Design and Performance Analysis of a Novel Group of Translational Parallel Robots for a Three-Axis Grinding Machine

Abstract

There are limited parallel robots applicable to three-axis grinding machines due to the restricted reachable workspace originating from multiple kinematic chains with spatial kinematic joints. The parallel robot will gain significant potential in the industry if a larger workspace can be achieved. This research introduces a special relationship between upper triangular matrix and parallel robot structures for the purpose of designing a group of novel parallel robots without spatial kinematic joints. The detailed inverse kinematic solution of the selected parallel manipulator is derived in accordance with its straightforward architecture. Several singularity configurations are found on the basis of the first-order kinematic relation. The translational reachable workspace is close to a triangular prism. The dexterity and stiffness performances based on the Jacobian matrix are explored for the chosen parallel manipulator. Both indices display a downward trend as the mobile platform rises higher.

1. Introduction

An industrial three-axis grinding machine requires the spindle to own large sliding strokes along three directions in the Cartesian coordinate system. This makes a serial robot with open kinematic chains a suitable candidate. However, the serial robot suffers from lower speed and stiffness and payload capacity, with higher inertia compared to a parallel robot with several kinematic chains [1]. The employment of multiple kinematic chains and spatial kinematic joints (e.g., spherical joint, university joint) leads to a smaller reachable workspace and various singularity configurations for the parallel robot, which significantly restricts its popularity in industrial applications [2].

It is essential to design novel parallel robot prototypes with pure translations in three-dimensional space (planar translational parallel mechanisms are not in the scope of this research due to insufficient degrees-of-freedom (DOFs)). Translational parallel robots are widely used in many applications, e.g., micro-positioning, machining, drilling, assorting, stacking, painting and high-speed picking-and-placing tasks [3]. It can be employed to deal with light-weight objects or heavy objects. Besides these applications, translational parallel robots can also be integrated into a hybrid robot with four- or five-DOFs for more complicated and flexible applications [4].

The famous and widely employed Delta robot [5] invented by Reymond Clavel in the early 1980s was equipped with three identical chains. Each kinematic chain included one rotational joint and one parallelogram mechanism with four spherical joints. A linear Delta robot could be assembled if the first rotational joint is replaced by a prismatic joint (prismatic joint denotes two parts that have relative translational movement) [6]. The Delta robot was popular in fast picking-and-placing operations. There were also many variants (for example, H4 [7] and I4 [8] robots) with enhanced performance for this kind of operation.

A systematic procedure on the basis of screw theory was implemented to design a family of translational parallel robots [9]. The configurable mobile platform was also integrated into these designs to enhance their translational performance. Hassani et al. [10] introduced a translational parallel manipulator with three kinematic limbs. The joints in each kinematic limb included a universal joint, prismatic joint, and universal joint. A novel translational parallel robot with three active sliding actuators was described in [11] based on position and orientation characteristic formulations. Both the forward and inverse kinematic models could be resolved conveniently due to their unique configurations. The partial decoupling feature existed in this design. Hamida et al. [12] presented four different kinds of translational parallel architectures including Delta robot, 3-UPU robot (P and U stand for prismatic joint and universal joint, respectively), tri-pyramid robot, and Romdhane-Affi-Fayet robot. This research developed multi-objective optimization for the predefined reachable workspace and the second robot had the smallest size. Tsai and Stamper [13] developed the University of Maryland Manipulator with symmetrical features and three identical kinematic chains. Each kinematic chain was composed of revolute–revolute–parallelogram mechanism with four revolute joints–revolute type (revolute joint indicates two parts have relative rotational movement). The authors in [14] compared three translational parallel manipulators, Delta robot, 3-UPU robot, and 3-PUU robot. An asymmetrical parallel manipulator with partial decoupled movement was proposed in [15]. There were two parallelogram mechanism units for the sake of pure translational motion of the mobile platform.

