﻿ 基于变形场不同离散方法的柔性机器人动力学建模与仿真
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 力学学报  2016, Vol. 48 Issue (4): 843-856  DOI: 10.6052/0459-1879-16-163 0

### 引用本文 [复制中英文]

[复制中文]
Fan Jihua , Zhang Dingguo . DYNAMIC MODELING AND SIMULATION OF FLEXIBLE ROBOTS BASED ON DIFFERENT DISCRETIZATION METHODS[J]. Chinese Journal of Ship Research, 2016, 48(4): 843-856. DOI: 10.6052/0459-1879-16-163.
[复制英文]

### 文章历史

2016-06-06 收稿
2016-06-15网络版发表

1. 江苏科技大学机电与汽车工程学院, 江苏 张家港 215600 ;
2. 南京理工大学理学院, 南京 210094 ;
3. 江苏科技大学苏州理工学院, 江苏 张家港 215600

0 引言

1 多杆柔性杆柔性铰机器人动力学方程

 图 1 空间多杆链式柔性机器人 Fig. 1 Multi-link spatial flexible robot

 $J\ddot{z}=R$ (1)

2 变形场的描述

 $\left. \matrix{ u_x^i = u_{x1}^i + u_{x2}^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {x_{ij}}\left( x \right) - {1 \over 2}\sum\limits_{k = 1}^{{N_i}} {\sum\limits_{l = 1}^{{N_i}} {{\delta _{ik}}{\delta _{il}}} } {H_{kl}}\left( x \right) \hfill \cr u_y^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {y_{ij}}\left( x \right) \hfill \cr u_z^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {z_{ij}}\left( x \right) \hfill \cr} \right\}$ (2)

 ${{H}_{kl}}\left( x \right)=\int_{0}^{x}{\left( \frac{\partial {{y}_{ik}}\left( \xi \right)}{\partial \xi }\cdot \frac{\partial {{y}_{il}}\left( \xi \right)}{\partial \xi }+\frac{\partial {{z}_{ik}}\left( \xi \right)}{\partial \xi }\cdot \frac{\partial {{z}_{il}}\left( \xi \right)}{\partial \xi } \right)}d\xi$ (3)

2.1 假设模态法

 \left. \begin{align} & u_{x1}^{i}(x,t)={{x}_{i}}(x){{\delta }_{i}}{{(t)}^{\text{T}}} \\ & u_{y}^{i}(x,t)={{y}_{i}}(x){{\delta }_{i}}{{(t)}^{\text{T}}} \\ & u_{z}^{i}(x,t)={{z}_{i}}(x){{\delta }_{i}}{{(t)}^{\text{T}}} \\ \end{align} \right\} (4)

 $\left. \matrix{ {\theta _{2m}} = {{\partial {w_2}} \over {\partial \bar x}}(0),{\theta _{2k}} = {{\partial {w_2}} \over {\partial \bar x}}({l_j}) \hfill \cr {\theta _{3m}} = {{\partial {w_3}} \over {\partial \bar x}}(0),{\theta _{3k}} = {{\partial {w_3}} \over {\partial \bar x}}({l_j}) \hfill \cr} \right\}$ (15)

Bj为由单位编号决定的定位矩阵， P为总体变形位移列阵，则有

 ${ P}_j = { B}_j { P}$ (16)

 \eqalign{ & P = [{w_{11}}{w_{21}}{\theta _{21}}{w_{31}}{\theta _{31}} \cdots \cr & {w_{1s + 1}}{w_{2s + 1}}{\theta _{2s + 1}}{w_{3s + 1}}{\theta _{3s + 1}}{]^T} \cr & \matrix{ {{\rm{No}}.} & 1 & 2 & \cdots & m & \cdots & k & \cdots & {s + 1} \cr } \cr & {B_j} = \left[ {\matrix{ 0 & 0 & \cdots & {{I_{5 \times 5}}} & \cdots & 0 & \cdots & 0 \cr 0 & 0 & \cdots & 0 & \cdots & {{I_{5 \times 5}}} & \cdots & 0 \cr } } \right] \cr} (17)

 $P = Rp$ (18)

 $\left. \matrix{ R = {\left[ {\matrix{ {{{\bf{0}}_{5 \times 5s}}} \hfill \cr {{I_{5s \times 5s}}} \hfill \cr } } \right]_{5(s + 1) \times 5s}} \hfill \cr p = {[{w_{12}}{w_{22}}{\theta _{22}}{w_{32}}{\theta _{32}} \cdots {w_{1s + 1}}{w_{2s + 1}}{\theta _{2s + 1}}{w_{3s + 1}}{\theta _{3s + 1}}]^{\rm{T}}} \hfill \cr} \right\}$ (19)

