The application of displaced trajectories in tasks of aircraft navigation in time

Modern requirements in accuracy of aircraft flight trajectory following is done with specifications to the systems of zone navigation in the international air space RNAV (Area Navigation) and RNP (Required Navigation Performance) may apply tight restrictions for the possibilities to correct for the aircraft arrival time to the points of route. Applying the flight speed varying is often insufficient to be on the time chart because of restricted range of available speeds. Application of standard waiting zones in route and in the vicinity of the airport to eliminate the time excess is restricted to the value of minimal flight time which is around 4 minutes or more. As an alternative for arrival time excess elimination to the necessary route point you may use the method of parallel displaced trajectories which is supported functionally with RNP systems. This method is done in the on-board navigation system navigation in time. Depending on the value of the time arrival excess to the necessary route point the navigation coil in time creates the recommendations for the crew how to use displayed trajectories to form the necessary point of exiting to the displaced trajectory to define the side displacement and to evaluate the arrival time correction.


Introduction
Steering an aircraft with parallel displaced trajectories is foreseen as special functions being realized with RNP systems. In its definition, an RNP system is a navigation system which helps to stay with necessary accuracy the necessary 3D flight trajectory [1,2]. Additionally, to 3D trajectories the flights with displacement is known to be applied for tactical tasks of air traffic control when the related services are sanctioned to control the air space. In particular, when two aircrafts hold the same air road [3,4].
The displaced trajectory mode functionality in modern systems of flight control requires to choose the value of side displacement from the current flight route with increment in 1 sea mile, to start moving to the parallel route in accordance with land services of control and then keep following the route of that displaced value [5,6]. If you need to cancel the flight of displaced trajectory and to return to the primary route is done normally automatically if it is not possible to keep that route or manually by the crew command. In this case the requirements of RNP specifications is done by holding the fixed value of side displacement from the primary route [7,8].
Having implemented the concept of 4 th flight trajectory which requires not only holding the necessary 3D trajectory, route time flight evaluation and control it may expand the functionality of displaced trajectory application [9,10].
It is more than clear the application of displaced trajectory to correct the arrival time to the route points by elimination of time excess. This article describes the method of displaced trajectory application to solve the task of the necessary arrival time control. This task is solved by navigation in time of the on-board navigation system. The navigation in time coil is done automatizedly from the primary coil of aircraft steering and may prognosticate the arrival time to the points of route on the primary route and also may form and display the recommendations of flight plan change and its altitude and speed parameters in route to hold the aircraft on the necessary time chart.
The received recommendations can be used by aircraft crew to hold the aircraft on the primary route in time or by correction of the flight plan, or by changing of the flight parameters. And also to coordinate and verify the aircraft position with the help of land services of air traffic control.
To eliminate the arrival time excess by using a displaced trajectory in the coil of navigation in time they form the recommended point of exiting to the displaced trajectory by defining the side displacement and evaluation of corrected arrival time to the necessary point of route.

