A MODEL FOR ESTIMATING THE LATERAL OVERLAP PROBABILITY OF AIRCRAFT WITH RNP ALERTING CAPABILITY IN PARALLEL RNAV ROUTES

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1 6 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES A MODEL FOR ESTIMATING THE LATERAL OVERLAP PROBABILITY OF AIRCRAFT WITH RNP ALERTING CAPABILITY IN PARALLEL RNAV ROUTES Sakae NAGAOKA* *Eectronic Navigation Research Institute/The Universit of Toko Kewords: coision risk, RNP, overap probabiit, RNAV Abstract The Latera overap probabiit is a critica parameter for estimating the coision risk of aircraft pairs fing on parae tracks. This is usua estimated from the distribution modes of cross track errors. As a mode of the distribution, a mixed probabiit densit function (pdf) mode consisting of the generaized Lapace (GL) core and doube exponentia (DE) tai is proposed in this paper. Based on this mode, the characteristics of the distribution are discussed in reation to the maximum atera overap probabiit. In addition to this, simpe expressions of maximum atera overap probabiit for an aircraft with RNP aerting capabiit fing on parae tracks are indicated in this paper. RNP RNAV was the term to describe sstems that were compiant with the requirements of the MASPS [3] in addition to the accurac requirement. This is current mentioned as RNP. The containment integrit was specified b the maximum aowabe probabiit for the event that TSE is greater than the containment imit ( RNP vaue) and the condition has not been detected. The probabiit is 1-5 /hr. Containment Limit (1-1x1-5 ) Desired path d S =4R+d R 4R 1 Introduction R 4R Most modern RNAV (Area Navigation) sstem provides on board performance monitoring and aerting. The navigation specifications deveoped for use b these sstems are current designated RNP (Required Navigation Performance) [1]. The concept of RNP has changed significant in the past ten ears [1],[]. For simpicit, we discuss the probem on the framework of the RNP manua []. In Ref. [], RNP is one of the ke factors in the determination of separation minima used in the airspace where an RNP tpe is specified. The RNP tpe was defined on in terms of navigation performance accurac [], which was specified as a imit where tota sstem error (TSE) must not exceed for 95% of fight time. RNP vaue: R (95%) Fig.1 Configuration of parae route. Fig.1 shows the configuration of parae route for RNP RNAV aircraft. The centerine corresponds to the desired path. It seems to be reasonabe for us to determine the route spacing on the basis of coision risk assessment. The atera overap probabiit is one of the ke parameters of the coision risk mode. This is usua estimated from the distribution of cross track errors of aircraft fing on the tracks under consideration. The shape of cross track error distribution for RNP RNAV aircraft ma be constrained b two conditions, one is the 95% accurac (RNP 1

2 Sakae NAGAOKA vaue) and the other is reated to the containment imit. Though we have no empirica distribution of cross track errors at present, we ma estimate a maximum atera overap probabiit for the RNP RNAV aircraft taking into account the conditions. In the previous papers [4][5], the atera overap probabiit of RNP RNAV aircraft fing on parae tracks was estimated on the basis of TSE (or cross track error) distribution modes taking into account both the 95% accurac and containment integrit. Modes of the probabiit densit functions describing the distributions of cross track errors have been proposed. The modes describing the distributions of the core region and that for tai region were assumed separate. The modes assumed the Gaussian-ike core and the doube exponentia or uniform tai. Under severa assumptions the characteristics of the atera overap probabiit for RNP RNAV parae tracks have been investigated. However, the proposed modes[3][4] have resuted in discontinuit in its probabiit densit at a point associated with the containment imit. In order to avoid this unnaturaness, a continuous probabiit densit function mode composed of a mixture of two distribution modes is proposed in this paper. In the new mode, a mixture of the generaized Lapace(GL) distribution and doube exponentia(de) distribution is used for describing the probabiit densit function of cross track errors. This paper first describes the methods of modeing the distribution and estimating the maximum atera overap probabiit. Then exampes of numerica cacuation are shown. Fina, simpe expressions for cacuating the maximum atera overap probabiit are derived. 3. Capabiit of RNP RNAV Aircraft An RNP RNAV equipped aircraft sstem aerts the piot when the TSE exceeds the containment imit. Fig. shows the possibe states of the navigationa sstem of an RNP RNAV equipped aircraft.. The states are as foows E1: TSE <R and no aert of oss of RNP capabiit E: TSE >R and no aert of oss of RNP capabiit E3: Aert of oss of RNP capabiit where R is the RNP vaue (the 95% accurac requirement). The frequenc of arge cross track error (TSE) of RNP RNAV aircraft wi be reduced significant due to the aerting capabiit. The piot can take remedia actions when the sstem issues an aert of oss of RNP capabiit. Fina, in most cases (E1 and E3) the TSE wi be bounded within the interva [-R,R]. On the case of E aows arge cross track errors beond the containment imit. In the foowing sections, we consider the atera overap probabiit of RNP RNAV aircraft fing on parae tracks under severa assumptions. The schematic view of the tracks under consideration is shown in Fig.1. E1 E3 TSE R & No aert of oss of RNP capabiit E TSE >R & No aert of oss of RNP capabiit Aert of oss of RNP capabiit Pr[E3]<1-4 /hr Pr[E]<1-5 /hr Fig. Probabiit of each event [3] 3 Modes of Cross Track Errors 3.1 Requirements for Probabiit Densit Function Let us denote the probabiit densit function (pdf) of the cross track error, X, of RNP-RNAV aircraft b f (x). The probabiit distribution function F(x) is defined b F ( x) = x f ( x) dx (1) From the 95% containment for RNP vaue, R satisfies F ( R) F( R) 1.5 () From the requirement for the containment imit, et the probabiit that x >R be γ (=

