﻿ 基于支持向量机的水下航行器操纵运动在线建模
 舰船科学技术  2016, Vol. 38 Issue (6): 81-85 PDF

Online modeling of underwater vehicles' maneuvering motion by using support vector machine
DU Wei, XU Feng, XIAO Tao, YANG Lu-chun, WANG Ming-jian, DU Xiu-qun
Wuhan Second Ship Design and Research Institute, Wuhan 430205, China
Abstract: This document presents the online maneuvering modeling for underwater vehicles. The hydrodynamic derivatives of MARIUSAUV are identified by using the incremental support vector machines. Besides, to improve the identification precision, integral sample structure for identification is applied for the construction of the in-out sample pairs. The data samples, i.e., surge velocities, sway velocities, yaw rates and rudder angles, are generated from the maneuvering simulation by using the hydrodynamic derivatives which are obtained from captive model tests. The comparison between the experimental hydrodynamic derivatives and the identified hydrodynamic derivatives demonstrates the validation of the parametric identification algorithm. This research is of great significance for underwater vehicles' on-line maneuvering modeling and control application.
Key words: system identification     underwater vehicles     hydrodynamic derivatives     maneuvering simulation
0 引言

1 MARIUS AUV 的操纵运动数学模型

 图 1 水下航行器的坐标系统 Fig. 1 Coordinate systems for underwater vehicles in maneuvering motion

 $\left\{ \begin{array}{l} m(\dot u-vr-{y_G}\dot r-{x_G}{r^2}) = {X_H} + {X_P}{\rm{ + }}{X_{\delta r}} \text{，} \\[5pt] m(\dot v + ur + {x_G}\dot r-{y_G}{r^2}) = {Y_H} + {Y_P} + {Y_{\delta r}} \text{，} \\[5pt] {I_z}\dot r \!+\! m\left[{{x_G}(\dot v \!+\! ur) \!-\!{y_G}(\dot u-vr)} \right] \!=\! {N_H} \!+ \!{N_P}\! + \!{N_{\delta r}} \text{。} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} \dot x = u\cos \psi + v\sin \psi \text{，} \\[5pt] \dot y = v\cos \psi-u\sin \psi \text{，} \\[5pt] \dot \psi = r \text{。} \end{array} \right.$ (2)

 $\begin{array}{l} {X_H} \!\!=\!\! \displaystyle\frac{\rho }{2}{L^3}{{X{'}}_{\dot u}}\dot u \!\!+\!\! \displaystyle\frac{\rho }{2}{L^2}\left( {{{X{'}}_{uu}}{u^2} \!\!+\!\! {{X{'}}_{vv}}{v^2}} \right) \!+\! \displaystyle\frac{\rho }{2}{L^4}{{X{'}}_{rr}}{r^2}{\rm{ \!\!+\!\! }}\displaystyle\frac{\rho }{2}{L^3}{{X{'}}_{vr}}vr \text{，} \\[10pt] {Y_H} \!\! = \!\! \displaystyle\frac{\rho }{2}{L^3}{{Y{'}}_r}ur \!\! + \!\! \displaystyle\frac{\rho }{2}{L^2}{{Y{'}}_v}uv + \displaystyle\frac{\rho }{{2u}}{L^5}{{Y{'}}_{rrr}}{r^3} +\displaystyle \frac{\rho }{2}{L^4}{{Y{'}}_{r\left| r \right|}}r\left| r \right| + \\[10pt] \;\;\;\;\;\;\; \displaystyle\frac{\rho }{2}{L^2}{{Y{'}}_{v\left| v \right|}}v\left| v \right| + \displaystyle\frac{\rho }{2}{L^3}{{Y{'}}_{\dot v}}\dot v + \displaystyle\frac{\rho }{2}{L^4}{Y_{\dot r}}^\prime \dot r \text{，} \\[10pt] {N_H} \!\! = \!\! \displaystyle\frac{\rho }{2}{L^4}{{N{'}}_r}ur \! + \! \displaystyle\frac{\rho }{2}{L^3}{{N{'}}_v}uv \! + \! \displaystyle\frac{\rho }{{2u}}{L^6}{{N{'}}_{rrr}}{r^3} + \displaystyle\frac{\rho }{2}{L^5}{{N{'}}_{r\left| r \right|}}r\left| r \right| + \\[10pt] \;\;\;\;\;\;\;\; \displaystyle\frac{\rho }{2}{L^3}{{N{'}}_{v\left| v \right|}}v\left| v \right| + \displaystyle\frac{\rho }{2}{L^4}{{N{'}}_{\dot v}}\dot v + \displaystyle\frac{\rho }{2}{L^5}{{N{'}}_{\dot r}}\dot r \text{。} \end{array}$ (3)

