Forward kinematics modeling and optimal design of multi-segment stacked hybrid mechanism

doi:10.3785/j.issn.1006-754X.2026.05.185

Chinese Journal of Engineering Design

2026, Vol. 33

Issue (2): 190-203 DOI: 10.3785/j.issn.1006-754X.2026.05.185

Robotic and Mechanism Design

Forward kinematics modeling and optimal design of multi-segment stacked hybrid mechanism

Yang QI(

),Yuanhang LOU(

)

School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin 300222, China

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Abstract

The multi-segment stacked hybrid mechanism consists of multiple parallel segments connected in series, combining the advantages of serial and parallel configurations. However, such mechanisms feature complex structures and exhibit distinct configurations with different numbers of segments, making it difficult to directly obtain unified kinematic position and velocity solutions valid for any number of segments, thereby hindering their design, analysis and optimization. To solve this problem, the multi-segment stacked hybrid mechanism 3-R₁S(RS) _N_-1R₂ is selected as the research object, and its forward kinematics modeling and optimal design are carried out based on the finite and instantaneous screw theory. Firstly, the configuration feature of the hybrid mechanism was described, which achieved multi-segment superimposition through a common motion platform and R joints. Meanwhile, the multi-segment motion superimposition and transmission principle of the mechanism was derived using the finite and instantaneous screw theory. Then, the forward kinematics model of a single segment was deduced. By further extension based on the principle of multi-segment motion superposition and transmission, the unified forward kinematics model and velocity Jacobian matrix valid for any number of segments were obtained, and the correctness of the position and velocity models was verified through simulation. Finally, the optimal number of segments was determined by defining standardized equal-weight performance comparison indicator. The multi-objective optimization was conducted using NSGA-Ⅱ (non-dominated sorting genetic algorithm-II), and the optimal solution was selected from the Pareto frontier by using the entropy-weighted TOPSIS (technique for order preference by similarity to an ideal solution) method. The comparison results before and after optimization indicated that the optimal solution improved significantly in four performance indicators, making the optimized hybrid mechanism more suitable for practical engineering applications. The research results provide new ideas for the analysis and optimization of multi-segment stacked hybrid mechanisms.

Key words： hybrid mechanism forward kinematics finite and instantaneous screw theory optimal design

Received: 03 September 2025 Published: 28 April 2026

CLC:

TH 112

Corresponding Authors: Yang QI E-mail: qiyang@tju.edu.cn;1402356217@qq.com

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Cite this article:

Yang QI,Yuanhang LOU. Forward kinematics modeling and optimal design of multi-segment stacked hybrid mechanism. Chinese Journal of Engineering Design, 2026, 33(2): 190-203.

URL:

https://www.zjujournals.com/gcsjxb/10.3785/j.issn.1006-754X.2026.05.185 OR https://www.zjujournals.com/gcsjxb/Y2026/V33/I2/190

多体节叠加型混联机构正运动学建模与优化设计

多体节叠加型混联机构由多个并联体节串联叠加构成，同时具备串联和并联两种构型的优点。但这类机构结构复杂，且在体节数不同时呈现出相异构型，难以直接获取任意体节数下均成立的运动学位置解和速度解，从而制约了其设计、分析和优化。为解决这一问题，以多体节叠加型混联机构3-R₁S(RS) _N_-1R₂为研究对象，基于有限与瞬时旋量理论对该机构进行了正运动学建模和优化设计。首先，描述了混联机构通过共用运动平台和R关节进行多体节叠加的构型特征，并利用有限与瞬时旋量理论得到了其多体节运动叠加传递原理。然后，推导了单体节的正运动学模型，基于多体节运动叠加传递原理进一步推广，得到了任意体节数下均成立的正运动学模型和速度雅可比矩阵，并通过仿真验证了位置和速度模型的正确性。最后，通过定义标准化等权重的性能对比指标选取了最优体节数，并利用NSGA-Ⅱ（non-dominated sorting genetic algorithm-II，非支配排序遗传算法-Ⅱ）开展多目标优化，通过熵权-逼近理想解排序法从Pareto前沿中选取了最优方案。优化前后的对比结果表明，最优方案在4项性能指标上均得到较大改善，优化后的混联机构更加符合实际工程应用的需求。研究结果为多体节叠加型混联机构的分析和优化提供了新思路。

关键词： 混联机构, 正运动学, 有限与瞬时旋量理论, 优化设计

Fig.1 Schematic diagram of 3-R₁S(RS) _N_-1R₂hybrid mechanism

Fig.2 3-R₁S(RS) _N_-1R₂hybrid mechanism prototype (N=3)

Fig.3 Schematic diagram of single-segment 3-RSR mechanism

Fig.4 Comparison of theoretical and simulated values of position solution of hybrid mechanism

Fig.5 Comparison of theoretical and simulated values of velocity solution of hybrid mechanism

Fig.6 Schematic diagram of minimum spherical joint distance

Fig.7 Distribution of performance comparison indicator K_u under different numbers of segments

性能指标	阶数	RAAE	RMAE	RMSE^①	R²
$β m a x$	1	0.029 78	0.081 56	0.034 75	0.981 85
	2	0.009 97	0.021 86	0.011 62	0.998 55
	3	0.008 63	0.016 62	0.009 93	0.999 18
	4	0.005 47	0.013 79	0.006 47	0.999 51
d $S, ? m i n G$	1	0.027 38	0.082 29	0.037 39	0.983 39
	2	0.001 15	0.002 96	0.001 41	0.999 97
	3	0.001 48	0.004 49	0.001 89	0.999 97
	4	0.001 22	0.005 33	0.001 84	0.999 94
η_Q	1	0.262 13	0.581 35	0.299 39	0.453 32
	2	0.047 49	0.163 75	0.058 92	0.952 49
	3	0.021 48	0.036 05	0.023 35	0.995 16
	4	0.007 82	0.025 23	0.010 59	0.999 14
$η F$	1	0.038 09	0.112 03	0.049 11	0.975 08
	2	0.001 72	0.005 13	0.002 11	0.999 95
	3	0.001 13	0.005 18	0.001 78	0.999 95
	4	0.000 60	0.001 56	0.000 74	0.999 99

Table 1 Accuracy evaluation results of response surface models

Fig.8 Distribution cloud maps of performance indicators of hybrid mechanism

Table 2 Parameter setting of NSGA-Ⅱ

Fig.9 Pareto frontier for multi-objective optimization of hybrid mechanism

性能指标	优化前	优化后		相对提升率^①/%
性能指标	优化前	未取整	取整后	相对提升率^①/%
$β m a x$ /(°)	34.33	45.70	46.08	34.23
d $S, m i n G$ /mm	97.73	116.22	114.08	16.73
η_Q	0.557 9	0.620 3	0.614 4	10.13
$η F$	136.41	175.53	173.66	27.31

Table 3 Comparison of performance indicators of hybrid mechanism before and after optimization

Fig.10 Comparison of end posture angle of hybrid mechanism before and after optimization

Fig.11 Comparison of minimum spherical joint distance of hybrid mechanism before and after optimization

Fig.12 Comparison of virtual power transmissibility of hybrid mechanism before and after optimization

Fig.13 Comparison of average load-bearing capacity of hybrid mechanism before and after optimization

Fig.14 Physical diagram of optimized hybrid mechanism prototype


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