Approximations to composite GPS protection ... - Semantic Scholar
GPS Solut DOI 10.1007/s10291-010-0193-5
ORIGINAL ARTICLE
Approximations to composite GPS protection levels for aircraft precision approach and landing Dean Bruckner • Frank van Graas Trent Skidmore
•
Received: 29 December 2009 / Accepted: 3 November 2010 Ó Springer-Verlag 2010
Abstract Persistent pseudorange biases constitute a serious potential integrity problem for differential GPS systems used in aircraft precision approach and landing. Various approaches to solve this problem are documented in the literature, including composite protection levels (PLs) that incorporate an explicit bias term in their mathematical expressions. A statistical characterization for such a PL was previously presented in this journal. Modeling GPS error in the position domain as multivariate normal with nonzero mean resulted in the definition of vertical, horizontal, and radial composite PLs, termed VPLc, HPLc, and RPLc, respectively. In the present effort, approximations to these computationally intensive PLs are presented for possible use in real time. Two of these are shown to be over-bounding approximations to exact quantities VPLc and HPLc. An approximation to RPLc is also presented, as well as a method of quantitative evaluation for each of these composite PLs. Monte-Carlo simulations for a single GPS measurement epoch are then developed to illustrate the exact PLs and their approximations and demonstrate that the approximations to VPLc and HPLc over-bound the exact PLs. The approximation to RPLc is shown to be far simpler computationally than the exact PL, but demonstrating that the approximation is an over-bound is left to future research. This paper makes available to the reader both the methods and the MatlabÒ simulation code needed to evaluate computationally efficient PL approximations. Thus, it fosters further research into the use of GPS in safety critical applications.
D. Bruckner (&) F. van Graas T. Skidmore Ohio University, Athens, OH, USA e-mail: [email protected] URL: www.ohio.edu/avionics
Keywords VPL HPL RPL SPL EPL Composite protection level Radial protection level Conspiring bias Bias bound GPS LAAS Monte-Carlo
Introduction The problem of persistent pseudorange biases in differential Global Positioning System (GPS) applications for aircraft precision approach and landing has been discussed extensively in the literature. The traditional method for setting a protection level (PL) in aviation uses has been to leave explicit bias terms out of mathematical expressions for normal (i.e., fault-free) GPS operation and to assume that residual biases behave enough like noise to include them conceptually in the noise terms. It is also assumed that the use of multiple antenna-receiver sets will detect any unexpected biases large enough to be a safety threat and label them as system faults. The GPS Local Area Augmentation System (LAAS) and Wide Area Augmentation System (WAAS) are well-known examples of this traditional approach (RTCA 2004, 2006, 2007). More recently, composite PLs with separate bias terms and noise terms have been suggested in the literature for normal GPS operation (van Graas et al. 2004; Rife and Pullen 2005). For further development of these composite PLs, a statistical characterization was previously presented for Global Positioning System (GPS) user range error as a correlated, normally distributed random variable with nonzero mean over the length of the aircraft precision approach operation (Bruckner et al. 2010b). This characterization includes modeling GPS error in the position domain as multivariate normal with nonzero mean. Based on that model, a vertical composite protection level VPLc
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and a horizontal composite protection level HPLc are implemented as univariate normal distributions with nonzero means. In setting the composite PL, the maximum possible value of the bias in the position domain is used to create a worst-case bound. The assumption of correlation is intended to model the correlated behavior of pseudorange bias between satellites observed by the authors in actual flight data over the past 15 years. The same observations indicate that pseudorange noise remains uncorrelated between satellites, and so the propagation of pseudorange noise in the present effort is the same as that used for the uncorrelated assumption. In Bruckner et al. (2010b), a method is also presented by which exact values, i.e. values accurate to a user-defined error tolerance and consistent with statistical assumptions, of VPLc and HPLc are obtained and by which more computationally efficient approximations may be evaluated. A statistical quadratic form under the multivariate normal distribution is then used to derive a new class of PLs based on the probability enclosed within a radius defined in two or more dimensions. A central chi-square representation of the quadratic form is also presented and is incorporated into a six-step computational procedure for the twodimensional composite radial protection level RPLc. This procedure is extended to the composite spherical protection level (SPLc) and the ellipsoidal protection level (EPLc). Computing this new class of PLs is shown to require a search through all possible combinations of noise and conspiring bias at each GPS measurement epoch. The difficulty of implementing these intensive search-based algorithms in real time and the desirability of substituting more computationally efficient approximations for them are noted. The present effort is an extension of this previous research. Approximations to VPLc, HPLc, and RPLc are presented for possible use in real time during aircraft precision approach and landing. Similar operations envisioned in the future, such as relative navigation between uninhabited aerial systems (UASs) or spacecraft, may also benefit. The VPLc approximation and one of the two HPLc approximations developed here are derived from PLs described in the literature for a satellite pseudorange bias that has been detected and identified as a system fault. However, we interpret the pseudorange bias as a characteristic of normal GPS operation, and the maximum bound of this bias is incorporated into the PL (Bruckner et al. 2010b). Under this new assumption, the two PLs previously described in the literature are shown to be overbounding approximations to exact quantities VPLc and HPLc. This use of exact PLs as a baseline for comparing candidate PL approximations, particularly for error distributions with nonzero mean values, was not found elsewhere in the literature.
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A Monte-Carlo simulation is developed to illustrate the exact PLs and their approximations. The simulation confirms that the approximations to VPLc and HPLc overbound the exact PLs. The approximation to RPLc is shown to be computationally simpler than the exact radial PL, but demonstrating that the approximation is an over-bound is left to future research. Improved computational efficiency of PL calculations, relative to the search-based exact VPLc and HPLc algorithms developed in Bruckner et al. (2010b), is a motivation for this paper. In the present effort, however, the comparison in computational efficiency remains qualitative, and measuring the improvement is left for future research. All of the MatlabÒ code used in this simulation is available from the corresponding author listed on the first page.
An approximation to VPLc An approximation to VPLc already exists in the original expressions for VPL-C and VPLH1 (van Graas et al. 2004; RTCA 2004). The primary difference between the approximation developed in this section and VPL-C and VPLH1 is that bounds Bmax,i are used here rather than observed value Bi. This is because the largest vector sum of conspiring pseudorange biases, translated from the range domain into the position domain and measured relative to the origin along the direction of interest, is desired. The observed bias value relative to the average bias computed simultaneously between a set of reference receivers, as computed in LAAS, is not used. Since the integrity problem of a single-receiver fault in a differential GPS architecture may be handled using three receivers in a mid-value select scheme, the reference receiver index j found in the LAAS specification is suppressed. This approximation is termed VPLc1 and is as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M M uX X s3;i Bmax;i ; VPLc1 ¼ Kffmd t s2 r2 þ ð1Þ 3;i i
i¼1
i¼1
where Kffmd is the fault-free missed detection multiplier, directly obtained from the required probability level; M is the number of user-to-satellite range measurements in the solution; ri is the i-th satellite sigma for range domain error in meters; and Bmax,i is the theoretical bound on the range domain pseudorange bias for the i-th satellite, in meters. The term s3,i is unit less and is the projection, onto the vertical axis in the locally level coordinate frame, of the unit vector from the user toward the i-th satellite. It is also element (3, 3) of the projection matrix S defined in the previous paper for the x, y, z, and t dimensions in the east– north–up locally level coordinate frame (Bruckner et al.
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lines and by the sum of the two line segments for bias and noise aligned on the vertical axis. Since the noise apart from the bias is still characterized completely by the multiplier Kffmd, which is retained here without modification, this construction is an over-bounding approximation. This is because the bias error is always less than or equal to the worst-case bias vector sum. In addition, the analogous form of the single-receiver-fault lateral protection level LPLH1 is an over-bounding approximation for the same reason (RTCA 2004).
