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Geometric and Numerical Foundations of Movements pp 13–42Cite as

Several Geometries for Movements Generations

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Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 117))

Abstract

In previous works we reanalyzed the kinematics of hand movements and locomotion, and suggested that several geometries are used conjointly by the brain for according the shape and the duration along trajectories; this was done in collaboration with Tamar Flash and her collaborators [10, 64, 67], and with Quang-Cuong Pham [79]. The variety of geometries which were implied in this process, were associated to sub-groups of the affine group of a plane: full affine, equi-affine and Euclidean. Other studies have shown how the above geometries constrain the production of the movements [92], or began to use the affine geometry in Robotics [80]. In this article, we propose to use a new variety of geometries which extends the preceding series in another direction, to cover wider contexts and more complex movements, like prehension, initiation of walking, locomotion, navigation, imagined motion. The new spaces adapted to those geometries have no points; they come from topos theory, which is an extension of set theory replacing sets by fields and graphs of dynamics. Any given topos generates a variety of different geometries, which can be mixed as in the preceding studies. Such geometries take into account efforts, forces and dynamics; they do not neglect them aside as does traditional geometry. In this preliminary report we indicate the simplest characteristics of spaces which underly the above examples. The hypothesis is also that these spaces are implemented in different, although overlapping, central nervous system networks in the brain, corresponding to the different action spaces mentioned above. Here, as for the known classical geometries, the most concrete suggestion concerns the timing of movement: we predict that different components of the controlled system are using different intrinsic time courses, and that the mapping between these different internal durations is an important part of the dynamic under geometrical control. This reminds us of a well known psychological observation, for instance that time in imagination does not flow as ordinary clocks time, but this also suggests that reaching an object with the hand has its own time, or that equilibrium control in walking works within a specific time, which is different from the walking trajectory displacement time.

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Notes

  1. 1.

    Recently, Q.-C. Pham and Y. Nakamura developed a new trajectory deformation algorithm based on affine transformations. Reference [80]. The idea is to apply a set of predefined affine transformations to a set of trajectory segments, to avoid unexpected obstacles or to achieve a new objective goal. They also conjugate this idea with optimization algorithms for better accuracy, respecting \(C^{1}\) continuity, keeping fixed final configuration and avoiding joint limits. The method was tested on a virtual planar three-links manipulator, and compared to polynomial interpolations; the main result is a considerable gain in computation time for equal accuracy. This can also be efficiently applied for minimizing curvature’s changes in 3D point to point deformation. The method was applied to rapid motion transfer from humans to robots, with better performance in kinematics than polynomial interpolation methods.

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  1. Institut de Mathématiques de Jussieu, Université Paris VII, Paris, France

    Daniel Bennequin

  2. Collège-de-France, Paris, France

    Alain Berthoz

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  1. Daniel Bennequin
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Bennequin, D., Berthoz, A. (2017). Several Geometries for Movements Generations. In: Laumond, JP., Mansard, N., Lasserre, JB. (eds) Geometric and Numerical Foundations of Movements . Springer Tracts in Advanced Robotics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-51547-2_2

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