Structure Design and Kinematic Analysis of a Partially-Decoupled 3T1R Parallel Manipulator

Abstract

A partially-decoupled three-translation and one-rotation (3T1R) parallel manipulator is proposed and analyzed in this paper. The new mechanism features with a symmetric layout and simple kinematic structure for generation of 3T1R motion in pick-and-place operations. The kinematics of this mechanism, including forward and inverse position problems, are analyzed. The closed-form solution of direct kinematics is obtained and verified by inverse kinematics. Moreover, the partially-decoupled performance can be found through the forward kinematics. The proposed mechanism is mainly composed of revolute joints and presents better performance. Hence, more potential applications of this manipulator can be expected.

1 Introduction

The parallel manipulators (PMs) with three translations and one rotation (3T1R), also known as Schönflies-motion parallel robots, have a great potential of various applications in the manufacturing industry. Many 3T1R robots have been proposed and developed. Examples include H4 [1], I4 [2], I4R [3], Par4 [4], Heli4 [5]. These robots have similar architectures but with different designs of the moving platform. A parallel Schönflies-motion robot admitting a rectangular workspace, which allows to utilize the shop-floor space efficiently for flexible pick-and-place applications, was recently proposed by Wu et al. [6]. Many other 3T1R robots constructed adopting quite different kinematic structures are available in literature. Zhao, Huang et al. [7] proposed a 4-URU-type 3T1R parallel robot. Jin and Yang [8] proposed a family of 3T1R parallel mechanisms based on single-open-chain structures. Kong et al. [9] synthesized a group of PMs with the same sub-chains based on screw theory. Huang et al. [10] developed a 3T1R-type high-speed parallel manipulator called as Cross-IV. Liu et al. [11] developed a X4 parallel robot prototype with one moving platform.

It is noted that the above stated 3T1R parallel mechanisms all have higher coupling degree, and are not input-output motion decoupled, which lead to their forward kinematics and dynamics analysis, motion control and trajectory planning are more complex. Partially or fully decoupled parallel manipulators are desirable. There are already some parallel mechanisms with partial motion decoupling or complete decoupling properties proposed, for example, mechanisms of decoupled two-rotation DOF [12], one-translation and two-rotation DOF [13], three-translation [14], etc. Examples include also a XYZ parallel elasticity mechanism [15], a partially decoupled 3-PPR robot with U-shape base [16]. In spite of the above proposed partially decoupled parallel mechanisms, very few 3T1R PMs with motion decoupling are reported. The SCARA parallel robot of FlexPicker [17] has partial motion decoupling, which is based on the Delta mechanism, attached with a RUPU kinematic chain, in order to achieve the rotation around the normal of the moving platform.

In this paper, a novel partially-decoupled 3T1R parallel manipulator is proposed. The kinematics analysis of this 3T1R PM is also the subject of this paper. The paper is organized as follows: Sect. 2 illustrates this 3T1R PM of 2-(RPa||3R)3R with type synthesis method based on POC equations [18]. In Sect. 3, the closed-form equations are established and the direct kinematics of this mechanism is solved. The reported work is concluded in Sect. 4.

2 Structure Design of a 2-(RPa||3R)3R Manipulator

The 3T1R parallel manipulator proposed here is shown in Fig. 1. The base platform 0 is connected to the moving platform 1 by left and right two identical hybrid chains. Each hybrid chain contains a sub-parallel mechanism (sub-PM) and a 3R serial kinematic chain. The manipulator is symmetrical about the plane x−y = 0. The intersecting line of the plane and the base is the line t-t.

Fig. 1.

3D CAD model and kinematic structure of 3T1R robot

With reference to the symmetric plane, the left hybrid chain composed of links 2, 3, 4, 4′, 5, 6, 7, 8 and 9, as shown in Fig. 1(b), is selected to illustrate the structure of manipulator. The CAD design of the hybrid chain is shown in Fig. 2(a). The shorter link 3 of a parallelogram composed of four spherical pairs (S₁, S₂, S₃ and S₄) is connected by actuated arm 2, to the base 0 by a revolute joint R₁₁, which is denoted as RPa. The extended part of the opposite link 5 of the parallelogram is connected in parallel to a sub-chain composed of two links 6 and 7 and three parallel revolute joints (3R, i.e., R₂₁||R₂₂||R₂₃). The connection line of the two spherical joints S₃ and S₄ is collinear with the axis of the rotation joints R₁₂ but perpendicular to the axis of the revolute joint R₂₃. Thus, a sub-parallel mechanism (sub-PM) is generated, as shown in Fig. 2(b), and denoted as RPa||3R.