3 A MODEL FOR ESTIMATING THE LATERAL OVERLAP PROBABILITY 1-5 ) assuming that the average fight period of deviated aircraft without the aert of RNP capabiit is one hour. Then, the requirement for the distribution function is given b F ( R) F( R) 1 γ (3) Herein, the pdf, f (x) satisfies F ( ) = f ( x) dx = 1 (4) In this paper we soe consider the case that the equaities in Eq.() and Eq.(3) are satisfied as a simpe exampe. In the previous papers [4],[5], separate modes were defined to satisf the constraints () and (3). This resuted in discontinuit at X = R. In this paper, we consider a continuous mixture distribution as shown in Fig3. Log 1 [1-Cumuative frequenc] % Core RNP vaue Containment Integrit (1-γ -5 )1% Tai γ = 1 5 R R Latera path-keeping error X Fig.3 Required positions on the distribution. 3. Mode of Probabiit Densit Function Let us assume the pdf of the cross track error X, f (x), has the foowing characteristics. (i) zero mean and smmetric around X=. (ii) unimoda (has a unique peak at X=) (iii) most heavi weighted in the core region ( X < K ), K ( < R) is an appropriate vaue ) and sow varing in magnitude in the tai ( X > K) region. Herein, we assume the foowing continuous mixture distribution f ( x) = (1 α ) f c ( x) + α f t ( x) (5) where α is the weighting coefficient associated with the tai region ( < α << 1 ). f c (x) and f t (t) are the pdfs associated with the core and tai, respective. 3.3 Mode for Core Distribution As the candidate of the f c (x), the generaized Lapace (GL) distribution ma be appropriate because it changes the shape according to choice of the shape parameter. The pdf of the GL distribution is given b 1/ b fc ( x) = (abγ( b)) exp( x / a ) (6) where a (> ) is the scae parameter, b (> ) the shape parameter and Γ (b) is the Gamma function which is defined b b 1 Γ( b) = t exp( t) dt (7) The shape of the GL distribution varies with its shape parameter b. In the case of b = 1, Eq.(6) becomes the doube exponentia (DE) distribution. The case of b =. 5 corresponds to the norma (or Gaussian) distribution. Fig.4 shows the shape of GL distribution for different shape parameter vaues. The previous papers [4],[5] indicated that the weighting coefficient 1 α of f c (x) for the Gaussian is greater than 1. This fact contradicts with the condition < α <<1. This suggests that the vaue of b which meets the distribution requirements described in Section 3.1 must be ess than.5. Log 1 f(x) b=.1 f ( x) = 1 abγ( b) b=.5 x / a 1/ b a= x Fig.4 Shapes of GL distribution (a=1) for different shape parameter vaues. e b=5 b= b=1 3.4 Distribution Mode for Tai Region 3

