Abstract
We investigate a class of linear discrete control systems, modeling the controlled dynamics of planar manipulators as well as the skeletal dynamics of human fingers and bird's toes. A self-similarity assumption on the phalanxes allows to reinterpret the control field ruling the whole dynamics as an Iterated Function System. By exploiting this relation, we apply results coming from self-similar dynamics in order to give a geometrical description of the control system and, in particular, of its reachable set. This approach is then applied to the investigation of the zygodactyl phenomenon in birds, and in particular in parrots. This arrangement of the toes of a bird's foot, common in species living on trees, is a distribution of the foot with two toes facing forward and two back. Reachability and grasping configurations are then investigated. Finally an hybrid system modeling the owl's foot is introduced.
Highlights
The aim of this paper is to introduce a class of linear discrete control systems, modeling the controlled dynamics of planar manipulators as well as the skeletal dynamics of human fingers and bird’s toes
We show that the recursively of the scaling relation, as well as its contractivity, allows to reinterpret the control field ruling the whole dynamics as an Iterated Function System, namely a set of contractive maps
We note that the connection between the biomechanics of avian feet and robotics is an active research domain, mostly motivated by the fact that the locomotion of birds turned out to be more efficient with respect with human locomotion - see for instance the project described in [18], where the locomotion of birds is mimicked in a robotic device
Summary
The aim of this paper is to introduce a class of linear discrete control systems, modeling the controlled dynamics of planar manipulators as well as the skeletal dynamics of human fingers and bird’s toes. We show that the recursively of the scaling relation, as well as its contractivity, allows to reinterpret the control field ruling the whole dynamics as an Iterated Function System, namely a set of contractive maps This opens the way to the wide theoretical background of fractal geometry and, in particular, to the branch devoted to the investigation of self-similar structures. ANNA CHIARA LAI AND PAOLA LORETI self-similar dynamics (like the attractor of an iterated function systems or the celebrated Open Set Condition) are used to describe the topology of the reachable set and other properties of the dynamical systems – see for instance [9] for an investigation on the left invertibility of discrete control systems via Iteration Function Systems This approach was explored in [8] and in [15], for a particular class of selfsimilar structures related to the theory of expansions in non-integer bases. We note that the connection between the biomechanics of avian feet and robotics is an active research domain, mostly motivated by the fact that the locomotion of birds turned out to be more efficient with respect with human locomotion - see for instance the project described in [18], where the locomotion of birds is mimicked in a robotic device
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