Mechanical Design and Analysis of a Novel Symmetrical 2T1R Parallel Robot

Abstract

The planar parallel robots are widely employed in industrial applications due to simple geometry, few linkage interferences, and a large, reachable workspace. The symmetric geometry can bring significant convenience to parallel robots. The complexity of the mathematic models can be simplified since only one calculation method can be proposed to deal with various kinematic limbs in a parallel manipulator. The symmetric geometry can ease the assembly and maintenance procedures due to the modular design of linkages/joints. A novel 2-translation and 1-rotation (2T1R) parallel robot with symmetric geometry is proposed in this research. There is one closed loop in each kinematic limb, and 18 revolute joints are applied in its planar structure. Both the inverse and direct kinematic models are explored. The first-order relationship between robot inputs and outputs are constructed. Various singularity configurations are obtained based on the Jacobian matrix. The reachable workspace is resolved by the discrete spatial searching methodology, followed by the impacts originating from various linkages. The dexterity analysis of the parallel robot is conducted.

1. Introduction

The parallel robot consists of fixed platform and mobile platform connected by several kinematic chains. Compared with its counterpart, the serial robot contains one open kinematic chain to bridge the fixed platform and mobile platform (also called as end-effector in a serial robot), while the parallel robot generally owns high speed, high acceleration, large stiffness and payload performances, low inertia, and high precision. The intrinsic shortcomings include coupled movements among several chains, limited and irregular reachable workspace, complicated singularities that may divide the workspace into several sub-workspaces, and high demands for the controller [1,2]. The parallel robot is widely employed for pick-and-place tasks [3], tactile sensors [4], a spot welding operation [5], flight simulator [6], parallel kinematic machine [7], and an exoskeleton [8].

The planar parallel robot has gained significant interest in recent years. The screw theory was implemented to obtain a family of planar parallel robots with a constrained moving platform [9]. Unlike the common planar parallel robots with prismatic joints or revolute joints, the spatial joints (university joints, spherical joints) were also adopted to enhance the diversity, and five different constraints on the mobile platform were categorized for the purpose of all feasible parallel structures. A family of redundant 2T1R parallel manipulators with closed-loop modules was designed according to the Atlas approach and the Grassmann line geometry in [10]. Only the planar kinematic joints, prismatic (P) joint, and revolute (R, R can stand for revolute joint or rotational motion) joints were adopted for parallel robot architectures. A planar pure-translational parallel robot 2-P(R-R)₂ was integrated into a five-degrees-of-freedom (DOFs) hybrid machine [11]. Two kinematic chains were identical, and the sliding shafts of both active prismatic joints were parallel. A novel 2T1R parallel robot with two kinematic chains was proposed in [12], and only two active sliding joints were required to actuate its whole movements. A 3-DOF parallel mechanism [13] was constructed with different kinematic limbs. Two identical chains were in the configuration of (RPR-RR)R, and the remaining chain was in the configuration of RPR. The screw theory was utilized to develop a group of planar parallel mechanisms with a single kinematic loop [14].

The common symmetric parallel robots are composed of several identical chains, and the first kinematic joint locations of all kinematic chain could form equilateral polygons where there are at least three kinematic limbs (a straight line can be formed where there are only two kinematic limbs). Only one methodology was sufficient to solve the kinematic/dynamic models of all kinematic chains within the symmetric robot, which saved significant computation cost for the robot. Symmetrical parallel robots tend to possess a symmetric reachable workspace due to the same kinematic chains. For example, the reachable workspace diagrams of the linear Delta robot [15], the 1-translation and 2-rotation parallel structure within the hybrid machine RAVASH [16] was symmetrical. The same kinematic joints and linkages among various kinematic chains simplified the exchange and maintenance work.

There are four primary functions of the closed-loop unit employed in parallel robots: (i) more diverse configurations [17], (ii) simple calculations and control t [18], (iii) special spatial relation among linkages and proper constraints on the mobile platform [19], (iv) and scaled (expanded or reduced) motion [20].

Kinematic performance evaluations are critical and fundamental once a parallel robot structure is selected. A metamorphic parallel robot [U-(R-R)₂]-R-[U-(R-R)₂] (U indicates universal joint) with two kinematic limbs was reported in [21]. Its singularity configurations were discussed on the basis of the Jacobian matrix obtained by the screw theory. Its singularity-free reachable workspace was determined based on the kinematic model. The singularity configuration of a redundant parallel robot was outlined comprehensively based on the Jacobian matrix [22]. Its singularity loci were further compared with those of a non-redundant parallel robot. The novel hybrid robot 3-RRR+3-CPR (C indicates cylindrical joint) with high orientational capability was proposed in [23]. Multiple motion/force transmission-related indices, including good-transmission workspace and local transmission index, were implemented and compared for two parallel modules of this robot. Cuan–Urquizo and Rodriguez–Leal [24] mentioned a symmetric parallel robot 3-CUP. The two unexpected parasite movements were discovered. All types of singularity configurations were explored comprehensively. Both the orientational workspace and translational workspace were investigated.

