Precise point positioning (PPP) has been proved as an effective approach for providing precise positioning services. However, satellite observation outages occur inevitably in the real dynamic application, which leads to partial or complete satellite signal outages and will significantly degrade the accuracy and continuity of PPP positioning. A commonly used approach is to integrate PPP and inertial measurement units (IMU) to enhance the positioning performance. Previous works on this topic are usually based on high sampling rate IMU data. This paper will make an initial assessment of the impacts of different sampling rate IMU data on the tightly coupled integration (TCI) between global positioning system (GPS) and inertial navigation system (INS). After providing the mathematical model of GPS PPP/INS TCI, we investigate the impact of the sampling rate of consumer-grade IMU data under two scenarios (1) original observation scenario; (2) simulation scenario by setting partial satellite signal outage. Results show that (1) the positioning accuracy of the PPP/INS TCI depends on the sampling rate of low-cost IMU data insignificantly when there are enough available satellites; (2) during satellite partial outage periods, the divergent speed of position is visibly affected by IMU sampling rate, and higher positioning accuracy is obtained while setting the IMU sampling rate to 20Hz and 50Hz; (3) the effect of IMU sampling rate on velocity and attitude solutions is also visible, compared with the 200Hz sampling rate, the accuracy of velocity and attitude are improved at 20Hz.
The emerging applications, such as autonomous driving and smart city, present an increasingly
urgent demand for high-precision and high-frequency spatial-temporal data information. The Precise
Point Positioning (PPP) technique [
To overcome the drawbacks of GPS, the Inertial Navigation System (INS) is considered to be an effective complementary method. Because INS can provide a stand-alone solution by using the measurements of carrier motion from the Inertial Measurement Unit (IMU). IMU data will not be affected by the surrounding environments. Such character makes INS can cover the shortage of GPS PPP during challenging environments. Nonetheless, it is difficult to maintain high precision positioning for a long time because of the accumulation of IMU sensors’ errors over time. Because of the complementary characteristics of GPS and INS, the integration of the two systems results in a more reliable, continuous, and accurate higher-rate positioning performance in complex environments.
Generally, the integration of PPP and INS can be classified as Loosely Coupled Integration (LCI)
and Tightly Coupled Integration (TCI). As the LCI is based on the solutions of PPP and INS, its
performance is determined by INS only during the periods of GNSS satellite partial- and
completeoutage. In the TCI models, the pseudo-range and carrier phase observations as range observations are
directly combined with the predicted distance of INS, which still can work during GNSS satellite
partial-outage periods. Wherein, the TCI performs much better than that of LCI, especially under the
GPS signal blocked conditions. Last decades, many works have been done on PPP/INS tight
integration and found it can provide decimeter positioning accuracies even when GNSS signals are
partially or completely interrupted [
To meet the demand of the mass-market on location services, it is necessary to evaluate the performance of the positioning method with low-cost hardware, small data storage, and non-time consuming operation. Therefore, this paper is to investigate the effect of consumer-grade IMU data sampling rate on GPS PPP/INS tight integration in terms of positioning accuracy, velocity accuracy, and attitude accuracy. In this contribution, the uncombined undifferenced PPP is introduced into the tightly coupled integration, and it is evaluated by a set of vehicle-borne data.
In this paper, the PPP/INS tight integration system is applied based on Extend Kalman Filter (EKF)
[
The state function of PPP/INS tight integration can be expressed as
X k k,k1 X k1 μk1, μk1 ~ N0,Qk-1 where k,k1 denotes the state transition matrix for the state parameter vector X k ; μk1 is the state noise vector with the apriori variance Qk-1 .
