﻿ 基于共形几何代数的空间并联机构位置正解<sup>*</sup>
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1. 北方工业大学 机械与材料工程学院, 北京 100144;
2. 北方工业大学 柔性变截面辊弯成形北京市工程技术研究中心, 北京 100144

Direct kinematics of a spatial parallel mechanism based on conformal geometric algebra
HUANG Xiguang1,2, HUANG Xu1
1. School of Mechanical and Materials Engineering, North China University of Technology, Beijing 100144, China;
2. Beijing Engineering Research Center of Flexible Roll Forming, North China University of Technology, Beijing 100144, China
Received: 2016-12-06; Accepted: 2017-03-06; Published online: 2017-04-20 17:25
Foundation item: National Natural Science Foundation of China (51105003); Beijing Natural Science Foundation (3172010)
Corresponding author. HUANG Xiguang, E-mail:marchbupt@126.com
Abstract: An algorithm is proposed for the direct kinematics analysis of a spatial general 3-RPS parallel mechanism based on conformal geometric algebra (CGA). The angle between the axis of an arbitrary kinematic chain and the plane of the fixed platform can be regarded as the unknown variable. The mathematical expression of the position of the spherical joint connecting the moving platform with the kinematic chain can be expressed in the unknown variable based on CGA. The outer product of two space balls and a flat surface are constructed two times, and the corresponding points of the remaining two vertices of the moving platform are obtained respectively. The 16th degree input-output polynomial equation in the unknown variable is straightforwardly obtained by distance formula and all 16 sets of closed-form solutions can be achieved. The algorithm avoids the use of rational angles or matrices, and complex computations for nonlinear and multivariable equations. A numerical example is given to demonstrate geometric characteristics of the motion and the algorithm is intuitive.
Key words: conformal geometric algebra (CGA)     spatial parallel mechanism     direct kinematics     input and output equation     analytical solution

1 共形几何代数

CGA包括内积、外积和几何积3种运算。对于矢量uv[12]

 (1)

 (2)

CGA中，几何元素的表达式如表 1所示。INPS和OPNS分别为内积零空间和外积零空间[12]

 几何元素 表达式1 (IPNS) 表达式2 (OPNS) 点 P=X+X2e∞/2+e0 球 S=P－r2e∞/2 S*=X1∧X2∧X3∧X4 平面 π=n+te∞ π*=X1∧X2∧X3∧e∞ 圆 Z=S1∧S2 Z*=X1∧X2∧X3 直线 l=π1∧π2 l*=X1∧X2∧e∞ 点对 PP=S1∧S2∧S3 P*P=X1∧X2

 (3)

 (4)

 (5)

 (6)

 (7)

2 机构几何模型

 图 1 一种空间3-RPS并联机构 Fig. 1 A spatial 3-RPS parallel mechanism
3 位置正解

 (8)
 (9)
 (10)

 (11)

 (12)

 (13)

 (14)

 (15)

 (16)

 (17)

n12n22代入式(17)，展开可得关于cos θ1的一元十六次方程:

 (18)

4 实例验证

 组号 P1 P2 P3 1 (0, 4.115, -0.840) (-1.320, 3.795, 0.628) (0.650, 3.545, 0.386) 2 (0, 4.909, 1.051) (-1.491, 3.616, 0.727) (0.455, 3.710, 0.273) 3 (0, 3.688, -1.376) (1.456, 5.000, -0.975) (0.150, 3.929, 0.097) 4 (0, 4.833, 0.719) (1.234, 4.991, -0.847) (0.668, 3.529, 0.399) 5 (0, 3.566, -1.505) (1.240, 4.991, -0.850) (-0.548, 4.281, -0.306) 6 (0, 4.861, 0.830) (1.153, 4.984, -0.799) (-0.693, 4.331, -0.390)

5 结论

1) 将共形几何代数引入空间并联机构位置正解分析中，提出了一种空间3-RPS并联机构位置正解解析解新算法。

2) 通过构造空间球、平面等几何体的外积运算，只需要简单的距离公式即可得到关于该问题的一元十六次输入输出方程，进而获得了全部的16组解析解，求解过程没有传统并联机构运动学理论必须处理的复杂旋转角和矩阵运算以及多元高次非线性方程组求解，过程简洁明了。

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#### 文章信息

HUANG Xiguang, HUANG Xu

Direct kinematics of a spatial parallel mechanism based on conformal geometric algebra

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(12): 2377-2381
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0917