Metrics is a science which quantifies the parameters to take decision on the quality of things, like a meter is used to define the length in a similar manner RNP parameters are used to evaluate the performance of avionics navigation.

As per ICAO definition RNP is a statement of the navigation performance for operation within a defined airspace, independent of the equipment utilized (ICAO Doc 9613). These RNP parameters give a statement of navigation, but it doesn’t give any information about area of airspace utilization. In order to give information about the area of airspace utilization to the user RNP specifications are defined for different phases of flight such as en-route to surface landing. This specification is defined in terms of RNP-x; it means that for a particular phase of flight, x is the accuracy specification of the navigation equipments. The RNP-x parameter specifies that the aircraft navigation system must provide the vertical position of an aircraft with a required accuracy (defined by x) for 95% of the time and the reference for this accuracy is the true position. RNAV (Area Navigation) is defined as a method of navigation which permits the aircraft to fly in a containment region and the aircraft navigation system can cross the containment region with a predefined probability. RNP is simply the requirements necessary to keep the aircraft inside the aircraft containment surface (Kelly, et al., 1994). These requirements are defined by RNP parameters namely accuracy, integrity, continuity and availability. These RNP parameters are translated into system specifications. Using these four RNP parameters the user (pilot) decides about the quality of a user position. The technology necessary to satisfy the RNP is expected to be developed, jointly by government agencies and avionics industry associations (Kelly, et al., 1994). RNP parameters are used by airspace planners to determine the airspace utilization and as an input in defining route widths and traffic separation requirements, these parameters are dependent of airspace system and is independent of navigation sensor (Kelly, et al., 1994).

3.2 Significance of RNP Parameters

The significance of RNP parameters is that, the accurate levels of these parameters will help in maintaining a predictable and orderly air traffic flow and helps in separation of two aircraft which are within the acceptable limit during various phases of flight. It is the only one factor to resolve the aircraft separation standard (Kelly, et al., 1994). RNP provides a mean of improvement to airspace utilization by incorporating its parameters. These RNP parameters are developed and standardized by ICAO (International Civil Aviation Organization), FAA (Federal Aviation Administration) and RTCA (Radio Technical Commission and Aeronautics). Therefore, these RNP parameters are uniform across the globe and are compatible between airplane user and ground station. These RNP parameters provide the user or pilot to intercommunicate with the fellow aircraft to avoid collision by exchanging its RNP parameters (Bhatti, 2007).

Table 3.1 shows about LAAS Guidance quality requirements (Jeffrey, 2004).

Table 3.1 LAAS Guidance quality requirements

3.3 Performance parameters

The minimum number of parameters required to define the navigation in particular avionic navigations are accuracy, continuity, integrity and availability (Kelly, et al, 1994). In this section these parameters and their calculations are discussed briefly.

3.3.1. Accuracy

Accuracy is defined as the degree of satisfaction of measured position at a given time to a defined reference value. Due to the presence of error sources the position of user deviate from the original position. It shows about the variation of user position with respect to estimated or original position. This information gives the user about how much will original position deviate from estimated position. This is calculated from the probability density function (pdf) of all error sources with threshold as 95%. If the probability of position error is within the threshold limit then accuracy is satisfied for that instant epoch. Stationary ground based systems such as the ILS or VOR have relatively stationary characteristics so that once the performance is measured during flight inspection, it is assumed that the accuracy does not change until the next inspection. The situation is different for GNSS, due to the continuous orbiting of satellites and propagation of signals in the atmosphere, the system error characteristics change with respect to location and time (Bhatti, 2007).

a) Accuracy calculation (Jeffrey, 2004)