Two approximations to HPLc
Fig. 1 Illustration of VPLc1 in x–z plane (dashed bold horizontal lines above and below ellipse), with head-to-tail bias vectors (white solid segments), and bias (red) and noise (blue) projected on vertical axis (vertical dashed lines)
2010b). This projection matrix is the same as the LAAS projection matrix except that in LAAS, the locally level coordinate frame is not always aligned with east and north (RTCA 2004, 2007). The VPLc1 approximation defined in Eq. (1) above is illustrated in the x–z plane in Fig. 1 for a sample noise ellipse offset by a bias from a 6-satellite simulated combination. The position domain bias components, mapped in the worst-case way from biases of 0.2 m uniform magnitude on each satellite, are shown in solid white, stretching ‘‘head-to-tail’’ from the origin to the center of the ellipse. Thus, this ellipse has been displaced from the origin in the vertical dimension as far as it can be with this bias magnitude and satellite geometry. The projection of this vector sum onto the vertical axis is shown by the shorter red dashed line segment, which starts at the origin. The length of the noise vector projected onto the vertical axis, as shown by the longer blue dashed line segment, is obtained by multiplying the square root of element (3, 3) of variance–covariance matrix R by Kffmd. In this simulated example, the probability of hazardously misleading information reaching the aircraft pilot is defined as PHMI = 0.00001. Then, the probability of error being contained by the protection level is Ppl = 1 PHMI = 0.99999. This in turn corresponds to Kffmd = 4.4172, which is obtained through the inverse of the Normal distribution. The value of VPLc1 = 0.78 m ? 2.19 m = 2.97 m is shown by the black horizontal dashed
No HPLc approximation implemented with a bias bound, rather than a bias value, was found in the literature. However, the similar form of HPLH1 found in the LAAS specification can be modified slightly to serve this purpose (RTCA 2004). A second approximation is obtained from prior unpublished research by van Graas at Ohio University. HPLc approximation 1 The LAAS positioning service HPL for the singlereceiver-fault scenario, HPLH1, may be adapted to serve as an approximation to exact HPLc. As before, bound Bmax,i is substituted for observed value Bi, and reference receiver subscript j is suppressed. Additionally, the absolute value is taken of all the components of projection matrix S defined in the LAAS specification. This is necessary to ensure a worst-case combination without the requirement of a search. This absolute value operator is necessary because of the four-dimensional least-squares definition of matrix S. Specifically, if all Bmax,i are identical, then computing the sum by multiplying by S without taking the absolute value results in projecting all error into the clock dimension. This would leave the resultant bias vector in the horizontal plane unusable, with zero length. The revised form is as follows: HPLc1 ¼ Kffmd dmajor vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u M M u X X t þ s1;i Bmax;i þ s2;i Bmax;i i¼1
i¼1
ð2Þ where terms Kffmd, M, ri, s3,i, and Bmax,i are as defined above; s1,i and s2,i are unit less components of matrix S and are the projections onto the east and north axes, respectively, of the unit vector from the user toward the i-th satellite; and dmajor is the 1-sigma uncertainty of error in
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the direction of the error ellipse’s semi-major axis, as defined below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ud 2 þ d 2 dx dy2 2 t x y 2 dmajor ¼ þ þdxy ð3Þ 2 2 s21;i r2i
ð4Þ
s22;i r2i
ð5Þ
i¼1
dy2 ¼
N X
0
North, m
dx2 ¼
N X
0.1
−0.1
−0.2
i¼1
dxy ¼
N X
−0.3
s1;i s2;i r2i
ð6Þ
i¼1
The term dmajor implements the worst-case horizontal error when multiplied by Kffmd and has units of m. The three component terms defined above are the error variance in the x (east) direction, the error variance in the y (north) direction, and error covariance between x and y, respectively. All three of these terms have units of m2. The bias vectors are summed head-to-tail in the worst-case scenario, and the resultant vector is the position domain bias in the horizontal plane. The remaining part of HPLc is approximated by the semi-major axis of the 1-sigma noise ellipse, multiplied by Kffmd (RTCA 2004). This combination is illustrated in Fig. 2, followed by a close-up view in Fig. 3.
Fig. 2 Illustration of HPLc1 in horizontal plane, shown by bold dashed lines at either end of the ellipse, obtained by adding the vector sum of head-to-tail bias vectors to ellipse’s semi-major axis length. Figure 3 is a close-up view of the ellipse’s center
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−0.4
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
East, m Fig. 3 Close-up of worst-case horizontal bias combinations: true (white), the HPLc1 approximation (black), and as magnitude of HPLc1 (red dashed)
In the close-up view of the bias vector combinations of Fig. 3, the white head-to-tail vector combination is the worst-case physically realizable bias combination obtained by searching through all possibilities. The black vector combination is worst-case bias set produced by HPLc1. The dashed line is the resultant of the vector sum of the bias components, here arranged collinearly with the noise ellipse’s semi-major axis to illustrate the protection level computation. As indicated above, this approximation technique strips out all sign information and thus allows vector orientations beyond the ‘‘range of motion’’ actually permitted by the projection matrix. This is because the sign combinations of rows one and two of the projection matrix (i.e., for the x and y axes that define the horizontal) are generally different. Therefore, one can set the signs of bias vector Bmax to match only one of them at a time to perform the absolute value operation. Artificially taking the absolute values of both rows at the same time and ignoring opposite signs that may exist for the same satellite in effect asserts that a given satellite can lie on reciprocal bearings from the user at the same time. However, the technique is clearly an overbound because the length of the new resultant bias vector (shown in black) must always be greater than or equal to the original one (shown in white). Moreover, a search is avoided. In this figure, the resultant bias vector is shown in its third-quadrant version, which is formed by making all x values positive and all y values negative. Of course, the resultant length is the same in all four quadrants, and the
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value of HPLc1 remains the same when the bias and noise magnitudes are added to form the PL.
calculation. It is also demonstrably an over-bound due to the same geometric arguments.
HPLc approximation 2 The second approximation to HPLc is essentially a simpler version of the first. The physical limits on vector orientation imposed by projection matrix S are further disregarded by ‘‘stretching’’ the bias vector combination until all the components are collinear. The following form is used: M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X HPLc2 ¼ Kffmd dmajor þ ðs1;i Bmax;i Þ2 þ ðs2;i Bmax;i Þ2 i¼1
ð7Þ The terms of this equation are the same as those defined above. If the bias bounds for each satellite all have the same magnitude, the form is even simpler: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M X HPLc2 ¼ Kffmd dmajor þ Bmax s21;i þ s22;i ð8Þ i¼1
The second term in this simplified expression is similar to the form of the GPS Horizontal Dilution of Precision (HDOP), but is not the same. In the HDOP form, the summation operator is located inside the square root operator, not outside it as defined here. Approximation HPLc2 is illustrated by the dashed line in Fig. 4. This approximation is slightly more conservative than HPLc1, with a larger bias term and a simpler
0.1
North, m
0
−0.1
An approximation to RPLc An approximation to RPLc with reasonable computational efficiency may be obtained by eliminating the full conspiring bias search and its evaluation of the probability integral at each iteration. Instead, the maximum bias length is derived using a less complex search that avoids performing the integral within each iteration. Array processing software capabilities can considerably speed this reduced search. For a maximum of M = 12 satellites, the number of bias sign combinations L = 2M = 4,096. The maximum length found among the resultant vectors in the x–y plane in the direction of the noise ellipse semi-major axis is selected as the worst-case horizontal bias. This bias is then added collinearly to the 1-sigma error ellipse semi-major axis, multiplied by Kffmd. This procedure may be summarized in the following equation that uses the inner product to obtain the j-th combination’s projection along the x and y axes: RPLc1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Kffmd dmajor þ max ðs1;j Bmax Þ2 þ ðs2;j Bmax Þ2 ;
j ¼ 1; 2; . . .; L
ð9Þ
Here vector s1,j contains the first row of matrix S with the j-th sign combination applied. Vector s2,j is the similarly defined second row of S, and Bmax is the bias vector. As illustrated in Bruckner et al. (2010b), use of a univariate sigma value in the definition of a radial measure creates an under-bound. However, the collinear geometric arrangement of the maximum bias combination with the 1-sigma semi-major axis, multiplied by Kffmd, reverses this effect to some degree. An analysis of the bounding qualities of this approximation is offered for future research.
−0.2
Summary of protection level equations −0.3
−0.4
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
East, m Fig. 4 Close-up of worst-case horizontal bias combinations: true (white), approximation HPLc1 (black), and approximation HPLc2 (red dashed)
The equations for the LAAS baseline PLs and the approximations to the composite PLs for the fault-free or H0 hypothesis are shown in Table 1. Weighting the leastsquares position and protection level calculations by the inverse of variance, as performed in LAAS, could be applied to the composite protection levels. However, it has been observed that the unweighted least-squares method is superior where significant biases exist (van Graas et al. 1995). Where biases are several times that of noise sigmas,
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GPS Solut Table 1 Equations for LAAS and composite protection level approximations
LAAS protection levels
VPLH0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M P ¼ Kffmd s23;i r2i i¼1
HPLH0 = Kffmddmajor
Approximations to composite protection levels sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M P P VPLc1 ¼ Kffmd s23;i r2i þ s3;i Bmax;i i¼1
i¼1
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ffi M 2 2 P s1;i Bmax;i þ P s2;i Bmax;i HPLc1 ¼ Kffmd dmajor þ i¼1
i¼1
M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P HPLc2 ¼ Kffmd dmajor þ ðs1;i Bmax;i Þ2 þ ðs2;i Bmax;i Þ2 i¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1;j Bmax Þ2 þ ðs2;j Bmax Þ2 ; RPLc1 ¼ Kffmd dmajor þ max W=
diag(1/r2i )
weighting by the inverse of bias is also an option. If this third option is selected, noise propagation corrections of the type described in (Bruckner et al. 2010b) must be included.
