Nominal parameters describing the setup
In recent years, there has been an increasing interest in remote sensing with bistatic SAR. Among the advantages bistatic SAR imaging offers in comparison to monostatic SAR imaging are that additional information can be exploited (specific bistatic angles can be chosen, and additional information is obtained from the bistatic reflectivity of targets and because of a reduction of di- and polyhedral effects), that SAR imaging along along-track direction will be feasible, that a cost reduction, as well as reduced size, weight, energy consumption can be achieved for passive receive-only systems, and that passive systems have a reduced vulnerability.
Several experiments have been carried out to prove the feasibility of remote sensing with bistatic SAR. While in the first experiments a stationary receiver or transmitter was involved or two airplanes (with almost parallel flight trajectories) have been used, so called hybrid experiments with Germany’s national remote sensing satellite TerraSAR-X as illuminator and with an airborne SAR receiver system have been performed recently, and first processing results have been presented (cf. (Rodriguez-Cassola et al. 2008) or (Ender et al. 2006, Walterscheid et al. 2009) (where a larger bandwidth and a double sliding spotlight mode have been used)).
Among the research challenges in remote sensing with bistatic SAR are the derivation of proper processing algorithms (that yield focused images even in the most general case of arbitrary flight trajectories) and the
Accurate position and attitude knowledge of the involved SAR antennas is required at several steps: It is an important issue in
Furthermore, accurate absolute position and attitude information is required for
Absolute position and attitude information with highest accuracy is (in several applications) required for
Moreover, in SAR interferometry the
The aforementioned position and attitude information can be obtained using inertial navigation or global navigation satellite systems (GNSS). In the following section, some fundamentals regarding these kinds of navigation are introduced, and the data fusion of corresponding measurements is considered. Example data fusion approaches for low cost position and attitude determination are given and analyzed in Section 3.
2. Introduction to GNSS/INS integration
2.1. Inertial navigation, satellite navigation, and its complementary characteristics
In inertial navigation, usually,
Global navigation satellite systems (GNSS) such as the Global Positioning System (GPS) can be used to determine position, velocity, and time of a point on a platform. By measuring the time of arrival (TOA) and by using the transmission time, which can be extracted from the received signal, the propagation time and finally the range, the receiver-satellite distance, can be derived. By trilateration the receiver position can be determined. Typically, four or more simultaneous measurements are required to solve for the 3D receiver position and clock bias. The carrier phase/frequency and the code phase of the received signals are tracked by appropriate phase/frequency and delay lock loops. There is a correlation of the inphase (
GNSS based navigation is non-autonomous. Usually, at least 4 GNSS satellites have to be continuously in view. It is depending on the environment and there is a high vulnerability. On the other hand, one obtains absolute position, velocity and time information which is long-term stable. The output rate is relatively low (e.g., 1 Hz), and regarding the carrier phase measurements, the ambiguity resolution and the cycle slip detection and repair is challenging.
As indicated above, GNSS based navigation and inertial navigation and the corresponding navigation solutions have
2.2. GNSS/INS integration approaches
Raw measurements and derived observations available from inertial navigation and GNSS based navigation, respectively, have been briefly mentioned in the preceding section.
From an information-theoretical point of view, it would be optimal if the raw measurements would be processed in a single centralized filter using, for example, a total state space model for the state space modelling. Especially if the
On the other hand, a decentralized, a distributed, estimation architecture can be considered which exploits the outputs of a GNSS receiver and of an INS. It is called a
Finally, one can distinguish a
Regarding the state space modelling, an
The estimated errors are usually fed back to correct the IMU measurements yielding errors at the output of the INS which do not grow unbounded with time but remain small so that the assumption of a linear system model remains reasonable.
An exemplary GNSS/INS indirect feedback tightly coupled integration architecture is shown in Figure 3.
In general, the development of a proper GNSS/INS integration approach depends on the application. For example, the dynamics of the platform have to be taken into consideration (e.g., orbiting satellite versus unmanned aerial vehicle (UAV)) – not only for a possible modelling of the dynamics but also for a proper derivation and consideration of expected errors (e.g., with respect to multipath), required bandwidth and update rates, possible positioning or initialization methods, etc. Furthermore, among others, the structure of the platform has to be considered, the environment has to be considered, constraints can de derived and incorporated, and the grade of the inertial sensors and the GNSS receivers available has to be taken into consideration.
More details regarding GNSS/INS integration are given in the literature (e.g., (Farrell and Barth 1999).
3. Low-cost GPS/INS integration with multiple GPS antennas
In this section, an example for GNSS/INS integration is given. GNSS/INS integration for position, velocity, and attitude estimation of an antenna mounted on an aircraft will be considered. The focus is on low cost. That is, a low-cost microelectromechanical system (MEMS) based IMU and L1 GPS receivers (that can output pseudorange, delta range, and carrier phase measurements) are supposed to be available.