 \begin{aligned} {X_{\delta r}} = & \frac{\rho }{2}{L^2}{u^2}{X{'}_{\delta r\delta r}}\delta _r^2\;{Y_{\delta r}} = \\ & \frac{\rho }{2}{L^2}{u^2}{Y{'}_{\delta r}}\delta _r^{}\;{N_{\delta r}} \!=\! \frac{\rho }{2}{L^2}{u^2}{N{'}_{\delta r}}\delta _r^{} \text{，} \end{aligned} (4)

 ${X_p} = \frac{1}{2}\rho {L^2}{u^2}{X{'}_p} \text{。}$ (5)

 ${Y{'}_{r\left| r \right|}} =-\frac{{{F_2}}}{{{L^2}{F_0}}}{Y{'}_{v\left| v \right|}},\;\;{N{'}_{r\left| r \right|}} =-\frac{{{F_3}}}{{{L^2}{F_2}}}{Y{'}_{v\left| v \right|}} \text{。}$ (6)

MARIUS AUV 的物理参数和水动力导数分别如表 1表 2 所示。

2 增量式最小二乘支持向量机

 $\left[\begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\overrightarrow 1 \\ {\overrightarrow 1 ^{\rm{T}}}\;\;\;\varOmega + {C^{-1}}I \end{array} \right]\left[\begin{array}{l} b\\ \alpha \end{array} \right] = \left[\begin{array}{l} \overrightarrow 0 \\ y \end{array} \right] \text{。}$ (7)

 $y(x) = \sum\limits_{i = 1}^n {{\alpha _i}K} ({x_i},x) + b \text{。}$ (8)

 \begin{aligned} { \varOmega} (i,j) = & K({x_i},{x_j}),\;\;i,j = 1,2,\cdots ,t,\;\;\;\alpha (t) = \\ & {({\alpha _1},{\alpha _2},\cdots {\alpha _t})^{]rm T}},\;\;\;b(t) = {b_t}\text{。} \end{aligned}

 $y(x,t) = \sum\limits_{i = 1}^t {{\alpha _i}(t)K({x_i},x)} + b(t) \text{，}$ (9)

 $\left[\begin{gathered} 0\;\;\;\;\;\;\;\;e(t) \hfill \\ e{(t)^{\text{T}}}\;\;\Gamma (t) \hfill \\ \end{gathered} \right]\left[\begin{gathered} b(t) \hfill \\ \alpha (t) \hfill \\ \end{gathered} \right] = \left[\begin{gathered} \;\;\overrightarrow 0 \hfill \\ y(t) \hfill \\ \end{gathered} \right] \text{，}$ (10)

 ${e^{\text{T}}}\alpha (t) = 0 \text{，}$ (11)
 $eb(t) + \varGamma (t)\alpha (t) = y(t) \text{，}$ (12)

 ${e^{\text{T}}}P(t)eb(t) + {e^{\text{T}}}\alpha (t) = {e^{\text{T}}}U(t)y(t) \text{，}$ (13)

 ${e^{\text{T}}}P(t)eb(t) = {e^{\text{T}}}U(t)y(t) \text{，}$ (14)

 $b(t) = \frac{{{e^{\text{T}}}P(t)y(t)}}{{{e^{\text{T}}}P(t)e}} \text{，}$ (15)