Monte-Carlo simulation results The results of a Monte-Carlo simulation are provided to illustrate the exact PLs and their approximations in a differential GPS architecture for aircraft precision approach and landing. The primary purpose of this section is to help the reader visualize these PLs geometrically and appreciate their merits relative to each other. Although error is quantified, it serves as an illustration rather than a formal analysis of these techniques. In this simulation, it is also assumed that severe ionospheric and tropospheric gradients, as well as other effects such as ionospheric scintillation, are not present. The intent is to illustrate the implementation of composite PLs and their approximations without unnecessary complications. Of course, these effects must be considered in actual implementation, and designers may readily adapt these composite PLs to such conditions by increasing bias bounds by appropriate amounts. Additionally, approaches such as tropospheric modeling or an ionosphere-free differential architecture may also be employed. The amount of increase to PL bias bounds needed is left to further research. A more detailed presentation of the formation of the bias and noise terms is also provided in Bruckner et al. (2010a). Simulation parameters The Monte-Carlo simulation was performed in Matlab for a single GPS measurement epoch using the parameters shown in Table 2. Results for all of the one- and twodimensional PLs and their approximations are obtained for the same simulation run. The error plots shown in the
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j ¼ 1; 2; . . .; L
W = diag(li) or W = diag(1/Bmax,i)
previous figures were all produced using this simulation code. However, several parameters in those previous illustrations differ from the values used here. The 1-sigma values of the components of range domain noise ri prior to multiplication by Kffmd are shown below in Fig. 5, as a function of elevation (RTCA 2004). The root-sum-square total value of the noise components is shown as well. The 20 cm value of range domain bias bound lmax is approximately the same as the 1-sigma value for total noise error for higher satellite elevation values. The position domain navigation system error (NSE) of a sample 100,000-iteration simulation projected successively along the x, y, and z axes appears in Fig. 6. Since the purpose of this simulation is to illustrate protection levels, no smoothing has been performed. The ellipse plots in other figures show simulation NSE in the position domain. Combining conspiring biases at reference station and aircraft Range domain bias bounds are combined and transformed into the position domain at both the reference station and the aircraft using a worst-case vector sum instead of the root-sum-square procedure employed for noise. This is based on the general principle that probabilistic measures for safety of life cannot be diluted or averaged across an ensemble; every aircraft approach must be fully protected. Nothing is assumed about the probability distribution of the range domain biases other than their orientations and maximum possible magnitudes, which are the bias bounds. In effect, we assume the biases at a single location are perfectly correlated. The one exception to this worst-case bias procedure is that the simulated resultant bias bounds from the reference station and the aircraft are combined in a root-sum-square sense. In this simulation, the primary bias component is assumed to be antenna group delay. If the antennas at the reference station and the aircraft are assumed to be of
GPS Solut Table 2 Parameters of Monte-Carlo simulation used to illustrate and evaluate PL approximations Quantity
Value(s)
Remarks
Satellite geometry
Chosen from the 24-slot constellation, 2 h into GPS day (IS-GPS 2004)
One of seven satellites available is removed to illustrate a more problematic geometry with six satellites
User location
30 deg N, 90 deg W
New Orleans
Number of iterations in 100,000 the simulation, Nsim Probability of 1 9 10-4 = 1 - 0.9999, corresponding hazardously misleading to Kffmd = 3.8906 information, PHMI
Maximum number that a Pentium 4 processor with 2 GB memory can handle in this Matlab implementation Set so that PHMI 9 Nsim [ 1, to allow a few outliers outside PL in the simulation
Maximum probability error tolerance, c
1 9 10-9
Must be smaller than PHMI, but this very small value is used to ensure an accurate estimate of probability contained by estimated PLs
Maximum range domain bias, Bmax
20 cm
Applied equally to all satellites as an upper bound. Includes group delay and does not include multipath or residual iono or tropo errors
Noise, ri
r2i = (r2pr_gnd ? r2air ? r2iono ? r2tropo) (RTCA 2004)
These terms assumed to be Gaussian noise only for simplicity’s sake in this simulation. For flight test results in Bruckner et al. (2010a), all components but thermal noise are shifted into the bias term and bounded by Bmax.
Ground noise and multipath, rpr_gnd
From C4 ground accuracy designator (GAD-C4) Includes thermal noise, multipath, etc. curve (RTCA 2004)
Airborne noise and multipath, rair
Noise from AAD-B curve; multipath from ‘A’ airborne accuracy designator (AAD-A) curve (RTCA 2004)
AAD-B curve for multipath is still under development, so AAD-A curve is used
Residual ionospheric delay, riono Residual tropospheric error, rtropo
0.02 m
Assuming dual-frequency receivers remove first-order ionosphere effects, this is for higher order iono terms As recommended in App. F for availability analyses (RTCA 2004)
Least-squares weighting technique
Unweighted
Best available technique for bias bounds and noise sigmas of comparable size. This is not the case for post-CNMP bias and noise in Bruckner et al. (2010a), where noise is much smaller than bias bounds.
Smoothing
None
Simulation illustrates PLs, not smoothing
Vertical dilution of precision (VDOP)
1.762
Using unweighted LS
Horizontal dilution of precision (HDOP)
1.942
Using unweighted LS
Zero
different types, antenna group delay at reference station and aircraft can be assumed to be independent, thereby justifying this exception. If components such as multipath, residual ionospheric error, and residual tropospheric error were to be shifted into the bias term, their independence between reference station and aircraft would need to be assumed or established in order to preserve this exception. Another justification would be to show that the likelihood of the total worst-case combination of bias and noise is less than the probability allowed for hazardously misleading information. The strictest probability requirement is currently 1 9 10-9 for Ground Based Augmentation System (GBAS) Service Levels (GSLs) D through F (RTCA 2004). Since the composite PL is typically smallest and therefore most
useful when these components are included in the bias term instead of the noise term, an example of such a justification is included below even though not implemented here. For the symmetric bias combinations analyzed here, such that two bias combinations in each epoch have the same magnitude, the probability of the worst-case total bias combination being realized at a single location, namely either reference station or aircraft, is 1/2(M - 1), where M is the number of satellites in the least-squares solution. It is assumed that the two largest bias components are those due to multipath and antenna group delay, that they are of the same order of magnitude, and that they are also independent. The probability of both of these biases conspiring together in a worst-case way at a single location, either reference station or aircraft, therefore, is [1/2(M - 1)]2, or
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GPS Solut Fig. 5 Noise sigma values (noise components and total) versus satellite elevation
0.7 Total noise sigma Ground sigma Airborne sigma Residual iono sigma
0.6
Error rms, m
0.5
0.4
0.3
0.2
0.1
0 0
10
20
30
40
50
60
70
80
90
Satellite elevation, degrees
Kffmd 9 r, which for illustration purposes is also assumed to be 2 9 10-9, equivalent to Kffmd = 6 in the standard normal distribution. Obviously, the overall probability of a worstcase bias and noise combination under this set of assumptions is vanishingly small, and some effort should be taken to reduce the bias or noise bounds in a practical implementation. Further research is needed to establish the independence of the bias components at a single location and between different locations. VPLc and its approximation
Fig. 6 Simulation NSE (noise and bias) for east, north, and up axes (unsmoothed; 100,000 points)
1/2(2M - 2). For M = 6, this probability is 1/1,024. For M = 8, it is 1/16,384, and for M = 10, it is 1/262,144. If antenna group delay and multipath are also assumed to be independent between the reference station and aircraft, then the probability of the total worst-case bias combination between the two locations is 1/(262,144)2 = 1.46 9 10-11. It is recognized that bias is independent of thermal noise (the only remaining component of the noise term), and thermal noise measurements at two locations are also independent of each other. The quantity 1.46 9 10-11 is then twice multiplied by the probability of the noise being outside
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Figure 7 shows the error ellipse projected onto the x– z plane. Exact and approximate VPLc values are shown as horizontal solid and dashed lines. The vertical bias of 1.083 m is apparent in the upward shift of the error ellipse—again the worst-case vertical displacement possible for this epoch. A confidence interval was not determined for either VPLc in this illustration, but several runs of the simulation placed 99.99 ± 0.01% of the errors inside the exact protection level value of 7.661 m. The performance of VPLc and its approximation VPLc1 in simulation for this single GPS epoch is shown in Table 3 below: HPLc and its approximations Similarly, errors projected onto the horizontal x–y plane and then onto the rotated horizontal axis, along with HPLc and its two approximations, are shown in Fig. 8. The angle
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Fig. 7 Simulated biased, unsmoothed error in x–z plane, projected onto vertical axis, with ± VPLc (solid horizontal lines) and approximation VPLc1 (dashed horizontal lines). The displacement of the error ellipse by the worst-case bias is shown by the small circle and vector extending from the origin. The projection of error from each simulation iteration onto the vertical axis is also shown
Fig. 8 Simulated biased, unsmoothed error in x–y plane, projected onto rotated axis with ± HPLc (solid lines), and two approximations, HPLc1 and HPLc2 (dashed lines) Table 4 Simulated performance of HPLc methods HPLc method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Table 3 Simulated performance of VPLc methods VPLc method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Exact
4.683
99.991
HPLc1 approximation
4.968
99.996
Exact
7.661
99.992
HPLc2 approximation
5.049
99.996
VPLc1 approximation
7.962
99.997
between the x axis and the rotated horizontal axis is -41 degrees. The resultant bias vector shown is the one that produced maximum displacement from the origin when projected onto the rotated horizontal axis. The simulated performance of HPLc and its two approximations appears below in Table 4. As expected, HPLc2 is slightly larger and therefore more conservative than HPLc1. RPLc and its approximation The set of conspiring pseudorange bias combinations available in the horizontal plane of the position domain appears in Fig. 9. The bias ‘‘cloud’’ exhibits symmetry through the origin, as one might expect for positive and negative pairs of bias combinations. The angle between the x axis and the approximate ‘‘major axis’’ of the bias cloud is approximately -25 degrees, sixteen degrees offset from the -41-degree angle of the noise ellipse’s major axis. In general, until the noise ellipse is considered, it is not known
which of these bias combinations will produce the largest protection level radius. Because the bias cloud and noise error ellipse in the horizontal plane do not have the same orientation, the worst-case bias combination considering bias alone may be different from the worst-case combination of bias and noise used for HPLc. Indeed, the greater the difference in orientation between the bias cloud and noise error ellipse, the greater the benefit that will be realized from a full search in a reduction of RPLc. After the worst-case set of conspiring biases is obtained by search, the noise ellipse is added to this plot, as shown in Fig. 10. Results appear in Table 5 below. The same bias combination was selected by both methods, but RPLc1 arranges noise and bias collinearly, yielding a larger PL. Summary of simulation results The quantity RPLc (99.99%) equals 4.732 m, which is slightly larger than the HPLc (99.99%) value of 4.683 m, as expected. The difference between these measures increases
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Fig. 9 Bias combinations available in the horizontal plane
for smaller values of enclosed probability Ppl and decreases for larger Ppl values. For example, RPLc (95%) is 2.592 m while HPLc (95%) is 2.500 m, a difference of 0.092 m instead of the 0.049 difference seen when Ppl is 99.99%. At probability levels of 99.999% and higher, this 100,000point simulation could not reliably provide results. The approximations for VPLc and HPLc are observed to be over-bounds, also as expected. At the 99.99% probability level used in this simulation, the use of the approximations added about a half order of magnitude (i.e., 0.005%) of excess overbounding error. Depending on the application and specified probability level, these approximations will continue to prove quite useful. The percentage of points enclosed in the PL for a given simulation run follows a sampling distribution whose mean was not verified, but which appears to be the nominal value of the intended probability. This is the case for all MonteCarlo results presented here, regardless of the PL type. The simulation helps to illustrate the differences between the various PL approximations and to verify the algorithms used to obtain them. However, for the very high probability levels used in LAAS similar applications, the simulation cannot yet support meaningful conclusions. Further development of the simulation is left to future research. The application of approximations VPLc1 and HPLc1 to flight test data, including a significant performance improvement obtained over LAAS PLs at PHMI = 2 9 10-9 (Kffmd = 6), is available to the reader in Bruckner et al. (2010a).