There is increasing interest in the fundamental kinematic performance evaluations of parallel robots. The authors in [16] designed a two-dimensional parallel structure (2PRR)-R + 2RRR (R stands for prismatic joint). The error-related indices are explained and compared using different sets of linkage dimensions. The evolution-based intelligent algorithm was employed to pursue a set of linkage lengths with the largest reachable workspace volume and lowest error-related performance. Damine et al. [17] proposed a 2UPS-U (S indicates spherical joint) for the purpose of otologic surgery. A common shape inside of the reachable workspace was predefined for this surgical procedure and the singularity locus under various sets of dimensions were comprehensively studied for future avoidance. The metamorphic kinematic joint was integrated into a three-limb reconfigurable parallel mechanism [18]. The reachable workspaces under various operation modes were explored based on the comprehensive inverse kinematic model. Its singularity configurations are revealed according to the Jacobian matrix obtained based on screw theory. The local transmission index was also measured for different operation modes. The kinematic models of both the 3-RRR and 3-PRR mechanisms were computed [19]. These two parallel robots owned various performances on rotation dexterity, prescribed flexible orientation workspace, dexterity, velocity, static stiffness, and motion/force transmission properties. The Rec4 parallel structures with three translations and one rotation was proposed in [20]. According to the calculations of the local transmission index, global input/output transmission indices, the effective transmission positional workspace was determined and used as the optimization objective for this parallel robot. Three kinds of redundant parallel robots with six-UPS kinematic chains were mentioned in [21]. This study compared the translational and orientational workspaces and singularity configurations of three robots. It was proved that the redundant design could significantly improve the workspace volume for all three robots, which guaranteed their potential applications in surgical operation. The parallel mechanism with one translation and two rotations inside of a hybrid kinematic machine was developed in [22]. It was determined that there was no parasite motion within the reachable workspace. The motion/force transmission property of this robot was evaluated based on screw theory. It is still a key topic to systematically propose a group of parallel robots instead of only designing one single parallel robot. It is essential to develop simple and straightforward type synthesis approaches for novel parallel architectures. There are few symmetrical parallel candidates for the three-axis grinding operation due to the small reachable workspace and unexpected singularity configurations.

The main contributions of this research include the following:

• This research designs a novel class of parallel manipulators with only translational movements in three-dimensional space and planar kinematic joints to be suitable for the three-axis griding machine.
• The special upper triangular matrix is employed to map with the parallel manipulator structures for the convenience of the design procedure.

The design process utilizes the unique upper triangular matrix. The algebraic approach based on the vector-loop equation is used for detailed kinematic analysis. The spatial search algorithm based on the inverse kinematic model is employed for the workspace determination. Two performance indices are evaluated based on the Jacobian matrix. The layout of this research is organized as follows: The mechanical design of this class of parallel robots is mentioned in Section 2, followed by the detailed inverse kinematic model in Section 3. The first-order kinematic relationship is expressed in Section 4. The reachable workspace of the selected robot is reported in Section 5. Two different kinematic performance indices are analyzed in Section 6 and Section 7. Section 8 and Section 9 conclude the overall work.

2. Mechanism Design

The design process is illustrated in Figure 1. Each step is explained in detail in the following contents. To construct the parallel manipulator with pure translations, priority should not be given to the spatial kinematic joints (spherical, cylindrical, and universal joints) due to their spatial movements and limited ranges of motion compared with planar kinematic joints (prismatic and revolute joints). The prismatic (P) joint can be selected in each kinematic chain since it possesses pure translational movement. The revolute (R) joint can be added if some special geometrical relations are realized. The parallelogram mechanism with parallel motion between any two opposite rods is commonly employed in serial or parallel robots for the purpose of special geometry. The R joint can be integrated into the parallelogram mechanism for pure translational movement. Hence, only a prismatic join and parallelogram mechanism are chosen for this group of 3-DOF translational parallel robots. The criteria are listed below.

i.

There are three kinematic chains. There is only one prismatic joint and two parallelogram units in each kinematic branch.

ii.

The prismatic joint connected to the fixed platform is the active joint in each chain. The prismatic joint axis can be with axis X or Y or Z. Different axes of prismatic joints between any two branches are required.

iii.

The plane for the parallelogram mechanism can be parallel to plane XOY or YOZ or XOZ. The same plane of parallelogram units in any branch is refrained.

According to the abovementioned conditions, all feasible kinematic chains can be classified into three categories, as shown below.

$$group A\left\{\begin{array}{l}{P}_X-P_{XY}-P_{XZ}\\ P_X-P_{XY}-P_{YZ}\\ P_X-P_{XZ}-P_{YZ}\end{array}\right.$$

(1)

$$group B\left\{\begin{array}{l}{P}_Y-P_{XY}-P_{XZ}\\ P_Y-P_{XY}-P_{YZ}\\ P_Y-P_{XZ}-P_{YZ}\end{array}\right.$$

(2)

$$group C\left\{\begin{array}{l}{P}_Z-P_{XY}-P_{XZ}\\ P_Z-P_{XY}-P_{YZ}\\ P_Z-P_{XZ}-P_{YZ}\end{array}\right.$$

(3)

where the prismatic joint along the X or Y or Z direction is denoted as P_X or P_Y or P_Z, respectively. The parallelogram unit in a plane that is parallel to the XOY or XOZ or YOZ plane is separately indicated as P_XY or P_XZ or P_YZ.

