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 浙江大学学报(工学版)  2018, Vol. 52 Issue (1): 125-132  DOI:10.3785/j.issn.1008-973X.2018.01.017 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2018.01.017
[复制中文]
PAN Li, BAO Guan-jun, XU Fang, ZHANG Li-bin. Dynamic compliant control of six DOF assembly robot[J]. Journal of Zhejiang University(Engineering Science), 2018, 52(1): 125-132.
dx.doi.org/10.3785/j.issn.1008-973X.2018.01.017
[复制英文]

### 通信联系人

orcid.org/0000-0003-0486-9312.
Email: lbz@zjut.edu.cn

### 文章历史

Dynamic compliant control of six DOF assembly robot
PAN Li , BAO Guan-jun , XU Fang , ZHANG Li-bin
Key Laboratory of E & M, Ministry of Education and Zhejiang Province, Zhejiang University of Technology, Hangzhou 310014, China
Abstract: A dynamic complaint control was presented in order to improve assembly accuracy and flexibility for 6 DOF assembly industrial robot. Then the industrial robot can not only track reference trajectories in working spaces, but also can be dynamically switched to contact force control with high flexibility. The dynamic model of the 6 DOF assembly industrial robot was established in joint space and was transformed into the working space of the end effector. The overall over control framework of the proposed control strategy consisted of reference tracking module, inner tracking control module, dynamic parameter identification module. The reference tracking module was designed based on sliding mode control while the trajectory of contact force was given by using an impedance filter. The switching condition between space tracking and contact force control was designed using sigmoid function. The dynamic parameter identification module was designed by using least square algorithm. All the control modules were verified through Lyapunov function to converge to stable region over wide working ranges. The proposed control was validated through simulations based on an industrial robot platform. Comparative results demonstrate that the proposed dynamic complaint control can significantly improve reference tracking accuracy and contact force control flexibility over wide working range as compared to typical proportional derivative (PD) control. The average relative tracking error of the complaint control can be well maintained within -4% and +4%.
Key words: assembly robot    compliant control    dynamic parameter identification    motion trajectory tracking    impedance filtering

1 系统描述 1.1 动态特性

 $\mathit{\boldsymbol{M}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\ddot \theta }} + \mathit{\boldsymbol{C}}\left( {\mathit{\boldsymbol{\theta }},\mathit{\boldsymbol{\dot \theta }}} \right)\mathit{\boldsymbol{\dot \theta }} + \mathit{\boldsymbol{G}}\left( \mathit{\boldsymbol{\theta }} \right) = \tau .$ (1)

 $\left. \begin{array}{l} \mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{J}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\dot \theta }},\\ \mathit{\boldsymbol{\dot \theta }} = {\mathit{\boldsymbol{J}}^{ - 1}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\dot x}}. \end{array} \right\}$ (2a)
 $\left. \begin{array}{l} \mathit{\boldsymbol{\ddot x = J}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\ddot \theta }} + \mathit{\boldsymbol{\dot J}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\dot \theta }},\\ \mathit{\boldsymbol{\ddot \theta }} = {\mathit{\boldsymbol{J}}^{ - 1}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\ddot x}} + \frac{{\rm{d}}}{{{\rm{d}}t}}\left[ {{\mathit{\boldsymbol{J}}^{ - 1}}\left( \mathit{\boldsymbol{\theta }} \right)} \right]\mathit{\boldsymbol{\dot x}}. \end{array} \right\}$ (2b)
 $\left. \begin{array}{l} \mathit{\boldsymbol{\tau }} = {\mathit{\boldsymbol{J}}^{\rm{T}}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{F}},\\ \mathit{\boldsymbol{F = }}{\mathit{\boldsymbol{J}}^{ - {\rm{T}}}}\left( \mathit{\boldsymbol{\theta }} \right)\mathit{\boldsymbol{\tau }}. \end{array} \right\}$ (2c)

 ${\mathit{\boldsymbol{M}}_{\rm{e}}}\left( \mathit{\boldsymbol{x}} \right)\mathit{\boldsymbol{\ddot x}} + {\mathit{\boldsymbol{C}}_{\rm{e}}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{\dot x}}} \right)\mathit{\boldsymbol{\dot x}} + {\mathit{\boldsymbol{G}}_{\rm{e}}}\left( \mathit{\boldsymbol{x}} \right) = \mathit{\boldsymbol{F}}.$ (3)

1) Me(x)为正定对称矩阵，即有Me(x)=MeT(x).

