Position-Adaptive Scatterer Localization for Radar Imaging Applications
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1 Position-Adaptive Scatterer Localization for Radar Imaging Applications Sean Young a, Atindra K. Mitra a, Tom Morton a, Raul Ordonez b a Air Force Research Laboratory, Wright-Patterson AFB, Ohio b ECE Department, University of Dayton, Dayton, Ohio, ABSTRACT Position-Adaptive Radar concepts have been formulated and investigated at the AFRL within the past few years. Adopting a position-adaptive approach to the design of distributed radar systems shows potential for the development of future radar systems that function under a variety of new and challenging environments. Specifically, we investigate notional control geometries and trajectories for multi-platform SUAV applications by integrating additional electromagnetic scattering-based metrics within more generic overall objective functions for multi-suav controls systems. We show that the formulation of these new categories of objective functions lead to realizations of multiplatform SUAV trajectories that position adaptively converge to a set of RF leakage points. After position-adaptive convergence to a set of leakage points, we show that an embedded scatterer (i.e. a metal cylinder) can be imaged by applying radar processing techniques derived for sparse apertures. Keywords: Position-Adaptive Radar, Small Unmanned Aerial Vehicles, Radar Imaging 1. INTRODUCTION Position-Adaptive Radar concepts have been formulated [1] - [4] and investigated at the AFRL within the past few years. For example, Figure 1 illustrates one of the original position-adaptive radar concepts [2] formulated and analyzed at the AFRL. This figure depicts a bistatic/multistatic radar system concept that is developed specifically for purposes of interrogating difficult and obscured targets via the application of low-altitude smart or robotic-type UAV platforms. Under this concept, a high-altitude radiating platform is denoted as a HUAV and a low-altitude position-adaptive platform is denoted as a LUAV. Derivation and analysis of approximate electromagnetic signal models for this notional urban scenario [2] indicates that onboard real-time computation of a phase parameter denoted as signal differential path length allows the LUAV to position-adaptively converge to a location between two buildings. Once the LUAV is hovering in front of an obscuration channel between two buildings, the LUAV can measure a leakage signal from the channel to determine characteristics of non-line-of-sight objects that are embedded between the buildings. This technique, denoted as exploitation of leakage signals via path trajectory diversity (E-LS- PTD) [2], is based on modulating scattering centers on embedded objects by implementing a fast trajectory on the HUAV while the LUAV is hovering in front of the obscuration channel. Analysis of computations and measurements of simulated LUAV receive signals, as a function of HUAV transmitter angle, lead to the characterization and modeling of this modulation effect and yield results that show potential for enabling the detection of a number of interesting parameters associated with deeply embedded targets and other objects of interest. Adopting this type of position-adaptive approach to the design of distributed radar systems shows potential for the development of future radar systems that function under new and challenging environments that contain large clutter discretes and require co-functionality within multi-signal RF environments. For purposes of the investigation that is outlined in this paper, we define a set of distributed airborne sensor geometries that are compatible for radar imaging via the employment of a swarm of SUAV (small unmanned aerial vehicle) platforms. Under this distributed robotic sensor concept, each of these SUAV platforms is equipped with a miniature radar sensor node that position-adaptively *atindra.mitra@wpafb.af.mil; phone x4331; AFRL/RYRR, Building 620, 2241 Avionics Circle, Wright-Patterson AFB, OH 45433;
2 converges to the vicinity of a set of signal leakage points. This type of radar systems formulation is based on the modeling and exploitation of RF leakage channels between large clutter discretes (i.e. buildings, mountains, etc.) in order to develop a novel approach to addressing challenging categories of multi-platform radar problems. Specifically, we investigate notional control geometries and trajectories for multi-platform SUAV applications by integrating additional electromagnetic scattering-based metrics within more generic overall objective functions for multi-suav controls systems. We show that the formulation of these new categories of objective functions lead to realizations of multi-platform SUAV trajectories that position adaptively converge to a set of RF leakage points. After position-adaptive convergence to a set of leakage points, we show that an embedded scatterer (i.e. a metal cylinder) can be imaged by applying radar processing techniques derived for sparse apertures. We illustrate this approach to position-adaptive radar system development by synthesizing radar signals for a notional set of SUAV geometries via electromagnetic simulation of scattering environments that contain large clutter discretes, leakage points between the clutter discretes, and embedded scatterers. We tabulate our results by injecting these synthesized radar signals into our multi-uav controls environment and present a set of sample output radar images for our embedded scatterers. Figure 2 illustrates the notional geometry that is adopted for this investigation. In Section II, we discuss the electromagnetic simulation of this geometry for purposes of synthesizing RF radar signals for each SUV platform. In Section III, electromagnetic scattering-based metrics that are developed via processing the synthesized signals from Section II are integrated within an overall objective function for a Multi-SUAV controls system such that the SUV s position adaptively converge to RF leakage points L1, L2, and L3. After position-adaptive convergence to the leakage points, the embedded scatterer (i.e. in this case a simulated metal cylinder) is imaged by using a radar imaging technique for sparse apertures that is discussed in Section IV. Figure 1: Position-Adaptive UAV Radar Concept for Urban Environments [2]