For the TCI model, the corresponding state parameters used in our work can be written as [
δdwet , δDCBr , δN1, δN2 , δIr ]T where δpInNS , δvInNS , and δ are the correction vectors of position, velocity, and attitude in the navigation (n) frame respectively;
δBa and Bg represent the bias error of accelerometer and gyroscope; δSa and δSg
represent the scale factor error of accelerometer and gyroscope. The remaining part is the state parameters related to PPP; δtr and δtr are the correction of clock offset and drift of receiver; δdwet is the residual of the zenith total delay of the tropospheric wet component; δDCBr is the correction of receiver hardware time delay in terms of Differential Code Bias (DCB); (1) (2) 1 and 2 denote the ambiguity error on frequency L1 and L2; δI r represent the ionospheric delay correction along the signal transmitting path.
The so-called PSI angle model [
vc f c ψ Cbp f b ωice ωicc vc gc
ψ ωicc ψ Cbp ωibb where pc , vc , and ψ indicate the position, velocity, and attitude error in the computing frame (c-frame) respectively; ωecc is the angular rate of the c-frame relative to the earth frame (e-frame) projected in c-frame; ωice is the rotation angular rate of e-frame with respect to the inertial frame (i-frame) projected in the c-frame; gc represents the gravity error in c-frame; f c is the specific force from accelerometers; Cbp represents the attitude direction cosine matrix for transforming from body frame (b-frame) to platform frame (p-frame), and “ ”denotes the cross-product; ωibb and f b represent the sensor errors of accelerometers and gyroscopes.
In addition, the first-order Gauss-Markov process [
δS δS / TS ωS where δB and δS represent the bias and scale factor, TB and TS are the correlation time of IMU sensor errors; ωB and ωS are the driving white noise of bias and scale factor errors respectively.
For the parameters related to GNSS, the random walk process is adopted to represent the dynamic behavior of the receiver clock offset/drift, the residual of zenith troposphere delay in the wet component, the receiver hardware time delays, and the ionospheric delays, and the random constant model is used to describe the changes of ambiguities. Then, the state transition matrix of the TCI can be achieved finally. 2.2.
(3) (4) (6) The measurement model for the TCI can be written as
Zk Hk X k k ,k ~ N0, Rk where Zk and Hk denote the innovation vector and the design matrix at epoch k for state parameter vector (5) X k ; k stands for the observation noise vector with the covariance matrix Rk .
The innovation vector is formed by making differential operations between GPS observations and INS predicted values
Pj Pj,INS Pj ps pINS pINS,l Pj ZTCI,k Lj Lj,INS Lj ps pINS pINS,l Lj
D vs vINS vINS,l Dj
Dj Dj,INS j
where P, L, and D indicate the raw observations of pseudo-range, carrier-phase, and Doppler at
frequency j, respectively; ‘INS’ represents the INS predicted values; ps and vs are the satellite
position and velocity calculated by GPS precise orbit/clock products; pINS and vINS are the position and
velocity at the IMU center predicted by INS mechanization; △ P, △ L, and △ D refer to the sum of
corrections of pseudo-range, carrier-phase, and Doppler, respectively; pINS ,l and vINS,l are the
leverarm between the measuring centers of GNSS receiver and that of IMU, which can be written as [
Then, we can get the designed coefficient matrix Hk can be obtained by making an error perturbation operation on the innovation vector in Eq. (6) around the initial state parameters provided by INS mechanization. 2.3.
According to the above mathematical models, the PPP/INS tightly integration solution can be received by adopting EKF. Figure 1 shows the implementation of TCI. After the initialization for EKF, the INS mechanization worked by using the compensated IMU outputs and provided INS-updated position, velocity, and attitude. Then, the time update phase is run based on the state function to achieve the predicted state parameter and the corresponding variance. Afterward, time synchronization between GPS observations and IMU data at the current IMU epoch is adopted to determine whether the Kalman measurement update working or going to the next IMU epoch. Finally, the navigation solutions from INS are then corrected with the estimated navigation error parameters by a closed-loop feedback process.