= standard deviation of vertical position error =

= standard deviation of lateral position error =

Vert(n) vertical projection of range to position for satellite

Lat (n) = horizontal projection of range to position for satellite

= Standard deviation of Pseudo-Range (PR) error for satellite n

VAC Vertical Accuracy

LAC Lateral Accuracy

= Standard deviation of ground receiver pseudo-range error

= Standard deviation of air born receiver pseudo-range error

= Standard deviation of residual ionosphere pseudo-range error

= Standard deviation of residual troposphere pseudo-range error

3.3.2. Integrity

In real life navigation system always suffer from failure and fault that can cause the degradation of its performance. If this performance degrades beyond the alert limits (discussed in next section) then navigation system become unusable to the user. If the navigation system doesn’t provide a warning signal to the user then it could proved to be disastrous, particularly if the user is with respect to aviation. To avoid this type of situation the remedy is to make the user aware of the failure in a timely manner by incorporating the parameters known as integrity parameters. These parameters will help the user to revert back to backup navigation systems such as LORAN-C or ILS to stay within alert limit. Therefore it can be said that integrity is the performance measure of navigation system that can perform failure detection successfully.

3.3.2.1 Classification of integrity provision

Integrity provision may be classified as (RTCM paper 97-2004/SC104-342)

a) Position domain alerts in this domain alarms are generated when the position error calculated by integrity monitoring receiver exceeds the DGPS (WAAS/LAAS) service provider protection levels (threshold values) for more than predefined period of time. The Reference Station (RS) will set the station healthy field in the header of each broadcast message as defined in RTCM SC 104 to indicate the unhealthy condition.

b) Pseudorange domain alert in this domain alarm is generated when the calculated residual error for a given pseudorange exceeds the pre-set threshold value for more than predefined period of time. Upon receiving a pseudorange alarm, the RS will set the PRC (PseudoRange Correction) and RRC (Range Rate Correction) values for a satellite as defined in RTCM SC 104.

3.3.2.2 Terms related to integrity

Various terms related to integrity are briefly discussed in this section (Bhatti, 2007).

a) Alert limits

Alert limits are defined for position errors for different phases of flight. These represent the largest position error values that should not be exceeded by the navigation system. The system should generate a warning if the limit is exceeded. Further this warning must be generated within an acceptable maximum time to alert and this time to alert varies with the phase of flight (En-route to surface landing). Vertical alert limit is a maximum allowable error in vertical direction. Horizontal alert limit is a maximum allowable error in horizontal direction. Position Failure is defined to occur whenever the difference between the true position and the derived position exceeds the applicable alert limit. The Probability of Missed Detection is the probability of not detecting a position failure. The Probability of False Alert is the probability of the indication of a positioning failure when a positioning failure has not occurred.

b) Protection Levels

Vertical Protection Level (VPL) is half the length of a segment on the vertical axis (perpendicular to the horizontal plane of WGS-84 ellipsoid), with its center being at the true position, which describes the region that is assured to contain the indicated vertical position. It is based upon the error estimates provided by WAAS/LAAS. Horizontal Protection Level (HPL) is the radius of a circle in the horizontal plane (the plane tangent to the WGS-84 ellipsoid), with its center being at the true position, which describes the region that is assured to contain the indicated horizontal position. It is based upon the error estimates provided by WAAS/LAAS.

c) Integrity and continuity risk

Integrity risk relates to the probability that a position failure occurs without generation of an alert within the TTA. For GNSS, the integrity risk requirement is more stringent than for the ground based systems. This is due to the fact that failures in the GNSS signals can affect a much larger number of aircraft simultaneously in comparison to a stationary ground based navigation system (Bhatti, 2007). Continuity risk is related to the probability of false alert and the probability of a failure being detected but not identified (does not include wrong exclusion). The integrity risk and continuity risk can be expressed in mathematical form as

Where IR is the integrity risk, CR is the continuity risk, MD is missed detection, MI is an incorrect isolation, DF is a detected failure, FA is a false alert, UF represents an unscheduled failure, NI represents the case where a failure has been detected but isolation is impossible within the time to alert (Shaojun, et al., 2006).

d) Stanford – ESA Integrity Graph

One can visualize the integrity risk using Stanford graph

Figure 3.1 Relationship between PL, PE, AL and integrity

If Position Error (PE) is less than alert limit (AL) and greater than protection levels (PL) (PL<PE<AL) then the system is available but not safe. This is not leading to a hazardous situation known as Misleading Information (MI). If AL is less than PE but it is greater than PL (PL<AL<PE) then the system is available but not safe leading to a hazardous situation known as Hazardous Misleading Information (HMI) (Boriana, et al., 2006).

