Stereo 3D Simulation of Rigid Body Inertia Ellipsoid for The Purpose of Unmanned Helicopter Autopilot Tuning

Transcription

1 nternatonal Journal of Engneerng Scence nventon SSN (Onlne): , SSN (Prnt): Volume 3 ssue 8 ǁ August 14 ǁ PP.8-35 Stereo 3D Smulaton of Rgd Bod nerta Ellpsod for The Purpose of Unmanned Helcopter Autoplot Tunng 1 Petar Getsov, Svetoslav Zabunov, 3 Maa Gadarova 1, Space Research and Technolog nsttute at the Bulgaran Academ of Scences, Bulgara 3 Sofa Unverst, Bulgara ABSTRACT : The current paper ams at presentng the capabltes and benefts of an onlne stereo 3D smulaton for the purpose of unmanned helcopter autoplot tunng. The parameters of the helcopter arframe are mportant for tunng the gans n the autoplot. The arframe s modelled as a rgd bod whose nertal propertes are full descrbed b the nerta ellpsod. The nerta ellpsod s another form of presentng the moment of nerta tensor of rgd bodes but nstead of usng a numercal approach the descrbed method mplements 3D graphcal vsualaton. The current paper focuses on the benefts from stereoscopc graphcal 3D presentaton of the nerta ellpsod and how such a method helps desgners and researchers analse, snthesse and tune unmanned helcopter autoplot algorthms. The smulaton, subject to the current materal, ma be observed at the followng web address: KEYWORDS: nerta ellpsod n stereo 3D smulaton, Rgd bod nerta ellpsod, Unmanned helcopter autoplot tunng.. NTRODUCTON The current artcle shows a new method of presentng the nerta ellpsod of rgd bodes and ts propertes usng a vsual onlne envronment. A smulaton that has unrestrcted access on the nternet demonstrates the nerta tensor of varous rgd bodes to researchers n unverstes and nsttutes around the world. The major features of the advsed smulaton are the stereo 3D envronment, n whch the nerta ellpsod of varous rgd bodes s demonstrated along wth translatons of the ellpsod and ts moment of nerta tensor to an pont n the bod reference frame. Along wth the nerta ellpsod, the prncpal aes of nerta are also dsplaed. The smulaton prnts the dagonaled translated moment of nerta tensor and the dagonalng rotaton matr (Fgure 1). The smulaton, descrbed n the current paper, ma be observed on web address: Fgure 1. Translated moment of nerta tensor and nerta ellpsod to the verte of a rectangular paralleleppedshaped homogeneous rgd bod and 3D graphcal vsualaton 8 Page

2 Stereo 3d Smulaton Of Rgd Bod nerta.... MATHEMATCAL FOUNDATONS OF THE DESCRBED 3D SMULATON To prove the consstenc and fdelt of the presented smulaton, a concse ntroducton to the nertal propertes of rgd bodes follows. Moment of nerta of a rgd bod about a gven as descrbes quanttatvel the bod nerta behavour durng a rotaton about that as. f a rgd bod has volume V then ts moment of nerta of about as OO s: OO r dm V, V r dv where s the bod denst at the locaton of elementar mass dm dv and r s the perpendcular radusvector from the as of rotaton OO to the elementar mass. About an gven as the rgd bod has a certan and generall dfferent moment of nerta. Each moment of nerta could be calculated usng the equaton above. An alternatve wa of calculatng the moment of nerta s b usng the relaton between the angular momentum and angular veloct: L, Here vector L s the angular momentum, and vector s the angular veloct of the rotatonal moton. Although useful, ths equaton s applcable onl to rotatons that are realed about a prncpal as of nerta. Onl then L and are parallel to each other and the relaton between them s presented usng a product wth a scalar value, as n the eample above. Such a rotaton s the rotaton of an aall smmetrc homogenous rgd bod about ts as of smmetr. Generall, vectors L and are not parallel, but a relaton connectng them stll ests and t s a tensor of second rang called moment of nerta tensor: O Ths tensor s n respect to a pont O, havng that as of rotaton passes through ths pont. To defne the relaton between L and, these two vectors should be presented n a matr form (one-row-matres): L ω O t follows that components of vector L are: L L L n ths general case, the momentar as of rotaton concdes wth. The nerta about ths as of rotaton creates an angular momentum about ths same as that s the projecton of the angular momentum vector L along vector. n matr form we have: Lω L ω ( ω s the transposed matr of ω ) Substtutng the angular momentum wth the product of angular veloct and the moment of nerta tensor elds: ωω Lω nωn ω ω, or the projecton of the L along the as of rotaton s derved from the magntude of and the scalar quantt. t s called a reduced moment of nerta from the moment of nerta tensor about a gven as of rotaton ω 9 Page