Fig. 2.

Kinematic structures of chains

This sub-PM is then further connected with a 3R serial kinematic chain composed of two links 8 and 9 and 3R serial kinematic chain, i.e., R₁₂||R₁₃⊥R₁₄, as shown in Fig. 2(c), which leads to a hybrid chain. Since the PM has two identical hybrid chains, the whole manipulator is recorded as 2-(RPa||3R)3R. The four rotation joints R₁₁, R₂₁, R₃₁ and R₄₁ on the base platform 0 are active and mutually perpendicular to each other, which means R₁₁⊥R₄₁, R₁₁⊥R₂₁ and R₄₁⊥R₃₁. Two rotation joints R₁₄ and R₄₄ on the moving platform are all parallel to the normal of the moving platform 1, i.e., R₁₄||R₄₄. Here, the symbols || and ⊥ stand respectively for being parallel and vertical, the same hereinafter.

3 Kinematic Analysis

3.1 Establishment of the Coordinate System and Parameterization

Without losing of generality, let the base platform 0 be a square. The four actuated joints R₁₁, R₂₁, R₃₁ and R₄₁ are located on the midpoint of its each side, i.e., A₁, A₂, A₃, A₄, as shown in Fig. 3. Furthermore, the frame coordinate system o-xyz is established on the base platform 0. The origin is located at the geometric center, point o, of the base platform 0. The axes x and y are collinear with and vertical to the connection line A₁A₃, respectively. On the moving platform 1, the moving coordinate system p-uvw is established at point p that is the midpoint of the connection line between the points F₁ and F₂. u axis is perpendicular to the line F₁F₂, while v axis coincides with this line F₁F₂. Both z and w axes are determined by the right-hand Cartesian coordinate rule, as shown in Fig. 3(a). For ease of understanding, the 2-(RPa||3R)3R PM is redrawn as stretched to a planar view, as shown in Fig. 3(b).

Fig. 3.

Parameterizations of the 2-(RPa||3R)3R PM

The structure parameters of the PM are denoted in the following way. The side length of the square base platform 0 is noted by 2l₁, and the length of the moving platform 1, i.e., F₁F₂ = 2l₂. For other link lengths, we have

$$ \begin{array}{*{20}c} {A_{1} B_{1} = A_{2} B_{2} = A_{3} B_{3} = A_{4} B_{4} = l_{3} (l_{3} \ne l_{1} ) ,\,B_{1} C_{1} = B_{2} C_{2} = B_{3} C_{3} = B_{4} C_{4} = l_{4} } \\ {D_{1} C_{1} = C_{2} D_{1} = C_{3} D_{2} = C_{4} D_{2} = l_{5} ,\,C_{1} E_{1} = C_{4} E_{2} = F_{1} E_{1} = F_{2} E_{2} = l_{6} } \\ \end{array} $$

The four input angles are defined as θ₁, θ₂, θ₃ and θ₄ for the four active joints, as shown in Fig. 3(a). The moving platform position is defined by the coordinates of point p as (x, y, z), and its orientation by angle $\alpha$, the angle from the forward direction of u-axis to x-axis or from the forward direction of v-axis to y-axis, as shown in Fig. 3(b).

3.2 Solution to Forward Position Problem

In the forward position analysis, we need to obtain output parameters, i.e., the coordinates of point p of the moving platform, defined by (x, y, z), and the orientation angle $\alpha$, as a function of the known input angles θ₁, θ₂, θ₃ and θ₄.

Solve Coordinates of Points C₁ and C₄.

The coordinates of four points A₁, A₂, A₃ and A₄ on the base platform 1 are (l₁,0,0), (0,−l₁,0), (−l₁,0,0) and (0,l₁,0), respectively. The coordinates of each end point of the four actuated arms 2, 6, 15 and 17, i.e., points B₁, B₂, B₃ and B₄, are easily calculated as respectively, (l₁ + l₃cosθ₁,0, l₃sinθ₁), (0,−l₁ + l₃cosθ₂, l₃sinθ₂), (−l₁ + l₃cosθ₃,0, l₃sinθ₃) and (0, l₁ + l₃cosθ₄, l₃sinθ₄). As stated in Sect. 2.2.1, the output links 5 and 12 of the left and right sub-PMs can only produce a motion in the plane o-yz and o-xz respectively. That is, x_C1 = 0 and y_C4 = 0. Hence, the coordinates of points C₂ and C₃ are found as $(0 ,y_{C1} - 2l_{5} ,z_{C1} )$ and $(x_{C4} - 2l_{5} ,0 ,z_{C4} )$ respectively. The link length constraints defined by $B_{1} C_{1} = B_{2} C_{2} = B_{3} C_{3} = B_{4} C_{4} = l_{4}$ imply that