 ${u_{x1}} = {S_1}p,{u_y} = {S_2}p,{u_z} = {S_3}p$ (20)

 ${S_1} = {N_{j,1}}{B_j}R,{S_2} = {N_{j,2}}{B_j}R,{S_3} = {N_{j,3}}{B_j}R$ (21)

 $\left. \matrix{ {\delta _i}(t) = [{\delta _{i1}}{\delta _{i2}} \cdots {\delta _{i{N_i}}}] = {p_i} = [w_{12}^iw_{22}^i\theta _{22}^iw_{32}^i \hfill \cr \theta _{32}^i \cdots w_{1s + 1}^iw_{2s + 1}^i\theta _{2s + 1}^iw_{3s + 1}^i\theta _{3s + 1}^i] \hfill \cr {x_i}(x) = [{x_{ij}}] = {S_1} \hfill \cr {y_i}(x) = [{y_{ij}}] = {S_2} \hfill \cr {z_i}(x) = [{z_{ij}}] = {S_3} \hfill \cr} \right\}$ (22)

i代表系统中的第i根杆，Ni 为有限元法对第i杆离散的广义坐标数，其为第i杆离散单元数的5倍.非线性耦合项为

 $u_{x2}^i = - {1 \over 2}\sum\limits_{k = 1}^{{N_i}} {\sum\limits_{l = 1}^{{N_i}} {{\delta _{ik}}} } {\delta _{il}}\left[ {{H_{kl}}(x - \sum\limits_{r = 1}^{j - 1} {{l_r}} ) + \sum\limits_{m = 1}^{j - 1} {{H_{kl}}} ({l_m})} \right]$ (23)

 ${H_{kl}}(\eta ) = \int_0^\eta {\left( {{{\partial {y_{ik}}} \over {\partial \xi }} \cdot {{\partial {y_{il}}} \over {\partial \xi }} + {{\partial {z_{ik}}} \over {\partial \xi }} \cdot {{\partial {z_{il}}} \over {\partial \xi }}} \right)} d\xi$ (24)

2.3 Bezier插值方法

Bezier插值离散方法通过特征控制多边形顶点的变形与伯恩斯坦基函数的线性组合来描述柔性梁变形场[31-32]. 本文选取5次Bezier插值，则令第i根杆的广义坐标和广义形函数为

 $\left. {\matrix{ \matrix{ {\delta _i}(t) = [{\delta _{ij}}] = [V_1^iV_2^iV_3^iV_4^iV_5^iV_6^iV_7^i \hfill \cr V_8^iV_9^iV_{10}^iV_{11}^iV_{12}^i] \hfill \cr} \hfill \cr \matrix{ {x_i}(x) = [{x_{ij}}] = [{J_{5,2}}({x \over {{L_i}}}){J_{5,3}}({x \over {{L_i}}}){J_{5,4}}({x \over {{L_i}}}){J_{5,5}}({x \over {{L_i}}}) \hfill \cr 00000000] \hfill \cr {y_i}(x) = [{x_{ij}}] = [0000{J_{5,2}}({x \over {{L_i}}}){J_{5,3}}({x \over {{L_i}}}) \hfill \cr {J_{5,4}}({x \over {{L_i}}}){J_{5,5}}({x \over {{L_i}}})0000] \hfill \cr} \hfill \cr \matrix{ {z_i}(x) = [{x_{ij}}] = [00000000 \hfill \cr {J_{5,2}}({x \over {{L_i}}}){J_{5,3}}({x \over {{L_i}}}){J_{5,4}}({x \over {{L_i}}}){J_{5,5}}({x \over {{L_i}}})] \hfill \cr} \hfill \cr } } \right\}$ (25)

 $u_{x2}^i = - {1 \over 2}\sum\limits_{k = 1}^{12} {\sum\limits_{l = 1}^{12} {{\delta _{ik}}} } {\delta _{il}}{H_{kl}}$ (26)

 ${H_{kl}}(x) = \int_0^x {\left( {{{\partial {y_{ik}}} \over {\partial \xi }} \cdot {{\partial {y_{il}}} \over {\partial \xi }} + {{\partial {z_{ik}}} \over {\partial \xi }} \cdot {{\partial {z_{il}}} \over {\partial \xi }}} \right)} d\xi$ (27)