Summary
Fig. 10 RPLc (large, bold black circle) with bias errors (yellow dots clustered near origin) and biased noise errors (blue dots) shown. The bias offset is also shown (bold black vector and small black circle symbol). The approximation RPLc1 (dashed circle) is slightly larger than the exact value, since noise and bias are arranged collinearly
Approximations to VPLc, HPLc, and RPLc are obtained and are compared to exact PL values developed in previous research. Results from a Monte-Carlo simulation for a single GPS measurement epoch are presented to illustrate these exact PLs and their approximations. It is shown that previously published composite vertical and horizontal PL formulations for biased measurements are approximations of the exact methods. In the case of VPLc and HPLc, these approximations are shown to be over-bounds of the exact values. The approximation to RPLc is shown to be far simpler than the exact PL, but a demonstration or proof that the approximation is an over-bound is left to future research. This paper makes available to the reader both the methods and the MatlabÒ simulation code needed to evaluate computationally efficient PL approximations. Thus, it fosters further research into the use of GPS in safety critical applications.
Table 5 Simulated performance of RPLc methods RPLc Method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Exact
4.732
99.991
RPLc1 approximation
4.865
99.992
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Future work A great deal of fruitful work remains to be accomplished in this research area. Suggested tasks include the following:
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1. 2.
3.
4.
5.
6.
7.
Characterize the bounding performance of the RPLc approximation. Quantify the improvements in computational efficiency for the PL approximations, relative to the exact PL methods. Examine the statistical independence of the multipath and antenna group delay bias components. If independence is proven, reduce the magnitude of their contributions to the composite PL formulated in Bruckner et al. (2010a, b). Examine the statistical independence between biases at the reference station and those in the airborne system. If dependence is found, calculate the probability that these will simultaneously occur along with the worstcase noise values at both locations, and update the algorithm and PL formulation used in this study. Determine appropriate maximum bias magnitudes for various effects, including ionospheric scintillation and severe ionospheric and tropospheric gradients. Use flight test data to examine composite PLs versus observed errors in more detail, deriving a confidence interval for the observed errors and comparing it to the PL. Develop and test additional PL approximations.
Acknowledgments This research was supported, in part, by the Federal Aviation Administration under Aviation Cooperative Research Agreement 98-G-002.
References Bruckner D, van Graas F, Skidmore T (2010a) Algorithm and flight test results to exchange code noise and multipath for biases in dual frequency differential GPS for precision approach. Navigation 57(3):213–229 Bruckner D, van Graas F, Skidmore T (2010b) Statistical characterization of composite protection levels for GPS. GPS Solut. doi: 10.1007/s10291-010-0188-2 IS-GPS (2004) Interface specification: navstar GPS space segment/ navigation user interfaces/IS-GPS-200D. Available from U.S. Coast Guard Navig Ctr, Alexandria, Virginia Rife J, Pullen S (2005) Impact of measurement biases on availability for category III LAAS. Navigation 50(4):215–228 RTCA (2004) Minimum aviation systems performance standards (MASPS) for the local area augmentation system (LAAS) RTCA DO-245A RTCA (2006) Minimum operational performance standards (MOPS) for GPS wide area augmentation system (WAAS) airborne equipment, RTCA DO-229D RTCA (2007) Minimum operational performance standards (MOPS) for GPS LAAS airborne equipment RTCA DO-253B van Graas F, Diggle D, Uijt de Haag M, Wullschleger V, Velez r, Lamb D, Dimeo M, Kuehl G, Hilb R (1995) FAA/Ohio University/United Parcel Service DGPS autoland flight test demonstration. Proc Inst Navig GPS 727–737. doi:9/1995
van Graas F, Krishnan V, Suddapalli R, Skidmore T (2004) Conspiring biases in the local area augmentation system. Proc Inst Navig Annu Meet 300–307. doi:6/2004
Author Biographies Dr. Dean Bruckner is the Assistant Director (Technical) of the Avionics Engineering Center at Ohio University in Athens, Ohio. He received his Ph.D. degree in Electrical Engineering from Ohio University in 2010 with a dissertation that included flight test results for dual-frequency GPS precision approach and landing. During a career in the United States Coast Guard, he earned a B.S.E.E. from the Coast Guard Academy and an M.S.E.E. from the Naval Postgraduate School, both with honors and with specialization in electronic navigation. He served as Commanding Officer of the Coast Guard Loran-C transmitter station near Istanbul, Turkey, and was the nationwide electronics support manager for Coast Guard ships, boats, aids to navigation and shore facilities. He has recently provided navigation and engineering expertise to the F-35 Joint Strike Fighter navigation sensor test program. Dr. Frank van Graas is the Fritz J. and Dolores H. Russ Professor of Electrical Engineering at Ohio University. He earned his Ph.D. from Ohio University in 1988. He holds a BSEE and MSEE degrees with specialization in avionics from Delft University of Technology, The Netherlands. His extensive involvement in satellite navigation included conducting the first real-time GPS attitude and heading flight experiment on a DC-3 in 1991, and the first kinematic dual-frequency GPS autoland flight tests using NASA Langley Research Center’s Boeing 737 in 1993. He received the Johannes Kepler Award for ‘‘sustained and significant contributions to satellite navigation’’ from the Satellite Division of the Institute of Navigation in 1996, and served as President of the Institute of Navigation from 1998 to 1999. At Ohio University’s Avionics Engineering Center, his roles include Principal Investigator for the FAAsponsored Global Positioning System (GPS) research grant to investigate the Local Area Augmentation System for Aircraft Precision Approach & Landing and Aircraft Surface Movement Guidance. He has lectured extensively throughout the United States, Canada, Europe and Russia for such organizations as the International Air Transport Association, the NATO Advisory Group on Aerospace Research and Development (AGARD), Navtech Seminars, Inc., and the International Society for Optical Engineering (SPIE).
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GPS Solut Dr. Trent Skidmore is a Senior Research Engineer and Adjunct Assistant Professor in the Avionics Engineering Center at Ohio University. He is involved in a variety of projects involving applications of GPS, including the use of differential GPS for aircraft precision approach applications. Recent projects include the Department of Defense Joint Precision Approach and Landing System (JPALS), the Automated Aerial Refueling (AAR) program, as well as the Federal Aviation Administration Local Area Augmentation System (LAAS). Trent received his B.Sc. in 1986 (with Honors) and M.Sc. in 1988 from Michigan Technological University and his Ph.D. from Ohio University in 1991, all in electrical engineering.
