# Development of a Dynamic Model of a Small High-Speed Autonomous Underwater Vehicle

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1 lp i Ml Sll Hi-Sp Uw Vil Hi N., il J. Silwll Bl p Elil p Eii Viii Pli Ii S Uii Bl, V 1 Eil: {, ilwll P : i p, Wii, KS 777 W L. N p p O Eii Viii Pli Ii S Uii Bl, V 1 i l i lp ll, ip w il. il l pp wii p p i ipl, w i qi - li ip il wi. p ii i l, wi i p i il, ili pl i w i il p i li l i wi pill i i ii. I. INROUION Wil ipi i l il i li, i pp, wi i il i, pil pl ip li qi i i li l ipi i ii i pil w-i UV. I ii il li i -- OF i l, w ii - i qi ip UV wi, wi l. i 7.9 l. ipl. UV li i wi l l ppli i. ii p p i l I i i Fi 1. II. RIGI-BOY KINEMIS. i i L X, Y, Z iill i i wi i 1,, 3 l,, -i G Fi. 1. L R B Si pi UV. 3 1 BLE I VLUES OF HE SYSEM BSE IMENSIONS N PRMEERS. P Vl Ui P Vl Ui L 39. i F 7. l R 1. i l i S = 1 i i ρ 1.9 l/ i 3. / i wi i i, j,. w i l q El i l Φ = {φ, θ, ψ} Fi. O q i i i { 1,, 3 } = Φ {i, j, } ψ θ ψ φ + ψ θ φ ψ φ + ψ θ φ ψ θ ψ φ + ψ θ φ ψ φ + ψ θ φ θ θ φ θ φ 1 w i. W = I, ii i, i i 1 =., -i li = {,, w} UV i li p li iil i l Ṙ = Φ w R = {X, Y, Z}. M, l li Ω = {p, q, } i -i i p Ω = φ 1 + θ + ψ 3 = φ θ ψi + φ θ + θ φ ψj + φ θ + θ φ ψ pi + qj + 3, -i l li El Φ = { φ, θ, ψ} l i Φ = GΦΩ w G =. 1 φ θ φ θ φ φ φ / θ φ / θ,g 1 = 1 θ φ θ φ φ φ θ

2 X 1 X X q G B Y Y 1,Y III. KIRHHOFF S EQUIONS OF MOION FOR BOY IN INVISI FLUI. Si-OF Ml q Z Z,Z 1 li P l H l i i UV B i Fi.. Si pi UV wi iil i i. B. Gi l I ili i--- i UV ill l l i i, l i l pl. W i l i li p,, w, ill i Fi 3, i llwi : 1 Wi l χ l σ: F = V χ, = V i χ i σ, w = V i χ σ, w χ = + w σ = w Pi-i i l l β: F = V β, = V i β, w = V β i, w = w β = + w 7 3 Yw-i i i-lip l γ l ν: F = V ν γ, = V ν i γ, w = V i ν, w γ = ν = w B ^ ^ w + Fi. 3. Si pi UV illi l i l i liip li. P = M + Ω H = + JΩ 9 F ii i iii li, Ki qi i i -i [1]: P = P Ω + FΦ, 1 H = H Ω + P + MΦ, 11 w F M l i il i i i. F ii, l il wi,,, l, i l. S l il G B pi, l, i, ppll ll. Ii 9 i 1 11 il: M + Ω =[M + Ω Ω]+FΦ, 1 + J Ω =[ + JΩ Ω]+[M + Ω ] + MΦ, 13 I ii i M, J,, llwi pi : 1 F pp ii ii, w ppi UV li il li w piipl B. H, ll - ii i. il i li i R =1. i i l L = 3 i, wi l i i il i liil l. F pp ii, UV i ll p pl., ll il i. ii il ll i l il l pl pi w piipl p il B, ill i Fi.,, ii, pli i