The main contributions of this research consist of

• Propose a novel planar parallel robot with three identical kinematic chains and pure rotational joints. The parallelogram linkage unit in each kinematic chain contributes two pairs of parallel rods and mitigates the complexity of kinematic models.
• Ten inverse singularity configurations are resolved for singularity avoidance.
• Both the reachable workspace and local condition index are conducted. The impacts from various linkages or directions are, respectively, studied.

It is a crucial task to design a proper parallel robot structure since it is the fundamental step in a robotic system, and all the following mathematical kinematic/dynamic models and controllers are established based on this robot structure. More parallel robot structures will enhance the richness in future industrial options and widen the applications of parallel robots. The remainder of this paper is organized as follows. Section 2 proposes this parallel robot, and both inverse and direct position problems are computed. The velocity mapping relationship is constructed, and the singularities are described in Section 3. The detailed workspace analysis is conducted in Section 4, followed by Section 5, which investigates the dexterity performance. The whole research work is discussed in Section 6 and concluded in Section 7.

2. Structural Description and Kinematic Models

The novel parallel robot illustrated in Figure 1 is composed of three identical kinematic chains. The global coordinate system is constructed for this parallel robot, as demonstrated in Figure 1. Each kinematic chain contains four linkages, and only the R joint is utilized in this design. Compared to spatial joints (spherical joint, universal joint), revolute joints are beneficial to a larger reachable workspace, higher stiffness, and payload capacity for parallel robots. The revolute joint can lower the manufacturing cost compared to spatial joints. All linkages lie in the XOY plane, and the revolute joint shafts are parallel to the Z direction (perpendicular to the XOY plane). The X axis is along the horizontal direction, and the Y axis is along the vertical direction. The origin O coincides with point A₁ in Figure 1. In each kinematic limb, there are two categories of linkages, one category with three kinematic pairs (linkage A_iB_iC_i and D_iE_iF_i, i = 1, 2, 3), while another category includes two kinematic pairs (linkage B_iD_i and C_iE_i). The revolute joint at point A_i is attached to the fixed platform, while the revolute joint at point F_i is connected to the mobile platform. The single-loop four-bar linkage mechanism B_iC_iE_iD_i is included in the i-th kinematic chain. This single-loop mechanism, B_iC_iE_iD_i, is in the shape of a parallelogram for the purpose of parallel relations between two opposite linkages. (If this characteristic is not preferred in an application, a general four-bar linkage mechanism can be utilized in this parallel robot, and the mobile platform can still possess planar movements.) The proposed parallel robot is in the architecture of 3-R(R-R)₂R, considering its joint configuration.

The linkage dimensions are provided as follows: A_iB_i = L_i1, B_iC_i = L_i2, B_iD_i = L_i3, C_iE_i = L_i4, D_iE_i = L_i5, and E_iF_i = L_i6. The moving platform is an equilateral triangle, and each side is denoted by L₄. The position of point A_i is defined as (x_Ai, y_Ai). (The z components of all points are zero and, therefore, are neglected.)

The modified Grübler–Kutzbach formula is utilized to explore the degrees of freedom of this parallel structure, as seen below:

$$M=(6-\lambda )(n-g-1)+{\displaystyle \sum _{i=1}^g{f}_i}+\upsilon =d(n-g-1)+{\displaystyle \sum _{i=1}^g{f}_i}+\upsilon =3(14-18-1)+18\times 1+0=3$$

(1)

where M indicates the degrees of freedom. $\lambda$ describes the total number of common constraints of all kinematic limbs, and it is equal to 3 for planar mechanisms and 6 for spatial mechanisms in general. (The common constraints of special parallel mechanisms can be checked by the screw theory [25]). d is the order of the parallel mechanism. n represents the total number of fixed and moving linkages. g denotes the total number of kinematic joints. f_i is the degree of freedom for the i-th kinematic joint. $\upsilon$ means the number of independent redundant constraints besides the common constraints mentioned in $\lambda$. $\upsilon$ is zero for this symmetric parallel structure.

The degrees of freedom of this parallel mechanism is 3 on the basis of Equation (1). The mobile platform can achieve two translations (2T) along X/Y axes and one rotation (1R) about the Z direction. The point P represents the geometric center of the mobile platform F₁F₂F₃, and its location is (x_P, y_P). The rotational motion is defined by the angle α between the linkage F₁F₂ and the positive X direction. Three actuators mounted for the revolute joints at points A₁, A₂, and A₃ are sufficient for its movement. The i-th actuator’s angular position is defined as θ_i, and the reference is the positive X-axis. All angles are measured in a counter-clockwise direction.

2.1. Inverse Kinematic Model

The input and output of this parallel robot are described as $Q={[{\theta }_1,{\theta }_2,{\theta }_3]}^{\mathrm{T}}$ and $X={[x_P,y_P,\alpha ]}^{\mathrm{T}}$, separately. The inverse kinematic model is established to solve for the input Q with a predefined output X.