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ORIGINAL ARTICLE
Approximations to composite GPS protection levels for aircraft precision approach and landing Dean Bruckner • Frank van Graas Trent Skidmore
•
Received: 29 December 2009 / Accepted: 3 November 2010 Ó Springer-Verlag 2010
Abstract Persistent pseudorange biases constitute a serious potential integrity problem for differential GPS systems used in aircraft precision approach and landing. Various approaches to solve this problem are documented in the literature, including composite protection levels (PLs) that incorporate an explicit bias term in their mathematical expressions. A statistical characterization for such a PL was previously presented in this journal. Modeling GPS error in the position domain as multivariate normal with nonzero mean resulted in the definition of vertical, horizontal, and radial composite PLs, termed VPLc, HPLc, and RPLc, respectively. In the present effort, approximations to these computationally intensive PLs are presented for possible use in real time. Two of these are shown to be over-bounding approximations to exact quantities VPLc and HPLc. An approximation to RPLc is also presented, as well as a method of quantitative evaluation for each of these composite PLs. Monte-Carlo simulations for a single GPS measurement epoch are then developed to illustrate the exact PLs and their approximations and demonstrate that the approximations to VPLc and HPLc over-bound the exact PLs. The approximation to RPLc is shown to be far simpler computationally than the exact PL, but demonstrating that the approximation is an over-bound is left to future research. This paper makes available to the reader both the methods and the MatlabÒ simulation code needed to evaluate computationally efficient PL approximations. Thus, it fosters further research into the use of GPS in safety critical applications.
D. Bruckner (&) F. van Graas T. Skidmore Ohio University, Athens, OH, USA e-mail: [email protected] URL: www.ohio.edu/avionics
Keywords VPL HPL RPL SPL EPL Composite protection level Radial protection level Conspiring bias Bias bound GPS LAAS Monte-Carlo
Introduction The problem of persistent pseudorange biases in differential Global Positioning System (GPS) applications for aircraft precision approach and landing has been discussed extensively in the literature. The traditional method for setting a protection level (PL) in aviation uses has been to leave explicit bias terms out of mathematical expressions for normal (i.e., fault-free) GPS operation and to assume that residual biases behave enough like noise to include them conceptually in the noise terms. It is also assumed that the use of multiple antenna-receiver sets will detect any unexpected biases large enough to be a safety threat and label them as system faults. The GPS Local Area Augmentation System (LAAS) and Wide Area Augmentation System (WAAS) are well-known examples of this traditional approach (RTCA 2004, 2006, 2007). More recently, composite PLs with separate bias terms and noise terms have been suggested in the literature for normal GPS operation (van Graas et al. 2004; Rife and Pullen 2005). For further development of these composite PLs, a statistical characterization was previously presented for Global Positioning System (GPS) user range error as a correlated, normally distributed random variable with nonzero mean over the length of the aircraft precision approach operation (Bruckner et al. 2010b). This characterization includes modeling GPS error in the position domain as multivariate normal with nonzero mean. Based on that model, a vertical composite protection level VPLc
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and a horizontal composite protection level HPLc are implemented as univariate normal distributions with nonzero means. In setting the composite PL, the maximum possible value of the bias in the position domain is used to create a worst-case bound. The assumption of correlation is intended to model the correlated behavior of pseudorange bias between satellites observed by the authors in actual flight data over the past 15 years. The same observations indicate that pseudorange noise remains uncorrelated between satellites, and so the propagation of pseudorange noise in the present effort is the same as that used for the uncorrelated assumption. In Bruckner et al. (2010b), a method is also presented by which exact values, i.e. values accurate to a user-defined error tolerance and consistent with statistical assumptions, of VPLc and HPLc are obtained and by which more computationally efficient approximations may be evaluated. A statistical quadratic form under the multivariate normal distribution is then used to derive a new class of PLs based on the probability enclosed within a radius defined in two or more dimensions. A central chi-square representation of the quadratic form is also presented and is incorporated into a six-step computational procedure for the twodimensional composite radial protection level RPLc. This procedure is extended to the composite spherical protection level (SPLc) and the ellipsoidal protection level (EPLc). Computing this new class of PLs is shown to require a search through all possible combinations of noise and conspiring bias at each GPS measurement epoch. The difficulty of implementing these intensive search-based algorithms in real time and the desirability of substituting more computationally efficient approximations for them are noted. The present effort is an extension of this previous research. Approximations to VPLc, HPLc, and RPLc are presented for possible use in real time during aircraft precision approach and landing. Similar operations envisioned in the future, such as relative navigation between uninhabited aerial systems (UASs) or spacecraft, may also benefit. The VPLc approximation and one of the two HPLc approximations developed here are derived from PLs described in the literature for a satellite pseudorange bias that has been detected and identified as a system fault. However, we interpret the pseudorange bias as a characteristic of normal GPS operation, and the maximum bound of this bias is incorporated into the PL (Bruckner et al. 2010b). Under this new assumption, the two PLs previously described in the literature are shown to be overbounding approximations to exact quantities VPLc and HPLc. This use of exact PLs as a baseline for comparing candidate PL approximations, particularly for error distributions with nonzero mean values, was not found elsewhere in the literature.
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A Monte-Carlo simulation is developed to illustrate the exact PLs and their approximations. The simulation confirms that the approximations to VPLc and HPLc overbound the exact PLs. The approximation to RPLc is shown to be computationally simpler than the exact radial PL, but demonstrating that the approximation is an over-bound is left to future research. Improved computational efficiency of PL calculations, relative to the search-based exact VPLc and HPLc algorithms developed in Bruckner et al. (2010b), is a motivation for this paper. In the present effort, however, the comparison in computational efficiency remains qualitative, and measuring the improvement is left for future research. All of the MatlabÒ code used in this simulation is available from the corresponding author listed on the first page.
An approximation to VPLc An approximation to VPLc already exists in the original expressions for VPL-C and VPLH1 (van Graas et al. 2004; RTCA 2004). The primary difference between the approximation developed in this section and VPL-C and VPLH1 is that bounds Bmax,i are used here rather than observed value Bi. This is because the largest vector sum of conspiring pseudorange biases, translated from the range domain into the position domain and measured relative to the origin along the direction of interest, is desired. The observed bias value relative to the average bias computed simultaneously between a set of reference receivers, as computed in LAAS, is not used. Since the integrity problem of a single-receiver fault in a differential GPS architecture may be handled using three receivers in a mid-value select scheme, the reference receiver index j found in the LAAS specification is suppressed. This approximation is termed VPLc1 and is as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M M uX X s3;i Bmax;i ; VPLc1 ¼ Kffmd t s2 r2 þ ð1Þ 3;i i
i¼1
i¼1
where Kffmd is the fault-free missed detection multiplier, directly obtained from the required probability level; M is the number of user-to-satellite range measurements in the solution; ri is the i-th satellite sigma for range domain error in meters; and Bmax,i is the theoretical bound on the range domain pseudorange bias for the i-th satellite, in meters. The term s3,i is unit less and is the projection, onto the vertical axis in the locally level coordinate frame, of the unit vector from the user toward the i-th satellite. It is also element (3, 3) of the projection matrix S defined in the previous paper for the x, y, z, and t dimensions in the east– north–up locally level coordinate frame (Bruckner et al.
GPS Solut
lines and by the sum of the two line segments for bias and noise aligned on the vertical axis. Since the noise apart from the bias is still characterized completely by the multiplier Kffmd, which is retained here without modification, this construction is an over-bounding approximation. This is because the bias error is always less than or equal to the worst-case bias vector sum. In addition, the analogous form of the single-receiver-fault lateral protection level LPLH1 is an over-bounding approximation for the same reason (RTCA 2004).