$$ \left\{ {\begin{array}{*{20}l} {x_{{B_{1} }}^{2} + y_{{C_{1} }}^{2} + (z_{{B_{1} }} - z_{{C_{1} }} )^{2} = l_{4}^{2} } \hfill \\ { (y_{{B_{2} }} - y_{{C_{1} }} + 2l_{5} )^{2} + (z_{{B_{2} }} - z_{{C_{1} }} )^{2} = l_{4}^{2} } \hfill \\ \end{array} } \right. $$

(1)

$$ \left\{ {\begin{array}{*{20}l} {x_{{C_{4} }}^{2} + y_{{B_{4} }}^{2} + (z_{{B_{4} }} - z_{{C_{4} }} )^{2} = l_{4}^{2} } \hfill \\ { (x_{{B_{3} }} - x_{{C_{4} }} + 2l_{5} )^{2} + (z_{{B_{3} }} - z_{{C_{4} }} )^{2} = l_{4}^{2} } \hfill \\ \end{array} } \right. $$

(2)

Equation (1) leads to

$$ a_{1} y_{{C_{1} }} + b_{1} z_{{C_{1} }} = c_{1} $$

(3)

here, $a_{1} = 2 (y_{B2} + 2l_{5} ),\,$$b_{1} = 2 (z_{B2} - z_{B1} ) ,$ $c_{1} = (y_{B2} + 2l_{5} )^{2} + z_{B2}^{2} - x_{B1}^{2} - z_{B1}^{2} .\,$ If a₁ = 0 and b₁ = 0, then $c_{1} = - x_{B1}^{2} = 0.\,$ However, due to $l_{3} \ne l_{1} ,$ i.e., $x_{B1} \ne 0,$ a₁ and b₁ are not zero at the same time. Hence, we have two cases as follows

$$ \left\{ {\begin{array}{*{20}l} {z_{{C_{1} }} = \frac{{c_{1} }}{{b_{1} }},y_{{C_{1} }} = \pm \sqrt {l_{4}^{2} - (z_{{B_{1} }} - z_{{C_{1} }} )^{2} - x_{{B_{1} }}^{2} } \;if\;a_{1} = 0} \hfill \\ {z_{{C_{1} }} = \frac{{e_{1} \pm \sqrt {e_{1}^{2} - 4d_{1} f_{1} } }}{{2d_{1} }},y_{{C_{1} }} = \frac{{c_{1} - b_{1} z_{{C_{1} }} }}{{a_{1} }}\;if\;a_{1} \ne 0} \hfill \\ \end{array} } \right. $$

(4)

here, $d_{1} = a_{1}^{2} + b_{1}^{2}$, $e_{1} = 2 (b_{1} c_{1} + z_{B1} a_{1}^{2} )$, $f_{1} = a_{1}^{2} (x_{B1}^{2} + z_{B1}^{2} - l_{4}^{2} )+ c_{1}^{2}$. Similarly, Eq. (2) leads to $a_{2} x_{C4} + b_{2} z_{C4} = c_{2}$ and coordinates of point C₄ can be obtained.

Solve Coordinates of Point p and Orientation α.

Once the coordinates of points C₁ and C₄ are obtained, the upper parts of each hybrid chain, i.e., links 8, 9 and 10, 11, and the moving platform 1 can be treated as a special single loop chain 6R mechanism, as shown in Fig. 4(a). A planar view is shown in Fig. 4(b).

Fig. 4.