It is necessary to relate three vertices’ coordinates and the center point location of the mobile platform considering its geometric shape. The mobile platform is indicated in Figure 2. Three medians, F₁G₁, F₂G₂, and F₃G₃, are perpendicular medians. The center point P is the common intersection of these three medians. According to Figure 2, the following vector-loop equations can be obtained,

$$O{F}_i=OP-F_{i}P(i=1,2,3)$$

(2)

Equation (2) can be split into two orthogonal directions (parallel to the X axis and Y axis), and the relationships are expressed as

$$\left\{\begin{array}{l}{x}_{F1}=x_P-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)\\ y_{F1}=y_P-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{x}_{Fi}=x_P-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)+L_4\cos \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]\\ y_{Fi}=y_P-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)+L_4\sin \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]\end{array}\right.(i=2,3)$$

(4)

A generic closed-loop formula can be employed for each kinematic limb,

$$C_i{E}_i=O{E}_i-O{C}_i(i=1,2,3)$$

(5)

where $O{E}_i=O{F}_i-E_i{F}_i$ and $O{C}_i=O{A}_i+A_i{B}_i+B_i{C}_i$.

Taking the first kinematic limb as an example, Equation (5) can be generated as

$${\left[M_{11}-(L_{11}+L_{12}+L_{16})\cos {\theta }_1\right]}^2+{\left[M_{12}-(L_{11}+L_{12}+L_{16})\sin {\theta }_1\right]}^2=L_{13}^2$$

(6)

where $M_{11}=x_{F1}-x_{A1}$ and $M_{12}=y_{F1}-y_{A1}$.

Equation (6) can be further arranged in the following format

$$N_{11}\sin {\theta }_1+N_{12}\sin {\theta }_1+N_{13}=0$$

(7)

where

$$\left\{\begin{array}{l}{N}_{11}=-2M_{12}(L_{11}+L_{12}+L_{16})\\ N_{12}=-2M_{11}(L_{11}+L_{12}+L_{16})\\ N_{13}=M_{11}^2+{(L_{11}+L_{12}+L_{16})}^2+M_{12}^2-L_{13}^2\end{array}\right.$$

The active joint angle ${\theta }_1$ can be expressed as a function of the parameter $t_1$, as seen below

$$t_1=\tan {\displaystyle \frac{{\theta }_1}{2}},\sin {\theta }_1={\displaystyle \frac{2t_1}{1+t_1^2}},\cos {\theta }_1={\displaystyle \frac{1-t_1^2}{1+t_1^2}}$$

(8)

Utilizing $t_1$ to substitute ${\theta }_1$ in Equation (7) leads to the following expression

$$(N_{13}-N_{12})t_1^2+2N_{11}{t}_1+(N_{12}+N_{13})=0$$

(9)

Therefore, the active joint angle ${\theta }_1$ can be resolved as

$${\theta }_1=2{\tan }^{-2}{\displaystyle \frac{-N_{11}\pm \sqrt{N_{11}^2+N_{12}^2-N_{13}^2}}{N_{13}-N_{12}}}$$

(10)

The calculation procedures for the remaining kinematic limbs are similar to that (Equations (6)–(10)) of the first kinematic limb. Equation (5) can be expressed in the following format corresponding to the second and third kinematic chains,

$${\left[M_{i1}-(L_{i1}+L_{i2}+L_{i6})\cos {\theta }_i\right]}^2+{\left[M_{i2}-(L_{i1}+L_{i2}+L_{i6})\sin {\theta }_i\right]}^2=L_{i3}^2(i=2,3)$$

(11)

where $M_{i1}=x_{Fi}-x_{Ai}$ and $M_{i2}=y_{Fi}-y_{Ai}$ (i = 2, 3).

The actuation joint angles ${\theta }_2$ and ${\theta }_3$ are formulated as follows

$${\theta }_i=2{\tan }^{-2}{\displaystyle \frac{-N_{i1}\pm \sqrt{N_{i1}^2+N_{i2}^2-N_{i3}^2}}{N_{i3}-N_{i2}}}(i=2,3)$$

(12)

where

$$\left\{\begin{array}{l}{N}_{i1}=-2M_{i2}(L_{i1}+L_{i2}+L_{i6})\\ N_{i2}=-2M_{i1}(L_{i1}+L_{i2}+L_{i6})\\ N_{i3}=M_{i1}^2+{(L_{i1}+L_{i2}+L_{i6})}^2+M_{i2}^2-L_{i3}^2\end{array}\right.(i=2,3)$$

It is noteworthy that there are generally two solutions corresponding to Equations (10) and (12). There are at most eight (2³ = 8) sets of solutions for a predefined mobile platform pose X.

2.2. Direct Kinematic Model

The mobile platform pose X is to be determined with the predefined actuation joint angles Q in the direct kinematic problem.

