改进多目标差分算法应用于机械臂多目标轨迹规划研究

doi:10.16182/j.issn1004731x.joss.25-0644

摘要/Abstract

摘要：

为解决单一目标的轨迹规划方法已难以满足机械臂精确性、多样性和复杂性的要求，提出一种基于改进多目标差分进化算法（guided multi-objective differential evolution， GMODE）的轨迹规划模型。利用三次多项式插值与B样条曲线构建多目标函数，并通过GMODE克服传统算法在种群多样性不足、易陷入局部最优和收敛缓慢等问题。引入分组策略、参数生成机制、基于模糊C均值聚类的精英变异优化B样条控制节点。仿真实验结果表明：该方法在时间、能量与冲击方面均取得优异表现；在结合自适应权重后，多目标性能得到明显提升。

关键词: 轨迹规划, 多目标优化, 机械臂, 多目标差分进化算法, 自适应权重

Abstract:

It is difficult for single-objective trajectory planning methods to meet the requirements of precision, diversity and complexity of robotic arms. A trajectory planning model based on an improved multi-objective differential evolution algorithm (guided multi-objective differential evolution, GMODE) algorithm is proposed. Cubic polynomial interpolation and B-spline curves are employed to construct multi-objective functions, while GMODE is adopted to overcome the limitations of traditional algorithms, such as insufficient population diversity, the tendency to fall into local optima, and slow convergence. A grouping strategy, parameter generation mechanism, and elite mutation based on fuzzy C-means clustering are introduced to optimize B-spline control nodes. Simulation experiments demonstrate that the proposed method achieves excellent performance in terms of time, energy, and impact. Furthermore, the incorporation of adaptive weights significantly enhances multi-objective performance.

Key words: trajectory planning, multi-objective optimization, robotic arm, multi-objective differential evolution algorithm, adaptive weights

中图分类号:

TP242.2

刘满强,尚自强 . 改进多目标差分算法应用于机械臂多目标轨迹规划研究[J]. 系统仿真学报, 2026, 38(6): 1647-1668.

Liu Manqiang,Shang Ziqiang . Application of Improved Multi-objective Differential Algorithm in Robotic Arm Multi-objective Trajectory Planning[J]. Journal of System Simulation, 2026, 38(6): 1647-1668.

