# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 6 Simulation Studies
### Analyses of KF RAIM  
In this paper, the following parameters are considered for analyses of KF RAIM. The continuity budget for false alert, \(P_{FA}\) is \(3.33\times 10^{-7}\)/sample [31]. The allocated probability of hazardously misleading information (HMI), \(P_{HMI}^{alloc}\) is \(10^{-7}\)/h during aircraft descent and initial and intermediate approach, and \(10^{-7}\)/approach during the final approach [40]. The prior probability of narrow fault of GPS and NavIC satellites is \(10^{-4}\)/satellite/h. As  
Figure 7: EKF and SKF carrier smoothed-pseudorange measurement innovations and position estimation errors in the NED frame. The ramp error in innovations is due to a low elevation satellite below the mask angle, which is not considered for estimation. suming a one-hour mean time to notify by the constellation service provider, the prior probability of fault is \(10^{-4}\)/satellite as the simulation duration is less than an hour [41]. Constellation-wide faults are not taken into account. \(P_{HMI}^{alloc}\) for the horizontal position, \(P_{HMI,\,H}^{alloc}\), is 10% of \(P_{HMI}^{alloc}\). \(P_{HMI}^{alloc}\) for the vertical position, \(P_{HMI,\,V}^{alloc}\), is the remaining 90%. For the prior fault probability and allocated integrity budget, at most two simultaneous independent satellite failures need to be monitored, as justified in [27]. The probability of three or more non-monitored faults \(P_{nm}\), being less than the \(P_{HMI}^{alloc}\), is subtracted from the latter. During the final approach phase, vertical and horizontal alert limits are considered to be 50 m and 40 m.  
Next, the test statistics and PLs of KF RAIM obtained using the SKF are shown alongside those of WLS RAIM in Figure 8. WLS RAIM follows the method of [25] but accounts for the parameter uncertainty of the measurement error model, as discussed in [29]. It is used for comparison purposes for the following reasons: First, it performs fault detection in the range domain, as is with the developed KF RAIM. Second, it is computationally simple. The presented KF RAIM takes 5.5 min in Matlab on an old, low-end IBM Thinkpad laptop for the 19.48 min simulation. The laptop runs on Linux and has 1 GB RAM, 2 GB swap memory, and a Core 2 Duo processor (CPU @ 2 GHz). WLS RAIM, on the other hand, requires 18 s.  
In order to gain a good understanding of the properties of KF RAIM, a few salient features are also analyzed. In this regard, Figure 9 depicts the following: \(M\) (number  
Figure 8: WLS and KF RAIM performance after an initial transient period of 15 s.  
of terms of the first test statistic \(\alpha_{1}\)), \(N\) of Equation (7), maximum eigenvalues, \(\lambda^{\max}_{\mathbf{\Omega},\,1}\) and DOF of the test statistics, and the eigenvalue ratios, \(\sqrt{\lambda^{\max}_{\mathbf{\bar{\Theta}}}}/\lambda^{\min}_{\mathbf{\bar{ \Theta}}}\). It is evident that \(\lambda^{\max}_{\mathbf{\Omega},\,1}\) of the two statistics is less than one, as desired. Initially, eigenvalue ratios are large because of the addition of white noise during the transient period. The initial transient period is determined by checking the filter innovation sequence. As the eigenvalue ratios have different orders of magnitude, they are subsequently plotted on different scales, each given on either side. The figure also shows adaptive calculations of \(M\) and \(N\) and the corresponding changes in the DOF. It is apparent that \(M\) and \(N\) must be calculated adaptively as they may change over time. Mean VPE bounds under faults are presented in Figure 10 for \(\alpha_{1}\) and \(\alpha_{2}\). They are plotted on different y-axes, each on either side. This shows separately the contribution of mean VPE bound corresponding to each test statistic. While \(N\) is generally much larger than \(M\), the second bound with \((N-M)\) terms is much less than the first with \(M\) terms. Although not shown in the figure, the third bound from ignored terms before \(N\) recent epochs has an even smaller contribution.