Attitude Error Cancellation for Strip-map SAR by Controlling Steering Angles of the Antenna Platform

Yasser Elkoteshya, Shuyuan Yangb, and Fathy Abdelkaderc

aSchool of International Education, Xidian University, Xi’an, 710071, China; bSchool of Electronic Engineering, Xidian University, Xi’an, 710071, China;

cMilitary Technical College, Cairo, Egypt.

Abstract— Attitude error is the change in the object attitude (roll, pitch and yaw) over the ideal. Attitude error of the SAR-carrying vehicle causes a translation and attitude errors of the SAR-antenna platform. Pitch and roll of the SAR-antenna move the antenna footprint in the flight path direction and transversely to flight path respectively. Yaw of SAR-antenna introduces a squint of the antenna footprint. The change in the footprint positions due to attitude error causes a degradation of the resultant SAR image. The attitude error can be canceled by controlling the steering angles of the antenna platform in azimuth and elevation directions. This paper describes in details how the angles that the antenna platform should be steered with in the azimuth and elevation directions to cancel the attitude error can be calculated, these angles are called reference angles. In the simulation study, an actual navigation data of a SAR-carrying vehicle are used to show the attitude error effect on the footprint positions for strip-map operation, then the reference azimuth and elevation angles are calculated to cancel this error. If the antenna platform tracks the reference angles with zero tracking errors, the attitude error will be ideally cancelled. So, in the simulation study also, the antenna platform tracking of the reference angles is simulated to check the effect of the tracking errors on the footprint positions. The simulation results and their related comparisons show that the proposed work effectively cancels the attitude error.

Keywords—strip-map SAR;; attitude error; coordinate transformations; SAR-antenna platform; adaptive neural control

I. INTRODUCTION Strip-map is a very popular imaging mode of synthetic

aperture radar (SAR). In strip-map mode, a long strip terrain is imaged during the full transit time by the antenna which is oriented parallel to the flight path and looking sideward to the ground. The broadside looking of the antenna causes a wide and broad width of the antenna footprint in the flight path direction and the transverse direction to the flight path respectively. In addition, the typical SAR antenna has a brad beam in a plane transverse to the flight path and a narrow beam in the other direction [1].

The ideal data collection of SAR assumes that the SAR-carrying vehicle moves at a constant speed in a straight line without any attitude error [2]. In actual data collection case and due to navigation conditions, the flight vehicle experiences a variety of deviations from the ideal path

(translation error) with different attitude (rotation or attitude error). Since the antenna platform in general is not on the rotational axes of the flying vehicle, rotation of the flight vehicle causes a translation and rotation of the antenna platform. The translational error results in the target scene changing in range during data collection while the attitude error introduces errors of a different kind.

The attitude error can be described as the change in the three Euler angles; roll, pitch, and yaw over the ideal. Roll of the SAR-antenna moves the antenna footprint transversely to the flight path which changes the antenna gain pattern over the target area. Roll doesn’t affect the SAR processing because the antenna beam is broad in the direction of footprint movement, only large amount of roll is apt to be significant. Pitch of the SAR-antenna moves the antenna footprint in the direction of the flight path and duo to narrow beam in that direction, the desired area may not be illuminated for some transmitted pulses. Yaw of the SAR-antenna cause a squint of the antenna footprint which causes a weak illumination of the resolution cell [3].

It is clear that the attitude error causes an errors in the footprint positions and consequently degrades the SAR image, so it should be cancelled instantaneously for each antenna position. The attitude error effect can be cancelled simply by steering the antenna platform in azimuth and elevation directions with angles called reference azimuth and elevation angles respectively. If the antenna platform tracks the reference angles with zero tracking errors, the attitude error will be ideally cancelled but actually there are certain tracking errors and their values depends on the controller performance.

The antenna platform is a dual-axis servo-mechanism which can perform two rotational motions in the azimuth and elevation directions. The modeling of a dual-axis servo-mechanism was presented in [4] and it was shown that the antenna platform is a second order multi-input multi-output (MIMO) nonlinear dynamic system.

The adaptive backstepping neural controller (ABNC) is one of the well known control techniques for uncertain MIMO nonlinear systems such as antenna platforms. In the ABNC, the adaptive backstepping technique is combined with neural networks (NNs) to approximate the unknown nonlinear

978-1-4799-2993-1/14/$31.00 ©2014 IEEE

functions in the control laws, where the NNs parameters are updated based on Lyapunov theory to guarantee semi-global boundedness of all signals in the closed-loop system [5, 6].

In this work, online cancellation of the attitude error of SAR-antenna by controlling the antenna platform steering angles in azimuth and elevation directions is described in details for strip-map SAR. The calculated reference azimuth and elevation steering angles are used are tracked by the antenna platform via an ABNC with a predefined tracking accuracy. The footprint positions (after canceling attitude error) with and without considering the controller error are compared with the ideal footprint positions to show the effectiveness of the proposed algorithm.

