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Motion Planning Method of Redundant Dual-Chain Manipulator with Multiple Constraints

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Abstract

Redundant and dual-arm manipulators exhibit good flexibility and excellent cooperation ability, respectively. Therefore, both these manipulators are used to design a redundant dual-chain manipulator with two kinematic chains and a fixed base. In addition, a motion planning method is proposed using a manifold analysis based on the characteristics of this designed manipulator to resolve the difficulty associated with motion planning for a dual-chain manipulator under multiple constraints. Firstly, the paths of the two end-effectors of the dual-chain manipulator are planned synchronously by the Rapidly-exploring Random Tree Star (RRT*) in the Cartesian space. Then, joint trajectories that correspond to the paths are generated by the manifold analysis in the joint space. In this method, the manifold continuity constraint is added to the end-effector path planning of RRT* to ensure that the planned paths have continuous joint trajectories. The joint trajectories are optimised by analysing the performance of the manipulator on the manifolds to improve their motion performance. The prototype experiments of loose coordination and tight coordination are provided in the experimental section; these experiments prove the feasibility and practicability of the motion planning method proposed in this study.

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Data Availability

The raw evaluation data can be viewed at: https://www.scidb.cn/anonymous/SkozWVZ6.

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Acknowledgements

This work has been supported by the National Natural Science Foundation of China (Grant numbers 51975008 and 52275001).

Funding

This work was supported by National Natural Science Foundation of China (Grant numbers 52275001 and 51975008).

Author information

Authors and Affiliations

  1. Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, 100124, China

    Zhenyong Zhou, Jing Zhao & Ziqiang Zhang

  2. Beijing Key Laboratory of Long-Life Technology of Precise Rotation and Transmission Mechanisms, Beijing Institute of Control Engineering, Beijing, 100094, China

    Xiaohui Li

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  1. Zhenyong Zhou
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  2. Jing Zhao
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  4. Xiaohui Li
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Contributions

All authors contributed to the study conception and design. Data collection and analysis were performed by Zhenyong Zhou and Ziqiang Zhang. The experimental scheme was formulated by Xiaohui Li. The first draft of the manuscript was written by Jing Zhao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Jing Zhao.

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Appendices

Appendix 1: The D-H Parameters of the Leader and Follower Chains

Table 4

Table 4 D-H parameters of the leader chain manipulator
Full size table

Table 5

Table 5 D-H parameters of the follower chain manipulator
Full size table

Appendix 2: Derivation of Analytical Expressions for Inverse Kinematics of Follower Chain

The end-effector poses of the dual-chain manipulator can be calculated when the angle of each joint is known. The end-effector pose \({}_{t1}{}^{0}{\varvec{T}}\) of the leader chain can be expressed as

$${}_{t1}{}^{0}{\varvec{T}}\text{=}{}_{1}{}^{0}{\varvec{T}}{}_{2}{}^{1}T{}_{3}{}^{2}{\varvec{T}}{}_{4}{}^{3}T{}_{5}{}^{4}{\varvec{T}}{}_{6}{}^{5}{\varvec{T}}{}_{7}{}^{6}{\varvec{T}}{}_{t1}{}^{7}{\varvec{T}}$$
(15)

where \({}_{i-1}{}^{i}{\varvec{T}}\in {{\varvec{R}}}^{4\times 4}\) represents the transformation relation between adjacent coordinate systems.

The end-effector pose \({}_{t2}{}^{0}{\varvec{T}}\) of the follower chain can be expressed as

$${}_{t2}{}^{0}{\varvec{T}}\text{=}{}_{4}{}^{0}{\varvec{T}}{}_{8}{}^{4}{\varvec{T}}{}_{9}{}^{8}{\varvec{T}}{}_{10}{}^{9}{\varvec{T}}{}_{11}{}^{10}{\varvec{T}}{}_{12}{}^{11}{\varvec{T}}{}_{13}{}^{12}{\varvec{T}}{}_{t2}{}^{13}{\varvec{T}}$$
(16)

where \({}_{4}{}^{0}{\varvec{T}}\) represents the pose of joint 4 relative to the world coordinate system, which is obtained by calculating Eq. 15.

A closed solution exists because the axes of joints 9–11 of the follower chain manipulator are parallel and satisfy the Pieper criterion. According to Eq. 16, the pose of the follower chain end-effector relative to the coordinate system of joint 4 can be expressed as

$${}_{13}{}^{4}{\varvec{T}}\text{=}{{}_{4}{}^{0}{\varvec{T}}}^{-1}{}_{t2}{}^{0}{\varvec{T}}{{}_{t2}{}^{13}{\varvec{T}}}^{-1}\text{=}{}_{8}{}^{4}{\varvec{T}}{}_{9}{}^{8}{\varvec{T}}{}_{10}{}^{9}{\varvec{T}}{}_{11}{}^{10}{\varvec{T}}{}_{12}{}^{11}{\varvec{T}}{}_{13}{}^{12}{\varvec{T}}$$
(17)

where \({}_{13}{}^{4}{\varvec{T}}\) can be denoted as \({}_{13}{}^{4}{\varvec{T}}\text{=}\left[\begin{array}{cccc}{n}_{x}& {o}_{x}& {a}_{x}& {p}_{x}\\ {n}_{y}& {o}_{y}& {a}_{y}& {p}_{y}\\ {n}_{z}& {o}_{z}& {a}_{z}& {p}_{z}\\ 0& 0& 0& 1\end{array}\right]\).