2) ${{\mathit{\boldsymbol{\dot M}}}_{\rm{e}}}$(x)-2Ce(x, ${\mathit{\boldsymbol{\dot x}}}$)为反对称矩阵，即有xT${{\mathit{\boldsymbol{\dot M}}}_{\rm{e}}}$(x)-2Ce(x, ${\mathit{\boldsymbol{\dot x}}}$)x=0.

1.2 控制策略

 图 1 六自由度机器人动态柔顺性控制原理 Fig. 1 Dynamic compliant control principle of 6 DOF assembly industrialrobot

2 动力学参数辨识

 ${\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\gamma }} = \tau \left( t \right).$ (4)

 ${\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( t \right) = \left[ {\mathit{\boldsymbol{\ddot \theta }},\mathit{\boldsymbol{\dot \theta }},1} \right],$ (5)
 $\mathit{\boldsymbol{\gamma }} = \left[ \begin{array}{l} \mathit{\boldsymbol{M}}\left( \mathit{\boldsymbol{\theta }} \right)\\ \mathit{\boldsymbol{C}}\left( \mathit{\boldsymbol{\theta }} \right)\\ \mathit{\boldsymbol{G}}\left( \mathit{\boldsymbol{\theta }} \right) \end{array} \right].$ (6)

 $\left. \begin{array}{l} \mathit{\Psi }\left( \mathit{\boldsymbol{\gamma }} \right) = \int\limits_0^t {{\varepsilon ^{\lambda \left( {t - \upsilon } \right)}}{{\left[ {\tau \left( \upsilon \right) - {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( \upsilon \right)\mathit{\boldsymbol{\gamma }}} \right]}^2}{\rm{d}}\upsilon } ;\\ 0 < \lambda < 1. \end{array} \right\}$ (7)
 $P\left( t \right) = {\left[ {\int\limits_0^t {{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( \upsilon \right)\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( \upsilon \right){\rm{d}}\upsilon } } \right]^{ - 1}}.$ (8)

 $\mathit{\boldsymbol{\dot {\hat \gamma} }} = \mathit{\boldsymbol{P}}\left( t \right)\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( t \right)\varepsilon \left( t \right).$ (9)

 $\varepsilon \left( t \right) = \tau \left( t \right) - {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\hat \gamma }}.$ (10)

 $\dot P\left( t \right) = \lambda P\left( t \right) - P\left( t \right)\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( t \right){\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\left( t \right)P\left( t \right).$ (11)

3 控制器设计

3.1 轨迹跟踪控制

 $\mathit{\boldsymbol{s}} = \mathit{\boldsymbol{\dot e}} + \mathit{\boldsymbol{ \boldsymbol{\varLambda} e}}.$ (12)

 $\mathit{\boldsymbol{e}} = {\mathit{\boldsymbol{x}}_{\rm{r}}} - \mathit{\boldsymbol{x}}.$ (13)

 $\mathit{\boldsymbol{F}} = {\mathit{\boldsymbol{U}}_{\rm{a}}} + {\mathit{\boldsymbol{U}}_{{\rm{s1}}}} + {\mathit{\boldsymbol{U}}_{{\rm{s2}}}}.$ (14)

 ${\mathit{\boldsymbol{U}}_{\rm{a}}} = {{\mathit{\boldsymbol{\hat M}}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\hat G}}}_{\rm{e}}} + {{\mathit{\boldsymbol{\hat M}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {{\mathit{\boldsymbol{\hat C}}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\hat C}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}}.$ (15)

 $V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = \frac{1}{2}{\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{M}}_{\rm{e}}}\mathit{\boldsymbol{s}}.$ (16)

