Integrity and continuity allocation for the RAIM with multiple constellations

Abstract

The allocation of integrity and continuity is crucial for the performance of integrity monitoring when multiple failure modes from multiple constellations have to be monitored. Using the four-state Markov model to link the integrity parameters, an optimal allocation algorithm for the slope-based receiver autonomous integrity monitoring is developed under the minimization criterion of protection level. The test based on a simulated dual constellation with various failure rates has shown that, when compared with other typical algorithms, the proposed algorithm can achieve a worldwide protection level decrement by at least 12.1% when the mean time between failures of each system is less than 10³ h.

Appendices

Appendix 1: Convergence of the proposed Markov model

The integrity and continuity should be allocated before the navigational operation. If the resulting integrity and continuity cannot be guaranteed to converge, the resulting integrity and continuity can be larger than the predefined integrity and continuity requirement with more integrity and continuity checks. Thus, for one failure mode, if the resulting integrity and continuity of the Markov model are guaranteed to converge, then

$$\begin{aligned} &P_{2} \left( k \right) < P_{2} \left( {k - 1} \right) \\ &P_{1} \left( k \right) + P_{3} \left( k \right) < P_{1} \left( {k - 1} \right) + P_{3} \left( {k - 1} \right) \\ \end{aligned}$$

Based on the established Markov model (1), the increment of states P ₂, P ₁ and P ₃ is

$$\Delta P_{2} \left( k \right) = P_{f} P_{\text{md}} P_{0} \left( {k - 1} \right) + \left( {P_{\text{md}} - 1} \right)P_{2} \left( {k - 1} \right) < 0$$

(19)

$$\begin{aligned} \Delta P_{1} \left( k \right) + \Delta P_{3} \left( k \right) &= \left( {P_{\text{fa}} - P_{f} P_{\text{fa}} + P_{f} - P_{f} P_{\text{md}} } \right)P_{0} \left( {k - 1} \right) + \left( {P_{\text{fa}} - 1} \right)P_{1} \left( {k - 1} \right) \\ &\quad + \left( {1 - P_{\text{md}} } \right)P_{2} \left( {k - 1} \right) - P_{r} P_{3} \left( {k - 1} \right) < 0 \\ \end{aligned}$$

(20)

Rearranging (19), we have

$$\frac{{P_{f} P_{\text{md}} }}{{1 - P_{\text{md}} }} < \frac{{P_{2} \left( {k - 1} \right)}}{{P_{0} \left( {k - 1} \right)}}$$

(21)

With satisfying (21), and the constraint from P ₀(k) + P ₁(k) + P ₂(k) + P ₃(k) = 1,

$$\left( {P_{\text{fa}} + P_{r} - 1} \right)P_{1} \left( {k - 1} \right) + \left\{ {\left( {\frac{1}{{P_{\text{md}} }} - 1} \right)\left[ {P_{fa} \left( {\frac{1}{{P_{f} }} - 1} \right) + \left( {1 + \frac{{P_{r} }}{{P_{f} }}} \right)} \right] - P_{r} } \right\}P_{2} \left( {k - 1} \right) - P_{r} < 0$$

(22)

If the initial state of the Markov model satisfies the constraints of (21) and (22), the required integrity and continuity have to be satisfied only at the initial epoch due to the decreasing trend of integrity risk and continuity risk.

Appendix 2: Derivation of differential terms

From (13), we have

$$\frac{{\partial L\left( {{\text{IR}}_{i} ,{\text{CR}}_{i} } \right)}}{{\partial T_{\text{fa}} }} = x{\text{Slope}}_{i}$$

(23)

Based on (9), let $F\left( {P_{\text{fa}} ,T_{\text{fa}} } \right) = 1 - P_{\text{fa}} - \frac{1}{{\varGamma \left( {\text{dof}} \right)}}\int_{0}^{{T_{\text{fa}}^{2} }} {e^{ - t} t^{{{\text{dof}} - 1}} {\text{d}}t}$, and we can implicitly differentiate to get,

$$\frac{{\partial T_{\text{fa}} }}{{\partial P_{\text{fa}} }} = - \frac{{\frac{\partial F}{{\partial P_{\text{fa}} }}}}{{\frac{\partial F}{{\partial T_{\text{fa}} }}}}{ = } - \frac{1}{2}\varGamma \left( {\text{dof}} \right)e^{{T_{\text{fa}}^{2} }} T_{\text{fa}}^{{1 - 2{\text{dof}}}}$$

