1. Why Positioning Accuracy Collapses the Moment Satellite Signals Are Blocked
By summarizing the control segment errors, transmission errors, and measurement errors, we can identify four notable characteristics of GPS (GNSS) errors: long-term stability, environment dependence, strong spatial correlation, and strong temporal correlation. When GPS signals are blocked, signal interruption occurs, and in environments with severe reflection, multipath errors increase — so GPS errors are environment-dependent. The long-term stability and environment dependence of GPS errors are exactly the starting point and one of the purposes for building a GPS/inertial integrated navigation system. The long-term stability of GPS errors can help correct the undamped divergence of inertial navigation errors, while the environment dependence of GPS errors can be overcome with the aid of the inertial navigation system.
For GNSS outage navigation, the navigation accuracy and coasting capability of a GPS and strapdown inertial navigation integrated system are greatly affected by the operating environment and dynamics. The operating environment mainly affects GPS performance. For example, tall buildings, tunnels, bridges, and trees can block part or all of the GPS satellite signals, leading to degraded positioning accuracy or complete loss of positioning.
Coasting capability refers to the length of time an inertial navigation system can operate on its own when GPS signals are interrupted — the same principle behind dead reckoning navigation. For a tactical-grade inertial navigation system, because its sensors are relatively low-performing, the time it can operate alone without GPS assistance is limited.
2. Satellite and Inertial Error Characteristics Are Naturally Complementary
GPS generally outputs only position and velocity data, and its velocity is the average velocity between two measurement updates; strapdown inertial navigation can output position, velocity, and attitude data, and its velocity and attitude are instantaneous quantities rather than averages. GPS has a low measurement update rate, typically between 1 Hz and 10 Hz, while strapdown inertial navigation can reach 1 kHz or even higher.
An optimal estimation approach such as Kalman filtering can not only correct the navigation solution of the strapdown inertial navigation system, but also estimate and compensate the inertial sensor errors in real time. Real-time compensation of inertial sensor errors improves the solving accuracy of the inertial navigation system itself, and in turn improves the accuracy of the integrated navigation system — especially the system accuracy when GPS signals are interrupted.
3. Obtaining Millimeter-Level Velocity Observations: Carrier-Phase Time Differencing
The control segment errors and transmission errors of GPS have strong temporal correlation on the order of millimeters per second. Assuming the update rate of GPS pseudorange and carrier phase is 1 Hz, then differencing the pseudorange or carrier-phase observations between two consecutive measurement epochs can reduce the control segment errors and transmission errors to the millimeter level. Especially for carrier-phase observations, since their measurement error is also on the millimeter level, this time-differencing operation between consecutive measurement epochs can reduce the carrier-phase error entirely to the millimeter level.
Using the carrier-phase observations of a single GPS receiver (with an update rate of 1 Hz), and while avoiding integer ambiguity resolution, precise velocity at the mm/s level can be derived in real time. This velocity-solving method proceeds in four steps: (1) perform a differencing operation between consecutive carrier-phase measurements; (2) compensate for the changes in ionospheric error, tropospheric error, satellite clock error, and satellite orbit error; (3) compensate for the changes in satellite Doppler and satellite geometry; and (4) use the least-squares method to solve for velocity. Among these four steps, the most important is the differencing operation in step (1), which first removes the integer ambiguity from the carrier-phase observations and then removes the temporally correlated errors in the carrier-phase measurements.
4. Carrier-Phase Time Differencing Alone Is Not Enough: Slow Position Error Accumulation
Carrier-phase time differencing is a high-precision, high-resolution observation, and it has clear advantages when combined with strapdown inertial navigation in an integrated navigation system: it improves both the accuracy of the integrated navigation solution and the compensation accuracy of inertial sensor errors. However, experimental studies also found that a combined system of carrier-phase time differencing and strapdown inertial navigation suffers from slow position error accumulation. This is caused by the fact that carrier-phase time differencing is an incremental measurement rather than an absolute measurement.
5. The Solution: Introduce Pseudorange and Isolate Noise with a Dual-Rate Kalman Filter
Because the measurement noise of pseudorange is far greater than that of carrier-phase time differencing, during error propagation the high precision of carrier-phase time differencing is completely “drowned out” by the high noise of pseudorange. Therefore, the basic requirement that the combination of carrier-phase time differencing, pseudorange, and strapdown inertial navigation places on the mathematical tool is the ability to isolate the measurement noise of carrier-phase time differencing from that of pseudorange — and an ordinary Kalman filter cannot meet this requirement.
The simplest method is to split the single Kalman filter driven jointly by carrier-phase time differencing and pseudorange into two Kalman filters driven separately by carrier-phase time differencing and pseudorange. Considering that pseudorange only serves to damp position errors, while the real-time correction of position, velocity, and attitude errors and the real-time compensation of inertial sensor errors are achieved through carrier-phase time differencing, the update rate of the filter driven by pseudorange can be far lower than that of the filter driven by carrier-phase time differencing. These two Kalman filters running at different update rates are named the dual-rate Kalman filter.
Because the update rate of the pseudorange-driven Kalman filter is far lower than that of the carrier-phase-time-difference-driven Kalman filter, the influence of pseudorange noise on system accuracy is greatly weakened, giving the system’s error covariance matrix enough time to converge to a small value.
Three Advantages of the Dual-Rate Kalman Filter
Compared with an ordinary Kalman filter, the dual-rate Kalman filter has the following three advantages: (1) it can isolate the noise between observations of different measurement precision; (2) it can fuse non-synchronous observation information; and (3) it splits the computation of one high-dimensional matrix into the computations of two low-dimensional matrices, thus improving computational efficiency.
