1,687 research outputs found
Input to State Stability of Bipedal Walking Robots: Application to DURUS
Bipedal robots are a prime example of systems which exhibit highly nonlinear
dynamics, underactuation, and undergo complex dissipative impacts. This paper
discusses methods used to overcome a wide variety of uncertainties, with the
end result being stable bipedal walking. The principal contribution of this
paper is to establish sufficiency conditions for yielding input to state stable
(ISS) hybrid periodic orbits, i.e., stable walking gaits under model-based and
phase-based uncertainties. In particular, it will be shown formally that
exponential input to state stabilization (e-ISS) of the continuous dynamics,
and hybrid invariance conditions are enough to realize stable walking in the
23-DOF bipedal robot DURUS. This main result will be supported through
successful and sustained walking of the bipedal robot DURUS in a laboratory
environment.Comment: 16 pages, 10 figure
Relative Critical Points
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are
critical points of appropriate scalar functions parametrized by the Lie algebra
(or its dual) of the symmetry group. Setting aside the structures - symplectic,
Poisson, or variational - generating dynamical systems from such functions
highlights the common features of their construction and analysis, and supports
the construction of analogous functions in non-Hamiltonian settings. If the
symmetry group is nonabelian, the functions are invariant only with respect to
the isotropy subgroup of the given parameter value. Replacing the parametrized
family of functions with a single function on the product manifold and
extending the action using the (co)adjoint action on the algebra or its dual
yields a fully invariant function. An invariant map can be used to reverse the
usual perspective: rather than selecting a parametrized family of functions and
finding their critical points, conditions under which functions will be
critical on specific orbits, typically distinguished by isotropy class, can be
derived. This strategy is illustrated using several well-known mechanical
systems - the Lagrange top, the double spherical pendulum, the free rigid body,
and the Riemann ellipsoids - and generalizations of these systems
On the stratification of a class of specially structured matrices
We consider specially structured matrices representing optimization problems with quadratic objective functions and (finitely many) affine linear equality constraints in an n-dimensional Euclidean space. The class of all such matrices will be subdivided into subsets ['strata'], reflecting the features of the underlying optimization problems. From a differential-topological point of view, this subdivision turns out to be very satisfactory: Our strata are smooth manifolds, constituting a so-called Whitney Regular Stratification, and their dimensions can be explicitly determined. We indicate how, due to Thom's Transversality Theory, this setting leads to some fundamental results on smooth one-parameter families of linear-quadratic optimization problems with ( finitely many) equality and inequality constraints
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