图/表 26

图1

图2

图3

图4

图5

表1

各算法的IGD性能指标对比

问题	指标	MODE	CMOPSO	IMMOEAD	MOPSO	MPSOD	NMPSO	NSGA-II	SMPSO	GMODE
DTLZ1	Mean	5.02E-01(-)	1.44E+01(-)	1.69E+00(-)	9.80E+00(-)	1.91E+00(-)	3.35E+00(-)	2.22E-01(-)	2.35E+00(-)	1.07E-01
DTLZ1	Std.	2.97E-01	4.44E+00	6.65E-01	3.56E+00	6.53E-01	1.64E+00	2.49E-01	3.33E+00	1.61E-01
DTLZ2	Mean	6.90E-02(-)	5.79E-02(-)	7.54E-02(-)	7.61E-02(-)	5.51E-02(-)	7.65E-02(-)	6.98E-02(-)	9.00E-02(-)	5.45E-02
DTLZ2	Std.	1.56E-03	8.13E-04	7.06E-04	5.11E-03	1.49E-04	2.89E-03	2.59E-03	9.41E-03	1.20E-03
DTLZ3	Mean	6.20E+01(-)	1.42E+02(-)	3.04E+01(-)	2.06E+02(-)	7.02E+01(-)	9.56E+01(-)	7.41E+00(-)	3.75E+01(-)	1.24E+00
DTLZ3	Std.	1.33E+01	3.43E+01	7.17E+00	5.65E+01	1.01E+01	2.24E+01	4.12E+00	4.24E+01	1.22E+00
DTLZ4	Mean	7.36E-02(-)	9.04E-02(-)	7.83E-02(-)	1.36E-01(-)	5.53E-02(-)	7.77E-02(-)	1.16E-01(-)	4.46E-01(-)	5.37E-02
DTLZ4	Std.	2.88E-03	1.62E-01	4.04E-03	2.20E-01	3.42E-04	2.44E-03	1.44E-01	2.39E-01	6.63E-04
DTLZ5	Mean	1.31E-02(-)	7.29E-03(-)	2.73E-02(-)	8.30E-03(-)	3.83E-02(-)	1.22E-02(-)	6.01E-03(-)	5.91E-03(-)	4.17E-03
DTLZ5	Std.	1.38E-03	5.99E-04	4.60E-04	1.14E-03	2.45E-03	1.62E-03	3.75E-04	5.02E-04	3.84E-05
DTLZ6	Mean	4.63E-03(-)	1.70E-01(-)	3.52E+00(-)	4.87E+00(-)	3.21E-02(-)	1.40E-02(-)	5.93E-03(-)	1.17E+00(-)	4.15E-03
DTLZ6	Std.	5.66E-05	4.36E-01	1.55E-01	5.39E-01	1.27E-03	2.67E-03	2.67E-04	8.72E-01	3.74E-05
DTLZ7	Mean	7.42E-02(-)	1.42E-01(-)	2.00E-01(-)	1.90E+00(-)	3.04E-01(-)	6.89E-02(-)	1.04E-01(-)	6.63E-01(-)	5.83E-02
DTLZ7	Std.	2.64E-03	1.65E-01	6.17E-03	5.34E-01	1.16E-01	3.87E-03	5.15E-02	3.94E-01	9.79E-04
WFG1	Mean	1.05E+00(-)	1.62E+00(-)	1.08E+00(-)	1.77E+00(-)	1.59E+00(-)	1.05E+00(-)	5.75E-01(+)	1.55E+00(-)	8.81E-01
WFG1	Std.	8.57E-02	3.57E-02	6.29E-02	1.28E-01	8.25E-02	6.88E-02	6.99E-02	4.02E-02	6.20E-02
WFG2	Mean	1.92E-01(-)	1.85E-01(=)	3.26E-01(-)	2.21E-01(-)	2.13E-01(-)	3.80E-01(-)	2.25E-01(-)	2.74E-01(-)	1.83E-01
WFG2	Std.	4.53E-03	6.00E-03	1.16E-02	1.40E-02	9.44E-03	2.56E-02	1.34E-02	2.09E-02	3.78E-03
WFG3	Mean	2.35E-01(-)	1.69E-01(=)	2.26E-01(-)	3.10E-01(-)	2.85E-01(-)	8.46E-02(+)	1.30E-01(+)	2.05E-01(-)	1.63E-01
WFG3	Std.	1.90E-02	1.80E-02	1.34E-02	8.61E-02	2.60E-02	6.96E-03	2.56E-02	3.53E-02	1.07E-02
WFG4	Mean	2.95E-01(-)	2.67E-01(-)	3.57E-01(-)	4.06E-01(-)	2.73E-01(-)	2.96E-01(-)	2.81E-01(-)	3.28E-01(-)	2.32E-01