3 B Vi F = V ^ M F ^ B ^ ^ Fi.. Si pi UV pl pi i. /li il M = X Y, Zẇ J = J Kṗ J M q, J Nṙ = 1 w J = 1 R, J = J = 1 1 3R + L. Spii F UV, i l F ρ i ll =, w ρ i i w i iil li. i = R, pil, j i i- pi qi l = π 3 R = il = p [] 1 3F ρπr., X = 1, Y = Zẇ =, Kṗ =, M q = Nṙ = 3 J 1 w = ρ i ipl li, J = J = 1 + R ii pil li, 1 =, = β β, 3 = β [ β, ] = 1 [ ] l, 1 β = l,= B. S-S Mi - i UV i i i i ii P H = =i I i, UV i llw l p. i i i P Ω + FΦ, = 17 H Ω + P + MΦ, = 1 1 N p i il i =1 i R. []. Fi.. Si pi UV wi pill.. Sli S- Mi F li - i, i i l li Ω UV l ii li. ql, 17 1 llwi: w P = M. FΦ, = 19 P + MΦ, = IV. FORES N MOMENS. UV B Hi w- Nl Hi L UV plll l li wll pi l B. UV w iil i w-i, w l ppll. UV w ll G w pl B., UV ll i l i pp UV. F ll -OF l, li p l ll ppil,,. ql, l l ll l, pill F ξ li ˆF ζ l l w ξ ˆζ, wi li i pl B wi li l li, ill i Fi. M, pill pi ˆMη i l i ˆη i ppil pl B. i ii UV w i pill χ [, ] pi []. I Fi -, pi, ˆζ-F ˆF ζ, ˆMη i χ w p,, 1. l l B. I Fi, l χ = ii w. Fllwi 19, i i l i li l l i UV, wi il

4 l 3 1 l ^ F ζ 3 -F ^ ζ 1 V = V = V = 1 ζ. ^.1 i i i -.1 L -. ξ -. i i V = V = V = 1 Plil Fi ị -1 l η ^ - M -3 - M- ^η χ χ 1 1 l χ 1 1 V η ^.1 i.1 i. i P M l χ l χ.1 i i. M-. l -.1 M l χ l χ Fi.. NHL pil. Fi. 7. Hi ii pil i pi plil i. wi., i li F ξ = F ξ i ξ = F ξ χi +iχ i σj +iχ σ 1 ˆF ζ = ˆF ζ iˆζ = ˆF ζ i χi +χ i σj +χ σ w F ξ = ξ χp L ˆF ζ = ĈζχP L. ii ξ l ξ = πr L i p P i l p V P = 1 ρv = 1 ρ + + w 3 L pill, -p UV p i pii, i ˆM η = ˆM η iˆη = ˆM η σj i σ l l i pi UV i M = M i + M j + M., i llw Ni ˆM η = P + M P P, i + P, j + P, = M =[ + 1 ] i +[ + ] j +[ + w] P i P = 1 V χ i χ σj 1 V χ i χ i σ i li ip i il iii li l. W χ, i i i i 1 =.., p. F, p pi l M =i.., ll w [ M = ˆMη 1 V χ i χ] σ = χp L 3 σ 7 [ ] M = ˆMη 1 V χ i χ i σ = χp L 3 i σ χ =Ĉηχ 1 χ i χ ρl 3 9 F i Fi, i ii p p i Fi 7 wi p χ. W w χ>, i pi pw w χ<, i pi ww. Fi 7 ii Ĉ ζ < χ> ˆF ζ < ; i, pw li i. i l χ<. Fi 7 ii ξ < χ 1 F ξ < ; i, i p. Fi 7 7 w ii pi Ĉη l ii. F Fi 7, 7, 7, i ii Ĉζ, Ĉη, ip p w, Fi 7, ii ξ i lil wi p. i i ppi i ii wi χ, l l w i p < χ ll w i i i- i plil i. F ξ, pi V = w i i i p l pi w pp V =1 lw V =. i l pl i Fi 7 l χ 1.

