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

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.

Graphical abstract

Keywords

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

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Nomenclature

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)
\left\{D\right\}	6-dimensional rigid motion
\left\{E\right\}	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
\left\{I{P}_i\right\}	Displacement submanifold of ith intermediate platform
\left\{I{P}_i^{\prime }\right\}	Subgroup in \left\{I{P}_i\right\} excluding \left\{T_2(\mathit{u})\right\}
J	6×6 Jacobian matrix
K	Stiffness matrix
k_i	Stiffness constant (i = 1, 2, …, n)
li	Distance the ith actuator moves
\left\{L_i\right\}	Displacement submanifold of the ith limb
\left\{M\right\}	Displacement submanifold of the terminal body relative to the base body
\left\{M_{ij}\right\}	Additional subgroup expanding from the three- to four-dimensional displacement submanifold
\left\{M(u_i)\right\}	Displacement subgroup associated with the ith joint

n	Moments of constraint
\left\{N_{ij}\right\}	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
\left\{P\right\}	Output motions of the configurable moving platform relative to the fixed base
\left\{Pa(\mathit{v})\right\}	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
{\dot{q}}_{ij}	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(Oi, u)} and {R(Oi, v)}	Rotations about the axis determined by the point O_i and the unit vector u and v, respectively
{R(Ni, u)}, {R(Ni, v)}, and {R(Ni, w)}	Rotations about the axis determined by the point N_i and the unit vector u, v, and w, respectively
S_{in}	nth subchain in limb i
\left\{S_{in}\right\}	Motions of the corresponding subchains (i = 1, 2, …, 4, and n = 1, 2, 3)
\left\{T\right\}	3-dimensional translation in space
\left\{T(Pl)\right\}	Two-dimensional displacement subgroup that is a subset of \left\{G(\mathit{w})\right\}
{T(u)}, {T(v)}, and {T(w)}	Translation along the unit vector u, v, and w, respectively
\left\{T_2(\mathit{u})\right\}	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
{\alpha }_i	Angles between the link C_i{D}_i (i = 1, 2) and z-axis
{\theta }_x, {\theta }_y	Angles the moving plate rotates around x- and y-axis, respectivley
{\gamma }_1, {\gamma }_3	Angles between the link A_i{B}_i (i = 1, 3) and x-axis
{\gamma }_2, {\gamma }_4	Angles between the link A_i{B}_i (i = 2, 4) and y-axis
{\kappa }_1, {\kappa }_2	Reference planes parallel to the plane xoz and yoz respectively
{\mathit{\text{\$}}}_{\text{L}}, {\mathit{\text{\$}}}_{\text{R}}	Instantaneous twist of the left and right serial chain, respectively
{\mathit{\text{\$}}}_{\text{P}}	Velocity of the moving platform
{\mathit{\text{\$}}}_{ij}	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)
{\mathit{\text{\$}}}_{\text{r}i1}	Reciprocal wrench in the left serial chain (i = 1, 2, …, 4)

{\mathit{\text{\$}}}_{\text{r}i2}	Reciprocal wrench in the right serial chain (i = 1, 2)
\stackrel{\cdot }{(\cdot )}	First derivative of (\cdot ) with respect to time
\mathit{\tau }	Vector of the actuated joint force or torques
\mathrm{\Delta }\mathit{q}	Joint deflection
\mathrm{\Delta }\mathit{x}	Displacement of the end effector

Acknowledgements

This research work was supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2020YJS153), and the National Natural Science Foundation of China (Grant No. 51975039).

Electronic Supplementary Material

The supplementary material can be found in the online version of this article at https://doi.org/10.1007/s11465-021-0655-1 and is accessible to authorized users.

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