To evaluate the performance of PPP/INS tightly integration as well as the impacts of IMU sampling rate on the TCI results, a set of vehicle-borne data collected around Wuhan China is processed. Figure 2 shows the detailed trajectory of the experiment. The test is arranged in an urban environment with a maximum speed of 10 m/s. A Trimble NetR9 receiver, a high-grade IMU POS1100 (gyro bias instability of 8 °/h ), a consumer-grade IMU ICM-20602 (gyro bias instability of 50 °/h ) were equipped. The sampling rates of the GPS observations and IMU measurements were 1Hz and 200Hz, respectively. In addition, the basic information about the IMU sensor is listed in Table 1.
In this paper, the raw GPS observations and the data from four IMU were processed in PPP/INS TCI mode. Meanwhile, the raw 200Hz IMU data is re-sampled to 100Hz, 50Hz, and 20Hz. To further evaluate the impacts of IMU sampling rate on the accuracy of positioning, velocity, and attitude determination while suffering GNSS signals outages, two groups of 30-second partial satellite signal outages were simulated. To assess the performance of the above schemes, the smoothed solutions from RTK/POS1100 tight integration were employed as reference values. The number of available satellites and the PDOP value for GPS are shown in Figure 3. The number of available satellites is almost about 6~7 and it drops to 4 under some GPS difficult environments. Velocity random walk ( / / ℎ)
The position differences of PPP/INS tightly integration calculated by four types of IMU data (200Hz, 100Hz, 50Hz, and 20hz) compared to the reference values are shown in Figure 4. For comparison, the PPP position solutions based on uncombined undifferenced GPS data were also provided. The position accuracy in terms of Root Mean Square (RMS) values is listed in Table 2. Significantly, the results of PPP/INS TCI are more accurate than that of PPP. Using IMU data at a 200 Hz sampling rate, the TCI position RMSs are 7.59 cm, 19.53 cm, and 55.51 cm in east, north, and up directions, with the improvements percentages of 26.2%, 51.1%, and 27.7%, respectively. While using IMU measurements at different sampling rates, this brings 0.63 cm, 2.34 cm, and 3.79 cm position differences in terms of RMS in the north, east, and vertical directions among the four IMU data based on PPP/INS TCI. According to Table 2, there are 0.8%, 5.8%, and 3.5% three-dimensional position improvements for the PPP/INS TCI based on 100Hz, 50 Hz, and 20Hz IMU data, respectively compared with those based on 200Hz IMU data. However, there no significant relationship exists between position accuracy and the reducing the IMU data sampling rate.
Besides, we also analyze the accuracy of velocity and attitude while using IMU data at different sampling rates. The time series of velocity and attitude differences with respect to the references are shown in Figure 5 and Figure 6, respectively. The corresponding RMSs are shown in Table 2. Significantly, compared with the results using IMU data at a 200Hz sampling rate, the accuracy of velocity and attitude are both degraded by using IMU data at 100Hz and 50Hz. There is an improvement while using 20Hz IMU data. According to the statistics, the velocity and attitude accuracies at 50Hz are the lowest. Wherein, the velocity RMSs are about 6.12 cm/s, 5.62 cm/s, and 2.02 cm/s in east, north, and up components, and that of attitude are about 0.602°, 0.805°, and 3.911° for roll, pitch, and heading, respectively. In addition, the velocity RMSs based on 20Hz IMU data are 2.10 cm/s, 1.85 cm/s, and 2.08 cm/s, and that of attitude are 0.103°, 0.129°, and 1.021° for roll, pitch, and heading, respectively.
