Forward Kinematics and Singularity of Kinematically Decoupled Stewart-type Parallel Mechanisms

doi:10.3969/j.issn.1004-132X.2025.09.006

Abstract

Abstract:

The forward kinematics equation of the six-degree-of-freedom parallel mechanisms was nonlinear and strongly coupled， and generally did not have a symbolic positive solution， which was not conducive to the real-time feedback control of the robots. Thus， a “7-4” Stewart-type parallel mechanism was designed with weak coupling in structures but decoupled in motions. The forward kinematics equation and link length coordination equation were solved analytically， and the singularity research was carried out. Firstly， based on the “2-1” kinematic links， a six-degree-of-freedom “7-4” Stewart-type parallel mechanism was synthesized， and the structural coupling characteristics of the mechanisms were analyzed based on the azimuth feature set theory. Secondly， based on 13 compatible equations and the theory of tetrahedral geometry， an analytical algorithm for solving the forward kinematics equation was proposed. At the same time， it was proved that the number of real solutions under general configuration was 8（they were symmetrical about the same plane ）. Then， according to the geometric constraint relationship between the moving ball hinges， the link length coordination equation was constructed. It is found that the equation also has a symbolic solution. The Jacobian matrix of the mechanisms was derived， and various singular types were analyzed. Finally， the internal relationship between the forward kinematics and singularity of the parallel mechanisms was analyzed.

Key words: parallel mechanism, forward kinematics, type synthesis, singularity, kinematically decoupled

摘要：

六自由度并联机构的正运动学方程非线性且强耦合，一般不具有符号式正解，不利于机器人的实时反馈控制。设计了一种在结构上弱耦合但在运动上可解耦的“7-4”式Stewart型并联机构，解析求解了正运动学方程和杆长协调方程，并开展了奇异性研究。基于“2-1”式运动链综合出六自由度“7-4”式Stewart型并联机构，并基于方位特征集理论，分析了机构的结构耦合特性。基于13个相容方程并运用四面体几何理论，提出了正运动学方程的一种解析求解算法，同时还证明出一般位形下实数解的个数为8（它们两两关于同一平面对称）。根据动球铰之间的几何约束关系，构建了杆长协调方程，研究发现，该方程也具有符号解。推导了机构的Jacobian矩阵，并分析了各种奇异类型。剖析了并联机构正运动学与奇异性之间的内在联系。

关键词: 并联机构, 正运动学, 型综合, 奇异性, 运动解耦

CLC Number:

TH112

Ningning HUANG, Jingjing YOU, Pengda YE, Huiping SHEN, Chenggang LI, Hongtao WU. Forward Kinematics and Singularity of Kinematically Decoupled Stewart-type Parallel Mechanisms[J]. China Mechanical Engineering, 2025, 36(9): 1951-1960.

黄宁宁, 尤晶晶, 叶鹏达, 沈惠平, 李成刚, 吴洪涛. 一种运动可解耦的Stewart型并联机构的正运动学及奇异性[J]. 中国机械工程, 2025, 36(9): 1951-1960.

Figures/Tables 21

Fig.1 “2-1” kinamatic link

Fig.2 “6-3” Stewart-type parallel mechanism

Fig.3 “7-4” Stewart-type parallel mechanism

Fig.4 Tetrahedron Q2O2Q3Q1

Fig.5 Process of forward position solution

Tab.1 Multiple solutions of forward kinematics

组别	Q₄	Q₁	Q₂
1	^（1）Q₄	^（1）Q_1（1）	^（1）Q_2（1）
2		^（1）Q_1（1）	^（1）Q_2（2）
3		^（1）Q_1（2）	^（1）Q_2（3）
4		^（1）Q_1（2）	^（1）Q_2（4）
5	^（2）Q₄	^（2）Q_1（1）	^（2）Q_2（1）
6		^（2）Q_1（1）	^（2）Q_2（2）
7		^（2）Q_1（2）	^（2）Q_2（3）
8		^（2）Q_1（2）	^（2）Q_2（4）

Tab.2 Mechanism parameters and input parameters mm

参数	数值	参数	数值	参数	数值
M₁	20.000	M₄	50.000	L₄	67.823
M_1*	40.000	M_4*	70.000	L_4*	86.023
x′	0.000	L₁	38.356	N₁	30.000
y′	30.000	L_1*	35.323	N₂	42.426
z′	0.000	L₂	35.868	N₃	42.426
M₃	40.000	L₃	43.111
M_3*	60.000	L_3*	56.878

