Single-frequency L5 attitude determination from IRNSS/NavIC and GPS: a single- and dual-system analysis

Abstract

The Indian Regional Navigation Satellite System (IRNSS) has recently (May 2016) reached its full operational capability. In this contribution, we provide the very first L5 attitude determination analyses of the fully operational IRNSS as a standalone system and also in combination with the fully operational GPS Block IIF along with the corresponding ambiguity resolution results. Our analyses are carried out for both a linear array of two antennas and a planar array of three antennas at Curtin University, Perth, Australia. We study the noise characteristics (carrier-to-noise density, measurement precision, time correlation), the integer ambiguity resolution performance (LAMBDA, MC-LAMBDA) and the attitude determination performance (ambiguity float and ambiguity fixed). A prerequisite for precise and fast IRNSS attitude determination is the successful resolution of the double-differenced integer carrier-phase ambiguities. In this contribution, we will compare the performance of the unconstrained and the multivariate-constrained LAMBDA method. It is therefore also shown what improvements are achieved when the known body geometry of the antenna array is rigorously incorporated into the ambiguity objective function. As our ambiguity-fixed outcomes show consistency between empirical and formal results, we also formally assess the precise attitude determination performance for several locations within the IRNSS service area.

Appendix: Nonlinearity of the GNSS attitude model

The orthonormality constraint of the rotation matrix $R_{q}$ in (6) is a nonlinear constraint. Here, through simulation, we explain the impact of this nonlinearity on the estimation of the attitude angles. Considering the linear array of CUCC–CUBB, we simulated two sets of $10^4$ baseline solutions, corresponding to the unconstrained ambiguity-float and ambiguity-fixed scenarios. They were generated from the normal distribution with the same mean (CUCC–CUBB baseline ground truth $\bar{b}$), but different variances. The variance matrix of the first set $Q_{1}$ is equal to the average formal variance matrix of the blue dots in Fig. 9a over the first 15,000 epochs, while the variance matrix of the second set is given by $Q_{2}=\frac{\sigma _{\phi _{I}}}{\sigma _{p_{I}}}Q_{1}$.

Fig. 14

Histograms of the heading (left) and elevation (right) estimation errors (blue) and their linearized formal counterparts (red). These results are computed based on $10^4$ normally distributed baseline samples with the mean value of the CUCC–CUBB baseline ground truth and with two variance matrices $Q_{1}$ (a) and $Q_{2}=\frac{\sigma _{\phi _{I}}}{\sigma _{p_{I}}}Q_{1}$ (b). $Q_{1}$ is equal to the average formal variance matrix of the blue dots in Fig. 9a over the first 15,000 epochs

As was previously mentioned, for the single-baseline scenario, the orthonormality constraint of the rotation matrix is equivalent to the length constraint on the baseline vector, i.e., $||x||=l$. It indicates that the baseline vector is constrained to lie on a sphere with known radius of l. Imposing the baseline-length constraint, we estimated the heading and elevation of the CUCC–CUBB baseline based on the two sets of simulated data. Figure 14 shows the corresponding histograms of the estimated attitude angles, corrected for the ground truth, on the basis of the samples with the variance matrix $Q_{1}$ (a) and samples with the variance matrix $Q_{2}$ (b). Given the linearized formal standard deviations of the estimated angles, we computed the corresponding zero-mean normal PDF (probability density function) which are indicted by the red curves in Fig. 14.

Fig. 15

Mapping the unconstrained ambiguity-float baseline solutions to their baseline-length-constrained counterparts. The scatter plot of $10^4$ normally distributed baseline samples with the mean value of the CUCC–CUBB baseline ground truth and with the variance matrix of $Q_{1}$ (the average formal variance matrix of the blue dots in Fig. 9a over the first 15,000 epochs). The gray vector and the sphere denote, respectively, the baseline ground truth and the sphere with the radius of the CUCC–CUBB baseline length l. The scatter plot is split into two clusters. light brown/dark brown the points that upon constraining the baseline length result in negative/positive $\{*\}$ estimation errors. $\{*\}$: heading (a); elevation (b)

The histograms in Fig. 14a demonstrate an asymmetric behavior. From these two histograms, it can be seen that the empirical density of the errors of the estimated angles at negative values is not the same as that at positive values, implying that the estimated angles are biased. This bias is called the nonlinearity bias which was already recognized, in the baseline domain, in the gray scatter plots in Fig. 10. Also, the deviation of these histograms from the red normal curve indicates that looking at only the standard deviations of the attitude angles is not enough to find out the complete probabilistic behavior of their estimators. The histograms in Fig. 14b, in contrast, show a very good consistency with their corresponding normal PDF. This is due to the fact that these estimations are based on the very precise samples with the precision ($Q_{2}$) at the level of phase precision, where the nonlinearity of the attitude model can be neglected.

