AC 20-191 Appendix C C.1.3 Second-Order Filter of GBAS NSE Generator.
The second order filter to be implemented in the GBAS NSE Generator is characterized by:
ω 2 H (s)= n [2] s2 + 2ωn s+ω 2n where ωn is the natural frequency given by For GAST C: ωn= 0.01 rad/sec For GAST D: ωn = 0.033 rad/sec C.1.4 Noise Scale Factor.
The model accounts for the variation in accuracy due to satellite geometry by setting the noise scale factor to a constant which is sampled from a distribution. For each run, the value of Kvert is first determined by selecting a sample, x, from a uniform distribution between 0 and 1. The value of Kvert is then given by the following function of x:
Kvert f(x) a= = a1 + a2 x− 3 [3] x−1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-1.
Table C-1: GAST Dependent Parameters for Kvert Service Type a1 a2 a3 Kvert_max GAST C 0.4 0.2 0.006 10/5.762=1.736 GAST D 0.52 0.47 0.005 10/5.762=1.736 C.1.4.1 If the random pick from a distribution between 0 and 1 results in K vert >Kvert_max, then the value should be discarded and another sample should be selected from the uniform distribution set to the maximum value from the table 2. An alternative acceptable means for computing the NSE scale factors is given in section C.2 below.
C.1.4.2 For each run, the value of K xtrk is determined by selecting a sample, x1, from a uniform distribution between 0 and 1. For each run, the value of Katrk is determined by selecting a sample, x2, from a uniform distribution between 0 and 1. The cross-track and along track scale factors are then computed by:
3This corresponds to the case where VPL>10m and therefore the system is not available.
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AC 20-191 Appendix C aK f ( ) = 3xtrk = x1 a1 + a2 x1 − [4] x1 −1 = ( ) aK f x 3atrk 2 = a1 + a2 x2 − [5] x2 −1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-2:
Table C-2: GAST Dependent Parameters for Kxtrk and Katrk Service Type a 1 a2 a3 Kxtrk_max, Katrk max GAST C 0.2 0.1 0.003 40/5.762=6.942 GAST D 0.21 0.12 0.003 17/5.762=2.951 C.1.4.3. If the random pick from a distribution between 0 and 1 results in Kxtrk>Kxtrk_max or Kaxtrk>Kaxtrk_max, then the value should be discarded and another sample should be selected from the uniform distribution.
C.1.5 NSE Step Generator.
The NSE step generator is illustrated in Figure C-3. Step errors occur when the satellite geometry changes, either due to the removal of an individual satellite from the position solution (e.g. a satellite fails and stops transmitting or the user receiver stops tracking a satellite for any reason) or due to an individual satellite rising or setting. The step generator produces representative NSE step errors in the vertical, along-track and crosstrack directions in meters. This is accomplished by scaling a unit step function by factors that are derived from representative statistical distributions. First, three random samples, one for each axis, are selected from a zero mean unit variance normal distribution. Then these samples are multiplied by scale factors that are chosen to simulate the statistical variation in the size of an error that would result from normal variations in the relative geometry between the user and the satellites. Finally, the resultant constant factors are multiplied with a unit step function time sequence.
Injecting one step in each simulated approach at a random time should reasonably model the effects of geometry changes.
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AC 20-191 Appendix C Figure C-3 NSE Step Generator
NSE Step_atrk NSE Step_xtrk NSE Step_vert GWN Generator
Seq Scale Factor Computation X X X K Step_atrk K Step_xtrk K Step_vert Unit Step Function Generator Uniform Random Number Generator Step Event Time X X X Seed Seed x Kstep = f(x) One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 t = event time 0 t = event time 0 One sample from Uniform distribution, 0≤x≤1 C.1.6 NSE Step Generator Scale Factor Computation.
For each run, the value of σ step_vert is determined by selecting a sample, x, from a uniform distribution between 0 and 1. The value of σstep_vert is then given by the following function of x:
( ) bσ step _ vert = f x = b1 + b2 x− 3 [6] x−1.01 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-3.
Table C-3: GAST Dependent Parameters for σ step _ vert Service Type b 1 b2 b3 max_ _vertstepσ GAST C 0.4 0.8 0.07 FASVAL/2 GAST D 0.5 0.8 0.05 3.5
C.1.6.1. If the random pick from a distribution between 0 and 1 results in σ step_vert >σstep_vert_max, then the value should be discarded and another sample should be selected from the uniform distribution.