Two approximations to HPLc
Fig. 1 Illustration of VPLc1 in x–z plane (dashed bold horizontal lines above and below ellipse), with head-to-tail bias vectors (white solid segments), and bias (red) and noise (blue) projected on vertical axis (vertical dashed lines)
2010b). This projection matrix is the same as the LAAS projection matrix except that in LAAS, the locally level coordinate frame is not always aligned with east and north (RTCA 2004, 2007). The VPLc1 approximation defined in Eq. (1) above is illustrated in the x–z plane in Fig. 1 for a sample noise ellipse offset by a bias from a 6-satellite simulated combination. The position domain bias components, mapped in the worst-case way from biases of 0.2 m uniform magnitude on each satellite, are shown in solid white, stretching ‘‘head-to-tail’’ from the origin to the center of the ellipse. Thus, this ellipse has been displaced from the origin in the vertical dimension as far as it can be with this bias magnitude and satellite geometry. The projection of this vector sum onto the vertical axis is shown by the shorter red dashed line segment, which starts at the origin. The length of the noise vector projected onto the vertical axis, as shown by the longer blue dashed line segment, is obtained by multiplying the square root of element (3, 3) of variance–covariance matrix R by Kffmd. In this simulated example, the probability of hazardously misleading information reaching the aircraft pilot is defined as PHMI = 0.00001. Then, the probability of error being contained by the protection level is Ppl = 1 PHMI = 0.99999. This in turn corresponds to Kffmd = 4.4172, which is obtained through the inverse of the Normal distribution. The value of VPLc1 = 0.78 m ? 2.19 m = 2.97 m is shown by the black horizontal dashed
No HPLc approximation implemented with a bias bound, rather than a bias value, was found in the literature. However, the similar form of HPLH1 found in the LAAS specification can be modified slightly to serve this purpose (RTCA 2004). A second approximation is obtained from prior unpublished research by van Graas at Ohio University. HPLc approximation 1 The LAAS positioning service HPL for the singlereceiver-fault scenario, HPLH1, may be adapted to serve as an approximation to exact HPLc. As before, bound Bmax,i is substituted for observed value Bi, and reference receiver subscript j is suppressed. Additionally, the absolute value is taken of all the components of projection matrix S defined in the LAAS specification. This is necessary to ensure a worst-case combination without the requirement of a search. This absolute value operator is necessary because of the four-dimensional least-squares definition of matrix S. Specifically, if all Bmax,i are identical, then computing the sum by multiplying by S without taking the absolute value results in projecting all error into the clock dimension. This would leave the resultant bias vector in the horizontal plane unusable, with zero length. The revised form is as follows: HPLc1 ¼ Kffmd dmajor vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u M M u X X t þ s1;i Bmax;i þ s2;i Bmax;i i¼1
i¼1
ð2Þ where terms Kffmd, M, ri, s3,i, and Bmax,i are as defined above; s1,i and s2,i are unit less components of matrix S and are the projections onto the east and north axes, respectively, of the unit vector from the user toward the i-th satellite; and dmajor is the 1-sigma uncertainty of error in
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the direction of the error ellipse’s semi-major axis, as defined below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ud 2 þ d 2 dx dy2 2 t x y 2 dmajor ¼ þ þdxy ð3Þ 2 2 s21;i r2i
ð4Þ
s22;i r2i
ð5Þ
i¼1
dy2 ¼
N X
0
North, m
dx2 ¼
N X
0.1
−0.1
−0.2
i¼1
dxy ¼
N X
−0.3
s1;i s2;i r2i
ð6Þ
i¼1
The term dmajor implements the worst-case horizontal error when multiplied by Kffmd and has units of m. The three component terms defined above are the error variance in the x (east) direction, the error variance in the y (north) direction, and error covariance between x and y, respectively. All three of these terms have units of m2. The bias vectors are summed head-to-tail in the worst-case scenario, and the resultant vector is the position domain bias in the horizontal plane. The remaining part of HPLc is approximated by the semi-major axis of the 1-sigma noise ellipse, multiplied by Kffmd (RTCA 2004). This combination is illustrated in Fig. 2, followed by a close-up view in Fig. 3.
Fig. 2 Illustration of HPLc1 in horizontal plane, shown by bold dashed lines at either end of the ellipse, obtained by adding the vector sum of head-to-tail bias vectors to ellipse’s semi-major axis length. Figure 3 is a close-up view of the ellipse’s center
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−0.4
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
East, m Fig. 3 Close-up of worst-case horizontal bias combinations: true (white), the HPLc1 approximation (black), and as magnitude of HPLc1 (red dashed)
In the close-up view of the bias vector combinations of Fig. 3, the white head-to-tail vector combination is the worst-case physically realizable bias combination obtained by searching through all possibilities. The black vector combination is worst-case bias set produced by HPLc1. The dashed line is the resultant of the vector sum of the bias components, here arranged collinearly with the noise ellipse’s semi-major axis to illustrate the protection level computation. As indicated above, this approximation technique strips out all sign information and thus allows vector orientations beyond the ‘‘range of motion’’ actually permitted by the projection matrix. This is because the sign combinations of rows one and two of the projection matrix (i.e., for the x and y axes that define the horizontal) are generally different. Therefore, one can set the signs of bias vector Bmax to match only one of them at a time to perform the absolute value operation. Artificially taking the absolute values of both rows at the same time and ignoring opposite signs that may exist for the same satellite in effect asserts that a given satellite can lie on reciprocal bearings from the user at the same time. However, the technique is clearly an overbound because the length of the new resultant bias vector (shown in black) must always be greater than or equal to the original one (shown in white). Moreover, a search is avoided. In this figure, the resultant bias vector is shown in its third-quadrant version, which is formed by making all x values positive and all y values negative. Of course, the resultant length is the same in all four quadrants, and the
GPS Solut
value of HPLc1 remains the same when the bias and noise magnitudes are added to form the PL.
calculation. It is also demonstrably an over-bound due to the same geometric arguments.
HPLc approximation 2 The second approximation to HPLc is essentially a simpler version of the first. The physical limits on vector orientation imposed by projection matrix S are further disregarded by ‘‘stretching’’ the bias vector combination until all the components are collinear. The following form is used: M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X HPLc2 ¼ Kffmd dmajor þ ðs1;i Bmax;i Þ2 þ ðs2;i Bmax;i Þ2 i¼1
ð7Þ The terms of this equation are the same as those defined above. If the bias bounds for each satellite all have the same magnitude, the form is even simpler: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M X HPLc2 ¼ Kffmd dmajor þ Bmax s21;i þ s22;i ð8Þ i¼1
The second term in this simplified expression is similar to the form of the GPS Horizontal Dilution of Precision (HDOP), but is not the same. In the HDOP form, the summation operator is located inside the square root operator, not outside it as defined here. Approximation HPLc2 is illustrated by the dashed line in Fig. 4. This approximation is slightly more conservative than HPLc1, with a larger bias term and a simpler
0.1
North, m
0
−0.1
An approximation to RPLc An approximation to RPLc with reasonable computational efficiency may be obtained by eliminating the full conspiring bias search and its evaluation of the probability integral at each iteration. Instead, the maximum bias length is derived using a less complex search that avoids performing the integral within each iteration. Array processing software capabilities can considerably speed this reduced search. For a maximum of M = 12 satellites, the number of bias sign combinations L = 2M = 4,096. The maximum length found among the resultant vectors in the x–y plane in the direction of the noise ellipse semi-major axis is selected as the worst-case horizontal bias. This bias is then added collinearly to the 1-sigma error ellipse semi-major axis, multiplied by Kffmd. This procedure may be summarized in the following equation that uses the inner product to obtain the j-th combination’s projection along the x and y axes: RPLc1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Kffmd dmajor þ max ðs1;j Bmax Þ2 þ ðs2;j Bmax Þ2 ;
j ¼ 1; 2; . . .; L
ð9Þ
Here vector s1,j contains the first row of matrix S with the j-th sign combination applied. Vector s2,j is the similarly defined second row of S, and Bmax is the bias vector. As illustrated in Bruckner et al. (2010b), use of a univariate sigma value in the definition of a radial measure creates an under-bound. However, the collinear geometric arrangement of the maximum bias combination with the 1-sigma semi-major axis, multiplied by Kffmd, reverses this effect to some degree. An analysis of the bounding qualities of this approximation is offered for future research.
−0.2
Summary of protection level equations −0.3
−0.4
−0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
East, m Fig. 4 Close-up of worst-case horizontal bias combinations: true (white), approximation HPLc1 (black), and approximation HPLc2 (red dashed)
The equations for the LAAS baseline PLs and the approximations to the composite PLs for the fault-free or H0 hypothesis are shown in Table 1. Weighting the leastsquares position and protection level calculations by the inverse of variance, as performed in LAAS, could be applied to the composite protection levels. However, it has been observed that the unweighted least-squares method is superior where significant biases exist (van Graas et al. 1995). Where biases are several times that of noise sigmas,
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GPS Solut Table 1 Equations for LAAS and composite protection level approximations
LAAS protection levels
VPLH0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M P ¼ Kffmd s23;i r2i i¼1
HPLH0 = Kffmddmajor
Approximations to composite protection levels sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M P P VPLc1 ¼ Kffmd s23;i r2i þ s3;i Bmax;i i¼1
i¼1
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ffi M 2 2 P s1;i Bmax;i þ P s2;i Bmax;i HPLc1 ¼ Kffmd dmajor þ i¼1
i¼1
M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P HPLc2 ¼ Kffmd dmajor þ ðs1;i Bmax;i Þ2 þ ðs2;i Bmax;i Þ2 i¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1;j Bmax Þ2 þ ðs2;j Bmax Þ2 ; RPLc1 ¼ Kffmd dmajor þ max W=
diag(1/r2i )
weighting by the inverse of bias is also an option. If this third option is selected, noise propagation corrections of the type described in (Bruckner et al. 2010b) must be included.