WFG4	Std.	6.50E-03	5.61E-03	7.44E-03	4.37E-02	5.50E-03	1.71E-02	9.29E-03	1.43E-02	2.86E-03
WFG5	Mean	2.48E-01(-)	2.48E-01(-)	3.54E-01(-)	6.95E-01(-)	2.55E-01(-)	2.93E-01(-)	2.85E-01(-)	3.22E-01(-)	2.31E-01
WFG5	Std.	6.51E-03	6.26E-03	7.91E-03	5.18E-02	1.20E-03	1.12E-02	1.00E-02	2.19E-02	2.36E-03
WFG6	Mean	2.82E-01(-)	2.56E-01(-)	3.67E-01(-)	3.51E-01(-)	2.82E-01(-)	4.07E-01(-)	3.27E-01(-)	3.80E-01(-)	2.39E-01
WFG6	Std.	2.25E-02	1.19E-02	7.42E-03	3.57E-02	1.13E-02	1.12E-02	1.59E-02	2.44E-02	3.83E-03
WFG7	Mean	2.72E-01(-)	2.37E-01(+)	3.61E-01(-)	3.85E-01(-)	2.66E-01(-)	3.10E-01(-)	2.82E-01(-)	3.62E-01(-)	2.52E-01
WFG7	Std.	6.52E-03	5.14E-03	5.46E-03	4.11E-02	4.47E-03	9.96E-03	9.71E-03	2.57E-02	5.28E-03
WFG8	Mean	3.76E-01(-)	3.64E-01(-)	3.95E-01(-)	6.05E-01(-)	4.04E-01(-)	3.64E-01(-)	3.78E-01(-)	5.03E-01(-)	3.04E-01
WFG8	Std.	8.97E-03	1.29E-02	7.00E-03	3.11E-02	1.41E-02	1.25E-02	9.37E-03	3.56E-02	4.56E-03
WFG9	Mean	3.51E-01(-)	2.26E-01(+)	3.43E-01(-)	3.26E-01(-)	2.68E-01(-)	3.28E-01(-)	2.84E-01(-)	2.97E-01(-)	2.44E-01
WFG9	Std.	3.61E-02	5.29E-03	7.09E-03	3.67E-02	1.23E-02	5.94E-02	1.30E-02	1.80E-02	5.12E-03
ZDT1	Mean	6.13E-03(-)	5.11E-03(-)	1.49E-02(-)	5.40E-01(-)	4.42E-03(-)	3.11E-02(-)	1.29E-02(-)	8.66E-02(-)	3.79E-03
ZDT1	Std.	5.78E-04	4.21E-04	6.84E-04	1.05E-01	6.79E-04	2.06E-02	1.99E-03	1.02E-01	3.37E-05
ZDT2	Mean	6.63E-03(-)	4.62E-03(-)	1.87E-02(-)	1.44E+00(-)	3.93E-03(-)	1.87E-02(-)	2.46E-02(-)	2.27E-02(-)	3.83E-03
ZDT2	Std.	6.32E-04	3.95E-04	9.67E-04	2.59E-01	1.94E-04	4.15E-03	3.40E-02	6.54E-02	2.85E-05
ZDT3	Mean	9.90E-03(-)	5.47E-03(-)	1.32E-02(-)	4.53E-01(-)	1.07E-02(-)	1.01E-01(-)	1.18E-02(-)	1.53E-01(-)	4.50E-03
ZDT3	Std.	1.33E-03	6.02E-04	4.20E-04	7.81E-02	4.66E-04	8.19E-04	6.04E-03	9.00E-02	4.67E-05
ZDT4	Mean	1.52E+01(-)	5.99E+00(-)	9.19E-03(=)	1.03E+01(-)	2.37E+01(-)	1.30E+01(-)	2.70E-01(-)	8.76E+00(-)	9.46E-02
ZDT4	Std.	3.81E+00	3.40E+00	1.02E-02	4.49E+00	4.37E+00	1.23E+01	2.20E-01	5.66E+00	9.54E-02
ZDT6	Mean	3.09E-03(-)	3.24E-03(-)	1.13E+00(-)	2.14E-01(-)	3.09E-03(-)	4.45E-03(-)	5.31E-02(-)	3.86E-03(-)	3.05E-03
ZDT6	Std.	1.74E-05	1.32E-04	5.30E-02	2.99E-01	9.11E-06	5.54E-04	2.80E-02	1.77E-04	1.47E-05
+/-/=		0/21/0	2/17/2	0/20/1	0/21/0	0/21/0	1/20/0	2/19/0	0/21/0
Best/All		0/21	2/21	1/21	0/21	0/21	1/21	1/21	0/21	16/21