4 Sakae NAGAOKA As mentioned in 3., we assume a distribution which decreases sow with the magnitude of the cross track error and which is ver ow in the tai region. Though it is not impossibe to assume the GL distribution for the tai, it increases the compexit of anasis significant because of too man mode parameters. Therefore, et us assume the DE tai as used in the previous paper [5]. f t ( x) = (λ) exp( x ) (8) where λ (>) is the scae parameter. 4. Latera Overap Probabiit Let us consider the aircraft pairs fing on the parae tracks of which track spacing is S at the same fight eve as shown in Fig.1. The atera overap probabiit of aircraft pairs is given b S + λ P ( S ) = C( z) dz S λ (9) where λ is the average wing span of aircraft, and C ( z) f ( x) f ( x z) dx (1) is the atera overap densit function. Herein, the track spacing S is assumed to be S =4R+d. d( ) is the distance between the containment imits and λ =.31NM(=59m). Under the assumption of 3., the foowing approximation[6] can be used C( S ) f ( S ) (11) At about z = S, b the assumption (iii) and Eq.(11), C (z) varies ver sow. Then, the foowing approximation can be made P ( S ) λ C( S ) (1) Fina, we obtain P S ) 4λ f ( S ) (13) ( 5 Soving Equations First of a, we have to obtain the parameters of f (x) which satisfies the equaities in Eq.() and Eq.(3) simutaneous. In the GL-DE mixture mode, there are four parameters f ( x) = f ( x a, b, λ, α) (14) On the other hand, since the number of equations is two, the parameters cannot be determined unique. Therefore, we ook for a set of parameters, which approximate maximizes the atera overap probabiit. Let us consider the probabiit densit at z = S f ( S ) = (1 α ) f c ( S ) + α f t ( S ) (15) Herein, f S ) << f ( S ) for S R and c( t < α <<1. Then, we can use the foowing approximation. P ( S ) 4λ f ( S ) 4λ α f t ( S ) (16) 4λ α = exp( S ) (17) λ Eq.(17) increases monotonica with α if α is independent of λ. The condition of maximization with respect to λ is given b P ( S ) / λ = (18) for λ = S, Eq.(17) becomes maxima. P ( S ) (λ α / S ) e (19) However, in our case α is a function of λ and dependent on it as described ater. 6. Determination of Parameters The conditions of equaities in Eq.() and Eq.(3) for the mixture distribution can be rewritten as ( 1 α ) I c, 1 + αi t, 1 = 1.5 () ( 1 α) I c, + αi t, = 1 γ (1) where mr I = f ( x dx (m=1,) () c, m mr c ) mr t, m f x mr t ( ) I = dx (m=1,) (3) For the GL core and DE tai modes, we obtain mr I t = e (4), m 1 = mr 1 / b 1 x / a c, m [ abγ( b)] e dx (5) mr I For the convenience of deaing with x, et us normaize the foowing quantities with the RNP vaue R. x = k x R (6) a = k a R (7) λ = k R (8) 4

