June  2015, 10(2): 401-419. doi: 10.3934/nhm.2015.10.401

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

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  September 2014 Revised  December 2014 Published  April 2015

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.
Citation: Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
References:
[1]

J. Baillieul, Avoiding obstacles and resolving kinematic redundancy, IEEE International Conference on Robotics and Automation, 3 (1986), 1698-1704. Google Scholar

[2]

M. F. Barnsley, Fractals Everywhere: New Edition, Courier Dover Publications, 2013. Google Scholar

[3]

M. F. Barnsley and K. Leśniak, On the continuity of the Hutchinson operator, preprint, arXiv:1202.2485, 2012. Google Scholar

[4]

W. J. Bock, Functional and evolutionary morphology of woodpeckers, Ostrich, 70 (1999), 23-31. Google Scholar

[5]

J. W. Burdick, Kinematic Analysis and Design of Redundant Robot Manipulators, Diss. Stanford University, 1988. Google Scholar

[6]

G. S. Chirikjian and J. W. Burdick, An obstacle avoidance algorithm for hyper-redundant manipulators, IEEE International Conference on Robotics and Automation, 1 (1990), 625-631. doi: 10.1109/ROBOT.1990.126052.  Google Scholar

[7]

G. S. Chirikjian and J. W. Burdick, The kinematics of hyper-redundant robot locomotion, IEEE Transactions on Robotics and Automation, 11 (1995), 781-793. doi: 10.1109/70.478426.  Google Scholar

[8]

Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set, Mathematics of Control, Signals and Systems, 14 (2001), 173-193. doi: 10.1007/PL00009881.  Google Scholar

[9]

N. Dubbini, B. Piccoli and A. Bicchi, Left invertibility of discrete systems with finite inputs and quantised output, International Journal of Control, 83 (2010), 798-809. doi: 10.1080/00207170903438069.  Google Scholar

[10]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[11]

E.-U. Imme and G. S. Chirikjian, Inverse kinematics of discretely actuated hyper-redundant manipulators using workspace densities, in Proceedings of 1996 IEEE International Conference on Robotics and Automation. Vol. 1, IEEE, 1996, 139-145. Google Scholar

[12]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Mathematical Research Letters, 8 (2001), 535-543. doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[13]

V. Komornik, A. C. Lai and M. Pedicini, Generalized golden ratios of ternary alphabets, J. Eur. Math. Soc., 13 (2011), 1113-1146. doi: 10.4171/JEMS/277.  Google Scholar

[14]

V. Komornik and P. Loreti, Unique developments in non-integer bases, American Mathematical Monthly, 105 (1998), 636-639. doi: 10.2307/2589246.  Google Scholar

[15]

A. C. Lai and P. Loreti, Robot's finger and expansions in non-integer bases, Networks and Heterogeneus Media, 7 (2012), 71-111. doi: 10.3934/nhm.2012.7.71.  Google Scholar

[16]

A. C. Lai and P. Loreti, From discrete to continuous reachability for a robot's finger model, Communications in Industrial and Applied Mathematics, 3 (2012), e-439, 22pp.  Google Scholar

[17]

M. D. Lichter, V. A. Sujan and S. Dubowsky, Computational issues in the planning and kinematics of binary robots, in Proceedings of ICRA'02 IEEE International Conference on Robotics and Automation. Vol. 1, IEEE, 2002, 341-346. doi: 10.1109/ROBOT.2002.1013384.  Google Scholar

[18]

L. Mederreg, et al., The RoboCoq Project: Modelling and Design of Bird-like Robot, 6th International Conference on Climbing and Walking Robots, CLAWAR, 2003. Google Scholar

[19]

A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 11 (2009), 21-32.  Google Scholar

[20]

B. Mishra, Grasp Metrics: Optimality and Complexity, Proceedings of the Workshop on Algorithmic Foundations of Robotics, AK Peters, Ltd., 1995. Google Scholar

[21]

R. Norberg, Treecreeper climbing, mechanics, energetics, and structural adaptations, Ornis Scandinavica, (1986), 191-209. Google Scholar

[22]

M. Pedicini, Greedy expansions and sets with deleted digits, Theoretical Computer Science, 332 (2005), 313-336. doi: 10.1016/j.tcs.2004.11.002.  Google Scholar

[23]

Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proceedings of the American Mathematical Society, 129 (2001), 2689-2699. doi: 10.1090/S0002-9939-01-05969-X.  Google Scholar

[24]

T. Quinn and J. Baumel, The digital tendon locking mechanism of the avian foot (Aves), Zoomorphology, 109 (1990), 281-293. doi: 10.1007/BF00312195.  Google Scholar

[25]

N. A. Secelean, Masura si Fractali, Univ. Lucian Blaga, Sibiu, 2002. Google Scholar

[26]

D. Sustaita, et al., Getting a Grip on Tetrapod Grasping: Form, Function, and Evolution, Biological Reviews, 88 (2013), 380-405. Google Scholar

[27]

A. V. Zinoviev and F. Ya Dzerzhinsky, Some general notes on the avian hindlimb biomechanics, Bulletin of Moscow Society of Naturalists, 105 (2000), p5. Google Scholar

show all references

References:
[1]

