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Physical Reservoir Computing in Robotics

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Reservoir Computing

Part of the book series: Natural Computing Series ((NCS))

Abstract

In recent years, there has been an increasing interest in using the concept of physical reservoir computing in robotics. The idea is to employ the robot’s body and its dynamics as a computational resource. On one hand, this has been driven by the introduction of mathematical frameworks showing how complex mechanical structures can be used to build reservoirs. On the other hand, with the recent advances in smart materials, novel additive manufacturing techniques, and the corresponding rise of soft robotics, a new and much richer set of tools for designing and building robots is now available. Despite the increased interest, however, there is still a wide range of unanswered research questions and a rich area of under-explored applications. We will discuss the current state of the art, the implications of using robot bodies as reservoirs, and the great potential and future directions of physical reservoir computing in robotics.

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Notes

  1. 1.

    The relation between Volterra series and dynamical systems is discussed in Boyd (1985).

  2. 2.

    Linear regression minimizes the quadratic error against the target. Since the error landscape is quadratic, there is only one (global) minimum. Note that this doesn’t necessarily mean that reservoir computing setups are always performing better than RNN networks.

  3. 3.

    Note that this also means that one could use biological bodies directly as reservoirs as well. One just has to find a way to read out the (at least partial) dynamic state of the system.

  4. 4.

    Note that the term feedforward highlights here the fact that there are no explicit feedback loops from the output back into the reservoir. However, the reservoir naturally will still have stable feedback loops inside the reservoir to achieve the fading memory that is a prerequisite for reservoir computing.

  5. 5.

    Or they approximate only the close neighbourhood of one equilibrium point, i.e. local approximation. Note that this is different from linear approximation approaches, e.g. expressed through a Jacobian matrix, since a Volterra series is still capturing nonlinearities.

  6. 6.

    For example, we can achieve separation through different integration time constants.

  7. 7.

    For more rigorous and mathematically sound descriptions, please refer to Boyd and Chua (1985).

  8. 8.

    Note that there are more mathematical details, but for the sake of simplicity, we discuss only the main points. For a more detailed discussion, we refer to Boyd (1985) and Boyd and Chua (1985).

  9. 9.

    Note that, interestingly, Shim and Husbands (2007) suggested a setup which looks very similar and is used for a feathered flyer and which predates Hauser et al.’s work.

  10. 10.

    There can be one or multiple feedback loops.

  11. 11.

    Note that one can argue that combing multiple such systems would then again allow us to approximate systems of higher order as well, see Maass et al. (2007).

  12. 12.

    Note that not every nonlinear system is feedback linearizable. However, there is a formal process to check for that by applying Lie derivatives. Note that this is equivalent to constructing the controllability matrix \(\mathbb {C} = [B,BA,BA^2,\dots ,BA^n] \) in linear systems. We refer to the reader to Slotine and Lohmiller (2001) for an excellent introduction. For a more in-depth discussion, we refer to Isidori (2001).

  13. 13.

    Note that this is directly connected to the degradation of the signal separation property discussed in Sect. 2.1, see Hauser et al. (2014) for a discussion.

  14. 14.

    Note that linear regression is solving the optimization problem to find a set of optimal weights \({{\mathbf {w}}}^{{*}}\) that minimize the quadratic error between the target output \(y_t(t)\) the produced output y(t).

  15. 15.

    The reason being that the theoretical model from Hauser et al. (2011) is based on Volterra series and it’s a generic way to approximated nonlinear, exponentially stable sets of differential equations.

  16. 16.

    Note that there had been previous work on physical reservoir computing, e.g. the previously mentioned “bucket in the water” setup by Fernando and Sojakka (2003), but none in the context of robotics.

  17. 17.

    Note that a limitation is that the different sensors should provide linearly independent information.

  18. 18.

    In the context of Sect. 6, we discuss only soft robotics structures. However, all the points are also true for biological systems or hybrid structures, e.g. the mixture of soft artificial structures and biological tissue.

  19. 19.

    And noise in the case of external feedback loops.

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Acknowledgements

This publication has been written with the support of the Leverhulme Trust Research Project RPG-2016-345.

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Authors and Affiliations

  1. Department of Engineering Mathematics, University of Bristol, Woodland Road, Bristol, BS8 1UB, UK

    Helmut Hauser

  2. Bristol Robotics Lab, Coldharbour Ln, Bristol, BS16 1QY, UK

    Helmut Hauser

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  1. Helmut Hauser
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Correspondence to Helmut Hauser .

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  1. University of Tokyo, Tokyo, Japan

    Kohei Nakajima

  2. Consejo Superior de Investigaciones Científicas, IFISC (UIB-CSIC), Palma, Spain

    Ingo Fischer

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Hauser, H. (2021). Physical Reservoir Computing in Robotics. In: Nakajima, K., Fischer, I. (eds) Reservoir Computing. Natural Computing Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-1687-6_8

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