EFFECT OF AN IN-FLIGHT VERTICAL ACCELEROMETER CALIBRATION ON LANDING ACCURACY AFTER BARO-INERTIAL SYSTEM FAILURE

. An issue of improving flight safety during landing with an inertial navigation system (INS) and a failed barometric altimeter is considered. In this paper, we propose a specific algorithm for in-flight calibration of the vertical channel of INS. Accordingly, the baro-inertial integration algorithm using a discrete five-state Kalman filter will be performed during a particular flight maneuver before landing. As a result, it is possible to estimate not only the bias of vertical accelerometer but also its scale factor, which is too small to be defined by a usual in-flight calibration algorithm. After applying the proposed algorithm, the flight management system can provide a safe landing with a standalone INS. The algorithm’s performance is assessed by simulating complete mathematical models of aircraft motion and control systems. The impact of calibrated bias and scale factor of vertical accelerometer on the altitude estimation error is provided through an analysis. failure, Kalman filter, automatic landing.


Introduction
Inertial navigation system (INS) is one of the most effective devices that provide an estimation of aircraft state to ensure the accuracy and safety of its motion. In contrast to other navigation and guidance systems, INS is completely autonomous and free of external influences such as weather and electromagnetic interferences. However, the major disadvantage of an INS is the unlimited growth of its errors over time, especially in the vertical channel (Babich, 1991;O'Donnel, 1964), making the long-term application of standalone INS inefficient. In order to overcome this drawback, integration of INS and other position and velocity measurements such as Global Navigation Satellite System (GNSS), radio systems, or barometric altimeter (BA) (Schmidt, 2010;Siouris, 1993) is used. By using signals from such systems, it is possible to obtain more accurate estimates of aircraft state even when the characteristics of INS are poorly known. Nevertheless, the advantage of INS and noninertial navigation sensors integration does not eliminate the need for in-flight inertial sensors calibration, since it allows us to reduce position errors when only a standalone INS is available due to external measurement failure.
As is well known, GNSS is the most common aiding system that can give an absolute drift-free position estimate with high accuracy (Groves, 2013;Mohinder et al., 2007). However, due to the effect of satellite geometry and aircraft maneuver, GNSS provides less accurate altitude estimation (Farrel & Barth, 1998;Kim & Sukkarieh, 2003). Hence, an additional aiding system, such as BA, is needed to improve the vertical channel (Ausman, 1991;Babich, 1991;Farrell & Barth, 1998;Kim & Sukkarieh, 2003;O'Donnel, 1964;Sobolev, 1994). Many researchers have developed various algorithms for baro-inertial integration, using simple vertical channel damping-loop mechanizations or complex forms of Kalman filter. The results show that using BA additionally to integrated navigation systems allows us to obtain a more reliable and accurate navigation solution. Furthermore, it is shown that baro-inertial integration meets the FAA requirement for a Category I precision approach (Gray & Maybeck, 1995).
However, in most publications, the authors used incomplete mathematical models of integrated systems. For example, the dynamic error of BA, i.e., lag in static pressure transmission, is not taken into account (Bevermeier et al., 2010;Kim & Sukkarieh, 2003;Sokolovi et al., 2014), or the accelerometer error model including only bias is considered (Babich, 1991;Sobolev, 1994). In some cases, additional information from a non-standard atmosphere condition (Jafar et al., 2018) or map-matching with a topographic map (Bevermeier et al., 2010) is included for compensating the BA bias caused by deviation of sea-level temperature and static pressure from their nominal values used in BA computation. This leads to the use of a more complex structure of integration algorithm, which can be avoided by a suitable selection of measurement information from BA output, as done in this work. Besides that, the main issue of these investigations is to provide a more stable and accurate altitude estimation with BA aiding. For this purpose, it is sufficient to define only the accelerometer bias since the rest of its characteristics are quite small (for aviation-grade INS). As a result, in the event of BA failure, the altitude estimation is not exact enough to ensure the safety of aircraft motion, especially in such complicated and dangerous phase of flight as landing, as shown in simulation results of this paper.
It is very important to have accurate altitude information during landing because any hazardous situation could occur if either the pilot or automatic control system receives erroneous altitude estimates. Numerous aviation disasters have been caused by the air data computer failure (Jeb, 2019;Luiz, 2013 (AAIU, 2016;BEA, 2012;KNKT, 2018), among which the BA failure due to the static port blockage by water or airframe icing is more dangerous (FAA, 2016).
Consequently, improving flight safety in the event of BA failure is always highly relevant. In this paper, we propose an algorithm that provides calibration of the vertical accelerometer of INS during a specific flight maneuver before landing. This method allows us to estimate not only the accelerometer bias but also its scale factor, which is too small (aviation-grade INS) to be defined by a "normal" inflight calibration (i.e., without any calibration maneuver). Using the calibrated bias and scale factor for additional compensation of INS error, the flight management system can provide a safe landing when the BA is failed.
Fully observable information of the measurement model including vertical accelerometer and BA is derived for calibration using a discrete Kalman filter, one of the most effective and widely used methods for realizing baro-inertial integration. The effectiveness of the proposed calibration algorithm follows an analysis of mathematical simulation results. The reliability of the obtained results is based on the fact that the investigation in this paper is performed using complete models of related dynamic systems (aircraft and landing control loops). In addition, an example of analyzing the impact of the calibrated bias and scale factor of a vertical accelerometer on altitude estimation error during landing is given.

