﻿ Kresling和Miura折痕混合型三指机械手的运动学分析及其设计
 机器人 2022, Vol. 44 Issue (1): 35-44 0

YANG Hui, WANG Xiang, QIAO Shangling, LIU Rongqiang. Design and Kinematics Analysis of a Three-finger Manipulator with Kresling and Miura Hybrid Origami Crease[J]. ROBOT, 2022, 44(1): 35-44.

Kresling和Miura折痕混合型三指机械手的运动学分析及其设计

1. 安徽大学电气工程与自动化学院, 安徽 合肥 230601;
2. 哈尔滨工业大学机器人技术与系统国家重点实验室, 黑龙江 哈尔滨 150001

Design and Kinematics Analysis of a Three-finger Manipulator with Kresling and Miura Hybrid Origami Crease
YANG Hui1,2 , WANG Xiang1 , QIAO Shangling2 , LIU Rongqiang2
1. School of Electrical Engineering and Automation, Anhui University, Hefei 230601, China;
2. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China
Abstract: A three-finger manipulator with a Kresling and Miura hybrid origami crease is proposed, which has large grasping range, simple structure and high flexibility. Firstly, geometrical analysis on the Kresling origami crease is performed, and the relation equation between the parameters and the strain energy of the Kresling origami crease is established. The equation between the moment and the knuckle length-width ratio of the multilayer Miura origami element is established by the coordinate method. Then, the mathematical model of the relationship among the parameters of the hybrid element is established by using the principle of virtual work, and the distribution of the end points is calculated by using the D-H (Denavit-Hartenberg) coordinate method to determine the workspace of the manipulator. The method of torque balance is used to analyze the situations that the manipulator grasps the cylinder and cuboid objects in fingertip manner and in envelope manner respectively, and the relation between the contact forces and the joint angles is derived.
Keywords: robot    hybrid origami manipulator    Kresling origami crease    Miura origami crease    dynamics analysis

1 引言（Introduction）

2 机械手结构（Manipulator structure）

 图 1 混合型机械手几何示意图 Fig.1 Geometric sketch of the hybrid manipulator
3 几何分析（Geometric analysis） 3.1 基本单元层几何分析 3.1.1 Kresling折痕几何分析

Kresling平面折痕结构和柱状结构如图 2所示，Kresling多边形外接圆半径$R$

 \begin{align} R=\dfrac{a}{2\sin \dfrac{\pi}{n}} \end{align} (1)
 图 2 Kresling折痕结构和柱状结构 Fig.2 Kresling crease structure and columnar structure

 \begin{align} H=b\sin \delta \end{align} (2)

 \begin{align} b=\sqrt{4R^{2}\sin^{2}\dfrac{2{\pi} -n\theta}{2n}+(b\sin \delta)^{2}} \end{align} (3)

 \begin{align} \dfrac{b\cos \delta}{2R} \leqslant 1 \end{align} (4)

 \begin{align} \dfrac{b}{a}\leqslant \frac{1}{\sin \dfrac{\pi}{n}\cos \delta} \end{align} (5)

 \begin{align} \dfrac{c}{a}=\sqrt{I_{1}^{2} +\arcsin (I_{1} \cos \delta \dfrac{b}{a})/I_{1} +I_{2}^{2}} \end{align} (6)

 图 3 不同条件下的比值$b/a$、角度$\delta$与应变能$W$ Fig.3 Ratio $b/a$, angle $\delta$ and strain energy $W$ in different conditions

Kresling折痕运动的平面高度差$\Delta H$

 \begin{align} \Delta H=h_{0} -H=b-b\sin \delta \end{align} (8)

 \begin{align} W=\dfrac{1}{2}nk(b_{1} -b_{0})^{2}+\dfrac{1}{2}nk(c_{1} -c_{0})^{2} \end{align} (9)

 图 5 Miura的折痕图 Fig.5 Crease pattern of Miura

 图 6 Miura折痕单元 Fig.6 Miura crease unit

 \begin{align} \begin{cases} \beta_{\rm m} =\arccos (2\cos c-1)-{\pi} \\ \beta_{\rm v} =2(\angle CAB+\angle EAB-{\pi} /2) \end{cases} \end{align} (11)

 图 7 山谷折痕角$\beta_{\rm v}$、山峰折痕角$\beta_{\rm m}$与输入角度$\theta_{\rm M}$关系 Fig.7 Relationship among the valley crease angle $\beta_{\rm v}$, the peak crease angle $\beta_{\rm m}$, and the input angle $\theta_{\rm M}$

