Inertial navigation technology composed of inertial sensors is widely used in foot-mounted pedestrian positioning. However, inertial sensors are susceptible to noise, which affects the performance of the system. The zero-velocity update (ZUPT) as a traditional method is utilized to suppress the cumulative error. Unfortunately, the walking distance calculated by a Kalman filter still has position error. To improve the positioning accuracy, a nonlinear Kalman filter with spatial distance inequality constraint for single foot is proposed in this work. Since the stride distance between adjacent stance phases has an upper bound in plane and height, an inertial navigation system (INS) established by one inertial measurement unit (IMU) is adopted to constrain the stride process. Eventually, the performance of the proposed method is verified by experiments. Compared to the single foot-mounted ZUPT method, the proposed method suppresses the plane error and the height error by 46.04% and 65.48%, respectively. For the dual foot constraint method, the proposed constraint method can reduce the number of sensors while ensuring the positioning accuracy. Moreover, the height error is reduced by 59.98% on average by optimizing the constraint algorithm. The experimental results show that the trajectory estimated by the proposed method is closer to the actual path.
Multimineral log analysis is a quantitative formation evaluation tool for geological and petrophysical reservoir characterization. Rock composition can be estimated by solving equations that relate log measurements to the petrophysical endpoints of minerals and fluids. Due to errors in log data and uncertainties in petrophysical endpoints of constituents, we propose using effective medium models from rock physics as additional independent information to validate or constrain the results. In this paper, we examine the Voigt-Reuss (VR) bound model, self-consistent approximation (SCA), and differential effective medium (DEM). The VR bound model provides the first-order quality control of multimineral results. We first show a conventional carbonate reservoir study with intervals where the predicted effective medium models from multimineral results are inconsistent with the measured elastic properties. We use the VR bound model as an inequality constraint in multimineral analysis for plausible alternative solutions. SCA and DEM models provide good estimates in low porosity intervals and imply geological information for the porous intervals. Then, we show a field case of the Bakken and Three Forks formations. A linear interpolation of the VR bound model helps validate multimineral results and approximate the elastic moduli of clay. There are two major advantages to use our new method (a) rock physics effective medium models provide independent quality control of petrophysical multimineral results, and (b) multimineral information leads to realistic rock physics models.
Статья посвящена разработке математических соотношений для построения алгоритма оценивания параметров сигналов в условиях ограничений. При работе транспортной системы возникают довольно сложные проблемы, которые связаны с необходимостью проведения оценки принятых параметров с требованиями соблюдения имеющихся ограничений. Ограничения могут представлять собой как равенства, так и неравенства. Поскольку ограничения-неравенства могут быть сведены путём добавления фиктивных переменных к условиям, а также их можно проверить по шагам, переводя в состав равенства, в статье разработан алгоритм, позволяющий иметь ограничения-равенства. Данная задача относится к классу статистических проблем оптимизации. Для ее решения использованы стандартные функции из подкаталога "optimization" вычислительной среды MatLAB. Построение такого алгоритма даст возможность не только уменьшить складские расходы, но и сократить основное производственное время.
The article is devoted to the development of mathematical relationships for constructing an algorithm for estimating signal parameters under constraints. During the operation of the transport system, rather complex problems arise, which are associated with the need to assess the adopted parameters with the requirements of compliance with the existing restrictions. Constraints can be either equality or inequality. Since the inequality constraint can be reduced by adding dummy variables to the equality conditions, and they can also be checked step by step, transforming them into equality, we will develop an algorithm that allows us to have equality constraints. This task belongs to the class of statistical optimization problems. To solve it, standard functions from the "optimization" subdirectory of the MatLAB computing environment will be used. The construction of such an algorithm will make it possible not only to reduce storage costs, but also to reduce the main production time.