Monte-Carlo simulation results The results of a Monte-Carlo simulation are provided to illustrate the exact PLs and their approximations in a differential GPS architecture for aircraft precision approach and landing. The primary purpose of this section is to help the reader visualize these PLs geometrically and appreciate their merits relative to each other. Although error is quantified, it serves as an illustration rather than a formal analysis of these techniques. In this simulation, it is also assumed that severe ionospheric and tropospheric gradients, as well as other effects such as ionospheric scintillation, are not present. The intent is to illustrate the implementation of composite PLs and their approximations without unnecessary complications. Of course, these effects must be considered in actual implementation, and designers may readily adapt these composite PLs to such conditions by increasing bias bounds by appropriate amounts. Additionally, approaches such as tropospheric modeling or an ionosphere-free differential architecture may also be employed. The amount of increase to PL bias bounds needed is left to further research. A more detailed presentation of the formation of the bias and noise terms is also provided in Bruckner et al. (2010a). Simulation parameters The Monte-Carlo simulation was performed in Matlab for a single GPS measurement epoch using the parameters shown in Table 2. Results for all of the one- and twodimensional PLs and their approximations are obtained for the same simulation run. The error plots shown in the
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j ¼ 1; 2; . . .; L
W = diag(li) or W = diag(1/Bmax,i)
previous figures were all produced using this simulation code. However, several parameters in those previous illustrations differ from the values used here. The 1-sigma values of the components of range domain noise ri prior to multiplication by Kffmd are shown below in Fig. 5, as a function of elevation (RTCA 2004). The root-sum-square total value of the noise components is shown as well. The 20 cm value of range domain bias bound lmax is approximately the same as the 1-sigma value for total noise error for higher satellite elevation values. The position domain navigation system error (NSE) of a sample 100,000-iteration simulation projected successively along the x, y, and z axes appears in Fig. 6. Since the purpose of this simulation is to illustrate protection levels, no smoothing has been performed. The ellipse plots in other figures show simulation NSE in the position domain. Combining conspiring biases at reference station and aircraft Range domain bias bounds are combined and transformed into the position domain at both the reference station and the aircraft using a worst-case vector sum instead of the root-sum-square procedure employed for noise. This is based on the general principle that probabilistic measures for safety of life cannot be diluted or averaged across an ensemble; every aircraft approach must be fully protected. Nothing is assumed about the probability distribution of the range domain biases other than their orientations and maximum possible magnitudes, which are the bias bounds. In effect, we assume the biases at a single location are perfectly correlated. The one exception to this worst-case bias procedure is that the simulated resultant bias bounds from the reference station and the aircraft are combined in a root-sum-square sense. In this simulation, the primary bias component is assumed to be antenna group delay. If the antennas at the reference station and the aircraft are assumed to be of
GPS Solut Table 2 Parameters of Monte-Carlo simulation used to illustrate and evaluate PL approximations Quantity
Value(s)
Remarks
Satellite geometry
Chosen from the 24-slot constellation, 2 h into GPS day (IS-GPS 2004)
One of seven satellites available is removed to illustrate a more problematic geometry with six satellites
User location
30 deg N, 90 deg W
New Orleans
Number of iterations in 100,000 the simulation, Nsim Probability of 1 9 10-4 = 1 - 0.9999, corresponding hazardously misleading to Kffmd = 3.8906 information, PHMI
Maximum number that a Pentium 4 processor with 2 GB memory can handle in this Matlab implementation Set so that PHMI 9 Nsim [ 1, to allow a few outliers outside PL in the simulation
Maximum probability error tolerance, c
1 9 10-9
Must be smaller than PHMI, but this very small value is used to ensure an accurate estimate of probability contained by estimated PLs
Maximum range domain bias, Bmax
20 cm
Applied equally to all satellites as an upper bound. Includes group delay and does not include multipath or residual iono or tropo errors
Noise, ri
r2i = (r2pr_gnd ? r2air ? r2iono ? r2tropo) (RTCA 2004)
These terms assumed to be Gaussian noise only for simplicity’s sake in this simulation. For flight test results in Bruckner et al. (2010a), all components but thermal noise are shifted into the bias term and bounded by Bmax.
Ground noise and multipath, rpr_gnd
From C4 ground accuracy designator (GAD-C4) Includes thermal noise, multipath, etc. curve (RTCA 2004)
Airborne noise and multipath, rair
Noise from AAD-B curve; multipath from ‘A’ airborne accuracy designator (AAD-A) curve (RTCA 2004)
AAD-B curve for multipath is still under development, so AAD-A curve is used
Residual ionospheric delay, riono Residual tropospheric error, rtropo
0.02 m
Assuming dual-frequency receivers remove first-order ionosphere effects, this is for higher order iono terms As recommended in App. F for availability analyses (RTCA 2004)
Least-squares weighting technique
Unweighted
Best available technique for bias bounds and noise sigmas of comparable size. This is not the case for post-CNMP bias and noise in Bruckner et al. (2010a), where noise is much smaller than bias bounds.
Smoothing
None
Simulation illustrates PLs, not smoothing
Vertical dilution of precision (VDOP)
1.762
Using unweighted LS
Horizontal dilution of precision (HDOP)
1.942
Using unweighted LS
Zero
different types, antenna group delay at reference station and aircraft can be assumed to be independent, thereby justifying this exception. If components such as multipath, residual ionospheric error, and residual tropospheric error were to be shifted into the bias term, their independence between reference station and aircraft would need to be assumed or established in order to preserve this exception. Another justification would be to show that the likelihood of the total worst-case combination of bias and noise is less than the probability allowed for hazardously misleading information. The strictest probability requirement is currently 1 9 10-9 for Ground Based Augmentation System (GBAS) Service Levels (GSLs) D through F (RTCA 2004). Since the composite PL is typically smallest and therefore most
useful when these components are included in the bias term instead of the noise term, an example of such a justification is included below even though not implemented here. For the symmetric bias combinations analyzed here, such that two bias combinations in each epoch have the same magnitude, the probability of the worst-case total bias combination being realized at a single location, namely either reference station or aircraft, is 1/2(M - 1), where M is the number of satellites in the least-squares solution. It is assumed that the two largest bias components are those due to multipath and antenna group delay, that they are of the same order of magnitude, and that they are also independent. The probability of both of these biases conspiring together in a worst-case way at a single location, either reference station or aircraft, therefore, is [1/2(M - 1)]2, or
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GPS Solut Fig. 5 Noise sigma values (noise components and total) versus satellite elevation
0.7 Total noise sigma Ground sigma Airborne sigma Residual iono sigma
0.6
Error rms, m
0.5
0.4
0.3
0.2
0.1
0 0
10
20
30
40
50
60
70
80
90
Satellite elevation, degrees
Kffmd 9 r, which for illustration purposes is also assumed to be 2 9 10-9, equivalent to Kffmd = 6 in the standard normal distribution. Obviously, the overall probability of a worstcase bias and noise combination under this set of assumptions is vanishingly small, and some effort should be taken to reduce the bias or noise bounds in a practical implementation. Further research is needed to establish the independence of the bias components at a single location and between different locations. VPLc and its approximation
Fig. 6 Simulation NSE (noise and bias) for east, north, and up axes (unsmoothed; 100,000 points)
1/2(2M - 2). For M = 6, this probability is 1/1,024. For M = 8, it is 1/16,384, and for M = 10, it is 1/262,144. If antenna group delay and multipath are also assumed to be independent between the reference station and aircraft, then the probability of the total worst-case bias combination between the two locations is 1/(262,144)2 = 1.46 9 10-11. It is recognized that bias is independent of thermal noise (the only remaining component of the noise term), and thermal noise measurements at two locations are also independent of each other. The quantity 1.46 9 10-11 is then twice multiplied by the probability of the noise being outside
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Figure 7 shows the error ellipse projected onto the x– z plane. Exact and approximate VPLc values are shown as horizontal solid and dashed lines. The vertical bias of 1.083 m is apparent in the upward shift of the error ellipse—again the worst-case vertical displacement possible for this epoch. A confidence interval was not determined for either VPLc in this illustration, but several runs of the simulation placed 99.99 ± 0.01% of the errors inside the exact protection level value of 7.661 m. The performance of VPLc and its approximation VPLc1 in simulation for this single GPS epoch is shown in Table 3 below: HPLc and its approximations Similarly, errors projected onto the horizontal x–y plane and then onto the rotated horizontal axis, along with HPLc and its two approximations, are shown in Fig. 8. The angle
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Fig. 7 Simulated biased, unsmoothed error in x–z plane, projected onto vertical axis, with ± VPLc (solid horizontal lines) and approximation VPLc1 (dashed horizontal lines). The displacement of the error ellipse by the worst-case bias is shown by the small circle and vector extending from the origin. The projection of error from each simulation iteration onto the vertical axis is also shown
Fig. 8 Simulated biased, unsmoothed error in x–y plane, projected onto rotated axis with ± HPLc (solid lines), and two approximations, HPLc1 and HPLc2 (dashed lines) Table 4 Simulated performance of HPLc methods HPLc method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Table 3 Simulated performance of VPLc methods VPLc method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Exact
4.683
99.991
HPLc1 approximation
4.968
99.996
Exact
7.661
99.992
HPLc2 approximation
5.049
99.996
VPLc1 approximation
7.962
99.997
between the x axis and the rotated horizontal axis is -41 degrees. The resultant bias vector shown is the one that produced maximum displacement from the origin when projected onto the rotated horizontal axis. The simulated performance of HPLc and its two approximations appears below in Table 4. As expected, HPLc2 is slightly larger and therefore more conservative than HPLc1. RPLc and its approximation The set of conspiring pseudorange bias combinations available in the horizontal plane of the position domain appears in Fig. 9. The bias ‘‘cloud’’ exhibits symmetry through the origin, as one might expect for positive and negative pairs of bias combinations. The angle between the x axis and the approximate ‘‘major axis’’ of the bias cloud is approximately -25 degrees, sixteen degrees offset from the -41-degree angle of the noise ellipse’s major axis. In general, until the noise ellipse is considered, it is not known
which of these bias combinations will produce the largest protection level radius. Because the bias cloud and noise error ellipse in the horizontal plane do not have the same orientation, the worst-case bias combination considering bias alone may be different from the worst-case combination of bias and noise used for HPLc. Indeed, the greater the difference in orientation between the bias cloud and noise error ellipse, the greater the benefit that will be realized from a full search in a reduction of RPLc. After the worst-case set of conspiring biases is obtained by search, the noise ellipse is added to this plot, as shown in Fig. 10. Results appear in Table 5 below. The same bias combination was selected by both methods, but RPLc1 arranges noise and bias collinearly, yielding a larger PL. Summary of simulation results The quantity RPLc (99.99%) equals 4.732 m, which is slightly larger than the HPLc (99.99%) value of 4.683 m, as expected. The difference between these measures increases
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GPS Solut
Fig. 9 Bias combinations available in the horizontal plane
for smaller values of enclosed probability Ppl and decreases for larger Ppl values. For example, RPLc (95%) is 2.592 m while HPLc (95%) is 2.500 m, a difference of 0.092 m instead of the 0.049 difference seen when Ppl is 99.99%. At probability levels of 99.999% and higher, this 100,000point simulation could not reliably provide results. The approximations for VPLc and HPLc are observed to be over-bounds, also as expected. At the 99.99% probability level used in this simulation, the use of the approximations added about a half order of magnitude (i.e., 0.005%) of excess overbounding error. Depending on the application and specified probability level, these approximations will continue to prove quite useful. The percentage of points enclosed in the PL for a given simulation run follows a sampling distribution whose mean was not verified, but which appears to be the nominal value of the intended probability. This is the case for all MonteCarlo results presented here, regardless of the PL type. The simulation helps to illustrate the differences between the various PL approximations and to verify the algorithms used to obtain them. However, for the very high probability levels used in LAAS similar applications, the simulation cannot yet support meaningful conclusions. Further development of the simulation is left to future research. The application of approximations VPLc1 and HPLc1 to flight test data, including a significant performance improvement obtained over LAAS PLs at PHMI = 2 9 10-9 (Kffmd = 6), is available to the reader in Bruckner et al. (2010a).
