Chow's Theorem Research Articles

Motion of two rigid bodies with rolling constraint

The motion of two rigid bodies under rolling constraint is considered. In particular, the following two problems are addressed: (1) given the geometry of the rigid bodies, determine the existence of an admissible path between two contact configurations; and (2) assuming that an admissible path exists, find such a path. First, the configuration space of contact is defined, and the differential equations governing the rolling constraint are derived. Then, a generalized version of Frobenius's theorem, known as Chow's theorem, for determining the existence of motion is applied. Finally, an algorithm is proposed that generates a desired path with one of the objects being flat. Potential applications of this study include adjusting grasp configurations of a multifingered robot hand without slipping, contour following without dissipation or wear by the end-effector of a manipulator, and wheeled mobile robotics.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Sufficient conditions for controllability

The problem is to find sufficient conditions for the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t)= f(x((t))+ \sum_{i = 1}^{m} u_{i}(t)g_{i}(x(t)), x(0)=x_{0} \in M</tex> to be controllable. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is a connected <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal C}^{\infty}</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional manifold, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_{1}, \cdots ,g_{m}</tex> are complete <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal C}^{\infty}</tex> vector fields on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u_{1}, \cdots , u_{m}</tex> are real-valued controls. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = n - 1, M,f, g_{1}, \cdots ,g_{n-1}</tex> are real-analytic, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is simply connected, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_{1}, \cdots ,g_{n-1}</tex> are linearly independent on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> , then necessary and sufficient conditions are known. For the case of our <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal C}^{\infty}</tex> system with general <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , we assume that the space spanned by the Lie algebra <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{A}</tex> generated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f, g_{1}, \cdots ,g_{m}</tex> and successive Lie brackets has constant dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> and the algebra <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{A}^{\prime}</tex> generated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_{1}, \cdots ,g_{m}</tex> and successive Lie brackets has constant dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p^{\prime} \leq p</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p^{\prime}=p</tex> , Chow's Theorem implies controllability for a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> -dimensional submanifold of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> containing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{0}</tex> . If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p^{\prime}&lt; p</tex> , sufficient conditions are found involving the computation of certain Lie brackets at points where the vector field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> is tangent to the integral manifolds of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{A}^{\prime}</tex> . Here we assume that every integral manifold of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{A}^{\prime}</tex> contains such a point.

Controllability criterion for systems in a Banach space (generalization of Chow's theorem)

Controllability criterion for systems in a Banach space (generalization of Chow's theorem)

On the Threshold Order of a Boolean Function

The notion of a threshold function is generalized to a Boolean function of threshold order r. Two characterizations of a Boolean function of threshold order r are presented, which are generalizations of the results of Kaplan and Winder and of Chow, for the case r = 1. Kaplan and Winder's characterization of a threshold function by means of Chebyshev approximation is generalized to a Boolean function of threshold order r. This results in classifying any Boolean function as a threshold function of some order r less than or equal to the number of variables. Chow's theorem on the ``equivalence'' between threshold functions and statistical recognition with independent distributions is generalized to the case of a Boolean function of threshold order r and of statistical recognition with a certain kind of dependence, called dependence of order r.