3.1.4 多层Miura单元建模

 图 8 8多层Miura单元 Fig.8 Multilayer Miura units

 \begin{align} M_{\rm T} =Fd\frac{\sin \dfrac{{\pi} -\alpha_{1}}{2}}{\sqrt{\lambda^{2}+2\lambda \cos \dfrac{{\pi} -\alpha_{1}}{2}+1}} \end{align} (13)

$F=$ 1 N，$d=$ 20 mm，由式(13) 可得$M_{\rm T}$与指节长宽比$\lambda$、绕$X$轴转角$\alpha_{1}$之间的关系，如图 9所示。当$0<\lambda <1$时，$M_{\rm T} $$\alpha_{1} \in (0, {\pi}) 范围内单调递增；当 \alpha_{1}={\pi} -2\arccos(- d_{\rm m}/ \lambda) 时， M_{\rm T} 取最大值 M_{\rm T\max}=F d 。当 \lambda \geqslant 1 时， M_{\rm T}$$ \alpha_{1} \in$ $(0, {\pi} -$ $2 \arccos$ $(-1/ \lambda))$范围内单调递增，在$\alpha_{1} \in$ $({\pi}$ $- 2\arccos$ $(-1/ \lambda) , {\pi})$范围内单调递减。当$\alpha_{1}=$ $\arccos$ $(- \lambda)$时，$M_{\rm T}$取最大值$M_{\rm T\max}=F d / \lambda$

 图 9 力矩$M_{\rm T}$与指节长宽比$\lambda$、角度$\alpha_{1}$的关系曲线 Fig.9 Relationship curve among the torque $M_{\rm T}$, the knuckle length-width ratio $\lambda$, and the angle $\alpha_{1}$

$\alpha_{1} \in (\Delta \alpha_{1}, \Delta \alpha_{2})$时，$F$所做的功$W_{ F 1}$

 \begin{align} W_{F{1}} =Fd(\varsigma_{1} -\varsigma_{2})/\lambda \end{align} (14)

 \begin{align} \begin{cases} \bar{M}_{F{1}} =Fd({\varsigma_{1} +\varsigma_{2}})/({\lambda \Delta \alpha_{2} -\lambda \Delta \alpha_{1}}) \\ W_{F{1}} =\bar{M}_{F{1}} ({\Delta \alpha_{2} -\Delta \alpha_{1}}) \end{cases} \end{align} (15)

$\alpha_{1} \in (0, {\pi})$时，根据式(15) 得到

 \begin{align} W_{F{1}} = \begin{cases} 2Fd/\lambda, & \lambda \geqslant 1 \\ 2Fd, & 0<\lambda <1 \end{cases} \end{align} (16)

$k_{ F}= {\rm d} F /{\rm d} \beta$，将式(17) 代入式(18) 可得

 \begin{align} \frac{k_{ F} b_{ F}} {k_{\rm T}} =\frac{ab\sin^{2}\beta} {1+a^{2}b^{2}-2ab\cos \beta} \end{align} (19)

 图 12 单位刚度比$k_{ F}b_{ F}/ k_{\rm T}$与边长比$b/a$、角度$\theta_{\rm K}$的关系 Fig.12 Relation among the unit stiffness $k_{F}b_{ F}/ k_{\rm T}$, the side length ratio $b / a$, and the angle $\theta_{\rm K}$

 \begin{align} \alpha_{1} =nb_{\rm m} \cdot \sin (\varphi /2)\cdot \varphi {\pi} /(90d) \end{align} (20)

Miura单元层数为$n$，当$n$不变时，弯曲角度$\alpha_{1}$随着$\varphi$增大而增大；当$\varphi$不变时，$\alpha_{1} $$n 增大而增大。取 b_{\rm m}=4 mm可以得到最大弯曲角度值为 \alpha_{1\max}= 18 ^{\circ} ，故 \alpha_{1} \in (15 ^{\circ} , 18 ^{\circ}) 。根据机器人技术的基本特性数据，每个手指关节的转动角应该在0~90 ^{\circ} 范围以内。因此，Miura折痕的数目至少为5层，最小单元角 \varphi 与弯曲角度 \alpha_{1} 、层数 n 之间的关系如图 13所示。  图 13 13最小单元角\varphi 与弯曲角度\alpha_{1}、层数 n 间关系 Fig.13 Relation among the minimum element angle \varphi , the bending angle \alpha_{1}, and the number of layers n 3.3 机械手工作空间分析 该研究中提出的机械手结构是由Kresling和Miura单元组成，如图 14所示。Kresling折痕构成的指段可以简化看作杆件，简化后的杆件长度分别对应指段长度 l_{i}$$ i= 1, 2, \cdots, m$）。第$i$个Miura结构的弯曲角度为$\alpha_{i}$