To further evaluate the performance of PPP/INS tightly coupled integration in complex conditions, we simulate the GNSS raw observations by setting two 30-second partial satellite signal outages at 532768 s and 533798 s. During each partial outage period, only three satellites are available. Simultaneously, IMU data with different sampling rates (200Hz, 100Hz, 50Hz, and 20hz) are processed to furtherly investigate the impacts of IMU sampling rates on TCI during GNSS challenging environments. Figure 7 illustrates the position solutions of PPP and TCI using four types of IMU data, and the corresponding RMSs are listed in Table 3. Visibly, the PPP/INS tightly coupled integration can obtain continuous positioning results even if there are not enough GNSS observations for PPP (during 532768 s~532798 s and 533798 s~533828 s). According to the statistics in Table 3, the position accuracy enhancements are 38.9%, 56.8%, and 35.4% in the east, north, and up while using IMU data at 200Hz sampling rate (10.35 cm, 20.26 cm, and 51.60 cm) compared to that of PPP (16.94 cm4.88 cm, and 80.82 cm). Similar position enhancements compared with the solutions of PPP can also be found while using the IMU data at a different rate. Comparing the position RMSs calculated by using different rate IMU data, the RMS differences are not remarkable with the maximum values of 0.09 cm, 0.16 cm, and 4.98 cm. Similarly, there is also no strong relationship between positioning accuracy and the IMU data rate. However, obvious position convergence can be found during the GPS outages.
Figure 8 and Figure 9 show the solutions of velocity and attitude calculated by PPP/INS TCI using the satellite partial outage simulation GPS data and different sampling rate IMU data. The corresponding RMSs are listed in Table 3. Results indicate that there are no significant impacts of the 30-second GPS partial outages on the velocity and attitude solutions can be found. It is mainly due to the high-accuracy character of INS during a short time. Besides, the influences of different IMU sampling rates on the velocity and attitude accuracy are much more visible than that of position.
Position drifts calculated by four types of IMU data during the outages are given in Figure 10, the maximum position errors are shown in Figure 11. Significantly, it can be seen that the position drifts of PPP/INS TCI diverge along with outage time. The position RMS while using 200Hz IMU data is degraded to 143.6 cm, 37.30 cm, and 141.3cm in east, north, and vertical directions while partial outage time lasts 30 s. However, the position drifts are restrained while using lower sampling rates. Among the results calculated from different rate IMU data, the position drifts after 30 s outage are improved to 34.20 cm, 20.03 cm, and 53.67 cm in east, north, and up directions while using IMU data at 100Hz. The maximum position drifts for 50Hz IMU data are 80.30 cm, 21.170 cm, and 115.30 cm and those for 20Hz are 55.02 cm, 25.09 cm, and 60.99 cm, respectively. It shows significantly that IMU sampling rate affects position drifts of PPP/INS TCI when there are satellite partial outages. In general, for consumer-grade IMU ICM-20602, the positioning results will diverge rapidly over time. Nevertheless, it is equivalent to reducing IMU sensor cumulative error by reducing INS mechanization frequency while downsampling IMU data rate. In addition, although there are position improvements when using low sampling rate (100Hz, 50Hz, and 20Hz) IMU data, such improvements do not present a significant positive correlation with IMU data rate. What is also evident is that the position drifts of PPP/INS TCI during the GNSS partial outage show a better positioning accuracy in the north component than in the other two directions.
In this paper, the impacts of IMU data interval on the performance of the tightly coupled integration between PPP and INS are evaluated by a set of vehicle-borne data. Moreover, we provide the GPS partial satellite outages simulation to furtherly evaluate the performance of PPP/INS tightly coupled integration in unexpected urban conditions. According to the results, the positioning accuracy of PPP/INS TCI is significantly improved compared with that of the stand-alone PPP approach when using both original data and simulation data. In addition, the effect of the IMU sampling rate on position accuracy is inconspicuous when the satellite signals are available for PPP. It is due to that the absolute positioning accuracy of PPP/INS TCI mainly depends on PPP. However, the position error drifts along with the increasing outage time, and such divergent speed is visibly affected by the IMU sampling rate.
Many thanks to GNSS Research Center, Wuhan University for providing GPS/INS vehicle-borne data. This study is funded by the National Key Research and Development Program of China (Grant No. 2020YFB0505802).