Accuracy prediction for highly non-linear kinematic parameter estimation

Highly accurate trajectory estimation is required in many surveying applications, along with a quality assessment of the derived parameters like positions and velocities. Quality prediction supports the planning of the measurements whereas quality control of the measurements and results supports the interpretation and decision making. However, assessing the accuracy of the estimated trajectory of a moving object is a problem where the traditional approach to quality prediction and assessment (i.e. variance propagation) may not be applicable, due to the required assumptions made for processing of the observations, whose impact is not reflected in the propagated uncertainty.

We propose an approach for the assessment of the accuracy of kinematic position and velocity solutions estimated from GNSS observables using a Kalman filter. The approaches substitutes repeated measurements by numerical simulation, in particular by a Monte Carlo-based numerical simulation replacing real measurements and ground truth by synthetic observations and assumptions about the reference trajectory.

Two real case studies requiring the estimation of accurate velocities are used to demonstrate the proposed approach for accuracy prediction and assessment. On the one hand, the trajectory of a sport athlete performing a training session in alpine skiing is chosen as an example of a smooth trajectory with high-variability in the dynamics. On the other hand, a dozer is chosen as representative for objects moving slowly, under strong vibrations, and performing sharp turns.

The developed Monte Carlo-based numerical simulation is than applied to investigate the impact of variations of the system noise on the estimated trajectory and to identify the optimal values of the system noise variance for each epoch. A proposal for a (near) real-time adaptivity of the system noise variance is developped, which allows adapting the system noise variance, based on the magnitude of the acceleration of the analyzed trajectory.

The speed of the investigated moving objects can be determined with a standard deviation of few mm/s to cm/s and negligible bias. The investigations confirm that the proposed simulation approach can be applied to predict and assess the accuracy of estimated trajectories and the corresponding velocities without requiring ground truth information.