﻿ 水下滑翔机稳态螺旋运动的时空尺度分析
 舰船科学技术  2019, Vol. 41 Issue (1): 71-75, 106 PDF

Analysis of the steady spiral motion of an underwater glider on temporal and spatial scales
YU Peng-yao, SHEN Cong, ZHEN Chun-bo, WANG Tian-lin
Naval Architecture and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China
Abstract: The underwater glider is a kind of energy saving underwater vehicle, and its turning motion can be approximately regarded as a part of steady spiral motion. In order to guide the turning motion of underwater glider in restricted waters (near the bottom or near the wall), the temporal and spatial scale characteristics of the steady spiral motion of an underwater glider are analyzed. Based on the kinematics and dynamics equations of the underwater glider, the simulation method of steady spiral motion is established, and it is verified by the comparison with the published results. Through the analysis of the steady spiral trajectory, the description method of temporal and spatial scales is given, and the research ideas of the temporal and spatial scales are established. Taking the Seawing underwater glider as an example, the variation law of the temporal and spatial scales with control parameters is revealed quantitatively, and the differences between the upward spiral motion and the downward spiral motion are found.
Key words: underwater glider     motion simulation     steady spiral motion     temporal and spatial scales
0 引　言

1 运动学建模

 图 1 坐标系与质量分布 Fig. 1 Coordinate systems and mass distribution
 ${{\dot b}} = {{{R}}_{{\rm{EB}}}}{{\upsilon }},{{\dot \varOmega }} = {{{R}}_{\Omega {\rm{B}}}}{{\omega }}\text{。}$ (1)

 $\begin{split} & {{{R}}_{{\rm{EB}}}} \!=\! \left(\!{\begin{array}{*{20}{c}} \!\!\!{{\rm{c}}\theta {\rm{c}}\psi }&{{\rm{s}}\varphi {\rm{s}}\theta {\rm{c}}\psi - {\rm{c}}\varphi {\rm{s}}\psi }&{{\rm{c}}\varphi {\rm{s}}\theta {\rm{c}}\psi + {\rm{s}}\varphi {\rm{s}}\psi }\\ \!\!\!{{\rm{c}}\theta {\rm{s}}\psi }&{{\rm{c}}\varphi {\rm{c}}\psi + {\rm{s}}\varphi {\rm{s}}\theta {\rm{s}}\psi }&{ - {\rm{s}}\varphi {\rm{c}}\psi + {\rm{c}}\varphi {\rm{s}}\theta {\rm{s}}\psi }\\ \!\!\!{ - {\rm{s}}\theta }&{{\rm{s}}\varphi {\rm{c}}\theta }&{{\rm{c}}\varphi {\rm{c}}\theta } \end{array}} \!\!\!\right)\text{，}\\ & {{{R}}_{{\rm{B}}\Omega }} = \left( {\begin{array}{*{20}{c}} 1&0&{ - {\rm{s}}\theta }\\ 0&{{\rm{c}}\varphi }&{{\rm{s}}\varphi {\rm{c}}\theta }\\ 0&{ - {\rm{s}}\varphi }&{{\rm{c}}\varphi {\rm{c}}\theta } \end{array}} \right)\text{。} \end{split}$ (2)
2 动力学建模

 图 2 螺旋转向运动的力学机理 Fig. 2 Mechanisms of turning spiraling motion

 $T{\rm{ = }}\frac{1}{2}{\left(\!\! {\begin{array}{*{20}{c}} {{\upsilon }} \\ {{\omega }} \end{array}}\!\! \right)^{\rm{T}}}{{M}}\left(\!\! {\begin{array}{*{20}{c}} {{\upsilon }} \\ {{\omega }} \end{array}} \!\!\right){\rm{ = }}\frac{1}{2}{\left(\!\! {\begin{array}{*{20}{c}} {{\upsilon }} \\ {{\omega }} \end{array}} \!\!\right)^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{{{M}}_t}}&{{{{C}}_t}} \\ {{{C}}_t^T}&{{{{I}}_t}} \end{array}} \right]\left(\!\! {\begin{array}{*{20}{c}} {{\upsilon }} \\ {{\omega }} \end{array}} \!\!\right)\text{，}$ (3)
 $\begin{gathered} {{{M}}_t} = {{{M}}_{{A}}} + ({m_{{\rm{rb}}}} + {m_{\rm{p}}} + {m_{\rm{b}}}){{I}} \text{，} \\ {{{C}}_t} = {{{C}}_{{A}}} - {m_{{\rm{rb}}}}{{{{\hat r}}}_{{\rm{rb}}}} - {m_{\rm{p}}}{{{{\hat r}}}_{\rm{p}}} - {m_{\rm{b}}}{{{{\hat r}}}_{\rm{b}}} \text{，}\\ {{{I}}_t} = {{{J}}_{{A}}} + {{{J}}_{{\rm{rb}}}} - {m_{\rm{p}}}{{{{\hat r}}}_{\rm{p}}}{{{{\hat r}}}_{\rm{p}}} - {m_{\rm{b}}}{{{{\hat r}}}_{\rm{b}}}{{{{\hat r}}}_{\rm{b}}} \text{。}\\ \end{gathered}$ (4)

