High-precision Computation of Inverse Kinematics for Redundant Robots Based on Flow Model

doi:10.3969/j.issn.1004-132X.2025.12.019

Abstract

Abstract:

To improve the accuracy of deep neural networks in solving inverse kinematics for redundant robots and reduce the probability of self-collision solutions， a solution method was proposed based on the conditional normalizing flows model. An improved L-M algorithm was employed to perform secondary optimization on the initial solutions generated by the conditional normalizing flows model to enhance computational accuracy. Furthermore， by training a multi-layer perceptron with extracted self-collision prior knowledge， a self-collision solution detector was constructed to filter out self-collision solutions. The results demonstrate that the calculated positional and angular errors remain below 0.01 mm and 0.1° respectively， the self-collision rate is maintained under 0.1%， and the single computation time is consistently within 10 ms. This method enables efficient and stable solutions for inverse kinematics problems in redundant robots.

Key words: inverse kinematics, redundant robot, conditional normalizing flow, Levenberg-Marquardt（L-M） algorithm

摘要：

为提高深度神经网络计算冗余机器人逆运动学解的精度，并降低自碰撞解的出现概率，提出了一种基于条件标准化流模型的求解方法。采用改进列文伯格-马夸尔特（L-M）算法对条件标准化流模型生成的初始解进行二次优化以提高计算精度。通过提取自碰撞先验信息训练多层感知机，构造自碰撞解检测器以剔除自碰撞解。结果表明，求解的位置和角度误差分别小于0.01 mm和0.1°，自碰撞率低于0.1%，并且单次计算的时间稳定在10 ms以内。该方法可用于对冗余机器人逆运动学问题的高效稳定求解。

关键词: 逆运动学, 冗余机器人, 条件标准化流, 列文伯格-马夸尔特算法

CLC Number:

TP241.2

Feng YIN, Xin HUANG, Jiayi ZHOU. High-precision Computation of Inverse Kinematics for Redundant Robots Based on Flow Model[J]. China Mechanical Engineering, 2025, 36(12): 2960-2967.

印峰, 黄欣, 周佳义. 基于流模型的冗余机器人逆运动学解高精度计算[J]. 中国机械工程, 2025, 36(12): 2960-2967.

Figures/Tables 15

Fig.1 The first three dimensions of Panda robot training conditional dataset

Fig.2 Network architecture diagram of CNF module

Fig.3 Robot collision detection models

Fig.4 The minimum distance Δr between capsules

Fig.5 Multi-layer perceptron

Fig.6 Self collision detection flowchart

Tab.1 MMD scores of generative networks

模型	MMD 得分
INN	0.061
cINN	0.058
CGAN（Cycle IK）	0.037
CNL（IKFLOW）	0.033
CNF（NOC-IK）	0.031

Tab.2 IKFLOW inverse kinematics solution results

机器人	隐藏层数	每个解的时间/ms	平均L2位置误差/cm	平均角度误差/（º）	自碰撞概率/%
Panda	6	3.3376	1.4405	8.4781	6.052
Panda	12	6.3605	0.6826	2.2817	4.564
Fetch	12	6.1903	1.4507	0.7864	2.976
Fetch	16	7.29508	0.9792	3.6491	2.192
Iiwa7	12	10.0659	2.0168	3.6801	0.028

Tab.3 NOC-IK（CNF）inverse kinematics solution results

机器人	隐藏层数	每个解的时间/ms	平均L2位置误差/cm	平均角度误差/（º）	自碰撞概率/%
Panda	6	2.0621	1.0177	5.0036	2.330
Panda	12	3.9418	0.6808	2.1554	1.584
Fetch	12	3.9242	1.4843	0.6724	3.004
Fetch	16	5.1982	0.8601	1.8602	1.660
Iiwa7	12	7.9731	1.3409	3.1826	0

Fig.7 Comparison of average pose errors of robots

Tab.4 Self-collision detection results

方法	每个解的时间/ms	输出自碰撞解个数
FCL	1.5855	0
$c o l (Θ)$	7.4157	0
MLP	0.2651	0

Tab.4 Self-collision detection results

方法	每个解的时间/ms	输出自碰撞解个数
FCL	1.5855	0
$c o l (Θ)$	7.4157	0
MLP	0.2651	0

Tab.5 Experimental results of ablation

模块

均方误差/cm

平均自碰撞

概率/%

每个解的

时间/ms

Fig.8 Total running time of NOC-IK

Fig.9 The self-collision probability

Tab.6 The results of comparison with other models

模型	均方误差/cm	平均自碰撞概率/%	每个解的时间/ms
B-NAFs	3.871	0.300	0.6800
BP	0.980		16.7800
ResNet-BP	0.600		8.7000
IKFLOW	0.772	6.732	6.7322
CycleIK^［15］	0.570		0.4100
CycleIK-Gaikpy^［15］	0.051		20.0071
CNN	0.012		12.9600
NOC-IK	0.001	0.077	9.9425

