Position-domain integrity risk-based ambiguity validation for the integer bootstrap estimator

Abstract

Integrity monitoring for ambiguity resolution is of significance for utilizing the high-precision carrier phase differential positioning for safety–critical navigational applications. The integer bootstrap estimator can provide an analytical probability density function, which enables the precise evaluation of the integrity risk for ambiguity validation. In order to monitor the effect of unknown ambiguity bias on the integer bootstrap estimator, the position-domain integrity risk of the integer bootstrapped baseline is evaluated under the complete failure modes by using the worst-case protection principle. Furthermore, a partial ambiguity resolution method is developed in order to satisfy the predefined integrity risk requirement. Static and kinematic experiments are carried out to test the proposed method by comparing with the traditional ratio test method and the protection level-based method. The static experimental result has shown that the proposed method can achieve a significant global availability improvement by 51% at most. The kinematic result reveals that the proposed method obtains the best balance between the positioning accuracy and the continuity performance.

Appendix: bounding the ambiguity bias

Since P(H₁) + P(H₀) = 1, thus,

$$P\left( {H_{1,i} } \right) = \prod\limits_{j = 1}^{n} {\left[ {\varPhi \left( {\frac{{1 - 2{\mathbf{l}}_{j}^{T} \Delta {\mathbf{z}}_{i} }}{{2\sigma_{\left. j \right|I} }}} \right) + \varPhi \left( {\frac{{1 + 2{\mathbf{l}}_{j}^{T} \Delta {\mathbf{z}}_{i} }}{{2\sigma_{\left. j \right|I} }}} \right) - 1} \right]} = 1 - P\left( {H_{0} } \right)$$

(11)

Let us define Δb _i,j  = ${\mathbf{l}}_{j}^{T} \Delta {\mathbf{z}}_{i}$ to be the bias under the ith ambiguity resolution failure mode. Because any single failure rate of ambiguity resolution in (11) should be no less than 1 − P(H₀),

$$\varPhi \left( {\frac{{1 - 2\Delta b_{j,i} }}{{2\sigma_{\left. j \right|I} }}} \right) + \varPhi \left( {\frac{{1 + 2\Delta b_{j,i} }}{{2\sigma_{\left. j \right|I} }}} \right) - 1 \ge 1 - P\left( {H_{0} } \right){\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j \in \left[ {1,n} \right]$$

(12)

Because the left-hand terms in (12) are monotonous with Δb_j,i, the threshold value Δb _th can be obtained iteratively by applying the Newton–Raphson method to (12), which yields,

$$\Delta b_{j,i} = \sum\limits_{k = 1}^{n} {{\mathbf{l}}_{j,k}^{T} \Delta {\mathbf{z}}_{i,k} } \in \left[ { - \Delta b_{th} ,\Delta b_{th} } \right]$$

(13)

According to the Cauchy–Schwartz inequality, the bias Δb_j,i in (13) should satisfy

$$n{\mathbf{l}}_{j,\hbox{min} } \Delta {\mathbf{z}}_{\hbox{min} } \le \sum\limits_{k = 1}^{n} {{\mathbf{l}}_{j,k}^{T} \Delta {\mathbf{z}}_{i,k} } \le n{\mathbf{l}}_{j,\hbox{max} } \Delta {\mathbf{z}}_{\hbox{max} }$$

(14)

where l_j,min and l_j,max are the minimum and the maximum element of L _j , Δz_min and Δz_max are the lower and upper bounds for ith failure mode, respectively. According to (14), in order to maintain the satisfaction of (13), the bias value should satisfy

$$- \text{abs} \left[ {\frac{{\Delta b_{th} }}{{nl_{j,\hbox{min} } }}} \right] \le \Delta {\mathbf{z}}_{i} \le \left[ {\frac{{\Delta b_{th} }}{{nl_{j,\hbox{max} } }}} \right]$$

(15)

in which abs[•] indicates the rounding operation and the absolute value sequentially. From (15), the lower and upper bounds of ambiguity bias as well as the searching space of ambiguity bias can be determined. It is noted that the searching space of ambiguity bias can be reduced when the success rate is higher.