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 智能系统学报  2020, Vol. 15 Issue (6): 1058-1067  DOI: 10.11992/tis.202005031 0

### 引用本文

NIU Guochen, ZHANG Yunxiao. Kinematics simulation and control system design of continuous robot[J]. CAAI Transactions on Intelligent Systems, 2020, 15(6): 1058-1067. DOI: 10.11992/tis.202005031.

### 文章历史

Kinematics simulation and control system design of continuous robot
NIU Guochen , ZHANG Yunxiao
Robotics Institute, Civil Aviation University of China, Tianjin 300300, China
Abstract: To enable robotic adaptation to increasingly complex unstructured environments, we designed a wire-driven continuous manipulator that combines a spherical joint and flexible support rod. To study the drive-mapping relation of the continuous robot, we established a kinematics model based on the assumptions of the constant curvature model, and we used MATLAB to simulate the kinematics and drive mapping. The spatial superiority of the continuous robot is demonstrated by the simulation results. We built a prototype platform for the three-joint continuous robot, and designed the handle operation mode for the end joint based on the characteristics of the robot. Experimental verification was performed on the prototype platform. Our experimental results verify both the rationality and correctness of the kinematic model and drive mapping relationship and the feasibility of the manipulation method.
Key words: continuous robot    follow terminal    control mode    kinematic model    flexible manipulator    space transformation    handle control    three joints

1 连续型机器人的结构设计和运动学分析 1.1 结构设计

1.2 运动学分析

1.2.1 运动学建模

 $\begin{gathered} {{T}} = {{P}}({{L}},{{\theta}} ,{{\varphi }}){{{R}}_{{Z}}}({{\varphi}} ){{{R}}_{{Y}}}({{\theta}} ){{{R}}_{{Z}}}( - {{\varphi}} ) = \left[ {\begin{array}{*{20}{c}} {{R}}&{{P}} \\ 0&1 \end{array}} \right] =\\ \left(\!\! {\begin{array}{*{20}{c}} {c\theta {c^2}\varphi + {s^2}\varphi }&{c\theta s\varphi c\varphi - c\varphi s\varphi }&{s\theta c\varphi }&{\dfrac{L}{\theta }c\varphi (1 - c\theta )} \\ {c\theta c\varphi s\varphi - c\varphi s\varphi }&{{c^2}\varphi + {s^2}\varphi c\theta }&{s\theta s\varphi }&{\dfrac{L}{\theta }s\varphi (1 - c\theta )} \\ { - s\theta c\varphi }&{ - s\theta s\varphi }&{c\theta }&{\dfrac{L}{\theta }s\theta } \\ 0&0&0&1 \!\! \end{array}} \right) \end{gathered}$ (1)

 ${}_{{{n}} + 1}^1{{T}} = {}_2^1{{T}} \times {}_3^2{{T}} \times {{L}} \times {}_{{{n}} + 1}^{{n}}{{T}}$ (2)
1.2.2 关节空间到绳长空间的转换

 Download: 图 5 关节弯曲引起绳长变化示意图 Fig. 5 Schematic of rope length changes caused by joint bending

 ${L_{ij}} \!=\! n \left[H \!+ \!{l_0} \cos \dfrac{{{\theta _i}}}{{2n}} - d \sin \dfrac{{{\theta _i}}}{{2n}} \cos ({\varphi _i} + (j - 1) \alpha )\right] \!\!\!$ (3)

 $\Delta {L_{ij}} = n\left\{ {{l_0} - \left[ {{l_0}\cos \frac{{{\theta _i}}}{{2n}} - d\sin \frac{{{\theta _i}}}{{2n}}\cos ({\varphi _i} + (j - 1)\alpha )} \right]} \right\}$ (4)

 $\Delta {L_j} = \mathop \sum \limits_{i = 1}^n \Delta {L_{ij}}$ (5)
1.2.3 ZYZ欧拉角求解

 $\begin{gathered} {{R}}({{\alpha}} ,{{\beta }},{{\gamma}} ) = {{{R}}_{{Z}}}({{\alpha}} ){{{R}}_{{Y}}}({{\beta}} ){{{R}}_{{Z}}}({{\gamma}} ) = \left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right) =\\ \left( {\begin{array}{*{20}{c}} {c\beta c\alpha }&{s\beta s\gamma c\alpha - c\gamma s\alpha }&{s\beta c\gamma c\alpha + s\gamma s\alpha } \\ {c\beta s\alpha }&{s\beta s\gamma s\alpha + c\gamma c\alpha }&{s\beta c\gamma s\alpha - s\gamma c\alpha } \\ { - s\beta }&{s\gamma c\beta }&{c\gamma c\beta } \end{array}} \right) \\ \end{gathered} \!\!\!\!\!\!$ (6)

 $\theta = \left\{ {\begin{array}{*{20}{l}} {A\tan 2(\sqrt {a_{31}^2 + a_{32}^2} ,{a_{33}}),}\;\;{s\theta \ne 0}\\ {{0^ \circ }}\\ {{{180}^ \circ }} \end{array}} \right.$ (7)
 $\varphi = \left\{ {\begin{array}{*{20}{l}} { - A\tan 2({a_{32}}, - {a_{31}}),}\;\;{s\theta \ne 0}\\ {{0^ \circ }{\rm{,}}}\;\;{\theta = {0^ \circ }}\\ {{0^ \circ }{\rm{,}}}\;\;{\theta = {{180}^ \circ }} \end{array}} \right.$ (8)
1.2.4 绳长空间到驱动信号空间的映射

 $\left\{ \begin{gathered} m = \dfrac{s}{c} \cdot \Delta l \\ f = \dfrac{s}{c} \cdot v \\ \end{gathered} \right.$ (9)

2 操作方法研究 2.1 系统设计

 Download: 图 7 连续型机器人系统框图 Fig. 7 Block diagram of continuous robot system

2.2 末端跟随控制

3 实验验证 3.1 工作空间仿真

 Download: 图 8 连续型机器人工作空间视图 Fig. 8 Views of the continuous robot workspace

3.2 绳长变化仿真

 Download: 图 9 关节姿态和绳长的仿真 Fig. 9 Simulation diagrams of joint attitude and line length

3.3 单关节弯曲实验

 Download: 图 10 单关节姿态改变实验图 Fig. 10 Experimental diagrams of attitude changes of single joint

3.4 三关节操作实验

1)操控手柄使第一关节弯曲角度为57.32°，旋转角度为37.83°，结果如图12(a)所示；

2)点击前进按钮实现第一关节的姿态跟随，图12(b)为姿态跟随的中间过程图；

3)操作第一关节弯曲角度为34.39°，结果如图12(c)所示；

4)继续点击前进按钮，完成第二关节姿态跟随，此时连续型机器人姿态为弯曲角度 $\varTheta =$ $\{ 0,57.32,34.39\}$ ，旋转角度 $\varPhi = \{ 0,37.83,37.83\}$ ，结果如图12(d)所示；

5)继续点击前进按钮，直到关节姿态参数为 $\varTheta = \{ 22.93,34.39,34.39\}$ $\varPhi = \{ 15.13,37.83,37.83\}$ ，其姿态传感器的结果为 $\varTheta = \{ 22.93,34.39,34.39\}$ $\varPhi =$ $\{ 17.19,37.83,36.69\}$ ，样机平台图如图12(e)所示，仿真图如图12(f)所示。