Measurement of the stereosopi rangefinder beam angular veloity using the digital image proessing method ROMAN VÍTEK Department of weapons and ammunition University of defense Kouniova 65, 62 Brno CZECH REPUBLIC roman.vitek@unob.z http://www.unob.z Abstrat: - This paper deals with the measurement of the mehanial onstrution kinematial harateristis using the digital image proessing method. This method has been utilized to determine the angular veloity and the angular position of the stereosopi rangefinder beam during its operation. Key-Words: - vibration, measurement, digital, image, proessing, ross-orrelation Introdution In last years a projet aimed on the researh of the passive traking systems has been solved at the University of Defense in Brno, Czeh Republi []. This projet is foused on the researh and development in the area of the weapons and fire ontrol systems design (e.g. [2], [3], [4] and [5]). During the solution of this projet the funtional model of the passive optoeletroni rangefinder has been developed, whih has been using for determining the position of the target and for traking of this target as well. The design of the rangefinder mentioned above is obvious from Fig.. 2 4 5 Fig. Optoeletroni rangefinder Two digital ameras and 2 are plaed on the beam 3 of the rangefinder. The beam is onneted by means of the rotational part 4 to the head of the tripod 6. The spatial position of this beam is established by means of two synhronous brushless servomotors. One servomotor 5 is used for rotating the beam around the horizontal axis (so alled elevation motion); the other servomotor (plaed inside of the tripod head 6) is used for rotating the beam around the vertial axis (so alled the traversal motion). 3 6 7 The whole rangefinder struture is mounted on the stationary tripod 7. The position of the target with respet to the rangefinder is determined from the pair of images, aquired synhronously from the digital ameras, using the methods of the digital stereophotogrammetry [6], [7]. The traking of the target is ahieved by setting the appropriate angular veloity to the servomotors so as the angular motion of the rangefinder was in aord with the angular motion of the target with respet to the rangefinder position, whereas the angular veloity of the servomotors was determined from ontrol deviation α in both vertial and horizontal diretion. For the ontrol deviation in the horizontal diretion the following formula is valid (lens distortion is not onsidered) xi x α x = artan, () x where α x is the horizontal ontrol deviation, x i is the horizontal image oordinate of the target, x is the horizontal oordinate of the prinipal point and x is the prinipal distane of the amera in the horizontal diretion. For vertial ontrol deviation α y the relation is analogial. To ontrol the angular veloity of the both servomotors the proportional-integral (PI) ontroller is used in the basi form [8] t () () () ωd t = Kα t + α t dt τ, (2) where ω d (t) is desired angular veloity of the beam in the partiular diretion, α(t) is the ontrol deviation determined from the image information, K is the proportional onstant of the ontroller and τ is the time onstant of the integral part of the ontroller. ISSN: 792-64 23 ISBN: 978-96-474-243-
2 Problem Formulation The onstants K and τ of the PI ontroller were determined using the software simulation of the ontroller for extreme values of the probable target veloity and the target distane to ahieve the time optimal traking of the seleted target. Furthermore, the traked target has to be kept in the field of view of the used amera, whereas this field of view an be further redued to derease the amount of omputational operations needed for target position determination, therefore to minimize the time onsumption and maximize the measurement rate respetively. The example of the simulation result for the target moving with veloity 2 m.s - (72 km.h - ) at distane 2 m is shown in Fig. 2. α [rad].4.3.2. -. -.2.5.5 2 2.5 3 3.5 4 4.5 5 Fig. 2 Control deviation ourse Unfortunately, after applying the determined values of the K and τ into the funtional model of the rangefinder it has been fond out that the target ran out from the seleted area of interest, therefore the traking algorithm rashed immediately after marking the target. From visual inspetion the reason seems to be the higher real angular veloity of the rangefinder beam than it was set to servomotor aording to the estimated ontrol deviation. Further effort was aimed on finding the soures of this disrepany. 3 Problem Solution Preliminary analysis of the target image oordinates reord showed that despite of the set angular veloity the real angular veloity ould be signifiantly higher than the desired one and it hanges periodially in addition. The following soures of this periodial motion were assumed: vibrations of the rangefinder beam as of the elasti body, vibrations of the rangefinder beam as of the solid body elastially mounted to the servo (output shaft of the transmission), elastially mounted rangefinder as the solid body on the basis (tripod) and inorretly ontrolled servo. To onfirm the atual reason of the parasiti motion the series of the angular motion measurements was prepared. 