Navigation Equations Research Articles

Error equations of inertial navigation

This paper derives basic error equations of inertial navigation which apply to any properly constructed inertial navigator. The equations are deduced from the integral equations of inertial navigation by a vectorial analysis. A major result of this analysis is a set of fundamental error propagation equations that has apparently been missed. These equations regard the absolute navigational errors. The conventional velocity and position errors are shown to be transfer errors.

A Direct Solution to GPS-Type Navigation Equations

One solution to the navigation equations involves iteration on the 4 by 4 augmented range-direction-cosine matrix beginning with an assumed position and so assumed direction cosines, of which there are 12 for 4 satellites. An algebraic, direct solution to this same basic equation set has recently been published. Both of these methods are reviewed. We offer a direct solution using modified functions of the range magnitude data from four satellites to yield user's clock bias correction, user's position, and true range vectors if desired. The highest order of matrix inversion used is 2 by 2. The highest order, nonlinear equation is a numeric square root. The principle of the formulation is use of differences among the range magnitudes and range magnitudes squared. An additional auxiliary difference equation is formed. A computation basis uses the ephimeride differences and an orthogonal vector. The method offers convenience, speed, simplicity, low dimensionality, and precision, with no operational constraints.

The inertial positioning problem in Hamiltonian mechanics

The general inertial positioning problem—to determine the actual coordinates of a vehicle, moving with respect to the inertial frame, from its measured inertial accelerations—is discussed in Hamiltonian mechanics. The use of the Hamiltonian formalism allows the avoidance of sophisticated geometrical considerations. Starting from the canonical equations of motion for the general problem, the navigation equations of some special problems (motion on a rotating sphere, motion on a rotating ellipsoid) are derived in linearized form. Using the simplicity of the linearized form, the solutions can be easily computed in analytical form. This is done for the above problems.

自由表面による慣性航法-III : 方式INと方式I(2)の理論ならびに若干の補足事項

In this paper, two more applications of the FS (Free Surface) mechanization of inertial navigation are described. Each utilizes its own set of two spatial references, which is different from that of another. One of the applications, the type IN mechanization, uses as the references the north-south axis Y_n and the earth axis Z_E. The other, the Type I (2) mechanization, uses the axis Z_E and an inertial axis Y_I that is in the equatorial plane pointing to Z_E at the initial time t=0. In each case, the navigation equations are derived, referring to the respective references, from the fundamental equations of inertial navigation, and the solutions are effected by means of the block diagram method. Also how to find the geographic coordinates is described. This is necessary for obtaining the attitude angles (roll, pitch and azimuth). Moreover, to help understanding the general aspects of FS mechanization, several supplementary but substantial topics are disscussed. They are as follows; countermeasures to the non-uniformity of the specific force vector, ranges took by the angles a_i (i=1〜4) where a_i is the angle between the platform axis Z_p and the i-th reference axis, ranges of altitude time rates h and h which the altimeter must cope with, the relations between the outputs of the inertial navigator and the vehicle's motion, and so on.

Inertial navigation equations based on relativistic effects

The problem of inertial navigation for the case of movement of a small deformable particle is considered as a problem of fluid mechanics concerning—in the framework of STR and GTR—calculation of mechanical characteristics in the coordinate system of an observer in accordance with the data of measurements of kinematic characteristics in the proper inertial coordinate system. The general solution of this problem for an arbitrary movement of the center of a small particle with rotation and deformation is given.

Drift Compensation and Acceleration Resolution for a Rotating Platform IMU

Theme T Delco Carousel IMU in the Titan IIIC vehicle is similar to the conventional inertial platform except that two sets of the inertial instruments are mounted on a platform which rotates at 1 rpm about an inertially fixed axis. Derivations of navigation equations for acceleration resolution and drift compensation for the IMU are presented. The equations have been checked out and implemented in the flight computer.

A Simplified Satellite Navigation System for an Autonomous Mars Roving Vehicle

The use of a retroflecting satellite and a laser rangefinder to navigate a Martian roving vehicle is considered in this paper. It is shown that a simple system can be employed to perform this task. An error analysis is performed on the navigation equations and it is shown that the error inherent in the scheme proposed can be minimized by the proper choice of measurement geometry. A non-linear programming approach is used to minimize the navigation error subject to constraints that are due to geometric and laser requirements. The problem is solved for a particular set of laser parameters and the optimal solution is presented.

Navigation formula to A. Busemann's variation problem of space travel

The indicatrix of the variation problem is given (1) by the perturbation differential equation for the semi-major axis and eccentricity, and (2) by a relation between the momentum and eccentric anomaly. The Hamiltonian equations of an extremal solution, therefore, reduce to a navigation equation. The remaining perturbation equations for the longitude of perihelion and the mean longitude at epoch, finally, yield the time dependence that gives the most economic fuel consumption.

Mechanization of SST Inertial Navigation Computations in ALL-DDA Computer

an all-dda computer is chosen for mechanization of SST inertial navigation equations to help make the cost of the inertial system more competitive. The ease with which the DDA handles dead-reckoning navigation is discussed but attention is focused on the operations required for initialization, computation of display and control functions and timing and switching. Techniques are presented for improving effective resolution of computations when computer scaling is coarse.

On the general equations of inertial navigation

Derived and investigated are the equations of ideal (unperturbed) and perturbed operation of a generalized system of inertial navigation which determines the coordinates of a moving object from the measurements of three linear accelerometers and three angular velocity meters measuring absolute angular velocity. The instruments are located along the axes of an orthogonal trihedron, the origin of which coincides with a certain point in the moving object. In contrast to [1 and 2], the orientation of the trihedron axes is arbitrary. The equations of ideal operation take into account the non-central nature of the Earth's gravitational field. The equations of perturbed operation include instrument error effects. Error equations are derived and investigated not only for small but also for the finite values of the variables of the perturbed operating conditions. The particular cases are the equations and the results of investigation of the particular control systems considered in [1 to 7].