3.3.2.2 Calculation of integrity

Integrity is calculated by user, based on hypothesis known as integrity hypothesis. Integrity risk is allocated across these hypotheses to calculate integrity. Integrity hypothesis is divided into three categories based on the following conditions.

a) Fault free reference receivers (H0 Hypothesis)

In fault free reference receiver condition, it is assumed that there is no fault in reference receivers . This condition is known as H0 hypothesis and is addressed through protection levels (VPL and HPL) calculation.

b) Single reference receiver failure (H1 Hypothesis)

In single reference receiver failure condition, it is assumed that there is one reference receiver failure. This condition is known as H1 hypothesis and is addressed through protection levels (VPL and HPL) calculation.

c) Other source of HMI (H2 Hypothesis)

The H2 hypothesis occurs due to satellite failure, signal deformation, Low signal power, Code Carrier divergence, atmosphere anomalies and ground equipment failure. H2 is addressed through design, analysis and monitoring (Jeffrey, 2004).

3.3.2.3 VPL calculation

a) Fault-Free reference receivers

For fault free reference receiver the errors are assumed to be Gaussian

Where

= multiplier which determines the probability of fault-free missed detection.

= projection of the vertical component for the ith ranging source (Stephen, et al, 1998).

= ith element of first column of S

= ith element of third column of S

= glide path angle for the final approach path

N = number of ranging sources used in the position solution

i = ranging source index

is the root mean square (RMS) of total errors

is the fault free error associated with the differential correction for satellite i

is the residual ionospheric error for satellite i

is the residual tropospheric error for satellite i

is the fault free error associated with airborne receiver

b) Single Fault Reference receiver

For single fault reference receiver model the VPL is evaluated as follows

Where

is the vertical projection of estimated error position due to faulty reference receiver j.

is a multiplier which determines the probability of missed detection given that the ground subsystem is faulted

is the standard deviation of vertical error position due to single fault reference receiver

Where

is the B value for the ith satellite and the jth reference receiver

j = ground subsystem reference failure index

Where is the variance of vertical position error due to single fault reference receiver.

Where

M[i] = number of ground subsystem reference receivers whose pseudo-range measurement was used to determine the differential correction for the ith ranging source for which the B value was transmitted to the airborne subsystem

is the variance of total range error due to single fault reference receiver

3.3.2.4) HPL calculation

a) Fault-Free reference receivers

For fault free reference receiver the errors are assumed to be Gaussian

Where

= multiplier which determines the probability of fault-free missed detection.

is the projection of the horizontal component for the ith ranging source.

= ith element of second column of S

N = number of ranging sources used in the position solution

i = ranging source index

b) Single Fault Reference receiver

For H1 hypothesis model

Where

is a Single faulty reference receiver missed detection inflation factor.

=horizontal projection of estimated error position due to faulty reference receiver j

is the standard deviation of horizontal error position due to single fault reference receiver.

Where

j = ground subsystem reference failure index

B[i,j] = the B value for the ith satellite and the jth reference receiver

Where

is the variance of horizontal position error due to single fault reference receiver.