Upper part of the robot

As shown in Fig. 4(a), let δ be the angle between vector C₁E₁ and x-axis. We assume two planes m and n pass through the points C₁, E₁ and F₁, and the points C₄, E₂ and F₂, respectively, as shown in Fig. 4(b). Thus, motions of two groups of points (C₁, E₁, F₁) and (C₄, E₂, F₂) always keep in the planes m and n, respectively. Then, we get

$$ y_{{E_{1} }} = y_{{C_{1} }} ,x_{{E_{2} }} = x_{{C_{4} }} $$

(5)

Hence the coordinates of points E₁ and E₂ are calculated. The constraint equation is expressed as $y_{E2}^{2} + (z_{C4} - z_{E2} )^{2} = l_{6}^{2}$. It is obtained, due to $z_{E1} = z_{E2}$ and y_C4 = 0, by

$$ y_{{E_{2} }}^{{}} = \pm \sqrt {l_{6}^{2} - (z_{{C_{4} }} - z_{{C_{1} }} - l_{6} { \sin }\delta )^{2} } $$

(6)

From Fig. 3(b), it is known that F₁F₂ = E₁E₂ = 2l₂, and hence we establish another constraint equation as

$$ (x_{{C_{4} }} - l_{6} { \cos }\delta )^{2} + (y_{{E_{2} }} - y_{{C_{1} }} )^{2} = 4l_{2}^{2} $$

(7)

Let $u = \tan {\delta \mathord{\left/ {\vphantom {\delta 2}} \right. \kern-0pt} 2}$. By expanding Eq. (7), we obtain a high-order polynomial equation with only one variable u as following

$$ f (u )= \sum\limits_{i = 0}^{8} {g_{i} u^{i} } $$

(8)

$$ \begin{array}{*{20}c} {g_{0} = { 16}l_{2}^{4} - 16l_{2}^{2} l_{6}^{2} + 4l_{6}^{4} + 8l_{2}^{2} t^{2} - 4l_{6}^{2} t^{2} + t^{4} + x_{{C_{4} }} (16l_{2}^{2} l_{6} - 8l_{6}^{3} + 4l_{6} t^{2} - 8l_{2}^{2} x_{{C_{4} }} } \\ { + 8l_{6}^{2} x_{{C_{4} }} - 2t^{2} x_{{C_{4} }} - 4l_{6} x_{{C_{4} }}^{2} + x_{{C_{4} }}^{3} )- y_{{C_{1} }}^{2} (8l_{2}^{2} - 2t^{2} + 4l_{6} x_{{C_{4} }} - 2x_{{C_{4} }}^{2} - y_{{C_{1} }}^{2} ),} \\ {g_{1} = \, 8l_{6} t (2l_{6}^{2} - 4l_{2}^{2} l_{6} - 2l_{6} t^{2} - 2l_{6} x_{{C_{4} }} + l_{6} x_{{C_{4} }}^{2} - l_{6} y_{{C_{1} }}^{2} ),} \\ {g_{2} = { 4(}1 6l_{2}^{4} - 4l_{6}^{4} \, + 8l_{2}^{2} t^{2} + 4l_{6}^{2} t^{2} + t^{4} )+ 4x_{{C_{4} }} (8l_{2}^{2} l_{6} + 4l_{6}^{3} + 2l_{6} t^{2} - 8l_{2}^{2} x_{{C_{4} }} - 2t^{2} x_{{C_{4} }} } \\ { \, - 2l_{6} x_{{C_{4} }}^{2} \, + x_{{C_{4} }}^{3} ) { } + 4y_{{C_{1} }}^{2} ( 2t^{2} - 8l_{2}^{2} - 2l_{6} x_{{C_{4} }}^{{}} + 2x_{{C_{4} }}^{2} + y_{{C_{1} }}^{2} ),} \\ {g_{3} = 8l_{6} t (3x_{{C_{4} }}^{2} - 12l_{2}^{2} - 2l_{6}^{2} - 2l_{6} x_{{C_{4} }} - 3y_{{C_{1} }}^{2} - 3t^{2} ),} \\ {g_{4} = \, 9 6l_{2}^{4} + 32l_{2}^{2} l_{6}^{2} + 24l_{6}^{4} + 48l_{2}^{2} t^{2} + 40l_{6}^{2} t^{2} + 6t^{4} - 2x_{{C_{4} }}^{2} (24l_{2}^{2} - 8l_{6}^{2} + 6t^{2} + 3x_{{C_{4} }}^{2} )} \\ { + 6y_{{C_{1} }}^{2} (2t^{2} + 6x_{{C_{4} }}^{2} - 8l_{2}^{2} + y_{{C_{1} }}^{2} ),} \\ {g_{5} = { 8}l_{6} t (2l_{6} x_{{C_{4} }} - 12l_{2}^{2} - 2l_{6}^{2} + 3 x_{{C_{4} }}^{2} - 3t^{2} - 3y_{{C_{1} }}^{2} ),} \\ {g_{6} = 64l_{2}^{4} - 16l_{6}^{4} + 32l_{2}^{2} t^{2} + 16l_{6}^{2} t^{2} + 4t^{4} - 4x_{{C_{4} }} (8l_{2}^{2} l_{6} + 4l_{6}^{3} + 4l_{6} t^{2} + 8l_{2}^{2} x_{{C_{4} }} } \\ { + 2t^{2} x_{{C_{4} }} + 2l_{6} x_{{C_{4} }}^{2} + x_{{C_{4} }}^{3} )+ 4y_{{C_{1} }}^{2} (2t^{2} - 8l_{2}^{2} + 2l_{6} x_{{C_{4} }} + 2x_{{C_{4} }}^{2} + y_{{C_{1} }}^{2} ),} \\ {g_{7} = 8l_{6} t (2l_{6}^{2} - 4l_{2}^{2} + 2l_{6} x_{{C_{4} }} + x_{{C_{4} }}^{2} - y_{{C_{1} }}^{2} - t^{2} ) ,} \\ {g_{8} = { 4(}4l_{2}^{4} - 4l_{2}^{2} l_{6}^{2} + l_{6}^{4} + 2l_{2}^{2} t^{2} - l_{6}^{2} t^{2} )^{2} + t^{4} + x_{{C_{4} }} (8l_{6}^{3} - 16l_{2}^{2} l_{6} - 4l_{6} t^{2} - 8l_{2}^{2} x_{{C_{4} }} } \\ { + 8l_{6}^{2} x_{{C_{4} }} - 2t^{2} x_{{C_{4} }} + 4l_{6} x_{{C_{4} }}^{2} + x_{{C_{4} }}^{3} )+ y_{{C_{1} }}^{2} (4l_{6} x_{{C_{4} }}^{{}} + 2t^{2} - 8l_{2}^{2} + 2x_{{C_{4} }}^{2} + y_{{C_{1} }}^{2} ),} \\ {t = z_{{C_{4} }} - z_{{C_{1} }} .} \\ \end{array} $$