Rearranging Equation (5) in the following format

$${(x_P+S_{i1})}^2+{(y_P+S_{i2})}^2-L_{i3}^2=0(i=1,2,3)$$

(13)

where

$$\left\{\begin{array}{l}{S}_{11}=-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)-x_{A1}-(L_{11}+L_{12}+L_{16})\cos {\theta }_1\\ S_{12}=-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)-y_{A1}-(L_{11}+L_{12}+L_{16})\sin {\theta }_1\end{array}\right.$$

$$\left\{\begin{array}{l}{S}_{i1}=-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)+L_4\cos \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]-x_{Ai}-(L_{i1}+L_{i2}+L_{i6})\cos {\theta }_i\\ S_{i2}=-{\displaystyle \frac{\sqrt{3}}{3}}{L}_4\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)+L_4\sin \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]-y_{Ai}-(L_{i1}+L_{i2}+L_{i6})\sin {\theta }_i\end{array}\right.(i=2,3)$$

Equation (13) can be further derived as

$$x_P^2+y_P^2+T_{i1}{x}_P+T_{i2}{y}_P+T_{i3}=0(i=1,2,3)$$

(14)

where

$$\left\{\begin{array}{l}{T}_{i1}=2S_{i1}\\ T_{i2}=2S_{i2}\\ T_{i3}=S_{i1}^2+S_{i2}^2-L_{i3}^2\end{array}\right.$$

The elements $x_P^2$ and $y_P^2$ can be eliminated from three expressions in Equation (14). For example, the expression when i = 1 separately subtracts the expressions when i = 2 and i = 3 yields

$$U_{i1}{x}_P+U_{i2}{y}_P+U_{i3}=0(i=1,2)$$

(15)

where

$$\left\{\begin{array}{l}{U}_{1j}=T_{1j}-T_{2j}\\ U_{2j}=T_{1j}-T_{3j}\end{array}\right.(j=1,2,3)$$

Two expressions in Equation (15) are sufficient to substitute parameters, $x_P$ and $y_P$, with the rotational parameter $\alpha$, as seen below,

$$\left\{\begin{array}{l}{x}_P={\displaystyle \frac{V_1}{V_3}}\\ y_P=-{\displaystyle \frac{V_2}{V_3}}\end{array}\right.$$

(16)

where

$$\left\{\begin{array}{l}{V}_1=U_{12}{U}_{23}-U_{13}{U}_{22}\\ V_2=U_{11}{U}_{23}-U_{13}{U}_{21}\\ V_3=U_{11}{U}_{22}-U_{12}{U}_{21}\end{array}\right.$$

Combining Equation (16) and the first expression in Equation (14) leads to

$$V_1^2+V_2^2+T_{11}{V}_1{V}_3-T_{12}{V}_2{V}_3+T_{13}{V}_3^2=0$$

(17)

The half-tangent angle formulas can be employed into Equation (17), which can be deduced as an eighth-degree equation. It indicates the maximal possible solutions of $\alpha$ is eight for a set of particular actuation joint angles. There are, at most, eight different configurations in the direct kinematic model.

The comparison between inverse and forward kinematic models can also be revealed from Equations (6) and (11). The forward kinematic model is more challenging since there are three desired elements for the mobile platform pose and one particular input actuation angle in each equation, compared with the inverse kinematic model where three particular variables for the mobile platform pose, as well as only one unknown input actuation angle.

3. Velocity Relationship and Singularity Configurations

The section aims to explore the relation between the actuation joints angular velocities $\dot{Q}={[{\dot{\theta }}_1,{\dot{\theta }}_2,{\dot{\theta }}_3]}^{\mathrm{T}}$ and the mobile platform linear and angular velocities $\dot{X}={[{\dot{x}}_P,{\dot{y}}_P,\dot{\alpha }]}^{\mathrm{T}}$ for this planar parallel robot. Taking the time differential on both sides of Equation (13) results in

$$W_{i1}{\dot{x}}_P+W_{i2}{\dot{y}}_P+W_{i3}\dot{\alpha }-W_{i4}{\dot{\theta }}_i=0(i=1,2,3)$$

(18)

where

$$\left\{\begin{array}{l}{W}_{i1}=x_p+S_{i1}\\ W_{i2}=y_p+S_{i2}\\ W_{13}={\displaystyle \frac{\sqrt{3}}{3}}{W}_{11}{L}_4\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)-{\displaystyle \frac{\sqrt{3}}{3}}{W}_{12}{L}_4\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)\\ \begin{array}{cc}\hfill W_{i3}& =W_{i1}{L}_4\left\{{\displaystyle \frac{\sqrt{3}}{3}}\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)-\sin \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]\right\}\hfill \\ \hfill & +W_{i2}{L}_4\left\{-{\displaystyle \frac{\sqrt{3}}{3}}\cos \left(\alpha +{\displaystyle \frac{\pi }{6}}\right)+\cos \left[\alpha +(i-2){\displaystyle \frac{\pi }{3}}\right]\right\}(i=2,3)\hfill \end{array}{W}_{i4}=(L_{i1}+L_{i2}+L_{i6})(W_{i2}\cos {\theta }_i-W_{i1}\sin {\theta }_i) \end{array}\right.$$