The rest of the paper is organized in the following: The strip-map SAR imaging scheme is presented in section II. In section III, the coordinate systems used in this work are defined. The reference angles calculations for strip-map SAR are presented in Section IV. Simulation results are explored in Section V. Section VI contains the conclusion.

Notations: In this paper, · and · denote the cosine and sine functions respectively. , , and denote the components of a point in , y, and diections respectively.

II. STRIP-MAP SAR It is a conventional mapping mode of SAR which assumes

a fixed pointing direction of the radar antenna broadside to the flight track as shown in Fig. 1. The fixed pointing direction is defined by look angle and squint angle . The look angle is the angle that the antenna center beam makes with vertical line while the squint angle is the angle that the antenna center beam makes with the axis perpendicular to the flight path. As a result of strip-map mode, an image for a strip of earth is formed in the width by the swath width and follows the length contour of the antenna-platform [7].

In Fig. 1, the antenna travels with velocity at altitude , is the slant range between the antenna center and a point

target at center of the antenna footprint, and are the antenna dimensions in the along and across track directions respectively, and the mid-swath is the line contour of the footprint centers along the whole flight path.

Fig. 1. Imaging geometry of strip-map SAR.

III. COORDINATE SYSTEMS DEFINITIONS In this work, three coordinate systems are defined, which

are reference frame {R}, body frame {P} and antenna frame {A} coordinate systems as shown in Fig. 2 [8].

A. Reference Coordinate System {R} It is stationary and non-time dependent coordinate system

with axis , , and The reference axis , , and are assumed to be positive in the east, north, and up directions respectively. The position of any ground target is measured w.r.t. the reference frame {R}.

B. Body Frame Coordinate System {P} This coordinate system is fixed to the aircraft structure at

its center of gravity and its axis are , , and . The axis is positive along the heading of the aircraft and the roll angle is defined about this axis (roll axis). The axis is positive along the left wing of the aircraft and the pitch angle

is defined about this axis (pitch axis). The axis is positive in the up direction and the yaw angle is defined about this axis (yaw axis). The position of the body frame {P} is measured w.r.t. the reference frame {R}.

C. Antenna Frame Coordinate System {A} This coordinate system is fixed to the antenna structure at

its center and its axis are , , and . The axis is positive in the up direction and the antenna azimuth angle is defined about this axis. The axis is positive in the direction from the antenna center into the perpendicular to its across-track side W and the antenna elevation angle is defined about this axis. The axis is positive in the direction determined by the right hand rule and no rotation is permissible about this axis. The position of the antenna frame {A} are measured w.r.t. the body frame {P}.

+ Elevation

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Fig. 2. Coordinate systems definitions.

IV. REFERENCE ANGLES CALCULATIONS FOR STRIP-MAP SAR

The antenna reference angles are the angles that the antenna platform should be steered with in the azimuth and elevation directions to cancel the attitude error. Algorithms I and II defined in the following first two subsections are used to calculate the reference azimuth and elevation angles as in the following third subsection.

A. Algorithm I This algorithm is designed to calculate the required

azimuth angle and elevation angle of the airborne antenna to make the antenna center beam points directly to a certain ground target T with the following given data:

• The position of the point target T w.r.t. the reference frame {R}.

• The position of the body frame {P} w.r.t. the reference frame {R}.

• The position of the antenna frame {A} w.r.t. the body frame {P}.

• The attitude of the body frame defined by Euler angles: roll , pitch , and yaw .

• The initial azimuth angle and elevation angle of the antenna-platform.

The algorithm I details can be summarized in the following points:

1) Find the target position w.r.t. the body frame as follows

, , , , , , (1) where is the direction cosine matrix from frame {R} into frame {P} which is defined as follows

2) Find the target position w.r.t. the antenna frame {A}

as follows , , , , , , (2)

where is the direction cosine matrix from frame {P} into frame {A} which is defined as follows 0

3) The azimuth angle and elevation angle that is required to make the antenna center beam points directly to a certain ground target T can be calculated as follows [9, 10].

(3)

(4)

4) In many situations and according to the values of

and , the calculated angles and are required to be modified to a new values and respectively as given by Table 1.

B. Algorithm II This algorithm is designed to calculate the position of the

ground point that is directly pointed by the antenna center beam (i.e. the point of intersection between the antenna center beam and the ground) given the following data:

• The same first three givens of algorithm I.

• The azimuth angle and elevation angle of the airborne antenna.

To achieve the goal of algorithm II, two points 0,0,0 and 0,0, are defined w.r.t. the antenna frame {A}. The algorithm II details can be summarized in the following points:

1) Find the position of points and w.r.t. the body frame {P} as follows

, , , , , , (5) , , , , , , (6)

where is the direction cosine matrix from frame {A} into frame {P} ( is the transpose of the matrix ).

2) Find the position of points and w.r.t. the reference frame {R} as follows

, , , , , , (7) , , , , , , (8)

where is the direction cosine matrix from the body frame {P} into the reference frame {R} ( is the transpose of the matrix ).

