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 浙江大学学报(工学版)  2018, Vol. 52 Issue (1): 1-7  DOI:10.3785/j.issn.1008-973X.2018.01.001 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2018.01.001
[复制中文]
WANG Wei, WANG Jin, LU Guo-dong. Reliability analysis of manipulator based on fourth-moment estimation[J]. Journal of Zhejiang University(Engineering Science), 2018, 52(1): 1-7.
dx.doi.org/10.3785/j.issn.1008-973X.2018.01.001
[复制英文]

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orcid.org/0000-0003-3106-021X.
Email: dwjcom@zju.edu.cn

### 文章历史

Reliability analysis of manipulator based on fourth-moment estimation
WANG Wei , WANG Jin , LU Guo-dong
State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China
Abstract: The trajectory is decomposed into a series of discrete path points in order to analyze the effect originated from linkage dimension deviations and joint clearances of the manipulator to the kinematic reliability. The safe trajectory can be obtained since all the positional errors of the path points are less than the required tolerance. The positional error of each independent discrete point was considered as the random variable. The extreme value distribution of the positional error of all the discrete points in the trajectory was analyzed. Then the performance function of the manipulator was established based on the maximum entropy principle. The fourth-moment reliability method (FMRM) was applied to estimate the kinematic reliability. The results obtained from the first-order reliability method (FORM), the first-order second-moment method (FOSM) and Monte Carlo simulations (MCS) were used as the benchmarks for a comparative study. The efficiency and accuracy of the FMRM were improved, and the computation time was shortened.
Key words: manipulator    kinematic reliability    fourth-moment    maximum entropy principle

1 机器人建模 1.1 运动学

 图 1 PUMA560机器人 Fig. 1 PUMA560 robot

 $A_i^{i - 1} = \left[ {\begin{array}{*{20}{c}} {{\rm{C}}{\theta _i}}&{ - {\rm{S}}{\theta _i}{\rm{C}}{\alpha _i}}&{{\rm{S}}{\theta _i}{\rm{S}}{\alpha _i}}&{{a_i}{\rm{C}}{\theta _i}}\\ {{\rm{S}}{\theta _i}}&{{\rm{C}}{\theta _i}{\rm{C}}{\alpha _i}}&{ - {\rm{C}}{\theta _i}{\rm{S}}{\alpha _i}}&{{a_i}{\rm{S}}{\theta _i}}\\ 0&{{\rm{S}}{\alpha _i}}&{{\rm{C}}{\alpha _i}}&{{d_i}}\\ 0&0&0&1 \end{array}} \right].$ (1)

 图 2 连杆坐标系 Fig. 2 Link coordinate of robot

 图 3 运动学模型 Fig. 3 Kinematic model of robot

 $A_n^0 = \prod\limits_{i = 1}^n {A_i^{i - 1}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{R}}&\mathit{\boldsymbol{P}}\\ 0&1 \end{array}} \right].$ (2)

1.2 机器人本体误差分析

 ${\theta _i} = {{\bar \theta }_i} + {\Delta _i}.$ (3)

1.3 极值分布与系统可靠性

 ${\rm{Pr}}\left\{ {\bigcap\limits_{i = 1}^n {\left( {{x_i} < k} \right)} } \right\} = \Pr \left\{ {{X_{\max }} < k} \right\}.$ (4)

2 可靠性分析

 ${\varepsilon _i} = \left\| {{{\mathit{\boldsymbol{P'}}}_i} - {\mathit{\boldsymbol{P}}_i}} \right\|.$ (5)

 图 4 单个路径点的可靠性 Fig. 4 Reliability at point in trajectory

 图 5 机器人末端运动路径 Fig. 5 Trajectory of manipulators' end-effector

 $P_{\rm{r}}^{\rm{s}} = \Pr \left\{ {\bigcap\limits_{i = 1}^N {\left( {{\varepsilon _i} < {h_{\rm{f}}}} \right)} } \right\}.$ (6)

 $P_{\rm{f}}^{\rm{s}} = 1 - P_{\rm{r}}^{\rm{s}}.$ (7)

 $\gamma = {\rm{max}}\left\{ {{\varepsilon _1},{\varepsilon _2}, \cdots ,{\varepsilon _N}} \right\}.$ (8)

 $Z = g\left( \gamma \right) = {h_{\rm{f}}} - \gamma .$ (9)

 $P_{\rm{f}}^{\rm{s}} = \Pr \left\{ {Z \le 0} \right\}.$ (10)