These qualified kinematic branches can be written in the form of the three-by-three upper triangular matrix (the elements under the main diagonal of a square matrix are set as zero) [23,24]. There are six elements in this matrix and each element is for one specific prismatic joint or parallelogram mechanism. The three elements in the first row are P_X, P_XY, and P_XZ from left to right. The two elements in the second row are P_Y and P_YZ from left to right. The only element in the third row is P_Z. The ‘_’ symbol in the matrix indicates that the corresponding element does not exist.

The kinematic branches in Equations (1)–(3) can be rearranged in the following upper triangular matrix form:

$$group A\left\{\begin{array}{l}\left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]\\ \left[\begin{array}{ccc}{P}_X& P_{XY}& –\\ & –& P_{YZ}\\ & & –\end{array}\right]\\ \left[\begin{array}{ccc}{P}_X& –& P_{XZ}\\ & –& P_{YZ}\\ & & –\end{array}\right]\end{array}\right.$$

(4)

$$group B\left\{\begin{array}{l}\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & P_Y& –\\ & & –\end{array}\right]\\ \left[\begin{array}{ccc}–& P_{XY}& –\\ & P_Y& P_{YZ}\\ & & –\end{array}\right]\\ \left[\begin{array}{ccc}–& –& P_{XZ}\\ & P_Y& P_{YZ}\\ & & –\end{array}\right]\end{array}\right.$$

(5)

$$group \mathrm{C}\left\{\begin{array}{l}\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & P_Z\end{array}\right]\\ \left[\begin{array}{ccc}–& P_{XY}& –\\ & –& P_{YZ}\\ & & P_Z\end{array}\right]\\ \left[\begin{array}{ccc}–& –& P_{XZ}\\ & –& P_{YZ}\\ & & P_Z\end{array}\right]\end{array}\right.$$

(6)

These nine kinematic legs classified into three groups (group A, group B, and group C) are fundamental for translational parallel mechanisms with three DOFs. The enumeration of the possible parallel mechanisms can be implemented by combing these kinematic chains from different groups. The matrix for the parallel robots can be expressed by the augmented matrix of an upper triangular matrix. The written rule is that each element should be filled in the current three-by-three matrix after the corresponding position in the left matrix is already filled, for the purpose of simplicity. There are in total 3 × 3 × 3 = 27 matrices. However, there are only five unique matrices among them. The distinguishment process of different matrices provides the advantage using the upper triangular matrix. It is more convenient and time-saving than only listing these mechanism in a similar manner as Equations (1)–(3). The obtained final five matrices are provided below.

$$\mathrm{First} \mathrm{matrix}: \left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & P_Y& –\\ & & P_Z\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]$$

(7)

$$\mathrm{Second} \mathrm{matrix}: \left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & P_Y& P_{YZ}\\ & & P_Z\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& –\\ & –& –\\ & & –\end{array}\right]$$

(8)

$$\mathrm{Third} \mathrm{matrix}: \left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & P_Y& P_{YZ}\\ & & P_Z\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]\left[\begin{array}{ccc}–& –& P_{XZ}\\ & –& –\\ & & –\end{array}\right]$$

(9)

$$\mathrm{Fourth} \mathrm{matrix}: \left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & P_Y& P_{YZ}\\ & & P_Z\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& –\\ & & –\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& –\\ & –& –\\ & & –\end{array}\right]$$

(10)

$$\mathrm{Fifth} \mathrm{matrix}: \left[\begin{array}{ccc}{P}_X& P_{XY}& P_{XZ}\\ & P_Y& P_{YZ}\\ & & P_Z\end{array}\right]\left[\begin{array}{ccc}–& P_{XY}& P_{XZ}\\ & –& P_{YZ}\\ & & –\end{array}\right]$$

(11)

A group of feasible parallel robots corresponding to matrices in Equations (7)–(11) is demonstrated in Figure 2. On the basis of the construction conditions of this family of parallel robots, there are three kinds of prismatic joints (three joint axes) in each matrix. The planes for the parallelogram mechanism range from two to three. Since only the prismatic joint and parallelogram mechanism are utilized in this group of parallel robots, the modular linkages and joints can be designed for the quick fabrication of a specific parallel robot. Although these parallel structures have the same three degrees of freedom, they may have various performances.