Mathematical model of Kalman filter calibrating the vertical accelerometer
To describe the vertical channel of INS, we consider that the residual error (after all compensations) in the estimate of vertical acceleration, measured by the vertical accelerometer, is caused by the following error sources: 1) an inaccuracy of setting the bias 0 a ∆ ; 2) an inaccuracy of setting the scale factor a k δ ; 3) measurement noise ξ with zero mean and standard deviation a σ . Respectively, a mathematical model for the vertical accelerometer of INS expresses the output a m as: where y a -vertical acceleration of an aircraft. Hence, the vertical acceleration can be estimated by the following equation: since a k δ 1  . The model of BA output is adopted as (Babich, 1991): where h m -output signal; h -aircraft altitude; 0 h ∆ -bias; τ -time constant, defined by (4); µ -measurement noise with zero mean and standard deviation b σ . The time constant τ of BA depends on the static pressure and temperature of the atmosphere (Lawford & Nippress, 1983;Sobolev, 1994): where 0 τ -time constant at sea-level; 0 P , 0 T -sea-level static pressure and temperature of the atmosphere; h P , h Tstatic pressure and temperature of the atmosphere at h.
From (3), the discrete model of BA output can be represented in the following form (Sobolev, 1994): here T ∆ -sample time. Essentially, the algorithm calibrating a vertical accelerometer by integrating its output and BA output is based on the comparison of the flight altitude changes estimated by these sensors separately. Since BA measures an absolute value of aircraft altitude, it is necessary to remove the bias 0 h ∆ from the output h m (5) to estimate the altitude change. For this purpose, the difference 0 m m m h h h ∆ = − between current BA output and the initial one is used as a measurement for the Kalman filter: h -altitude at the initial moment; µ -a sample of BA measurement noise at the initial moment. The Kalman filter state vector is established as: where y V -vertical speed; b h ∆ -predicted variation of BA output since initial moment (without consideration of measurement noise).
Thus, the evolution of Kalman filter state is described by the following system of equations: Taking into account (2) and (7-12) we have: The measurement for Kalman filter can be formed as follows, according to (6) and (7): ( where The block diagram of the Kalman filter using (13) and (14) in a recursive loop is shown in Figure 1.

Mathematical model of an aircraft
A nonlinear mathematical model of longitudinal motion of a passenger aircraft is used for modeling the flight during landing. Due to the limited flight time, the aircraft motion model with a flat earth is considered. Thus, the flight dynamics equations are as follow (Zaporozhets & Kostiukov, 1992): where T -thrust; X, Y -drag and lift forces; M z -pitching moment; V -speed; α -angle of attack; θ -flight path angle; z ω -rate of pitch angle; ϑ -pitch angle; h -altitude; m -mass; I z -inertial moment. Figure 2 provides a definition of all forces and moments acting on an aircraft, as well as angles indicated in the flight dynamics equations (15)(16)(17)(18)(19)(20).

Calibration maneuver planning
In order to design a maneuver for in-flight accelerometer calibration let's rewrite equation (2) for vertical acceleration estimation: where 0 a a k ∆ δ is omitted because it is very small in comparison with 0 a ∆ . The essence of an algorithm calibrating the accelerometer with output structure (1) is to find such values of 0 a ∆ and a k δ , which maximally reduce the difference between altitude estimate obtained by double integration of vertical acceleration a y estimated by (24) and the one received directly from BA output. According to (24), the bias 0 a ∆ can be well defined in a normal flight condition without any vertical acceleration, i.e., when 0 m a ≈ , and vice versa, the scale factor a k δ can be calibrated only under a significant vertical maneuver. Therefore, in this paper, to study the effect of a flight maneuver on the accuracy of vertical accelerometer calibration, we suggest changing the aircraft trajectory with the command illustrated in Figure 3  To perform such maneuver as given in Figure 3, an altitude/speed hold autopilot is used (see Figure 4 for detail).
Respectively, the control signals are:

Analysis of results
To assess the feasibility of the proposed in-flight calibration algorithm, a simulation was performed, using the aviation-grade accelerometer (Groves, 2015) with 0 a ∆ = 0.001 m/s 2 , a k δ = 1000 ppm, a σ = 0.0062 m/s 2 and the BA with b σ = 1 m and bias 0 h ∆ modeled as random constant with zero mean and standard deviation 0 h σ = 30 m. The accelerometer sampling rate is 1000 Hz, whereas the BA sampling rate is 100 Hz.
It is considered that, at the initial moment, the aircraft is in a steady-state level flight (altitude 500 m and speed 80 m/s) heading forward to the glide-slope line. From here, the initial state of Kalman filter is taken as zero for all its elements. The initial value of the covariance matrix of estimation errors is set to (29): Given flight dynamics equations (15-20) with aerodynamic model (21-23), as well as automatic control system models (25-28) and (31-34), the aircraft motion was simulated in Matlab/Simulink environment.

Simulation results without calibration maneuver
Figures 5 and 6 present the calibration results obtained in a stabilized level flight, i.e., without any maneuver. One can find that the accelerometer bias 0 a ∆ can be quite accurately calibrated, while the scale factor a k δ can not be identified. Th e time history plots in Figure 5 show that the estimation error fairly converges towards zero. Aft er 30 s the estimated value of bias almost approaches its actual value. At the same time, as shown in Figure 6, the estimated value of scale factor does not change over time and is always close to zero, while its actual value is given as 1000 ppm. Th ese simulation results are consistent with the conclusion theoretically made in section 3 by analyzing equation (24). Th us, as expected, any maneuver is required in order to increase the calibration accuracy.

Selection of maneuver optimizing vertical accelerometer calibration
To study the infl uence of fl ight maneuver on calibration accuracy, a calibration algorithm was simulated with maneuvers controlled by the altitude/speed hold autopilot (see Figure 4) = 7, 8, 9, 10, 11, 12, 13, 14 s. To examine calibration accuracy, we introduce the relative calibration error calculated from the steady-state values of bias and scale factor, which are estimated by Kalman fi lter, in the form of: where x -mean of steady-state estimates of parameter x, true x -true value of x. Th e simulation results (with fl ight duration 200 s, t 0 = 20 s) are presented in Figures 7 and 8.
As shown in Figure 7, the bias calibration accuracy for any of the planned maneuvers is quite high (the error is not more than 1.1 %, the standard deviation of error is less than 1.5%). Th is is very important because any desirable maneuver should aff ect only the scale factor calibration process, but not the bias calibration, which is well performed in a level fl ight. Figure 8 presents the results of scale factor calibration. According to them, the command durations cmd T ∆ a) b) Figure 6. Scale factor estimation during stabilized level fl ight: a -estimated value; b -estimation error and the corresponding ranges of reference altitude change r h ∆ , at which the scale factor calibration error (30) is guaranteed no more than 2%, are shown in Table 1.
As can be seen from the obtained results, the longer the command duration cmd T ∆ is, the smaller the altitude change r h ∆ is required, and therefore less energy is spent on maneuvering. However, when cmd T ∆ ≥ 11 s, the calibration accuracy is too sensitive to r h ∆ : the scale factor ∆ is 38 m, see Figure 8). Figure 9 illustrates the calibration result achieved from one of the appropriate maneuvers.

