix and structurematrix) that are highly robust to noise. It is simple to implement and gives very good results for objects viewed from large distances. The assumptions originally taken byTomasi andKanade (1991), Poelman and Kanade (1997) are summarized below: • The camera model is orthographic; the position of n image points (ufp,vfp) has been tracked in F frames (F ­ 3); n image points corre- sponding to scene points P. • The problem can be stated: given (ufp,vfp), the positions of n image points that have been tracked in F frames 1 ¬ f ¬ F, 1 < p ¬ n, compute motion of the camera from one frame to another. • Theaim is to track (ufp,vfp) in f frames for p: points.After subtracting themean 2D position, themeasurement equations can bewritten in the following form ufp = i ⊤ f sp vfp = j ⊤ f sp (3.1) where if – rotation sp – position. Themeasurement matrix can be calculated as follows W̃=RS (3.2) where R – rotation matrix, R=(i1, . . . ,iF ,j1, . . . ,jF) ⊤ R – shape matrix (in a coordinate system attached to the object cen- troid), S=(s1, . . . ,sP) The size of the measurement matrix is as follows: W̃=R2F×3S3×P . 862 P. Kohut, P. Kurowski The factorization method algorithm is based on SVD decomposition W̃=UΛV (3.3) in which Λ must be of rank 3. When noise is present, the adjusted matrices are defined Λ′ =Λ(1 : 3,1 : 3) U′ =U(:,1 : 3) V′ =V(:,1 : 3) (3.4) and constructed R̂=U′ √ Λ′ Ŝ= √ Λ′V ′⊤ (3.5) The factorization of W̃ is straightforward via SVD but is not unique W=UΛV W= R̂Ŝ= R̂(QQ−1)Ŝ=(R̂Q)(Q−1Ŝ) (3.6) (R̂Q)(Q−1Ŝ)= R̂(QQ−1)Ŝ)= M̂Ŝ= Ŵ Fortunately, two constrains can be added: • the norms of 3D vectors forming the rows of Rmust be unit, • in R, the i⊤i must be orthogonal to the corresponding j⊤i . The rows of the matrix R do not satisfy the constraints mentioned above but when looking for a (correction) matrix Q such that |mf|2 = i⊤i QQi⊤i =1 |nf|2 = j⊤i QQj⊤i =1 (3.7) mfnf = i ⊤ i QQj ⊤ i =0 new matrices R = R̂Q and S = Q−1R still factorize W̃, and the rows of R satisfy the constraints. In the case of a orthographic model of a camera, the method does not determine cameramotion along theoptic axis.Therefore, shape reconstruction of the resulting object is usually deformed.To avert the problem, twomethods are applied (Christy and Horaud, 1996; Poelman and Kanade, 1997; Tomasi andKanade, 1991) that enable approximationofperspective camera character. The first method adapts scaled orthographic model of a camera, also referred to as weak perspective; whereas the other one para-perspective projection. In the case of weak perspective, it is assumed that diversity in the object depth in the direction of optic axis is negligible compared to the distance fromwhich Application of modal analysis... 863 the object is recorded. The method introduces an effect of scaling the image coordinates by the ratio of focal length-to-depth. The second method much better constitutes approximation of the camera model, since apart from the scaling effect the effect of location is introduced.Thisalsomeans that itmodels nearer or farther objects as an effect of observation at different angles. The two methods impose other metric limitations on the matrix Q. Let xf, yf, and zf designate relative camera locations. Three models of the camera can be distinguished with the limitations presented in Table 1. Table 1.Three models of metric limitations |mf|2 =1 Orthographic |nf|2 =1 mfnf =0 |mf|2 = |nf|2 =1/z2f Weak perspective mfnf =0 |m1|2 =1 |mf|2/(1+x2f)= |nf|2/(1+y2f)= 1/z2f Para-perspective mfnf =0 |m1|2 =1 where zf – depth to the object center of mass xf,yf – components of translation between the origins of the camera and fixed coordinate system f – index indicating f-th frame. The next step of the before-mentioned methods is, based on metric limi- tations presented in Table 1, determination of thematrix Q. In this research, the correction matrix Q was determined by means of the Newton-Raphson method. The research provided algorithms and procedures enabling determi- nation of all three models, the purpose of which can be specified as follows: 1. Given, in the sequence of m image frames (m­ 3), n mutually corre- sponding image points xij, j=1, . . . ,n, i=1, . . . ,m 2. Objective: determine affinite cameramatrices Mi, ti and 3D points Xj such that the reprojection error is minimized for all Mi, ti,Xj min Mi,ti,Xj ∑ ij ‖xij − x̃ij‖2 = min Mi,ti,Xj ∑ ij ‖xij − (MiXj + ti)‖2 864 P. Kohut, P. Kurowski The following algorithm was developed and implemented for all the three cases: orthographic, scaled orthographic and para-perspective, according to metric limitations spelled out in Table 1: 1. Calculation of SVD of W̃=UDV 2. Specification of geometry and orientation: R̂=U′ √ D ′, Ŝ= √ D ′ V ′⊤ 3. Determination of the correction matrix Q by imposing metric limita- tions. Application of the Newton-Rasphonmethod 4. Determination of the motion matrix M and structure S 5. Setting the first camera as the reference system relative to global coor- dinates. 