 图 14 机械手单根手指坐标图 Fig.14 Coordinate diagram of a single finger of the manipulator

 \begin{align} \begin{cases} \varepsilon_{\rm d} =l_{i} -\dfrac{2al_{i} {\pi} -a^{2}l_{i} \varepsilon_{\theta}} {2a{\pi} } \\[5pt] \phi_{i}^{\max} =2\arcsin \dfrac{l_{i} -\varepsilon_{\rm d}} {4R} \\[5pt] d_{i} =l_{i} -2R\sin \dfrac{\phi_{i}} {2} \end{cases} \end{align} (21)

 \begin{align} {\rm rot} (x, \varphi_{i} )& = \begin{bmatrix} {c_{1}} & {-s_{1}} & 0 & 0 \\ {s_{1}} & {c_{1}} & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align} (22)
 \begin{align} {\rm rot} (y, a_{i} )& = \begin{bmatrix} {c_{1}} & 0 & {-s_{1}} & 0 \\ 0 & 1 & 0 & 0 \\ {-s_{1}} & 0 & {c_{1}} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align} (23)
 \begin{align} \mathit{\boldsymbol{P}} & = \begin{bmatrix} 1 & 0 & 0 & {d_{i} s_{i} c_{i} f_{i} /2} \\ 0 & 1 & 0 & {d_{i} s_{i} s_{i} f_{i} /2} \\ 0 & 0 & 1 & {d_{i} c_{i} f_{i} /2} \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align} (24)
 \begin{align} (x, y, z)&=\prod\limits_{i=1}^m {{\mathit{\boldsymbol{D}}}_{i}} {\mathit{\boldsymbol{\kappa}}} \end{align} (25)

 \begin{align} F_{1} =\frac{T_{1} \sin \phi_{3{\rm A}}} {r_{1} \cos \phi_{1{\rm A}} +l_{1} (\cos \phi_{2{\rm A}} +\cos \phi_{3\rm A})} \end{align} (28)

3.4.4 包络抓取长方体时的动力学分析

 \begin{align} \begin{cases} \tan \alpha_{1} =\dfrac{a}{\mu_{1} -\psi_{1}} \\[5pt] \tan \zeta_{12} =\dfrac{a-l_{1} \sin \alpha_{1}} {l_{1} \cos \alpha_{1} -\mu_{2} +\psi_{1}} \\[5pt] \tan \zeta_{13} =\dfrac{a+b_{0}} {l_{1} \cos \alpha_{1} -\mu_{3} +\psi_{1}} \end{cases} \end{align} (29)
 图 17 手指包络抓取长方体 Fig.17 A finger grasping a cuboid in an envelope manner

 图 18 物体长度$h$与角度间关系 Fig.18 Relation between the length $h$ of the object and the angle
3.4.5 包络抓取圆柱体时的动力学分析

 图 19 手指包络抓取圆柱体截面图 Fig.19 Section diagram of a finger grasping a cylinder in an envelope manner

 \begin{align} \begin{cases} \tan \alpha_{1} =\dfrac{a}{\mu_{1} -{b_{0}} /2+r/\sin \alpha_{1}} \\[8pt] \varTheta_{12} =\dfrac{l_{1} \sin \alpha_{1} -a-\omega} {\xi -\mu_{2} -\chi} \\[8pt] \varTheta_{123} =-\dfrac{(a+\omega)(2P-rQ)}{\psi -h_{0}} \end{cases} \end{align} (30)

 图 21 接触力与角度关系 Fig.21 Relationship between the contact force and the angle

$N= \cos( \alpha_{2} - \alpha_{3})$不变时，$F_{1}$随着$\alpha_{1}$增大而增大；当$\alpha_{1}$不变时，$F_{1}$$N= \cos( \alpha_{2} - \alpha_{3})$增大而增大。第2个手指关节接触力$F_{2}$先增大后减小，接触力$F_{3}$随着$N= \cos( \alpha_{2} - \alpha_{3})$增大而增大。当$N=$ 0.5时，$F_{2\max}=$ 46 N。

4 结论（Conclusion）

(1) 提出一种折痕混合型三指机械手，通过对Kresling折痕与Miura折痕开展几何分析，确定了以Kresling折痕为手指指段、以Miura折痕作为手指关节的折纸机械手。

(2) 通过建立混合型单元，确定了采用1层Kresling折痕作为手指的指段，采用5层Miura折痕作为手指的关节，利用D-H法确定了机械手的抓取空间。

(3) 分析了机械手在包络抓取长方体与圆柱体对象时各接触点的压力与关节角度之间的关系，对比发现各接触点的压力随着关节角度的增大而增大。

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