3. The angular veloity of the rangefinder beam measurement analysis onsidering the usage of image information The angular displaement and angular veloity of the rangefinder beam was determined by means of two methods: a) evaluating the signal from the servomotor s resolver and b) omputing the rotational motion from the image aquired from digital amera, mounted on the appropriate plae on the rangefinder onstrution. Ad a) The servomotor is equipped with so alled resolver, whih is the rotary eletrial transformer used for measuring degrees of rotation. The instantaneous position of the servomotor shaft an be read throughout the ontroller bus as the integer number, whereas this number represents the number of inrements with respet to the set absolute zero. The angular displaement of the rangefinder beam α an be omputed using the formula 2iπ α =, (3) nr where i is the number of inrements read from the resolver, n is the number if inrements for one turn of the servomotor shaft, r is the ratio number of the gear plaed between the servomotor and the beam and π is the irular onstant. The resolver is relatively sensitive and preise sensor of angular displaement. For n = 2 6 and r = 35 the minimum theoretially distinguishable value of the beam angular displaement is approximately.7 μrad. On the other hand, the resolver signal is affeted by motion of the rangefinder struture in the feedbak and it annot be used for estimating the angular motion of the struture espeially in the ase of the elasti beam vibrations. Therefore, the resolver signal was used only as the supplemental information and the atual angular motion of the rangefinder struture was determined by the optial method presented hereafter. Ad b) If the digital amera is plaed on the moving struture, the relative motion of the amera with respet ISSN: 792-64 23 ISBN: 978-96-474-243-
to the observed objet takes effet in the hange of the image oordinates of the objet. In ase of the general spatial motion the reonstrution of the struture motion is relatively ompliated proess, whih usually requires the usage of greater number of the ameras. Also the alibration of this measurement is quite ompliated with emphasis on the preise determination their relative position. If the struture motion is simple (e.g. rotation around one axis only), the situation signifiantly simplifies and we an reonstrut the struture motion using only one amera, plaed on the appropriate plae. Hereafter we will suppose only rotating motion of the rangefinder around the vertial axis. The amera is mounted on the rangefinder so that the olumns of the amera detetor are parallel to the rotation axis. If the rotation axis goes through the projetion entre of the amera, the angle of the beam rotation α an be determined using the formula ( pi+ pi) α = artan 2, (4) + pi+ pi where is the prinipal distane of the amera, p i is the horizontal image oordinate of the observed objet in the beginning of the rotation and p i+ is the horizontal image oordinate of the observed objet in the end of the rotation. If the projetion entre of the amera lies off the rotation axis, the angle of the beam rotation α an be determined solving the formula in its impliit form pi+ ( ( Xi b) sinα + ( Zi a) osα + a) = (5) = X b osα + Z a sin α + b, (( i ) ( i ) ) where b is the distane between the projetion entre and the rotation axis in diretion parallel to the detetor plane, a is the distane between the projetion entre and the rotation axis in diretion perpendiular to the detetor plane, X i is the observed objet oordinate in the diretion parallel to the detetor plane in the beginning of the motion and Z i is the observed objet oordinate in the diretion perpendiular to the detetor plane in the beginning of the motion. The other variables are idential with formula (4). Comparing the formula (5) to formula (4) it is obvious that the value α depends not only on the position of the projetion entre with respet to the rotation axis, but also on the position of the observed objet with respet to the amera. Therefore, even if we know the oordinates of the projetion entre a and b (for example from the design doumentation) and if we use the relation between X i and Z i aording to the formula pi Xi = Zi, (6) we still have only one onditional equation (5) for two unknown variables α and Z i. It means that at least on more onditional equation has to be obtained, e.g. using some supplemental measurement. Substituting (6) into (5) we get the modified form of onditional equation pi pi+ Zi b sin ( Zi a) os a α + α + = (7) pi = Zi b osα + ( Zi a) sin α + b, and if we extrat the Z i at the both sides of this equation, we get the final formula pi b a a pi+ sinα + osα + Zi Zi Z = i (8) pi b a b = osα + sin α +. Zi Zi Z i It an be learly seen that for Z i >> a and Z i >> b it is valid a b and, (9) Zi Zi therefore the formula (8) an be written in form pi pi+ sinα + osα = () pi = osα + sin α, whih leads diretly to the formula (4). In the other words, if the observed referene objet is plaed far enough omparing to the design dimensions of the amera with