It is worth noting that the mapping relationship between an augmented matrix and a parallel robot structure is not one-to-one due to the complexity of the matrix or the parallel manipulators. One matrix may represent multiple parallel robots. The matrix in Equation (11) is taken as one example, and two additional parallel robot structures can be constructed in Figure 3 based on this matrix.

3. Position Analysis

The parallel robot denoted in Figure 2a is selected for the following calculations because it has only two planes for the parallelogram units. The schematic diagrams of this mechanism are provided in Figure 4. Three active prismatic joints are marked as $A_{11},B_{11},C_{11}$ and their positions are represented by parameters $s_1,s_2,s_3$, respectively. The center position of the moving platform is indicated by P(x,y,z). The points A₁₂, B₁₂, and C₁₂ separately denote the distal end of the first parallelogram mechanism and the proximal end of the second parallelogram mechanism in three kinematic chains. The distal ends of three kinematic chains are listed as $A_{13}$, $B_{13}$, and $C_{13}$, respectively.

The linkage dimensions of A₁₁A₁₂, A₁₁A₁₃, B₁₁B₁₂, B₁₁B₁₃, C₁₁C₁₂, and C₁₁C₁₃, are represented as L₁₁, L₁₂, L₂₁, L₂₂, L₃₁, and L₃₂, respectively. The depth, width, and height of the moving platform are separately denoted as 2L₄₁, 2L₄₂, and 2L₄₃. Three connection points attached on the moving platform are indicated as $A_{13}(x,y-L_{42},z-L_{43})$, $B_{13}(x-L_{41},y,z)$, and $C_{13}(x,y-L_{42},z+L_{43})$.

The joint angles should be measured with reference to various directions considering the spatial configuration. Angles ${\theta }_{11}$, ${\theta }_{12}$, ${\theta }_{22}$, and ${\theta }_{32}$ are measured from the positive direction of the X axis. Angle ${\theta }_{21}$ is measured from the positive direction of the Y axis while angle ${\theta }_{31}$ is measured from the positive direction of the Z axis.

The inverse kinematic problem is to obtain the positions of three prismatic actuators with the position of P. According to the loop-closure equation in the first branch, the following occurs:

$$O{A}_{11}+A_{11}{A}_{12}+A_{12}{A}_{13}=O{A}_{13}$$

(12)

Equation (12) can be split into three orthogonal directions, as shown below.

$$\left\{\begin{array}{l}{s}_1+L_{11}\cos {\theta }_{11}+L_{12}\cos {\theta }_{12}=X_{A13}\\ L_{12}\sin {\theta }_{12}=Y_{A13}\\ L_{11}\sin {\theta }_{11}=Z_{A13}\end{array}\right.$$

(13)

Equation (13) is solved as follows:

$$\left\{\begin{array}{l}{\theta }_{12}=\arcsin {\displaystyle \frac{y-L_{42}}{L_{12}}}\\ {\theta }_{11}=\arcsin {\displaystyle \frac{z-L_{43}}{L_{11}}}\\ s_1=x-L_{11}\cos {\theta }_{11}-L_{12}\cos {\theta }_{12}\end{array}\right.\mathrm{or} \left\{\begin{array}{l}{\theta }_{12}=\pi -\arcsin {\displaystyle \frac{y-L_{42}}{L_{12}}}\\ {\theta }_{11}=\pi -\arcsin {\displaystyle \frac{z-L_{43}}{L_{11}}}\\ s_1=x-L_{11}\cos {\theta }_{11}-L_{12}\cos {\theta }_{12}\end{array}\right.$$

(14)

The vector-loop equation of the second branch is arranged as follows:

$$O{B}_{11}+B_{11}{B}_{12}+B_{12}{B}_{13}=O{B}_{13}$$

(15)

Decompose Equation (15) into three perpendicular directions as follows:

$$\left\{\begin{array}{l}{L}_{21}\cos {\theta }_{21}+L_{22}\cos {\theta }_{22}=X_{B13}\\ s_2+L_{21}\sin {\theta }_{21}=Y_{B13}\\ L_{22}\sin {\theta }_{22}=Z_{B13}\end{array}\right.$$