3) The position of the ground point T can be calculated by finding the intersection between the ray and the ground as follows:

0 (9)

TABLE I. MODEFICATIONS OF REFERENCE ANGLES

No Condition Modification 1 0 0 0 2 0 0 3 0 0 4 0 0 5 0 0

C. Reference Angles Calculations The following algorithm summaries the procedures of

calculating the antenna reference azimuth and elevation angles for strip-map SAR.

1) Define required look angle , squint angle , and the ideal attitude , , and .

2) For the first antenna position, algorithm II is applied with , , , , and to find the ground target that is pointed by the antenna center beam.

3) At the same sensor position, algorithm I is applied with actual roll angle , actual pitch angle , actual yaw angle

, 0, and 0 to find the required azimuth and elevation angles of the antenna to make the antenna center beam points the ground point calculated from step 2.

4) Steps 2 and 3 are repeated for every sensor position along the flight path.

V. SIMULATION RESULTS In simulation study, an actual and ideal navigation data for

30 seconds with sampling rate of 80 MHZ are used to test the proposed algorithm.

The actual and ideal navigation data are shown in Figures 1 - 4. Fig. 1 shows the actual and ideal flight paths in plane of the frame {R}(i.e. the position of frame {P}). Fig. 2 shows the ideal and actual altitude of frame {P}. The translation error between the actual and ideal flight paths are shown in Fig. 3. The attitude error of the body frame is shown in Fig. 4, where the ideal roll, pitch, and yaw angles are 0 , 0 , and 57.92 respectively.

Fig. 3. The actual and ideal flight paths in the plane.

Fig. 4. The actual and ideal altitude of the SAR-carrying vehicle.

Fig. 5. Translation error of the SAR-carrying vehicle.

Fig. 6. Attitude error of the SAR-carrying vehicle.

Fig. 7 shows the ideal and actual mid swath lines for a strip-map operation with 75 look angle and 0 squint angle under the assumption that body and antenna frames are coincident. It is clear from this figure that the attitude error causes a large shift in the footprint positions especially in the across track direction so the attitude error should be cancelled.

Now, the proposed algorithm is applied to cancel the attitude error of the flying vehicle and the antenna reference azimuth and elevation angles are calculated. The reference azimuth and elevation angles are shown in Fig. 8 and Fig. 9 respectively. It is clear from theses figures that without canceling the attitude error the reference azimuth and elevation angles are the same as the squint and look angles respectively.

By setting the azimuth and elevation angles of the antenna platform equal to the reference azimuth and elevation angles respectively (i.e. the antenna platform tracks the reference azimuth and elevation angles with zero tracking errors), the attitude error will be ideally cancelled. The errors in and directions of the actual mid-swath lines with attitude error and with ideally canceling attitude error against the ideal mid-swath line are shown in Fig. 10 and Fig. 11 respectively.

In Fig. 10 and Fig. 11, the error between ideal mid-swath line and the actual mid-swath line with ideally canceled attitude error is duo to translation error of the flying vehicle over the ideal path.

Fig. 7. The actual mid-swath lines with ideal and actual attitude.

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Fig. 8. Reference azimuth angle of rotation.

Fig. 9. Reference elevation angle of rotation.

Fig. 10. Mid-swath errors in X direction.

Fig. 11. Mid-swath errors in Y direction.

Since the antenna platform can track the reference azimuth and elevation angles with certain tracking errors, so the tracking error of the antenna platform will be considered. The model of the antenna platform controlled by an ABNC with a predefined tracking accuracy, as descried in our previous work [6], is used here to check the effect of the controller error on the actual mid-swath line. The same model and controller parameters as in [6] are used in this simulation except that predefined tracking accuracy is set to 0.005.

Fig. 12 and Fig. 13 show the output response of the antenna platform to track the reference azimuth and elevation angles respectively, where the initial azimuth and elevation angles are set to 0 and 75 respectively. Fig. 14 and Fig. 15 show the tracking errors of the azimuth and elevation angles respectively. From these figures, large tracking errors can be noticed during the first few seconds (training period of the controller) then converge to the required accuracy after 1.2 second for the azimuth angle and less than 1 second for the elevation angle.

Fig. 12. Output azimuth angle follows the refernce azimuth angle.

Fig. 13. Output elevation angle follows the refernce elevation angle.

Fig. 14. Azimuth angle error.

Fig. 15. Elevation angle error.

The error in and directions of the actual mid-swath lines with and without controller error against ideal mid-swath line are shown in Fig. 16 and Fig. 17 respectively. These figures show the data after the first 2 seconds due to large tracking errors at the first 2 seconds.

VI. CONCLUSION This article describes in details the attitude error

cancellation for a strip-map SAR by steering the antenna platform in azimuth and elevation directions with certain angles called reference azimuth and elevation angles respectively. The ABNC technique with a predefined tracking accuracy is used to control the antenna platform for tracking the reference azimuth and elevation angles. The simulation results show that the proposed algorithm is perfectly cancel the attitude error even the antenna platform tracks the reference angle with certain tracking error. The same technique with a little modifications can be applied to other operating modes of SAR system. The cancellation of attitude error depends on the accuracy of the attitude measuring devices.

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Fig. 17. Mid-swath error in direction.

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