respet to the rotation axis, the angle of struture rotation α an be omputed diretly by means of the formula (4) using the evaluated image oordinates of the observed objet from one amera, whih signifiantly simplifies the measurement. The simulation of the measurement has been made, when for the known rangefinder design dimensions and given range of the referene objet oordinates the image oordinates of the referene objet were omputed as the funtion of the angle α. Then the angle α was inversely omputed from these image oordinates using the both formulas (4) and (5). The differene of these omputations is shown in Fig. 3 as the Z i oordinate dependene. From this diagram it an be seen that if the objet distane is longer than 5 m, the relative error in angle α evaluation is lower than.%. If we know the ourse of the angle α with respet to time as the vetor of disrete values α with respet to the vetor of disrete values t, the angular veloity an be omputed as the numerial derivation of the vetor α with respet to time. Beause the angular displaement of the rangefinder beam between two onseutive frames is important, the simplest method of numerial ISSN: 792-64 232 ISBN: 978-96-474-243-
α/α [-].2..8.6.4.2 2 4 6 8 2 4 6 8 2 Z [m] Fig. 3 Relative error in angle α evaluation derivation was used to ompute the angular veloity ω in form α αi =, for i=.. n 2, () t i+ ω i ti+ i where α i and α i+ are the elements of the vetor α, t i is the element of the vetor t and n is the size of these vetors. 3.2 The angular veloity of the rangefinder beam measurement implementation To verify the hypotheses mentioned in the beginning of the Chapter 3 of this artile the following measurements has been arranged. A) Hypothesis: The rangefinder beam vibrates as the elasti body. Two ameras are mounted at the ends of the rangefinder beam. If the beam behaves as the elasti body (the motion in basi natural shape of vibration is supposed mainly), the reord of the angle α from the first amera will show the phase shift with respet to the reord of the other amera. Otherwise, the beam an be onsidered solid. B) Hypothesis: The rangefinder beam vibrates as the solid body elastially mounted at the output shaft of the transmission. One amera is mounted on the end of the beam; the other is mounted on the output shat of the transmission. Again, if the reord of the angle α is phase-shifted with respet to the reord of the angle α from the other amera, the onnetion between the output shaft of the servo is elasti. Otherwise, the onnetion is solid and the struture behaves as the solid body. C) Hypothesis: Elastially mounted rangefinder as the solid body on the basis (tripod). One amera is mounted on the end of the beam; the other is mounted on the head of the tripod. Theoretially, the head of the tripod should stay still during the rotation motion of the rangefinder. If the head of the tripod moves synhronously with the rangefinder, the reason of the parasite vibrations ould be the elastiity of the tripod. D) Hypothesis: Inorretly ontrolled servomotor. The rangefinder beam is dismounted from the output shaft of the transmission, so as it annot retroatively effet the servo behavior by its inertia. Camera is mounted on the output shaft of the transmission. If the servo response, measured by means of the amera, is phase or amplitude shifted with respet to the desired value of the angular veloity, the servomotor is ontrolled inorretly. Otherwise, the servo motor is ontrolled orretly, or the non-linear response of the servo is aused only by inertia of the rangefinder beam. The series of measurements has been made for onfigurations mentioned above, whereas for eah onfiguration the desired angular veloity of the beam has been set to values -.5 rad.s -, -. rad.s -, -.5 rad.s - and -. rad.s -, whih represents the range of the angular veloities used for traking of the ground targets. As the example, in the following pitures the results of the evaluated angular veloity of the rangefinder beam analysis for desired angular veloity ω =. rad.s - are shown for variant when the amera is mounted at one end of the beam and the amera is mounted at the other end of beam (variant A). ω [rad.s - ] -5 - -5 5 x -3-2 5 5 Fig. 4 Angular veloity of amera - A In Fig. 5 the angular veloity of the beam evaluated using the image information from amera is shown. ISSN: 792-64 233 ISBN: 978-96-474-243-