(16)

The result for the second kinematic limb is derived as follows:

$$\left\{\begin{array}{l}{\theta }_{22}=\arcsin {\displaystyle \frac{z}{L_{22}}}\\ {\theta }_{21}=\arccos {\displaystyle \frac{x-L_{41}-L_{22}\cos {\theta }_{22}}{L_{21}}}\\ s_2=y-L_{21}\sin {\theta }_{21}\end{array}\right.\mathrm{or} \left\{\begin{array}{l}{\theta }_{22}=\pi -\arcsin {\displaystyle \frac{z}{L_{22}}}\\ {\theta }_{21}=-\arccos {\displaystyle \frac{x-L_{41}-L_{22}\cos {\theta }_{22}}{L_{21}}}\\ s_2=y-L_{21}\sin {\theta }_{21}\end{array}\right.$$

(17)

The loop-closure equation for the last branch is written as follows:

$$O{C}_{11}+C_{11}{C}_{12}+C_{12}{C}_{13}=O{C}_{13}$$

(18)

Equation (18) can be derived as follows:

$$\left\{\begin{array}{l}{L}_{31}\sin {\theta }_{31}+L_{32}\cos {\theta }_{32}=X_{C13}\\ L_{32}\sin {\theta }_{32}=Y_{C13}\\ s_3+L_{31}\cos {\theta }_{31}=Z_{C13}\end{array}\right.$$

(19)

The solution for the third branch can be calculated as follows:

$$\left\{\begin{array}{l}{\theta }_{32}=\arcsin {\displaystyle \frac{y-L_{42}}{L_{32}}}\\ {\theta }_{31}=\arcsin {\displaystyle \frac{x-L_{32}\cos {\theta }_{32}}{L_{31}}}\\ s_3=z+L_{43}-L_{31}\cos {\theta }_{31}\end{array}\right.\mathrm{or} \left\{\begin{array}{l}{\theta }_{32}=\pi -\arcsin {\displaystyle \frac{y-L_{42}}{L_{32}}}\\ {\theta }_{31}=\pi -\arcsin {\displaystyle \frac{x-L_{32}\cos {\theta }_{32}}{L_{31}}}\\ s_3=z+L_{43}-L_{31}\cos {\theta }_{31}\end{array}\right.$$

(20)

The last equation in Equations (14), (17), and (20) are the solution for its inverse kinematic problem. It is worth noting that there are multiple sets of solutions (at most 2 × 2 × 2 = 8) given the mobile platform position. Eight sets of solutions can represent the mechanism in eight kinds of initial assembly configurations. These assembly configurations cannot be converted during operation since the transformation will pass the singularity configurations, as mentioned in Section 4.

The direct kinematic problem is to calculate P(x,y,z) with predefined driving joint positions $s_1,s_2,s_3$. Three equations containing three unknown parameters (x,y,z) will be left when eliminating all passive joints variables (${\theta }_{11}$, ${\theta }_{12}$, ${\theta }_{21}$, ${\theta }_{22}$, ${\theta }_{31}$, and ${\theta }_{32}$) from Equations (14), (17), and (20). If the first set of inverse kinematic solutions in Equations (14), (17), and (20) are selected, the following equations are obtained using this strategy:

$$\left\{\begin{array}{l}{s}_1+L_{11}\cos \left(\arcsin {\displaystyle \frac{z-L_{43}}{L_{11}}}\right)+L_{12}\cos \left(\arcsin {\displaystyle \frac{y-L_{42}}{L_{12}}}\right)=x\\ s_2=y-L_{21}\sin \left\{\arccos {\displaystyle \frac{x-L_{41}-L_{22}\cos [\arcsin (z/L_{22})]}{L_{21}}}\right\}\\ s_3=z+L_{43}-L_{31}\cos \left[\arcsin {\displaystyle \frac{x-L_{32}\cos \left(\arcsin \frac{y-L_{42}}{L_{32}}\right)}{L_{31}}}\right]\end{array}\right.$$

(21)

Each expression in Equation (21) has three unknown variables and trigonometric functions. This calculation process is complicated and time-consuming, which is not suitable for industrial applications. The alternative methodology is to employ numerical methods (e.g., Newton–Raphson approach) [25] or artificial intelligent (AI) algorithms (e.g., support vector machines) [26] to obtain the approximate direct kinematic solution The most important concern using alternative methods is accuracy. There are also other concerns utilizing the numerical approaches. For example, high computational cost, it is not possible to resolve all sets of kinematic solutions, and it is not clear which set of kinematic solutions is suitable for the direct kinematic problem. The other challenges for the AI algorithms include a large portion of time spent on the training and validation periods, and a heavy dependence on the network architecture and training dataset.

