Lie group based nonlinear state errors for MEMS-IMU/GNSS/magnetometer integrated navigation

In the integrated navigation system using extended Kalman filter (EKF), the state error conventionally uses linear approximation to tackle the commonly nonlinear problem. However, this error definition can diverge the filter in some adverse situations due to significant distortion of the linear approximation. By contrast, the nonlinear state error defined in the Lie group satisfies the autonomous equation, which thus has distinctively better convergence property. This work proposes a novel strapdown inertial navigation system (SINS) nonlinear state error defined in the Lie group and derives the SINS equations of the Lie group EKF (LG-EKF) for the MIMU/GNSS/magnetometer integrated navigation system. The corresponding measurement equations are also derived. A land vehicle field test has been conducted to evaluate the performance of EKF, ST-EKF (state transformation extended Kalman filter) and LG-EKF, which verifies LG-EKF's superior estimation accuracy of the heading angle as well as the other two horizontal angles (pitch and roll). The LG-EKF proposed in this paper is unlimited in the choice of sensors, which means it can be applied with both high-end and low-end inertial sensors.

Vehicle coupled bifurcation analysis of steering angle and driving torque

The mechanism of vehicle dynamics steering bifurcation has almost been confirmed. But the present steering bifurcation mechanism cannot explain the bifurcation phenomena caused by the driving torque. As a result, the vehicle coupled bifurcation analysis of the steering angle and driving torque has not been studied. Based on the five degrees of freedom (5DOF) vehicle system dynamics model with driving torque involved, the vehicle dynamics equilibriums under different driving torque and driving mode were searched by a hybrid method in this paper. The hybrid method combined the real-coded Genetic Algorithm with Quasi-Newton gradient method. According to the definition of static bifurcation of nonlinear systems, the equilibrium bifurcation of 5DOF vehicle system was confirmed. Then, the 5DOF vehicle system model was transformed into autonomous equation with the front wheel steering angle as intermediate variable. From the two aspects of constant steering angle amplitude and constant driving torque, the bifurcation diagrams of different driving mode were calculated. The vehicle coupled bifurcation characteristics of steering angle and driving torque were analyzed. The results show that the values of the driving torque will directly affect the bifurcation characteristics of vehicle dynamics system. The coupled feature of the front wheel steering angle and driving torque effect on vehicle bifurcation is obvious.

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus {\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, {C}_{10,5} and {C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes {C}_{8,3} and {C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula {\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class {C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

Periodic solutions of Liénard–Mathieu differential equation with a small parameter

The Liénard–Mathieu equation with a small parameter ε is examined. The existence of periodic and quasiperiodic solutions is proved using the averaging method, the method of complexification and the phase space analysis of an associated autonomous equation. The results extend and generalize those of Kalas and Kadeřábek [6].

The effects of symmetry-breaking functions on the Ermakov-Pinney equation

The Ermakov-Pinney Equation, x?? + ?2x = h2/x3, has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the ?2 and h2. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the sl(2,R) symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases.

The Modelling of Crater Wear in Cutting with TiN Coated High Speed Steel Tool

The flow zone of the chip in contact with the tool reaches a high temperature in cutting. According to chip hardening experiments α-γ transformation may occur in steel, so the tool is in contact with a high-temperature γ phase at high pressure. The microscopic examination of worn surfaces showed that the degradation of the tool is the result of adhesive/abrasive and thermally activated processes, therefore both friction length and temperature must be taken into consideration in the modelling of crater wear. Wear rate can be described by a non-linear autonomous equation. TiN coating, which increases tool life in high speed steel, changes and slows down the wear of the tool. The activation energy of wear can be calculated from the constants of the wear equation determined by cutting experiments. The deoxidation products to be found in the workpiece in cutting may form a protective layer on the TiN layer that blocks or slows down wear.

On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

In this study we apply partial Noether andλ-symmetry approaches to a second-order nonlinear autonomous equation of the formy′′+fyy′+g(y)=0, called Liénard equation corresponding to some important problems in classical mechanics field with respect tof(y)andg(y)functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on theλ-symmetry method, we analyzeλ-symmetries for the case thatλ-function is in the form ofλ(x,y,y′)=λ1(x,y)y′+λ2(x,y). Finally, a classification problem for the conservation forms and invariant solutions are considered.