2.4 B样条插值方法

B样条插值方法是现代函数逼近中一个十分活跃的分支，是计算方法的重要基础，应用很广泛，可以利用它创造出结构分析的新方法. 在杆轴上沿s方向作一个样条分划，即

 $\left. \matrix{ 0 = {x_0} ＜ {x_1} ＜ {x_2} ＜ \cdots ＜ {x_m} ＜ l \hfill \cr {x_i} = {x_0} + ih,h = {x_{i + 1}} - {x_i} = l/m \hfill \cr} \right\}$ (28)

 $\left. {\matrix{ {{u_x} = \phi {a_x}} \hfill \cr {{u_y} = \phi {a_y}} \hfill \cr {{u_z} = \phi {a_z}} \hfill \cr } } \right\}$ (29)

 $\left. \matrix{ {a_x} = {[{w_{0x}}{{w'}_{0x}}{a_{1x}} \cdots {a_{m - 1x}}{w_{mx}}{{w'}_{mx}}]^{\rm{T}}} \hfill \cr {a_y} = {[{w_{0y}}{{w'}_{0y}}{a_{1y}} \cdots {a_{m - 1y}}{w_{my}}{{w'}_{my}}]^{\rm{T}}} \hfill \cr {a_z} = {[{w_{0z}}{{w'}_{0z}}{a_{1z}} \cdots {a_{m - 1z}}{w_{mz}}{{w'}_{mz}}]^{\rm{T}}} \hfill \cr \phi = [{\phi _{ - 1}}{\phi _0}{\phi _1} \cdots {\phi _{m - 1}}{\phi _m}{\phi _{m + 1}}] \hfill \cr} \right\}$ (30)

 $\left. \matrix{ {\delta _i}(t) = [{\delta _{ij}}] = \hfill \cr [a_{1x}^i \cdots a_{m - 1x}^iw_{mx}^iw_{mx}^{'i}a_{1y}^i \cdots a_{m - 1y}^iw_{my}^iw_{my}^{'i}a_{1z}^i \hfill \cr \cdots a_{m - 1z}^iw_{mz}^iw_{mz}^{'i}] \hfill \cr {x_i}(x) = [{x_{ij}}] = \hfill \cr [{\phi _1}\;{\phi _2}\;\; \cdots \;\;{\phi _{m + 1}}\;0\;\;0\;\; \cdots \;\;0\;\;0\;\;0\;\; \cdots \;\;0] \hfill \cr {y_i}(x) = [{y_{ij}}] = \hfill \cr [0\;\;0\;\; \cdots 0\;\;{\phi _1}\;\;{\phi _2}\;\; \cdots \;\;{\phi _{m + 1}}\;\;0\;\;0\;\; \cdots \;\;0] \hfill \cr {z_i}(x) = [{z_{ij}}] = \hfill \cr [0\;\;0\;\; \cdots 0\;\;0\;\;0\;\; \cdots \;\;0\;\;{\phi _1}\;\;{\phi _2}\;\; \cdots \;\;{\phi _{m + 1}}] \hfill \cr} \right\}$ (31)

 $\left. {\matrix{ {u_x^i = u_{x1}^i + u_{x2}^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {x_{ij}}\left( x \right) - } \hfill \cr {{1 \over 2}\sum\limits_{k = 1}^{{N_i}} {\sum\limits_{l = 1}^{{N_i}} {{\delta _{ik}}{\delta _{il}}} } {H_{kl}}\left( x \right)} \hfill \cr {u_y^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {y_{ij}}\left( x \right)} \hfill \cr {u_z^i = \sum\limits_{j = 1}^{{N_i}} {{\delta _{ij}}} {z_{ij}}\left( x \right)} \hfill \cr } } \right\}$ (32)

 ${H_{kl}}(x) = \int_0^x {\left( {{{\partial {y_{ik}}} \over {\partial \xi }} \cdot {{\partial {y_{il}}} \over {\partial \xi }} + {{\partial {z_{ik}}} \over {\partial \xi }} \cdot {{\partial {z_{il}}} \over {\partial \xi }}} \right)} d\xi$ (33)
3 动力学仿真 3.1 4种变形场离散方法适用性比较

