﻿ 树形多体系统动力学约束力算法
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 力学学报  2016, Vol. 48 Issue (1): 201-212  DOI: 10.6052/0459-1879-15-201 0

### 引用本文 [复制中英文]

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Liu Fei, Hu Quan, Zhang Jingrui. CONSTRAINT FORCE ALGORITHM FOR TREE-LIKE MULITBODY SYSTEM DYNAMICS[J]. Chinese Journal of Ship Research, 2016, 48(1): 201-212. DOI: 10.6052/0459-1879-15-201.
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### 文章历史

2015-06-03 收稿
2015-09-08 录用
2015-09-14 网络版发表

1 树形多体系统的描述

 图 1 树形多体系统的拓扑结构 Fig.1 Topology of tree-like multibody system

 图 2 树形系统中的任意刚体 Fig.2 Arbitrary rigid-body in tree-like system

$\begin{array}{l} H_j^T \cdot {H_j} = {U^{{N_j} \times {N_j}}}{\mkern 1mu} ,\;W_j^T \cdot {W_j} = {U^{(6 - {N_j}) \times (6 - {N_j})}}\\ W_j^T \cdot {H_j} = {{\bf{0}}^{(6 - {N_j}) \times {N_j}}}{\mkern 1mu} ,\;H_j^T \cdot {W_j} = {{\bf{0}}^{{N_j} \times (6 - {N_j})}} \end{array}$

 $\widehat P = \left[ \begin{array}{l} U{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array} \right.\left. \begin{array}{l} \widetilde P\\ U \end{array} \right] \in {R^{6 \times 6}}$ (1)

 $\widetilde P = \left[ \begin{array}{l} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {p_3}{\kern 1pt} {\kern 1pt} \\ {p_3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ - {p_2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_1}{\kern 1pt} \end{array} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. \begin{array}{l} {p_2}\\ - {p_1}\\ 0 \end{array} \right]$ (2)

 ${I_{Cj}} = \left[ \begin{array}{l} {J_{Cj}}\\ {\bf{0}} \end{array} \right.\left. \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\bf{0}}\\ {m_j}U \end{array} \right] \in {R^{6 \times 6}}$ (3)

2 任意树形系统约束力算法的一般形式 2.1 铰链处受力分解

 ${\pmb M}\dot {\pmb x} + {\pmb b}(\theta ,{\pmb Q},{\pmb F}_{\rm E} ) = {\pmb T}$ (4)

 ${\pmb F}_j = {\pmb H}_j \cdot {\pmb F}_{{\rm T}j} + {\pmb W}_j \cdot {\pmb F}_{{\rm S}j}$ (5)

 ${\pmb F}_j = {\pmb H}_j \cdot {\pmb F}_{{\rm T}j}$ (6)

2.2 运动学与动力学递推关系

 $\left. {\begin{array}{*{20}{l}} {{{\mathop V\limits^. }_j} = \widehat P_{c(j)}^{\rm{T}}{{\mathop V\limits^. }_{c(j)}} + {H_j}{{\mathop Q\limits^. }_j}{\rm{ }}{F_j} = {I_j}{{\mathop V\limits^. }_j} + {\rm{ }}\sum\limits_{m = 1}^M {{{\widehat P}_{j,o({j_m})}}{F_{o({j_m})}}} } \end{array}} \right\}$ (7)

 ${F_j} + {\rm{ }}\sum\limits_{m = 1}^M {{{\widehat P}_{j,o({j_m})}}} \cdot ( - {F_{o({j_m})}}) + ( - {I_j}{\mathop V\limits^. _j}) = 0$ (8)

2.3 系统递推关系的组集 2.3.1 单链结构的递推关系组集

 $\left. \begin{array}{l} {F_j} = {I_j}{\mathop V\limits^. _j} + {\widehat P_{j,o({j_m})}}{F_{o(j)}}\\ {\mathop V\limits^. _j} = {\widehat P^T}_{c(j)} + {H_j}{\mathop Q\limits^. _j} \end{array} \right\}$ (9)