3 Figure 2: Notional Multi-Platform Position-Adaptive Geometry for Radar Imaging 2. RF SIGNAL SYNTHESIS FOR MULTI-SUAV GEOMETRY Figure 3 illustrates the CAD file for purposes of conducting electromagnetic simulations of the notional Multi-SUAV geometry depicted in Figure 2. Electromagnetic simulations for purposes of this investigation are conducted using a software package denoted as FEKO [5]. FEKO offers several options for performing electromagnetic simulations including a method-of-moments code (MoM), a physical optics code (PO), and hybridized versions of these two codes. We selected the PO code due to time constraints and the need for computational efficiency. In general, in order to simulate scattering from the perfect electric conductors (PEC) within the notional geometry of Figure 3, we observe the form of the following surface scattering integral in order to justify implementing a computationally efficient PO approximation. = 04 ( ), (1) In Eqn. (1), J is the surface current density on the PEC materials within our simulated environment, represents localized coordinates on the surface of the PEC scatterers, represents field evaluation coordinates (i.e. the synthesized receive waveform), η o is the free-space impedance, k is the free-space wavenumber, and g is the free-space Green s function:, = exp ( )4 R (2) The well-known PO or Kirchhoff approximation for Eqn. (1) is as follows:
4 = x ( + ) 2 x (3) This condition can greatly simplify the numerical computations (i.e. allows us to justify using a PO code) and implies that the surface magnetic field,, is approximately equal to the incident magnetic field, from the SUAV source. Meeting this surface boundary condition requirement motivates the selection of relatively large (or smooth) structures in relation to the wavelengths for the radar waveforms that we select for our multi-platform SUAV simulation. In order to continue generating the parameters and pre-conditions for this simulation set, we selected a design simulation frequency for each of the three notional radars on the three SUAV platforms of 2 GHz. This frequency corresponds to a wavelength of 15 cm at the selected center frequency of 2 GHz. We also selected a bandwidth of 100 MHz to simulate each of the three SUAV radars. For our case, this corresponds to 100 frequency points from 1.95 to 2.05 GHz. Next, we select the PEC structures for our scattering environment to be relatively large in relation to our characteristic wavelength of 15 cm. With this constraint/approximation in mind, we selected dimensions for the cylindrical scatterer at the center of the simulation space in Figure 3 as 2 meters in height and 1 meter in diameter. The three shields, used to define the leakage points under this position-adaptive concept, each have a height of three meters. The holes or leakage points between the three shields are 25.5 degrees each and the radius from the center of the cylinder to the boundary of any given shield is 3 meters. With these structural dimensions in mind along with the physical constraints for accurate PO simulation, we selected a simulation grid size of one-third wavelengths or 5 cm. This selection was partially justified by the fact that all the structures in our simulation space are smooth and all the structures along with all the other gaps between structures are large compared to 5 cm. In order to verify the validity of this approach, we ran a few sample simulations at one-sixth the wavelength for a grad spacing and compared outputs. The resulting outputs from two initial sample runs were identical to within over a 95 percent accuracy level. Figure 3: CAD Model for Synthesizing Muti-SUAV Radar Signals Using FEKO Electromagnetic Simulation
5 After selecting the above-mentioned simulation parameters, we conducted a set of simulations for the PEC-based scattering environment of Figure 3 with the FEKO PO code as a function of azimuth angle and elevation angle. We simulated scattering output corresponding to 100 frequency points from 1.95 to 2.05 GHz for a series of spatial points within the simulation space where we varied the elevation angle from 30 degrees to 3 degrees at three degree increments and the azimuth angle from 0 to 120 degrees at 1 degree increments. We generated 360 degrees of simulation data in azimuth due to the symmetries within our simulation space. After generating this set of synthetic data, we ported this data into a Matlab workspace and conditioned the data to simulate scattering versus range and azimuth by applying the appropriate link budget based terms for propagation loss due to (1/R 4 ) terms from the basic radar range equation. All SUAV platform altitudes are selected at 3 meters (across all azimuth and elevation simulation points) for these signal conditioning computations. While the resulting signal simulation is relatively coarsely spaced in terms of the spacial increments between neighboring points, we have a large enough density of points to meet the basic objectives of our basic investigation: The insertion of scattering-based metrics within a general objective function for Multi-SUAV Control in order to access the potential for developing multi-platform radar imaging applications via the implementation of position-adaptive techniques that spatially adapt to signal leakage points between large clutter discretes. 