Simulation of aircraft landing with failed BA
To examine the appropriateness of the proposed calibration algorithm, we simulated the landing (from glideslope capture to flare) of a passenger aircraft introduced in section 2 with BA failure. The aircraft motion is driven by an automatic control system using only feedback signals from a standalone INS. Three simulation scenarios were performed: 1. Scenario 1: calibration was not performed until BA failure. In this case, after BA failure, the initial error model of the vertical accelerometer is used in INS.
2. Scenario 2: before BA failure, the calibration was performed in a normal stabilized level flight, i.e., without any calibration maneuver. In this case, after BA failure, only the estimated value of bias is used in INS to provide additional error compensation. 3. Scenario 2: before BA failure, the calibration was performed with calibration maneuver allowing to estimate not only the bias but also scale factor. After BA failure, the estimated values of both bias and scale factor are used in INS to provide additional error compensation. The level flight at 500 m altitude and speed 80 m/s before glide-slope capture is stabilized by the altitude/speed hold autopilot shown in Figure 4. For glide-slope capture, an autopilot whose block diagram illustrated in Figure 10 is used.   Figure 10. Glide-slope coupler Accordingly, control inputs are formed as: where 0 T u , 0 e u -program controls corresponding to the trim condition (flight path angle -3 deg, speed 80 m/s); (Brian & Frank, 2003); V ,â ,ˆz ω ,θ ,δ -INS based estimate of velocity, acceleration, pitch angle rate, flight path angle and glide path deviation; V k = 9.26, a k = -22.27, V k θ = -0.44, k ω = 1.22, k θ = -2.12, k δ = 0.08. The automatic flare control system is turned on at an altitude of 15 m and forms the following control inputs (Brian & Frank, 2003):  (Brian & Frank, 2003); ˆz ω ,θ ,ĥ -INS based estimate of pitch angle rate, flight path angle and altitude; k 1 = -918.8, k 2 = 999.89, k 3 = -76.62, k 4 = -6.64, k 5 = -77.28, k 6 = 0.054.
A block diagram of the presented automatic flare control is shown in Figure 11.
The simulation ends when the aircraft touches the runway, either the based INS estimate of altitude is equal to zero. The simulation results are shown in Figure 12 and Table 2. For convenience of analysis, landing trajectories are presented only in the last seconds.
As can be seen from Figure 12, additional INS error compensation using the calibration results (lines h 1 and h 2 ) is needed to prevent a crash-landing (line h 3 ). From the data in Table 2, we find that in scenario 1 the vertical speed and the flightpath angle at touchdown are too large and therefore cause the aircraft to fall to the ground. It can be explained, focusing to lines h 1 and INS1 h : when the aircraft is at about 5 m above the ground, due to inaccurate altitude information (estimation error is about 7 m) from INS, the automatic flare control system still forces the aircraft to track the straight-line path of desired flare trajectory (which is very close to line INS3 h ). As a result, the aircraft continues to descend at high vertical speed and hits the ground.
In contrast, when the error model of vertical accelerometer is adjusted by the calibration results (scenarios 2 and 3), a soft landing can be expected based on behaviors of aircraft trajectories in the last moment (see lines h 2 and h 3 ). Indeed, Table 2 shows that, when the aircraft is almost close to the ground, its vertical speed is about -1.3 m/s, the flightpath angle is less than 1 degree and the aircraft is  pitching up ( z ω ≈ -2.3 deg/s). This is a significant performance improvement in comparison to scenario 1. Now let's see the impact of the calibrated bias and scale factor of vertical accelerometer on altitude estimation error. To simplify the analysis, altitude estimation errors in three scenarios were taken at the moment of crash-landing occurring in scenario 1 (i.e., at t = 148.2 s): 1 h ∆ ≈ 7 m, 2 h ∆ ≈ 3.5 m, 3 h ∆ ≈ 1 m. In scenario 1 the altitude estimation error is largest because no error compensation is made. In scenario 2 the bias compensation is provided, reducing the altitude estimation error from 7 m to 3.5 m. The effectiveness of the proposed calibration algorithm follows the comparison between scenarios 2 and 3: additional scale factor compensation in scenario 3 reduces the altitude estimation error by another more 2.5 m (from 3.5 m to 1 m). Thus, if the calibration is performed with a designed calibration maneuver, it is possible to significantly reduce the altitude estimation error, which means that a more successful landing can be achieved.

Conclusions
An approach to additional (in-flight) vertical accelerometer calibration, based on refining its characteristics through a specific aircraft maneuver, was investigated. Validity and performance of the proposed algorithm were assessed by simulations of flight dynamics and aerodynamic models of a passenger aircraft, and baro-inertial integration using Kalman filter and its associated models. Applying designed calibration maneuver before capturing the glide-slope, simulation results show acceptable landing accuracy and safety level with only feedback signal from standalone INS (assuming a BA failure). Moreover, the proposed scheme for the measurement signal generation used in Kalman filter confirms that the influence of BA bias on the calibration accuracy can be eliminated, allowing a relevant calibration of the vertical accelerometer without the need to estimate the BA bias.
In this paper, the time constant τ of BA is assumed known exactly, using equation (4). However, in practice, it is required to adjust τ for obtaining acceptably accurate estimate of altitude changes. This requires a more complicated integrated system for calibrating τ using GNSS and optical navigation system in further work. A nonlinear estimation with extended or unscented Kalman filter could also be an option for the next development. Investigation on the Pitot port blockage will also be considered in the future work to better cover the flight safety, in particular for incidents from air data computer failures.