 ${P_{{\rm{chain}}}} = \left[ \begin{array}{l} U\\ - {\widehat P_{n - 1}}\\ \\ \\ \end{array} \right.\begin{array}{*{20}{c}} {}\\ U\\ { - {{\widehat P}_{n - 2}}}\\ \begin{array}{l} \\ \end{array} \end{array}\begin{array}{*{20}{c}} {}\\ {}\\ U\\ \begin{array}{l} \ddots \\ \end{array} \end{array}\begin{array}{*{20}{c}} {}\\ {}\\ {}\\ \begin{array}{l} \ddots \\ - {\widehat P_1} \end{array} \end{array}\left. \begin{array}{l} \\ \\ \\ \\ U \end{array} \right] \in {R^{6n \times 6n}}$ (10)
 $\left. \!\! \begin{array}{l} {H }= {\rm diag}\{{\pmb H}_i \} \in {\pmb R}^{6n\times N} \cr { W }= {\rm diag}\{{\pmb W}_i \} \in {\pmb R}^{6n\times (6n - N)}\cr { I} = {\rm diag}\{{\pmb I}_i \} \in {\pmb R}^{6n\times 6n}\cr { F} = {\rm col}\{{\pmb F}_i \} \in {\pmb R}^{6n} \cr { F}_{\rm T}= {\rm col}\{{\pmb F}_{{\rm T}i} \} \in {\pmb R}^N \cr { F}_{\rm S} = {\rm col}\{{\pmb F}_{{\rm S}i} \} \in {\pmb R}^{(6n - N)} \cr \dot { V} = {\rm col}\{\dot{\pmb V}_i \} \in {\pmb R}^{6n}\cr \dot { Q} = {\rm col}\{ \dot{\pmb Q}_i \} \in {\pmb R}^N \end{array}\!\! \right\}$ (11)

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} {P_{{\rm{chain}}}}F = I\mathop V\limits^. \\ P_{{\rm{chain}}}^{\rm{T}}\mathop V\limits^. = H\mathop Q\limits^. \end{array} \end{array}} \right\}$ (12)
2.3.2 节点的递推关系组集

 ${ P}_{\rm node } = \left[\begin{array}{ccc|c} {P}_{o(j_1 )} & & & \\ & \ddots & & {\bf 0} \\ & & { P}_{o(j_ M )} & \\ \hline { P}_{{\rm n}j,o(j_1 )} & \cdots & { P}_{{\rm n}j,o(j_ M )} & { P}_{{\rm n}j} \end{array} \right] =\\ \qquad \left[\begin{array}{ccc|c} {\pmb U} & & & \\ & \ddots & & {\bf 0} \\ & & {\pmb U} & \\ \hline -\hat{\pmb P}_{j,o(j_1 )} & \cdots &- \hat {\pmb P}_{j,o(j_ M )} & {\pmb U} \end{array} \right]$ (13)

 $\left. \!\! \begin{array}{l} { H }= {\rm diag}\{{\pmb H}_i \} \in {\pmb R}^{6( M + 1)\times N} \\ { W }= {\rm diag}\{{\pmb W}_i \} \in {\pmb R}^{6( M + 1)\times (6( M + 1) - N)} \\ { I }= {\rm diag}\{{\pmb I}_i \} \in {\pmb R}^{6( M + 1)\times 6( M + 1)} \\ { F} = {\rm col}\{{\pmb F }_i \} \in {\pmb R}^{6( M + 1)} \\ { F}_{\rm T} = {\rm col}\{{\pmb F }_{{\rm T} i} \} \in {\pmb R}^N \\ { F}_{\rm S} = {\rm col}\{{\pmb F }_{{\rm S} i} \} \in {\pmb R }^{(6( M + 1) - N)} \\ \dot{ V} = {\rm col}\{\dot{\pmb V}_i \} \in {\pmb R }^{6( M + 1)} \\ \dot{ Q} = {\rm col}\{ \dot{\pmb Q}_i \} \in {\pmb R}^N \end{array}\!\! \right\}$ (14)

 $\left.\!\! \begin{array}{l} { P}_{\rm node}{ F} ={ I}\dot{ V} \\ { P}_{\rm node}^{\rm T} \dot { V} = { H}\dot { Q} \\ \end{array} \right \}$ (15)

2.3.3 系统级递推关系组集

 $\left. \!\! \begin{array}{l} { P}{ F} = { I}\dot{ V} \\ { P}^{\rm T} \dot{ V} = { H}\dot{ Q} \\ \end{array} \!\! \right \}$ (16)

