Design and analysis of the gripper mechanism based on generalized parallel mechanisms with configurable moving platform

doi:10.1007/s11465-021-0655-1

Front. Mech. Eng.

2021, Vol. 16

Issue (4) : 765-781 https://doi.org/10.1007/s11465-021-0655-1

RESEARCH ARTICLE

Design and analysis of the gripper mechanism based on generalized parallel mechanisms with configurable moving platform

Lin WANG, Yuefa FANG(

), Luquan LI

Department of Mechanical Engineering, Beijing Jiaotong University, Beijing 100044, China

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Abstract

Generalized parallel mechanisms with a configurable moving platform have become popular in the research field of parallel mechanism. This type of gripper mechanism can be applied to grasp large or heavy objects in different environments that are dangerous and complex for humans. This study proposes a family of novel (5 + 1) degrees of freedom (three translations and two rotations plus an additional grasping motion) gripper mechanisms based on the generalized parallel mechanisms with a configurable moving platform. First, the configurable moving platform, which is a closed loop, is designed for grasping manipulation. The hybrid topological arrangement is determined to improve the stiffness of the manipulator and realize high load-to-weight ratios. A sufficient rule based on Lie group theory is proposed to synthesize the mechanism. The hybrid limb structure is also enumerated. A family of novel gripper mechanisms can be assembled through the hybrid limbs by satisfying the rule. Two examples of the gripper mechanisms with and without parallelogram pairs are shown in this study. A kinematic analysis of the example mechanism is presented. The workspace shows that the mechanism possesses high rotational capability. In addition, a stiffness analysis is performed.

Keywords generalized parallel mechanism configurable moving platform gripper mechanism type synthesis kinematic analysis

Corresponding Author(s): Yuefa FANG

Just Accepted Date: 28 September 2021 Online First Date: 16 November 2021 Issue Date: 28 January 2022

Cite this article:

Lin WANG,Yuefa FANG,Luquan LI. Design and analysis of the gripper mechanism based on generalized parallel mechanisms with configurable moving platform[J]. Front. Mech. Eng., 2021, 16(4): 765-781.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0655-1
https://academic.hep.com.cn/fme/EN/Y2021/V16/I4/765

Fig.1 Structure of the configurable traveling platform with different configurations: (a) initial configuration, (b) grasping configuration, (c) rotation around x-axis, and (d) rotation around y-axis.

Fig.2 Evolution of the topological structure of the generalized parallel mechanism.

Fig.3 Topological arrangement based on Lie group theory.

Type	Equivalent kinematic chain
Type	Three-dimensional displacement	Four-dimensional displacement	Five-dimensional displacement submanifold
1	$u P u R N 1 u R N 2$	$u P u R N 1 u R N 2 u R N 3$	$u P u R N 1 u R N 2 u R N 3 ⊥ R N 4$
2	$u P ⊥ P a u R N$	$u R N 1 u P a u R N 2 u R N 3$	$u P a u R N 1 u R N 2 u R N 3 ⊥ R N 4$
3	$v P v R N 1 v R N 2$	$u P ⊥ P a u R N 1 u R N 2$	$u P ⊥ P a u R N 1 u R N 2 ⊥ R N 3$
4	$v P ⊥ P a v R N$	$u R N 1 u R N 2 u C$	$u R N 1 u R N 2 u C ⊥ R N 3$
5	$w P w R N 1 w R N 2$	$⊥ P u R N u C$	$⊥ P u R N 1 u C ⊥ R N 2$
6	$w P ⊥ P a w R N$	$⊥ P a u R N u C$	$⊥ P u R N 1 u C ⊥ R N 2$
7	?	$v P v R N 1 v R N 2 v R N 3$	$v P v R N 1 v R N 2 v R N 3 ⊥ R N 4$
8	?	$v R N 1 v P a v R N 2 v R N 3$	$v P a v R N 1 v R N 2 v R N 3 ⊥ R N 4$
9	?	$v P ⊥ P a v R N 1 v R N 2$	$v P ⊥ P a v R N 1 v R N 2 ⊥ R N 3$
10	?	$v R N 1 v R N 2 v C$	$v R N 1 v R N 2 v C ⊥ R N 3$
11	?	$⊥ P v R N v C$	$⊥ P v R N 1 v C ⊥ R N 2$
12	?	$⊥ P a v R N v C$	$⊥ P v R N 1 v C ⊥ R N 2$
13	?	$w P w R N 1 w R N 2 w R N 3$	$w P w R N 1 w R N 2 w R N 3 ⊥ R N 4$
14	?	$w R N 1 w P a w R N 2 w R N 3$	$w P a w R N 1 w R N 2 w R N 3 ⊥ R N 4$
15	?	$w P ⊥ P a w R N 1 w R N 2$	$w P ⊥ P a w R N 1 w R N 2 ⊥ R N 3$
16	?	$w R N 1 w R N 2 w C$	$w R N 1 w R N 2 w C ⊥ R N 3$
17	?	$⊥ P w R N w C$	$⊥ P w R N 1 w C ⊥ R N 2$
18	?	$⊥ P a w R N w C$	$⊥ P w R N 1 w C ⊥ R N 2$

Tab.1 Structures of subchains corresponding to different dimensional displacement submanifolds

Fig.4 4-[PPaR-PRRR] gripper mechanism.

Fig.5 4-[PRRR-PRRR] gripper mechanism.

Fig.6 Geometrical model of 4-[PPaR-PRRR] gripper mechanism.

