Dynamics Modeling and Base Dynamics Parameter Determination of 6-UPRU Parallel Manipulators

doi:10.3969/j.issn.1004-132X.2025.12.013

Abstract

Abstract:

To address the challenges in applying the dynamics model of parallel manipulators， the 6-UPRU parallel manipulators were focused on to investigate the precise dynamics modeling and base dynamics parameter determination. Firstly， a detailed kinematics analysis of the manipulator's moving components was conducted. Subsequently， the precise dynamics model was derived using the Newton-Euler method， and through the proposed model linearization rules， which was transformed into a linear form with respect to the dynamics parameters. Then， a fifth-order Fourier series-based exciting trajectory satisfying physical constraints was designed， with the goal of minimizing the condition number of the observation matrix. Furthermore， by performing QR decomposition on the observation matrix， the base dynamics parameters were successfully extracted， reducing the number of dynamics parameters from 29 to 17， effectively addressing the issue of parameter redundancy. Finally， the correctness of the theoretical model was validated through force-fitting experiments conducted on the SimMechanics and ADAMS simulation platforms and the prototype's base dynamics parameter identification experiments.

Key words: parallel manipulator, dynamics modeling, base dynamics parameter, Newton-Euler method

摘要：

为克服并联机器人动力学模型应用难题，以6-UPRU型并联机器人作为研究对象，对其精确动力学建模及基本动力学参数确定问题进行了研究。首先对机器人的各运动构件进行详细的运动学分析；然后基于牛顿-欧拉法推导机器人的精确动力学模型，并结合提出的模型线性化运算规则，将其转化为关于动力学参数的线性表达形式；随后，以最小化观测矩阵条件数为优化目标，设计了一条满足物理约束的五阶傅里叶级数形式的激励轨迹；进一步，通过对观测矩阵进行QR分解，成功提取了基本动力学参数，将动力学参数从29个减少至17个，有效解决了参数冗余问题；最后，通过SimMechanics和ADAMS仿真平台下的力拟合实验以及样机基本动力学参数辨识实验，验证了理论模型的正确性。

关键词: 并联机器人, 动力学建模, 基本动力学参数, 牛顿-欧拉法

CLC Number:

TP242

Tao NI, Yahui ZHAO, Zeren ZHAO, Kaiqiang YANG. Dynamics Modeling and Base Dynamics Parameter Determination of 6-UPRU Parallel Manipulators[J]. China Mechanical Engineering, 2025, 36(12): 2911-2919.

倪涛, 赵亚辉, 赵泽仁, 杨凯强. 6-UPRU并联机器人动力学建模及基本动力学参数确定[J]. 中国机械工程, 2025, 36(12): 2911-2919.

Figures/Tables 16

Fig.1 The CAD model of the 6-UPRU parallel manipulator

Fig. 2 The structure and parameters of the UPRU branch

Fig.3 The force analysis of the link Li1

Fig.4 The force analysis of the entire branch

Fig.5 The force analysis of moving platform

Tab.1 Dynamic parameters

编号	动力学参数	编号	动力学参数
1	$I a 1$	10	$m 3$
2	$I n 1$	11	$m 3 l c 3$
3	$I a 2 + I a 3 + I a 4$	12	$m 4 l c 4$
4	$I n 2 + I n 3 + I n 4$	13	$m 2 l c 2 2 + m 3 l c 3 2 + m 4 l c 4 2$
5	$m 1$	14	$m p$
6	$m 1 l c 1$	15	$m p r c {p}$
7	$m 1 l c 1 2$	16	$I c {p} ¯$
8	$m 2$	17	$m p r c {p} (r c {p}) T ¯$
9	$m 2 l c 2$

Tab.1 Dynamic parameters

编号	动力学参数	编号	动力学参数
1	$I a 1$	10	$m 3$
2	$I n 1$	11	$m 3 l c 3$
3	$I a 2 + I a 3 + I a 4$	12	$m 4 l c 4$
4	$I n 2 + I n 3 + I n 4$	13	$m 2 l c 2 2 + m 3 l c 3 2 + m 4 l c 4 2$
5	$m 1$	14	$m p$
6	$m 1 l c 1$	15	$m p r c {p}$
7	$m 1 l c 1 2$	16	$I c {p} ¯$
8	$m 2$	17	$m p r c {p} (r c {p}) T ¯$
9	$m 2 l c 2$

