Design and analysis of partially decoupled translational parallel mechanisms with single-loop structures

Abstract

This study presents a family of novel translational parallel mechanisms (TPMs) with single-loop topological structures. The proposed mechanism consists of only revolute and prismatic joints. The novel TPMs are simpler in structure and have fewer joints and components than the well-known Delta Robot. Four types of 2-degree of freedom driving systems are applied to different limb structures to avoid the moving actuator that causes the problem of increased moving mass. Four sample TPMs are constructed using the synthesized limbs, and one of them is investigated in terms of kinematic performance. First, a position analysis is performed and validated through numerical simulation to reveal the characteristics of partially decoupled motion, which improves the controllability of TPM. Second, singular configurations are identified, and the resulting singularity curve is obtained. Lastly, the workspace of TPM is analyzed, and the relationship between the singular configurations and the reachable workspace is explored. The workspace of the 3-CRR (C denotes the cylindrical joint and R denotes the revolute joint) translational mechanism is also presented to prove that the proposed TPM has a fairly large workspace.

Key words： translational parallel mechanism; single loop; multiple driving system; workspace analysis; singularity

Nomenclature

a_1, a_2, a_3	Lengths of links O{A}_{\text{1}}, O{A}_2, and A_{\text{1}}{A}_{\text{3}}, respectively
b_i	Length of link A_i{B}_i (i = 1, 2, 3)
c_i	Length of link B_i{C}_i (i = 1, 2, 3)
{\dot{d}}_{ij}	Intensity associated with the jth prismatic joint in the ith limb (i = 1, 2, and j = 1, 2)
\left\{D\right\}	6-dimensional rigid motion
\left\{G(\mathit{u})\right\}, \left\{G(\mathit{v})\right\}	Planar motion determined by the normal vectors u and v, respectively
\left\{G_2(\mathit{v})\right\}	A subset of \left\{G(\mathit{v})\right\}
h	Distance between point o and the center point of the end effector
k	Nonzero coefficient
l_i	Length of different links
m	Component of distance between points o and C_i in the y-direction (i = 1, 2, 3)
\left\{M\right\}	Displacement submanifold of the terminal body relative to the base body
\left\{M_{\text{S}i}\right\}	Motion subgroup of the ith serial kinematic chain (i = 1, 2)
\left\{M({\mathit{u}}_i)\right\}	Displacement subgroup associated with the ith joint (i = 1, 2, …, n)
n	Component of distance between points o and C_i in the x-direction (i = 1, 2, 3)
\left\{Pa(\mathit{u})\right\}, \left\{Pa(\mathit{v})\right\}, \left\{Pa(\mathit{w})\right\}	Translations along the direction perpendicular to the long side of the parallelogram with the axes of revolute joints in the composite joint parallel to the u, v, and w direction, respectively
\left\{R(N,\mathit{u})\right\}, \left\{R(N,\mathit{v})\right\}	Rotations about the axis determined by the point N and the unit vectors u and v, respectively
\left\{R(N_i,\mathit{u})\right\}, \left\{R(N_i,\mathit{v})\right\}, \left\{R(N_i,\mathit{w})\right\}	Rotations about the axis determined by the point N_i and the unit vectors u, v, and w, respectively (i = 1, 2, …, 6)
S_i	ith serial kinematic chain (i = 1, 2)
{\mathit{s}}_{ij}	Unit vector associated with the jth joint of the ith limb (i = 1, 2, and j = 1, 2, …, 4)
{\mathit{s}}_{\text{r}ij}	Unit vector associated with the jth reciprocal wrench in the ith limb (i = 1, 2, and j = 1, 2)
\left\{T\right\}	3-dimensional translation in space
\left\{T(\mathit{u})\right\}, \left\{T(\mathit{v})\right\}, \left\{T(\mathit{w})\right\}	Translations along the unit vectors u, v, and w, respectively
{\mathit{v}}_{\text{p}}^{}	Linear velocity of the center of mass of the mobile platform
{\mathit{w}}_{\text{p}}^{}	Angular velocity of the center of mass of the mobile platform
x_{A\text{1}}	Motion parameter of actuated prismatic pair in the first limb
\left\{X(\mathit{u})\right\}, \left\{X(\mathit{v})\right\}, \left\{X(\mathit{w})\right\}	3-dimensional translation and one rotation about the unit vectors u, v, and w, respectively
y_{A\text{2}}	Motion parameter of actuated prismatic pair in the second limb
\alpha	Output rotational angle around one axis in the 2-DOF driving system
{\alpha }_1	Angle between the Y axis and the long rod of the parallelogram joint in the first limb
{\alpha }_2	Angle between the X axis and the long rod of the parallelogram joint in the second limb
\theta	Output rotational angle around the other axis in the 2-DOF driving system
{\theta }_i	Rotational angles of the first revolute pair in the ith limb (i = 1, 2)
{\dot{\theta }}_{ij}	Intensity associated with the jth revolute joint in the ith limb (i = 1, 2, and j = 1, 2)
{\mathit{\text{\$}}}_{\text{p}}	Instantaneous twist of the end effector
{\mathit{\text{\$}}}_{ij}	Unit screw of the jth joint in the ith limb (i = 1, 2, and j = 1, 2, …, 4)
{\mathit{\text{\$}}}_{\text{r}ij}	jth reciprocal wrench in the ith limb (i = 1, 2, and j = 1, 2)
{\gamma }_i	Rotational angle of the cylindrical joint in the ith limb (i = 1, 2, 3)
{\phi }_i	Angle between links A_i{B}_i and B_i{C}_i (i = 1, 2, 3)

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). Then, the first author would like to acknowledge the China Scholarship Council (Grant No. 202007090138) for financial support and the use of the research facilities at Lassonde School of Engineering at York University, Canada.

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