 $\dot \theta = \left\{ {\matrix{ {{{{\Omega _0}} \over T}t - {{{\Omega _0}} \over {2\pi }}\sin ({{2\pi } \over T}t),} & {0 \le t \le T} \cr {{\Omega _0},} & {t > T} \cr } } \right.$ (34)

 图 3 柔性杆末端的横向变形(Ω0 = 4 rad/s) Fig. 3 >Tip deformation of the link along y direction (Ω0 = 4 rad/s)
 图 4 柔性杆末端的横向变形(Ω0 = 10$rad/s) Fig. 4 Tip deformation of the link along y direction (Ω0 = 10 rad/s)  图 5 柔性杆末端的横向变形速度(Ω0 = 4 rad/s) Fig. 5 Time rate of tip deformation of the link along y direction (Ω0 = 4 rad/s)  图 6 柔性杆末端的横向变形速度(Ω0 = 10 rad/s) Fig. 6 Time rate of tip deformation of the link along y direction \\vskip -1mm (Ω0 = 10 rad/s)  图 7 柔性杆末端的纵向变形(Ω0 = 4 rad/s) Fig. 7 Tip deformation of the link along x direction (Ω0 = 4 rad/s) 表 1表 2分别给出了Ω0 = 4 rad/s和Ω0 =10 rad/s时4种离散方法的计算效率、误差以及大范围运动恒定时的响应频率、振幅，表 3表示在不同大范围运动角速度时假设模态法、Bezier插值方法和B样条插值方法的仿真结果与有限元结果之间的误差.该算例中，有限元法把杆分成5个单元；假设模态法则取悬臂梁自由振动的前4阶固有振型；Bezier插值方法取5次伯恩斯坦基函数；B样条插值方法选取3次B样条函数构成的混合参数法.如表 1表 2所示，4种离散方法在大范围运动恒定下的响应频率以及振幅基本一致.计算效率方面，有限元法的计算效率最低，Bezier插值方法的最高，B样条插值方法的其次.如表 3所示，在大范围运动角速度Ω0变大时，假设模态法的计算误差变化趋势相对Bezier插值方法和B样条插值方法的要大. 因此随着角速度Ω0的增大，若要保证假设模态法的计算精度，有必要增加模态阶数. 然而，增加模态阶数则又会降低计算效率.另外，研究发现，当柔性杆处于较大变形时，假设模态法的精度大大降低，以致引起数值发散[31].  图 8 柔性杆末端的纵向变形(Ω0 = 10 rad/s) Fig. 8 Tip deformation of the link along x direction (Ω0 = 10 rad/s) 表 1 >4种离散方法的计算时间、误差和大范围运动恒定时响应频率、振幅(Ω0 rad/s) Table 1 The relative time,the relative error,the amplitude and the frequency of these four discrete methods (Ω0 = 4 rad/s) 表 2 4种离散方法的计算时间、误差和大范围运动恒定时响应频率、振幅(Ω0 =10 rad/s) Table 2 The relative time,the relative error,the amplitude and the frequency of these four discrete methods (Ω0 = 10 rad/s) 表 3 不同大范围运动角速度时计算误差变化趋势 Table 3 The trend of calculation error with different angular velocity 3.2 双杆柔性机械臂自由落体动力学仿真 图 9所示为双杆柔性机械臂动力学模型，已知柔性机械臂在重力作用下做自由落体运动，杆的参数取为：杆长L1= L2= 7.11 m，杆的质量L1=L2 = 314.88 kg，抗拉压刚度为E1A1=E2A2 = 2.84×107 N，抗弯刚度E1I1=E2I2 = 3.8× 106 N⋅m2.  图 9 双杆柔性机械臂自由落体运动初始位置 Fig. 9 The initial position of freefall flexible manipulators 图 10分别表示第1根以及第2根柔性杆横向弯曲变形，4种离散方法的计算结果基本一致.图 11表示系统的能量，从4张小图也能看到4种离散方法的能量变化是一致的，主要是重力势能与动能的转换，弹性势能相对较小；且从图中可知机械能守恒，符合保守系统机械能始终守恒定律，这也说明了本文软件的正确性.  图 10 柔性杆横向弯曲变形 Fig. 10 Tip deformation of the link along y direction}  图 11 系统能量 Fig. 11 System energy 图 12表示杆1和杆2末端水平、竖直方向位移.图 13表示双杆柔性机械臂末端点轨迹，从图 13可知，杆1运动轨迹近似圆弧，而杆2的运动轨迹则无明显规律.图 14给出了系统在不同时刻的整体位形. 在4种离散方法的精度和效率的比较中得出了与算例1同样的结论.  图 12 柔性杆末端水平、竖直方向位移 Fig. 12 Tip displacement of flexible links along s and y direction  图 13 杆末端点轨迹 Fig. 13 The trajectories of the end points  图 14 柔性机械臂整体位形 Fig. 14 The configuration of the flexible manipulator 3.3 空间五根柔性杆机械臂自由落体动力学仿真 为了展示本文算法和软件对复杂系统的适应性，考虑图 15所示空间五根柔性杆机械臂物理模型，已知系统在重力的作用下做自由落体运动. 杆1与杆2之间存在0.2 m的偏置距离.本仿真算例考虑柔性杆轴向拉伸变形、yz方向两维横向弯曲变形，以及杆的扭转变形，并计入柔性杆横向弯曲变形引起的纵向缩短.  图 15 空间5根柔性杆机械臂物理模型 Fig. 15 The physical model of a 5-link spatial flexible manipulator 杆的参数选取参考Canadarm2[25]机械臂的参数，如表 4所示.本仿真算例选取计算效率和计算精度相对较高的Bezier插值离散法对柔性杆变形场进行描述. 表 4 柔性杆物理模型参数 Table 4 The physical model parameters of flexible links 图 16表示模型第5根柔性杆y方向和z方向的横向弯曲变形，其柔性效应明显，y方向横向弯曲变形已经达到杆长的30%，属较大变形范围. 图 17表示柔性杆末端扭转角.由图 17所知，杆1、杆2、杆3和杆4末端产生了扭转变形，而杆5末端没有产生扭转变形.系统运动不是平面运动而是空间运动，柔性杆产生了三维空间变形. 图 18表示系统柔性杆末端点$XY\$平面轨迹图，从图中能明显地看出杆5是做旋转运动.图 19表示系统中第五根柔性杆整体横向弯曲变形图. 图 20表示系统在不同时刻的整体空间位形.图 21表示系统的能量变化，如图可知机械能守恒，同时系统的动能、弹性势能、重力势能相互转化.