 $\begin{array}{l} \dot V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{M}}_{\rm{e}}}\left( {\mathit{\boldsymbol{\ddot e}} + \mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}}} \right) + \frac{1}{2}{\mathit{\boldsymbol{s}}^{\rm{T}}}{{\mathit{\boldsymbol{\dot M}}}_{\rm{e}}}\mathit{\boldsymbol{s = }}\\ {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{M}}_{\rm{e}}}\left( {\mathit{\boldsymbol{\ddot e}} + \mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}}} \right) + {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{C}}_{\rm{e}}}\mathit{\boldsymbol{s = }}\\ {\mathit{\boldsymbol{s}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{M}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} - \mathit{\boldsymbol{F}} + {\mathit{\boldsymbol{C}}_{\rm{e}}}\mathit{\boldsymbol{\dot x}} + {\mathit{\boldsymbol{G}}_{\rm{e}}} + {\mathit{\boldsymbol{M}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {\mathit{\boldsymbol{C}}_{\rm{e}}}\mathit{\boldsymbol{s}}} \right) = \\ {\mathit{\boldsymbol{s}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{M}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} - \mathit{\boldsymbol{F}} + {\mathit{\boldsymbol{G}}_{\rm{e}}} + {\mathit{\boldsymbol{M}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {\mathit{\boldsymbol{C}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + {\mathit{\boldsymbol{C}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}}} \right). \end{array}$ (17)

 $\begin{array}{l} \dot V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = {\mathit{\boldsymbol{s}}^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde G}}}_{\rm{e}}} + {{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {{{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}} - {\mathit{\boldsymbol{U}}_{{\rm{s1}}}} - {\mathit{\boldsymbol{U}}_{{\rm{s2}}}}} \right) = - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{U}}_{{\rm{s1}}}}\mathit{\boldsymbol{s}} + {\mathit{\boldsymbol{s}}^{\rm{T}}} \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde G}}}_{\rm{e}}} + {{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}} - {\mathit{\boldsymbol{U}}_{{\rm{s2}}}}} \right). \end{array}$ (18)

 $\mathit{\boldsymbol{ \boldsymbol{\varDelta} }} = {{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde G}}}_{\rm{e}}} + {{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}}.$ (19)

 $\left| \mathit{\boldsymbol{ \boldsymbol{\varDelta} }} \right| \le \delta .$ (20)

 $\left. \begin{array}{l} {\mathit{\boldsymbol{U}}_{{\rm{s1}}}} = {\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}},\\ {\mathit{\boldsymbol{U}}_{{\rm{s2}}}} = {\mathit{\boldsymbol{k}}_2}{\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right). \end{array} \right\}$ (21)

k1k2分别为6×1正常数控制增益矩阵，k2可以定义为

 ${\mathit{\boldsymbol{k}}_2} = \mathit{\boldsymbol{\alpha }}\left( {\mathit{\boldsymbol{\delta }} + {\mathit{\boldsymbol{\eta }}_{\rm{s}}}} \right).$ (22)

 $\mathit{\boldsymbol{\alpha }} \ge 1,{\mathit{\boldsymbol{\eta }}_{\rm{s}}} \ge 0.$ (23)

 $\begin{array}{l} \dot V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = {\mathit{\boldsymbol{s}}^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}{{\mathit{\boldsymbol{\ddot x}}}_{\rm{r}}} + {{\mathit{\boldsymbol{\tilde G}}}_{\rm{e}}} + {{\mathit{\boldsymbol{\tilde M}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} \dot e}} + {{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}{{\mathit{\boldsymbol{\dot x}}}_{\rm{r}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {{{\mathit{\boldsymbol{\tilde C}}}_{\rm{e}}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} e}} - {\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} - {\mathit{\boldsymbol{k}}_2}{\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right)} \right) = - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} + \\ \;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{s}}^{\rm{T}}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }} - {\mathit{\boldsymbol{k}}_2}{\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right)} \right) = - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\left[ {\mathit{\boldsymbol{\alpha }}\left( {\mathit{\boldsymbol{\delta }} + {\mathit{\boldsymbol{\eta }}_{\rm{s}}}} \right) - {\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right)\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}} \right]. \end{array}$ (24)

 $\mathit{\boldsymbol{\alpha \delta }} - {\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right)\mathit{\boldsymbol{ \boldsymbol{\varDelta} }} \ge 0.$ (25)

 $\begin{array}{*{20}{c}} {\dot V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) \le - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{\eta }}_{\rm{s}}}{\mathop{\rm sgn}} \left( \mathit{\boldsymbol{s}} \right) \le }\\ { - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} - {\mathit{\boldsymbol{\eta }}_{\rm{s}}}\left| \mathit{\boldsymbol{s}} \right|.} \end{array}$ (26)

 $W\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = - {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{k}}_1}\mathit{\boldsymbol{s}} - {\mathit{\boldsymbol{\eta }}_{\rm{s}}}\left| \mathit{\boldsymbol{s}} \right| \le - \dot V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right).$ (27)

