360
5 X-ray Pulsar-Based Navigation: Theories and Experiments
+
2μ s
c 2 ln
n i · ˜
r SC,k + n i · b +
˜
r SC,k + b
n i · b + b
;
n i =
cos α i cos λ i cos α i sin λ i sin α i
T ;
r SC,k = ˜
r SC,k + δr SC,k ;
δr SC,k is the correction of approximate position vector of the spacecraft at epoch
t k ; δt k is the onboard clock bias under the timescale TCB; η i is the measurement
noise of the pulse arrival-time; λ i and α i are, respectively, the right ascension and
declination of pulsar i in the BCRS; r SC,k and ˜
r SC,k are, respectively, the real and
approximate position vectors of the spacecraft relative to the SSB at epoch t k .
It should be pointed out that only for the same pulse signal can the measured pulse
arrival-time be compared with the predicted one, and then the difference of arrivaltime obtained. If the distance difference between the approximate and real positions of
the spacecraft is within a pulse cycle wavelength in the time transformation process,
there will not be the problem of integer ambiguity, and thus it is ensured that the
arrival-time of the same pulse is comparable. In the application of autonomous navigation of the spacecraft using the X-ray pulsars, the ground-based TT&C stations
or other navigation ways can be used to assist the determination of system state
parameters in the initial stage of navigation system. Taking the pulse arrival-time
as the basic observables, the Kalman filter is designed to estimate the system state
parameters. The high-precision orbit model is used to propagate for short-term orbit
prediction, and hence the spacecraft’s approximate position accuracy can be reached
1 km, while the pulse wavelength of millisecond pulsar is more than several hundred
kilometers, so there is generally not the problem of integer ambiguity. In addition,
the pulse phase is measurable within a pulse cycle. According to the relationship
between the pulse phases and the ranging observables, the phase measurement equation can be established. In this way, the multiple phase measurements can be obtained
within a pulse cycle, besides the phase measurement at the peak point (the measurement of arrival-time), so as to increase the redundant observations and improve the
parameter-estimating accuracy.
5.6.5.4 Measurement Equation for Relative Navigation
The relative navigation is the process of determining the navigation parameters of
the spacecraft relative to a reference object frame, which is a linear combination of
range or phase measurements, also known as differencing measurement. Similar to
the differencing measurement of the GNSS, there are generally three types of differencing linear-combinations: single difference, double difference and triple difference. The single difference, for the same pulsar, refers to the difference between
the phase measured at the spacecraft and that predicted at the position of defining
the pulsar timing model. The double difference refers to the difference between
two single differences obtained by the spacecraft observing two pulsars at the same
5 X-ray Pulsar-Based Navigation: Theories and Experiments
+
2μ s
c 2 ln
n i · ˜
r SC,k + n i · b +
˜
r SC,k + b
n i · b + b
;
n i =
cos α i cos λ i cos α i sin λ i sin α i
T ;
r SC,k = ˜
r SC,k + δr SC,k ;
δr SC,k is the correction of approximate position vector of the spacecraft at epoch
t k ; δt k is the onboard clock bias under the timescale TCB; η i is the measurement
noise of the pulse arrival-time; λ i and α i are, respectively, the right ascension and
declination of pulsar i in the BCRS; r SC,k and ˜
r SC,k are, respectively, the real and
approximate position vectors of the spacecraft relative to the SSB at epoch t k .
It should be pointed out that only for the same pulse signal can the measured pulse
arrival-time be compared with the predicted one, and then the difference of arrivaltime obtained. If the distance difference between the approximate and real positions of
the spacecraft is within a pulse cycle wavelength in the time transformation process,
there will not be the problem of integer ambiguity, and thus it is ensured that the
arrival-time of the same pulse is comparable. In the application of autonomous navigation of the spacecraft using the X-ray pulsars, the ground-based TT&C stations
or other navigation ways can be used to assist the determination of system state
parameters in the initial stage of navigation system. Taking the pulse arrival-time
as the basic observables, the Kalman filter is designed to estimate the system state
parameters. The high-precision orbit model is used to propagate for short-term orbit
prediction, and hence the spacecraft’s approximate position accuracy can be reached
1 km, while the pulse wavelength of millisecond pulsar is more than several hundred
kilometers, so there is generally not the problem of integer ambiguity. In addition,
the pulse phase is measurable within a pulse cycle. According to the relationship
between the pulse phases and the ranging observables, the phase measurement equation can be established. In this way, the multiple phase measurements can be obtained
within a pulse cycle, besides the phase measurement at the peak point (the measurement of arrival-time), so as to increase the redundant observations and improve the
parameter-estimating accuracy.
5.6.5.4 Measurement Equation for Relative Navigation
The relative navigation is the process of determining the navigation parameters of
the spacecraft relative to a reference object frame, which is a linear combination of
range or phase measurements, also known as differencing measurement. Similar to
the differencing measurement of the GNSS, there are generally three types of differencing linear-combinations: single difference, double difference and triple difference. The single difference, for the same pulsar, refers to the difference between
the phase measured at the spacecraft and that predicted at the position of defining
the pulsar timing model. The double difference refers to the difference between
two single differences obtained by the spacecraft observing two pulsars at the same