Kinematic Analysis and Performance Optimization of a 2UPU-2SPU Parallel Mechanism with 2R2T Motion Capabilities

The existence of parallel mechanisms has a long history, but it is fair to say that research on the kinematics of parallel mechanisms only gradually began in the 1960s. In 1962, Gough and Whitehall [1] invented a 6-DOF tire testing device using a system of jacks connected by joints; Stewart [2] first conducted a kinematic study of this mechanism in 1965, and also built a simulated flight environment for aircraft flight training based on this mechanism model, making it the most widely used parallel mechanism to date.

Compared to 6-DOF parallel mechanisms, parallel mechanisms with fewer DOF cannot perform some complex motions, but they have advantages such as simple structure, low cost, fewer drive components, easy control, and large workspace. Rolland [3] proposed two types of 4-DOF parallel mechanisms for material handling: Manta & Kanuk. Tsai and Joshi [4] analyzed and studied the 3-UPU parallel mechanism through research on the structural characteristics of parallel manipulators. Chen et al. [5] proposed a 4-DOF parallel platform manipulator with 2 Rotational Prismatic Spherical (PRS) and 2 Prismatic Spherical Spherical (PSS) support chains, and conducted a kinematic analysis of this 2T2R parallel mechanism. Li and Huang [6] used the method of constraint synthesis for a systematic type synthesis of 4-DOF parallel robots, and proposed a 2R2T parallel mechanism with 2 symmetrical support chains. Yoon and Ryu [7] proposed two types of 4-DOF spatial parallel mechanisms with two platforms: 1T3R and 2T2R. These are relatively early studies on parallel mechanisms with fewer DOF.

As parallel mechanisms are widely applied in various fields, scholars have paid more attention to parallel mechanisms with fewer DOF. Research on parallel mechanisms such as 3R, 3T, 2R1T, 3T2R, 2T1R, and 2R2T has increased.

Xie et al. [8] considered force transmission and analyzed the kinematic optimization design of 2-DOF parallel mechanisms using a 4R parallel mechanism as an example; combining configuration evolution with Lie group theory, Fan et al. [9] proposed a type synthesis method applied to 2T2R, 1T2R, and 2R parallel mechanisms. Ye et al. [10] used the motion equivalent chain method to obtain a type synthesis of a class of 2R2T parallel mechanisms, and Araujo-Gómez et al. [11] designed a 3UPS-1RPU parallel manipulator with 2R2T DOF. Nurahmi et al. [12] studied the constraint-based design, optimization design, and reconstruction strategy of the 3-RPS parallel robotic arm. Chai et al. [13] proposed system dynamics modeling and analysis of a 2PRU-UPR parallel robot based on the screw theory.

By 2022, research on parallel mechanisms with fewer DOF has increased significantly. Zhang et al. [14] proposed a 3-DOF parallel mechanism based on an antenna support structure. Yan et al. [15] introduced a novel 5-DOF redundantly actuated 4PUS-PPPU parallel mechanism with a large tilting ability, including its design, analysis, and control. Song et al. [16] performed dynamic modeling and generalized force analysis of the 3-RPS parallel mechanism based on five-dimensional geometric algebra space, and Zhou et al. [17] proposed a new 5-DOF weakly coupled compliant parallel mechanism. Ye et al. [18] proposed a type synthesis of a 4-DOF non-overconstrained parallel mechanism with a symmetrical structure. Qin et al. [19] analyzed the workspace of the 3-RPS parallel mechanism based on the significant characteristics of the rotational branching mechanism. Song et al. [20] analyzed symmetric 3-UPU parallel mechanisms and their variant mechanisms under different geometric and assembly conditions. Shi et al. ;21] proposed a novel bionic 5-DOF parallel actuation mechanism by establishing a mapping relationship between local musculoskeletal systems of organisms and mechanism configurations, and Pan et al. [22] proposed and studied a new 3R parallel compliant mechanism.

In the aforementioned research on parallel mechanisms with fewer DOF, due to the variety of 2R2T parallel mechanisms, the literature related to the kinematic analysis and research of 2R2T parallel mechanisms is relatively limited. Analyzing and studying new types of 2R2T parallel mechanisms has significant practical engineering application implications. Therefore, this paper proposes a 2UPU-2SPU parallel mechanism, conducts a solution for the inverse kinematics of this mechanism, and analyzes the workspace. The analysis results show that this mechanism has great potential in fields like machine tools, agriculture, aerospace, and medical, where there is not a high demand for motion dimensions but a high requirement for mechanical performance.