Fig.7 Kinematics of the first part motion.

Parameter	Value/mm
$a$	880
$b$	180
$c$	280
$d$	290
$k$	780
l	1500
m	250

Tab.2 Architectural parameters of the example mechanism

Fig.8 Orientation workspace corresponding to the motion of (a) the first part and (b) the second part.

Fig.9

θ y

at various z-positions: (a) z = −500 mm, (b) z = −700 mm, (c) z = −900 mm, (d) z = −1100 mm.

Fig.10

θ y

at various x-positions: (a) x = −100 mm, (b) x = 0 mm, (c) x = 100 mm, (d) x = 200 mm.

Fig.11

θ x

at various z-positions: (a) z = −500 mm, (b) z = −700 mm, (c) z = −900 mm, (d) z = −1100 mm.

Fig.12

θ x

at various y-positions: (a) y = −100 mm, (b) y = 0 mm, (c) y = 100 mm, (d) y = 200 mm.

Fig.13 Infinitesimal screws and wrenches of the hybrid limb.

Fig.14 Stiffness distribution of the mechanism: (a) linear stiffness along x-axis, (b) linear stiffness along y-axis, (c) linear stiffness along z-axis, (d) torsion stiffness around x-axis, and (e) torsion stiffness around y-axis.

a	Length of link $A i B i$ (i = 1, 2, …, 4)
b	Length of link $B i C i$ (i = 1, 2, …, 4)
c	Length of link $C i D i$ (i = 1, 2, …, 4)
d	Length of link $D i E i$ (i = 1, 2, …, 4)
${D}$	6-dimensional rigid motion
${E}$	Rigid connection, no relative motion
F	Force applied to the moving platform
f	Force of constraint
{G(u)}, {G(v)}, and {G(w)}	Planar motion determined by the normal u, v and w, respectively
h	Distance the moving plate moves along z-axis
${I P i}$	Displacement submanifold of ith intermediate platform
${I P i ′}$	Subgroup in ${I P i}$ excluding ${T 2 (u)}$
J	6×6 Jacobian matrix
K	Stiffness matrix
$k i$	Stiffness constant (i = 1, 2, …, n)
l_i	Distance the ith actuator moves
${L i}$	Displacement submanifold of the ith limb
${M}$	Displacement submanifold of the terminal body relative to the base body
${M i j}$	Additional subgroup expanding from the three- to four-dimensional displacement submanifold
${M (u i)}$	Displacement subgroup associated with the ith joint

n	Moments of constraint
${N i j}$	Additional subgroup expanding from the four- to five-dimensional displacement submanifold
P	Prismatic joint
$P a$	Composite joint of a planar hinged parallelogram that produces circular translation between two opposite bars
${P}$	Output motions of the configurable moving platform relative to the fixed base
${P a (v)}$	Translation along the direction perpendicular to the long side of the parallelogram with the axes of revolutes in the composite joint parallel to the v-direction
$q ˙ i j$	Intensity associated with jth joint of the ith limb (i = 1, 2, …, 4, and j = 1, 2, 3 in the left chain and i = 1, 2, and j = 4, 5, …, 7 in the right chain)
R	Revolute joint
R	Rotaion matrix
{R(N, u)} and {R(N, v)}	Rotations about the axis determined by the point N and the unit vector u and v, respectively
{R(O_i, u)} and {R(O_i, v)}	Rotations about the axis determined by the point $O i$ and the unit vector u and v, respectively
{R(N_i, u)}, {R(N_i, v)}, and {R(N_i, w)}	Rotations about the axis determined by the point $N i$ and the unit vector u, v, and w, respectively
$S i n$	nth subchain in limb i
${S i n}$	Motions of the corresponding subchains (i = 1, 2, …, 4, and n = 1, 2, 3)
${T}$	3-dimensional translation in space
${T (P l)}$	Two-dimensional displacement subgroup that is a subset of ${G (w)}$
{T(u)}, {T(v)}, and {T(w)}	Translation along the unit vector u, v, and w, respectively
${T 2 (u)}$	Planar motion in plane determined by the normal u
v	Linear velocity of the body
w	Angular velocity of the body
{X(u)}, {X(v)}, and {X(w)}	3-dimensional translation and one rotation about the unit vector u, v, and w, respectively
$α i$	Angles between the link $C i D i$ (i = 1, 2) and z-axis
$θ x$ , $θ y$	Angles the moving plate rotates around x- and y-axis, respectivley
$γ 1$ , $γ 3$	Angles between the link $A i B i$ (i = 1, 3) and x-axis
$γ 2$ , $γ 4$	Angles between the link $A i B i$ (i = 2, 4) and y-axis
$κ 1$ , $κ 2$	Reference planes parallel to the plane xoz and yoz respectively
$$ L$ , $$ R$	Instantaneous twist of the left and right serial chain, respectively
$$ P$	Velocity of the moving platform
$$ i j$	Unit screw associated with jth joint of the ith limb (i = 1, 2, …, 4, and j = 1, 2, 3 in the left chain and i = 1, 2, and j = 4, 5, …, 7 in the right chain)
$$ r i 1$	Reciprocal wrench in the left serial chain (i = 1, 2, …, 4)

$$ r i 2$	Reciprocal wrench in the right serial chain (i = 1, 2)
$(?) ?$	First derivative of $(?)$ with respect to time
$τ$	Vector of the actuated joint force or torques
$Δ q$	Joint deflection
$Δ x$	Displacement of the end effector

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