5 A MODEL FOR ESTIMATING THE LATERAL OVERLAP PROBABILITY where k x, k a and k are the normaized (dimensioness) cross track error, scae parameter of GL mode and that of DE mode, respective. Using these, we obtain m / k I t = e (9), m 1 = m 1 / b 1 kx / ka c, m k Γ m ab ( b)) e dk x I ( (3) We dea with the probem of maximizing a function consisting of four parameters, k a, b, k,α. Eiminating α from Eq.() and Eq.(1) under a given k, we obtain the foowing equation G( k, b, k ) ( I / k + e )( I (1.5)) a c, c,1 / k ( I c, (1 γ ))( I c,1 + e ) = (31) Once b and k are given, k a = k a ( b, k ) can be obtained b soving the above equation. Then, α can be cacuated b / k α ( b, k ) (,1 (1.5)) /(,1 (1 = I c I c e )) (3) 7 Exampes of Cacuation 7.1 Method of Soving Equations First, et the tai parameter be λ = k R with k =4. Second, we set a vaue for b. Third we obtain soutions of k a which satisf Eq.(31) numerica b the Newton method. Then we cacuate the candidates of the soution for α ( < α < 1 ) using Eq.(3). The set of soutions can be obtained b the above-mentioned procedures. Among these soutions, the set which maximizes P ( S ) woud be the fina soution. The cacuation was made b a mathematica software package caed MATHCAD. Fig.5 indicates an exampe of evauated vaues of the function G ( k a,.1,4) for b =.1 and k = 4. Soutions for k a can be seen at approximate 1 and 6. The substitution of soutions of k a (b) obtained from Eq.(31) for Eq.(3) ieds the possibe α vaues. If the cacuated α exists within (,1), it woud be an effective soution. G(k a,b,k ). -. b=.1 k = λ / R = Normaized scae parameter k a =a/r Fig.5 Possibe soutions of k a for a given b(=.1). λ = S = 4R. Log 1 α(b,k ) -5 Non-effective soution Effective soution < α <1..4 Shape parameter b Fig.6 Possibe soutions for α. k = λ / R = 4. Fig.6 pots the soutions of α = α( b, k ) for various vaues of b. b is changed discrete within [.5,.45] b each.5 interva. In this case, we took λ = 4R. As seen in the figure, the effective soution ( < α < 1) can be obtained for the region where b is up to a vaue between.4 and.45. Detaied numerica computation showed that an effective soution existed unti b =.475. In the region where b is smaer than a vaue of about.3, the vaue of α asmptotica approaches to a vaue. No effective soution exists at b =.5. This means that it is impossibe to fit the mixed distribution b the Gaussian core. Fig.7 shows exampes of distributions b soutions for various vaues of b when k = 4. The cases for b =.1,.3 and.4 are indicated. α ooks constant for sma vaues of b and starts decreasing at about b =.3. Since f t( x) >> 5

6 Sakae NAGAOKA f c (x) and since the tai probabiit does not depend on b, one can expect that the mixture probabiit for arge deviations is a function of apha aone. This is wh the tai probabiities for b<.3 seem to have equa vaues. Fig.8 shows the simiar distributions for the case k = 5. Log 1 [1-Cummuative frequenc] Log 1 [1-Cummuative frequenc] b =.1 k = 4( λ = 4R) b =.3 b = Normaized atera deviation k x = x /R Fig.7 Mixture distribution for λ = 4R b =.1 k = 5( λ = 5R) b =.3 b = Normaized atera deviation k x = x /R Fig.8 Mixture distribution for λ = 5R. 7. Maximum vaue for P (S ) From Eq.(1), α can be obtained b 1 γ I c, α = (33) I t, I c, where and I = 1 f ( x dx (34) c, R c ) I t = 1 f x dx R t ( ) (35) Using Eq.(34) and Eq.(35), Eq.(33) can be rewritten as, γ f R = c ( x) dx α (36) ( f t ( x) dx f ( x) dx) R R c As seen in Fig.7 and Fig.8, the core rapid decreases in the region 1 < k x < (R<x<R) as the shape parameter b deceases from about.3. In the region k >, i.e., x>r, the reation x f c ( x) << f t ( x) woud be satisfied. Then, α becomes a function of λ, independent of b, if R f c ( x) dx is negigib sma compared to γ and f x dx R t ( ). The estimate of α, name, α can be approximated b = γ /[ f R t ( x) dx] α (37) Substituting the DE tai mode into Eq.(37), we obtain R α = γ e (38) The conditions < a < 1 are fufied for an choice of λ >. Substituting Eq.(38) for α of Eq.(17), we obtain ( S R) γ e P ( S ) λ (39) λ From the condition P ( S ) / λ =, we can see that Eq.(39) has a oca maximum at λ = S R (assuming S 4R ). The maximum atera overap probabiit for the DE tai mode, P ( ) can be given b P ( S ) S DE DE γ e λ (4) S R 7.3 Exampe of Cacuated P (S ) Fig.9 pots the quantities, i.e., q1 f t ( S, λ), q α max ( λ), and q 3 P ( S, λ) as a function of the DE tai parameter λ for the case S = 4R and R=1NM. As shown in the figure, q 1 becomes maxima at k = 4 ( λ = S ) (not visibe) and q 3 has a maximum at k = ( λ =S -R) as shown in Section 7.. 6