Summary
Fig. 10 RPLc (large, bold black circle) with bias errors (yellow dots clustered near origin) and biased noise errors (blue dots) shown. The bias offset is also shown (bold black vector and small black circle symbol). The approximation RPLc1 (dashed circle) is slightly larger than the exact value, since noise and bias are arranged collinearly
Approximations to VPLc, HPLc, and RPLc are obtained and are compared to exact PL values developed in previous research. Results from a Monte-Carlo simulation for a single GPS measurement epoch are presented to illustrate these exact PLs and their approximations. It is shown that previously published composite vertical and horizontal PL formulations for biased measurements are approximations of the exact methods. In the case of VPLc and HPLc, these approximations are shown to be over-bounds of the exact values. The approximation to RPLc is shown to be far simpler than the exact PL, but a demonstration or proof that the approximation is an over-bound is left to future research. This paper makes available to the reader both the methods and the MatlabÒ simulation code needed to evaluate computationally efficient PL approximations. Thus, it fosters further research into the use of GPS in safety critical applications.
Table 5 Simulated performance of RPLc methods RPLc Method
Value in m (99.99%)
Percent of simulated errors enclosed (%)
Exact
4.732
99.991
RPLc1 approximation
4.865
99.992
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Future work A great deal of fruitful work remains to be accomplished in this research area. Suggested tasks include the following:
GPS Solut
1. 2.
3.
4.
5.
6.
7.
Characterize the bounding performance of the RPLc approximation. Quantify the improvements in computational efficiency for the PL approximations, relative to the exact PL methods. Examine the statistical independence of the multipath and antenna group delay bias components. If independence is proven, reduce the magnitude of their contributions to the composite PL formulated in Bruckner et al. (2010a, b). Examine the statistical independence between biases at the reference station and those in the airborne system. If dependence is found, calculate the probability that these will simultaneously occur along with the worstcase noise values at both locations, and update the algorithm and PL formulation used in this study. Determine appropriate maximum bias magnitudes for various effects, including ionospheric scintillation and severe ionospheric and tropospheric gradients. Use flight test data to examine composite PLs versus observed errors in more detail, deriving a confidence interval for the observed errors and comparing it to the PL. Develop and test additional PL approximations.
Acknowledgments This research was supported, in part, by the Federal Aviation Administration under Aviation Cooperative Research Agreement 98-G-002.
References Bruckner D, van Graas F, Skidmore T (2010a) Algorithm and flight test results to exchange code noise and multipath for biases in dual frequency differential GPS for precision approach. Navigation 57(3):213–229 Bruckner D, van Graas F, Skidmore T (2010b) Statistical characterization of composite protection levels for GPS. GPS Solut. doi: 10.1007/s10291-010-0188-2 IS-GPS (2004) Interface specification: navstar GPS space segment/ navigation user interfaces/IS-GPS-200D. Available from U.S. Coast Guard Navig Ctr, Alexandria, Virginia Rife J, Pullen S (2005) Impact of measurement biases on availability for category III LAAS. Navigation 50(4):215–228 RTCA (2004) Minimum aviation systems performance standards (MASPS) for the local area augmentation system (LAAS) RTCA DO-245A RTCA (2006) Minimum operational performance standards (MOPS) for GPS wide area augmentation system (WAAS) airborne equipment, RTCA DO-229D RTCA (2007) Minimum operational performance standards (MOPS) for GPS LAAS airborne equipment RTCA DO-253B van Graas F, Diggle D, Uijt de Haag M, Wullschleger V, Velez r, Lamb D, Dimeo M, Kuehl G, Hilb R (1995) FAA/Ohio University/United Parcel Service DGPS autoland flight test demonstration. Proc Inst Navig GPS 727–737. doi:9/1995
van Graas F, Krishnan V, Suddapalli R, Skidmore T (2004) Conspiring biases in the local area augmentation system. Proc Inst Navig Annu Meet 300–307. doi:6/2004
Author Biographies Dr. Dean Bruckner is the Assistant Director (Technical) of the Avionics Engineering Center at Ohio University in Athens, Ohio. He received his Ph.D. degree in Electrical Engineering from Ohio University in 2010 with a dissertation that included flight test results for dual-frequency GPS precision approach and landing. During a career in the United States Coast Guard, he earned a B.S.E.E. from the Coast Guard Academy and an M.S.E.E. from the Naval Postgraduate School, both with honors and with specialization in electronic navigation. He served as Commanding Officer of the Coast Guard Loran-C transmitter station near Istanbul, Turkey, and was the nationwide electronics support manager for Coast Guard ships, boats, aids to navigation and shore facilities. He has recently provided navigation and engineering expertise to the F-35 Joint Strike Fighter navigation sensor test program. Dr. Frank van Graas is the Fritz J. and Dolores H. Russ Professor of Electrical Engineering at Ohio University. He earned his Ph.D. from Ohio University in 1988. He holds a BSEE and MSEE degrees with specialization in avionics from Delft University of Technology, The Netherlands. His extensive involvement in satellite navigation included conducting the first real-time GPS attitude and heading flight experiment on a DC-3 in 1991, and the first kinematic dual-frequency GPS autoland flight tests using NASA Langley Research Center’s Boeing 737 in 1993. He received the Johannes Kepler Award for ‘‘sustained and significant contributions to satellite navigation’’ from the Satellite Division of the Institute of Navigation in 1996, and served as President of the Institute of Navigation from 1998 to 1999. At Ohio University’s Avionics Engineering Center, his roles include Principal Investigator for the FAAsponsored Global Positioning System (GPS) research grant to investigate the Local Area Augmentation System for Aircraft Precision Approach & Landing and Aircraft Surface Movement Guidance. He has lectured extensively throughout the United States, Canada, Europe and Russia for such organizations as the International Air Transport Association, the NATO Advisory Group on Aerospace Research and Development (AGARD), Navtech Seminars, Inc., and the International Society for Optical Engineering (SPIE).
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GPS Solut Dr. Trent Skidmore is a Senior Research Engineer and Adjunct Assistant Professor in the Avionics Engineering Center at Ohio University. He is involved in a variety of projects involving applications of GPS, including the use of differential GPS for aircraft precision approach applications. Recent projects include the Department of Defense Joint Precision Approach and Landing System (JPALS), the Automated Aerial Refueling (AAR) program, as well as the Federal Aviation Administration Local Area Augmentation System (LAAS). Trent received his B.Sc. in 1986 (with Honors) and M.Sc. in 1988 from Michigan Technological University and his Ph.D. from Ohio University in 1991, all in electrical engineering.
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