4. First-Order Kinematics

The velocity mapping relationship between the driving joints and moving platform is fundamental for a robot; this relationship can be connected by the Jacobian matrix. Only one set of inverse kinematic solutions is considered in this section for simplification. The left equation of Equations (14), (17), and (20) is selected. It is also assumed that expressions $\cos {\theta }_{11}$, $\cos {\theta }_{12}$, $\cos {\theta }_{21}$, $\cos {\theta }_{22}$, $\sin {\theta }_{31}$, and $\cos {\theta }_{32}$ are positive variables.

The first step is to only keep the last expression of Equations (14), (17), and (20). The third expression of Equations (14), (17), and (20) can be, respectively, arranged as follows:

$$s_1+L_{11}\cos (\arcsin {\displaystyle \frac{z-L_{43}}{L_{11}}})+L_{12}\cos (\arcsin {\displaystyle \frac{y-L_{42}}{L_{12}}})=x$$

(22)

$$L_{21}\cos (\arccos {\displaystyle \frac{x-L_{41}-L_{22}\cos {\theta }_{22}}{L_{21}}})+L_{22}\cos (\arcsin {\displaystyle \frac{z}{L_{22}}})=x-L_{41}$$

(23)

$$L_{31}\sin (\arcsin {\displaystyle \frac{x-L_{32}\cos {\theta }_{32}}{L_{31}}})+L_{32}\cos (\arcsin {\displaystyle \frac{y-L_{42}}{L_{32}}})=x$$

(24)

The second step is to take the time derivative. Differentiate Equations (22)–(24) with respect to time.

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

(25)

where

$\dot{X}={[\dot{x},\dot{y},\dot{z}]}^{\mathrm{T}}$

$\dot{Q}={[{\dot{s}}_1,{\dot{s}}_2,{\dot{s}}_3]}^{\mathrm{T}}$

$J_X=\left[\begin{array}{ccc}-1& {\displaystyle \frac{y-L_{42}}{\sqrt{L_{12}^2-{(y-L_{42})}^2}}}& {\displaystyle \frac{z-L_{43}}{\sqrt{L_{11}^2-{(z-L_{43})}^2}}}\\ -1& {\displaystyle \frac{s_2-y}{\sqrt{L_{11}^2-{(s_2-y)}^2}}}& {\displaystyle \frac{z}{\sqrt{L_{12}^2-z^2}}}\\ -1& {\displaystyle \frac{L_{42}-y}{\sqrt{L_{12}^2-{(L_{42}-y)}^2}}}& {\displaystyle \frac{-(L_{43}-s_3+z)}{\sqrt{L_{11}^2-{(L_{43}-s_3+z)}^2}}}\end{array}\right]$

$J_Q=\left[\begin{array}{ccc}-1& 0& 0\\ 0& {\displaystyle \frac{s_2-y}{\sqrt{L_{11}^2-{(s_2-y)}^2}}}& 0\\ 0& 0& {\displaystyle \frac{-(L_{43}-s_3+z)}{\sqrt{L_{11}^2-{(L_{43}-s_3+z)}^2}}}\end{array}\right]$

The Jacobian matrix for this translational parallel robot is derived as follows:

$$J={J_Q}^{-1}{J}_X$$

(26)

When the parallel robot is in singularity configurations and their adjacent areas/regions, the robot maybe cannot keep the expected smooth movement or cannot move. The robot may not be under control, or the degree-of-freedom might change. The force relationship is distorted, and it may be harmful for the robot and actuators. The whole reachable workspace may be divided into several subspaces by the singularity loci. The singularity configuration of this spatial parallel robot can be explored by the obtained Jacobian matrix. The scenario when Det(J_X) = 0 is too complicated to be solved. However, some straightforward solutions can be found by observing this matrix. For example, $y=L_{42}=s_2$ will lead to the second column of this matrix, which contains only element zero and Det(J_X) = 0 under this condition.