The analytical equation of the follower chain joint 8 is

$${\theta }_{8}=\pm 2{\text{arctan}}\left(\sqrt{\frac{1-{m}_{1}}{1+{m}_{1}}}\right)-{\text{atan}}2\left(\frac{{p}_{x}}{{n}_{1}},\frac{{p}_{y}}{{n}_{1}}\right)$$
(18)

where \({n}_{1}=\sqrt{{p}_{x}^{2}+{p}_{y}^{2}}\) and \({m}_{1}=\frac{{d}_{9}+{d}_{10}-{d}_{11}}{{n}_{1}}\).

The analytical equation of the follower chain joint 13 is

$${\theta }_{13}={\text{atan}}2\left({n}_{y}{c}_{8}-{n}_{x}{s}_{8},{o}_{y}{c}_{8}-{o}_{x}{s}_{8}\right)\pm \frac{\pi }{2}$$
(19)

where \({c}_{8}\) is shorthand for \(\mathrm{cos}{\theta }_{8}\), and \({s}_{8}\) is shorthand for \(\mathrm{sin}({\theta }_{8})\).

The analytical equation of the follower chain joint 12 is

$${\theta }_{12}={\text{atan}}2\left({m}_{2},{n}_{2}\right)$$
(20)

where \({m}_{2}={n}_{y}{c}_{8}{c}_{13}-{n}_{x}{s}_{6}{c}_{13}-{o}_{y}{c}_{8}{s}_{13}+{o}_{x}{s}_{8}{s}_{13}\) and \({n}_{2}={a}_{x}{s}_{8}-{a}_{y}{c}_{8}\).

The analytical equation of the follower chain joint 10 is

$${\theta }_{10}=\pm 2{\text{arctan}}\left(\sqrt{\frac{1-{o}_{3}}{1+{o}_{3}}}\right)$$
(21)

where \({o}_{3}=\frac{{m}_{3}^{2}+{n}_{3}^{2}-{a}_{10}^{2}-{a}_{11}^{2}}{2{a}_{10}{a}_{11}}\), \({m}_{3}={p}_{x}{c}_{8}+{p}_{y}{s}_{8}-{d}_{12}\left({c}_{13}\left({o}_{x}{c}_{8}+{o}_{y}{s}_{8}\right)+{s}_{13}\left({n}_{x}{c}_{8}+{n}_{y}{s}_{8}\right)\right)\), and \({n}_{3}={d}_{12}\left({n}_{z}{s}_{13}+{o}_{z}{c}_{13}\right)+{d}_{8}-{p}_{z}\).

The analytical equation of the follower chain joint 9 is

$${\theta }_{9}={\text{atan}}2\left({m}_{4},{n}_{4}\right)$$
(22)

where \({m}_{4}=\frac{{n}_{3}\left({a}_{10}+{a}_{11}{c}_{10}\right)-{m}_{3}{a}_{11}{s}_{10}}{{\text{(}{a}_{10}+{a}_{11}{c}_{10}\text{)}}^{2}+{\left({a}_{11}{s}_{10}\right)}^{2}}\) and \({n}_{4}=\frac{{m}_{3}\left({a}_{10}+{a}_{11}{c}_{10}\right)+{n}_{3}{a}_{11}{s}_{10}}{{\left({a}_{10}+{a}_{11}{c}_{10}\right)}^{2}+{\left({a}_{11}{s}_{10}\right)}^{2}}\).

The analytical equation of the follower chain joint 11 is

$${\theta }_{11}={\theta }_{9}+{\theta }_{10}-{\text{atan}}2\left({m}_{5},{n}_{5}\right)$$
(23)

where \({m}_{5}={o}_{x}{c}_{8}{c}_{13}+{o}_{y}{s}_{8}{c}_{13}+{n}_{x}{c}_{8}{s}_{13}+{n}_{y}{s}_{8}{s}_{13}\) and \({n}_{5}={o}_{z}{c}_{13}+{n}_{z}{s}_{13}\).

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Zhou, Z., Zhao, J., Zhang, Z. et al. Motion Planning Method of Redundant Dual-Chain Manipulator with Multiple Constraints. J Intell Robot Syst 108, 69 (2023). https://doi.org/10.1007/s10846-023-01921-3

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