 ${ P }= \left[\begin{array}{ccc|c} { P}_{{\rm ch}o(j_1 )} & {\bf 0} & {\bf 0} & {\bf 0} \\ {\bf 0} & \cdots & {\bf 0} & \cdots\\ {\bf 0} & {\bf 0} & {{ P}_{{\rm ch}o(j_ M )} } & {\bf 0} \\ \hline {{ P}_{{\rm n }j,{\rm ch} o(j_1 )} } & \cdots & {{ P}_{{\rm n }j,{\rm ch}o(j_ M )} } & {{ P}_{{\rm n }j} } \end{array} \right]$ (17)
 图 3 节点外接单链结构 Fig.3 Node with outboard chains

 ${ P}_{{\rm ch }o(j_m )} = \left[\!\! \begin{array}{ccccc} {\pmb U} & {\bf 0} & {\bf 0} & \cdots & {\bf 0} \\ { - \hat{\pmb P}^{{\rm chain }m}_{n_{ cm } - 1} } & {\pmb U} & {\bf 0} & \cdots & {\bf 0} \\ {\bf 0} & { - \hat{\pmb P}^{{\rm chain }m}_{n_{ cm } - 2} } & {\pmb U} & \cdots & {\bf 0} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {\bf 0} & {\bf 0} & \cdots & - \hat{\pmb P}^{{\rm chain }m}_1 & {\pmb U} \end{array} \!\! \right]$ (18)

 ${ P}_{{\rm n }j,{\rm ch}o(j_m )} = \left[ {\bf 0} \ \ \cdots \ \ {\bf 0} \ \ - \hat {\pmb P}_{j,o(j_m )} \right] \in {\pmb R}^{6\times 6n_{{\rm c}m} }$ (19)

 ${ P} = \left[\begin{array}{cc} { P}_{{\rm node }j} & {\bf 0} \\ { P}_{{\rm ch }c (j),{\rm node }j} & { P}_{{\rm ch }c(j)} \end{array} \right] =\\ \left[\begin{array}{ccc|c|c} { P}_{o(j_1 )} & {\bf 0} & {\bf 0} & {\bf 0} & {\bf 0} \\ {\bf 0} & \ddots & {\bf 0} & \vdots & \vdots \\ {\bf 0} & {\bf 0} & { P}_{o(j_ M )} & {\bf 0} & {\bf 0} \\ \hline { P}_{{\rm n }j,o(j_{1} )} & \cdots & { P}_{{\rm n }j,o(j_ M )} & { P}_{{\rm n }j} & {\bf 0} \\ \hline {\bf 0} & \cdots & {\bf 0} & { P}_{{\rm ch} c(j),{\rm n}j} & { P}_{{\rm ch}c(j)} \end{array} \right]$ (20)
 图 4 单链结构外接节点 Fig.4 Chain with outboard node

 ${ P}_{{\rm ch} c(j),{\rm n}j} = [-\hat {\pmb P }^{\rm T}_{c(j),j} \ \ {\bf 0} \ \ \cdots \ \ {\bf 0} ]^{\rm T}$ (21)

(1)由内到外逐层分辨出节点与单链结构，并将每个部分的各自的矩阵${ P}_{{\rm ch }x}$与${ P}_{{\rm n }y}$由外到内排列，作为整个矩阵${ P}$的对角线：其中${ P}_{{\rm ch }x}$按照式(10)进行定义，而${ P}_{{\rm n }y} = {\pmb U}$；

(2)在每个节点刚体所在行和列的对应位置补充由于耦合所产生的矩阵，并将所补充的矩阵具体化，得到整个系统的矩阵${ P}$：所补充的矩阵${ P}_{{\rm n }y,{\rm ch }x}$为分块行矩阵，其列数与矩阵${ P}_{{\rm ch }x}$的维度相同，且其中唯一非零项为$- \hat{\pmb P}_{{\rm n }y,{\rm ch }x}$，位于最后一列；所补充的矩阵${ P}_{{\rm ch }x,{\rm n }y}$为分块列矩阵，其行数与矩阵${ P}_{{\rm ch }x}$的维度相同，且其中唯一的非零项为$- \hat{\pmb P}_{{\rm ch }x,{\rm n }y}$，位于第一行；

(3)按照式(14)以及矩阵${ P}$中分块对角线上的矩阵下标顺序，定义余下8个组集矩阵；

(4)系统的动力学递推关系组集为式(16).