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AC 20-191 Appendix C C.1.6.2. For each run, the value of σ step_xtrk is determined by selecting a sample, x1, from a uniform distribution between 0 and 1. The value of σstep_atrk is determined by selecting a sample, x2, from a uniform distribution between 0 and 1. The cross-track and along track NSE step scale factors are then computed by:
σ step _ xtrk = f( bx1)= b + b x− 31 2 1 [7] x1 −1 σ step trk ( ) b_ a = f x2 = b1 + b 32 x2 − [8] x2 −1 where the parameters of the function are dependent on the GBAS Approach Service Type as given in Table C-4.
Table C-4: GAST Dependent Parameters for σ step _ xtrk or σ step _ atrk Service Type b 1 b2 b3 max_ _axtrk stepσ GAST C 0.32 0.32 0.05 20m GAST D 0.35 0.35 0.35 5.5m C.1.6.3. If the random pick from a distribution between 0 and 1 results in σstep_xtrk >σstep_xtrk_max, or σstep_atrk>σstep_axtrk_max, then the value should be discarded and another sample should be selected from the uniform distribution.
C.1.6.4. The scale factor, Kstep_vert, of the magnitude of the vertical step error is then selected from a normal distribution with 0 mean and a standard deviation of σstep_vert. The scale factor, Kstep_xtrk, of the cross-track deviation is selected from a normal distribution with 0 mean and a standard deviation of σstep_xtrk.
C.1.7 Latency Model.
The latency of the GLS output should be delayed for 400 msec.
C.1.8 Fault Mode Generator.
The fault mode generator as depicted in Figure C-4 can be used for modeling undetected, non-aircraft system error conditions (monitor “limit” cases) or any worstcase undetected error (monitor “malfunction” cases). The fault mode generator produces a ramp error with characteristics as illustrated in Figure C-5. The effect of a malfunction is modeled as a ramp, with a start time, a ramp rate and a total exposure time, Tmax. The maximum value of the ramp depends on the ramp rate and time to alert.
The ramp is assumed to increase to the level of the maximum value and then to exceed that value for a period equal to the time-to-detect and mitigate the failure. The erroneous satellite is isolated and the error returns to the nominal value (i.e. the fault error is set to zero). The model may alternatively produce step errors where the C-6
AC 20-191 Appendix C maximum change in error due to the step is specified rather than the ramp rate. (See reference [7] for more details regarding GLS fault modeling).
Figure C-4 Limit or Malfunction Generator or both NSE LMF_atrk NSE LMF_xtrk NSE LMF_vert GWN Generator
Seq Set Limits X X X K Limit_atrk K Limit_xtrk K Limit_vert Unit Ramp Function Generator Uniform Random Number Generator Ramp Event Time X X X Seed Seed x Kother = f(x) One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 One sample from normal distribution, µ=0, σ = 1 t = event time 0 e 0 One sample from Uniform distribution, 0≤x≤1 Limit or Malfunction Axis to Apply Ramp Rate Ramp Duration From Figure C-5, it can be seen that for the ramp:
EffectiveVALTMAX = TTD+ [9] RR EMAX and the Effective VAL depend on the type of malfunction. For Satellite Ranging sources, the Effective VAL is a function of the maximum error allowable with probability greater than 1x10-9 by the Pmd performance constraint with conditional probability (ICAO Annex 10 appendix B Section 3.6.7.3.3.3 ), (i.e., 1.6 meters) multiplied by the geometry screening limit.
EMAX =1.6⋅ Svert _ max [10] Where:
Svert _ max is the maximum vertical projection for any satellite allowed by geometry screening. This airframer limits the size of EMAX by specifying a maximum Svert for satellites used in the position solution as described in reference [8].
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AC 20-191 Appendix C C.1.8.1. For ground segment reference receiver failures, the effective VAL will depend on the geometry screening applied in the airborne equipment. If no additional geometry screening is applied other than VPL<VAL, the maximum effective VAL is 9.35 [m] (see Table C-5). If additional geometry screening is applied, a lower effective VAL may result. Reference [8] explains how to compute the effective VAL given additional geometry screening. Figure C-6 shows a plot of maximum vertical and lateral errors as a function of vertical and lateral Alert Limit screening. The calculations to produce the plot in Figure C-6 are described in detail in Reference [9].
Figure C-5 Malfunction Transient time Position Error on Axis Of Choice Effective VAL TTD Slope Characterized by Ramp Rate [ft/s] Total Exposure Time - Tmax EMAX C.1.8.2. For ionospheric anomalies, the maximum vertical error Emax is limited to a specified maximum allowable position error for the airborne installation for each axis, vertical (maxEV) and lateral (maxEL), as a part of the satellite geometry screening in the avionics [10]. These values along with broadcast information provided by the ground station determine the geometry screening.
