Kinematic characteristics and dynamics analysis of 3-P(2-SS) robot with coplanar parallel driving pairs

doi:10.3785/j.issn.1006-754X.2025.05.177

Chinese Journal of Engineering Design

2025, Vol. 32

Issue (5): 646-654 DOI: 10.3785/j.issn.1006-754X.2025.05.177

Robotic and Mechanism Design

Kinematic characteristics and dynamics analysis of 3-P(2-SS) robot with coplanar parallel driving pairs

Jin LI(

)

Department of Mechanical Engineering, Shanxi Institute of Technology, Yangquan 045000

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Abstract

The 3-P(2-SS) parallel robot driven by three parallel and coplanar prismatic pairs is more suitable for operation in narrow and long areas compared to the traditional configurations. For this robot, its kinematic equations and Jacobian matrix were established, and the multi-solution nature of the kinematic equations was analyzed. By using the kinematic equations and the screw theory, it was confirmed that under normal working conditions, the robot's moving platform always had only three translational degrees of freedom, thereby simplifying the robot's kinematic equations and Jacobian matrix. The dexterity of the robot was analyzed using the condition number of the Jacobian matrix. The robot exhibited better dexterity when the sliders on both sides were located on the same side of the moving platform; the robot's dexterity deteriorated when it was near the singular points and the boundaries of the motion range. A robot dynamic model was established based on the Lagrange equation. The issue of dynamic uncertainty caused by the local degrees of freedom of the link was resolved using the minimum angular velocity assumption, the problem of the non-constant inertia tensor of the ink relative to the static coordinate system was addressed through transformation of the moving coordinate system, and the robot dynamic equation was obtained. By comparing the results of the dynamics calculation and motion simulation, the correctness of the dynamic equation was confirmed, with the error in the driving force calculation results caused by the minimum angular velocity assumption being within an acceptable range. The research results indicate that the 3-P(2-SS) parallel robot with coplanar parallel driving pairs has excellent operability, and the proposed dynamics modeling method provides a theoretical foundation for the construction of control systems for this type of robot.

Key words： 3-P(2-SS) parallel robot degree of freedom dexterity dynamics

Received: 20 March 2025 Published: 31 October 2025

CLC:

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Cite this article:

Jin LI. Kinematic characteristics and dynamics analysis of 3-P(2-SS) robot with coplanar parallel driving pairs. Chinese Journal of Engineering Design, 2025, 32(5): 646-654.

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https://www.zjujournals.com/gcsjxb/10.3785/j.issn.1006-754X.2025.05.177 OR https://www.zjujournals.com/gcsjxb/Y2025/V32/I5/646

驱动副共面平行的3-P(2-SS)型并联机器人运动特性及动力学分析

由3个平行且共面的移动副驱动的3-P(2-SS)型并联机器人相对传统构型更适合在狭长区域作业。针对该机器人，构建了其运动学方程和雅可比矩阵，并分析了其运动学方程的多解性。利用其运动学方程和螺旋理论，证实了在正常工况下机器人动平台始终只有3个平移自由度，并简化了机器人的运动学方程和雅可比矩阵。利用雅可比矩阵条件数分析了机器人的灵巧性，当2侧的滑块位于动平台同侧时，机器人的灵巧性较好；机器人在奇异点和运动范围边界附近时灵巧性较差。基于拉格朗日方程构建了机器人动力学模型，利用最小角速度假设解决了连杆的局部自由度带来的动力学不确定问题，通过动坐标系变换解决了连杆惯性张量相对静坐标系的不恒定问题，得到了机器人动力学方程。通过动力学计算结果与运动仿真结果的对比，证实了动力学方程的正确性，最小角速度假设导致的驱动力计算结果误差在可控范围内。研究结果表明，驱动副共面平行的3-P(2-SS)并联机器人具有良好的操作性，提出的动力学建模方法为该型机器人力控制系统的构建提供了理论基础。

关键词： 3-P(2-SS), 并联机器人, 自由度, 灵巧性, 动力学

Fig.1 Motion mechanism schematic of 3-P(2-SS) robot with coplanar parallel driving pairs

参数	含义
$L 1$	连杆 $A 1 A 2$ 、 $A 3 A 4$ 的长度
$L 2$	连杆 $B 1 B 2$ 、 $B 3 B 4$ 的长度
$L 3$	连杆 $C 1 C 2$ 、 $C 3 C 4$ 的长度
$M$	$B 5 O D$ 、 $A 5 O D$ 的长度
$N$	$C 5 O D$ 的长度
$E$	相邻移动副滑轨的间距
$α$	$A 1 A 3$ 、 $B 1 B 3$ 与X₀轴之间的夹角 $A 2 A 4$ 、 $B 2 B 4$ 与X_D轴之间的夹角
$T$	$A 0 A 1$ 、 $A 0 A 3$ 、 $A 5 A 2$ 、 $A 5 A 4$ 、 $B 0 B 1$ 、 $B 0 B 3$ 、 $B 5 B 2$ 、 $B 5 B 4$ 、 $C 0 C 1$ 、 $C 0 C 3$ 、 $C 5 C 2$ 、 $C 5 C 4$ 的长度

Table 1 Structural parameters of 3-P(2-SS) robot with coplanar parallel driving pairs

Fig.2 Different installation poses of moving platform

Fig.3 Condition number of Jacobi matrix with different kinematic inverse solutions

参数	含义
$g$	重力加速度
$m D$	动平台质量
$m S 1$	滑块A质量
$m S 2$	滑块B质量
$m S 3$	滑块C质量
$m l 1$	连杆 $A 1 A 2$ 、 $A 3 A 4$ 的质量
$m l 2$	连杆 $B 1 B 2$ 、 $B 3 B 4$ 的质量
$m l 3$	连杆 $C 1 C 2$ 、 $C 3 C 4$ 的质量
$v l 1$	连杆 $A 1 A 2$ 、 $A 3 A 4$ 沿 $O 0$ 各轴向的平动速度
$v l 2$	连杆 $B 1 B 2$ 、 $B 3 B 4$ 沿 $O 0$ 各轴向的平动速度
$v l 3$	连杆 $C 1 C 2$ 、 $C 3 C 4$ 沿 $O 0$ 各轴向的平动速度
$A l 1$	连杆 $A 1 A 2$ 、 $A 3 A 4$ 相对 $O A$ 的惯性张量
$B l 2$	连杆 $B 1 B 2$ 、 $B 3 B 4$ 相对 $O B$ 的惯性张量
$C l 3$	连杆 $C 1 C 2$ 、 $C 3 C 4$ 相对 $O C$ 的惯性张量
$A l 1$	连杆 $A 1 A 2$ 、 $A 3 A 4$ 相对 $O A$ 的转动角速度
$B l 2$	连杆 $B 1 B 2$ 、 $B 3 B 4$ 相对 $O B$ 的转动角速度
$C l 3$	连杆 $C 1 C 2$ 、 $C 3 C 4$ 相对 $O C$ 的转动角速度

Table 2 Dynamics parameters for 3-P(2-SS) robot with coplanar parallel driving pairs

Fig.4 Coordinate system

{O A}

attached to link

A 1 A 2

Fig.5 Comparison of dynamics calculation results with simulation results of mobile auxiliary drive force of 3-P(2-SS) robot with coplanar parallel driving pairs


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