The signature of the attitude angles histograms is driven by the variance matrix of the simulated samples, hence the size, shape and orientation of their scatter plot. The asymmetric signature in Fig. 14a can therefore be explained through the specific orientation of the first set of simulated data scatter plot. Figure 15a, b shows the scatter plot of the first set of the simulated samples each of which is split into two clusters (light/dark brown) based on two different criteria. Clusters in panel (a), upon constraining the baseline length, result in heading estimation errors either of negative (light brown) or positive (dark brown) values, while clusters in panel (b) are the counterparts of those in panel (a) for the elevation estimation errors. It can be seen that the two clusters are not symmetric in any of the two panels. This explains the asymmetric behavior of the histograms in Fig. 14a.

In Fig. 15a/b, the light brown and the dark brown clusters are separated by a two-dimensional manifold being the locus of the points which, upon constraining the baseline length, result in the heading/elevation estimation errors equal to zero. The intersection of these two manifolds then accommodates the points which, upon constraining the baseline length, are mapped to the baseline ground truth $\bar{b}$. This intersection is described by a straight line given by

$$\begin{aligned} \begin{array}{l} b_{o}=\left( I_{3}+k\,Q_{\hat{b}\hat{b}}\right) \;\bar{b};\\ \\ k\in \mathbb {R}:\; \left| \left| b_{o}-\bar{b}\right| \right| ^{2}_{Q_{\hat{b}\hat{b}}}\le \left| \left| b_{o}-b\right| \right| ^{2}_{Q_{\hat{b}\hat{b}}}\qquad \forall \; b\in \mathbb {S}^{3}(l) \end{array} \end{aligned}$$

(17)

where $\mathbb {S}^{3}(l)$ is the set of points on the circumference of a three-dimensional zero-centered sphere with the radius of l. Proof is as follows. Given the unconstrained ambiguity-float baseline solution $\hat{b}$, its baseline-length-constrained counterpart is given by

$$\begin{aligned} \hat{\hat{b}}=\underset{{\begin{array}{c} \left| \left| b\right| \right| =l\\ b\in \mathbb {R}^{3} \end{array}}}{\mathrm{argmin}}~ \left| \left| \hat{b}-b\right| \right| ^{2}_{Q_{\hat{b}\hat{b}}} \end{aligned}$$

(18)

which, from geometrical point of view, is the point where the ellipsoid $\mathbb {E}^{3}=\{b\in \mathbb {R}^{3}|~||\hat{b}-b||^{2}_{Q_{\hat{b}\hat{b}}}={\mathrm{constant}}\}$ just touches the sphere $\mathbb {S}^{3}(l)$ (Teunissen 2010). This indicates that the gradient vectors of the mentioned ellipsoid and sphere will be parallel at point $\hat{\hat{b}}$. Therefore to find the locus of the points $b_{o}\in \mathbb {R}^{3}$ for which $\hat{\hat{b}}=\bar{b}$, the gradient vector of the corresponding ellipsoid $\mathbb {E}^{3}$ and sphere $\mathbb {S}^{3}(l)$ at $\bar{b}$ should be set parallel to each other, i.e.,

$$\begin{aligned} Q_{\hat{b}\hat{b}}^{-1}\left( b_{o}-\bar{b}\right) = k~ \bar{b} \end{aligned}$$

(19)

where k is a scalar. Equation (19) can be worked out to

$$\begin{aligned} b_{o}=\left( I_{3}+k\,Q_{\hat{b}\hat{b}}\right) \;\bar{b} \end{aligned}$$

(20)

With changing k, (20) describes a straight line which is parallel to $Q_{\hat{b}\hat{b}}\bar{b}$ and passes through $\bar{b}$. Note the values of k should result in $\bar{b}$ being the solution of (18) for $\hat{b}=b_{o}$ given by (20). Therefore, k follows from

$$\begin{aligned} k\in \mathbb {R}:\; \left| \left| b_{o}-\bar{b}\right| \right| ^{2}_{Q_{\hat{b}\hat{b}}}\le \left| \left| b_{o}-b\right| \right| ^{2}_{Q_{\hat{b}\hat{b}}}\qquad \forall \; b\in \mathbb {S}^{3}(l)\qquad \end{aligned}$$

(21)

$\square$