3.1. Preliminary considerations
The data fusion approaches are formulated with respect to the
The error is defined as usual, that is, as observed or estimated value – true value.
The state vector for the tightly coupled integration is chosen to be
That is, the state vector comprises 17 states.
Accelerometer levelling can be used to determine the initial bank angle (roll) and elevation angle (pitch) of the platform from the accelerometer measurements as follows
The MEMS-based IMU with a typical bias instability between several 100 /h and several 10000 /h can not sense Earth’s rotation. Hence, gyro-compassing for alignment in azimuth can not be performed. Other sensors, such as a magnetometer, can be used to derive heading. Heading information can, for example, also be derived from GPS velocity measurements according to
It can be shown by an observability analysis that only with additional redundant attitude information the states are completely observable (independent of the manoeuvre of the platform, dependent on the number of satellites in view).
This redundant attitude information can be provided by a multi-antenna GNSS receiver system. Here we use a non-dedicated system consisting of the master GPS receiver (including antenna) and two more (independent) GPS receivers (with antennas).
Because of an approximately straight and level flight, that can be assumed for a remote sensing experiment, the size effect related to the accelerometer triad can be neglected. Moreover, because of the low-cost IMU, in addition to other approximations (e.g., no Euler acceleration, that is, Earth’s rotation rate assumed to be constant), the transport rate and Coriolis terms can be neglected in the strapdown processing and in the system model.
In subsequent sections, continuous-time models are provided. The appropriate discrete-time models can be derived as shown in the literature, e.g., in case of a time-variant system the state transition is described as
3.2. Strapdown mechanization
With aforementioned simplifications due to the characteristic of a low-cost IMU, the mechanization can be expressed as
It is performed with the specific force vector
3.3. Error state system model
Because of the low-cost IMU, the uncompensated systematic error Δb
With the aforementioned constraints and simplifications, the
3.4. Observation models
In the tightly coupled integration the measurement vector contains the differences in predicted (based on strapdown solution for position and velocity) and measured pseudorange
In the tightly coupled integration, the states are
The matrix has the dimension (3 +
Sequential processing has to be performed to avoid inversion of a huge matrix.
If the redundant attitude information is obtained from a multi-antenna GPS receiver system we can, instead of first computing the attitude, directly exploit the double-difference carrier phase measurements of the system. This kind of integration of the attitude information from a GPS multiple-antenna system has been proposed in (Hirokawa and Ebinuma 2009). In the measurement vector we have then rather differences of predicted and present double-difference carrier phase measurements (related to antennas A0, A1, and A2) than the difference in attitude. In case of two satellites,
where the symbols in parentheses represent satellites (and where the time index has again been neglected for convenience). The true double-difference carrier phase observation can be expressed as shown by the following example:
and the measured double-difference carrier phase can be modelled as
where besides the integer ambiguity
As soon as there is another satellite in view, the measurement vector is augmented by four more rows (pseudorange and delta range differences to the new satellite, respectively, and new double-difference carrier phase measurements related to the master satellite
In case of
3.5. Further remarks to the integration approach
Derivations of the Kalman filter and its augmentations for nonlinear models as well as the resulting algorithms are not repeated here. The reader is for example referred to (Maybeck 1982), (Simon 2006).
The measurements from the IMU or from the inertial navigation system can be expected with a high data rate (e.g., 100 Hz), which is much higher than the data rate of the measurements derived from GPS. Hence, to reduce the computational burden, to obtain a filter that operates at a relatively low rate, the measurements from the inertial sensors are not incorporated as measurements (cf. Section 3.4). Instead, they are directly used in the strapdown processing to predict position, velocity, and attitude. In addition, only the
As soon as a new measurement vector can be computed, that is, as soon as information from GPS is available, the error state is estimated, and the state vector is
3.6 Simulation setup and results
For the following experiments a
|Velocity (ENU) [m/s]|
|Initial Position (φ, λ, h)|
|Angles (φ, θ, ψ) [°]|
|Start Time (UTC)||October 29, 2006, 00:11:27|
|Update rate||1 Hz|
|Measurements||Depending on integration approach: code pseudorange, delta range, (double-differenced carrier phase)|
|Method||Point positioning, and attitude determination from a non-dedicated multiple antenna system (3 antenna-receiver pairs with baselines of 2 m each)|
|Error modelling||Tropospheric delay is estimated and corrected for;|
The small ionospheric delay is estimated and corrected for; No multipath (calibrated out); Ambiguities resolved and cycle slips repaired (assumed)
|Satellites in view||2–8 (as depicted in Figs. 4–5)|
|Elevation angle||≥ 5°|
|Update rate||100 Hz|
|Error modelling||Gyroscope (Angular rates)|
|Bias stability [°/h]||360|
|Scale factor [ppm]||10000|
|Bias stability [μg]||2400|
|Scale factor [ppm]||10000|
An extended Kalman filter has been used for the GPS/INS integration which is based on the state space modelling provided in Sections 3.1–3.5. An ideal time-synchronization has been assumed, time-delayed measurements have not been taken into consideration. An initialization has been performed as described in Section 3.1.