J. Baillieul, Avoiding obstacles and resolving kinematic redundancy, IEEE International Conference on Robotics and Automation, 3 (1986), 1698-1704. Google Scholar

[2]

M. F. Barnsley, Fractals Everywhere: New Edition, Courier Dover Publications, 2013. Google Scholar

[3]

M. F. Barnsley and K. Leśniak, On the continuity of the Hutchinson operator, preprint, arXiv:1202.2485, 2012. Google Scholar

[4]

W. J. Bock, Functional and evolutionary morphology of woodpeckers, Ostrich, 70 (1999), 23-31. Google Scholar

[5]

J. W. Burdick, Kinematic Analysis and Design of Redundant Robot Manipulators, Diss. Stanford University, 1988. Google Scholar

[6]

G. S. Chirikjian and J. W. Burdick, An obstacle avoidance algorithm for hyper-redundant manipulators, IEEE International Conference on Robotics and Automation, 1 (1990), 625-631. doi: 10.1109/ROBOT.1990.126052.  Google Scholar

[7]

G. S. Chirikjian and J. W. Burdick, The kinematics of hyper-redundant robot locomotion, IEEE Transactions on Robotics and Automation, 11 (1995), 781-793. doi: 10.1109/70.478426.  Google Scholar

[8]

Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set, Mathematics of Control, Signals and Systems, 14 (2001), 173-193. doi: 10.1007/PL00009881.  Google Scholar

[9]

N. Dubbini, B. Piccoli and A. Bicchi, Left invertibility of discrete systems with finite inputs and quantised output, International Journal of Control, 83 (2010), 798-809. doi: 10.1080/00207170903438069.  Google Scholar

[10]

J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[11]

E.-U. Imme and G. S. Chirikjian, Inverse kinematics of discretely actuated hyper-redundant manipulators using workspace densities, in Proceedings of 1996 IEEE International Conference on Robotics and Automation. Vol. 1, IEEE, 1996, 139-145. Google Scholar

[12]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Mathematical Research Letters, 8 (2001), 535-543. doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[13]

V. Komornik, A. C. Lai and M. Pedicini, Generalized golden ratios of ternary alphabets, J. Eur. Math. Soc., 13 (2011), 1113-1146. doi: 10.4171/JEMS/277.  Google Scholar

[14]

V. Komornik and P. Loreti, Unique developments in non-integer bases, American Mathematical Monthly, 105 (1998), 636-639. doi: 10.2307/2589246.  Google Scholar

[15]

A. C. Lai and P. Loreti, Robot's finger and expansions in non-integer bases, Networks and Heterogeneus Media, 7 (2012), 71-111. doi: 10.3934/nhm.2012.7.71.  Google Scholar

[16]

A. C. Lai and P. Loreti, From discrete to continuous reachability for a robot's finger model, Communications in Industrial and Applied Mathematics, 3 (2012), e-439, 22pp.  Google Scholar

[17]

M. D. Lichter, V. A. Sujan and S. Dubowsky, Computational issues in the planning and kinematics of binary robots, in Proceedings of ICRA'02 IEEE International Conference on Robotics and Automation. Vol. 1, IEEE, 2002, 341-346. doi: 10.1109/ROBOT.2002.1013384.  Google Scholar

[18]

L. Mederreg, et al., The RoboCoq Project: Modelling and Design of Bird-like Robot, 6th International Conference on Climbing and Walking Robots, CLAWAR, 2003. Google Scholar

[19]

A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. (Bucur.), 11 (2009), 21-32.  Google Scholar

[20]

B. Mishra, Grasp Metrics: Optimality and Complexity, Proceedings of the Workshop on Algorithmic Foundations of Robotics, AK Peters, Ltd., 1995. Google Scholar

[21]

R. Norberg, Treecreeper climbing, mechanics, energetics, and structural adaptations, Ornis Scandinavica, (1986), 191-209. Google Scholar

[22]

M. Pedicini, Greedy expansions and sets with deleted digits, Theoretical Computer Science, 332 (2005), 313-336. doi: 10.1016/j.tcs.2004.11.002.  Google Scholar

[23]

Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proceedings of the American Mathematical Society, 129 (2001), 2689-2699. doi: 10.1090/S0002-9939-01-05969-X.  Google Scholar

[24]

T. Quinn and J. Baumel, The digital tendon locking mechanism of the avian foot (Aves), Zoomorphology, 109 (1990), 281-293. doi: 10.1007/BF00312195.  Google Scholar

[25]

N. A. Secelean, Masura si Fractali, Univ. Lucian Blaga, Sibiu, 2002. Google Scholar

[26]

D. Sustaita, et al., Getting a Grip on Tetrapod Grasping: Form, Function, and Evolution, Biological Reviews, 88 (2013), 380-405. Google Scholar

[27]

A. V. Zinoviev and F. Ya Dzerzhinsky, Some general notes on the avian hindlimb biomechanics, Bulletin of Moscow Society of Naturalists, 105 (2000), p5. Google Scholar

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