September  2020, 12(3): 395-420. doi: 10.3934/jgm.2020022

Control of locomotion systems and dynamics in relative periodic orbits

1. 

Università di Padova Dipartimento di Matematica "Tullio Levi-Civita", Via Trieste 63, 35121 Padova, Italy

2. 

Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Dedicated to James Montaldi

Received  December 2019 Revised  June 2020 Published  July 2020

Fund Project: FF has been partially supported by the MIUR-PRIN project 20178CJA2B New Frontiers of Celestial Mechanics: theory and applications. MZ gratefully acknowledges support from the MIUR grant Dipartimenti di Eccellenza 2018-2022 (CUP: E11G18000350001)

The connection between the dynamics in relative periodic orbits of vector fields with noncompact symmetry groups and periodic control for the class of control systems on Lie groups known as '(robotic) locomotion systems' is well known, and has led to the identification of (geometric) phases. We take an approach which is complementary to the existing ones, advocating the relevance——for trajectory generation in these control systems——of the qualitative properties of the dynamics in relative periodic orbits. There are two particularly important features. One is that motions in relative periodic orbits of noncompact groups can only be of two types: either they are quasi-periodic, or they leave any compact set as $ t\to\pm\infty $ ('drifting motions'). Moreover, in a given group, one of the two behaviours may be predominant. The second is that motions in a relative periodic orbit exhibit 'spiralling', 'meandering' behaviours, which are routinely detected in numerical integrations. Since a quantitative description of meandering behaviours for drifting motions appears to be missing, we provide it here for a class of Lie groups that includes those of interest in locomotion (semidirect products of a compact group and a normal vector space). We illustrate these ideas on some examples (a kinematic car robot, a planar swimmer).

Citation: Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. Journal of Geometric Mechanics, 2020, 12 (3) : 395-420. doi: 10.3934/jgm.2020022
References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616.  doi: 10.1088/0951-7715/10/3/002.  Google Scholar

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Alberto Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete Contin. Dyn. Syst., 20 (2008), 1-35.  doi: 10.3934/dcds.2008.20.1.  Google Scholar

[4]

Aldo Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 19 (1990), 197-246.   Google Scholar

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B. BittnerR. L. Hatton and S. Revzen, Geometrically optimal gaits: A data-driven approach, Nonlinear Dynamics, 94 (2018), 1933-1948.  doi: 10.1007/s11071-018-4466-9.  Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Controls, Springer–Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[7]

A. V. BorisovA. A. Kilin and I. S. Mamaev, How to control the Chaplygin ball using rotors Ⅱ, Regul. Chaotic Dyn., 18 (2013), 144-158.  doi: 10.1134/S1560354713010103.  Google Scholar

[8]

T. Bröcker and T. Dieck, Representations of Compact Lie Groups, Springer, New York, 1985. doi: 10.1007/978-3-662-12918-0.  Google Scholar

[9]

R. W. Brockett, Systems theory on group manifolds and coset spaces, SIAM J. Control, 10 (1972), 265-284.  doi: 10.1137/0310021.  Google Scholar

[10]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[11]

A. Cabrera, Base-controlled mechanical systems and geometric phases, Journal of Geometry and Physics, 58 (2008), 334-367.  doi: 10.1016/j.geomphys.2007.11.009.  Google Scholar

[12]

T. Chambrion and A. Munnier, Generalized scallop theorem for linear swimmers, preprint, INRIA-00508646 (2010).  Google Scholar

[13]

T. Chambrion and A. Munnier, Generic controllability of 3D swimmers in a perfect fluid, SIAM J. Control Optim., 50 (2012), 2814-2835.  doi: 10.1137/110828654.  Google Scholar

[14]

G. Cicconofri and A. DeSimone, Modelling biological and bio-inspired swimming at microscopic scales: Recent results and perspectives, Comput. & Fluids, 179 (2019), 799-805.  doi: 10.1016/j.compfluid.2018.07.020.  Google Scholar

[15]

R. Cushman, J. J. Duistermaat and J. Śnyaticki, Geometry of Nonholonomically Constrained Systems, World Scientific, Singapore, 2010. doi: 10.1142/7509.  Google Scholar

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J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Springer, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

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F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 12 pp. doi: 10.3842/SIGMA.2007.051.  Google Scholar

[18]