 $\alpha \left( t \right) = P\left( t \right)\left( {y\left( t \right)-\frac{{e{e^{\text{T}}}P\left( t \right)y\left( t \right)}}{{{e^{\text{T}}}P\left( t \right)e}}} \right) \text{。}$ (16)

t 时刻，核函数矩阵式 t×t 方阵，表示如下：

 ${\textstyle{{\varGamma}} \left( t \right) \!\! = \!\! {{{\varOmega}} _t} + {{{C}}^{-1}}I \!\! = \!\! \left[\begin{gathered} k\!\!\left( {{x_1},\!\!{x_1}} \right) \!\!+\!\! {1 \mathord{\left/ {\vphantom {1 {C\; \cdots \;\;\;\;\;\;\;k\left( {{x_t},\;{x_1}} \right)}}} \right. } {C\;\; \cdots \;\;\;k\left( {{x_t},\;{x_1}} \right)}} \hfill \\ \;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\; \ddots \;\;\;\;\;\;\;\; \vdots \hfill \\ \;k\left( {{x_1},\;{x_1}} \right)\;\;\;\; \cdots \;k\left( {{x_t},\!\!{x_t}} \right) \!\!+\!\! {1 \mathord{\left/ {\vphantom {1 C}} \right. } C} \hfill \\ \end{gathered} \right] \text{，}}$ (17)

 ${{\varGamma}} \left( {t + 1} \right) = {{{\varOmega}} _{t + 1}} + {{{C}}^{-1}}I = \left[\begin{gathered} k\left( {{x_1},\;{x_1}} \right) + {1 \mathord{\left/ {\vphantom {1 {C\;\; \cdots \;\;\;\;\;\;\;\;k\left( {{x_t},\;{x_1}} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k\left( {{x_{t + 1}},\;{x_1}} \right)}}} \right. } {C\;\; \cdots \;\;\;\;\;\;\;\;k\left( {{x_t},\;{x_1}} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k\left( {{x_{t + 1}},\;{x_1}} \right)}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ \;\;\;\;k\left( {{x_1},\;{x_1}} \right)\;\;\;\;\;\;\; \cdots \;\;\;\;k\left( {{x_t},\;{x_t}} \right) + {1 \mathord{\left/ {\vphantom {1 {C\;}}} \right. } {C\;}}\;\;\;\;\;\;\;\;\;\;k\left( {{x_{t + 1}},\;{x_t}} \right) \hfill \\ \;\;\;k\left( {{x_1},\;{x_{t + 1}}} \right)\;\;\;\;\;\; \cdots \;\;\;\;\;\;\;\;\;k\left( {{x_t},\;{x_{t + 1}}} \right)\;\;\;\;\;\;k\left( {{x_{t + 1}},\;{x_{t + 1}}} \right) + {1 \mathord{\left/ {\vphantom {1 {C\;}}} \right. } {C\;}}\; \hfill \\ \end{gathered} \right] \text{。}$ (18)

 $\varGamma \left( {t + 1} \right) = \left[\begin{gathered} \varGamma \left( t \right)\;\;\;\;\;\;\;\;\;\;W\left( {t + 1} \right) \\ W{\left( {t + 1} \right)^{\rm T}}\;\;\;w\left( {t + 1} \right) \\ \end{gathered} \right]\text{。}$ (19)

 \begin{aligned} {A} = & {\left[\begin{array}{l} {A_{11}}\;\;E\\ {E^{\rm T}}\;\;d \end{array} \right]^{-1}} = \left[\begin{array}{l} {A_{11}}\;\;0\\ 0\;\;\;\; 0 \end{array} \right] + \left[\begin{array}{l} A_{11}^{-1}{A_{12}}\times\\ \;\;-I \end{array} \right]\\[5pt] & {\left( {d-{E^{\rm T}}A_{11}^{-1}E} \right)^{-1}}\left[{{E^{\rm T}}A_{11}^{-1}\;\;-I} \right] \text{。} \end{aligned} (20)

 $\begin{array}{l} P\left( {t + 1} \right) = \Gamma {\left( {t + 1} \right)^{-1}} = \\ \quad\quad\,{\left[\begin{array}{l} \;\;\;\Gamma \left( t \right)\;\;\;\;\;\;\;\;\;W\left( {t + 1} \right)\\[10pt] W{\left( {t + 1} \right)^T}\;\;\;\;\;w\left( {t + 1} \right) \end{array} \right]^{-1}} \!\!= \left[\begin{array}{l} \Gamma {\left( t \right)^{-1}}\;\;0\\ \;\;\;\;0\;\;\;\;\;0 \end{array} \right] +\\ \quad\,\,\begin{array}{l} \left. {\left[{\begin{array}{*{20}{c}} {\Gamma {{\left( t \right)}^{-1}}W\left( {t + 1} \right)}\\[10pt] {-I} \end{array}} \right.} \right]\left[{w\left( {t + 1} \right)-W{{\left( {t + 1} \right)}^{\rm{T}}}} \right.\\[10pt] \,\,\left. {\Gamma {{\left( t \right)}^{-1}}W\left( {t + 1} \right)} \right]\left[{W{{\left( {t + 1} \right)}^T}\Gamma {{\left( t \right)}^{-1}}-I} \right]\text{。} \end{array} \end{array}$ (21)

