Observability of satellite launcher navigation with INS, GPS, attitude sensors and reference trajectory

Highlights

• Roll error and gyroscope biases are only observable with attitude sensors.

• Modelling biases as Markov processes or random walks have limited impact.

• GPS position bias is unobservable with the tested navigation solutions.

• Reference trajectory improve roll estimation but not its observability.

• Sensor quality must be taken into account along with observability analysis.

Abstract

The navigation system of a satellite launcher is of paramount importance. In order to correct the trajectory of the launcher, the position, velocity and attitude must be known with the best possible precision. In this paper, the observability of four navigation solutions is investigated. The first one is the INS/GPS couple. Then, attitude reference sensors, such as magnetometers, are added to the INS/GPS solution. The authors have already demonstrated that the reference trajectory could be used to improve the navigation performance. This approach is added to the two previously mentioned navigation systems. For each navigation solution, the observability is analyzed with different sensor error models. First, sensor biases are neglected. Then, sensor biases are modelled as random walks and as first order Markov processes.

The observability is tested with the rank and condition number of the observability matrix, the time evolution of the covariance matrix and sensitivity to measurement outlier tests. The covariance matrix is exploited to evaluate the correlation between states in order to detect structural unobservability problems. Finally, when an unobservable subspace is detected, the result is verified with theoretical analysis of the navigation equations.

The results show that evaluating only the observability of a model does not guarantee the ability of the aiding sensors to correct the INS estimates within the mission time. The analysis of the covariance matrix time evolution could be a powerful tool to detect this situation, however in some cases, the problem is only revealed with a sensitivity to measurement outlier test. None of the tested solutions provide GPS position bias observability. For the considered mission, the modelling of the sensor biases as random walks or Markov processes gives equivalent results. Relying on the reference trajectory can improve the precision of the roll estimates. But, in the context of a satellite launcher, the roll estimation error and gyroscope bias are only observable if attitude reference sensors are present.

Introduction

The navigation is a critical element of a satellite launcher. Attitude, velocity and position must be known with the best possible precision in order to correct the launcher trajectory. For decades, purely inertial means of navigation were exploited with success [1]. However, inertial navigation estimates are prone to drift. Therefore, high quality units are needed to provide the required precision. Since those units are expensive, aiding sensors are used to reduce the cost and improve precision [1], [2], [3]. To ensure that the INS error is reduced by the aiding sensors, the observability of the navigation model must be verified.

For space vehicles, several navigation systems are proposed. Among others, the solution of an INS combined with a GPS receiver is often put forward [4], [5] and is considered as usable [1], [2], [6]. Some tests were also performed on the Space Shuttle and have demonstrated the viability of this solution [7]. However, it is already known that this solution may suffer from observability problems [8], [9], [10]. The observability of this approach involves some maneuvers [11], [12], [13], [14], [15]. For example, on a system with a low-grade INS and a single-antenna GPS, the gyroscope bias in the direction of the specific force is unobservable if the vehicle moves with constant attitude and acceleration [12]. Another example is the yaw, which is non-observable, during the hovering of a helicopter [16]. In the context of a satellite launcher, the trajectory is optimized to minimize the fuel consumption. That implies that the trajectory is mostly aimed in one direction. Consequently, there is no flexibility to perform the needed maneuvers. Even though the GPS integration type can improve precision and prevents jamming, it has no influence on the observability [14], [17]. Therefore, to simplify the analysis, only the loosely coupled integration is studied. A long lever arm between the INS and the GPS antenna may increase the observability problem [12]. Considering that this aspect has already been evaluated, it will not be treated here and the INS and GPS antenna are considered collocated.

To ensure proper attitude estimation, attitude measurements may be needed [18]. For an airplane, the observability of the INS/GPS combination is not guaranteed unless 3 non-aligned GPS antennas with sufficient lever arm are used [19]. On a helicopter, the yaw observability benefits from the addition of a magnetometer [16]. A star tracker can be employed to solve the attitude observability problem on a space vehicle [20]. It was exploited to improve the INS/GPS precision for the SHEFEX-2 mission [21], [22].

The reference trajectory data could be used to increase the navigation solution precision and robustness to GPS outages [23]. The underlying idea of this approach is that, on the launch pad, the attitude of the launcher is perfectly known. As the mission progresses, the confidence that the launcher is following the predicted attitude reduces. If this confidence can be quantified, it allows exploiting the reference trajectory attitude data to better estimate the attitude of the launcher. However, the observability of this approach has been evaluated only with the rank of the observability matrix on a INS/GPS navigation system which neglects the sensor biases.

