Scientific Journls of the Mritime University of Szczecin Zeszyty Nukowe Akdemii Morskiej w Szczecinie 216, 45 (117), 29 ISSN 17-867 (Printed) Received: 1.8.215 ISSN 292-78 (Online) Accepted: 18.2.216 DOI: 1.1742/82 Published: 25..216 Determining the sptil orienttion of remote sensing sensors on the bsis of incomplete coordinte systems Ew Zls, Józef Snecki, Andrzej Klewski, Grzegorz Stępień Mritime University of Szczecin, Fculty of Nvigtion 1 2 Wły Chrobrego St., 7-5 Szczecin, Polnd e-mils: ew.zls@gmil.com, profjs@wp.pl, {.klewski; g.stepien}@m.szczecin.pl corresponding uthor Key words: remote sensing, Euler ngles, incomplete coordinte system, exterior orienttion of sensor, isometric trnsformtion, xes rottions Abstrct This rticle presents method of determining the sptil orienttion of mesuring sensors. This method is bsed on isometric spce trnsformtion of rigid body registered in n oblique coordinte system nd is dopted for photogrmmetric purposes. The pproch is bsed on incomplete coordinte systems used for determintion of trnsformtion prmeters. In this publiction n incomplete coordinte system is one without one of the xes nd in which the mtching points connected to primry nd secondry coordinte systems re on the two other xes. On the bsis of ngulr momentum, trnsloction of the beginning of coordinte system is determined first. The next step is to clculte the Euler ngles exterior orienttion of sensor. In this method the beginning (the center) of the coordinte system is ssocited with the sensor itself. This pproch, in comprison with the methods known from photogrmmetry nd remote sensing, llows one to reduce the points needed for trnsformtion. In cse of determining the Euler ngles two points re indispensble nd, in cse of moving the beginning of coordinte system, three points re essentil. At the end of this pper the nlysis of trnsformtion, bsed on independent control points (ICP), ws completed. Introduction Determining the sptil position of the elements of the world round us, set of points defined generlly by mens of distnces nd ngles, is crried out using vrious types of reference systems. These cn be creted with plnes, surfces or defined xes. Geodesy nd photogrmmetry use polr, sphericl nd rectngulr (Crtesin) coordinte systems. In these systems externl orienttion elements of the mesuring sensors re clculted generlly in simplified mnner by using so-clled sptil (smll-ngles) rottion mtrix (Sitek, 1991). In rigid body mechnics slightly different pproch is used. Clcultion of the sptil orienttion of the sensor is performed bsed on the points locted on the xis of rottion. Generlly, in this wy the Euler or Tit-Bryn ngles re determined nd clerly define the sptil orienttion of sensors. This pproch is lso free from the limittion of rotting coordinte systems by only smll ngles, to pply the smll-ngle functions (Brnowski, 21). In this publiction, the uthors set themselves the objective of combining the bove-mentioned pproches. The im is to use sptil trnsformtion of inclined systems (oblique coordinte system) known in rigid body mechnics nd their ppliction to the orienttion sensors nd instruments for surveng, photogrmmetry nd remote sensing, where different pproch in trnsformtion is used. This pproch cn be very useful in defining the positions of sensors mounted on Unmnned Aeril Systems (UAS). Simultneously, the uthors of this thesis show tht the orienttion of sensors nd instruments cn be Zeszyty Nukowe Akdemii Morskiej w Szczecinie 45 (117) 29