Materials and methods
The parameters of recommended displayed trajectory being applied in the navigation in time to eliminate the time excess are given in figure 1.  To satisfy the necessary accuracy in time calculation using the displaced trajectory the displaced trajectory flight is studied with particular aircraft specifications and the current parameters of wind.
To choose the recommended side displacement is done with preliminary calculation on the condition of arrival time excess elimination provision to the route point with time control. Time excess value is calculated in navigation in time by comparison of the estimated time of arrival to the necessary one. The estimated time of arrival is calculated after the results of flight modelling on the primary route.
Generally, the arrival time excess (∆t) is calculated as: where RTAnecessary time of arrival to the route point with time control; ETAestimated time of arrival.
The necessary side displacement ( ) is calculated with the following approximate dependence: where Rcalculated radius of turn with maximum permitted careen; Vtrue air speed; α 0way line angle of the transition between the primary and the displaced trajectories.
Standard value of the transition way line angle is 45°. But with small values of displacement this angle could not be reached. The conditions when it is not possible to reach the transition way line angle of the value 45°.
In this case the value of transition way line angle with sufficient degree of accuracy can be accepted as: ).
The received value of side displacement (Z) is rounded up to the lower value with equality to 1 sea mile. The value of side displacement cannot be more than 20 sea miles and subsequently can be restricted by the crew up to the flight plan, air space characteristics, requirements of air traffic land control services.
The calculation of time excess being eliminated value and minimal length of the displaced trajectory can be done by division of the displaced trajectory in particular the entry point and return to the primary route in separate distances. This how the necessary level of flight accuracy is granted.
Type scheme of the displaced trajectory division of the coordinate system of the primary line route including the entry point and return to the primary route is given in figure 2. if the careen angle is changed in the range until 30° the value of course change in this section (∆ 1 ) can be represented as: In the calculated radius of turn: where  -the angle of course;  -the angle of careen; γ maxthe maximum angle of careen; gthe acceleration of free fall; Vthe true air speed; tthe current time.
The exit point to the way line is defined with the accepted law of the careen angle control being done as the aircraft control as [4]: where Zside displacement from the way line, k z , and k ztransmitting coefficients of speed and displacement.
This section is characterized with small aircraft displacement from the necessary way line and small dissonance value of way angle, careen angle control law () can be represented as: where ktransmitting coefficients in course,  -current course calculated from the necessary way line,  0slide angle on the primary trajectory.
Transmitting coefficients value can be obtained from the accepted law of the careen angle control. If we accept the careen angle restriction in the range until 30° course change value can be represented as: Exiting time to the way line (t 3,6 ) can be found with the expression: 3,6 R . The turn section is characterized with maximum careen angle for the current flight mode. While the turn the course value changes in the range between the course received at the end of the turn section and exiting course to the way line.
Flight time in the turn section (t 2,5 ): The sign of the slide angle is defined with the turn direction relatively to the direction of wind: in turn with the wind they use the sign «plus».
The transition way line is characterized with keeping the course angle value in this line of way () which is different from the necessary flight course in route with 45° or less if the corresponding dissonance cannot be reached because of insufficient value of side displacement.
Course angle of the transition way line () can be found after the solution of given equation system. With sufficient degree of accuracy, the calculated value of the transition way line course angle relatively to the current way line can be defined as: The angle value should not be higher than: So having chosen the recommended value of side displacement they should find the course angles of the transition way line from the primary route to the displaced trajectory and back. Then they should find the time of flight in the transition sections and summary time spent for transition from the primary route to the displaced trajectory and back (t). They find the minimal distance of displaced trajectory utilizing (S) as sum of projections to the necessary way line all stages of transition.
The value of time excess being eliminated when they take the chosen displaced trajectory should be: If the calculated value of the time excess being eliminated exceeds the given value, they correct the value of recommended side displacement.
As recommendations how to use side displacement to eliminate the time excess which are given to the aircraft crew are accepted:  the position of calculated point of exiting to the displaced trajectory;  the value of side displacement from the necessary way line;  to evaluate the time of arrival to the route point with given arrival time when they use the displaced trajectory.
The calculated point of exiting to the displaced trajectory is established from the minimal distance from the current way point provided that the return to the primary flight route is before a change of the current point. The distance is calculated as calculated minimal distance of displaced trajectory using. If it is not possible to return to the primary flight route before a change of the current way point displacement trajectory calculation is done for the following flight route sections and of course for the next way point.
The calculated value of side displacement is displayed and can be corrected if necessary. After its correction the value of the time excess being eliminated must be re-calculated and minimal distance of the trajectory displacement.
Arrival time evaluation to the route point with given arrival time is done by subtracting from the calculated arrival time value and the time excess being eliminated on the displaced trajectory. Obtained so the correctable arrival time is displayed to make a decision if to take the displaced trajectory or not.
With the displayed information the crew makes a decision if they take the displaced trajectory and also in which direction to exit on that trajectory and if necessary that procedure must be sanctioned with the air traffic land control services and without passing by the calculated point they initiate the exit to the parallel route. Having reached the necessary value of side displacement and they initiate the cancel of flight on the displaced trajectory and the aircraft returns to the primary route.

Evaluation results and realization
The evaluation of utilizing the parallel displaced trajectories to eliminate the arrival time excess is done by aircraft flight modelling on the primary route with different given values of true air speed and different values of wind speed components.
Dependence of side displacement value (Z) and minimal distance of the displaced trajectory (S) from the value of the excess time being eliminated for some values of air speed is given in figure 3.

Conclusion
The flights with 4 th trajectories and parallel displaced trajectories which is supported as a function by the RNP system may help to eliminate the arrival time excess to the necessary route point and also keeping the necessary navigation characteristics and realization of a little predictability. And the value of the time excess being eliminated by that option can be 4 minutes of flight.
Creation of recommendations to take the displaced trajectory is proposed as navigation in time of the aircraft steering system. The given recommendations can be used by the aircraft crew to keep the flight time chart in route and to coordinate and concord the aircraft position in cooperation with traffic control land services.