3 Stereo 3d Smulaton Of Rgd Bod nerta... defned b ts drecton unt vector n ω or n n vector form. Thus the reduced moment of nerta ma be epressed usng the components of the unt vector n and the moment of nerta tensor: n ωn ω n n n nn nn nn, ω Because vector n has unt length, ts components could be substtuted wth the drecton cosnes defnng the as of rotaton: ω cos cos cos cos cos cos cos cos cos At the same tme, the reduced moment of nerta ma be epressed through L and as follows: ω ω Lω L The nertal ellpsod enables the researcher to observe n stereoscopc 3D graphcal scene the moment of nerta tensor propertes, whch s essental for the followng autoplot snthess labour. The ellpsod s defned n such a manner that ts three aes are the recprocals of the square roots of the prncpal moments of nerta (Fgure 1,, 3 and 4): 1 1 a b c Fgure. The nerta ellpsod of the non-dsplaced moment of nerta tensor What the smulaton offers to the researcher The smulaton was developed n order to facltate researchers and scentsts whle studng the mechancal propertes of unmanned helcopter arframes and ther nput n autoplot algorthms. Further the smulaton helps engneers n unmanned helcopter autoplot analss and snthess. The smulaton helps the constructon of the nerta ellpsod of dfferent rgd bodes and the observaton of the prncpal moments of nerta. To make clear the effectveness of the smulaton whle solvng mechancs problems, several tasks are dsclosed as eamples. The solutons could be observed n 3D stereo mode. Each step of a soluton has ts 3 Page

4 Stereo 3d Smulaton Of Rgd Bod nerta... graphcal representaton, whch clarfes the noton of the used mathematcal formulae and presents the acqured results and solutons n graphcal manner. Hence the smulaton s useful n drawng practcal understandng among researchers. Task 1 Let s have a homogenous rgd bod. ts shape s a rectangular parallelepped and ts mass s 1 kg (Fgure ). Bod reference frame orgn concdes wth the centre of mass and ts aes are parallel to the bod edges. The bod dmensons are 1.,.5,. m along the O, O, O aes respectvel. The task s to calculate the moment of nerta tensor for the centre of mass c. Note: Wthout applng smlart transformaton, the moment of nerta tensor s dagonal c., and correspond to the prncple moments of nerta of the centre of mass along the bod reference frame coordnate aes. Ths follows from the bod homogenous and smmetrcal propertes. Fgure 3. Dsplacement of the moment of nerta tensor along the O and O aes. Soluton to task 1: Homogenous rectangular parallelepped moment of nerta tensor at ts centre of mass n a reference frame orented along the bod edges s derved b the followng formula: b c 1 c m a c, 1 a b m kg s the bod mass, and a 1m, b. 5 m and c. Here 1 the answer follows to be:.4 c m are the bod dmensons. Hence, 31 Page

5 Stereo 3d Smulaton Of Rgd Bod nerta... Task Calculate the translated moment of nerta tensor from task 1 at the verte of the bod that has onl postve coordnates n the bod reference frame. Soluton to task : The parallel as theorem (Hugens-Stener theorem) solves ths problem. (1) mrr 1 r r t t c m r mr r mr r r mr r m r mr r r mr r mr r m r r n the above equaton matr 1 s the 3 3 dentt matr, vector r s the translaton vector. The bod s homogenous and ts centre of mass and geometrcal centre concde. t follows that the translaton vector pontng to the verte of the bod wth onl postve coordnates s equal to (see Fgure 1). Fgure 3 shows a translaton of the moment of nerta tensor along both the O and O aes. Note that n the case of translaton along a sngle coordnate as, a translated tensor s dagonal. n the case of moment of nerta tensor translaton along two aes, four of the translated tensor s non-dagonal components are eroes. Task 3 Dagonale the translated moment of nerta tensor from task. Soluton to task 3: Usng the theor of smmetrc matres one could alwas dagonale a 3 3 smmetrc matr or smmetrc rang tensor. The dagonaled tensor wll have onl three non-ero values and the wll be found n the man dagonal. The other three values (products of nerta) found n the non-dagonaled tensor wll be represented b the transformaton used to dagonale the tensor. Ths transformaton s a smlart transformaton realed through applng a rotaton matr R. We can prove ths assumpton, takng nto account that under a certan rotaton R the prncple aes of the moment of nerta tensor concde wth the aes of the bod reference frame and the rotated tensor becomes dagonaled: D () D R R D D n D D D D and egenvalues of the pursued dagonaled tensor we have: (3) n DD nd ndd nd From the egenvectors Ths matr equaton shows a homogenous sstem of three lnear equatons havng a non-trval soluton onl f the determnant of ts coeffcents s ero: 1 (4) D D D Formula (4) has three real roots: 1 D, D and 3 D, hence the dagonaled tensor s: (5) D 1 3 D After we appl the smlart transformaton () to equaton (3) we get: n D D n D R R n D (6) n n n n n D R n D R 3 Page