Ew Zls, Józef Snecki, Andrzej Klewski, Grzegorz Stępień mde using incomplete coordinte systems (without one dimension) without decline of finl ccurcy. Sptil orienttion of mesuring sensors The obliquity of sensors during mesurement mens tht, in prctice, the problem of trnsformtion between the two coordinte systems often occurs. With control points which re known in both coordinte systems, clcultion of the trnsformtion prmeters nd ppliction of conversion function is possible. As result, the coordintes of points in the secondry system re clculted on the bsis of coordintes of the sme points in the primry system. Due to the good legibility nd ese of describing to the vrious trnsformtions, the vector nd mtrix re pplied in clcultion (Czrnecki, 214). Sptil orienttion of two mutully ngled systems cn be determined on the bsis of Euler ngles (Figure 1). with combintion of three mtrix rottions round ech xis, which cn be written (Kurczyński, 214): X X 1 Y Y cos sin Z Z sin cos cos sincos sin x 1 sin cos y sin cos 1 z (2) ω rottion round x-xis; φ rottion round the y-xis; κ rottion round the z-xis; X, Y, Z coordintes in secondry system (field); x, y, z coordintes in primry (sensor); X, Y, Z vector of trnsltion (shift of the system); λ the coefficient of chnge of scle (in isometric trnsformtion). Nrrowed down to only the ngulr elements, the trnsformtion is written in the form (Sitek, 1991; Kurczyński, 214): X Y Z 11 21 1 12 22 2 1 2 x y z () Figure 1. The orienttion of sptil systems using Euler ngles (Czrnecki, 214) Rottions tht define the reltionship between the xes of systems, my be expressed through cosines of ngles which re creted together by the vrious xes (herefter successive rottions xes re mrked with ' nd "), nd represented by mtrix B: x, x cosx, y cosx, z y, x cosy, y cosy, z z, x cos z, y cos z, z cos B cos (1) cos In photogrmmetry nd remote sensing, determintion of the sptil orienttion of sensor is performed In this mtrix elements contin the ngulr elements of the externl orienttion of the sensor in the implicit form. For this reson, these equtions re often linerized by expnding the function using the Tylor series nd pplng smll rottion mtrix, which uses the functions of smll-ngles (Czrnecki, 214; Kędzierski, Fryśkowsk & Wierzbicki, 215): 1 B 1 (4) 1 This pproch is resonble in the cse of smll-ngle rottions of systems (inclintion of sensor), nd in the opposite sitution the use of interpoltion or itertion lgorithms becomes necessry (Preuss, & Kurczyński, 211; Snecki et l., 215). The ppliction of sptil trnsformtions to describe the bsolute displcements of the spce structures (for exmple, the hull structure of vessel), bsed on increments of smll ngles, were presented by Niebylski & Klewski (215) (Preuss, Kurczyński, 211): Scientific Journls of the Mritime University of Szczecin 45 (117)
Determining the sptil orienttion of remote sensing sensors on the bsis of incomplete coordinte systems u u u xi zi u u u xo yo zo zx xx yx zx yy In the shortened nottion: yz xz xi yz zz zi U i = U ō + Φ B i +ε B i zx xi yz z i (5) U i the displcement vector of the observed point P i ; U ō vector of trnsltion; Φ rottion mtrix ; ε deformtion mtrix ; B i coefficient mtrix (of coordintes of P i points ). A common feture of the listed forms of trnsformtion is tht they re bsed on smll ngles of rottion. Menwhile, in the cse of the sensors mounted on, for exmple, UAV (Unmnned Aeril Vehicles), the ngles of externl orienttion my exceed 1, which leds to the need for dditionl processing (interpoltion or itertion) (Kędzierski, Fryśkowsk & Wierzbicki, 215). Incomplete coordinte systems Depending on whether the xes representing the coordinte system re determined on the plne (x, y) or in the spce (x, y, z), the system cn be flt or three-dimensionl. In this cse of using sptil system (without one xis), we would not hve their full record, nd hve to del with n incomplete