Equation (18) is written in the following format

$$J_X\dot{X}=J_Q\dot{Q}$$

(19)

where

$$\left\{\begin{array}{l}{J}_X=\left[\begin{array}{ccc}{W}_{11}& W_{12}& W_{13}\\ W_{21}& W_{22}& W_{23}\\ W_{31}& W_{32}& W_{33}\end{array}\right]\\ J_Q=\left[\begin{array}{ccc}{W}_{14}& 0& 0\\ 0& W_{24}& 0\\ 0& 0& W_{34}\end{array}\right]\end{array}\right.$$

Equation (19) can be rewritten as

$$\dot{Q}=J\dot{X}$$

(20)

where J is the Jacobian matrix and $J=J_Q^{-1}{J}_X$.

The singularity configurations can be explored by the Jacobian matrix. The first singularity type is the inverse kinematic singularity when det(J_Q) = 0. There are 10 cases under this condition.

(i) $\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}-{\theta }_1\right)\ne 0$ and $L_4=\sqrt{3}{\displaystyle \frac{(y_{A1}-y_P)\cos {\theta }_1-(x_{A1}-x_P)\sin {\theta }_1}{\sin \left({\theta }_1-\alpha -\frac{\pi }{6}\right)}}$. The first expression means $\alpha +{\displaystyle \frac{\pi }{6}}-{\theta }_1\ne 0 or \pi$, which implies that line F₁P is not permitted to be parallel to line A₁C₁.

(ii) ${\theta }_1\ne 0 or \pi$ and ${\theta }_1=\alpha +{\displaystyle \frac{\pi }{6}}$ and $\tan {\theta }_1={\displaystyle \frac{y_P-y_{A1}}{x_P-x_{A1}}}$.

(iii) ${\theta }_1=0 or \pi$ and $\alpha +{\displaystyle \frac{\pi }{6}}=0 or \pi$ and $y_P=y_{A1}$.

(iv) $\sqrt{3}\sin \left({\theta }_2-\alpha \right)\ne \sin \left(\alpha +{\displaystyle \frac{\pi }{6}}-{\theta }_2\right)$ and $L_4={\displaystyle \frac{\sqrt{3}\left[(y_{A2}-y_P)\cos {\theta }_2-(x_{A2}-x_P)\sin {\theta }_2\right]}{\sqrt{3}\sin \left({\theta }_2-\alpha \right)-\sin \left(\alpha +\frac{\pi }{6}-{\theta }_2\right)}}$.

(v) ${\theta }_2=0 or \pi$ and $\alpha ={\displaystyle \frac{\pi }{6}}$ and $y_P=y_{A2}$.

(vi) $\sqrt{3}\cos \left(\alpha -{\displaystyle \frac{\pi }{6}}-{\theta }_3\right)\ne \sin \left(\alpha +{\displaystyle \frac{\pi }{6}}-{\theta }_3\right)$ and $L_4={\displaystyle \frac{\sqrt{3}\left[(y_{A3}-y_P)\cos {\theta }_3-(x_{A3}-x_P)\sin {\theta }_3\right]}{\sqrt{3}\cos \left(\alpha -\frac{\pi }{6}-{\theta }_3\right)-\sin \left(\alpha +\frac{\pi }{6}-{\theta }_3\right)}}$.

(vii) ${\theta }_3=\pm {\displaystyle \frac{\pi }{2}}$ and $\alpha =0 or \pi$ and $x_P=x_{A3}$.

(viii) ${\theta }_3=0 or \pi$ and $\tan \alpha =\sqrt{3}+1$ and $y_P=y_{A3}$.

(ix) ${\theta }_3\ne \pm {\displaystyle \frac{\pi }{2}}$ and $\sqrt{3}\cos \left(\alpha -{\displaystyle \frac{\pi }{6}}-{\theta }_3\right)=\sin \left(\alpha +{\displaystyle \frac{\pi }{6}}-{\theta }_3\right)$ and $\tan {\theta }_3={\displaystyle \frac{y_P-y_{A3}}{x_P-x_{A3}}}$.

(x) $L_{i1}+L_{i2}+L_{i6}=0(i=1,2,3)$.

The second singularity type is the direct kinematic singularity when det(J_X) = 0. There are some simple cases by observing the matrix J_X although the equation det(J_X) = 0 is complicated. For example, there are dependent relationships between any two rows or columns (e.g., W₁₁/W₂₁ = W₁₂/W₂₂ = W₁₃/W₂₃), or three elements in the same row or column are zero simultaneously.

All the above singularity configurations and their adjacent areas should be circumvented for the purpose of smooth and safe movement.

4. Workspace Determination

The reachable workspace of this planar parallel manipulator is defined as all possible poses $X={[x_P,y_P,\alpha ]}^{\mathrm{T}}$ the mobile platform center point P can reach. The widely used discrete spatial searching algorithm [26,27] indicated in Figure 3 is utilized to explore its reachable workspace. The overall procedure is summarized as

(I) Identify the current robot configuration (as seen in Figure 1), all linkage dimensions, and the locations of all fixed points.