6. System Scheme: Inertial Solving + High-Rate Filtering + Low-Rate Filtering
The integrated navigation scheme based on the dual-rate Kalman filter consists of three parts: the inertial solving part, the high-rate Kalman filtering part, and the low-rate Kalman filtering part. In the system actually built, the inertial solving period is 0.01 s; the high-rate Kalman filtering period is the same as the GPS signal update period, which is 1 s; and the low-rate Kalman filtering period is the same as the update period of the down-sampled GPS signal, which is 100 s. Unlike a loosely coupled architecture, which treats GNSS position and velocity as independent measurements, this scheme works directly on the raw observations.
The role of the high-rate Kalman filter is to realize the tight combination (tightly coupled INS) of carrier-phase time differencing and strapdown inertial navigation. The position, velocity, and attitude errors it outputs are fed back to the output of the inertial navigation system to correct the inertial navigation solution, and the inertial sensor drift estimates it outputs are fed back to the input of the inertial solving algorithm to compensate the drift errors of the inertial sensors online. The role of the low-rate Kalman filter is to realize the tight combination of pseudorange and strapdown inertial navigation; the position it outputs is fed back to the inertial navigation solution corrected by the high-rate Kalman filter, to damp the accumulation of position error. The low-rate filter only damps the position error of the system, while the correction and compensation of other errors in the system are mainly accomplished by the high-rate Kalman filter.
7. Enabling Online Compensation of Inertial Sensor Random Errors
Inertial sensor error is one of the main error sources of a GPS and strapdown inertial navigation integrated system. According to their characteristics, inertial sensor errors can be divided into systematic errors and random errors. Systematic errors mainly include the scale factor error, the misalignment error of the sensitive axis, and the constant drift of the inertial sensor; these errors can be pre-calibrated before the system runs, using a turntable through multi-position tests and rate tests. Random errors, however, cannot be pre-calibrated, and therefore must be compensated online with the aid of GPS — the general approach covered in MEMS IMU error compensation methods.
According to the testing standards for inertial sensors, the five main random errors that may exist in an inertial sensor are: quantization noise, white noise, bias instability, random walk, and ramp error. Among them, white noise can be directly used as the driving noise of the system and does not need to be modeled; quantization noise can be transformed into equivalent white noise by transforming the inertial navigation error equation, so it also does not need to be modeled; and to compensate colored noise in real time during the Kalman filtering process, the colored noise needs to be modeled.
For colored noise, by incorporating its stochastic differential equation into the inertial navigation error equation, the drift of the inertial sensor caused by it can be estimated during the Kalman filtering process of integrated navigation, and the credibility of this estimate depends on the observability of the inertial sensor drift. Identifying these noise terms in practice is done with Allan variance analysis. For a general navigation environment such as vehicle navigation, swaying motion and linear motion are very common, so the system states — including the inertial sensor drift — are fully observable.
For a 100-second GPS signal interruption, the coasting capability of the dual-rate-Kalman-filter-based integrated navigation scheme is more than doubled compared with the classic integrated navigation scheme based on the tight combination of pseudorange and strapdown inertial navigation. For a 60-second GPS signal interruption, the coasting capability is improved by nearly two times. For a 20-second GPS signal interruption, the coasting capability is improved by nearly four times, and the absolute value of the coasting error drops below 10 m, reaching the positioning accuracy of GPS Standard Positioning Service (SPS).
The main factors affecting the coasting capability of an integrated navigation system are three: the position error and velocity error at the start of coasting, and the calibration error of the inertial sensors. The position error forms the constant bias in the coasting error, the velocity error forms the first-order term in the coasting error, and the inertial sensor calibration error forms the second-order term in the coasting error. In the dual-rate-Kalman-filter-based integrated navigation scheme, because high-precision carrier-phase time differencing is introduced to correct the system errors, high-precision velocity error compensation and inertial sensor calibration can be achieved.
Observing the experimental results in three different test environments leads to the following conclusion: compared with the classic integrated navigation scheme based on the tight combination of pseudorange and strapdown inertial navigation, the dual-rate-Kalman-filter-based integrated navigation scheme improves position accuracy, significantly improves velocity accuracy and attitude accuracy, and can effectively suppress the negative impact of increases and jumps in DOP on integrated navigation accuracy. The significant improvement in velocity and attitude accuracy comes mainly from three reasons: (1) the improvement of the filtering mechanism — the high-rate Kalman filter uses high-precision carrier-phase time differencing as its observation while providing correction of position, velocity, and attitude as well as inertial sensor calibration; (2) the improvement of observation accuracy — carrier-phase time differencing has millimeter-level measurement accuracy, far higher than that of pseudorange; and (3) the improvement of the inertial sensor calibration model — an accurate random error calibration model markedly improves the estimation and calibration accuracy of random errors.
9. Conclusion
The dual-rate-Kalman-filter-based integrated navigation scheme significantly improves the coasting capability of the system. For the short coasting times common in vehicle navigation (within 20 seconds), the coasting error drops below 10 m, reaching the navigation accuracy of GPS Standard Positioning Service (SPS); for longer coasting times (60 to 100 seconds), the coasting capability is at least more than doubled.
[Optional editor’s note — DAISCH perspective, removable]: For engineering teams using a high-precision MEMS IMU and integrated inertial navigation, these conclusions mean that for GNSS outage navigation — in tunnels, underground garages, and urban canyons where satellite signals are degraded or even denied (see also GNSS-denied navigation) — system availability depends not only on the bias and noise specifications of the IMU itself, but also on the choice of observations and the design of the filtering structure. Only by processing the millimeter-level carrier-phase time differencing separately from the absolute pseudorange measurement, and compensating the random errors of the inertial sensors online, can coasting capability and long-term accuracy both be guaranteed.