**1. Introduction**

Most navigation systems are aimed at minimizing slow-varying position and velocity errors of the object. These methods aim to keep the errors at a low level for as long as possible. However, there are certain applications that impose specific demands on navigation systems, e.g., synthetic aperture radar (SAR) terrain imagery.

Synthetic aperture radars allow for high-resolution terrain imaging with similar quality to optical methods. In contrast, however, this technique is independent of weather and lighting conditions. Currently, unmanned aerial vehicles (UAV) are commonly used as carriers of SAR systems. In side-looking aerial radar (SLAR), the direction of flight is called *azimuth* while the direction of observation is called *range* [1].

The advantages of SAR systems are accompanied by high requirements and limitations. To ge<sup>t</sup> focused and geometrically correct images, the UAV must move in a strictly defined manner, most often with uniform rectilinear motion, which ensures proper conditions for receiving and processing echo signals [2–9]. However, in real-world scenarios it is impossible to fulfill these requirements.

In [10], Fornaro distinguished three types of SAR-carrier trajectories: nominal, real, and measured. Due to measurement errors, the measured trajectory does not coincide with the real one. The discrepancy between the real and the nominal trajectories result from atmospheric disturbances and autopilot controls [3]. According to [6,11], the unfavorable effects of the atmosphere on the UAV lead to the range (radar to object) distance instabilities, nonuniform UAV motion in the azimuth direction, and changes

in the UAV's attitude. Groundspeed variations can be compensated by adjusting the radar pulse repetition frequency (PRF) so the distance traveled by a UAV in one pulse period is constant [12]. The second method of groundspeed instability compensation consists in resampling the received echo signals [2,3]. The UAV attitude variation can be compensated by the radar antenna stabilization system [13]. As a result, the flightpath curvature has the most negative impact on the quality of radar images. According to Kirk [4] and Fornaro [3], UAV movement instabilities can be divided into lowand high-frequency errors. Low-frequency errors (with a period longer than a duration of the synthetic aperture) lead to a shift in the center frequency of the echo signal, which results in geometric distortions of the image. High-frequency errors (with a period shorter than the synthetic aperture) increase the amplitude of the side lobes, which blurs the image. Therefore, the compensation of these errors has the greatest impact on the quality of radar images [9]. As a result, in SAR systems it is necessary to compensate the influence of UAV motion instabilities on the received echo signal phase history. These procedures are called motion compensation (MOCO) [14]. MOCO algorithms can be divided into procedures using navigation equipment, as well as procedures based on the analysis of the echo signal (autofocus) [10,14].

In navigation procedures, the echo signal phase correction is performed based on the measured and calculated discrepancy between the nominal and actual trajectories. However, the navigation system is a source of additional errors, which could decrease the quality of SAR images. The analysis of the inertial measurement unit (IMU) and INS errors in the imaging process is presented in [15]. The paper [16] presents the SAR MOCO procedure using INS navigation corrections, which sugges<sup>t</sup> that in the case of the limited duration of the measurement session (approx. 30 s), it is possible to significantly improve the image quality.

In the case of the navigation systems dedicated to cooperation with SAR radars, it is necessary to consider their accuracy in the short and long term. The INS system ensures high short-term accuracy, but in the long-term perspective an increase of the positioning errors is the main disadvantage of the inertial navigation [17]. This drawback can be compensated for using additional sensors (e.g., Doppler radar, GPS receiver), ensuring a high long-term accuracy [18]. The disadvantage of this approach, however, is high-frequency measurement errors, which transfer to the MOCO procedure (through a data fusion algorithm), causing errors in the initial phase of the echo signals.

In the article [19], Fuxiang and Zheng noticed that navigation systems proposed in the MOCO literature, and in particular the methods of INS and GPS data integration, are predominantly focused on reducing the long-term errors, which makes them more suitable for navigating than for SAR MOCO. For example, in [20] Gong and Fang presented the navigation system used in MOCO, consisting of INS and GPS with real time kinematic (RTK). They presented four strategies for integrating the measurement data: a Kalman filter, an extended Kalman filter (EKF), a combination of the unscented Kalman filter (UKF) with an EKF, and a model predictive filter (MPF) with an EKF. The authors focused on long-term error minimalization and did not consider the impact of the short-term errors on the radar images. Focusing on the navigation system, and disregarding its influence on the SAR subsystem, is a common practice in the literature [19,21,22].

In the work [19], Fuxiang and Zheng presented a proposal to solve the high-frequency navigation system errors caused by GPS. According to the authors, the period of the Kalman filter correction phase (the period of GPS data used in the filter) should be longer than the duration of a synthetic aperture. In such a system, the INS errors grow during the synthetic aperture, which, however, is much less disadvantageous than random changes introduced by the GPS. Unfortunately, this limits the duration of the synthetic aperture, which may be especially problematic in the case of UAVs moving with low speeds in the azimuthal direction. A short synthetic aperture decreases the spatial resolution of the radar image.