表1

表2

各算法HV性能指标对比

问题	指标	MODE	CMOPSO	IMMOEAD	MOPSO	MPSOD	NMPSO	NSGA-II	SMPSO	GMODE
DTLZ1	Mean	1.69E-01(-)	0(-)	0(-)	0(-)	0(-)	1.76E-06(-)	4.82E-01(-)	1.57E-01(-)	6.60E-01
DTLZ1	Std.	3.25E-01	0	0	0	0	9.62E-06	3.69E-01	2.23E-01	3.08E-01
DTLZ2	Mean	5.18E-01(-)	5.41E-01(-)	5.25E-01(-)	5.11E-01(-)	5.56E-01(=)	5.60E-01(+)	5.29E-01(-)	4.62E-01(-)	5.56E-01
DTLZ2	Std.	3.98E-03	2.47E-03	2.52E-03	1.10E-02	4.78E-04	1.03E-03	3.86E-03	2.17E-02	4.34E-03
DTLZ3	Mean	0(-)	0(-)	0(-)	0(-)	0(-)	0(-)	0(-)	1.34E-02(-)	1.80E-01
DTLZ3	Std.	0	0	0	0	0	0	0	3.74E-02	2.47E-01
DTLZ4	Mean	5.12E-01(-)	5.20E-01(-)	5.25E-01(-)	4.62E-01(-)	5.55E-01(-)	5.60E-01(=)	5.14E-01(-)	3.15E-01(-)	5.59E-01
DTLZ4	Std.	6.29E-03	8.11E-02	3.52E-03	1.04E-01	1.18E-03	1.03E-03	5.71E-02	1.20E-01	2.53E-03
DTLZ5	Mean	1.91E-01(-)	1.97E-01(-)	1.84E-01(-)	1.96E-01(-)	1.70E-01(-)	1.96E-01(-)	1.99E-01(-)	1.98E-01(-)	2.00E-01
DTLZ5	Std.	1.26E-03	5.07E-04	4.21E-04	2.21E-03	3.15E-03	6.38E-04	1.92E-04	4.96E-04	4.36E-05
DTLZ6	Mean	2.00E-01(-)	1.69E-01(-)	0(-)	0(-)	1.83E-01(-)	1.96E-01(-)	1.99E-01(-)	4.77E-02(-)	2.00E-01
DTLZ6	Std.	4.24E-05	7.25E-02	0	0	8.88E-04	1.04E-03	1.48E-04	8.54E-02	3.75E-05
DTLZ7	Mean	2.63E-01(-)	2.57E-01(-)	2.21E-01(-)	6.97E-03(-)	1.53E-01(-)	2.75E-01(-)	2.46E-01(-)	1.28E-01(-)	2.79E-01
DTLZ7	Std.	2.55E-03	1.56E-02	4.09E-03	1.02E-02	4.79E-02	1.18E-03	7.59E-03	7.03E-02	5.13E-04
WFG1	Mean	4.72E-01(-)	2.26E-01(-)	3.74E-01(-)	1.72E-01(-)	2.20E-01(-)	4.37E-01(-)	6.93E-01(=)	2.59E-01(-)	7.08E-01
WFG1	Std.	4.94E-02	1.62E-02	3.73E-02	4.07E-02	3.39E-02	3.34E-02	4.05E-02	1.34E-02	4.34E-02
WFG2	Mean	9.06E-01(-)	9.18E-01(-)	8.64E-01(-)	8.84E-01(-)	8.81E-01(-)	8.65E-01(-)	9.04E-01(-)	8.41E-01(-)	9.30E-01
WFG2	Std.	2.49E-03	2.83E-03	7.23E-03	1.07E-02	8.17E-03	1.33E-02	5.01E-03	1.54E-02	1.08E-03
WFG3	Mean	3.10E-01(-)	3.45E-01(=)	3.13E-01(-)	2.66E-01(-)	2.62E-01(-)	3.79E-01(+)	3.75E-01+	3.34E-01(-)	3.48E-01
WFG3	Std.	8.74E-03	9.32E-03	6.12E-03	3.10E-02	1.55E-02	4.23E-03	1.00E-02	1.73E-02	5.99E-03
WFG4	Mean	4.68E-01(-)	4.84E-01(-)	4.87E-01(-)	4.27E-01(-)	4.92E-01(-)	5.41E-01(+)	5.03E-01(-)	4.52E-01(-)	5.37E-01