Fig.6 Theoretical calculation and virtual test results

Tab.3 Schematic diagram of mechanism configuration

第1、5组解对称	第2、6组解对称

第3、7组解对称	第4、8组解对称

Tab.4 Theoretical calculation and experimental measurement data

	任意选取的数据/mm					理论计算/mm	试验测量/mm
	L_1*	L₃	L_3*	L₄	L_4*	L₁
1	35.322 97	43.1105 4	56.877 73	67.823 30	86.023 25	38.353 63	38.353 63
2	40.176 93	59.772 10	72.626 50	75.952 01	94.417 73	34.857 32	34.857 32
3	46.904 13	103.453 64	114.832 72	100.749 83	117.764 71	43.452 25	43.452 25

Tab.5 Some elements in J

a_j	$2 x + 2 n c β c γ - 2 M j$
b₁	$2 y + 2 n c β s γ$
d₁	$2 z - 2 n s β$
a₂	$2 (x - x' + n c γ s α s β - n c α s γ)$
b₂	$2 (y - y' + n c α c γ + n s α s β s γ)$
d₂	$2 (z - z' + n c β s α)$
a_e	$2 (M e + x - n c β c γ)$
b₃	$2 (y - n c β s γ)$
d₃	$2 (z + n s β)$
a₄	$2 x$
b_h	$2 (M h + y)$
d₄	$2 z$
B_j	$- 2 n {z c β + s β [(x - M j) c γ + y s γ]}$
D_j	$2 n c β [y c γ + (M j - x) s γ]$
B_e	$2 n {z c β + s β [(M e + x) c γ + y s γ]}$
D_e	$2 n c β [- y c γ + (M e + x) s γ]$
A₂	$- 2 n {s α [(y - y') c γ + (x' - x) s γ] + c α {(z' - z) c β + s β [(x' - x) c γ + (y' - y) s γ]}}$
B₂	$2 n s α {(z' - z) s β + c β [(x - x') c γ + (y - y') s γ]}$
D₂	$2 n {s α s β [(y - y') c γ + (x' - x) s γ] +$ $c α [(x' - x) c γ + (y' - y) s γ]}$

Tab.5 Some elements in J

a_j	$2 x + 2 n c β c γ - 2 M j$
b₁	$2 y + 2 n c β s γ$
d₁	$2 z - 2 n s β$
a₂	$2 (x - x' + n c γ s α s β - n c α s γ)$
b₂	$2 (y - y' + n c α c γ + n s α s β s γ)$
d₂	$2 (z - z' + n c β s α)$
a_e	$2 (M e + x - n c β c γ)$
b₃	$2 (y - n c β s γ)$
d₃	$2 (z + n s β)$
a₄	$2 x$
b_h	$2 (M h + y)$
d₄	$2 z$
B_j	$- 2 n {z c β + s β [(x - M j) c γ + y s γ]}$
D_j	$2 n c β [y c γ + (M j - x) s γ]$
B_e	$2 n {z c β + s β [(M e + x) c γ + y s γ]}$
D_e	$2 n c β [- y c γ + (M e + x) s γ]$
A₂	$- 2 n {s α [(y - y') c γ + (x' - x) s γ] + c α {(z' - z) c β + s β [(x' - x) c γ + (y' - y) s γ]}}$
B₂	$2 n s α {(z' - z) s β + c β [(x - x') c γ + (y - y') s γ]}$
D₂	$2 n {s α s β [(y - y') c γ + (x' - x) s γ] +$ $c α [(x' - x) c γ + (y' - y) s γ]}$

Fig.7 Orientation singular surface

Fig.8 Orientation singular configuration

Fig.9 Position singular configuration

Fig.10 The first Hunt’s singularity

Fig.11 The second Hunt’s singularity

Fig.12 The third Hunt’s singularity

Fig.13 Colinearly actuation singularity

Fig.14 Points Q1、Q3 trajectory of motion

Fig.15 Unique solution at point Q2

Tab.6 Relationship between the number of forward position solutions and singularity

四组位置正解			一组位置正解
构型
奇异	无奇异位形		Hunt奇异，Q₁Q₃为瞬时旋转轴
两组位置正解
构型
奇异	若O₂点位于面H₂Q₄Q₂上，则为Hunt奇异，Q₄H₂为瞬时旋转轴	Hunt奇异，Q₁Q₃为瞬时旋转轴

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