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Underwater Vehicles Part 3

Tham khảo tài liệu 'underwater vehicles part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Guidance Laws for Autonomous Underwater Vehicles 71 and cosu 0 sinup R Up 0 1 0 - sinup 0 cosup 53 respectively. Hence the full rotation can be represented by R Xp Up R Xp R Up 54 such that E t R Xp up T p t - Pp. - . 55 where e t s t e t h t T e R3 represents the along-track cross-track and vertical-track errors relative to Pp m decomposed in the path-fixed reference frame. The path-following control objective is identical to 32 and e t can be reduced to zero by assigning an appropriate steering law to the velocity of p t as well as a purposeful collaborative behavior to Pp m . Specifically the steering law involves Xr e arctan 56 which is equivalent to 21 with A 0 used to shape the convergence behavior toward the xz-plane of the path-fixed frame and vr h arctan h t Ựe t 2 A2 p 57 used to shape the convergence behavior toward the xy-plane of the path-fixed frame see Fig. 8. Also Pp m moves collaboratively toward the direct projection of p t onto the x-axis of the path-fixed reference frame by m U t cos Xr e cosưr h ỵs t lPp m l 58 where Y 0 and pp m ựxp m 2 yp m 2 zp m 2 . In sum four angular variables 50 51 56 and 57 are used to specify the 3D steering law required for path-following purposes. Fortunately these variables can be compactly represented by the azimuth angle X Xp Up Xr Ur atan2 f Xp Up Xr Ur g Xp Up Xr Ur 59 where 72 Underwater Vehicles Fig. 8. The main variables associated with steering for regularly parameterized 3D paths f Xp Up XrU cos Xp sin Xr cosUr - sin Xp sin Up sin Ur sin Xp cos up cos Xr cos ur 60 and g Xp Up Xr ur - sin Xp sin Xr cos Ur - cos Xp sin Up sin Ur cos Xp cos up cos Xr cos up 61 and the elevation angle u up Xr ur arcsin sin up cos Xr cos ur cos up sin ur . 62 Through the use of trigonometric addition formulas it can be shown that 59 is equivalent to 19 in the 2D case i.e. when up ur 0 . 5.1 Path parameterizations Applicable arc-length parameterizations of straight lines and helices are now given. 5.1.1 Parameterization of .