F. FassòL. C. García-Naranjo and A. Giacobbe, Quasi-periodicity in relative quasi-periodic tori, Nonlinearity, 28 (2015), 4281-4301.  doi: 10.1088/0951-7715/28/11/4281.  Google Scholar

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F. FassòL. C. García-Naranjo and J. Montaldi, Integrability and dynamics of the n-dimensional symmetric Veselova top, J. Nonlinear Sci., 29 (2019), 1205-1246.  doi: 10.1007/s00332-018-9515-5.  Google Scholar

[20]

V. Fedonyuk and P. Tallapragada, Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh, Nonlinear Dynamics, 93 (2018), 835-846.  doi: 10.1007/s11071-018-4230-1.  Google Scholar

[21]

Y. N. FedorovL. C. García-Naranjo and J. Vankerschaver, The motion of the 2d hydrodynamical Chaplygin sleigh in the presence of circulation, Discrete Contin. Dyn. Syst., 33 (2013), 4017-4040.  doi: 10.3934/dcds.2013.33.4017.  Google Scholar

[22]

B. FiedlerB. SandstedeA. Scheel and C. Wulff, Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts, Doc. Math., 1 (1996), 479-505.   Google Scholar

[23]

B. Fiedler and D. Turaev, Normal Forms, Resonances, and Meandering Tip Motions near Relative Equilibria of Euclidean Group Actions, Arch. Rational Mech. Anal., 145 (1998), 129-159.  doi: 10.1007/s002050050126.  Google Scholar

[24]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[25]

M. J. Field, Local structure for equivariant dynamics, in Singularity Theory an its Applications. Part II (eds. M. Roberts and I. Stewart), Lecture Notes in Mathematics, 1463, Springer, Berlin, 1991,142–166. doi: 10.1007/BFb0085430.  Google Scholar

[26] M. J. Field, Dynamics and Symmetry, Imperial College Press, London, 2007.  doi: 10.1142/p515.  Google Scholar
[27]

L. Giraldi and F. Jean, Periodical body's deformations are optimal strategies for locomotion, preprint, hal-02266220, 2019. doi: 10.1137/19M1280120.  Google Scholar

[28]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, Birhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[29]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[30]

V. Jurdjevic and H. Sussmann, Control systems on Lie groups, J. Differential Equations, 12 (1972), 313-329.  doi: 10.1016/0022-0396(72)90035-6.  Google Scholar

[31]

S. D. Kelly and R. M. Murray, Geometric phases and robotic locomotion, Journal of Robotic Systems, 12 (1995), 417-431.  doi: 10.1002/rob.4620120607.  Google Scholar

[32]

V. V. Kozlov and S. M. Ramodanov, On the motion of a variable body through an ideal fluid, J. Appl. Math. Mech., 65 (2001), 579-587.  doi: 10.1016/S0021-8928(01)00063-6.  Google Scholar

[33]

M. Krupa, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), 1453-1486.  doi: 10.1137/0521081.  Google Scholar

[34]

J. Lohéac and A. Munnier, Controllability of 3D low Reynolds number swimmers, ESAIM Control Optim. Calc. Var., 20 (2014), 236-268.  doi: 10.1051/cocv/2013063.  Google Scholar

[35]

C. M. Marle, Géométrie des systèmes mécaniques à liaisons actives, in Symplectic Geometry and Mathematical Physics (eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman), Birkhäuser, Boston, 1991,260–287.  Google Scholar

[36]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, $2^nd$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[37]

J. E. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990), no. 436. doi: 10.1090/memo/0436.  Google Scholar

[38]

R. Mason and J. Burdick, Propulsion and control of deformable bodies in a ideal fluid, Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C), 1 (1999), 773–780. doi: 10.1109/ROBOT.1999.770068.  Google Scholar

[39]

R. M. Murray and S. S. Sastry, Nonholonomic motion planning: Steering using sinusoids, IEEE Trans. Automat. Control, 38 (1993), 700-716.  doi: 10.1109/9.277235.  Google Scholar

[40]

E. M. Purcell, Life at low Reynolds number, American Journal of Physics, 45 (1977), 3-11.  doi: 10.1063/1.30370.  Google Scholar

[41]

N. Sansonetto and M. Zoppello, On the trajectory generation of the hydrodynamic Chaplygin sleigh, IEEE Control System Letters, 4 (2020), 922-927.  doi: 10.1109/LCSYS.2020.2996763.  Google Scholar