Real roots of the equation yield the corresponding angle δ, then y_E2 (with two values) by Eq. (6). By putting y_E2 values into Eq. (7), we obtain the real value of y_E2 or y_C4. Finally, the coordinates (x, y, z) of point p on the moving platform 1 and rotation angle $\alpha$ can be easily obtained. The equations show that the translation motion of point F₁ on the moving platform 1, along y axis, is determined only by the two joints R₁₁ and R₂₁ (i.e., θ₁ and θ₂), and the translation motion of point F₂ on the moving platform 1, along x axis, only by the two joints R₃₁ and R₄₁ (i.e., θ₃ and θ₄), while the translation along z axis and orientation angle $\alpha$ are determined by four input angles θ₁, θ₂, θ₃ and θ₄. In this light, we say the PM has partial motion decoupling property.

3.3 Inverse Position Solution

The purpose of the inverse position solutions is to obtain the input angles θ₁, θ₂, θ₃ and θ₄ as a function of the known output variables, i.e., the coordinates of point p of the moving platform 1, defined by (x, y, z), and the orientation angle of the end-effector $\alpha$. In the moving coordinate system p-uvw, the coordinates of points E₁ and E₂ are (0, −l₂, −l₆) and (0, l₂, −l₆) respectively. Since the coordinates of points B₁, B₂, B₃ and B₄ are known in the forward position problems afore-mentioned, we may establish the four constraint equations below according to the link length constraints and inverse kinematics can be obtained

$$ \theta_{i} = 2{ \arctan }\frac{{2l_{3} z_{i} \pm \sqrt {4z_{i}^{2} l_{3}^{2} - h_{i} t_{i} } }}{{h_{i} }},\left( {i = 1,2,3,4} \right), $$

(9)

$$ z_{1} = z_{{C_{1} }} ,\,z_{2} = z_{{C_{2} }} ,\,z_{3} = z_{{C_{3} }} ,\,z_{4} = z_{{C_{4} }} $$

$$ h_{1} = l_{1}^{2} - 2l_{1} l_{3} + l_{3}^{2} - l_{4}^{2} + y_{{C_{1} }}^{2} + z_{{C_{1} }}^{2} $$

$$ t_{1} = l_{1}^{2} + 2l_{1} l_{3} + l_{3}^{2} - l_{4}^{2} + y_{{C_{1} }}^{2} + z_{{C_{1} }}^{2} $$

