Positioning, Navigation and Timing (PNT)

Scientific staff: Peter Teunissen (0.2 fte), Christian Tiberius (1.0 fte), Hans van der Marel (0.7 fte), Sandra Verhagen (0.9 fte).

Introduction This decade will bring a proliferation of Global Navigation Satellite Systems (GNSS) that are likely to revolutionize society the same way as the mobile phone did. The promise of a broader multi-frequency, multi-signal GNSS ‘system of systems’ will enable a much wider range of demanding applications compared to the current situation. This PNT-programme aims to address these GNSS challenges by developing the theory and models necessary to fulfil high-accuracy and -integrity requirements (cf. Fig 9). During the review period, we published 154 peer reviewed papers and 1 patent on this theme.

Theory for PNT
Integer Inference and Validation: We are the founders of the geodetic theory of integer inference, a key area for high-precision GNSS. Examples of our contributions are: integer bootstrapping, with its popular closed form success-rate; invention of integer-aperture (IA) and integer-equivariance (IE) estimation; and an extension of collocation- and Kriging-theory to the mixed-integer case. Although Kriging is one of the most popular spatial prediction methods, we have shown that it is suboptimal when the model of the spatio-temporal field contains an integer-parametrized multivariate trend. Within our new class of IE-predictors we were able to determine the best predictor and show that its mean squared prediction error is smaller than that of Kriging. We have also demonstrated that current integer ambiguity selection methods are not sustainable for future GNSSs and that they need to be replaced by our IA-theory based validation test procedures.

Stochastic model: Variance component estimation (VCE) is an important topic as our knowledge of the stochastic model is still at a rather rudimentary level in many modern geodetic measurement techniques. We have introduced Least-Squares (LS)-VCE as a method that unifies many of the existing VCE methods. LS-VCE is attractive as it allows one to directly apply the existing body of knowledge of least-squares theory, such as statistical hypothesis testing, nonlinear estimation and curvature, and estimability with S-transformations. Our LS-VCE has been successfully applied to various GNSS models and measurement scenarios, from short-baseline to global network based models.

Figure 9 Global Navigation Satellite Systems (GNSS) are linked to a wide range of Positioning, Navigation, and Timing (PNT) applications through system models and theory.

Attitude Determination: GNSS attitude determination has a wide variety of challenging (terrestrial, sea, air, and space) applications. Until recently, the well-known LAMBDA method was universally used as the ambiguity domain attitude determination method. We have shown however that this method is not optimal for the highly nonlinear GNSS attitude model and that it should be replaced by our newly developed multivariate constrained (MC-) LAMBDA method. With its new ambiguity objective function, in which the nonlinear body-geometry is fully integrated, the method is currently the best performing attitude determination method as was demonstrated in land, sea and air experiments. As a maritime spin-off, an improved sinkage-monitoring method was also developed.

Array processing: The Array-aided Precise Point Positioning (A-PPP) concept was patented and introduced as a generalization of Precise Point Positioning (PPP). It is an array processing concept that uses data from multiple antennas in known formation to realize improved GNSS parameter estimation (position, time, equipment delays and atmospheric delays). The improvements can be exploited in different ways, e.g. to improve accuracy, to reduce convergence time, to achieve higher success rates, or to improve between-platform positioning. To enable fast, efficient and accurate A-PPP, a novel orthonormality constrained mixed integer least-squares problem was introduced and solved. The A-PPP principle is generally applicable. It applies to single-, dual-, and multi frequency GNSS receivers, as well as to any current and future GNSS, standalone or in combination. It is also not restricted to GNSS, as it applies for instance to acoustic phase-based positioning and other interferometric techniques as well.

Figure 10 A 2D illustration of three different cases of integer ambiguity ratio-test validation. The green and red dots result in correct and incorrect integer outcomes respectively, while blue dots result in a float solution as an outcome. The left panel shows poor performance (as the aperture pull-in region was chosen too large). The middle and the right panel show good performance; the fixed failure-rate approach was used, resulting the validation to adapt to the strength of the underlying model (they both do have the same guaranteed small failure-rate).

