## Controlled Lagrangians and stabilization of Euler-Poincaré mechanical systems with broken symmetry. II: Potential shaping.(English)Zbl 1494.93080

Summary: We apply the method of controlled Lagrangians by potential shaping to Euler-Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group $$\mathsf{G}\ltimes V$$ with a particular interest in those systems whose configuration space is the special Euclidean group $$\mathsf{SE}(3) = \mathsf{SO}(3)\ltimes\mathbb{R}^3$$. The key idea behind the work is the use of representations of $$\mathsf{G}\ltimes V$$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting the representations and advected parameters. Our motivating examples are a heavy top spinning on a movable base and an underwater vehicle with non-coincident centers of gravity and buoyancy. We consider a few different control problems for these systems, and show that our results give a general framework that reproduces our previous work on the former example and also those of Leonard on the latter. Also, in one of the latter cases, we demonstrate the advantage of our representation-based approach by giving a simpler and more succinct formulation of the problem.
For Part I, see [the authors, “Controlled Lagrangians and stabilization of Euler-Poincaré mechanical systems with broken symmetry”, Preprint, arXiv:2003.10584].

### MSC:

 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 70Q05 Control of mechanical systems 34H15 Stabilization of solutions to ordinary differential equations 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics

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### References:

 [1] Blankenstein, G.; Ortega, R.; van der Schaft, AJ, The matching conditions of controlled Lagrangians and IDA-passivity based control, Int J Control, 75, 9, 645-665 (2002) · Zbl 1018.93006 [2] Bloch AM, Leonard NE, Marsden JE (1999) Potential shaping and the method of controlled lagrangians. In: Proceedings of the 38th IEEE conference on decision and control (Cat. No.99CH36304), vol. 2, pp 1652-1657 [3] Bloch, AM; Leonard, NE; Marsden, JE, Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Int J Robust Nonlinear Control, 11, 3, 191-214 (2001) · Zbl 0980.93065 [4] Bloch, AM; Leonard, NE; Marsden, JE, Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem, IEEE Trans Autom Control, 45, 12, 2253-2270 (2000) · Zbl 1056.93604 [5] Bloch, AM; Chang, DE; Leonard, NE; Marsden, JE, Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping, IEEE Trans Autom Control, 46, 10, 1556-1571 (2001) · Zbl 1057.93520 [6] Borum AD, Bretl T (2014) Geometric optimal control for symmetry breaking cost functions. In: 53rd IEEE conference on decision and control, pp 5855-5861 [7] Borum, AD; Bretl, T., Reduction of sufficient conditions for optimal control problems with subgroup symmetry, IEEE Trans Autom Control, 99, 3209-3224 (2016) · Zbl 1370.49010 [8] Bullo F, Lewis AD (2004) Geometric control of mechanical systems, volume 49 of texts in applied mathematics. Springer · Zbl 1066.70002 [9] Cendra, H.; Holm, DD; Marsden, JE; Ratiu, TS, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, Amer Math Soc Trans, 186, 1-25 (1998) · Zbl 0989.37052 [10] Chang, DE; Marsden, JE, Reduction of controlled Lagrangian and Hamiltonian systems with symmetry, SIAM J Control Optim, 43, 1, 277-300 (2004) · Zbl 1076.70020 [11] Chang, DE; Bloch, AM; Leonard, NE; Marsden, JE; Woolsey, CA, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: COCV, 8, 393-422 (2002) · Zbl 1070.70013 [12] Chyba M, Haberkorn T, Smith RN, Wilkens GR (2007) Controlling a submerged rigid body: a geometric analysis. In: Bullo F, Fujimoto K (eds), Lagrangian and Hamiltonian methods for nonlinear control 2006. Springer, Berlin, pp 375-385 · Zbl 1140.70479 [13] Contreras C, Ohsawa T (2020) Controlled Lagrangians and stabilization of Euler-Poincaré mechanical systems with broken symmetry I: Kinetic shaping [14] Hamberg J (1999) General matching conditions in the theory of controlled Lagrangians. In: Proceedings of the 38th IEEE conference on decision and control, 1999, vol 3, pp 2519-2523 [15] Hamberg J (2000) Controlled Lagrangians, symmetries and conditions for strong matching. In IFAC Lagrangian and Hamiltonian methods for nonlinear control [16] Holm, DD; Marsden, JE; Ratiu, TS, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv Math, 137, 1, 1-81 (1998) · Zbl 0951.37020 [17] Holm DD, Schmah T, Stoica C (2009) Geometric mechanics and symmetry: from finite to infinite dimensions. Oxford texts in applied and engineering mathematics. Oxford University Press · Zbl 1175.70001 [18] Leonard, NE, Stability of a bottom-heavy underwater vehicle, Automatica, 33, 3, 331-346 (1997) · Zbl 0872.93061 [19] Leonard, NE, Stabilization of underwater vehicle dynamics with symmetry-breaking potentials, Syst Control Lett, 32, 1, 35-42 (1997) · Zbl 0901.93057 [20] Leonard, NE; Marsden, JE, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry, Physica D, 105, 1-3, 130-162 (1997) · Zbl 0963.70528 [21] Marsden, JE; Ratiu, TS, Introduction to mechanics and symmetry (1999), New York: Springer, New York [22] Marsden, JE; Ratiu, TS; Weinstein, A., Semidirect products and reduction in mechanics, Trans Am Math Soc, 281, 1, 147-177 (1984) · Zbl 0529.58011 [23] Marsden JE, Ratiu TS, Weinstein A (1984) Reduction and Hamiltonian structures on duals of semidirect product Lie algebras. In Fluids and plasmas: geometry and dynamics, volume 28 of Contemporary Mathematics. American Mathematical Society · Zbl 0546.58025 [24] Nijmeijer, H.; van der Schaft, A., Nonlinear dynamical control systems (1990), New York: Springer, New York · Zbl 0701.93001 [25] Ortega R, Perez J, Nicklasson P, Sira-Ramirez H (1998) Passivity-based Control of Euler-Lagrange systems: mechanical. Electrical and electromechanical applications. Communications and control engineering. Springer, London [26] Ortega, R.; van der Schaft, AJ; Mareels, I.; Maschke, B., Putting energy back in control, IEEE Control Syst, 21, 2, 18-33 (2001) [27] Ortega, R.; Spong, MW; Gomez-Estern, F.; Blankenstein, G., Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans Autom Control, 47, 8, 1218-1233 (2002) · Zbl 1364.93662 [28] Smith, RN; Chyba, M.; Wilkens, GR; Catone, CJ, A geometrical approach to the motion planning problem for a submerged rigid body, Int J Control, 82, 9, 1641-1656 (2009) · Zbl 1190.93021 [29] Spong, MW; Bullo, F., Controlled symmetries and passive walking, IEEE Trans Autom Control, 50, 7, 1025-1031 (2005) · Zbl 1365.93329 [30] van der Schaft, AJ, Stabilization of Hamiltonian systems, Nonlinear Anal Theory Methods Appl, 10, 10, 1021-1035 (1986) · Zbl 0613.93049 [31] Woolsey, CA; Leonard, NE, Stabilizing underwater vehicle motion using internal rotors, Automatica, 38, 12, 2053-2062 (2002) · Zbl 1015.93044 [32] Woolsey, CA; Techy, L., Cross-track control of a slender, underactuated AUV using potential shaping, Ocean Eng, 36, 1, 82-91 (2009)
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