SUMMARYWe address general filtering problems on the Euclidean groupSE(3). We first generalize, to stochastic nonlinear systems evolving onSE(3), the particle filter of Liu and West for simultaneous estimation of the state and covariance. The filter is constructed in a coordinate-invariant way, and explicitly takes into account the geometry ofSE(3) andP(n), the space of symmetric positive definite matrices. Some basic results for bilinear systems onSE(3) with linear and quadratic measurements are also derived. Three examples—GPS attitude estimation, needle tip location, and vision-based robot end-effector pose estimation—are presented to illustrate the framework.
This paper examines an alternative approach to interest rate modeling, in which the nonlinear and random behavior of interest rates is captured by a stochastic differential equation evolving on a curved state space. We consider as candidate state spaces the matrix Lie groups; these offer not only a rich geometric structure, but—unlike general Riemannian manifolds—also allow for diffusion processes to be constructed easily without invoking the machinery of stochastic calculus on manifolds. After formulating bilinear stochastic differential equations on general matrix Lie groups, we then consider interest rate models in which the short rate is defined as linear or quadratic functions of the state. Stochastic volatility is also augmented to these models in a way that respects the Riemannian manifold structure of symmetric positive-definite matrices. Methods for numerical integration, parameter identification, pricing, and other practical issues are addressed through examples.Financial mathematics, Financial engineering, Interest rate modelling, Affine term structure models,
This paper formulates the developable surface design problem in an optimal control setting. Given a regular curve bt on the unit sphere corresponding to a one-parameter family of rulings, and two base curve endpoints a0,a1∈R3, we consider the problem of constructing a base curve at such that at0=a0,at1=a1, and the resulting surface fs,t=at+sbt is developable. We formulate the base curve design problem as an optimal control problem, and derive solutions for objective functions that reflect various practical aspects of developable surface design, e.g., minimizing the arc length of the base curve, keeping the line of regression distant from the base curve, and approximating a given arbitrary ruled surface by a developable surface. By drawing upon the large body of available results for the optimal control of linear systems with quadratic criteria, our approach provides a flexible method for designing developable surfaces that are optimized for various criteria.
In this paper, a light-weight and wearable haptic glove system is introduced which is designed for virtual environments. In order to reduce the weight of the system, micro McKibben artificial muscles are used and 2-port solenoid valve pneumatic system was developed for faster response. Hydraulic system was also developed to actuate artificial muscles, so that we can overcome softness and inaccuracy of pneumatic system due to the compressibility of air. We verify the functionality and usefulness of the proposed system by synchronizing it with a virtual environment.
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