Bulletin of the Japan Society for Industrial and Applied Mathematics

In this paper, we discuss a control scheme for a biped robot. There are two ways for designing a control law for a biped robot. One is a method based on a ZMP(Zero moment point), and the other is a method based on PDW(Passive Dynamic Walking). In this paper, we claim that the PDW is very important and essential in walking, and introduce a design concept based on the idea of PDW. The construction of this paper is the following. In Chapter 1, we briefly introduce the motivation of our research. In Chapter 2, we explain the idea of the ZMP. In Chapter 3, we show a history of the research of PDW. In Chapter 4, we show some interesting aspects in PDW. In Chapter 5, some design issues in a walking robot based onPDW. In Chapter 6, we show some future themes. Finally in Chapter 7, we conclude.

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We consider the vicious walker model, which was introduced by Michael Fisher in his Boltzmann medal lecture in 1983 as a mathematical model of wetting and melting phenomena. It is a system of particles performing noncolliding random walk in one dimension. Using nonintersecting property of the paths of vicious walkers and by elementary calculus of deter minants, we show that the Green function of the system is equal to the Schur function, which plays an important role in the representation theory of symmetric group, and its two kinds of determinantal expressions are derived. MacMahon conjecture, Bender-Knuth conjecture and Macdonald equality for the summations of Schur functions are discussed from the viewpoint of vicious walker model. By taking the diffusion scaling limit of the vicious walker model, a system of noncolliding Brownian particles is constructed and its relation to the distribution of eigenvalues of real symmetric random matrices is clarified.

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In this paper, we consider a method for finding certain eigenvalues of generalized eigenvalue problems in a given domain of the complex plane. We also discuss the relation between the presented method and the Lanczos method briefly. The presented method provides a good performance in parallel computations. A numerical example that was obtained on a PC cluster is included.

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We give an overview of the studies of stochastic control and filtering theory, tracing the historical situation from Kalman-Bucy filtering, LQG stochastic control theory and their mathematical generalization to nonlinear systems and nonlinear filtering to H^∞ control and risk-sensitive stochastic control. Then we explain how we could formulate portfolio optimization problems for Merton's ICAPM, which are typical ones on mathematical finance, as risk-sensitive stochastic control problems based on understanding the situation, and analyze them by employing the methods established through such studies. Dynamic programming approach to stochastic control and the methods of measure change in nonlinear filtering apply to obtain explicit representation of optimal strategies for the portfolio optimization problems. More other aspects could be seen.

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