Method and apparatus of single epoch position bound
A method for determining a protection level of a position estimate using a single epoch of GNSS measurements, the method includes: specifying a prior probability density P(x) of a state x; specifying a system model h(x) that relates the state x to observables z of the measurements; quantifying quality metrics q associated with the measurements; specifying a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ using a set of experimental data; and defining a posterior probability density P(x|z, q, θ); estimating the state x; and computing the protection level by integrating the posterior probability density P(x|z, q, θ) over the state x.
1. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein defining the posterior probability density is based on the formula:
P
(
x
|
z
,
q
,
θ
)
∝
P
(
x
)
∏
i
f
(
z
i
-
h
i
(
x
)
|
θ
,
q
i
)
where i is a natural number.
2. The method of claim 1 , wherein the system model is specified as:
z=h ( x )+ r;
where r is the residual error.
3. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein quantifying the quality metrics q associated with the measurements comprises excluding one or more outliers of the measurements using a random sample consensus (RANSAC) technique.
4. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein the non-Gaussian residual error probability density model is a student-t distribution function, and specifying the residual error probability density model ƒ(r|θ, q) further comprises:
identifying a model for a pseudorange using the student-t distribution function as:
f
pr
(
r
)
=
Γ
(
v
+
1
2
)
v
π
σ
2
Γ
(
v
2
)
(
1
+
r
2
σ
2
v
)
-
v
+
1
2
where r is the residual error, ν is a degrees of freedom parameter, σ is a scaling parameter, and Γ is the gamma function.
5. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein the non-Gaussian residual error probability density model is a student-t distribution function, and specifying the residual error probability density model ƒ(r|θ, q) further comprises:
identifying a model for a carrier phase using the student-t distribution function as:
f
p
h
a
s
e
(
r
)
=
w
+
(
1
-
w
)
Γ
(
v
+
1
2
)
v
π
σ
2
Γ
(
v
2
)
(
1
+
r
2
σ
2
v
)
-
v
+
1
2
where r is the residual error, ν is a degrees of freedom parameter, σ is a scaling parameter, w is a weight, and Γ is the gamma function.
6. The method of claim 4 , wherein fitting the model parameters θ comprises:
defining the degrees of freedom parameter and the scaling parameter using unconstrained student-t parameters;
defining the unconstrained student-t parameters as a function of the quality metrics; and
finding maximum likelihood values for the unconstrained student-t parameters.
7. The method of claim 5 , wherein fitting the model parameters θ comprises:
defining the degrees of freedom parameter and the scaling parameter and the uniform weight parameter using unconstrained parameters;
defining the unconstrained parameters as a function of the quality metrics; and
finding maximum likelihood values for the unconstrained parameters.
8. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein integrating the posterior probability density over the state comprises:
applying Markov chain Monte Carlo (MCMC) numerical integration using a plurality of interacting chains with modified density functions.
9. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x; and
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein integrating the posterior probability density over the state x comprises:
performing a first round of sampling by drawing a first set of samples from an unbiased distribution of the posterior probability density and estimating a first location of a quantile on the first set of samples;
performing a second round of sampling by drawing a second set of samples from a first constrained distribution of the posterior probability density that is determined based on the first location of the quantile;
estimating a second location of the quantile on the second set of samples, wherein the second location of the quantile is used to set a second constrained distribution for a third round of sampling;
repeating the sampling using a constrained distribution set in previous sampling until nth round of sampling is completed, where n is a natural number greater than 2;
determining whether samples of the n round samplings are adequate for computing the protection level; and
combining the samples of n rounds sampling with appropriate weights when the samples of n rounds are adequate for computing the protection level.
10. A computer-implemented method for global navigation satellite system (GNSS) measurements, the method comprising:
specifying, in advance, a prior probability density P(x) of a state x;
specifying, in advance, a system model h(x) that relates the state x to observables z of the measurements;
quantifying, during the measurements, quality metrics q associated with the measurements;
specifying, in advance, a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ determined off-line a priori, where r is a residual error;
defining, during the measurements, a posterior probability density P(x|z, q, θ);
estimating, during the measurements, the state x;
computing, during the measurements, a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x; and
performing bound propagation using phase change error distribution or data obtained by one or more sensors.
11. The method of claim 10 , wherein performing the bound propagation further comprises:
modelling an error distribution on a phase change using a non-Gaussian distribution function and evaluating bounds on a position change numerically using a Markov chain Monte Carlo technique; or
modelling the error distribution on the position change using a Gaussian over-bound method.
12. The method of claim 1 , wherein the state x is defined as:
x
=
[
p
c
Δ
d
t
r
Δ
k
P
,
r
Δ
k
L
,
r
Δ
T
z
e
n
i
t
h
]
where p is the position of a rover, c is the speed of light in vacuum, Δdt r is a difference in receiver clock bias between the rover and a base station, Δk P,r is a difference in receiver code instrumental delays on pseudorange, Δk L,r is a difference in receiver code instrumental delays on carrier phase, and ΔT zenith is a difference in zenith tropospheric delays.
13. An apparatus for global navigation satellite system (GNSS) measurements, the apparatus comprising:
a receiver configured to receive a GNSS signal and process the received signal for the measurements; and
a processor configured to:
specify a prior probability density P(x) of a state x;
specify a system model h(x) that relates the state x to observables z of the measurements;
quantify quality metrics q associated with the measurements;
specify a non-Gaussian residual error probability density model ƒ(r|θ, q) and fitting model parameters θ using a set of experimental data, where r is the residual error;
define a posterior probability density P(x|z, q, θ); and
estimate the state x; and
compute a protection level of a position estimate by integrating the posterior probability density P(x|z, q, θ) over the state x,
wherein defining the posterior probability density is based on the formula:
P
(
x
|
z
,
q
,
θ
)
∝
P
(
x
)
∏
i
f
(
z
i
-
h
i
(
x
)
|
θ
,
q
i
)
where i is a natural number.
14. The apparatus of claim 13 , wherein the processor is further configured to:
communicate with an external sensor to obtain position change data tracked by the sensor to determine a bound propagation.