3.1. Feature tracking The feature tracking algorithm with pyramid decomposition (Harris and Stephens, 1988; Ma et al., 2004; Tomasi andKanade, 1991; Trucco andVerri, 1998) was based on the translation model in which the displacement h of the feature point x between two consecutive frames can be calculated byminimi- zing the sumof squaredifferences between two images Ii(x) and Ii+1(x+h) in a small window W(x) around the feature point x. Theminimization problem for the displacement h can be described as follows min h E(h)= ∑ x̃∈W(x) [I2(x̃+h)− I1(x̃)]2 (3.8) The closed-form solution is given by h=−G−1b (3.9) where G=   ∑ W(x) I2x ∑ W(x) IxIy ∑ W(x) IxIy ∑ W(x) I2y   b=   ∑ W(x) IxIt ∑ W(x) IyIt   (3.10) where: Ix, Iy are image gradients, It = I2− I1 – temporal image derivative. 4. Experiment – application of ’Structure from motion’ to modal analysis techniques The estimation ofmodalmodel parameters was carried out based on determi- ned vibration characteristics. For the estimation, a timemethodwas selected, Application of modal analysis... 865 whose input was fed with functions of internal and mutual correlations. For the purpose of improving data quality and smoothing the correlation func- tion, the average of several measuring sessions was applied. The correlation functionswere subsequently used for conductingmodal analysis. Since nome- asurement of exerting force was carried out, the only solution to the problem was execution of operational modal analysis. The Balanced Realization (BR) algorithm was implemented. The analysis was carried out in the whole me- asurement band, i.e. 0-200Hz. For the purpose of determining a stabilization diagram, models from the 2nd to 50th row were estimated. Figure 6 presents an exemplary stabilization diagram in the discussed case. It is noticeable that in spite of poor quality of time characteristics (Fig.5a) being analysed, the stabilization of poles is remarkably good, which means that for a given fre- quency, the poles for individual rows form stabilized vertical lines. In every analysis, a pole of frequency of 100Hz and relatively low damping coefficient (not exceeding 0.2%) occurs. In a classic measurement with the application of accelerometer sensors, this frequency could be the evidence of the adverse influence of interference coming from electrical mains. In the case of vision measurement, the sole explanation of the occurence of this frequency is the negative lighting effect on the conducted measurement. The developed method of ”structure from motion” with para-perspective model was tested on the data obtained from a series of simulations. After verification, experimental tests were conducted on the special laboratory steel frame stand (Fig.1). The upper part of the framewas considered. In order to enhance the accuracy of vibrationmeasurements (Kohut andKurowski, 2005, 2006), the upper beam of the frame was divided into three segments (Fig.3). In eachmeasurement session, themeasurementswere carried out on adifferent frame part. For each segment, 4 measurement sessions were carried out. The excitation was exerted by means of random noise. Based on the developed method, vibration amplitude characteristics were obtained for all measurement points of the segmented upper beam of the frame. Examplary vibration characteristic for a separatedmeasurement point is shown in Fig.5a. Additionally, by means of the developed method of ’structure from mo- tion’ with para-perspective model, geometry of the whole framewas obtained (Fig.5b). As a result of the conducted experiment, vibration characteristics were obtained for all measurement points (all parts of the examined frame). The algorithm of ’structure frommotion’ correctly specified the amplitude charac- teristics of vibrating object displacements. 