According to Kennedy's work [15], the INS used in a MOCO algorithm should not be reinitialized during the synthetic aperture. A proposed solution consists of the main INS (aircraft navigation system) and the auxiliary INS (IMU mounted on the radar antenna). A correction of the auxiliary INS

(reinitialization) is performed after the synthesis of the aperture has been completed. This method is therefore analogous to that proposed by Fuxiang and Zheng [18]; they di ffer only in the source of the correction information (main INS or GPS receiver).

In [21], Fang and Gong presented a navigation system divided into two INS blocks using a common IMU. Errors of the main INS are estimated by a Predictive Iterated Kalman Filter (PIKF) by comparing results from the INS and GPS. As proposed by the authors, the main INS is only used during the approach to the radar scanning area. At the start of the measurement session, the output from the main INS is used to initialize the secondary INS (not corrected by the GPS), whose output is used in the SAR MOCO algorithm. The proposed mechanism has two goals. The first one is to reduce the operating time (and errors) of the secondary INS. The second goal is to reduce the influence of the GPS errors on the MOCO by isolating two INS blocks. The authors emphasize that the radar measurement session should last no longer than 15 s to keep INS errors at an acceptable level.

The second group of MOCO methods, called autofocus, consists in the analysis of the echo signals received by the radar sensor [14,23–33]. A grea<sup>t</sup> variety of autofocus methods exists due to their individual limitations, e.g., the Phase Gradient Autofocus (PGA) needs the presence of highly reflective objects in the imaged area [24], while in the Map-Drift method the area should be diversified (presence of edges, shadows, etc.) [32]. A majority of autofocus methods are iterative, which extends the time needed to obtain a result (image) [23]. For comparison, in the navigational MOCO procedures the image synthesis is performed once, and the corrections are computed independently in a dedicated subsystem. As a result, in navigational MOCO it is possible to reduce the computational e ffort (no iterations) and time consumption.

In summary, there are currently several navigational MOCO procedures using data from INS or INS/GPS systems. However, in many cases the navigation system is not optimized for the MOCO algorithm—these methods focus on the long-term accuracy of the system, ignoring the adverse impact of fast-changing errors contributed by the GPS receiver. In papers dealing with this problem, the authors have proposed methods limiting the duration of the synthetic aperture (shorter than the GPS update period) or limiting the duration of the measurement session (INS-only systems).

Based on this analysis, the authors propose a multi-instance INS (MINS) system, combining the advantages of the classic INS and INS/GPS systems. This paper is related to the works presented in [16,34–36]. An application of pure inertial navigation for SAR MOCO is presented in [16] and using INS/GPS for MOCO is described in [34]. A method of calculating position deviations from a theoretical, nominally rectilinear trajectory for a UAV-based SAR is explained in [36]. The MOCO method presented in this paper, however, is new and has not been presented before, both by the authors of this paper nor by other authors. The results presented in the further parts of this paper, based on real measurements obtained during UAV flights, show that the proposed MINS system is in many aspects superior with respect to the INS and the INS/GPS systems used for SAR MOCO applications.

The layout of the further parts of this paper is as follows: the idea of a multi-instance inertial navigation system is explained in Section 2, selected results of testing the algorithm with the use of real navigation data are presented in Section 3, whereas in Section 4 the influence of the MINS-based MOCO on SAR images is compared with the results obtained with other MOCO methods. Conclusions are provided in Section 5.

#### **2. Materials and Methods**

The proposed MOCO method, using the MINS system, is based on a combination of INS and INS/GPS systems described in [16,34,36]. The results presented in the mentioned papers show that each proposed solution improves the quality of SAR images, but in both cases the improvement is only partial and complementary. The INS system improves the image contrast (IC), entropy (E), spatial resolution (SR), peak to sidelobe ratio (PSLR), and integrated sidelobe ratio (ISLR), while the INS/GPS system better reduces the image geometric distortions.

The purpose of INS instances proposed in MINS is to keep INS errors limited in such a way that the corrections calculated by a navigation correction algorithm (NCA) [36] do not disturb the initial phases of radar echo signals in a single synthetic aperture. Therefore, in the proposed system, the INS error corrections are not performed during an aperture synthesis. During any single synthetic aperture, only the uncorrected INS system is used as a data source for an NCA, and a periodic INS correction, using the INS/GPS system, is conducted in a special manner that takes into account the specificity of the aperture synthesis algorithm.

The proposed method assumes a parallel work of the MINS and INS/GPS systems. A diagram illustrating the concept of such a system is shown in Figure 1.

**Figure 1.** Data flow in multi-instance inertial navigation system (MINS)-based motion correction (MOCO).

The INS/GPS subsystem [34] determines the UAV's position, velocity, and attitude. Its output is then used to initialize the *i-th* INS instance, *INSi*, in the MINS system. The integrated system also provides the position reference used to calculate MINS errors. Data obtained from the MINS are then used by the NCA to determine the navigation corrections for the SAR system. The INS/GPS system is initialized at the system launch and works until its power is <sup>o</sup>ff.

