The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Variational Principles in Mechanics

Temperature drift modeling of MEMS gyroscope based on genetic-Elman neural network

Dual-optimization for a MEMS-INS/GPS system during GPS outages based on the cubature Kalman filter and neural networks

Introduction: Global Positioning Systems (GPS) offer many benefits in assisting school transportation operations achieve their mission. A variety of systems exist with varying capabilities. These systems offer not only positioning information in nearly real time and/or delayed time, but also vehicle position history, travel route, with speed and time, student identification, emergency alert, and automated accident notification. Recognizing that many systems are available that provide this information, and further, that this information is valuable in efficiently managing the school bus fleet, and further, that standardization of minimum information across the State of Michigan would be beneficial to school districts, the Pupil Transportation Advisory Committee recommends the Michigan Department of Education adopt the following as a recommended guideline for purchasing GPS equipment. ________________________________________________________________

Global Positioning Systems

In this article, we investigate the neural adaptive quantized control problem for microelectromechanical system (MEMS) gyroscope with full-state constraints and lumped disturbances. With two different kinds of one-to-one nonlinear mappings, the traditional gyroscope model with matched disturbances is transformed into an unconstrained one with both unmatched and matched disturbances, thus the predefined time-varying state constraints imposed on MEMS gyroscope can be achieved. To compensate for the lumped disturbances, a state estimator-based minimal learning parameter neural network is proposed to obtain fast and smooth disturbance estimates for both position and velocity control loops, which not only can eliminate the poor transient behaviors that widely appear in the available neural adaptive control with a large adaptive gain, but also greatly reduce the number of leaning parameters updated online. Furthermore, by employing a hysteresis logarithmic quantizer, the neglected difficulty, named as constrained data bandwidth of actuator can be overcome with less chattering in control signal, which is more convenient to implement. Finally, the neural adaptive control for MEMS gyroscope is developed such that satisfactory tracking performance is achieved despite of large disturbances, full-state constraints as well as quantized input. The effectiveness and advantages of the proposed control method are demonstrated through extensive simulations.

Neural Adaptive Control for MEMS Gyroscope With Full-State Constraints and Quantized Input

Fuzzy adaptive integration scheme for low-cost SINS/GPS navigation system

Special approaches are required for integration of global navigation satellite system (GNSS) with a strap-down inertial navigation system (INS), particularly based on low-cost micro-electro mechanical system (MEMS)-grade inertial sensors. The proposed approach should be computationally efficient, mathematically nonsingular, and executable on small digital signal processor (DSP) modules. This paper presents a new INS/GNSS navigation system based on a direct decentralized integration scheme. Based on the proposed QR-factorized cubature Kalman filter (CKF) structure, two cascade filters are implemented for separate estimation of the orientation attitude-heading angles and the 3-D position/velocity components. Owing to the QR-factorization, the numerical errors in the update process of estimation covariance matrix are removed. Considering the nonlinear dynamics of the strap-down INS as well as the large uncertainties included in stochastic model of the MEMS-grade inertial sensors, the QR-factorized CKF yields enhanced accuracy and reliability compared with the pure Kalman filter. The decentralized integration scheme provides the separate estimation of orientation components from the position and velocity vectors. Therefore, the propagation of the position and velocity estimation errors into the orientation filter section does not occur. The performance of the presented system is assessed through real data of vehicular field tests.

Decentralized INS/GNSS System With MEMS-Grade Inertial Sensors Using QR-Factorized CKF

In this paper, two knowledge based controllers are proposed to overcome the difficulties of a computed torque nonlinear controller (NC) in perfect trajectory tracking of nonholonomic wheeled mobile robots (WMRs). First, the effects of different dynamic models developed in angular and Cartesian coordinate systems are fully examined on the persistent excitation condition and consequently on the trajectory tracking performance of WMRs. Using the dynamic model coordinated in the Cartesian frame as the base of the NC results in perfect compensation of large position off-tracks and unbiased estimation of the plant's unknown parameters. However, using the WMR's dynamic model with rotation angles of driving wheels as the base of nonlinear and fuzzy controllers leads to accurate orientation tracking. Through replacing the proportional and differential terms of the NC by fuzzy functions, a fuzzy nonlinear controller (FNC) is generated. Due to the complicated dynamics of the WMR in which the center of mass does not coincide with the center of rotation, the expert knowledge of fuzzy controllers is extracted considering the rotation angles and rates of driving wheels as input variables. Fuzzy tuning of the NC results in a superior tracking performance against measurement noises, though the control torques are decreased and smoothed significantly. Second, a complete fuzzy controller (FC) is generated to make perfect tracking of the WMR's position and orientation. The local stability analysis of fuzzy controllers is examined considering the corresponding analytical structures as nonlinear controllers. The superior performances of the proposed fuzzy controllers compared to those of the NCs are evaluated through simulations.

From Nonlinear to Fuzzy Approaches in Trajectory Tracking Control of Wheeled Mobile Robots

Design and implementation of SMO for a nonlinear MIMO AHRS

A new adaptive fuzzy neuro-observer is designed.Proposed observer is implemented to MEMS INS/GPS for land vehicle positioning.Nonlinearities and uncertainties are approximated based on fuzzy neural network.Positioning accuracy is increased owing to the fuzzy neural network approximation.Stability of the observer is guaranteed in the sense of Lyapunov. A combined MEMS Inertial Navigation System (INS) with GPS is used to provide position and velocity data of land vehicles. Data fusion of INS and GPS measurements are commonly achieved through a conventional Extended Kalman filter (EKF). Considering the required accurate model of system together with perfect knowledge of predefined error models, the performance of the EKF is decreased due to unmodeled nonlinearities and unknown bias uncertainties of MEMS inertial sensors. Universal knowledge based approximators comprising of neural networks and fuzzy logic methods are capable of approximating the nonlinearities and the uncertainties of practical systems. First, in this paper, a new fuzzy neural network (FNN) function approximator is used to model unknown nonlinear systems. Second, the process of design and real-time implementation of an adaptive fuzzy neuro-observer (AFNO) in integrated low-cost INS/GPS positioning systems is proposed. To assess the long time performance of the proposed AFNO method, wide range tests of a real INS/GPS with a car vehicle have been performed. The unbiased estimation results of the AFNO show the superiority of the proposed method compared with the classic EKF and the adaptive neuro-observer (ANO) including a pure artificial neural network (ANN) function approximator.

Jafar Keighobadi

Papers

Fuzzy adaptive integration scheme for low-cost SINS/GPS navigation system

Decentralized INS/GNSS System With MEMS-Grade Inertial Sensors Using QR-Factorized CKF

From Nonlinear to Fuzzy Approaches in Trajectory Tracking Control of Wheeled Mobile Robots

Design and implementation of SMO for a nonlinear MIMO AHRS

Adaptive fuzzy neuro-observer applied to low cost INS/GPS