 ${{P}} = \frac{{\partial T}}{{\partial {{\upsilon }}}},{{\varPi }} = \frac{{\partial T}}{{\partial {{\omega }}}}\text{，}$ (5)

 $\left( {\begin{array}{*{20}{c}} {{P}} \\ {{\varPi }} \end{array}} \right){\rm{ = }}{{M}}\left( {\begin{array}{*{20}{c}} {{\upsilon }} \\ {{\omega }} \end{array}} \right)\text{。}$ (6)

 $\left( {\begin{array}{*{20}{c}} {{{\dot P}}} \\ {{{\dot \Pi }}} \end{array}} \right){\rm{ = }}{{M}}\left( {\begin{array}{*{20}{c}} {{{\dot \upsilon }}} \\ {{{\dot \omega }}} \end{array}} \right)\text{。}$ (7)

 ${{\dot P}} = {{P}} \times {{\omega }} + {{R}}_{{EB}}^{\rm{T}}({{G}} + {{B}}) + {{{F}}_{\rm{B}}}\text{，}$ (8)
 ${{\dot \Pi }} = {{\Pi }} \times {{\omega }} + {{P}} \times {{\upsilon }} + {{{r}}_{{\rm{CG}}}} \times {{R}}_{{EB}}^{\rm{T}}{{G}} + {{{M}}_{\rm{B}}}\text{。}$ (9)

 $\left(\!\! {\begin{array}{*{20}{c}} {{{\dot \upsilon }}} \\ {{{\dot \omega }}} \end{array}} \!\!\right){\rm{ = }}{{{M}}^{ - 1}}\left(\!\! {\begin{array}{*{20}{c}} \begin{gathered} ({{{M}}_t}{{\upsilon }} + {{{C}}_t}{{\omega }}) \times {{\omega }}+ \\ {{R}}_{{\rm{EB}}}^{\rm{T}}({{G}} + {{B}}) + {{{F}}_{\rm{B}}} \\ \end{gathered} \\ \begin{gathered} ({{C}}_t^T{{\upsilon }} + {{{I}}_t}{{\omega }}) \times {{\omega }} + ({{{M}}_t}{{\upsilon }} + {{{C}}_t}{{\omega }}) \times {{\upsilon }}+ \\ \begin{array}{*{20}{c}} {}&{}&{} \end{array} {{{r}}_{{\rm{CG}}}} \times {{R}}_{{\rm{EB}}}^{\rm{T}}{{G}} + {{{M}}_{\rm{B}}} \\ \end{gathered} \end{array}} \!\!\right)\text{。}\!\!\!\!\!\!\!\!$ (10)
3 稳态螺旋运动的时空描述

 图 3 水下滑翔机的螺旋运动轨迹 Fig. 3 The spiral motion trajectory of an underwater glider

 图 4 运动速度随时间的变化规律 Fig. 4 Variation of the velocity with the time

 图 5 本文方法的验证 Fig. 5 Validation of the method presented in this paper
4 时空尺度的分析

 图 6 时空尺度随转角的变化规律 Fig. 6 Relationship between the temporal and spatial scales and the rotation angle

 图 7 时空尺度随纵向位置的变化规律 Fig. 7 Relationship between the temporal and spatial scales and the longitudinal position of the movable block

 图 8 时空尺度随净浮力的变化规律 Fig. 8 Relationship between the temporal and spatial scales and the net buoyancy
5 结　语

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