The detailed algorithm of MINS is presented in Figure 2. It has two main branches presented in Figure 2 using red and blue colored boxes and lines. The red branch is related to the overlapping INS instances, i.e., a situation when two INS instances work in parallel. The blue branch is executed when an INS overlap is finished or impossible to create and is related to a situation when only a single INS instance is working.

The SAR system synthesizes the image line for the central measurement within the synthetic aperture of length *MSA*, where *MSA* is the number of soundings that make up the synthetic aperture. Therefore, the IMU and radar data have to be buffered and the navigation algorithm's output is delayed by the duration of the aperture (usually less than 1 s). The switching of INS instances can be performed if the buffer contains enough IMU data to calculate navigation corrections for the new synthetic aperture. During the MINS system operation, the IMU measurements are used both by the INS/GPS and an *i–th* instance *INSi*. Both systems calculate the position and velocity of the UAV's center of gravity (the origin of the body frame, *b-frame*). The results from the *INSi* are then compared with the current INS/GPS estimates.

**Figure 2.** MINS algorithm.

When *INSi* exceeds a position error threshold and data buffer contains enough IMU data, the instance number *i* is increased and the next INS instance is started (the red part of the algorithm becomes active). If there is not enough data in the buffer (e.g., due to the end of the measurement session), then no new INS instance is created and the current instance runs until all the buffered data have been processed. A new INS instance is initialized using the INS/GPS data determined

for the moment when the position error threshold was exceeded. The *INSi* position error, denoted as ||Δ**r***nb*,*INS*,*INS*/*GPS*||, is a norm of the vector describing the position difference of the *b-frame* origin expressed in the *n-frame* navigation system (North East Down, NED), determined by the *i-th* INS instance *INSi* (**r***nb*,*INS*) and the INS/GPS system (**r***nb*,*INS*/*GPS*):

$$\|\|\mathbf{\bar{\mathbf{r}}}\_{b,\text{INS},\text{INS},\text{CPS}}^{n}\|\| = \|\|\mathbf{\bar{\mathbf{r}}}\_{b,\text{INS}}^{n} - \mathbf{\bar{\mathbf{r}}}\_{b,\text{INS},\text{CPS}}^{n}\|\| = \left\|\begin{bmatrix} n\\c\\d\\d \end{bmatrix}\_{b,\text{INS}} - \begin{bmatrix} n\\c\\d\\d \end{bmatrix}\_{b,\text{INS},\text{CPS}}\right\|\tag{1}$$

where *n*, *e*, *d* are position coordinates expressed in the *n-frame*. After exceeding the position error threshold ||Δ**r***ntr*||, i.e., when:

$$\left\| \Delta \mathbf{r}\_{b,INS,INS,GPS}^{n} \right\| \geq \left\| \Delta \mathbf{r}\_{tr}^{n} \right\|\tag{2}$$

the instances *INSi*−<sup>1</sup> and *INSi* run in parallel, as shown in Figure 3. The purpose of the simultaneous operation of both instances is to enable their switching in a way that does not occur within a single synthetic aperture, which could lead to an abrupt change of the calculated position. Switching the INS instance (the source for the NCA) is performed after the current aperture is synthesized and before moving to the next aperture, which ensures smooth navigation data used in each of them.

**Figure 3.** INS instances switching in MINS.

At the end of the overlap, the set of navigational corrections determined on the basis of the new *INSi* instance is sufficiently large to allow for an aperture synthesis. As a result, after instance switching, the pixels that make up a given line of the image are created using the *INSi* data, while the previous line was created using the *INSi*−<sup>1</sup> instance. The length of the overlap can be expressed in terms of the number of radar soundings. The parallel work of instances begins at the time of the first IMU measurement after exceeding the INS error threshold, continues through *MSA* − 1 radar soundings, and ends when the first IMU measurement occurs after processing the given number of soundings. In the radar system used in the further described experiments, *MSA* is an odd number:

$$M\_{SA} = 2a - 1\tag{3}$$

where *a* is a rounded down number of soundings making up half of the synthetic aperture.

The navigational corrections are calculated using results of the INS instance. During the overlap, a pair of corrections are determined using *INSi*−<sup>1</sup> and *INSi*. The image synthesis algorithm, thanks to the knowledge about the length of the synthetic aperture shared with MINS, can detect the end of the correction data set and switch to the next source—the next INS instance.

In the presented algorithm, the INS instance initialization procedure can be interpreted as a form of INS error correction; however, the INS/GPS data are not used to correct the current INS instance (the errors of which continue to increase) but to initialize a new one. The initial INS errors correspond to small errors of the INS/GPS system. Thus, the proposed method combines the advantages of the INS and the INS/GPS systems. Thanks to the initialization of the instances, it is possible to keep INS errors at a low level, depending on the established error threshold, and at the same time the NCA uses the uncorrected INS results. According to the conclusions presented in [16] and [34], this should have a positive impact on the quality of the radar terrain images.