WFG4	Std.	3.77E-03	3.64E-03	2.81E-03	1.41E-02	4.99E-03	2.37E-03	5.89E-03	9.84E-03	3.10E-03
WFG5	Mean	4.88E-01(-)	4.81E-01(-)	4.59E-01(-)	2.58E-01(-)	4.82E-01(-)	5.12E-01(+)	4.82E-01(-)	4.40E-01(-)	5.08E-01
WFG5	Std.	5.72E-03	4.92E-03	5.35E-03	1.75E-02	2.09E-03	3.57E-03	5.30E-03	1.51E-02	2.47E-03
WFG6	Mean	4.79E-01(-)	4.98E-01(-)	4.64E-01(-)	4.49E-01(-)	4.85E-01(-)	4.17E-01(-)	4.52E-01(-)	4.13E-01(-)	5.22E-01
WFG6	Std.	2.11E-02	1.33E-02	1.22E-02	1.63E-02	1.14E-02	9.86E-04	1.57E-02	2.00E-02	6.24E-03
WFG7	Mean	4.88E-01(-)	5.17E-01(-)	4.91E-01(-)	4.23E-01(-)	5.12E-01(-)	5.52E-01(+)	5.13E-01(-)	4.22E-01(-)	5.23E-01
WFG7	Std.	4.91E-03	4.00E-03	4.43E-03	1.92E-02	4.87E-03	1.15E-03	5.20E-03	1.94E-02	4.51E-03
WFG8	Mean	4.06E-01(-)	4.12E-01(-)	4.21E-01(-)	3.14E-01(-)	3.79E-01(-)	4.58E-01(=)	4.26E-01(-)	3.36E-01(-)	4.58E-01
WFG8	Std.	5.84E-03	7.17E-03	3.54E-03	9.70E-03	8.60E-03	3.42E-03	5.42E-03	1.61E-02	2.80E-03
WFG9	Mean	4.05E-01(-)	5.05E-01(+)	4.81E-01(-)	4.59E-01(-)	4.80E-01(-)	4.66E-01(=)	4.85E-01(-)	4.60E-01(-)	5.02E-01
WFG9	Std.	2.96E-02	5.28E-03	5.29E-03	1.08E-02	1.03E-02	5.62E-02	1.09E-02	1.14E-02	3.74E-03
ZDT1	Mean	7.15E-01(-)	7.17E-01(-)	7.03E-01(-)	1.71E-01(-)	7.18E-01(-)	6.88E-01(-)	7.06E-01(-)	6.14E-01(-)	7.21E-01
ZDT1	Std.	9.23E-04	8.16E-04	8.65E-04	7.35E-02	1.38E-03	2.23E-02	2.63E-03	1.20E-01	3.06E-05
ZDT2	Mean	4.39E-01(-)	4.43E-01(-)	4.18E-01(-)	0(-)	4.45E-01(-)	4.36E-01(-)	4.16E-01(-)	4.23E-01(-)	4.45E-01
ZDT2	Std.	9.16E-04	9.34E-04	1.54E-03	0	2.47E-04	2.30E-03	3.32E-02	7.11E-02	3.46E-05
ZDT3	Mean	5.93E-01(-)	5.99E-01(-)	5.94E-01(-)	2.91E-01(-)	5.97E-01(-)	5.67E-01(-)	5.95E-01(-)	5.55E-01(-)	6.00E-01
ZDT3	Std.	1.35E-03	1.09E-03	1.05E-03	5.72E-02	1.44E-03	3.90E-04	1.60E-02	8.55E-02	2.79E-05
ZDT4	Mean	0(-)	0(-)	7.11E-01(=)	0(-)	0(-)	0(-)	4.80E-01(-)	0(-)	6.09E-01
ZDT4	Std.	0	0	1.09E-02	0	0	0	1.59E-01	0	1.14E-01
ZDT6	Mean	3.89E-01(-)	3.89E-01(-)	0(-)	2.64E-01(-)	3.88E-01(-)	3.88E-01(-)	3.22E-01(-)	3.88E-01(-)	3.89E-01
ZDT6	Std.	1.67E-05	1.44E-04	0	1.57E-01	1.02E-05	4.68E-04	3.40E-02	1.48E-04	1.34E-05
+/-/=		0/21/0	1/19/1	0/201	0/21/0	0/20/1	5/15/3	1/19/1	0/21/0	~
Best/All		0/21	1/21	1/21	0/21	0/21	7/21	0/21	0/21	12/21