To obtain the coordinates of the motion pair connected to the motion platform in the fixed coordinate system, use a transformation matrix to convert the coordinates from the moving coordinate system to the fixed coordinate system. Assuming the rotation angles around the X, Y and Z axes are $\alpha, \beta$ and $\gamma$. respectively, the transformation matrix is:

$\begin{gathered}{ }_o^{o_1} \boldsymbol{R}=\boldsymbol{R}_x \boldsymbol{R}_y \boldsymbol{R}_z= {\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & c \alpha & -s \alpha \\ 0 & s \alpha & c \alpha\end{array}\right]\left[\begin{array}{ccc}c \beta & 0 & s \beta \\ 0 & 1 & 0 \\ -s \beta & 0 & c \beta\end{array}\right]\left[\begin{array}{ccc}c \gamma & -s \gamma & 0 \\ s \gamma & c \gamma & 0 \\ 0 & 0 & 1\end{array}\right]}\end{gathered}$       (5)

where, c represents cos, s represents sin. Due to the restriction of rotation around the Z-axis in the 2UPU-2SPU parallel mechanism (lack of freedom to rotate around the Z-axis), so $\gamma$ = 0.

To perform inverse kinematic analysis based on the 2UPU2SPU parallel mechanism with 2R2T DOF, let's assume the moving platform's rotation angle $\alpha^{\circ}$ around the $\mathrm{X}$-axis, the rotation angle $\beta^{\circ}$ around the $\mathrm{Y}$-axis, and its center coordinates in the base coordinate system, are known. By calculating the coordinates of the U-joint on the moving platform in the base coordinate system, we can determine the distance between the two ends of each branch. Let $M_i$ represent the center of the Ujoint or S-joint on the fixed platform, $N_i$ represent the center of the U-joint on the moving platform, $O$ represent the center of the fixed platform, and $O_1$ represent the center of the moving platform. The length of the branch can be represented as $\left|M_i N_i\right|$ :

$\boldsymbol{M}_i \boldsymbol{N}_i=-\boldsymbol{O} \boldsymbol{M}_i+\boldsymbol{O} \boldsymbol{O}_1+\boldsymbol{O}_1 \boldsymbol{N}_i$        (6)

Given $M_i O$ and $O O_1$, we only need to calculate the vector representation of $O_1 N_i$ in the base coordinate system. Using the rotation matrix formula for rotation about a fixed axis, we get:

${ }_{o_1}^o \boldsymbol{R}=\boldsymbol{R}_x \boldsymbol{R}_y=\left[\begin{array}{ccc}c \beta & 0 & s \beta \\ 0 & c \alpha & -s \alpha \cdot c \beta \\ 0 & s \alpha & c \alpha \cdot c \beta\end{array}\right]$      (7)

Finally, we can get $O N_i$:

$\boldsymbol{O} \boldsymbol{N}_i={ }_{o_1}^O \boldsymbol{R} \boldsymbol{N}_i+\boldsymbol{O} \boldsymbol{O}_1=\left[\begin{array}{lll}x_i & y_i & z_i\end{array}\right]^{\mathrm{T}}$       (8)

Let $O M_i$ be $\left[\begin{array}{lll}X_i & Y_i & Z_i\end{array}\right]^T$, then the length $\left|M_i N_i\right|$ of the $\mathrm{P}-$ joint is:

$L_i=\sqrt{\left(x_i-X_i\right)^2+\left(y_i-Y_i\right)^2+\left(z_i-Z_i\right)^2}$      (9)

In summary, the inverse kinematics solution for the 2UPU-2SPU parallel mechanism is completed.

Given that the output force transmission index calculates the power coefficient between the driving force and the instantaneous movement of the moving platform, and the instantaneous movement of the moving platform can only be determined through calculation, an evaluation function is set to assess the output transmission performance of the mechanism in various poses, and PSO is utilized for optimization.

The formula for the flight velocity of each particle in a particle swarm is given by:

$\begin{aligned} v_i^{\prime}= & v_i+c_1 \times \operatorname{rand}() \times\left(\text { pbest }_i-x_i\right) +c_2 \times \operatorname{rand}() \times\left(\text { gbest }_i-x_i\right)\end{aligned}$         (15)

where, $v_i$ is the velocity of the $i$-th particle, to introduce randomness into the velocity calculation, use $\operatorname{rand}()$ generated uniformly between (0, 1), $x_i$ represents the current position of the particle, $c_1$ and $c_2$ represent the cognitive and social learning factors, respectively, pbest $_i$ represents the best position found by the particle during its flight process, and gbest $_i$ represents the best position found by any particle during the flight process of all particles.