(II) Define the upper and lower boundaries for all three elements in mobile platform pose X. All elements start from the lower boundaries.

(III) Use the analytic inverse kinematic model from Section 2.1 to obtain the corresponding solution. If the solution is within the predefined active joint strokes, this pose is marked. Otherwise, this pose is not feasible and should be discarded.

(IV) Move to the next pose by $x_P=x_P+x_P\_stepsize$ or $y_P=y_P+y_P\_stepsize$ or $\alpha =\alpha +\alpha \_stepsize$.

(V) Repeat steps (III) and (IV) until all poses are thoroughly examined and all three elements reach their upper boundaries respectively.

The robot configuration in Figure 1 is employed in this section. Three fixed points A_i (i = 1, 2, 3) can form any two-dimensional shape for its motion, and an equilateral triangle with a side length of 1300 mm is taken as an example in this section. The positions of point A_i are solved as (0, 0), (1300, 0), and (650, 650$\sqrt{3}$) (unit is millimeter), separately. The linkage dimensions of three kinematic chains of this planar parallel robot are identical for the purpose of symmetrical features and modular design. The virtual prototype size is defined as: L_i1 = 250 mm, L_i2 = L_i5 = 200 mm, L_i3 = L_i4 = 300 mm, and L_i6 = 150 mm (i = 1, 2, 3). The side length L₄ of the moving platform F₁F₂F₃ is defined as 150 mm.

The initial estimated boundaries of three elements in X are predefined. The lower and upper boundaries of the element $x_P$ are [−200 mm, 1500 mm]. The element $y_P$ ranges from 100 mm to (650$\sqrt{3}$ − 100) mm. The orientational angle $\alpha$ of the mobile platform ranges from $-\pi /2$ to $\pi /2$. Both the $x_P\_stepsize$ and $y_P\_stepsize$ are set as 4 mm. $\alpha \_stepsize$ is defined as $\pi /36$.

The angular position ranges of three actuation joints are predefined with the consideration of singularity configurations avoidance mentioned in Section 3. The range $(0,\pi )$ is for the first and second actuation joints. The third driving joint ranges from $\pi$ to $2\pi$.

The reachable workspace of this planar parallel manipulator is determined and illustrated in Figure 4 according to the aforementioned discrete approach, parameters, and constraints. From Figure 4, each cross section along rotational angle $\alpha$ is close to a Reuleaux triangle, and the size of each cross-section shape parallel to the XOY plane is not identical. The feasible $x_P$ ranges from 336 mm to 964 mm. The qualified $y_P$ ranges from 140 mm to 732 mm. The feasible $\alpha$ ranges from $-\pi /2$ to $\pi /2$. When $x_P$ or $y_P$ is close to the midpoint of the corresponding feasible range, the mobile platform is capable of rotating half a round. When $x_P$ or $y_P$ approaches the boundary of the corresponding feasible range, the mobile platform tends to have reduced rotational capacity.

Parametric analysis is provided in this section to compare the impacts from various linkages. The initial set of linkage dimensions (predefined dimensions) is mentioned in the second paragraph of this section and employed as a reference. The other workspace plots are obtained by changing one linkage length, while the other dimensions remain the same. Four different values are selected for L_i1, L_i2 (or L_i5), L_i3 (or L_i4), L_i6, and L₄ (i = 1, 2, 3). The various three-dimensional reachable workspaces are demonstrated in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. The feasible strokes of the mobile platform movement under various conditions are listed in

Table 1. Researchable workspace comparisons under various conditions.

Parameter (Unit: mm)	X Range (Unit: mm)	Y Range (Unit: mm)	Alpha Range (Unit: radian)	Workspace Volume	Workspace Volume Ratio
Predefined dimensions	(336, 964)	(140, 732)	(−1.57, 1.57)	414,254	100%
Li1 = 150	(456, 844)	(240, 596)	(−1.57, 1.57)	116,978	28.24%
Li1 = 200	(396, 904)	(192, 664)	(−1.57, 1.57)	243,416	58.76%
Li1 = 250	(312, 988)	(100, 732)	(−1.57, 1.57)	595,874	143.84%
Li1 = 300	(288, 1012)	(100, 728)	(−1.57, 1.57)	686,938	165.83%
Li2 = 100	(456, 844)	(240, 596)	(−1.57, 1.57)	116,978	28.24%
Li2 = 150	(396, 904)	(192, 664)	(−1.57, 1.57)	243,416	58.76%
Li2 = 250	(312, 988)	(100, 732)	(−1.57, 1.57)	595,874	143.84%
Li2 = 300	(288, 1012)	(100, 728)	(−1.57, 1.57)	686,938	165.83%
Li3 = 100	(596, 704)	(340, 440)	(−0.79, 0.79)	4334	1.05%
Li3 = 200	(456, 844)	(240, 596)	(−1.57, 1.57)	116,978	28.24%
Li3 = 400	(228, 1072)	(100, 844)	(−1.57, 1.57)	854,244	206.21%
Li3 = 500	(124, 1176)	(100, 956)	(−1.57, 1.57)	1,295,064	312.63%
Li6 = 50	(456, 844)	(240, 596)	(−1.57, 1.57)	116,978	28.24%
Li6 = 100	(396, 904)	(192, 664)	(−1.57, 1.57)	243,416	58.76%
Li6 = 200	(312, 988)	(100, 732)	(−1.57, 1.57)	595,874	143.84
Li6 = 250	(288, 1012)	(100, 728)	(−1.57, 1.57)	686,938	165.83%
L4 = 10	(432, 868)	(224, 628)	(−1.57, 1.57)	245,554	59.28%
L4 = 80	(384, 916)	(180, 684)	(−1.57, 1.57)	325,982	78.69%
L4 = 220	(316, 984)	(100, 732)	(−1.57, 1.57)	495,540	119.62%
L4 = 290	(296, 1004)	(100, 724)	(−1.57, 1.57)	539,242	130.17%