Tab.2 The expression corresponding to the symbol ψik

符号	表达式
$ψ i 1$	$- [(ω p i ∙ n i) n i × ω p i × n i] / l i - [(ε p i · n i) r p c i × n i] /$ $(r p c i · l i n i)$
$ψ i 2$	$[ε p i × n i + (ω p i · n i) n i × ω p i × n i] / l i$
$ψ i 3$	$- [(ω b i · n i) n i × ω b i × n i] / l i - [(ε b i · n i) r b c i × n i] /$ $(r b c i · l i n i)$
$ψ i 4$	$[ε b i × n i + (ω b i · n i) n i × ω b i × n i] / l i$
$ψ i 5$	$a p i - g$
$ψ i 6$	$[n i × (a p i - g) × n i] / l i - b p i$
$ψ i 7$	$(n i × b p i × n i) / l i$
$ψ i 8$	$a b i - g$
$ψ i 9$	$- [n i × (a b i - g) × n i] / l i - b b i$
$ψ i 10$	$n i × (a b i - g) × n i$
$ψ i 11$	$- [n i × (a b i - g) × n i] / l i - n i × b b i × n i$
$ψ i 12$	$- (n i × g × n i) / l i$
$ψ i 13$	$(n i × b b i × n i) / l i$

Tab.2 The expression corresponding to the symbol ψik

符号	表达式
$ψ i 1$	$- [(ω p i ∙ n i) n i × ω p i × n i] / l i - [(ε p i · n i) r p c i × n i] /$ $(r p c i · l i n i)$
$ψ i 2$	$[ε p i × n i + (ω p i · n i) n i × ω p i × n i] / l i$
$ψ i 3$	$- [(ω b i · n i) n i × ω b i × n i] / l i - [(ε b i · n i) r b c i × n i] /$ $(r b c i · l i n i)$
$ψ i 4$	$[ε b i × n i + (ω b i · n i) n i × ω b i × n i] / l i$
$ψ i 5$	$a p i - g$
$ψ i 6$	$[n i × (a p i - g) × n i] / l i - b p i$
$ψ i 7$	$(n i × b p i × n i) / l i$
$ψ i 8$	$a b i - g$
$ψ i 9$	$- [n i × (a b i - g) × n i] / l i - b b i$
$ψ i 10$	$n i × (a b i - g) × n i$
$ψ i 11$	$- [n i × (a b i - g) × n i] / l i - n i × b b i × n i$
$ψ i 12$	$- (n i × g × n i) / l i$
$ψ i 13$	$(n i × b b i × n i) / l i$

Fig.6 6-UPRU parallel manipulator SimMechanics simulation model

Fig.7 6-UPRU parallel manipulator SimMechanics model logic blocks

Fig.8 The manipulator excitation trajectory increment

Tab.3 Base dynamic parameters

编号	基本动力学参数	编号	基本动力学参数
1	$I a 1$	10	$m p r c y {p}$
2	$I a 2 + I a 3 + I a 4$	11	$m p r c z {p}$
3	$∑ i = 1 4 (I n i + m i l c i 2)$	12	$I c 11 {p} + m p [(r c y {p}) 2 + (r c z {p}) 2 - r p 2 / 2]$
4	$m 1 + m 2 + m p / 6$	13	$I c 12 {p} - m p r c x {p} r c y {p}$
5	$m 1 l c 1 + m 2 l c 2$	14	$I c 13 {p} - m p r c x {p} r c z {p}$
6	$m 3$	15	$I c 22 {p} + m p [(r c x {p}) 2 + (r c z {p}) 2 - r p 2 / 2]$
7	$m 3 l c 3$	16	$I c 23 {p} - m p r c y {p} r c z {p}$
8	$m 4 l c 4$	17	$I c 33 {p} + m p [(r c x {p}) 2 + (r c y {p}) 2 - r p 2 / 2]$
9	$m p r c x {p}$

Tab.3 Base dynamic parameters

编号	基本动力学参数	编号	基本动力学参数
1	$I a 1$	10	$m p r c y {p}$
2	$I a 2 + I a 3 + I a 4$	11	$m p r c z {p}$
3	$∑ i = 1 4 (I n i + m i l c i 2)$	12	$I c 11 {p} + m p [(r c y {p}) 2 + (r c z {p}) 2 - r p 2 / 2]$
4	$m 1 + m 2 + m p / 6$	13	$I c 12 {p} - m p r c x {p} r c y {p}$
5	$m 1 l c 1 + m 2 l c 2$	14	$I c 13 {p} - m p r c x {p} r c z {p}$
6	$m 3$	15	$I c 22 {p} + m p [(r c x {p}) 2 + (r c z {p}) 2 - r p 2 / 2]$
7	$m 3 l c 3$	16	$I c 23 {p} - m p r c y {p} r c z {p}$
8	$m 4 l c 4$	17	$I c 33 {p} + m p [(r c x {p}) 2 + (r c y {p}) 2 - r p 2 / 2]$
9	$m p r c x {p}$

Fig.9 Comparison between measured values and theoretical analytical values in the SimMechanics model

Fig.10 6-UPRU parallel manipulator ADAMS simulation model

Fig.11 Comparison between measured values and theoretical analytical values in the ADAMS model

Fig.12 6-UPRU parallel manipulator prototype

Fig.13 Comparison between force sensor measurement and fitted values

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