2.4 约束力算法的一般形式

 ${ F} = { H} \cdot { F}_{\rm T} + { W} \cdot { F}_{\rm S}$ (22)

 $\dot { Q} = { H}^{\rm T}{ P}^{\rm T}\dot{ V}$ (23)
 ${ W}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P}{ F }= {\bf 0}$ (24)

 ${ W}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P}{ W} \cdot { F}_{\rm S} = - { W}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P}{ H} \cdot { F}_{\rm T}$ (25)

 $\left.\begin{array}{l} { A}={ W}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P W} \\ { B}={ W}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P H} \\ { C }= { H}^{\rm T}{ P}^{\rm T}{ I}^{ - 1}{ P H } \end{array} \right \}$ (26)

 ${ A F}_{\rm S} = - { B F}_{\rm T}$ (27)

 $\dot{ Q} = { C} \cdot { F}_{\rm T} - { B}^{\rm T}{ A}^{ - 1}{ B} \cdot { F}_{\rm T} =\\ \qquad { H}^{\rm T}{ P}^{\rm T}({ I}^{ - 1}{ P}{ H }\cdot { F}_{\rm T} + { I}^{ - 1}{ P}{ W} \cdot { F}_{\rm S} )$ (28)

 ${ A} \cdot { F}_{\rm S} =\dot{ X}$ (29)

(1)计算系统的等效控制力${ F}_{\rm T}$；

(2)解方程(29)，得到系统约束力${ F}_{\rm S}$；

(3)根据公式(28)得到动力学解算结果.

3 任意树形系统约束力算法的串行化

3.1 关于矩阵${ A}$及方程组解法的讨论

 ${ A} = \left[\begin{array}{ccccc} {\pmb A}_n & {\pmb B}_{n - 1}^{\rm T} & & & \\ {\pmb B}_{n - 1} & {\pmb A}_{n - 1} & {\pmb B}_{n - 2}^{\rm T} & & \\ & {\pmb B}_{n - 2} & {\pmb A}_{n - 2} &{\pmb B}_{n - 3}^{\rm T} & \\ & & \cdots & \cdots & \cdots \\ & & & {\pmb B}_1 & {\pmb A}_1 \end{array} \right]$ (30)

 $\left.\begin{array}{l} {\pmb A}_i = {\pmb W}_i^{\rm T} ({\pmb I}_i^{ - 1} + \hat {\pmb P}_{i - 1}^{\rm T} {\pmb I}_{i - 1}^{ - 1} \hat {\pmb P }_{i - 1} ){\pmb W}_i {\rm }\\ \qquad i = n,n - 1,\cdots,1 \\ {\pmb B}_i = - {\pmb W}_i^{\rm T} {\pmb I}_i^{ - 1} \hat {\pmb P}_i {\pmb W}_{i + 1} \\ \qquad i = n -1,n- 2,\cdots,1 \end{array} \right\}$ (31)

 $\left.\begin{array}{l} {\pmb A}_i = {\pmb W}_i^{\rm T} \cdot ({\pmb I}_i^{ - 1} + {\pmb I}_{i - 1,O_i }^{ - 1} ) \cdot {\pmb W}_i \quad i = n,n-1,\cdots,1 \\ {\pmb B}_i = - {\pmb W}_i^{\rm T} \cdot {\pmb I}_i ^{ - 1} \cdot \hat{\pmb P}_i \cdot {\pmb W}_{i + 1} \quad i = n- 1,n- 2,\cdots,1 \end{array} \right\}$ (32)

(1) ${\pmb A}_j^{\rm t}$：位于矩阵${\pmb A}_j$处，代表节点j与其内接体$c(j)$间的耦合对节点j的影响，有

 ${\pmb A}_j^{\rm t} = {\pmb W}_j^{\rm T} {\pmb I}_{c(j),O_j }^{ - 1} {\pmb W}_j$ (33)

(2) ${\pmb B}_{c(j)}^{\rm t}$和 $({\pmb B}_{c(j)}^{\rm t} )^{\rm T}$：分别位于矩阵${\pmb A}_j$的下方和右方，表达了节点j与其内接体$c(j)$间的耦合对内接体$c(j)$的影响，有

 ${\pmb B}_{c(j)}^{\rm t} = - {\pmb W}_{c(j)}^{\rm T} {\pmb I}_{c(j)}^{ - 1} \hat{\pmb P}_{c(j)} {\pmb W}_j$ (34)

