By using very simple considerations related to the mechanical Lagrangian itself, one obtains a new general class of parameter adaptation algorithms for robot manipulators, which provides such approaches as PID adaptation schemes. These models could apply to random structural mechanical Systems, subject to the conditon that they are defined by Lagrangians.

A large deviation theorem of the Cramér–Petrov type and a ranking limit theorem of Loève are used to derive an approximation for the statistical distribution of the failure time of fibrous materials. For that, fibrous materials are modeled as a series of independent and identical bundles of parallel filaments and the asymptotic distribution of their failure time is determined in terms of statistical characteristics of the individual filaments, as both the number of filaments in each bundle and the number of bundles in the chain grow large simultaneously. While keeping the number n of filaments in each bundle fixed and increasing only the chain length k leads to a Weibull limiting distribution for the failure time, letting both increase in such a way that log k(n) = o(n), we show that the limit distribution is for . Since fibrous materials which are both long and have many filaments prevail, the result is of importance in the materials science area since refined approximations to failure-time distributions can be achieved.