6 Stereo 3d Smulaton Of Rgd Bod nerta... t follows that vectors (7) n n R n n R D n are the egenvectors of. D We see that D and have the same egenvalues. Calculatng the egenvalues of and substtutng them n (5) gves the wanted dagonaled tensor D. Analse equaton (6) helps n fndng the egenvalues of. Smlarl to (3), ths matr equaton derves a homogenous sstem of three lnear equatons. Ths sstem has non-trval soluton onl f the determnant of ts coeffcents s ero: (8) 1 The cubc equaton (8) elds three roots for , 3.98 and Substtutng these egenvalues n (5) leads to the sought soluton:.37 D 3.98 (see Fgure 1). 4.5 Task 4 Calculate the rotaton matr R that dagonales the tensor from task 3. Soluton to task 4: Each egenvalue of substtuted n (3) elds a correspondng soluton for to a unt egenvector: n (9) 1 3 D D D n n D1 D D3 1 1 j 1 k n D, f n D s constrant From (7) and (9) t follows that the rotaton matr R transforms vectors n to the orthogonal bass of the bod reference frame, j, k. However, the rotaton matr s defned b the drecton cosnes of one bass rotated to another bass: cos cos cos (1) R cos cos cos cos cos cos Vectors (11) n are unt vectors and ther components are equal to ther drecton cosnes, so: n1 R n n3 The last step s to fnd the egenvectors n. We use equaton (6) along wth the unt length of the egenvectors: 33 Page

7 Stereo 3d Smulaton Of Rgd Bod nerta... (1) 1 Sstem (1) enables the last equaton: and to be epressed through, usng the frst three equatons, and substtutng n (13) 1 1 From the last equaton of sstem (13) s found as follows: (14) 1 We fnd the values of 1 and b substtutng n the frst two equatons of (13). Note that the components n do not have defned sgn. There are two possbltes eldng an egenvector,, of egenvector n along a defned lne, but wth two possble opposte drectons. The correct drecton wll be found later. After fndng all three egenvectors n 1, n and n 3 ther components are substtuted n (11) to generate the sought rotaton matr R (see Fgure 1):.91 R However, f the ncorrect drectons (sgns n (14)) were chosen, matr R would eld an nverse (mproper) rotaton. Ths fact can be verfed b calculatng the determnant of R, whch should be 1: R 1 Ths means that matr R s a proper rotaton. n the case when the determnant s 1, matr R should be corrected to a proper rotaton b multplng t wth a negatve dentt (changng drecton of all egenvectors) or b changng the sgn of one of ts rows (changng drecton of one of the egenvectors). Vst to smulate other eamples.. CONCLUSON The descrbed n ths artcle smulaton of rgd bod propertes n a stereo 3D vrtual envronment enables researchers and engneers to observe setups that are mpossble to be created n laborator condtons. Such an approach reveals mportant nsghts for the scentsts workng on unmanned helcopter autoplot analss, snthess and tunng. Authors of the presented materal epress ther grattude to Assoc. Professor Vesela Decheva Page

8 Stereo 3d Smulaton Of Rgd Bod nerta... REFERENCES [1] Arnold, V.. Mathematcal Methods of Classcal Mechancs Second Edton. Sprnger-Verlag, 1989 [] Beer, P. F. & Johnston, E. R. Jr. Vector Mechancs for Engneers ffth edton. McGraw-Hll, nc. SBN , 1988 [3] Goldsten, H., Pool, C., and Safko, J. Classcal Mechancs thrd edton. Addson Wesle, Mam, 1 [4] Zabunov, S. Stereo 3-D Vson n Teachng Phscs, Phs. Teach., 5, 3, pp. 163, 1 [5] Zabunov, S. Rgd bod moton n stereo 3D smulaton. Eur. J. Phs., 31, 1345, 1 [6] Zabunov, S. Mathematcal Methods n E-Learnng Mechancs for Graduates Rotaton Matr. Phscs Educaton Journal, Vol. 7, ssue 4, pp. 1, 1 [7] Getsov P., D. Yordanov and S. Zabunov. Unmanned Arplane Autoplot Tunng. Journal of Engneerng Research and Applcatons, Vol. 4, ssue 7 (v.1), 14, pp.1-7 [8] Getsov P., D. Yordanov and S. Zabunov. Unmanned Aeral Vehcle Flght Control over a Crcular Path b Means of Manual Takeoff and Automatc Landng. Research nvent: nternatonal Journal of Engneerng And Scence, Vol.4, ssue 7 (14), pp [9] Getsov, P., D. Yordanov and S. Zabunov. Unmanned Aeral Vehcle Falure Modes Algorthm Modelng. OSR Journal of Engneerng (OSRJEN), Vol. 4, ssue 7 (14), Part, PP [1] Zabunov, S., P. Getsov and M. Gadarova. The Rgd Bod Moton Table n a Matr Form. nternatonal Journal of Scence and Research, Volume 3 ssue 7 Jul Page