There are two cases when Det(J_Q) is equal to zero. The first case is $s_3=z+L_{43}$, which leads to ${\theta }_{31}=\pi /2$ based on Equation (20). Any two adjacent linkages of the first parallelogram mechanism in the third kinematic chain are perpendicular in this situation. The second case happens when $s_2=y$, which means that ${\theta }_{21}=0$ on the basis of Equation (17). All four linkages of the first parallelogram mechanism in the second kinematic chain are collinear with the Y axis. The mentioned singularity configurations can also be determined by considering the generalized joint–parallelogram mechanism, as seen below.

Considering there are six parallelogram units, this robot lies in the singularity configuration when any parallelogram is in a singularity configuration. The first singularity configuration happens when any two adjacent linkages are perpendicular to each other, as demonstrated in Figure 5a. Under such situation, the right linkage in white could bear a force along the horizontal direction (the shaft direction of the red linkages) although there is no driving force/torque from the corresponding actuator. Another singularity configuration occurs when the angle between any two adjacent linkages is zero, as shown in Figure 5b. In this case, the right linkage in white could also resist a force along the horizontal direction (the shaft direction of the red linkages) even if there is no active force/torque from the relevant active joint. The rotational degree-of-freedom of the parallelogram mechanism is lost at this instant since it is uncertain that the right linkage in white moves upward or downward. These kinds of singularities can be prevented by restricting the range of motion of passive joints.

5. Reachable Workspace

The reachable workspace of this translational parallel robot in Figure 4a is the set of all the positions its mobile platform can reach in three-dimensional space. Parallel robots generally have smaller reachable workspaces compared with serial robots in a similar scale. There are three factors affecting its reachable workspace: linkage length, strokes of active prismatic joints and passive revolute joints, and linkage interference.

It is assumed that the same kind of linkages have the same length. Each driven revolute joint can rotate no more than $\pi /2$ radians to avoid singularity of the parallelogram module, as mentioned in Figure 5. The detailed dimensions are provided in

Table 1. The parameters for the translational parallel robot.

Parameter	Value
L11, L21, L31	50 mm
L12, L22, L32	55 mm
2L41, 2L42, 2L43	10 mm
s1	(0, 210) mm
s2, s3	(0, 105) mm
${\theta }_{11}$$, {\theta }_{12}$$, {\theta }_{21}$$, {\theta }_{31}$	$(\pi /2,\pi )$ radians
${\theta }_{22}$$, {\theta }_{32}$	$(0,\pi /2)$ radians

.

The spatial search approach [27] is employed for this three-DOF miniature parallel robot, and the obtained reachable workspace is demonstrated in Figure 6. Its reachable workspace is similar to a triangular prism with a small notch, as seen in the upper left corner in front view and lower left corner in top view. The workspace has a downward trend as the position along the Z direction increases.

6. Dexterity Index

The most widely used kinematic indices for parallel robots are the manipulability index and the condition number on the basis of Jacobian matrix. The latter is selected in this section and defined as follows:

$$LCI={\displaystyle \frac{1}{‖J‖\cdot ‖J^{-1}‖}}$$

(27)

where LCI is short for the local condition index.

The range of LCI is [0,1]. The parallel robot tends to possess better kinematic performance when LCI approaches the maximum value 1.

The dexterity distributions are illustrated based on different layers to better represent its layout: z = 10 mm; z = 20 mm; z = 30 mm; z = 40 mm; and z = 50 mm. These illustrations are shown in Figure 7. According to Figure 7, the local condition index layouts are complicated and it is complex to describe a simple positive/negative proportional between the index and X/Y value. Figure 7a–e is close to the shapes of two-, three-, and one-tunnel. The LCI distributions on these four layers are similar. The layer z = 50 mm is much smaller and seems different from the other layers; however, the performance distribution has resemblance to other layers in this smaller area. According to the five plots of Figure 7, the LCI value has a downward trend as the height (z value) of the mobile platform increases. The LCI of Figure 7e is the minimal among these figures since this height is close to the upper boundary of the reachable workspace.

