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A Geometric Algebra Based Higher Dimensional Approximation Method for Statics and Kinematics of Robotic Manipulators

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A Correction to this article was published on 07 August 2020

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Abstract

Evaluation of a robotic (serial or parallel) manipulator’s static and kinematics performance requires the solution of closed and complicated non-closed form systems of equations for both the forward and inverse problem types. In this work, a robotic system or manipulator is represented as a network whose motion generating kinematic pairs are represented as the network’s inter-connected nodes. Within this network theoretic context, we develop a formulation employing higher dimensional multivectors defined in Clifford Algebra that approximates the computational outcomes of such complicated systems of equations, for both inverse and forward problem types. The statics and kinematics performance of these mechanical networks (serial and parallel robots) are rapidly evaluated without the need to solve their respective traditional systems of equations using what we call ‘screw hypervolumes’, which capture the algebro-geometric structures generated by a robot’s physical design parameters and end-effector position in terms of kinematics and statics. These dimensionless hypervolume functions enable the specialist to produce a very computationally efficient function/index that approximates the kinematic and static output spaces of robots. We find that this formulation inadvertently presents a curious mathematical analogue to these systems of equations. A 6R serial robot, the Delta robot and the 3-RRR manipulator are analysed using the hypervolume methods for the case studies. This approach can be adapted and applied to other multi-agent physical systems displaying non-commutative interactions as well.

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  • 07 August 2020

    Typographic error in Eq. (41c): the subscript.

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Acknowledgements

The authors graciously acknowledge Debaditya Biswas for his assistance in the preparation of the figures.

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Authors and Affiliations

  1. The University of Western Australia, BlueStem Pty. Ltd., 35 Stirling Highway, Crawley, Perth, WA, 6009, Australia

    Sudharsan Thiruvengadam & Karol Miller

  2. BlueStem Pty. Ltd., 128 Parry Avenue, Bull Creek, Perth, WA, 6149, Australia

    Sudharsan Thiruvengadam

Authors
  1. Sudharsan Thiruvengadam
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  2. Karol Miller
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Correspondence to Sudharsan Thiruvengadam.

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Communicated by Hongbo Li

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Thiruvengadam, S., Miller, K. A Geometric Algebra Based Higher Dimensional Approximation Method for Statics and Kinematics of Robotic Manipulators. Adv. Appl. Clifford Algebras 30, 17 (2020). https://doi.org/10.1007/s00006-019-1039-z

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