coordinte system. Niebylski nd Klewski (215) use the concept of the incomplete system, s Crtesin system, which overcomes one or two coordinte xes. In prctice, the photogrmmetric pproch to the determintion of exterior orienttion of sensors using the eqution (2) (except the previously described simplifictions) elimintes the verticl xis Z, ssuming for ll points vlue Z =. By configuring next the eqution the spce trnsformtion to the plne is obtined, this is known s DLT (Direct Liner Trnsformtion) trnsformtion, nd cn be written: A1 x B1 y C1z D1 X A x B y Cz 1 (6) A2 x B2 y C2z D2 Y A x B y C z 1 The eqution (6) is n lterntive to isometric trnsformtion (by similrity) nd is used for the processing of imges which come from the non-metric sensors, for the flt ground. It contins eleven unknowns (A1, B1, C1,... D2), so to trnsform coordintes from the originl (x, y, z) to the secondry system (X, Y incomplete) six control points re needed. In prctice, in the cse of sensors mounted on the pltform (UAVs), the incomplete plne orthogonl coordinte system is often used to determine the position nd orienttion of the sensor (externl orienttion elements). The method of determining exterior orienttion The method of determining the ngulr exterior orienttion elements of the mesuring sensors consists of using n pproch known in rigid body mechnics, in which points re locted on the xes (before trnsformtion) (Jeżewski, 197; Brnowski, 21). YAW PITCH Top View Profile Rottion ROLL Front View Figure 2. Rottion of coordinte systems: Pitch, Roll, Yw Tit-Bryn ngles (Wikipedi, 215) Defining the next three rottions with Tit-Bryn ngles (s in Figure 2). With two points situted on the xes, providing n incomplete coordinte system, the ngulr orienttion of the sensor elements cn be determined through the following trnsformtions: 1. Pitch (φ) Z-xis plne ZX X Z tn (7) ZZ The point on the xis Z. After the ngle is clculted, the trnsformtion is performed for the φ ngle nd then the new coordintes re clculted. 2. Roll (ω) Z-xis plne YZ Y Z tn (7b) Z Z Zeszyty Nukowe Akdemii Morskiej w Szczecinie 45 (117) 1
Ew Zls, Józef Snecki, Andrzej Klewski, Grzegorz Stępień The point on the xis Z. After the ngle is clculted, the trnsformtion is performed for the ω ngle nd then the new coordintes re clculted.. Yw (ω) X-xis plne XY Y X tn (7c) X X The point on the xis X. After the ngle is clculted, the trnsformtion is performed for the κ ngle nd then the new coordintes re clculted. In this method, the setting of the xis (choice of plnes) is free. It is enough to tke the motion of the sensor s shown in Figure 2 nd pply the order of clcultion s shown in the reltionships (7). It is connected with the pproprite setting (reorienttion) of the xis nd subsequently with clculting ngles: Pitch, Roll, Yw. With this pproch individul ngles re not dependent on ech other nd there is the possibility of their independent clcultion. The verifiction of the method The method of determining exterior orienttion of the mesuring sensors hs been verified in the lbortory nd in the field. In the lbortory, the coordintes of the ten points (in [m]) nd ngles of rottion were ssumed, s follows: 2 2 15 15 4 4 1 p : 2 2 1 4 2 1 (8) 1 1 2 1 4 4 1 where the first row of the mtrix p re X, the second Y, nd the third Z coordintes; rottion ngles: ω := 2, φ := 15, κ := 25 (9) After rottions for the coordintes s dopted bove, the mtrix took the form (1). Where A is the mtrix clculted on the bsis of rottion ccording to (2) nd () for the ssumed vlues of the ngles (9), nd tkes the form (11). As the result of reverse trnsformtion, the set of points were found (12). Compring the coordintes of the points obtined by the mtrix (8) nd (12) it