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Controlled invariants and trajectory planning for underactuated mechanical systems
Norwegian Univ Sci & Technol NTNU, Dept Engn Cybernet, Trondheim, Norway.
Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.
Univ Texas Dallas, Erik Jonsson Sch Engn & Comp Sci, Richardson, TX 75080 USA.
2014 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 59, no 9, 2555-2561 p.Article in journal (Refereed) Published
Abstract [en]

We study the problem of motion planning for underactuated mechanical systems. The idea is to reduce complexity by imposing via feedback a sufficient number of invariants and then to compute a projection of the dynamics onto an induced invariant sub-manifold of the closed-loop system. The inspiration comes from two quite distant methods, namely the method of virtual holonomic constraints, originally invented for planning and orbital stabilization of gaits of walking machines, and the method of controlled Lagrangians, primarily invented as a nonlinear technique for stabilization of (relative) equilibria of controlled mechanical systems. Both of these techniques enforce the presence of particular invariants that can be described as level sets of conserved quantities induced in the closed-loop system. We link this structural feature of both methods to a procedure to transform a Lagrangian system with control inputs via a feedback action into a control-free Lagrangian system with a sufficient number of first integrals for the full state space or an invariant sub-manifold. In both cases, this transformation allows efficient (analytical) description of a new class of trajectories of forced mechanical systems appropriate for further orbital stabilization. For illustration purposes, we approach the challenging problem for a controlled mechanical system with two passive degrees of freedom: planning periodic (or bounded) forced upperhemisphere trajectories of the spherical pendulum on a puck. Another example of the technique is separately reported in [21].

Place, publisher, year, edition, pages
2014. Vol. 59, no 9, 2555-2561 p.
Keyword [en]
Controlled Lagrangians, feedback equivalence, motion and trajectory planning, spherical pendulum a puck, underactuated mechanical systems, virtual holonomic constraints
National Category
Electrical Engineering, Electronic Engineering, Information Engineering
URN: urn:nbn:se:umu:diva-95875DOI: 10.1109/TAC.2014.2308641ISI: 000342924100028OAI: diva2:770903
Available from: 2014-12-11 Created: 2014-11-06 Last updated: 2016-02-15Bibliographically approved

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Shiriaev, Anton S.Freidovich, Leonid B.
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