The fitness calculation formula for the i-th particle can be expressed as:

$f_i=\frac{\iiint \tau d y d z d \alpha}{\iiint d y d z d \alpha}$        (16)

Then the evaluation function is defined as:

$f_i=\frac{\iiint \lambda_0 d y d z d \alpha}{2 \iiint d y d z d \alpha}+\frac{1-\sum_j^8\left(\left|X_{\text {limit }}^j / 2-X^j\right| / 8 X_{\text {limit }}^j\right)}{6}$         (17)

where, $y$ and $z$ are the displacements of the moving platform along the $\mathrm{Y}$ and $\mathrm{Z}$ axes, respectively; $\alpha$ is the rotation angle around the $\mathrm{X}$ axis; $\lambda_O$ is the minimum output transmission performance index of the mechanism in this pose; $X_{\text {limit }}^j$ represents the extreme values for the optimization object, and $X^j$ represents the current values of the optimization parameters. The second term of this polynomial function is used to assess the extent to which the installation positions of the joints are distant from the extreme positions, aiming to reduce the difficulty in the design of specific devices.

Table 1. Optimization parameters

From the analysis in Chapter 3, it is known that the coordinates of $M_i$ and $N_i$ should be solved as parameters for the PSO. Here, $M_2$ and $M_4$ are symmetrical about the YOZ plane, with the same $\mathrm{Y}$-axis coordinates and opposite $\mathrm{X}$-axis coordinates. Similarly, $N_2$ and $N_4$ are symmetrical about the $\mathrm{Y}_1 \mathrm{O}_1 \mathrm{Z}_1$ plane, with their $\mathrm{Y}$-axis and $\mathrm{X}$-axis coordinates following the same symmetry as $M_2$ and $M_4$. Ultimately, optimization is performed on eight parameters of the motion joints, with specific position parameters as shown in Table 1. Analyzing the principles of the PSO, an optimization program for the 2UPU-2SPU parallel mechanism was developed, and a flow chart of its running process is given in Figure 9, with the program stopping when the fitness variation is less than $10^4$ more than 20 times.

9.png

Figure 9. Flowchart of the PSO

The number of individuals is set to 30 , with 100 iterations, and the restrictions on the P-joint's displacement length $L$, the rotation angle range Theta for axes not parallel to the P-joint, and the range of the joint installation coordinate parameters $X_i$, $Y_i, x_i, y_i$ are:

$\left\{\begin{array}{l}750 \mathrm{~mm} \leq L \leq 1100 \mathrm{~mm} \\ \mid \text { Theta } \mid \leq 60^{\circ} \\ 0 \mathrm{~mm}<X_i, Y_i<300 \mathrm{~mm} \\ 0 \mathrm{~mm}<x_i, y_i<200 \mathrm{~mm}\end{array}\right.$         (18)

The relationship between the final convergence curve fitness and the number of iterations is shown in Figure 10. Analyzing the fitness value, the number of times the optimization fell into local optima was 5, achieving satisfactory optimization results.

10.png

Figure 10. Optimization convergence curve

Analyzing the parameters of the optimized individuals, rounding the results, and fine-tuning to determine the installation positions of the joints, as shown in Table 2.

The installation of the joints and the overall structure after optimization is shown in Figure 11.

Table 2. Final joint installation positions

11.png

Figure 11. Joint arrangement after optimization

12a.png

(a) Before optimization

12b.png

(b) After optimization

Figure 12. Comparison before and after optimization

Before optimization, the fitness of the installation points shown in Figure 8(a) was 0.2316. After optimization, the fitness improved to 0.598. Comparing the performance maps as shown in Figure 12, there was a significant increase in the minimum motion/force transmission performance after optimization.

The images and calculation results reveal the following points:

(1) The output transmission performance of the mechanism has significantly improved after optimization;

(2) At the same Z-axis height, the output transmission performance decreases with an increase in displacement along the Y-axis;

(3) At the same Y-axis position, the output transmission performance decreases as the height on the Z-axis increases.

These results indicate that the mechanism is more suitable for operating within the second quadrant of the Y-Z coordinate system and that the working height needs to be carefully considered.