[42]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585.  doi: 10.1017/S002211208900025X.  Google Scholar

[43]

G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London Ser. A, 209 (1951), 447-461.  doi: 10.1098/rspa.1951.0218.  Google Scholar

[44]

M. Zoppello and F. Cardin, Swim-like motion of bodies immersed in an ideal fluid, ESAIM Control Optim. Calc. Var., 25 (2019), 38 pp. doi: 10.1051/cocv/2017028.  Google Scholar

show all references

References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616.  doi: 10.1088/0951-7715/10/3/002.  Google Scholar

[3]

Alberto Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete Contin. Dyn. Syst., 20 (2008), 1-35.  doi: 10.3934/dcds.2008.20.1.  Google Scholar

[4]

Aldo Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 19 (1990), 197-246.   Google Scholar

[5]

B. BittnerR. L. Hatton and S. Revzen, Geometrically optimal gaits: A data-driven approach, Nonlinear Dynamics, 94 (2018), 1933-1948.  doi: 10.1007/s11071-018-4466-9.  Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Controls, Springer–Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[7]

A. V. BorisovA. A. Kilin and I. S. Mamaev, How to control the Chaplygin ball using rotors Ⅱ, Regul. Chaotic Dyn., 18 (2013), 144-158.  doi: 10.1134/S1560354713010103.  Google Scholar

[8]

T. Bröcker and T. Dieck, Representations of Compact Lie Groups, Springer, New York, 1985. doi: 10.1007/978-3-662-12918-0.  Google Scholar

[9]

R. W. Brockett, Systems theory on group manifolds and coset spaces, SIAM J. Control, 10 (1972), 265-284.  doi: 10.1137/0310021.  Google Scholar

[10]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[11]

A. Cabrera, Base-controlled mechanical systems and geometric phases, Journal of Geometry and Physics, 58 (2008), 334-367.  doi: 10.1016/j.geomphys.2007.11.009.  Google Scholar

[12]

T. Chambrion and A. Munnier, Generalized scallop theorem for linear swimmers, preprint, INRIA-00508646 (2010).  Google Scholar

[13]

T. Chambrion and A. Munnier, Generic controllability of 3D swimmers in a perfect fluid, SIAM J. Control Optim., 50 (2012), 2814-2835.  doi: 10.1137/110828654.  Google Scholar

[14]

G. Cicconofri and A. DeSimone, Modelling biological and bio-inspired swimming at microscopic scales: Recent results and perspectives, Comput. & Fluids, 179 (2019), 799-805.  doi: 10.1016/j.compfluid.2018.07.020.  Google Scholar

[15]

R. Cushman, J. J. Duistermaat and J. Śnyaticki, Geometry of Nonholonomically Constrained Systems, World Scientific, Singapore, 2010. doi: 10.1142/7509.  Google Scholar

[16]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Springer, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[17]

F. Fassò and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 12 pp. doi: 10.3842/SIGMA.2007.051.  Google Scholar

[18]

F. FassòL. C. García-Naranjo and A. Giacobbe, Quasi-periodicity in relative quasi-periodic tori, Nonlinearity, 28 (2015), 4281-4301.  doi: 10.1088/0951-7715/28/11/4281.  Google Scholar

[19]

F. FassòL. C. García-Naranjo and J. Montaldi, Integrability and dynamics of the n-dimensional symmetric Veselova top, J. Nonlinear Sci., 29 (2019), 1205-1246.  doi: 10.1007/s00332-018-9515-5.  Google Scholar

[20]

V. Fedonyuk and P. Tallapragada, Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh, Nonlinear Dynamics, 93 (2018), 835-846.  doi: 10.1007/s11071-018-4230-1.  Google Scholar

[21]

Y. N. FedorovL. C. García-Naranjo and J. Vankerschaver, The motion of the 2d hydrodynamical Chaplygin sleigh in the presence of circulation, Discrete Contin. Dyn. Syst., 33 (2013), 4017-4040.  doi: 10.3934/dcds.2013.33.4017.  Google Scholar

[22]

B. FiedlerB. SandstedeA. Scheel and C. Wulff, Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts, Doc. Math., 1 (1996), 479-505.   Google Scholar

[23]

