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  September 2020 Early access  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.

[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.

[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.

[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. 

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[24]

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

[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.

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

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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. 

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[24]

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

[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.

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

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

[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.

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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