C.1.8.3. Table C-5 and Table C-6 give the characteristics for transient errors in the vertical and horizontal directions respectively for each of the three major identified fault types.
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AC 20-191 A ppendix C Table C-5 Malfunction Transient Characteristics in the Vertical Direction Fault Type Service Type Ramp Rates [m/s] Effective VAL [m] Emax [m] Time to Detect (TTD) [s] Ranging Source Failures GAST C 0 - ∞ 10 Dependent 6 GAST D 0 - ∞ 1.6 × Svert Dependent 2.5 Iono Anomaly GAST C 0 – 4 N/A N/A N/A GAST D 0 – 4 N/A maxEV
N/A Single Ref Receiver Failure GAST C 0 - ∞ 10 Dependent 6 GAST D 0 - ∞ 9.35 [note] Dependent 2.5 Note: This value is an absolute worst case assuming no additional geometry screening is afforded based on reference receiver fault monitoring using TBAC. Only geometry screening of VPL<VAL=10m is assumed here. Smaller maximum values can be obtained by using additional geometry screening per reference [9].
Table C-6 Malfunction Transient Characteristics in the Lateral Direction Fault Type Service Type Ramp Rates [m/s] Effective LAL [m] Emax [m] Time to Detect (TTD) [s] Ranging Source Failures GAST C 0 - ∞ 40 Dependent 6 GAST D 0 - ∞ 1.6 × Slat Dependent 2.5 Iono Anomaly GAST C 0 – 4 N/A N/A N/A GAST D 0 – 4 N/A maxEL
N/A Single Ref Receiver Failure GAST C 0 - ∞ 40 Dependent 6 GAST D 0 - ∞ 35.9 [note] Dependent 2.5 Note: This value is an absolute worst case assuming no additional geometry screening is afforded based on reference receiver fault monitoring using TBAC. Only geometry screening of LPL<LAL=40m is assumed here. Smaller maximum values can be obtained by using additional geometry screening per reference [9].
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AC 20-191 Appendix C Figure C-6 Maximum Error (E) and Monitor Threshold (TBAC) as a function of Alert Limits for Various Numbers of Ground Reference Receivers (M).
C.1.9 Compatibility with Rare Undetected Non-Aircraft System Error Conditions.
Demonstration of compliance with section 5.2.9.6, of this AC may be done by analysis to show that for all possible sizes of navigation error the joint probability that the error is not detected and that the error results in the aircraft landing outside of the safe landing box as defined in section 5.2.10.1.6 is less than 1x10-5. The analysis uses the nominal touchdown distributions (lateral and longitudinal) along with the geometry factors (Svert and Slat), and the maximum allowable Pmd performance of the monitors for satellite ranging source failures and for the Reference Receiver Fault Monitor (RRFM). The nominal touchdown distribution is used to compute a probability of an unsuccessful landing given a particular size of error PUL|E (E). This probability is then multiplied by the probability of an error not being detected as a function of E, Pmd(E).
The probability of an unsuccessful landing given an error is the joint probability that the fault that causes an error, E, is not detected and the landing will be unsuccessful given an error, E:
𝑃𝑃𝑈𝑈𝑈𝑈(𝐸𝐸) =𝑃𝑃 −𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸)∙𝑃𝑃𝑚𝑚𝑚𝑚(𝐸𝐸) < 10 5 [11] To form the conditional unsuccessful landing probability,𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸), a conditional touchdown distribution must be used that would result from a constant bias error in C-10
AC 20-191 Appendix C addition to the fault-free NSE and FTE distributions. This must be done for the full range of relevant error sizes to form the total conditional probability of an unsuccessful landing as a function of the error. The conditional unsuccessful landing probability is expressed as follows for the land short and land long cases:
Land Short ∫ LSC 𝑃𝑃(𝑈𝑈𝑈𝑈|𝐸𝐸)(𝐸𝐸) = p TSE_LON|E(x, E)dx−∞ [12] Land Long 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) =∫ ∞ p (x, E)dxLLC TSE_LON|E [13] Land with wheels less than 5 ft from the edge of the runway:
( ) ∫ ∞ ( ) +∫ −RWE+GW/2 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸 𝐸𝐸 = p x, E dx p (x, E)dxRWE−GW/2 TSE_LAT|E −∞ TSE_LAT|E [14] Where:
LSC – is the land short criteria of section 5.2.9.6 (i.e. 200 ft) LLC – is the land long criteria of section 5.2.9.6 (i.e. 3000 ft) RWE – is the lateral landing criteria as defined in section 5.2.9.6. (i.e. 70 ft) GW – is the lateral distance between the main landing gear pTSE_LON|E(x, E) - is the probability density function for the longitudinal touchdown given a bias of magnitude E.
pTSE_LAT|E(x, E) - is the probability density function for the lateral touchdown given a bias of magnitude E.
