A Comparison Between Hard- and Soft-Constrained RTK Positioning With Multiple Reference Antennas

Real-time kinematic (RTK) ambiguity resolution can be improved by engaging multiple reference antennas. The baseline information between the antennas can be taken as true values and used as a hard-constraint; or it can be regarded as weighted, a priori measurements, and used as a softconstraint. In this contribution, a comparison between the two constraints is made to explicate their difference in RTK ambiguity resolution for some specific applications, in which the coordinates of the reference antennas cannot be accurately calibrated in advance. The functional and stochastic models of the hard and soft-constrained RTK positioning are given. The float ambiguity precision of the unconstrained, the hard and the soft-constrained cases is compared. The closed-formulae of the ambiguity dilution of precision (ADOP) for the unconstrained and the hard-constrained models are derived, and they are proven to be the upper and lower bound of the soft-constrained ADOP, respectively. A simulated and a real data show that, with the hard or soft-constraints, the single-frequency ambiguity resolution success rate, as well as the performance in time-to-first-fix (TTFF), is improved significantly. When there is a bias smaller than 6 cm in the a priori baseline vectors between the reference antennas, the two constraints have a comparable performance in ambiguity success rate. Also, a sharp increase in the hard-constrained TTFF is observed at the same time. When the bias gets larger, the hard-constrained ambiguity success rates are decreased drastically. In contrast, the soft-constraint maintains a relatively good performance, showing a much greater tolerance for bias.