7 A MODEL FOR ESTIMATING THE LATERAL OVERLAP PROBABILITY Log 1 q(λ) q 1 = ft ( S ) = (λ) q S = 4R R = 1NM λ =. 31NM = α ( 3 = P S ) = 4 ( λ q q Normaized scae parameter k (=λ/r) Fig.9 Reation between tai parameter λ and P (S ). 8 Discussion q e λ) 1 S In the previous chapter we on deat with the DE tai mode. However, we have no information about the true tai distribution at this moment. As a most conservative mode for the non-increasing (fattish or decreasing with the magnitude of cross track error) tai distribution, et us consider a uniform distribution for f t (x). In the case of a uniform tai distribution, P ( S ) depends on the ength of the tais. From the consideration in the previous paper [3], we can understand intuitive that the most conservative case is that one tai on one track covers the whoe core region of the other track. Then, the pdf mode of the uniform tai can be given b 1/(L) for x L = S + R f t ( x) = (41) otherwise Eq.() and Eq.(1) can be satisfied regardess of the distribution tpes for the core and the tai. Eq.(37) is aso not imited to the DE tai mode. According, pugging Eq.(41) into Eq.(37), we obtain α for the above uniform tai. This is given b α = L γ / S (4) Then, the vaue of P ( S ) for the uniform tai mode is given b P ( S ) uniform 4λ α f ( S ) = λ γ / S t (43) Tabe 1 compares the simpe expressions of the maximum atera overap probabiit for each mode. These can be expressed as a function of four essentia parameters, i.e., the route spacing S, RNP vaue R, the probabiit of arge cross track errors associated with the containment imit γ and the average wing span of aircraft λ. According, the ratio of the maximum atera overap probabiities for the DE and uniform tai modes is given b P ( S ) uniform ( S R) e = (44) P ( S ) DE S In case of S = 4R, the ratio becomes ver simpe, i.e., e / This means that the maxima overap probabiit for t P ( S ) the uniform tai mode is about 1.4 times greater than for the DE tai mode. Tabe 1 Comparison of maximum atera overap probabiit for each mode. Tai Doube Uniform with mode Exponentia L = S + R (DE) x f t (x) e λ 1/(L) for x L otherwise α R γ e γ L S P ( S ) ( S λ γ R) e 9. Concuding Remarks λ γ This paper proposes a continuous mixture distribution mode for cross track errors of the aircraft with RNP aerting capabiit. Its components are the generaized Lapace (GL) distribution and the doube exponentia (DE) distribution. For arge deviations, a simpe expression for the maximum atera overap probabiit was derived. This approximation S 7

8 Sakae NAGAOKA was compared with that for a uniform tai mode. The anasis of the atera overap probabiit indicated the foowing characteristics: (1) a Gaussian distribution for the overap probabiit is not accurate and () for arge deviations, the maximum atera overap probabiit for DE tai mode is up to 7 % smaer than that for the uniform tai mode. Our resuts are initia. The invove a arge number of parameters. But the motivate the foowing future work: an anasis of the robustness to other tai distributions than the doube exponentia distribution. This answers how conservative the current modes are. And the exact soutions of the maximization probem wi be investigated. This requires forma soutions to non-inear constrained optimization probems, but gives the confidence that the resuts are appicabe. Copright Statement The authors confirm that the, and/or their compan or institution, hod copright on a of the origina materia incuded in their paper. The aso confirm the have obtained permission, from the copright hoder of an third part materia incuded in their paper, to pubish it as part of their paper. The authors grant fu permission for the pubication and distribution of their paper as part of the ICAS8 proceedings or as individua off-prints from the proceedings. ACKNOWLEDGEMENT The author is ver gratefu to Dr. Caus Gwiggner and Dr. Masato Fujita for his discussion and comments on a draft of this paper. References [1] Performance Based Navigation Manua [Draft 5.1], Vo.1, ICAO, March 7 [] Manua on Required Navigation Performance (RNP), Second Edition, ICAO Doc AN/937, 1999 [3] Minimum Aviation Sstem Performance Standards: Required Navigation Performance for Area Navigation, RTCA DO-36A, RTCA Inc., Sept. 13,. [4] Nagaoka S., Estimating the atera overap probabiit for RNP RNAV parae tracks, ICAO SASP- WG/WHL/5-WP/9, Toko, Ma 4. [5] Nagaoka S., An aternative doube exponentia tai mode for estimating the atera overap probabiit of aircraft for RNP RNAV parae tracks, ICAO SASP- WG/WHL/6-WP/5, Washington D.C., November, 4 [6] Air Traffic Services Panning Manua, ICAO Doc 946-AN/94,Part II,p.II--4-19,

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