The attitude estimation results are depicted in Figure 4.
Because of the proper initialization (including an estimation of the gyroscope bias instability and an appropriate correction) of the IMU, the errors are relatively small, that is, after 3 minutes (180 s) the heading error is in case of the stand-alone IMU still smaller than 3 . Without initialization and calibration the error would be larger than 20 . As expected, in case of the stand-alone IMU, the largest error can be observed in heading.
In the tightly coupled GPS/INS integration without redundant attitude information the attitude errors (and sensor biases) are not directly mapped into observation space. However, these error states will also be updated with every new position and velocity measurement. These quantities are related to each other as shown by the system model, and this is reflected in the predicted error covariance matrix (used for computing the Kalman gain). It is therefore no surprise that the attitude estimation results using a tightly coupled GPS/INS integration (represented by the black, dashed curves in Figure 4) are better than using a similar INS alone.
A much more accurate attitude estimation result is obtained using additional redundant attitude information (shown by blue dotted lines in Figure 4). The measurement vector is then given by Eq. (12). The redundant attitude information is here obtained from a multiple antenna GPS receiver system. In case of less than 4 satellites in view, the system can not provide attitude information. Hence, the errors become much larger during such periods.
Finally, if the double-difference carrier phase measurements from the non-dedicated multi-antenna system are used (with the observation matrix given by Eqs. (17) (for the case of two satellites)) the results are comparable in case of four or more satellites in view, but if only two or three satellites are in view still a robust and accurate attitude information is obtained.
In Figure 5, the obtained position, velocity, and attitude errors are shown.
The large stand-alone IMU related position and velocity errors (about 1 km and 15 m/s after 3 minutes, respectively) are not included in the figure. In general, if only two satellites are in view, the position and velocity errors grow remarkably. During the period where only three satellites are in view, the position and velocity errors are bounded (which is not the case in a loosely coupled GPS/INS integration). Again, the incorporation of the double-difference carrier phase measurements into the integration approach yields the best results.
Tightly coupled indirect feedback GNSS/INS integration approaches have been compared in Section 3. Required additional redundant attitude information can be derived from GNSS using three antenna-receiver pairs. In that case, the double-difference carrier phase measurements should be directly incorporated into the integration approach, as has been shown by looking at approaches for low-cost GPS/INS integration and appropriate simulation results. Such approaches and low-cost sensors can for example be used for footprint chasing in bistatic SAR.
Carrier phase measurements have to be exploited to achieve the required attitude estimation accuracy. The detection and repair of cycle slips and the rapid integer ambiguity resolution are challenging. It is easier if shorter baselines are used (because of a reduced search space), however, the longer the baselines the better the expected attitude accuracy.
The lever arm between specific force origin of the IMU and phase center of the main GNSS antenna can be easily incorporated.
Depending on the application, the assumptions have to be proven. For example, delayed measurements have to be considered, and depending on the platform (e.g., UAV) multipath errors that can not be calibrated are an issue, and unconsidered flexure and vibrations can remarkably decrease the accuracy.
Constraints, such as that an airplane is in straight and level flight during a remote sensing experiment, can be easily incorporated into the estimation approach.
A novel GNSS/INS direct integration approach that exploits direction cosine matrix orthogonality constraints and that incorporates a model of the dynamics has been proposed recently (Edwan et al. 2009).
The redundant attitude information required can possibly also be derived from a comparison of a quick-look SAR image with an existing digital map.
To obtain more accurate position and attitude estimates, an IMU of a higher grade and precise point positioning can be exploited.
The importance of accurate and reliable attitude and position information for remote sensing with bistatic SAR has been pointed out in the introduction. In the following section, inertial navigation and navigation based on global navigation satellite systems have been mentioned. Reasons for GNSS/INS integration have been provided, and appropriate data fusion architectures have been briefly introduced and discussed. In the main section, low-cost tightly coupled GPS/INS integration approaches with and without additional redundant attitude information have been presented, and simulation results have been discussed.
Part of the work reported herein has been funded by the German Research Foundation (DFG), grant number KN 876/1-2, which is gratefully acknowledged.