3 矩估计方法 3.1 一阶可靠性法及一次二阶矩法

 $\left. {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\min }\\ {{\rm{s}}{\rm{.}}\;{\rm{t}}{\rm{.}}} \end{array}}&{\begin{array}{*{20}{c}} {\beta = {{\left( {{\mathit{\boldsymbol{U}}^{\rm{T}}}\mathit{\boldsymbol{U}}} \right)}^{1/2}},}\\ {g\left( \mathit{\boldsymbol{X}} \right) = 0.} \end{array}} \end{array}} \right\}$ (11)

 $P_{\rm{f}}^{\rm{s}} = \phi \left( { - \beta } \right).$ (12)

 $Z = g\left( {{\mu _X}} \right) + \sum\limits_{i = 1}^n {{g_i}\left( {{x_i} - {\mu _i}} \right)} .$ (13)

 $\beta = {\mu _Z}/{\sigma _Z}.$ (14)

3.2 四阶矩估计

 ${\upsilon _{Xi}} = E\left( {{x^i}} \right) = \int\limits_X {{x^i}{f_X}\left( x \right){\rm{d}}x} .$ (15)

 ${\mu _{Xi}} = E\left\{ {{{\left( {x - {\mu _X}} \right)}^i}} \right\} = \int\limits_X {{{\left( {x - {\mu _X}} \right)}^i}{f_X}\left( x \right){\rm{d}}x} .$ (16)

4 基于最大熵的概率分布

Shannon在1948年将热力学熵的概念引入到信息研究理论中，从统计学角度对事件的不确定进行度量，对系统功能函数Z的Shannon熵定义如下：

 $H\left[ {{f_Z}} \right] = - \int\limits_z {{f_Z}\left( z \right)\ln \left[ {{f_Z}\left( z \right)} \right]{\rm{d}}z} .$ (17)

Jaynes[16]提出最大熵原理，指出：给定外在条件，针对随机变量存在多个可能的概率分布，符合已知的约束条件并使随机变量的熵取得最大值的分布是最佳概率分布.

 $\begin{array}{*{20}{c}} {L = - \int_z {{f_Z}\left( z \right)\ln \left[ {{f_Z}\left( z \right)} \right]{\rm{d}}z} - }\\ {\sum\limits_{i = 0}^4 {{\lambda _i}\left[ {\int_z {{z^i}{f_Z}\left( z \right){\rm{d}}z} - {\upsilon _{Zi}}} \right]} .} \end{array}$ (18)

 $Y = \frac{{Z - {\mu _Z}}}{{{\sigma _Z}}}.$ (19)

 $\begin{array}{l} P_{\rm{f}}^{\rm{s}} = \Pr \left\{ {Z \le 0} \right\} = \Pr \left\{ {Y \le \frac{{ - {\mu _Z}}}{{{\sigma _Z}}}} \right\} = \\ \int\limits_{ - \infty }^{ - {\mu _Z}/{\sigma _Z}} {\exp \left( { - \sum\limits_{i = 0}^4 {{\lambda _i}{y^i}} } \right){\rm{d}}y} . \end{array}$ (20)
5 仿真分析

 图 6 含关节间隙的仿真路径 Fig. 6 Trajectory of simulations with joint clearance

Monte Carlo模拟(MCS)作为系统可靠性测量最重要的方法之一，能够在模型未知的情况下精确求解系统可靠性[6, 10].通过MCS计算机器人的运动可靠性为

 $P_{\rm{f}}^{\rm{s}} = \frac{{{N_{\rm{f}}}}}{{{N_{\rm{t}}}}}.$ (21)

 图 7 运动失效概率 Fig. 7 Probability of failure

 $\xi = \tilde P_{\rm{f}}^{\rm{s}} - P_{\rm{f}}^{\rm{s}}.$ (22)

 图 8 各矩估计法相对于MCS的估计偏差 Fig. 8 Estimation error of each moment estimation method relative to MCS

 ${T_{\rm{t}}} = {T_{\rm{s}}} + {T_{\rm{c}}}.$ (23)

 ${\omega _{\rm{s}}} = \frac{{{T_{\rm{s}}}}}{{{T_{\rm{t}}}}} \times 100\% ,$ (24)
 ${\omega _{\rm{c}}} = \frac{{{T_{\rm{c}}}}}{{{T_{\rm{t}}}}} \times 100\% .$ (25)

 图 9 样本数量对结果的影响 Fig. 9 Effect of sample size to results

6 结论

(1) 在一定的轨迹精度要求下，机器人运动的可靠性主要取决于路径中位置误差最大的插补点的可靠性.

