# NEXA: New Extensible State-Space Approach

NEXA is a robust non-linear least-squares generic sequential estimation engine. With NEXA, trajectories of any type –i.e., time series of parameters– can be computed from given measurements (input data) and from the mathematical models that relate the measurements to the [unknown] parameters (input software).

A typical application of NEXA is the fusion of inertial measurements and satellite-to-receiver distances, a.k.a. INS/GNSS integration. NEXA can be used in real-time or post-mission modes. It can be used with real or simulated data for trajectory determination and system design respectively.

**Design principles.** NEXA has been designed to be extensible and adaptative.

NEXA is a software platform and system built on GeoNumerics' ITAVERA framework. The main component of NEXA is the NEXA runtime platform, a directly executable or callable programme that, provided with the appropriate toolboxes, does the actual robust non-linear least-squares trajectory determination. The NEXA runtime platform integrates all necessary input/output, organizational and mathematical machinery. The NEXA runtime, by itself, knows nothing about models. Mathematical models (non-differential and differential observation equations) that relate measurements, unknowns to be calculated (parameters) and instrument constants (instruments) are provided to NEXA by plug-&-play model toolboxes. The toolboxes, on the contrary, know nothing about trajectory estimation; they just provide modelling information. Typically, a toolbox bundles together models, measurement types, parameter types and instrument types that are domain-related. The NEXA runtime comes with the BASIC toolbox for the fundamental geomatic and navigation modelling entities.

NEXA is extensible in the sense that model toolboxes enable it to deal with any type of sensor and time-dependent parameter determination task. It is also extensible in the sense that its input/output functions and formats are not hardwired to specific sensors and measurements. In other words, NEXA is extensible because it is generic and it is generic because of its standardized input/output interface and of its plug-&-play interface to model toolboxes.

NEXA is adaptative to trajectory determination scenarios through strategists. Examples of three different scenarios for the same set of sensors are: land vehicle, airplane or helicopter navigation. In NEXA, the trajectory estimation process follows a sequence of phases where the computation evolves from an initial situation, for example, a lost situation –NEXA knows nothing about the value of its parameters– to a normal navigation situation where the trajectory states are reasonably well known and updated, epoch by epoch. From lost to normal navigation NEXA usually goes through a series of intermediate navigation modes where, roughly speaking, the trajectory unknown parameters are incrementally introduced in the computations in order to guarantee numerical convergence to the correct navigation parameters according to a scenario-dependent computation strategy. A strategist is an interchangeable software component that contains a particular trajectory computation strategy to be used by NEXA.

The NEXA runtime, by itself, knows nothing about trajectory determination scenarios. The navigation modes are provided to NEXA by plug-&-play strategists. The strategists, on the contrary, know nothing about trajectory estimation; they just provide trajectory computation strategy information.

With NEXA's Software Development Kit (NEXA SDK) toolboxes with new models and strategists with computation strategies can be designed and programmed.

**Application to INS/GNSS position-velocity-attitude (PVA) determination.** The integration of inertial measurements with range measurements to satellites –a.k.a. INS/GNSS integration– is not the only application of NEXA and, in general, of the sequential least-squares estimation inspired in the Kalman filter. However, it is one of the most important ones as it is the core of vehicle navigation, guidance and control, particularly of aircraft and spacecraft.

INS/GNSS integration is the use of GNSS ranging signals or of positions computed thereof, to calibrate the errors of an inertial measurement unit (IMU) and thus, mitigate the drifts of a pure inertial measurement-based determination of trajectories as computed by an inertial navigation system (INS).

In an INS/GNSS PVA-trajectory determination, NEXA expects to be provided, as input, with time sorted (1) frequent –a few hundred Hz– inertial measurements (angular and velocity rates) and (2) less frequent –a few Hz– GNSS measurements. The numerical integrator of the NEXA platform, using the field of the differential inertial equations, will compute, in a prediction step, a PVA trajectory. The field or model is provided by the dynamicSURVEY model toolbox. When, from time to time, GNSS measurements arrive, a weighted adjustment of the GNSS measurements –through their models– and the PVA prediction is computed by NEXA, in an update step. The model for the GNSS measurements is provided by the dynamicSURVEY toolbox. The mechanism for the adjustment is provided by NEXA.

**Features.**

- sequential robust non-linear least-squares parameter estimation with stochastic and stochastic differential models –i.e., least-squares Kalman-filter and simultaneous prediction-&-filtering
- real-time and post-mission solution modes
- outlier detection-and-removal
- computation of parameter and residual covariances and correlations
- computation of observation internal and external reliability parameters
- INS/GNSS and, in general, multi-sensor trajectory determination (with appropriate model toolboxes)
- separation between estimation platform (NEXA) and modelling (model toolboxes)
- separation between models and solution strategy (strategist component)
- standardized, generic text (XML) and binary file interface

**History.** NEXA is the result of the long experience of GeoNumerics’ researchers and developers, dating back to the mid 1990s at the former Institute of Cartography of Catalonia (ICC) and, more recently in the period 2000-2010, at the former Institute of Geomatics (IG). The classical methods of mathematics, numerical analysis, geodesy, sensor orientation and accurate navigation have shaped NEXA. In those years the ideas, theory and prototypes of generic robust non-linear least-squares sequential estimation behind NEXA matured along with their implemented in various prototypes and were tested against a number of use cases (land vehicle navigation, manned and unmanned aircraft, spacecraft navigation, airborne strapdown INS/GNSS gravimetry among other). NEXA is the current embodiment of the experience accumulated in the past more than twenty years; it has been developed at GeoNumerics since 2014, partly supported by EU Horizon 2020 Research and Innovation Programme, the projects mapKITE (grant 641518 ), COREGAL (grant 641585) and, more recently, GIMS (grant 776335).