5 B. UV B F M wi F w, F, F F w = F w 3 = F w i θi +θi φj +θ φ 3 F = F 3 = F i θi +θi φj +θ φ 31 F = F i 3 w F w =. L = i + j + pii B G., wi M w B ql M w = F w. M, ppll ll -i i M p = M p i. UV l S Hi w i Fi, w -i i i i plll wi l., l li i pl il i l li i pl il i. l pil N1-3 iil [3] wi l pl S. _ ~ V ~ _ V w B Fi.. Si pi UV lw-p il i wi i l. 1 Hil Fi: L δ 1 δ l li il i 1. Fi i i i B wi i 1,, 3. l -i i i 1 3 = 3 δ i δ 1 i δ δ 1 i j, =1, 33 W i i l i.., δ 1 = δ =, i li wi,, -i i. i li L il i, w pj li pl i p V = V β, wi li l -i. ii 1 = κ 1 + δ 1 = κ + δ i l i, w κ 1 = κ =. i il lpi il i i l, i Fi 9 L = L PS = PS 3 w = 1, P = 1 ρ V. pi F = 1 N 3 = [ κ L iκ ] i [ iκ +L κ ] 3 L 1 = i + j = i j B i i 1, pil., i B M = F =1,. Vil Fi: L δ 3 δ l li il i 3 w i Fi. Fi i i i wi i 1,, 3. l -i i i 1 3 = δ i δ i δ δ 1 i j, =3, 3 W i l i.., δ 3 = δ =, i li wi,, -i i. i li L il i, w pj li pl i p Ṽ = V ν, wi li l γ -i. Li γ 3 = κ 3 γ δ 3 γ = κ γ δ i l i w κ 3 = κ =., i Fi 9 L = L γ PS = γ PS 37 = +,, Fi V _ L N 3 _ Hil Fi ~ 1 ~ V L N = -, 3, ~ Vil Fi Si pi l. ~

6 w P = 1 ρṽ. pi F = 1 N = [ κ γ L iκ γ] i [ iκ γ+l κ γ] j 3 L 3 = i + = i B i i 3 i, pil., i B M = F =3,.. l F M Si ll l wll l B i F = F i + F j + F = F ξ + ˆF ζ + F w + F + F + F 1 + F + F 3 + F 39 M = M i + M j + M = M + M p + M w + M 1 + M + M 3 + M V. SEY-SE PIH-XIS MOION Eqi 7 i wi 3 ii ii pili i l pii i, w il i l wi l δ. B i i i pl, l χ =. - pl i l, G li l -i, l, i li δ i p V. I Fi 1, w p ii wi V - i ii δ =. l UV wi l. F Fi 1, w i wi il ii p 7 wi UV l pii i l. O, i i wi l il i p. F Fi 1. w l wi <, il i pl i li qi l wi V 1. Siil l p i Fi 1 w UV i l. w iil i Fi 11 wi i i. 1 1 l B 1 1 l Epill U-l Ri =.1 i =. i =.3 i =. i =. i Sp V l i F δ i l l 1 i F - Ri Fi Sll Sp V Fi. 11. Il i i l UV w i llw l. 1 l B l Epill U-l Ri =.1 i =. i =.3 i =. i =. i Sp V l i F 1 1 δ i 3 l l 1 i F -1 Ri Fi Sll Sp V Fi. 1. Il i i l UV w i llw l. l 1 1 -l H 3-l H -l H Sp V l Sp V Fi. 1. Il i UV wi-- i l i ii i li I Fi 11, w w ii,,, δ wi V l G pii B, w UV i l. W i δ l lw p lil i p. B, i l ii pii p. VI. KNOWLEGMEN i M. Bill B P. G Wi Nl Hi L l pii l w ili i i pp. l P. i Wl Viii i pi i i il. i w w pp i p Oi Nl R i N131. REFERENES [1] H. L, Hi, Uii i P, Si Eii, 193. []. I. F, Gi l O Vil, Wil, i, El, 199. [3] I. H.. E., Wi Si,, Nw Y, 199. [] W. B, pl p,.

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