(3) ${\pmb D}_{j,o(j_m )}^{\rm t}$和$({\pmb D}_{j,o(j_m )}^{\rm t} )^{{\rm T}}$：分别位于矩阵${\pmb A}_j$所在行与矩阵${\pmb A}_{o(j_m )}$所在列的交点处和其转置的位置($m = 1,2,\cdots ,M$)，表达了节点j与其各个外接体o(jM间的耦合，且

 ${\pmb D}_{j,o(j_m )}^{\rm t} = - {\pmb W}_j^{\rm T} {\pmb I}_j^{ - 1} \cdot \hat{\pmb P}_{j,o(j_m )} {\pmb W}_{o(j_m )}$ (35)

(4) ${\pmb b}_{o(j_a ),o(j_b )}^{\rm t}$：位于节点j任意两个外接体$o(j_a )$和$o(j_b )$所对应的矩阵${\pmb A}_{o(j_a )}$所在行和矩阵${\pmb A}_{o(j_b )}$所在列的耦合项($a$和$b$遍历$1,2,\cdots ,M$中任选两数的所有数对)，表达了节点j的各个外接结构之间的耦合，且

 ${\pmb b }_{o(j_a ),o(j_b )}^{\rm t} = {\pmb W}_{o(j_a )}^{\rm T} \hat{\pmb P}_{j,o(j_a )}^{\rm T} {\pmb I}_j^{ - 1} \hat{\pmb P }_{j,o(j_b )} {\pmb W}_{o(j_b )}$ (36)

 ${\pmb A }_j = {\pmb W}_j^{\rm T} {\pmb I}_j^{ - 1} {\pmb W }_j + {\pmb A }_j^{\rm t}$ (37)

 ${\pmb B }_{c(j)} = {\pmb B }_{c(j)}^{\rm t}$ (38)

 $\left.\!\!\begin{array}{l} {\pmb E}_i^{(j)} = {\pmb B}_i^{(j - 1)} \cdot ({\pmb A}_{i + 2^{j - 1}}^{(j - 1)} )^{ - 1} \\ {\pmb G}_i^{(j)} = ({\pmb B}_{i - 2^{j - 1}}^{(j - 1)} )^{\rm T} \cdot ({\pmb A}_{i - 2^{j - 1}}^{(j - 1)} )^{ - 1} \end{array} \right \}$ (39)

 $\left.\!\!\begin{array}{l} {\pmb A}_i^{(j)} = {\pmb A}_i^{(j - 1)} - {\pmb E}_i^{(j)} \cdot ({\pmb B}_i^{(j - 1)} )^{\rm T} - {\pmb G}_i^{(j)} \cdot {\pmb B}_{i - 2^{j - 1}}^{(j - 1)} \\ {\pmb B}_i^{(j)} = - {\pmb E}_i^{(j)} \cdot {\pmb B}_{i + 2^{j - 1}}^{(j - 1)} \\ {\pmb D}_{i,o(i_m )}^{(j)} = {\pmb D}_{i,o(i_m )}^{(j - 1)} - {\pmb D}_{i - 2^{j - 1},o(i_m )}^{(j - 1)} \cdot {\pmb G}_i^{(j){\rm T}} \\ {\pmb b}_{i,i_b }^{(j)} = {\pmb b}_{i,i_b }^{(j - 1)} - {\pmb b}_{i - 2^{j - 1},i_b }^{(j - 1)} \cdot {\pmb G}_i^{(j){\rm T}} \\ \dot {\pmb X}_i^{(j)} = \dot {\pmb X}_i^{(j - 1)} - {\pmb E}_i^{(j)} \cdot \dot {\pmb X}_{i + 2^{j - 1}}^{(j - 1)} - {\pmb G}_i^{(j)} \cdot \dot {\pmb X}_{i - 2^{j - 1}}^{(j - 1)} \end{array} \right\}$ (40)

 $\bar { A} = \left[\begin{array}{cc} {\bar { A}_{\rm chain} } & {\bar { A}_{{\rm chain\_node}} } \\ {\bf 0} & {\bar { A}_{\rm node} } \end{array} \right]$ (41)
 $\bar {\dot { X}} = \left[\begin{array}{l} \bar {\dot { X}}_{\rm chain} \\ \bar {\dot { X}}_{\rm node} \end{array} \right]$ (42)