Note: Care must be taken to insure consistency of units when making these calculations.
C.1.10 Computing Pmd for Ranging Source Errors.
A bound on the probability of missed detection for the ranging source error, 𝑷𝑷𝒎𝒎𝒎𝒎(𝑬𝑬𝑹𝑹) is defined by the performance constraint region given in Annex 10, appendix B, section
3.6.7.3.3.2. The Pmd performance must lie below the curve defined by Table B-76A in the SARPs repeated here in Table C-7 for convenience.
Table C-7 Pmd_limit Parameters Probability of Missed Detection Pseudorange Error (meters) Pmd_limit ≤ 1 0 ≤ |Er| < 0.75 Pmd_limit ≤ 10(-2.56×|Er| + 1.92) 0.75 ≤ |Er| < 2.7 Pmd_limit ≤ 10-5 2.7 ≤ |Er| < ∞ C-11
AC 20-191 Appendix C For example, in the case of the longitudinal touchdown requirement the vertical position error has the largest effect on the touchdown location. The worst-case projection of a range error into vertical error max (|SAprvert.i|), may be used to determine the resulting limit on Pmd(EV) by substituting ER = EV /max(|S Aprvert.i|).
Figure C-7 illustrates the relationship between Pmd_limit and the Pmd (EV) for max(|SAprvert.i| )= 5.
Figure C-7 Example of the Satellite Ranging Source Pmd in the Range domain and Position Domain.
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AC 20-191 Appendix C Computing Pmd for Reference Receiver Fault Monitoring (RRFM).
The Pmd for the RRFM is given by:
T ( V ) BAC PmdApprox E = ∫ pBmd(x, EV )dx 0 [15] Where:
TBAC is the maximum threshold for the RRFM monitor given by:
0.0842×VAL 2 TBAC = 5.5× + 0.42 Meters [16] m−1 Where:
VAL – is the vertical alert limit that is used by airborne equipment to screen geometry expressed in meters.
And pBmd(x,EV) is the probability density function (pdf) of |Bj,vert(EV)| in the faulted circumstance given by:
0.0842×VAL dnorm x, EV , M −1 M 0.0842×VAL ( ) x ≥ 0; + dnorm− x, EBmd x E = V , p , V M −1 M [17] x< 0; 0 Where dnorm(x,µ,σ) is the Gaussian pdf ( ) 2x−µ− 2 ( ) e 2σdnorm x,µ,σ = [18] 2πσ Bj,vert(EV) is the broadcast difference between the broadcast pseudo-range corrections and the corrections obtained by excluding the jth reference receiver measurement projected in the vertical direction across all of the ranging sources used in the position solution. In this case, it is written as a function of the size of the bias in the vertical error, Ev, due to the single reference receiver fault/ For a derivation of these expressions see Reference [9].
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AC 20-191 Appendix C C.1.11 Example Assessments.
Figure C-8 illustrates a landing short assessment for a hypothetical aircraft with a nominal longitudinal touchdown point of 1500 ft from the threshold and a dispersion that can be bounded by a Gaussian distribution with σ=220 ft. Also, a max( SAprvert. i ) of 5, VAL of 10 meters and GPA of 3 degrees is used. Rearranging equation [11]:
10−5 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) < [19] 𝑃𝑃𝑚𝑚𝑚𝑚(𝐸𝐸) Hence by dividing 10-5 by the Pmd curves for satellite ranging sources and RRFM, the grey “keep out regions” shown in Figure C-8 can be obtained. The assessment is then simple. If the curve for 𝑃𝑃𝑈𝑈𝑈𝑈|𝐸𝐸(𝐸𝐸) does not enter the keep out regions, then the requirement that 𝑃𝑃 (𝐸𝐸) < 10−5𝑈𝑈𝑈𝑈 is met for all values of E.
An alternative approach to the analysis is illustrated in Figure C-9 where the probability of an unsuccessful landing is explicitly calculated for both monitor types (ranging sources and RRFM).
Extension of these examples to the land long and lateral cases is straight forward.
Figure C-8 Example Assessment of Landing Short Performance C-14