cn be seen tht the differences in the coordintes re less thn 1 15 m. These differences occur only for points lng on the xes of the coordinte system, nd re t negligible level. The terrin verifiction ws performed in similr mnner. For this purpose, the control points were founded nd the coordintes before nd fter rottion of the sensor (the totl sttion) were clculted in the sme wy. Subsequently, the inverse trnsformtion nd the origin coordintes were compred. Finlly, ssuming tht the distribution of errors is ccidentlly determined, the men error (stndrd devition) of mesured points nd the men error with which it hs been clculted, gve: [ VV ] m m.12 m, M.4 m (1) n 1 n V the difference between the length of the vector (loction of points) in the compred systems; n the number of points of nlyzed spce. 2.588.85 5.155 1.11 4.627 15.1 4.82 6.17 12.71 15.425 P 1 : A p.4 1.4.977 7.16 2.2.727 8.142 2.669 15.697 6.672 (1) 1 9.77.88 8.99 1.18 1.257.47 4.128 1.676 4.928 4.19 11 21 1 12 22 2 1 1 cos sin cos sin 2 cos sin 1 sin cos sin cos sin cos 1 11 12 1.875.48.259 A : 21 22 2.477.814..76.41.98 1 2 (11) 16 16 1.6651 2 2 15 15 2.9141 4 4 1 16 16 M 1 P 1 4.4411 2 2 2.591 1 4 2 1 (12) 16 16 1 2 1 1.111 4.4411 4 4 1 2 Scientific Journls of the Mritime University of Szczecin 45 (117)
Determining the sptil orienttion of remote sensing sensors on the bsis of incomplete coordinte systems Conclusions The presented method is bsed on rigid body trnsformtion of Tit-Bryn ngles. This pproch reduces the number of control points needed for trnsformtion (in both coordinte systems) to two. These points cn be locted t ny two xes of the coordinte system. The method thus llows determintion of the ngulr exterior orienttion of the sensor using n incomplete coordinte system, while simultneously reducing djustment points (compred to the trnsformtions commonly used in surveng, photogrmmetry nd remote sensing). In ddition: the presented method cn be used for ny ngles nd is not limited to smll ngles nd their functions; there is no need for trnsforming function lineriztion; there is no need for n dditionl interpoltion nd itertion. In nlyzing the ccurcy of this method, it should be emphsized tht in the lbortory, the comptibility of points fter the inverse trnsformtion ws t the level of numericl ccurcy of 1 15 m. Under field conditions the ccurcy ws within.12 m. Hence, the ccurcy is typicl for mesurements by totl sttion s result of instrumentl, environmentl nd personl errors. Tking this into ccount the results should lso be recognized s stisfctory. According to the uthors, this method my be n lterntive to those currently used in photogrmmetry nd remote sensing for the determintion of the sptil orienttion of the mesuring sensors. References 1. Brnowski, L. (21) Equtions of motion of spin-stbilized projectile for flight stbility testing. Journl of Theoreticl nd Applied Mechnics 51, 1. pp. 25 246. 2. Czrnecki, K. (214) Geodezj współczesn. Wrszw PWN.. Jeżewski, M. (197) Fizyk. Wrszw: PWN. 4. Kędzierski, M., Fryśkowsk, A. & Wierzbicki, D. (215) Oprcowni fotogrmetryczne z niskiego pułpu. Wrszw: WAT. 5. Kurczyński, Z. (214) Fotogrmetri. Wrszw: PWN. 6. Preuss, R. & Kurczyński, Z. (211) Podstwy fotogrmetrii. Wrszw: Wydwnictwo Politechniki Wrszwskiej. 7. Snecki, J., Stępień, G., Konieczny, J., Niebylski, J. & Klewski A. (215) Teledetekcj Wykorzystnie zdlnej informcji. Szczecin: Wydwnictwo Nukowe Akdemii Morskiej w Szczecinie. 8. Sitek, Z. (1991) Fotogrmetri ogóln i inżynieryjn. Wrszw Wrocłw: PPWK. 9. Wikipedi (215) [Online] Avilble from: https://en.wikipedi.org/wiki/ship_motions#/medi/file:rottions.png, [Access: 26 th August 215]. Zeszyty Nukowe Akdemii Morskiej w Szczecinie 45 (117)