References 23

[1]	欧群文，贠超，杨学兵，等.基于神经网络的冗余机械臂运动学逆解研究［J］.机电工程，2016，33（6）：663-667.
	Qunwen OU， YUN Chao， YANG Xuebing， et al. Algorithm Based on Neural Network for Inverse Kinematics of Redundant Manipulator［J］ Journal of Mechanical & Electrical Engineering， 2016，33 （6）： 663-667.
[2]	李进，刘璇，张建华，等.基于RBF神经网络间接求取运动学逆解的研究［J］.机床与液压，2019，47（23）：32-37.
	LI Jin， LIU Xuan， ZHANG Jianhua， et al. Research on Indirect Solution of Inverse Kinematics Based on RBF Neural Network［J］. Machine Tool & Hydraulics， 2019，47（23）：32-37.
[3]	刘世平，曹俊峰，孙涛，等.基于BP神经网络的冗余机械臂逆运动学分析［J］.中国机械工程， 2019，30（24）： 2974-2977.
	LIU Shiping， CAO Junfeng， SUN Tao， et al. Inverse Kinematics Analysis of Redundant Manipulators Based on BP Neural Network［J］. China Mechanical Engineering，2019，30（24）：2974-2977.
[4]	刘世平，夏文杰，陈萌，等.基于卷积神经网络的冗余机械臂运动学逆解求解［J］.载人航天，2022，28（1）：10-15.
	LIU Shiping， XIA Wenjie， CHEN Meng， et al. Inverse Kinematics Solution of Redundant Manipulator Based on Convolution Neural Networks［J］. Manned Spaceflight，2022，28（1）：10-15.
[5]	石建平.基于粒子群优化算法的冗余机械臂运动学逆解［J］.贵阳学院学报（自然科学版），2020，15（3）：48-52.
	SHI Jianping. Inverse Kinematics Solution of Redundant Manipulator Based on Particle Swarm Optimization Algorithm［J］. Journal of Guiyang University（Natural Sciences），2020，15（3）：48-52.
[6]	ARDIZZONE L， KRUSE J， WIRKERT S， et al. Analyzing Inverse Problems with Invertible Neural Networks［C］∥International Conference on Learning Representations （ICLR）.New Orleans， 2019：1808.04730.
[7]	GARG S， DUDEJA S， RASTOGI V. Inverse Kinematics of Tendon Driven Continuum Robots Using Invertible Neural Network［C］∥2022 2nd International Conference on Computers and Automation （CompAuto）. Paris， 2022： 82-86.
[8]	AMES B， MORGAN J， KONIDARIS G. IKFLOW： Generating Diverse Inverse Kinematics Solutions［J］. IEEE Robotics and Automation Letters， 2022， 7（3）： 7177-7184.
[9]	PAPAMAKARIOS G， NALISNICK E， JIMENEZ REZENDE D， et al. Normalizing Flows for Probabilistic Modeling and Inference［J］. Journal of Machine Learning Research，2019， 57（22）：64.
[10]	KINGMA D P， DHARIWAL P. GLOW： Generative Flow with Invertible 1×1 Convolutions［C］∥The 32nd International Conference on Neural Information Processing Systems （NIPS'18）. Red Hook， 2018： 10236-10245.
[11]	BEESON P， AMES B. TRAC-IK： an Open-source Library for Improved Solving of Generic Inverse Kinematics［C］∥2015 IEEE-RAS 15th International Conference on Humanoid Robots （Humanoids）. Seoul， 2015： 928-935.
[12]	HABEKOST J G， STRAHL E， ALLGEUER P， et al. CycleIK： Neuro-inspired Inverse Kinematics［C］∥ International Conference on Artificial Neural Networks. Heraklion， 2023： 457-470.
[13]	ZHU J Y， PARK T， ISOLA P， et al. Unpaired Image-to-image Translation Using Cycle-consistent Adversarial Networks ［C］ ∥Proceedings of the IEEE International Conference on Computer Vision. Venice， 2017： 2223-2232.
[14]	KERZEL M， ALLGEUER P， STRAHL E， et al. Nicol： a Neuro-inspired Collaborative Semi-humanoid Robot that Bridges Social Interaction and Reliable Manipulation［J］.IEEE Access，2023， 11： 123531-123542.
[15]	DEMBY’S J， GAO Y， DESOUZA G N. A Study on Solving the Inverse Kinematics of Serial Robots Using Artificial Neural Network and Fuzzy Neural Network［C］∥2019 IEEE International Conference on Fuzzy Systems （FUZZ-IEEE）.New Orleans， 2019： 1-6.
[16]	GILBERT E G， FOO C P. Computing the Distance between General Convex Objects in Three-dimensional Space［J］. IEEE Transactions on Robotics and Automation， 1990， 6（1）： 53-61.
[17]	SAFEEA M， NETO P， BEAREE R. Efficient Calculation of Minimum Distance Between Capsules and Its Use in Robotics［J］. IEEE Access， 2018， 7： 5368-5373.
[18]	NVIDIA GPU Technology Conference GTC）. Warp： a High-performance Python Framework for GPU Simulation and Graphics［EB/OL］.（2022-03-31）［2025-01-03］..
[19]	KIM H， LEE H， KANG W H， et al. Softflow： Probabilistic Framework for Normalizing Flow on Manifolds［J］. Neural Information Processing Systems， 2020， 33： 16388-16397.
[20]	PADMANABHA G A， ZABARAS N. Solving Inverse Problems Using Conditional Invertible Neural Networks［J］. Journal of Computational Physics， 2021， 433： 110194.
[21]	李佳，邱伟，邹劭芬.基于改进LM-GN算法的磁性目标定位方法研究［J］.数字海洋与水下攻防， 2023，6（5）： 552-561.
	LI Jia， QIU Wei， ZOU Shaofen. Research on Magnetic Target Localization Based on Improved LM-GN Algorithm［J］. Digital Ocean & Underwater Warfare， 2023，6（5）：552-561.
[22]	李国民，宿梦瑶，朱代先.光束平差法中的一种改进LM算法［J］.西安科技大学学报，2022，42（1）：152-159.
	LI Guoming， SU Mengyao， ZHU Daixian. An Improved LM Algorithm in Bundle Adjustment［J］. Journal of Xi'an University of Science and Technology， 2022，42（1）： 152-159.
[23]	RAKITA D， MUTLU B， GLEICHER M. RelaxedIK： Real-time Synthesis of Accurate and Feasible Robot Arm Motion［C］∥Robotics： Science and Systems （RSS 2018.Pittsburgh， 2018：46968269.