Self-similar control systems and applications to zygodactyl bird's foot

Anna Chiara Lai; Paola Loreti; Anna Chiara Lai; Paola Loreti

doi:10.3934/nhm.2015.10.401

Networks and Heterogeneous Media

2015, Volume 10, Issue 2: 401-419. doi: 10.3934/nhm.2015.10.401

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Self-similar control systems and applications to zygodactyl bird's foot

Anna Chiara Lai ^{1
,},
Paola Loreti ^{2
,}

1.
Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione Matematica, Via A. Scarpa n.16 00161 Roma
2.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma

Received: 01 September 2014 Revised: 01 December 2014
Primary: 58F12, 93A30; Secondary: 92B05.

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.
- Discrete control theory,
- iterated function systems,
- avian feet,
- zygodactyl feet,
- planar manipulator
Citation: Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot[J]. Networks and Heterogeneous Media, 2015, 10(2): 401-419. doi: 10.3934/nhm.2015.10.401

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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.

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