Where

M[i] = number of ground subsystem reference receivers whose pseudo-range measurement was used to determine the differential correction for the ith ranging source for which the B value was transmitted to the airborne subsystem

is the variance of total range error due to single fault reference receiver

Integrity is said to be satisfied if it meets both HPL and VPL as shown by eq 3.16

3.3.3. Continuity

Continuity is the capability of the system to perform its function without unscheduled interruptions during the intended operation. The function of the navigation system includes not only accuracy but also integrity. Hence, the continuity of service relates to the capability of the navigation system to provide navigation output, with the specified accuracy and integrity, throughout the intended operation, assuming that it was available at the start of the operation. Continuity depends on many factors such as traffic density, complexity of airspace and availability of alternative navigation aids such as ILS or LORAN-C (Bhatti, 2007).

a) Continuity calculation

While calculating continuity the user calculates the predicted protection levels such as PVPL (Predicted Vertical protection levels) and PHPL (Predicted Horizontal Protection levels) for H0 and H1 hypothesis. The calculations of PVPL & PHPL are similar to VPL and HPL calculation, but these predicted protection levels take accounts of fault free detection (false alarm) of reference receiver. Predicted protection levels (PVPL and PHPL) are compared against Alert Limits (VAL and HAL). Verify the constellations that support the continuity of the protection levels. In next step determine the number of critical satellites (Ncrit) whose failure will cause the protection levels to exceed the alert limits. Compare the Ncrit with the maximum number of critical satellite that can be tolerated (Ncrit-max). Ncrit-max is determined by the satellite failure rate (mean time between unscheduled outages) and allocated continuity risk. The continuity is said to meet if it satisfies the following eq 3.17 (Jeffrey, 2004).

3.3.4. Availability

The availability of a system is characterized by the portion of time during which reliable navigation information is presented to the crew, autopilot or other flight management system. The navigation service is said to be available if accuracy, integrity and continuity requirements are satisfied. The availability of GNSS depends on the relative geometry of the coverage area and available satellites and the potentially long time to restore a satellite in the event of a failure.

a) Availability calculation

For a given location and time, instantaneous availability for GPS is given in eq 3.18 (Jeffrey, 2004).

Where

Prob (f,N) = probability of f failure in N visible satellites

Avail (f,p) = Accuracy (f,p) Integrity(f, p) continuity(f,p)

F = number of satellite failures to consider., N = Number of visible satellites.

3.4 Statistical parameters required for estimating RNP parameters

The performance of GPS and its augmentation are affected by the errors. These errors are sum of random errors and deterministic error. The deterministic errors can be predicted at any instant of time. The random errors can be predicted by applying statistical parameters to series of previous values. Therefore it is necessary to study about the statistical parameters which describe the behaviour of errors. These errors are added with original range to give pseudorange. To estimate the original range the errors present in pseudorange should be estimated. This error estimation can be done by using mathematical and statistical model. These models represent the noise in mathematical form and by careful analysis of this model it is possible to estimate the error present in pseudorange. To model these errors the following steps are followed

Step 1 The probability of errors is calculated as

Where is number of favourable outcome

n is total number of trials.

Step 2 The mean is calculated as

Where represents a sample errors

= represents the probability of errors.

The mean is denoted as. It gives the average of all errors and is also known as expected value of errors. The mean alone is unable to characterize the errors therefore another parameter known as variance is used.

Step 3 The variance is calculated as

It is denoted by symbol

Step 4 The standard deviation is calculated as follows

The standard deviation is denoted by. This standard deviation gives the dispersion (deviation) from the mean. The larger the standard deviations of the random variables, then the random variables are far from the mean and the smaller the standard deviations of the random variables, then the random variables are closer towards the mean (Papoulis, 1984).

3.5 Distribution and density function applicable to RNP parameters

The various errors associated with GPS are Ionospheric error, receiver error, tropospheric error, multipath error, clock bias error etc. All these errors are random and their values are random in nature. These errors can be analyzed if its distribution and density function are calculated. The distribution function gives the cumulative probability of the event (xi) and is denoted as. The distribution function gives the relationship between the value of random variable and its associated probability. The density function gives the probability of the event (xi). These density functions are used for evaluating the integrity parameters such as protection levels.

3.6 Conclusion

This chapter describes about various performance parameters (RNP parameters) of LAAS that are required for guiding the airborne user from en-route to surface landing. The RNP parameters discussed in this chapter is accuracy, integrity, continuity and availability.

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