W(s(t))对时间积分，可以表达为

 $\int\limits_0^t {W\left( {\mathit{\boldsymbol{s}}\left( \tau \right)} \right){\rm{d}}\tau } = V\left( {\mathit{\boldsymbol{s}}\left( 0 \right)} \right) - V\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right).$ (28)

 $\mathop {\lim }\limits_{t \to \infty } \int\limits_0^t {W\left( {\mathit{\boldsymbol{s}}\left( \tau \right)} \right){\rm{d}}\tau } < \infty .$ (29)

 $\mathop {\lim }\limits_{t \to \infty } W\left( {\mathit{\boldsymbol{s}}\left( t \right)} \right) = 0.$ (30)

 ${\rm{sat}}\left( \mathit{\boldsymbol{s}} \right) = \left\{ \begin{array}{l} 1,s > \mu ;\\ \frac{s}{\mu }, - \mu < s \le \mu ;\\ - 1,s < \mu . \end{array} \right.$ (31)

3.2 阻抗滤波器设计

 ${\mathit{\boldsymbol{F}}_{\rm{d}}} = {\mathit{\boldsymbol{M}}_{\rm{d}}}\left( {{{\mathit{\boldsymbol{\ddot x}}}_{\rm{d}}} - \mathit{\boldsymbol{\ddot x}}} \right) + {\mathit{\boldsymbol{B}}_{\rm{d}}}\left( {{{\mathit{\boldsymbol{\dot x}}}_{\rm{d}}} - \mathit{\boldsymbol{\dot x}}} \right) + {\mathit{\boldsymbol{K}}_{\rm{d}}}\left( {{\mathit{\boldsymbol{x}}_{\rm{d}}} - \mathit{\boldsymbol{x}}} \right).$ (32)

 ${\rm{sigm}}\left( {{F_{\rm{e}}}} \right) = \left\{ {\begin{array}{*{20}{l}} {1,{F_{\rm{e}}} > \bar \omega ;}\\ {0,{F_{\rm{e}}} \le \bar \omega .} \end{array}} \right.$ (33)

4 仿真实验验证与分析

 图 2 六自由度装配机器人实验平台 Fig. 2 Experimental platform ofsix DOF assembly robot

 $\mathit{\boldsymbol{ \boldsymbol{\varLambda} }} = {\rm{diag}}\left[ {6.1,6,6.2,6.8,7.1,8} \right],$
 ${\mathit{\boldsymbol{k}}_1} = {\rm{diag}}\left[ {12.8,12.2,10,16,15,18} \right],$
 ${\mathit{\boldsymbol{k}}_2} = {\rm{diag}}\left[ {2.8,2.2,1.6,12,1.5,1.8} \right],\mu = 0.5,$

 ${\rm{AREJ}} = \frac{1}{N}\sum\limits_{j = 1}^N {\frac{{{\theta _j} - {\theta _{j{\rm{d}}}}}}{{{\theta _{j{\rm{d}}}}}} \times 100\% } .$ (34)

 ${\rm{AREE}} = \frac{1}{{6N}}\sum\limits_{i = 1}^6 {\sum\limits_{j = 1}^N {\frac{{{x_{ij}} - {x_{{\rm{r}}ij}}}}{{{x_{{\rm{r}}ij}}}} \times 100\% } } .$ (35)

 ${\rm{AREF}} = \frac{1}{{6N}}\sum\limits_{i = 1}^6 {\sum\limits_{j = 1}^N {\frac{{{F_{ij}} - {F_{{\rm{d}}ij}}}}{{{F_{{\rm{d}}ij}}}} \times 100\% } } .$ (36)

4.1 空间轨迹跟踪

 图 3 工业装配机器人的各个关节角位移轨迹跟踪的平均相对误差 Fig. 3 Relative tracking errors for joint angle in industrial robot

 图 4 工业装配机器人末端机械手位置轨迹跟踪的平均相对误差 Fig. 4 Average relative tracking error of end effector of industrial assembling robot
4.2 接触力柔顺性控制

 图 5 工业装配机器人的末端执行器位置轨迹跟踪的平均相对误差 Fig. 5 Average relative tracking error of end effector of industrial assembling robot

 图 6 工业装配机器人的末端执行器的接触力平均相对误差 Fig. 6 Average relative tracking error of contact force of end effector in industrial assembling robot
5 结语

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