866 P. Kohut, P. Kurowski Fig. 3. Examined upper beam of the frame divided into three parts. Each part of the frame was measured in a separatemeasurement session Fig. 4. Segment II of the upper beam along with a separatemeasurement point Fig. 5. (a) Vibration amplitude of the examined object for a selectedmeasurement point. Displacement components X, Y ,Z presented in the camera coordinate system. (b) Reconstruction of model geometry 4.1. Modal analysis of the upper section of the frame The results of experimental measurements were applied as input data to the algorithm of estimation of modal parameters based on visual datameasu- remnts.Modal analysis was carried out bymeans of the Balanced Realization algorithm in the frequency range of 2-200Hz. Order and poles of the modal Application of modal analysis... 867 modelwere selected on thebasis of the stabilization diagram.Thediagramwas created for the estimation of modal models from the 2nd to 50th order with a step 1. The obtained stabilization diagram is presented in Fig.6. Analysis of the diagram shows clear stabilization with strong vertical lines. Basing on the diagram, modal model parameters were estimated. Results are collected in Table 2. In Fig.7, a few selected modeshapes obtained during estimation process are additionally presented. According to the measurement data, the presented modeshapes are related only with the top beam of the measured frame. Fig. 6. Stabilization diagram obtained during estimation Fig. 7. Selection of the obtained modeshapes Table 2.Modal coefficients obtained after parameters estimation No. Frequency Damping No. Frequency Damping [Hz] coefficient [%] [Hz] coefficient [%] 1 10.63 3.02 8 99.92 0.23 2 33.01 2.76 9 109.72 1.38 3 43.91 0.99 10 121.46 0.32 4 55.46 2.22 11 131.87 1.23 5 74.19 1.53 12 146.05 1.53 6 76.48 1.32 13 165.37 0.29 7 87.90 1.19 14 191.60 1.93 868 P. Kohut, P. Kurowski 5. Conclusions In the project,methodology and software for realisation ofmodal analysis and estimation of quantities characterising dynamic properties of the structure based on visual data were developed. The carried out research proved that it is possible to apply vision systems to modal analysis. The developed tool facilitates vibration detection based on vision images andmodal analysis. The structure was created and a prototype of vision system dedicated for modal analysis purposes was manufactured. A laboratory test stand integrated with the vision system was designed and made. For realisation of the experiment, software, user’s graphical interface and algorithms controlling motion of the camera along two axes were elaborated. In relation to new demands of modal analysis, new methodology algori- thms and procedures enabling automation of the preliminary stage of modal analysis were designed, i.e. algorithms for automatic geometry representation andmeasurement points localization. As a result of the investigation and estimation of the modal model, it can be stated that frequencies anddamping factors found from the visionmethods correspondwith the results obtainedbymeansof classicalmethods (Kohutand Kurowski, 2005, 2006). The representation of 3D vibration forms indicates the necessity of takingup further researchon the improvement of spatial resolution of vibrations obtained from developedmethods and algorithms. References 1. 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Przedstawiono opracowaną metodykę pomiarów am- plitudy drgań oraz narzędzie programowedo realizacji analizymodalnej bazującej na danych wizyjnych. Opracowano dedykowane procedury i algorytmy wykorzystujące metody wizyjne. Amplituda drgań została obliczona w wybranych punktach pomia- rowych struktury. Trójwymiarowe pomiary drgań oraz geometrię struktury uzyskano poprzez zastosowanie i opracowanie trójwymiarowych pasywnych metod wizyjnych. Jako dane wejściowe do opracowanego narzędzia programowego użyto sygnały wizyj- ne otrzymane z „szybkiej” kamery cyfrowej X-StreamVision w postaci plików ’avi’. Uzyskana z systemu wizyjnego amplituda drgań stanowiła dane wejściowe do algo- rytmu operacyjnej analizy modalnej.W tej dziedzinie wykorzystany został algorytm zbilansowanej realizacji (ang.Balanced Realization). Manuscript received November 6, 2008; accepted for print May 11, 2009