B. Fiedler and D. Turaev, Normal Forms, Resonances, and Meandering Tip Motions near Relative Equilibria of Euclidean Group Actions, Arch. Rational Mech. Anal., 145 (1998), 129-159.  doi: 10.1007/s002050050126.  Google Scholar

[24]

M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205.  doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[25]

M. J. Field, Local structure for equivariant dynamics, in Singularity Theory an its Applications. Part II (eds. M. Roberts and I. Stewart), Lecture Notes in Mathematics, 1463, Springer, Berlin, 1991,142–166. doi: 10.1007/BFb0085430.  Google Scholar

[26] M. J. Field, Dynamics and Symmetry, Imperial College Press, London, 2007.  doi: 10.1142/p515.  Google Scholar
[27]

L. Giraldi and F. Jean, Periodical body's deformations are optimal strategies for locomotion, preprint, hal-02266220, 2019. doi: 10.1137/19M1280120.  Google Scholar

[28]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, Birhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[29]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[30]

V. Jurdjevic and H. Sussmann, Control systems on Lie groups, J. Differential Equations, 12 (1972), 313-329.  doi: 10.1016/0022-0396(72)90035-6.  Google Scholar

[31]

S. D. Kelly and R. M. Murray, Geometric phases and robotic locomotion, Journal of Robotic Systems, 12 (1995), 417-431.  doi: 10.1002/rob.4620120607.  Google Scholar

[32]

V. V. Kozlov and S. M. Ramodanov, On the motion of a variable body through an ideal fluid, J. Appl. Math. Mech., 65 (2001), 579-587.  doi: 10.1016/S0021-8928(01)00063-6.  Google Scholar

[33]

M. Krupa, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), 1453-1486.  doi: 10.1137/0521081.  Google Scholar

[34]

J. Lohéac and A. Munnier, Controllability of 3D low Reynolds number swimmers, ESAIM Control Optim. Calc. Var., 20 (2014), 236-268.  doi: 10.1051/cocv/2013063.  Google Scholar

[35]

C. M. Marle, Géométrie des systèmes mécaniques à liaisons actives, in Symplectic Geometry and Mathematical Physics (eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman), Birkhäuser, Boston, 1991,260–287.  Google Scholar

[36]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, $2^nd$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[37]

J. E. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990), no. 436. doi: 10.1090/memo/0436.  Google Scholar

[38]

R. Mason and J. Burdick, Propulsion and control of deformable bodies in a ideal fluid, Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C), 1 (1999), 773–780. doi: 10.1109/ROBOT.1999.770068.  Google Scholar

[39]

R. M. Murray and S. S. Sastry, Nonholonomic motion planning: Steering using sinusoids, IEEE Trans. Automat. Control, 38 (1993), 700-716.  doi: 10.1109/9.277235.  Google Scholar

[40]

E. M. Purcell, Life at low Reynolds number, American Journal of Physics, 45 (1977), 3-11.  doi: 10.1063/1.30370.  Google Scholar

[41]

N. Sansonetto and M. Zoppello, On the trajectory generation of the hydrodynamic Chaplygin sleigh, IEEE Control System Letters, 4 (2020), 922-927.  doi: 10.1109/LCSYS.2020.2996763.  Google Scholar

[42]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585.  doi: 10.1017/S002211208900025X.  Google Scholar

[43]

G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London Ser. A, 209 (1951), 447-461.  doi: 10.1098/rspa.1951.0218.  Google Scholar

[44]

M. Zoppello and F. Cardin, Swim-like motion of bodies immersed in an ideal fluid, ESAIM Control Optim. Calc. Var., 25 (2019), 38 pp. doi: 10.1051/cocv/2017028.  Google Scholar

Figure 1.  The phase
Figure 2.  Images of gaits
Figure 3.  The car robot
Figure 4.  Four trajectories of a point of the car's frame in the $ (x,y) $-plane. The gaits have $ \dot\psi_2^\ell = 1 $ and $ \phi^\ell $ as shown in the insets. The coordinates in the insets' plots are time (horizontal) and $ \phi^\ell $ (vertical). In all cases $ \lambda = 2.5 $, $ a = 0.4 $ and the initial configuration of the car is $ (\theta_0,x_0,y_0) = (\pi/4,0,0) $. The value of $ \theta^\ell(T) $ is $ 0 $ in (a), $ 2\pi $ in (b), approximately $ 0.262\,\pi $ in (c) and approximately $ 0.727\pi $ in (d)
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