7. Stiffness Index

This parallel robot can bear external forces along three orthogonal axes in the global Cartesian coordinate system of Figure 3. The resistance of the mobile platform is transferred into its three sliding actuators. This relationship can be transformed by the Jacobian matrix. The stiffness matrix is defined as follows:

$$K=J^{\mathrm{T}}{K}_{J}J$$

(28)

where the superscript T indicates transpose and the following occurs:

$$K_J=\left[\begin{array}{ccc}{K}_x& 0& 0\\ 0& K_y& 0\\ 0& 0& K_z\end{array}\right]$$

Three active sliding motors have identical stiffness because they are the same. It is assumed that $K_x=K_y=K_z=1000$ N/mm. The stiffness distributions in five layers along axis z are shown in Figure 8. The overall stiffness performance is decreasing as the mobile platform rises (a larger z value). Akin to the dexterity layout in Section 6, the fifth stiffness distribution in Figure 8e is similar to those in Figure 8a–d. The largest stiffness performance is realized when the maximal x value and y value are reached in Figure 8a–d.

8. Discussion

The selected parallel mechanism is in singularity configuration once any parallelogram mechanism is in singularity configurations. The singularity configuration by evaluating J_X is complicated. However, the reachable workspace is still a solid three-dimensional shape without hollow parts.

The reachable workspace shapes of other three-dimensional translational parallel structures are provided for comparison. The symmetrical workspace of the robot in [24] has a rectangular section along its vertical direction and the rectangular section is larger with a larger height. The reachable workspace of the parallel architecture in [28] is close to the shape of a cuboid with a symmetrical horizontal plane. The reachable workspace of the Delta robot is an irregular shape with no symmetrical plane and the largest inscribed shape is similar to an inverted cone [29].

The current reachable workspace is close to a regular three-dimensional shape—a triangular prism. The operational workspace generally is an inscribed regular shape within the reachable workspace, which indicates that the available region is smaller, especially as the reachable workspace is a truncated triangular prism. Another possible issue lies in the various sizes in the different layers, which could restrict the maximum grinding capacity of the workpiece size.

The layouts of the dexterity and stiffness indices are illustrated in Figure 7 and Figure 8. When the mobile platform rises along the Z direction, both indices show a downward trend. There is no straightforward proportional relationship between the index and the X or Y parameter. The dexterity index is dependent on the robot configuration and no direct or simple methods can be used for improvement. However, the stiffness index can be promoted by several approaches. For example, using stronger material for linkages, designing the linkage sectional area to make sure it can bear higher external force/torque, adding extra chains to support the moving platform while meeting the motion requirement can affect the stiffness index. The grinding operation can always choose a region with high indices inside of the reachable workspace. With the stiffness index taken as an example, the maximum stiffness in the chosen region inside of the reachable workspace limits the highest stiffness the machine tool can bear. If the grinding operation can be chosen in a region where the highest stiffness is larger, the grinding operation can be safer, and its capacity is larger.

The balance among the reachable workspace and the dexterity and stiffness indices is a challenging task. The optimal design is generally conducted by intelligent algorithms (e.g., genetic algorithm). This research is limited to kinematic analysis, which does not include dynamic characteristics related to real implementation. More error-related analyses including joint clearance are not disclosed. Future work includes the optimal design, fabrication, and assembly of the selected parallel robot. A proper intelligent control strategy will be applied for its smooth and accurate movement.

9. Conclusions

• A novel class of pure translational parallel robots is obtained through the mapping relation with the augmented matrix of the upper triangular matrix in this research. The proposed mapping relation can serve as a reference for the enumeration of other types of parallel mechanisms under prescribed conditions. Only prismatic joints (prismatic joint, revolute joint within parallelogram mechanism) are employed for the sake of larger range of motion of the kinematic joints.
• One translational parallel structure with simple architecture was chosen. Its analytical inverse kinematic model is constructed while the analytical explicit solution to the direct kinematic model cannot be found due to complexity. The alternative direction kinematic solution can be realized by numerical approaches.
• This parallel robot is in singularity configuration once any parallelogram mechanism is in singularity. The miniature reachable workspace and three standard views are obtained to understand its distribution. Both the overall dexterity and stiffness indices drop as the mobile platform rises (the z value increases).
• The chosen parallel mechanism is suitable for a three-axis grinding machine. The intrinsic characteristics of the parallel mechanism include high precision and payload capacity compared with the serial mechanism in similar conditions. The employment of only the prismatic joint and revolute joint can ensure high stiffness. The overall stiffness is illustrated by the stiffness analysis. The reachable workspace is suitable as it is not split into several subspaces due to singularity configurations. The four-axis or five-axis hybrid grinding machine can be constructed with the addition of an extra one or two rotations on the current moving platform or the workpiece.