The addition of attitude reference sensors to an INS can improve the precision by reducing the attitude uncertainties. But it does not provide velocity and position observability [20], [24], [25]. Since this research is seeking for the observability of all states within the navigation model, the solution combining only these two sensor types is rejected.

Different error models could be exploited depending on the sensors used and the possibility of estimating the sensor errors [26]. With low end sensors, the bias could be a significant source of error. The bias drift of an inertial sensor can be modelled as a first order Markov process [16], [27], [28], [29]. A time constant of 100 s is often used [30], [31], [32]. However, the time constant could be as long as 1 h [2]. In some cases, when the time constant is of the same order of magnitude as the mission time, the bias is simply modelled as a random walk [11], [12], [13], [14], [33], [34]. The effects of the error model on the observability have not been evaluated before and will be explored in this paper.

The observability is often evaluated with the help of the observability matrix rank [11], [12], [13], [14], [15], [35]. Unfortunately, only considering the rank of this matrix may not be sufficient. Due to the limited digital precision of computers (around 15 digits with double precision in Matlab^®), a near singular matrix might not be detected by the rank of the observability matrix. Therefore, the singular-value decomposition or the condition number of the observability matrix provides a better evaluation of the observability [36]. The order of magnitude of the condition number gives an estimate of the digital precision loss. A rough rule of thumb is that an increase of 10 in the condition number leads to the loss of one significant digit in the estimates [37].

On the other hand, the time evolution of the estimate covariance matrix gives information which can be overlooked by the observability matrix analysis [11], [12], [35]. The idea is that the variance of an unobservable (or barely observable) state evolves in the same manner either if the Kalman correction is applied or not. This analysis is highly recommended, if not essential, to evaluate the performance of the Kalman filter [38]. But, with higher order systems, this approach can be cumbersome and relations between states may be difficult to analyze [35]. Fortunately, in the studied cases, the state relationships are evident. Thus, there is no need to rely on more complex techniques, such as evaluating the normalized eigenvalues and eigenvectors of the estimate covariance matrix [35]. The covariance matrix may also be exploited to evaluate the correlation between states. A perfect correlation coefficient (negative or positive) can indicate a structural unobservability, and one should be suspicious of a correlation coefficient which exceeds 0.9 in absolute value [36].

Evaluating the observability of a non-linear system may be complicated. However, the model can be approximated by a piecewise constant model and the observability be evaluated locally for each constant segment [12], [14], [39], [40], [41]. The observability can also be assessed with theoretical analyses of the navigation equation and observability matrix [11], [12], [13], [14], [39], [42].

The first contribution of this work is the observability analysis of four navigation solutions in the context of a satellite launcher mission. The first solution is an INS combined with a single antenna GPS. The second solution adds attitude reference sensors to the INS/GPS couple. For the third approach, the reference trajectory data is added to the INS/GPS couple as suggested in Ref. [23]. The last solution exploits the GPS, INS, attitude reference sensors and the reference trajectory. The observability of the navigation solution with reference trajectory has only been verified with the rank of the observability matrix on a simplified model which includes only the GPS and INS as sensors. In this paper, a more complete evaluation is done using many observability analysis tools and different sensor models.

The second contribution is the evaluation of the effects on the observability of the sensor error equations in the navigation model. Each navigation system is tested with two different sets of inertial sensors. Relatively good sensors are first employed. For these sensors, the bias is considered low enough to be neglected in the navigation model. Then, low quality sensors, which require the bias to be estimated, are used. Modelling sensor biases as a random walk and as a first order Markov process is explored.

For the third contribution, the observability analysis is done using the rank and the condition number of the observability matrix. Then, the time evolution of the covariance matrix is exploited to evaluate the observability quality and the ability of the aiding sensors to correct the INS estimates within the mission time. Afterward, the correlation between the states is analyzed to detect structural unobservability. Next, sensitivity to measurement outlier tests are performed to confirm the results obtained from the previous approaches. Finally, unobservable subspaces are investigated with theoretical analysis of the navigation equations.

The paper is structured as follows: section 2 introduces the navigation solutions. Then, the methodology and the observability evaluation technique are presented in section 3. Section 4 shows the observability results obtained with the studied navigation solutions.