(2) 基于大量的样本，MCS能够获得比较准确的结果，精度最高，但效率最低.四阶矩估计法的效率接近于一阶可靠性法和一次二阶距法，但仅需较少的样本，同时计算精度较高，在综合考虑效率及精度的情况下，四阶矩估计方法具有较明显的优势.

(3) 传统的一阶可靠性法和一次二阶距法主要针对结构简单并且输出满足正态分布的系统；四阶矩估计方法可以适用于任意分布类型的系统，能够避免传统估计方法中的假设偏差，进一步拓展应用的领域.

 [1] LAI X, HE H, LAI Q, et al. Computational prediction and experimental validation of revolute joint clearance wear in the low-velocity planar mechanism[J]. Mechanical Systems and Signal Processing, 2017, 85(5): 963-976. [2] LIAN B, SUN T, SONG Y. Parameter sensitivity analysis of a 5-DoF parallel manipulator[J]. Robotics and Computer-Integrated Manufacturing, 2017, 46(14): 1-14. [3] LI Y, CHEN G, SUN D, et al. Dynamic analysis and optimization design of a planar slider-crank mechanism with flexible components and two clearance joints[J]. Mechanism and Machine Theory, 2016, 99(10): 37-57. [4] GENG X, WANG X, WANG L, et al. Non-probabilistic time-dependent kinematic reliability assessment for function generation mechanisms with joint clearances[J]. Mechanism and Machine Theory, 2016, 104(6): 202-221. [5] ERKAYA S. Investigation of joint clearance effects on welding robot manipulators[J]. Robotics and Computer-Integrated Manufacturing, 2012, 28(4): 449-457. DOI:10.1016/j.rcim.2012.02.001 [6] RAO S S, BHATTI P K. Probabilistic approach to manipulator kinematics and dynamics[J]. Reliability Engineering and System Safety, 2001, 72(1): 47-58. DOI:10.1016/S0951-8320(00)00106-X [7] 宋月娥, 吴林, 戴明. 机器人关节间隙误差分析[J]. 机械工程学报, 2003, 39(4): 11-14. SONG Yue-e, WU Lin, DAI Ming. Error analysis of robot joint clearance[J]. Chinese Journal of Mechanical Engineering, 2003, 39(4): 11-14. [8] KIM J, SONG W, KANG B. Stochastic approach to kinematic reliability of open-loop mechanism with dimensional tolerance[J]. Applied Mathematical Modelling, 2010, 34(5): 1225-1237. DOI:10.1016/j.apm.2009.08.009 [9] WANG J, ZANG J, DU X. Hybrid dimension reduction for mechanism reliability analysis with random joint clearances[J]. Mechanism and Machine Theory, 2011, 46(10): 1396-1410. DOI:10.1016/j.mechmachtheory.2011.05.008 [10] HAFEZIPOUR M, KHODAYGAN S. An uncertainty analysis method for error reduction in end-effector of spatial robots with joint clearances and link dimension deviations[J]. International Journal of Computer Integrated Manufacturing, 2016, 30(8): 1-11. [11] BOWLING A P, RENUAD J E, NEWKIRK J T, et al. Reliability-based design optimization of robotic system dynamic performance[J]. IEEE/RSJ International Conference on Intelligence, 2007, 129(4): 3611-3617. [12] LI J, CHEN J, FAN W. The equivalent extreme-value event and evaluation of the structural system reliability[J]. Structural Safety, 2007, 29(2): 112-131. DOI:10.1016/j.strusafe.2006.03.002 [13] ZHANG Z, XU L, FLORES P, et al. A Kriging model for dynamics of mechanical systems with revolute joint clearances[J]. Journal of Computational and Nonlinear Dynamics, 2014, 9(3): 1-13. [14] 王锋, 陈凯, 陈小平. 一种含间隙机械臂的在线校准方法[J]. 机器人, 2013, 35(5): 521-526. WANG Feng, CHEN Kai, CHEN Xiao-ping. An online calibration method for manipulator with joint clearance[J]. Robot, 2013, 35(5): 521-526. [15] 李云贵, 赵国藩. 结构可靠度的四阶矩分析法[J]. 大连理工大学学报, 1992, 18(4): 455-459. LI Yun-gui, ZHAO Guo-fan. Reliability analysis of structures based on maximum entropy theory[J]. Journal of Dalian University of Technology, 1992, 18(4): 455-459. [16] JAYNES E T. Information theory and statistical mechanics[J]. Physical Review, 1957, 106(4): 171-190.