$$ h_{2} = l_{1}^{2} + 2l_{1} l_{3} + l_{3}^{2} - l_{4}^{2} + 2l_{1} y_{{C_{2} }} + 2l_{3} y_{{C_{2} }} + y_{{C_{2} }}^{2} + z_{{C_{2} }}^{2} $$

$$ t_{2} = l_{1}^{2} - 2l_{1} l_{3} + l_{3}^{2} - l_{4}^{2} + 2l_{1} y_{{C_{2} }} - 2l_{3} y_{{C_{2} }} + y_{{C_{2} }}^{2} + z_{{C_{2} }}^{2} $$

$$ h_{3} = l_{1}^{2} - l_{4}^{2} + 2l_{1} l_{3} + l_{3}^{2} + 2l_{1} x_{{C_{3} }} + 2l_{3} x_{{C_{3} }} + x_{{C_{3} }}^{2} + z_{{C_{3} }}^{2} $$

$$ t_{3} = l_{1}^{2} - l_{4}^{2} - 2l_{1} l_{3} + l_{3}^{2} + 2l_{1} x_{{C_{3} }} - 2l_{3} x_{{C_{3} }} + x_{{C_{3} }}^{2} + z_{{C_{3} }}^{2} $$

$$ h_{4} = l_{1}^{2} + l_{3}^{2} - 2l_{1} l_{3} - l_{4}^{2} + x_{{C_{4} }}^{2} + z_{{C_{4} }}^{2} $$

$$ t_{4} = l_{1}^{2} + l_{3}^{2} + 2l_{1} l_{3} - l_{4}^{2} + x_{{C_{4} }}^{2} + z_{{C_{4} }}^{2} $$

It is easy to find that when the coordinates of the point p and orientation angle of the moving platform 1 are known, the input angles θ₁, θ₂, θ₃ and θ₄ have two sets of solutions, and the points C₁ and C₄ have two sets of the coordinate solutions. Hence the number of the inverse solutions is 64(4 × 16), which leads to the PM has totally 64 configurations.

3.4 Verification of Forward and Inverse Kinematics

The structure parameters are l₁ = 300 mm, l₂ = 70 mm, l₃ = 350 mm, l₄ = 800 mm, l₅ = 100 mm and l₆ = 50 mm, respectively. The four input angles θ₁, θ₂, θ₃ and θ₄ take the values of 58.6839°, 150.8342°, 144.9223° and 61.2457°, respectively. Considering the actual configuration of the PM, we take z_C1>0 and z_C4>0, according to Eqs. (1)–(4), the coordinates of points C₁ and C₄ are obtained, which are (0, –134.5310, 923.2359) and (–99.1480, 0, 947.7786), respectively. We substitute these values into Eq. (8) and obtain

$$ f(u) = 10^{8} \times ({ 3} . 7 5 2 5 2- 1. 3 3 0 7 8u + 4. 4 5 4 5 2u^{2} $$

$$ - 7. 9 0 2 4 6u^{3} + { 8} . 2 8 8 5 9u^{4} - 9. 8 4 9 1 4u^{5} $$

$$ + 2.29185u^{6} - 3. 2 7 7 4 7u^{7} - 1. 2 9 4 7 4u^{8} ) = 0 $$

The real roots of above equation are found as: u₁ = −3.71915, u₂ = 0.867349. The forward solutions are obtained and shown in Table 1.

Table 1. Forward position solutions

Substituting this group of the forward solutions into the inverse solutions Eq. (9), and considering that z_E1>z_C1, z_E2>z_C4, the sets of inverse solutions are reduced to 16, one of which is just one of the given inputs, i.e., θ₁ = 58.684°, θ₂ = 150.8342°, θ₃ = 144.9223° and θ₄ = 61.2457°. The values are consistent with the four known input angles, which verifies the forward and inverse solutions.

4 Conclusion

A novel parallel manipulator 2-(RPa||3R)3R generating three-translation and one-rotation output is proposed. In this work, the kinematics analysis of the new PM was conducted. The close-form forward solutions of the PM are obtained and verified by inverse kinematics. In addition, the PM features motion decoupling which can be found through kinematics.

The contributions of this work are the development of the new 3T1R parallel manipulator 2-(RPa||3R)3R and establishment of its kinematics. The proposed manipulator is symmetric in structure and mainly composed of revolute joints. The new design will lead to less material use and lightweight, and consequentially, a reduced manufacturing cost. Hence, more potential applications can be expected.