. The workspace volume is defined as the total feasible poses using the discrete approach and is summarized in Table 1 for different sets of linkage dimensions. To further compare the impact originated from various linkages, the workspace volume ratio indicating the volume ratio using selected set of linkage dimensions to using predefined dimensions is supplemented in Table 1.

According to Table 1 and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, the reachable workspace shapes are similar to Figure 4b. In each group of plots from Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, the last two reachable workspace plots (the chosen linkage dimension is longer than that in predefined dimensions) are truncated in three side directions, but the workspace ranges along both the X and Y directions are larger than that of the predefined dimensions. The corresponding workspace volume ratios are higher than 100%, while the ratios of the first two scenarios (the chosen linkage dimension is shorter than that in predefined dimensions) are smaller than 100%. The longer the selected linkage in the single variable comparison, the larger the workspace, as observed from four subplots in each figure. Among all these scenarios, the scenario where L_i3 = 100 mm has the smallest orientational range [−0.79, 0.79], while the orientational range of the other scenarios are the same. The workspace in Figure 6a is the smallest since it has 4334 feasible poses, and the workspace volume ratio is 1.05%. The workspace in Figure 6d is the largest because it owns 1,295,064 qualified poses, and the workspace volume ratio is 312.63%.

5. Local Condition Index Evaluation

The singularity analysis in Section 3 provides these singularity configurations the robot should avoid during its motion. In addition, the robot should also prevent the poses close to the singularity configurations. The dexterity analysis has the ability to describe the motion distortion between active joints and the mobile platform, which is suitable to identify the motion transmission performance of each single pose and singularities, and their adjacent regions could be refrained with predefined requirements in applications. The most widely employed index for dexterity is the local condition index (LCI) [28], as defined below

$$Local Condition Index={\displaystyle \frac{1}{\kappa (J)}}$$

(21)

where $\kappa (J)$ is the condition number of Jacobian matrix. $\kappa (J)={‖J‖}_f\cdot {‖J^{-1}‖}_f={\sigma }_{\max }/{\sigma }_{\min }$. The symbol ${‖\cdot ‖}_f$ indicates the Frobenius norm of a square matrix. ${\sigma }_{\max }$ represents the maximal singular value of the Jacobian matrix J, and ${\sigma }_{\min }$ is the multiplicative inverse of the minimal singular value of the Jacobina matrix J.

The maximum and minimum values of the local condition index are 1 and 0, respectively. The larger this index, the better the motion transmission performance between the mobile platform and three driving joints.

The distributions of the local condition index within the reachable workspace are demonstrated in Figure 10. One element in the mobile platform pose $X={[x_P,y_P,\alpha ]}^{\mathrm{T}}$ is set as a constant number in each subplot for the purpose of illustration. $\alpha$ = 0 in Figure 10a, y_P is set as 400 mm in Figure 10b, and x_P is 700 mm in Figure 10c.

Three subplots in Figure 10 are not regular surfaces, and there are no symmetrical planes. The minimal value in any subplot is larger than zero, which indicates there are no singularity configurations within its reachable workspace due to predefined dimensions. In Figure 10a, the maximal value happens when x_P =648 mm and y_P = 724 mm. Another peak point occurs when both the x_P and y_P values are smallest. In the right portion of Figure 10b, the performance improves when the x_P value is larger and the orientational angle is close to zero. In the remaining portion of Figure 10b, the performance is better when the x_P value approaches 730 mm. The highest performance happens when y_P = 330 mm and $\alpha$ = 0.87 in Figure 10c.

