Markov model of data measurement complex for track geometry car

. A stochastic model of a nonadjustable data measurement complex platform for track geometry cars is introduced. A state vector evaluation algorithm based on the approximation of a posteriori probability density by the system of a posteriori moments is also offered.


Introduction
At present an advanced transportation infrastructure plays a vital part in the general economic progress of any nation. Railroads account for a substantial share of this infrastructure. The transportation industry is continuously struggling with growing cargo and passenger traffic, so the arising issues of time efficiency, safety and reliability cannot be solved without maintaining a high operational quality level of the rolling stock and railroad tracks. In order to ensure safe and reliable railroad functionality a variety of regular safety measures is applied, involving track fitness parameter control and registration activities. Out of all existing track fitness control equipment only track geometry cars (TGC) provide for an overall precision track sub-and superstructure monitoring. The TGC measurement capabilities are based on data measurement complexes (DMC) built upon nonadjustable platforms (NP). These complexes feature a number of apparent advantages. Continually active sensors allow for the recording of some crucial characteristics that otherwise may escape detection by initial orientation systems with adjustable platforms. Current platform orientation data is preserved during the sensor calibration process. There are no additional specific elements of the initial orientation system. Meanwhile the NP TGC performance efficiency is highly dependent on its model's adequacy to the real operational environments. That is why NP TGC movement model designers should remember that the NP own drift rate relates nonlinearly to the accelerations axially directed within the relative gyroscopic coordinate system as well as the fact that the NP sensor measurements bear a stochastic nature. For that matter the objective of this research is to design an efficient NP TGC movement model together with a measurement interference filter algorithm.

NP Movement Equations
In order to develop an NP movement model the following right-hand coordinate systems T , originating at the platform suspension center; OXYZplatform-related For better NP movement analysis visualization it is assumed that the three corresponding basis vectors of the first three CSs are coincident at the reference time.
Each gyro unit's own drift rate i  is correlated by the second degree trimetric polynomial from the axially directed accelerations within the related CS [2]: where K is a block matrix,  is the block multiplication sign, The platform drift rate projections may be represented with a system of differential equations in relation to the Euler-Krylov angles    , , [3]: ( Equations (1), (2) allow to correlate the actual NP orientation, i.e, time variation of Euler-Krylov angles    , , , to the current overloads along its axes, but for the moment in parametrical form only. In order to identify the acceleration projections z y x g , g , g on the GCS axis with known acceleration projections on the ACS axis a matrix rotations correlation will be employed. Trihedron T of the ACS rotates in relation to trihedron I of the ICS around Axis mundi at the rate з  , hence where The rotation of the platform-related trihedron J of the GCS, in relation to trihedron I may be described in its turn with the help of the first degree rotation matrix D : Consequently, the rotation of the platform trihedron J in relation to the Earth trihedron T may be represented by the following matrix equality: where T V is the transposed matrix V .
It follows from (4) and (5), that the acceleration projections The construction of a stochastic model of the NP it is acknowledged that due to internal and external disturbances accompanying the NP operation accelerations inevitably present in the astronomical coordinate system bear a random nature. With reference to the above equation (6) takes the following form: where w is the vector of random disturbing accelerations, directed along the Earth trihedron axes. This vector is described by white Gaussian vector-noise (WGN) with zero mathematical expectation and known intensities matrix By combining equations (1), (2) and (7), the NP movement model may be represented as follows: is the NP noise vector, also described as the WGN with zero mathematical expectations and the intensities matrix

Construction of Observation Equations
In order to acquire a posteriori evaluations of the actual platform orientation we need a proper measuring instrument model that will take into account all the array of possible random impacts [2]. Further on the term measuring instrument will denote the complex of accelerometers positioned on a gyrostabilized platform. The relationship of acceleration projections along the platform trihedron axes may be established from Equation (6) by linearization of Matrix D . In this case Equation (6) takes the following form where Physical accelerometers inevitably feature their own random measurement noise generated by their random operational environment impacts such as heat, electrical magnetism, gravity, etc. [2]. In this case the model of the signal measured by the accelerometer triad may be represented by the following equation In the vector-matrix form Equation (12) may look as follows . Equations (9) and (14) represent an NP movement model in the object-observer form. The use thereof allows for evaluating State vector (9) with the help of modern nonlinear stochastic evaluation techniques [4]. The a posteriori probability density      t , Y of this process has the following form [4]: And now we will establish an a posteriori moments system that will equip us with the required approximation precision for Equation (15) in the given time interval of the system operation (9). In order to solve the set task we will employ the technique outlined in [4,5]. We will proceed by multiplying the both sides of Equation (15)  . With consideration that changing the integration and differentiation order in the left side of the equality is admissible we achieve the following: (16) Using the technique of integration by parts, considering the borderline conditions and drift and diffusion coefficients Taylor series expansion, the first addend (drift coefficient) in the right hand side of Equality (15) will be represented as In order to find the central moments Equality (15) will be multiplied by normal distribution density depending on the first two moments only, the resulting equation system reduces to Kalman-type filtering.