3 结果和分析 3.1 操纵运动仿真

 图 2 10°/10°Z 形仿真试验 Fig. 2 10°/10° zigzag test in horizontal plane
3.2 样本对构造

 $\text{纵向运动}:\text{ }\left\{ \begin{matrix} \text{输入}:\left\{ \begin{matrix} {{u}_{k}}\left| {{u}_{k}} \right|+{{u}_{k+1}}\left| {{u}_{k+1}} \right|,v_{k}^{2}+v_{k+1}^{2},r_{k}^{2}+r_{k+1}^{2},{{v}_{k}}{{r}_{k}}+{{v}_{k+1}}{{r}_{k+1}}, \\ \delta _{r,k+1}^{2}u_{k+1}^{2}+\delta _{r,k+1}^{2}u_{k+1}^{2},\ 1/{{u}_{k+1}}-{{u}_{k+1}}-1/{{u}_{k}}{{u}_{k}} \\ \end{matrix} \right. \\ \text{输出}:\left\{ {{u}_{k+1}}-{{u}_{k}} \right\} \\ \end{matrix} \right.$
 $\text{横向云动}{\text{:}} \left\{ \!\! \begin{gathered} \text{输入}{\text{:}}\left\{ \begin{gathered} {u_k}{r_k} + {u_{k + 1}}{r_{k + 1}},{u_k}{v_k} + {u_{k + 1}}{v_{k + 1}},r_k^3 + r_{k + 1}^3,{r_k}\left| {{r_k}} \right| + {r_{k + 1}}\left| {{r_{k + 1}}} \right|,\hfill \\ {v_k}\left| {{v_k}} \right| + {v_{k + 1}}\left| {{v_{k + 1}}} \right|,{\delta _{r,k}}u_k^2 + {\delta _{r,k + 1}}u_{k + 1}^2 \hfill \\ \end{gathered} \right\} \hfill \\ \text{输出}{\text{:}}\left\{ {{v_{k + 1}}-{v_k}} \right\} \hfill \\ \end{gathered} \right. \text{；}$
 $\text{摇首运动}{\text{:}}\left\{ \!\!\begin{gathered} \text{输入}{\text{:}}\left\{ \begin{gathered} {u_k}{r_k} + {u_{k + 1}}{r_{k + 1}},{u_k}{v_k} + {u_{k + 1}}{v_{k + 1}},r_k^3 + r_{k + 1}^3,{r_k}\left| {{r_k}} \right| + {r_{k + 1}}\left| {{r_{k + 1}}} \right|,\hfill \\ {v_k}\left| {{v_k}} \right| + {v_{k + 1}}\left| {{v_{k + 1}}} \right|,{\delta _{r,k}}u_k^2 + {\delta _{r,k + 1}}u_{k + 1}^2 \hfill \\ \end{gathered} \right\} \hfill \\ \text{输出}{\text{:}}\left\{ {{r_{k + 1}}-{r_k}} \right\} \hfill \\ \end{gathered} \right. \text{；}$
3.3 参数辨识

 图 3 Xu∣u∣, Xvv, Xrr 的在线辨识结果 Fig. 3 On-line identification results of Xu∣u∣, Xvv, Xrr

 图 4 Xδrδr, Xvr, Yr 的在线辨识结果 Fig. 4 On-line identification results of Xδrδr, Xvr, Yr

 图 5 Yv, Yrrr, Yδr 的在线辨识结果 Fig. 5 On-line identification results of Yv, Yrrr, Yδr

 图 6 Yv∣v∣, Xr∣r∣, Nr 的在线辨识结果 Fig. 6 On-line identification results of Yv∣v∣, Xr∣r∣, Nr

 图 7 Nv, Nrrr, Nδr 的在线辨识结果 Fig. 7 On-line identification results of Nv, Nrrr, Nδr

 图 8 Nv∣v∣, Nr∣r∣的在线辨识结果 Fig. 8 On-line identification results of Nv∣v∣, Nr∣r∣
4 结语

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