表2

表3

KUKA KR3 R540机械臂D-H参数

连杆 $i$	$θ i / r a d$	$d i / m$	$a i / m$	$α i / (∘)$
1	$θ 1$	0.345	0.02	-90
2	$θ 2$	0	0.26	0
3	$θ 3$	0	0.02	-90
4	$θ 4$	0.260	0	90
5	$θ 5$	0	0	-90
6	$θ 6$	0.075	0	0

表3

图6

图7

表4

图8

图9

图10

图11

图12

图13

表5

图14

表6

RS007LD-H参数

连杆 $i$	$θ i / r a d$	$d i / m$	$a i / m$	$α i / (∘)$
1	$θ 1$	0.180	0.148 7	-90
2	$θ 2$	0	0.180 0	0
3	$θ 3$	0	0.455 0	-90
4	$θ 4$	0.475	0	90
5	$θ 5$	0	0	-90
6	$θ 6$	0.078	0	0

表6

图15

表7

图16

表8

图17

表9

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关节编号	起始点	路径点1	路径点2	终止点
1	-0.523 6	0.523 6	1.047 2	1.570 8
2	0	1.047 2	-0.523 6	0
3	0.261 8	0.785 4	1.047 2	1.309 0
4	-0.523 6	0.314 2	-0.942 5	-0.523 6
5	0.418 9	1.152 0	1.518 4	1.885 0
6	-0.785 4	0.157 1	-1.256 6	-0.785 4

实验场景	时间权重	冲击权重	能量权重	评价指标
自设场景I	0.50	0.35	0.15	59.079 1
自设场景II	0.4	0.4	0.2	64.200 7
自设场景III	0.8	0.1	0.1	25.633 1
平衡权重	1/3	1/3	1/3	60.373 8
时间优先	0.6	0.2	0.2	45.012 0
能量优先	0.2	0.2	0.6	52.256 4
冲击优先	0.2	0.6	0.2	95.544 5

关节编号	起点	路径点1	路径点2	路径点3	终点
1	-0.057 7	-0.057 7	-0.057 7	-0.057 7	-0.057 7
2	-0.957 7	-0.586 5	-0.239 7	-0.261 5	-0.565 1
3	2.144	2.144	2.189	2.020	1.492
4	-0.261 2	-0.261 2	-0.261 2	-0.261 2	-0.261 2
5	0.052 7	0.052 7	0.052 7	0.052 7	0.052 7
6	3.200	3.210	3.210	3.210	3.210

关节编号	起始点	路径点1	路径点2	终止点
1	0	-0.184 6	-0.368 6	-0.392 7
2	-1.570 8	-1.827 9	-1.554 5	-1.047 2
3	-0.785 4	-0.066 4	0.558 2	0.523 6
4	1.047 2	1.168 4	0.773 1	0.392 7
5	0.523 6	0.734 1	0.740 5	0.523 6
6	-0.523 6	-0.767 1	-0.974 7	-0.785 4

指标		GMODE	GMODE-1	GMODE-2	GMODE-3
时间复杂度/s	Mean	499.25	661.91	466.64	110.77
时间复杂度/s	Std	11.34	7.92	13.14	7.12
空间复杂度/MB	Mean	5 129.92	2 379.24	5 969.78	6 264.91
空间复杂度/MB	Std	6.51	100.46	0.05	12.72
综合指标	Mean	0.489 8	0.428 3	0.512 8	0.353 3
综合指标	Std	0.227 0	0.251 4	0.152 8	0.244 3