The minimal, maximal, and mean values of local condition index under the condition of a single variable (x_P, y_P, and $\alpha$) are provided in Figure 11 to further understand this robot’s local condition index distributions. In each subplot of Figure 11, one element of $X={[x_P,y_P,\alpha ]}^{\mathrm{T}}$ is chosen as the horizontal axis, while the other two elements can have any combination within the reachable workspace, which is not the same as in Figure 10 where the missing element is set as a constant value. The maximal value in Figure 11a, minimal values in Figure 11b, and minimal values in Figure 11c are constant numbers, respectively. The maximal values in both Figure 11a,b display an increasing trend and then remain constant. The minimal value in Figure 11a decreases firstly and then keeps a constant value. The layouts of other values do not follow simple trends. The integration of Figure 10 and Figure 11 equips designers with detailed local condition index layouts. Due to detailed operation requirements for an application, a specific operational workspace can be found with proper local condition index performances inside of the reachable workspace.

6. Discussions

The proposed parallel robot contains three kinematic limbs, and there are 18 revolute joints in total. Revolute joints can facilitate larger reachable workspace, as well as higher stiffness and payload capacity for parallel robots compared to spatial joints (spherical joint, universal joint). Revolute joints can lower the manufacturing cost compared to spatial joints. This robot structure is conventional, except that there is a closed loop in each kinematic limb. This symmetric robot is composed of more revolute joints compared to the conventional 3-RRR planar parallel mechanism. In general, more joints will cause higher expenses, lower reliability, and mechanical efficiency. In addition, it is preferred to systematically generate a family of parallel robots with similar intrinsic features. The possible methodologies include the Lie group and Lie algebra, screw theory, single open-chain approach, and generalized function (G_F) set theory.

The inverse kinematic solution is simple, while an explicit analytical solution for the direct position problem could not be obtained, even with the application of parallelogram units in all kinematic chains. These belong to the inherent features of parallel robots, compared with serial robots where the direct kinematic problem is simple and the inverse kinematic problem is challenging. Alternative approaches toward the direct position problem include numerical methods [29] and artificial intelligent (AI) algorithms [30]. A proper approach tailored for the direct position problem is required to achieve high accuracy and high computation efficiency in real-time applications. It is not crucial that there are large computation costs in the training and validation periods using AI algorithms.

All 10 cases of the inverse kinematic singularity are derived based on the Jacobian matrix, while the direct kinematic singularity configurations are complex. In real applications, the linkage dimensions are provided together with the pose information, and the calculation is transferred to a simple numerical calculation process. A verification process can be adopted by employing det(J_X) = 0 to ensure the chosen pose is not a direct singularity configuration.

The large reachable workspace of this parallel robot originates from its simple structure and the applications of pure revolute joints. According to Table 1 and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it is possible to improve the reachable workspace volume by changing linkage dimensions appropriately. It is also worth noticing that the local condition index is not close to the best value 1. A systematic dimensional synthesis can be employed to maximize this robot’s various performance indices once an application is determined, along with its detailed design requirements [31,32,33]. It is noteworthy that the workspace accuracy can be improved with much smaller step sizes ($x_P\_stepsize$, $y_P\_stepsize$, and $\alpha \_stepsize$), while the computation cost will grow significantly. The workspace comparison should be implemented under the identical step size settings.

The proposed 3-DOF parallel robot is suitable for the prototype of a planar spraying robot or laser cutting machine, considering its simple geometry, large workspace, and straightforward and convenient inverse kinematic solutions. A more sophisticated operation (e.g., spatial pick-and-place task, welding task) can be realized when integrating with additional movements. Additional motions can be supplemented on the robot side or the workpiece side to conform to various design criteria.

In future work, a group of planar parallel robot variants of this robot will be explored. A proper parallel robot structure will be selected for a specific application with detailed requirements. The dimensional synthesis will be implemented for the purpose of optimal performance. Its applications can be further expanded. For example, the series elastic actuator (SEA) can be utilized for changing stiffness performances under various scenarios [34]. Multiple performance comparisons with other parallel robots will be implemented, e.g., to compare its stiffness with other parallel robots with spatial joints (spherical joint, universal joint), to compare its speed and acceleration with other parallel architectures with active sliding joints.

7. Conclusions

(i) A planar three-degrees-of-freedom parallel robot with identical kinematic chains and pure revolute joints is designed in this research. There are three active kinematic joints attached to the fixed platform. An explicit solution can be resolved for the inverse kinematic model, but not for the direct kinematic model due to its complexity. There are, at most, eight sets of solutions in both kinematic problems. The singularity configurations are determined based on the Jacobian matrix. There are 10 different scenarios for the inverse singularity configurations.

(ii) The reachable workspace of this parallel manipulator possesses a large orientation capacity. A half-round rotation could be conducted in most positions within a reachable workspace. The parametric analysis with a single variable demonstrates longer linkage, which tends to possess a larger workspace volume in a particular range. The workspace volume ratio can achieve 312.63% when L_i3 = 500 mm. The smallest workspace volume ratio is 1.05% when L_i3 = 100 mm. There is no simple positive or negative correlation between any component of $X={[x_P,y_P,\alpha ]}^{\mathrm{T}}$ and the local condition index due to its irregular distributions.