3. POSITION-ADAPTIVE CONTROL Figure 4 and Figure 5 illustrate outputs from a sample Matlab-Based Multi-SUAV controls system simulation based on two techniques, known as Sliding Mode Control Using Artificial Potentials [6] [7] and Extremum-Seeking Control [9]. The three green circles in Figure 4 (i.e. three agents in a triangular formation) represent the initial position of the three SUAV platforms. The tips of the triangle in Figure 5 show the final locations of the three SUAV platforms for this sample run. This sample run demonstrates the successful convergence of the three SUAV platforms to the three leakage points for the simulation geometry depicted in Figures 2 and 3. This position-adaptive radar control technique (as depicted in Figures 4 and 5) was developed by adding an additional scattering term to an objective function, or potential P, as follows. Raúl Ordóñez 3/17/09 11:33 Deleted: a Raúl Ordóñez 3/17/09 11:33 Deleted:. = (4) The first term in Eqn. 4 is a weighted target-tracking term, the second term is a weighted scenario-avoidance term, the third term is a formation-keeping (or collision-avoidance) term, and the forth term is our new RF scattering term. For the set of sample runs that are generated for this investigation, =0 since we are not modeling any target tracking at this point, and the scenario-avoidance, formation-keeping, position-adaptive scattering terms are equally weighted as follows: = = =.1. With this approach to Multi-SUAV control, the target-tracking, scenario-avoidance, and formation-keeping components of the potential in Eqn. 4 are associated with the following forms. = ( )2 (5) = ( )2 (6) = (7) Where x i and y i denote the coordinates of the i-th agent or i-th SUAV platform and x t and y t denote the location of a target for control system simulations that incorporate target tracking. Also, the numerical value of 30 in Eqn. 6 represents desired standoff distance for collision avoidance with the scene in order to prevent the SUAV platforms from crossing the scene.
6 Figure 4: Initial Multi-Platform SUAV Configuration Figure 5: Joint Multi-Platform Convergence to Signal Leakage Points Via Development of Position-Adaptive Control Algorithm with RF-Based Objective Function
7 Next, the first step toward formulating our initial RF-Scattering-Based component into the overall potential function for this sliding-mode control scheme is a global filtering computation that can be described in pseudo-code as follows:, = ( _,, )(8), =,,, 9 (9) Eqn. 8 converts the raw synthetic data (discussed in Section II) from the frequency, azimuth, and elevation domain into the time domain via Fourier-transforming each point-by-point frequency slice into a synthesized temporal domain and then extracting the maximum value from each temporal slice of data. This operation transforms the raw data from a basic three-dimensional simulation space into a two-dimensional array of range-dependent maximum values as a function of azimuth and elevation. Eqn. 9 is a 9-element sliding low-pass filtering operation across azimuth. Eqns. 8 and 9 have the effect of generating a smoothed RF data array with relatively larger azimuth-dependent scattering trends from the shield structures and relatively smaller azimuth-dependent scattering trends from within the leakage points. A RF-scattering-based component of the potential function is formulated by performing the following computation on the re-conditioned data array in Eqn. 9. = ( (, ) (, )4 (10) The resulting four-component potential function is integrated within an Matlab-based Multi-SUAV simulation code with Sliding Mode Control Using Artificial Potentials and the new position-adaptive control input, u i, is implemented within this Matlab simulation environment via the observation of the following sliding mode control paradigm [6] [7]. _ = +, + (11) [ (, ] (12) = 0 _ + (, ) (13) In Eqn. 11, s i_new is known as a sliding manifold functional and the eq subscript in Eqn. 12 represents a filtered (or bounded) version of the quantity in the brackets. (, ) in Eqn. 13 represents additive disturbances. 4. SPARSE APERTURE IMAGING VIA POSITION-ADAPTIVE LEAKAGE POINTS In this section, we present a set of sample radar images from the application of a SAR (synthetic aperture radar) imaging algorithm known as the polar re-formatting algorithm (PFA). We apply the PFA algorithm over discrete synthetic rfdata intervals that are within the angular sector covered by the position-adaptive leakage points detected via the methods described in the last section. Figure 6 illustrates the basic functionality of the PFA [8]. The left half of this figure represents the raw data domain in the sense that the data points are in a synthetic measurement space that varies with radial distance and angle. These radial distances and angles are defined between a SUAV platform and a selected focusing point on the ground (i.e. a number of pulses of synthetic data that are encompassed by the leakage intervals.) The right half of Figure 8 is the result for performing a 2D sinc interpolation on the raw data that is represented in the left half of the figure. After the data is interpolated from polar to Cartesian coordinates, a 2D inverse Fourier transform operation is performed on the gridded Cartesian data to compute a radar image on a per platform basis. For purposes of generating multi-platform images, we rotate two of the images onto the localized coordinate system of a third image and then perform a summing operation to generate a final three-platform sample image. An approximate model for the PFA-based radar image processing steps employed in this investigation can be mathematically represented as follows:
8 (, ), [ 2( cos ( ) [ 2( sin ] (14), represents raw data in the r and θ based measurement space, (, ) is the re-sampled data over a Cartesian grid, L represents the diameter of a circle that defines an annular region of the ground for purposes of radar imaging, r and θ are sampling intervals that are relatively coarsely defined due to the constraints associated with this particular synthetic rfdata set. The sample radar images in Figures 7 and 8 are generated by modeling, as follows:, =, exp exp ( (, ) (15) where arg is a precomputed simulation phase between the radar transmitter and receiver on a per SUAV platform basis. The second exponential term in Eqn. 15 is a phase correction term that is integrated within the PFA processing. Since this particular phase correction term is based on the actual phase values of the simulated rfdata, we are denoting this approach as first order data driven approach. The sample radar images in Figures 9 and 10 are generated by modeling, as follows:, =, exp exp ( 4 0 )(16) The second exponential term in Eqn. 16 is also denoted as a phase correction term. Since this phase correction term is a function of 0 (i.e. the distance from a SUAV platform to the scene center) and the embedded cylinder in our simulation geometry in located at scene center, we denote this approach as focusing to embedded scatter. As a more novel approach to Multi-SUAV image formation, it is possible to develop iterative approaches where the data driven approach is Eqn. 15 is used to iteratively form updated images and estimate values for 0 as part of an intelligent position-adaptive control system development process where sample images of the type in Figures 7 and 8 converge to sample images of the type in Figures 9 and 10.
9 Figure 6: Illustration of Single-Platform Image Processing with Polar-to-Cartesian Interpolation [8] Figure 7: Result of Single-Platform Imaging Over Position-Adaptive Leakage Points and First-Order Data-Driven Phase Compensation
10 Figure 8: Result of Three-Platform Imaging Over Position-Adaptive Leakage Points and First-Order Data-Driven Phase Compensation Figure 9: Result of Single-Platform Imaging Over Position-Adaptive Leakage Points and Focusing to Embedded Scatterer
11 Figure 10: Result of Three-Platform Imaging Over Position-Adaptive Leakage Points and Focusing to Embedded Scatterer 5. SUMMARY AND FUTURE PLANS We have outlined an approach to designing position-adaptive radar systems for Multi-SUAV platforms and have shown some potential for pursuing this approach as an alternative approach to performing radar imaging under certain challenging conditions. There are many potential avenues for follow-up research activities related to a number of aspects of this preliminary study. For example, due to time constraints, the simulation space for generating the synthetic rfdata for this investigation was relatively coarsely sampled. In terms of the further advancement potential of this particular positive-adaptive technique, there is much more room for further research towards developing additional more localized and advanced potential function components for more detailed investigations of RF-based scattering metrics that can be integrated into control systems and associated objective functions. (The preliminary RF-based scattering potential component formulated and investigated in this paper requires global processing of the data which, from a general systems concept point of view, may require additional SUV encirclements of a region of interest.) Also, there is always more room to incorporate joint robotics and intelligent processing techniques to develop more mature Multi-Platform SUAV Radar Systems for more futuristic applications. ACKNOWLEDGEMENTS The authors would like to thank Keith Loree, the AFRL/RYRR Branch Chief, and Larrell Walters, at IDCAST (Institute for Development Commercialization of Advanced Sensor Technology), for support. REFERENCES 1. Atindra K. Mitra, Position-Adaptive UAV Radar for Low-Altitude Sensing Applications, Proceedings of the 2003 IEEE Aerospace Conference, Big Sky, Montana, March Atindra K. Mitra, Position-Adaptive UAV Radar for Urban Environments, Proceedings of the 2003 International Radar Conference, Adelaide, Australia, September Atindra K. Mitra, "Passive Position-Adaptive Radar Modes for non-los Interrogation of Embedded Objects," Radar Sensor Technology X, SPIE Defense and Security Symposium, Orlando, Florida, March Atindra K. Mitra, "Leakage Signal Analysis for Position-Adaptive UAV Radar Applications," Radar Sensor Technology XI, SPIE Defense and Security Symposium, Orlando, Florida, April Ulrich Jakobus, Review of Advanced EM Modeling Techniques in the Computer Code FEKO, Applied Computational Electromagnetics Society Newsletter, Vol. 18, No. 2, July Veysel Gazi, Raul Ordonez, Target Tracking Using Artificial Potentials and Sliding Mode Control, Proceedings of American Control Conference, Boston, Massachusetts, June-July Jingyi Yao, Raul Ordonez, Veysel Gazi, Swarm Tracking Using Artificial Potentials and Sliding Mode Control, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 129, No. 5, September Steve Plimpton, Image Processing - Synthetic Aperture Radar Analysis,
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