 $\left[\begin{array}{cc} {\bar { A}_{\rm chain} } & {\bar { A}_{{\rm chain\_node}} } \\ {\bf 0} & {\bar { A}_{\rm node} } \end{array} \right]\left[\begin{array}{l} { F}_{{\rm S} {\rm chain}} \\ { F}_{{\rm S} {\rm node}} \end{array} \right] = \left[\begin{array}{l} \bar {\dot { X}}_{\rm chain} \\ \bar {\dot { X}}_{\rm node} \end{array} \right]$ (43)

 $\left. \begin{array}{l} \bar { A}_{\rm node}{ F}_{{\rm S} \rm node} = \bar {\dot { X}}_{\rm node} \\ \bar { A}_{\rm chain} { F}_{{\rm S} \rm chain} = \bar {\dot { X}}_{\rm chain} - \bar { A}_{{\rm chain\_node}} \cdot { F}_{{\rm S} \rm node} \end{array} \right \}$ (44)

3.2 任意树形系统约束力算法的串行化

(1)以串行方式计算单个刚体的等效控制力${\pmb F }_{{\rm T }i}$；

(2)解方程${ A} \cdot { F}_{\rm S} = \dot { X}$，得到各约束力${\pmb F }_{{\rm S }i}$：

① 以串行方式依次计算各体的$\dot {\pmb X}_i$和矩阵${ A}$中各项；

② 按照式(39)和式(40)，以串行方式对方程${ A} \cdot { F}_{\rm S} = \dot { X}$中各单链部分进行${K}$轮分块奇偶消元(${K}$为大于等于$\log _2 [\max (n_{{\rm chain }m} )]$的最小整数)，得到方程组${ A}^{(K)} { F}_{\rm S} = \dot { X}^{(K)}$；

③ 对线性方程组${ A}^{(K)} { F}_{\rm S} = \dot { X}^{(K)}$进行换法变换，得到符合要求的$\bar { A}$,$\bar { F}_{\rm S}$和$\bar {\dot { X}}$；

④ 解方程$\bar { A}_{\rm node} { F}_{{\rm S} \rm node} = \bar {\dot { X}}_{\rm node}$得到约束力${ F}_{{\rm S} \rm node}$，并带入方程$\bar { A}_{\rm chain} { F}_{{\rm S} \rm chain} = \bar {\dot { X}}_{\rm chain} - \bar { A}_{{\rm chain\_node}} \cdot { F}_{{\rm S} \rm node}$，以串行方式求解各单链刚体的约束力${\pmb F}_{{\rm S }i}$；

(3) 根据式(28)，以串行方式计算得到动力学解算结果.

4 仿真算例及结果

 图 5 空间机器人结构和尺寸 Fig.5 Structure and size of space robot

 图 6 算例的拓扑结构 Fig.6 Topology of example
 图 7 机械臂1 结构 Fig.7 Structure of arm 1
4.1 算法正确性验证

 图 8 本体姿态角 Fig.8 Main-body’s attitude angles
 图 9 机械臂1 各转角 Fig.9 Arm1’s rotational angles
 图 10 臂杆L11 约束力 Fig.10 L11’s constraint forces & moments

 图 11 本体姿态角误差 Fig.11 Errors of main-body’s attitude angles
 图 12 机械臂1 各转角误差 Fig.12 Errors of arm1’s rotational angles
4.2 对算法效率的讨论

5 结论与展望

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CONSTRAINT FORCE ALGORITHM FOR TREE-LIKE MULITBODY SYSTEM DYNAMICS
Liu Fei, Hu Quan, Zhang Jingrui
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
Abstract: The e cient dynamics algorithm for multibody system has always been a research hotspot. In recent years, many e cient dynamics algorithms made progresses in improving the computational e ciency, but most of them cannot provide an explicit expression of the system's dynamics or resolve the constraint forces. Based on the above issue, a constraint force algorithm (CFA) and its serialized application for the dynamics modelling of an arbitrary multibody system in tree-topology are studied in this paper. The constraint forces for the system can be directly obtained by solving for the system motion through CFA. The presented algorithm is an extension of the original CFA which is only applicable to a single-chained system. It is obtained by analysing the dynamics and kinematics of an arbitrary node in a treelike system. Serialization of the algorithm is developed to reduce the computational cost to a linear relationship with the number of degrees of freedoms (DOFs). The accuracy of the proposed method is validated through a numerical simulation of a space robot with multiple manipulators. The e ciency of the serialized CFA is verified by comparing the CPU time of di erent algorithms.
Key words: constraint force algorithm    multibody system    dynamics    tree-topology    serialization