(24)

Similarly, based on (13),

$$\frac{{\partial L\left( {{\text{IR}}_{i} ,{\text{CR}}_{i} } \right)}}{{\partial K_{\text{md}} }}{ = }\sigma_{0}$$

(25)

With respect to (11), we have

$$\frac{{\partial K_{\text{md}} }}{{\partial P_{\text{md}} }} = - \sqrt {\frac{\pi }{2}} e^{{\frac{{K_{\text{md}}^{2} }}{2}}}$$

(26)

Based on the definition of integrity risk and continuity risk from (4) and (5), the following can be derived from (1),

$$\begin{aligned} &\frac{{\partial P_{\text{md}} }}{{\partial {\text{IR}}_{i} }} = \frac{1}{{p_{2} + P_{f} p_{0} }} \hfill \\ &\frac{{\partial P_{\text{md}} }}{{\partial {\text{CR}}_{i} }} = - \frac{1}{{p_{2} + P_{f} p_{0} }} \hfill \\ &\frac{{\partial P_{\text{fa}} }}{{\partial {\text{CR}}_{i} }} = \frac{1}{{\left( {1 - P_{f} } \right)p_{0} + p_{1} }} \hfill \\ \end{aligned}$$

(27)

The differential terms in (14) and (15) can be obtained the equations from (23) to (27).

Appendix 3: Search range determination for the Lagrangian coefficients

For each failure mode, with allocating the total integrity risk requirement to the failure mode i, the maximum of P _md can be obtained from the Markov model as

$$P_{{{\text{md}},i}}^{\hbox{max} } = \frac{{{\text{IR}}_{\text{req}} }}{{P_{{{\text{f2}},i}} }}$$

Thus, the minimum of K _md is

$$K_{{{\text{md}},i}}^{\hbox{min} } = \sqrt 2 {\text{erfinv}}\left( {0.5 - 0.5P_{{{\text{md}},i}}^{\hbox{max} } } \right)$$

(28)

in which erfinv is the Normal inverse error function. Setting up a maximum allowable position error (MAPE), e.g., the xAL, the upper bound of missed detection threshold can be determined as

$$K_{\text{md}}^{\hbox{max} } = \frac{\text{MAPE}}{{\sigma_{0} }}$$

(29)

Given the minimum and maximum of K _md from (28) and (29), the searching range of λ can be obtained from (16).

As induced from (18), the searching range of μ can be achieved only if the variation of fault detection threshold T _fa can be determined. Since K _md can be derived first by the allocated integrity risk, the maximum probability of false-alert can be confirmed from allocating the total continuity risk to one failure mode,

$$P_{{{\text{fa}},i}}^{\hbox{max} } = \frac{{{\text{CR}}_{\text{req}} - \left( {1 - P_{{{\text{md}},i}} } \right)p_{{{\text{f2}},i}} - \left( {1 - P_{r,i} } \right)p_{3}^{i} }}{{p_{{{\text{f1}},i}} }}$$

Based on (9), the minimum of T _fa can be achieved as

$$T_{{{\text{fa}},i}}^{\hbox{min} } = \sqrt {\text{chi} 2inv(1 - P_{{{\text{fa}},i}}^{\hbox{max} } ,{\text{dof}})}$$

(30)

in which chi2inv is the Chi-square inverse cumulative distribution function.

Since xPL should be within the protection of the alarm limit xAL under the fault-free condition, the maximum of T _fa can be obtained as

$$T_{{{\text{fa}},i}}^{\hbox{max} } = \frac{{{\text{MAPE}} - K_{{{\text{md}},i}} \sigma_{0} }}{{x{\text{Slope}}_{\hbox{max} } }}$$

(31)

With the determined range of T _fa from the above analysis, the searching range of μ can be derived from (18).