Modelling for PNT
New and Multi-GNSS: We have been at the forefront of modelling, integrating and analysing the use of new and multi GNSSs for PNT. This includes the characterization and initial assessment of the Chinese BeiDou (BDS) system and the first results of mixing GPS with Galileo and BDS, respectively. This work was also recognized by the European Space Agency (ESA) by means of a Galileo award. We also studied quadruple integration of GPS, BDS, Galileo and QZSS for long baseline RTK and single-frequency RTK. Our results show a remarkable robustness in satellite-deprived environments (urban canyons; open pit mines). And recently we successfully demonstrated how to L5-integrate the new Indian IRNSS signals with GPS, Galileo and QZSS for precise positioning.

GNSS Bias Modelling: New GNSS constellations come with system-specific bias characteristics. Inter-system biases (ISBs) need precise modelling if one is to take advantage of multi-GNSS ‘mixing’ to improve PNT solutions. We were the first to analyse un-differenced GPS-Galileo mixing and to determine their mixed receiver ISBs. We also discovered a new bias-type among the GEO, IGSO and MEO satellites of BDS. We have shown that this mixed receiver half-cycle inter-satellite-type-bias (ISTB) severely affects ambiguity resolution and we demonstrated how it could be calibrated. As acknowledged by the Chair of the Radio Technical Commission for Maritime Services (RTCM) SCl04, our discovery has enabled GNSS receiver manufacturers to modify and align their receivers such that now mutually consistent BDS extraction procedures are realized.

PPP-RTK: PPP-RTK is a relatively new positioning concept that combines the single-receiver positioning benefit of PPP with the ambiguity-fixing capability of network-RTK. We have used S-system theory to compare the different mechanizations that have been proposed in the literature. This enabled us also to identify the methods that cannot be accepted as proper PPP-RTK methods. The PPP-RTK development is still in its infancy, as studies focused on ionosphere-free, dual-frequency GPS so far. To do justice to the many different observation types of multi-GNSS, we have taken an un-differenced approach throughout. This allows for much greater flexibility, and, together with the S-system theory, for a proper PPP-RTK incorporation of next generation GNSSs in PNT. Our approach also applies to single-frequency users, thus enabling PPP-RTK even for low-grade GNSS receivers.

Figure 11 Horizontal East-North instantaneous position scatter plots (upper panels), and vertical position time series (lower panels), for single frequency B1 BDS (at left), single frequency L1 GPS (middle), and B1+L1 BDS+GPS combined (at right). These results pertain to a 1 kilometer baseline, observed over a 3-day period, with a cut-off elevation angle of 25 degrees.

Future Plans for PNT

Theory and modelling: Our work on multi-frequency, multi-GNSS, with associated theory and model development, will be continued and extended. Provision of a rigorous quality description for the ambiguity resolved GNSS-parameters is possible for integer-estimation, but not yet for the more complex integer-aperture-estimation. With regard to bias-robustness, we will work on model reduction methods to improve on the success-rates and partial ambiguity resolution performance. New theory is needed for mixed-integer model testing. This lack of appropriate theory has as a consequence that in practice still classical methods (e.g. statistical theory of hypothesis testing for linear models) are applied to the mixed-integer model. With the advent of more demanding GNSS applications, the identified lack of testing theory is not acceptable, as one will have to satisfy the need for corresponding tighter quality control requirements.

GNSS-complements: Despite GNSS’ huge success, the concept of satellite-based positioning also has its limitations (e.g. urban canyons, indoor, multipath).  Indoor radio positioning and SuperGPS are the two GNSS-complementary concepts on which our research will be focussed. Recently we already started with the topic of indoor positioning, using wide band radio signals. This concept deserves also from the algorithmic point of view attention, as -unlike in satellite navigation- the problem is highly non-linear, due to the short ranges between transmitters and receiver, and the highly varying geometry. The second GNSS-complementary concept is SuperGPS, a new terrestrial radio-positioning concept in which an extremely accurate time reference is distributed over an optical network, and the ‘last mile’ to the user is covered by a very wide bandwidth radio signal (order of GHz). This should deliver pico-second timing and cm-positioning accuracy, and can be used for instance, to support vehicles driving autonomously on highways (see Fig 12). The extremely accurate time reference may also offer new opportunities in geodesy, beyond GNSS, as determining gravitational potential differences across the optical network.

Figure 12 The concept of SuperGPS: a hybrid optical-wireless system. A network of (existing) glass-fibre (in blue) distributes a highly accurate time reference (in the red square), to terrestrial wide-band transmitters (in green squares) providing radio-signals which allow users to position with cm-accuracy.