ω [rad.s - ] 5 x -3-5 - between the beam and the output shaft of the transmission behaves as absolutely stiff in the operational range of the traking veloities. The reord of the angular veloity ω of the amera mounted at the end of the beam and the angular veloity ω of the amera mounted at the tripod head is shown in Fig. 7 and Fig. 8. 5 x -3-5 -2 5 5 Fig. 5 Angular veloity of amera - A ω [rad.s - ] -5 - The similarity of these ourses was ompared using the normalized ross-orrelation method aording to the formula [9] R ωω ( Tj) = n 2 i= ( ) ( ( ti) ) ( Tj ti) ω ω ω + ω ( ) n 2 2 2 n 2 ( ( ti) ) ( Tj ti) ω ω ω + ω i= i= (2) where ω and ω are mean values of the vetors ω and ω respetively. For omparison only the motion parts of the reords were extrated. The result of the rossorrelation is shown in Fig. 6..8 ω [rad.s - ] -5-2 5 5 2.5 x -3 2.5.5 -.5 - Fig. 7 Angular veloity of amera - C R ω ω [-].6.4.2 -.5-2 -2.5 5 5 Fig. 8 Angular veloity of amera - C -.2 -.4 -.6 -.8-6 -4-2 2 4 6 Fig. 6 Cross-orrelation of ω and ω - A In an be seen from Fig. 6 that the ross-orrelation response displays almost perfet math in both amplitude and phase part of the ω and ω reords. It means that the rangefinder beam behaves as the solid body and the parasiti vibrations, displayed in the Fig. 4 and Fig. 5, are aused by elastiity of another part of the rangefinder design. The B) measurement onfiguration was analyzed with the same results, i.e. the onnetion Also in this ase only the motion part of these reords was used for analysis. Already from these time reords it is obvious that the tripod, whih should be absolutely stiff theoretially, shows the vibrations of relatively signifiant level. The orrelation response for the measurement onfiguration C) is displayed in Fig. 9. It an be seen from this diagram that even if the maximum of the orrelation response is in the zero point, the linear dependeny between ω and ω is weaker than in ase of the variants A) and B). ISSN: 792-64 234 ISBN: 978-96-474-243-
R ω ω [-].8.6.4.2 presented in this artile. This method an be used for the fast evaluation of the angular displaement and veloities of the ivil and mehanial strutures, where the diretion of the motion vetor an be estimated in advane and when the added amera mass does not effet the dynami harateristis of the struture signifiantly. -.2 -.4 -.6 -.8-6 -4-2 2 4 6 Fig. 9 Cross-orrelation of ω and ω - C 3.3 The angular veloity of the rangefinder beam measurement evaluation Taking into aount the results of the measurements mentioned above the following partial onlusions an be made. Variant A) Beause the reords of the rangefinder beam angular veloities at the ends of the beam do not show the differene in the both amplitude and phase part, it an be said that the beam behaves as the solid body in the range of the operating traking veloities. Variant B) Beause the reords of the angular veloities at the end of the beam and at the output shaft of the transmission do not show the signifiant differene in the both amplitude and phase part, it an be said that the onnetion of the rangefinder beam to the output shaft of the transmission an be onsidered solid in the range of the operating traking veloities. Variant C) Beause the reord of the angular veloity of the tripod head is not zero, it is obvious that the tripod is not stiff enough. Furthermore, the reord of the angular veloity at the end of the rangefinder beam shows signifiantly higher amplitude swing than the reord of the angular veloity at the head of the tripod and these reords show the phase shift in the transition part. Therefore, it an be said that the low stiffness of the tripod an be onsidered as the main reason of the parasiti vibrations of the rangefinder beam. In addition, the amplitude and phase shift between the motion of the beam and tripod head an mean that another flexible onnetion exists between the output shaft of the transmission and the tripod head (it ould be likely the baklash of the transmission). 4 Conlusion The simple method of the vibration harateristis measurement using the digitized image information is Referenes: [] Institutional researh plan MOFVT42 - Researh into passive opti-eletroni systems of automati target traking for fire-ontrol systems. The Researh and Development Innovation Information System of the Czeh Republi. http://www.isvav.z/researhplandetail.do?rowid= MOFVT42 [2] Jedlika L., Beer S., Videnka M.: Modelling of pressure gradient in the spae behind the projetile. In Proeedings Of The 7th Wseas International Conferene On System Siene And Simulation In Engineering (ICOSSSE '8). WSEAS Press, 28. p. -4. ISBN: 978-96-474-27-7. [3] Balla J., Popelinsky L., Krist Z.: Drive of Weapon with Together Bound Barrels and Breehes. In Proeedings Of The 5th Wseas International Conferene On Applied And Theoretial Mehanis (MECHANICS '9). WSEAS Presss, 29. p. 48-53. ISBN: 978-96-474-4-3. [4] Vitek R.: Influene of the small arm barrel bore length on the angle of jump dispersion. In Proeedings Of The 7th Wseas International Conferene On System Siene And Simulation In Engineering (ICOSSSE '8). WSEAS Press, 28. p. 4-8. ISBN: 978-96-474-27-7. [5] Mako M., Balaz T.: The design of the operator measuring in fire ontrol system. In Mathematial Methods And Computational Tehniques In Researh And Eduation. WSEAS Press, 27. p.27-275. ISBN: 978-96-6766-8-4. [6] Kraus K.: Photogrammetry, Volume, Fundamentals and Standard Proesses, Fourth Edition, Dümmler, 2, s. 397, ISBN: 978-3427786849 [7] Vitek, R. The mathematial model of spatial objet position determination using the image information. [Researh report of the projet 42]. University of Defense in Brno, 25. p. 32. [8] Balate, J.: Automati ontrol. Prague: BEN, 24. p. 664. ISBN: 978-8-73-48-3. [9] Orfanidis, S.J.: Optimum Signal Proessing. An Introdution. 2nd Edition, Prentie-Hall, Englewood Cliffs, NJ, 996. p. 59. ISBN: 978-2389383. ISSN: 792-64 235 ISBN: 978-96-474-243-