 图 16 第5根杆末端横向弯曲变形(y方向和z方向) Fig. 16 Tip deformation of the fifth link along y and z direction
 图 17 柔性杆轴向扭转角 Fig. 17 The axial twist angles of the flexible links
 图 18 杆末端点XY平面轨迹图 Fig. 18 The trajectories of the links on XY plane
 图 19 第5根杆整体横向弯曲变形(y,z方向) Fig. 19 The deformation of the fifth link along y and zdirection
 图 20 系统整体位形 Fig. 20 The configuration of the flexible robot
 图 21 系统能量 Fig. 21 System energy
4 4 结论

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DYNAMIC MODELING AND SIMULATION OF FLEXIBLE ROBOTS BASED ON DIFFERENT DISCRETIZATION METHODS
Fan Jihua1,3, Zhang Dingguo2
1. School of Mechatronics-Engineering and Automotive, Jiangsu University of Science and Technology, Zhangjiagang 215600, Jiangsu, China ;
2. School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China ;
3. Suzhou Institute of Technology, Jiangsu University of Science and Technology, Zhangjiagang 215600, Jiangsu, China
Abstract: Dynamic modeling and simulation of flexible robots based on different discretization methods are investigated in this paper. Firstly, the physical model of flexible robots consisting of n links and n revolute joints is established. Secondly, the assumed mode method, finite element method, Bezier interpolation method and B-spline interpolation method are used to describe the deformation of the flexible link. Then, kinematics of both rotary-joint motion and link deformation is described by 4×4 homogenous transformation matrices, and the Lagrangian equations are used to derive the governing equations of motion of the system. The longitudinal deformation and the transverse deformation of the flexible link are considered, and the coupling term of the deformation which is caused by the transverse deformation is included in the total longitudinal deformation. Then, a software package for the dynamic simulation of the flexible robots based on the different discretization methods is developed, after that, the dynamic analysis for flexible robots are studied by three examples. The simulation results show that the computational efficiency of finite element method is the lowest, and the Bezier interpolation method and B-spline interpolation method are better than the assumed mode method in dealing with the large deformation dynamic problem. As new discretization methods, Bezier interpolation method and B-spline interpolation method can be used to describe the deformation of the flexible link, and they can be extended to the dynamic modeling for spatial flexible robots.
Key words: assumed mode method    finite element method    Bezier interpolation method    B-spline interpolation method    flexible robots